ɍɀɊɆɉɏɃɂɃɒɀɌɅɃɀ ɉɌɈɉȽɖ ɌɉȽɋɀɇɀɈɈɖɐ ȽɖɌɉɅɉɍɀɇɊɀɋȻɍɎɋɈɖɐ ɍɀɐɈɉɆɉȾɃɄ Ȼ.Ⱦ. Ʌɨɺɢɠɝɛ

ɎȿȾȿɊȺɅɖɇɈȿ ȺȽȿɇɌɋɌȼɈ ɉɈ ɈȻɊȺɁɈȼȺɇɂɘ
Ƚɨɫɭɞɚɪɫɬɜɟɧɧɨɟ ɨɛɪɚɡɨɜɚɬɟɥɶɧɨɟ ɭɱɪɟɠɞɟɧɢɟ ɜɵɫɲɟɝɨ ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɨɝɨ ɨɛɪɚɡɨɜɚɧɢɹ
«ɍɉɇɌɅɃɄ ɊɉɆɃɍɀɐɈɃɒɀɌɅɃɄ ɎɈɃȽɀɋɌɃɍɀɍ»
Ȼ.Ⱦ. Ʌɨɺɢɠɝɛ
ɍɀɊɆɉɏɃɂɃɒɀɌɅɃɀ ɉɌɈɉȽɖ ɌɉȽɋɀɇɀɈɈɖɐ
ȽɖɌɉɅɉɍɀɇɊɀɋȻɍɎɋɈɖɐ ɍɀɐɈɉɆɉȾɃɄ
Ɋɟɤɨɦɟɧɞɨɜɚɧɨ ɜ ɤɚɱɟɫɬɜɟ ɭɱɟɛɧɨɝɨ ɩɨɫɨɛɢɹ
Ɋɟɞɚɤɰɢɨɧɧɨ-ɢɡɞɚɬɟɥɶɫɤɢɦ ɫɨɜɟɬɨɦ
Ɍɨɦɫɤɨɝɨ ɩɨɥɢɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ
ɂɡɞɚɬɟɥɶɫɬɜɨ
Ɍɨɦɫɤɨɝɨ ɩɨɥɢɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ
2009
ɍȾɄ 536.5(075.8)
ȻȻɄ 22.36ɹ73
Ʉ54
Ʉ54
Ʉɧɹɡɟɜɚ Ⱥ.Ƚ.
Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɨɫɧɨɜɵ ɫɨɜɪɟɦɟɧɧɵɯ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ
ɬɟɯɧɨɥɨɝɢɣ: ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ / Ⱥ.Ƚ. Ʉɧɹɡɟɜɚ; Ɍɨɦɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ. – Ɍɨɦɫɤ: ɂɡɞ-ɜɨ Ɍɨɦɫɤɨɝɨ ɩɨɥɢɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ, 2009. – 357 ɫ.
ISBN 978-5-98298-804-1
ȼ ɩɨɫɨɛɢɢ ɞɚɧɵ ɨɫɧɨɜɧɵɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨ ɡɚɤɨɧɚɯ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ (ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɤɨɧɜɟɤɰɢɟɣ ɢ ɢɡɥɭɱɟɧɢɟɦ) ɢ ɫɩɨɫɨɛɚɯ ɢɯ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ
ɨɩɢɫɚɧɢɹ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɦɨɞɟɥɢɪɨɜɚɧɢɸ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɢɯ ɫɬɚɞɢɣ. ɍɞɟɥɟɧɨ ɜɧɢɦɚɧɢɟ ɢɧɠɟɧɟɪɧɵɦ ɩɨɞɯɨɞɚɦ ɤ
ɪɟɲɟɧɢɸ ɡɚɞɚɱ. Ɉɩɢɫɚɧɵ ɫɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ, ɢɦɟɸɳɢɟ ɬɟɩɥɨɮɢɡɢɱɟɫɤɭɸ ɩɪɢɪɨɞɭ (ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ, ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ, ɞɢɮɮɭɡɢɹ).
Ɋɚɡɪɚɛɨɬɚɧɨ ɜ ɪɚɦɤɚɯ ɪɟɚɥɢɡɚɰɢɢ ɂɧɧɨɜɚɰɢɨɧɧɨɣ ɨɛɪɚɡɨɜɚɬɟɥɶɧɨɣ
ɩɪɨɝɪɚɦɦɵ Ɍɉɍ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ «Ɇɚɬɟɪɢɚɥɨɜɟɞɟɧɢɟ, ɧɚɧɨɦɚɬɟɪɢɚɥɵ ɢ ɧɚɧɨɬɟɯɧɨɥɨɝɢɢ» ɢ ɩɪɟɞɧɚɡɧɚɱɟɧɨ ɞɥɹ ɫɬɭɞɟɧɬɨɜ 5 ɤɭɪɫɚ, ɨɛɭɱɚɸɳɢɯɫɹ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ 150900 «Ɍɟɯɧɨɥɨɝɢɹ, ɨɛɨɪɭɞɨɜɚɧɢɟ ɢ ɚɜɬɨɦɚɬɢɡɚɰɢɹ ɦɚɲɢɧɨɫɬɪɨɢɬɟɥɶɧɵɯ ɩɪɨɢɡɜɨɞɫɬɜ», ɫɩɟɰɢɚɥɢɡɚɰɢɹɦ 151001.01 «Ɍɟɯɧɨɥɨɝɢɹ ɚɜɬɨɦɚɬɢɡɢɪɨɜɚɧɧɨɝɨ ɩɪɨɢɡɜɨɞɫɬɜɚ», 150900.17 «Ɏɢɡɢɤɚ ɜɵɫɨɤɢɯ ɬɟɯɧɨɥɨɝɢɣ ɜ
ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ», ɚ ɬɚɤɠɟ ɚɫɩɢɪɚɧɬɨɜ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ.
ɍȾɄ 536.5(075.8)
ȻȻɄ 22.36ɹ73
Ɋɟɰɟɧɡɟɧɬɵ
Ⱦɨɤɬɨɪ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ ɌȽɍ
Ⱥ.ɘ. Ʉɪɚɣɧɨɜ
Ⱦɨɤɬɨɪ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ
ɩɪɨɪɟɤɬɨɪ ɌȽȺɋɍ, ɡɚɜ. ɤɚɮɟɞɪɨɣ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ
ɋ.ɇ. Ʉɨɥɭɩɚɟɜɚ
ISBN 978-5-98298-804-1
© ȽɈɍ ȼɉɈ «Ɍɨɦɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ
ɭɧɢɜɟɪɫɢɬɟɬ», 2009
© Ʉɧɹɡɟɜɚ Ⱥ.Ƚ., 2009
© Ɉɮɨɪɦɥɟɧɢɟ. ɂɡɞɚɬɟɥɶɫɬɜɨ Ɍɨɦɫɤɨɝɨ
ɩɨɥɢɬɟɯɧɢɱɟɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ, 2009
ɈȽɅȺȼɅȿɇɂȿ
ɉɊȿȾɂɋɅɈȼɂȿ..........................................................................................8
ɑȺɋɌɖ 1. ȼɜɟɞɟɧɢɟ ɜ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ......10
1.1.
Ɍɟɦɩɟɪɚɬɭɪɚ, ɬɟɩɥɨɮɢɡɢɤɚ, ɬɟɩɥɨɨɛɦɟɧ..................................10
1.2.
Ʉɨɥɢɱɟɫɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ..............11
1.3.
Ɇɟɯɚɧɢɡɦɵ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ: ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ,
ɤɨɧɜɟɤɰɢɹ, ɢɡɥɭɱɟɧɢɟ................................................................12
1.4.
ɂɫɬɨɪɢɱɟɫɤɢɣ ɷɤɫɤɭɪɫ...............................................................13
1.5.
ȼɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ ɬɟɯɧɨɥɨɝɢɢ ..........................................15
1.6.
Ɍɟɩɥɨɜɚɹ ɡɚɳɢɬɚ........................................................................22
1.7.
Ɍɟɩɥɨɨɛɦɟɧɧɢɤɢ........................................................................25
1.8.
ɏɢɦɢɱɟɫɤɢɟ ɬɟɯɧɨɥɨɝɢɢ ...........................................................27
1.9.
Ɉɬɧɨɲɟɧɢɟ ɬɟɩɥɨɨɛɦɟɧɚ ɤ ɬɟɪɦɨɞɢɧɚɦɢɤɟ.............................28
1.10
Ɋɨɥɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɜ ɪɚɡɪɚɛɨɬɤɟ
ɫɨɜɪɟɦɟɧɧɵɯ ɬɟɯɧɨɥɨɝɢɣ ..........................................................33
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ......................................................35
ɑȺɋɌɖ 2. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɢ ɭɪɚɜɧɟɧɢɹ...............................................36
2.1.
ɂɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ....................................................................36
2.2.
ɏɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ.....................................40
2.3.
Ɍɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ .....................................................................42
2.4.
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ .............................................43
2.5.
Ɂɚɞɚɱɚ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ............................................................46
2.6.
ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ..................................................48
2.7.
Ɉ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱ ɜ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.....................50
2.8.
Ɇɧɨɝɨɫɥɨɣɧɚɹ ɫɬɟɧɤɚ................................................................53
2.9.
ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ ɞɥɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ .....................55
2.10.
Ʉɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ .................................58
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ......................................................60
Ɂɚɞɚɧɢɹ.......................................................................................61
ɑȺɋɌɖ 3. Ʉɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ ɢ ɬɟɩɥɨɩɟɪɟɞɚɱɚ..........................62
3.1.
Ʉɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ .......................................................62
3.2.
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɩɟɪɟɧɨɫɚ............................................................................69
3.3.
Ɂɚɞɚɱɚ ɨɛ ɨɛɬɟɤɚɧɢɢ ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɵ ..................................76
3.4.
Ɍɟɩɥɨɩɟɪɟɞɚɱɚ ɱɟɪɟɡ ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ......................................83
3.5.
Ɇɧɨɝɨɫɥɨɣɧɚɹ ɫɬɟɧɤɚ................................................................85
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ......................................................88
3
Ɂɚɞɚɧɢɹ.......................................................................................88
ɑȺɋɌɖ 4. ɗɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɢ ɚɧɚɥɢɡɚ ɪɚɡɦɟɪɧɨɫɬɟɣ………...90
4.1.
ɉɨɧɹɬɢɟ ɨ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ........................................................90
4.2.
Ɍɟɨɪɟɦɵ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ..........................................................92
4.3.
Ɉɫɧɨɜɧɵɟ ɤɪɢɬɟɪɢɢ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ .......................................93
4.4.
ɏɚɪɚɤɬɟɪɧɵɟ ɦɚɫɲɬɚɛɵ ............................................................97
4.5.
ɉɪɢɦɟɪɵ ɤɪɢɬɟɪɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ .......................................98
4.6.
ɇɟɤɨɬɨɪɵɟ ɷɦɩɢɪɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ .........................................99
4.7.
ɉɪɢɛɥɢɠɟɧɧɵɣ ɫɩɨɫɨɛ ɪɚɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɬɟɩɥɨɨɬɞɚɱɢ ɢ ɬɪɟɧɢɹ..............................................................105
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................110
Ɂɚɞɚɧɢɹ.....................................................................................111
ɑȺɋɌɖ 5. Ɂɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦɚɯ
ɤɨɨɪɞɢɧɚɬ ...............................................................................112
5.1.
ɍɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦɚɯ
ɤɨɨɪɞɢɧɚɬ.................................................................................112
5.2.
ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɟɥ
ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ ..............................................................114
5.3.
ɉɪɨɫɬɟɣɲɢɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ...................................116
5.4.
ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ .........................................................119
5.5.
Ɇɧɨɝɨɫɥɨɣɧɚɹ ɰɢɥɢɧɞɪɢɱɟɫɤɚɹ ɫɬɟɧɤɚ..................................121
5.6.
Ʉɪɢɬɢɱɟɫɤɢɣ ɞɢɚɦɟɬɪ ɬɟɩɥɨɢɡɨɥɹɰɢɢ....................................122
5.7.
ɒɚɪɨɜɚɹ ɫɬɟɧɤɚ .......................................................................125
5.8.
Ɋɟɲɟɧɢɟ ɩɪɨɫɬɟɣɲɢɯ ɡɚɞɚɱ ɜ ɛɟɡɪɚɡɦɟɪɧɨɣ ɮɨɪɦɟ ..............128
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................130
Ɂɚɞɚɧɢɹ.....................................................................................131
ɑȺɋɌɖ 6. Ɂɚɞɚɱɢ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ ..................................132
6.1.
Ɂɚɞɚɱɚ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ..........................................................132
6.2.
ɐɢɥɢɧɞɪ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ................................138
6.3.
ɉɪɨɜɨɞ ɫ ɢɡɨɥɹɰɢɟɣ.................................................................141
6.4.
ɒɚɪ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ .......................................144
6.5.
Ɂɚɞɚɱɚ ɨɛ ɨɲɢɛɤɚɯ ɬɟɪɦɨɩɚɪɵ ...............................................146
6.6.
Ɉɯɥɚɠɞɟɧɢɟ ɪɚɛɨɱɢɯ ɥɨɩɚɬɨɤ ɬɭɪɛɢɧɵ .................................152
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................155
Ɂɚɞɚɧɢɹ.....................................................................................155
4
ɑȺɋɌɖ 7. ɇɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ..........................157
7.1.
Ɉɛɡɨɪ ɡɚɞɚɱ ɢ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɡɚɞɚɱ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ...................................................................157
7.2.
ɉɪɨɫɬɟɣɲɢɟ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ...................................................................158
7.3.
ɉɪɨɫɬɟɣɲɢɟ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ ɬɟɨɪɢɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɰɢɨɧɧɨɝɨ
ɦɟɬɨɞɚ ......................................................................................162
7.3.1. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ .....................162
7.3.2. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜɬɨɪɨɝɨ ɪɨɞɚ .....................166
7.3.3. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ....................171
7.4.
Ɇɟɬɨɞ Ⱦɸɚɦɟɥɹ .......................................................................173
7.5.
ɉɪɢɦɟɪɵ ɫɨɩɪɹɠɟɧɧɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ...................................................................178
7.5.1. Ɍɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ ɜ ɫɪɟɞɟ............................178
7.5.2. Ɉɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ ɫ ɩɨɤɪɵɬɢɟɦ ........................................181
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................188
Ɂɚɞɚɧɢɹ.....................................................................................189
ɑȺɋɌɖ 8. Ʉɥɚɫɫɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɵɯ
ɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ........................................................190
8.1.
Ɋɟɲɟɧɢɟ ɤɪɚɟɜɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ ................................................190
8.2.
Ɇɟɬɨɞ ɢɫɬɨɱɧɢɤɨɜ ...................................................................191
8.3.
Ɇɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ..............................................197
8.3.1. Ɂɚɞɚɱɚ ɞɥɹ ɤɪɭɝɚ ......................................................................197
8.3.2. ɂɧɬɟɝɪɚɥ ɉɭɚɫɫɨɧɚ..................................................................201
8.3.3. ɇɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ ..........................................................202
8.4.
Ɇɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ .....................207
8.5.
Ɂɚɞɚɱɚ ɨɛ ɨɫɬɵɜɚɧɢɢ ɛɟɫɤɨɧɟɱɧɨɣ ɩɥɚɫɬɢɧɵ ........................210
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................212
Ɂɚɞɚɧɢɹ.....................................................................................213
ɑȺɋɌɖ 9. ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ: ɬɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ..............214
9.1.
Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ ..................214
9.2.
Ɉɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ.................................218
…9.2.1. Ɂɚɤɨɧ ɉɥɚɧɤɚ ...........................................................................218
9.2.2. Ɂɚɤɨɧ ɫɦɟɳɟɧɢɹ ȼɢɧɚ .............................................................218
9.2.3. Ɂɚɤɨɧ ɋɬɟɮɚɧɚ – Ȼɨɥɶɰɦɚɧɚ ...................................................219
9.2.4. Ɂɚɤɨɧ Ʌɚɦɛɟɪɬɚ .......................................................................220
5
9.2.5.
9.3.
9.4.
9.5.
9.6.
9.7.
9.8.
Ɂɚɤɨɧ Ʉɢɪɯɝɨɮɚ .......................................................................221
Ɇɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ .......................223
ɉɨɧɹɬɢɟ ɫɟɪɨɝɨ ɬɟɥɚ................................................................225
Ʌɭɱɢɫɬɵɣ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɬɟɥɚɦɢ ....................................226
ɇɚɩɪɚɜɥɟɧɧɵɟ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ .................................228
ɉɟɪɟɧɨɫ ɢɡɥɭɱɟɧɢɹ ɜ ɩɨɝɥɨɳɚɸɳɢɯ
ɩɪɨɩɭɫɤɚɸɳɢɯ ɫɪɟɞɚɯ.............................................................231
Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ ɩɪɢ ɧɚɥɢɱɢɢ ɞɪɭɝɢɯ
ɜɢɞɨɜ ɷɧɟɪɝɢɢ ..........................................................................234
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................240
Ɂɚɞɚɧɢɹ.....................................................................................241
ɑȺɋɌɖ 10. ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ: ɮɚɡɨɜɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ...............243
10.1.
ɍɫɥɨɜɢɹ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ................................................243
10.2.
ɍɪɚɜɧɟɧɢɟ Ʉɥɚɩɟɣɪɨɧɚ – Ʉɥɚɭɡɢɭɫɚ ......................................247
10.3.
ɋɥɟɞɫɬɜɢɹ ɭɫɥɨɜɢɹ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ...............................248
10.4.
Ɂɚɞɚɱɚ ɋɬɟɮɚɧɚ........................................................................252
10.5.
ɉɪɨɫɬɟɣɲɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ..................255
10.6.
Ɋɚɡɦɵɬɵɟ ɢ ɬɨɱɟɱɧɵɟ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ..............................260
10.7.
ɗɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɞɜɭɯɮɚɡɧɨɣ ɡɨɧɵ .......................................262
10.8.
ɗɥɟɦɟɧɬɵ ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɮɚɡɨɜɵɯ
ɩɪɟɜɪɚɳɟɧɢɣ............................................................................264
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................267
Ɂɚɞɚɧɢɹ.....................................................................................268
ɑȺɋɌɖ 11. ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ: ɞɢɮɮɭɡɢɹ
ɢ ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ..........................................................269
11.1.
ɉɪɨɫɬɟɣɲɢɟ ɩɨɧɹɬɢɹ ɨ ɤɢɧɟɬɢɤɟ
ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ................................................................269
11.2.
ɉɪɢɦɟɪɵ ɨɩɢɫɚɧɢɹ ɤɢɧɟɬɢɤɢ ɝɨɦɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ ..............272
11.3.
Ʉɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ ..........................................279
11.4.
Ɉɬ ɱɟɝɨ ɡɚɜɢɫɢɬ ɫɤɨɪɨɫɬɶ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ?...................281
11.5.
Ɋɟɚɤɰɢɢ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ....................................284
11.5.1. Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ ........................284
11.5.2. Ɉɫɨɛɟɧɧɨɫɬɢ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ ............................288
11.6.
Ʉɢɧɟɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɬɜɟɪɞɨɮɚɡɧɵɯ
ɩɪɟɜɪɚɳɟɧɢɣ............................................................................291
11.7.
Ⱦɢɮɮɭɡɢɨɧɧɚɹ ɤɢɧɟɬɢɤɚ.........................................................295
11.7.1. Ɉɛɳɢɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨ ɞɢɮɮɭɡɢɢ ........................................295
11.7.2. Ɋɨɥɶ ɞɢɮɮɭɡɢɢ ɜ ɧɟɤɨɬɨɪɵɯ ɮɢɡɢɤɨ–ɯɢɦɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɚɯ ................................................................................298
6
11.7.3.
11.8
Ɇɚɤɪɨɤɢɧɟɬɢɱɟɫɤɢɟ ɨɛɥɚɫɬɢ ɩɪɨɬɟɤɚɧɢɹ
ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ..........................................................300
ɏɢɦɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ ɜ ɭɪɚɜɧɟɧɢɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ...................................................................304
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ ....................................................307
Ɂɚɞɚɧɢɹ.....................................................................................307
ɑȺɋɌɖ 12. Ɋɚɡɧɵɟ ɡɚɞɚɱɢ .......................................................................309
12.1.
Ɉɯɥɚɠɞɟɧɢɟ ɩɨɪɢɫɬɨɣ ɩɥɚɫɬɢɧɵ ............................................309
12.2.
Ⱦɜɢɠɭɳɢɟɫɹ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ................................................311
12.3.
ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɧɚɝɪɟɜɚɧɢɟ ɩɪɨɜɨɥɨɤɢ ...................................314
12.4.
əɞɟɪɧɵɣ ɬɟɩɥɨɜɵɞɟɥɹɸɳɢɣ ɷɥɟɦɟɧɬ ....................................317
12.5.
Ɍɟɩɥɨɨɛɦɟɧ ɩɪɢ ɧɚɥɢɱɢɢ ɜɹɡɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ..............319
12.6.
ɇɚɝɪɟɜ ɬɟɥ ɢɡɥɭɱɟɧɢɟɦ ɈɄȽ...................................................321
12.7.
Ɉɛɨɥɨɱɤɚ, ɨɯɥɚɠɞɚɟɦɚɹ ɢɡɥɭɱɟɧɢɟɦ, ɫ ɪɟɡɤɢɦ
ɩɟɪɟɩɚɞɨɦ ɪɚɜɧɨɜɟɫɧɵɯ ɬɟɦɩɟɪɚɬɭɪ ......................................328
12.8.
Ɇɨɞɟɥɶ ɩɪɨɰɟɫɫɚ ɰɟɦɟɧɬɚɰɢɢ ɤɨɦɩɚɤɬɧɨɝɨ
ɦɚɬɟɪɢɚɥɚ.................................................................................330
12.9.
ɇɚɝɪɟɜ ɢɡɥɭɱɟɧɢɟɦ ɈȽɄ ɪɚɡɥɚɝɚɸɳɟɣɫɹ
ɩɨɥɢɦɟɪɧɨɣ ɩɥɟɧɤɢ .................................................................336
Ɂɚɞɚɧɢɹ.....................................................................................341
ɉɊɂɅɈɀȿɇɂə .......................................................................................344
1.
Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɢ ɩɪɚɜɢɥɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
Ʌɚɩɥɚɫɚ ....................................................................................344
2.
ɂɡɨɛɪɚɠɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɮɭɧɤɰɢɣ .........................................347
3.
Ɏɢɡɢɱɟɫɤɢɟ ɜɟɥɢɱɢɧɵ.............................................................349
ɉɊɂɇəɌɕȿ ɈȻɈɁɇȺɑȿɇɂə................................................................352
ɊȿɄɈɆȿɇȾɍȿɆȺə ɅɂɌȿɊȺɌɍɊȺ .....................................................355
7
ɉɊȿȾɂɋɅɈȼɂȿ
ɍɱɟɛɧɨɟ ɩɨɫɨɛɢɟ «Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɨɫɧɨɜɵ ɫɨɜɪɟɦɟɧɧɵɯ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɣ», ɩɪɟɞɧɚɡɧɚɱɟɧɨ, ɩɪɟɠɞɟ ɜɫɟɝɨ, ɞɥɹ ɦɚɝɢɫɬɪɚɧɬɨɜ ɢ ɚɫɩɢɪɚɧɬɨɜ, ɨɛɭɱɚɸɳɢɯɫɹ ɜ ɬɟɯɧɢɱɟɫɤɢɯ ɜɵɫɲɢɯ ɭɱɟɛɧɵɯ ɡɚɜɟɞɟɧɢɹɯ ɩɨ ɫɩɟɰɢɚɥɶɧɨɫɬɹɦ «Ɏɢɡɢɤɚ ɜɵɫɨɤɢɯ ɬɟɯɧɨɥɨɝɢɣ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ», «Ɍɟɩɥɨɮɢɡɢɤɚ ɢ ɬɟɩɥɨɬɟɯɧɢɤɚ» ɢ ɞɪ.
ȼ ɩɨɫɨɛɢɢ ɤɪɚɬɤɨ ɢɡɥɨɠɟɧɵ ɨɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɢ ɦɚɫɫɵ, ɚ ɬɚɤɠɟ ɫɨɩɭɬɫɬɜɭɸɳɢɯ ɢɦ ɹɜɥɟɧɢɣ, ɧɟɤɨɬɨɪɵɟ ɫɩɨɫɨɛɵ ɜɵɜɨɞɚ
ɭɪɚɜɧɟɧɢɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɢ ɞɥɹ ɠɢɞɤɨɫɬɢ, ɭɞɟɥɟɧɨ ɜɧɢɦɚɧɢɟ ɢɧɠɟɧɟɪɧɵɦ ɩɨɞɯɨɞɚɦ ɤ ɪɟɲɟɧɢɸ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɡɚɞɚɱ.
ȼ ɩɟɪɜɨɣ ɝɥɚɜɟ ɞɚɧɨ ɤɪɚɬɤɨɟ ɨɩɢɫɚɧɢɟ ɩɪɢɪɨɞɧɵɯ ɢ ɬɟɯɧɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ, ɝɞɟ ɹɜɥɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢɝɪɚɟɬ ɛɨɥɶɲɭɸ ɪɨɥɶ, ɨɛɨɫɧɨɜɵɜɚɟɬɫɹ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ, ɱɬɨ ɦɨɠɟɬ
ɫɩɨɫɨɛɫɬɜɨɜɚɬɶ ɤɚɤ ɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɸ ɬɟɯɧɨɥɨɝɢɣ, ɬɚɤ ɢ ɭɝɥɭɛɥɟɧɧɨɦɭ ɩɨɧɢɦɚɧɢɸ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ, ɩɪɢɜɨɞɹɬɫɹ ɨɫɧɨɜɧɵɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɫɨɨɬɧɨɲɟɧɢɹ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɞɪɭɝɢɯ ɝɥɚɜɚɯ. ȼɬɨɪɚɹ ɢ ɬɪɟɬɶɹ
ɝɥɚɜɵ ɩɨɫɜɹɳɟɧɵ ɨɩɢɫɚɧɢɸ ɦɟɯɚɧɢɡɦɨɜ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɢ ɜɵɜɨɞɭ ɨɫɧɨɜɧɵɯ ɭɪɚɜɧɟɧɢɣ. Ɂɞɟɫɶ ɠɟ ɩɪɢɜɨɞɹɬɫɹ ɩɪɢɦɟɪɵ ɩɪɨɫɬɵɯ ɦɨɞɟɥɟɣ, ɤɨɬɨɪɵɟ ɚɧɚɥɢɡɢɪɭɸɬɫɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɚɱɟɫɬɜɟɧɧɵɯ ɢɧɠɟɧɟɪɧɵɯ
ɩɨɞɯɨɞɨɜ; ɜɜɨɞɹɬɫɹ ɮɢɡɢɱɟɫɤɢɟ ɩɨɧɹɬɢɹ (ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɛɦɟɧɚ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ, ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɢ ɞɪ.). ɋɨɛɫɬɜɟɧɧɨ ɤɨɧɜɟɤɬɢɜɧɨɦɭ ɬɟɩɥɨɨɛɦɟɧɭ ɩɨɫɜɹɳɟɧɚ ɝɥɚɜɚ 3 ɢ ɧɟɤɨɬɨɪɵɟ ɪɚɡɞɟɥɵ ɝɥɚɜɵ 4, ɝɞɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨ ɩɪɢɜɨɞɢɬɫɹ ɫɜɨɞɤɚ ɨɫɧɨɜɧɵɯ ɬɟɨɪɟɦ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ, ɞɚɟɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟ
ɨ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɢ ɦɟɬɨɞɟ ɚɧɚɥɢɡɚ ɪɚɡɦɟɪɧɨɫɬɟɣ, ɤɨɬɨɪɵɟ ɧɚ ɩɪɚɤɬɢɤɟ
ɫ ɭɫɩɟɯɨɦ ɩɪɢɦɟɧɹɸɬɫɹ ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɢ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɞɚɧɧɵɯ ɬɟɩɥɨɮɢɡɢɱɟɫɤɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ. ɉɨɫɤɨɥɶɤɭ ɩɪɨɰɟɫɫɵ ɨɛɪɚɛɨɬɤɢ ɩɪɢɦɟɧɹɸɬɫɹ ɤ ɞɟɬɚɥɹɦ ɫɚɦɵɯ ɪɚɡɥɢɱɧɵɯ ɮɨɪɦ, ɬɨ ɞɥɹ ɢɡɭɱɟɧɢɹ ɜ ɤɭɪɫɟ ɬɟɩɥɨɮɢɡɢɤɢ ɩɨɥɟɡɧɨɣ ɛɭɞɟɬ ɝɥɚɜɚ 5, ɝɞɟ ɩɪɢɜɨɞɢɬɫɹ ɫɜɨɞɤɚ ɭɪɚɜɧɟɧɢɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦ ɤɨɨɪɞɢɧɚɬ, ɞɚɸɬɫɹ ɩɪɢɦɟɪɵ ɩɪɚɤɬɢɱɟɫɤɢ ɩɨɥɟɡɧɵɯ ɡɚɞɚɱ, ɜ ɬɨɦ ɱɢɫɥɟ ɞɥɹ ɩɪɨɛɥɟɦ ɬɟɩɥɨɜɨɣ ɡɚɳɢɬɵ,
ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. ɒɟɫɬɚɹ ɝɥɚɜɚ ɜɤɥɸɱɚɟɬ ɱɚɫɬɧɵɟ ɡɚɞɚɱɢ ɫ
ɨɛɴɟɦɧɵɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɬɟɩɥɚ ɪɚɡɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɩɪɢɪɨɞɵ, ɧɚɩɪɢɦɟɪ,
ɫɜɹɡɚɧɧɵɟ ɫ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɧɚɝɪɟɜɨɦ. ɂɧɬɟɪɟɫ ɞɥɹ ɛɭɞɭɳɢɯ ɢɧɠɟɧɟɪɨɜ
ɢ ɢɫɫɥɟɞɨɜɚɬɟɥɟɣ ɩɪɟɞɫɬɚɜɥɹɟɬ, ɧɚɩɪɢɦɟɪ, ɡɚɞɚɱɚ ɨɛ ɨɲɢɛɤɚɯ ɬɟɪɦɨɩɚɪɵ. ɋɨɛɫɬɜɟɧɧɨ ɚɧɚɥɢɬɢɱɟɫɤɢɦ ɦɟɬɨɞɚɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨɫɜɹɳɟɧɵ 7-ɹ ɢ 8-ɹ ɝɥɚɜɵ. ɇɨ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɢɡɜɟɫɬɧɵɯ ɭɱɟɛɧɢɤɨɜ
ɩɨ ɦɟɬɨɞɚɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱ, ɡɞɟɫɶ ɧɟ ɞɨɤɚɡɵɜɚɸɬɫɹ ɬɟɨɪɟɦɵ. Ɋɟɤɨɦɟɧɞɚɰɢɢ ɩɨ ɩɪɚɤɬɢɱɟɫɤɨɦɭ ɢɫɩɨɥɶɡɨɜɚɧɢɸ ɬɟɯ ɢɥɢ ɢɧɵɯ ɦɟɬɨɞɨɜ ɞɚɸɬɫɹ ɧɚ
ɩɪɢɦɟɪɚɯ ɡɚɞɚɱ, ɢɦɟɸɳɢɯ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɨɬɧɨɲɟɧɢɟ ɤ ɫɨɜɪɟɦɟɧɧɵɦ
8
ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɦ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦ ɩɪɨɰɟɫɫɚɦ. Ɉɫɧɨɜɧɵɟ ɡɚɤɨɧɵ
ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ ɢ ɪɹɞ ɱɚɫɬɧɵɯ ɡɚɞɚɱ ɬɟɨɪɢɢ ɩɟɪɟɧɨɫɚ ɢɡɥɭɱɟɧɢɹ
ɞɚɧɵ ɜ ɝɥɚɜɟ 9. ɋɨɩɭɬɫɬɜɭɸɳɢɦ ɹɜɥɟɧɢɹɦ – ɮɚɡɨɜɵɦ ɩɟɪɟɯɨɞɚɦ, ɯɢɦɢɱɟɫɤɢɦ ɩɪɟɜɪɚɳɟɧɢɹɦ ɢ ɹɜɥɟɧɢɹɦ ɞɢɮɮɭɡɢɢ ɩɨɫɜɹɳɟɧɵ ɝɥɚɜɵ 10 ɢ 11,
ɝɞɟ ɩɨɦɢɦɨ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ, ɩɪɢɜɨɞɹɬɫɹ ɱɚɫɬɧɵɟ ɡɚɞɚɱɢ, ɜɚɪɢɚɧɬɵ
ɤɨɬɨɪɵɯ ɜɫɬɪɟɱɚɸɬɫɹ ɜ ɩɪɨɰɟɫɫɚɯ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ, ɪɹɞ ɩɪɨɫɬɵɯ,
ɧɨ ɩɨɥɟɡɧɵɯ ɡɚɞɚɱ ɯɢɦɢɱɟɫɤɨɣ ɤɢɧɟɬɢɤɢ ɢ ɬɟɨɪɢɢ ɞɢɮɮɭɡɢɢ. ȼɫɸɞɭ
ɭɞɟɥɹɟɬɫɹ ɜɧɢɦɚɧɢɟ ɮɢɡɢɱɟɫɤɨɣ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɩɨɫɬɚɧɨɜɨɤ ɡɚɞɚɱ ɢ ɩɨɥɭɱɚɟɦɵɯ ɪɟɲɟɧɢɣ. Ɂɚɤɥɸɱɢɬɟɥɶɧɨɣ ɝɥɚɜɨɣ ɜ ɤɭɪɫɟ ɹɜɥɹɟɬɫɹ 12-ɹ ɝɥɚɜɚ,
ɝɞɟ ɦɨɠɧɨ ɧɚɣɬɢ ɫɚɦɵɟ ɪɚɡɧɵɟ ɡɚɞɚɱɢ, ɹɜɥɹɸɳɢɟɫɹ ɩɨɜɬɨɪɟɧɢɟɦ ɜɫɟɝɨ
ɩɪɨɣɞɟɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ȼ ɩɨɫɨɛɢɟ (ɜ ɪɚɡɧɵɟ ɝɥɚɜɵ) ɜɤɥɸɱɟɧɵ ɪɹɞ ɡɚɞɚɱ, ɤɨɬɨɪɵɯ ɧɟɬ ɜ
ɤɥɚɫɫɢɱɟɫɤɢɯ ɭɱɟɛɧɢɤɚɯ ɩɨ ɬɟɩɥɨɮɢɡɢɤɟ. Ɉɧɢ ɡɚɢɦɫɬɜɨɜɚɧɵ ɢɡ ɧɚɭɱɧɨɣ
(ɚ ɢɧɨɝɞɚ ɢ ɢɡ ɠɭɪɧɚɥɶɧɵɯ ɫɬɚɬɟɣ) ɥɢɬɟɪɚɬɭɪɵ. ȼ ɧɟɤɨɬɨɪɵɯ ɪɚɡɞɟɥɚɯ
ɭɱɟɛɧɨɝɨ ɤɭɪɫɚ ɞɚɧɵ ɫɧɨɫɤɢ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɭɝɥɭɛɥɟɧɢɹ ɜ ɞɚɧɧɭɸ ɧɚɭɱɧɭɸ ɩɪɨɛɥɟɦɭ. Ʉɚɠɞɵɣ ɪɚɡɞɟɥ (ɤɪɨɦɟ ɩɨɫɥɟɞɧɟɝɨ)
ɫɨɞɟɪɠɢɬ ɨɤɨɥɨ ɞɟɫɹɬɢ ɜɨɩɪɨɫɨɜ ɧɚ ɩɨɜɬɨɪɟɧɢɟ ɩɪɨɣɞɟɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ȼ ɤɨɧɰɟ ɤɚɠɞɨɣ ɝɥɚɜɵ ɩɪɟɞɫɬɚɜɥɟɧɵ ɡɚɞɚɱɢ, ɤɨɬɨɪɵɟ ɪɟɤɨɦɟɧɞɭɸɬɫɹ
ɞɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɣ ɩɪɨɪɚɛɨɬɤɢ ɢɥɢ ɞɥɹ ɪɟɲɟɧɢɹ ɜɨ ɜɪɟɦɹ ɩɪɚɤɬɢɱɟɫɤɢɯ ɢɥɢ ɥɚɛɨɪɚɬɨɪɧɵɯ ɡɚɧɹɬɢɣ. ɑɢɫɥɟɧɧɵɦ ɦɟɬɨɞɚɦ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɞɚɧɧɨɦ ɤɭɪɫɟ ɨɫɨɛɨɝɨ ɜɧɢɦɚɧɢɹ ɧɟ ɭɞɟɥɹɟɬɫɹ, ɬɚɤ ɤɚɤ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɢɦɟɸɬɫɹ ɦɧɨɝɨɱɢɫɥɟɧɧɵɟ ɫɩɟɰɢɚɥɶɧɵɟ ɭɱɟɛɧɵɟ ɤɭɪɫɵ.
ɋɩɢɫɨɤ ɥɢɬɟɪɚɬɭɪɵ, ɜ ɬɨɦ ɱɢɫɥɟ ɢ ɩɨ ɱɢɫɥɟɧɧɵɦ ɦɟɬɨɞɚɦ, ɪɟɤɨɦɟɧɞɨɜɚɧɧɨɣ ɞɥɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ ɢɡɭɱɟɧɢɹ, ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɤɨɧɰɟ ɩɨɫɨɛɢɹ.
Ɂɚɦɟɱɭ, ɱɬɨ ɜ ɞɚɧɧɨɦ ɤɭɪɫɟ ɜɨ ɜɫɟɯ ɱɚɫɬɧɵɯ ɢ ɱɢɫɥɟɧɧɵɯ ɩɪɢɦɟɪɚɯ,
ɚ ɬɚɤɠɟ ɜ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɡɚɞɚɧɢɹɯ ɢɫɩɨɥɶɡɨɜɚɧɚ ɫɦɟɲɚɧɧɚɹ ɫɢɫɬɟɦɚ
ɟɞɢɧɢɰ ɢɡɦɟɪɟɧɢɹ, ɜɤɥɸɱɚɹ ɫɢɫɬɟɦɭ ɟɞɢɧɢɰ ɋɂ ɢ ɪɚɡɧɵɟ ɬɟɯɧɢɱɟɫɤɢɟ
ɟɞɢɧɢɰɵ. Ɍɚɤ ɤɚɤ ɜ ɩɪɚɤɬɢɤɟ ɢɧɠɟɧɟɪɚɦ ɢ ɢɫɫɥɟɞɨɜɚɬɟɥɹɦ (ɜ ɬɨɦ ɱɢɫɥɟ,
ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɫɩɪɚɜɨɱɧɨɣ ɥɢɬɟɪɚɬɭɪɵ) ɩɪɢɯɨɞɢɬɫɹ ɫɬɚɥɤɢɜɚɬɶɫɹ ɫ
ɪɚɡɥɢɱɧɵɦɢ ɫɢɫɬɟɦɚɦɢ ɟɞɢɧɢɰ, ɩɪɢ ɪɟɲɟɧɢɢ ɱɚɫɬɧɵɯ ɡɚɞɚɱ ɩɨɥɟɡɧɨ
ɛɭɞɟɬ ɨɫɜɨɢɬɶ, ɤɚɤ ɪɚɡɧɵɟ ɫɢɫɬɟɦɵ ɟɞɢɧɢɰ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨɛɨɣ.
ɉɪɟɞɫɬɚɜɥɟɧɧɨɟ ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ ɹɜɥɹɟɬɫɹ ɥɢɲɶ ɜɜɟɞɟɧɢɟɦ ɜ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɟɪɜɭɸ
ɱɚɫɬɶ ɡɚɞɭɦɚɧɧɨɝɨ ɚɜɬɨɪɨɦ ɤɭɪɫɚ. ȼɨ ɜɬɨɪɭɸ ɱɚɫɬɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ
ɜɤɥɸɱɢɬɶ ɤɨɦɩɥɟɤɫɧɵɟ ɦɨɞɟɥɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɨɛɪɚɛɨɬɤɢ
ɦɚɬɟɪɢɚɥɨɜ ɢ ɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɯ
ɢɫɬɨɱɧɢɤɨɜ ɫ ɩɨɞɪɨɛɧɵɦ ɨɩɢɫɚɧɢɟɦ ɮɢɡɢɱɟɫɤɨɣ ɨɫɧɨɜɵ ɩɨɫɬɚɧɨɜɨɤ ɡɚɞɚɱ ɢ ɦɟɬɨɞɨɜ ɢɯ ɪɟɲɟɧɢɹ, ɱɬɨ ɡɚɱɚɫɬɭɸ ɜɵɩɭɫɤɚɟɬɫɹ ɢɡ ɧɚɭɱɧɨɣ ɢ
ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɵ.
Ⱦ.ɮ.-ɦ.ɧ., ɩɪɨɮɟɫɫɨɪ
Ʉɧɹɡɟɜɚ Ⱥɧɧɚ Ƚɟɨɪɝɢɟɜɧɚ
9
ɑȺɋɌɖ 1
ȼ ɜ ɟ ɞ ɟ ɧ ɢ ɟ ɜ ɦ ɨ ɞ ɟ ɥ ɢ ɪ ɨ ɜ ɚɧ ɢ ɟ
ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ
1.1. Ɍɟɦɩɟɪɚɬɭɪɚ, ɬɟɩɥɨɮɢɡɢɤɚ, ɬɟɩɥɨɨɛɦɟ ɧ
Ɍɟɦɩɟɪɚɬɭɪɚ – ɨɞɧɨ ɢ ɨɫɧɨɜɧɵɯ ɩɨɧɹɬɢɣ, ɢɝɪɚɸɳɢɯ ɜɚɠɧɭɸ ɪɨɥɶ ɧɟ
ɬɨɥɶɤɨ ɜ ɬɟɪɦɨɞɢɧɚɦɢɤɟ, ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɧɨ ɢ ɜ ɮɢɡɢɤɟ ɜ ɰɟɥɨɦ. Ɍɟɦɩɟɪɚɬɭɪɚ ɬɟɥɚ ɟɫɬɶ ɦɟɪɚ ɟɝɨ ɧɚɝɪɟɬɨɫɬɢ.
ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɦɨɥɟɤɭɥɹɪɧɨ-ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ, ɬɟɦɩɟɪɚɬɭɪɚ ɟɫɬɶ
ɦɟɪɚ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɞɜɢɠɟɧɢɹ ɦɨɥɟɤɭɥ. Ɍɟɦɩɟɪɚɬɭɪɚ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɱɚɫɬɢɰ ɪɚɛɨɱɟɝɨ ɬɟɥɚ. ȿɟ ɱɢɫɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɫɜɹɡɚɧɨ ɫ ɜɟɥɢɱɢɧɨɣ ɫɪɟɞɧɟɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɦɨɥɟɤɭɥ ɜɟɳɟɫɬɜɚ
mV 2 3
kT ,
2
2
ɝɞɟ k – ɩɨɫɬɨɹɧɧɚɹ Ȼɨɥɶɰɦɚɧɚ 1,380662•10-23 Ⱦɠ/Ʉ, m – ɦɚɫɫɚ ɷɥɟɦɟɧɬɚɪɧɵɯ ɱɚɫɬɢɰ (ɦɨɥɟɤɭɥ), V – ɫɪɟɞɧɟɤɜɚɞɪɚɬɢɱɧɚɹ ɫɤɨɪɨɫɬɶ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ. Ɍɟɦɩɟɪɚɬɭɪɚ, ɨɩɪɟɞɟɥɟɧɧɚɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ɧɚɡɵɜɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨɣ.
ȼ ɫɢɫɬɟɦɟ ɋɂ ɟɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɹɜɥɹɟɬɫɹ Ʉɟɥɶɜɢɧ
(Ʉ). ɋɭɳɟɫɬɜɭɸɬ ɞɪɭɝɢɟ ɲɤɚɥɵ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ: Ɏɚɪɟɧɝɟɣɬɚ, Ɋɟɨɦɸɪɚ ɢ ɐɟɥɶɫɢɹ.
Ⱦɥɹ ɧɢɯ ɫɩɪɚɜɟɞɥɢɜɵ ɫɨɨɬɧɨɲɟɧɢɹ
T 273K t qC t qR t 32qF ,
5
5
4
9
ɝɞɟ K – ɝɪɚɞɭɫɵ ɩɨ Ʉɟɥɶɜɢɧɭ, C – ɩɨ ɐɟɥɶɫɢɸ, R – ɩɨ Ɋɟɨɦɸɪɭ, F –
ɩɨ Ɏɚɪɟɧɝɟɣɬɭ.
ɉɨɧɹɬɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɫɜɹɡɚɧɨ ɫ ɜɟɫɶɦɚ ɧɟɨɩɪɟɞɟɥɟɧɧɵɦɢ ɩɨɧɹɬɢɹɦɢ ɬɟɩɥɚ ɢ ɯɨɥɨɞɚ. ɉɨɤɚ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚɭɱɢɥɢɫɶ ɢɡɦɟɪɹɬɶ, ɢ ɩɪɢɲɥɢ ɤ
ɩɨɧɹɬɢɸ ɚɛɫɨɥɸɬɧɨɣ ɲɤɚɥɵ ɬɟɦɩɟɪɚɬɭɪɵ, ɩɪɨɲɥɨ ɧɟɦɚɥɨ ɜɪɟɦɟɧɢ – ɨɬ
Ƚɚɥɢɥɟɹ ɞɨ ɐɟɥɶɫɢɹ1. ɉɨɧɹɬɢɹ «ɬɟɩɥɨ» ɢ «ɬɟɦɩɟɪɚɬɭɪɚ» ɪɚɡɞɟɥɢɬɶ ɛɵɥɨ
ɟɳɟ ɫɥɨɠɧɟɟ. Ʉɨɝɞɚ ɧɚɝɪɟɜɚɸɬ ɬɟɥɨ, ɬɟɦɩɟɪɚɬɭɪɚ ɟɝɨ ɩɨɜɵɲɚɟɬɫɹ. Ʉɨɝɞɚ ɬɟɩɥɨ ɩɟɪɟɬɟɤɚɟɬ ɨɬ ɨɞɧɨɝɨ ɬɟɥɚ ɤ ɞɪɭɝɨɦɭ, ɬɟɦɩɟɪɚɬɭɪɚ ɨɞɧɨɝɨ ɬɟɥɚ
ɩɚɞɚɟɬ, ɚ ɞɪɭɝɨɝɨ ɩɨɜɵɲɚɟɬɫɹ. ȼɟɥɢɤɢɦ ɡɚɤɨɧɨɦ ɩɪɢɪɨɞɵ ɧɭɠɧɨ ɫɱɢɬɚɬɶ ɬɨɬ ɮɚɤɬ, ɱɬɨ ɬɟɩɥɨ ɜɫɟɝɞɚ ɩɟɪɟɬɟɤɚɟɬ ɨɬ ɝɨɪɹɱɟɝɨ ɬɟɥɚ ɤ ɯɨɥɨɞɧɨɦɭ, ɢ ɬɟɦɩɟɪɚɬɭɪɚ ɫɨɩɪɢɤɚɫɚɸɳɢɯɫɹ ɬɟɥ ɫɬɪɟɦɢɬɫɹ ɜɵɪɨɜɧɹɬɶɫɹ. ɗɬɨ
1
ɋɦɨɪɨɞɢɧɫɤɢɣ ə.Ⱥ. Ɍɟɦɩɟɪɚɬɭɪɚ. Ɇ.: ɇɚɭɤɚ, 1987. 192 c.
10
ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɟɬ ɩɪɨɰɟɫɫ ɩɟɪɟɞɚɱɢ ɬɟɩɥɚ ɨɬ ɦɟɯɚɧɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɩɪɨɬɟɤɚɬɶ ɜ ɪɚɡɧɵɯ ɧɚɩɪɚɜɥɟɧɢɹɯ.
ɋɚɦɨɩɪɨɢɡɜɨɥɶɧɵɣ ɧɟɨɛɪɚɬɢɦɵɣ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ (ɬɨɱɧɟɟ, ɷɧɟɪɝɢɢ
ɜ ɮɨɪɦɟ ɬɟɩɥɨɬɵ) ɦɟɠɞɭ ɬɟɥɚɦɢ ɢɥɢ ɭɱɚɫɬɤɚɦɢ ɜɧɭɬɪɢ ɬɟɥɚ ɫ ɪɚɡɥɢɱɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɨɣ, ɧɚɡɵɜɚɸɬ ɬɟɩɥɨɨɛɦɟɧɨɦ. ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɦɨɠɟɬ ɜɵɡɵɜɚɬɶɫɹ ɬɚɤɠɟ ɧɟɨɞɧɨɪɨɞɧɨɫɬɶɸ ɩɨɥɟɣ ɢɧɵɯ ɮɢɡɢɱɟɫɤɢɯ
ɜɟɥɢɱɢɧ, ɧɚɩɪɢɦɟɪ, ɝɪɚɞɢɟɧɬɨɦ ɤɨɧɰɟɧɬɪɚɰɢɣ (ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɞɢɮɮɭɡɢɨɧɧɵɣ ɬɟɪɦɨɷɮɮɟɤɬ). Ɍɟɩɥɨɨɛɦɟɧ ɫɭɳɟɫɬɜɟɧɟɧ ɜɨ ɦɧɨɝɢɯ ɩɪɨɰɟɫɫɚɯ
ɧɚɝɪɟɜɚɧɢɹ, ɨɯɥɚɠɞɟɧɢɹ, ɤɨɧɞɟɧɫɚɰɢɢ, ɤɢɩɟɧɢɹ, ɜɵɩɚɪɢɜɚɧɢɹ, ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ, ɩɥɚɜɥɟɧɢɹ ɢ ɨɤɚɡɵɜɚɟɬ ɡɧɚɱɢɬɟɥɶɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɦɚɫɫɨɨɛɦɟɧɧɵɟ (ɚɛɫɨɪɛɰɢɹ, ɞɢɫɬɢɥɥɹɰɢɹ, ɪɟɤɬɢɮɢɤɚɰɢɹ, ɫɭɲɤɚ ɢ ɞɪ.) ɢ ɯɢɦɢɱɟɫɤɢɟ
ɩɪɨɰɟɫɫɵ.
Ⱦɜɢɠɭɳɢɟɫɹ ɫɪɟɞɵ, ɭɱɚɫɬɜɭɸɳɢɟ ɜ ɬɟɩɥɨɨɛɦɟɧɟ ɢ ɢɧɬɟɧɫɢɮɢɰɢɪɭɸɳɢɟ ɟɝɨ, ɧɚɡɵɜɚɸɬɫɹ ɬɟɩɥɨɧɨɫɢɬɟɥɹɦɢ (ɨɛɵɱɧɨ ɤɚɩɟɥɶɧɵɟ ɠɢɞɤɨɫɬɢ, ɝɚɡɵ ɢ ɩɚɪɵ, ɪɟɠɟ - ɫɵɩɭɱɢɟ ɦɚɬɟɪɢɚɥɵ). ɂɡɜɟɫɬɧɵ ɞɜɚ ɨɫɧɨɜɧɵɯ
ɫɩɨɫɨɛɚ ɩɪɨɜɟɞɟɧɢɹ ɬɟɩɥɨɜɵɯ ɩɪɨɰɟɫɫɨɜ: ɩɭɬɟɦ ɬɟɩɥɨɨɬɞɚɱɢ ɢ ɬɟɩɥɨɩɟɪɟɞɚɱɟɣ. Ɍɟɩɥɨɨɬɞɚɱɚ – ɷɬɨ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɪɚɡɞɟɥɚ
ɮɚɡ (ɱɚɳɟ ɬɜɟɪɞɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ) ɢ ɬɟɩɥɨɧɨɫɢɬɟɥɟɦ. Ɍɟɩɥɨɩɟɪɟɞɚɱɚ –
ɷɬɨ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɞɜɭɦɹ ɬɟɩɥɨɧɨɫɢɬɟɥɹɦɢ ɢɥɢ ɢɧɵɦɢ ɫɪɟɞɚɦɢ ɱɟɪɟɡ ɪɚɡɞɟɥɹɸɳɭɸ ɢɯ ɬɜɟɪɞɭɸ ɫɬɟɧɤɭ ɥɢɛɨ ɦɟɠɮɚɡɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ.
Ɍɟɨɪɢɟɣ ɬɟɩɥɨɨɛɦɟɧɚ ɢɥɢ ɬɟɩɥɨɮɢɡɢɤɨɣ ɧɚɡɵɜɚɟɬɫɹ ɧɚɭɤɚ, ɢɡɭɱɚɸɳɚɹ ɩɪɨɰɟɫɫɵ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɬɟɦɩɟɪɚɬɭɪɧɵɦ ɩɨɥɟɦ ɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɜɟɳɟɫɬɜ.
1.2. Ʉɨɥɢɱɟɫ ɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ
ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɥɨɬɧɨɫɬɶɸ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɬ.ɟ. ɤɨɥɢɱɟɫɬɜɨɦ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɜɚɟɦɨɣ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɞɢɧɢɱɧɭɸ ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ. ɗɬɚ ɜɟɥɢɱɢɧɚ ɢɡɦɟɪɹɟɬɫɹ
ɜ ȼɬ/ɫɦ2 ɢɥɢ Ⱦɠ/(ɫɦ2ɫ). Ȼɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɟɟ ɛɭɤɜɨɣ q .
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɜɚɟɦɨɟ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɩɪɨɢɡɜɨɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ F , ɜ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ ɦɨɳɧɨɫɬɶɸ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢɥɢ ɩɪɨɫɬɨ ɬɟɩɥɨɜɵɦ ɩɨɬɨɤɨɦ - Q . ȿɞɢɧɢɰɟɣ
ɟɟ ɢɡɦɟɪɟɧɢɹ ɫɥɭɠɢɬ Ⱦɠ/ɫ ɢɥɢ ȼɬ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɜɚɟɦɨɟ ɡɚ ɩɪɨɢɡɜɨɥɶɧɵɣ ɩɪɨɦɟɠɭɬɨɤ
ɜɪɟɦɟɧɢ W ɱɟɪɟɡ ɩɪɨɢɡɜɨɥɶɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ F ɟɫɬɶ QW . ɗɬɚ ɜɟɥɢɱɢɧɚ
ɢɡɦɟɪɹɟɬɫɹ ɜ Ⱦɠ.
ɗɬɢ ɜɟɥɢɱɢɧɵ ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
q Q F QW F W .
(1.1)
11
Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɢ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɹɜɥɹɸɬɫɹ ɩɪɟɞɦɟɬɨɦ ɢɡɭɱɟɧɢɹ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ
(ɬɟɩɥɨɩɟɪɟɞɚɱɢ).
1.3. Ɇ ɟɯɚ ɧɢ ɡɦ ɵ ɩɟ ɪɟ ɧɨɫɚ ɬɟ ɩɥ ɚ :
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, ɤɨɧɜɟɤɰɢɹ, ɢɡɥ ɭɱɟɧɢɟ
Ɍɟɩɥɨɬɚ ɦɨɠɟɬ ɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɜ ɥɸɛɵɯ ɜɟɳɟɫɬɜɚɯ ɢ ɞɚɠɟ ɱɟɪɟɡ
ɜɚɤɭɭɦ.
ɋɭɳɟɫɬɜɭɟɬ ɬɪɢ ɨɫɧɨɜɧɵɯ ɦɟɯɚɧɢɡɦɚ ɬɟɩɥɨɩɟɪɟɧɨɫɚ – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, ɤɨɧɜɟɤɰɢɹ ɢ ɢɡɥɭɱɟɧɢɟ.
ȼɨ ɜɫɟɯ ɜɟɳɟɫɬɜɚɯ ɬɟɩɥɨɬɚ ɩɟɪɟɞɚɟɬɫɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɡɚ ɫɱɟɬ
ɩɟɪɟɧɨɫɚ ɷɧɟɪɝɢɢ ɦɢɤɪɨɱɚɫɬɢɰɚɦɢ. Ɇɨɥɟɤɭɥɵ, ɚɬɨɦɵ, ɷɥɟɤɬɪɨɧɵ ɢ ɞɪɭɝɢɟ ɦɢɤɪɨɱɚɫɬɢɰɵ, ɢɡ ɤɨɬɨɪɵɯ ɫɨɫɬɨɢɬ ɜɟɳɟɫɬɜɨ, ɞɜɢɠɭɳɢɟɫɹ ɫɨ ɫɤɨɪɨɫɬɹɦɢ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɦɢ ɢɯ ɬɟɦɩɟɪɚɬɭɪɟ, ɩɟɪɟɧɨɫɹɬ ɷɧɟɪɝɢɸ ɢɡ ɡɨɧɵ
ɫ ɛɨɥɟɟ ɜɵɫɨɤɢɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ ɜ ɡɨɧɭ ɫ ɛɨɥɟɟ ɧɢɡɤɢɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ.
ȼ ɠɢɞɤɨɫɬɹɯ, ɧɚɪɹɞɭ ɫ ɞɜɢɠɟɧɢɟɦ ɦɢɤɪɨɱɚɫɬɢɰ, ɦɟɠɞɭ ɡɨɧɚɦɢ ɫ
ɪɚɡɧɵɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ, ɜɨɡɦɨɠɧɨ ɩɟɪɟɦɟɳɟɧɢɟ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɯ
ɨɛɴɟɦɨɜ. ȼ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ, ɤɚɤ ɢ ɜ ɝɢɞɪɨɦɟɯɚɧɢɤɟ, ɬɟɪɦɢɧɨɦ
«ɠɢɞɤɨɫɬɶ» ɨɛɨɡɧɚɱɚɟɬɫɹ ɥɸɛɚɹ ɫɩɥɨɲɧɚɹ ɫɪɟɞɚ, ɨɛɥɚɞɚɸɳɚɹ ɫɜɨɣɫɬɜɨɦ ɬɟɤɭɱɟɫɬɢ. ɉɨɞɪɚɡɞɟɥɟɧɢɟ ɧɚ ɤɚɩɟɥɶɧɭɸ ɠɢɞɤɨɫɬɶ ɢ ɝɚɡ ɢɫɩɨɥɶɡɭɟɬɫɹ ɬɨɥɶɤɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɚɝɪɟɝɚɬɧɨɟ ɫɨɫɬɨɹɧɢɟ ɜɟɳɟɫɬɜɚ ɢɝɪɚɟɬ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɪɨɰɟɫɫɟ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ. ɉɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɜɦɟɫɬɟ ɫ
ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɦɢ ɨɛɴɟɦɚɦɢ ɜɟɳɟɫɬɜɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɩɟɪɟɧɨɫɚ ɢɥɢ ɩɪɨɫɬɨ ɤɨɧɜɟɤɰɢɢ.
Ʉɨɧɜɟɤɰɢɟɣ ɦɨɠɧɨ ɩɟɪɟɞɚɜɚɬɶ ɬɟɩɥɨɬɭ ɧɚ ɛɨɥɶɲɢɟ ɪɚɫɫɬɨɹɧɢɹ. ɇɚɩɪɢɦɟɪ, ɨɬ Ɍɗɐ (ɬɟɩɥɨɷɥɟɤɬɪɨɰɟɧɬɪɚɥɢ) ɬɟɩɥɨɬɚ ɩɟɪɟɞɚɟɬɫɹ ɩɨ ɬɪɭɛɚɦ
ɜɦɟɫɬɟ ɫ ɞɜɢɠɭɳɟɣɫɹ ɝɨɪɹɱɟɣ ɜɨɞɨɣ ɧɚ ɞɟɫɹɬɤɢ ɤɢɥɨɦɟɬɪɨɜ ɞɥɹ ɨɬɨɩɥɟɧɢɹ ɠɢɥɵɯ ɢ ɩɪɨɦɵɲɥɟɧɧɵɯ ɡɞɚɧɢɣ. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɬɟɩɥɨɧɨɫɢɬɟɥɟɦ
ɹɜɥɹɟɬɫɹ ɜɨɞɚ.
ɑɚɫɬɨ ɩɪɢɯɨɞɢɬɫɹ ɪɚɫɫɱɢɬɵɜɚɬɶ ɤɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ
ɠɢɞɤɨɫɬɶɸ ɢ ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɜɟɪɞɨɝɨ ɬɟɥɚ. ɗɬɨɬ ɩɪɨɰɟɫɫ ɩɨɥɭɱɢɥ ɫɩɟɰɢɚɥɶɧɨɟ ɧɚɡɜɚɧɢɟ – ɤɨɧɜɟɤɬɢɜɧɚɹ ɬɟɩɥɨɨɬɞɚɱɚ (ɬɟɩɥɨɬɚ ɨɬɞɚɟɬɫɹ ɨɬ
ɠɢɞɤɨɫɬɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɢɥɢ ɧɚɨɛɨɪɨɬ).
Ɍɪɟɬɶɢɦ ɫɩɨɫɨɛɨɦ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɹɜɥɹɟɬɫɹ ɢɡɥɭɱɟɧɢɟ. Ɂɚ ɫɱɟɬ ɢɡɥɭɱɟɧɢɹ ɬɟɩɥɨɬɚ ɩɟɪɟɞɚɟɬɫɹ ɜɨ ɜɫɟɯ ɥɭɱɟɩɪɨɡɪɚɱɧɵɯ ɫɪɟɞɚɯ, ɜ ɬɨɦ ɱɢɫɥɟ, ɢ ɜ ɜɚɤɭɭɦɟ, ɧɚɩɪɢɦɟɪ, ɜ ɤɨɫɦɨɫɟ, ɝɞɟ ɷɬɨ – ɟɞɢɧɫɬɜɟɧɧɵɣ ɫɩɨɫɨɛ
ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɦɟɠɞɭ ɬɟɥɚɦɢ. ɇɨɫɢɬɟɥɹɦɢ ɷɧɟɪɝɢɢ ɩɪɢ ɬɟɩɥɨɨɛɦɟɧɟ
ɢɡɥɭɱɟɧɢɟɦ ɹɜɥɹɸɬɫɹ ɮɨɬɨɧɵ, ɢɡɥɭɱɚɟɦɵɟ ɢ ɩɨɝɥɨɳɚɟɦɵɟ ɬɟɥɚɦɢ, ɭɱɚɫɬɜɭɸɳɢɦɢ ɜ ɬɟɩɥɨɨɛɦɟɧɟ.
12
ɑɚɫɬɨ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɪɚɡɥɢɱɧɵɦɢ
ɫɩɨɫɨɛɚɦɢ (ɫɥɭɱɚɣ ɫɥɨɠɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ). ɇɚɩɪɢɦɟɪ, ɤɨɧɜɟɤɬɢɜɧɚɹ
ɬɟɩɥɨɨɬɞɚɱɚ ɨɬ ɝɚɡɚ ɤ ɫɬɟɧɤɟ ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɩɚɪɚɥɥɟɥɶɧɵɦ ɩɟɪɟɧɨɫɨɦ ɬɟɩɥɨɬɵ ɢɡɥɭɱɟɧɢɟɦ.
ɉɟɪɟɧɨɫ ɦɚɫɫɵ ɜɟɳɟɫɬɜɚ ɢɡ ɨɞɧɨɣ ɬɨɱɤɢ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜ ɞɪɭɝɭɸ ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɧɚɥɢɱɢɢ ɪɚɡɧɨɫɬɢ ɤɨɧɰɟɧɬɪɚɰɢɣ ɞɚɧɧɨɝɨ ɜɟɳɟɫɬɜɚ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɬɨɱɤɚɯ. ɇɚɩɪɢɦɟɪ, ɤɨɦɩɨɧɟɧɬɵ ɝɚɡɨɜɨɣ ɫɦɟɫɢ, ɧɟɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɟ ɩɨ ɨɛɴɟɦɭ, ɛɭɞɭɬ ɩɟɪɟɧɨɫɢɬɶɫɹ ɢɡ ɡɨɧ, ɝɞɟ ɢɯ
ɤɨɧɰɟɧɬɪɚɰɢɹ ɩɨɜɵɲɟɧɚ, ɜ ɡɨɧɵ ɫ ɧɢɡɤɨɣ ɤɨɧɰɟɧɬɪɚɰɢɟɣ.
ɋɩɨɫɨɛɵ ɩɟɪɟɧɨɫɚ ɦɚɫɫɵ, ɤɚɤ ɢ ɬɟɩɥɨɬɵ, ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵɦɢ.
ȿɫɥɢ ɦɚɫɫɚ ɩɟɪɟɧɨɫɢɬɫɹ ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ ɞɜɢɠɟɧɢɹ ɚɬɨɦɨɜ ɢɥɢ ɦɨɥɟɤɭɥ,
ɬɨ ɬɚɤɨɣ ɩɪɨɰɟɫɫ ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɭɡɢɟɣ. ɇɚɢɛɨɥɟɟ ɢɧɬɟɧɫɢɜɧɨ ɞɢɮɮɭɡɢɢ ɩɪɨɬɟɤɚɟɬ ɜ ɝɚɡɚɯ, ɩɨɫɤɨɥɶɤɭ ɦɨɥɟɤɭɥɵ ɜ ɧɢɯ ɛɨɥɟɟ ɩɨɞɜɢɠɧɵ, ɱɟɦ
ɜ ɠɢɞɤɨɫɬɹɯ ɢ ɬɜɟɪɞɵɯ ɬɟɥɚɯ. ȼ ɠɢɞɤɨɫɬɹɯ ɢ ɝɚɡɚɯ, ɧɚɪɹɞɭ ɫ ɞɢɮɮɭɡɢɟɣ,
ɜɨɡɦɨɠɟɧ ɢ ɤɨɧɜɟɤɬɢɜɧɵɣ ɦɚɫɫɨɩɟɪɟɧɨɫ ɡɚ ɫɱɟɬ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɢɯ ɨɛɴɟɦɨɜ.
ɉɪɢ ɫɭɛɥɢɦɚɰɢɢ, ɫɭɲɤɟ, ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɹɯ ɩɪɢɯɨɞɢɬɫɹ ɪɚɫɫɱɢɬɵɜɚɬɶ ɤɨɧɜɟɤɬɢɜɧɵɣ ɩɟɪɟɧɨɫ ɦɚɫɫɵ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɜ ɠɢɞɤɭɸ ɢɥɢ
ɝɚɡɨɜɭɸ ɮɚɡɭ. Ɍɚɤɨɣ ɩɪɨɰɟɫɫ ɧɚɡɵɜɚɟɬɫɹ ɤɨɧɜɟɤɬɢɜɧɨɣ ɦɚɫɫɨɨɬɞɚɱɟɣ.
Ⱥɧɚɥɨɝɢɱɧɵɟ ɩɪɨɰɟɫɫɵ ɬɟɩɥɨ- ɢ ɦɚɫɫɨɩɟɪɟɧɨɫɚ ɨɩɢɫɵɜɚɸɬɫɹ ɨɞɢɧɚɤɨɜɵɦɢ ɩɨ ɮɨɪɦɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɭɪɚɜɧɟɧɢɹɦɢ, ɩɨɷɬɨɦɭ ɦɧɨɝɢɟ
ɜɵɜɨɞɵ ɬɟɨɪɢɢ ɬɟɩɥɨɩɟɪɟɧɨɫɚ ɛɭɞɭɬ ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɞɥɹ ɬɟɨɪɢɢ ɩɟɪɟɧɨɫɚ ɦɚɫɫɵ.
ɋɭɳɟɫɬɜɭɸɬ ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɦɟɯɚɧɢɡɦɵ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɢ ɦɚɫɫɵ, ɤɨɬɨɪɵɟ ɬɚɤɠɟ ɧɚɛɥɸɞɚɸɬɫɹ ɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ (ɧɚɩɪɢɦɟɪ,
ɞɢɮɮɭɡɢɨɧɧɚɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢ ɬɟɪɦɨɞɢɮɮɭɡɢɹ, ɦɧɨɝɨɤɨɦɩɨɧɟɧɬɧɚɹ ɞɢɮɮɭɡɢɹ ɢ ɞɪ.)
1.4. ɂɫɬɨɪɢɱɟɫɤɢɣ ɷɤɫɤɭɪɫ 2
Ɉɫɧɨɜɵ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɛɵɥɢ ɡɚɥɨɠɟɧɵ
ɟɳɟ ɬɪɭɞɚɦɢ Ʌɨɦɨɧɨɫɨɜɚ, ɇɶɸɬɨɧɚ, Ʌɚɦɛɟɪɬɚ, Ȼɢɨ, Ɏɭɪɶɟ, Ʌɚɩɥɚɫɚ,
ɉɭɚɫɫɨɧɚ ɢ ɞɪɭɝɢɯ ɭɱɟɧɵɯ. Ⱦɥɢɬɟɥɶɧɨɟ ɜɪɟɦɹ ɷɬɚ ɬɟɨɪɢɹ ɨɫɬɚɜɚɥɚɫɶ
ɞɨɫɬɨɹɧɢɟɦ ɬɟɨɪɟɬɢɤɨɜ ɢ ɬɨɥɶɤɨ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɧɚɯɨɞɢɥɚ ɩɪɚɤɬɢɱɟɫɤɨɟ ɩɪɢɦɟɧɟɧɢɟ. Ɉɞɧɚɤɨ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɩɨɥɨɠɟɧɢɟ ɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɢɥɨɫɶ.
2
Ʉɚɪɬɚɲɨɜ ɗ.Ɇ. Ⱥɧɚɥɢɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ ɜ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɬɜɟɪɞɵɯ ɬɟɥ:
ɭɱɟɛɧɨɟ ɩɨɫɨɛɢɟ. Ɇ.: ȼɵɫɲ. ɲɤ., 1985. 480 c.; Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɚɜɢɚɰɢɨɧɧɨɣ
ɢ ɪɚɤɟɬɧɨ-ɤɨɫɦɢɱɟɫɤɨɣ ɬɟɯɧɢɤɟ, ɩɨɞ ɨɛɳɟɣ ɪɟɞ. Ⱥɜɞɭɟɜɫɤɨɝɨ ȼ.ɋ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1992. 528 c. ɢ ɞɪ.
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Ɉɫɧɨɜɨɩɨɥɨɠɧɢɤɨɦ ɭɱɟɧɢɹ ɨ ɬɟɩɥɨɬɟ ɛɵɥ ɜɟɥɢɤɢɣ ɪɭɫɫɤɢɣ ɭɱɟɧɵɣ
Ɇ.ȼ.Ʌɨɦɨɧɨɫɨɜ. ȼ ɫɟɪɟɞɢɧɟ XVIII ɜɟɤɚ, ɨɩɟɪɟɞɢɜ ɧɚ ɫɬɨ ɫ ɥɢɲɧɢɦ ɥɟɬ
ɧɚɭɤɭ Ɂɚɩɚɞɧɨɣ ȿɜɪɨɩɵ, Ʌɨɦɨɧɨɫɨɜ ɫɨɡɞɚɥ ɟɞɢɧɭɸ ɬɟɨɪɢɸ ɬɟɩɥɨɬɵ ɢ
ɜɟɳɟɫɬɜɚ, ɢɡɥɨɠɢɜ ɨɫɧɨɜɵ ɟɟ ɜ ɪɚɛɨɬɟ «Ɋɚɡɦɵɲɥɟɧɢɹ ɨ ɩɪɢɱɢɧɟ ɬɟɩɥɨɬɵ ɢ ɯɨɥɨɞɚ» (1744). ɗɬɚ ɬɟɨɪɢɹ ɬɟɩɥɨɬɵ ɫɨɞɟɪɠɚɥɚ ɜ ɫɟɛɟ ɜɫɟ ɨɫɧɨɜɧɵɟ
ɷɥɟɦɟɧɬɵ ɫɨɜɪɟɦɟɧɧɨɣ ɬɟɨɪɢɢ: ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɚɫɫɵ ɢ ɷɧɟɪɝɢɢ,
ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɬɟɩɥɨɬɟ ɤɚɤ ɨ ɪɟɡɭɥɶɬɚɬɟ ɞɜɢɠɟɧɢɹ ɷɥɟɦɟɧɬɚɪɧɵɯ ɱɚɫɬɢɰ ɬɟɥɚ; ɨ ɫɬɟɩɟɧɢ ɧɚɝɪɟɜɚ ɤɚɤ ɦɟɪɵ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɞɜɢɠɟɧɢɹ ɷɬɢɯ
ɱɚɫɬɢɰ; ɨ ɦɟɯɚɧɢɡɦɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɤɚɤ ɨɛɦɟɧɟ ɷɧɟɪɝɢɢ ɞɜɢɠɟɧɢɹ
ɦɟɠɞɭ ɨɬɞɟɥɶɧɵɦɢ ɱɚɫɬɢɰɚɦɢ; ɨɛ ɚɛɫɨɥɸɬɧɨɦ ɧɭɥɟ ɬɟɦɩɟɪɚɬɭɪɵ.
ȼ 1807 ɝ. Ɏɪɚɧɰɭɡɫɤɢɣ ɭɱɟɧɵɣ ɀ. Ɏɭɪɶɟ ɫɮɨɪɦɭɥɢɪɨɜɚɥ ɨɫɧɨɜɧɭɸ
ɝɢɩɨɬɟɡɭ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɩɨɥɨɠɢɜɲɭɸ ɧɚɱɚɥɨ ɪɚɡɜɢɬɢɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɂɦ ɠɟ ɜ 1822 ɝ. ɢɡɥɨɠɟɧɚ ɬɟɨɪɢɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɨɬɵ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ ɜ ɬɪɭɞɟ «Ⱥɧɚɥɢɬɢɱɟɫɤɚɹ ɬɟɨɪɢɹ
ɬɟɩɥɚ».
ȼ 1831 ɝɨɞɭ ɡɧɚɦɟɧɢɬɵɣ ɪɭɫɫɤɢɣ ɦɚɬɟɦɚɬɢɤ Ɇ.Ȼ. Ɉɫɬɪɨɝɪɚɞɫɤɢɣ
ɨɩɭɛɥɢɤɨɜɚɥ ɪɚɛɨɬɭ «Ɂɚɦɟɱɚɧɢɹ ɩɨ ɬɟɨɪɢɢ ɬɟɩɥɨɬɵ», ɜ ɤɨɬɨɪɨɣ ɞɚɥ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɜɟɪɞɨɝɨ, ɨɞɧɨɪɨɞɧɨɝɨ ɢ
ɢɡɨɬɪɨɩɧɨɝɨ ɬɟɥɚ.
ɋ ɪɚɡɜɢɬɢɟɦ ɬɟɯɧɢɤɢ ɪɨɥɶ ɩɪɨɰɟɫɫɨɜ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɜ ɪɚɡɥɢɱɧɵɯ
ɬɟɩɥɨɜɵɯ ɭɫɬɪɨɣɫɬɜɚɯ ɢ ɦɚɲɢɧɚɯ ɜɨɡɪɨɫɥɚ, ɢ ɬɟɩɥɨɜɵɦ ɩɪɨɰɟɫɫɚɦ ɫɬɚɥɨ ɭɞɟɥɹɬɶɫɹ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ ɜɧɢɦɚɧɢɹ, ɨɫɨɛɟɧɧɨ ɫɨ ɜɬɨɪɨɣ ɩɨɥɨɜɢɧɵ XIX ɜɟɤɚ.
ȼ ɧɚɭɱɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɜɫɟ ɲɢɪɟ ɫɬɚɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɦɟɬɨɞɵ
ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ. ȼ 1881–1882 ɝɨɞɚɯ ɜ Ɇɨɫɤɜɟ Ⱥ.Ƚ. ɋɬɨɥɟɬɨɜ
ɩɪɨɱɟɥ ɫɜɨɢ ɡɧɚɦɟɧɢɬɵɟ ɥɟɤɰɢɢ ɩɨ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ȼ ɤɧɢɝɟ
«Ɍɟɨɪɢɹ ɬɟɩɥɨɬɵ» (1882 ɝ.) ɨɧ ɩɢɫɚɥ: «ɋ ɢɫɬɨɪɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ
ɭɱɟɧɢɟ ɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɟɫɬɶ ɧɚɱɚɥɨ ɬɟɨɪɢɢ ɬɟɩɥɨɬɵ ɢ ɜɨɨɛɳɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ». ɋɬɨɥɟɬɨɜɚ Ⱥ.Ƚ. ɫɱɢɬɚɸɬ ɩɟɪɜɵɦ ɫɨɡɞɚɬɟɥɟɦ ɤɭɪɫɚ
ɫɨɜɪɟɦɟɧɧɨɣ ɬɟɩɥɨɮɢɡɢɤɢ, ɤɨɬɨɪɚɹ ɜɵɞɟɥɹɟɬɫɹ ɜ ɫɚɦɨɫɬɨɹɬɟɥɶɧɭɸ ɧɚɭɤɭ ɜ ɤɨɧɰɟ XIX – ɧɚɱɚɥɟ XX ɜ.
Ɋɭɫɫɤɢɣ ɭɱɟɧɵɣ ȼ.Ⱥ. Ɇɢɯɟɥɶɫɨɧ (1890) ɛɵɥ ɩɟɪɜɵɦ ɢɫɫɥɟɞɨɜɚɬɟɥɟɦ, ɩɨɫɬɚɜɢɜɲɢɦ ɜɨɩɪɨɫ ɨ ɡɚɜɢɫɢɦɨɫɬɢ ɥɭɱɟɢɫɩɭɫɤɚɧɢɹ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɞɥɢɧɵ ɜɨɥɧɵ.
ȼ 20-ɟ ɝɨɞɵ XX ɜɟɤɚ ɪɚɡɜɢɬɢɟ ɭɱɟɧɢɹ ɨ ɬɟɩɥɨɨɛɦɟɧɟ ɜɨɡɝɥɚɜɢɥ ɚɤɚɞɟɦɢɤ Ɇ.ȼ. Ʉɢɪɩɢɱɟɜ, ɲɤɨɥɚ ɤɨɬɨɪɨɝɨ ɡɚɥɨɠɢɥɚ ɨɫɧɨɜɵ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɢ ɟɟ ɩɪɢɥɨɠɟɧɢɹ ɤ ɜɨɩɪɨɫɚɦ ɬɟɩɥɨɩɟɪɟɞɚɱɢ.
Ȼɨɥɶɲɨɟ ɜɥɢɹɧɢɟ ɧɚ ɪɚɡɜɢɬɢɟ ɬɟɨɪɢɢ ɬɟɩɥɨ- ɢ ɦɚɫɫɨɨɛɦɟɧɚ ɨɤɚɡɚɥɢ
ɪɚɛɨɬɵ Ⱥ.ȼ. Ʌɵɤɨɜɚ ɢ ɟɝɨ ɲɤɨɥɵ. ɒɢɪɨɤɨɟ ɩɪɢɡɧɚɧɢɟ ɩɨɥɭɱɢɥɢ ɢɫɫɥɟɞɨɜɚɧɢɹ ɋ.ɋ Ʉɭɬɚɬɟɥɚɞɡɟ, ɪɚɡɜɢɜɲɟɝɨ ɬɟɨɪɢɸ ɩɨɞɨɛɢɹ ɜ ɩɪɨɰɟɫɫɚɯ ɢɡɦɟɧɟɧɢɹ ɚɝɪɟɝɚɬɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜɟɳɟɫɬɜɚ.
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ɂɡ ɪɚɛɨɬ ɡɚɪɭɛɟɠɧɵɯ ɭɱɟɧɵɯ, ɩɨɫɜɹɳɟɧɧɵɯ ɬɟɨɪɢɢ ɬɟɩɥɨɬɵ, ɤɪɨɦɟ
ɧɚɡɜɚɧɧɵɯ ɜɵɲɟ, ɲɢɪɨɤɨ ɢɡɜɟɫɬɧɵ ɬɪɭɞɵ Ʉɢɪɯɝɨɮɚ, ɉɭɚɫɫɨɧɚ, ȼɟɛɟɪɚ,
Ɍɨɦɫɨɧɚ, ɉɥɚɧɤɚ, Ʌɚɦɟ, ɉɭɚɧɤɚɪɟ, Ʉɚɪɫɥɨɭ, ȿɝɟɪɚ, ɗɤɤɟɪɬɚ,
Ⱦɪɟɣɤɚ ɢ ɞɪ.
Ɍɟɨɪɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɨɣ ɛɨɥɶɲɢɧɫɬɜɚ ɫɨɜɪɟɦɟɧɧɵɯ ɦɟɬɨɞɨɜ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ, ɩɨɡɜɨɥɹɟɬ ɪɚɡɪɚɛɚɬɵɜɚɬɶ ɫɩɨɫɨɛɵ ɭɩɪɚɜɥɟɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ ɨɛɪɚɛɨɬɤɢ ɢ
ɩɨɥɭɱɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ, ɪɚɫɫɱɢɬɵɜɚɬɶ ɤɨɧɫɬɪɭɤɰɢɢ ɭɫɬɚɧɨɜɨɤ ɢ ɫɬɪɨɢɬɟɥɶɧɵɯ ɨɛɴɟɤɬɨɜ, ɨɛɴɹɫɧɹɬɶ ɢ ɩɪɟɞɫɤɚɡɵɜɚɬɶ ɬɟ ɢɥɢ ɢɧɵɟ ɩɪɢɪɨɞɧɵɟ
ɹɜɥɟɧɢɹ. Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ.
1.5. ȼɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ ɬɟɯɧɨɥɨɝɢɢ
ɉɨɞ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɦɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ (ȼɌɉ3)
ɡɚɱɚɫɬɭɸ ɩɨɧɢɦɚɸɬ ɬɚɤɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ, ɤɨɬɨɪɵɟ ɪɟɚɥɢɡɭɸɬɫɹ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɷɧɟɪɝɢɢ.
ɍɫɥɨɜɧɨ ɤ ȼɌɉ ɨɬɧɨɫɹɬ ɬɟ, ɩɪɢ ɤɨɬɨɪɵɯ ɞɨɫɬɢɝɚɸɬɫɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜɵɲɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɥɚɜɥɟɧɢɹ ɠɟɥɟɡɚ.
Ɍɢɩɢɱɧɵɦɢ ɩɪɢɦɟɪɚɦɢ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ (ȼɌɉ) ɹɜɥɹɸɬɫɹ: ɥɚɡɟɪɧɚɹ ɢ ɷɥɟɤɬɪɨɧɧɚɹ ɬɟɯɧɨɥɨɝɢɢ (ɫɜɚɪɤɚ,
ɪɟɡɤɚ, ɬɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ); ɞɭɝɨɜɚɹ ɫɜɚɪɤɚ ɢ ɞɪɭɝɢɟ ɫɩɨɫɨɛɵ ɫɨɟɞɢɧɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ (ɞɢɮɮɭɡɢɨɧɧɚɹ ɩɚɣɤɚ; ɬɟɪɦɢɬɧɚɹ ɢ ɋȼɋ – ɫɜɚɪɤɚ);
ɩɥɚɡɦɟɧɧɵɟ ɬɟɯɧɨɥɨɝɢɢ ɧɚɧɟɫɟɧɢɹ ɩɨɤɪɵɬɢɣ ɢ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɨɛɪɚɛɨɬɤɢ; ɢɨɧɧɵɟ ɬɟɯɧɨɥɨɝɢɢ; ɤɢɫɥɨɪɨɞɧɚɹ ɪɟɡɤɚ; ɫɨɜɦɟɳɟɧɧɵɟ ɬɟɯɧɨɥɨɝɢɢ
ɪɟɡɤɢ, ɫɜɚɪɤɢ, ɧɚɩɥɚɜɤɢ; ɩɪɨɰɟɫɫɵ ɩɨɥɭɱɟɧɢɹ ɬɨɧɤɢɯ ɩɥɟɧɨɤ ɢ ɜɵɪɚɳɢɜɚɧɢɟ ɦɨɧɨɤɪɢɫɬɚɥɥɨɜ; ɩɪɨɰɟɫɫɵ ɯɢɦɢɱɟɫɤɨɣ ɢ ɞɢɮɮɭɡɢɨɧɧɨɣ ɨɛɪɚɛɨɬɤɢ ɩɨɜɟɪɯɧɨɫɬɟɣ ɦɚɬɟɪɢɚɥɨɜ ɢ ɞɪ. Ʉ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɦ ɬɟɯɧɨɥɨɝɢɹɦ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɢ ɦɧɨɝɢɟ ɬɟɯɧɨɥɨɝɢɢ ɩɨɥɭɱɟɧɢɹ ɧɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ ɜ
ɯɢɦɢɱɟɫɤɨɣ ɩɪɨɦɵɲɥɟɧɧɨɫɬɢ; ɩɟɪɟɪɚɛɨɬɤɢ ɢ ɫɠɢɝɚɧɢɹ ɩɪɢɪɨɞɧɨɝɨ ɬɨɩɥɢɜɚ; ɪɚɡɥɢɱɧɵɟ ɦɟɬɚɥɥɭɪɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ.
ɇɨ ɥɸɛɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɨɤɚɡɵɜɚɟɬɫɹ ɜɟɫɶɦɚ ɭɫɥɨɜɧɵɦ.
ȼɨ ɜɫɟɯ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɹɯ ɩɪɨɢɫɯɨɞɢɬ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɪɚɡɥɢɱɧɵɯ ɜɢɞɨɜ ɷɧɟɪɝɢɢ ɜ ɬɟɩɥɨɜɭɸ ɷɧɟɪɝɢɸ ɢ (ɢɥɢ) ɟɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɞɥɹ ɩɨɥɭɱɟɧɢɹ, ɩɟɪɟɪɚɛɨɬɤɢ ɢ ɦɨɞɢɮɢɤɚɰɢɢ ɦɚɬɟɪɢɚɥɨɜ ɢ ɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ.
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Ɍɚɤɨɝɨ ɬɢɩɚ ɫɨɤɪɚɳɟɧɢɹ ɢɫɩɨɥɶɡɭɸɬɫɹ, ɧɚɩɪɢɦɟɪ, ɜ ɤɧɢɝɟ Ɋɵɤɚɥɢɧ ɇ.ɇ., ɍɝɥɨɜ Ⱥ.Ⱥ., Ⱥɧɢɳɟɧɤɨ Ʌ.Ɇ. ȼɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ. Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɨɫɧɨɜɵ. Ɇ.: ɇɚɭɤɚ, 1986. – 174 c
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Ʉɚɠɞɵɣ ɢɡ ɧɚɡɜɚɧɧɵɯ ȼɌɉ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɜɤɥɸɱɚɟɬ ɦɧɨɠɟɫɬɜɨ
ɱɚɫɬɧɵɯ ɬɟɯɧɨɥɨɝɢɣ, ɡɚɜɢɫɹɳɢɯ ɨɬ ɤɨɧɤɪɟɬɧɨɝɨ ɬɟɯɧɢɱɟɫɤɨɝɨ ɪɟɲɟɧɢɹ,
ɭɫɥɨɜɢɣ ɢ ɦɚɬɟɪɢɚɥɨɜ, ɞɥɹ ɤɨɬɨɪɵɯ ɩɪɨɰɟɫɫ ɩɪɟɞɧɚɡɧɚɱɟɧ.
ɇɚɩɪɢɦɟɪ, ɢɡ ɬɟɪɦɢɱɟɫɤɢɯ ɫɩɨɫɨɛɨɜ ɪɟɡɤɢ (ɧɟɡɚɦɟɧɢɦɵɯ ɩɪɢ ɪɚɡɞɟɥɤɟ ɭɫɬɚɪɟɜɲɟɣ ɬɟɯɧɢɤɢ ɧɚ ɦɟɬɚɥɥɨɥɨɦ, ɜ ɡɚɝɨɬɨɜɢɬɟɥɶɧɵɯ ɰɟɯɚɯ ɦɚɲɢɧɨɫɬɪɨɢɬɟɥɶɧɵɯ ɢ ɫɭɞɨɫɬɪɨɢɬɟɥɶɧɵɯ ɩɪɟɞɩɪɢɹɬɢɣ, ɜ ɪɟɦɨɧɬɧɨɦ,
ɫɬɪɨɢɬɟɥɶɧɨɦ ɩɪɨɢɡɜɨɞɫɬɜɟ ɢ ɩɪ.) ɜɵɞɟɥɹɸɬ ɝɚɡɨɤɢɫɥɨɪɨɞɧɭɸ (ȽɄɊ);
ɜɨɡɞɭɲɧɨ-ɩɥɚɡɦɟɧɧɭɸ (ȼɉɊ); ɷɥɟɤɬɪɨɞɭɝɨɜɭɸ ɩɨɥɭɚɜɬɨɦɚɬɢɱɟɫɤɭɸ
ɪɟɡɤɭ ɩɨɪɨɲɤɨɜɨɣ ɩɪɨɜɨɥɨɤɨɣ (ɗȾɊ); ɥɚɡɟɪɧɭɸ ɪɟɡɤɢ.
ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɫɭɳɟɫɬɜɭɟɬ ɦɧɨɝɨ ɪɚɡɥɢɱɧɵɯ ɫɩɨɫɨɛɨɜ ɫɜɚɪɤɢ 4,
ɤɨɬɨɪɵɟ ɤɥɚɫɫɢɮɢɰɢɪɭɸɬɫɹ ɩɨ ɪɚɡɥɢɱɧɵɦ ɩɪɢɡɧɚɤɚɦ: ɤɭɡɧɟɱɧɚɹ (ɝɨɪɧɨɜɚɹ) ɫɜɚɪɤɚ; ɝɚɡɨɩɪɟɫɫɨɜɚɹ ɫɜɚɪɤɚ; ɤɨɧɬɚɤɬɧɚɹ ɫɜɚɪɤɚ; ɬɟɪɦɢɬɧɚɹ
ɫɜɚɪɤɚ; ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɞɭɝɨɜɚɹ ɫɜɚɪɤɚ; ɷɥɟɤɬɪɨɲɥɚɤɨɜɚɹ ɫɜɚɪɤɚ; ɞɭɝɨɜɚɹ
ɫɜɚɪɤɚ ɜ ɫɪɟɞɟ ɡɚɳɢɬɧɨɝɨ ɝɚɡɚ; ɚɬɨɦɧɨɜɨɞɨɪɨɞɧɚɹ ɫɜɚɪɤɚ; ɝɚɡɨɜɚɹ ɫɜɚɪɤɚ
ɢ ɬ.ɞ.
Ⱦɥɹ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɣ ɯɚɪɚɤɬɟɪɧɵ ɫɥɟɞɭɸɳɢɟ ɨɫɨɛɟɧɧɨɫɬɢ:
– ɫɭɳɟɫɬɜɟɧɧɚɹ ɧɟɪɚɜɧɨɜɟɫɧɨɫɬɶ ɩɪɨɰɟɫɫɨɜ, ɫɜɹɡɚɧɧɚɹ ɫ ɧɟɨɞɧɨɪɨɞɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɟɟ ɢɡɦɟɧɟɧɢɟɦ ɜɨ ɜɪɟɦɟɧɢ;
– ɜɵɫɨɤɢɟ ɫɤɨɪɨɫɬɢ ɧɚɝɪɟɜɚ ɢ ɨɯɥɚɠɞɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɷɥɟɦɟɧɬɨɜ ɫɢɫɬɟɦɵ;
– ɧɚɥɢɱɢɟ ɫɥɨɠɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ;
– ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɧɟɫɤɨɥɶɤɢɯ ɪɚɡɥɢɱɧɵɯ ɮɚɡ, ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ
ɤɨɬɨɪɵɦɢ ɢɡɦɟɧɹɟɬɫɹ;
– ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ, ɫɨɩɭɬɫɬɜɭɸɳɢɟ ɧɚɝɪɟɜɭ ɢ ɨɯɥɚɠɞɟɧɢɸ ɢɥɢ ɥɟɠɚɳɢɟ ɜ ɨɫɧɨɜɟ ɬɟɯɧɨɥɨɝɢɢ.
ȼ ɩɨɫɥɟɞɧɢɟ ɝɨɞɵ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ ɬɟɯɧɨɥɨɝɢɢ, ɢɫɩɨɥɶɡɭɸɳɢɟ
ɜɵɫɨɤɨɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɟ ɢɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ (ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɷɥɟɤɬɪɨɧɧɵɣ ɥɭɱ, ɥɚɡɟɪɧɨɟ ɢɡɥɭɱɟɧɢɟ; ɩɨɬɨɤɢ ɩɥɚɡɦɵ, ɤɢɫɥɨɪɨɞɧɚɹ ɫɬɪɭɹ ɢ
ɞɪ.), ɧɚɲɥɢ ɲɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɜ ɷɥɟɤɬɪɨɧɧɨɣ ɩɪɨɦɵɲɥɟɧɧɨɫɬɢ ɢ
ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ ɞɥɹ ɪɟɲɟɧɢɹ ɬɚɤɢɯ ɩɪɨɛɥɟɦ, ɤɚɤ ɫɜɚɪɤɚ, ɪɟɡɤɚ, ɩɨɥɭɱɟɧɢɟ ɨɬɜɟɪɫɬɢɣ, ɬɟɪɦɨɨɛɪɚɛɨɬɤɚ, ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ, ɨɩɥɚɜɥɟɧɢɟ ɩɨɜɟɪɯɧɨɫɬɧɨɝɨ ɫɥɨɹ, ɥɚɡɟɪɧɨ-ɩɥɚɡɦɟɧɧɚɹ ɨɛɪɚɛɨɬɤɚ, ɨɫɚɠɞɟɧɢɟ ɩɥɟɧɨɤ, ɡɨɧɧɚɹ ɨɱɢɫɬɤɚ, ɜɵɪɚɳɢɜɚɧɢɟ ɦɨɧɨɤɪɢɫɬɚɥɥɨɜ ɢ ɞɪ.
Ɍɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɫɯɟɦɵ, ɩɨɤɚɡɚɧɧɨɣ ɧɚ ɪɢɫ. 1.1. Ɍɟɯɧɨɥɨɝɢɱɟɫɤɚɹ ɤɚɦɟɪɚ (ɜɚɤɭɭɦɧɚɹ ɢɥɢ ɝɚɡɨɧɚɩɨɥɧɹɟɦɚɹ ɤɚɦɟɪɚ, ɪɟɚɤɬɨɪ, ɩɟɱɶ ɢ ɬ.ɩ.) ɩɪɟɞɧɚɡɧɚɱɟɧɧɚɹ ɞɥɹ ɩɨɞɞɟɪɠɚɧɢɹ
ɡɚɞɚɧɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɪɟɠɢɦɨɜ ɜ ɡɚɦɤɧɭɬɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɞɥɹ ɢɫ4
ɋɜɚɪɤɨɣ ɧɚɡɵɜɚɟɬɫɹ ɩɪɨɰɟɫɫ ɩɨɥɭɱɟɧɢɹ ɧɟɪɚɡɴɟɦɧɨɝɨ ɫɨɟɞɢɧɟɧɢɹ ɞɟɬɚɥɟɣ ɩɭɬɟɦ
ɩɪɢɦɟɧɟɧɢɹ ɦɟɫɬɧɨɝɨ ɧɚɝɪɟɜɚ.
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ɤɥɸɱɟɧɢɹ ɜɥɢɹɧɢɹ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɧɚ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ; ɞɥɹ
ɫɛɨɪɚ ɢ ɨɬɜɨɞɚ ɨɬɯɨɞɨɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ. Ɍɟɯɧɨɥɨɝɢɱɟɫɤɚɹ
ɤɚɦɟɪɚ ɢɦɟɟɬ ɫɢɫɬɟɦɭ ɞɚɬɱɢɤɨɜ ɤɨɧɬɪɨɥɹ ɢ ɭɩɪɚɜɥɟɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦ ɩɪɨɰɟɫɫɨɦ.
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɞɥɹ ɪɹɞɚ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ (ɧɚɩɪɢɦɟɪ, ɬɚɤɢɯ, ɤɚɤ ɫɜɚɪɤɚ ɜ ɨɬɤɪɵɬɨɣ ɚɬɦɨɫɮɟɪɟ) ɬɟɯɧɨɥɨɝɢɱɟɫɤɚɹ ɤɚɦɟɪɚ
ɨɬɫɭɬɫɬɜɭɟɬ. Ɉɫɧɨɜɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɨɩɟɪɚɰɢɹɯ (ɧɚɩɪɢɦɟɪ, ɩɪɢ ɫɜɚɪɤɟ, ɪɟɡɤɟ,
ɢɫɩɚɪɟɧɢɢ ɢ ɞɪ.).
ȼɫɩɨɦɨɝɚɬɟɥɶɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɫɨɡɞɚɧɢɹ
ɨɩɪɟɞɟɥɟɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɧɵɯ ɪɟɠɢɦɨɜ ɧɟɤɨɬɨɪɵɯ ɬɨɱɟɤ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɚɩɩɚɪɚɬɚ (ɧɚɩɪɢɦɟɪ, ɞɥɹ ɩɨɞɨɝɪɟɜɚ ɩɨɞɥɨɠɟɤ ɩɪɢ ɧɚɧɟɫɟɧɢɢ ɬɨɧɤɢɯ
ɩɥɟɧɨɤ). Ɇɚɬɟɪɢɚɥɵ ɢ ɡɚɝɨɬɨɜɤɢ ɞɥɹ
ɩɨɥɭɱɟɧɢɹ ɝɨɬɨɜɨɝɨ ɢɡɞɟɥɢɹ ɨɛɪɚɡɭɸɬ ɦɚɬɟɪɢɚɥɶɧɵɟ ɩɨɬɨɤɢ. Ɍɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɨɬɯɨɞɵ ɫɤɥɚɞɵɜɚɸɬɫɹ ɢɡ ɷɧɟɪɊɢɫ. 1.1. Ȼɥɨɱɧɚɹ ɫɬɪɭɤɬɭɪɚ ɬɟɯɝɟɬɢɱɟɫɤɢɯ ɢ ɦɚɬɟɪɢɚɥɶɧɵɯ.
ɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ: 1 – ɬɟɯɈɫɧɨɜɧɵɟ ɢ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɟ
ɧɨɥɨɝɢɱɟɫɤɚɹ ɤɚɦɟɪɚ; 2 – ɨɫɧɨɜɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ; 3 ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ, ɦɚɬɟɪɢ– ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɟ ɷɧɟɪɝɟɬɢɱɟ- ɚɥɶɧɵɟ ɩɨɬɨɤɢ ɢɦɟɸɬ ɫɜɨɢ ɚɜɬɨɧɨɦɫɤɢ ɢɫɬɨɱɧɢɤɢ; 4 – ɦɚɬɟɪɢɚɥɶɧɵɟ ɧɵɟ ɫɢɫɬɟɦɵ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɢ ɭɩɪɚɜɩɨɬɨɤɢ; 5 – ɝɨɬɨɜɵɣ ɩɪɨɞɭɤɬ; 6. ɥɟɧɢɹ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɫ
ɫɢɫɬɟɦɨɣ ɤɨɧɬɪɨɥɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ
– ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɨɬɯɨɞɵ
ɩɪɨɰɟɫɫɚ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɨɫɭɳɟɫɬɜɥɹɬɶ ɭɩɪɚɜɥɹɟɦɵɣ ɜɨ ɜɪɟɦɟɧɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ. ȼ ɩɪɨɬɢɜɧɨɦ
ɫɥɭɱɚɟ, ɟɫɥɢ ɬɚɤɚɹ ɫɜɹɡɶ ɨɬɫɭɬɫɬɜɭɟɬ, ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ ɤɨɧɬɪɨɥɢɪɭɟɬɫɹ ɩɨ ɤɨɧɟɱɧɨɦɭ ɩɪɨɞɭɤɬɭ.
ɉɪɢɧɰɢɩɢɚɥɶɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɣ
ɹɜɥɹɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɫɨɜɪɟɦɟɧɧɵɯ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɢɫɬɨɱɧɢɤɨɜ. ȼɜɢɞɭ ɛɨɥɶɲɨɣ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɡɧɚɱɢɦɨɫɬɢ, ɨɫɬɚɧɨɜɢɦɫɹ ɧɚ ɨɫɨɛɟɧɧɨɫɬɹɯ ɩɪɢɦɟɧɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɜɢɞɨɜ ȼɄɂɗ.
ɗɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɵɟ ɬɟɯɧɨɥɨɝɢɢ5. Ʉɚɤ ɢɫɬɨɱɧɢɤ ɷɧɟɪɝɢɢ ɞɥɹ ȼɌɉ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɷɥɟɤɬɪɨɧɧɵɣ ɥɭɱ (ɌɗɅ) ɩɪɢɦɟɧɹɟɬɫɹ ɫ ɤɨɧɰɚ 50-ɯ ɝɨɞɨɜ 20-ɝɨ ɜɟɤɚ.
5
ɒɢɩɤɨ Ⱥ.Ⱥ., ɉɨɛɨɥɶ ɂ.Ʌ. ɍɪɛɚɧ ɂ.Ƚ. ɍɩɪɨɱɧɟɧɢɟ ɫɬɚɥɟɣ ɢ ɫɩɥɚɜɨɜ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɨɝɨ ɧɚɝɪɟɜɚ. Ɇɢɧɫɤ: ɇɚɭɤɚ ɢ ɬɟɯɧɢɤɚ, 1995. 280 c.
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ɋɭɳɧɨɫɬɶ ɩɪɨɰɟɫɫɚ ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɨɣ ɨɛɪɚɛɨɬɤɢ ɡɚɤɥɸɱɚɟɬɫɹ ɜ
ɬɨɦ, ɱɬɨ ɤɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɧɨɜ, ɭɫɤɨɪɟɧɧɵɯ ɜ ɷɥɟɤɬɪɢɱɟɫɤɨɦ
ɩɨɥɟ, ɩɪɢ ɫɨɭɞɚɪɟɧɢɢ ɫ ɨɛɪɚɛɚɬɵɜɚɟɦɵɦ ɢɡɞɟɥɢɟɦ, ɩɨɦɟɳɟɧɧɵɦ ɜ ɜɚɤɭɭɦɧɭɸ ɤɚɦɟɪɭ, ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɬɟɩɥɨɜɭɸ. Ƚɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ
ɷɥɟɤɬɪɨɧɨɜ (ɞɥɢɧɚ ɩɪɨɛɟɝɚ) ɜ ɬɜɟɪɞɨɟ ɬɟɥɨ ɡɚɜɢɫɢɬ ɨɬ ɭɫɤɨɪɹɸɳɟɝɨ ɧɚɩɪɹɠɟɧɢɹ ɢ ɩɥɨɬɧɨɫɬɢ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɜɟɳɟɫɬɜɚ. Ɍɚɤ, ɞɥɹ ɭɫɤɨɪɹɸɳɟɝɨ ɧɚɩɪɹɠɟɧɢɹ ɜ ɞɟɫɹɬɤɢ ɤɢɥɨɜɨɥɶɬ ɝɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɷɥɟɤɬɪɨɧɧɨɝɨ ɩɨɬɨɤɚ ɜ ɦɟɬɚɥɥɵ ɫɨɫɬɚɜɥɹɟɬ ɞɟɫɹɬɤɢ ɦɢɤɪɨɦɟɬɪɨɜ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ
ɨɬ ɞɢɚɦɟɬɪɚ ɩɹɬɧɚ ɧɚɝɪɟɜɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɬɟɩɥɨɜɨɣ ɢɫɬɨɱɧɢɤ, ɨɛɭɫɥɨɜɥɟɧɧɵɣ ɞɟɣɫɬɜɢɟɦ ɷɥɟɤɬɪɨɧɧɨɝɨ ɥɭɱɚ, ɦɨɠɟɬ ɛɵɬɶ ɤɚɤ ɩɨɜɟɪɯɧɨɫɬɧɵɦ, ɬɚɤ ɢ ɨɛɴɟɦɧɵɦ.
Ɍɚɤ ɤɚɤ ɞɢɚɩɚɡɨɧ ɦɨɳɧɨɫɬɢ ɢ ɩɥɨɬɧɨɫɬɢ ɷɧɟɪɝɢɢ ɜ ɷɥɟɤɬɪɨɧɧɨɦ ɥɭɱɟ ɜɟɥɢɤ (ɞɨ ɟɞɢɧɢɰ ɦɟɝɚɜɚɬɬ ɢ 10 ȼɬ/ɫɦ2), ɜɨɡɦɨɠɧɨ ɩɨɥɭɱɟɧɢɟ ɜɫɟɯ
ɜɢɞɨɜ ɬɟɪɦɢɱɟɫɤɨɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɌɗɅ ɧɚ ɦɚɬɟɪɢɚɥɵ: ɧɚɝɪɟɜ ɞɨ ɡɚɞɚɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪ, ɩɥɚɜɥɟɧɢɟ ɩɪɚɤɬɢɱɟɫɤɢ ɥɸɛɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɢɫɩɚɪɟɧɢɟ
ɫ ɜɟɫɶɦɚ ɛɨɥɶɲɢɦɢ ɫɤɨɪɨɫɬɹɦɢ.
ɗɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɚɹ ɬɟɯɧɨɥɨɝɢɹ ɪɚɡɜɢɜɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɢɯ ɧɚɩɪɚɜɥɟɧɢɹɯ: ɩɥɚɜɤɚ ɢ ɢɫɩɚɪɟɧɢɟ ɜ ɜɚɤɭɭɦɟ, ɫɜɚɪɤɚ ɢ ɩɪɟɰɢɡɢɨɧɧɚɹ ɨɛɪɚɛɨɬɤɚ. Ɉɛɵɱɧɨ ɨɛɪɚɛɨɬɤɭ ɩɪɨɜɨɞɹɬ ɜ ɫɪɟɞɧɟɦ ɜɚɤɭɭɦɟ, p | 10 4 ɉɚ, ɤɨɝɞɚ
ɩɨɬɟɪɢ ɦɨɳɧɨɫɬɢ ɌɗɅ ɧɚ ɪɚɫɫɟɹɧɢɟ ɧɟɜɟɥɢɤɢ. ȼ ɧɟɤɨɬɨɪɵɯ ɢɫɫɥɟɞɨɜɚɬɟɥɶɫɤɢɯ ɰɟɧɬɪɚɯ ɢɦɟɟɬɫɹ ɨɩɵɬ ɩɨ ɫɜɚɪɤɟ ɢ ɨɛɪɚɛɨɬɤɟ ɌɗɅ, ɜɵɜɟɞɟɧɧɵɦ ɜ ɚɬɦɨɫɮɟɪɭ (ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɫɢɫɬɟɦɵ ɲɥɸɡɨɜɚɧɢɹ).
ɉɪɢ ɩɥɚɜɤɟ ɢ ɢɫɩɚɪɟɧɢɢ ɜ ɜɚɤɭɭɦɟ, ɞɥɹ ɧɚɧɟɫɟɧɢɹ ɩɥɟɧɨɤ ɢ ɩɨɤɪɵɬɢɣ ɢɫɩɨɥɶɡɭɸɬ ɦɨɳɧɵɟ (ɞɨ ɧɟɫɤɨɥɶɤɢɯ ɦɟɝɚɜɚɬɬ) ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɵɟ
ɩɟɱɢ ɩɪɢ ɭɫɤɨɪɹɸɳɟɦ ɧɚɩɪɹɠɟɧɢɢ 20–30 ɤȼ. ɉɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ
ɡɞɟɫɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɧɟɜɟɥɢɤɚ ɢ ɨɛɵɱɧɨ ɧɟ ɩɪɟɜɵɲɚɟɬ 104–105 ȼɬ/ɫɦ2.
ɉɪɟɢɦɭɳɟɫɬɜɨ ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɨɣ ɩɟɪɟɩɥɚɜɤɢ ɩɟɪɟɞ ɞɭɝɨɜɨɣ ɨɛɭɫɥɨɜɥɟɧɨ ɱɚɫɬɢɱɧɵɦ ɭɞɚɥɟɧɢɟɦ ɜɪɟɞɧɵɯ ɩɪɢɦɟɫɟɣ ɢɡ ɩɟɪɟɩɥɚɜɥɹɟɦɨɝɨ ɦɟɬɚɥɥɚ ɢ ɩɨɜɵɲɟɧɢɟɦ ɨɞɧɨɪɨɞɧɨɫɬɢ ɫɥɢɬɤɨɜ.
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɌɗɅ ɞɥɹ ɧɚɧɟɫɟɧɢɹ ɩɥɟɧɨɤ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɥɭɱɢɬɶ ɫɤɨɪɨɫɬɢ ɧɚɩɵɥɟɧɢɹ, ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɞɨɫɬɢɠɢɦɵɟ ɞɥɹ ɞɪɭɝɢɯ ɦɟɬɨɞɨɜ ɧɚɧɟɫɟɧɢɹ ɩɥɟɧɨɤ ɢ ɩɨɤɪɵɬɢɣ.
Ⱦɥɹ ɫɜɚɪɤɢ ɦɟɬɚɥɥɨɜ – ɨɫɧɨɜɧɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ ɜ ɩɪɢɦɟɧɟɧɢɢ ɌɗɅ
ɞɥɹ ȼɌɉ – ɢɫɩɨɥɶɡɭɸɬ ɭɫɬɚɧɨɜɤɢ ɬɪɟɯ ɤɥɚɫɫɨɜ: ɧɢɡɤɨ– ɫɪɟɞɧɟ– ɢ ɜɵɫɨɤɨɜɨɥɶɬɧɵɟ, ɤɨɬɨɪɵɟ ɨɯɜɚɬɵɜɚɸɬ ɞɢɚɩɚɡɨɧɵ ɭɫɤɨɪɹɸɳɢɯ ɧɚɩɪɹɠɟɧɢɣ
ɨɬ 20 ɞɨ 150 ɤȼ. Ɇɨɳɧɨɫɬɶ ɫɜɚɪɨɱɧɵɯ ɭɫɬɚɧɨɜɨɤ ɨɛɵɱɧɨ ɫɨɫɬɚɜɥɹɟɬ ɨɬ
1 ɞɨ 120 ɤȼɬ ɢ ɛɨɥɟɟ ɩɪɢ ɦɚɤɫɢɦɚɥɶɧɨɣ ɭɞɟɥɶɧɨɣ ɦɨɳɧɨɫɬɢ 105 – 106
ȼɬ/ɫɦ2. ɍɫɬɚɧɨɜɤɢ ɫ ɭɫɤɨɪɹɸɳɢɦ ɧɚɩɪɹɠɟɧɢɟɦ ɜɵɲɟ 150 ɤȼ ɜ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɩɪɨɰɟɫɫɚɯ, ɤɚɤ ɩɪɚɜɢɥɨ, ɧɟ ɩɪɢɦɟɧɹɸɬɫɹ, ɩɨɫɤɨɥɶɤɭ ɨɧɢ
18
ɬɪɟɛɭɸɬ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɡɚɳɢɬɵ ɨɛɫɥɭɠɢɜɚɸɳɟɝɨ ɩɟɪɫɨɧɚɥɚ ɨɬ ɞɥɢɧɧɨɜɨɥɧɨɜɨɝɨ ɪɟɧɬɝɟɧɨɜɫɤɨɝɨ ɢɡɥɭɱɟɧɢɹ.
ȼ ɩɨɫɥɟɞɧɢɟ ɝɨɞɵ ɢɫɫɥɟɞɭɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɞɥɹ ȼɌɉ
ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɷɥɟɤɬɪɨɧɧɵɯ ɩɨɬɨɤɨɜ ɫ ɷɧɟɪɝɢɟɣ 20 Ɇɷȼ. Ɉɫɧɨɜɧɨɟ ɩɪɟɢɦɭɳɟɫɬɜɨ ɬɚɤɢɯ ɩɭɱɤɨɜ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɛɨɥɶɲɨɣ ɝɥɭɛɢɧɟ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɷɥɟɤɬɪɨɧɧɨɝɨ ɩɭɱɤɚ ɜ ɨɛɴɟɦ ɜɟɳɟɫɬɜɚ.
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɌɗɅ ɜ ɫɜɚɪɨɱɧɵɯ ɩɪɨɰɟɫɫɚɯ ɜɨɡɦɨɠɧɨ ɩɨɥɭɱɟɧɢɟ ɜ ɦɟɬɚɥɥɚɯ ɝɥɭɛɨɤɢɯ ɩɪɨɩɥɚɜɥɟɧɢɣ, ɧɚɡɵɜɚɟɦɵɯ ɬɚɤɠɟ ɩɨ ɮɨɪɦɟ
ɲɜɚ ɤɢɧɠɚɥɶɧɵɦɢ, ɞɥɹ ɤɨɬɨɪɵɯ ɯɚɪɚɤɬɟɪɧɨ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɨɬɧɨɲɟɧɢɟ ɝɥɭɛɢɧɵ ɩɪɨɩɥɚɜɥɟɧɢɹ h f ɤ ɯɚɪɚɤɬɟɪɧɨɦɭ ɞɢɚɦɟɬɪɭ ɡɨɧɵ ɩɪɨɩɥɚɜɥɟɧɢɹ d f . Ⱦɥɹ ɌɗɅ ɞɨɫɬɢɝɧɭɬɨ ɫɨɨɬɧɨɲɟɧɢɢ h f d f ɢ ɛɨɥɟɟ ɩɪɢ ɩɪɨɩɥɚɜɥɟɧɢɢ ɦɟɬɚɥɥɨɜ. ɗɬɨ ɨɬɤɪɵɜɚɟɬ ɲɢɪɨɤɢɟ ɜɨɡɦɨɠɧɨɫɬɢ ɞɥɹ ɨɞɧɨɩɪɨɯɨɞɧɨɣ ɫɜɚɪɤɢ ɢɡɞɟɥɢɣ ɛɨɥɶɲɨɣ ɬɨɥɳɢɧɵ ɜ ɪɚɡɥɢɱɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɩɨɥɨɠɟɧɢɹɯ. ɌɗɅ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɞɥɹ ɫɜɚɪɤɢ ɨɫɨɛɨ ɜɚɠɧɵɯ
ɢɡɞɟɥɢɣ ɜ ɪɚɤɟɬɧɨɣ ɢ ɤɨɫɦɢɱɟɫɤɨɣ ɬɟɯɧɢɤɟ, ɜ ɫɚɦɨɥɟɬɨɫɬɪɨɟɧɢɢ ɢ ɷɧɟɪɝɨɦɚɲɢɧɨɫɬɪɨɟɧɢɢ, ɝɞɟ ɬɪɟɛɭɟɬɫɹ ɜɵɫɨɤɚɹ ɧɚɞɟɠɧɨɫɬɶ ɫɜɚɪɧɵɯ ɲɜɨɜ.
ȼɚɠɧɵɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɧɟɞɪɟɧɢɹ ɌɗɅ ɹɜɥɹɟɬɫɹ ɩɪɟɰɢɡɢɨɧɧɚɹ ɨɛɪɚɛɨɬɤɚ ɢɡɞɟɥɢɣ ɢɡ ɫɜɟɪɯɬɜɟɪɞɵɯ ɢ ɬɭɝɨɩɥɚɜɤɢɯ ɦɟɬɚɥɥɨɜ, ɞɟɬɚɥɢ ɢɡ ɤɨɬɨɪɵɯ ɜɫɟ ɲɢɪɟ ɢɫɩɨɥɶɡɭɸɬ ɩɪɢ ɫɨɡɞɚɧɢɢ ɧɨɜɨɣ ɬɟɯɧɢɤɢ. Ⱦɥɹ ɩɪɟɰɢɡɢɨɧɧɨɣ ɨɛɪɚɛɨɬɤɢ ɬɚɤɢɯ ɞɟɬɚɥɟɣ (ɫɜɟɪɥɟɧɢɹ, ɪɟɡɤɢ, ɮɪɟɡɟɪɨɜɚɧɢɹ ɢ ɞɪ.),
ɜ ɨɫɧɨɜɧɨɦ, ɩɪɢɦɟɧɹɸɬ ɜɵɫɨɤɨɜɨɥɶɬɧɵɟ ɭɫɬɚɧɨɜɤɢ (ɫ ɭɫɤɨɪɹɸɳɢɦ ɧɚɩɪɹɠɟɧɢɟɦ ɞɨ 150 ɤȼ) ɧɟɛɨɥɶɲɨɣ ɦɨɳɧɨɫɬɢ (ɞɨ 1 ɤȼɬ), ɨɛɟɫɩɟɱɢɜɚɸɳɢɟ ɭɞɟɥɶɧɭɸ ɦɨɳɧɨɫɬɶ ɞɨ 108 ȼɬ/ɫɦ2. Ɉɫɨɛɟɧɧɨɫɬɶɸ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ
ɹɜɥɹɟɬɫɹ ɜɵɫɨɤɚɹ ɥɨɤɚɥɶɧɨɫɬɶ ɌɗɅ (ɞɢɚɦɟɬɪ ɡɨɧɵ ɜɨɡɞɟɣɫɬɜɢɹ – ɧɟɫɤɨɥɶɤɨ ɦɢɤɪɨɦɟɬɪɨɜ), ɚ ɬɚɤɠɟ ɜɨɡɦɨɠɧɨɫɬɶ ɭɩɪɚɜɥɹɬɶ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɥɭɱɚ ɩɪɚɤɬɢɱɟɫɤɢ ɛɟɡɢɧɟɪɰɢɨɧɧɨ,
ɱɬɨ ɧɚ ɞɟɥɟ ɧɟɞɨɫɬɭɩɧɨ ɞɥɹ ɞɪɭɝɢɯ ȼɄɂɗ.
ɗɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɚɹ ɩɨɜɟɪɯɧɨɫɬɧɚɹ ɡɚɤɚɥɤɚ - ɩɟɪɫɩɟɤɬɢɜɧɵɣ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ. Ⱦɥɹ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɨɧɚ ɢɦɟɟɬ ɬɚɤɢɟ
ɪɟɲɚɸɳɢɟ ɩɪɟɢɦɭɳɟɫɬɜɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɞɪɭɝɢɦɢ ɦɟɬɨɞɚɦɢ (ɥɚɡɟɪɧɚɹ,
ɢɨɧɧɚɹ, Ɍȼɑ), ɤɚɤ ɛɨɥɶɲɚɹ ɝɥɭɛɢɧɚ ɡɚɤɚɥɤɢ, ɜɚɤɭɭɦɧɚɹ ɡɚɳɢɬɚ ɨɛɪɚɛɚɬɵɜɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɛɨɥɟɟ ɜɵɫɨɤɚɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɩɪɨɰɟɫɫɚ,
ɥɭɱɲɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɨɝɨ ɩɨɜɟɪɯɧɨɫɬɧɨɝɨ ɫɥɨɹ.
ɗɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɭɸ ɡɚɤɚɥɤɭ ɩɨɜɟɪɯɧɨɫɬɟɣ ɪɚɡɜɢɜɚɸɬ, ɩɪɟɠɞɟ ɜɫɟɝɨ, ɜ
ɚɜɬɨɦɨɛɢɥɟɫɬɪɨɟɧɢɢ, ɩɪɨɢɡɜɨɞɫɬɜɟ ɫɟɥɶɫɤɨɯɨɡɹɣɫɬɜɟɧɧɨɣ ɬɟɯɧɢɤɢ ɢ
ɦɚɲɢɧ ɞɥɹ ɩɟɪɟɪɚɛɨɬɤɢ ɫɟɥɶɫɤɨɯɨɡɹɣɫɬɜɟɧɧɨɣ ɩɪɨɞɭɤɰɢɢ, ɜ ɢɧɫɬɪɭɦɟɧɬɚɥɶɧɨɣ ɢ ɥɟɝɤɨɣ ɩɪɨɦɵɲɥɟɧɧɨɫɬɹɯ, ɬɭɪɛɨɫɬɪɨɟɧɢɢ. ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ
ɷɥɟɤɬɪɨɧɧɨɥɭɱɟɜɚɹ ɡɚɤɚɥɤɚ ɡɚɦɟɧɹɟɬ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɧɚɧɟɫɟɧɢɹ ɭɩɪɨɱɧɹɸɳɢɯ ɩɨɤɪɵɬɢɣ, ɧɟɫɦɨɬɪɹ ɧɚ ɢɯ ɫɪɚɜɧɢɬɟɥɶɧɭɸ
ɞɟɲɟɜɢɡɧɭ.
19
Ʌɚɡɟɪɧɚɹ ɬɟɯɧɨɥɨɝɢɹ (ɅɌ)6. ȼɡɚɢɦɨɞɟɣɫɬɜɢɟ ɢɡɥɭɱɟɧɢɹ ɥɚɡɟɪɚ ɫ ɜɟɳɟɫɬɜɨɦ ɩɪɢɜɨɞɢɬ ɤ ɩɨɝɥɨɳɟɧɢɸ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɷɧɟɪɝɢɢ ɜ ɬɨɧɤɨɦ
ɩɨɜɟɪɯɧɨɫɬɧɨɦ ɫɥɨɟ ɦɟɬɚɥɥɚ, ɚ ɜ ɫɥɭɱɚɟ ɞɢɷɥɟɤɬɪɢɤɨɜ ɢ ɩɨɥɭɩɪɨɜɨɞɧɢɤɨɜ ɥɚɡɟɪɧɨɟ ɢɡɥɭɱɟɧɢɟ ɦɨɠɟɬ ɩɪɨɧɢɤɚɬɶ ɜɝɥɭɛɶ ɨɛɴɟɦɚ, ɫɨɡɞɚɜɚɹ ɨɛɴɟɦɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɨɬɵ.
ɉɨɝɥɨɳɟɧɢɟ ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɪɚɡɜɢɬɢɸ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨ – ɢ ɦɚɫɫɨɨɛɦɟɧɚ, ɩɪɨɬɟɤɚɸɳɢɯ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɜ ɨɛɴɟɦɟ ɦɚɬɟɪɢɚɥɚ. Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɢɝɪɚɸɬ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ ɜ ɪɚɡɜɢɬɢɢ
ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɢ ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɨɩɪɟɞɟɥɹɸɬ ɪɟɡɭɥɶɬɚɬ ɜɨɡɞɟɣɫɬɜɢɹ ɢɡɥɭɱɟɧɢɹ ɧɚ ɜɟɳɟɫɬɜɨ.
Ɋɚɡɜɢɬɢɟ ɅɌ ɧɚɱɚɥɨɫɶ ɩɪɚɤɬɢɱɟɫɤɢ ɨɞɧɨɜɪɟɦɟɧɧɨ ɫ ɫɨɡɞɚɧɢɟɦ ɩɟɪɜɵɯ ɥɚɡɟɪɨɜ (1960 ɝ.). ȼ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɥɚɡɟɪɧɚɹ ɬɟɯɧɨɥɨɝɢɹ ɨɬɧɨɫɢɬɫɹ ɤ ɱɢɫɥɭ ɧɚɢɛɨɥɟɟ ɪɚɡɜɢɜɚɸɳɢɯɫɹ ɬɟɯɧɨɥɨɝɢɣ. Ɉɧɚ ɩɪɢɦɟɧɹɟɬɫɹ ɩɪɢ
ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɡɚɤɚɥɤɟ, ɥɟɝɢɪɨɜɚɧɢɢ, ɩɥɚɜɤɟ, ɫɜɚɪɤɟ, ɩɪɨɛɢɜɚɧɢɢ ɨɬɜɟɪɫɬɢɣ ɢ ɞɪ.
ɂɫɫɥɟɞɨɜɚɧɢɹ ɩɨɤɚɡɵɜɚɸɬ, ɱɬɨ ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɦɚɬɟɪɢɚɥɚ ɨɫɧɨɜɧɨɟ
ɡɧɚɱɟɧɢɟ ɢɦɟɸɬ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ – ɷɧɟɪɝɢɹ, ɦɨɳɧɨɫɬɶ, ɞɥɢɬɟɥɶɧɨɫɬɶ ɜɨɡɞɟɣɫɬɜɢɹ, ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ – ɜɪɟɦɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɩɥɨɬɧɨɫɬɢ ɦɨɳɧɨɫɬɢ, ɭɫɥɨɜɢɹ ɮɨɤɭɫɢɪɨɜɤɢ, ɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ
ɦɚɬɟɪɢɚɥɨɜ
(ɤɨɷɮɮɢɰɢɟɧɬɵ
ɩɨɝɥɨɳɟɧɢɹ,
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ,
ɩɥɨɬɧɨɫɬɶ ɢ ɬ.ɞ.).
ȼ ɩɪɨɦɵɲɥɟɧɧɨɣ ɬɟɯɧɨɥɨɝɢɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɥɚɡɟɪɵ ɧɟɫɤɨɥɶɤɢɯ ɬɢɩɨɜ, ɪɚɡɥɢɱɚɸɳɢɟɫɹ ɞɥɢɧɨɣ ɜɨɥɧɵ (ɨɬ 0,7 ɞɨ 10,6 ɦɤɦ) ɢ ɜɢɞɨɦ ɝɟɧɟɪɚɰɢɢ ɢɡɥɭɱɟɧɢɹ (ɧɟɩɪɟɪɵɜɧɨɟ, ɢɦɩɭɥɶɫɧɨ-ɩɟɪɢɨɞɢɱɟɫɤɨɟ, ɢɦɩɭɥɶɫɧɨɟ).
ɇɚɢɛɨɥɶɲɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɩɨɥɭɱɢɥɢ ɢɦɩɭɥɶɫɧɵɟ ɥɚɡɟɪɵ ɞɥɹ ɷɥɟɤɬɪɨɧɧɨɣ ɩɪɨɦɵɲɥɟɧɧɨɫɬɢ, ɪɚɞɢɨɩɪɨɦɵɲɥɟɧɧɨɫɬɢ, ɩɪɢɛɨɪɨɫɬɪɨɟɧɢɹ ɢ
ɞɪ., ɫɪɟɞɧɹɹ ɦɨɳɧɨɫɬɶ ɤɨɬɨɪɵɯ ɧɟ ɩɪɟɜɵɲɚɟɬ 1 ɤȼɬ, ɯɨɬɹ ɜ ɢɦɩɭɥɶɫɟ
ɦɨɳɧɨɫɬɶ ɦɨɠɟɬ ɩɪɟɜɨɫɯɨɞɢɬɶ ɫɨɬɧɢ ɢ ɬɵɫɹɱɢ ɤɢɥɨɜɚɬɬ. Ʌɚɡɟɪɧɵɟ ɭɫɬɚɧɨɜɤɢ ɞɥɹ ɢɦɩɭɥɶɫɧɨɣ ɫɜɚɪɤɢ, ɬɟɪɦɨɨɛɪɚɛɨɬɤɢ ɢ ɩɨɥɭɱɟɧɢɹ ɨɬɜɟɪɫɬɢɣ ɜɧɟɞɪɟɧɵ ɜ ɩɪɨɦɵɲɥɟɧɧɨɫɬɶ ɫ ɷɤɨɧɨɦɢɱɟɫɤɢɦ ɷɮɮɟɤɬɨɦ ɜ ɞɟɫɹɬɤɢ
ɦɢɥɥɢɨɧɨɜ ɪɭɛɥɟɣ.
ɉɪɨɦɵɲɥɟɧɧɵɟ ɥɚɡɟɪɵ ɫ ɛɨɥɶɲɨɣ ɧɟɩɪɟɪɵɜɧɨɣ ɦɨɳɧɨɫɬɶɸ ɛɨɥɟɟ
1-5 ɤȼɬ ɜɟɫɶɦɚ ɩɟɪɫɩɟɤɬɢɜɧɵ ɞɥɹ ɫɜɚɪɤɢ, ɬɟɪɦɨɨɛɪɚɛɨɬɤɢ ɢ ɞɪɭɝɢɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ, ɫɬɚɧɤɨɫɬɪɨɟɧɢɢ, ɚɜɬɨɦɨɛɢɥɶɧɨɣ ɢ ɞɪɭɝɢɯ ɨɬɪɚɫɥɹɯ ɩɪɨɦɵɲɥɟɧɧɨɫɬɢ.
ȼ ɩɨɫɥɟɞɧɟɟ ɜɪɟɦɹ ɢɧɬɟɧɫɢɜɧɨ ɪɚɡɪɚɛɚɬɵɜɚɸɬɫɹ ɤɨɧɫɬɪɭɤɰɢɢ ɢɦɩɭɥɶɫɧɨ – ɩɟɪɢɨɞɢɱɟɫɤɢɯ ɥɚɡɟɪɨɜ, ɩɟɪɫɩɟɤɬɢɜɧɵɯ ɞɥɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ ɩɨɥɭɱɟɧɢɹ ɢ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ. ɂɦɩɭɥɶɫɧɨ6
Ɋɵɤɚɥɢɧ ɇ.ɇ., ɍɝɥɨɜ Ⱥ.Ⱥ., Ʉɨɤɨɪɚ Ⱥ.ɇ. Ʌɚɡɟɪɧɚɹ ɨɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɨɜ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1975. 296 c.
20
ɩɟɪɢɨɞɢɱɟɫɤɢɟ ɜɨɡɞɟɣɫɬɜɢɹ ɩɨɡɜɨɥɹɸɬ ɭɦɟɧɶɲɢɬɶ ɪɚɡɦɟɪ ɡɨɧɵ ɬɟɪɦɢɱɟɫɤɨɝɨ ɜɥɢɹɧɢɹ ɢ ɬɟɦ ɫɚɦɵɦ ɩɨɜɵɫɢɬɶ ɄɉȾ ɩɪɨɰɟɫɫɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɷɬɨ
ɩɪɢɜɨɞɢɬ ɤ ɭɫɤɨɪɟɧɢɸ ɨɛɪɚɡɨɜɚɧɢɹ ɨɬɜɟɪɫɬɢɣ, ɬɚɤ ɤɚɤ ɜ ɩɪɨɦɟɠɭɬɤɚɯ
ɦɟɠɞɭ ɢɦɩɭɥɶɫɚɦɢ ɤɚɧɚɥ ɜ ɜɟɳɟɫɬɜɟ ɨɫɜɨɛɨɠɞɚɟɬɫɹ ɨɬ ɩɪɨɞɭɤɬɨɜ ɪɚɡɪɭɲɟɧɢɢ. ȼɚɠɧɵɦɢ ɞɥɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɢɦɟɧɟɧɢɣ ɹɜɥɹɸɬɫɹ ɚɜɬɨɤɨɥɟɛɚɧɢɹ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɚɜɬɨɤɨɥɟɛɚɬɟɥɶɧɵɯ ɪɟɠɢɦɨɜ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɥɭɱɲɟ
ɤɨɧɬɪɨɥɢɪɨɜɚɬɶ ɯɨɞ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ.
ȼ ɩɨɫɥɟɞɧɢɟ ɝɨɞɵ ɩɨɥɭɱɢɥɚ ɪɚɡɜɢɬɢɟ ɥɚɡɟɪɧɨ-ɩɥɚɡɦɟɧɧɚɹ ɦɟɬɚɥɥɭɪɝɢɹ. ȿɟ ɝɥɚɜɧɨɟ ɨɬɥɢɱɢɟ ɨɬ ɥɚɡɟɪɧɨɣ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɡɚɤɥɸɱɚɟɬɫɹ
ɜ ɬɨɦ, ɱɬɨ ɥɚɡɟɪɧɨɦɭ ɜɨɡɞɟɣɫɬɜɢɸ ɩɨɞɜɟɪɝɚɸɬɫɹ ɦɚɬɟɪɢɚɥɵ ɢ ɢɡɞɟɥɢɹ,
ɧɚɯɨɞɹɳɢɟɫɹ ɜ ɫɪɟɞɟ ɚɤɬɢɜɧɵɯ ɝɚɡɨɜ (ɚɡɨɬɚ, ɭɝɥɟɤɢɫɥɨɝɨ ɝɚɡɚ ɢ ɞɪ.) ɩɪɢ
ɩɨɜɵɲɟɧɧɨɦ ɢ ɜɵɫɨɤɨɦ ɞɚɜɥɟɧɢɢ (ɧɟɫɤɨɥɶɤɨ ɞɟɫɹɬɤɨɜ ɚɬɦɨɫɮɟɪ). ɉɪɢɫɭɬɫɬɜɢɟ ɝɚɡɨɜ ɫɩɨɫɨɛɫɬɜɭɟɬ ɨɛɪɚɡɨɜɚɧɢɸ ɩɥɚɡɦɟɧɧɨɝɨ ɫɝɭɫɬɤɚ ɜɛɥɢɡɢ
ɩɨɜɟɪɯɧɨɫɬɢ ɦɟɬɚɥɥɚ, ɟɫɥɢ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɩɪɟɜɵɲɚɟɬ 106
ȼɬ/ɫɦ2. Ɉɞɧɨɜɪɟɦɟɧɧɨɟ ɜɨɡɞɟɣɫɬɜɢɟ ɩɥɚɡɦɟɧɧɨɝɨ ɫɝɭɫɬɤɚ ɢ ɩɪɨɲɟɞɲɟɝɨ
ɱɟɪɟɡ ɧɟɝɨ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɮɨɪɦɢɪɭɟɬ ɡɨɧɭ ɨɛɪɚɛɨɬɤɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɞɟɥɢɹ, ɜɵɡɵɜɚɹ ɧɚɝɪɟɜ, ɩɨɫɥɟɞɭɸɳɟɟ ɩɥɚɜɥɟɧɢɟ ɜɟɳɟɫɬɜɚ ɢ
ɫɢɧɬɟɡ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɚɤɢɯ ɜɟɳɟɫɬɜ ɤɚɤ ɤɚɪɛɢɞɵ ɢ ɧɢɬɪɢɞɵ. ȿɫɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɚɬɦɨɫɮɟɪɭ ɜɨɫɫɬɚɧɨɜɢɬɟɥɶɧɵɯ ɝɚɡɨɜ, ɧɚɩɪɢɦɟɪ, ɜɨɞɨɪɨɞɚ, ɬɨ
ɜɨɡɦɨɠɧɨ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ ɬɭɝɨɩɥɚɜɤɢɯ ɢ ɞɪɭɝɢɯ ɦɟɬɚɥɥɨɜ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɨɤɢɫɥɨɜ (ɝɟɬɟɪɨɝɟɧɧɨɟ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ). ȼ ɥɚɡɟɪɧɨɦ ɷɪɨɡɢɨɧɧɨɦ ɮɚɤɟɥɟ ɜɨɡɦɨɠɧɨ ɬɚɤɠɟ ɝɨɦɨɝɟɧɧɨɟ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ.
ɉɥɚɡɦɟɧɧɚɹ ɬɟɯɧɨɥɨɝɢɹ. ɉɪɢɦɟɧɟɧɢɟ ɩɥɚɡɦɵ ɞɥɹ ɦɟɬɚɥɥɭɪɝɢɢ ɢ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ ɢ ɞɪ. ɨɬɪɚɫɥɹɯ ɚɤɬɢɜɧɨ ɧɚɱɚɥɨɫɶ
ɜ ɤɨɧɰɟ 50-ɯ ɝɨɞɨɜ 20-ɝɨ ɜɟɤɚ. ɉɟɪɜɨɧɚɱɚɥɶɧɨ ɨɫɧɨɜɧɨɟ ɜɧɢɦɚɧɢɟ ɛɵɥɨ
ɫɨɫɪɟɞɨɬɨɱɟɧɨ ɧɚ ɬɚɤɢɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɤɚɤ ɫɜɚɪɤɚ ɢ ɪɟɡɤɚ
ɦɟɬɚɥɥɨɜ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɩɥɚɡɦɟɧɧɚɹ ɬɟɯɧɨɥɨɝɢɹ ɧɚɱɚɥɚ ɚɤɬɢɜɧɨ ɩɪɢɦɟɧɹɬɶɫɹ ɜ ɦɟɬɚɥɥɭɪɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ, ɜɤɥɸɱɚɹ ɩɪɨɰɟɫɫɵ ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɢ ɫɢɧɬɟɡɚ, ɩɟɪɟɩɥɚɜɤɢ ɢ ɪɚɮɢɧɢɪɨɜɚɧɢɹ ɬɭɝɨɩɥɚɜɤɢɯ ɦɟɬɚɥɥɨɜ, ɩɨɥɭɱɟɧɢɹ ɩɨɪɨɲɤɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɫɮɟɪɨɢɞɢɡɚɰɢɢ ɱɚɫɬɢɰ, ɧɚɧɟɫɟɧɢɹ ɩɨɤɪɵɬɢɣ ɢ ɞɪ.
Ƚɟɧɟɪɚɰɢɹ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɨɣ ɩɥɚɡɦɵ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɞɜɭɦɹ ɨɫɧɨɜɧɵɦɢ ɫɩɨɫɨɛɚɦɢ – ɫ ɩɨɦɨɳɶɸ ɞɭɝɢ ɩɪɹɦɨɝɨ ɞɟɣɫɬɜɢɹ (ɤɨɝɞɚ ɨɞɢɧ ɢɡ
ɷɥɟɤɬɪɨɞɨɜ, ɨɛɵɱɧɨ – ɚɧɨɞ, ɹɜɥɹɟɬɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɨɛɪɚɛɚɬɵɜɚɟɦɵɦ
ɦɚɬɟɪɢɚɥɨɦ) ɢɥɢ ɫ ɩɨɦɨɳɶɸ ɢɨɧɢɡɨɜɚɧɧɨɝɨ ɝɚɡɚ, ɤɨɬɨɪɵɣ ɩɨɞɚɟɬɫɹ ɱɟɪɟɡ ɨɬɜɟɪɫɬɢɟ ɜ ɚɧɨɞɟ ɜ ɫɜɨɛɨɞɧɨɟ ɩɪɨɫɬɪɚɧɫɬɜɨ. Ɉɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ
ɩɪɨɢɡɜɨɞɢɬɫɹ ɩɪɢ ɜɜɟɞɟɧɢɢ ɜ ɩɥɚɡɦɭ ɞɢɫɩɟɪɫɧɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɩɪɨɜɨɥɨɤɢ, ɤɨɬɨɪɚɹ ɩɥɚɜɢɬɫɹ, ɞɪɨɛɢɬɫɹ ɧɚ ɦɟɥɤɢɟ ɤɚɩɥɢ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɚɡɨɜɨɝɨ
ɩɨɬɨɤɚ ɢ ɩɨɞɜɟɪɝɚɟɬɫɹ ɩɥɚɡɦɟɧɧɨɣ ɨɛɪɚɛɨɬɤɟ.
21
ɇɚɪɹɞɭ ɫ ɞɭɝɨɜɨɣ ɩɥɚɡɦɨɣ ɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɢɫɩɨɥɶɡɭɟɬɫɹ ȼɑ – ɢ ɋȼɑ – ɩɥɚɡɦɚ. ȼɑ – ɩɥɚɡɦɚ ɡɚɠɢɝɚɟɬɫɹ ɜɧɭɬɪɢ ɤɜɚɪɰɟɜɨɝɨ ɰɢɥɢɧɞɪɚ ɢ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɤɨɧɬɚɤɬɢɪɭɟɬ ɫɨ ɫɬɟɧɤɚɦɢ ɤɚɦɟɪɵ, ɱɬɨ ɞɟɥɚɟɬ
ɟɟ «ɫɬɟɪɢɥɶɧɨɣ», ɬ.ɟ. ɧɟɡɚɝɪɹɡɧɟɧɧɨɣ ɩɪɨɞɭɤɬɚɦɢ ɢɫɩɚɪɟɧɢɹ. ȼɑ – ɩɥɚɡɦɚ ɨɛɵɱɧɨ ɧɟɪɚɜɧɨɜɟɫɧɚ, ɢ ɷɬɨ ɞɚɟɬ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɥɭɱɟɧɢɹ ɜɟɳɟɫɬɜ ɫ
ɭɧɢɤɚɥɶɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ.
ɉɥɚɡɦɟɧɧɵɣ ɩɨɬɨɤ ɪɚɫɩɪɟɞɟɥɟɧ ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɡɚɤɨɧɭ, ɩɪɢ ɷɬɨɦ
ɦɚɤɫɢɦɭɦ ɦɨɳɧɨɫɬɢ ɩɪɢɯɨɞɢɬɫɹ ɧɚ ɰɟɧɬɪɚɥɶɧɭɸ ɱɚɫɬɶ ɡɨɧɵ ɜɨɡɞɟɣɫɬɜɢɹ – ɚɧɨɞɧɨɟ ɩɹɬɧɨ. ȼ ɧɟɤɨɬɨɪɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɢɫɬɨɱɧɢɤɚ ɪɚɫɩɪɟɞɟɥɟɧɧɨɣ ɩɨ ɧɨɪɦɚɥɶɧɨɦɭ ɡɚɤɨɧɭ.
ɉɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɜ ɚɧɨɞɧɨɦ ɩɹɬɧɟ ɦɨɠɟɬ ɞɨɫɬɢɝɚɬɶ 106 ȼɬ/ɫɦ2, ɚ ɜ ɫɜɨɛɨɞɧɨɣ ɩɥɚɡɦɟɧɧɨɣ ɫɬɪɭɟ, ɢɫɬɟɤɚɸɳɟɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɨ, ɨɧɚ ɧɟɫɤɨɥɶɤɨ
ɧɢɠɟ – ɞɨ 104 ȼɬ/ɫɦ2.
Ɉɫɨɛɟɧɧɨ ɩɟɪɫɩɟɤɬɢɜɧɵɦ ɹɜɥɹɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɩɥɚɡɦɟɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɜ ɦɟɬɚɥɥɭɪɝɢɢ ɢ ɬɟɯɧɨɥɨɝɢɢ ɧɟɨɪɝɚɧɢɱɟɫɤɢɯ ɦɚɬɟɪɢɚɥɨɜ. Ⱦɨɫɬɨɢɧɫɬɜɚ ɩɥɚɡɦɟɧɧɨɣ ɬɟɯɧɨɥɨɝɢɢ ɨɛɭɫɥɨɜɥɟɧɵ ɧɟ ɬɨɥɶɤɨ ɜɵɫɨɤɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɩɨɬɨɤɨɜ, ɧɨ ɢ ɜɨɡɦɨɠɧɨɫɬɶɸ ɩɪɨɜɟɞɟɧɢɹ
ɩɥɚɡɦɨ – ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ, ɩɪɢɜɨɞɹɳɢɯ ɤ ɫɢɧɬɟɡɭ ɧɨɜɵɯ ɜɟɳɟɫɬɜ.
1.6. Ɍɟɩɥɨɜɚɹ ɡɚɳɢɬɚ
Ɉɫɨɛɟɧɧɨ ɫɥɨɠɧɵɟ ɢ ɜɚɠɧɵɟ ɡɚɞɚɱɢ ɫɬɨɹɬ ɜ ɨɛɥɚɫɬɢ ɢɡɭɱɟɧɢɹ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɚɜɢɚɰɢɨɧɧɨɣ ɢ ɪɚɤɟɬɧɨ-ɤɨɫɦɢɱɟɫɤɨɣ ɬɟɯɧɢɤɟ 7.
ɉɪɢ ɫɜɟɪɯɡɜɭɤɨɜɵɯ ɫɤɨɪɨɫɬɹɯ ɩɨɥɟɬɚ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɧɹɸɬɫɹ ɭɫɥɨɜɢɹ
ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɨɬɞɟɥɶɧɵɯ ɷɥɟɦɟɧɬɚɯ ɤɨɧɫɬɪɭɤɰɢɣ ɥɟɬɚɬɟɥɶɧɵɯ ɚɩɩɚɪɚɬɨɜ. ȼɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɟɝɨ ɨɯɥɚɠɞɟɧɢɹ ɢɥɢ ɡɚɳɢɬɵ ɨɬ ɚɷɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɧɚɝɪɟɜɚ, ɹɜɥɹɸɳɟɝɨɫɹ ɫɥɟɞɫɬɜɢɟɦ ɬɪɟɧɢɹ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɥɟɬɚɬɟɥɶɧɨɝɨ ɚɩɩɚɪɚɬɚ ɢ ɧɚɛɟɝɚɸɳɢɦ ɩɨɬɨɤɨɦ ɜɨɡɞɭɯɚ ɢɥɢ ɩɨɬɨɤɨɦ ɤɚɤɢɯ-ɥɢɛɨ ɞɪɭɝɢɯ ɝɚɡɨɜ, ɫɨɫɬɚɜɥɹɸɳɢɯ ɚɬɦɨɫɮɟɪɵ ɩɥɚɧɟɬ. ɉɪɨɛɥɟɦɚ ɬɟɩɥɨɜɨɣ ɡɚɳɢɬɵ ɥɟɬɚɬɟɥɶɧɨɝɨ ɚɩɩɚɪɚɬɚ ɨɬ ɜɵɫɨɤɢɯ ɭɞɟɥɶɧɵɯ ɬɟɩɥɨɜɵɯ ɩɨɬɨɤɨɜ ɢ ɜɵɫɨɤɢɯ ɬɟɦɩɟɪɚɬɭɪ ɪɚɡɪɚɛɚɬɵɜɚɟɬɫɹ ɫ 50-60-ɯ ɝɨɞɨɜ
XX ɜɟɤɚ. Ɂɚ ɷɬɨ ɜɪɟɦɹ ɩɪɨɜɟɞɟɧɨ ɢɫɫɥɟɞɨɜɚɧɢɟ ɪɚɡɥɢɱɧɵɯ ɜɢɞɨɜ ɬɟɩɥɨɡɚɳɢɬɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɢ ɬɟɩɥɨɡɚɳɢɬɧɵɯ ɩɨɤɪɵɬɢɣ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ
ɧɚɞɟɠɧɭɸ ɡɚɳɢɬɭ ɥɟɬɚɬɟɥɶɧɨɝɨ ɚɩɩɚɪɚɬɚ. Ɋɚɡɪɚɛɨɬɚɧɚ ɬɟɨɪɢɹ ɢ ɢɫɫɥɟɞɨɜɚɧɵ ɨɫɧɨɜɧɵɟ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɢ ɬɟɩɥɨɨɛɦɟɧɚ ɩɪɨɰɟɫɫɨɜ ɜɨɡɞɟɣɫɬɜɢɹ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɢ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɝɚɡɨɜɵɯ ɩɨɬɨɤɨɜ ɧɚ ɪɚɡɥɢɱɧɵɟ ɤɨɧɫɬɪɭɤɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ. ɇɟ ɦɟɧɟɟ
7
ɉɨɥɟɠɚɟɜ ɘ.ȼ., ɘɪɟɜɢɱ Ɏ.Ȼ. Ɍɟɩɥɨɜɚɹ ɡɚɳɢɬɚ. Ɇ.: ɗɧɟɪɝɢɹ, 1976. 392 c.;
ɉɨɥɟɠɚɟɜ ɘ.ȼ., Ɏɪɨɥɨɜ Ƚ.Ⱥ. Ɍɟɩɥɨɜɨɟ ɪɚɡɪɭɲɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ. Ʉɢɟɜ: Ⱥɤɚɞɟɦɩɟɪɢɨɞɢɤɚ, 2006. 354 c.
22
ɫɥɨɠɧɵɟ ɢ ɜɚɠɧɵɟ ɩɪɨɛɥɟɦɵ ɜɨɡɧɢɤɚɸɬ ɩɪɢ ɤɨɧɫɬɪɭɢɪɨɜɚɧɢɢ ɫɨɜɪɟɦɟɧɧɵɯ ɚɜɢɚɰɢɨɧɧɵɯ ɢ ɪɚɤɟɬɧɵɯ ɞɜɢɝɚɬɟɥɟɣ. ȼɵɫɨɤɚɹ ɬɟɩɥɨɜɚɹ ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɪɟɚɤɬɢɜɧɵɯ ɞɜɢɝɚɬɟɥɟɣ, ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɫɨɜɪɟɦɟɧɧɵɯ ɬɨɩɥɢɜ
ɬɪɟɛɭɸɬ ɨɬ ɤɨɧɫɬɪɭɤɬɨɪɚ ɭɦɟɧɢɟ ɩɪɨɜɟɫɬɢ ɫɥɨɠɧɵɣ ɢɧɠɟɧɟɪɧɵɣ ɪɚɫɱɟɬ ɬɟɩɥɨɨɛɦɟɧɚ. Ⱦɥɹ ɩɪɨɜɟɞɟɧɢɹ ɬɚɤɨɝɨ ɪɨɞɚ ɪɚɫɱɟɬɨɜ ɜɚɠɧɨ ɡɧɚɬɶ ɦɟɬɨɞɵ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɢ ɪɚɞɢɚɰɢɨɧɧɨɝɨ ɧɚɝɪɟɜɚ, ɦɟɬɨɞɵ
ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɫɜɨɣɫɬɜɚ ɬɟɩɥɨɡɚɳɢɬɧɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɢ ɤɢɧɟɬɢɤɭ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɧɚ ɩɨɜɟɪɯɧɨɫɬɹɯ ɢ ɜ ɨɛɴɟɦɟ (ɪɚɡɞɟɥ 11).
Ȼɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɬɟɨɪɢɹ ɬɟɩɥɨɨɛɦɟɧɚ ɢɦɟɟɬ ɜ ɪɚɫɱɟɬɚɯ ɬɟɩɥɨɜɵɯ
ɪɟɠɢɦɨɜ ɥɟɬɚɬɟɥɶɧɵɯ ɚɩɩɚɪɚɬɨɜ, ɤɚɛɢɧ ɚɩɩɚɪɚɬɨɜ, ɫɢɫɬɟɦ ɠɢɡɧɟɨɛɟɫɩɟɱɟɧɢɹ ɢ ɤɨɧɞɢɰɢɨɧɢɪɨɜɚɧɢɹ, ɧɚɞɟɠɧɨɣ ɪɚɛɨɬɵ ɪɚɞɢɨɷɥɟɤɬɪɨɧɧɨɣ ɚɩɩɚɪɚɬɭɪɵ, ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɚɬɨɦɧɨɣ ɷɧɟɪɝɟɬɢɤɟ, ɜ ɨɛɟɫɩɟɱɟɧɢɢ ɬɟɩɥɨɜɵɯ
ɪɟɠɢɦɨɜ ɹɞɟɪɧɵɯ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɭɫɬɚɧɨɜɨɤ ɢ ɢɯ ɛɟɡɨɩɚɫɧɨɫɬɢ.
ɉɪɢ ɩɪɨɟɤɬɢɪɨɜɚɧɢɢ ɝɪɚɠɞɚɧɫɤɢɯ ɢ ɩɪɨɦɵɲɥɟɧɧɵɯ ɡɞɚɧɢɣ 8 ɜɨɡɧɢɤɚɟɬ ɦɧɨɝɨ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱ, ɤɨɬɨɪɵɟ ɫɯɨɞɧɵ ɫ ɩɪɨɛɥɟɦɚɦɢ ɬɟɩɥɨɜɨɣ
ɡɚɳɢɬɵ ɢ ɞɥɹ ɪɟɲɟɧɢɹ ɤɨɬɨɪɵɯ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɪɟɠɢɦɵ ɪɚɛɨɬɵ ɢ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɫɢɫɬɟɦɵ ɤɨɧɞɢɰɢɨɧɢɪɨɜɚɧɢɹ ɜ ɭɫɥɨɜɢɹɯ ɝɨɞɨɜɨɣ ɢɡɦɟɧɱɢɜɨɫɬɢ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨ- ɢ ɦɚɫɨɨɛɦɟɧɚ ɦɟɠɞɭ ɩɨɦɟɳɟɧɢɟɦ, ɷɥɟɦɟɧɬɚɦɢ
ɫɢɫɬɟɦɵ ɢ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
ȼ ɫɬɪɨɢɬɟɥɶɧɨɣ ɬɟɩɥɨɮɢɡɢɤɟ ɜɨɡɞɭɲɧɵɦ ɪɟɠɢɦɨɦ ɡɞɚɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɮɚɤɬɨɪɨɜ ɢ ɹɜɥɟɧɢɣ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɨɛɳɢɣ ɩɪɨɰɟɫɫ
ɨɛɦɟɧɚ ɜɨɡɞɭɯɚ ɦɟɠɞɭ ɜɫɟɦɢ ɟɝɨ ɩɨɦɟɳɟɧɢɹɦɢ, ɧɚɪɭɠɧɵɦ ɜɨɡɞɭɯɨɦ,
ɜɤɥɸɱɚɹ ɩɟɪɟɦɟɳɟɧɢɟ ɜɨɡɞɭɯɚ ɜɧɭɬɪɢ ɩɨɦɟɳɟɧɢɣ, ɞɜɢɠɟɧɢɟ ɟɝɨ ɱɟɪɟɡ
ɤɚɧɚɥɵ ɢ ɜɨɡɞɭɯɨɜɨɞɵ ɢ ɨɛɬɟɤɚɧɢɟ ɡɞɚɧɢɹ ɩɨɬɨɤɨɦ ɜɨɡɞɭɯɚ. Ɉɬɞɟɥɶɧɵɟ
ɩɪɨɰɟɫɫɵ ɜɟɫɶɦɚ ɫɥɨɠɧɵ. ɂɯ ɨɩɢɫɚɧɢɟ ɛɚɡɢɪɭɟɬɫɹ ɧɚ ɤɥɚɫɫɢɱɟɫɤɢɯ
ɭɪɚɜɧɟɧɢɹɯ ɩɟɪɟɧɨɫɚ ɦɚɫɫɵ, ɷɧɟɪɝɢɢ, ɢɦɩɭɥɶɫɚ ɜ ɬɭɪɛɭɥɟɧɬɧɵɯ ɩɨɬɨɤɚɯ. ȿɫɬɟɫɬɜɟɧɧɵɟ ɫɢɥɵ, ɜɥɢɹɸɳɢɟ ɧɚ ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɜɨɡɞɭɯɚ, ɷɬɨ –
ɝɪɚɜɢɬɚɰɢɹ ɢ ɜɟɬɟɪ. ȼɨɡɞɭɯɨɨɛɦɟɧ ɫ ɭɱɟɬɨɦ ɷɬɢɯ ɫɢɥ ɬɪɭɞɧɨ ɪɚɫɫɱɢɬɵɜɚɬɶ ɢ ɩɪɨɝɧɨɡɢɪɨɜɚɬɶ.
ȼɨɡɞɭɲɧɵɣ ɪɟɠɢɦ ɫɜɹɡɚɧ ɫ ɬɟɩɥɨɜɵɦ ɪɟɠɢɦɨɦ ɡɞɚɧɢɣ. ɂɧɮɢɥɶɬɪɚɰɢɹ ɧɚɪɭɠɧɨɝɨ ɜɨɡɞɭɯɚ ɩɪɢɜɨɞɢɬ ɤ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦ ɬɪɚɬɚɦ ɬɟɩɥɚ ɧɚ ɟɝɨ
ɩɨɞɨɝɪɟɜ. ɗɤɫɮɢɥɶɬɪɚɰɢ ɜɥɚɠɧɨɝɨ ɜɧɭɬɪɟɧɧɟɝɨ ɜɨɡɞɭɯɚ ɭɜɥɚɠɧɹɟɬ ɫɬɟɧɵ ɢ ɩɟɪɟɤɪɵɬɢɹ ɢ ɫɧɢɠɚɟɬ ɬɟɩɥɨɡɚɳɢɬɧɵɟ ɫɜɨɣɫɬɜɚ ɨɝɪɚɠɞɟɧɢɣ.
ɇɚɪɭɠɧɵɟ ɨɝɪɚɠɞɟɧɢɹ ɡɞɚɧɢɣ ɞɨɥɠɧɵ ɩɪɟɞɨɯɪɚɧɹɬɶ ɩɨɦɟɳɟɧɢɹ
ɡɞɚɧɢɣ ɨɬ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɯ ɚɬɦɨɫɮɟɪɧɵɯ ɜɨɡɞɟɣɫɬɜɢɣ. ɗɬɢ ɮɭɧɤɰɢɣ
ɨɝɪɚɠɞɟɧɢɣ ɨɤɚɡɵɜɚɸɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɜɚɠɧɵɦɢ ɜ ɨɛɳɟɣ ɡɚɞɚɱɟ ɫɨɡɞɚɧɢɹ
ɦɢɤɪɨɤɥɢɦɚɬɚ ɩɨɦɟɳɟɧɢɣ. ɇɚ ɜɧɭɬɪɟɧɧɸɸ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɪɭɠɧɨɝɨ ɨɝ8
Ȼɨɝɨɫɥɨɜɫɤɢɣ ȼ.ɇ. ɋɬɪɨɢɬɟɥɶɧɚɹ ɬɟɩɥɨɮɢɡɢɤɚ. Ɇ.: ȼɵɫɲ. ɲɤ., 1982. 415 c.;
ɒɢɥɶɞ ȿ., Ʉɢɫɫɟɥɶɦɚɧ ɏ.Ɏ. ɋɬɪɨɢɬɟɥɶɧɚɹ ɮɢɡɢɤɚ. Ɇ.: ɋɬɪɨɣɢɡɞɚɬ, 1982. 296 c.
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ɪɚɠɞɟɧɢɹ ɢɡɥɭɱɟɧɢɟɦ ɢ ɤɨɧɜɟɤɰɢɟɣ ɩɟɪɟɞɚɟɬɫɹ ɨɩɪɟɞɟɥɟɧɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ. ȼ ɯɨɥɨɞɧɵɣ ɩɟɪɢɨɞ ɝɨɞɚ ɷɬɨ ɬɟɩɥɨ ɬɟɪɹɟɬɫɹ ɱɟɪɟɡ ɨɝɪɚɠɞɟɧɢɹ ɜ ɫɬɨɪɨɧɭ ɜɧɟɲɧɟɣ ɫɪɟɞɵ. Ɉɫɧɨɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɨɬɟɪɹɦ ɬɟɩɥɚ
ɨɤɚɡɵɜɚɸɬ ɬɟɩɥɨɡɚɳɢɬɧɵɟ ɫɜɨɣɫɬɜɚ ɦɚɬɟɪɢɚɥɨɜ ɨɝɪɚɠɞɟɧɢɹ. ȼ ɫɬɚɰɢɨɧɚɪɧɵɯ ɭɫɥɨɜɢɹɯ ɜɫɟ ɬɟɩɥɨ, ɜɨɫɩɪɢɧɹɬɨɟ ɩɨɜɟɪɯɧɨɫɬɶɸ ɨɝɪɚɠɞɟɧɢɹ ɜ
ɩɨɦɟɳɟɧɢɢ, ɩɟɪɟɞɚɟɬɫɹ ɧɚɪɭɠɧɨɦɭ ɜɨɡɞɭɯɭ. Ɍɚɤɨɣ ɪɟɠɢɦ ɯɚɪɚɤɬɟɪɟɧ
ɞɥɹ ɡɢɦɧɢɯ ɭɫɥɨɜɢɣ ɩɪɢ ɧɟɛɨɥɶɲɢɯ ɤɨɥɟɛɚɧɢɹɯ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɭɬɪɢ ɢ
ɫɧɚɪɭɠɢ ɡɞɚɧɢɣ. ȼ ɥɟɬɧɢɣ ɩɟɪɢɨɞ ɨɝɪɚɠɞɟɧɢɹ ɞɨɥɠɧɵ ɡɚɳɢɬɢɬɶ ɩɨɦɟɳɟɧɢɹ ɨɬ ɩɨɥɭɞɟɧɧɨɝɨ ɡɧɨɹ, ɜɨɫɩɪɟɩɹɬɫɬɜɨɜɚɬɶ ɪɟɡɤɨɦɭ ɤɨɥɟɛɚɧɢɸ
ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɬɟɱɟɧɢɟ ɫɭɬɨɤ.
Ⱦɥɹ ɫɬɪɨɢɬɟɥɟɣ ɜɚɠɧɵ ɦɧɨɝɢɟ ɜɨɩɪɨɫɵ, ɨɬɧɨɫɹɳɢɟɫɹ ɤ ɨɛɥɚɫɬɢ
ɫɬɪɨɢɬɟɥɶɧɨɣ ɬɟɩɥɨɮɢɡɢɤɢ. ɗɬɨ – ɩɪɨɦɟɪɡɚɧɢɟ, ɩɭɱɟɧɢɟ ɝɪɭɧɬɨɜ ɢ ɢɯ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɫ ɢɧɠɟɧɟɪɧɵɦɢ ɫɨɨɪɭɠɟɧɢɹɦɢ ɜ ɪɚɣɨɧɚɯ ɫɟɡɨɧɧɨɝɨ
ɩɪɨɦɟɪɡɚɧɢɹ ɢ ɜ ɪɚɣɨɧɚɯ «ɜɟɱɧɨɣ» ɦɟɪɡɥɨɬɵ, ɬɟɩɥɨɜɥɚɠɧɨɫɬɧɵɣ ɪɟɠɢɦ
ɝɢɞɪɨɬɟɯɧɢɱɟɫɤɢɯ ɫɨɨɪɭɠɟɧɢɣ, ɨɫɨɛɟɧɧɨ ɜ ɡɨɧɟ ɩɟɪɟɦɟɧɧɨɝɨ ɝɨɪɢɡɨɧɬɚ
ɜɨɞɵ ɢ ɮɢɥɶɬɪɚɰɢɢ ɝɪɭɧɬɨɜɵɯ ɜɨɞ; ɜɨɩɪɨɫɵ ɦɨɪɨɡɨɫɬɨɣɤɨɫɬɢ ɦɚɬɟɪɢɚɥɨɜ, ɫɭɲɤɢ ɢɡɞɟɥɢɣ, ɩɪɨɰɟɫɫɵ ɬɟɩɥɨ- ɢ ɦɚɫɫɨɨɛɦɟɧɚ ɩɪɢ ɬɜɟɪɞɟɧɢɢ ɛɟɬɨɧɚ ɢ ɢɡɝɨɬɨɜɥɟɧɢɢ ɫɬɪɨɢɬɟɥɶɧɵɯ ɞɟɬɚɥɟɣ ɢ ɤɨɧɫɬɪɭɤɰɢɣ ɧɚ ɡɚɜɨɞɚɯ.
ɉɪɨɛɥɟɦɵ ɬɟɩɥɨɜɨɣ ɡɚɳɢɬɵ ɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɧɟɪɚɡɪɵɜɧɨ ɫɜɹɡɚɧɵ ɫ
ɪɚɡɪɚɛɨɬɤɨɣ ɢ ɭɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɟɦ ɧɨɜɵɯ ɦɚɬɟɪɢɚɥɨɜ. Ɍɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɞɥɹ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɡɞɚɧɢɣ (ɫɨɨɪɭɠɟɧɢɣ), ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɨɛɨɪɭɞɨɜɚɧɢɹ, ɭɡɥɨɜ ɚɜɢɚɰɢɨɧɧɨɣ ɢ ɪɚɤɟɬɧɨɣ
ɬɟɯɧɢɤɢ ɢ ɞɪ., ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɜɵɫɨɤɨɣ ɩɨɪɢɫɬɨɫɬɶɸ ɢ ɧɢɡɤɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ. ȼɵɫɨɤɚɹ ɩɨɪɢɫɬɨɫɬɶ ɷɬɢɯ ɦɚɬɟɪɢɚɥɨɜ ɞɨɫɬɢɝɚɟɬɫɹ ɞɨɛɚɜɥɟɧɢɟɦ ɜ ɦɚɬɟɪɢɚɥ ɩɨɪɢɫɬɨɝɨ ɧɚɩɨɥɧɢɬɟɥɹ (ɩɪɢɪɨɞɧɨɝɨ ɢɥɢ ɢɫɤɭɫɫɬɜɟɧɧɨɝɨ), ɜɫɩɭɱɢɜɚɧɢɟɦ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ, ɜɜɟɞɟɧɢɟɦ ɢ ɩɨɫɥɟɞɭɸɳɢɦ
ɭɞɚɥɟɧɢɟɦ ɪɚɡɥɢɱɧɵɯ ɞɨɛɚɜɨɤ (ɨɛɵɱɧɨ ɜɵɝɨɪɚɸɳɢɯ), ɜɜɟɞɟɧɢɟɦ ɜɨɡɞɭɯɚ ɜ ɫɭɫɩɟɧɡɢɸ ɢɥɢ ɪɚɫɩɥɚɜ, ɜɵɞɟɥɟɧɢɟɦ ɝɚɡɨɨɛɪɚɡɧɵɯ ɩɪɨɞɭɤɬɨɜ
ɜɫɥɟɞɫɬɜɢɟ ɩɪɨɬɟɤɚɧɢɹ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ, ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɜɨɥɨɤɨɧ.
ɉɨ ɬɟɦɩɟɪɚɬɭɪɟ ɩɪɢɦɟɧɟɧɢɹ ɪɚɡɥɢɱɚɸɬ ɧɟɨɝɧɟɭɩɨɪɧɵɟ ɢ ɨɝɧɟɭɩɨɪɧɵɟ
ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ. Ⱦɥɹ ɧɟɨɝɧɟɭɩɨɪɧɵɯ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɢɫɩɨɥɶɡɭɟɦɵɯ ɨɛɵɱɧɨ ɜ ɫɬɪɨɢɬɟɥɶɫɬɜɟ ɠɢɥɵɯ ɡɞɚɧɢɣ ɢ
ɩɪɨɦɵɲɥɟɧɧɵɯ ɫɨɨɪɭɠɟɧɢɣ, ɪɟɝɥɚɦɟɧɬɢɪɭɟɬɫɹ ɩɥɨɬɧɨɫɬɶ, ɩɪɨɱɧɨɫɬɶ,
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ – ɜɨɞɨ-, ɛɢɨ-, ɦɨɪɨɡɨɫɬɨɣɤɨɫɬɶ ɢ ɫɩɨɫɨɛɧɨɫɬɶ ɤ ɜɨɡɝɨɪɚɧɢɸ. Ɉɝɧɟɭɩɨɪɧɵɟ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ
ɩɪɢɦɟɧɹɸɬ ɜ ɬɟɩɥɨɜɵɯ ɚɝɪɟɝɚɬɚɯ, ɚɝɪɟɫɫɢɜɧɵɯ ɫɪɟɞɚɯ ɢɥɢ ɩɪɢ ɡɧɚɱɢɬɟɥɶɧɨɦ ɩɟɪɟɩɚɞɟ ɬɟɦɩɟɪɚɬɭɪɵ. Ʉ ɧɢɦ ɩɪɟɞɴɹɜɥɹɸɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ
ɬɪɟɛɨɜɚɧɢɹ. ɋɨɛɫɬɜɟɧɧɨ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ ɩɪɟɞɧɚɡɧɚɱɟɧɵ
ɞɥɹ ɭɦɟɧɶɲɟɧɢɹ ɩɨɬɟɪɶ ɬɟɩɥɚ ɨɛɴɟɤɬɨɦ, ɚ ɬɟɩɥɨɡɚɳɢɬɧɵɟ – ɝɥɚɜɧɵɦ
ɨɛɪɚɡɨɦ, ɞɥɹ ɡɚɳɢɬɵ ɩɟɪɫɨɧɚɥɚ ɢ ɨɛɨɪɭɞɨɜɚɧɢɹ ɨɬ ɬɟɩɥɚ, ɩɨɫɬɭɩɚɸɳɟɝɨ
ɢɡɜɧɟ.
24
1.7. Ɍɟɩɥɨɨɛɦɟɧɧɢɤɢ
Ɍɟɩɥɨɨɛɦɟɧɧɵɟ ɚɩɩɚɪɚɬɵ, ɢɥɢ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ, ɩɪɟɞɧɚɡɧɚɱɟɧɵ ɞɥɹ
ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɨɬ ɨɞɧɢɯ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ ɤ ɞɪɭɝɢɦ ɢ ɩɨɞɪɚɡɞɟɥɹɸɬɫɹ
ɧɚ ɪɟɤɭɩɟɪɚɬɢɜɧɵɟ, ɫɦɟɫɢɬɟɥɶɧɵɟ ɢ ɪɟɝɟɧɟɪɚɬɢɜɧɵɟ. Ɋɚɫɱɟɬ ɬɟɩɥɨɨɛɦɟɧɧɨɣ ɚɩɩɚɪɚɬɭɪɵ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɜɹɡɚɧ ɫ ɬɟɨɪɢɟɣ ɬɟɩɥɨɨɛɦɟɧɚ ɢ
ɬɟɪɦɨɞɢɧɚɦɢɤɨɣ9.
ȼ ɪɟɤɭɩɟɪɚɬɢɜɧɵɯ ɚɩɩɚɪɚɬɚɯ, ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɯ ɜ ɯɢɦɢɱɟɫɤɨɣ ɬɟɯɧɨɥɨɝɢɢ, ɬɟɩɥɨɧɨɫɢɬɟɥɢ ɩɪɨɯɨɞɹɬ ɩɨ ɪɚɡɥɢɱɧɵɦ ɨɛɴɟɦɚɦ,
ɪɚɡɞɟɥɟɧɧɵɦ ɬɜɟɪɞɨɣ ɫɬɟɧɤɨɣ, ɱɟɪɟɡ ɤɨɬɨɪɭɸ ɩɪɨɢɫɯɨɞɢɬ ɬɟɩɥɨɨɛɦɟɧ.
Ⱦɥɹ ɭɦɟɧɶɲɟɧɢɹ ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɬɟɧɤɚ ɜɵɩɨɥɧɹɟɬɫɹ ɢɡ
ɦɚɬɟɪɢɚɥɚ ɫ ɯɨɪɨɲɟɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ: ɦɟɞɢ, ɥɚɬɭɧɢ, ɫɩɥɚɜɨɜ ɚɥɸɦɢɧɢɹ ɢ ɬ.ɞ. ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɪɭɛɱɚɬɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ, ɜ
ɤɨɬɨɪɵɯ ɨɞɢɧ ɬɟɩɥɨɧɨɫɢɬɟɥɶ ɞɜɢɠɟɬɫɹ ɜ ɬɪɭɛɚɯ, ɚ ɞɪɭɝɨɣ – ɜ ɦɟɠɬɪɭɛɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. ȼ ɬɚɤɢɯ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚɯ ɫɦɟɲɟɧɢɹ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ
ɧɟ ɩɪɨɢɫɯɨɞɢɬ, ɢ ɨɧɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɥɹ ɫɚɦɵɯ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɫɨɱɟɬɚɧɢɣ ɝɪɟɸɳɟɝɨ ɢ ɧɚɝɪɟɜɚɟɦɨɝɨ ɜɟɳɟɫɬɜ.
ȼ ɫɦɟɫɢɬɟɥɶɧɵɯ ɚɩɩɚɪɚɬɚɯ ɨɛɚ ɬɟɩɥɨɧɨɫɢɬɟɥɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɩɨɫɬɭɩɚɸɬ ɜ ɨɞɢɧ ɨɛɴɟɦ ɢ ɨɛɦɟɧɢɜɚɸɬɫɹ ɬɟɩɥɨɬɨɣ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɱɟɪɟɡ
ɩɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ ɮɚɡ. ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɵ ɫɦɟɫɢɬɟɥɶɧɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ, ɜ ɤɨɬɨɪɵɯ ɫɦɟɲɢɜɚɸɬɫɹ ɬɟɩɥɨɧɨɫɢɬɟɥɢ, ɧɟ ɬɪɟɛɭɸɳɢɟ
ɞɚɥɶɧɟɣɲɟɝɨ ɪɚɡɞɟɥɟɧɢɹ, ɧɚɩɪɢɦɟɪ, ɩɪɢ ɩɨɞɨɝɪɟɜɟ ɜɨɞɵ ɩɚɪɨɦ ɢɥɢ ɝɨɪɹɱɟɣ ɜɨɞɨɣ. Ⱦɥɹ ɩɨɞɞɟɪɠɚɧɢɹ ɡɚɞɚɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪ ɜ ɫɢɫɬɟɦɟ ɝɨɪɹɱɟɝɨ
ɜɨɞɨɫɧɚɛɠɟɧɢɹ (60 ºɋ d T d 75 ºɋ) ɢ ɜ ɪɚɞɢɚɬɨɪɚɯ ɨɬɨɩɥɟɧɢɹ (T < 95ºɋ)
ɫɦɟɲɢɜɚɸɬ ɜɨɞɭ, ɢɞɭɳɭɸ ɢɡ ɤɨɬɟɥɶɧɨɣ ɢɥɢ Ɍɗɐ ( T d 150 ºɋ), ɫ ɜɨɞɨɣ
( T d 20 y 70 ºɋ), ɜɨɡɜɪɚɳɚɸɳɟɣɫɹ ɨɬ ɩɨɬɪɟɛɢɬɟɥɹ ɬɟɩɥɚ.
ɂɫɩɨɥɶɡɭɸɬ ɫɦɟɫɢɬɟɥɶɧɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ ɢ ɞɥɹ ɥɟɝɤɨ ɪɚɡɞɟɥɹɸɳɢɯɫɹ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ: ɝɚɡ – ɠɢɞɤɨɫɬɶ, ɝɚɡ – ɞɢɫɩɟɪɫɧɵɣ ɬɜɟɪɞɵɣ ɦɚɬɟɪɢɚɥ, ɜɨɞɚ – ɦɚɫɥɨ ɢ ɬ.ɞ.
ȼ ɪɟɝɟɧɟɪɚɬɢɜɧɵɯ ɚɩɩɚɪɚɬɚɯ ɜ ɟɞɢɧɫɬɜɟɧɧɵɣ ɪɚɛɨɱɢɣ ɨɛɴɟɦ ɫɧɚɱɚɥɚ
ɩɨɫɬɭɩɚɟɬ ɝɨɪɹɱɢɣ ɬɟɩɥɨɧɨɫɢɬɟɥɶ, ɧɚɝɪɟɜɚɸɳɢɣ ɦɚɫɫɭ ɬɜɟɪɞɨɝɨ ɦɚɬɟɪɢɚɥɚ (ɤɢɪɩɢɱɧɭɸ ɤɥɚɞɤɭ ɢɥɢ ɦɚɫɫɭ ɦɟɬɚɥɥɚ), ɚ ɡɚɬɟɦ ɜ ɬɨɬ ɠɟ ɨɛɴɟɦ
ɩɨɞɚɟɬɫɹ ɧɚɝɪɟɜɚɟɦɚɹ ɫɪɟɞɚ, ɤɨɬɨɪɚɹ ɜɨɫɩɪɢɧɢɦɚɟɬ ɬɟɩɥɨɬɭ ɨɬ ɧɚɝɪɟɬɨɝɨ
ɦɚɬɟɪɢɚɥɚ. ȼ ɪɟɝɟɧɟɪɚɬɢɜɧɵɯ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚɯ ɜ ɤɚɱɟɫɬɜɟ ɩɪɨɦɟɠɭɬɨɱɧɨɝɨ ɬɟɩɥɨɧɨɫɢɬɟɥɹ ɢɫɩɨɥɶɡɭɟɬɫɹ ɬɜɟɪɞɵɣ ɞɨɫɬɚɬɨɱɧɨ ɦɚɫɫɢɜɧɵɣ ɦɚɬɟɪɢɚɥ – ɥɢɫɬɵ ɦɟɬɚɥɥɚ, ɤɢɪɩɢɱɢ, ɪɚɡɥɢɱɧɵɟ ɡɚɫɵɩɤɢ. Ɋɟɝɟɧɟɪɚɬɢɜɧɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ ɧɟɡɚɦɟɧɢɦɵ ɞɥɹ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɞɨɝɪɟɜɚ ɝɚɡɨɜ
( T ! 1000 ºɋ), ɩɨɫɤɨɥɶɤɭ ɠɚɪɨɫɬɨɣɤɨɫɬɶ ɦɟɬɚɥɥɨɜ ɨɝɪɚɧɢɱɟɧɚ, ɚ ɧɚɫɚɞɤɚ
9
Ʉɟɣɫ ȼ.Ɇ., Ʌɨɧɞɨɧ Ⱥ.Ʌ. Ʉɨɦɩɚɤɬɧɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ. Ɇ.; Ʌ.: Ƚɨɫɷɧɟɪɝɨɢɡɞɚɬ,
1962. 160 c.; Ɍɟɩɥɨɬɟɯɧɢɤɚ, ɩɨɞ ɨɛɳɟɣ ɪɟɞ. Ʉɪɭɬɨɜɚ ȼ.ɂ., 1986, 432 c.
25
ɢɡ ɨɝɧɟɭɩɨɪɧɵɯ ɤɢɪɩɢɱɟɣ ɦɨɠɟɬ ɪɚɛɨɬɚɬɶ ɩɪɢ ɨɱɟɧɶ ɜɵɫɨɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ. ɂɧɨɝɞɚ ɪɟɝɟɧɟɪɚɬɢɜɧɵɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɢ ɜɵɝɨɞɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢ
ɞɥɹ ɨɯɥɚɠɞɟɧɢɹ ɡɚɩɵɥɟɧɧɵɯ ɝɚɡɨɜ, ɤɨɬɨɪɵɟ ɫɩɨɫɨɛɧɵ ɛɵɫɬɪɨ ɢɡɧɚɲɢɜɚɬɶ ɢ ɡɚɛɢɜɚɬɶ ɬɪɭɛɤɢ ɪɟɤɭɩɟɪɚɬɢɜɨɜ.
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɧɚɡɧɚɱɟɧɢɹ ɪɚɡɥɢɱɚɸɬ ɬɪɢ ɬɢɩɚ
ɬɟɩɥɨɨɛɦɟɧɧɢɤɨɜ: ɚ) ɧɚɝɪɟɜɚɬɟɥɢ (ɨɯɥɚɞɢɬɟɥɢ), ɜ ɤɨɬɨɪɵɯ ɬɟɩɥɨɧɨɫɢɬɟɥɢ ɧɟ ɢɡɦɟɧɹɸɬ ɮɚɡɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ; ɛ) ɢɫɩɚɪɢɬɟɥɢ (ɤɢɩɹɬɢɥɶɧɢɤɢ) ɢ
ɤɨɧɞɟɧɫɚɬɨɪɵ, ɩɪɟɞɧɚɡɧɚɱɟɧɧɵɟ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɮɚɡɨɜɨɝɨ ɫɨɫɬɨɹɧɢɹ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ; ɜ) ɞɥɹ ɨɫɭɳɟɫɬɜɥɟɧɢɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɬɟɩɥɨɨɛɦɟɧɚ ɢ ɯɢɦɢɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ (ɜɵɩɚɪɧɵɟ ɚɩɩɚɪɚɬɵ, ɤɪɢɫɬɚɥɥɢɡɚɬɨɪɵ, ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɬɨɪɵ ɢ ɞɪ.).
ȼ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚɯ ɫ ɜɧɭɬɪɟɧɧɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɷɧɟɪɝɢɢ ɩɪɢɦɟɧɹɸɬ
ɧɟ ɞɜɚ, ɤɚɤ ɨɛɵɱɧɨ, ɬɟɩɥɨɧɨɫɢɬɟɥɹ, ɚ ɨɞɢɧ ɬɟɩɥɨɧɨɫɢɬɟɥɶ, ɤɨɬɨɪɵɣ ɨɬɜɨɞɢɬ ɬɟɩɥɨɬɭ, ɜɵɞɟɥɢɜɲɭɸɫɹ ɜ ɫɚɦɨɦ ɚɩɩɚɪɚɬɟ. ɉɪɢɦɟɪɨɦ ɬɚɤɢɯ ɚɩɩɚɪɚɬɨɜ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɹɞɟɪɧɵɟ ɪɟɚɤɬɨɪɵ, ɷɥɟɤɬɪɨɧɚɝɪɟɜɚɬɟɥɢ ɢ ɞɪɭɝɢɟ
ɭɫɬɪɨɣɫɬɜɚ. ɇɟɡɚɜɢɫɢɦɨ ɨɬ ɩɪɢɧɰɢɩɚ ɞɟɣɫɬɜɢɹ, ɬɟɩɥɨɨɛɦɟɧɧɵɟ ɚɩɩɚɪɚɬɵ, ɩɪɢɦɟɧɹɟɦɵɟ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ ɬɟɯɧɢɤɢ, ɢɦɟɸɬ ɫɜɨɢ ɧɚɡɜɚɧɢɹ,
ɨɩɪɟɞɟɥɹɟɦɵɟ ɧɚɡɧɚɱɟɧɢɹɦɢ ɢ ɬɟɯɧɢɱɟɫɤɢɦɢ ɨɫɨɛɟɧɧɨɫɬɹɦɢ.
Ɉɛɳɢɦ ɭɪɚɜɧɟɧɢɟɦ ɩɪɢ ɪɚɫɱɟɬɟ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚ ɥɸɛɨɝɨ ɬɢɩɚ ɹɜɥɹɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ Q1 , ɩɨɥɭɱɟɧɧɨɣ
ɜ ɬɟɩɥɨɨɛɦɟɧɧɢɤɟ ɩɪɢ ɨɯɥɚɠɞɟɧɢɢ ɝɨɪɹɱɟɝɨ ɬɟɩɥɨɧɨɫɢɬɟɥɹ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ T1' ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ T1'' , ɪɚɜɧɨ ɪɚɡɧɨɫɬɢ ɷɧɬɚɥɶɩɢɣ ɬɟɩɥɨɧɨɫɢɬɟɥɹ
ɧɚ ɜɯɨɞɟ ɜ ɬɟɩɥɨɨɛɦɟɧɧɢɤ H1' ɢ ɧɚ ɜɵɯɨɞɟ ɢɡ ɧɟɝɨ H1'' :
Q1
H1' H1"
M 1 c 'p1T1' c 'p' 1T1'' ,
ɝɞɟ M 1 – ɦɚɫɫɨɜɵɣ ɪɚɫɯɨɞ ɬɟɩɥɨɧɨɫɢɬɟɥɹ, c p – ɢɡɨɛɚɪɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ
(ɪɚɡɞɟɥ 1.10).
ɇɟɫɤɨɥɶɤɨ ɩɪɨɰɟɧɬɨɜ (ɨɬ 1 ɞɨ 10) ɬɟɩɥɚ ɬɟɪɹɟɬɫɹ ɜ ɨɤɪɭɠɚɸɳɭɸ
ɫɪɟɞɭ ɱɟɪɟɡ ɫɬɟɧɤɢ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚ, ɚ ɨɫɧɨɜɧɚɹ ɱɚɫɬɶ Q2 KQ1 (ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɥɟɡɧɨɝɨ ɞɟɣɫɬɜɢɹ ɬɟɩɥɨɨɛɦɟɧɧɢɤɚ
K ɭɱɢɬɵɜɚɟɬ
ɬɟɩɥɨɩɨɬɟɪɢ) – ɩɟɪɟɞɚɟɬɫɹ ɜɬɨɪɨɦɭ ɬɟɩɥɨɧɨɫɢɬɟɥɸ. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɥɭɱɚɟɦɨɟ ɯɨɥɨɞɧɵɦ ɬɟɩɥɨɧɨɫɢɬɟɥɟɦ, ɟɫɬɶ
Q2
H 2'' H 2'
M 2 c 'p' 2T2'' c 'p 2T1'
KQ1 .
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ ɩɨɡɜɨɥɹɟɬ ɧɚɣɬɢ ɨɞɢɧ ɧɟɢɡɜɟɫɬɧɵɣ ɩɚɪɚɦɟɬɪ: ɥɢɛɨ ɪɚɫɯɨɞ ɨɞɧɨɝɨ ɢɡ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ, ɥɢɛɨ ɨɞɧɭ ɢɡ
ɬɟɦɩɟɪɚɬɭɪ. ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ ɞɨɥɠɧɵ ɛɵɬɶ ɢɡɜɟɫɬɧɵ ɥɢɛɨ ɢɡ
ɷɤɫɩɟɪɢɦɟɧɬɚ, ɥɢɛɨ ɢɡ ɫɩɟɰɢɚɥɶɧɵɯ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ.
26
1.8. ɏɢɦɢɱɟɫɤɢɟ ɬɟɯɧɨɥɨɝɢɢ
Ɍɟɩɥɨɨɛɦɟɧ ɜ ɯɢɦɢɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɱɚɫɬɨ ɨɩɪɟɞɟɥɹɟɬ
ɨɫɧɨɜɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɪɚɛɨɬɵ ɚɩɩɚɪɚɬɭɪɵ. Ɍɚɤ, ɬɟɦɩɟɪɚɬɭɪɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɧɫɬɚɧɬɵ ɫɤɨɪɨɫɬɢ k ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ, k k0exp Ea RT (ɪɚɡɞɟɥ 11), ɝɞɟ k 0 - ɩɪɟɞɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɣ ɦɧɨɠɢɬɟɥɶ, E a - ɷɧɟɪɝɢɹ
ɚɤɬɢɜɚɰɢɢ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ, R - ɭɧɢɜɟɪɫɚɥɶɧɚɹ ɝɚɡɨɜɚɹ ɩɨɫɬɨɹɧɧɚɹ,
ɨɩɪɟɞɟɥɹɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɭɫɬɚɧɚɜɥɢɜɚɸɳɭɸɫɹ
ɜ ɯɨɞɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɬɟɦɩɟɪɚɬɭɪɭ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚ
ɫɬɟɩɟɧɶ ɡɚɜɟɪɲɟɧɧɨɫɬɢ ɪɟɚɤɰɢɢ.
ɉɟɪɟɦɟɲɢɜɚɧɢɟ ɠɢɞɤɢɯ ɫɪɟɞ ɫ ɩɨɦɨɳɶɸ ɦɟɯɚɧɢɱɟɫɤɢɯ ɦɟɲɚɥɨɤ
ɩɪɢɦɟɧɹɸɬ ɞɥɹ ɜɵɪɚɜɧɢɜɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪ ɢ ɤɨɧɰɟɧɬɪɚɰɢɣ ɜ ɨɛɴɟɦɟ ɪɟɚɤɰɢɨɧɧɨɣ ɦɚɫɫɵ ɢ ɞɥɹ ɢɧɬɟɧɫɢɮɢɤɚɰɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɫɨ ɫɬɟɧɤɚɦɢ ɚɩɩɚɪɚɬɨɜ.
ɉɪɢ ɪɚɛɨɬɟ ɩɪɨɦɵɲɥɟɧɧɵɯ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɬɨɪɨɜ ɫɬɚɰɢɨɧɚɪɧɵɣ
ɩɪɨɰɟɫɫ ɬɟɩɥɨɨɛɦɟɧɚ ɦɨɠɟɬ ɩɪɨɬɟɤɚɬɶ ɜ ɪɹɞɟ ɫɥɭɱɚɟɜ ɬɨɥɶɤɨ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɟɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɪɟɚɤɬɨɪɚ ɧɟɩɪɟɪɵɜɧɨɝɨ ɞɟɣɫɬɜɢɹ ɫ ɢɧɬɟɧɫɢɜɧɵɦ ɩɟɪɟɦɟɲɢɜɚɧɢɟɦ ɪɟɚɤɰɢɨɧɧɨɣ ɦɚɫɫɵ ɢ
ɜɧɟɲɧɢɦ ɨɬɜɨɞɨɦ ɬɟɩɥɨɬɵ, ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɧɟɨɛɪɚɬɢɦɚɹ ɷɤɡɨɬɟɪɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɩɨ ɤɨɧɰɟɧɬɪɚɰɢɢ ɋ ɨɫɧɨɜɧɨɝɨ
ɤɨɦɩɨɧɟɧɬɚ, ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ:
M 1C1T1 QchVk 0 exp Ea RT C
M 2C2T2 /F T Tx .
Ʌɟɜɚɹ ɱɚɫɬɶ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɬɟɩɥɨɬɟ, ɩɨɫɬɭɩɚɸɳɟɣ
ɜ ɪɟɚɤɬɨɪ ɫ ɦɚɫɫɨɜɵɦ ɩɨɬɨɤɨɦ Ɇ1 ɢɫɯɨɞɧɵɯ ɤɨɦɩɨɧɟɧɬɨɜ (M1, ɋ1, T1) ɢ
ɬɟɩɥɨɜɵɞɟɥɟɧɢɸ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɚɤɰɢɢ (Qch – ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɬɚ ɪɟɚɤɰɢɢ,
V – ɨɛɴɟɦ ɚɩɩɚɪɚɬɚ). ȼ ɩɪɚɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ – ɬɟɩɥɨɬɚ, ɨɬɜɨɞɢɦɚɹ ɫ ɩɪɨɞɭɤɬɚɦɢ ɪɟɚɤɰɢɢ (Ɇ2, ɋ2, Ɍ2), ɢ ɬɟɩɥɨɬɚ, ɩɟɪɟɞɚɜɚɟɦɚɹ ɱɟɪɟɡ ɬɟɩɥɨɨɛɦɟɧɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ F ɯɥɚɞɚɝɟɧɬɭ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ
Tx. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ / [ȼɬ/(ɦ2·Ʉ)] (ɪɚɡɞɟɥ 3) ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ ɜɟɥɢɱɢɧɭ, ɡɚɜɢɫɹɳɭɸ ɨɬ ɦɧɨɝɢɯ ɮɚɤɬɨɪɨɜ), ɢ ɪɚɫɫɱɢɬɵɜɚɟɬɫɹ ɧɚ
ɨɫɧɨɜɟ ɬɟɨɪɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ.
ȼ ɪɹɞɟ ɯɢɦɢɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɧɚɩɪɢɦɟɪ, ɤɚɬɚɥɢɬɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜ ɧɟɩɨɞɜɢɠɧɨɦ ɫɥɨɟ ɞɢɫɩɟɪɫɧɨɝɨ ɤɚɬɚɥɢɡɚɬɨɪɚ, ɜɚɠɧɭɸ
ɪɨɥɶ ɢɝɪɚɟɬ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɨɬɜɨɞɚ (ɩɨɞɜɨɞɚ) ɬɟɩɥɨɬɵ ɯɢɦɢɱɟɫɤɨɝɨ ɩɪɟɜɪɚɳɟɧɢɹ ɨɬ ɜɧɭɬɪɟɧɧɢɯ ɭɱɚɫɬɤɨɜ ɫɥɨɹ ɤ ɟɝɨ ɩɟɪɢɮɟɪɢɢ, ɬɟɩɥɨɨɬɜɨɞɚ ɨɬ
ɫɥɨɹ ɤ ɬɟɩɥɨɨɛɦɟɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɪɟɚɤɬɨɪɚ ɢ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ
ɮɢɥɶɬɪɭɸɳɢɦɫɹ ɱɟɪɟɡ ɫɥɨɣ ɩɨɬɨɤɨɦ ɪɟɚɝɟɧɬɨɜ ɢ ɩɨɜɟɪɯɧɨɫɬɶɸ ɱɚɫɬɢɰ.
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɦɟɠɮɚɡɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɧɟɩɨɞɜɢɠɧɨɦ ɫɥɨɟ ɬɚɤɠɟ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɢ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. Ⱥɧɚɥɨ27
ɝɢɱɧɵɟ ɩɪɨɰɟɫɫɵ ɬɟɩɥɨɨɛɦɟɧɚ ɩɪɨɢɫɯɨɞɹɬ ɜ ɚɩɩɚɪɚɬɚɯ ɫ ɞɜɢɠɭɳɢɦɢɫɹ
ɫɥɨɹɦɢ ɦɚɬɟɪɢɚɥɨɜ, ɩɪɟɞɧɚɡɧɚɱɟɧɧɵɦɢ ɞɥɹ ɧɟɩɪɟɪɵɜɧɨɝɨ ɤɨɧɬɚɤɬɚ
ɮɢɥɶɬɪɭɸɳɟɝɨɫɹ ɩɨɬɨɤɚ ɫ ɞɢɫɩɟɪɫɧɵɦ ɦɚɬɟɪɢɚɥɨɦ.
ɇɟɤɨɬɨɪɵɟ ɯɢɦɢɤɨ-ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ (ɧɚɝɪɟɜɚɧɢɟ, ɩɪɨɤɚɥɢɜɚɧɢɟ, ɫɭɲɤɚ, ɤɪɢɫɬɚɥɥɢɡɚɰɢɹ, ɪɚɫɬɜɨɪɟɧɢɟ) ɨɫɭɳɟɫɬɜɥɹɸɬɫɹ ɜ ɩɨɬɨɤɟ
ɫɩɥɨɲɧɨɣ ɮɚɡɵ (ɝɚɡ, ɩɚɪ ɢɥɢ ɤɚɩɟɥɶɧɚɹ ɠɢɞɤɨɫɬɶ), ɧɟɫɭɳɟɦ ɫ ɫɨɛɨɣ
ɦɟɥɤɢɟ ɬɜɟɪɞɵɟ ɱɚɫɬɢɰɵ. Ɉɬɧɨɫɢɬɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɮɚɡ ɜ ɜɟɪɬɢɤɚɥɶɧɵɯ
ɚɩɩɚɪɚɬɚɯ, ɜ ɤɨɬɨɪɵɯ ɩɪɨɜɨɞɹɬɫɹ ɭɤɚɡɚɧɧɵɟ ɩɪɨɰɟɫɫɵ, ɦɨɠɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɢɡɦɟɧɹɬɶɫɹ (ɨɬ ɫɤɨɪɨɫɬɢ ɫɩɥɨɲɧɨɣ ɮɚɡɵ ɞɨ ɫɤɨɪɨɫɬɢ ɨɫɚɠɞɟɧɢɹ
ɱɚɫɬɢɰ), ɱɬɨ ɜɥɢɹɟɬ ɧɚ ɯɚɪɚɤɬɟɪ ɬɟɩɥɨɨɛɦɟɧɚ.
1.9. Ɉɬɧɨɲɟɧɢɟ ɬɟɩɥɨɨɛɦɟɧɚ ɤ ɬɟɪɦɨɞɢɧɚɦɢɤɟ
ȼ ɤɭɪɫɟ ɬɟɪɦɨɞɢɧɚɦɢɤɢ, ɤɚɤ ɨɧ ɩɪɟɩɨɞɚɟɬɫɹ ɞɥɹ ɫɬɭɞɟɧɬɨɜ ɢɧɠɟɧɟɪɧɵɯ ɫɩɟɰɢɚɥɶɧɨɫɬɟɣ, ɢɦɟɸɬɫɹ ɩɨɫɬɨɹɧɧɵɟ ɫɫɵɥɤɢ ɧɚ ɬɟɩɥɨɨɛɦɟɧ
ɦɟɠɞɭ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɦɢ ɫɢɫɬɟɦɚɦɢ. ɑɬɨɛɵ ɭɫɬɚɧɨɜɢɬɶ ɫɜɹɡɶ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɬɟɪɦɨɞɢɧɚɦɢɤɨɣ, ɜɫɩɨɦɧɢɦ ɤɪɚɬɤɨ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ.
ɉɟɪɜɵɣ ɡɚɤɨɧ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɞɥɹ ɡɚɤɪɵɬɵɯ ɫɢɫɬɟɦ ɦɨɠɟɬ ɛɵɬɶ
ɫɮɨɪɦɭɥɢɪɨɜɚɧ ɜ ɫɥɟɞɭɸɳɟɣ ɮɨɪɦɟ: ɬɟɩɥɨɬɚ, ɫɨɨɛɳɚɟɦɚɹ ɫɢɫɬɟɦɟ,
ɢɞɟɬ ɧɚ ɩɪɢɪɚɳɟɧɢɟ ɟɟ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢ ɧɚ ɫɨɜɟɪɲɟɧɢɟ ɪɚɛɨɬɵ
G QW
dU GA ,
(1.2)
ɝɞɟ dU ! 0 , ɟɫɥɢ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ ɜɨɡɪɚɫɬɚɟɬ; G A ! 0 , ɟɫɥɢ
ɪɚɛɨɬɚ ɫɨɜɟɪɲɚɟɬɫɹ ɫɚɦɨɣ ɫɢɫɬɟɦɨɣ. ȿɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ ɜɟɥɢɱɢɧ,
ɜɯɨɞɹɳɢɯ ɜ (1.2), ɹɜɥɹɟɬɫɹ Ⱦɠɨɭɥɶ (Ⱦɠ, ɜ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ ɋɂ) ɢɥɢ ɤɚɥɨɪɢɹ (ɤɚɥ)
ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɦɨɠɧɨ ɩɟɪɟɩɢɫɚɬɶ ɞɥɹ ɭɞɟɥɶɧɵɯ ɜɟɥɢɱɢɧ (ɨɬɧɟɫɟɧɧɵɯ ɤ ɟɞɢɧɢɰɟ ɦɚɫɫɵ)
Gq
W
du Gw,
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɫɟ ɜɟɥɢɱɢɧɵ ɢɡɦɟɪɹɸɬɫɹ ɜ ɟɞɢɧɢɰɚɯ ɷɧɟɪɝɢɢ, ɞɟɥɟɧɧɵɯ ɧɚ ɟɞɢɧɢɰɭ ɦɚɫɫɵ, ɧɚɩɪɢɦɟɪ, Ⱦɠ/ɤɝ, Ⱦɠ/ɝ ɢ.ɬ.ɞ.
ɉɨɞ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɟɣ ɜ ɬɟɪɦɨɞɢɧɚɦɢɤɟ ɩɨɧɢɦɚɸɬ ɷɧɟɪɝɢɸ ɯɚɨɬɢɱɟɫɤɨɝɨ ɞɜɢɠɟɧɢɹ ɦɨɥɟɤɭɥ ɢ ɚɬɨɦɨɜ, ɜɤɥɸɱɚɸɳɭɸ ɷɧɟɪɝɢɸ ɩɨɫɬɭɩɚɬɟɥɶɧɨɝɨ, ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɢ ɤɨɥɟɛɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɣ, ɤɚɤ ɦɨɥɟɤɭɥɹɪɧɨɝɨ, ɬɚɤ ɢ ɜɧɭɬɪɢɦɨɥɟɤɭɥɹɪɧɨɝɨ, ɚ ɬɚɤɠɟ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɷɧɟɪɝɢɸ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɟɠɞɭ ɦɨɥɟɤɭɥɚɦɢ. Ʉɢɧɟɬɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɦɨɥɟɤɭɥ ɹɜɥɹɟɬɫɹ
ɨɞɧɨɡɧɚɱɧɨɣ ɮɭɧɤɰɢɟɣ ɬɟɦɩɟɪɚɬɭɪɵ; ɡɧɚɱɟɧɢɟ ɩɨɬɟɧɰɢɚɥɶɧɨɣ ɷɧɟɪɝɢɢ
ɡɚɜɢɫɢɬ ɨɬ ɫɪɟɞɧɟɝɨ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɦɨɥɟɤɭɥɚɦɢ, ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɬ
ɡɚɧɢɦɚɟɦɨɝɨ ɨɛɴɟɦɚ. ɉɨɷɬɨɦɭ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ ɟɫɬɶ ɧɟɤɨɬɨɪɚɹ ɨɞɧɨɡɧɚɱɧɚɹ ɮɭɧɤɰɢɹ ɫɨɫɬɨɹɧɢɹ.
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Ɋɚɛɨɬɚ ɜ ɬɟɪɦɨɞɢɧɚɦɢɤɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɞɟɣɫɬɜɭɸɳɟɣ ɫɢɥɵ ɧɚ ɩɭɬɶ ɟɟ ɞɟɣɫɬɜɢɹ. Ɍɚɤ, ɪɚɛɨɬɚ ɩɪɨɬɢɜ ɫɢɥ ɜɧɟɲɧɟɝɨ ɞɚɜɥɟɧɢɹ ɟɫɬɶ ɪɚɛɨɬɚ ɪɚɫɲɢɪɟɧɢɹ
GA p ³ dFdn ,
F
ɝɞɟ pdFd n – ɷɥɟɦɟɧɬɚɪɧɚɹ ɪɚɛɨɬɚ, ɡɚɬɪɚɱɟɧɧɚɹ ɧɚ ɩɟɪɟɦɟɳɟɧɢɟ ɤɚɠɞɨɣ ɷɥɟɦɟɧɬɚɪɧɨɣ ɩɥɨɳɚɞɤɢ, ɢɡ ɤɨɬɨɪɵɯ ɫɨɫɬɨɢɬ ɩɥɨɳɚɞɶ F , ɨɝɪɚɧɢɱɢɜɚɸɳɚɹ ɨɛɴɟɦ V , ɢɥɢ
G A pdV .
Ɍ.ɟ., ɪɚɛɨɬɚ ɟɫɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ ɞɚɜɥɟɧɢɹ ɧɚ ɩɪɢɪɚɳɟɧɢɟ ɨɛɴɟɦɚ. ȿɫɥɢ dV ! 0 , ɪɚɛɨɬɚ ɩɨɥɨɠɢɬɟɥɶɧɚ, ɬ.ɟ. ɬɟɥɨ ɫɨɜɟɪɲɚɟɬ ɪɚɛɨɬɭ. ȿɫɥɢ
dV 0 , ɨɛɴɟɦ ɭɦɟɧɶɲɚɟɬɫɹ, ɬ.ɟ. ɪɚɛɨɬɚ ɫɨɜɟɪɲɚɟɬɫɹ ɧɚɞ ɬɟɥɨɦ. ȿɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ ɪɚɛɨɬɵ ɹɜɥɹɟɬɫɹ, ɤɚɤ ɫɤɚɡɚɧɨ ɜɵɲɟ, Ⱦɠ.
ȼ ɬɟɪɦɨɞɢɧɚɦɢɤɟ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɪɚɜɧɨɜɟɫɧɵɯ ɩɪɨɰɟɫɫɨɜ
ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬ p V ɞɢɚɝɪɚɦɦɭ, ɜ ɤɨɬɨɪɨɣ ɨɫɶɸ ɚɛɫɰɢɫɫ
ɫɥɭɠɢɬ
ɭɞɟɥɶɧɵɣ
ɨɛɴɟɦ
ɚ
ɛ
Ɋɢɫ. 1.2. Ʉ ɨɩɪɟɞɟɥɟɧɢɸ ɪɚɛɨɬɵ ɪɚɫ- J V M (ɪɢɫ. 1.2). ɋɨɫɬɨɹɧɢɟ
ɲɢɪɟɧɢɹ: ɚ) ɬɟɥɨ ɪɚɫɲɢɪɹɟɬɫɹ; ɛ) ɬɟɥɨ ɬɟɥɚ ɧɚ ɷɬɨɣ ɞɢɚɝɪɚɦɦɟ ɢɡɨɛɪɚɫɠɢɦɚɟɬɫɹ
ɠɚɟɬɫɹ ɬɨɱɤɨɣ. Ɋɚɛɨɬɚ – ɪɚɛɨɬɚ
ɪɚɫɲɢɪɟɧɢɹ – ɩɪɢ ɩɟɪɟɯɨɞɟ ɫɢɫɬɟɦɵ ɢɡ ɫɨɫɬɨɹɧɢɹ 1 ɜ ɫɨɫɬɨɹɧɢɟ 2 ɢɡɨɛɪɚɠɚɟɬɫɹ ɩɥɨɳɚɞɶɸ ɩɨɞ ɤɪɢɜɨɣ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɩɭɬɢ ɩɪɨɰɟɫɫɚ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɟɥɢɱɢɧɚ ɪɚɛɨɬɵ ɡɚɜɢɫɢɬ ɨɬ ɭɫɥɨɜɢɣ ɩɪɨɬɟɤɚɧɢɹ ɩɪɨɰɟɫɫɚ ɢɥɢ ɭɫɥɨɜɢɣ ɫɨɜɟɪɲɟɧɢɹ ɪɚɛɨɬɵ.
ɂɬɚɤ, ɬɟɩɥɨɬɚ ɢ ɪɚɛɨɬɚ – ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨɜɨɝɨ ɢ ɦɟɯɚɧɢɱɟɫɤɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɢɫɬɟɦɵ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
Ɉɬɧɨɲɟɧɢɟ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɨɬɵ G QW , ɩɨɥɭɱɟɧɧɨɝɨ ɬɟɥɨɦ ɩɪɢ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɢɡɦɟɧɟɧɢɢ ɟɝɨ ɫɨɫɬɨɹɧɢɹ, ɤ ɫɜɹɡɚɧɧɨɦɭ ɫ ɷɬɢɦ ɩɪɨɰɟɫɫɨɦ
ɢɡɦɟɧɟɧɢɸ ɬɟɦɩɟɪɚɬɭɪɵ dT ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɧɨɣ ɬɟɩɥɨɟɦɤɨɫɬɶɸ ɬɟɥɚ ɜ
ɞɚɧɧɨɦ ɩɪɨɰɟɫɫɟ
C G QW d T .
(1.3)
Ɉɛɵɱɧɨ ɜɟɥɢɱɢɧɭ ɬɟɩɥɨɟɦɤɨɫɬɢ ɨɬɧɨɫɹɬ ɤ ɟɞɢɧɢɰɟ ɤɨɥɢɱɟɫɬɜɚ ɜɟɳɟɫɬɜɚ ɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɩɪɢɧɹɬɨɣ ɟɞɢɧɢɰɵ ɢɡɦɟɪɟɧɢɹ ɪɚɡɥɢɱɚɸɬ:
1) ɭɞɟɥɶɧɭɸ ɦɚɫɫɨɜɭɸ ɬɟɩɥɨɟɦɤɨɫɬɶ c , ɨɬɧɟɫɟɧɧɭɸ ɤ 1 ɤɝ ɢ ɢɡɦɟɪɹɟɦɭɸ ɜ Ⱦɠ/(ɤɝ.Ʉ);
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2) ɭɞɟɥɶɧɭɸ ɨɛɴɟɦɧɭɸ ɬɟɩɥɨɟɦɤɨɫɬɶ cc , ɨɬɧɟɫɟɧɧɭɸ ɤ ɤɨɥɢɱɟɫɬɜɭ
ɜɟɳɟɫɬɜɚ, ɫɨɞɟɪɠɚɳɟɦɭɫɹ ɜ 1 ɦ3 ɨɛɴɟɦɚ ɩɪɢ ɧɨɪɦɚɥɶɧɵɯ ɮɢɡɢɱɟɫɤɢɯ
ɭɫɥɨɜɢɹɯ ɢ ɢɡɦɟɪɹɟɦɭɸ ɜ Ⱦɠ/(ɦ3.Ʉ);
3) ɭɞɟɥɶɧɭɸ ɦɨɥɶɧɭɸ ɬɟɩɥɨɟɦɤɨɫɬɶ P c , ɨɬɧɟɫɟɧɧɭɸ ɤ ɨɞɧɨɦɭ ɤɢɥɨɦɨɥɸ ɢ ɢɡɦɟɪɹɟɦɭɸ ɜ Ⱦɠ/(ɤɦɨɥɶ.Ʉ).
ɋɜɹɡɶ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɬɟɩɥɨɟɦɤɨɫɬɹɦɢ ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ
c P c P ; cc P c 2 2,4; cc cU ,
(1.4)
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ɝɞɟ 22.4 ɦ – ɨɛɴɟɦ ɨɞɧɨɝɨ ɤɢɥɨɦɨɥɹ ɢ U – ɩɥɨɬɧɨɫɬɶ ɜ ɧɨɪɦɚɥɶɧɵɯ
ɭɫɥɨɜɢɹɯ.
ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ ɩɪɢ ɨɞɧɨɦ ɢ ɬɨɦ ɠɟ ɤɨɥɢɱɟɫɬɜɟ ɫɨɨɛɳɚɟɦɨɣ ɬɟɩɥɨɬɵ ɡɚɜɢɫɢɬ ɨɬ ɯɚɪɚɤɬɟɪɚ ɩɪɨɢɫɯɨɞɹɳɟɝɨ ɩɪɢ ɷɬɨɦ ɩɪɨɰɟɫɫɚ, ɩɨɷɬɨɦɭ ɬɟɩɥɨɟɦɤɨɫɬɶ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɩɪɨɰɟɫɫɚ. ɗɬɨ ɨɡɧɚɱɚɟɬ,
ɱɬɨ ɨɞɧɨ ɢ ɬɨ ɠɟ ɬɟɥɨ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɩɪɨɰɟɫɫɚ (ɢɥɢ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɭɫɥɨɜɢɣ) ɬɪɟɛɭɟɬ ɞɥɹ ɫɜɨɟɝɨ ɧɚɝɪɟɜɚɧɢɹ ɧɚ 1 ɝɪɚɞɭɫ ɪɚɡɥɢɱɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɨɬɵ.
Ɍɟɩɥɨɟɦɤɨɫɬɶ ɢ ɟɫɬɶ ɬɚɤɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɤɨɬɨɪɨɟ ɜ ɞɚɧɧɵɯ
ɭɫɥɨɜɢɹɯ ɬɪɟɛɭɟɬɫɹ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ ɧɚ ɨɞɢɧ ɝɪɚɞɭɫ.
ȼ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɸɬ
ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ
c p G qW d T p
(1.5)
ɢ ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ
cJ GqW dT J .
(1.6)
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɩɟɪɜɵɦ ɡɚɤɨɧɨɦ ɬɟɪɦɨɞɢɧɚɦɢɤɢ, ɡɚɩɢɫɚɧɧɵɦ ɞɥɹ
ɭɞɟɥɶɧɵɯ ɜɟɥɢɱɢɧ, ɢɦɟɟɦ
GqW du pdJ .
(1.7)
Ɍɚɤ ɤɚɤ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ – ɮɭɧɤɰɢɹ ɫɨɫɬɨɹɧɢɹ u u T , J (ɬ.ɟ.
ɮɭɧɤɰɢɹ ɩɟɪɟɦɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹ T , J ), ɞɥɹ ɧɟɟ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
ɪɚɜɟɧɫɬɜɨ
du wu wT J dT wu wJ T dJ .
ɂɡ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɪɚɜɟɧɫɬɜ ɧɚɯɨɞɢɦ
GqW wu wT J dT >wu wJ T p @ dJ .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɢɡɨɯɨɪɧɨɝɨ ɩɪɨɰɟɫɫɚ ( J const )
GqW J wu wT J dT ;
cJ
GqW
dT J
wu
wT J –
(1.8)
ɬɟɩɥɨɟɦɤɨɫɬɶ ɬɟɥɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ ɪɚɜɧɚ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɟɝɨ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɜ ɢɡɨɯɨɪɧɨɦ ɩɪɨɰɟɫɫɟ.
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ȼ ɢɡɨɛɚɪɧɨɦ ɩɪɨɰɟɫɫɟ ( p const ) ɢɦɟɟɦ
GqW
dT p
ɬ.ɟ.
cp
wu
dT J >wu dJ T p @ wJ wT p ,
c J >wu wJ T p @ wJ wT p .
(1.9)
ȼ ɬɟɪɦɨɞɢɧɚɦɢɤɟ ɜɚɠɧɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ ɬɚɤɚɹ ɮɭɧɤɰɢɹ ɫɨɫɬɨɹɧɢɹ
ɤɚɤ ɷɧɬɚɥɶɩɢɹ, ɤɨɬɨɪɚɹ ɫɜɹɡɚɧɚ ɫ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɟɣ ɫɨɨɬɧɨɲɟɧɢɟɦ
H U pV
ɢɥɢ
h u pJ .
(1.10)
Ɍɚɤ ɠɟ ɤɚɤ ɢ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ, ɢ ɬɟɩɥɨɬɚ, ɜ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ ɋɂ
ɷɧɬɚɥɶɩɢɹ ɢɡɦɟɪɹɟɬɫɹ ɜ ɞɠɨɭɥɹɯ ɢɥɢ ɞɠɨɭɥɹɯ ɧɚ ɤɝ. Ɍɚɤ ɤɚɤ
dh du pdJ Jdp ,
ɬɨ ɜɦɟɫɬɨ (1.7) ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
(1.11)
GqW dh Jdp .
Ɍɨɝɞɚ ɞɥɹ ɢɡɨɛɚɪɧɨɝɨ ɩɪɨɰɟɫɫɚ
c p GqW dT p
wh
wT p .
(1.12)
Ɍ.ɟ., ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ ɪɚɜɧɚ ɱɚɫɬɧɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɷɧɬɚɥɶɩɢɢ ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ.
ȼ ɬɚɛɥɢɰɟ 3.1. ɉɪɢɥɨɠɟɧɢɹ 3 ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɩɪɢɜɟɞɟɧɵ ɬɟɩɥɨɟɦɤɨɫɬɢ ɢ ɩɥɨɬɧɨɫɬɢ ɪɚɡɥɢɱɧɵɯ ɜɟɳɟɫɬɜ ɩɪɢ ɤɨɦɧɚɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɚɬɦɨɫɮɟɪɧɨɦ ɞɚɜɥɟɧɢɢ.
ɍɪɚɜɧɟɧɢɹ ɩɟɪɜɨɝɨ ɡɚɤɨɧɚ ɬɟɪɦɨɞɢɧɚɦɢɤɢ (1.7) ɢɥɢ (1.11) ɦɵ ɦɨɠɟɦ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɢɧɨɣ ɮɨɪɦɟ
dqW du
dq
dJ
dh
dp
p
ɢɥɢ W
J
.
(1.13)
dt
dt
dt
dt
dt
dt
Ɍɚɤ ɤɚɤ
§ du ·
§ du · dT § dh ·
§ dh · dT
ɢ¨ ¸
,
¨ ¸
¨
¸
¨
¸
© dt ¹J © dT ¹J dt
© dt ¹ p © dT ¹ p dt
ɬɨ
dT § dh ·
dT § du ·
(1.14)
cJ
ɢ cp
¨ ¸ .
¨ ¸
dt © dt ¹ p
dt © dt ¹J
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ ɜɯɨɞɹɳɢɯ ɜ (1.14) ɜɟɥɢɱɢɧ, ɪɚɡɦɟɪɧɨɫɬɶ du dt J ɜ ɫɢɫɬɟɦɟ ɟɞɢɧɢɰ ɋɂ ɟɫɬɶ Ⱦɠ/(ɤɝ ɫ)=ȼɬ/ɤɝ.
ɍɪɚɜɧɟɧɢɹ (1.14) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɞɥɹ ɟɞɢɧɢɰɵ ɨɛɴɟɦɚ
31
cJ U
dT
dt
§ du ·
U¨ ¸
© dt ¹J
ɢ
cpU
dT
dt
§ dh ·
U¨ ¸
© dt ¹ p
(1.15)
Ɋɚɡɧɨɫɬɶ c p c v M ' T , ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɪɚɜɧɚ ɪɚɛɨɬɟ ɜɧɟɲɧɟɝɨ
ɞɚɜɥɟɧɢɹ ɩɨ ɢɡɦɟɧɟɧɢɸ ɨɛɴɟɦɚ p' V ; M - ɦɚɫɫɚ ɫɠɢɦɚɟɦɨɝɨ ɜɟɳɟɫɬɜɚ
ɜ ɨɛɴɟɦɟ ' V .
ȼɬɨɪɨɣ ɡɚɤɨɧ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɬɚɤɨɣ
ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɫɨɫɬɨɹɧɢɹ ɤɚɤ ɷɧɬɪɨɩɢɹ s , ɬɚɤ ɱɬɨ ɞɥɹ
ɪɚɜɧɨɜɟɫɧɵɯ ɩɪɨɰɟɫɫɨɜ ɢɦɟɟɦ
G q W T ds ɢɥɢ G QW T d S .
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɨɩɪɟɞɟɥɟɧɢɟ ɭɞɟɥɶɧɨɝɨ ɩɨɬɨɤɚ ɬɟɩɥɚ ɜ (1.7) ɢ (1.11),
ɧɚɣɞɟɦ
Tds du pdJ ɢ Tds dh Jdp
(1.16)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɦɟɫɬɨ (1.13) ɩɨɥɭɱɢɦ
ds du
dJ
ds dh
dp
T
p
ɢɥɢ T
J
(1.17)
dt dt
dt dt
dt
dt
ɍɪɚɜɧɟɧɢɹ (1.16), (1.17) ɟɫɬɶ ɱɚɫɬɧɵɟ ɮɨɪɦɵ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ, ɡɚɩɢɫɚɧɧɨɝɨ ɜɞɨɥɶ ɬɪɚɟɤɬɨɪɢɢ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɦɚɫɫ.
ȼɬɨɪɨɣ ɡɚɤɨɧ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧ ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨɛɚɦɢ.
Ⱦɥɹ ɧɟɨɛɪɚɬɢɦɵɯ ɩɪɨɰɟɫɫɨɜ ɷɬɨɬ ɡɚɤɨɧ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɬɨɥɶɤɨ ɜɨɡɦɨɠɧɨɫɬɶ ɢ ɧɚɩɪɚɜɥɟɧɢɟ ɢɯ ɩɪɨɬɟɤɚɧɢɹ.
Ɂɚɤɨɧɵ ɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɧɟ ɦɨɝɭɬ ɭɫɬɚɧɨɜɢɬɶ, ɩɨɱɟɦɭ
ɩɪɨɬɟɤɚɸɬ ɧɟɨɛɪɚɬɢɦɵɟ ɩɪɨɰɟɫɫɵ, ɩɨɱɟɦɭ ɜɫɟ ɪɟɚɥɶɧɵɟ ɩɪɨɰɟɫɫɵ – ɧɟɨɛɪɚɬɢɦɵ.
Ⱦɥɹ ɧɟɨɛɪɚɬɢɦɵɯ ɩɪɨɰɟɫɫɨɜ ɷɧɬɪɨɩɢɹ ɧɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɤɚɤ
ɮɭɧɤɰɢɹ ɫɨɫɬɨɹɧɢɹ. ɑɬɨɛɵ ɨɩɢɫɚɬɶ ɧɟɨɛɪɚɬɢɦɵɟ ɩɪɨɰɟɫɫɵ, ɜ ɫɨɜɪɟɦɟɧɧɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɟ ɫɭɳɟɫɬɜɭɸɬ ɪɚɡɧɵɟ ɫɩɨɫɨɛɵ, ɤɨɬɨɪɵɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɰɟɥɟɣ ɤɨɧɤɪɟɬɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ.
Ɍɪɟɬɢɣ ɡɚɤɨɧ ɬɟɪɦɨɞɢɧɚɦɢɤɢ, ɭɫɬɚɧɚɜɥɢɜɚɸɳɢɣ ɫɜɨɣɫɬɜɚ ɫɢɫɬɟɦ
ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ, ɫɬɪɟɦɹɳɟɣɫɹ ɤ ɚɛɫɨɥɸɬɧɨɦɭ ɧɭɥɸ, ɢɦɟɟɬ ɧɟ ɫɬɨɥɶ
ɩɪɢɧɰɢɩɢɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɤɚɤ ɩɟɪɜɵɟ ɞɜɚ.
Ⱦɥɹ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɢ ɞɥɹ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɛɨɥɶɲɭɸ ɪɨɥɶ ɢɝɪɚɸɬ ɟɳɟ ɞɜɟ ɮɭɧɤɰɢɢ ɫɨɫɬɨɹɧɢɹ – ɷɧɟɪɝɢɹ Ƚɟɥɶɦɝɨɥɶɰɚ f ɢ ɷɧɟɪɝɢɹ Ƚɢɛɛɫɚ g , ɫɜɹɡɚɧɧɵɟ ɫ h ɢ u ɫɨɨɬɧɨɲɟɧɢɹɦɢ
f u T s ; g h Ts
(1.18)
Ɍɚɤ ɤɚɤ
df du Tds sdT ,
ɬɨ ɜɦɟɫɬɨ (1.16) ɧɚɣɞɟɦ
32
df
Ⱥɧɚɥɨɝɢɱɧɨ ɢɦɟɟɦ
dg
sdT pdJ .
(1.19)
dh Tds sdT
ɢ
dg sdT Jdp .
(1.20)
ɍɪɚɜɧɟɧɢɹ (1.19) ɢ (1.20) – ɬɨɠɟ ɟɫɬɶ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɫɤɨɪɨɫɬɶ ɬɟɩɥɨɩɟɪɟɧɨɫɚ, ɦɵ ɞɨɥɠɧɵ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɨɜɵɟ ɮɢɡɢɱɟɫɤɢɟ ɩɪɢɧɰɢɩɵ, ɚ ɢɦɟɧɧɨ ɜɜɟɫɬɢ ɡɚɤɨɧɵ ɩɟɪɟɧɨɫɚ, ɤɨɬɨɪɵɟ ɧɟ ɹɜɥɹɸɬɫɹ ɫɨɫɬɚɜɧɨɣ ɱɚɫɬɶɸ ɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɢ. ɗɬɨ, ɧɚɩɪɢɦɟɪ, ɡɚɤɨɧɵ ɬɟɩɥɨɨɛɦɟɧɚ Ɏɭɪɶɟ, ɇɶɸɬɨɧɚ, ɋɬɟɮɚɧɚȻɨɥɶɰɦɚɧɚ ɢ ɞɪ. ɇɨ ɨɱɟɧɶ ɜɚɠɧɨ ɩɨɦɧɢɬɶ, ɱɬɨ ɨɩɢɫɚɧɢɟ ɬɟɩɥɨɩɟɪɟɧɨɫɚ
ɬɪɟɛɭɟɬ, ɱɬɨɛɵ ɧɨɜɵɟ (ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ) ɮɢɡɢɱɟɫɤɢɟ ɩɪɢɧɰɢɩɵ ɧɟ ɩɪɨɬɢɜɨɪɟɱɢɥɢ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɦ ɡɚɤɨɧɚɦ.
1.10. Ɋɨɥɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ
ɜ ɪɚɡɪɚɛɨɬɤɟ ɫɨɜɪɟɦɟɧɧɵɯ ɬɟɯɧɨɥɨɝɢɣ
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɬɚɤɨɝɨ ɬɢɩɚ, ɤɚɤ ɥɚɡɟɪɧɚɹ, ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɚɹ ɢ ɩɥɚɡɦɟɧɧɚɹ ɬɟɯɧɨɥɨɝɢɢ ɩɪɢɜɨɞɢɬ ɤ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɪɟɲɚɬɶ ɫɩɟɰɢɚɥɶɧɵɟ ɡɚɞɚɱɢ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ, ɫɨɫɬɚɜɧɨɣ ɱɚɫɬɶɸ ɤɨɬɨɪɨɝɨ ɹɜɥɹɟɬɫɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ. Ⱦɥɹ ɫɥɨɠɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ
ɦɨɞɟɥɶ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɵɦ ɢɧɫɬɪɭɦɟɧɬɨɦ, ɩɨɡɜɨɥɹɸɳɢɦ ɩɪɨɜɨɞɢɬɶ ɤɚɤ
ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɟ ɢɫɫɥɟɞɨɜɚɧɢɹ, ɬɚɤ ɢ ɨɩɬɢɦɢɡɢɪɨɜɚɬɶ ɪɚɡɪɚɛɨɬɚɧɧɭɸ
ɬɟɯɧɨɥɨɝɢɸ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜ ɨɛɥɚɫɬɢ ɫɨɜɪɟɦɟɧɧɵɯ ɬɟɯɧɨɥɨɝɢɣ
ɜɤɥɸɱɚɟɬ:
– ɢɫɫɥɟɞɨɜɚɧɢɟ ɢ ɪɚɡɪɚɛɨɬɤɭ ɮɢɡɢɱɟɫɤɢɯ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ;
– ɪɚɡɪɚɛɨɬɤɭ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɢ ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɦɨɞɟɥɹɦ ɪɚɡɧɵɯ
ɬɟɯɧɨɥɨɝɢɣ;
– ɩɨɥɭɱɟɧɢɟ ɢɧɠɟɧɟɪɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɧɵɯ ɢ ɤɨɧɰɟɧɬɪɚɰɢɨɧɧɵɯ ɩɨɥɟɣ ɜ ɩɪɨɰɟɫɫɚɯ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ;
– ɢɫɫɥɟɞɨɜɚɧɢɟ ɢ ɪɚɡɪɚɛɨɬɤɭ ɢ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɵɯ ɡɚɞɚɱ (ɜ
ɬɨɦ ɱɢɫɥɟ, ɬɟɩɥɨɨɛɦɟɧɚ) ɤɚɤ ɫɪɟɞɫɬɜɚ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ;
– ɢɡɭɱɟɧɢɟ ɫɨɩɪɹɠɟɧɧɵɯ ɢ ɫɜɹɡɚɧɧɵɯ ɡɚɞɚɱ ɞɥɹ ɩɨɥɭɱɟɧɢɹ ɛɨɥɟɟ
ɩɨɥɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨ ɬɟɩɥɨ– ɢ ɦɚɫɫɨɩɟɪɟɧɨɫɟ ɜ ɩɪɨɰɟɫɫɚɯ ɨɛɪɚ33
ɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ, ɧɚɯɨɠɞɟɧɢɟ ɭɫɥɨɜɢɣ ɨɩɬɢɦɢɡɚɰɢɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɦɟɬɨɞɨɜ ɢɯ ɪɟɚɥɢɡɚɰɢɢ;
– ɧɚɯɨɠɞɟɧɢɟ ɭɫɥɨɜɢɣ ɤɨɧɬɪɨɥɹ, ɭɩɪɚɜɥɟɧɢɹ ɢ ɪɟɝɭɥɢɪɨɜɚɧɢɹ ɞɥɹ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ.
ȼɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɹɜɥɹɟɬɫɹ ɨɞɧɨɣ ɢɡ ɜɚɠɧɟɣɲɢɯ ɮɨɪɦ
ɜɧɟɞɪɟɧɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɨɣ ɬɟɯɧɢɤɢ ɜ ɩɪɨɟɤɬɢɪɨɜɚɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ. Ɉɧ ɢɫɩɨɥɶɡɭɟɬɫɹ ɤɚɤ ɜɨ ɜɪɟɦɹ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ (ɩɪɢ ɢɞɟɧɬɢɮɢɤɚɰɢɢ ɩɚɪɚɦɟɬɪɨɜ ɦɨɞɟɥɢ –
ɤɚɤ ɫɨɫɬɚɜɧɚɹ ɱɚɫɬɶ ɪɟɲɟɧɢɹ ɨɛɪɚɬɧɵɯ ɡɚɞɚɱ, ɩɪɢ ɩɪɨɜɟɪɤɟ ɚɞɟɤɜɚɬɧɨɫɬɢ ɢ ɩɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ), ɬɚɤ ɢ ɜ ɯɨɞɟ ɫɢɧɬɟɡɚ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ – ɞɥɹ ɩɪɨɜɟɪɤɢ ɢ ɫɪɚɜɧɟɧɢɹ ɩɪɨɟɤɬɧɵɯ
ɪɟɲɟɧɢɣ.
ɉɨɞ ɜɵɱɢɫɥɢɬɟɥɶɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɨɦ ɩɨɧɢɦɚɟɬɫɹ ɬɚɤɚɹ ɨɪɝɚɧɢɡɚɰɢɹ
ɢɫɫɥɟɞɨɜɚɧɢɹ, ɤɨɝɞɚ ɧɚ ɨɫɧɨɜɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɩɪɨɜɨɞɢɬɫɹ ɢɡɭɱɟɧɢɟ ɭɫɬɪɨɣɫɬɜ ɢ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɫ ɩɨɦɨɳɶɸ ɗȼɆ, ɩɪɨɢɝɪɵɜɚɟɬɫɹ ɢɯ ɩɨɜɟɞɟɧɢɟ ɜ ɪɚɡɥɢɱɧɵɯ ɭɫɥɨɜɢɹɯ, ɧɚɯɨɞɹɬɫɹ ɨɩɬɢɦɚɥɶɧɵɟ ɩɚɪɚɦɟɬɪɵ ɢ ɪɟɠɢɦɵ ɞɟɣɫɬɜɭɸɳɢɯ ɢ ɩɪɨɟɤɬɧɵɯ ɤɨɧɫɬɪɭɤɰɢɣ.
ɇɟɨɛɯɨɞɢɦɨɫɬɶ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɤɚɤ
ɦɟɬɨɞɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɜɵɡɜɚɧɚ ɬɟɦ, ɱɬɨ ɪɟɲɟɧɢɟ ɫɨɜɪɟɦɟɧɧɵɯ ɧɚɭɱɧɨɬɟɯɧɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɨɬɥɢɱɚɸɳɢɯɫɹ ɱɪɟɡɜɵɱɚɣɧɨ ɫɥɨɠɧɵɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦ ɨɩɢɫɚɧɢɟɦ, ɬɪɚɞɢɰɢɨɧɧɵɦɢ ɦɟɬɨɞɚɦɢ ɫɬɚɧɨɜɢɬɫɹ ɡɚɬɪɭɞɧɢɬɟɥɶɧɵɦ, ɚ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɜɨɨɛɳɟ ɧɟɜɨɡɦɨɠɧɵɦ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɢ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɤɚɤ
ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɚɯɨɞɹɬ ɩɪɢɦɟɧɟɧɢɟ ɜ ɫɚɦɵɯ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ,
ɬɚɤɢɯ ɤɚɤ ɷɧɟɪɝɟɬɢɤɚ, ɚɷɪɨɤɨɫɦɢɱɟɫɤɚɹ ɬɟɯɧɢɤɚ, ɨɛɪɚɛɨɬɤɚ ɞɚɧɧɵɯ ɧɚɬɭɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɷɤɨɧɨɦɢɱɟɫɤɢɟ ɩɪɨɛɥɟɦɵ, ɝɟɨ – ɢ ɚɫɬɪɨɮɢɡɢɤɚ,
ɯɢɦɢɹ, ɛɢɨɥɨɝɢɹ ɢ ɞɪ. ɑɚɫɬɨ ɩɨɞɝɨɬɨɜɤɚ ɢ ɩɪɨɜɟɞɟɧɢɟ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ
ɷɤɫɩɟɪɢɦɟɧɬɚ ɨɤɚɡɵɜɚɸɬɫɹ ɷɤɜɢɜɚɥɟɧɬɧɵɦɢ ɫɨɡɞɚɧɢɸ ɤɪɭɩɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɭɫɬɚɧɨɜɨɤ. ȼɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɜɤɥɸɱɚɟɬɫɹ ɜ
ɩɪɨɝɪɚɦɦɭ ɩɪɨɜɟɞɟɧɢɹ ɤɪɭɩɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ. Ɉɛɥɚɞɚɹ
ɨɛɳɢɦɢ ɫ ɧɚɬɭɪɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɨɦ ɫɜɨɣɫɬɜɚɦɢ, ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɢɦɟɟɬ ɢ ɧɟɤɨɬɨɪɵɟ, ɩɪɢɫɭɳɢɟ ɬɨɥɶɤɨ ɟɦɭ, ɨɫɨɛɟɧɧɨɫɬɢ.
ȼɨ-ɩɟɪɜɵɯ, ɨɤɚɡɵɜɚɟɬɫɹ ɜɨɡɦɨɠɧɵɦ ɩɪɨɜɟɞɟɧɢɟ «ɷɤɫɩɟɪɢɦɟɧɬɚ» ɜ
ɞɨɫɬɚɬɨɱɧɨ ɲɢɪɨɤɨɦ ɞɢɚɩɚɡɨɧɟ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ ɩɪɨɰɟɫɫɚ ɢ ɭɫɬɚɧɨɜɤɢ ɛɟɡ ɦɨɞɢɮɢɤɚɰɢɢ ɫɭɳɟɫɬɜɭɸɳɢɯ ɭɫɬɚɧɨɜɨɤ ɢɥɢ ɪɚɡɪɚɛɨɬɤɢ ɧɨɜɵɯ. Ȼɥɚɝɨɞɚɪɹ ɷɬɨɦɭ ɜɨɡɦɨɠɧɨ ɩɪɨɜɟɞɟɧɢɟ ɛɨɥɶɲɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɡɚ ɫɪɚɜɧɢɬɟɥɶɧɨ ɧɟɛɨɥɶɲɨɟ ɜɪɟɦɹ.
ȼɨ-ɜɬɨɪɵɯ, ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɭɩɪɚɜɥɹɬɶ ɞɟɬɚɥɶɧɨɫɬɶɸ ɚɧɚɥɢɡɚ ɩɪɨɰɟɫɫɚ. ɗɬɨ ɜɚɠɧɨ, ɟɫɥɢ ɪɚɡɦɟɪɵ ɨɛɥɚɫɬɢ, ɝɞɟ ɩɪɨɬɟɤɚɟɬ ɩɪɨɰɟɫɫ, ɢ
ɟɝɨ ɞɥɢɬɟɥɶɧɨɫɬɶ – ɦɚɥɵ, ɱɬɨ ɬɢɩɢɱɧɨ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɥɚɡɟɪɧɨɣ ɢ ɩɥɚɡɦɟɧɧɨɣ ɬɟɯɧɨɥɨɝɢɣ.
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ȼ-ɬɪɟɬɶɢɯ, ɢɡɭɱɚɟɦɵɟ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɱɚɫɬɨ ɢɦɟɸɬ ɢɫɤɥɸɱɢɬɟɥɶɧɨ ɫɥɨɠɧɵɣ ɯɚɪɚɤɬɟɪ ɢɡ-ɡɚ ɤɨɦɩɥɟɤɫɧɨɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɪɚɡɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ. ȼɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɩɨɡɜɨɥɹɟɬ ɢɡɭɱɚɬɶ
ɜɥɢɹɧɢɟ ɤɚɠɞɨɝɨ ɹɜɥɟɧɢɹ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
ȼ-ɱɟɬɜɟɪɬɵɯ, ɜ ɫɥɭɱɚɟ ɡɚɜɢɫɢɦɨɫɬɢ ɮɢɡɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɨɬ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɩɚɪɚɦɟɬɪɨɜ, ɜɥɢɹɧɢɟ ɤɚɠɞɨɝɨ ɢɡ ɧɢɯ ɬɚɤɠɟ ɦɨɠɧɨ ɢɫɫɥɟɞɨɜɚɬɶ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
ȼ-ɩɹɬɵɯ, ɜɨɡɦɨɠɧɨ ɩɪɨɜɟɞɟɧɢɟ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɛɟɡ
ɦɨɞɢɮɢɤɚɰɢɢ ɭɫɬɚɧɨɜɤɢ ɧɚ ɨɫɧɨɜɟ ɛɚɧɤɚ ɦɨɞɟɥɟɣ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ.
ȼ-ɲɟɫɬɵɯ, ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɩɪɢɱɢɧɵ ɧɟɫɨɨɬɜɟɬɫɬɜɢɹ ɪɟɡɭɥɶɬɚɬɨɜ ɧɚɬɭɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɬɟɨɪɟɬɢɱɟɫɤɢɦ ɩɪɨɝɧɨɡɚɦ ɩɨɫɪɟɞɫɬɜɨɦ ɦɧɨɝɨɤɪɚɬɧɨɝɨ «ɩɪɨɢɝɪɵɜɚɧɢɹ ɩɪɨɰɟɫɫɚ»
ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɭɫɥɨɜɢɣ ɟɝɨ ɩɪɨɜɟɞɟɧɢɹ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ, ɜ ɬɨɦ ɱɢɫɥɟ ɜɵɱɢɫɥɢɬɟɥɶɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ, ɹɜɥɹɸɬɫɹ ɢɫɤɥɸɱɢɬɟɥɶɧɨ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦɢ.
Ɉɫɧɨɜɨɣ ɜɫɹɤɨɝɨ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɞɨɥɠɧɨ ɛɵɬɶ ɮɢɡɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɹɜɥɟɧɢɹ, ɢɫɩɨɥɶɡɭɸɳɟɟ ɬɨɱɧɵɟ ɢ ɩɪɢɛɥɢɠɟɧɧɵɟ
ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɦɟɬɨɞɵ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. Ʉɚɤɢɟ ɦɟɯɚɧɢɡɦɵ ɬɟɩɥɨɩɟɪɟɧɨɫɚ ȼɵ ɡɧɚɟɬɟ?
2. ɑɬɨ ɩɨɧɢɦɚɸɬ ɩɨɞ ȼɌɌɉ? ɉɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪɵ.
3. ȼ ɱɟɦ ɫɨɫɬɨɹɬ ɨɫɨɛɟɧɧɨɫɬɢ ɗɅ-ɬɟɯɧɨɥɨɝɢɣ?
4. ɉɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪɵ ɥɚɡɟɪɧɵɯ ɬɟɯɧɨɥɨɝɢɣ.
5. ȼ ɱɟɦ ɫɨɫɬɨɹɬ ɨɫɨɛɟɧɧɨɫɬɢ ɩɥɚɡɦɟɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɷɧɟɪɝɢɢ?
6. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɪɨɥɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɞɥɹ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ?
7. Ʉɚɤɭɸ ɪɨɥɶ ɢɝɪɚɸɬ ɩɪɨɰɟɫɫɵ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɫɬɪɨɢɬɟɥɶɫɬɜɟ?
8. ɑɬɨ ɬɚɤɨɟ «ɬɟɩɥɨɜɚɹ ɡɚɳɢɬɚ»?
9. ȼ ɤɚɤɢɯ ɩɪɢɪɨɞɧɵɯ ɫɢɫɬɟɦɚɯ ɜɚɠɧɵ ɩɪɨɰɟɫɫɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ?
10. ɇɚɡɨɜɢɬɟ ɤɨɥɢɱɟɫɬɜɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɢ
ɟɞɢɧɢɰɵ ɢɯ ɢɡɦɟɪɟɧɢɹ
11. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɨɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɬɟɪɦɨɞɢɧɚɦɢɤɢ.
12. ɑɬɨ ɬɚɤɨɟ «ɬɟɩɥɨɟɦɤɨɫɬɶ»?
13. ȼɵɩɢɲɢɬɟ ɨɫɧɨɜɧɭɸ ɮɨɪɦɭ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ.
14. Ʉɚɤɢɟ ȼɵ ɡɧɚɟɬɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɮɭɧɤɰɢɢ ɫɨɫɬɨɹɧɢɹ?
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ɑȺɋɌɖ 2
Ɉ ɫ ɧ ɨ ɜ ɧ ɵ ɟ ɩ ɨ ɧ ɹ ɬ ɢ ɹ ɢ ɭ ɪ ɚ ɜɧ ɟ ɧ ɢ ɹ
2.1. ɂɫ ɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ
ɂɫɬɨɱɧɢɤɢ ɬɟɩɥɚ, ɢɫɩɨɥɶɡɭɟɦɵɟ ɜ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ (ȼɌɉ), ɦɨɝɭɬ ɛɵɬɶ ɩɨɜɟɪɯɧɨɫɬɧɵɦɢ ɢɥɢ ɨɛɴɟɦɧɵɦɢ;
ɧɟɩɪɟɪɵɜɧɵɦɢ, ɢɦɩɭɥɶɫɧɵɦɢ ɢ ɢɦɩɭɥɶɫɧɨ-ɩɟɪɢɨɞɢɱɟɫɤɢɦɢ; ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɦɢ ɢ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦɢ; ɧɟɩɨɞɜɢɠɧɵɦɢ ɢ ɞɜɢɠɭɳɢɦɢɫɹ.
Ɍɚɤɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɢɫɬɨɱɧɢɤɨɜ ɹɜɥɹɟɬɫɹ ɭɫɥɨɜɧɨɣ ɢ ɡɚɜɢɫɢɬ ɤɚɤ
ɨɬ ɪɟɚɥɶɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ, ɬɚɤ ɢ ɨɬ ɫɜɨɣɫɬɜ ɦɚɬɟɪɢɚɥɨɜ.
Ɉɞɢɧ ɢ ɬɨɬ ɠɟ ɢɫɬɨɱɧɢɤ ɩɪɢ ɪɚɡɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɯ ɦɚɫɲɬɚɛɚɯ ɦɨɠɟɬ ɛɵɬɶ ɨɬɧɟɫɟɧ ɤ ɪɚɡɧɵɦ ɬɢɩɚɦ.
ȿɫɥɢ ɜɵɞɟɥɟɧɢɟ ɬɟɩɥɚ ɩɪɨɢɫɯɨɞɢɬ, ɜ ɨɫɧɨɜɧɨɦ, ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ (ɧɚ
ɝɪɚɧɢɰɟ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɬɟɥɚ), ɬɨ ɬɚɤɨɣ ɢɫɬɨɱɧɢɤ ɩɪɢɧɹɬɨ ɧɚɡɜɚɬɶ ɩɨɜɟɪɯɧɨɫɬɧɵɦ. Ʉ ɱɢɫɥɭ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɷɧɟɪɝɢɢ ɨɬɧɨɫɹɬ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɣ ɷɥɟɤɬɪɨɧɧɵɣ ɥɭɱ; ɥɚɡɟɪɧɨɟ ɢɡɥɭɱɟɧɢɟ ɪɚɡɥɢɱɧɵɯ ɞɥɢɧ
ɜɨɥɧ, ɞɟɣɫɬɜɭɸɳɟɟ ɧɚ ɦɟɬɚɥɥɵ; ɩɨɬɨɤɢ ɩɥɚɡɦɵ, ɝɟɧɟɪɢɪɭɟɦɵɟ ɩɥɚɡɦɨɬɪɨɧɚɦɢ ɢɥɢ ɞɪɭɝɢɦɢ ɦɟɬɨɞɚɦɢ; ɫɜɚɪɨɱɧɭɸ ɞɭɝɭ; ɫɜɟɬɨɜɨɟ ɢɡɥɭɱɟɧɢɟ
ɲɢɪɨɤɨɝɨ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɞɢɚɩɚɡɨɧɚ (ɧɚɩɪɢɦɟɪ, ɫɮɨɤɭɫɢɪɨɜɚɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɤɫɟɧɨɧɨɜɵɯ ɥɚɦɩ). ɉɨɜɟɪɯɧɨɫɬɧɵɟ ɢɫɬɨɱɧɢɤɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ
ɦɨɳɧɨɫɬɶɸ Q ɢ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɭɞɟɥɶɧɨɣ ɦɨɳɧɨɫɬɶɸ q (ɬ.ɟ., ɦɨɳɧɨɫɬɶɸ ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ), ɚ ɬɚɤɠɟ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ.
ȼ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɜɵɞɟɥɟɧɢɟ ɬɟɩɥɨɬɵ ɩɪɨɢɫɯɨɞɢɬ, ɜ ɨɫɧɨɜɧɨɦ, ɜɧɭɬɪɢ
ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɬɟɥɚ, ɢɫɬɨɱɧɢɤɢ ɧɚɡɵɜɚɸɬɫɹ ɨɛɴɟɦɧɵɦɢ. Ʉ ɱɢɫɥɭ ɨɛɴɟɦɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɨɬɧɨɫɹɬ ɪɟɡɢɫɬɢɜɧɵɟ ɢ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɟ ɧɚɝɪɟɜɚɬɟɥɢ, ɢɫɩɨɥɶɡɭɸɳɢɟ ɬɟɩɥɨɬɭ Ⱦɠɨɭɥɹ–Ʌɟɧɰɚ. Ɉɛɴɟɦɧɵɟ ɢɫɬɨɱɧɢɤɢ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɦɨɳɧɨɫɬɶɸ ɢ ɨɛɴɟɦɧɨɣ ɭɞɟɥɶɧɨɣ ɦɨɳɧɨɫɬɶɸ (ɬ.ɟ. ɦɨɳɧɨɫɬɶɸ, ɩɪɢɯɨɞɹɳɟɣɫɹ ɧɚ ɟɞɢɧɢɰɭ ɨɛɴɟɦɚ).
ȼ ɪɹɞɟ ɫɥɭɱɚɟɜ ɩɨɜɟɪɯɧɨɫɬɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ ɦɨɝɭɬ ɩɪɟɜɪɚɳɚɬɶɫɹ
ɜ ɨɛɴɟɦɧɵɟ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɞɟɣɫɬɜɢɢ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɧɚ ɧɟɤɨɬɨɪɵɟ
ɞɢɷɥɟɤɬɪɢɤɢ ɜɵɞɟɥɟɧɢɟ ɢ ɩɨɝɥɨɳɟɧɢɟ ɷɧɟɪɝɢɢ ɩɪɨɢɫɯɨɞɢɬ ɧɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɚ ɜ ɨɛɴɟɦɟ ɜɟɳɟɫɬɜɚ. ɉɪɢ ɭɜɟɥɢɱɟɧɢɢ ɦɨɳɧɨɫɬɢ ɷɥɟɤɬɪɨɧɧɨɝɨ ɥɭɱɚ ɦɚɤɫɢɦɭɦ ɷɧɟɪɝɨɜɵɞɟɥɟɧɢɹ ɬɚɤɠɟ ɫɦɟɳɚɟɬɫɹ ɜ ɨɛɴɟɦ ɦɚɬɟɪɢɚɥɚ. Ɍɚɤ, ɩɨɬɨɤ ɪɟɥɹɬɢɜɢɫɬɫɤɢɯ ɷɥɟɤɬɪɨɧɨɜ ɦɨɠɧɨ ɨɬɧɟɫɬɢ, ɫɤɨɪɟɟ, ɤ
ɨɛɴɟɦɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɪɚɫɩɪɟɞɟɥɟɧɧɵɦ ɢɫɬɨɱɧɢɤɚɦ.
ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ
(ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɷɧɟɪɝɢɢ ɩɨ ɨɛɴɟɦɭ ɢɥɢ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɜɪɟɦɟɧɧɵɟ ɩɚɪɚɦɟɬɪɵ) ɢɝɪɚɸɬ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ ɜ ȼɌɉ. ɇɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧ36
ɧɵɦ ɬɢɩɨɦ ɢɫɬɨɱɧɢɤɨɜ ɹɜɥɹɸɬɫɹ ɝɚɭɫɫɨɜɵ, ɷɧɟɪɝɢɹ ɜ ɤɨɬɨɪɵɯ ɪɚɫɩɪɟɞɟɥɟɧɨ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ ɧɨɪɦɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ.
Ɍɪɚɞɢɰɢɨɧɧɵɟ ɨɛɴɟɦɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɨɬɵ (ɧɚɩɪɢɦɟɪ, ɪɟɡɢɫɬɢɜɧɵɟ) ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɛɨɥɶɲɢɦ ɜɪɟɦɟɧɟɦ ɧɚɝɪɟɜɚ ɞɨ ɪɚɛɨɱɟɣ ɬɟɦɩɟɪɚɬɭɪɵ (ɨɬ ɞɨɥɟɣ ɫɟɤɭɧɞɵ ɞɨ ɧɟɫɤɨɥɶɤɢɯ ɦɢɧɭɬ ɢɥɢ ɞɚɠɟ
ɱɚɫɨɜ). ȼ ɧɟɤɨɬɨɪɵɯ ɜɵɫɨɤɨɱɚɫɬɨɬɧɵɯ ɩɥɚɡɦɨɬɪɨɧɚɯ ɜɵɯɨɞ ɧɚ ɪɚɛɨɱɢɣ
ɪɟɠɢɦ ɦɨɠɟɬ ɫɨɫɬɚɜɥɹɬɶ ɬɵɫɹɱɧɵɟ ɞɨɥɢ ɫɟɤɭɧɞɵ. Ɇɨɳɧɨɫɬɶ ɨɛɴɟɦɧɵɯ
ɢɫɬɨɱɧɢɤɨɜ ɞɨɫɬɢɝɚɟɬ ɧɟɫɤɨɥɶɤɢɯ ɫɨɬɟɧ ɤɢɥɨɜɚɬɬ ɢɥɢ ɞɚɠɟ ɦɟɝɚɜɚɬɬ, ɜ
ɬɨ ɠɟ ɜɪɟɦɹ ɤɨɧɰɟɧɬɪɚɰɢɹ ɷɧɟɪɝɢɢ ɨɛɵɱɧɨ ɧɟɜɟɥɢɤɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɬɚɤ
ɧɚɡɵɜɚɟɦɵɦɢ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɦɢ ɢɫɬɨɱɧɢɤɚɦɢ.
Ɉɫɧɨɜɧɵɦɢ ɨɛɥɚɫɬɹɦɢ ɩɪɢɦɟɧɟɧɢɹ ɨɛɴɟɦɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɨɬɵ ɜ
ȼɌɉ ɹɜɥɹɸɬɫɹ ɩɥɚɜɤɚ ɦɟɬɚɥɥɨɜ, ɜɵɪɚɳɢɜɚɧɢɟ ɦɨɧɨɤɪɢɫɬɚɥɥɨɜ, ɩɨɥɭɱɟɧɢɟ ɩɥɟɧɨɤ ɦɟɬɨɞɨɦ ɬɟɪɦɢɱɟɫɤɨɝɨ ɢɫɩɚɪɟɧɢɹ ɜ ɜɚɤɭɭɦɟ, ɯɢɦɢɤɨɬɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɨɜ ɢ ɞɪ.
ɒɢɪɨɤɨɟ ɩɪɢɦɟɧɟɧɢɟ ɜ ȼɌɉ ɩɨɥɭɱɢɥɢ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɟ ɢɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ (ɪɚɡɞɟɥ 1.4).
Ɉɛɵɱɧɨ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɷɧɟɪɝɢɢ ɧɚɡɵɜɚɸɬ ɬɟ,
ɡɨɧɵ ɜɨɡɞɟɣɫɬɜɢɹ (ɩɹɬɧɨ ɧɚɝɪɟɜɚ) ɤɨɬɨɪɵɯ ɧɚ ɨɛɪɚɛɚɬɵɜɚɟɦɨɟ ɬɟɥɨ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɯɚɪɚɤɬɟɪɧɵɦɢ ɪɚɡɦɟɪɚɦɢ ɬɟɥɚ. ɉɨɫɤɨɥɶɤɭ ɝɥɚɜɧɵɦ
ɧɚɡɧɚɱɟɧɢɟɦ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɹɜɥɹɟɬɫɹ ɧɚɝɪɟɜ ɨɩɪɟɞɟɥɟɧɧɨɣ ɡɨɧɵ ɬɟɥɚ ɞɨ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚ ɦɚɥɵɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ, ɭɞɟɥɶɧɚɹ ɦɨɳɧɨɫɬɶ ɢɫɬɨɱɧɢɤɨɜ ɞɨɥɠɧɚ ɛɵɬɶ ɛɨɥɶɲɨɣ
(103–106 ȼɬ/ɫɦ2 ɢ ɜɵɲɟ).
ɂɡ ɱɢɫɥɚ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɭɫɥɨɜɧɨ ɜɵɞɟɥɹɸɬ ɜɵɫɨɤɨɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɟ ɢɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ (ȼɄɂɗ), ɭɞɟɥɶɧɚɹ ɦɨɳɧɨɫɬɶ
ɤɨɬɨɪɵɯ ɧɚ ɨɩɪɟɞɟɥɟɧɧɨɦ ɭɱɚɫɬɤɟ ɩɪɟɜɵɲɚɟɬ 104 ȼɬ/ɫɦ2. Ʉ ȼɄɂɗ ɨɬɧɨɫɹɬ ɩɨɬɨɤɢ ɷɥɟɤɬɪɨɧɨɜ ɢ ɢɨɧɨɜ, ɫɮɨɤɭɫɢɪɨɜɚɧɧɵɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥ,
ɫɬɪɭɢ ɢ ɫɝɭɫɬɤɢ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɨɣ ɩɥɚɡɦɵ, ɝɟɧɟɪɢɪɭɟɦɵɟ ɫ ɩɨɦɨɳɶɸ
ɫɩɟɰɢɚɥɶɧɵɯ ɭɫɬɪɨɣɫɬɜ – ɞɭɝɨɜɵɯ ɩɥɚɡɦɨɬɪɨɧɨɜ, ɜɡɪɵɜɧɵɯ ɩɥɚɡɦɟɧɧɵɯ
ɝɟɧɟɪɚɬɨɪɨɜ, ɫɮɨɤɭɫɢɪɨɜɚɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɥɚɡɟɪɨɜ ɪɚɡɥɢɱɧɵɯ ɬɢɩɨɜ. ɉɨɞ
ɞɟɣɫɬɜɢɟɦ ȼɄɂɗ ɧɚ ɭɱɚɫɬɤɚɯ ɦɟɬɚɥɥɢɱɟɫɤɢɯ ɬɟɥ ɬɟɦɩɟɪɚɬɭɪɭ, ɛɥɢɡɤɭɸ
ɤ ɬɟɦɩɟɪɚɬɭɪɟ ɩɥɚɜɥɟɧɢɹ, ɩɨɥɭɱɚɸɬ ɡɚ ɧɟɫɤɨɥɶɤɨ ɦɢɥɥɢɫɟɤɭɧɞ.
ȼɄɂɗ ɦɨɝɭɬ ɛɵɬɶ ɤɚɤ ɢɦɩɭɥɶɫɧɵɦɢ (ɢɦɩɭɥɶɫɧɨ–ɩɟɪɢɨɞɢɱɟɫɤɢɦɢ),
ɬɚɤ ɢ ɧɟɩɪɟɪɵɜɧɵɦɢ. Ɇɨɳɧɨɫɬɶ ɢɦɩɭɥɶɫɧɵɯ ȼɄɂɗ ɦɨɠɟɬ ɫɭɳɟɫɬɜɟɧɧɨ
ɩɪɟɜɵɲɚɬɶ ɟɞɢɧɢɰɵ ɦɟɝɚɜɚɬɬ, ɚ ɧɟɩɪɟɪɵɜɧɵɯ – ɞɨɫɬɢɝɚɬɶ ɦɨɳɧɨɫɬɟɣ
ɟɞɢɧɢɰ ɢ ɞɚɠɟ ɞɟɫɹɬɤɨɜ ɦɟɝɚɜɚɬɬ.
Ɇɧɨɝɢɟ ɜɨɩɪɨɫɵ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɢɫɬɨɱɧɢɤɨɜ
ɫ ɜɟɳɟɫɬɜɨɦ ɢɫɫɥɟɞɨɜɚɧɵ ɧɟ ɞɨɫɬɚɬɨɱɧɨ ɩɨɥɧɨ, ɱɬɨ ɜ ɨɩɪɟɞɟɥɟɧɧɨɣ ɫɬɟɩɟɧɢ ɫɞɟɪɠɢɜɚɟɬ ɢɯ ɩɪɢɦɟɧɟɧɢɟ ɜ ɩɪɨɦɵɲɥɟɧɧɨɣ ɬɟɯɧɨɥɨɝɢɢ. Ɉɞɧɚɤɨ
ɭɠɟ ɫɟɣɱɚɫ ɹɫɧɨ, ɱɬɨ ɨɩɪɟɞɟɥɹɸɳɭɸ ɪɨɥɶ ɜ ɩɪɨɰɟɫɫɚɯ ɢɝɪɚɸɬ ɹɜɥɟɧɢɹ,
37
ɫɜɹɡɚɧɧɵɟ ɫ ɩɟɪɟɧɨɫɨɦ ɬɟɩɥɚ ɢ ɦɚɫɫɵ, ɩɨɷɬɨɦɭ ɞɥɹ ɧɚɩɪɚɜɥɟɧɧɨɝɨ ɮɨɪɦɢɪɨɜɚɧɢɹ ɡɨɧ ɨɛɪɚɛɨɬɤɢ ɢ ɨɩɬɢɦɚɥɶɧɨɝɨ ɭɩɪɚɜɥɟɧɢɹ ɜɜɨɞɨɦ ɷɧɟɪɝɢɢ ɜ
ɜɟɳɟɫɬɜɨ ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɢɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ.
ɂɬɚɤ, ɛɨɥɶɲɨɟ ɪɚɡɧɨɨɛɪɚɡɢɟ ɫɢɫɬɟɦ ɢ ɩɨɞɫɢɫɬɟɦ ɜ ɫɨɜɪɟɦɟɧɧɵɯ
ɬɟɯɧɨɥɨɝɢɹɯ ɩɪɢɜɨɞɢɬ ɤ ɦɧɨɝɨɨɛɪɚɡɢɸ ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ ɬɟɩɥɨɬɵ,
ɜɨɡɧɢɤɚɸɳɢɯ ɜ ɪɟɚɥɶɧɵɯ ɩɪɨɰɟɫɫɚɯ. Ɉɞɧɚɤɨ, ɧɟ ɫɦɨɬɪɹ ɧɚ ɷɬɨ, ɜɫɟ ɢɫɬɨɱɧɢɤ ɢ ɫɬɨɤɢ ɦɨɝɭɬ ɛɵɬɶ ɤɥɚɫɫɢɮɢɰɢɪɨɜɚɧɵ ɩɨ ɨɫɧɨɜɧɵɦ ɩɪɢɡɧɚɤɚɦ,
ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɫɯɟɦɚɬɢɡɢɪɨɜɚɬɶ ɢ ɜ ɢɡɜɟɫɬɧɨɣ ɦɟɪɟ ɭɧɢɮɢɰɢɪɨɜɚɬɶ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɬɟɩɥɨɮɢɡɢɤɢ. Ʉ ɨɫɧɨɜɧɵɦ ɩɪɢɡɧɚɤɚɦ ɨɬɧɨɫɹɬɫɹ ɮɨɪɦɚ ɢ ɪɚɡɦɟɪɵ ɢɫɬɨɱɧɢɤɚ; ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɩɥɨɬɧɨɫɬɢ
ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ; ɫɤɨɪɨɫɬɶ ɩɟɪɟɦɟɳɟɧɢɹ; ɞɥɢɬɟɥɶɧɨɫɬɶ ɞɟɣɫɬɜɢɹ. ȼ ɪɟɚɥɶɧɵɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ ɢɫɬɨɱɧɢɤɢ ɢ ɫɬɨɤɢ ɬɟɩɥɨɬɵ ɢɦɟɸɬ
ɮɨɪɦɭ ɢ ɪɚɡɦɟɪɵ, ɤɨɬɨɪɵɟ ɧɟ ɜɫɟɝɞɚ ɬɨɱɧɨ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɢ. ɉɪɢ ɚɧɚɥɢɡɟ ɬɟɩɥɨɜɵɯ ɩɪɨɰɟɫɫɨɜ ɪɟɚɥɶɧɵɟ ɢɫɬɨɱɧɢɤɢ ɡɚɦɟɧɹɸɬ
ɢɞɟɚɥɢɡɢɪɨɜɚɧɧɵɦɢ, ɮɨɪɦɚ ɤɨɬɨɪɵɯ ɜ ɬɨɣ ɢɥɢ ɢɧɨɣ ɫɬɟɩɟɧɢ ɩɪɢɛɥɢɠɟɧɚ ɤ ɮɚɤɬɢɱɟɫɤɨɣ. ɂɞɟɚɥɢɡɢɪɨɜɚɧɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ ɦɨɝɭɬ ɛɵɬɶ
ɬɪɟɯɦɟɪɧɵɦɢ, ɞɜɭɦɟɪɧɵɦɢ, ɨɞɧɨɦɟɪɧɵɦɢ ɢ ɬɨɱɟɱɧɵɦɢ. Ɍɪɟɯɦɟɪɧɵɦɢ
(ɨɛɴɟɦɧɵɦɢ) ɢɫɬɨɱɧɢɤɚɦɢ ɧɚɡɵɜɚɸɬ ɬɚɤɢɟ, ɷɧɟɪɝɢɹ ɜ ɤɨɬɨɪɵɯ ɪɚɫɩɪɟɞɟɥɟɧɨ ɩɨ ɧɟɤɨɬɨɪɨɦɭ ɨɛɴɟɦɭ. ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɜɫɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɨɬɵ ɬɪɟɯɦɟɪɧɵɟ, ɬɚɤ ɤɚɤ ɜ ɥɸɛɨɦ ɩɪɨɰɟɫɫɟ (ɜ ɩɪɨɰɟɫɫɚɯ ɬɪɟɧɢɹ, ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ, ɪɟɡɤɢ, ɫɜɚɪɤɢ, ɧɚɩɥɚɜɤɢ ɢ ɬ.ɞ.) ɜɵɞɟɥɟɧɢɟ ɷɧɟɪɝɢɢ ɩɪɨɢɫɯɨɞɢɬ ɜ
ɧɟɤɨɬɨɪɨɦ ɨɛɴɟɦɟ. Ɉɞɧɚɤɨ ɟɫɥɢ ɪɚɡɦɟɪ ɨɛɴɟɦɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɞɧɨɣ ɢɡ ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ ɦɧɨɝɨ ɦɟɧɶɲɟ ɪɚɡɦɟɪɨɜ ɜ ɞɪɭɝɢɯ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɢɫɬɨɱɧɢɤ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɞɜɭɦɟɪɧɵɦ. Ɉɞɧɨɦɟɪɧɵɦɢ ɧɚɡɵɜɚɸɬ ɢɫɬɨɱɧɢɤɢ, ɨɞɢɧ ɢɡ ɪɚɡɦɟɪɨɜ ɤɨɬɨɪɵɯ ɧɚɫɬɨɥɶɤɨ ɩɪɟɜɵɲɚɟɬ ɞɪɭɝɢɟ, ɱɬɨ ɡɧɚɱɟɧɢɟɦ ɩɨɫɥɟɞɧɢɯ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. ɇɚɤɨɧɟɰ, ɟɫɥɢ ɜɫɟ
ɪɚɡɦɟɪɵ ɢɫɬɨɱɧɢɤɚ ɜɟɫɶɦɚ ɦɚɥɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɪɚɡɦɟɪɚɦɢ ɨɛɥɚɫɬɢ
ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜ ɤɨɬɨɪɨɣ ɨɧ ɞɟɣɫɬɜɭɟɬ, ɢɫɬɨɱɧɢɤ ɦɨɠɧɨ ɩɨɥɚɝɚɬɶ ɬɨɱɟɱɧɵɦ. Ɍɚɤɨɣ ɢɫɬɨɱɧɢɤ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɤɚɱɟɫɬɜɟ ɧɟɤɨɬɨɪɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɚɛɫɬɪɚɤɰɢɢ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɣ ɭɞɨɛɧɨ ɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɮɨɪɦɭɥɵ
ɞɥɹ ɨɩɢɫɚɧɢɹ ɩɪɨɰɟɫɫɚ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɨɬɵ ɨɬ ɢɫɬɨɱɧɢɤɨɜ
ɪɟɚɥɶɧɨɣ ɮɨɪɦɵ.
Ɂɨɧɚ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɪɟɚɥɶɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɜɫɟɝɞɚ ɢɦɟɟɬ ɨɝɪɚɧɢɱɟɧɧɵɟ ɪɚɡɦɟɪɵ. ɑɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ QW , ɜɵɞɟɥɢɜɲɟɟɫɹ
ɜ ɨɛɥɚɫɬɢ ɞɟɣɫɬɜɢɹ ɢɫɬɨɱɧɢɤɚ V ɡɚ ɜɪɟɦɹ W , ɧɭɠɧɨ ɜɵɱɢɫɥɢɬɶ ɢɧɬɟɝɪɚɥ
Wª
º
«
QW
q x, y , z , t dV »dt ,
«
»
0 ¬« V
¼»
³³
38
ɝɞɟ ɮɭɧɤɰɢɹ q x, y , z , t ɨɩɢɫɵɜɚɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɷɧɟɪɝɢɢ ɜ ɨɛɥɚɫɬɢ
ɞɟɣɫɬɜɢɹ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɨɬɵ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
Ɇɨɠɧɨ ɩɪɢɧɹɬɶ,
q x, y , z , t q 0 f x, y , z , t ,
ɝɞɟ q0 - ɦɚɤɫɢɦɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ, Ⱦɠ/(ɦ2ɫ).
Ⱦɥɹ ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ, ɞɟɣɫɬɜɭɸɳɢɯ ɜ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ, ɜɢɞ ɮɭɧɤɰɢɣ f x , y , z , t ɨɱɟɧɶ ɫɥɨɠɟɧ ɢ, ɤɚɤ ɩɪɚɜɢɥɨ, ɡɚɪɚɧɟɟ ɧɟɢɡɜɟɫɬɟɧ. ȼ ɫɜɹɡɢ ɫ ɷɬɢɦ ɩɪɢ ɫɯɟɦɚɬɢɡɚɰɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɨɫɧɨɜɟ ɚɧɚɥɢɡɚ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɢ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɨɩɪɟɞɟɥɹɸɳɢɯ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟ, ɫɨɫɬɚɜɥɹɸɬ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɜɨɡɦɨɠɧɨɦ ɜɢɞɟ ɷɬɨɣ ɮɭɧɤɰɢɢ.
ɇɚɩɪɢɦɟɪ, ɩɪɢ ɦɨɞɟɥɢɪɨɜɚɧɢɢ ɩɪɨɰɟɫɫɨɜ ɪɟɡɚɧɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɧɚ ɩɥɨɳɚɞɤɟ ɤɨɧɬɚɤɬɚ ɦɟɠɞɭ ɫɬɪɭɠɤɨɣ ɢ
ɩɟɪɟɞɧɟɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɪɟɡɰɚ, ɤɚɤ ɩɪɚɜɢɥɨ, ɨɩɢɫɵɜɚɸɬ ɨɞɧɨɦɟɪɧɵɦɢ
ɡɚɤɨɧɚɦɢ:
f ( x ) 1,0 x x s ,
f ( x ) exp kx ,0 x xs ,
f ( x ) exp kx 2 ,0 x xs ɢ ɬ.ɞ.
ɉɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɨɞɢɧɨɱɧɨɝɨ ɢɦɩɭɥɶɫɚ ɧɟɩɨɞɜɢɠɧɨɝɨ ɥɚɡɟɪɧɨɝɨ ɥɭɱɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɡɚɤɨɧ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ
ɜ ɜɢɞɟ
ª x a 2 y a 2 º
f ( x , y ) exp «
» ,t d ti ɢ f ( x, y ) 0, t ! ti .
R2
»¼
«¬
ȿɫɥɢ ɦɚɤɫɢɦɭɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɨɛɴɟɦɟ ɦɚɬɟɪɢɚɥɚ, ɬɨ
ɮɭɧɤɰɢɹ f ɫɬɚɧɨɜɢɬɫɹ ɬɪɟɯɦɟɪɧɨɣ
ª x a 2 y a 2 z c º
f ( x , y , z ) exp «
» ,t d t i .
2
d
R
¼»
¬«
ȼ ɩɪɨɰɟɫɫɚɯ ɫɜɚɪɤɢ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɪɚɡɥɢɱɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɷɧɟɪɝɢɢ, ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɫɤɚɧɢɪɭɸɳɢɦ ɷɥɟɤɬɪɨɧɧɵɦ ɥɭɱɨɦ ɢ ɞɪ.
ɜɢɞ ɷɮɮɟɤɬɢɜɧɨɣ ɮɭɧɤɰɢɢ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɜ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɪɟ ɡɚɜɢɫɢɬ
ɨɬ ɯɚɪɚɤɬɟɪɚ ɞɜɢɠɟɧɢɹ ɢɫɬɨɱɧɢɤɚ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɜɨɡɞɟɣɫɬɜɢɹ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɩɨɜɟɪɯɧɨɫɬɢ ɞɟɬɚɥɢ, ɩɟɪɟɦɟɳɚɸɳɟɣɫɹ ɜɞɨɥɶ ɨɫɢ Ox ɫɨ ɫɤɨɪɨɫɬɶɸ V , ɫɤɚɧɢɪɭɸɳɢɦ ɷɥɟɤɬɪɨɧɧɵɦ ɥɭɱɨɦ, ɱɚɫɬɨ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɮɭɧɤɰɢɸ ɜɢɞɚ
ª x 2 y Vt 2 z c º
f ( x , y , z ) exp «
» , x d hx ,
2
d
R
¼»
¬«
39
ɝɞɟ h x - ɩɨɥɭɲɢɪɢɧɚ ɫɤɚɧɢɪɨɜɚɧɢɹ. Ⱦɥɹ ɧɟɜɵɫɨɤɨɣ ɩɥɨɬɧɨɫɬɢ ɦɨɳɧɨɫɬɢ q 0 ɷɥɟɤɬɪɨɧɧɨ-ɥɭɱɟɜɵɯ ɭɫɬɚɧɨɜɨɤ, ɲɢɪɨɤɨ ɩɪɢɦɟɧɹɟɦɵɯ ɜ ɩɪɨɦɵɲɥɟɧɧɨɫɬɢ, ɢɫɬɨɱɧɢɤ ɫɬɚɧɨɜɢɬɫɹ ɞɜɭɦɟɪɧɵɦ
ª x 2 y Vt 2 º
f ( x , y , z ) exp «
» , x d hx .
R2
»¼
«¬
ɋ ɭɜɟɥɢɱɟɧɢɟɦ ɫɥɨɠɧɨɫɬɢ ɯɚɪɚɤɬɟɪɚ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɜɨɡɪɚɫɬɚɟɬ ɢ
ɱɢɫɥɨ ɩɚɪɚɦɟɬɪɨɜ, ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɨɩɢɫɚɧɢɹ ɯɚɪɚɤɬɟɪɚ ɜɨɡɞɟɣɫɬɜɢɹ.
ɂɫɬɨɱɧɢɤɢ ɬɟɩɥɚ, ɫɜɹɡɚɧɧɵɟ ɫ ɯɢɦɢɱɟɫɤɢɦɢ ɪɟɚɤɰɢɹɦɢ, ɬɚɤɠɟ ɦɨɝɭɬ
ɛɵɬɶ ɩɨɜɟɪɯɧɨɫɬɧɵɦɢ ɢ ɨɛɴɟɦɧɵɦɢ. ɂɯ ɨɫɨɛɟɧɧɨɫɬɢ ɛɭɞɭɬ ɪɚɫɫɦɨɬɪɟɧɵ ɜ ɪɚɡɞɟɥɟ 11.8.
2.2. ɏɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ
ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɩɪɨɰɟɫɫ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ, ɤɚɤ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬɚɤ ɢ
ɜɨ ɜɪɟɦɟɧɢ. Ⱥɧɚɥɢɬɢɱɟɫɤɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɫɜɨɞɢɬɫɹ ɤ ɢɡɭɱɟɧɢɸ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ-ɜɪɟɦɟɧɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ, ɬ.ɟ. ɤ ɧɚɯɨɠɞɟɧɢɸ ɭɪɚɜɧɟɧɢɹ
T T x, y , z , t .
(2.1)
ɗɬɨ ɟɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ.
Ɋɚɡɥɢɱɚɸɬ ɫɬɚɰɢɨɧɚɪɧɵɟ ɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ ɬɟɦɩɟɪɚɬɭɪɧɵɟ ɩɨɥɹ.
ȿɫɥɢ ɬɟɩɥɨɜɨɣ ɪɟɠɢɦ ɹɜɥɹɟɬɫɹ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ, ɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɜ
ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɨɥɹ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɢ ɬɚɤɨɟ ɬɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ
T T1 x, y , z ; w T w t 0 .
(2.2)
Ɍɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ ɦɨɠɟɬ ɛɵɬɶ ɬɪɟɯɦɟɪɧɵɦ (2.1), ɞɜɭɦɟɪɧɵɦ
T f 2 x, y , t ; w T w z 0
(2.3)
ɢ ɨɞɧɨɦɟɪɧɵɦ
T f 3 x, t ; w T w z 0 ; w T w y 0 .
(2.4)
ɗɬɨ ɡɚɜɢɫɢɬ ɨɬ ɤɨɧɤɪɟɬɧɨɣ ɮɢɡɢɱɟɫɤɨɣ ɫɢɬɭɚɰɢɢ.
ȼɵɛɟɪɟɦ ɜ ɬɜɟɪɞɨɦ ɬɟɥɟ ɩɨɜɟɪɯɧɨɫɬɶ ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɜ ɤɚɤɨɣɥɢɛɨ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɬɟɦɩɟɪɚɬɭɪɚ ɜɫɟɯ ɟɟ ɬɨɱɟɤ ɛɵɥɚ ɨɞɢɧɚɤɨɜɨɣ ɢ
ɪɚɜɧɨɣ Ti . Ɍɚɤɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɡɵɜɚɟɬɫɹ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɟɦɩɟɪɚɬɭɪɵ Ti ; ɦɨɠɧɨ ɫɤɚɡɚɬɶ, ɱɬɨ ɷɬɚ ɩɨɜɟɪɯɧɨɫɬɶ ɨɬɞɟɥɹɟɬ
ɱɚɫɬɢ ɬɟɥɚ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɛɨɥɶɲɟɣ Ti , ɨɬ ɱɚɫɬɟɣ ɫ ɦɟɧɶɲɟɣ ɬɟɦɩɟɪɚɬɭɪɨɣ. Ɇɵ ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɫɟɛɟ ɢɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɪɨɜɟ40
ɞɟɧɧɵɟ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ Ti , ɨɬɥɢɱɚɸɳɢɯɫɹ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɧɚ ɰɟɥɵɟ ɝɪɚɞɭɫɵ ɢɥɢ ɧɚ ɞɨɥɢ ɝɪɚɞɭɫɚ. ɗɬɢ
ɢɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɦɨɝɭɬ ɪɚɫɩɨɥɚɝɚɬɶɫɹ ɥɸɛɵɦ ɨɛɪɚɡɨɦ. ɇɨ
ɞɜɟ ɬɚɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟ ɦɨɝɭɬ ɩɟɪɟɫɟɤɚɬɶɫɹ, ɢɛɨ ɧɢɤɚɤɚɹ ɱɚɫɬɶ ɬɟɥɚ ɧɟ
ɦɨɠɟɬ ɢɦɟɬɶ ɞɜɟ ɬɟɦɩɟɪɚɬɭɪɵ ɨɞɧɨɜɪɟɦɟɧɧɨ.
ɉɟɪɟɫɟɱɟɧɢɟ ɢɡɨɬɟɪɦɢɱɟɫɤɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɩɥɨɫɤɨɫɬɶɸ ɞɚɟɬ ɧɚ ɷɬɨɣ
ɩɥɨɫɤɨɫɬɢ ɫɟɦɟɣɫɬɜɨ ɢɡɨɬɟɪɦ. Ɉɧɢ ɨɛɥɚɞɚɸɬ ɬɟɦɢ ɠɟ ɫɜɨɣɫɬɜɚɦɢ, ɱɬɨ ɢ
ɢɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ. Ɍɚɤ, ɧɚ ɪɢɫ. 2. 1 ɩɨɤɚɡɚɧɵ ɢɡɨɬɟɪɦɵ, ɬɟɦɩɟɪɚɬɭɪɵ ɤɨɬɨɪɵɯ ɨɬɥɢɱɚɸɬɫɹ ɧɚ ' T .
Ɍɟɦɩɟɪɚɬɭɪɚ ɜ ɬɟɥɟ ɦɟɧɹɟɬɫɹ ɬɨɥɶɤɨ ɜ
ɧɚɩɪɚɜɥɟɧɢɹɯ, ɩɟɪɟɫɟɤɚɸɳɢɯ ɢɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ. ɉɪɢ ɷɬɨɦ ɧɚɢɛɨɥɶɲɢɣ ɩɟɪɟɩɚɞ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɟɞɢɧɢɰɭ
ɞɥɢɧɵ ɩɪɨɢɫɯɨɞɢɬ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ ɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ.
Ɋɢɫ. 2.1. ɂɡɨɬɟɪɦɵ
ȼɨɡɪɚɫɬɚɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ ɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɝɪɚɞɢɟɧɬɨɦ ɬɟɦɩɟɪɚɬɭɪɵ. Ƚɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ ɟɫɬɶ ɜɟɤɬɨɪ, ɧɚɩɪɚɜɥɟɧɧɵɣ
ɩɨ ɧɨɪɦɚɥɢ ɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɫɬɨɪɨɧɭ ɜɨɡɪɚɫɬɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɱɢɫɥɟɧɧɨ ɪɚɜɧɵɣ ɩɪɨɢɡɜɨɞɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɷɬɨɦɭ ɧɚɩɪɚɜɥɟɧɢɸ
& wT
,
(2.5)
’ T { gradT n 0
wn
&
ɝɞɟ n 0 – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ, ɧɨɪɦɚɥɶɧɵɣ ɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ
ɢ ɧɚɩɪɚɜɥɟɧɧɵɣ ɜ ɫɬɨɪɨɧɭ ɜɨɡɪɚɫɬɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ; w T w n – ɩɪɨɢɡɜɨɞɧɚɹ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɧɨɪɦɚɥɢ.
ɋɤɚɥɹɪɧɚɹ ɜɟɥɢɱɢɧɚ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɝɪɚɞɢɟɧɬɚ ɧɟ ɨɞɢɧɚɤɨɜɚ ɞɥɹ
ɪɚɡɧɵɯ ɬɨɱɟɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. Ɉɧɚ ɛɨɥɶɲɟ ɬɚɦ, ɝɞɟ ɪɚɫɫɬɨɹɧɢɟ ' n ɦɟɠɞɭ ɢɡɨɬɟɪɦɢɱɟɫɤɢɦɢ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɦɟɧɶɲɟ.
ɉɪɨɟɤɰɢɢ ɜɟɤɬɨɪɚ ɝɪɚɞɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɵ ’ T ɧɚ ɤɨɨɪɞɢɧɚɬɧɵɟ ɨɫɢ
Ox, O y, O z ɨɩɪɟɞɟɥɹɸɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ
wT
wT
;
˜ c os n , x ’ T x
wn
dx
wT
wT
;
(2.6)
˜ cosn, y
’ T y
dy
wn
wT
wT
.
˜ c os n , z ’ T z
wn
dz
Ɉɫɧɨɜɧɵɦ ɭɫɥɨɜɢɟɦ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɨɬɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɹɜɥɹɟɬɫɹ ɧɚɥɢɱɢɟ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɜ ɪɚɡɥɢɱɧɵɯ ɟɝɨ ɬɨɱɤɚɯ.
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2.3. Ɍɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ
Ʉɚɤ ɢ ɞɪɭɝɢɟ ɜɢɞɵ ɬɟɩɥɨɨɛɦɟɧɚ, ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢɦɟɟɬ ɦɟɫɬɨ
ɬɨɥɶɤɨ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɜ ɪɚɡɥɢɱɧɵɯ ɱɚɫɬɹɯ (ɬɨɱɤɚɯ) ɬɟɥɚ ɬɟɦɩɟɪɚɬɭɪɚ
ɧɟɨɞɢɧɚɤɨɜɚ.
Ɉɫɧɨɜɧɵɦ ɡɚɤɨɧɨɦ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɹɜɥɹɟɬɫɹ
ɝɢɩɨɬɟɡɚ Ɏɭɪɶɟ (1768–1830), ɫɨɝɥɚɫɧɨ ɤɨɬɨɪɨɣ ɷɥɟɦɟɧɬɚɪɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ dQ (Ⱦɠ), ɩɪɨɯɨɞɹɳɟɟ ɱɟɪɟɡ ɷɥɟɦɟɧɬ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ dF ɡɚ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ dt , ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɬɟɦɩɟɪɚɬɭɪɧɨɦɭ ɝɪɚɞɢɟɧɬɭ
dT
dQ O
dFdt .
(2.7)
dn
Ɉɩɵɬɧɵɦ ɩɭɬɟɦ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɜ (2.7) ɟɫɬɶ ɮɢɡɢɱɟɫɤɢɣ ɩɚɪɚɦɟɬɪ ɜɟɳɟɫɬɜɚ. Ɉɧ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɩɨɫɨɛɧɨɫɬɶ ɜɟɳɟɫɬɜɚ ɩɪɨɜɨɞɢɬɶ ɬɟɩɥɨɬɭ ɢ ɧɚɡɵɜɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɪɨɯɨɞɹɳɟɟ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɞɢɧɢɰɭ
ɩɥɨɳɚɞɢ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɬ.ɟ. ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ,
ɟɫɬɶ ɜɟɤɬɨɪ, ɨɩɪɟɞɟɥɹɟɦɵɣ ɫɨɨɬɧɨɲɟɧɢɟɦ
&
q
& wT
O n 0
wn
(2.8)
ȼɟɤɬɨɪ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚɩɪɚɜɥɟɧ ɩɨ ɧɨɪɦɚɥɢ ɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ȿɝɨ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ ɫɨɜɩɚɞɚɟɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɭɛɵɜɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ, ɬ.ɟ. ɬɟɩɥɨɬɚ ɜɫɟɝɞɚ ɩɟɪɟɞɚɟɬɫɹ ɨɬ ɝɨɪɹɱɢɯ ɬɨɱɟɤ ɤ ɯɨɥɨɞɧɵɦ.
Ʌɢɧɢɢ, ɤɚɫɚɬɟɥɶɧɵɟ ɤ ɤɨɬɨɪɵɦ
&
ɫɨɜɩɚɞɚɸɬ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɜɟɤɬɨɪɚ q ,
Ɋɢɫ. 2.2. ɂɡɨɬɟɪɦɵ ɢ ɥɢɧɢɢ ɬɨɤɚ ɧɚɡɵɜɚɸɬɫɹ ɥɢɧɢɹɦɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ. Ʌɢɧɢɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɨɪɬɨɝɨɧɚɥɶɧɵ ɢɡɨɬɟɪɦɢɱɟɫɤɢɦ ɩɨɜɟɪɯɧɨɫɬɹɦ (ɪɢɫ. 2. 2). ɋɤɚɥɹɪɧɚɹ ɜɟɥɢɱɢɧɚ
ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɟɫɬɶ
wT
.
(2.9)
q O
wn
ɂɡɦɟɪɹɟɬɫɹ ɷɬɚ ɜɟɥɢɱɢɧɚ ɜ ɫɢɫɬɟɦɟ ɋɂ ɜ Ⱦɠ/(ɫ•ɦ2)
Ƚɢɩɨɬɟɡɚ Ɏɭɪɶɟ ɛɵɥɚ ɩɨɞɬɜɟɪɠɞɟɧɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ.
42
ȿɫɥɢ ɝɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ ɞɥɹ ɪɚɡɧɵɯ ɬɨɱɟɤ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɥɢɱɟɧ, ɬɨ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɩɪɨɣɞɟɬ ɱɟɪɟɡ ɜɫɸ
ɢɡɨɬɟɪɦɢɱɟɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ F ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɛɭɞɟɬ
wT
(2.10)
Q ³ qdF ³ O
dF
wn
F
F
ȿɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ Q ɜ ɫɢɫɬɟɦɟ ɋɂ ɫɥɭɠɢɬ Ⱦɠ/ɫ.
ɉɨɥɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ (Ⱦɠ), ɩɪɨɲɟɞɲɟɟ ɡɚ ɜɪɟɦɹ W ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ F , ɟɫɬɶ
W
QW
³ ³ O
0F
wT
dFdt .
wn
(2.11)
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɪɚɜɟɧɫɬɜɚɦɢ
wT
wT
wT
; q y O
; q z O
,
(2.12)
q x O
wx
wz
wy
ɬɚɤ ɱɬɨ
q iqx jq y kq z ,
ɝɞɟ i , j,k ɟɞɢɧɢɱɧɵɟ ɜɟɤɬɨɪɵ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ.
2.4. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ʉɚɤ ɭɠɟ ɛɵɥɨ ɫɤɚɡɚɧɨ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɜɟɳɟɫɬɜɚ ɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ, ɞɚɜɥɟɧɢɹ ɢ ɬɢɩɚ ɜɟɳɟɫɬɜɚ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ
ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɩɪɨɯɨɞɢɬ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɟɞɢɧɢɰɭ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɧɨɦ ɝɪɚɞɢɟɧɬɟ, ɪɚɜɧɨɦ
ɟɞɢɧɢɰɟ:
&
q
.
O
’T
ȼ ɱɢɫɬɨɦ ɜɢɞɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɧɚɛɥɸɞɚɟɬɫɹ ɬɨɥɶɤɨ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ. Ʉɨɧɤɪɟɬɧɵɣ ɦɟɯɚɧɢɡɦ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɡɚɜɢɫɢɬ ɨɬ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɫɪɟɞɵ.
ȼ ɝɚɡɚɯ ɩɪɢ ɨɛɵɱɧɵɯ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɞɚɜɥɟɧɢɢ ɩɟɪɟɧɨɫ ɷɧɟɪɝɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɡɚ ɫɱɟɬ ɯɚɨɬɢɱɧɨɝɨ ɦɨɥɟɤɭɥɹɪɧɨɝɨ
ɞɜɢɠɟɧɢɹ, ɞɢɮɮɭɡɢɢ ɦɨɥɟɤɭɥ, ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɤɨɬɨɪɨɣ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ
ɬɟɦɩɟɪɚɬɭɪɟ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɝɚɡɨɜ ɦɟɧɹɟɬɫɹ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0,006 ɞɨ 0,6 ȼɬ/(ɦ•Ʉ). ɇɚɢɛɨɥɶɲɟɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɨɛɥɚɞɚɟɬ
ɫɚɦɵɣ ɥɟɝɤɢɣ ɝɚɡ – ɜɨɞɨɪɨɞ. ɉɪɢ ɤɨɦɧɚɬɧɵɯ ɭɫɥɨɜɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟ43
ɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɨɞɨɪɨɞɚ O | 0,2 ȼɬ/(ɦ•Ʉ). ɍ ɛɨɥɟɟ ɬɹɠɟɥɵɯ ɝɚɡɨɜ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɦɟɧɶɲɟ. Ɍɚɤ ɭ ɞɢɨɤɫɢɞɚ ɭɝɥɟɪɨɞɚ – O | 0,02 ȼɬ/(ɦ•Ʉ), ɚ ɭ
ɜɨɡɞɭɯɚ – O | 0,025 ȼɬ/(ɦ•Ʉ).
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɝɚɡɨɜ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ
ɢ ɡɚɦɟɬɧɨ ɧɟ ɦɟɧɹɟɬɫɹ ɫ ɞɚɜɥɟɧɢɟɦ
(ɪɢɫ. 2.3).
Ɇɟɯɚɧɢɡɦ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɨɬɵ ɜ ɤɚɩɟɥɶɧɵɯ ɠɢɞɤɨɫɬɹɯ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɩɟɪɟɧɨɫ ɷɧɟɪɝɢɢ
ɩɭɬɟɦ ɧɟɫɬɪɨɣɧɵɯ ɭɩɪɭɝɢɯ ɤɨɥɟɛɚɧɢɣ.
Ɍɚɤɨɟ ɬɟɨɪɟɬɢɱɟɫɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɦɟɯɚɧɢɡɦɟ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɜ ɠɢɞɤɨɫɬɹɯ, ɜɵɞɜɢɧɭɬɨɟ
Ⱥ.ɋ. ɉɪɟɞɜɨɞɢɬɟɥɟɜɵɦ, ɛɵɥɨ ɢɫɩɨɥɶɡɨɜɚɧɨ ɇ.Ȼ. ȼɚɪɝɚɮɬɢɤɨɦ ɞɥɹ
ɨɩɢɫɚɧɢɹ ɨɩɵɬɧɵɯ ɞɚɧɧɵɯ ɩɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɪɚɡɥɢɱɧɵɯ ɠɢɞɤɨɫɬɟɣ. ɗɬɚ ɬɟɨɪɢɹ ɧɚɲɥɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɟ ɩɨɞɬɜɟɪɠɞɟɧɢɟ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɤɚɩɟɥɶɧɵɯ
ɠɢɞɤɨɫɬɟɣ ɥɟɠɢɬ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0,07
ɞɨ 0,7 ȼɬ/(ɦ•Ʉ). Ⱦɥɹ ɛɨɥɶɲɢɧɫɬɜɚ
ɠɢɞɤɨɫɬɟɣ O ɭɛɵɜɚɟɬ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ (ɢɫɤɥɸɱɟɧɢɟ ɫɨɫɬɚɜɥɹɸɬ ɜɨɞɚ ɢ
ɝɥɢɰɟɪɢɧ) ɢ ɪɚɫɬɟɬ ɫ ɞɚɜɥɟɧɢɟɦ.
ȼ ɦɟɬɚɥɥɚɯ ɢ ɫɩɥɚɜɚɯ ɨɫɧɨɜɧɵɦ
ɩɟɪɟɞɚɬɱɢɤɨɦ ɬɟɩɥɨɬɵ ɹɜɥɹɸɬɫɹ Ɋɢɫ . 2.3. ɉɪɢɦɟɪɵ ɡɚɜɢɫɢɦɨɫɬɟɣ
ɫɜɨɛɨɞɧɵɟ ɷɥɟɤɬɪɨɧɵ, ɤɨɬɨɪɵɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɦɨɠɧɨ ɭɩɨɞɨɛɢɬɶ ɢɞɟɚɥɶɧɨɦɭ ɨɞɧɨ- ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ10: 1 – ɜɨɡɞɭɯ; 2 –
ɚɬɨɦɧɨɦɭ ɝɚɡɭ. ɉɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ ɦɢɧɟɪɚɥɶɧɚɹ ɜɚɬɚ, U 150 ɤɝ/ɦ3; 3
ɩɨɫɪɟɞɫɬɜɨɦ ɤɨɥɟɛɚɬɟɥɶɧɵɯ ɞɜɢɠɟ- – ɦɢɧɟɪɚɥɶɧɚɹ ɜɚɬɚ, U 400 ɤɝ/ɦ3;
ɧɢɣ ɚɬɨɦɨɜ ɢɥɢ ɜ ɜɢɞɟ ɭɩɪɭɝɢɯ ɡɜɭ- 4 – ɫɭɯɨɣ ɩɨɪɢɫɬɵɣ ɤɪɚɫɧɵɣ ɤɢɪɤɨɜɵɯ ɜɨɥɧ ɧɟ ɢɫɤɥɸɱɚɟɬɫɹ, ɧɨ ɟɟ ɩɢɱ; 5 – ɜɨɞɚ; 6. – ɠɟɥɟɡɨ, 99,9 %;
ɞɨɥɹ, ɤɚɤ ɩɪɚɜɢɥɨ, ɧɟɡɧɚɱɢɬɟɥɶɧɚ ɩɨ 7 – ɥɚɬɭɧɶ (67 % Cu , 33 % Z n ); 8 –
ɫɪɚɜɧɟɧɢɸ ɫ ɩɟɪɟɧɨɫɨɦ ɷɧɟɪɝɢɢ ɦɟɞɶ, 99,9 %; 9 – ɫɟɪɟɛɪɨ, 9,9 %.
ɷɥɟɤɬɪɨɧɧɵɦ ɝɚɡɨɦ.
10
Ɍɟɩɥɨɬɟɯɧɢɤɚ, ɩɨɞ ɪɟɞ. Ȼɚɫɤɚɤɨɜɚ Ⱥ.ɉ. Ɇ.: ɗɧɟɪɝɨɚɬɨɦɢɡɞɚɬ, 1982. 264 ɫ.
44
Ɍɚɤ ɤɚɤ ɜ ɦɟɬɚɥɥɚɯ ɧɨɫɢɬɟɥɹɦɢ ɬɟɩɥɨɜɨɣ ɢ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ
ɹɜɥɹɸɬɫɹ ɷɥɟɤɬɪɨɧɵ, ɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɞɪɭɝ ɞɪɭɝɭ. ɉɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜɫɥɟɞɫɬɜɢɟ ɭɫɢɥɟɧɢɹ ɪɨɥɢ ɬɟɩɥɨɜɵɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɟɣ ɪɚɫɫɟɢɜɚɧɢɟ
ɷɥɟɤɬɪɨɧɨɜ ɭɜɟɥɢɱɢɜɚɟɬɫɹ; ɷɬɨ ɜɥɟɱɟɬ ɡɚ ɫɨɛɨɣ ɭɦɟɧɶɲɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨ– ɢ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɢ ɭ ɱɢɫɬɵɯ ɦɟɬɚɥɥɨɜ. ɉɪɢ ɧɚɥɢɱɢɢ
ɪɚɡɧɨɝɨ ɪɨɞɚ ɩɪɢɦɟɫɟɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɦɟɬɚɥɥɨɜ ɪɟɡɤɨ ɭɛɵɜɚɟɬ. ɗɬɨ
ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɢɫɤɚɠɟɧɢɹ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɪɟɲɟɬɤɢ ɩɪɢ ɧɚɥɢɱɢɢ
ɩɪɢɦɟɫɟɣ ɩɪɟɩɹɬɫɬɜɭɸɬ ɞɜɢɠɟɧɢɸ ɷɥɟɤɬɪɨɧɨɜ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɱɢɫɬɵɯ
ɦɟɬɚɥɥɨɜ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɩɥɚɜɨɜ ɪɚɫɬɭɬ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ.
ȼ ɬɜɟɪɞɵɯ ɬɟɥɚɯ – ɞɢɷɥɟɤɬɪɢɤɚɯ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɪɨɫɬɨɦ ɬɟɦɩɟɪɚɬɭɪɵ. Ʉɚɤ ɩɪɚɜɢɥɨ, ɞɥɹ ɦɚɬɟɪɢɚɥɨɜ ɫ
ɛɨɥɶɲɟɣ ɩɥɨɬɧɨɫɬɶɸ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢɦɟɟɬ ɛɨɥɟɟ ɜɵɫɨɤɨɟ ɡɧɚɱɟɧɢɟ. Ʉɨɷɮɮɢɰɢɟɧɬ O ɡɚɜɢɫɢɬ ɨɬ ɫɬɪɭɤɬɭɪɵ ɦɚɬɟɪɢɚɥɚ.
Ɇɧɨɝɢɟ ɫɬɪɨɢɬɟɥɶɧɵɟ ɢ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɟ ɦɚɬɟɪɢɚɥɵ ɢɦɟɸɬ ɩɨɪɢɫɬɨɟ ɫɬɪɨɟɧɢɟ (ɤɢɪɩɢɱ, ɛɟɬɨɧ, ɚɫɛɟɫɬ, ɲɥɚɤ…), ɢ ɩɪɢɦɟɧɟɧɢɟ ɡɚɤɨɧɚ
Ɏɭɪɶɟ ɤ ɬɚɤɢɦ ɬɟɥɚɦ ɞɨɫɬɚɬɨɱɧɨ ɭɫɥɨɜɧɨ; ɭɫɥɨɜɧɵɦ (ɷɮɮɟɤɬɢɜɧɵɦ) ɹɜɥɹɟɬɫɹ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨɪɨɲɤɨɨɛɪɚɡɧɵɯ ɢ ɩɨɪɢɫɬɵɯ ɬɟɥ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɢɬ ɨɬ ɢɯ
ɩɥɨɬɧɨɫɬɢ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɜɨɡɪɚɫɬɚɧɢɢ ɩɥɨɬɧɨɫɬɢ ɨɬ 400 ɞɨ 800 ɤɝ/ɦ3
ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɚɫɛɟɫɬɚ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɨɬ 0,105 ɞɨ
0,214 ȼɬ/(ɦ•Ʉ). ɗɮɮɟɤɬɢɜɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɢɥɶɧɨ
ɡɚɜɢɫɢɬ ɨɬ ɜɥɚɠɧɨɫɬɢ. Ⱦɥɹ ɜɥɚɠɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɭɳɟɫɬɜɟɧɧɨ ɜɵɲɟ, ɱɟɦ ɞɥɹ ɫɭɯɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢ ɜɨɞɵ ɜ ɨɬɞɟɥɶɧɨɫɬɢ. ɇɚɩɪɢɦɟɪ, ɞɥɹ ɫɭɯɨɝɨ ɤɢɪɩɢɱɚ O 0,35 ȼɬ/(ɦ•Ʉ); ɞɥɹ ɜɨɞɵ
O 0,6 , ɚ ɞɥɹ ɜɥɚɠɧɨɝɨ ɤɢɪɩɢɱɚ O 1,0 ȼɬ/(ɦ•Ʉ). ɗɬɨɬ ɷɮɮɟɤɬ ɦɨɠɟɬ
ɛɵɬɶ ɨɛɴɹɫɧɟɧ ɬɟɦ, ɱɬɨ ɚɛɫɨɪɛɰɢɨɧɧɨ ɫɜɹɡɚɧɧɚɹ ɜɥɚɝɚ ɢɦɟɟɬ ɞɪɭɝɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɫɜɨɛɨɞɧɨɣ ɜɨɞɨɣ. ɍɜɟɥɢɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɡɟɪɧɢɫɬɵɯ ɦɚɬɟɪɢɚɥɨɜ ɫ ɩɨɜɵɲɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɦɨɠɧɨ ɨɛɴɹɫɧɢɬɶ ɬɟɦ, ɱɬɨ ɫ ɩɨɜɵɲɟɧɢɟɦ T ɜɨɡɪɚɫɬɚɟɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɫɪɟɞɵ, ɡɚɩɨɥɧɹɸɳɟɣ ɩɨɪɵ ɦɟɠɞɭ ɡɟɪɧɚɦɢ, ɚ ɬɚɤɠɟ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɬɟɩɥɨɩɟɪɟɞɚɱɚ ɢɡɥɭɱɟɧɢɟɦ ɜ ɩɨɪɚɯ ɡɟɪɧɢɫɬɨɝɨ ɦɚɫɫɢɜɚ.
ɉɪɢ ɨɱɟɧɶ ɧɢɡɤɢɯ ɢ ɨɱɟɧɶ ɜɵɫɨɤɢɯ ɞɚɜɥɟɧɢɹɯ ɢ ɬɟɦɩɟɪɚɬɭɪɚɯ ɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɦɨɝɭɬ ɦɟɧɹɬɶɫɹ ɨɱɟɧɶ ɫɢɥɶɧɨ. Ɍɚɤ, ɤɜɚɪɰ S i O 2 ɩɪɢ
ɬɟɦɩɟɪɚɬɭɪɟ
–260 ɨɋ
ɢɦɟɟɬ
ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
O ! 1000 ȼɬ/(ɦ•Ʉ), ɚ ɩɪɢ ɤɨɦɧɚɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ O | 7 ȼɬ/(ɦ•Ʉ).
ɉɨɪɹɞɨɤ ɜɟɥɢɱɢɧ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɪɚɡɥɢɱɧɵɯ
ɦɚɬɟɪɢɚɥɨɜ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ. 2.4. ɇɟɤɨɬɨɪɵɟ ɞɪɭɝɢɟ ɞɚɧɧɵɟ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɜ ɉɪɢɥɨɠɟɧɢɢ 3.
45
Ɋɢɫ. 2.4. ɉɨɪɹɞɨɤ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɧɟɤɨɬɨɪɵɯ ɦɚɬɟɪɢɚɥɨɜ11
2.5. Ɂɚ ɞɚɱɚ ɨ ɩɥɨɫ ɤɨɣ ɫ ɬɟɧ ɤɟ
ɉɪɨɫɬɟɣɲɢɦ ɩɪɢɦɟɪɨɦ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɡɚɤɨɧɚ Ɏɭɪɶɟ ɹɜɥɹɟɬɫɹ ɡɚɞɚɱɚ
ɨ ɬɟɩɥɨɩɟɪɟɧɨɫɟ ɱɟɪɟɡ ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ ɬɨɥɳɢɧɨɣ L (ɪɢɫ. 2.5), ɞɥɢɧɚ ɢ
ɲɢɪɢɧɚ ɤɨɬɨɪɨɣ ɛɟɫɤɨɧɟɱɧɨ ɜɟɥɢɤɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɬɨɥɳɢɧɨɣ.
Ɋɢɫ. 2.5. ɉɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɱɟɪɟɡ
ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ
ȿɫɥɢ ɨɛɟ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ ɢɦɟɸɬ
ɩɨɫɬɨɹɧɧɵɟ, ɧɨ ɪɚɡɥɢɱɧɵɟ ɬɟɦɩɟɪɚɬɭɪɵ
( T1 ! T2 ), ɬɨ ɬɟɩɥɨ ɛɭɞɟɬ ɩɟɪɟɧɨɫɢɬɶɫɹ
ɬɨɥɶɤɨ ɜ ɨɞɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ – ɩɨ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɹɦ ɫɬɟɧɤɢ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ɏɭɪɶɟ
ɢɦɟɟɦ
dT
dQ O F
dx .
dx
ɂɧɬɟɝɪɢɪɭɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ O , ɧɚɣɞɟɦ
OF
T T Q
T1 T2 ɢɥɢ q 1 2 (2.13)
L
LO
ȼɟɥɢɱɢɧɚ, ɫɬɨɹɳɚɹ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɫɨ-
11
ɒɧɟɣɞɟɪ ɉ. ɂɧɠɟɧɟɪɧɵɟ ɩɪɨɛɥɟɦɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɇ.: ɂɡɞ-ɜɨ ɢɧɨɫɬɪɚɧɧɨɣ
ɥɢɬɟɪɚɬɭɪɵ, 1960. 480 ɫ.
46
ɨɬɧɨɲɟɧɢɹ (2.13), ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢɥɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ.
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɫɬɟɤɥɹɧɧɚɹ ɜɢɬɪɢɧɚ ɦɚɝɚɡɢɧɚ ɢɦɟɟɬ ɩɥɨɳɚɞɶ 12 ɦ2 ɢ
ɬɨɥɳɢɧɭ 1 ɫɦ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɟɤɥɚ O 0,8 ȼɬ/(ɦ•Ʉ).
ȼ ɯɨɥɨɞɧɵɣ ɞɟɧɶ ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɤɥɚ ɫɨɫɬɚɜɥɹɟɬ
272 Ʉ (–1 ɋ), ɚ ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ 296 Ʉ (+3 ɋ). Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɤɥɨ ɢ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɫɪɟɞɧɟɦ ɫɟɱɟɧɢɢ ɦɟɠɞɭ ɜɧɟɲɧɟɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɫɬɟɤɥɚ.
Ɋɟɲɟɧɢɟ. Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɤɥɨ ɪɚɜɟɧ
OF
T1 T2 0,8 ˜12 ˜ 4 3840 ȼɬ.
Q
0,01
L
Ɍɟɦɩɟɪɚɬɭɪɚ ɜ ɫɪɟɞɧɟɦ ɫɟɱɟɧɢɢ x L 2 ɪɚɜɧɚ 274 Ʉ, ɬɚɤ ɤɚɤ ɜ
ɫɬɟɤɥɟ ɫɨɡɞɚɟɬɫɹ ɥɢɧɟɣɧɵɣ ɩɪɨɮɢɥɶ ɬɟɦɩɟɪɚɬɭɪɵ:
Qx
qx
T T1 T1 .
OF
O
Ʉɚɤ ɭɠɟ ɢɡɜɟɫɬɧɨ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɧɨɝɢɯ ɦɚɬɟɪɢɚɥɨɜ ɧɟ ɹɜɥɹɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. ȼ ɨɩɪɟɞɟɥɟɧɧɵɯ ɢɧɬɟɪɜɚɥɚɯ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɨɧ ɦɨɠɟɬ ɛɵɬɶ ɚɩɩɪɨɤɫɢɦɢɪɨɜɚɧ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɶɸ
O O 0 1 E T ,
ɝɞɟ O 0 - ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɧɟɤɨɬɨɪɨɣ ɯɚɪɚɤɬɟɪɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɟ; E - ɷɦɩɢɪɢɱɟɫɤɚɹ ɩɨɫɬɨɹɧɧɚɹ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɬɨɤ ɬɟɩɥɚ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬɧɨɲɟɧɢɸ
Q
O 0F ª
E
T1 T2 T12 T22 º»
«
L ¬
2
¼
ɢɥɢ
Q
(2.14)
O mF
T1 T2 ,
L
ɝɞɟ O m – ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɫɪɟɞɧɟɣ ɬɟɦɩɟɪɚɬɭɪɟ Tm T1 T2 2 .
ɗɬɢ ɢ ɞɪɭɝɢɟ ɡɚɞɚɱɢ ɦɨɝɭɬ ɛɵɬɶ ɪɟɲɟɧɵ ɞɨɫɬɚɬɨɱɧɨ ɫɬɪɨɝɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɫɨɜɪɟɦɟɧɧɵɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ, ɱɬɨ ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ ɞɚɥɟɟ.
47
2.6. ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɉɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ, ɫɜɹɡɚɧɧɵɯ ɫ ɧɚɯɨɠɞɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ,
ɧɟɨɛɯɨɞɢɦɨ ɢɦɟɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
ȼɵɜɟɫɬɢ ɟɝɨ ɦɨɠɧɨ ɪɚɡɥɢɱɧɵɦɢ ɫɩɨɫɨɛɚɦɢ.
ȼ ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ ɞɟɥɚɸɬ ɫɥɟɞɭɸɳɢɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ: ɬɟɥɨ ɨɞɧɨɪɨɞɧɨ ɢ ɢɡɨɬɪɨɩɧɨ; ɮɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɩɨɫɬɨɹɧɧɵ; ɞɟɮɨɪɦɚɰɢɹ
ɨɛɴɟɦɚ, ɫɜɹɡɚɧɧɚɹ ɫ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ, ɹɜɥɹɟɬɫɹ ɨɱɟɧɶ ɦɚɥɨɣ ɜɟɥɢɱɢɧɨɣ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɚɦɢɦ ɷɥɟɦɟɧɬɚɪɧɵɦ ɨɛɴɟɦɨɦ; ɜɧɭɬɪɟɧɧɢɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɵɟ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɨɝɭɬ ɛɵɬɶ ɡɚɞɚɧɵ ɜ ɜɢɞɟ
qv
q v x , y , z ,t ,
(2.15)
ɪɚɫɩɪɟɞɟɥɟɧɵ ɪɚɜɧɨɦɟɪɧɨ.
ȼ ɨɫɧɨɜɭ ɜɵɜɨɞɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɩɨɥɨɠɟɧ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ d Q1 , ɜɜɟɞɟɧɧɨɟ ɜ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɢɡɜɧɟ ɡɚ
ɜɪɟɦɹ dt ɜɫɥɟɞɫɬɜɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ ɬɚɤɠɟ ɨɬ ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ d Q 2 , ɪɚɜɧɨ ɢɡɦɟɧɟɧɢɸ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢɥɢ ɷɧɬɚɥɶɩɢɢ
dQ ɜɟɳɟɫɬɜɚ (ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɢɡɨɯɨɪɢɱɟɫɤɢɣ ɢɥɢ ɢɡɨɛɚɪɢɱɟɫɤɢɣ ɩɪɨɰɟɫɫ), ɫɨɞɟɪɠɚɳɢɯɫɹ ɜ ɷɥɟɦɟɧɬɚɪɧɨɦ ɨɛɴɟɦɟ:
d Q1 d Q 2
dQ .
(2.16)
ȼɵɞɟɥɢɦ ɜ ɬɟɥɟ ɷɥɟɦɟɧɬɚɪɧɵɣ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞ (ɪɢɫ. 2.6) ɫɨ ɫɬɨɪɨɧɚɦɢ dx , dy , dz ɢ ɫ ɝɪɚɧɹɦɢ, ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɤɨɨɪɞɢɧɚɬɧɵɦ ɩɥɨɫɤɨɫɬɹɦ. Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɩɨɞɜɨɞɢɬɫɹ ɤ ɝɪɚɧɹɦ
ɡɚ ɜɪɟɦɹ dt ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɟɣ Ox ,O y ,O z , ɨɛɨɡɧɚɱɢɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
dQ x , dQ y , dQ z . Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɤɨɬɨɪɨɟ ɛɭɞɟɬ ɨɬɜɨɞɢɬɶɫɹ ɱɟɪɟɡ
ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɟ ɝɪɚɧɢ, ɛɭɞɟɬ d Q x d x , dQ y dy , d Q z d z .
Ɉɱɟɜɢɞɧɵ ɫɨɨɬɧɨɲɟɧɢɹ
d Qx
q x d y d zd t ; d Q x d x
q x d xd y d z d t ,
ɝɞɟ d Q x – ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɞɜɟɞɟɧɧɨɟ ɤ ɝɪɚɧɢ d y d z ɡɚ ɜɪɟɦɹ dt ;
d Q x d x – ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɨɬɜɟɞɟɧɧɨɟ ɨɬ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɝɪɚɧɢ;
q x , q x d x – ɩɪɨɟɤɰɢɢ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ɧɨɪɦɚɥɢ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɩɨɜɟɪɯɧɨɫɬɹɦ.
48
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɟɪɟɧɨɫɢɦɨɟ ɜ
ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɟɫɬɶ
dQ x1
dQ x dQ x dx
q x dydzdt q x dx dydzdt
.
ɉɭɫɬɶ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ –
ɧɟɩɪɟɪɵɜɧɚɹ ɮɭɧɤɰɢɹ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ
ɢɧɬɟɪɜɚɥɟ dx . Ɍɨɝɞɚ ɨɧɚ ɦɨɠɟɬ ɛɵɬɶ ɪɚɡ- Ɋɢɫ. 2.6. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɜɵɥɨɠɟɧɚ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ
ɜɨɞɭ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨ2
ɜɨɞɧɨɫɬɢ
wq
w qx
1
q x d x q x x dx dxdx
.
..
.
wx
2!
wx 2
Ɉɝɪɚɧɢɱɢɜɚɹɫɶ ɩɟɪɜɵɦɢ ɱɥɟɧɚɦɢ ɪɹɞɚ, ɧɚɣɞɟɦ
wq
d Q x1 x d x dy d z d t .
wx
Ⱥɧɚɥɨɝɢɱɧɨ ɧɚɣɞɟɦ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɞɜɨɞɢɦɨɟ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ ɞɪɭɝɢɯ ɨɫɟɣ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɨɞɜɟɞɟɧɧɨɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦɭ ɨɛɴɟɦɭ, ɟɫɬɶ
wq y wqz ·
§ wq
(2.17)
dQ1 ¨ x ¸ dxdydz .
w
x
w
y
w
z
©
¹
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɜɵɞɟɥɹɟɦɨɟ ɜɧɭɬɪɟɧɧɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɟɫɬɶ qV , ȼɬ/ɦ3. Ɍɨɝɞɚ
d Q 2 qV dV d t ,
(2.18)
ɝɞɟ dV dxdydz .
ȼɢɞ ɬɪɟɬɶɟɝɨ ɫɥɚɝɚɟɦɨɝɨ ɡɚɜɢɫɢɬ ɨɬ ɬɢɩɚ ɩɪɨɰɟɫɫɚ.
ȼ ɢɡɨɯɨɪɧɨɦ ɩɪɨɰɟɫɫɟ ɜɫɹ ɬɟɩɥɨɬɚ, ɩɨɞɜɟɞɟɧɧɚɹ ɤ ɷɥɟɦɟɧɬɚɪɧɨɦɭ
ɨɛɴɟɦɭ, ɭɣɞɟɬ ɧɚ ɢɡɦɟɧɟɧɢɟ ɟɝɨ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ (ɫɦ. ɮɨɪɦɭɥɵ
(1.15))
dQ3 V dU U wu dtdV cJ U wT dtdV .
2.19)
wt
wt
Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ (2.16)-(2.19), ɧɚɣɞɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɜɢɞɟ
cJ U
wT
wt
ɢɥɢ
cJ U
wq y wq z ·
§ wq
¸ qV
¨¨ x ¸
w
w
w
x
y
z
©
¹
wT
wt
’ ˜ q qV .
49
(2.20)
ȼ ɢɡɨɛɚɪɧɨɦ ɩɪɨɰɟɫɫɟ ɜɫɹ ɬɟɩɥɨɬɚ, ɩɨɞɜɟɞɟɧɧɚɹ ɤ ɨɛɴɟɦɭ, ɭɣɞɟɬ ɧɚ
ɢɡɦɟɧɟɧɢɟ ɷɧɬɚɥɶɩɢɢ (ɫɦ. (1.15)). Ɍɨɝɞɚ
wh
wT
(2.21)
dtdV c pU
dtdV ,
wt
wt
ɢ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɢɡɨɛɚɪɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ ɩɪɢɧɢɦɚɟɬ
ɜɢɞ
wq y wqz ·
§ wq
wT
c pU
¨ x (2.22)
¸ qV
x
y
z
wt
w
w
w
©
¹
ɢɥɢ
wT
cpU
’ ˜ q qV .
wt
Ⱦɥɹ ɬɜɟɪɞɵɯ ɬɟɥ ɪɚɡɥɢɱɢɟɦ ɬɟɩɥɨɟɦɤɨɫɬɟɣ c p ɢ c v , ɤɚɤ ɩɪɚɜɢɥɨ,
dQ3 V
dH
U
ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. Ɍɨɝɞɚ, ɩɨɥɚɝɚɹ, ɱɬɨ ɞɥɹ ɬɜɟɪɞɵɯ ɬɟɥ ɫɩɪɚɜɟɞɥɢɜ ɡɚɤɨɧ Ɏɭɪɶɟ, ɩɪɢɞɟɦ ɤ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɢɞɚ
wT
w § wT · w § wT · w § wT ·
(2.23)
cU
O
O
O
q .
w t w x ¨© w x ¸¹ w y ©¨ w y ¹¸ w z ©¨ w z ¹¸ V
2.7. Ɉ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱ
ɜ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ȼ ɫɥɭɱɚɟ ɨɞɧɨɦɟɪɧɨɝɨ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ, ɤɨɝɞɚ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɜ ɨɞɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɧɚɩɪɢɦɟɪ, ɜɞɨɥɶ ɨɫɢ Ox , ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɭɩɪɨɳɚɟɬɫɹ. Ɇɵ
ɩɪɢɯɨɞɢɦ ɤ ɨɞɧɨɦɟɪɧɨɦɭ ɧɟɫɬɚɰɢɨɧɚɪɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
wT
w § wT ·
(2.24)
cU
¨O
¸ qV
wt wx © wx ¹
ȿɫɥɢ ɜ ɬɟɥɟ ɨɬɫɭɬɫɬɜɭɸɬ ɨɛɴɟɦɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ, ɚ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɵɦɢ ɜ ɧɟɤɨɬɨɪɨɦ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ, ɬɨ ɩɨɥɭɱɚɟɦ ɫɨɜɫɟɦ ɩɪɨɫɬɨɟ ɭɪɚɜɧɟɧɢɟ
wT
w 2T
(2.25)
a 2,
wt
wx
ɝɞɟ a O cU – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ. ɗɬɨɬ ɤɨɷɮɮɢɰɢɟɧɬ ɹɜɥɹɟɬɫɹ ɩɚɪɚɦɟɬɪɨɦ ɜɟɳɟɫɬɜɚ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜɨ ɜɪɟɦɟɧɢ. ȿɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɩɨɫɨɛɧɨɫɬɶ ɜɟɳɟɫɬɜɚ ɩɪɨɜɨɞɢɬɶ ɬɟɩɥɨɬɭ, ɬɨ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɦɟɪɨɣ ɬɟɩɥɨɢɧɟɪɰɢɨɧɧɵɯ ɫɜɨɣɫɬɜ
50
ɬɟɥɚ. ɉɪɢ ɩɪɨɱɢɯ ɪɚɜɧɵɯ ɭɫɥɨɜɢɹɯ ɛɵɫɬɪɟɟ ɧɚɝɪɟɜɚɟɬɫɹ ɬɨ ɬɟɥɨ, ɤɨɬɨɪɨɟ
ɨɛɥɚɞɚɟɬ ɛɨɥɶɲɢɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ. ȿɫɥɢ ɩɪɢɤɨɫɧɭɬɶɫɹ ɪɭɤɨɣ ɤ ɞɟɪɟɜɹɧɧɨɣ ɪɚɦɟ ɢ ɫɬɟɤɥɭ ɨɤɧɚ, ɬɨ ɩɨɤɚɠɟɬɫɹ, ɱɬɨ
ɫɬɟɤɥɨ – ɯɨɥɨɞɧɟɟ, ɯɨɬɹ ɬɟɦɩɟɪɚɬɭɪɵ ɢɯ ɨɞɢɧɚɤɨɜɵ. ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ
ɬɟɦ, ɱɬɨ ɫɬɟɤɥɨ, ɢɦɟɸɳɟɟ ɜ 4,5 ɪɚɡɚ ɛɨɥɶɲɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ, ɨɛɟɫɩɟɱɢɜɚɟɬ ɛɨɥɟɟ ɛɵɫɬɪɵɣ ɨɬɜɨɞ ɬɟɩɥɚ ɨɬ ɪɭɤɢ.
Ⱦɥɹ ɨɞɧɨɦɟɪɧɨɝɨ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ, ɬ.ɟ., ɤɨɝɞɚ
ɬɟɦɩɟɪɚɬɭɪɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ ( w T w t 0 ), ɨɞɧɨɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
d 2T
0.
(2.26)
dx 2
ɂɦɟɧɧɨ ɬɚɤɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɧɚɦ ɬɪɟɛɭɟɬɫɹ, ɱɬɨɛɵ
ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ. Ⱦɨɩɨɥɧɢɬɟɥɶɧɨ ɤ (2.26) ɦɵ ɢɦɟɟɦ 2
ɭɫɥɨɜɢɹ
x 0 : T T1 ɢ x L : T T2 .
(2.27)
Ɂɚɞɚɜ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɦɵ ɫɮɨɪɦɭɥɢɪɨɜɚɥɢ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (2.26) ɟɫɬɶ
T C1x C 2 .
(2.28)
ɉɨɞɫɬɚɜɥɹɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɜ (2.27) ɭɫɥɨɜɢɹ ɩɪɢ x 0 ɢ x L , ɦɵ
ɩɨɥɭɱɢɦ ɞɜɚ ɭɪɚɜɧɟɧɢɹ, ɩɨɡɜɨɥɹɸɳɢɟ ɨɩɪɟɞɟɥɢɬɶ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɧɚɣɞɟɦ
T T
T T1 1 2 x ,
L
ɨɬɤɭɞɚ ɫ ɭɱɟɬɨɦ ɡɚɤɨɧɚ Ɏɭɪɶɟ ɫɥɟɞɭɸɬ ɫɨɨɬɧɨɲɟɧɢɹ (2.13).
ȼ ɫɥɭɱɚɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɡɚɜɢɫɹɳɟɝɨ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ, ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (2.27) ɞɨɩɨɥɧɹɸɬ ɫɬɚɰɢɨɧɚɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɢɞɚ
d § dT ·
(2.29)
¨O
¸ 0.
dx © dx ¹
ɉɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ (2.29) ɟɫɬɶ
dT
dT
O
C1 ɢɥɢ O 0 1 E T C1 .
dx
dx
ɉɨɫɥɟɞɭɸɳɟɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɚɟɬ
T2
O 0T E
C1x C 2 ,
2
ɨɬɤɭɞɚ, ɢɫɩɨɥɶɡɭɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (2.27) ɢ ɨɩɪɟɞɟɥɟɧɢɟ ɩɨɬɨɤɚ ɬɟɩɥɚ
Q qF , ɧɚɯɨɞɢɦ ɭɪɚɜɧɟɧɢɟ (2.14).
51
ɍɪɚɜɧɟɧɢɟ (2.23) ɨɩɢɫɵɜɚɟɬ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜ ɫɚɦɨɦ ɨɛɳɟɦ ɜɢɞɟ. ɉɪɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɨɥɭɱɚɟɬɫɹ ɛɟɫɱɢɫɥɟɧɧɨɟ ɦɧɨɠɟɫɬɜɨ ɪɟɲɟɧɢɣ, ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɯ ɟɦɭ. Ⱦɥɹ ɪɟɲɟɧɢɹ
ɩɪɚɤɬɢɱɟɫɤɨɣ ɡɚɞɚɱɢ ɧɟɨɛɯɨɞɢɦɨ ɡɚɞɚɬɶ ɜɩɨɥɧɟ ɤɨɧɤɪɟɬɧɵɟ ɞɚɧɧɵɟ, ɬ.ɟ.
ɨɝɪɚɧɢɱɢɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɭɸ ɩɪɨɛɥɟɦɭ ɨɩɪɟɞɟɥɟɧɧɵɦɢ ɭɫɥɨɜɢɹɦɢ,
ɱɬɨɛɵ ɫɞɟɥɚɬɶ ɪɟɲɟɧɢɟ ɨɞɧɨɡɧɚɱɧɵɦ.
ɑɚɫɬɶ ɭɫɥɨɜɢɣ ɩɨɡɜɨɥɹɟɬ ɩɟɪɟɣɬɢ ɨɬ ɨɛɳɟɣ ɮɨɪɦɵ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɤ ɟɝɨ ɱɚɫɬɧɨɦɭ ɜɢɞɭ, ɫɩɪɚɜɟɞɥɢɜɨɦɭ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ
ɩɪɨɛɥɟɦɵ. ɗɬɨ – ɮɢɡɢɱɟɫɤɢɟ ɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɟ ɭɫɥɨɜɢɹ.
Ⱦɪɭɝɚɹ ɱɚɫɬɶ ɭɫɥɨɜɢɣ ɩɨɡɜɨɥɹɟɬ ɜɵɛɪɚɬɶ ɢɡ ɜɫɟɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɪɟɲɟɧɢɣ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɨɥɶɤɨ ɬɨ, ɤɨɬɨɪɨɟ ɷɬɢɦ ɭɫɥɨɜɢɹɦ ɨɬɜɟɱɚɟɬ. ɗɬɢ ɭɫɥɨɜɢɹ ɧɚɡɵɜɚɸɬɫɹ ɭɫɥɨɜɢɹɦɢ ɨɞɧɨɡɧɚɱɧɨɫɬɢ ɢɥɢ
ɤɪɚɟɜɵɦɢ ɭɫɥɨɜɢɹɦɢ.
Ʉɪɚɟɜɵɟ ɭɫɥɨɜɢɹ ɩɨɞɪɚɡɞɟɥɹɸɬ ɧɚ ɭɫɥɨɜɢɹ ɧɚɱɚɥɶɧɵɟ ɢ ɝɪɚɧɢɱɧɵɟ.
ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɨɩɪɟɞɟɥɹɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɬɟɥɟ ɜ
ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɡɚɞɚɟɬɫɹ ɡɧɚɱɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɞɥɹ ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ. ɂɦɟɧɧɨ ɬɚɤɢɟ
ɭɫɥɨɜɢɹ ɦɵ ɢɦɟɥɢ ɜ ɧɚɲɟɣ ɩɪɨɫɬɨɣ ɡɚɞɚɱɟ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɜɬɨɪɨɝɨ ɪɨɞɚ ɡɚɞɚɟɬɫɹ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɜ ɤɚɠɞɨɣ ɬɨɱɤɟ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɞɥɹ ɥɸɛɨɝɨ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɡɚɞɚɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɚ
ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɢ ɡɚɤɨɧ ɬɟɩɥɨɨɛɦɟɧɚ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɟɥɚ ɢ
ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɩɢɫɵɜɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɦɢ ɫɪɟɞɫɬɜɚɦɢ ɭɫɥɨɜɢɹ ɨɞɧɨɡɧɚɱɧɨɫɬɢ ɩɪɢ ɪɟɲɟɧɢɢ ɤɨɧɤɪɟɬɧɵɯ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ, ɜ ɬɨɦ ɱɢɫɥɟ
ɢ ɡɚɞɚɱ, ɨɬɧɨɫɹɳɢɯɫɹ ɤ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦ ɫɢɫɬɟɦɚɦ, ɩɪɢɛɟɝɚɸɬ ɤ ɬɨɣ ɢɥɢ
ɢɧɨɣ ɫɯɟɦɚɬɢɡɚɰɢɢ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɨɛɦɟɧɚ. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɮɚɤɬɢɱɟɫɤɢɟ
ɬɟɥɚ ɢ ɢɫɬɨɱɧɢɤɢ, ɞɟɣɫɬɜɭɸɳɢɟ ɜ ɪɟɚɥɶɧɨɦ ɩɪɢɪɨɞɧɨɦ ɢɥɢ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɦ ɩɪɨɰɟɫɫɟ, ɡɚɦɟɧɹɸɬ ɬɟɥɚɦɢ ɢ ɢɫɬɨɱɧɢɤɚɦɢ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɮɨɪɦɵ, ɩɪɢɛɥɢɠɟɧɧɨ ɨɬɪɚɠɚɸɳɢɦɢ ɨɪɢɝɢɧɚɥɵ. Ɋɟɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɬɟɥ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɢ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɬɚɤɠɟ ɫɯɟɦɚɬɢɡɢɪɭɸɬ, ɡɚɦɟɧɹɹ ɫɥɨɠɧɵɟ ɫɢɬɭɚɰɢɢ, ɢɦɟɸɳɢɟ ɦɟɫɬɨ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ, ɛɨɥɟɟ ɩɪɨɫɬɵɦɢ, ɢɞɟɚɥɢɡɢɪɨɜɚɧɧɵɦɢ.
ɋɯɟɦɚɬɢɡɚɰɢɹ ɞɟɥɚɟɬɫɹ ɫ ɰɟɥɶɸ ɨɛɥɟɝɱɟɧɢɹ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɨɛɦɟɧɚ ɜ ɤɨɧɤɪɟɬɧɵɯ ɡɚɞɚɱɚɯ. Ɉɞɧɚɤɨ ɞɥɹ ɢɫɫɥɟɞɨɜɚɧɢɹ ɨɫɧɨɜɧɵɯ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ ɩɪɢ ɪɟɲɟɧɢɢ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɧɟ
ɜɫɟɝɞɚ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɜɫɟ ɫɬɨɪɨɧɵ ɢ ɞɟɬɚɥɢ ɹɜɥɟɧɢɹ, ɬɟɦ ɛɨɥɟɟ
ɱɬɨ ɩɨɝɪɟɲɧɨɫɬɢ, ɜɵɡɜɚɧɧɵɟ ɭɫɥɨɠɧɟɧɢɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɚɩɩɚɪɚɬɚ,
ɦɨɝɭɬ ɫɜɟɫɬɢ ɧɚ ɧɟɬ ɭɬɨɱɧɟɧɢɹ, ɞɨɫɬɢɝɧɭɬɵɟ ɞɟɬɚɥɢɡɚɰɢɟɣ ɩɪɨɰɟɫɫɚ.
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ɋ ɭɫɥɨɠɧɟɧɢɟɦ ɭɫɥɨɜɢɣ ɨɞɧɨɡɧɚɱɧɨɫɬɢ ɱɢɫɬɨ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ ɪɟɲɟɧɢɹ ɧɚɪɚɫɬɚɸɬ ɨɱɟɧɶ ɛɵɫɬɪɨ, ɢɧɨɝɞɚ ɧɚɫɬɨɥɶɤɨ, ɱɬɨ ɫɚɦɨ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɢɧɬɟɪɟɫɭɸɳɟɝɨ ɧɚɫ ɩɪɨɰɟɫɫɚ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɜɨɡɦɨɠɧɵɦ. Ɋɚɡɥɢɱɧɵɟ ɫɩɨɫɨɛɵ ɫɯɟɦɚɬɢɡɚɰɢɢ ɮɨɪɦɵ ɬɟɥ, ɭɫɥɨɜɢɣ ɢɯ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ, ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ ɪɚɡɥɢɱɧɨɣ
ɩɪɢɪɨɞɵ ɢ ɩɨɡɜɨɥɹɸɬ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɢ ɢɫɫɥɟɞɨɜɚɬɶ ɰɟɥɵɣ ɪɹɞ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɵɯ ɡɚɞɚɱ, ɨ ɤɨɬɨɪɵɯ ɪɟɱɶ ɩɨɣɞɟɬ ɞɚɥɟɟ.
2.8. Ɇɧɨɝɨɫɥɨɣɧɚɹ ɫɬɟɧɤɚ
ȼ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɬɫɹ ɫɬɟɧɤɢ, ɫɨɫɬɨɹɳɢɟ ɢɡ ɧɟɫɤɨɥɶɤɢɯ ɫɥɨɟɜ ɪɚɡɥɢɱɧɵɯ ɦɚɬɟɪɢɚɥɨɜ (ɪɢɫ. 2.7). Ɋɚɫɫɦɨɬɪɢɦ, ɧɚɩɪɢɦɟɪ, ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɱɟɪɟɡ ɫɬɟɧɤɭ, ɫɨɫɬɨɹɳɭɸ ɢɡ ɬɪɟɯ ɩɥɨɬɧɨ ɩɪɢɥɟɝɚɸɳɢɯ ɞɪɭɝ ɤ
ɞɪɭɝɭ ɫɥɨɟɜ ɬɨɥɳɢɧɵ L1 , L2 , L3 . Ʉɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɷɬɢɯ
ɫɥɨɟɜ O 1 , O 2 , O 3 ɪɚɡɥɢɱɧɵ. Ɍɟɦɩɟɪɚɬɭɪɵ ɧɚɪɭɠɧɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ T1 ɢ
T2 ɡɚɞɚɧɵ, ɩɪɢɱɟɦ T1 ! T2 , ɬ.ɟ. ɡɚɞɚɧɵ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɟɪɜɨɝɨ ɪɨɞɚ.
Ɍɚɤ ɤɚɤ ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɦ ɬɟɦɩɟɪɚɬɭɪɧɨɦ ɩɨɥɟ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ,
ɩɪɨɯɨɞɹɳɢɣ ɱɟɪɟɡ ɦɧɨɝɨɫɥɨɣɧɭɸ ɫɬɟɧɤɭ, ɨɞɢɧɚɤɨɜ ɞɥɹ ɤɚɠɞɨɝɨ ɫɥɨɹ,
ɬɨ, ɜɜɨɞɹ ɨɛɨɡɧɚɱɟɧɢɹ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪ ɤɨɧɬɚɤɬɢɪɭɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ
T c ɢ T cc ɢ ɩɨɥɶɡɭɹɫɶ ɭɪɚɜɧɟɧɢɟɦ (2.13), ɡɚɩɢɲɟɦ
Q
Q
T T c F L1 O1 ;
T c T cc F L 2 O 2 ;
T cc T2 F L3 O 3 .
Q
ɗɬɢ ɪɚɜɟɧɫɬɜɚ ɥɟɝɤɨ ɪɚɡɪɟɲɚɸɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɪɚɡɧɨɫɬɟɣ ɬɟɦɩɟɪɚɬɭɪ
T1 T c Q L1 O 1 F ;
Ɋɢɫ. 2.7. ɉɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɱɟɪɟɡ
ɬɪɟɯɫɥɨɣɧɭɸ ɫɬɟɧɤɭ
ɢɥɢ
Q
T c T cc Q L2 O 2 F ;
T c T cc QL3 O3 F .
ɋɤɥɚɞɵɜɚɹ, ɩɪɚɜɵɟ ɢ ɥɟɜɵɟ ɱɚɫɬɢ ɪɚɜɟɧɫɬɜ, ɧɚɣɞɟɦ
T1 T2
Q L1 O 1 L2 O 2 L3 O 3 F
T1 T2 F
L1 O 1 L2 O 2 L3 O 3
.
ɉɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ ɫɬɟɧɤɭ ɟɫɬɶ
53
(2.30)
T1 T2
q
3
¦ Li
,
Oi i 1
ɬ.ɟ., ɩɨɬɨɤ ɬɟɩɥɚ ɢ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɡɚɜɢɫɹɬ ɨɬ ɫɭɦɦɵ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɜɫɟɯ ɫɥɨɟɜ.
ɗɬɭ ɡɚɞɚɱɭ ɦɨɠɧɨ ɪɟɲɢɬɶ ɧɟɦɧɨɝɨ ɢɧɚɱɟ, ɡɚɩɢɫɵɜɚɹ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ
ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (2.26) ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɤɨɧɬɚɤɬɟ ɫɥɨɟɜ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ɏɭɪɶɟ, ɩɨɬɨɤ ɬɟɩɥɚ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ
ɝɪɚɞɢɟɧɬɭ ɬɟɦɩɟɪɚɬɭɪɵ, ɬɚɤ ɱɬɨ ɪɚɜɟɧɫɬɜɨ ɩɨɬɨɤɨɜ ɬɟɩɥɚ ɢ ɬɟɦɩɟɪɚɬɭɪ
ɧɚ ɝɪɚɧɢɰɚɯ ɪɚɡɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɨɡɧɚɱɚɟɬ
§ wT ·
O1 ¨
¸
© wx ¹ x
L1 0
§ wT ·
O2 ¨
¸
© wx ¹ x
L1 0
; T x L 0
1
Tx
L1 0
ɢ
§ wT ·
O2¨
¸
© wx ¹ x
L1 L2 0
§ wT ·
O3¨
¸
© wx ¹ x
L1 L 2 0
; T x L L 0
1
2
Tx
L1 L2 0 .
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɚɤɨɝɨ ɬɢɩɚ ɧɚɡɵɜɚɸɬ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ
ɱɟɬɜɟɪɬɨɝɨ ɪɨɞɚ. Ɋɚɜɟɧɫɬɜɨ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɩɨɜɟɪɯɧɨɫɬɹɯ ɪɚɡɞɟɥɚ ɦɚɬɟɪɢɚɥɨɜ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɦɟɠɞɭ ɧɢɦɢ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɢɞɟɚɥɶɧɵɣ ɬɟɩɥɨɜɨɣ
ɤɨɧɬɚɤɬ.
Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɩɥɨɫɤɭɸ ɫɬɟɧɤɭ, ɫɨɞɟɪɠɚɳɭɸ n ɫɥɨɟɜ, ɟɫɬɶ
T T
(2.31)
q n 1 2 .
¦ Li
Oi i 1
ɂɧɨɝɞɚ ɜɜɨɞɹɬ ɜ ɪɚɫɫɦɨɬɪɟɧɢɟ ɷɤɜɢɜɚɥɟɧɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
O eq
n
n
¦ Li ¦ Li
i 1
Oi ,
(2.32)
i 1
ɤɨɬɨɪɵɣ ɪɚɜɟɧ ɤɨɷɮɮɢɰɢɟɧɬɭ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɮɢɤɬɢɜɧɨɣ ɨɞɧɨɫɥɨɣɧɨɣ ɫɬɟɧɤɢ ɫ ɬɨɥɳɢɧɨɣ, ɪɚɜɧɨɣ ɫɭɦɦɟ ɬɨɥɳɢɧ ɫɥɨɟɜ ɦɧɨɝɨɫɥɨɣɧɨɣ
ɫɬɟɧɤɢ. ɉɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɝɪɚɧɢɰɚɯ
ɨɞɧɨɫɥɨɣɧɨɣ ɢ ɦɧɨɝɨɫɥɨɣɧɨɣ ɫɬɟɧɨɤ ɨɞɢɧɚɤɨɜɵ, ɢ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ,
ɩɪɨɯɨɞɹɳɢɟ ɱɟɪɟɡ ɧɢɯ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɫɨɜɩɚɞɚɸɬ.
54
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɫɬɟɧɤɚ ɩɟɱɢ ɫɨɫɬɨɢɬ ɢɡ ɜɧɭɬɪɟɧɧɟɝɨ ɫɥɨɹ ɧɟɪɠɚɜɟɸɳɟɣ ɫɬɚɥɢ, ɬɨɥɳɢɧɨɣ 1,2 ɫɦ, ɩɨɤɪɵɬɨɝɨ ɜɧɟɲɧɢɦ ɫɥɨɟɦ ɚɫɛɟɫɬɨɜɨɣ
ɢɡɨɥɹɰɢɢ ɬɨɥɳɢɧɨɣ 5 ɫɦ. Ɍɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟɪɠɚɜɟɸɳɟɣ ɫɬɚɥɢ ɪɚɜɧɚ 800 Ʉ, ɚ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɚɫɛɟɫɬɚ ɪɚɜɧɚ 350 Ʉ. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ
ɫɬɟɧɤɭ ɩɟɱɢ ɢ ɬɟɦɩɟɪɚɬɭɪɭ ɤɨɧɬɚɤɬɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɚɥɢ ɢ ɚɫɛɟɫɬɚ.
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɫɬɚɥɢ ɢ ɚɫɛɟɫɬɚ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ O 1 1 9 ȼɬ/(ɦ•Ʉ) ɢ O2 0,7 ȼɬ/(ɦ•Ʉ).
Ɋɟɲɟɧɢɟ. ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɲɟ ɮɨɪɦɭɥɵ, ɧɚɣɞɟɦ ɬɟɩɥɨɜɨɣ
ɩɨɬɨɤ
T1 T2 F .
Q
L1 O 1 L 2 O 2
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɟɫɬɶ
800 350
T1 T2 6245 ȼɬ/ɦ2.
q
L1 O1 L2 O2 0,012 19 0,05 0,7
Ɍɟɩɟɪɶ ɧɚɯɨɞɢɦ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚ ɤɨɧɬɚɤɬɟ
0,012
L
796 Ʉ.
Tx T1 q 1 800 6245
19
O1
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɟɩɚɞ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɧɟɪɠɚɜɟɸɳɟɣ ɫɬɚɥɢ ɫɨɫɬɚɜɥɹɟɬ ɜɫɟɝɨ ɥɢɲɶ ɨɤɨɥɨ 4 Ʉ, ɚ ɩɟɪɟɩɚɞ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɚɫɛɟɫɬɟ 446 Ʉ.
2.9. ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ
ɞɥɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɰɟɥɟɫɨɨɛɪɚɡɟɧ ɢɧɨɣ ɩɨɞɯɨɞ ɤ ɬɟɩɥɨɩɟɪɟɞɚɱɟ.
ɗɬɨɬ ɩɨɞɯɨɞ, ɜ ɤɨɬɨɪɨɦ ɩɪɢɦɟɧɹɸɬɫɹ ɤɨɧɰɟɩɰɢɢ ɷɥɟɤɬɪɢɱɟɫɤɢɯ ɰɟɩɟɣ,
ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɚɧɚɥɨɝɢɟɣ ɦɟɠɞɭ ɩɟɪɟɧɨɫɨɦ ɬɟɩɥɚ ɢ ɷɥɟɤɬɪɢɱɟɫɬɜɚ.
ȿɫɥɢ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɚɧɚɥɨɝɢɱɟɧ ɷɥɟɤɬɪɢɱɟɫɤɨɦɭ ɬɨɤɭ,
ɤɨɦɩɥɟɤɫ L O F ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, ɚ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɤɚɤ ɚɧɚɥɨɝ ɪɚɡɧɨɫɬɢ ɩɨɬɟɧɰɢɚɥɨɜ, ɬɨ ɫɨɨɬɧɨɲɟɧɢɟ (2.13)
OF
Q
T T L 1 2
ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɮɨɪɦɟ, ɚɧɚɥɨɝɢɱɧɨɣ ɡɚɤɨɧɭ Ɉɦɚ
'T
,
Q
RT
ɝɞɟ ' T T1 T2 – ɩɟɪɟɩɚɞ ɬɟɦɩɟɪɚɬɭɪ (ɬɟɪɦɢɱɟɫɤɢɣ ɩɨɬɟɧɰɢɚɥ),
RT
L O F – ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ. Ɉɛɪɚɬɧɚɹ ɜɟɥɢɱɢɧɚ ɬɟɪɦɢ55
ɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɧɚɡɵɜɚɟɬɫɹ ɬɟɩɥɨɜɨɣ ɩɪɨɜɨɞɢɦɨɫɬɶɸ, ɚ ɨɬɧɨɲɟɧɢɟ O L – ɭɞɟɥɶɧɨɣ ɬɟɩɥɨɜɨɣ ɩɪɨɜɨɞɢɦɨɫɬɶɸ ɞɥɹ ɤɨɧɞɭɤɬɢɜɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ.
Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫɨɨɬɧɨɲɟɧɢɟ (2.30) ɞɥɹ
ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ ɬɪɟɯɫɥɨɣɧɭɸ ɫɬɟɧɤɭ (ɪɢɫ. 2.7 ɢ 2.8)
Q
ɝɞɟ ' T
T1 T2 , Rt ,i
'T
,
RT ,1 RT ,2 RT ,3
Li O i F , i 1, 2 , 3 .
ɗɥɟɤɬɪɢɱɟɫɤɭɸ ɚɧɚɥɨɝɢɸ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢ ɞɥɹ ɪɟɲɟɧɢɹ ɛɨɥɟɟ ɫɥɨɠɧɵɯ ɡɚɞɚɱ. ɇɚɩɪɢɦɟɪ, ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ
ɩɪɨɰɟɫɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɨɬɟɤɚɟɬ ɜ
ɦɚɬɟɪɢɚɥɚɯ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɚɪɚɥɥɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɭ.
Ɋɢɫ. 2.8. ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ ɤ ɪɢɫ. 2.7 ɞɥɹ ɡɚɞɚɱɢ ɫ
ɬɪɟɯɫɥɨɣɧɨɣ ɫɬɟɧɤɨɣ
ɇɚ ɪɢɫ. 2.9. ɩɨɤɚɡɚɧɚ ɩɥɢɬɚ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɞɜɭɯ ɦɚɬɟɪɢɚɥɨɜ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɚɪɚɥɥɟɥɶɧɨ ɢ ɢɦɟɸɳɢɯ ɩɨɩɟɪɟɱɧɵɟ ɫɟɱɟɧɢɹ F1 ɢ F2 . ɋɩɪɚɜɚ ɧɚ ɷɬɨɦ ɠɟ ɪɢɫɭɧɤɟ ɩɪɟɞɫɬɚɜɥɟɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɬɟɩɥɨɜɚɹ ɰɟɩɶ.
ɑɬɨɛɵ ɪɟɲɢɬɶ ɷɬɭ ɡɚɞɚɱɭ ɩɪɢ ɡɚɞɚɧɧɨɦ ɩɟɪɟɩɚɞɟ ɬɟɦɩɟɪɚɬɭɪ ɩɨɩɟɪɟɤ ɩɥɢɬɵ, ɤɚɠɞɵɣ ɫɥɨɣ ɫɨɫɬɚɜɧɨɣ ɤɨɧɫɬɪɭɤɰɢɢ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ
ɨɬɞɟɥɶɧɨ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɥɹ ɤɚɠɞɨɣ ɢɡ ɞɜɭɯ ɫɟɤɰɢɣ ɩɟɪɟɧɨɫ ɬɟɩɥɚ
ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɦɟɪɧɵɦ.
RT ,1
RT ,2
L
O 1F1
L
O 2 F2
ɚ
ɛ
Ɋɢɫ. 2.9. Ɍɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɱɟɪɟɡ ɫɨɫɬɚɜɧɭɸ ɫɬɟɧɤɭ ɢɡ ɞɜɭɯ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɟɤɰɢɣ
56
ȿɫɥɢ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɦɟɠɞɭ ɤɨɧɬɚɤɬɢɪɭɸɳɢɦɢ ɦɚɬɟɪɢɚɥɚɦɢ
ɦɚɥɚ, ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɜɞɨɥɶ ɫɥɨɟɜ ɛɭɞɟɬ ɧɚɦɧɨɝɨ ɛɨɥɶɲɟ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɜ ɩɨɩɟɪɟɱɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɢ ɡɚɞɚɱɭ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɦɟɪɧɨɣ
ɛɟɡ ɫɤɨɥɶɤɨ-ɧɢɛɭɞɶ ɫɟɪɶɟɡɧɨɣ ɩɨɬɟɪɢ ɬɨɱɧɨɫɬɢ.
ɉɨɫɤɨɥɶɤɭ ɬɟɩɥɨɜɵɟ ɩɨɬɨɤɢ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ, ɨɛɳɢɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
ɚɪɢɮɦɟɬɢɱɟɫɤɭɸ ɫɭɦɦɭ
§ 1
T1 T2
T T
1 ·
1 2
Q Q1 Q 2
¨
¸ T T .
L O 1F1 L O 2 F2 ¨© RT ,1 RT ,2 ¸¹ 1 2
Ɉɬɦɟɬɢɦ, ɱɬɨ ɨɛɳɚɹ ɩɥɨɳɚɞɶ, ɤɨɬɨɪɭɸ ɩɟɪɟɫɟɤɚɟɬ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ,
ɪɚɜɧɚ ɫɭɦɦɟ ɞɜɭɯ ɨɬɞɟɥɶɧɵɯ ɩɥɨɳɚɞɟɣ ɢ ɱɬɨ ɨɛɪɚɬɧɚɹ ɜɟɥɢɱɢɧɚ ɫɭɦɦɚɪɧɨɝɨ ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɪɚɜɧɚ ɫɭɦɦɟ ɨɛɪɚɬɧɵɯ ɜɟɥɢɱɢɧ
ɨɬɞɟɥɶɧɵɯ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ.
Ɍɟɩɥɨɜɚɹ ɰɟɩɶ ɞɥɹ ɷɬɨɣ ɡɚɞɚɱɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɚɪɚɥɥɟɥɶɧɨɟ ɫɨɟɞɢɧɟɧɢɟ ɞɜɭɯ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ RT ,1 ɢ RT ,2 .
Ȼɨɥɟɟ ɫɥɨɠɧɵɦ ɩɪɢɦɟɪɨɦ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɩɨɧɹɬɢɹ ɬɟɩɥɨɜɵɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɹɜɥɹɟɬɫɹ ɡɚɞɚɱɚ ɨ ɩɟɪɟɞɚɱɟ ɬɟɩɥɚ ɱɟɪɟɡ ɫɨɫɬɚɜɧɭɸ ɫɬɟɧɤɭ, ɤɨɬɨɪɚɹ ɞɨɥɠɧɚ ɩɪɟɞɫɬɚɜɥɹɬɶɫɹ ɫ ɩɨɦɨɳɶɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɢ ɩɚɪɚɥɥɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɯ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ (ɪɢɫ. 2.10). Ⱦɥɹ ɷɬɨɣ ɫɢɫɬɟɦɵ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɪɟɞɧɟɝɨ ɫɥɨɹ ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
RT ,2
RT ,B RT ,C
,
RT ,B RT ,C
ɚ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
'T
,
Q n N
¦ RT ,n
n 1
Ɋɢɫ. 2.10. Ɍɟɩɥɨɜɚɹ ɰɟɩɶ ɫ ɩɚɪɚɥɥɟɥɶɧɨ ɢ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɦɢ
ɷɥɟɦɟɧɬɚɦɢ.
57
ɝɞɟ N – ɱɢɫɥɨ ɫɥɨɟɜ; RT ,n –
ɬɟɪɦɢɱɟɫɤɨɟ
ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɤɚɠɞɨɝɨ ɫɥɨɹ; ' T – ɪɚɡɧɨɫɬɶ
ɬɟɦɩɟɪɚɬɭɪ ɜɧɟɲɧɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ;
L3
L1
RT ,1
, RT ,3
O 1F1
O 3 F3
LC
LB
RT ,B
; RT ,C
.
O B FB
O C FC
ȼ ɷɬɨɣ ɡɚɞɚɱɟ ɬɚɤɠɟ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɨɬɨɤ ɬɟɩɥɚ ɹɜɥɹɟɬɫɹ ɨɞɧɨɦɟɪɧɵɦ.
ȿɫɥɢ ɬɟɪɦɢɱɟɫɤɢɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ RT ,B ɢ RT ,C ɫɢɥɶɧɨ ɨɬɥɢɱɚɸɬɫɹ
ɞɪɭɝ ɨɬ ɞɪɭɝɚ, ɦɨɝɭɬ ɩɪɨɹɜɢɬɶɫɹ ɫɭɳɟɫɬɜɟɧɧɵɟ ɞɜɭɦɟɪɧɵɟ ɷɮɮɟɤɬɵ. Ɍɚɤɢɟ ɡɚɞɚɱɢ ɞɨɥɠɧɵ ɪɚɫɫɦɚɬɪɢɜɚɬɶɫɹ ɜ ɛɨɥɟɟ ɫɥɨɠɧɨɣ ɩɨɫɬɚɧɨɜɤɟ.
2.10. Ʉɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ȿɫɥɢ ɪɚɡɥɢɱɧɵɟ ɬɟɩɥɨɩɪɨɜɨɞɹɳɢɟ ɫɥɨɢ ɧɚɯɨɞɹɬɫɹ ɜ ɤɨɧɬɚɤɬɟ, ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɬɜɟɪɞɵɯ ɬɟɥ ɜɨɡɧɢɤɚɟɬ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ.
ɗɬɨ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, ɤɨɬɨɪɨɟ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɤɨɧɬɚɤɬɧɵɦ
ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ, ɜɨɡɧɢɤɚɟɬ, ɤɨɝɞɚ ɩɨɜɟɪɯɧɨɫɬɢ ɞɜɭɯ ɦɚɬɟɪɢɚɥɨɜ ɧɟɞɨɫɬɚɬɨɱɧɨ ɩɥɨɬɧɨ ɩɪɢɠɚɬɵ ɞɪɭɝ ɤ ɞɪɭɝɭ ɢ ɦɟɠɞɭ ɧɢɦɢ ɨɫɬɚɟɬɫɹ ɬɨɧɤɢɣ ɫɥɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɢɥɶɧɨ ɭɜɟɥɢɱɟɧɧɨɣ ɤɚɪɬɢɧɵ ɤɨɧɬɚɤɬɚ ɦɟɠɞɭ ɞɜɭɦɹ ɬɜɟɪɞɵɦɢ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɦɚɬɟɪɢɚɥɵ ɤɚɫɚɸɬɫɹ ɞɪɭɝ ɞɪɭɝɚ ɬɨɥɶɤɨ ɜɟɪɲɢɧɚɦɢ ɩɪɨɮɢɥɟɣ
ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɟɣ, ɚ ɜɩɚɞɢɧɵ ɤɨɧɬɚɤɬɢɪɭɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɡɚɩɨɥɧɟɧɵ ɢɧɨɪɨɞɧɨɣ ɫɪɟɞɨɣ, ɜɨɡɦɨɠɧɨ ɜɨɡɞɭɯɨɦ, ɠɢɞɤɨɫɬɶɸ ɢɥɢ
ɜɚɤɭɭɦɨɦ.
Ʉɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɡɚɜɢɫɢɬ, ɩɪɟɠɞɟ ɜɫɟɝɨ, ɨɬ
ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɟɣ; ɞɚɜɥɟɧɢɹ, ɩɪɢɠɢɦɚɸɳɟɝɨ ɞɜɟ ɩɨɜɟɪɯɧɨɫɬɢ ɞɪɭɝ ɤ ɞɪɭɝɭ; ɫɪɟɞɵ ɜ ɪɚɣɨɧɟ ɤɨɧɬɚɤɬɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɡɨɧɟ ɤɨɧɬɚɤɬɚ. Ɇɟɯɚɧɢɡɦ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɡɨɧɟ ɤɨɧɬɚɤɬɚ ɞɨɜɨɥɶɧɨ
ɫɥɨɠɟɧ. ȼ ɦɟɫɬɚɯ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɝɨ ɤɨɧɬɚɤɬɚ ɬɜɟɪɞɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ
ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɪɨɰɟɫɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ ɩɟɪɟɧɨɫ ɬɟɩɥɚ ɱɟɪɟɡ ɡɚɡɨɪɵ, ɡɚɩɨɥɧɟɧɧɵɟ ɠɢɞɤɨɫɬɶɸ ɢɥɢ ɝɚɡɨɦ, ɩɪɨɢɡɜɨɞɢɬɫɹ ɤɨɧɜɟɤɰɢɟɣ ɢɥɢ
ɢɡɥɭɱɟɧɢɟɦ.
ȿɫɥɢ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ ɞɜɟ ɤɨɧɬɚɤɬɢɪɭɸɳɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɫɨɫɬɚɜɥɹɟɬ Q F , ɚ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɩɨɩɟɪɟɤ ɡɚɩɨɥɧɟɧɧɨɝɨ ɠɢɞɤɨɫɬɶɸ ɢɥɢ ɝɚɡɨɦ ɡɚɡɨɪɚ, ɤɨɬɨɪɵɣ ɪɚɡɞɟɥɹɟɬ ɞɜɟ ɬɜɟɪɞɵɟ ɩɨɜɟɪɯɧɨɫɬɢ, ɪɚɜɧɚ ' Ti , ɬɨ ɤɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ RT ,i ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
' Ti
.
RT ,i
Q F
ɋɱɢɬɚɟɬɫɹ, ɱɬɨ ɞɜɟ ɩɨɜɟɪɯɧɨɫɬɢ ɧɚɯɨɞɹɬɫɹ ɜ ɢɞɟɚɥɶɧɨɦ ɬɟɩɥɨɜɨɦ
ɤɨɧɬɚɤɬɟ, ɤɨɝɞɚ ɤɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɬɪɟɦɢɬɫɹ ɤ
ɧɭɥɸ, ɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɧɟɬ ɩɟɪɟɩɚɞɚ ɬɟɦɩɟɪɚɬɭɪ. ɉɪɢ ɧɟɢɞɟɚɥɶɧɨɦ ɬɟɩɥɨɜɨɦ ɤɨɧɬɚɤɬɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɫɭɳɟɫɬɜɭɟɬ ɩɟɪɟɩɚɞ
ɬɟɦɩɟɪɚɬɭɪ. ɉɪɢ ɩɨɫɬɚɧɨɜɤɟ ɤɨɧɤɪɟɬɧɵɯ ɡɚɞɚɱ ɜɫɟɝɞɚ ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ
58
ɜɢɞɭ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɬɚɤɨɝɨ ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ. Ⱦɥɹ ɫɢɥɶɧɨ
ɲɟɪɨɯɨɜɚɬɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɨɧɨ ɦɨɠɟɬ ɛɵɬɶ ɡɧɚɱɢɬɟɥɶɧɵɦ.
ɉɪɨɛɥɟɦɚ ɤɨɧɬɚɤɬɧɨɝɨ ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɞɨɫɬɚɬɨɱɧɨ
ɫɥɨɠɧɚ, ɢ ɧɟ ɫɭɳɟɫɬɜɭɟɬ ɟɞɢɧɨɣ ɬɟɨɪɢɢ, ɩɨɡɜɨɥɹɸɳɟɣ ɞɨɫɬɚɬɨɱɧɨ ɬɨɱɧɨ
ɪɚɫɫɱɢɬɵɜɚɬɶ ɤɨɧɬɚɤɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱɚɯ.
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɫɬɟɧɚ ɡɞɚɧɢɹ ɫɨɫɬɨɢɬ ɢɡ ɫɥɨɹ ɨɛɵɱɧɨɝɨ ɤɢɪɩɢɱɚ
( L1 0,1 ɦ;
O1 0,7 ȼɬ/(ɦ•Ʉ)) ɢ ɫɥɨɹ ɝɢɩɫɨɜɨɣ ɲɬɭɤɚɬɭɪɤɢ
( L2 0,038 ɦ; O2 0,48 ȼɬ/(ɦ•Ʉ)). ɋɪɚɜɧɢɬɶ ɬɟɩɥɨɜɵɟ ɩɨɬɨɤɢ ɱɟɪɟɡ ɷɬɭ
ɫɬɟɧɭ ɢ ɱɟɪɟɡ ɬɚɤɭɸ ɠɟ ɫɬɟɧɭ ɫ ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɟɠɞɭ ɤɢɪɩɢɱɨɦ ɢ ɲɬɭɤɚɬɭɪɤɨɣ, ɪɚɜɧɵɦ RT ,k 0,1 Ʉ/ȼɬ
(ɪɢɫ. 2.11).
Ɋɟɲɟɧɢɟ. ɉɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɱɟɪɟɡ ɢɞɟɚɥɢɡɢɪɨɜɚɧɧɭɸ
ɫɬɟɧɤɭ ɩɪɢ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ' T T1 0 T2 0 ɜ 1 Ʉ ɪɚɜɧɚ
Q
F T10 T20 1
L1 O1 L2 O2
1
0,1 0,7 0,038 0,48
4,5 ȼɬ/(ɦ2.•Ʉ).
ɉɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɬɪɟɬɶɢɦ, ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɦ ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ, ɩɨɫɥɟ ɱɟɝɨ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
Q
F T10 T20 1
1
3,11 ȼɬ/(ɦ2.•Ʉ).
0,222 0,1
RT ,1 RT , 2 RT ,k
Ⱦɨɫɬɚɬɨɱɧɨ ɫɬɪɨɝɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɡɚɞɚɱɢ ɨ ɧɚɯɨɠɞɟɧɢɢ ɩɨɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɷɬɨɣ ɞɜɭɯɫɥɨɣɧɨɣ ɫɬɟɧɤɟ ɜɤɥɸɱɚɟɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (2.26) ɞɥɹ ɤɚɠɞɨɝɨ ɫɥɨɹ ɫɬɟɧɤɢ
d 2Ti
dx
0 , i 1, 2
2
ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
x 0 : T1 T10 ;
x
x
L1 L2 : T2 T20 ;
L1 : O 1
dT1
dx
T1 T2
59
O2
RT ,k
dT2
dx
Q
.
F
Q
;
F
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ
ɤɚɠɞɨɝɨ ɫɥɨɹ ɢɦɟɟɬ ɜɢɞ
Ti Ai x Bi ,
(2.33)
ɝɞɟ Ai , Bi – ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɤɨɬɨɪɵɟ ɧɚɯɨɞɢɦ ɫ ɩɨɦɨɳɶɸ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ.
ɉɨɞɫɬɚɜɥɹɹ ɫɨɨɬɧɨɲɟɧɢɟ (2.33) ɜ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (ɤɨɬɨɪɵɯ ɢɦɟɟɬɫɹ ɱɟɬɵɪɟ, ɤɚɤ ɢ
ɩɨɫɬɨɹɧɧɵɯ Ai , Bi ), ɧɚɯɨɞɢɦ ɫɢɫɬɟɦɭ ɱɟɬɵɪɟɯ
ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ
B1 T1 0 ;
Ɋɢɫ. 2.11. ɂɥɥɸɫɬɪɚɰɢɹ
ɤ ɡɚɞɚɱɟ ɫ ɬɟɪɦɢɱɟɫɤɢɦ
ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ
A2 L1 L2 B 2 T20 ;
O 1B1 O 2 B 2
Q
;
F
Q
.
F
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɩɪɟɞɟɥɹɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:
O
T2 0 1 T1 0
O1
O2
B1 T1 0 ; B 2
T10 ; A2
;
O2
L1 L 2
O
T20 1 T1 0
·
O2
O
T §
1 0 ¨ RT ,k O 1 1 1¸ .
A1
L1 L2
L1 ©
O2 ¹
Ɋɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
§
· x
§O
· x
O
T1 T1 0 ¨ T2 0 1 T1 0 ¸
T1 0 ¨ 1 RT ,k 1¸ ;
O2
©
¹ L1 L2
© O2
¹ L1
L1 A1 A2 B1 B 2 RT ,k
§
· x
O
O
T1 ¨ T20 1 T10 ¸
1 T10 .
O2
©
¹ L1 L2 O 2
ɗɬɢ ɭɪɚɜɧɟɧɢɹ ɞɚɸɬ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɬɨɱɤɟ ɫɬɟɧɵ
ɡɞɚɧɢɹ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. Ʉɚɤɢɟ ɬɢɩɵ ɢɫɬɨɱɧɢɤɨɜ ɷɧɟɪɝɢɢ ȼɵ ɡɧɚɟɬɟ?
2. ɇɚɡɨɜɢɬɟ ɨɫɧɨɜɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ.
3. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɡɚɤɨɧ Ɏɭɪɶɟ.
60
4. Ɉɩɢɲɢɬɟ ɮɢɡɢɱɟɫɤɢɟ ɦɟɯɚɧɢɡɦɵ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɜ ɝɚɡɚɯ, ɠɢɞɤɨɫɬɹɯ ɢ ɬɜɟɪɞɵɯ ɬɟɥɚɯ.
5. Ⱦɚɣɬɟ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
6. Ʉɚɤɢɟ ȼɵ ɡɧɚɟɬɟ ɜɢɞɵ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɜ ɡɚɞɚɱɚɯ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ?
7. Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɨɞɧɨɫɥɨɣɧɨɣ ɢ ɦɧɨɝɨɫɥɨɣɧɨɣ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ?
8. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ ɞɥɹ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ?
9. ɑɬɨ ɬɚɤɨɟ «ɤɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ»?
10. Ʉɚɤ ɮɨɪɦɭɥɢɪɭɸɬɫɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɤɨɧɬɚɤɬɟ ɩɪɢ ɧɚɥɢɱɢɢ ɤɨɧɬɚɤɬɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ?
Ɂɚɞɚɧɢɹ
1. ɉɨɥɶɡɭɹɫɶ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨɛ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɚɧɚɥɨɝɢɢ, ɪɚɫɫɱɢɬɚɬɶ ɬɟɩɥɨɜɨɣ
ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɧɤɭ, ɢɡɨɛɪɚɠɟɧɧɭɸ ɧɚ
ɪɢɫ. 2.12, ɩɪɢ ɭɫɥɨɜɢɹɯ:
h 75 ɦɦ;
H 50
ɦɦ;
G1 1 5 0 ɦɦ;
G 2 1 7 ɦɦ; Ti 1 0 0 ɋ; Te 0 ɋ; O1 0,744 ,
O2 1,488 ; O3 1,932 ɤɤɚɥ/(ɦ•ɱɚɫ•ɝɪɚɞ).
2. Ʉɚɤ ɢɡɦɟɧɢɬɫɹ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ
ɷɬɭ ɫɬɟɧɤɭ, ɟɫɥɢ ɦɟɠɞɭ ɦɚɬɟɪɢɚɥɚɦɢ 1 ɢ 2
ɢɦɟɟɬɫɹ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɟɥɢɱɢɧɨɣ 0,1 K/ȼɬ.
Ɋɢɫ. 2.12. ɂɥɥɸɫɬɪɚɰɢɹ
3. ȼ ɡɚɞɚɱɟ ɢɡ ɪɚɡɞɟɥɚ 2.9. ɪɚɫɫɱɢɬɚɬɶ ɡɚɤ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ 1
ɜɢɫɢɦɨɫɬɢ ɩɨɬɨɤɚ ɬɟɩɥɚ ɱɟɪɟɡ ɫɬɟɧɤɭ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɤɨɧɬɚɤɬɟ ɫɬɚɥɢ ɢ ɚɫɛɟɫɬɨɜɨɣ ɢɡɨɥɹɰɢɢ ɩɪɢ ɜɚɪɶɢɪɨɜɚɧɢɢ ɲɢɪɢɧɵ ɩɨɫɥɟɞɧɟɣ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 1 ɞɨ 7 ɫɦ.
61
ɑȺɋɌɖ 3
Ʉɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ ɢ ɬɟɩɥɨɩɟɪɟɞɚɱɚ
3.1. Ʉɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ
ɉɨɞ ɤɨɧɜɟɤɰɢɟɣ ɩɨɧɢɦɚɸɬ ɩɪɨɰɟɫɫ ɩɟɪɟɧɨɫɚ ɜɟɳɟɫɬɜɚ ɜ ɫɪɟɞɟ ɡɚ
ɫɱɟɬ ɩɟɪɟɦɟɳɟɧɢɹ ɦɚɤɪɨɱɚɫɬɢɰ ɠɢɞɤɨɫɬɢ (ɝɚɡɚ). ɉɟɪɟɦɟɳɟɧɢɟ ɷɬɢɯ ɱɚɫɬɢɰ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɧɟɜɨɨɪɭɠɟɧɧɵɦ ɝɥɚɡɨɦ, ɫɞɟɥɚɜ ɢɯ ɜɢɞɢɦɵɦɢ ɫ
ɩɨɦɨɳɶɸ ɤɪɚɫɢɬɟɥɟɣ ɢɥɢ ɞɵɦɚ. ȿɫɬɟɫɬɜɟɧɧɨ, ɱɬɨ ɤɨɧɜɟɤɰɢɹ ɜɨɡɦɨɠɧɚ
ɬɨɥɶɤɨ ɜ ɬɟɤɭɱɟɣ ɫɪɟɞɟ. ɉɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɤɨɧɜɟɤɰɢɟɣ ɜɫɟɝɞɚ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɩɪɢɱɟɦ ɝɥɚɜɧɭɸ ɪɨɥɶ ɜ ɫɨɜɦɟɫɬɧɨɣ ɩɟɪɟɞɚɱɟ ɬɟɩɥɨɬɵ ɢɝɪɚɟɬ ɤɨɧɜɟɤɬɢɜɧɵɣ ɩɟɪɟɧɨɫ.
ɋɨɜɦɟɫɬɧɵɣ ɩɪɨɰɟɫɫ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɤɨɧɜɟɤɰɢɟɣ ɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɧɚɡɵɜɚɟɬɫɹ ɤɨɧɜɟɤɬɢɜɧɵɦ ɬɟɩɥɨɨɛɦɟɧɨɦ.
Ɋɚɡɥɢɱɚɸɬ ɫɜɨɛɨɞɧɭɸ ɢ ɜɵɧɭɠɞɟɧɧɭɸ ɤɨɧɜɟɤɰɢɸ. Ʉɨɧɜɟɤɰɢɹ, ɫɨɡɞɚɜɚɟɦɚɹ ɩɪɢɧɭɞɢɬɟɥɶɧɵɦ ɫɩɨɫɨɛɨɦ (ɦɟɲɚɥɤɨɣ, ɜɟɧɬɢɥɹɬɨɪɨɦ ɢ ɬ.ɞ.)
ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɜɵɧɭɠɞɟɧɧɨɣ. ȿɫɥɢ ɠɟ ɞɜɢɠɟɧɢɟ ɷɥɟɦɟɧɬɨɜ ɨɛɴɟɦɚ ɫɪɟɞɵ ɜɵɡɜɚɧɨ ɧɚɥɢɱɢɟɦ ɜ ɧɟɣ ɬɟɦɩɟɪɚɬɭɪɧɵɯ ɝɪɚɞɢɟɧɬɨɜ, ɚ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɡɧɵɯ ɩɥɨɬɧɨɫɬɟɣ, ɬɨ ɬɚɤɚɹ ɤɨɧɜɟɤɰɢɹ ɧɚɡɵɜɚɟɬɫɹ ɫɜɨɛɨɞɧɨɣ ɢɥɢ
ɟɫɬɟɫɬɜɟɧɧɨɣ. Ɉɧɚ ɫɨɡɞɚɟɬɫɹ ɡɚ ɫɱɟɬ ɬɨɝɨ, ɱɬɨ ɛɨɥɟɟ ɯɨɥɨɞɧɵɟ ɱɚɫɬɢɰɵ
ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ, ɢɦɟɸɳɢɟ ɛɨɥɶɲɭɸ ɩɥɨɬɧɨɫɬɶ, ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɪɚɜɢɬɚɰɢɨɧɧɨɝɨ ɩɨɥɹ Ɂɟɦɥɢ ɨɩɭɫɤɚɸɬɫɹ ɜɧɢɡ, ɚ ɛɨɥɟɟ ɧɚɝɪɟɬɵɟ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɚɪɯɢɦɟɞɨɜɨɣ ɫɢɥɵ ɩɨɞɧɢɦɚɸɬɫɹ ɜɜɟɪɯ. Ɍɢɩɢɱɧɵɦɢ ɩɪɢɦɟɪɚɦɢ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɹɜɥɹɸɬɫɹ ɬɟɩɥɨɨɬɞɚɱɚ ɨɬ ɫɬɟɧ ɢɥɢ ɫ ɤɪɵɲ ɡɞɚɧɢɹ
ɜ ɛɟɡɜɟɬɪɟɧɧɵɣ ɞɟɧɶ; ɤɨɧɜɟɤɰɢɹ ɜ ɫɨɫɭɞɟ ɫ ɠɢɞɤɨɫɬɶɸ, ɜ ɤɨɬɨɪɭɸ ɩɨɝɪɭɠɟɧɚ ɧɚɝɪɟɜɚɬɟɥɶɧɚɹ ɫɩɢɪɚɥɶ (ɤɢɩɹɬɢɥɶɧɢɤ); ɬɟɩɥɨɨɬɞɚɱɚ ɨɬ ɫɨɥɧɟɱɧɨɝɨ ɤɨɥɥɟɤɬɨɪɚ ɜ ɛɟɡɜɟɬɪɟɧɧɭɸ ɩɨɝɨɞɭ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɟɫɬɟɫɬɜɟɧɧɚɹ
ɤɨɧɜɟɤɰɢɹ ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɬɟɩɥɨɨɛɦɟɧɟ ɡɚ ɫɱɟɬ ɬɟɩɥɨɜɨɝɨ ɪɚɫɲɢɪɟɧɢɹ
ɧɚɝɪɟɬɨɣ ɨɤɨɥɨ ɬɟɩɥɨɨɬɞɚɸɳɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɠɢɞɤɨɫɬɢ.
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɪɚɫɲɢɪɟɧɢɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɨɛɴɟɦɧɨɝɨ ɪɚɫɲɢɪɟɧɢɹ
1 § wJ ·
DT
,
(3.1)
¨
¸
J © wT ¹ p const
ɝɞɟ J U 1 – ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ.
Ⱦɥɹ ɢɞɟɚɥɶɧɵɯ ɝɚɡɨɜ ɤɨɷɮɮɢɰɢɟɧɬ ɭɞɟɥɶɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɪɚɫɲɢɪɟɧɢɹ
ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɭɪɚɜɧɟɧɢɟɦ ɫɨɫɬɨɹɧɢɹ ɢɞɟɚɥɶɧɨɝɨ
ɝɚɡɚ (ɭɪɚɜɧɟɧɢɟɦ Ʉɥɚɩɟɣɪɨɧɚ)
pJ m 1RT ,
(3.2)
ɝɞɟ R 8,31 – ɭɧɢɜɟɪɫɚɥɶɧɚɹ ɝɚɡɨɜɚɹ ɩɨɫɬɨɹɧɧɚɹ, m – ɦɨɥɹɪɧɚɹ ɦɚɫɫɚ.
62
ɇɚɯɨɞɢɦ ɢɡ (3.1) ɢ (3.2):
R
1
{ .
Jpm T
Ⱦɥɹ ɤɚɩɟɥɶɧɵɯ ɠɢɞɤɨɫɬɟɣ ɬɟɦɩɟɪɚɬɭɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɨɛɴɟɦɧɨɝɨ
ɪɚɫɲɢɪɟɧɢɹ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɧɶɲɟ, ɱɟɦ ɞɥɹ ɝɚɡɨɜ, ɚ ɢɧɨɝɞɚ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɧɭɥɟɜɵɟ ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ.
ɉɪɢ ɧɟɛɨɥɶɲɨɦ ɩɟɪɟɩɚɞɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɦɟɫɬɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɫɨɨɬɧɨɲɟɧɢɹ (3.1) ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
1 Js J f
1 U f Us
DT |
,
(3.3)
U f Ts T f
J f Ts T f
DT
ɝɞɟ ɢɧɞɟɤɫ " f " – ɨɬɧɨɫɢɬɫɹ ɤ ɠɢɞɤɨɫɬɢ ɜɞɚɥɢ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ; ɢɧɞɟɤɫ
" s" – ɤ ɠɢɞɤɨɫɬɢ ɜɛɥɢɡɢ ɬɟɩɥɨɨɬɞɚɸɳɟɣ ɫɬɟɧɤɢ ( Ts – ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ). ɇɚɥɢɱɢɟ ɪɚɡɧɨɫɬɢ ɩɥɨɬɧɨɫɬɟɣ U f U s ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ
ɧɚ ɥɸɛɨɣ ɟɞɢɧɢɱɧɵɣ ɨɛɴɟɦ ɛɭɞɟɬ ɞɟɣɫɬɜɨɜɚɬɶ ɩɨɞɴɟɦɧɚɹ ɫɢɥɚ, ɪɚɜɧɚɹ
ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɫɭɦɦɟ ɜɵɬɚɥɤɢɜɚɸɳɟɣ ɚɪɯɢɦɟɞɨɜɨɣ ɫɢɥɵ A U f g
( g – ɭɫɤɨɪɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ) ɢ ɫɢɥɵ ɬɹɠɟɫɬɢ G U s g :
F
A G g U f Us
D T U f g Ts T f .
(3.4)
ɉɨɞɴɟɦɧɚɹ ɫɢɥɚ F ɩɟɪɟɦɟɳɚɟɬ ɩɟɪɟɝɪɟɬɭɸ ɠɢɞɤɨɫɬɶ ɜɜɟɪɯ ɛɟɡ ɤɚɤɢɯ-ɥɢɛɨ
ɩɨɛɭɠɞɚɸɳɢɯ ɭɫɬɪɨɣɫɬɜ – ɜɨɡɧɢɤɚɟɬ ɟɫɬɟɫɬɜɟɧɧɚɹ ɤɨɧɜɟɤɰɢɹ (ɪɢɫ. 3.1).
ȼɫɟ ɪɚɫɫɭɠɞɟɧɢɹ ɨ ɜɨɡɧɢɤɧɨɜɟɧɢɢ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɫɩɪɚɜɟɞɥɢɜɵ ɢ ɞɥɹ
ɫɥɭɱɚɹ ɨɯɥɚɠɞɟɧɢɹ ɠɢɞɤɨɫɬɢ ɫ ɬɨɣ ɥɢɲɶ
ɪɚɡɧɢɰɟɣ, ɱɬɨ ɩɨɞɴɟɦɧɚɹ ɫɢɥɚ, ɤɚɤ ɢ ɜɟɤɊɢɫ. 3.1. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨ- ɬɨɪ g , ɛɭɞɟɬ ɧɚɩɪɚɜɥɟɧɚ ɜɧɢɡ, ɬɚɤ ɤɚɤ
ɫɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɨɤɨɥɨ ɜɟɪɬɢɤɚɥɶɧɨɣ
ɬɟɩɥɨɨɬɞɚɸɳɟɣ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɧɨɫɢɬɟɥɹ ɨɤɨɥɨ ɯɨɥɨɞɧɨɣ
ɫɬɟɧɤɢ ɩɪɢ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧ- ɩɨɜɟɪɯɧɨɫɬɢ ɛɭɞɟɬ ɛɨɥɶɲɟ, ɱɟɦ ɜɞɚɥɢ ɨɬ
ɜɟɤɰɢɢ
ɧɟɟ.
Ⱦɜɢɠɟɧɢɸ ɬɟɩɥɨɧɨɫɢɬɟɥɟɣ ɜɛɥɢɡɢ ɩɨɜɟɪɯɧɨɫɬɢ ɜɫɟɝɞɚ ɩɪɨɬɢɜɨɞɟɣɫɬɜɭɟɬ ɫɢɥɚ ɜɧɭɬɪɟɧɧɟɝɨ ɬɪɟɧɢɹ, ɜɨɡɧɢɤɚɸɳɚɹ ɢɡ-ɡɚ ɧɚɥɢɱɢɹ ɭ ɝɚɡɨɜ ɢ
ɠɢɞɤɨɫɬɟɣ ɫɢɥ ɜɹɡɤɨɫɬɢ. Ȼɥɚɝɨɞɚɪɹ ɜɹɡɤɨɦɭ ɬɪɟɧɢɸ, ɬɟɱɟɧɢɟ ɠɢɞɤɨɫɬɢ
ɨɤɨɥɨ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɬɨɪɦɚɠɢɜɚɟɬɫɹ, ɩɨɷɬɨɦɭ, ɧɟ ɫɦɨɬɪɹ ɧɚ ɬɨ, ɱɬɨ ɧɚɢɛɨɥɶɲɢɣ ɩɪɨɝɪɟɜ ɠɢɞɤɨɫɬɢ, ɚ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢ ɜɟɥɢɱɢɧɚ ɩɨɞɴɟɦɧɨɣ ɫɢɥɵ ɩɪɢ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɛɭɞɭɬ ɨɤɨɥɨ ɬɟɩɥɨɩɪɨɜɨɞɹɳɟɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɱɚɫɬɢɰ ɠɢɞɤɨɫɬɢ, «ɩɪɢɥɢɩɲɢɯ» ɤ ɫɚɦɨɣ ɩɨ63
ɜɟɪɯɧɨɫɬɢ, ɪɚɜɧɚ ɧɭɥɸ (ɪɢɫ. 3.1). ɇɭɥɟɜɚɹ ɫɤɨɪɨɫɬɶ ɠɢɞɤɨɫɬɢ ɭ ɫɚɦɨɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɢɦɟɟɬ ɦɟɫɬɨ ɢ ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ.
ɉɨɫɤɨɥɶɤɭ ɫɤɨɪɨɫɬɶ ɠɢɞɤɨɫɬɢ ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɛɨɥɶɲɟ,
ɱɟɦ ɩɪɢ ɫɜɨɛɨɞɧɨɣ, ɬɨ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɦɨɠɟɬ ɛɵɬɶ ɩɟɪɟɞɚɧɨ ɛɨɥɶɲɟ
ɬɟɩɥɚ, ɱɟɦ ɜɨ ɜɬɨɪɨɦ ɩɪɢ ɡɚɞɚɧɧɨɦ ɩɟɪɟɩɚɞɟ ɬɟɦɩɟɪɚɬɭɪ.
ɉɪɨɰɟɫɫ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɦɨɠɟɬ ɛɵɬɶ ɫɬɚɰɢɨɧɚɪɧɵɦ ɢ
ɧɟɫɬɚɰɢɨɧɚɪɧɵɦ. ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɬɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ ɠɢɞɤɨɫɬɢ ɧɟ ɢɡɦɟɧɹɟɬɫɹ ɜɨ ɜɪɟɦɟɧɢ, ɚ ɜɨ ɜɬɨɪɨɦ – ɩɟɪɟɦɟɧɧɨ.
Ɉ. Ɋɟɣɧɨɥɶɞɫ ɜ 1884 ɝɨɞɭ ɭɫɬɚɧɨɜɢɥ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɞɜɭɯ ɪɟɠɢɦɨɜ
ɞɜɢɠɟɧɢɹ ɠɢɞɤɨɫɬɢ, ɨɞɢɧ ɢɡ ɤɨɬɨɪɵɯ ɩɨɥɭɱɢɥ ɧɚɡɜɚɧɢɟ ɥɚɦɢɧɚɪɧɨɝɨ, ɚ
ɞɪɭɝɨɣ – ɬɭɪɛɭɥɟɧɬɧɨɝɨ.
ȿɫɥɢ ɠɢɞɤɨɫɬɶ ɢɥɢ ɝɚɡ ɜɫɬɭɩɚɸɬ ɜ ɤɨɧɬɚɤɬ ɫ ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɜɟɪɞɨɝɨ
ɬɟɥɚ, ɢɦɟɸɳɟɣ ɞɪɭɝɭɸ ɬɟɦɩɟɪɚɬɭɪɭ, ɩɪɨɰɟɫɫ ɨɛɦɟɧɚ ɬɟɩɥɨɜɨɣ ɷɧɟɪɝɢɟɣ
ɢɥɢ ɬɟɩɥɨɦ ɧɚɡɵɜɚɸɬ ɬɟɩɥɨɨɬɞɚɱɟɣ. Ɍɚɤɨɣ ɩɪɨɰɟɫɫ ɱɚɫɬɨ ɜɫɬɪɟɱɚɟɬɫɹ ɜ
ɠɢɡɧɢ, ɧɨ ɩɨɞɪɨɛɧɨ ɨɩɢɫɚɬɶ ɟɝɨ ɦɟɯɚɧɢɡɦ ɞɨɜɨɥɶɧɨ ɫɥɨɠɧɨ.
ɉɪɢ ɥɚɦɢɧɚɪɧɨɦ ɬɟɱɟɧɢɢ ɜɫɟ ɱɚɫɬɢɰɵ ɠɢɞɤɨɫɬɢ ɞɜɢɠɭɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɞɪɭɝ ɞɪɭɝɭ, ɧɟ ɩɟɪɟɦɟɲɢɜɚɹɫɶ ɩɨ ɧɨɪɦɚɥɢ n ɤ ɧɚɩɪɚɜɥɟɧɢɸ
ɞɜɢɠɟɧɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ (ɪɢɫ. 3.2,ɚ). ɂɡ-ɡɚ ɫɪɚɜɧɢɬɟɥɶɧɨ
ɦɚɥɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɠɢɞɤɨɫɬɟɣ (ɢɥɢ ɝɚɡɨɜ) ɬɟɩɥɨɬɚ
ɩɨ ɜɫɟɦɭ ɨɛɴɟɦɭ ɠɢɞɤɨɫɬɢ ɜ ɥɚɦɢɧɚɪɧɨɦ ɩɨɬɨɤɟ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ ɦɟɞɥɟɧɧɨ.
ɉɪɢ ɬɭɪɛɭɥɟɧɬɧɨɦ ɪɟɠɢɦɟ ɱɚɫɬɢɱɤɢ ɠɢɞɤɨɫɬɢ, ɭɱɚɫɬɜɭɹ ɜ ɨɛɳɟɦ
ɩɨɫɬɭɩɚɬɟɥɶɧɨɦ ɞɜɢɠɟɧɢɢ, ɩɟɪɟɦɟɳɚɸɬɫɹ ɯɚɨɬɢɱɧɨ, ɧɟɭɩɨɪɹɞɨɱɟɧɧɨ, ɫ
ɨɛɪɚɡɨɜɚɧɢɟɦ ɜɢɯɪɟɣ ɢ ɩɨɹɜɥɟɧɢɟɦ ɧɟɪɟɝɭɥɹɪɧɨɣ ɩɭɥɶɫɚɰɢɢ ɫɤɨɪɨɫɬɢ,
ɞɚɜɥɟɧɢɹ ɢ ɞɪɭɝɢɯ ɩɚɪɚɦɟɬɪɨɜ. ɑɟɦ ɱɚɳɟ ɨɛɪɚɡɭɸɬɫɹ ɜɢɯɪɢ, ɬɟɦ ɢɧɬɟɧɫɢɜɧɟɟ ɩɟɪɟɦɟɲɢɜɚɧɢɟ ɩɨɬɨɤɚ, ɬɟɦ ɛɨɥɶɲɟ ɟɝɨ ɬɭɪɛɭɥɟɧɬɧɨɫɬɶ. ɉɟɪɟɧɨɫ
ɬɟɩɥɨɬɵ ɜ ɜɨɡɦɭɳɟɧɧɨɦ ɜɢɯɪɹɦɢ ɩɨɬɨɤɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɤɨɧɜɟɤɰɢɟɣ,
ɧɨ ɜɛɥɢɡɢ ɫɬɟɧɤɢ ɷɬɨɝɨ ɧɟ ɧɚɛɥɸɞɚɟɬɫɹ ɢɡ-ɡɚ «ɩɪɢɥɢɩɚɧɢɹ» ɱɚɫɬɢɰ ɠɢɞɤɨɫɬɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ. ɇɚ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ
ɫɢɥ ɜɹɡɤɨɫɬɢ ɮɨɪɦɢɪɭɟɬɫɹ ɬɨɧɤɢɣ ɫɥɨɣ ɡɚɬɨɪɦɨɠɟɧɧɨɣ ɠɢɞɤɨɫɬɢ, ɩɨɥɭɱɢɜɲɢɣ ɧɚɡɜɚɧɢɟ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ (ɬ.ɟ., ɷɬɨ – ɨɛɥɚɫɬɶ ɬɟɱɟɧɢɹ ɨɤɨɥɨ ɨɛɬɟɤɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɜ ɤɨɬɨɪɨɣ ɫɤɨɪɨɫɬɶ ɠɢɞɤɨɫɬɢ ɡɚɦɟɞɥɹɟɬɫɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥ ɜɹɡɤɨɫɬɢ), ɤɨɬɨɪɵɣ ɞɜɢɠɟɬɫɹ ɩɚɪɚɥɥɟɥɶɧɨ ɫɬɟɧɤɟ (ɪɢɫ. 3.2,ɛ). Ƚɢɩɨɬɟɡɚ ɨ ɩɪɢɥɢɩɚɧɢɢ ɠɢɞɤɨɫɬɢ ɤ ɫɬɟɧɤɟ
ɢ ɨɛɪɚɡɨɜɚɧɢɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɛɵɥɚ ɪɚɡɪɚɛɨɬɚɧɚ Ʌ. ɉɪɚɧɞɬɥɟɦ ɜ
1904 ɝɨɞɭ. ɋɤɨɪɨɫɬɶ ɜɧɭɬɪɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ ɢɡɦɟɧɹɟɬɫɹ ɨɬ ɧɭɥɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɞɨ ɫɤɨɪɨɫɬɢ
ɜɧɟɲɧɟɝɨ ɩɨɬɨɤɚ, ɬ.ɟ. ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɛɨɥɶɲɢɦ ɩɨɩɟɪɟɱɧɵɦ ɝɪɚɞɢɟɧɬɨɦ. ȼ ɩɪɟɞɟɥɚɯ ɷɬɨɝɨ ɫɥɨɹ ɩɟɪɟɧɨɫ ɬɟɩɥɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɟɩɥɨɩɪɨɜɨɞ64
ɧɨɫɬɶɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɯɚɪɚɤɬɟɪ ɞɜɢɠɟɧɢɹ ɠɢɞɤɨɫɬɢ ɩɪɟɞɨɩɪɟɞɟɥɹɟɬ
ɦɟɯɚɧɢɡɦ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ (ɷɧɟɪɝɢɢ) ɜ ɩɨɬɨɤɟ.
ɚ
ɛ
Ɋɢɫ. 3.2. Ɍɟɩɥɨɨɬɞɚɱɚ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɪɟɠɢɦɚɯ ɬɟɱɟɧɢɹ: ɚ) ɩɪɢ ɥɚɦɢɧɚɪɧɨɦ;
ɛ) ɩɪɢ ɬɭɪɛɭɥɟɧɬɧɨɦ12
ɉɨɝɪɚɧɢɱɧɵɦ ɫɥɨɟɦ ɧɚɡɵɜɚɸɬ ɨɛɥɚɫɬɶ ɬɟɱɟɧɢɹ (ɜɛɥɢɡɢ ɫɬɟɧɤɢ)
ɜɹɡɤɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɣ ɠɢɞɤɨɫɬɢ, ɯɚɪɚɤɬɟɪɢɡɭɟɦɭɸ ɦɚɥɨɣ ɬɨɥɳɢɧɨɣ
ɢ ɛɨɥɶɲɢɦ ɩɨɩɟɪɟɱɧɵɦ ɝɪɚɞɢɟɧɬɨɦ ɫɤɨɪɨɫɬɢ ɢɥɢ ɬɟɦɩɟɪɚɬɭɪɵ, ɢɡɦɟɧɟɧɢɟɦ ɤɨɬɨɪɵɯ ɨɛɭɫɥɨɜɥɟɧɵ ɩɪɨɰɟɫɫɵ ɩɟɪɟɧɨɫɚ ɜɟɳɟɫɬɜɚ, ɤɨɥɢɱɟɫɬɜɚ
ɞɜɢɠɟɧɢɹ ɢ ɬɟɩɥɨɬɵ.
ɉɨɞ ɜɧɟɲɧɢɦ ɩɨɬɨɤɨɦ ɩɨɞɪɚɡɭɦɟɜɚɸɬ ɨɛɥɚɫɬɶ ɩɨɬɨɤɚ ɠɢɞɤɨɫɬɢ, ɜ
ɤɨɬɨɪɨɣ ɜɥɢɹɧɢɟ ɫɢɥ ɜɹɡɤɨɫɬɢ ɧɢɱɬɨɠɧɨ ɦɚɥɨ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɢɥɚɦɢ
ɢɧɟɪɰɢɢ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɜ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɫɢɥɵ ɜɹɡɤɨɫɬɢ ɢ ɢɧɟɪɰɢɢ
ɫɨɢɡɦɟɪɢɦɵ.
Ɍɨɥɳɢɧɚ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ G – ɜɟɥɢɱɢɧɚ ɭɫɥɨɜɧɚɹ, ɬɚɤ ɤɚɤ ɩɟɪɟɯɨɞ ɨɬ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɤ ɜɧɟɲɧɟɦɭ ɩɨɬɨɤɭ ɧɟ ɹɜɥɹɟɬɫɹ ɪɟɡɤɢɦ. Ɂɚ ɬɨɥɳɢɧɭ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ G ɩɪɢɧɢɦɚɸɬ ɪɚɫɫɬɨɹɧɢɟ
ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ ɞɨ ɫɥɨɹ ɠɢɞɤɨɫɬɢ, ɫɤɨɪɨɫɬɶ ɤɨɬɨɪɨɝɨ ɨɬɥɢɱɚɟɬɫɹ
ɨɬ ɫɤɨɪɨɫɬɢ ɜɧɟɲɧɟɝɨ ɩɨɬɨɤɚ ɧɚ ɦɚɥɭɸ, ɡɚɪɚɧɟɟ ɡɚɞɚɧɧɭɸ ɜɟɥɢɱɢɧɭ.
Ɍɨɥɳɢɧɚ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜɨɡɪɚɫɬɚɟɬ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɪɚɫɫɬɨɹɧɢɹ ɨɬ
ɩɟɪɟɞɧɟɣ ɤɪɨɦɤɢ. ɇɚ ɧɟɤɨɬɨɪɨɦ ɤɪɢɬɢɱɟɫɤɨɦ ɪɚɫɫɬɨɹɧɢɢ ɜɥɢɹɧɢɟ ɢɧɟɪɰɢɨɧɧɵɯ ɫɢɥ ɫɬɚɧɨɜɢɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ ɛɨɥɶɲɢɦ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɫɢɥɚɦɢ
ɜɹɡɤɨɫɬɢ, ɢ ɜ ɩɨɬɨɤɟ ɧɚɱɢɧɚɸɬ ɜɨɡɪɚɫɬɚɬɶ ɫɥɚɛɵɟ ɜɨɡɦɭɳɟɧɢɹ. ɉɨɫɬɟɩɟɧɧɨ ɨɧɢ ɭɫɢɥɢɜɚɸɬɫɹ, ɢ ɭɩɨɪɹɞɨɱɟɧɧɵɣ ɪɟɠɢɦ ɜɹɡɤɨɝɨ ɬɟɱɟɧɢɹ (ɥɚɦɢɧɚɪɧɵɣ) ɫɦɟɧɹɟɬɫɹ ɬɭɪɛɭɥɟɧɬɧɵɦ ɪɟɠɢɦɨɦ. ȼ ɨɛɥɚɫɬɢ ɬɭɪɛɭɥɟɧɬɧɨ-
12
Ʌɚɲɭɬɢɧɚ ɇ.Ƚ., Ɇɚɤɚɲɨɜɚ Ɉ.ȼ., Ɇɟɞɜɟɞɟɜ Ɋ.Ɇ. Ɍɟɯɧɢɱɟɫɤɚɹ ɬɟɪɦɨɞɢɧɚɦɢɤɚ ɫ ɨɫɧɨɜɚɦɢ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɢ ɝɢɞɪɚɜɥɢɤɢ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1988. 336 c.
65
ɝɨ ɬɟɱɟɧɢɹ ɩɪɨɢɫɯɨɞɢɬ ɢɧɬɟɧɫɢɜɧɵɣ ɩɟɪɟɧɨɫ ɤɚɤ ɬɟɩɥɨɜɨɣ ɷɧɟɪɝɢɢ, ɬɚɤ
ɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ.
ɉɪɢ ɬɟɩɥɨɨɛɦɟɧɟ ɦɟɠɞɭ ɫɬɟɧɤɨɣ ɢ ɫɪɟɞɨɣ ɜ ɨɛɥɚɫɬɢ, ɝɪɚɧɢɱɚɳɟɣ ɫ
ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɟɥɚ, ɜɨɡɧɢɤɚɟɬ ɬɟɩɥɨɜɨɣ ɩɨɝɪɚɧɢɱɧɵɣ ɫɥɨɣ, ɩɪɟɞɫɬɚɜɥɹɸɳɢɣ ɫɨɛɨɣ ɩɪɢɫɬɟɧɧɵɣ ɫɥɨɣ ɠɢɞɤɨɫɬɢ, ɜ ɤɨɬɨɪɨɦ ɬɟɦɩɟɪɚɬɭɪɚ ɦɟɧɹɟɬɫɹ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɫɬɟɧɤɢ Ts ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɟɲɧɟɝɨ ɩɨɬɨɤɚ Te .
Ɍɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ 13 rT G O ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜɨ ɦɧɨɝɨ
ɪɚɡ ɩɪɟɜɵɲɚɟɬ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɭɪɛɭɥɟɧɬɧɨɝɨ ɜɧɟɲɧɟɝɨ
ɩɨɬɨɤɚ ɢ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɹɸɳɢɦ ɜ ɩɪɨɰɟɫɫɚɯ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ. ɂɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ Te ɞɨ Ts ɩɪɨɢɫɯɨɞɢɬ, ɜ ɨɫɧɨɜɧɨɦ, ɜ
ɩɪɟɞɟɥɚɯ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ.
ɂ. ɇɶɸɬɨɧ ɜɩɟɪɜɵɟ ɨɛɪɚɬɢɥ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɹɜɥɹɟɬɫɹ ɪɟɲɚɸɳɢɦ ɮɚɤɬɨɪɨɦ ɜ ɩɪɨɰɟɫɫɟ ɬɟɩɥɨɨɛɦɟɧɚ ɦɟɠɞɭ ɬɟɥɨɦ ɢ ɫɪɟɞɨɣ. ȼ XVIII ɜɟɤɟ Ƚ. Ɋɢɯɦɚɧ ɩɟɪɜɵɦ ɞɚɥ ɨɛɫɬɨɹɬɟɥɶɧɵɣ ɚɧɚɥɢɡ
ɩɪɨɰɟɫɫɨɜ ɨɯɥɚɠɞɟɧɢɹ ɧɚɝɪɟɬɵɯ ɬɟɥ ɜ ɜɨɡɞɭɯɟ ɢ ɩɨɤɚɡɚɥ ɢɯ ɡɚɜɢɫɢɦɨɫɬɶ
ɧɟ ɬɨɥɶɤɨ ɨɬ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ, ɧɨ ɢ ɨɬ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɨɛɴɟɦɚ ɬɟɥɚ. ɉɨɫɥɟɞɭɸɳɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɜɵɹɜɢɥɢ ɛɨɥɶɲɭɸ ɫɥɨɠɧɨɫɬɶ
ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨɨɛɦɟɧɚ, ɬɟɫɧɨ ɩɟɪɟɩɥɟɬɚɸɳɢɯɫɹ ɫ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɦɢ
ɩɪɨɰɟɫɫɚɦɢ. Ȼɵɥɨ ɧɚɣɞɟɧɨ, ɱɬɨ ɜ ɩɪɨɰɟɫɫɟ ɬɟɩɥɨɨɛɦɟɧɚ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɨɬɞɚɜɚɟɦɨɣ ɢɥɢ ɩɨɥɭɱɚɟɦɨɣ ɬɟɥɨɦ ɨɬ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ, ɩɪɹɦɨ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ F , ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ
Te ɢ Ts , ɞɥɢɬɟɥɶɧɨɫɬɢ ɩɪɨɰɟɫɫɚ, ɚ ɬɚɤɠɟ ɡɚɜɢɫɢɬ ɨɬ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ
ɫɪɟɞɵ, ɯɚɪɚɤɬɟɪɚ ɟɟ ɞɜɢɠɟɧɢɹ, ɮɨɪɦɵ ɬɟɥɚ ɢ ɟɝɨ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɪɚɡɦɟɪɨɜ. Ⱦɥɹ ɷɥɟɦɟɧɬɚɪɧɨɣ ɩɥɨɳɚɞɤɢ ɢ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɜɪɟɦɟɧɢ ɩɪɨɰɟɫɫ
ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
dQ W D Te Ts dFd W ,
(3.5)
ɧɚɡɜɚɧɧɵɦ ɨɫɧɨɜɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɢɥɢ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ–Ɋɢɯɦɚɧɚ. ȼ ɭɪɚɜɧɟɧɢɢ (3.5) Te Ts – ɬɟɦɩɟɪɚɬɭɪɧɵɣ
ɧɚɩɨɪ; D – ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ, ɧɚɡɜɚɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɬɞɚɱɢ, ȼɬ/(ɦ2•Ʉ).
Ⱦɥɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɨɛɦɟɧɚ ɩɪɢ ɧɟɢɡɦɟɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɟ ɫɪɟɞɵ ɢ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ
Q D Te Ts F ,
(3.6)
ɚ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ –
q Q F D Te T s (3.7)
13
ȼ ɱɚɫɬɢ 2 ɜɜɟɞɟɧɨ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ RT
66
L O F .
ɂɡ (3.7) ɢ (3.5) ɢɦɟɟɦ
dQ W
q
,
Te Ts dFd W Te Ts ɬ.ɟ. ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɪɚɜɟɧ ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɨɬɵ, ɜɨɫɩɪɢɧɢɦɚɟɦɨɣ (ɢɥɢ ɨɬɞɚɜɚɟɦɨɣ) ɟɞɢɧɢɰɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɩɪɢ
ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɞɜɢɠɭɳɟɣɫɹ ɫɪɟɞɨɣ ɜ 1 Ʉ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɨɛɦɟɧɚ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɶɸ ɬɟɥɚ ɢ ɨɦɵɜɚɸɳɟɣ ɫɪɟɞɨɣ ɢ ɭɱɢɬɵɜɚɟɬ ɤɨɧɤɪɟɬɧɵɟ ɭɫɥɨɜɢɹ ɩɪɨɬɟɤɚɧɢɹ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ. ȼ ɨɬɥɢɱɢɟ ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɧɟ ɹɜɥɹɟɬɫɹ ɬɟɩɥɨɮɢɡɢɱɟɫɤɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɜɟɳɟɫɬɜɚ, ɢ ɟɝɨ
ɡɧɚɱɟɧɢɹ ɧɟ ɩɪɢɜɨɞɹɬɫɹ ɜ ɫɩɪɚɜɨɱɧɢɤɚɯ.
ɇɚɢɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɜɟɥɢɱɢɧɭ D ɨɤɚɡɵɜɚɸɬ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O , ɭɞɟɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ c , ɩɥɨɬɧɨɫɬɶ U ,
ɜɹɡɤɨɫɬɶ ɠɢɞɤɨɫɬɢ (ɞɢɧɚɦɢɱɟɫɤɚɹ P ɢɥɢ ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ Q P U ), ɤɨɷɮɮɢɰɢɟɧɬ ɨɛɴɟɦɧɨɝɨ ɪɚɫɲɢɪɟɧɢɹ DT .
ȼɫɟ ɪɟɚɥɶɧɵɟ ɠɢɞɤɨɫɬɢ ɨɛɥɚɞɚɸɬ ɜɹɡɤɨɫɬɶɸ. ȼɹɡɤɨɫɬɶ – ɫɜɨɣɫɬɜɨ
ɠɢɞɤɨɫɬɢ ɨɤɚɡɵɜɚɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɟɪɟɦɟɳɟɧɢɸ (ɫɞɜɢɝɭ) ɟɟ ɫɥɨɟɜ.
Ⱦɢɧɚɦɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ P ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɛɨɣ ɫɢɥɭ, ɤɨɬɨɪɚɹ ɜɨɡɧɢɤɚɟɬ
ɧɚ ɤɜɚɞɪɚɬɧɨɦ ɦɟɬɪɟ ɩɨɜɟɪɯɧɨɫɬɢ ɞɜɭɯ ɩɟɪɟɦɟɳɚɸɳɢɯɫɹ ɞɪɭɝ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɚ ɫɥɨɟɜ ɠɢɞɤɨɫɬɢ ɩɪɢ ɝɪɚɞɢɟɧɬɟ ɫɤɨɪɨɫɬɢ d w d n 1 .
D
Ɍɚɛɥɢɰɚ 3.1
Ɂɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ
ȼɢɞ ɤɨɧɜɟɤɰɢɢ ɢ ɫɪɟɞɚ
ɋɜɨɛɨɞɧɚɹ ɤɨɧɜɟɤɰɢɹ, ɜɨɡɞɭɯ
ɋɜɨɛɨɞɧɚɹ ɤɨɧɜɟɤɰɢɹ, ɜɨɞɚ
ȼɵɧɭɠɞɟɧɧɚɹ ɤɨɧɜɟɤɰɢɹ, ɜɨɡɞɭɯ
ȼɵɧɭɠɞɟɧɧɚɹ ɤɨɧɜɟɤɰɢɹ, ɜɨɞɚ
Ʉɢɩɹɳɚɹ ɜɨɞɚ
Ʉɨɧɞɟɧɫɢɪɭɸɳɢɣɫɹ ɜɨɞɹɧɨɣ ɩɚɪ
D , ȼɬ/(ɦ2.Ʉ)
5–25
20–100
10–200
50–10 000
3000–100 000
5000–100 000
ɋɨɨɬɧɨɲɟɧɢɟ (3.7) ɫɥɭɠɢɬ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɛɦɟɧɚ. ɉɪɢ ɤɚɠɭɳɟɣɫɹ ɩɪɨɫɬɨɬɟ ɫɨɨɬɧɨɲɟɧɢɹ, ɧɚɣɬɢ D – ɞɨɜɨɥɶɧɨ ɬɪɭɞɧɚɹ ɡɚɞɚɱɚ (ɩɪɢɦɟɪɵ ɫɦ. ɜ ɪɚɡɞɟɥɟ 3.3). Ⱥɧɚɥɢɬɢɱɟɫɤɨɟ ɨɩɪɟɞɟɥɟɧɢɟ D
ɧɚɬɚɥɤɢɜɚɟɬɫɹ ɧɚ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ, ɤɨɬɨɪɵɟ ɢɧɨɝɞɚ ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɩɪɟɨɞɨɥɢɦɵɦɢ. Ɋɟɡɭɥɶɬɚɬɵ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ ɨɩɪɟɞɟɥɟɧɢɹ D
ɫɩɪɚɜɟɞɥɢɜɵ ɬɨɥɶɤɨ ɞɥɹ ɞɚɧɧɨɝɨ ɤɨɧɤɪɟɬɧɨɝɨ ɫɥɭɱɚɹ. ȼ ɬɚɛɥɢɰɟ 3.1.
67
ɭɤɚɡɚɧɵ ɧɟɤɨɬɨɪɵɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ, ɜɤɥɸɱɚɹ ɫɥɭɱɚɢ ɤɢɩɟɧɢɹ ɢ ɤɨɧɞɟɧɫɚɰɢɢ, ɤɨɬɨɪɵɟ
ɨɛɵɱɧɨ ɨɬɧɨɫɹɬ ɤ ɨɛɥɚɫɬɢ ɤɨɧɜɟɤɰɢɢ.
Ɉɞɧɨɣ ɢɡ ɨɫɧɨɜɧɵɯ ɡɚɞɚɱ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɢ ɹɜɥɹɟɬɫɹ
ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ. ɋ ɧɟɤɨɬɨɪɵɦɢ ɩɪɢɦɟɪɚɦɢ
ɪɚɫɱɟɬɧɵɯ (ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɢɥɢ ɨɩɪɟɞɟɥɟɧɧɵɯ ɧɚ ɨɫɧɨɜɟ ɷɤɫɩɟɪɢɦɟɧɬɚ)
ɮɨɪɦɭɥ ɦɵ ɞɚɥɟɟ ɩɨɡɧɚɤɨɦɢɦɫɹ.
ɚ
ɍɤɚɠɟɦ ɨɫɧɨɜɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ
ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɤ ɩɨɬɨɤɭ ɠɢɞɤɨɫɬɢ. ɇɚ ɪɢɫ. 3.3, ɚ ɩɨɤɚɡɚɧɚ
ɧɚɝɪɟɬɚɹ ɩɥɨɫɤɚɹ ɩɥɚɫɬɢɧɚ, ɨɯɥɚɠɞɚɟɦɚɹ ɨɛɬɟɤɚɸɳɢɦ ɟɟ ɜɨɡɞɭɲɧɵɦ ɩɨɬɨɤɨɦ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɨɤɚɡɚɧɵ ɩɪɨɮɢɥɢ
ɫɤɨɪɨɫɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɵ. ɉɪɟɠɞɟ ɜɫɟɝɨ, ɨɬɦɟɬɢɦ, ɱɬɨ ɢɡ-ɡɚ ɞɟɣɫɬɜɢɹ ɫɢɥ
ɜɹɡɤɨɫɬɢ ɫɤɨɪɨɫɬɶ w y ɭɦɟɧɶɲɚɟɬɫɹ
ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɤ ɫɬɟɧɤɟ. ɉɨɫɤɨɥɶɤɭ
ɫɤɨɪɨɫɬɶ ɫɥɨɹ ɠɢɞɤɨɫɬɢ, ɩɪɢɦɵɤɚɸɳɟɝɨ ɤ ɫɬɟɧɤɟ, ɪɚɜɧɚ ɧɭɥɸ, ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ (ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ
ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ) ɨɬ ɫɬɟɧɤɢ ɤ ɷɬɨɦɭ
ɫɥɨɸ ɠɢɞɤɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ
wT
Q
D Ts T f ,f (3.8)
q O f
wy y 0
F
ɛ
Ɋɢɫ. 3.3. Ʉ ɨɩɢɫɚɧɢɸ ɨɫɨɛɟɧɧɨɫɬɟɣ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɩɟɪɟɧɨɫɚ: ɚ) ɩɪɨɮɢɥɢ ɫɤɨɪɨɫɬɢ ɢ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɣ
ɤɨɧɜɟɤɰɢɢ ɨɤɨɥɨ ɧɚɝɪɟɬɨɣ ɩɥɚɫɬɢɧɵ; ɛ) ɩɪɨɮɢɥɢ ɫɤɨɪɨɫɬɢ ɢ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ ɫɜɨɛɨɞɧɨɣ
ɤɨɧɜɟɤɰɢɢ ɨɤɨɥɨ ɧɚɝɪɟɬɨɣ ɩɥɚɫɬɢɧɵ, ɨɬɤɥɨɧɟɧɧɨɣ ɨɬ ɝɨɪɢɡɨɧɬɚɥɢ ɧɚ ɭɝɨɥ E
ɏɨɬɹ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɩɪɨɰɟɫɫɚ ɫ
ɬɚɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ,
ɱɬɨ ɩɟɪɟɞɚɱɚ ɬɟɩɥɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɝɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɫɬɟɧɤɟ w T w y ɨɩɪɟɞɟɥɹy 0
ɟɬɫɹ ɫɤɨɪɨɫɬɶɸ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɠɢɞɤɨɫɬɶɸ ɨɬ ɫɬɟɧɤɢ ɜ ɨɫɧɨɜɧɨɣ ɩɨɬɨɤ. ɉɨɷɬɨɦɭ ɝɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɫɬɟɧɤɟ
ɡɚɜɢɫɢɬ ɨɬ ɩɨɥɹ ɬɟɱɟɧɢɹ, ɢ ɱɟɦ ɜɵɲɟ
ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ, ɬɟɦ ɛɨɥɶɲɟ ɢ ɝɪɚɞɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɵ, ɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ. ȼ ɬɨ ɠɟ ɜɪɟɦɹ ɫɭɳɟɫɬɜɟɧɧɭɸ ɪɨɥɶ ɢɝɪɚɟɬ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɠɢɞɤɨɫɬɢ. ɇɚɩɪɢɦɟɪ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɨɞɵ
68
ɩɪɢɦɟɪɧɨ ɧɚ ɩɨɪɹɞɨɤ ɛɨɥɶɲɟ, ɱɟɦ ɜɨɡɞɭɯɚ. ɉɨɷɬɨɦɭ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɞɥɹ ɜɨɞɵ ɛɨɥɶɲɟ, ɱɟɦ ɞɥɹ ɜɨɡɞɭɯɚ.
Ⱥɧɚɥɨɝɢɱɧɵɟ ɨɫɨɛɟɧɧɨɫɬɢ ɢɦɟɟɬ ɢ ɫɜɨɛɨɞɧɚɹ ɤɨɧɜɟɤɰɢɹ (ɪɢɫ. 3.3,ɛ;
ɪɢɫ. 3.1). Ɉɫɧɨɜɧɨɟ ɨɬɥɢɱɢɟ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ. ȼ ɭɫɥɨɜɢɹɯ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɫɤɨɪɨɫɬɶ ɩɪɢ ɭɞɚɥɟɧɢɢ ɨɬ ɫɬɟɧɤɢ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ
ɫɤɨɪɨɫɬɢ ɧɚɛɟɝɚɸɳɟɝɨ ɩɨɬɨɤɚ, ɨɛɭɫɥɨɜɥɟɧɧɨɣ ɜɧɟɲɧɟɣ ɫɢɥɨɣ. ȼ ɭɫɥɨɜɢɹɯ ɫɜɨɛɨɞɧɨɣ ɤɨɧɜɟɤɰɢɢ ɫɤɨɪɨɫɬɶ ɩɪɢ ɭɞɚɥɟɧɢɢ ɨɬ ɩɥɚɫɬɢɧɵ ɫɧɚɱɚɥɚ
ɜɨɡɪɚɫɬɚɟɬ, ɚ ɡɚɬɟɦ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɜɹɡɤɨɫɬɢ ɞɨɜɨɥɶɧɨ ɛɵɫɬɪɨ ɫɧɢɠɚɟɬɫɹ
ɞɨ ɧɭɥɹ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɪɚɡɧɨɫɬɶ ɩɥɨɬɧɨɫɬɟɣ ɦɟɧɹɟɬɫɹ ɦɟɞɥɟɧɧɟɟ. Ɉɞɧɚɤɨ, ɜ ɤɨɧɰɟ ɤɨɧɰɨɜ, ɩɨɞɴɟɦɧɚɹ ɫɢɥɚ ɬɚɤɠɟ ɭɦɟɧɶɲɚɟɬɫɹ, ɩɨ ɦɟɪɟ ɬɨɝɨ,
ɤɚɤ ɩɥɨɬɧɨɫɬɶ ɠɢɞɤɨɫɬɢ ɩɪɢɛɥɢɠɚɟɬɫɹ ɤ ɩɥɨɬɧɨɫɬɢ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ;
ɷɬɨ ɜɵɡɵɜɚɟɬ ɩɨɜɵɲɟɧɢɟ ɫɤɨɪɨɫɬɢ ɞɨ ɧɟɤɨɬɨɪɨɝɨ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ, ɚ ɡɚɬɟɦ ɟɟ ɩɚɞɟɧɢɟ ɞɨ ɧɭɥɹ ɧɚ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ
ɧɚɝɪɟɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. Ɍɟɦɩɟɪɚɬɭɪɧɵɟ ɩɨɥɹ ɩɪɢ ɫɜɨɛɨɞɧɨɣ ɢ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɚɧɚɥɨɝɢɱɧɵ ɩɨ ɮɨɪɦɟ, ɢ ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɦɟɯɚɧɢɡɦɨɦ
ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɬɜɟɪɞɵɦ
ɬɟɥɨɦ ɹɜɥɹɟɬɫɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ.
ɉɪɢ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɞɜɢɠɟɧɢɟ ɫɪɟɞɵ ɨɛɵɱɧɨ ɫɨɡɞɚɟɬɫɹ ɧɚɫɨɫɨɦ ɢɥɢ ɜɟɧɬɢɥɹɬɨɪɨɦ ɢ ɟɝɨ ɫɤɨɪɨɫɬɶ ɦɨɠɧɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɡɦɟɪɢɬɶ. ɉɪɢ ɫɜɨɛɨɞɧɨɣ ɤɨɧɜɟɤɰɢɢ ɫɤɨɪɨɫɬɶ ɡɚɜɢɫɢɬ ɨɬ ɩɟɪɟɩɚɞɚ ɬɟɦɩɟɪɚɬɭɪ ɦɟɠɞɭ ɫɬɟɧɤɨɣ ɢ ɠɢɞɤɨɫɬɶɸ, ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɜɨɝɨ ɪɚɫɲɢɪɟɧɢɹ
ɠɢɞɤɨɫɬɢ (ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɹɟɬ ɢɡɦɟɧɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɧɚ ɟɞɢɧɢɰɭ ɩɟɪɟɩɚɞɚ ɬɟɦɩɟɪɚɬɭɪɵ) ɢ ɫɢɥɨɜɨɝɨ ɩɨɥɹ, ɤɨɬɨɪɨɟ ɜ ɫɢɫɬɟɦɚɯ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɧɚ Ɂɟɦɥɟ, ɨɛɭɫɥɨɜɥɟɧɨ ɩɪɨɫɬɨ ɫɢɥɨɣ ɬɹɠɟɫɬɢ.
3.2. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ
ɤ ɨ ɧ ɜ ɟ ɤ ɬ ɢ ɜ ɧ ɨ ɝ ɨ ɬ ɟ ɩ ɥ ɨ ɩ ɟ ɪ ɟ ɧ ɨ ɫ ɚ 14
ɉɨɫɤɨɥɶɤɭ ɤɨɧɜɟɤɬɢɜɧɵɣ ɩɟɪɟɧɨɫ ɬɟɩɥɚ ɜɫɟɝɞɚ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɩɟɪɟɧɨɫɟ ɦɚɫɫɵ ɢ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ, ɜ ɞɚɧɧɨɦ ɪɚɡɞɟɥɟ
ɩɪɟɞɫɬɚɜɥɟɧ ɨɞɢɧ ɢɡ ɦɧɨɝɨɱɢɫɥɟɧɧɵɯ ɫɩɨɫɨɛɨɜ ɜɵɜɨɞɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɭɪɚɜɧɟɧɢɣ. ɍɪɚɜɧɟɧɢɹ, ɜɵɪɚɠɚɸɳɢɟ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ ɦɚɫɫɵ,
ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɢ ɷɧɟɪɝɢɢ, ɦɨɠɧɨ ɜɵɜɟɫɬɢ, ɚɧɚɥɢɡɢɪɭɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɩɨɬɨɤɚ, ɜɬɟɤɚɸɳɟɝɨ ɢ ɜɵɬɟɤɚɸɳɟɝɨ ɢɡ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɝɨ ɤɨɧɬɪɨɥɶɧɨɝɨ ɨɛɴɟɦɚ, ɩɨɤɚɡɚɧɧɨɝɨ ɧɚ ɪɢɫ. 3.4. Ɂɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɜɟɥɢɱɢɧɵ A ɞɥɹ ɤɨɧɬɪɨɥɶɧɨɝɨ ɨɛɴɟɦɚ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
14
ȿɫɥɢ ɜɨɡɧɢɤɚɸɬ ɬɪɭɞɧɨɫɬɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɯɚɪɚɤɬɟɪɚ, ɩɪɢ ɢɡɭɱɟɧɢɢ ɤɭɪɫɚ ɩɪɢ
ɩɟɪɜɨɦ ɱɬɟɧɢɢ ɷɬɨɬ ɢ ɩɨɫɥɟɞɭɸɳɢɣ ɩɚɪɚɝɪɚɮɵ ɦɨɝɭɬ ɛɵɬɶ ɨɩɭɳɟɧɵ.
69
ɇɚɩɪɢɦɟɪ, ɱɬɨɛɵ ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟɧɢɟ ɧɟɪɚɡɪɵɜɧɨɫɬɢ, ɩɪɢɦɟɦ
A = ɩɥɨɬɧɨɫɬɶ × ɨɛɴɟɦ
ɢɥɢ
A U dx dy dz .
ɂɡɦɟɧɟɧɢɟ ɜɨ ɜɪɟɦɟɧɢ ɷɬɨɣ ɜɟɥɢɱɢɧɵ ɟɫɬɶ
wU
dx dy dz .
wt
ɉɨɬɨɤ ɦɚɫɫɵ ɱɟɪɟɡ ɝɪɚɧɶ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɧɚɩɪɚɜɥɟɧɢɸ Ox , ɪɚɜɟɧ ɩɪɨɢɡɜɟɞɟɧɢɸ ɩɥɨɬɧɨɫɬɢ ɧɚ ɫɨɫɬɚɜɥɹɸɳɭɸ ɫɤɨɪɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɭɸ ɝɪɚɧɢ, ɢ ɧɚ ɩɥɨɳɚɞɶ ɝɪɚɧɢ. Ɍɨɝɞɚ ɞɥɹ ɝɪɚɧɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ
ɬɨɱɤɭ P , ɢɦɟɟɦ
U udydz .
Ⱦɥɹ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɝɪɚɧɢ, ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɬɨɱɤɭ Q , ɩɨɬɨɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɜɟɧɫɬɜɨɦ
ª
w U u º
u
dx » dydz .
U
«
w
x
¬
¼
Ɋɢɫ. 3.4. Ȼɟɫɤɨɧɟɱɧɨ ɦɚɥɵɣ ɤɨɧɬɪɨɥɶɧɵɣ ɨɛɴɟɦ
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɡɭɥɶɬɢɪɭɸɳɢɣ ɩɨɬɨɤ ɦɚɫɫɵ ɜ ɷɬɨɦ ɧɚɩɪɚɜɥɟɧɢɢ
ɟɫɬɶ
w U u dxdydz .
wx
Ⱥɧɚɥɨɝɢɱɧɨ ɞɥɹ ɞɜɭɯ ɞɪɭɝɢɯ ɧɚɩɪɚɜɥɟɧɢɣ ɧɚɣɞɟɦ
70
w U w
dxdydz .
wz
wy
Ɇɚɫɫɚ ɜ ɨɛɴɟɦɟ ɧɟ ɩɪɨɢɡɜɨɞɢɬɫɹ, ɬ.ɟ. ɫɭɦɦɚ ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ
ɦɚɫɫɵ ɜ ɨɛɴɟɦɟ ɪɚɜɧɚ ɧɭɥɸ.
ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɵɟ ɜɵɪɚɠɟɧɢɹ ɜ ɨɛɳɟɟ ɭɪɚɜɧɟɧɢɟ «ɫɨɯɪɚɧɟɧɢɹ» ɜɟɥɢɱɢɧɵ Ⱥ, ɧɚɣɞɟɦ
ª w U u w U v w U w º
wU
d xd y d z «
» d x dy d z .
wt
w
w
w
x
y
z
¬
¼
ɉɨɫɤɨɥɶɤɭ ɜɵɞɟɥɟɧɧɵɣ ɨɛɴɟɦ – ɩɪɨɢɡɜɨɥɶɧɵɣ, ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
ª w U u w U v w U w º
wU
«
(3.9)
»
x
y
z
wt
w
w
w
¬
¼
ɢɥɢ
wU
div Uu 0 ,
(3.10)
wt
ɝɞɟ u - ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɫ ɤɨɦɩɨɧɟɧɬɚɦɢ u , v , w .
ȼɟɥɢɱɢɧɭ Uu ɧɚɡɵɜɚɸɬ ɩɥɨɬɧɨɫɬɶɸ ɩɨɬɨɤɚ ɠɢɞɤɨɫɬɢ (ɝɚɡɚ).
ɑɚɫɬɧɵɣ ɫɥɭɱɚɣ ɭɪɚɜɧɟɧɢɹ ɧɟɪɚɡɪɵɜɧɨɫɬɢ ɩɨɥɭɱɚɟɬɫɹ ɢɡ (3.9) ɢɥɢ
(3.10) ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɠɢɞɤɨɫɬɶ (ɝɚɡ) – ɧɟɫɠɢɦɚɟɦɵ, ɬ.ɟ.
wu wv ww
div u 0 ɢɥɢ
(3.11)
0.
wx wy wz
ɂɦɟɟɦ ɢɡ (3.10)
wU
wU
div Uu U div u u grad U wt
wt
(3.12)
wU
wU
wU
wU dU
u v w
0.
wt
wx
wy
wz dt
Ɋɚɫɫɦɚɬɪɢɜɚɹ ɚɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɤɨɧɬɪɨɥɶɧɵɣ ɨɛɴɟɦ, ɦɨɠɧɨ ɜɵɜɟɫɬɢ ɭɪɚɜɧɟɧɢɹ ɫɨɯɪɚɧɟɧɢɹ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ.
ȼ ɫɥɭɱɚɟ ɧɟɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɹ ɗɣɥɟɪɚ
ª wu
º
(3.13)
U« u grad u » grad p ,
t
w
¬
¼
ɜɩɟɪɜɵɟ ɭɫɬɚɧɨɜɥɟɧɧɵɟ Ʌ. ɗɣɥɟɪɨɦ ɜ 1755 ɝɨɞɭ.
ɉɪɟɞɫɬɚɜɢɦ ɛɨɥɟɟ ɩɨɞɪɨɛɧɭɸ ɡɚɩɢɫɶ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɗɣɥɟɪɚ:
du wu
wu
wu
wu
1 wp
,
u
v
w
dt wt
wx
wy
wz
U wx
dv wv
wv
wv
wv
1 wp
,
(3.14)
u
v
w
dt wt
wx
wy
wz
U wy
w U v dxdydz ɢ
71
dw ww
ww
ww
ww
1 wp
.
u
v
w
dt wt
wx
wy
wz
U wz
ɉɪɨɫɬɟɣɲɢɦ ɩɪɢɦɟɪɨɦ ɨɛɴɟɦɧɵɯ ɫɢɥ ɹɜɥɹɟɬɫɹ ɫɢɥɚ ɬɹɠɟɫɬɢ. ɀɢɞɤɨɫɬɶ, ɧɚɯɨɞɹɳɚɹɫɹ ɜ ɤɨɧɬɪɨɥɶɧɨɦ ɨɛɴɟɦɟ, ɢɫɩɵɬɵɜɚɟɬ ɜɨɡɞɟɣɫɬɜɢɟ ɫɢɥɵ ɬɹɠɟɫɬɢ, ɪɚɜɧɨɣ ɩɪɨɢɡɜɟɞɟɧɢɸ g ɧɚ ɦɚɫɫɭ, ɡɚɤɥɸɱɟɧɧɭɸ ɜ ɤɨɧɬɪɨɥɶɧɨɦ ɨɛɴɟɦɟ
Ug dxdydz .
ɉɪɢ ɭɱɟɬɟ ɫɢɥɵ ɬɹɠɟɫɬɢ ɭɪɚɜɧɟɧɢɟ (3.13) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ª wu
º
(3.15)
U« u grad u » grad p Ug .
¬ wt
¼
ȼ ɫɥɭɱɚɟ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ (ɝɚɡɚ) ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɹ ɇɚɜɶɟ–ɋɬɨɤɫɚ.
ȼ ɫɥɭɱɚɟ ɜɹɡɤɨɣ ɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ ɷɬɢ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ
O P
du
1
P
(3.16)
’U v
’’ ˜u 'u ,
dt
U
U
U
ɝɞɟ ɢɫɩɨɥɶɡɨɜɚɧɵ ɨɛɨɡɧɚɱɟɧɢɹ ’U { grad U ; ’ ˜ u { div u ; ' ... ’ ˜ ’... ɨɩɟɪɚɬɨɪ Ʌɚɩɥɚɫɚ, ɞɟɣɫɬɜɭɸɳɢɣ ɧɚ ɤɚɠɞɭɸ ɤɨɦɩɨɧɟɧɬɭ ɜɟɤɬɨɪɚ
ɫɤɨɪɨɫɬɢ u .
ȼ ɫɤɚɥɹɪɧɨɣ ɮɨɪɦɟ ɢɦɟɟɦ
§ w 2u w 2u w 2u ·
1 wp
wu
wu
wu
wu
u
v
w
Q¨ 2 2 2 ¸ ,
¨ wx
wt
wx
wy
wz
U wx
wy
w z ¸¹
©
§ w 2v w 2v w 2v ·
wv
wv
wv
wv
1 wp
u
v w
Q¨ 2 2 2 ¸ ,
¨ wx
wt
wx
wy
wz
U wy
wy
w z ¸¹
©
§ w 2w w 2w w 2w ·
1 wp
ww
ww
ww
ww
u
v
w
Q¨ 2 2 2 ¸,
¨
wt
wx
wy
wz
U wz
wy
w z ¹¸
© wx
ɝɞɟ Q P U – ɤɢɧɟɦɚɬɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɜɹɡɤɨɫɬɢ.
ȼ ɞɪɭɝɨɦ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɭɱɢɬɵɜɚɹ ɭɫɥɨɜɢɟ ɧɟɫɠɢɦɚɟɦɨɫɬɢ (3.11),
ɩɪɢɞɟɦ ɤ ɭɪɚɜɧɟɧɢɹɦ ɞɜɢɠɟɧɢɹ ɜɹɡɤɨɣ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ
du
1
P
(3.17)
’U 'u .
dt
U
U
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɨɞɧɨɦɟɪɧɨɝɨ ɞɜɢɠɟɧɢɹ ɠɢɞɤɨɫɬɢ (ɝɚɡɚ), ɤɨɝɞɚ
u z 0 , v w 0 , u u x ,t ,
ɭɪɚɜɧɟɧɢɟ ɗɣɥɟɪɚ (3.13) ɢ ɭɪɚɜɧɟɧɢɟ ɇɚɜɶɟ–ɋɬɨɤɫɚ (3.17) ɩɪɢɧɢɦɚɸɬ
ɜɢɞ
wu
wu
1 wp
,
u
wt
wx
U wx
72
wu
wu
1 wp
w 2u
u
Q 2 .
wt
wx
U wx
wx
ɍɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɜ ɮɨɪɦɟ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɥɟɞɭɟɬ ɢɡ ɮɨɪɦɭɥɵ
ɉɨɬɨɤ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ, ɨɝɪɚɧɢɱɢɜɚɸɳɭɸ ɤɨɧɬɪɨɥɶɧɵɣ ɨɛɴɟɦ,
ɦɨɠɟɬ ɛɵɬɶ ɫɜɹɡɚɧ ɤɚɤ ɫ ɤɨɧɜɟɤɰɢɟɣ, ɬɚɤ ɢ ɫ ɹɜɥɟɧɢɹɦɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɋɚɛɨɬɭ ɧɚɞ ɤɨɧɬɪɨɥɶɧɵɦ ɨɛɴɟɦɨɦ ɦɨɝɭɬ ɫɨɜɟɪɲɚɬɶ ɫɢɥɵ, ɢɦɟɸɳɢɟ
ɫɚɦɨɟ ɪɚɡɧɨɟ ɩɪɨɢɫɯɨɠɞɟɧɢɟ – ɫɜɹɡɚɧɧɵɟ ɤɚɤ ɫ ɹɜɥɟɧɢɟɦ ɜɹɡɤɨɫɬɢ, ɬɚɤ ɢ
ɫ ɜɨɡɦɨɠɧɵɦɢ ɯɢɦɢɱɟɫɤɢɦɢ ɩɪɟɜɪɚɳɟɧɢɹɦɢ. ȼɵɞɟɥɟɧɢɟ ɢɥɢ ɩɨɝɥɨɳɟɧɢɟ ɬɟɩɥɚ ɜ ɪɟɚɤɰɢɹɯ ɨɛɟɫɩɟɱɢɬ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɢɫɬɨɱɧɢɤɨɜɨɟ ɫɥɚɝɚɟɦɨɟ. Ʉ ɪɚɛɨɬɟ ɫɢɥ ɞɚɜɥɟɧɢɹ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɢ ɬɟɩɥɨɜɵɟ ɷɮɮɟɤɬɵ, ɫɜɹɡɚɧɧɵɟ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɦ ɪɚɫɲɢɪɟɧɢɟɦ. ɂɡɛɟɝɚɹ ɝɪɨɦɨɡɞɤɢɯ ɜɵɤɥɚɞɨɤ, ɩɪɟɞɫɬɚɜɢɦ ɞɜɟ ɧɚɢɛɨɥɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɵɟ ɮɨɪɦɵ ɭɪɚɜɧɟɧɢɹ
ɛɚɥɚɧɫɚ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ.
Ɍɚɤ, ɜ ɫɥɭɱɚɟ ɧɟɜɹɡɤɨɝɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɝɨ ɨɞɧɨɤɨɦɩɨɧɟɧɬɧɨɝɨ ɝɚɡɚ
ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ
ds
(3.18)
U T s U T
’ ˜ JT ,
dt
ɝɞɟ s – ɷɧɬɪɨɩɢɹ ɟɞɢɧɢɰɵ ɦɚɫɫɵ ɜɟɳɟɫɬɜɚ, s s T , J ɢɥɢ s s T , p (ɫɦ. ɪɚɡɞɟɥ 1.5). ȿɫɥɢ ɝɚɡ ɬɟɱɟɬ ɜ ɭɫɥɨɜɢɹɯ ɩɨɫɬɨɹɧɧɨɝɨ ɨɛɴɟɦɚ, ɬɨ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɭɪɚɜɧɟɧɢɟɦ Ƚɢɛɛɫɚ (1.19)
df pd J sdT ,
ɧɚɣɞɟɦ
cJ
ds
dT JT dJ ,
(3.19)
T
[
d p J T dT T d J ,
J
ɝɞɟ
73
cJ
§ w2 f ·
T ¨
–
¨ w T 2 ¸¸
©
¹J
§ ws ·
T¨
¸
© wT ¹J
ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ;
§ w2 f ·
§ ws ·
§ wU ·
JT T ¨
¨
–
¸
¨
¸
¨ wJw T ¸¸
© w T ¹T © w T ¹ J
©
¹
ɢɡɨɯɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɪɦɢɱɟɫɤɨɝɨ ɭɜɟɥɢɱɟɧɢɹ ɞɚɜɥɟɧɢɹ;
§ w2 f ·
§ wp ·
[ T J ¨ ¸
J¨ 2 ¸ –
¨ wJ ¸
© wJ ¹ T
©
¹T
ɢɡɨɬɟɪɦɢɱɟɫɤɢɣ ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ; ɚ [ T 1 – ɢɡɨɬɟɪɦɢɱɟɫɤɚɹ ɫɠɢɦɚɟɦɨɫɬɶ.
ɂɦɟɟɬ ɦɟɫɬɨ ɫɨɨɬɧɨɲɟɧɢɟ
J T D T [T ,
ɝɞɟ D T – ɨɛɴɟɦɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɪɦɢɱɟɫɤɨɝɨ ɪɚɫɲɢɪɟɧɢɹ (3.1)
1 § wJ ·
DT
. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɞɥɹ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɫ ɭɪɚɜɧɟɧɢɟɦ ɫɨJ ¨© w T ¸¹ p
ɫɬɨɹɧɢɹ (3.2) pJ
m 1RT ɧɚɯɨɞɢɦ
§ wp ·
J ¨ ¸
© wJ ¹ T
[T
JT
§ wp ·
¨ wT ¸
©
¹J
R
mJ
RT
mJ
p;
p
T
DT [T .
Ɂɚɩɢɫɵɜɚɹ ɩɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ (3.19) ɜ ɜɢɞɟ
ds c J dT
dJ
JT
,
dt T dt
dt
ɩɨɞɫɬɚɜɢɦ ɩɪɨɢɡɜɨɞɧɭɸ ɷɧɬɪɨɩɢɢ ɜ ɭɪɚɜɧɟɧɢɟ (3.18). ɇɚɣɞɟɦ
UcJ
dT
dt
’ ˜ JT U JTT
dJ
.
dt
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɟɦ ɧɟɪɚɡɪɵɜɧɨɫɬɢ (3.10)
dU
wU
div Uu U div u { 0 .
wt
dt
Ɍɚɤ ɤɚɤ U J 1 , ɬɨ
74
(3.20)
dJ
dt
1 dU
U 2 dt
div u ’ ˜ u
.
{
U
U
ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɡ (3.20) ɩɨɥɭɱɢɦ
dT
UcJ
’ ˜ J T J T T’ ˜ u
dt
ɢɥɢ
ª wT
wT
wT
wT º
UcJ « u
v
w »
wx
wy
wz ¼
¬ wt
(3.21)
§ wu wv ww ·
¸¸.
’ ˜ J T D T [T T ¨¨ w
x
w
y
w
z
¹
©
ȼ ɫɥɭɱɚɟ ɧɟɫɠɢɦɚɟɦɨɝɨ ɝɚɡɚ ɜɬɨɪɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɭɪɚɜɧɟɧɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. ɗɬɢɦ ɫɥɚɝɚɟɦɵɦ, ɤɚɤ ɩɪɚɜɢɥɨ, ɩɪɟɧɟɛɪɟɝɚɸɬ ɢ ɜ ɫɥɭɱɚɟ ɫɠɢɦɚɟɦɨɝɨ ɝɚɡɚ, ɩɨɥɚɝɚɹ, ɱɬɨ ɦɧɨɠɢɬɟɥɶ DT ˜ [T –
ɦɚɥ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ªwT
wT
wT
wT º
UcJ «
u
v
w
wx
wy
w z »¼
¬ wt
’ ˜ JT .
ɂɫɩɨɥɶɡɭɹ ɜɦɟɫɬɨ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ ɞɥɹ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ Ƚɟɥɶɦɝɨɥɶɰɚ ɭɪɚɜɧɟɧɢɟ Ƚɢɛɛɫɚ (1.20), ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɜ ɭɫɥɨɜɢɹɯ, ɤɨɝɞɚ
ɡɚɞɚɟɬɫɹ (ɢɥɢ ɢɳɟɬɫɹ) ɞɚɜɥɟɧɢɟ, ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ªwT
wT
wT
wT º
Uc p «
u
v
w
wx
wy
w z »¼
¬ wt
’ ˜ JT .
ɉɪɢ ɧɚɥɢɱɢɢ ɜɹɡɤɢɯ ɫɢɥ ɜɦɟɫɬɨ (3.18) ɢɦɟɟɦ
UT
ds
dt
’ ˜ J T IJ ˜˜ ’ u ,
(3.22)
ɝɞɟ IJ – ɬɟɧɡɨɪ ɜɹɡɤɢɯ ɧɚɩɪɹɠɟɧɢɣ. ɋɢɦɜɨɥɨɦ «··» ɨɛɨɡɧɚɱɟɧɨ ɞɜɨɣɧɨɟ
ɫɤɚɥɹɪɧɨɟ ɩɪɨɢɡɜɟɞɟɧɢɟ ɬɟɧɡɨɪɨɜ. ȼɢɞ ɜɬɨɪɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ (3.22) ɰɟɥɢɤɨɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɚɤɨɧɚɦɢ ɜɹɡɤɨɝɨ ɬɪɟɧɢɹ.
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɧɟɪɚɡɪɵɜɧɨɫɬɢ, ɞɜɢɠɟɧɢɹ ɢ ɷɧɟɪɝɢɢ
ɜɵɪɚɠɚɸɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɟ ɡɚɤɨɧɵ ɫɨɯɪɚɧɟɧɢɹ. Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɤɨɧɤɪɟɬɧɵɯ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɯ ɢ ɬɟɩɥɨɜɵɯ ɡɚɞɚɱ
ɧɟɨɛɯɨɞɢɦɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɤɪɚɟɜɵɟ ɭɫɥɨɜɢɹ. Ɂɚɞɚɧɢɟ ɤɪɚɟɜɵɯ ɭɫɥɨɜɢɣ ɫɜɨɞɢɬɫɹ ɤ ɭɠɟ ɢɡɜɟɫɬɧɵɦ ɧɚɦ ɭɫɥɨɜɢɹɦ ɱɟɬɵɪɟɯ ɬɢɩɨɜ. Ɉɛɥɚɫɬɶ
ɧɚɭɤɢ, ɧɚɡɵɜɚɟɦɚɹ ɬɟɨɪɢɟɣ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɜ ɧɚɫɬɨɹɳɟɟ
75
ɜɪɟɦɹ ɨɱɟɧɶ ɲɢɪɨɤɨ ɪɚɡɜɢɬɚ, Ɋɟɡɭɥɶɬɚɬɵ ɪɟɲɟɧɢɹ ɱɚɫɬɧɵɯ ɡɚɞɚɱ ɦɨɠɧɨ
ɩɨɱɟɪɩɧɭɬɶ ɜ ɫɩɟɰɢɚɥɶɧɨɣ ɥɢɬɟɪɚɬɭɪɟ, ɜ ɬɨɦ ɱɢɫɥɟ, ɭɱɟɛɧɨɝɨ ɯɚɪɚɤɬɟɪɚ.
Ⱦɥɹ ɡɚɞɚɱ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɨɛɪɚɛɨɬɤɢ ɢ ɫɢɧɬɟɡɚ ɦɚɬɟɪɢɚɥɨɜ ɜ ɭɫɥɨɜɢɹɯ ɜɨɡɞɟɣɫɬɜɢɹ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɛɨɥɶɲɭɸ ɪɨɥɶ ɦɨɝɭɬ ɢɝɪɚɬɶ ɡɚɞɚɱɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɩɨɡɜɨɥɹɸɳɢɟ ɪɚɫɫɱɢɬɚɬɶ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɨɬɞɚɱɢ. ȼɜɢɞɭ ɢɯ ɩɨɜɵɲɟɧɧɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ
ɫɥɨɠɧɨɫɬɢ ɨɝɪɚɧɢɱɢɦɫɹ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɵɦɢ ɩɪɢɦɟɪɚɦɢ.
3.3. Ɂɚɞɚɱɚ ɨɛ ɨɛɬɟɤɚɧɢɢ ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɵ
Ʉɚɤ ɜɢɞɧɨ, ɩɪɨɰɟɫɫɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɠɢɞɤɨɫɬɢ ɨɫɥɨɠɧɹɸɬɫɹ ɩɨ
ɫɪɚɜɧɟɧɢɸ ɫ ɬɟɩɥɨɩɟɪɟɞɚɱɟɣ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ, ɱɬɨ ɫɜɹɡɚɧɨ ɫ ɞɜɢɠɟɧɢɟɦ
ɠɢɞɤɨɫɬɢ. ɉɨɝɪɭɠɟɧɧɨɟ ɜ ɞɜɢɠɭɳɭɸɫɹ ɠɢɞɤɨɫɬɶ ɧɚɝɪɟɬɨɟ ɬɟɥɨ ɨɯɥɚɠɞɚɟɬɫɹ ɡɧɚɱɢɬɟɥɶɧɨ ɛɵɫɬɪɟɟ, ɱɟɦ ɜ ɧɟɩɨɞɜɢɠɧɨɣ ɠɢɞɤɨɫɬɢ, ɝɞɟ ɬɟɩɥɨɩɟɪɟɞɚɱɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ ɩɪɨɰɟɫɫɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
Ȼɭɞɟɦ ɩɨɥɚɝɚɬɶ, ɱɬɨ ɢɦɟɸɳɢɟɫɹ ɜ ɠɢɞɤɨɫɬɢ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ
ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵ ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɟɟ ɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɦɨɠɧɨ ɛɵɥɨ
ɫɱɢɬɚɬɶ ɩɨɫɬɨɹɧɧɵɦɢ (ɧɟ ɡɚɜɢɫɹɳɢɦɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ). ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɷɬɢ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɛɭɞɟɦ ɩɨɥɚɝɚɬɶ ɧɚɫɬɨɥɶɤɨ ɛɨɥɶɲɢɦɢ,
ɱɬɨɛɵ ɦɨɠɧɨ ɛɵɥɨ ɩɪɟɧɟɛɪɟɱɶ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚ ɫɱɟɬ ɞɢɫɫɢɩɚɰɢɢ ɷɧɟɪɝɢɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɜɧɭɬɪɟɧɧɟɝɨ ɬɪɟɧɢɹ, ɨɛɭɫɥɨɜɥɟɧɧɨɝɨ ɜɹɡɤɨɫɬɶɸ.
Ȼɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɬɚɤɢɟ ɭɫɥɨɜɢɹ, ɤɨɝɞɚ ɤɨɧɜɟɤɰɢɸ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɫɬɚɰɢɨɧɚɪɧɨɣ. Ɍɨɝɞɚ ɜɫɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ ɜɪɟɦɟɧɢ ɢɡ ɭɪɚɜɧɟɧɢɣ
ɜɵɩɚɞɚɸɬ, ɢ ɞɥɹ ɧɟɫɠɢɦɚɟɦɨɣ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɦɵ ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɹ
§ w 2u w 2u w 2u ·
1 wp
wu
wu
wu
u v w
Q¨ 2 2 2 ¸ ,
¨ wx
wx
wy
wy
U wx
wy ¸¹
wy
©
§ w 2 v w 2v w 2 v ·
wv
wv
1 wp
wv
Q¨ 2 2 2 ¸ ,
u v w
¨ wx
wy
U wy
wy
wx
wy
wy ¸¹
©
§ w2w w2w w2w ·
ww
ww
1 wp
ww
Q¨ 2 2 2 ¸ ,
w
v
u
¨ wx
wy
U wz
wy
wx
wy
wy ¸¹
©
§ w 2T w 2T w 2T ·
§ wT
wT
wT ·
Uc p ¨ u
v
w
O¨ 2 2 2 ¸,
¨
wy
w z ¹¸
wy
w z ¹¸
© wx
© wx
wu wv ww
0.
wx wy wz
Ɇɵ ɢɦɟɟɦ ɫɢɫɬɟɦɭ 5 ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɹɬɢ ɧɟɢɡɜɟɫɬɧɵɯ:
ɬɟɦɩɟɪɚɬɭɪɵ, ɤɨɦɩɨɧɟɧɬ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɢ ɞɚɜɥɟɧɢɹ. Ɋɟɲɟɧɢɟ ɷɬɨɣ
76
ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɡɚɜɢɫɢɬ ɨɬ ɞɜɭɯ ɩɨɫɬɨɹɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ – ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɜɹɡɤɨɫɬɢ. ɗɬɢ ɞɜɟ ɜɟɥɢɱɢɧɵ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɭɸ ɪɚɡɦɟɪɧɨɫɬɶ, ɢ ɢɡ ɧɢɯ ɦɨɠɧɨ ɫɨɫɬɚɜɢɬɶ
ɤɨɦɩɥɟɤɫ
Q
O
Pr
,a
.
a
c JU
ɗɬɨɬ ɤɨɦɩɥɟɤɫ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɦ ɉɪɚɧɞɬɥɹ.
Ʉɪɨɦɟ ɷɬɢɯ ɞɜɭɯ ɮɢɡɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ, ɪɟɲɟɧɢɟ ɛɭɞɟɬ ɡɚɜɢɫɟɬɶ ɨɬ
ɧɟɤɨɬɨɪɨɝɨ ɯɚɪɚɤɬɟɪɧɨɝɨ ɪɚɡɦɟɪɚ l , ɯɚɪɚɤɬɟɪɧɨɣ ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ u ɢ
ɯɚɪɚɤɬɟɪɧɨɝɨ ɩɟɪɟɩɚɞɚ ɬɟɦɩɟɪɚɬɭɪ T T0 .
Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɬɟɱɟɧɢɹ ɢ ɬɟɩɥɨɨɛɦɟɧɚ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɸɬ ɤɨɦɩɥɟɤɫɵ
DT l
u l
Re
ɢ Nu
–
O
Q
ɱɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ ɢ ɱɢɫɥɨ ɇɭɫɫɟɥɶɬɚ. ɋ ɢɯ ɮɢɡɢɱɟɫɤɢɦ ɫɦɵɫɥɨɦ ɦɵ ɩɨɡɧɚɤɨɦɢɦɫɹ ɩɨɡɠɟ.
Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɬɟɱɟɧɢɹ ɜ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɢ ɬɟɩɥɨɨɛɦɟɧɚ ɷɬɭ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɦɨɠɧɨ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɫɬɢɬɶ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɠɢɞɤɨɫɬɶɸ ɨɛɬɟɤɚɟɬɫɹ ɩɥɨɫɤɚɹ ɫɬɟɧɤɚ. ɉɭɫɬɶ ɨɫɶ
x ɧɚɩɪɚɜɥɟɧɚ ɜɞɨɥɶ ɫɬɟɧɤɢ, ɚ ɨɫɶ y - ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɚ ɤ ɧɟɣ.
ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɜɦɟɫɬɨ ɫɢɫɬɟɦɵ ɩɹɬɢ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɦ
§ w 2u w 2u ·
1 wp
wu
wu
v
Q¨ 2 2 ¸,
(3.23)
u
¨ wx
¸
wx
wy
U wx
w
y
©
¹
§ w 2v w 2v ·
1 wp
wv
wv
(3.24)
u
v
Q¨ 2 2 ¸,
¨ wx
¸
wx
wy
U wx
w
y
©
¹
§ w 2T w 2T ·
§ wT
wT ·
Uc p ¨ u
v
O¨ 2 2 ¸,
(3.25)
¨ wx
¸
w y ¸¹
w
y
© wx
©
¹
wu wv
(3.26)
0.
wx wy
Ɉɛɨɡɧɚɱɢɦ ɬɨɥɳɢɧɭ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɛɭɤɜɨɣ G , ɚ ɯɚɪɚɤɬɟɪɧɨɣ
ɞɥɢɧɨɣ ɜ ɩɪɨɞɨɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɛɭɞɟɬ ɪɚɫɫɬɨɹɧɢɟ ɨɬ ɧɚɱɚɥɚ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ L . Ɂɚ ɯɚɪɚɤɬɟɪɧɭɸ ɫɤɨɪɨɫɬɶ ɜ ɧɚɩɪɚɜɥɟɧɢɢ Ox , ɨɱɟɜɢɞɧɨ,
ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɫɤɨɪɨɫɬɶ ɡɚ ɩɪɟɞɟɥɚɦɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ u u f .
ɉɨɝɪɚɧɢɱɧɵɣ ɫɥɨɣ ɹɜɥɹɟɬɫɹ ɬɨɧɤɢɦ, ɬɚɤ ɱɬɨ G L .
Ɍɨɝɞɚ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ (3.26) ɫɢɫɬɟɦɵ ɩɨɡɜɨɥɹɟɬ ɨɰɟɧɢɬɶ ɫɤɨɪɨɫɬɶ ɜ ɧɚɩɪɚɜɥɟɧɢɢ O y
77
G
.
L
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɰɟɧɤɚ ɫɥɚɝɚɟɦɵɯ ɩɟɪɜɨɝɨ ɢɡ ɭɪɚɜɧɟɧɢɣ ɞɜɢɠɟɧɢɹ
ɞɚɟɬ
u f2
u ·
G uf
1 'p
§u
~
uf
Q¨ f2 f2 ¸ .
L
L G
U L
G ¹
©L
ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɥɚɝɚɟɦɵɟ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (3.23) ɫɪɚɜɧɢɦɵ ɩɨ ɜɟɥɢɱɢɧɟ, ɚ ɜɨɬ ɩɟɪɜɵɦ ɫɥɚɝɚɟɦɵɦ ɜ ɫɤɨɛɤɚɯ ɫɩɪɚɜɚ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɜɬɨɪɵɦ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɜɢɞɭ ɬɨɧɤɨɫɬɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɞɜɢɠɟɧɢɟ ɜ ɧɟɦ
ɛɭɞɟɬ ɩɪɨɢɫɯɨɞɢɬɶ, ɜ ɨɫɧɨɜɧɨɦ, ɩɚɪɚɥɥɟɥɶɧɨ ɨɛɬɟɤɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ,
ɬ.ɟ. ɫɤɨɪɨɫɬɶ v ɛɭɞɟɬ ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɫɤɨɪɨɫɬɶɸ u . ȼɞɨɥɶ ɧɚɩɪɚɜɥɟɧɢɹ O y ɫɤɨɪɨɫɬɶ ɦɟɧɹɟɬɫɹ ɛɵɫɬɪɨ – ɡɚɦɟɬɧɨɟ ɢɡɦɟɧɟɧɢɟ ɟɟ ɩɪɨɢɫɯɨɞɢɬ ɧɚ ɪɚɫɫɬɨɹɧɢɹɯ ɩɨɪɹɞɤɚ G . ȼ ɧɚɩɪɚɜɥɟɧɢɢ ɠɟ ɨɫɢ Ox ɫɤɨɪɨɫɬɶ
u ɦɟɧɹɟɬɫɹ ɦɟɞɥɟɧɧɨ; ɡɚɦɟɬɧɨɟ ɢɡɦɟɧɟɧɢɟ ɟɟ ɩɪɨɢɫɯɨɞɢɬ ɥɢɲɶ ɧɚ ɩɪɨɬɹɠɟɧɢɢ ɪɚɫɫɬɨɹɧɢɣ ɩɨɪɹɞɤɚ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɣ ɞɥɢɧɵ ɡɚɞɚɱɢ, ɧɚɩɪɢɦɟɪ, ɪɚɡɦɟɪɚ ɬɟɥɚ. ɉɨɷɬɨɦɭ ɟɟ ɩɪɨɢɡɜɨɞɧɵɟ ɩɨ y ɜɟɥɢɤɢ ɩɨ ɫɪɚɜɧɟɧɢɸ
ɫ ɩɪɨɢɡɜɨɞɧɵɦɢ ɩɨ x . Ⱥ ɫɪɚɜɧɢɜɚɹ ɭɪɚɜɧɟɧɢɟ (3.23) ɫ ɭɪɚɜɧɟɧɢɟɦ
(3.24), ɡɚɦɟɬɢɦ, ɱɬɨ ɩɪɨɢɡɜɨɞɧɚɹ w p w y ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɪɨɢɡɜɨɞɧɨɣ w p w x . Ɉɬɧɨɲɟɧɢɟ ɷɬɢɯ ɩɪɨɢɡɜɨɞɧɵɯ ɩɨɪɹɞɤɚ v u .
ȼɧɟ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɞɜɢɠɟɧɢɟ ɹɜɥɹɟɬɫɹ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ ɢ ɚɞɢɚɛɚɬɢɱɟɫɤɢɦ, ɢ ɞɥɹ ɧɟɝɨ ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɟ Ȼɟɪɧɭɥɥɢ
U2
h const
2
(ɡɞɟɫɶ h – ɷɧɬɚɥɶɩɢɹ, U – ɫɤɨɪɨɫɬɶ ɨɫɧɨɜɧɨɝɨ ɩɨɬɨɤɚ, U U x ).
ȼ ɚɞɢɚɛɚɬɢɱɟɫɤɢɯ ɭɫɥɨɜɢɹɯ ɢɡ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ (ɢɡ ɜɬɨɪɨɝɨ ɭɪɚɜɧɟɧɢɹ
(1.16)) ɫɥɟɞɭɟɬ:
1
g r a d p gr a d h .
U
Ɍɨɝɞɚ
1 dp
dU
,
U
dx
U dx
ɝɞɟ ɱɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɡɚɦɟɧɟɧɵ ɧɚ ɩɨɥɧɵɟ, ɬɚɤ ɤɚɤ ɢ ɞɚɜɥɟɧɢɟ, ɢ
ɫɤɨɪɨɫɬɶ ɨɫɧɨɜɧɨɝɨ ɩɨɬɨɤɚ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɥɢɲɶ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɵ.
Ɍɚɤɢɦ, ɨɛɪɚɡɨɦ, ɩɪɟɧɟɛɪɟɝɚɹ ɦɚɥɵɦɢ ɫɥɚɝɚɟɦɵɦɢ, ɩɪɢɞɟɦ ɤ ɭɪɚɜɧɟɧɢɹɦ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɉɪɚɧɞɬɥɹ:
v ~ u f
78
wu
wu
dU
w 2u
(3.27)
u
v
U
Q 2 ,
wx
wy
dx
wy
wu wv
(3.28)
0.
wx wy
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɤ ɷɬɢɦ ɭɪɚɜɧɟɧɢɹ ɬɪɟɛɭɸɬ ɨɛɪɚɳɟɧɢɹ ɜ ɧɭɥɶ
ɫɤɨɪɨɫɬɢ ɧɚ ɫɬɟɧɤɟ
v u 0 ɩɪɢ y 0 .
ɉɪɢ ɭɞɚɥɟɧɢɢ ɨɬ ɫɬɟɧɤɢ ɩɪɨɞɨɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɞɨɥɠɧɚ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɫɬɪɟɦɢɬɶɫɹ ɤ ɫɤɨɪɨɫɬɢ ɨɫɧɨɜɧɨɝɨ ɩɨɬɨɤɚ, ɬ.ɟ.
u U x ɩɪɢ y o f .
ɉɨɫɬɚɧɨɜɤɚ ɠɟ ɨɬɞɟɥɶɧɨɝɨ ɭɫɥɨɜɢɹ ɞɥɹ v ɧɚ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɧɟ ɬɪɟɛɭɟɬɫɹ.
ɉɪɢɛɥɢɠɟɧɢɟ ɬɨɧɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɫɩɪɚɜɟɞɥɢɜɨ ɢ ɞɥɹ ɬɟɩɥɨɜɨɣ ɮɭɧɤɰɢɢ, ɢ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ. ɉɨ ɬɟɦ ɠɟ ɫɚɦɵɦ ɫɨɨɛɪɚɠɟɧɢɹɦ, ɱɬɨ
ɢ ɜɵɲɟ, ɩɪɨɢɡɜɨɞɧɨɣ w 2T w x 2 ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɪɨɢɡɜɨɞɧɨɣ w 2T w y 2 , ɬɚɤ ɱɬɨ ɜ ɫɥɭɱɚɟ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɬɟɱɟɧɢɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (3.25) ɫɜɨɞɢɬɫɹ ɤ ɭɪɚɜɧɟɧɢɸ
u
wT
wT
v
wx
wy
O w 2T
.
Uc p wy 2
(3.29)
ɉɟɪɟɣɞɟɦ ɜ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ (3.27), (3.28) ɤ ɧɨɜɵɦ (ɛɟɡɪɚɡɦɟɪɧɵɦ) ɩɟɪɟɦɟɧɧɵɦ
U
u
v Re
x
y Re
; xc
,
uc
, vc
, Uc
, yc
L
L
uf
uf
uf
ufL
ɝɞɟ R e
(ɬ.ɟ. ɜ ɤɚɱɟɫɬɜɟ ɯɚɪɚɤɬɟɪɧɨɣ ɫɤɨɪɨɫɬɢ ɩɪɢɧɹɬɚ ɫɤɨɪɨɫɬɶ
Q
ɧɚ ɛɟɫɤɨɧɟɱɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɫɬɟɧɤɢ, ɚ ɜ ɤɚɱɟɫɬɜɟ ɯɚɪɚɤɬɟɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɝɨ ɦɚɫɲɬɚɛɚ – ɪɚɡɦɟɪ ɬɟɥɚ), ɩɪɢɞɟɦ ɤ ɭɪɚɜɧɟɧɢɹɦ
w uc
w uc
dU c w 2uc
vc
,
uc
Uc
w xc
w yc
dxc w yc 2
w u c w vc
0,
w xc w yc
ɤɨɬɨɪɵɟ ɧɟ ɫɨɞɟɪɠɚɬ ɜɹɡɤɨɫɬɢ, ɬ.ɟ. ɢɯ ɪɟɲɟɧɢɹ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ. ɗɬɨ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ ɩɪɢ ɢɡɦɟɧɟɧɢɢ R e ɜɫɹ ɤɚɪɬɢɧɚ ɞɜɢɠɟ79
ɧɢɹ ɜ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɩɨɞɜɟɪɝɚɟɬɫɹ ɥɢɲɶ ɩɨɞɨɛɧɨɦɭ ɩɪɟɨɛɪɚɡɨɜɚɧɢɸ,
ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɨɞɨɥɶɧɵɟ ɪɚɫɫɬɨɹɧɢɢ ɢ ɫɤɨɪɨɫɬɢ ɨɫɬɚɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ, ɚ ɩɨɩɟɪɟɱɧɵɟ – ɦɟɧɹɸɬɫɹ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɤɨɪɧɸ ɢɡ R e .
Ⱦɚɥɟɟ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɩɨɥɭɱɚɸɳɢɟɫɹ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɹ
ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɛɟɡɪɚɡɦɟɪɧɵɟ ɫɤɨɪɨɫɬɢ, ɤɚɤ ɧɟ ɡɚɜɢɫɹɳɢɟ ɨɬ R e ,
ɞɨɥɠɧɵ ɛɵɬɶ ɩɨɪɹɞɤɚ ɟɞɢɧɢɰɵ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
u
(3.30)
v~ f ,
Re
ɬ.ɟ. ɨɬɧɨɲɟɧɢɟ ɩɨɩɟɪɟɱɧɨɣ ɫɤɨɪɨɫɬɢ ɤ ɩɪɨɞɨɥɶɧɨɣ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɤɨɪɧɸ ɢɡ R e . Ɍɨ ɠɟ ɫɚɦɨɟ ɨɬɧɨɫɢɬɫɹ ɤ ɬɨɥɳɢɧɟ ɩɨɝɪɚɧɢɱɧɨɝɨ
ɫɥɨɹ: ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ xc , yc ɬɨɥɳɢɧɚ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ
Gc | 1, ɚ ɜ ɪɟɚɥɶɧɵɯ ɤɨɨɪɞɢɧɚɬɚɯ
L
.
(3.31)
G~
Re
ȼ ɫɥɭɱɚɟ ɪɟɚɥɶɧɨɣ ɩɥɚɫɬɢɧɤɢ ɜɫɟ ɜɟɥɢɱɢɧɵ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ
ɞɜɭɯ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ. ȼ ɫɥɭɱɚɟ ɠɟ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɣ ɩɥɚɫɬɢɧɤɢ ɧɟɬ ɧɢɤɚɤɢɯ ɯɚɪɚɤɬɟɪɧɵɯ ɪɚɡɦɟɪɨɜ ɞɥɢɧɵ. ɉɨɷɬɨɦɭ ɫɤɨɪɨɫɬɶ
uc u u f ɦɨɠɟɬ ɡɚɜɢɫɟɬɶ ɬɨɥɶɤɨ ɨɬ ɬɚɤɨɣ ɤɨɦɛɢɧɚɰɢɢ ɤɨɨɪɞɢɧɚɬ xc , yc ,
ɤɨɬɨɪɚɹ ɧɟ ɫɨɞɟɪɠɢɬ ɦɚɫɲɬɚɛɚ ɞɥɢɧɵ. ɇɚɩɪɢɦɟɪ,
yc
U
[
y
.
(3.32)
Qx
xc
ɑɬɨ ɠɟ ɤɚɫɚɟɬɫɹ v , ɬɨ ɡɞɟɫɶ ɮɭɧɤɰɢɟɣ [ ɞɨɥɠɧɨ ɛɵɬɶ ɩɪɨɢɡɜɟɞɟɧɢɟ
vc xc .
ȼɜɟɞɟɦ ɮɭɧɤɰɢɸ ɬɨɤɚ \ , ɬɚɤɭɸ, ɱɬɨ
w\
w\
,v .
u
wx
wy
ɍɤɚɡɚɧɧɵɦ ɫɜɨɣɫɬɜɚɦ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɮɭɧɤɰɢɹ ɬɨɤɚ ɜɢɞɚ
\
xQ U f [ .
Ɍɨɝɞɚ
· 1 QU
wf
1 QU § wf
[
f ¸{
u U
{ Uf c , v
[ f c f . (3.33)
¨
2 x © w[
2
x
w[
¹
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɫɧɨɜɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɞɜɢɠɟɧɢɹ ɜ ɩɨɝɪɚɧɢɱɧɨɦ
ɫɥɨɟ ɹɜɥɹɟɬɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜ ɧɟɦ ɩɪɨɞɨɥɶɧɨɣ ɫɤɨɪɨɫɬɢ u (ɩɨɫɤɨɥɶɤɭ
ɩɨɩɟɪɟɱɧɚɹ ɫɤɨɪɨɫɬɶ v – ɦɚɥɚ). ɗɬɚ ɫɤɨɪɨɫɬɶ ɜɨɡɪɚɫɬɚɟɬ ɨɬ ɧɭɥɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɬɟɤɚɟɦɨɣ ɩɥɚɫɬɢɧɤɢ ɞɨ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ
ɜɧɟɲɧɟɝɨ ɩɨɬɨɤɚ ɩɪɢ ɨɩɪɟɞɟɥɟɧɧɨɦ ɡɧɚɱɟɧɢɢ [ . ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɡɚɤɥɸɱɢɬɶ, ɱɬɨ ɬɨɥɳɢɧɚ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɨɛɬɟɤɚɟɦɨɣ
80
ɩɥɚɫɬɢɧɤɢ (ɨɩɪɟɞɟɥɟɧɧɚɹ ɤɚɤ ɡɧɚɱɟɧɢɟ y , ɩɪɢ ɤɨɬɨɪɨɦ u U ~ 1 ) ɢɦɟɟɬ
ɩɨɪɹɞɨɤ ɜɟɥɢɱɢɧɵ
G ~ xv U .
Ɍ.ɟ., ɬɨɥɳɢɧɚ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜɨɡɪɚɫɬɚɟɬ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɤɨɪɧɸ ɢɡ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɤɪɚɹ ɩɥɚɫɬɢɧɤɢ.
ɉɨɞɫɬɚɜɥɹɹ (3.33) ɜ (3.27) ɩɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ ɫɤɨɪɨɫɬɢ ɨɫɧɨɜɧɨɝɨ ɩɨɬɨɤɚ U c o n s t u f , ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɮɭɧɤɰɢɢ f [ ff cc 2 f ccc 0 .
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɞɥɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɭɬ
f 0 f c 0 0 ; f c f 1.
(ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɟɣ, ɨɱɟɜɢɞɧɨ, ɫɢɦɦɟɬɪɢɱɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɥɨɫɤɨɫɬɢ y 0 , ɩɨɷɬɨɦɭ ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɬɨɥɶɤɨ ɩɪɚɜɭɸ ɩɨɥɭɩɥɨɫɤɨɫɬɶ).
Ɋɟɲɟɧɢɟ ɩɨɥɭɱɟɧɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɧɚɣɞɟɧɨ ɱɢɫɥɟɧɧɨ ɢ ɩɨɞɪɨɛɧɨ
ɡɚɬɚɛɭɥɢɪɨɜɚɧɨ, ɱɬɨ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɥɢɬɟɪɚɬɭɪɟ 15. ɉɪɢ ɦɚɥɵɯ ɡɧɚɱɟɧɢɹɯ
[ ɫɩɪɚɜɟɞɥɢɜɚ ɚɩɩɪɨɤɫɢɦɚɰɢɹ
1
f [
D [ 2 O [ 5 , D 0 . 332 .
(3.34)
2
ɉɪɢ ɛɨɥɶɲɢɯ [ ɢɦɟɟɦ f [ [ 1 . 72 .
Ʉɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ ɥɟɝɤɨ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɫ ɩɨɦɨɳɶɸ
ɫɨɨɬɧɨɲɟɧɢɣ (3.33).
ɉɟɪɟɯɨɞɹ ɜ (3.29) ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ uc , vc , xc , yc ɢ
T T0
T
, ɩɪɢɞɟɦ ɤ ɡɚɞɚɱɟ ɞɥɹ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɜɢɞɚ
Ts T0
1 w 2T
,
P r w yc 2
wT
wT
uc
vc
w xc
w yc
(3.35)
xc 0 , T 0 ; yc 0 , T 1 ; yc o 0 , T 0 .
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɬɚɤɠɟ ɢɳɟɬɫɹ ɜ ɜɢɞɟ T T [ , ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ
ɨɛɵɤɧɨɜɟɧɧɨɦɭ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɞɥɹ T
d 2T 1
dT
P
r
0;
f
[
d[
d[ 2 2
[ 0 , T 1 ; [ o f , T o 1.
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ ɜɢɞɟ
15
ɒɥɢɯɬɢɧɝ Ƚ. Ɍɟɨɪɢɹ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, Ɇ.: ɇɚɭɤɚ, 1974. 711 ɫ.
81
f
T
³ ª¬ f cc [ º¼
Pr
f
d[
[
³ ª¬ f cc [ º¼
Pr
d[ .
0
ɂɫɩɨɥɶɡɭɹ ɚɩɩɪɨɤɫɢɦɚɰɢɸ (3.34), ɧɚɣɞɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɤɢ
B Pr § wT ·
,
(3.36)
J ¨
¸
c
y
w
c
x
©
¹ yc 0
ɝɞɟ
0 . 3 3 2 P r
B Pr f
³ ¬ª f cc [ ¼º
Pr
.
(3.37)
d[
0
ɉɪɢɜɟɞɟɦ ɚɫɢɦɩɬɨɬɢɤɢ ɞɥɹ ɮɭɧɤɰɢɢ B Pr ɞɥɹ ɦɚɥɵɯ ɢ ɛɨɥɶɲɢɯ
ɡɧɚɱɟɧɢɣ ɱɢɫɥɚ ɉɪɚɧɞɬɥɹ:
B Pr o
P r S ,
Pr o 0;
B Pr o 0 . 339Pr 1 3 , P r o f .
ȼɨ ɜɫɟɦ ɞɢɚɩɚɡɨɧɟ ɢɡɦɟɧɟɧɢɹ ɱɢɫɟɥ ɉɪɚɧɞɬɥɹ ɮɭɧɤɰɢɹ B Pr ɜ ɜɵɪɚɠɟɧɢɢ (3.36) ɚɩɩɪɨɤɫɢɦɢɪɭɟɬɫɹ ɜɵɪɚɠɟɧɢɟɦ
23
0 . 0817 ª1 72Pr 1º
¬«
¼»
B Pr 12
.
(3.38)
Ɂɚɩɢɲɟɦ ɬɟɩɟɪɶ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɥɨɤɚɥɶɧɨɝɨ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ ɢɥɢ
ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɥɨɤɚɥɶɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ
Nu x
DT x
O
x § wT ·
Ts T0 ¨© w y ¸¹ y
Re x B Pr ,
(3.39)
0
ɝɞɟ R e x xU Q – ɥɨɤɚɥɶɧɨɟ ɱɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ.
ɗɬɨ ɪɟɲɟɧɢɟ ɤɨɪɪɟɤɬɧɨ ɜ ɫɥɭɱɚɟ ɨɛɬɟɤɚɧɢɹ ɩɥɚɫɬɢɧɵ ɥɚɦɢɧɚɪɧɵɦ
ɩɨɬɨɤɨɦ.
ȼ ɫɥɭɱɚɟ ɨɛɬɟɤɚɧɢɹ ɩɥɚɫɬɢɧɵ ɬɭɪɛɭɥɟɧɬɧɵɦ ɩɨɬɨɤɨɦ ɚɧɚɥɨɝɢɱɧɨɟ
ɪɟɲɟɧɢɟ ɩɪɢɜɨɞɢɬ ɤ ɮɨɪɦɭɥɟ
Nu x
0 , 0 29 6P r 1 3R e 4x 5 .
(3.40)
ɗɬɢ ɪɟɡɭɥɶɬɚɬɵ ɯɨɪɨɲɨ ɫɨɝɥɚɫɭɸɬɫɹ ɫ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ.
82
3. 4. Ɍɟ ɩɥ ɨɩ ɟ ɪɟ ɞɚɱ ɚ ɱɟ ɪɟ ɡ ɩɥ ɨɫɤ ɭɸ ɫ ɬɟ ɧ ɤɭ
ɉɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ ɨɬ ɨɞɧɨɣ ɫɪɟɞɵ (ɠɢɞɤɨɫɬɢ ɢɥɢ ɝɚɡɚ) ɤ ɞɪɭɝɨɣ ɱɟɪɟɡ ɪɚɡɞɟɥɹɸɳɭɸ ɢɯ ɫɬɟɧɤɭ ɧɚɡɵɜɚɟɬɫɹ ɬɟɩɥɨɩɟɪɟɞɚɱɟɣ. ɉɪɢɦɟɪɚɦɢ
ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɹɜɥɹɸɬɫɹ ɩɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ ɨɬ ɝɨɪɹɱɟɣ ɜɨɞɵ ɤ ɜɨɡɞɭɯɭ
ɩɨɦɟɳɟɧɢɹ ɱɟɪɟɡ ɦɟɬɚɥɥɢɱɟɫɤɢɟ ɫɬɟɧɤɢ ɛɚɬɚɪɟɣ ɰɟɧɬɪɚɥɶɧɨɝɨ ɨɬɨɩɥɟɧɢɹ, ɩɟɪɟɞɚɱɚ ɬɟɩɥɨɬɵ ɨɬ ɝɨɪɹɱɢɯ ɩɚɪɨɜ ɯɥɚɞɚɝɟɧɬɚ ɤ ɜɨɞɟ ɢɥɢ ɜɨɡɞɭɯɭ
ɱɟɪɟɡ ɫɬɚɥɶɧɵɟ ɫɬɟɧɤɢ ɬɪɭɛ ɜ ɤɨɧɞɟɧɫɚɬɨɪɚɯ ɜɨɞɹɧɨɝɨ ɢɥɢ ɜɨɡɞɭɲɧɨɝɨ
ɨɯɥɚɠɞɟɧɢɹ ɢ ɬ.ɞ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ ɫɬɟɧɤɢ ɧɟ ɞɨɥɠɧɵ ɩɪɟɩɹɬɫɬɜɨɜɚɬɶ ɬɟɩɥɨɩɟɪɟɞɚɱɟ ɨɬ ɨɞɧɨɣ ɫɪɟɞɵ ɤ ɞɪɭɝɨɣ, ɢ ɩɨɷɬɨɦɭ ɨɧɢ ɢɡɝɨɬɚɜɥɢɜɚɸɬɫɹ
ɢɡ ɦɚɬɟɪɢɚɥɨɜ, ɢɦɟɸɳɢɯ ɜɵɫɨɤɭɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ.
Ʉɨɝɞɚ ɠɟ ɬɪɟɛɭɟɬɫɹ ɭɦɟɧɶɲɢɬɶ ɬɟɩɥɨɩɪɢɬɨɤɢ (ɧɚɩɪɢɦɟɪ, ɜ ɤɚɦɟɪɵ
ɯɨɥɨɞɢɥɶɧɢɤɚ, ɤ ɯɨɥɨɞɧɵɦ ɬɪɭɛɨɩɪɨɜɨɞɚɦ ɢ ɚɩɩɚɪɚɬɚɦ) ɢɥɢ ɭɦɟɧɶɲɢɬɶ
ɩɨɬɟɪɢ ɬɟɩɥɨɬɵ ɨɬ ɝɨɪɹɱɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɜ ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ, ɬɨ
ɫɬɟɧɤɢ ɢ ɚɩɩɚɪɚɬɵ ɩɨɤɪɵɜɚɸɬ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɦɢ ɦɚɬɟɪɢɚɥɚɦɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɰɟɫɫ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɱɟɪɟɡ ɨɞɧɨɪɨɞɧɭɸ ɩɥɨɫɤɭɸ
ɫɬɟɧɤɭ ɫ ɬɨɥɳɢɧɨɣ L ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ O
(ɪɢɫ. 3.5). ɋɬɟɧɤɚ ɪɚɡɞɟɥɹɟɬ ɞɜɟ ɫɪɟɞɵ – ɬɟɩɥɭɸ ɢ ɯɨɥɨɞɧɭɸ, ɢɦɟɸɳɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ Te1 ɢ Te 2 . ȼ ɤɚɱɟɫɬɜɟ ɬɟɦɩɟɪɚɬɭɪ Te1 ɢ Te 2 ɩɪɢɧɢɦɚɸɬ
ɬɟɦɩɟɪɚɬɭɪɵ ɫɪɟɞ ɧɚ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɫɬɟɧɤɢ. Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɪɟɞɚɯ ɢ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ ɜ ɭɫɥɨɜɢɹɯ ɫɬɚɰɢɨɧɚɪɧɨɣ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɩɨɤɚɡɚɧɨ ɧɚ ɷɬɨɦ ɠɟ ɪɢɫɭɧɤɟ.
ɍɞɟɥɶɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɤɨɬɨɪɵɣ ɩɨɥɭɱɚɟɬ ɫɬɟɧɤɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɚɤɨɧɨɦ ɇɶɸɬɨɧɚ-Ɋɢɯɦɚɧɚ
(3.41)
q D1 Te1 T1 .
ɂɡ ɭɫɥɨɜɢɹ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɨɧ
ɞɨɥɠɟɧ ɪɚɜɧɹɬɶɫɹ ɩɨɬɨɤɭ ɬɟɩɥɚ, ɨɬɜɨɞɢɦɨɦɭ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɜɝɥɭɛɶ
ɫɬɟɧɤɢ. Ʉɚɤ ɦɵ ɭɠɟ ɡɧɚɟɦ, ɷɬɨɬ ɩɨɬɨɤ
ɬɟɩɥɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧ ɜ ɜɢɞɟ
Ɋɢɫ. 3.5. Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟO
(3.42) ɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɧɤɟ ɢ ɜ
T1 T2 ,
L
ɨɦɵɜɚɸɳɢɯ ɫɪɟɞɚɯ
ɧɨ, ɜ ɨɬɥɢɱɢɟ ɨɬ ɩɪɟɞɵɞɭɳɟɝɨ, ɬɟɦɩɟɪɚɬɭɪɵ T1 ɢ T2 ɧɚɦ ɧɟɢɡɜɟɫɬɧɵ. Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɨɬɜɨɞɢɦɵɣ ɬɟɩɥɨɜɨɫɩɪɢɧɢɦɚɸɳɟɣ ɫɪɟɞɨɣ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɦ ɠɟ ɡɚɤɨɧɨɦ, ɦɨɠɟɬ ɛɵɬɶ
ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ
q
q D 2 T2 Te 2 83
(3.43)
ɢ ɬɚɤɠɟ ɪɚɜɟɧ ɩɨɬɨɤɭ ɬɟɩɥɚ, ɫɥɟɞɭɸɳɟɦɭ ɱɟɪɟɡ ɫɬɟɧɤɭ ɩɨɫɪɟɞɫɬɜɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɋɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (3.41) – (3.43) ɦɨɠɟɬ ɛɵɬɶ ɥɟɝɤɨ
ɪɟɲɟɧɚ. ɂɡ ɧɟɟ ɫɥɟɞɭɟɬ
§ 1 L 1 ·
q¨
¸ Te1 Te2
© D1 O D 2 ¹
ɢɥɢ
q / Te1 Te 2 ,
(3.44)
ɝɞɟ
1
§ 1 L 1 ·
(3.45)
/ ¨
¸ –
D
O
D
2¹
© 1
ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ (ɢɥɢ ɬɟɪɦɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ) ɩɥɨɫɤɨɣ
ɫɬɟɧɤɢ. ȿɝɨ ɪɚɡɦɟɪɧɨɫɬɶ – ȼɬ/(ɦ2.Ʉ).
Ɉɛɪɚɬɧɚɹ ɜɟɥɢɱɢɧɚ
1 L 1
rT
–
(3.46)
D1 O D 2
ɟɫɬɶ ɩɨɥɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫ ɪɚɡɦɟɪɧɨɫɬɶɸ (ɦ2.Ʉ)/ȼɬ (ɜ ɨɬɥɢɱɢɟ ɨɬ RT rT F , ɢɦɟɸɳɟɝɨ ɪɚɡɦɟɪɧɨɫɬɶ Ʉ/ȼɬ; ɢɧɞɟɤɫ «Ɍ» ɦɨɠɧɨ
ɨɩɭɫɬɢɬɶ). ȿɝɨ ɜɟɥɢɱɢɧɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɭɦɦɨɣ ɭɠɟ ɢɡɜɟɫɬɧɨɝɨ ɧɚɦ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɞɜɭɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɬɟɩɥɨɨɬɞɚɱɢ 1 D1
ɢ 1 D2 .
ɂɫɩɨɥɶɡɭɹ (3.41) ɢ (3.43), ɧɚɣɞɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɟɣ ɫɬɟɧɨɤ
q
q
T1 Te1 ; T2 Te 2 .
(3.47)
D1
D2
ɉɪɢɦɟɪ. Ʉɢɪɩɢɱɧɚɹ ɫɬɟɧɚ ɬɨɥɳɢɧɨɣ 0,1 ɦ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O 0,7 ȼɬ/(ɦ.Ʉ) ɨɛɞɭɜɚɟɬɫɹ ɯɨɥɨɞɧɵɦ ɜɟɬɪɨɦ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ 270 Ʉ ɩɪɢ ɤɨɷɮɮɢɰɢɟɧɬɟ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ 40 ȼɬ/(ɦ2.Ʉ). ɋ
ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ ɫɬɟɧɵ ɧɚɯɨɞɢɬɫɹ ɧɟɩɨɞɜɢɠɧɵɣ ɜɨɡɞɭɯ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ
330 Ʉ ɩɪɢ ɤɨɷɮɮɢɰɢɟɧɬɟ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ 10 ȼɬ/(ɦ2.Ʉ). Ɍɪɟɛɭɟɬɫɹ ɪɚɫɫɱɢɬɚɬɶ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ.
Ɋɟɲɟɧɢɟ. ɉɨ ɮɨɪɦɭɥɚɦ (3.46) ɜɵɱɢɫɥɹɟɦ ɬɪɢ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɹ.
rT 1
rT
1
D1
1
40
L
O
0,1
0,7
84
0 , 025 (ɦ2Ʉ)/ȼɬ;
0,143 (ɦ2Ʉ)/ȼɬ;
1
1
0,10 (ɦ2Ʉ)/ȼɬ.
D 2 10
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɪɚɜɧɚ
Te1 Te 2
330 270
q
224 ȼɬ/ɦ2.
rT
0,025 0,141 0,10
ɋɬɪɨɝɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɜɤɥɸɱɚɟɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
d 2T
0
2
dx
ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɬɶɟɝɨ ɪɨɞɚ
rT 2
x 0: O
dT
dx
D 1 Te1 T ; x
L: O
dT
dx
D 2 Te 2 T .
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɡɚɞɚɱɚɦ, ɪɚɫɫɦɨɬɪɟɧɧɵɦ ɜ ɩɪɟɞɵɞɭɳɟɦ ɪɚɡɞɟɥɟ, ɢɥɢ ɫɥɟɞɭɟɬ ɢɡ (2.28), (3.47) ɢ (3.45)
T Te1
ª
Te2 Te1 «1 /
¬
D1 D 2 º x /
.
» D 1D 2 ¼ L D 1
3.5. Ɇɧɨɝɨɫɥɨɣɧɚɹ ɫɬɟɧɤɚ
ɉɪɢ ɪɚɫɱɟɬɟ ɦɧɨɝɨɫɥɨɣɧɨɣ ɫɬɟɧɤɢ ɭɱɢɬɵɜɚɸɬɫɹ ɬɟɪɦɢɱɟɫɤɢɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜɫɟɯ ɟɟ ɫɥɨɟɜ. ɉɨ ɚɧɚɥɨɝɢɢ ɫ ɩɪɟɞɵɞɭɳɢɦ, ɩɨɥɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɫɬɟɧɤɢ ɢɡ n ɫɥɨɟɜ ɫ ɬɨɥɳɢɧɚɦɢ Gi
ɟɫɬɶ
G
1 G1 G 2
1
... n rT
D1 O1 O 2
On D2
ɢɥɢ
n
G
1
1
¦ i rT
.
(3.48)
D1 i 1 O i D 2
Ɍɨɝɞɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ (ɢɥɢ ɬɟɪɦɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ)
ɦɧɨɝɨɫɥɨɣɧɨɣ ɫɬɟɧɤɢ ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
1
/ 1 rT
,
(3.49)
n
Gi
1
1
¦ D1 i 1 O i D 2
ɚ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ – ɩɨ ɮɨɪɦɭɥɟ
85
q
/ T1 T2 .
(3.50)
ɍɪɚɜɧɟɧɢɟ (3.45) – ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɭɪɚɜɧɟɧɢɹ (3.49) ɞɥɹ n 1 .
Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɧɤɭ ɫ ɩɥɨɳɚɞɶɸ ɩɨɜɟɪɯɧɨɫɬɢ F ɟɫɬɶ
Q
F / T1 T2 .
Fq
(3.51)
ɗɬɨ – ɨɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɟɪɟɞɚɱɢ.
ɍɪɚɜɧɟɧɢɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ ɧɚ ɫɬɵɤɟ
ɞɜɭɯ ɥɸɛɵɯ i -ɝɨ ɢ i 1 -ɝɨ ɫɥɨɟɜ ɦɧɨɝɨɫɥɨɣɧɨɣ ɫɬɟɧɤɢ ɚɧɚɥɨɝɢɱɧɵ
(3.47)
i
ª 1
G º
Ti 1 Ti q « ¦ k » .
(3.52)
D
O
¬« 1 k 1 k ¼»
Ɋɢɫ. 3.6. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɡɚɞɚɱɟ
ɉɪɢɦɟɪ.
Ɍɟɩɥɨɨɛɦɟɧɧɢɤ
(ɪɢɫ. 3.6) ɩɚɪ–ɠɢɞɤɨɫɬɶ ɫ ɩɥɨɳɚɞɶɸ
ɥɢɰɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ 3200 ɦ2 ɢɡɝɨɬɨɜɥɟɧ ɢɡ ɫɥɨɹ ɧɢɤɟɥɹ ɬɨɥɳɢɧɨɣ
0,635 ɫɦ ɢ ɩɨɤɪɵɬ ɫɨ ɫɬɨɪɨɧɵ ɩɚɪɚ
ɫɥɨɟɦ ɦɟɞɢ ɬɨɥɳɢɧɨɣ 0,12 ɫɦ. ɋɨɩɪɨɬɢɜɥɟɧɢɟ ɫɥɨɹ ɧɚɤɢɩɢ ɜɨɞɵ ɫɨ
ɫɬɨɪɨɧɵ
ɩɚɪɚ
ɫɨɫɬɚɜɥɹɟɬ
2
rT 0,00205 (ɦ ɱɚɫ ɝɪɚɞ)/ɤɤɚɥ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɨɬɞɚɱɢ ɨɬ ɩɚɪɚ ɤ
ɫɬɟɧɤɟ ɢ ɨɬ ɫɬɟɧɤɢ ɤ ɠɢɞɤɨɫɬɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɪɚɜɧɵ 4700 ɢ 528
ɤɤɚɥ/(ɦ2ɱɚɫ ɝɪɚɞ).
Ɋɟɲɟɧɢɟ. Ƚɪɟɸɳɢɣ ɩɚɪ ɢɦɟɟɬ ɬɟɦɩɟɪɚɬɭɪɭ 110 ɨɋ, ɚ ɩɨɞɨɝɪɟɬɚɹ
ɠɢɞɤɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɭ 74 ɨɋ. Ɉɩɪɟɞɟɥɢɬɶ: ɨɛɳɭɸ ɬɟɩɥɨɨɬɞɚɱɭ ɨɬ ɩɚɪɚ ɤ
ɠɢɞɤɨɫɬɢ; ɩɚɞɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɥɨɟ ɧɚɤɢɩɢ ɢ ɬɟɦɩɟɪɚɬɭɪɭ ɝɪɚɧɢɰɵ
ɪɚɡɞɟɥɚ «ɦɟɞɶ – ɧɢɤɟɥɶ», ɟɫɥɢ ɢɯ ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɪɚɜɧɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 334 ɢ 50,6 ɤɤɚɥ/(ɦ ɱɚɫ ɝɪɚɞ).
ɋɨɝɥɚɫɧɨ ɭɪɚɜɧɟɧɢɸ (3.49) ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜɵɱɢɫɥɢɦ
ɩɨ ɮɨɪɦɭɥɟ
1
=
/
G1 G 2
1
1
r D1 T O1 O 2 D 2
1
1
0,0012 0,00635
1
0,00205 4700
334
50,6
528
86
1
0,000213 0,00205 0,0000036 0,000125 0,00189
1
234 ɤɤɚɥ/(ɦ2ɱɚɫ ɝɪɚɞ).
0,0042816
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɧɤɭ ɡɚɞɚɧɧɨɣ ɩɥɨɳɚɞɢ ɫɨɫɬɚɜɥɹɟɬ
Q 234 u 0,32 u 110 74 ɤɤɚɥ/ɱɚɫ
Ɂɧɚɹ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɥɨɹ ɧɚɤɢɩɢ, ɩɨɥɭɱɚɟɦ ɞɥɹ ɩɟɪɟɩɚɞɚ ɬɟɦɩɟɪɚɬɭɪɵ
0 ,00205
17 ,3 ɨɋ.
'T 2700 u
0 ,32
ɂ, ɧɚɤɨɧɟɰ, ɜɵɱɢɫɥɹɟɦ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ «ɦɟɞɶ–
ɧɢɤɟɥɶ»:
2700
0,000213 0,00205 0,0000036 110 19,2 ;
TCu Ni 110 0,32
TCu Ni 90,8 ɨɋ.
ɉɪɢ ɡɚɞɚɧɧɨɦ ɬɟɦɩɟɪɚɬɭɪɧɨɦ ɩɟɪɟɩɚɞɟ ɢ ɧɟɢɡɦɟɧɧɨɣ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɨɛɦɟɧɚ, ɜɟɥɢɱɢɧɨɣ, ɨɩɪɟɞɟɥɹɸɳɟɣ ɬɟɩɥɨɩɟɪɟɞɚɱɭ, ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ / . ɇɟɨɛɯɨɞɢɦɚɹ ɜɟɥɢɱɢɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɨɛɦɟɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ–ɫɨɫɬɚɜɧɨɣ ɫɬɟɧɵ, ɪɚɫɫɱɢɬɚɧɧɨɣ
ɞɥɹ ɡɚɞɚɧɧɨɣ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɦɟɠɞɭ ɧɚɪɭɠɧɵɦ ɢɫɬɨɱɧɢɤɨɦ ɢ
ɫɬɨɤɨɦ ɬɟɩɥɚ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɟɪɦɢɱɟɫɤɨɣ ɩɪɨɜɨɞɢɦɨɫɬɶɸ ɫɢɫɬɟɦɵ ɢɥɢ
ɜɟɥɢɱɢɧɨɣ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɟɪɟɞɚɱɢ, ɤɨɬɨɪɵɣ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɨɬɞɟɥɶɧɵɯ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ. ɂɡ ɚɧɚɥɢɡɚ ɡɚɜɢɫɢɦɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɨɬ ɜɯɨɞɹɳɢɯ ɜ ɧɟɝɨ ɜɟɥɢɱɢɧ
ɫɥɟɞɭɟɬ ɫɩɨɫɨɛ ɢɧɬɟɧɫɢɮɢɤɚɰɢɢ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɬɟɩɥɨɨɛɦɟɧɧɵɯ ɚɩɩɚɪɚɬɚɯ ɩɪɢ ɫɧɢɠɟɧɢɢ ɢɯ ɝɚɛɚɪɢɬɨɜ ɢ ɦɟɬɚɥɥɨɟɦɤɨɫɬɢ. Ɍɚɤ, ɢɡ (3.45) ɩɪɢ
ɦɚɥɨɣ ɬɟɪɦɢɱɟɫɤɨɣ ɬɨɥɳɢɧɟ ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɫɬɟɧɤɢ L O o 0 , ɫɥɟɞɭɟɬ
1
.
/
1
1
D1 D 2
ɉɨɷɬɨɦɭ ɩɪɢ ɫɭɳɟɫɬɜɟɧɧɨɦ ɪɚɡɥɢɱɢɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ D1 ɢ D 2 ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɪɢɦɟɪɧɨ ɜɨ ɫɬɨɥɶɤɨ ɠɟ ɪɚɡ,
ɜɨ ɫɤɨɥɶɤɨ ɪɚɡ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɦɟɧɶɲɢɣ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɨɬɞɚɱɢ.
ȼ ɡɚɤɥɸɱɟɧɢɟ ɪɚɡɞɟɥɚ ɟɳɟ ɪɚɡ ɩɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɫɨɨɬɧɨɲɟɧɢɹ (3.5) –
(3.8) ɧɟ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɡɚɤɨɧ ɩɟɪɟɞɚɱɢ ɬɟɩɥɚ, ɚ ɞɚɸɬ ɥɢɲɶ ɮɨɪɦɭɥɭ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ, ɟɫɥɢ ɢɡɜɟɫɬɧɵ ɪɚɡɧɨɫɬɶ
ɬɟɦɩɟɪɚɬɭɪ ɢ ɩɨɬɨɤ ɬɟɩɥɚ. ȼɟɥɢɱɢɧɚ D ɡɚɜɢɫɢɬ ɨɬ ɫɜɨɣɫɬɜ ɠɢɞɤɨɫɬɢ,
87
ɝɟɨɦɟɬɪɢɢ ɢ ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɬɟɤɚɧɢɹ, ɯɚɪɚɤɬɟɪɚ ɬɟɱɟɧɢɹ
ɢ ɬ.ɩ., ɬ.ɟ. ɨɬ ɜɫɟɯ ɩɚɪɚɦɟɬɪɨɜ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɝɢɞɪɨɞɢɧɚɦɢɤɭ ɬɟɱɟɧɢɹ ɢ ɬɟɩɥɨɨɛɦɟɧ. ȼ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɦɟɬɨɞɚɯ ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɟɩɥɨɨɬɞɚɱɢ ɢ ɬɟɯɧɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɷɬɢ ɮɨɪɦɭɥɵ ɹɜɥɹɸɬɫɹ ɤɥɸɱɟɜɵɦɢ ɞɥɹ
ɨɰɟɧɤɢ ɜɟɥɢɱɢɧɵ D , ɚ ɬɨɱɧɟɟ, ɟɟ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɚɧɚɥɨɝɚ – ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ
DL
Nu
O
ɤɚɤ ɮɭɧɤɰɢɢ ɫɤɨɪɨɫɬɢ ɩɨɬɨɤɚ, ɜɹɡɤɢɯ ɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɠɢɞɤɨɫɬɢ. ɗɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ ɢ ɫɨɫɬɚɜɢɥɢ ɨɫɧɨɜɭ ɬɟɨɪɢɢ ɬɟɩɥɨɩɟɪɟɧɨɫɚ ɤɚɤ ɢɧɠɟɧɟɪɧɨɣ ɢ ɧɚɭɱɧɨɣ ɞɢɫɰɢɩɥɢɧɵ.
3.7. ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɑɬɨ ɩɨɧɢɦɚɸɬ ɩɨɞ ɤɨɧɜɟɤɬɢɜɧɵɦ ɬɟɩɥɨɨɛɦɟɧɨɦ?
2. Ʉɚɤɢɟ ɜɢɞɵ ɤɨɧɜɟɤɰɢɢ ȼɵ ɡɧɚɟɬɟ?
3. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɦɵɫɥ ɩɨɧɹɬɢɹ «ɬɟɩɥɨɨɬɞɚɱɚ»?
4. Ɂɚɩɢɲɢɬɟ ɨɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ.
5. ȼ ɱɟɦ ɫɨɫɬɨɹɬ ɨɫɨɛɟɧɧɨɫɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɢ ɜ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ ɜ ɫɥɭɱɚɟ ɫɜɨɛɨɞɧɨɣ ɢ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ?
6. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɜɟɥɢɱɢɧɵ A
ɞɥɹ ɤɨɧɬɪɨɥɶɧɨɝɨ ɨɛɴɟɦɚ.
7. ɉɟɪɟɱɢɫɥɢɬɟ ɨɫɧɨɜɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɩɟɪɟɧɨɫɚ.
8. ɑɟɦ ɨɬɥɢɱɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɨɞɜɢɠɧɨɣ
ɫɪɟɞɵ ɨɬ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ?
9. ɑɬɨ ɬɚɤɨɟ «ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ»?
10. Ɉɩɪɟɞɟɥɢɬɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɦɧɨɝɨɫɥɨɣɧɨɣ ɫɬɟɧɤɢ.
3.8. Ɂɚɞɚɧɢɹ
1. Ⱦɥɹ ɩɪɢɦɟɪɚ ɢɡ ɪɚɡɞɟɥɚ 3.4 ɪɚɫɫɱɢɬɚɣɬɟ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɥɢɱɢɧɵ
ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢ ɬɟɦɩɟɪɚɬɭɪ T1 ɢ T2 ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɭɥɢɰɟ ɜ ɢɧɬɟɪɜɚɥɟ 230–280 Ʉ.
2. Ʉɚɤ ɢɡɦɟɧɢɬɫɹ ɪɟɲɟɧɢɟ, ɟɫɥɢ ɜ ɪɚɫɱɟɬɚɯ ɭɱɟɫɬɶ, ɱɬɨ ɫɨ ɫɬɨɪɨɧɵ
ɩɨɦɟɳɟɧɢɹ ɤɢɪɩɢɱɧɚɹ ɫɬɟɧɚ ɩɨɤɪɵɬɚ ɫɥɨɟɦ ɲɬɭɤɚɬɭɪɤɢ ɬɨɥɳɢɧɨɣ
L 0,035 ɦ ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O 0,48 ȼɬ/(ɦ.Ʉ)?
88
3. ɇɚ ɪɢɫ. 3.7 ɩɨɤɚɡɚɧɨ ɩɨɩɟɪɟɱɧɨɟ ɫɟɱɟɧɢɟ ɬɢɩɢɱɧɨɝɨ ɩɨɬɨɥɤɚ ɠɢɥɨɝɨ ɞɨɦɚ. ɋɨɫɬɚɜɢɬɶ ɬɟɩɥɨɜɭɸ
ɰɟɩɶ ɬɢɩɢɱɧɨɣ ɫɟɤɰɢɢ ɩɨɬɨɥɤɚ. Ɋɚɫɫɱɢɬɚɬɶ ɬɟɪɦɢɱɟɫɤɢɟ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɫɥɨɟɜ. Ƚɞɟ ɭɯɨɞɢɬ ɛɨɥɶɲɟ ɬɟɩɥɚ: ɱɟɪɟɡ ɩɟɪɟɦɵɱɤɢ ɢɥɢ ɱɟɪɟɡ ɢɡɨɥɹɰɢɸ?
Ɂɚɞɚɱɭ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɱɢɬɚɬɶ
ɨɞɧɨɦɟɪɧɨɣ.
ɂɫɩɨɥɶɡɨɜɚɬɶ ɫɥɟɞɭɸɳɢɟ ɞɚɧɧɵɟ ɞɥɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɚɬɟɪɢɚɥɨɜ: ɫɨɫɧɚ 0,15 ȼɬ/(ɦ Ʉ); ɫɬɟɤɥɨɜɨɥɨɤɧɨ 0,035 ȼɬ/(ɦ Ʉ); ɲɬɭɤɚɬɭɪɤɚ
0,814 ȼɬ/(ɦ Ʉ).
89
Ɋɢɫ. 3.7. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ 3
ɑȺɋɌɖ 4
ɗɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɢ
ɦɟɬɨɞɚ ɚɧɚɥɢɡɚ ɪɚɡɦɟɪɧɨɫɬɟɣ
4.1. ɉɨɧɹɬɢɟ ɨ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ
ȼ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɢ ɝɢɞɪɨɞɢɧɚɦɢɤɟ ɜɟɥɢɤɚ ɪɨɥɶ ɷɤɫɩɟɪɢɦɟɧɬɚ.
ɑɬɨɛɵ ɨɫɭɳɟɫɬɜɢɬɶ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɤɚɤɨɝɨ-ɥɢɛɨ ɩɪɨɰɟɫɫɚ, ɧɟɨɛɯɨɞɢɦɨ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ, ɡɧɚɬɶ, ɨɬ ɤɚɤɢɯ ɜɟɥɢɱɢɧ ɡɚɜɢɫɢɬ
ɢɫɤɨɦɚɹ ɜɟɥɢɱɢɧɚ (ɧɚɩɪɢɦɟɪ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ). Ƚɨɜɨɪɹ ɞɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɧɭɠɧɨ ɡɧɚɬɶ ɩɟɪɟɱɟɧɶ ɜɟɥɢɱɢɧ, ɫɭɳɟɫɬɜɟɧɧɵɯ ɞɥɹ ɢɡɭɱɚɟɦɨɝɨ ɩɪɨɰɟɫɫɚ. ȿɫɥɢ ɦɵ ɪɚɫɩɨɥɚɝɚɟɦ ɫɜɟɞɟɧɢɹɦɢ, ɩɨɡɜɨɥɹɸɳɢɦɢ
ɫɨɫɬɚɜɢɬɶ ɮɢɡɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɩɪɨɰɟɫɫɚ ɢ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɟɝɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɜ ɜɢɞɟ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɢ ɝɪɚɧɢɱɧɵɯ
ɭɫɥɨɜɢɣ, ɬɨ ɩɟɪɟɱɟɧɶ ɫɭɳɟɫɬɜɟɧɧɵɯ ɜɟɥɢɱɢɧ ɢɡɜɟɫɬɟɧ. ȿɫɥɢ ɠɟ ɩɨ ɬɟɦ
ɢɥɢ ɢɧɵɦ ɩɪɢɱɢɧɚɦ ɩɨɥɭɱɢɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɩɪɨɰɟɫɫɚ ɧɟɜɨɡɦɨɠɧɨ, ɬɨ ɩɟɪɟɱɟɧɶ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɢɡ ɮɢɡɢɱɟɫɤɢɯ ɫɨɨɛɪɚɠɟɧɢɣ, ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɯ ɧɚɛɥɸɞɟɧɢɣ ɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɤɚɱɟɫɬɜɟɧɧɨɝɨ ɯɚɪɚɤɬɟɪɚ.
ɉɟɪɟɱɟɧɶ ɫɭɳɟɫɬɜɟɧɧɵɯ ɜɟɥɢɱɢɧ ɜɤɥɸɱɚɟɬ ɡɚɜɢɫɢɦɵɟ ɩɟɪɟɦɟɧɧɵɟ
(ɢɫɤɨɦɵɟ ɜɟɥɢɱɢɧɵ), ɧɟɡɚɜɢɫɢɦɵɟ ɩɟɪɟɦɟɧɧɵɟ (ɤɨɨɪɞɢɧɚɬɵ, ɜɪɟɦɹ) ɢ
ɩɨɫɬɨɹɧɧɵɟ. ɑɟɦ ɫɥɨɠɧɟɟ ɢɡɭɱɚɟɦɵɣ ɩɪɨɰɟɫɫ, ɬɟɦ, ɤɚɤ ɩɪɚɜɢɥɨ, ɛɨɥɶɲɟɟ ɱɢɫɥɨ ɜɟɥɢɱɢɧ ɫɨɞɟɪɠɢɬ ɩɟɪɟɱɟɧɶ, ɚ ɩɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɱɢɫɥɚ ɫɭɳɟɫɬɜɟɧɧɵɯ ɜɟɥɢɱɢɧ ɦɧɨɝɨɤɪɚɬɧɨ ɜɨɡɪɚɫɬɚɸɬ ɬɪɭɞɧɨɫɬɢ ɜ ɩɪɨɜɟɞɟɧɢɢ
ɷɤɫɩɟɪɢɦɟɧɬɚ, ɩɪɟɞɫɬɚɜɥɟɧɢɢ ɟɝɨ ɪɟɡɭɥɶɬɚɬɨɜ, ɢɯ ɬɪɚɤɬɨɜɤɟ ɢ ɨɛɨɛɳɟɧɢɢ. ɉɨɷɬɨɦɭ ɜ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɜ ɨɛɥɚɫɬɢ ɝɢɞɪɨɞɢɧɚɦɢɤɢ, ɬɟɩɥɨɨɛɦɟɧɚ ɢ ɞɪɭɝɢɯ ɨɛɥɚɫɬɹɯ ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɢɡɵɫɤɚɬɶ ɫɩɨɫɨɛ ɩɨɫɬɚɧɨɜɤɢ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɟɝɨ ɪɟɡɭɥɶɬɚɬɨɜ, ɱɬɨɛɵ ɩɪɢ ɦɢɧɢɦɚɥɶɧɨɦ ɨɛɴɟɦɟ ɢɡɦɟɪɟɧɢɣ ɩɨɥɭɱɢɬɶ ɦɚɤɫɢɦɭɦ
ɢɧɮɨɪɦɚɰɢɢ.
ɗɬɚ ɰɟɥɶ ɦɨɠɟɬ ɛɵɬɶ ɞɨɫɬɢɝɧɭɬɚ ɩɭɬɟɦ ɤɚɱɟɫɬɜɟɧɧɨɝɨ ɚɧɚɥɢɡɚ ɩɟɪɟɱɧɹ ɫɭɳɟɫɬɜɟɧɧɵɯ ɜɟɥɢɱɢɧ ɦɟɬɨɞɨɦ ɪɚɡɦɟɪɧɨɫɬɟɣ ɢɥɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɩɪɨɰɟɫɫɚ ɦɟɬɨɞɨɦ ɩɨɞɨɛɢɹ.
ɋ ɩɨɧɹɬɢɟɦ ɩɨɞɨɛɢɹ ɜɩɟɪɜɵɟ ɜɫɬɪɟɱɚɸɬɫɹ ɜ ɤɭɪɟ ɝɟɨɦɟɬɪɢɢ ɩɪɢ
ɪɚɫɫɦɨɬɪɟɧɢɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɮɢɝɭɪ. ɉɪɹɦɨɭɝɨɥɶɧɢɤɢ ɩɨɞɨɛɧɵ, ɟɫɥɢ
ɨɬɧɨɲɟɧɢɹ ɥɢɧɟɣɧɵɯ ɪɚɡɦɟɪɨɜ ɫɯɨɞɫɬɜɟɧɧɵɯ ɫɬɨɪɨɧ ɪɚɜɧɵ ɦɟɠɞɭ ɫɨɛɨɣ, ɬ.ɟ. ɟɫɥɢ ɞɥɹ ɧɢɯ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ (ɪɢɫ. 4.1)
a1 a 2 a 3
C
b1 b2 b3
ɢɥɢ
90
a1 a 2 b1 b2
C,
a 2 a 3 b2 b3
ɝɞɟ ɜɟɥɢɱɢɧɚ C ɟɫɬɶ ɤɨɧɫɬɚɧɬɚ ɝɟɨɦɟɬɪɢɱɟɫɤɨɝɨ ɩɨɞɨɛɢɹ.
ɉɨ ɚɧɚɥɨɝɢɢ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ, ɱɬɨ ɩɪɹɦɵɟ ɬɪɭɛɵ ɩɨɞɨɛɧɵ, ɟɫɥɢ ɫɨɛɥɸɞɚɟɬɫɹ ɪɚɜɟɧɫɬɜɨ
l1 l 2 l 3
C,
d1 d 2 d 3
ɝɞɟ d i – ɞɢɚɦɟɬɪ ɬɪɭɛ, l i – ɢɯ
Ɋɢɫ. 4.1. ɉɨɞɨɛɧɵɟ ɮɢɝɭɪɵ
ɞɥɢɧɚ.
ɉɨɧɹɬɢɟ ɩɨɞɨɛɢɹ ɦɨɠɟɬ ɛɵɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɨ ɧɚ ɥɸɛɵɟ ɮɢɡɢɱɟɫɤɢɟ
ɜɟɥɢɱɢɧɵ, ɚ ɬɚɤɠɟ ɧɚ ɩɪɨɰɟɫɫɵ ɢ ɹɜɥɟɧɢɹ.
ɉɪɢɦɟɪ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɨɩɢɪɚɸɳɟɣɫɹ ɧɚ
ɬɟɨɪɢɸ ɩɨɞɨɛɢɹ, ɩɪɨɞɟɦɨɧɫɬɪɢɪɨɜɚɧ ɜ ɪɚɡɞɟɥɟ 3.3.
Ⱦɥɹ ɫɥɨɠɧɵɯ ɩɪɨɰɟɫɫɨɜ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯɫɹ ɦɧɨɝɢɦɢ ɮɢɡɢɱɟɫɤɢɦɢ
ɜɟɥɢɱɢɧɚɦɢ, ɤɚɠɞɚɹ ɩɟɪɟɦɟɧɧɚɹ ɜɟɥɢɱɢɧɚ ɢɦɟɟɬ ɫɜɨɸ ɤɨɧɫɬɚɧɬɭ ɩɨɞɨɛɢɹ. ȿɫɥɢ ɹɜɥɟɧɢɹ ɩɨɞɨɛɧɵ, ɬɨ ɤɨɧɫɬɚɧɬɵ ɩɨɞɨɛɢɹ ɧɚɯɨɞɹɬɫɹ ɦɟɠɞɭ ɫɨɛɨɣ ɜ ɨɩɪɟɞɟɥɟɧɧɨɦ ɫɨɨɬɧɨɲɟɧɢɢ, ɢ ɞɥɹ ɞɚɧɧɨɝɨ ɮɢɡɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ
(ɢɥɢ ɫɢɫɬɟɦɵ) ɢɯ ɜɵɛɨɪ ɨɛɭɫɥɨɜɥɟɧ ɭɫɥɨɜɢɟɦ ɩɨɞɨɛɢɹ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ. ɗɬɢ ɛɟɡɪɚɡɦɟɪɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɤɨɦɩɥɟɤɫɵ,
ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɷɬɨ ɹɜɥɟɧɢɟ ɢ
ɩɪɨɰɟɫɫ. ɇɚɡɵɜɚɸɬɫɹ ɨɧɢ ɤɪɢɬɟɪɢɹɦɢ ɢɥɢ ɱɢɫɥɚɦɢ ɩɨɞɨɛɢɹ. Ⱦɥɹ ɜɫɟɯ
ɩɨɞɨɛɧɵɯ ɹɜɥɟɧɢɣ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɱɢɫɥɨɜɵɟ ɡɧɚɱɟɧɢɹ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɪɢɬɟɪɢɟɦ ɩɨɞɨɛɢɹ ɧɚɡɵɜɚɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɵɣ ɤɨɦɩɥɟɤɫ, ɫɨɫɬɚɜɥɟɧɧɵɣ ɢɡ ɜɟɥɢɱɢɧ, ɫɭɳɟɫɬɜɟɧɧɵɯ ɞɥɹ ɞɚɧɧɨɝɨ ɩɪɨɰɟɫɫɚ.
ȼɫɟ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɵɣ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ, ɚ ɢɯ
ɧɭɥɟɜɚɹ ɪɚɡɦɟɪɧɨɫɬɶ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɩɪɨɜɟɪɤɨɣ ɩɪɚɜɢɥɶɧɨɫɬɢ ɢɯ ɫɨɫɬɚɜɥɟɧɢɹ. Ɉɛɵɱɧɨ ɢɯ ɧɚɡɵɜɚɸɬ ɢɦɟɧɚɦɢ ɭɱɟɧɵɯ, ɜɧɟɫɲɢɯ ɛɨɥɶɲɨɣ
ɜɤɥɚɞ ɜ ɢɡɭɱɟɧɢɟ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨɨɛɦɟɧɚ, ɝɢɞɪɨɞɢɧɚɦɢɤɢ ɢ ɞɪɭɝɢɯ ɧɚɭɤ, ɢ ɨɛɨɡɧɚɱɚɸɬ ɧɚɱɚɥɶɧɵɦɢ ɥɚɬɢɧɫɤɢɦɢ ɛɭɤɜɚɦɢ ɢɯ ɮɚɦɢɥɢɣ.
Ɇɟɬɨɞɵ ɩɨɞɨɛɢɹ ɢ ɪɚɡɦɟɪɧɨɫɬɟɣ ɥɟɠɚɬ ɜ ɨɫɧɨɜɟ ɦɨɞɟɥɢɪɨɜɚɧɢɹ. Ɍɚɤ
ɩɪɢɧɹɬɨ ɧɚɡɵɜɚɬɶ ɦɟɬɨɞ ɢɡɭɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɢɥɢ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ
ɩɪɨɰɟɫɫɨɜ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɤɚɤɨɣ-ɥɢɛɨ ɫɢɫɬɟɦɟ (ɨɛɪɚɡɰɟ), ɜ ɬɟɯɧɢɱɟɫɤɨɦ
ɭɫɬɪɨɣɫɬɜɟ ɢɥɢ ɫɨɨɪɭɠɟɧɢɢ ɩɨɫɪɟɞɫɬɜɨɦ ɢɯ ɜɨɫɩɪɨɢɡɜɟɞɟɧɢɹ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ ɧɚ ɦɨɞɟɥɢ ɫɢɫɬɟɦɵ. Ɇɨɞɟɥɶ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɨɛɪɚɡɰɭ ɦɨɠɟɬ ɛɵɬɶ ɜɵɩɨɥɧɟɧɚ ɜ ɦɚɫɲɬɚɛɟ, ɤɚɤ ɦɟɧɶɲɟɦ, ɬɚɤ ɢ
ɛɨɥɶɲɟɦ ɟɞɢɧɢɰɵ. Ɇɟɬɨɞ ɩɨɞɨɛɢɹ ɢɥɢ ɦɟɬɨɞ ɪɚɡɦɟɪɧɨɫɬɟɣ ɩɨɡɜɨɥɹɸɬ
ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɦ ɞɨɥɠɟɧ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɩɪɨɰɟɫɫ ɜ ɦɨ91
ɞɟɥɢ, ɱɬɨɛɵ ɪɟɡɭɥɶɬɚɬɵ ɟɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ, ɩɪɟɞɫɬɚɜɥɟɧɧɵɟ ɜ ɮɨɪɦɟ ɡɚɜɢɫɢɦɨɫɬɟɣ ɦɟɠɞɭ ɛɟɡɪɚɡɦɟɪɧɵɦɢ ɤɨɦɩɥɟɤɫɚɦɢ, ɛɵɥɢ ɫɩɪɚɜɟɞɥɢɜɵ ɢ
ɞɥɹ ɨɛɪɚɡɰɚ.
Ɇɨɞɟɥɢɪɨɜɚɧɢɟ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ ɬɟɯɧɢɤɢ: ɷɧɟɪɝɟɬɢɤɟ, ɚɜɢɚɰɢɨɧɧɨɣ ɢ ɪɚɤɟɬɧɨɣ ɬɟɯɧɢɤɟ, ɝɢɞɪɨɬɟɯɧɢɤɟ ɢ ɞɪ.
Ɂɧɚɱɟɧɢɟ ɦɟɬɨɞɨɜ ɪɚɡɦɟɪɧɨɫɬɟɣ ɢ ɩɨɞɨɛɢɹ ɧɟ ɢɫɱɟɪɩɵɜɚɟɬɫɹ ɢɯ
ɩɪɢɦɟɧɟɧɢɟɦ ɜ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ. ɗɬɢ ɦɟɬɨɞɵ ɨɱɟɧɶ
ɩɨɥɟɡɧɵ ɢ ɩɪɢ ɬɟɨɪɟɬɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ɡɚɞɚɱ ɝɢɞɪɨɞɢɧɚɦɢɤɢ, ɬɟɩɥɨɨɛɦɟɧɚ, ɦɚɤɪɨɤɢɧɟɬɢɤɢ, ɦɟɯɚɧɢɤɢ ɢ ɞ.ɬ. Ɉɫɨɛɟɧɧɨ ɩɥɨɞɨɬɜɨɪɧɨ ɫɨɱɟɬɚɧɢɟ
ɦɟɬɨɞɨɜ ɪɚɡɦɟɪɧɨɫɬɟɣ ɢ ɩɨɞɨɛɢɹ ɫ ɮɢɡɢɱɟɫɤɢɦɢ ɫɨɨɛɪɚɠɟɧɢɹɦɢ, ɨɩɵɬɧɵɦɢ ɞɚɧɧɵɦɢ ɢ ɪɟɡɭɥɶɬɚɬɚɦɢ ɬɟɨɪɟɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ.
ɂɬɚɤ, ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ, ɱɬɨ ɩɟɪɜɨɟ
ɱɚɫɬɧɨɟ ɨɩɢɫɚɧɢɟ ɹɜɥɟɧɢɹ (ɢɫɤɨɦɭɸ ɡɚɤɨɧɨɦɟɪɧɨɫɬɶ) ɩɨɥɭɱɚɸɬ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɧɚ ɨɫɧɨɜɟ ɦɨɞɟɥɶɧɨɣ ɫɢɫɬɟɦɵ (ɢɥɢ «ɦɨɞɟɥɶɧɨɝɨ ɹɜɥɟɧɢɹ»), ɚ ɪɟɡɭɥɶɬɚɬɵ ɩɪɟɞɫɬɚɜɥɹɸɬ ɜ ɤɪɢɬɟɪɢɚɥɶɧɨɦ ɜɢɞɟ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ
ɥɟɝɤɨ ɢ ɛɵɫɬɪɨ ɩɨɥɭɱɚɬɶ ɞɚɧɧɵɟ ɞɥɹ ɞɪɭɝɢɯ ɹɜɥɟɧɢɣ, ɩɨɞɨɛɧɵɯ ɦɨɞɟɥɶɧɨɦɭ. Ɍɟɨɪɢɹ ɩɨɞɨɛɢɹ ɞɚɟɬ ɦɟɬɨɞɢɱɟɫɤɢɟ ɭɤɚɡɚɧɢɹ ɩɨ ɜɵɛɨɪɭ ɜɟɥɢɱɢɧ, ɢɡɦɟɪɹɟɦɵɯ ɜ ɨɩɵɬɟ, ɩɨ ɨɛɪɚɛɨɬɤɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ, ɩɨ
ɨɛɨɛɳɟɧɢɸ ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ ɧɚ ɞɪɭɝɢɟ ɹɜɥɟɧɢɹ, ɩɨɞɨɛɧɵɟ ɢɫɫɥɟɞɨɜɚɧɧɨɦɭ, ɚ ɬɚɤɠɟ ɩɨɡɜɨɥɹɟɬ ɪɚɫɫɱɢɬɚɬɶ ɢ ɩɨɫɬɪɨɢɬɶ ɦɨɞɟɥɶ, ɩɨɞɨɛɧɭɸ ɧɚɬɭɪɟ.
4.2. Ɍɟɨɪɟɦɵ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ
Ɍɟɨɪɢɹ ɩɨɞɨɛɢɹ ɛɚɡɢɪɭɟɬɫɹ ɧɚ ɬɪɟɯ ɬɟɨɪɟɦɚɯ.
ȼ ɡɧɚɦɟɧɢɬɨɣ ɤɧɢɝɟ «Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɧɚɱɚɥɚ ɧɚɬɭɪɚɥɶɧɨɣ ɮɢɥɨɫɨɮɢɢ» ɂ. ɇɶɸɬɨɧ ɜ 1686 ɝɨɞɭ ɧɚ ɩɪɢɦɟɪɟ ɩɨɞɨɛɧɨɝɨ ɬɟɱɟɧɢɹ ɞɜɭɯ ɠɢɞɤɨɫɬɟɣ ɜɩɟɪɜɵɟ ɪɚɫɩɪɨɫɬɪɚɧɢɥ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɩɨɞɨɛɢɟ ɧɚ ɮɢɡɢɱɟɫɤɢɟ
ɹɜɥɟɧɢɹ. ɇɨ ɟɫɥɢ ɇɶɸɬɨɧ ɜɵɫɤɚɡɚɥ ɬɨɥɶɤɨ ɨɫɧɨɜɧɭɸ ɢɞɟɸ ɩɨɞɨɛɢɹ, ɬɨ
ɮɪɚɧɰɭɡɫɤɢɣ ɦɚɬɟɦɚɬɢɤ ɀ. Ȼɟɪɬɪɚɧ ɜ 1848ɝ. ɞɚɥ ɫɬɪɨɝɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ
ɢ ɭɫɬɚɧɨɜɢɥ ɨɫɧɨɜɧɨɟ ɫɜɨɣɫɬɜɨ ɩɨɞɨɛɧɵɯ ɹɜɥɟɧɢɣ, ɧɚɡɜɚɧɧɨɟ ɩɨɡɠɟ
ɩɟɪɜɨɣ ɬɟɨɪɟɦɨɣ ɩɨɞɨɛɢɹ. Ɂɜɭɱɢɬ ɨɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.
1. Ɍɟɨɪɟɦɚ ɩɨɞɨɛɢɹ. ɉɨɞɨɛɧɵɟ ɦɟɠɞɭ ɫɨɛɨɣ ɹɜɥɟɧɢɹ ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ.
ɗɬɚ ɬɟɨɪɟɦɚ ɩɨɡɜɨɥɹɟɬ ɜɵɜɟɫɬɢ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɤɪɢɬɟɪɢɟɜ ɩɨɞɨɛɢɹ ɢ
ɭɤɚɡɵɜɚɟɬ, ɱɬɨ ɜ ɨɩɵɬɚɯ ɧɭɠɧɨ ɢɡɦɟɪɹɬɶ ɥɢɲɶ ɬɟ ɜɟɥɢɱɢɧɵ, ɤɨɬɨɪɵɟ
ɫɨɞɟɪɠɚɬɫɹ ɜ ɤɪɢɬɟɪɢɹɯ ɩɨɞɨɛɢɹ ɢɡɭɱɚɟɦɨɝɨ ɩɪɨɰɟɫɫɚ.
ɋɥɟɞɭɸɳɢɣ ɲɚɝ ɜ ɪɚɡɜɢɬɢɢ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɛɵɥ ɫɞɟɥɚɧ ɜ ɧɚɱɚɥɟ XX
ɜɟɤɚ, ɤɨɝɞɚ ɪɨɫɫɢɣɫɤɢɣ ɭɱɟɧɵɣ Ⱥ. Ɏɟɞɟɪɦɚɧ ɜ 1911 ɝ. ɢ ɚɦɟɪɢɤɚɧɫɤɢɣ
ɮɢɡɢɤ Ⱦɠ. Ȼɭɤɢɧɝɟɦ ɜ 1914 ɝ. ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɩɪɟɞɥɨɠɢɥɢ
ɜɬɨɪɭɸ ɬɟɨɪɟɦɭ ɩɨɞɨɛɢɹ:
92
2. Ɍɟɨɪɟɦɚ ɩɨɞɨɛɢɹ. ɂɫɯɨɞɧɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɭɪɚɜɧɟɧɢɹ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɞɚɧɧɨɟ ɮɢɡɢɱɟɫɤɨɟ ɹɜɥɟɧɢɟ, ɜɫɟɝɞɚ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɤɪɢɬɟɪɢɹɦɢ ɩɨɞɨɛɢɹ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɦɢ ɷɬɨ ɹɜɥɟɧɢɟ.
ɗɬɢ ɮɭɧɤɰɢɨɧɚɥɶɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɤɪɢɬɟɪɢɹɦɢ ɩɨɞɨɛɢɹ ɧɚɡɵɜɚɸɬɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɩɨɞɨɛɢɹ ɢɥɢ ɤɪɢɬɟɪɢɚɥɶɧɵɦɢ ɭɪɚɜɧɟɧɢɹɦɢ. ɂɡ
ɬɟɨɪɟɦɵ ɫɥɟɞɭɟɬ, ɱɬɨ ɪɟɡɭɥɶɬɚɬɵ ɨɩɵɬɨɜ ɧɟɨɛɯɨɞɢɦɨ ɨɛɪɚɛɚɬɵɜɚɬɶ ɢ
ɩɪɟɞɫɬɚɜɥɹɬɶ ɜ ɜɢɞɟ ɤɪɢɬɟɪɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ.
Ɍɪɟɬɶɹ ɬɟɨɪɟɦɚ ɩɨɞɨɛɢɹ ɛɵɥɚ ɩɪɟɞɥɨɠɟɧɚ ɫɨɜɟɬɫɤɢɦɢ ɭɱɟɧɵɦɢ
Ⱥ.Ⱥ. Ƚɭɯɦɚɧɨɦ ɢ Ɇ.ȼ. Ʉɢɪɩɢɱɟɜɵɦ ɜ 1936 ɝɨɞɭ.
3. Ɍɟɨɪɟɦɚ ɩɨɞɨɛɢɹ. ɉɨɞɨɛɧɵ ɬɟ ɹɜɥɟɧɢɹ, ɭɫɥɨɜɢɹ ɨɞɧɨɡɧɚɱɧɨɫɬɢ
ɤɨɬɨɪɵɯ ɩɨɞɨɛɧɵ ɢ ɞɥɹ ɤɨɬɨɪɵɯ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ, ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ
ɭɫɥɨɜɢɹ ɨɞɧɨɡɧɚɱɧɨɫɬɢ, ɱɢɫɥɟɧɧɨ ɪɚɜɧɵ.
Ɍɪɟɬɶɹ ɬɟɨɪɟɦɚ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɩɪɢɡɧɚɤɢ, ɩɨ ɤɨɬɨɪɵɦ ɨɩɪɟɞɟɥɹɸɬ,
ɤɚɤɢɟ ɹɜɥɟɧɢɹ ɩɨɞɨɛɧɵ ɞɪɭɝ ɞɪɭɝɭ, ɬ.ɟ. ɨɧɚ ɩɨɡɜɨɥɹɟɬ ɜɵɹɜɢɬɶ ɬɟ ɹɜɥɟɧɢɹ, ɧɚ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɵ ɪɟɡɭɥɶɬɚɬɵ ɷɤɫɩɟɪɢɦɟɧɬɚ,
ɩɨɥɭɱɟɧɧɵɟ ɧɚ ɦɨɞɟɥɶɧɨɣ ɫɢɫɬɟɦɟ.
4.3. Ɉɫ ɧɨɜɧɵɟ ɤɪɢɬɟɪɢɢ ɬɟɨɪ ɢɢ ɩɨɞɨɛɢɹ
ȼɫɟ ɨɫɧɨɜɧɵɟ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ ɬɟɩɥɨɜɵɯ, ɦɟɯɚɧɢɱɟɫɤɢɯ ɢ ɝɢɞɪɨɦɟɯɚɧɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ ɩɨɥɭɱɚɸɬɫɹ ɢɡ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɢɯ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɩɪɨɰɟɫɫ. ɇɚɩɪɢɦɟɪ, ɫɨɨɬɧɨɲɟɧɢɟ ɫɢɥ
ɢɧɟɪɰɢɢ
U l 3w
Fi ma =
W
ɢ ɦɚɫɫɨɜɵɯ ɫɢɥ (ɫɢɥ ɬɹɠɟɫɬɢ)
Fm mg U l 3 g
ɜ ɩɨɬɨɤɟ ɠɢɞɤɨɫɬɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɵɦ ɤɨɦɩɥɟɤɫɨɦ
Fm
U l 3g
gl
gl
,
2
Fi
U l 3w W w l W w
ɤɨɬɨɪɵɣ ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɬɟɪɢɟɦ Ɏɪɭɞɚ:
gl
.
(4.1)
Fr
w2
ȼ ɷɬɢɯ ɭɪɚɜɧɟɧɢɹɯ g – ɭɫɤɨɪɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ; U – ɩɥɨɬɧɨɫɬɶ ɠɢɞɤɨɫɬɢ; l 0 – ɯɚɪɚɤɬɟɪɧɵɣ ɥɢɧɟɣɧɵɣ ɪɚɡɦɟɪ; w – ɫɤɨɪɨɫɬɶ; W –
ɜɪɟɦɹ (ɢɥɢ ɦɚɫɲɬɚɛ ɜɪɟɦɟɧɢ).
93
Ʉɪɢɬɟɪɢɣ Ɏɪɭɞɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ ɦɚɫɫɨɜɵɯ ɫɢɥ (ɫɢɥ
ɬɹɠɟɫɬɢ) ɢ ɫɢɥ ɢɧɟɪɰɢɢ ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ.
ɋɜɹɡɶ ɦɟɠɞɭ ɫɢɥɚɦɢ ɢɧɟɪɰɢɢ Fi ɢ ɫɢɥɚɦɢ ɞɚɜɥɟɧɢɹ F p ' p l 2
( ' p – ɩɟɪɟɩɚɞ ɞɚɜɥɟɧɢɹ) ɩɪɢ ɜɵɧɭɠɞɟɧɧɨɦ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ
Fp
' pl 2
'p
'p
Fi
U l 3w W
ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɤɪɢɬɟɪɢɟɦ ɗɣɥɟɪɚ
Eu
U lw W
Uw2
'p
.
(4.2)
Uw2
ɂɡ ɜɵɪɚɠɟɧɢɹ (4.2) ɫɥɟɞɭɟɬ, ɱɬɨ ɱɢɫɥɨ ɗɣɥɟɪɚ ɹɜɥɹɟɬɫɹ ɦɟɪɨɣ ɨɬɧɨɲɟɧɢɹ ɩɟɪɟɩɚɞɚ ɫɬɚɬɢɱɟɫɤɢɯ ɞɚɜɥɟɧɢɣ ɜ ɩɨɬɨɤɟ (ɝɢɞɪɚɜɥɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ) ɤ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɢ ɩɨɬɨɤɚ.
Ɉɱɟɧɶ ɜɚɠɧɵɦ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɝɢɞɪɨɞɢɧɚɦɢɤɢ ɢ ɜɵɧɭɠɞɟɧɧɨɣ
ɤɨɧɜɟɤɰɢɢ ɹɜɥɹɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɵɣ ɤɨɦɩɥɟɤɫ, ɩɨɤɚɡɵɜɚɸɳɢɣ ɫɜɹɡɶ ɦɟɠP l 2' w
ɞɭ ɫɢɥɚɦɢ ɢɧɟɪɰɢɢ Fi ɢ ɫɢɥɚɦɢ ɜɹɡɤɨɫɬɢ FP
:
'l
Fi U l 3 ' w 'W U l ' l 'W U wl
.
P
P
FP P l 2 ' w ' l
ɗɬɨɬ ɤɨɦɩɥɟɤɫ ɧɚɡɜɚɧ ɤɪɢɬɟɪɢɟɦ Ɋɟɣɧɨɥɶɞɫɚ
U w l0 w l 0
,
(4.3)
Re
P
Q
ɝɞɟ Q – ɤɢɧɟɦɚɬɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ.
ɑɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɫɢɥɚɦɢ
ɢɧɟɪɰɢɢ ɢ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɬɪɟɧɢɹ (ɜɹɡɤɨɫɬɢ), ɤɨɬɨɪɨɟ ɨɩɪɟɞɟɥɹɟɬ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɣ ɪɟɠɢɦ ɜɵɧɭɠɞɟɧɧɨɝɨ ɞɜɢɠɟɧɢɹ ɫɪɟɞɵ.
ɉɪɢ ɫɜɨɛɨɞɧɨɦ ɞɜɢɠɟɧɢɢ ɫɪɟɞɵ (ɟɫɬɟɫɬɜɟɧɧɚɹ ɤɨɧɜɟɤɰɢɹ), ɤɨɝɞɚ
ɞɜɢɠɟɧɢɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ ɪɚɡɧɨɫɬɢ ɩɥɨɬɧɨɫɬɟɣ, ɜɵɡɜɚɧɧɨɣ ɧɟɪɚɜɧɨɦɟɪɧɨɫɬɶɸ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ, ɤɪɢɬɟɪɢɟɦ ɩɨɞɨɛɢɹ, ɨɩɪɟɞɟɥɹɸɳɢɦ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟ ɬɟɩɥɨɬɵ ɜ ɫɪɟɞɟ, ɹɜɥɹɟɬɫɹ ɤɪɢɬɟɪɢɣ Ƚɪɚɫɝɨɮɚ. Ɉɧ ɧɚɯɨɞɢɬɫɹ ɢɡ ɩɪɨɢɡɜɟɞɟɧɢɹ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ ɧɚ ɨɬɧɨɲɟɧɢɟ
ɩɨɞɴɟɦɧɨɣ ɫɢɥɵ FU U gE T ' T l 3 , ɝɞɟ E T – ɬɟɦɩɟɪɚɬɭɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɨɛɴɟɦɧɨɝɨ ɪɚɫɲɢɪɟɧɢɹ, ɤ ɫɢɥɟ ɜɹɡɤɨɫɬɢ FP :
Re
FU
FP
U wl U gE T ' T l 3
P
Pl 2 w l
ɢɥɢ
94
gl 3
P U
2
ET ' T
Gr
g l 0 3E T ' T
(4.4)
Q2
ɑɢɫɥɨ Ƚɪɚɫɝɨɮɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɩɨɞɴɟɦɧɨɣ ɫɢɥɨɣ, ɜɨɡɧɢɤɚɸɳɟɣ ɜ ɫɪɟɞɟ ɜɫɥɟɞɫɬɜɢɟ ɪɚɡɧɨɫɬɢ ɩɥɨɬɧɨɫɬɟɣ, ɢ ɫɢɥɨɣ ɦɨɥɟɤɭɥɹɪɧɨɝɨ ɬɪɟɧɢɹ (ɜɹɡɤɨɫɬɢ).
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɟɫɥɢ ɪɟɱɶ ɢɞɟɬ ɨ ɝɢɞɪɨɦɟɯɚɧɢɱɟɫɤɨɦ ɩɨɞɨɛɢɢ ɩɨɬɨɤɨɜ, ɬɨ ɞɥɹ ɧɢɯ ɜ ɥɸɛɵɯ ɫɯɨɞɫɬɜɟɧɧɵɯ ɬɨɱɤɚɯ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ Fr
(ɢɥɢ Gr ), E u ɢ Re ɢɦɟɸɬ ɨɞɢɧɚɤɨɜɵɟ ɡɧɚɱɟɧɢɹ.
ȼɚɠɧɟɣɲɢɟ ɤɪɢɬɟɪɢɢ ɬɟɩɥɨɜɨɝɨ ɩɨɞɨɛɢɹ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɢɡ
ɨɫɧɨɜɧɵɯ ɭɪɚɜɧɟɧɢɣ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ.
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɧɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ Ɏɭɪɶɟ (2.7) ɟɫɬɶ
(4.5)
QO ¬ªT1 T2 / G O ¼º F W O ' T l l 2W .
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɧɧɨɣ ɜ ɩɪɨɰɟɫɫɟ ɬɟɩɥɨɨɬɞɚɱɢ, ɫɨɫɬɚɜɥɹɟɬ
QD D T1 T2 d Fd W D ' T l 2W .
(4.6)
Ɍɟɩɥɨɬɚ, ɜɨɫɩɪɢɧɹɬɚɹ ɬɟɥɨɦ ɦɚɫɫɨɣ M
Q Mc T2 T1 cU l 3' T
(4.7)
Ɋɚɡɞɟɥɢɜ (4.5) ɧɚ (4.7), ɩɨɥɭɱɚɟɦ ɛɟɡɪɚɡɦɟɪɧɵɣ ɤɨɦɩɥɟɤɫ
2
QO O ' Tl W 1 l aW
,
Q cU l 3' T
l2
ɹɜɥɹɸɳɢɣɫɹ ɨɞɧɢɦ ɢɡ ɜɚɠɧɟɣɲɢɯ ɤɪɢɬɟɪɢɟɜ ɩɨɞɨɛɢɹ – ɤɪɢɬɟɪɢɟɦ Ɏɭɪɶɟ:
F o aW l 0 2 ,
(4.8)
ɝɞɟ a – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ, a O / cU .
ɑɢɫɥɨ Ɏɭɪɶɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɛɟɡɪɚɡɦɟɪɧɨɟ ɜɪɟɦɹ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɫɤɨɪɨɫɬɶɸ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ, ɮɢɡɢɱɟɫɤɢɦɢ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɢ ɪɚɡɦɟɪɚɦɢ ɬɟɥɚ.
Ʉɪɢɬɟɪɢɣ Ɏɭɪɶɟ ɜɦɟɫɬɟ ɫ ɤɪɢɬɟɪɢɟɦ Ȼɢɨ
B i D l0 O w ,
(4.9)
ɝɞɟ O w – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɫɬɟɧɤɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ
ɩɪɨɰɟɫɫɵ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɨɬɵ.
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɱɢɫɥɚ Ȼɢɨ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɭɫɬɚɧɨɜɥɟɧɢɢ ɫɨɨɬɧɨɲɟɧɢɹ ɦɟɠɞɭ ɢɧɬɟɧɫɢɜɧɨɫɬɹɦɢ ɬɟɩɥɨɨɬɞɚɱɢ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɢ ɩɨɞɜɨɞɚ ɬɟɩɥɨɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɢɡ ɜɧɭɬɪɟɧɧɢɯ ɫɥɨɟɜ ɬɟɥɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ.
ȿɫɥɢ ɜɡɹɬɶ ɨɬɧɨɲɟɧɢɟ (4.7) ɤ (4.5), ɬɨ ɩɨɥɭɱɢɦ
95
cU l 3' T
cU wǻT w l
.
O l 2' T W l O ' T l a
ɗɬɨɬ ɤɨɦɩɥɟɤɫ ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɬɟɪɢɟɦ ɉɟɤɥɟ
w l0
Pe
.
(4.10)
a
Ʉɪɢɬɟɪɢɣ ɉɟɤɥɟ ɟɫɬɶ ɨɬɧɨɲɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɤɨɧɜɟɤɰɢɟɣ, ɤ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɩɟɪɟɞɚɜɚɟɦɨɝɨ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɬ.ɟ. ɱɢɫɥɨ ɉɟɤɥɟ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɩɟɪɟɧɨɫɨɦ ɬɟɩɥɨɬɵ ɤɨɧɜɟɤɰɢɟɣ ɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɜ ɩɨɬɨɤɟ.
Q
QO
ȼɚɠɧɵɣ ɜ ɬɟɨɪɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɤɪɢɬɟɪɢɣ ɇɭɫɫɟɥɶɬɚ
ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ ɜɵɪɚɠɟɧɢɣ (4.5) ɢ (4.6):
QD D l 2' T W
Dl
.
QO O l 2' T W 1 l O
ɑɢɫɥɨ ɇɭɫɫɟɥɶɬɚ ɹɜɥɹɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɬɞɚɱɢ
D l0
Nu
(4.11)
O
ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɝɪɚɧɢɰɟ «ɬɜɟɪɞɨɟ ɬɟɥɨ – ɠɢɞɤɨɫɬɶ». ɑɟɦ ɢɧɬɟɧɫɢɜɧɟɟ ɩɪɨɬɟɤɚɟɬ ɩɪɨɰɟɫɫ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɬɟɦ ɛɨɥɶɲɟ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ D , ɬɟɦ ɛɨɥɶɲɟ Nu .
ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɱɢɫɥɨ ɇɭɫɫɟɥɶɬɚ ɹɜɥɹɟɬɫɹ ɨɩɪɟɞɟɥɹɟɦɵɦ, ɬɚɤ
ɤɚɤ ɜ ɧɟɝɨ ɜɯɨɞɢɬ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ, ɤɨɬɨɪɵɣ ɩɪɢ ɢɫɫɥɟɞɨɜɚɧɢɢ
ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɹɜɥɹɟɬɫɹ ɨɛɵɱɧɨ ɢɫɤɨɦɨɣ ɜɟɥɢɱɢɧɨɣ. ɉɨ
ɜɧɟɲɧɟɦɭ ɜɢɞɭ ɤɪɢɬɟɪɢɣ Nu ɫɨɜɩɚɞɚɟɬ ɫ ɤɪɢɬɟɪɢɟɦ B i , ɧɨ ɦɟɠɞɭ ɧɢɦɢ ɫɭɳɟɫɬɜɭɟɬ ɩɪɢɧɰɢɩɢɚɥɶɧɨɟ ɪɚɡɥɢɱɢɟ. ȼ ɤɪɢɬɟɪɢɣ Ȼɢɨ ɜɯɨɞɢɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɫɬɟɧɤɢ, ɚ ɜ ɤɪɢɬɟɪɢɣ ɇɭɫɫɟɥɶɬɚ – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɫɪɟɞɵ, ɨɦɵɜɚɸɳɟɣ ɫɬɟɧɤɭ. Ʉɪɢɬɟɪɢɣ Ȼɢɨ ɢɫɩɨɥɶɡɭɸɬ ɞɥɹ ɨɩɢɫɚɧɢɹ ɜɧɭɬɪɟɧɧɟɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɚ ɤɪɢɬɟɪɢɣ ɇɭɫɫɟɥɶɬɚ – ɞɥɹ ɨɩɢɫɚɧɢɹ ɜɧɟɲɧɟɝɨ
ɬɟɩɥɨɨɛɦɟɧɚ.
ɂɡ ɤɪɢɬɟɪɢɟɜ ɉɟɤɥɟ ɢ Ɋɟɣɧɨɥɶɞɫɚ ɦɨɠɧɨ ɨɛɪɚɡɨɜɚɬɶ ɟɳɟ ɨɞɢɧ ɤɪɢɬɟɪɢɣ – ɤɪɢɬɟɪɢɣ ɉɪɚɧɞɬɥɹ:
Pe w l v
v
Pr
(4.12)
Re a wl a
Ʉɪɢɬɟɪɢɣ ɉɪɚɧɞɬɥɹ, ɫɨɞɟɪɠɚɳɢɣ ɬɨɥɶɤɨ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɠɢɞɤɨɫɬɢ, ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɜɥɢɹɧɢɟ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ
ɫɪɟɞɵ ɧɚ ɤɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ ɢ ɹɜɥɹɟɬɫɹ ɦɟɪɨɣ ɩɨɞɨɛɢɹ ɩɨɥɟɣ
ɬɟɦɩɟɪɚɬɭɪ ɢ ɫɤɨɪɨɫɬɟɣ. Ʉɢɧɟɦɚɬɢɱɟɫɤɚɹ ɜɹɡɤɨɫɬɶ ɫɭɳɟɫɬɜɟɧɧɨ
96
ɜɥɢɹɟɬ ɧɚ ɯɚɪɚɤɬɟɪ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ, ɚ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɶ – ɧɚ ɩɪɨɰɟɫɫ ɬɟɩɥɨɨɛɦɟɧɚ.
ɋ ɤɪɢɬɟɪɢɹɦɢ Ɋɟɣɧɨɥɶɞɫɚ, ɇɭɫɫɟɥɶɬɚ ɢ ɉɪɚɧɞɬɥɹ ɦɵ ɜɫɬɪɟɬɢɥɢɫɶ
ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɱɚɫɬɧɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ
(ɪɚɡɞɟɥ 3.3).
ɉɪɢ ɬɟɩɥɨɜɨɦ ɩɨɞɨɛɢɢ ɫɢɫɬɟɦ ɢɯ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ F o , Pe (ɢɥɢ
Re ), Nu ɞɨɥɠɧɵ ɢɦɟɬɶ ɨɞɢɧɚɤɨɜɵɟ ɱɢɫɥɨɜɵɟ ɡɧɚɱɟɧɢɹ.
Ⱦɥɹ ɩɨɞɨɛɧɵɯ ɫɢɫɬɟɦ, ɤɪɨɦɟ ɬɟɩɥɨɜɨɝɨ ɩɨɞɨɛɢɹ, ɞɨɥɠɧɨ ɫɨɛɥɸɞɚɬɶɫɹ ɬɚɤɠɟ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɟ ɢ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɩɨɞɨɛɢɟ.
4.4. ɏɚɪɚɤɬɟɪɧɵɟ ɦɚɫɲɬɚɛɵ
ȼɨ ɦɧɨɝɢɟ ɤɪɢɬɟɪɢɢ ɜɯɨɞɢɬ ɥɢɧɟɣɧɚɹ ɜɟɥɢɱɢɧɚ l 0 , ɧɚɡɵɜɚɟɦɚɹ ɨɩɪɟɞɟɥɹɸɳɢɦ ɥɢɧɟɣɧɵɦ ɪɚɡɦɟɪɨɦ. Ɉɩɪɟɞɟɥɹɸɳɢɦ ɧɚɡɵɜɚɟɬɫɹ ɪɚɡɦɟɪ, ɨɬ
ɤɨɬɨɪɨɝɨ ɜ ɛɨɥɶɲɟɣ ɫɬɟɩɟɧɢ ɡɚɜɢɫɢɬ ɪɚɡɜɢɬɢɟ ɩɪɨɰɟɫɫɚ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɨɛɦɟɧɚ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ ɜɧɭɬɪɢ ɤɪɭɝɥɵɯ ɝɥɚɞɤɢɯ ɬɪɭɛ ɡɚ ɨɩɪɟɞɟɥɹɸɳɢɣ ɪɚɡɦɟɪ ɩɪɢɧɢɦɚɸɬ ɜɧɭɬɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɬɪɭɛɵ l 0 d in . ȼ ɫɥɭɱɚɟ ɩɨɩɟɪɟɱɧɨɝɨ ɨɦɵɜɚɧɢɹ ɝɥɚɞɤɨɣ ɬɪɭɛɵ ɢɥɢ ɩɭɱɤɚ
ɬɪɭɛ ɡɚ ɨɩɪɟɞɟɥɹɸɳɢɣ ɪɚɡɦɟɪ ɩɪɢɧɢɦɚɸɬ ɧɚɪɭɠɧɵɣ ɞɢɚɦɟɬɪ ɬɪɭɛɵ
l 0 d ext . ɉɪɢ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ ɜ ɤɚɧɚɥɚɯ ɜ ɤɚɱɟɫɬɜɟ ɨɩɪɟɞɟɥɹɸɳɟɝɨ
ɪɚɡɦɟɪɚ ɢɫɩɨɥɶɡɭɸɬ ɷɤɜɢɜɚɥɟɧɬɧɵɣ ɞɢɚɦɟɬɪ
l 0 d eq 4F 3 ,
ɝɞɟ F – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ ɤɚɧɚɥɚ, 3 – ɩɟɪɢɦɟɬɪ ɫɟɱɟɧɢɹ,
ɱɟɪɟɡ ɤɨɬɨɪɵɣ ɩɟɪɟɞɚɟɬɫɹ ɬɟɩɥɨɬɚ.
Ɏɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɠɢɞɤɨɫɬɢ, ɜɯɨɞɹɳɢɟ ɜ ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ, ɡɚɜɢɫɹɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ. Ɏɢɡɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɫɪɟɞɵ ɞɥɹ ɪɚɫɱɟɬɚ
ɛɟɡɪɚɡɦɟɪɧɵɯ ɤɪɢɬɟɪɢɟɜ ɧɚɯɨɞɹɬɫɹ ɩɪɢ ɨɞɧɨɣ ɨɩɪɟɞɟɥɹɸɳɟɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɡɚ ɤɨɬɨɪɭɸ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɧɹɬɚ ɬɟɦɩɟɪɚɬɭɪɚ ɠɢɞɤɨɫɬɢ ɜ ɹɞɪɟ ɩɨɬɨɤɚ T f ɢɥɢ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɦɟɠɞɭ ɫɬɟɧɤɨɣ ɢ ɩɨɬɨɤɨɦ
'T
Ts T f . ɉɪɢ ɧɟɛɨɥɶɲɢɯ ɢɡɦɟɧɟɧɢɹɯ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɪɟɞɟɥɚɯ ɭɱɚ-
ɫɬɤɚ ɬɟɩɥɨɨɛɦɟɧɚ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɫɪɟɞɧɟɚɪɢɮɦɟɬɢɱɟɫɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɩɨɬɨɤɚ
T f ,1 T f ,2
Tm
.
2
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɚɯɨɞɹɬ ɫɪɟɞɧɟɥɨɝɚɪɢɮɦɢɱɟɫɤɭɸ ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɦɟɠɞɭ ɫɪɟɞɧɟɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ɫɬɟɧɤɢ ɢ ɬɟɦɩɟɪɚɬɭɪɨɣ ɩɨɬɨɤɚ
Te T f
,
Tm
2.3 ˜ lg T e T f
97
ɝɞɟ Te – ɪɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɠɢɞɤɨɫɬɢ ɢ ɫɬɟɧɤɢ ɧɚ ɜɯɨɞɟ, T f – ɧɚ ɜɵɯɨɞɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɝɨ ɭɱɚɫɬɤɚ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɭɤɚɡɚɬɶ, ɤɚɤɚɹ ɢɡ ɬɟɦɩɟɪɚɬɭɪ ɠɢɞɤɨɫɬɢ ɩɪɢɧɹɬɚ ɜ ɤɚɱɟɫɬɜɟ ɨɩɪɟɞɟɥɹɸɳɟɣ, ɪɹɞɨɦ ɫ ɤɪɢɬɟɪɢɟɦ ɩɪɨɫɬɚɜɥɹɸɬ ɢɧɞɟɤɫ P rS , N u f , R e m ɢ ɬ.ɞ.
ȼ ɤɪɢɬɟɪɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɹɯ ɡɚ ɫɤɨɪɨɫɬɶ ɩɨɬɨɤɚ ɩɪɢɧɢɦɚɸɬ ɭɫɪɟɞɧɟɧɧɨɟ ɟɟ ɡɧɚɱɟɧɢɟ ɜ ɡɚɞɚɧɧɨɦ ɩɨɩɟɪɟɱɧɨɦ ɫɟɱɟɧɢɢ
w V F M pF ,
ɝɞɟ V – ɨɛɴɟɦɧɵɣ ɪɚɫɯɨɞ ɠɢɞɤɨɫɬɢ; F – ɩɥɨɳɚɞɶ ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ
ɩɨɬɨɤɚ; U – ɩɥɨɬɧɨɫɬɶ ɠɢɞɤɨɫɬɢ.
4.5. ɉɪɢɦɟɪɵ ɤɪɢɬɟɪɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ
ɉɪɢ ɢɡɭɱɟɧɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚɢɛɨɥɶɲɢɣ ɩɪɚɤɬɢɱɟɫɤɢɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɩɪɟɞɟɥɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ D ,
ɤɨɬɨɪɵɣ ɜɯɨɞɢɬ ɬɨɥɶɤɨ ɜ ɤɪɢɬɟɪɢɣ Nu . ɉɨɷɬɨɦɭ ɭɪɚɜɧɟɧɢɹ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɪɟɲɚɸɬɫɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɬɨɝɨ ɱɢɫɥɚ. Ɍɟɨɪɢɹ ɩɨɞɨɛɢɹ
ɩɨɡɜɨɥɹɟɬ ɜ ɨɛɳɟɦ ɜɢɞɟ ɭɫɬɚɧɨɜɢɬɶ ɤɪɢɬɟɪɢɚɥɶɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ, ɞɨɫɬɚɬɨɱɧɨ ɩɨɥɧɨ ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɩɪɨɰɟɫɫ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ.
Ɉɛɨɛɳɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɢɦɟɟɬ ɜɢɞ
Nu f Ho , Fo ,Re ,Pr,Gr ,l l 0 ,
(4.13)
ɝɞɟ H o wW l 0 – ɤɪɢɬɟɪɢɣ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɣ ɝɨɦɨɯɪɨɧɧɨɫɬɢ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɫɤɨɪɨɫɬɶ ɢɡɦɟɧɟɧɢɹ ɩɨɥɹ ɫɤɨɪɨɫɬɟɣ ɞɜɢɠɭɳɟɣɫɹ ɫɪɟɞɵ ɜɨ
ɜɪɟɦɟɧɢ.
Ʉɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ Ho ɢ F o ɹɜɥɹɸɬɫɹ ɨɫɧɨɜɧɵɦɢ ɤɪɢɬɟɪɢɹɦɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɩɪɨɰɟɫɫɨɜ. Ɍɚɤ ɤɚɤ ɷɤɫɩɟɪɢɦɟɧɬɵ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɨɬɞɚɱɢ ɩɪɨɜɨɞɹɬ ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɦ ɪɟɠɢɦɟ, ɬɨ ɭɪɚɜɧɟɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɡɚɩɢɲɟɬɫɹ ɜ ɜɢɞɟ
N u f Re, Pr,Gr ,l l 0 .
(4.14)
ɉɪɢ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɭɱɟɬɚ ɜɥɢɹɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ
ɫɜɨɣɫɬɜɚ ɫɪɟɞɵ (ɜ ɭɦɟɪɟɧɧɨɦ ɞɢɚɩɚɡɨɧɟ ɢɯ ɢɡɦɟɧɟɧɢɹ), ɚ ɬɚɤɠɟ ɧɚɩɪɚɜɥɟɧɢɹ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɜ ɭɪɚɜɧɟɧɢɟ ɩɨɞɨɛɢɹ ɜɜɨɞɢɬɫɹ ɨɬɧɨɲɟɧɢɟ ɱɢɫɟɥ ɉɪɚɧɞɬɥɹ ɞɥɹ ɠɢɞɤɨɫɬɢ, ɜɵɱɢɫɥɟɧɧɵɯ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ɩɨɬɨɤɚ ɢ
ɫɬɟɧɤɢ
§
·
§ Pr f ·
,
N u f ¨ R e,P r,G r ,¨
l
l
¸ 0 ¸¸
¨
© Prs ¹
©
¹
ɢɥɢ
98
n
p§
0 ,2 5
Pr f ·
N u C Re Pr G r ¨
(4.15)
l l0 q .
¸
Pr
© s¹
Ɍɟɨɪɢɹ ɩɨɞɨɛɢɹ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɬɨɥɶɤɨ ɨɛɳɢɣ ɜɢɞ ɤɪɢɬɟɪɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɤɪɢɬɟɪɢɹ Nu ɨɬ ɨɩɪɟɞɟɥɹɸɳɢɯ ɤɪɢɬɟɪɢɟɜ ɧɚɯɨɞɢɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ, ɢ ɤɨɧɫɬɚɧɬɵ m , n , p , q ,C
ɹɜɥɹɸɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɨɩɪɟɞɟɥɹɟɦɵɦɢ.
ȼ ɨɬɞɟɥɶɧɵɯ ɫɥɭɱɚɹɯ ɭɪɚɜɧɟɧɢɟ ɤɪɢɬɟɪɢɚɥɶɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɭɩɪɨɳɚɟɬɫɹ. Ɍɚɤ, ɞɥɹ ɜɵɧɭɠɞɟɧɧɨɝɨ ɬɭɪɛɭɥɟɧɬɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜɥɢɹɧɢɟɦ ɫɜɨɛɨɞɧɨɣ ɤɨɧɜɟɤɰɢɢ ɩɪɟɧɟɛɪɟɝɚɸɬ, ɬ.ɟ. ɧɟ ɭɱɢɬɵɜɚɸɬ ɤɪɢɬɟɪɢɣ Ƚɪɚɫɝɨɮɚ
§
·
§ Pr f ·
,
N u f ¨ R e,P r,¨
l
l
¸ 0 ¸¸ .
¨
Pr
© s¹
©
¹
ɉɪɢ ɫɜɨɛɨɞɧɨɦ ɞɜɢɠɟɧɢɢ ɠɢɞɤɨɫɬɢ ɢɡ ɤɪɢɬɟɪɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
ɭɛɢɪɚɸɬ ɤɪɢɬɟɪɢɣ Ɋɟɣɧɨɥɶɞɫɚ ɜɜɢɞɭ ɨɬɫɭɬɫɬɜɢɹ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ
§
·
§ Pr f ·
,
N u f ¨ Pr ,G r ,¨
l
l
¸ 0 ¸¸ .
¨
Pr
© s¹
©
¹
ȿɫɥɢ ɨɦɵɜɚɸɳɟɣ ɫɪɟɞɨɣ ɹɜɥɹɟɬɫɹ ɝɚɡ, ɬɨ ɢɡ ɤɪɢɬɟɪɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɫɤɥɸɱɚɸɬ ɨɬɧɨɲɟɧɢɟ P r f Prs , ɩɨɫɤɨɥɶɤɭ ɞɥɹ ɝɚɡɨɜ ɱɢɫɥɨ ɉɪɚɧm
ɞɬɥɹ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɱɬɢ ɧɟ ɡɚɜɢɫɢɬ.
Ⱦɥɹ ɢɞɟɚɥɶɧɵɯ ɝɚɡɨɜ ɤɪɢɬɟɪɢɣ ɉɪɚɧɞɬɥɹ ɡɚɜɢɫɢɬ ɨɬ ɚɬɨɦɧɨɫɬɢ ɝɚɡɚ.
Ⱦɥɹ ɨɞɧɨɚɬɨɦɧɵɯ ɝɚɡɨɜ Pr 0,66 ; ɞɥɹ ɞɜɭɯɚɬɨɦɧɵɯ (ɧɚɩɪɢɦɟɪ, ɫɭɯɨɣ
ɜɨɡɞɭɯ) Pr 0,71 ; ɞɥɹ ɬɪɟɯɚɬɨɦɧɵɯ Pr 0,84 ; ɞɥɹ ɦɧɨɝɨɚɬɨɦɧɵɯ Pr 1,0 .
ɉɪɢɦɟɪɵ ɪɚɫɱɟɬɨɜ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɨɬɞɚɱɢ ɫɨɞɟɪɠɚɬɫɹ ɜ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɟ ɢ ɥɢɬɟɪɚɬɭɪɟ ɫɩɟɰɢɚɥɶɧɨɝɨ ɯɚɪɚɤɬɟɪɚ ɩɨ ɬɟɨɪɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ.
4.6. ɇɟɤɨɬɨɪɵɟ ɷɦɩɢɪɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ
Ⱦɨɦɢɧɢɪɨɜɚɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ, ɚ ɬɚɤɠɟ ɧɟɞɨɫɬɚɬɨɱɧɨɫɬɶ ɪɚɡɜɢɬɢɹ ɮɟɧɨɦɟɧɨɥɨɝɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɬɪɟɛɭɸɬ ɨɛɪɚɳɚɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɢɦɟɸɳɢɟɫɹ ɷɦɩɢɪɢɱɟɫɤɢɟ ɮɨɪɦɭɥɵ.
Ɋɚɛɨɬɚ ɫ ɷɦɩɢɪɢɱɟɫɤɢɦɢ ɮɨɪɦɭɥɚɦɢ, ɢɯ ɜɵɛɨɪ ɢ ɩɪɢɦɟɧɟɧɢɟ ɬɪɟɛɭɸɬ ɨɬ ɢɫɫɥɟɞɨɜɚɬɟɥɹ ɢ ɢɧɠɟɧɟɪɚ, ɜɟɫɶɦɚ ɨɛɴɟɦɧɵɯ ɢ ɩɪɨɜɟɪɟɧɧɵɯ
ɡɧɚɧɢɣ ɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɪɨɰɟɫɫɟ, ɚ ɬɚɤɠɟ ɨɩɪɟɞɟɥɟɧɧɨɝɨ ɨɩɵɬɚ ɨɛɳɟɧɢɹ ɫ ɷɬɢɦ ɤɚɩɪɢɡɧɵɦ ɢɧɫɬɪɭɦɟɧɬɨɦ. ɗɦɩɢɪɢɱɟɫɤɢɯ ɮɨɪɦɭɥ ɫɭɳɟ99
ɫɬɜɭɟɬ ɞɨɫɬɚɬɨɱɧɨ ɦɧɨɝɨ. Ɉɧɢ ɨɬɪɚɠɚɸɬ ɪɚɡɥɢɱɧɵɟ ɦɟɬɨɞɢɤɢ ɮɢɡɢɱɟɫɤɢɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ, ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɭɸ ɨɪɢɟɧɬɚɰɢɸ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɟ, ɞɥɹ ɧɢɯ ɫɜɨɣɫɬɜɟɧɧɵ ɫɭɳɟɫɬɜɟɧɧɵɟ ɪɚɫɯɨɠɞɟɧɢɹ. ɂ ɷɬɨ ɟɫɬɟɫɬɜɟɧɧɨ, ɬɚɤ ɤɚɤ ɭɫɥɨɜɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ, ɧɚ ɛɚɡɟ ɤɨɬɨɪɵɯ ɮɨɪɦɭɥɵ ɩɨɥɭɱɟɧɵ,
ɜɩɨɥɧɟ ɤɨɧɤɪɟɬɧɵ. ɉɨɷɬɨɦɭ ɩɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɨɞɟɥɟɣ ɷɦɩɢɪɢɱɟɫɤɢɟ
ɮɨɪɦɭɥɵ ɫɥɟɞɭɟɬ ɢɫɩɨɥɶɡɨɜɚɬɶ ɫ ɛɨɥɶɲɨɣ ɨɫɬɨɪɨɠɧɨɫɬɶɸ, ɛɨɥɟɟ ɞɨɜɟɪɹɹ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɡɚɤɨɧɚɦ ɫɨɯɪɚɧɟɧɢɹ.
Ɍɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɬɜɟɪɞɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɨɛɬɟɤɚɟɦɨɣ ɟɟ ɠɢɞɤɨɫɬɶɸ ɦɨɠɟɬ ɢɦɟɬɶ ɪɚɡɥɢɱɧɵɟ ɮɨɪɦɵ. ɗɬɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢɥɢ ɤɨɧɞɭɤɬɢɜɧɚɹ ɬɟɩɥɨɩɟɪɟɞɚɱɚ; ɤɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ ɩɪɢ ɨɛɬɟɤɚɧɢɢ ɩɨɜɟɪɯɧɨɫɬɢ; ɞɢɮɮɭɡɧɵɣ (ɜɢɯɪɟɜɨɣ) ɩɟɪɟɧɨɫ ɬɟɩɥɚ; ɬɟɩɥɨɨɛɦɟɧ ɩɪɢ ɤɨɧɞɟɧɫɚɰɢɢ ɩɚɪɨɜ ɧɚ ɯɨɥɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ; ɬɟɩɥɨɨɛɦɟɧ ɩɪɢ ɤɢɩɟɧɢɢ
ɠɢɞɤɨɫɬɢ ɧɚ ɧɚɝɪɟɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ; ɬɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ.
ȿɫɬɟɫɬɜɟɧɧɨ ɩɪɟɞɩɨɥɨɠɢɬɶ (ɷɬɨ ɫɥɟɞɭɟɬ ɢ ɢɡ ɬɟɨɪɟɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɝɨ ɜ ɪɚɡɞɟɥɟ 3.3), ɱɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ
ɨɛɬɟɤɚɟɦɨɣ ɝɪɚɧɢɰɟ ɨɛɭɫɥɨɜɥɟɧɚ ɫɤɨɪɨɫɬɶɸ ɢ ɯɚɪɚɤɬɟɪɨɦ ɬɟɱɟɧɢɹ, ɜɹɡɤɢɦɢ ɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ ɫɨɩɪɢɤɚɫɚɸɳɢɯɫɹ ɫɪɟɞ. ɉɨɷɬɨɦɭ
ɩɪɢ ɷɦɩɢɪɢɱɟɫɤɨɦ ɨɩɪɟɞɟɥɟɧɢɢ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ Nu ɩɪɢɧɢɦɚɸɬ ɫɥɟɞɭɸɳɭɸ ɮɨɪɦɭɥɭ ɞɥɹ ɜɵɪɚɠɟɧɢɹ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ
(4.16)
N u C Re D Pr E ,
ɝɞɟ D ,E ,C ! 0 – ɱɢɫɥɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɤɨɬɨɪɵɟ ɨɬɪɚɠɚɸɬ ɤɚɱɟɫɬɜɚ
ɬɟɱɟɧɢɹ ɢ ɬɟɩɥɨɨɛɦɟɧɚ, ɫɨɞɟɪɠɚɬ ɜɫɟɜɨɡɦɨɠɧɵɟ ɩɨɩɪɚɜɨɱɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɧɚ ɭɫɪɟɞɧɟɧɢɟ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ ɩɨ ɩɨɜɟɪɯɧɨɫɬɢ, ɭɱɟɬɚ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɢ ɬ.ɞ. Ⱦɚɧɧɵɣ ɜɢɞ ɡɚɜɢɫɢɦɨɫɬɢ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ ɨɬ
ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɧɟ ɫɜɹɡɚɧ ɫ ɤɚɤɨɣ-ɥɢɛɨ ɦɨɞɟɥɶɸ, ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɨ ɦɟɯɚɧɢɡɦɟ ɬɟɩɥɨɨɛɦɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɚ ɨɩɪɟɞɟɥɟɧ, ɩɪɟɠɞɟ ɜɫɟɝɨ,
ɠɟɥɚɧɢɟɦ ɩɨɧɢɡɢɬɶ ɪɚɡɦɟɪɧɨɫɬɶ ɡɚɞɚɱɢ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɬɚɤɚɹ ɜɨɡɦɨɠɧɨɫɬɶ ɫɭɳɟɫɬɜɭɟɬ.
ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɧɟɬ ɢɧɵɯ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɚɪɚɦɟɬɪɨɜ,
ɤɪɨɦɟ ɱɢɫɟɥ R e, P r , ɤɨɬɨɪɵɟ ɛɵ ɫɨɞɟɪɠɚɥɢ ɬɨɥɶɤɨ ɫɤɨɪɨɫɬɶ ɬɟɱɟɧɢɹ,
ɜɹɡɤɢɟ ɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɠɢɞɤɨɫɬɢ. ɂɦɟɧɧɨ ɷɬɢɦ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨɦ ɨɛɴɹɫɧɹɟɬɫɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɧɨɫɬɶ ɮɨɪɦɭɥɵ (4.16) ɞɥɹ ɤɚɱɟɫɬɜɟɧɧɨ
ɪɚɡɥɢɱɧɵɯ ɜɢɞɨɜ ɬɟɱɟɧɢɹ.
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɨɜ ɩɨɥɭɷɦɩɢɪɢɱɟɫɤɢɯ ɮɨɪɦɭɥ ɞɚɥɟɟ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɧɟɤɨɬɨɪɵɟ ɩɪɚɤɬɢɱɟɫɤɢ ɩɨɥɟɡɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɞɥɹ ɥɚɦɢɧɚɪɧɵɯ ɢ ɬɭɪɛɭɥɟɧɬɧɵɯ ɬɟɱɟɧɢɣ16
16
Ⱦɟɣɧɟɤɨ ȼ.ȼ. Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɞɢɧɚɦɢɤɢ ɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɢ ɬɟɩɥɨɨɛɦɟɧɚ.
ɇɨɜɨɫɢɛɢɪɫɤ: 1996. 360 ɫ.
100
ȼ ɩɪɟɞɫɬɚɜɥɟɧɧɵɯ ɧɢɠɟ ɮɨɪɦɭɥɚɯ ɡɧɚɱɟɧɢɹ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ ɨɩɪɟɞɟɥɟɧɵ ɞɥɹ ɫɪɟɞɧɟɝɨ ɢɥɢ «ɩɪɟɨɛɥɚɞɚɸɳɟɝɨ» ɡɧɚɱɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ. Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɨɩɪɟɞɟɥɟɧɵ ɩɪɢ ɫɪɟɞɧɟɚɪɢɮɦɟɬɢɱɟɫɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɦɟɠɞɭ ɫɬɟɧɤɨɣ ɢ ɬɟɱɟɧɢɟɦ. ɑɢɫɥɨ ɉɪɚɧɞɬɥɹ, ɨɩɪɟɞɟɥɹɸɳɟɟ
ɫɨɨɬɧɨɲɟɧɢɟ ɞɢɮɮɭɡɧɵɯ ɫɜɨɣɫɬɜ ɩɟɪɟɧɨɫɚ ɢɦɩɭɥɶɫɚ ɢ ɬɟɩɥɚ, ɢɡɦɟɧɹɟɬɫɹ ɜ ɩɪɟɞɟɥɚɯ, ɭɤɚɡɚɧɧɵɯ ɜ ɬɚɛɥ. 4.1.
Ɍɚɛɥɢɰɚ 4.1
ɂɧɬɟɪɜɚɥ ɢɡɦɟɧɟɧɢɹ ɱɢɫɥɚ ɉɪɚɧɞɬɥɹ
ɠɢɞɤɢɟ ɦɟɬɚɥɥɵ
ɝɚɡɵ
ɜɨɞɚ
Pr 1
Pr  >0,5; 1@
ɥɟɝɤɢɟ ɨɪɝɚɧɢɱɟɫɤɢɟ ɠɢɞɤɨɫɬɢ
P r  > 5 ; 60 @
ɦɚɫɥɚ
Pr ! 50
Pr  >1; 14@
ɉɪɨɞɨɥɶɧɨɟ ɨɛɬɟɤɚɧɢɟ ɩɥɚɫɬɢɧɵ17
ȼ ɫɥɭɱɚɟ ɨɛɬɟɤɚɧɢɹ ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɵ ɬɨɪɦɨɠɟɧɢɟ ɬɟɱɟɧɢɹ ɜɵɡɵɜɚɸɬ ɫɢɥɵ ɜɹɡɤɨɫɬɢ, ɡɚɜɢɫɹɳɢɟ ɨɬ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ W
wu
(4.17)
W P ,
wy
ɝɞɟ P Q U - ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ. ȿɫɥɢ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɜɵɪɚɠɟɧɨ ɜ ɇ/ɦ2, ɚ ɝɪɚɞɢɟɧɬ ɫɤɨɪɨɫɬɢ – ɜ ɫ-1, ɬɨ ɪɚɡɦɟɪɧɨɫɬɶɸ
P ɹɜɥɹɟɬɫɹ (ɇ.ɫ)/ɦ2 .Ʉɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ Re , ɩɪɢ
ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɨɬ ɥɚɦɢɧɚɪɧɨɝɨ ɬɟɱɟɧɢɹ ɤ ɬɭɪɛɭɥɟɧɬɧɨɦɭ,
ɡɚɜɢɫɢɬ ɨɬ ɲɟɪɨɯɨɜɚɬɨɫɬɢ ɩɨɜɟɪɯɧɨɫɬɢ, ɜɟɥɢɱɢɧɵ ɬɭɪɛɭɥɟɧɬɧɵɯ ɩɭɥɶɫɚɰɢɣ ɢ ɯɚɪɚɤɬɟɪɚ ɬɟɱɟɧɢɹ ɨɫɧɨɜɧɨɝɨ ɩɨɬɨɤɚ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ ɢɡɦɟɧɹɟɬɫɹ ɫ ɪɚɫɫɬɨɹɧɢɟɦ
ɨɬ ɩɟɪɟɞɧɟɣ ɤɪɨɦɤɢ ɩɥɚɫɬɢɧɵ ɢɥɢ ɨɬ ɜɯɨɞɚ ɜ ɬɪɭɛɭ ɢɥɢ ɜ ɤɚɧɚɥ; ɩɚɪɚɦɟɬɪɨɦ, ɨɬɪɚɠɚɸɳɢɦ ɷɬɨ ɢɡɦɟɧɟɧɢɟ, ɹɜɥɹɟɬɫɹ ɦɟɫɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɨɬɞɚɱɢ D T D T x D T ,x . ȿɫɥɢ ɧɟɨɛɯɨɞɢɦɨ ɪɚɫɫɱɢɬɚɬɶ ɨɛɳɭɸ
ɬɟɩɥɨɨɬɞɚɱɭ ɨɬ ɩɥɚɫɬɢɧɵ, ɫɥɟɞɭɟɬ ɡɧɚɬɶ ɫɪɟɞɧɢɣ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɨɬɞɚɱɢ
17
ȼ ɮɨɪɦɭɥɚɯ ɷɬɨɝɨ ɪɚɡɞɟɥɚ ɩɪɢɧɹɬɵ ɨɛɨɡɧɚɱɟɧɢɹ: ɢɧɞɟɤɫ «L» ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɟɥɢɱɢɧɚɦ, ɨɫɪɟɞɧɟɧɧɵɦ ɩɨ ɞɥɢɧɟ; «x» – ɪɚɫɫɬɨɹɧɢɸ ɨɬ ɩɟɪɟɞɧɟɣ ɤɪɨɦɤɢ ɨɛɬɟɤɚɟɦɨɣ
ɩɥɚɫɬɢɧɵ.
101
L
D T
1
D dx .
L ³ T ,x
(4.18)
0
Ⱦɪɭɝɢɟ ɩɚɪɚɦɟɬɪɵ ɬɟɱɟɧɢɹ ɬɚɤɠɟ ɡɚɜɢɫɹɬ ɨɬ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɩɟɪɟɞɧɟɣ
ɤɪɨɦɤɢ ɩɥɚɫɬɢɧɵ ɢɥɢ ɨɬ ɜɯɨɞɚ ɜ ɬɪɭɛɭ.
ɇɚɩɪɢɦɟɪ, ɦɟɫɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɪɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɚɤ
2W x
,
(4.19)
C f ,x
U uf 2
ɝɞɟ u f - ɫɤɨɪɨɫɬɶ ɧɚɛɟɝɚɸɳɟɝɨ ɩɨɬɨɤɚ.
ɋɪɟɞɧɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɪɟɧɢɹ ɞɥɹ ɩɥɚɫɬɢɧɵ ɞɥɢɧɨɣ
L ɪɚɜɟɧ
C
L
f
1
C dx .
L ³ f ,x
(4.20)
0
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ ɹɜɥɹɟɬɫɹ ɛɟɡɪɚɡɦɟɪɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɣ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɢɥɢ ɫɢɥɭ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɧɚ ɝɪɚɧɢɰɟ ɫ
ɬɜɟɪɞɵɦ ɬɟɥɨɦ.
ɂɫɩɨɥɶɡɭɟɦ ɞɜɚ ɜɢɞɚ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ
ufx
ufL
Rex
; ReL
.
Q
Q
ɉɪɢɛɥɢɠɟɧɧɵɟ (ɷɦɩɢɪɢɱɟɫɤɢɟ) ɮɨɪɦɭɥɵ ɞɥɹ ɫɥɭɱɚɹ ɨɛɬɟɤɚɧɢɹ ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɵ ɢɦɟɸɬ ɫɥɟɞɭɸɳɢɣ ɜɢɞ.
Ʌɚɦɢɧɚɪɧɨɟ ɬɟɱɟɧɢɟ ( R e
R e x 10 5 )
Nu x
0,323 Re 0,5 Pr 0,33 ; P r t 1 ;
Nu L
0,646 Re 0,5 Pr 0,33 ,
Nu x
0,66 Re 0,5 – ɩɪɢ ɬɟɱɟɧɢɢ ɜɨɡɞɭɯɚ;
Nu x
0,76 Re 0,5 – ɩɪɢ ɬɟɱɟɧɢɢ ɤɚɩɟɥɶɧɵɯ ɠɢɞɤɨɫɬɟɣ;
Nu L 0 ,58Gr ˜ Pr 0 ,25 , Gr ˜ P r 1 0 9 – ɫɜɨɛɨɞɧɚɹ ɤɨɧɜɟɤɰɢɹ ɜɞɨɥɶ
ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɚɫɬɢɧɵ.
Ɍɭɪɛɭɥɟɧɬɧɨɟ ɬɟɱɟɧɢɟ ( 5 ˜10 5 R e
Nu L
0 ,036 Re L 0 ,8 Pr 0 ,33 , Pr ! 0,5 ;
102
R e L 10 7 )
Nu L
0 ,036 Re L 0 ,8 Pr 0 ,43 17400 289 Pr 0 ,33 , Pr  >0 ,7; 380@
Nu L 0 ,0 ,21Gr ˜ Pr 0 ,66 , Gr ˜ Pr ! 1 0 9 – ɜ ɭɫɥɨɜɢɹɯ ɫɜɨɛɨɞɧɨɣ ɤɨɧɜɟɤɰɢɢ ɜɞɨɥɶ ɜɟɪɬɢɤɚɥɶɧɨɣ ɩɥɚɫɬɢɧɵ
Ɍɟɱɟɧɢɟ ɜ ɤɪɭɝɥɨɣ ɬɪɭɛɟ
ɉɪɢɛɥɢɡɢɬɟɥɶɧɵɟ ɮɨɪɦɵ ɩɪɨɮɢɥɟɣ ɫɤɨɪɨɫɬɟɣ ɩɪɢ ɥɚɦɢɧɚɪɧɨɦ ɢ
ɬɭɪɛɭɥɟɧɬɧɨɦ ɬɟɱɟɧɢɹɯ ɜ ɬɪɭɛɟ ɩɨɤɚɡɚɧɵ ɧɚ ɪɢɫ. 4.2. Ɂɚ ɜɯɨɞɧɵɦ ɫɟɱɟɧɢɟɦ ɧɚ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɪɭɛɵ ɮɨɪɦɢɪɭɟɬɫɹ ɩɨɝɪɚɧɢɱɧɵɣ
ɫɥɨɣ, ɤɨɬɨɪɵɣ ɩɨ ɦɟɪɟ ɭɞɚɥɟɧɢɹ ɨɬ ɜɯɨɞɚ ɜɫɟ ɛɨɥɟɟ ɡɚɩɨɥɧɹɟɬ ɩɥɨɳɚɞɶ
ɩɨɩɟɪɟɱɧɨɝɨ ɫɟɱɟɧɢɹ. ȼ ɫɥɭɱɚɟ ɥɚɦɢɧɚɪɧɨɝɨ ɬɟɱɟɧɢɹ ɩɪɨɮɢɥɶ ɫɤɨɪɨɫɬɢ
ɫɬɚɧɨɜɢɬɫɹ ɩɚɪɚɛɨɥɢɱɟɫɤɢɦ, ɩɪɢɱɟɦ ɫɪɟɞɧɹɹ ɫɤɨɪɨɫɬɶ ɫɨɫɬɚɜɥɹɟɬ ½ ɫɜɨɟɝɨ ɡɧɚɱɟɧɢɹ ɧɚ ɨɫɢ ɬɪɭɛɵ. ɉɪɢ ɬɭɪɛɭɥɟɧɬɧɨɦ ɬɟɱɟɧɢɢ ɩɪɨɮɢɥɶ ɫɬɚɧɨɜɢɬɫɹ ɛɨɥɟɟ ɧɚɩɨɥɧɟɧɧɵɦ, ɢ ɫɪɟɞɧɹɹ ɫɤɨɪɨɫɬɶ ɫɨɫɬɚɜɥɹɟɬ ~83% ɨɬ ɡɧɚɱɟɧɢɹ ɧɚ ɨɫɢ ɬɪɭɛɵ.
ɚ
ɛ
Ɋɢɫ. 4.2. ɉɪɨɮɢɥɢ ɫɤɨɪɨɫɬɟɣ ɩɪɢ ɥɚɦɢɧɚɪɧɨɦ (ɚ)
ɢ ɬɭɪɛɭɥɟɧɬɧɨɦ (ɛ) ɬɟɱɟɧɢɹɯ ɜ ɬɪɭɛɟ
Ⱦɥɹ ɪɚɫɱɟɬɚ ɱɢɫɥɚ Ɋɟɣɧɨɥɶɞɫɚ ɜ ɬɪɭɛɚɯ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɫɪɟɞɧɟɟ
ɡɧɚɱɟɧɢɟ ɩɨ ɫɟɱɟɧɢɸ ɫɤɨɪɨɫɬɢ ɬɟɱɟɧɢɹ ɝɚɡɚ (ɠɢɞɤɨɫɬɢ) ɜ ɬɪɭɛɟ. ȼ ɩɪɟɞu 0D
ɫɬɚɜɥɟɧɧɵɯ ɧɢɠɟ ɮɨɪɦɭɥɚɯ R e
; ɝɞɟ u 0 – ɪɚɫɯɨɞɧɚɹ ɫɤɨɪɨɫɬɶ;
Q
D 2r0 – ɜɧɭɬɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɬɪɭɛɵ (ɲɢɪɢɧɚ ɤɚɧɚɥɚ).
Ʌɚɦɢɧɚɪɧɨɟ ɬɟɱɟɧɢɟ ( Re 2000 )
Nu
4,36 , ɟɫɥɢ Ts – ɫɪɟɞɧɹɹ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɪɭɛɟ;
Nu
2,66 , ɟɫɥɢ Ts – ɦɚɤɫɢɦɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɪɭɛɟ;
Nu 1,86 Re ˜ Pr 0 ,33 D L 0 ,33 – ɭɫɪɟɞɧɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ( L – ɞɥɢɧɚ
ɬɪɭɛɵ)
103
Ɍɭɪɛɭɥɟɧɬɧɨɟ ɬɟɱɟɧɢɟ ( Re ! 2300 )
Nu 0,023Re 0 ,8 Pr 0 ,33 ; 0,5 Pr 100 , ɟɫɥɢ Ts – ɫɪɟɞɧɹɹ ɬɟɦɩɟɪɚɬɭɪɚ
ɜ ɬɪɭɛɟ
Nu 0,625 Pe 0 ,4 ; Pe Re ˜ Pr  >100;10000 @ ;
>
@
Nu 0,023Re 0 ,8 Pr 0 ,4 , Pr  >0,7; 2500@ ; Re  104 ; 105 ;
Nu 0,022Re 0 ,8 Pr 0 ,6 , Pr  >0,5; 1@ ;
Nu 0,0155Re 0 ,8 Pr 0 ,5 , Pr  >1; 20@.
ȿɫɥɢ ɩɪɢɜɟɫɬɢ ɷɬɢ ɮɨɪɦɭɥɵ ɤ ɪɚɡɦɟɪɧɨɦɭ ɜɢɞɭ, ɬɨ ɛɭɞɭɬ ɜɢɞɧɵ
ɫɬɟɩɟɧɢ ɜɥɢɹɧɢɹ ɪɚɡɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɮɚɤɬɨɪɨɜ ɧɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ.
Ⱦɥɹ ɤɚɧɚɥɨɜ ɧɟɤɪɭɝɥɨɝɨ ɫɟɱɟɧɢɹ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢɜɟɞɟɧɧɵɣ
(ɝɢɞɪɚɜɥɢɱɟɫɤɢɣ) ɞɢɚɦɟɬɪ D 4S / P , ɝɞɟ S – ɩɥɨɳɚɞɶ ɫɟɱɟɧɢɹ, P ɩɟɪɢɦɟɬɪ ɤɚɧɚɥɚ.
ȼ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱɚɯ ɨɛɵɱɧɨ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɹɬɶ ɢɡɦɟɧɟɧɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ ɬɟɱɟɧɢɢ ɠɢɞɤɨɫɬɢ ɜ ɤɚɧɚɥɟ ɫ ɡɚɞɚɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɩɪɢ
ɞɚɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɠɢɞɤɨɫɬɢ ɧɚ ɜɯɨɞɟ ɢ ɧɚ ɫɬɟɧɤɟ ɤɚɧɚɥɚ. Ⱦɥɹ ɬɟɱɟɧɢɹ ɜ ɬɪɭɛɟ ɞɥɢɧɨɣ L ɢ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ɫɬɟɧɤɢ Ts ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɜ
ɠɢɞɤɨɫɬɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
SD 2
q c pU u
T2 T1 ,
4
(4.21)
ɝɞɟ T2 ,T1 – ɫɪɟɞɧɢɟ ɩɨ ɫɟɱɟɧɢɹɦ ɬɟɦɩɟɪɚɬɭɪɵ (ɪɢɫ. 4.2) ɢɥɢ ɬɟɦɩɟɪɚɬɭɪɵ ɫɦɟɲɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ ɷɥɟɦɟɧɬɚɪɧɨɣ ɞɥɢɧɟ
dx ɛɭɞɟɬ ɫɜɹɡɚɧ ɫ ɢɡɦɟɧɟɧɢɟɦ ɫɪɟɞɧɟɦɚɫɫɨɜɨɣ ɬɟɦɩɟɪɚɬɭɪɵ T , ɚ ɬɚɤɠɟ
ɫ ɪɚɡɧɨɫɬɶɸ ɬɟɦɩɟɪɚɬɭɪ ɫɬɟɧɤɢ Ts x ɢ ɫɪɟɞɧɟɦɚɫɫɨɜɨɣ ɬɟɦɩɟɪɚɬɭɪɵ
ɠɢɞɤɨɫɬɢ T x ɧɚ ɷɬɨɣ ɞɥɢɧɟ ɫɥɟɞɭɸɳɟɣ ɡɚɜɢɫɢɦɨɫɬɶɸ
dq
m c pd T
D T S D d x T s T ,
ɝɞɟ m – ɦɚɫɫɨɜɵɣ ɪɚɫɯɨɞ ɠɢɞɤɨɫɬɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
D T
q
F Ts T
,
(4.22)
ɝɞɟ F – ɨɛɳɚɹ ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɧɬɚɤɬɚ ɠɢɞɤɨɫɬɢ ɫ ɬɟɩɥɨɩɟɪɟɞɚɸɳɟɣ ɩɨɜɟɪɯɧɨɫɬɶɸ. Ɍɚɤ ɤɚɤ ɢ Ts , ɢ T ɦɨɝɭɬ ɦɟɧɹɬɶɫɹ ɩɨ ɞɥɢɧɟ ɬɪɭ104
ɛɵ, ɬɨ ɞɥɹ ɩɪɚɤɬɢɱɟɫɤɨɝɨ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɭɪɚɜɧɟɧɢɹ (4.22) ɧɟɨɛɯɨɞɢɦɨ
ɪɚɡɪɚɛɨɬɚɬɶ ɭɞɨɛɧɵɣ ɫɩɨɫɨɛ ɨɫɪɟɞɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ, ɱɬɨ ɜɫɟɝɞɚ ɹɜɥɹɟɬɫɹ ɨɬɞɟɥɶɧɨɣ ɡɚɞɚɱɟɣ.
4.7. ɉɪɢɛɥɢɠɟɧɧɵɣ ɫɩɨɫɨɛ ɪɚɫɱɟɬɚ
ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɨɬɞɚɱɢ ɢ ɬɪɟɧɢɹ
ȼɟɪɧɟɦɫɹ ɤ ɡɚɞɚɱɟ ɨ ɬɟɩɥɨɨɛɦɟɧɟ ɦɟɠɞɭ ɭɫɬɚɧɨɜɢɜɲɢɦɫɹ ɥɚɦɢɧɚɪɧɵɦ ɩɨɬɨɤɨɦ ɢ ɩɥɨɫɤɨɣ ɫɬɟɧɤɨɣ. ȼ ɩɪɢɛɥɢɠɟɧɢɢ ɬɨɧɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ
ɫɥɨɹ ɬɨɥɳɢɧɨɣ G ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɭɪɚɜɧɟɧɢɹɦɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɜɵɜɟɞɟɧɧɵɦɢ ɜ ɪɚɡɞɟɥɟ 3.3, ɜ ɤɨɬɨɪɵɯ ɩɪɟɧɟɛɪɟɠɟɦ ɢɡɦɟɧɟɧɢɟɦ ɞɚɜɥɟɧɢɹ
(ɱɬɨ, ɤɨɧɟɱɧɨ, ɤɨɪɪɟɤɬɧɨ ɞɥɹ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɩɨɬɨɤɚ)
§ wu
wu ·
w 2u
(4.23)
¨u wx v wy ¸ P 2 ,
w
y
©
¹
wu wv
wx wy
0,
(4.24)
O w 2T
(4.25)
Uc p wy 2
ȼ ɤɚɱɟɫɬɜɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɪɢɦɟɦ
v u 0 ɩɪɢ y 0 ;
(4.26)
u U f ɩɪɢ y o G .
(4.27)
Ⱦɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɢɦɟɟɦ
T Ts ɩɪɢ y 0 ;
(4.28)
T Tf ɩɪɢ y o G ,
(4.29)
ɝɞɟ G T – ɬɨɥɳɢɧɚ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ.
ɉɟɪɜɵɦ ɲɚɝɨɦ ɜ ɩɪɢɛɥɢɠɟɧɧɨɦ ɦɟɬɨɞɟ18 ɹɜɥɹɟɬɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɫɤɨɪɨɫɬɟɣ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɜɢɞɟ ɫɬɟɩɟɧɧɵɯ ɪɹɞɨɜ – ɩɨɥɢɧɨɦɨɜ. ɉɪɢ ɜɵɛɨɪɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɷɬɢɯ ɩɨɥɢɧɨɦɨɜ ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶɫɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɟɣ ɨɩɢɫɵɜɚɟɬɫɹ ɪɹɞɨɦ
u y a by cy2 dy 3 .
(4.30)
ɂɡ ɭɫɥɨɜɢɹ (4.26) ɫɪɚɡɭ ɧɚɯɨɞɢɦ
a 0.
w 2u
ɂɡ ɷɬɨɝɨ ɠɟ ɭɫɥɨɜɢɹ ɫɥɟɞɭɟɬ
0 ɩɪɢ y 0 .
wy 2
wT
wT
u
v
wx
wy
18
Ʉɪɟɣɬ Ɏ., Ȼɥɷɤ ɍ. Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. Ɇ.: Ɇɢɪ, 1983. 512 ɫ.
105
ɂɡ ɭɫɥɨɜɢɹ (4.27) ɢ ɭɪɚɜɧɟɧɢɹ (4.23) ɢɦɟɟɦ
w u w y 0 ɩɪɢ y o G .
ɗɬɢɯ ɭɫɥɨɜɢɣ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɧɚɣɬɢ ɨɫɬɚɥɶɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ:
b
3 uf
; c 0; d
2 G
uf
2G 3
.
ɉɨɞɫɬɚɜɥɹɹ ɢɯ ɜ ɭɪɚɜɧɟɧɢɟ (4.30), ɧɚɣɞɟɦ
3
u
3 y 1§ y ·
.
(4.31)
u f 2 G 2 ¨© G ¸¹
Ʉɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɧɚ ɫɬɟɧɤɟ ɧɚɣɞɟɦ ɩɨ ɮɨɪɦɭɥɟ (4.17), ɢɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɢ
2 uf
W
P
.
(4.32)
3 G
Ⱦɚɥɟɟ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɤɨɥɢɱɟɫɬɜɚ
ɞɜɢɠɟɧɢɹ. ɉɭɫɬɶ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɧɚɱɢɧɚɟɬɫɹ ɧɚ ɫɬɟɧɤɟ ɢ ɩɪɨɫɬɢɪɚɟɬɫɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ O y , ɜɵɯɨɞɹ ɡɚ ɩɪɟɞɟɥɵ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ.
Ɉɧ ɢɦɟɟɬ ɬɨɥɳɢɧɭ dx ɜ ɧɚɩɪɚɜɥɟɧɢɢ Ox ɢ ɟɞɢɧɢɱɧɭɸ ɬɨɥɳɢɧɭ ɜ ɧɚɩɪɚɜɥɟɧɢɢ Oz (ɪɢɫ. 4.3).
ɑɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ȺB ɩɟɪɟɞɚɟɬɫɹ ɩɨɬɨɤ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
G
³Uu
2
dy ,
0
ɚ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɋD – ɩɨɬɨɤ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
G
G
·
d §
2
2 ¸
¨
u
dy
u
dy
dx .
U
U
³
¸
dx ¨ ³
0
©0
¹
ɀɢɞɤɨɫɬɶ ɩɨɫɬɭɩɚɟɬ ɜ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ȼD ɫ
ɪɚɫɯɨɞɨɦ
G
·
Ɋɢɫ. 4.3. ɗɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɜ
d §
¨
¸ dx .
U
udy
ɥɚɦɢɧɚɪɧɨɦ ɩɨɝɪɚɧɢɱɧɨɦ ɫɥɨɟ,
³
¨
¸
dx
ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɩɪɢ ɜɵɜɨɞɟ
©0
¹
ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɚɥɚɧɫɚ
ɗɬɚ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɪɚɡɧɨɫɬɶɸ
ɢɦɩɭɥɶɫɚ
ɦɟɠɞɭ ɪɚɫɯɨɞɨɦ, ɜɵɬɟɤɚɸɳɢɦ ɱɟɪɟɡ
ɩɨɜɟɪɯɧɨɫɬɶ ɋD ɢ ɪɚɫɯɨɞɨɦ, ɜɬɟɤɚɸɳɢɦ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ AB. Ɍɚɤ ɤɚɤ
ɠɢɞɤɨɫɬɶ, ɩɨɫɬɭɩɚɸɳɚɹ ɜ ɨɛɴɟɦ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ȼD, ɢɦɟɟɬ ɫɨɫɬɚɜɥɹɸɳɭɸ ɫɤɨɪɨɫɬɢ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ x , ɪɚɜɧɭɸ ɫɤɨɪɨɫɬɢ ɜɧɟɲɧɟɝɨ ɩɨ106
ɬɨɤɚ, ɬɨ ɱɟɪɟɡ ɷɬɭ ɜɟɪɯɧɸɸ ɩɨɜɟɪɯɧɨɫɬɶ ɜ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ x ɩɨɫɬɭɩɢɬ ɩɨɬɨɤ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ
G
·
d §
¨
uf
U udy ¸ dx .
¸
dx ¨ ³
©0
¹
ɋɭɦɦɢɪɭɹ ɜɫɟ ɫɨɫɬɚɜɥɹɸɳɢɟ ɤɨɥɢɱɟɫɬɜɚ ɞɜɢɠɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ
ɨɫɢ x , ɩɨɥɭɱɢɦ
G
G
§G
·
§G
·
¨
¸
¸
d ¨
d
2
2
2
Uu dy ¨ Uu dy ¸dx Uu dy uf ¨ Uudy ¸dx
dx ¨
dx ¨
¸
¸
0
0
©0
¹
©0
¹
G
§
·
¸
d ¨
¨ Uu uf u dy ¸dx .
dx ¨
¸
©0
¹
ɇɚ ɩɨɜɟɪɯɧɨɫɬɢ ȼD ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɨɬɫɭɬɫɬɜɭɟɬ, ɬɚɤ ɤɚɤ ɷɬɚ
ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɯɨɞɢɬɫɹ ɜɧɟ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɢ w u w y 0 . Ɉɞɧɚɤɨ
ɫɭɳɟɫɬɜɭɟɬ ɤɚɫɚɬɟɥɶɧɚɹ ɫɢɥɚ ɬɪɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɚɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɠɢɞɤɨɫɬɢ ɢ ɬɜɟɪɞɨɣ ɫɬɟɧɤɢ, ɬ.ɟ. ɩɪɢ y 0 . ɂɦɟɸɬɫɹ ɫɢɥɵ ɞɚɜɥɟɧɢɹ, ɞɟɣɫɬɜɭɸɳɢɟ ɧɚ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ ɫɨ ɫɬɨɪɨɧɵ ɩɨɜɟɪɯɧɨɫɬɟɣ AB
ɢ ɋD. ɋɭɦɦɢɪɭɹ ɜɫɟ ɫɢɥɵ, ɩɨɥɭɱɚɟɦ ɫɨɨɬɧɨɲɟɧɢɟ
³
³
³
³
³
dp
dp
§
·
p xG ¨ p x x dx ¸ G W dx G x dx W dx .
dx
dx
©
¹
(4.33)
ɉɪɢ ɨɛɬɟɤɚɧɢɢ ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɵ ɝɪɚɞɢɟɧɬɨɦ ɞɚɜɥɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ x ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɢ ɬɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɤɨɥɢɱɟɫɬɜɚ
ɞɜɢɠɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
G
·
d §
¨ U u u f u dy ¸ W .
(4.34)
¸
dx ¨ ³
©0
¹
ɉɨɞɫɬɚɜɥɹɹ ɜ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɧɚɣɞɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɫɤɨɪɨɫɬɢ
(4.31) ɢ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ (4.32), ɧɚɣɞɟɦ
G
3º ª
3
ª
3 y 1§ y · º
d
2 3 y 1§ y ·
U uf «
¨ ¸ » ˜ «1 ¨ ¸ » dy
2 G 2© G ¹ » « 2 G 2© G ¹ »
dx ³
«
0
¬
¼ ¬
¼
2 uf
P
.
3 G
ɉɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɩɪɢɯɨɞɢɦ ɤ ɭɪɚɜɧɟɧɢɸ
d § 2 39G ·
¨ U uf
¸
dx ©
280 ¹
107
2 uf
P
.
3 G
(4.35)
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɚɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɬɨɥɳɢɧɵ ɩɨɝɪɚɧɢɱɧɨɝɨ
ɫɥɨɹ ɨɬ ɜɹɡɤɨɫɬɢ, ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɩɟɪɟɞɧɟɣ ɤɪɨɦɤɢ ɢ ɫɤɨɪɨɫɬɢ ɧɟɜɨɡɦɭɳɟɧɧɨɝɨ ɩɨɬɨɤɚ
G 2 14 0Q x
C.
2
1 3u f
Ɍɚɤ ɤɚɤ ɧɚ ɩɟɪɟɞɧɟɣ ɤɪɨɦɤɟ (ɬ.ɟ. ɩɪɢ x 0 ) G 0 , ɬɨ ɩɨɫɬɨɹɧɧɚɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɪɚɜɧɚ ɧɭɥɸ, ɢ ɦɵ ɩɨɥɭɱɚɟɦ
G 4 ,64
.
(4.36)
x Re1x 2
ɑɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ, ɜɵɱɢɫɥɢɦ ɢɡ (4.32) ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ, ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ (4.36)
3 Puf 1 2
Re x ,
Wx
9 ,28 x
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɢɡ ɮɨɪɦɭɥɵ (4.19) ɧɚɯɨɞɢɦ
2W x 0 ,647
.
(4.37)
C f ,x
2
12
Uuf Re x
Ɉɛɪɚɬɢɦɫɹ ɬɟɩɟɪɶ ɤ ɭɪɚɜɧɟɧɢɸ ɷɧɟɪɝɢɢ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ, ɚɧɚɥɨɝɢɱɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɞɥɹ ɫɤɨɪɨɫɬɢ (4.30)
T y e fy gy 2 hy3
(4.38)
ɂɡ ɭɪɚɜɧɟɧɢɹ (4.25) ɢ ɭɫɥɨɜɢɣ (4.26), (4.28) ɫɥɟɞɭɟɬ, ɱɬɨ
d 2T
0 ɩɪɢ y 0 .
dy 2
ɂɡ ɭɫɥɨɜɢɹ (4.29) ɫɥɟɞɭɟɬ, ɱɬɨ
d T d y 0 ɩɪɢ y o G T .
Ɍɚɤ ɱɬɨ ɱɟɬɵɪɟɯ ɭɫɥɨɜɢɣ ɜɩɨɥɧɟ ɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɫɟɯ
ɱɟɬɵɪɟɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ (4.38). ɂɦɟɟɦ
3 Tf
T
e Ts ; f
, g 0 , h f2 ,
2 GT
2G T
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
3
T Ts
3 y 1§ y ·
(4.39)
¨
¸ .
Tf T s 2 G T 2 © G T ¹
Ⱦɚɥɟɟ ɧɚɦ ɬɪɟɛɭɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɷɧɟɪɝɢɢ, ɤɨɬɨɪɨɟ ɧɚɯɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɢɧɬɟɝɪɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɛɚɥɚɧɫɚ ɢɦɩɭɥɶɫɚ.
ɇɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɷɥɟɦɟɧɬɚɪɧɵɣ ɨɛɴɟɦ, ɜɵɯɨɞɹɳɢɣ ɡɚ ɩɪɟ108
ɞɟɥɵ ɤɚɤ ɬɟɩɥɨɜɨɝɨ, ɬɚɤ ɢ ɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɵɯ ɫɥɨɟɜ. ɉɪɢɜɨɞɢɦ
ɨɤɨɧɱɚɬɟɥɶɧɭɸ ɮɨɪɦɭɥɭ
GT
O § wT ·
(4.40)
0.
c pU ¨© w y ¸¹ y 0
0
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɨɛɵɱɧɨ ɢɡɜɟɫɬɧɨ ɤɚɤ ɢɧɬɟɝɪɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɞɥɹ ɥɚɦɢɧɚɪɧɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ ɩɪɢ ɧɢɡɤɢɯ ɫɤɨɪɨɫɬɹɯ ɬɟɱɟɧɢɹ.
ɉɨɞɫɬɚɜɥɹɹ ɜ ɥɟɜɭɸ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (4.40) ɫɤɨɪɨɫɬɶ ɢ ɬɟɦɩɟɪɚɬɭɪɭ
ɢɡ (4.39) ɢ (4.40), ɧɚɣɞɟɦ
d
dx
³ Tf T u d y GT
GT
0
0
³ Tf T u d y ³ ª¬Tf Ts T Ts º¼ u d y
GT
Tf Ts u f ³
0
ª 3 y 1 § y ·3 º ª 3 y 1 y 3 º
§ ·
«1 ¨
¨ ¸ » dy
¸ »˜«
« 2 G T 2 © G T ¹ » «¬ 2 G 2 © G ¹ »¼
¬
¼
§3G
3 G T 2 3 GT 2 1 G T 4 3 GT 4 1 GT 4 ·
T
Tf Ts u f ¨¨
¸.
3
3
3 ¸
4
4
20
8
20
28
G
G
G
G
G
G
©
¹
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
GT
§
T
T
u
G
T
T
u
d
y
f
s
f
¨
f
³
3 2
3 4·
] ] ¸,
280 ¹
© 20
0
ɝɞɟ ] G T G .
ɉɪɢ ɭɫɥɨɜɢɢ ] 1 (ɱɬɨ, ɤɨɧɟɱɧɨ, ɧɟ ɜɫɟɝɞɚ ɫɩɪɚɜɟɞɥɢɜɨ), ɜɬɨɪɵɦ
ɫɥɚɝɚɟɦɵɦ ɜ ɫɤɨɛɤɟ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɟɪɜɵɦ. Ɍɨɝɞɚ,
ɩɨɞɫɬɚɜɥɹɹ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɢɧɬɟɝɪɚɥɚ ɜ ɭɪɚɜɧɟɧɢɟ (4.40) ɢ ɢɫɩɨɥɶɡɭɹ (4.39), ɧɚɣɞɟɦ
wG
O § wT ·
3
3 O Tf Ts u f Tf T s ] 2
¨
¸
20
w x c p U © w y ¹ y 0 2 c pU
G]
ɢɥɢ
wG
O
1
u f ] 3G
.
10
w x c pU
ɂɡ ɭɪɚɜɧɟɧɢɹ (4.36) ɢɦɟɟɦ:
G
wG
Q
10,75
,
wx
uf
ɱɬɨ ɞɚɟɬ
109
]3
10 a
,a
10 ,75 Q
O
c pU
ɢɥɢ
GT 0,976 G Pr 1 3 .
(4.41)
Ɍɟɩɟɪɶ ɦɨɠɧɨ ɧɚɣɬɢ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɬɟɩɥɨɨɬɞɚɱɢ (ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ).
ɉɨɬɨɤ ɬɟɩɥɚ, ɩɨɫɬɭɩɚɸɳɢɣ ɜ ɨɛɬɟɤɚɟɦɭɸ ɩɥɚɫɬɢɧɭ ɢɥɢ ɨɬɜɨɞɢɦɵɣ
ɢɡ ɧɟɟ (ɞɥɹ ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ), ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɫɨɨɬɧɨɲɟɧɢɸ
wT
q T D T Tf O f
,
wy y 0
ɝɞɟ ɢɧɞɟɤɫ f ɨɬɧɨɫɢɬɫɹ ɤ ɠɢɞɤɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨɥɶɡɭɹɫɶ ɫɨɨɬɧɨɲɟɧɢɹɦɢ (4.39), (4.41), ɚ ɡɚɬɟɦ – ɫɨɨɬɧɨɲɟɧɢɟɦ (4.36), ɧɚɣɞɟɦ
q
2 OL
2 O L Pr 1 3
Tf Ts Tf Ts 3 GT
3 G 0 ,976
2 O L Pr 1 3 Re1x 2
Tf Ts 0,33 O L Pr 1 3 Re1x 2 Ts Tf .
x
3 x 0 ,976 ˜ 4 ,64
Ɍɨɝɞɚ ɥɨɤɚɥɶɧɨɟ ɱɢɫɥɨ ɇɭɫɫɟɥɶɬɚ ɦɨɠɧɨ ɛɭɞɟɬ ɜɵɱɢɫɥɢɬɶ ɩɨ ɮɨɪɦɭɥɟ
D xx
q
x
Nux
0,33 Pr1 3 Re1x 2 .
O L Ts Tf O L
Ɂɚɦɟɬɢɦ, ɱɬɨ ɷɬɨɬ ɩɪɢɛɥɢɠɟɧɧɵɣ ɦɟɬɨɞ, ɩɨɡɜɨɥɹɸɳɢɣ ɩɨɥɭɱɚɬɶ
ɭɞɨɜɥɟɬɜɨɪɢɬɟɥɶɧɵɟ ɪɟɡɭɥɶɬɚɬɵ, ɦɢɧɭɹ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɬɪɭɞɧɨɫɬɢ,
ɜɨɡɧɢɤɚɸɳɢɟ ɩɪɢ ɪɟɲɟɧɢɢ ɡɚɞɚɱ ɬɟɨɪɢɢ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɬɨɥɶɤɨ ɩɪɢ ɭɫɥɨɜɢɢ ɯɨɪɨɲɟɝɨ ɩɨɧɢɦɚɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɜ ɩɪɨɰɟɫɫɟ ɬɟɩɥɨɨɛɦɟɧɚ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
11. Ʉɚɤɨɝɨ ɬɢɩɚ ɜɟɥɢɱɢɧɵ ɜɯɨɞɹɬ ɜ ɱɢɫɥɨ ɫɭɳɟɫɬɜɟɧɧɵɯ ɩɚɪɚɦɟɬɪɨɜ
ɩɪɢ ɢɡɭɱɟɧɢɢ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ?
12. ɑɬɨ ɬɚɤɨɟ «ɤɪɢɬɟɪɢɢ ɩɨɞɨɛɢɹ»? ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɨɫɧɨɜɧɚɹ
ɢɞɟɹ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ?
13. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɬɟɨɪɟɦɵ ɩɨɞɨɛɢɹ.
14. Ɉɯɚɪɚɤɬɟɪɢɡɭɣɬɟ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɤɪɢɬɟɪɢɟɜ Ɋɟɣɧɨɥɶɞɫɚ,
ɉɟɤɥɟ ɢ ɱɢɫɥɚ ɇɭɫɫɟɥɶɬɚ.
15. ɑɟɦ ɨɬɥɢɱɚɸɬɫɹ ɱɢɫɥɨ ɇɭɫɫɟɥɶɬɚ ɢ ɤɪɢɬɟɪɢɣ Ȼɢɨ?
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16. Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɯɚɪɚɤɬɟɪɧɵɣ ɦɚɫɲɬɚɛ ɜ ɡɚɞɚɱɚɯ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɨɛɦɟɧɚ?
17. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɦɵɫɥ «ɤɪɢɬɟɪɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ»? ɉɪɢɜɟɞɢɬɟ
ɩɪɢɦɟɪɵ.
18. ȼ ɱɟɦ, ɩɨ ȼɚɲɟɦɭ ɦɧɟɧɢɸ, ɡɚɤɥɸɱɚɸɬɫɹ ɧɟɞɨɫɬɚɬɤɢ ɢ ɞɨɫɬɨɢɧɫɬɜɚ ɷɦɩɢɪɢɱɟɫɤɢɯ ɮɨɪɦɭɥ?
19. Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɫɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ ɢ
ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɪɟɧɢɹ?
20. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɦɟɬɨɞɚ ɪɚɫɱɟɬɚ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɬɞɚɱɢ?
Ɂɚɞɚɧɢɹ
1. ɉɨɥɶɡɭɹɫɶ ɫɯɟɦɨɣ ɪɟɲɟɧɢɹ, ɢɡɥɨɠɟɧɧɨɣ ɜ ɪɚɡɞɟɥɟ 4.7, ɧɚɣɬɢ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɬɟɩɥɨɜɵɦ ɢ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɢɦ ɩɨɝɪɚɧɢɱɧɵɦ ɫɥɨɟɦ,
ɩɪɢɝɨɞɧɨɟ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɜɟɥɢɱɢɧɵ ] G T / G .
2. ȼɨɡɞɭɯ ɩɪɢ ɚɬɦɨɫɮɟɪɧɨɦ ɞɚɜɥɟɧɢɢ ɨɛɬɟɤɚɟɬ ɩɥɨɫɤɭɸ ɩɥɚɫɬɢɧɭ
ɞɥɢɧɨɣ 3 ɦ ɢ ɲɢɪɢɧɨɣ 5 ɦ. Ɍɟɦɩɟɪɚɬɭɪɵ ɜɨɡɞɭɯɚ ɢ ɩɥɚɫɬɢɧɵ 15 ɢ 65 ɨɋ,
ɚ ɫɤɨɪɨɫɬɶ ɜɨɡɞɭɯɚ 35 ɦ/ɫ. Ɋɚɫɫɱɢɬɚɬɶ ɧɚ ɡɚɞɧɟɣ ɤɪɨɦɤɟ ɩɥɚɫɬɢɧɵ (ɬ.ɟ.,
ɩɪɢ x 3 ɦ) ɚ) ɫɪɟɞɧɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ; ɛ) ɫɢɥɭ ɜɹɡɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɥɚɫɬɢɧɵ. Ɉɩɪɟɞɟɥɢɬɶ ɜ) ɫɪɟɞɧɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ; ɩɨɥɧɵɣ ɩɨɬɨɤ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ.
3. ȼɨɞɚ ɨɛɬɟɤɚɟɬ ɤɜɚɞɪɚɬɧɭɸ ɩɥɚɫɬɢɧɭ ɫɨ ɫɬɨɪɨɧɨɣ 2 ɦ. Ɍɟɦɩɟɪɚɬɭɪɚ
ɜɨɞɵ ɫɨɫɬɚɜɥɹɟɬ 90 ɨɋ., ɚ ɟɟ ɫɤɨɪɨɫɬɶ 10 ɦ/ɫ. ɉɥɚɫɬɢɧɚ ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ 30 ɨɋ. Ɉɩɪɟɞɟɥɢɬɶ ɫɢɥɭ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɩɥɚɫɬɢɧɵ ɢ
ɩɨɬɨɤ ɬɟɩɥɚ ɫ ɟɟ ɩɨɜɟɪɯɧɨɫɬɢ.
111
ɑȺɋɌɖ 5
Ɂɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ
5.1. ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɪɚɡɥɢɱɧɵɯ ɫɢɫ ɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ
ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɮɨɪɦɟ (2.23)
cU
wT
wt
w § wT
O
w x ¨© w x
· w § wT
¸ wy ¨ O wy
¹
©
· w § wT
¸ ¨O
¹ wy © wy
·
¸ qV
¹
ɩɪɢɦɟɧɢɦɨ ɬɨɥɶɤɨ ɞɥɹ ɞɟɤɚɪɬɨɜɨɣ ɢɥɢ ɩɪɹɦɨɭɝɨɥɶɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ȼ ɩɪɚɤɬɢɤɟ ɱɚɫɬɨ ɜɫɬɪɟɱɚɸɬɫɹ ɬɚɤɢɟ ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɟ ɩɪɢɜɨɞɹɬ ɤ
ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɢɧɨɣ ɮɨɪɦɟ, ɛɨɥɟɟ ɭɞɨɛɧɨɣ ɞɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɟɲɟɧɢɹ ɢ ɟɝɨ ɮɢɡɢɱɟɫɤɨɣ ɬɪɚɤɬɨɜɤɢ.
ɗɬɨ ɨɬɧɨɫɢɬɫɹ, ɧɚɩɪɢɦɟɪ, ɤ ɬɟɥɚɦ, ɢɦɟɸɳɢɦ ɮɨɪɦɭ ɮɢɝɭɪ ɜɪɚɳɟɧɢɹ
(ɰɢɥɢɧɞɪ, ɞɢɫɤ, ɫɮɟɪɚ), ɧɚɝɪɟɜɚɟɦɵɦ ɫ ɩɨɜɟɪɯɧɨɫɬɢ. Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ
ɫɥɚɝɚɟɦɵɟ, ɜɵɪɚɠɚɸɳɢɟ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟ ɢ ɚɤɤɭɦɭɥɢɪɨɜɚɧɢɟ ɷɧɟɪɝɢɢ
ɢɧɜɚɪɢɚɧɬɧɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (ɬ.ɟ., ɧɟɢɡɦɟɧɧɵ); ɧɨ
ɫɥɚɝɚɟɦɵɟ, ɜɵɪɚɠɚɸɳɢɟ ɪɟɡɭɥɶɬɢɪɭɸɳɢɣ ɤɨɧɞɭɤɬɢɜɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɡɚɜɢɫɹɬ ɨɬ ɝɟɨɦɟɬɪɢɢ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɬ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ. ɇɚ
ɞɨɤɚɡɚɬɟɥɶɫɬɜɟ ɷɬɨɝɨ ɭɬɜɟɪɠɞɟɧɢɹ ɦɵ ɨɫɬɚɧɚɜɥɢɜɚɬɶɫɹ ɧɟ ɛɭɞɟɦ. ɉɨɡɧɚɤɨɦɢɦɫɹ ɬɨɥɶɤɨ ɫ ɡɚɩɢɫɶɸ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɢ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ ɜ ɞɨɩɨɥɧɟɧɢɟ ɤ (2.23).
Ɂɚɜɢɫɢɦɨɫɬɶ ɜɢɞɚ ɭɪɚɜɧɟɧɢɹ ɨɬ ɩɪɢɧɹɬɨɣ ɜ ɡɚɞɚɱɟ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɦɨɠɧɨ ɢɫɤɥɸɱɢɬɶ, ɟɫɥɢ ɜɵɪɚɡɢɬɶ ɤɨɧɞɭɤɬɢɜɧɵɟ ɱɥɟɧɵ ɫ ɩɨɦɨɳɶɸ
ɨɩɟɪɚɬɨɪɚ Ʌɚɩɥɚɫɚ:
1 wT
a wt
'T qV
,
O
(5.1)
ɝɞɟ a O cU . ȼɦɟɫɬɨ ɨɛɨɡɧɚɱɟɧɢɹ ' ɞɥɹ ɨɩɟɪɚɬɨɪɚ Ʌɚɩɥɚɫɚ ɱɚɫɬɨ ɢɫɩɨɥɶɡɭɸɬ ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɟɦɭ ɨɛɨɡɧɚɱɟɧɢɟ ’ 2 . Ɂɚɩɢɫɶ (5.1) ɜɩɨɥɧɟ ɤɨɪɪɟɤɬɧɚ, ɟɫɥɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɫɱɢɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ. ȿɫɥɢ ɷɬɨ ɭɫɥɨɜɢɟ
ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɬɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɡɚɩɢɫɵɜɚɟɬɫɹ ɜ ɜɢɞɟ
cU
wT
wt
div O grad T qV ,
(5.2)
ɝɞɟ div { ’ ˜ – ɨɛɨɡɧɚɱɟɧɢɟ ɨɩɟɪɚɬɨɪɚ ɞɢɜɟɪɝɟɧɰɢɢ («ɧɚɛɥɚ ɫ ɬɨɱɤɨɣ»);
grad { ’ – ɨɛɨɡɧɚɱɟɧɢɟ ɨɩɟɪɚɬɨɪɚ ɝɪɚɞɢɟɧɬɚ.
112
Ɏɨɪɦɚ ɥɚɩɥɚɫɢɚɧɚ, ɚ ɬɚɤɠɟ ɨɩɟɪɚɬɨɪɨɜ ɝɪɚɞɢɟɧɬɚ ɢ ɞɢɜɟɪɝɟɧɰɢɢ ɞɥɹ
ɪɚɡɥɢɱɧɵɯ ɫɢɫɬɟɦ ɤɨɨɪɞɢɧɚɬ ɪɚɡɥɢɱɧɚ. Ⱦɥɹ ɞɟɤɚɪɬɨɜɨɣ ɩɪɹɦɨɭɝɨɥɶɧɨɣ
ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 5.1) ɢɦɟɟɦ T T x , y , z ,t ɢ
'{
w2
w2
w2
;
wx 2 wy 2 wz 2
w
w
w
,
’{i
j k
wx
wy
wz
ɬ.ɟ.
Ɋɢɫ. 5.1. Ⱦɟɤɚɪɬɨɜɚ
ɫɢɫɬɟɦɚ ɤɨɨɪɞɢɧɚɬ
q
§ wT
wT
wT ·
O¨¨
i
j
k¸
wy
wz ¸¹
© wx
Ograd T .
T r , M , z ,t . Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ x ɢ y ɫɜɹɡɚɧɵ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ r ɢ M ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ɩɪɨɫɬɵɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
x r c o s M ; y r sin M .
(5.3)
Ɉɩɟɪɚɬɨɪɵ Ʌɚɩɥɚɫɚ ɢ ɝɪɚɞɢɟɧɬɚ ɢɦɟɸɬ ɜɢɞ
w2
1 w w
1 w2
'
r
;
r w r w r r 2 wM 2 w z 2
w
1 w
w
’ ir
iM
iz .
wr
wz
r wM
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (5.1) ɡɚɩɢɫɵɜɚɟɬɫɹ ɬɚɤ
w T 1 w § w T · 1 w 2T w 2T qV
,
(5.4)
a
r
w t r w r ¨© w r ¸¹ r 2 wM 2 w z 2
O
ɚ ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢɦɟɸɬ ɜɢɞ
wT
wT
1 wT
; q M O
; q z O
.
(5.5)
q r O
wr
wz
r wM
ȼ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɢɦɟɟɦ T T r , T , M (ɪɢɫ. 5.3) ɢ
ȼ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 5.2) T
w §
w ·
w2
1 w § 2 w ·
1
1
'
,
¨r
¸
¨ si n 4 w4 ¸ 2
¹ r sin 4 wM 2
r 2 w r © w r ¹ r 2sin 4 w4 ©
w
1 w
1
w
.
’ ir
i4
iM
r w4
r sin 4 wM
wr
ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɬɪɟɯɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɤɨɦɩɨɧɟɧɬɵ ɜɟɤɬɨɪɚ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɩɪɢɧɢɦɚɸɬ ɜɢɞ
113
w §
wT ·
w 2T qV
1 w § 2 wT ·
1
1
; (5.6)
¨r
¸
¨ sin 4 w4 ¸ 2
¹ r sin 4 wM2 O
r 2 w r © w r ¹ r 2sin 4 w4 ©
wT
1 wT
1 wT
; q 4 O
; q M O
.
(5.7)
q r O
wr
r w4
r si n 4 wM
Ⱦɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ x , y , z ɫɜɹɡɚɧɵ ɫ ɤɨɨɪɞɢɧɚɬɚɦɢ ɫɮɟɪɢɱɟɫɤɨɣ
ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ r ,T , M ɫɨɨɬɧɨɲɟɧɢɹɦɢ
x r sin 4 cos M ; y r sin 4 si n M ; z r cos 4
(5.8)
1 wT
a wt
Ɋɢɫ. 5.2. ɐɢɥɢɧɞɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ
ɤɨɨɪɞɢɧɚɬ
Ɋɢɫ. 5.3. ɋɮɟɪɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ
ɤɨɨɪɞɢɧɚɬ
Ɏɨɪɦɭɥɵ (5.3) ɢ (5.8) ɩɨɡɜɨɥɹɸɬ ɥɟɝɤɨ ɩɟɪɟɯɨɞɢɬɶ ɨɬ ɨɞɧɨɣ ɫɢɫɬɟɦɵ
ɤɨɨɪɞɢɧɚɬ ɤ ɞɪɭɝɨɣ, ɟɫɥɢ ɷɬɨ ɧɟɨɛɯɨɞɢɦɨ.
5.2. ɍɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɞɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ
ɍɪɚɜɧɟɧɢɹ (5.4) ɢ (5.6) ɨɫɨɛɟɧɧɨ ɭɞɨɛɧɵ, ɤɨɝɞɚ ɧɭɠɧɨ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɬɟɥɚɯ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ – ɜ ɰɢɥɢɧɞɪɟ ɢɥɢ
ɲɚɪɟ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ ɭɪɚɜɧɟɧɢɹ ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɸɬɫɹ ɩɪɢ ɡɚɞɚɧɢɢ
ɨɫɨɛɵɯ ɭɫɥɨɜɢɣ, ɤɨɝɞɚ ɩɨɥɟ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɤɨɨɪɞɢɧɚɬɵ. ɇɚɩɪɢɦɟɪ, ɜ ɫɥɭɱɚɟ ɞɥɢɧɧɨɝɨ ɰɢɥɢɧɞɪɚ, ɧɚ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɬɨɪɨɝɨ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ (ɪɢɫ. 5.4, ɚ), ɢɥɢ
ɜ ɫɥɭɱɚɟ ɲɚɪɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɪɚɜɧɨɦɟɪɧɨ ɩɪɨɝɪɟɬɨɣ ɠɢɞɤɨɫɬɢ
(ɪɢɫ. 5.4, ɛ). Ʉɚɤ ɢ ɜ ɫɥɭɱɚɟ ɩɥɚɫɬɢɧɵ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ (ɫ
ɨɞɧɨɪɨɞɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɨ ɜɫɟɣ ɟɟ ɩɨɜɟɪɯɧɨɫɬɢ, ɪɢɫ. 5.4, ɜ), ɭɪɚɜɧɟɧɢɹ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɚɧɨɜɹɬɫɹ ɨɞɧɨɦɟɪɧɵɦɢ.
Ɍɚɤ, ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɩɨɥɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɡɚɜɢɫɹɳɟɟ ɬɨɥɶɤɨ ɨɬ ɤɨɨɪɞɢɧɚɬɵ r , ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
114
w T 1 w § w T · qV
,
(5.9)
¨r
¸
wt r wr © wr ¹ O
ɫɥɟɞɭɸɳɢɦ ɢɡ (5.4), ɚ ɜ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ – ɭɪɚɜɧɟɧɢɟɦ
1 wT
1 w § 2 w T · qV
,
(5.10)
¨r
¸
a wt r 2 wr © wr ¹ O
ɫɥɟɞɭɸɳɢɦ ɢɡ (5.6).
ȼ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɦɵ ɢɦɟɥɢ ɭɪɚɜɧɟɧɢɟ (2.24), ɤɨɬɨɪɨɟ ɩɪɢ ɡɚɦɟɧɟ ɤɨɨɪɞɢɧɚɬɵ x ɧɚ r (ɞɥɹ ɟɞɢɧɨɨɛɪɚɡɢɹ ɡɚɩɢɫɢ) ɢ ɩɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
1 w T w 2T qV
.
(5.11)
a wt wr 2
O
a
z
qe
y
T
Ts
x
ɚ
ɛ
ɜ
Ɋɢɫ. 5.4. Ʉ ɡɚɩɢɫɢ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ
ȼɫɟ ɬɪɢ ɭɪɚɜɧɟɧɢɹ (5.9) – (5.10) ɦɨɠɧɨ ɨɛɴɟɞɢɧɢɬɶ ɜ ɨɞɧɨ
a
wT
wt
1 w § n w T · qV
,
¨r
¸
r n wr © wr ¹ O
(5.12)
ɝɞɟ n 0 – ɞɥɹ ɩɥɚɫɬɢɧɵ; n 1 – ɞɥɹ ɰɢɥɢɧɞɪɚ ɢ n 2 – ɞɥɹ ɫɮɟɪɵ
(ɪɢɫ. 5.4).
ȼ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
T
T T0
;W
T T0
t
;[
t
r
,
r
ɝɞɟ T – ɧɟɤɨɬɨɪɚɹ ɯɚɪɚɤɬɟɪɧɚɹ ɞɥɹ ɩɪɨɰɟɫɫɚ ɬɟɦɩɟɪɚɬɭɪɚ (ɧɚɩɪɢɦɟɪ,
ɬɟɦɩɟɪɚɬɭɪɚ ɨɦɵɜɚɸɳɟɣ ɬɟɥɨ ɠɢɞɤɨɫɬɢ; ɢɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɫɬɟɧɤɢ); T0 –
ɧɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ; r – ɯɚɪɚɤɬɟɪɧɵɣ ɞɥɹ ɡɚɞɚɱɢ ɪɚɡɦɟɪ (ɧɚɩɪɢɦɟɪ,
ɲɢɪɢɧɚ ɩɥɚɫɬɢɧɵ, ɪɚɞɢɭɫ ɰɢɥɢɧɞɪɚ ɢɥɢ ɫɮɟɪɵ), ɭɪɚɜɧɟɧɢɟ (5.12) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
1 wT
Fo wW
1 w § n wT ·
¨[
¸ qV ,
[ n w[ © w[ ¹
115
(5.13)
qV x2
– ɛɟɡɪɚɡɦɟɪɧɚɹ ɩɥɨɬɧɨɫɬɶ
ɝɞɟ Fo
– ɱɢɫɥɨ Ɏɭɪɶɟ; qV
O T T0 x2
ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɯɚɪɚɤɬɟɪɧɭɸ ɢɥɢ ɦɚɫɲɬɚɛɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɢɡ ɭɫɥɨɜɢɹ qV 1 . Ɍɨɝɞɚ
q
T T0 V x 2 .
O
at 5.3. ɉɪɨɫɬɟɣɲɢɟ ɫɬɚɰɢɨɧɚɪɧɵɟ
ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɰɢɥɢɧɞɪɢɱɟɫ ɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
Ɋɚɫɫɦɨɬɪɢɦ ɫɬɚɰɢɨɧɚɪɧɵɣ ɩɪɨɰɟɫɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɟ (ɬɪɭɛɟ) ɫ
ɜɧɭɬɪɟɧɧɢɦ ɪɚɞɢɭɫɨɦ d 1 2r1 ɢ ɧɚɪɭɠɧɵɦ
ɪɚɞɢɭɫɨɦ d 2 2r2 (ɪɢɫ. 5.5). ɉɪɟɞɩɨɥɨɠɢɦ,
ɱɬɨ ɰɢɥɢɧɞɪ ɢɥɢ ɬɪɭɛɚ ɢɦɟɸɬ ɞɨɫɬɚɬɨɱɧɨ
ɛɨɥɶɲɭɸ ɞɥɢɧɭ, ɱɬɨɛɵ ɬɟɩɥɨɨɬɜɨɞɨɦ ɫ ɬɨɪɰɨɜ
ɦɨɠɧɨ ɛɵɥɨ ɩɪɟɧɟɛɪɟɱɶ, ɢ ɱɬɨ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɧɚ ɜɧɭɬɪɟɧɧɟɦ ɢ ɜɧɟɲɧɟɦ ɪɚɞɢɭɫɚɯ ɰɢɥɢɧɞɪɚ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɤɨɨɪɞɢɧɚɬ M ɢ z , ɬɚɤ ɱɬɨ
ɡɚɞɚɱɚ ɨɛ ɨɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ
ɫɬɚɧɨɜɢɬɫɹ ɨɞɧɨɦɟɪɧɨɣ.
ɇɚɫ ɢɧɬɟɪɟɫɭɟɬ ɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɩɨɷɬɨɦɭ ɜɨɫɩɨɥɶɡɭɟɦɫɹ
ɫɬɚɰɢɨɧɚɪɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ,
ɤɨɬɨɪɨɟ ɫɥɟɞɭɟɬ ɢɡ (5.9), ɟɫɥɢ ɜ ɧɟɦ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɜɪɟɦɟɧɢ ɩɨɥɨɠɢɬɶ ɪɚɜɧɨɣ ɧɭɥɸ:
Ɋɢɫ. 5.5. Ʉ ɡɚɞɚɱɟ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
d 2T 1 dT
0.
(5.14)
ɜ ɬɪɭɛɟ
dx 2 r dr
Ɉɛɨɡɧɚɱɚɹ d T d r u , ɧɚɣɞɟɦ ɭɪɚɜɧɟɧɢɟ
du 1
u 0,
dr r
ɨɬɤɭɞɚ, ɪɚɡɞɟɥɹɹ ɩɟɪɟɦɟɧɧɵɟ, ɢɦɟɟɦ
du
dr
.
u
r
ɂɧɬɟɝɪɢɪɭɟɦ:
ln u ln r ln C1 ,
116
ɨɬɤɭɞɚ
dT
C1
.
u
dr
r
ɉɨɜɬɨɪɧɨɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɚɟɬ
T C1ln r C 2 .
(5.15)
ɂɡ ɩɨɥɭɱɟɧɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ ɫɪɚɡɭ ɜɢɞɧɨ, ɱɬɨ ɭɞɟɥɶɧɵɣ ɬɟɩɥɨɜɨɣ
ɩɨɬɨɤ ɧɟ ɩɨɫɬɨɹɧɟɧ ɩɨ ɬɨɥɳɢɧɟ ɢ ɭɛɵɜɚɟɬ ɩɨ ɧɚɩɪɚɜɥɟɧɢɸ ɤ ɜɧɟɲɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ (ɜɞɨɥɶ ɪɚɞɢɭɫɚ r )
dT
C
(5.16)
q r O
O 1 .
dr
r
ȼ ɫɬɚɰɢɨɧɚɪɧɵɯ ɭɫɥɨɜɢɹɯ ɞɨɥɠɟɧ ɛɵɬɶ ɩɨɫɬɨɹɧɧɵɦ ɩɨɥɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɩɪɨɯɨɞɹɳɢɣ ɱɟɪɟɡ ɭɱɚɫɬɨɤ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɬɪɭɛɵ ɡɚɞɚɧɧɨɣ
ɞɥɢɧɵ l ɢ ɪɚɜɧɵɣ
Q q r F q r 2S rl = con st
(5.17)
ɉɨɫɤɨɥɶɤɭ ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ F ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɫ ɪɚɞɢɭɫɨɦ, ɩɨɬɨɤ q r ɞɨɥɠɟɧ ɭɛɵɜɚɬɶ.
ȼɬɨɪɨɟ ɫɥɟɞɫɬɜɢɟ ɩɨɥɭɱɟɧɧɵɯ ɫɨɨɬɧɨɲɟɧɢɣ – ɬɟɦɩɟɪɚɬɭɪɚ ɩɨ ɬɨɥɳɢɧɟ ɬɪɭɛɵ ɢɡɦɟɧɹɟɬɫɹ ɧɟ ɥɢɧɟɣɧɨ, ɚ ɩɨ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɦɭ ɡɚɤɨɧɭ, ɜ
ɨɬɥɢɱɢɟ ɨɬ ɚɧɚɥɨɝɢɱɧɨɣ ɡɚɞɚɱɢ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ.
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ C1 ɢ C2 ɦɨɝɭɬ ɛɵɬɶ ɧɚɣɞɟɧɵ, ɟɫɥɢ ɛɭɞɭɬ ɡɚɞɚɧɵ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ.
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɟɪɜɨɝɨ ɪɨɞɚ. ɉɭɫɬɶ ɧɚ ɩɨɜɟɪɯɧɨɫɬɹɯ ɰɢɥɢɧɞɪɚ
r r1 ɢ r r2 ɡɚɞɚɧɵ ɬɟɦɩɟɪɚɬɭɪɵ T1 ɢ T2 . Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ (5.15), ɧɚɣɞɟɦ
T1 T r1 C1ln r1 C 2 ;
T2 T r2 C1ln r2 C 2 .
Ɉɩɪɟɞɟɥɹɹ ɤɨɧɫɬɚɧɬɵ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢɡ ɷɬɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ,
ɩɨɥɭɱɢɦ
T1ln r2 r T2ln r r1 T
,
(5.18)
ln r2 r1 ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɪɨɯɨɞɹɳɟɟ ɱɟɪɟɡ ɭɱɚɫɬɨɤ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ ɞɥɢɧɵ l ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɫɨɝɥɚɫɧɨ ɡɚɤɨɧɭ Ɏɭɪɶɟ,
ɟɫɬɶ
dT
O 'T
Q O
(5.19)
˜ l ˜ 2S r {
˜ 2S l , ȼɬ,
dr
ln r2 r1 ɬ.ɟ., ɞɟɣɫɬɜɢɬɟɥɶɧɨ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɪɚɞɢɭɫɚ.
117
ȼ ɩɪɚɤɬɢɤɟ ɬɟɯɧɢɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ ɢɫɩɨɥɶɡɭɸɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɩɨɝɨɧɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ (ɢɥɢ ɥɢɧɟɣɧɭɸ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ)
Q
2SO
ql
'T
(5.20)
l ln r2 r1 Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɬɶɟɝɨ ɪɨɞɚ. Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɦɵ ɢɦɟɟɦ, ɟɫɥɢ ɡɚɞɚɧɵ ɧɟ ɬɟɦɩɟɪɚɬɭɪɵ ɫɬɟɧɨɤ, ɚ ɬɟɦɩɟɪɚɬɭɪɵ ɫɪɟɞ,
ɨɦɵɜɚɸɳɢɯ ɬɪɭɛɭ, ɢ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɨɬɞɚɱɢ.
Ʉɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɬɪɭɛɵ ɫ ɧɚɪɭɠɧɨɣ
ɢ ɜɧɭɬɪɟɧɧɟɣ ɫɬɨɪɨɧ ɦɨɠɟɬ ɛɵɬɶ ɜɵɪɚɠɟɧ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɚ ɇɶɸɬɨɧɚ
ɢ ɞɨɥɠɟɧ ɪɚɜɧɹɬɶɫɹ ɩɨɝɨɧɧɨɦɭ ɬɟɩɥɨɜɨɦɭ ɩɨɬɨɤɭ ɜɫɥɟɞɫɬɜɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɱɟɪɟɡ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɫɬɟɧɤɭ (5.20). Ɍ.ɟ., ɦɵ ɢɦɟɟɦ ɫɢɫɬɟɦɭ
ɭɪɚɜɧɟɧɢɣ
q l D 1 Te1 T1 ˜ 2S r1 ;
ql
O
T T ˜ 2 S ;
ln r2 r1 1 2
q l D 2 T2 Te2 ˜ 2S r2 ,
ɪɟɲɚɹ ɤɨɬɨɪɭɸ, ɧɚɣɞɟɦ ɩɨɝɨɧɧɭɸ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ
ql
/ c S Te1 Te 2 ,
(5.21)
ɝɞɟ
1
–
(5.22)
1
1 § r2 ·
1
ln ¨ ¸ 2D 1r1 2O © r1 ¹ 2D 2 r2
ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ (ɢɥɢ ɬɟɪɦɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ) ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ, ȼɬ/(ɦ·Ʉ). ȿɝɨ ɪɚɡɦɟɪɧɨɫɬɶ ɨɬɥɢɱɚɟɬɫɹ ɪɚɡɦɟɪɧɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɞɥɹ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ. Ʉɨɷɮɮɢɰɢɟɧɬ / c ɱɢɫɥɟɧɧɨ ɪɚɜɟɧ ɤɨɥɢɱɟɫɬɜɭ ɬɟɩɥɚ, ɩɪɨɯɨɞɹɳɟɦɭ ɱɟɪɟɡ ɫɬɟɧɤɭ ɬɪɭɛɵ ɟɞɢɧɢɱɧɨɣ ɞɥɢɧɵ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɨɬ ɨɞɧɨɣ ɫɪɟɞɵ ɤ ɞɪɭɝɨɣ, ɟɫɥɢ ɬɟɦɩɟɪɚɬɭɪɧɵɣ ɧɚɩɨɪ ɦɟɠɞɭ ɧɢɦɢ ɫɨɫɬɚɜɥɹɟɬ 1 Ʉ.
Ɉɛɪɚɬɧɚɹ ɜɟɥɢɱɢɧɚ
/c
/ c 1
1
1 § r2 ·
1
ln ¨ ¸ 2D1r1 2O © r1 ¹ 2D 2 r2
(5.23)
ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɧɵɦ ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɬɪɭɛɵ; 1 / 2D i ri ,
i 1, 2 – ɬɟɪɦɢɱɟɫɤɢɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹɦɢ ɬɟɩɥɨɨɬɞɚɱɢ; 1 / 2O ln r2 r1 – ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫɬɟɧɤɢ ɬɪɭɛɵ. Ɍ.ɟ., ɜ
118
ɫɥɭɱɚɟ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɨɬɞɚɱɢ ɡɚɜɢɫɢɬ ɧɟ
ɬɨɥɶɤɨ ɨɬ ɜɟɥɢɱɢɧ D1 ɢ D 2 , ɧɨ ɢ ɨɬ ɞɢɚɦɟɬɪɨɜ (ɢɥɢ ɪɚɞɢɭɫɨɜ) ɫɬɟɧɤɢ.
ɂɡ ɬɨɣ ɠɟ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɦɵ ɦɨɠɟɦ ɧɚɣɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɫɬɟɧɨɤ
/ T T / c Te1 Te 2 T1 Te1 c e1 e 2 ; T2
Te 2 ,
2D 1r1
2D 2r2
ɩɨɞɫɬɚɜɥɹɹ ɤɨɬɨɪɵɟ ɜ (5.18), ɧɚɣɞɟɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ
ɬɨɥɳɢɧɟ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ.
5.4. ɗɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ
ȼɵɪɚɠɟɧɢɟ (5.19) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɮɨɪɦɟ ɡɚɤɨɧɚ Ɉɦɚ
Q
'T
,
ln r2 r1 2S l O ɝɞɟ ɡɧɚɦɟɧɚɬɟɥɶ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɨɥɨɝɨ
ɰɢɥɢɧɞɪɚ
ln r2 r1 RT
.
2SO l
ɉɪɢɧɰɢɩɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɝɨ ɢ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɫɨɟɞɢɧɟɧɢɣ ɬɟɪɦɢɱɟɫɤɢɯ
ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɜ ɰɟɩɶ, ɫɩɪɚɜɟɞɥɢɜɵɟ
ɞɥɹ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ ɜ ɩɪɹɦɨɭɝɨɥɶɧɨɣ
ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɦɨɠɧɨ ɩɪɢɦɟɧɢɬɶ ɢ
ɞɥɹ ɡɚɞɚɱɢ ɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɩɨɥɨɦ
ɰɢɥɢɧɞɪɟ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɧɚɩɪɢɦɟɪ, ɱɬɨ ɠɢɞɤɨɫɬɶ ɬɟɱɟɬ ɜ ɬɪɭɛɟ, ɩɨɤɪɵɬɨɣ ɬɟɩɥɨɢɡɨɥɹɰɢɨɧɧɵɦ ɦɚɬɟɪɢɚɥɨɦ (ɪɢɫ. 5.6).
Ɋɢɫ. 5.6. ɋɟɱɟɧɢɟ ɬɟɩɥɨɢɡɨɥɢɂɡɜɟɫɬɧɨ, ɱɬɨ ɫɪɟɞɧɹɹ ɬɟɦɩɟɪɚɬɭɪɚ
ɪɨɜɚɧɧɨɣ ɬɪɭɛɵ
ɠɢɞɤɨɫɬɢ ɪɚɜɧɚ T0 , ɚ ɬɟɦɩɟɪɚɬɭɪɚ
ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ – Ts . ɏɚɪɚɤɬɟɪɢɫɬɢɤɢ ɦɚɬɟɪɢɚɥɚ
ɬɪɭɛɵ ɨɛɨɡɧɚɱɢɦ ɢɧɞɟɤɫɨɦ 1, ɚ ɬɟɩɥɨɢɡɨɥɹɰɢɢ – ɢɧɞɟɤɫɨɦ 2. Ʉɨɧɜɟɤɬɢɜɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɠɢɞɤɨɫɬɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɮɨɪɦɭɥɨɣ
RT ,0
1
DF
119
1
.
D 2S r1l
ɝɞɟ F 2S r1l – ɩɥɨɳɚɞɶ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɪɭɛɵ.
Ʉɨɧɜɟɤɬɢɜɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɠɢɞɤɨɫɬɢ ɧɭɠɧɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɢɬɶ ɫ ɞɜɭɦɹ ɤɨɧɞɭɤɬɢɜɧɵɦɢ ɬɟɪɦɢɱɟɫɤɢɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢɹɦɢ ɞɥɹ ɞɜɭɯ ɬɜɟɪɞɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɩɨɫɤɨɥɶɤɭ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɪɚɫɩɪɨɫɬɪɚɧɹɟɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɱɟɪɟɡ ɤɚɠɞɵɣ ɢɡ ɷɬɢɯ ɦɚɬɟɪɢɚɥɨɜ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɜ ɷɬɨɣ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ
T0 T s
§ 'T ·
, (5.24)
Q ¨
¸
§ r2 ·
§ r3 ·
1
1
1
© R ¹ fu l l
ln ¨ ¸ ln ¨ ¸
2SD1r1l 2S lO 1 © r1 ¹ 2S l O 2 © r2 ¹
ɝɞɟ ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ ɜ ɡɧɚɦɟɧɚɬɟɥɟ ɟɫɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ.
Ɍɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, ɜɯɨɞɹɳɟɟ ɜ ɫɨɨɬɧɨɲɟɧɢɟ (5.24), ɹɜɥɹɟɬɫɹ ɫɭɦɨɣ ɜɫɟɯ ɬɟɪɦɢɱɟɫɤɢɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɦɟɠɞɭ ɞɜɭɦɹ ɢɡɜɟɫɬɧɵɦɢ
ɬɟɦɩɟɪɚɬɭɪɚɦɢ.
ȿɫɥɢ ɢɡɜɟɫɬɧɵ ɬɟɦɩɟɪɚɬɭɪɵ T1 ɢ Ts , ɬɨ ɩɨɥɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɞɨɥɠɧɨ ɪɚɜɧɹɬɶɫɹ ɫɭɦɦɟ ɬɨɥɶɤɨ ɞɜɭɯ ɤɨɧɞɭɤɬɢɜɧɵɯ ɫɨɩɪɨɬɢɜɥɟɧɢɣ
ɬɪɭɛɵ ɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ.
T1 Ts
§ 'T ·
.
(5.25)
Q ¨
¸
R
§
·
§
·
r
1
1
r
©
¹ fu l l
ln ¨ 2 ¸ ln ¨ 3 ¸
2S l O 1 © r1 ¹ 2S l O 2 © r2 ¹
ɉɪɢ ɢɡɜɟɫɬɧɨɦ ɬɟɩɥɨɜɨɦ ɩɨɬɨɤɟ ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɫɬɟɧɤɢ
ɬɪɭɛɵ ɥɟɝɤɨ ɧɚɯɨɞɢɬɫɹ ɢɡ ɷɬɨɝɨ ɫɨɨɬɧɨɲɟɧɢɹ.
ɉɪɢɦɟɪ. ȼ ɚɥɸɦɢɧɢɟɜɨɣ ɬɪɭɛɟ, ɢɦɟɸɳɟɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ
O 1 1 8 5 ȼɬ/(ɦ Ʉ), ɬɟɱɟɬ ɜɨɞɹɧɨɣ ɩɚɪ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T0 110 ɨɋ. ȼɧɭɬɪɟɧɧɢɣ ɞɢɚɦɟɬɪ ɬɪɭɛɵ – 10 ɫɦ, ɧɚɪɭɠɧɵɣ ɞɢɚɦɟɬɪ – 12 ɫɦ. Ɍɪɭɛɚ ɪɚɫɩɨɥɨɠɟɧɚ ɜ ɩɨɦɟɳɟɧɢɢ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ Te 30 ɨɋ; ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ ɨɬ ɬɪɭɛɵ ɤ ɜɨɡɞɭɯɭ D e ɪɚɜɟɧ 15 ȼɬ/(ɦ2Ʉ). Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɬɪɭɛɵ, ɟɫɥɢ ɬɪɭɛɚ ɧɟ
ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɚ.
ɑɬɨɛɵ ɫɧɢɡɢɬɶ ɬɟɩɥɨɜɵɟ ɩɨɬɟɪɢ ɨɬ ɬɪɭɛɵ, ɨɧɚ ɛɵɥɚ ɩɨɤɪɵɬɚ ɫɥɨɟɦ
ɬɟɩɥɨɢɡɨɥɹɰɢɢ ( O 2 0 ,2 ȼɬ/(ɦ Ʉ)) ɬɨɥɳɢɧɨɣ 5 ɫɦ. ɇɚɣɬɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɨɬ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ ɬɪɭɛɵ. ɉɪɟɞɩɨɥɨɠɢɬɶ,
ɱɬɨ ɤɨɧɜɟɤɬɢɜɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɚɪɚ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨ.
Ɋɟɲɟɧɢɟ. Ⱦɥɹ ɬɪɭɛɵ ɛɟɡ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɧɚɢɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɵɦɢ
ɹɜɥɹɸɬɫɹ ɤɨɧɞɭɤɬɢɜɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɚɦɨɣ ɬɪɭɛɵ ɢ ɤɨɧɜɟɤɬɢɜɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɤɨɦɧɚɬɧɨɝɨ ɜɨɡɞɭɯɚ. ɉɨɫɤɨɥɶɤɭ
120
ɤɨɧɜɟɤɬɢɜɧɵɦ ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɩɚɪɚ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ,
ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɪɭɛɵ ɪɚɜɧɚ ɬɟɦɩɟɪɚɬɭɪɟ ɩɚɪɚ.
Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɬɪɭɛɵ ɫɥɟɞɭɟɬ ɢɡ ɫɨɨɬɧɨɲɟɧɢɹ
T0 Te
80
110 30
,
q
4
lnr2 r1 ln6 5
1
1
1
,
57
˜
10
0
,
177
2SO1
2Sr2 D e 2S ˜185 2S ˜ 0 ,06 ˜15
q 452 ȼɬ/ɦ.
Ⱦɥɹ ɬɪɭɛɵ ɫ ɬɟɩɥɨɢɡɨɥɹɰɢɟɣ ɧɭɠɧɨ ɞɨɛɚɜɢɬɶ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɢɡɨɥɹɰɢɢ, ɢ ɫɨɨɬɧɨɲɟɧɢɟ ɞɥɹ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɩɪɢɦɟɬ
ɜɢɞ
q
T0 Te
lnr2 r1 lnr3 r2 1
2SO1
2Sr3D e
2SO 2
80
1,57 ˜10 4 0 ,096 0 ,482
,
q 138 ȼɬ/ɦ.
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɩɨɡɜɨɥɹɟɬ ɜɬɪɨɟ ɫɧɢɡɢɬɶ ɩɨɬɟɪɢ ɬɟɩɥɚ
ɩɚɪɨɦ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɜ ɨɛɨɢɯ ɫɥɭɱɚɹɯ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɤɨɧɞɭɤɬɢɜɧɵɦ
ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɚɥɸɦɢɧɢɟɜɨɣ ɬɪɭɛɵ ɛɟɡ ɫɤɨɥɶɤɨ-ɧɢɛɭɞɶ
ɡɚɦɟɬɧɨɝɨ ɫɧɢɠɟɧɢɹ ɬɨɱɧɨɫɬɢ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɫɱɟɬɚ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ.
5.5. Ɇɧɨɝɨɫɥɨɣɧɚɹ ɰɢɥɢɧɞɪɢɱɟɫɤɚɹ ɫɬɟɧɤɚ
Ɋɟɡɭɥɶɬɚɬɵ ɥɟɝɤɨ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɧɚ ɦɧɨɝɨɫɥɨɣɧɭɸ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɫɬɟɧɤɭ.
Ɍɚɤ, ɜ ɫɥɭɱɚɟ ɬɪɭɛɵ, ɫɨɫɬɨɹɳɟɣ ɢɡ n ɫɥɨɟɜ ɫ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ Oi , i 1,2, ..., n , ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ
ɧɚɣɞɟɦ
q
Tn T11 S
n
r
1
¦ 2O l n ir1
i
i
i 1
.
(5.26)
Ⱦɥɹ ɦɧɨɝɨɫɥɨɣɧɨɣ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ ɨɫɬɚɟɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦ
ɩɨɧɹɬɢɟ ɷɤɜɢɜɚɥɟɧɬɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɍɨɝɞɚ
q
2SO ɷɤɜ T1 Tn1 ,
ln rn1 r1 ɝɞɟ
121
O ɷɤɜ
ln rn 1 r1 n
1 §r ·
¦ O ln ¨ ir1 ¸
© i ¹
i 1 i
–
ɷɤɜɢɜɚɥɟɧɬɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɧɨɝɨɫɥɨɣɧɨɣ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ.
ɍɪɚɜɧɟɧɢɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪ ɦɟɠɞɭ i -ɦ ɢ i 1 -ɦ ɫɥɨɹɦɢ
ɩɪɢɦɭɬ ɜɢɞ
q § 1
r
r
1
1 r ·
Ti 1 Ti c ¨ ln 2 (5.27)
ln 3 . .. ln i 1 ¸ ,
ri ¹
2S © O 1 r1 O 2 r2
Oi
ɚ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɟɪɟɞɚɱɢ 1
.
(5.28)
/c
n
§ ri 1 ·
1
1
1
¦
ln ¨
¸
2D 1r1 i 1 2O i © ri ¹ 2D 2r2
5.6. Ʉ ɪ ɢ ɬ ɢ ɱ ɟ ɫ ɤ ɢ ɣ ɞ ɢ ɚ ɦ ɟ ɬ ɪ ɬ ɟ ɩ ɥ ɨ ɢ ɡ ɨ ɥ ɹ ɰ ɢ ɢ
ɂɬɚɤ, ɦɵ ɡɧɚɟɦ, ɱɬɨ ɪɚɞɢɚɥɶɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɜ ɬɪɭɛɟ ɨɛɪɚɬɧɨ
ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɥɨɝɚɪɢɮɦɭ ɧɚɪɭɠɧɨɝɨ ɪɚɞɢɭɫɚ, ɬɨɝɞɚ ɤɚɤ ɪɚɫɫɟɹɧɢɟ ɬɟɩɥɚ ɨɬ ɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɷɬɨɦɭ ɪɚɞɢɭɫɭ.
Ɍɚɤɨɝɨ ɪɨɞɚ ɞɜɨɣɧɨɣ ɷɮɮɟɤɬ ɜɨɡɪɚɫɬɚɸɳɟɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɪɚɞɢɚɥɶɧɨɣ
ɩɪɨɜɨɞɢɦɨɫɬɢ ɢ ɨɞɧɨɜɪɟɦɟɧɧɨɝɨ ɭɜɟɥɢɱɟɧɢɹ ɩɥɨɳɚɞɢ ɨɯɥɚɠɞɚɸɳɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɩɨ ɦɟɪɟ ɜɨɡɪɚɫɬɚɧɢɹ r ɨɡɧɚɱɚɟɬ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɨɩɪɟɞɟɥɟɧɧɵɣ ɪɚɞɢɭɫ, ɩɪɢ ɤɨɬɨɪɨɦ ɩɨɬɟɪɢ ɬɟɩɥɚ ɦɚɤɫɢɦɚɥɶɧɵ. ɋɨɨɬɧɨɲɟɧɢɟ
ɜɧɭɬɪɟɧɧɟɝɨ ɢ ɜɧɟɲɧɟɝɨ ɪɚɞɢɭɫɨɜ ɬɚɤɠɟ ɢɝɪɚɟɬ ɜɚɠɧɭɸ ɪɨɥɶ.
Ɍɚɤ, ɟɫɥɢ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ, ɧɨ ɧɟɛɨɥɶɲɨɦ r1 (ɜɧɭɬɪɟɧɧɟɦ ɪɚɞɢɭɫɟ) ɦɵ ɭɜɟɥɢɱɢɦ ɬɨɥɳɢɧɭ ɫɬɟɧɤɢ ɩɭɬɟɦ ɭɜɟɥɢɱɟɧɢɹ r2 , ɬɨ ɞɟɣɫɬɜɢɟ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɜ ɮɨɪɦɭɥɟ (5.22) ɨɤɚɠɟɬɫɹ ɛɨɥɟɟ ɫɢɥɶɧɵɦ,
ɱɟɦ ɩɪɢ ɬɨɦ ɠɟ ɭɜɟɥɢɱɟɧɢɢ ɬɨɥɳɢɧɵ ɫɬɟɧɤɢ ɬɪɭɛɵ ɫ ɛɨɥɶɲɢɦ ɜɧɭɬɪɟɧɧɢɦ ɪɚɞɢɭɫɨɦ. ȿɫɥɢ r1 ɮɢɤɫɢɪɨɜɚɧɨ, ɬɨ ɩɨɬɨɤ q l q l r2 (ɮɨɪɦɭɥɚ
(5.21)) ɛɭɞɟɬ ɦɚɤɫɢɦɚɥɶɧɵɦ ɩɪɢ ɬɚɤɨɦ ɡɧɚɱɟɧɢɢ r2 , ɩɪɢ ɤɨɬɨɪɨɦ
dq
dr2
§ 1
1 ·
2S' T ¨
¨ O r2 D r 2 ¸¸
©
2 2 ¹
§ 1
1 r
1 ·
ln 2 ¨
¸
© D 1r1 O r1 D 2r2 ¹
ɗɬɨ ɪɚɜɟɧɫɬɜɨ ɞɚɟɬ ɧɚɦ ɤɪɢɬɢɱɟɫɤɢɣ ɪɚɞɢɭɫ
122
2
0.
(5.29)
O
.
(5.30)
D2
Ɋɚɫɫɭɠɞɟɧɢɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨ ɢɥɥɸɫɬɪɢɪɭɸɬɫɹ ɤɪɢɜɵɦɢ ɧɚ ɪɢɫ 5.7.
ɞɥɹ ɨɞɧɨɫɥɨɣɧɨɣ ɬɪɭɛɵ (ɪɢɫ. 5.5) ɫ ɜɧɭɬɪɟɧɧɢɦ ɭɞɟɥɶɧɵɦ ɩɨɜɟɪɯɧɨɫɬɧɵɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ 1 D1 0 . ȼ ɬɚɤɨɦ ɫɥɭɱɚɟ T1 Te1 ɢ
1
.
/c
1 § r2 ·
1
ln ¨ ¸ 2O © r1 ¹ 2D 2 r2
ȼɜɨɞɹ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɢ ɩɚɪɚɦɟɬɪɵ
r2 x
y
E=1
0,8
y
E=0
0,6
r2
O
,E
,
r1
D 2r2
ql
,
2SO Te1 Te 2 0,4
ɩɪɢɞɟɦ ɤ ɫɨɨɬɧɨɲɟɧɢɸ
0,2
y
2
4
6
8
x
Ɋɢɫ. 5.7. ȼɥɢɹɧɢɟ ɨɬɧɨɲɟɧɢɹ ɪɚɞɢɭɫɨɜ ɧɚ
ɜɟɥɢɱɢɧɭ ɩɨɬɨɤɚ ɬɟɩɥɚ ɫɤɜɨɡɶ ɨɞɧɨɫɥɨɣɧɭɸ ɫɬɟɧɤɭ ɬɪɭɛɵ. Ʉɪɢɜɵɟ ɩɨɫɬɪɨɟɧɵ
ɞɥɹ ȕ = 0,1,2,3,4,5, 6,7,8 ɢ ɪɚɫɩɨɥɨɠɟɧɵ ɜ
ɩɨɪɹɞɤɟ ɜɨɡɪɚɫɬɚɧɢɹ ɩɚɪɚɦɟɬɪɚ. ɉɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ ɭɤɚɡɵɜɚɸɬ ɩɨɥɨɠɟɧɢɟ
ɦɚɤɫɢɦɭɦɚ ɧɚ ɤɪɢɜɵɯ, ɤɨɬɨɪɵɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɤɪɢɬɢɱɟɫɤɨɦɭ ɡɧɚɱɟɧɢɸ ɪɚɞɢɭɫɚ
1
E
ln x x
.
(5.31)
Ʉɪɢɜɚɹ ɞɥɹ E 0 ɢɥɥɸɫɬɪɢɪɭɟɬ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɤɚɤ
ɜɧɭɬɪɟɧɧɟɟ, ɬɚɤ ɢ ɧɚɪɭɠɧɨɟ
ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɪɚɜɧɵ ɧɭɥɸ. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ r2 o f . Ʉɪɢ-
ɜɚɹ ɞɥɹ E 1 ɩɪɟɞɫɬɚɜɥɹɟɬ ɬɨɬ
ɨɫɨɛɵɣ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɤɪɢɬɢɱɟɫɤɢɣ ɧɚɪɭɠɧɵɣ ɪɚɞɢɭɫ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɮɢɤɬɢɜɧɵɦ ɭɫɥɨɜɢɹɦ: ɬɨɥɳɢɧɚ
ɫɬɟɧɤɢ ɪɚɜɧɚ ɧɭɥɸ. Ⱦɥɹ ɞɚɧɧɨɝɨ ɜɧɭɬɪɟɧɧɟɝɨ ɪɚɞɢɭɫɚ ɜɟɥɢɱɢɧɚ ɤɪɢɬɢɱɟɫɤɨɝɨ ɧɚɪɭɠɧɨɝɨ ɪɚɞɢɭɫɚ ɭɜɟɥɢɱɢɜɚɟɬɫɹ, ɟɫɥɢ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɬɪɭɛɵ ɢɥɢ ɟɫɥɢ ɩɨɧɢɠɚɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ.
ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɤɪɢɬɢɱɟɫɤɨɝɨ ɧɚɪɭɠɧɨɝɨ ɪɚɞɢɭɫɚ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ,
ɱɬɨ ɩɪɢ ɧɟɤɨɬɨɪɵɯ ɪɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ, ɜɨɩɪɟɤɢ ɩɪɢɜɵɱɧɵɦ ɞɥɹ ɨɛɵɜɚɬɟɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦ, ɩɨɬɟɪɹ ɬɟɩɥɚ ɢɡɨɥɢɪɨɜɚɧɧɨɣ ɬɪɭɛɨɣ ɮɚɤɬɢɱɟɫɤɢ
ɦɨɠɟɬ ɛɵɬɶ ɫɧɢɠɟɧɚ ɩɭɬɟɦ ɭɦɟɧɶɲɟɧɢɹ ɬɨɥɳɢɧɵ ɬɟɩɥɨɢɡɨɥɹɰɢɢ.
Ɍɚɤ, ɞɥɹ ɞɜɭɯɫɥɨɣɧɨɣ ɬɪɭɛɵ, ɫɟɱɟɧɢɟ ɤɨɬɨɪɨɣ ɢɡɨɛɪɚɠɟɧɨ ɧɚ
ɪɢɫ. 5.6, ɩɨɥɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
123
RT ,C
§d ·
§d ·
1
1
1
1
,
ln ¨ 2 ¸ ln ¨ 3 ¸ D1d1 2O 1 © d1 ¹ 2O 2 © d 2 ¹ D 2 d 3
1
/C
(5.32)
ɝɞɟ d 1 2r1 ; d 1 2r1 ; d 1 2r1 ; O1 – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɬɪɭɛɵ;
O 2 – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢɡɨɥɹɰɢɨɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
Ɉɬ ɬɨɥɳɢɧɵ ɢɡɨɥɹɰɢɢ ɜ (5.32) ɡɚɜɢɫɹɬ ɞɜɚ ɩɨɫɥɟɞɧɢɯ ɫɥɚɝɚɟɦɵɯ.
Ɍɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢɡɨɥɹɰɢɢ (ɬɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ) ɪɚɫɬɟɬ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɬɨɥɳɢɧɵ ɢɡɨɥɹɰɢɨɧɧɨɝɨ ɩɨɤɪɵɬɢɹ, ɚ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɟɩɥɨɢɡɨɥɹɰɢɢ (ɱɟɬɜɟɪɬɨɟ ɫɥɚɝɚɟɦɨɟ) ɩɚɞɚɟɬ.
ɉɨɫɥɟɞɧɟɟ ɫɜɹɡɚɧɨ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɨɬɞɚɱɢ ɫ ɪɨɫɬɨɦ
ɪɚɞɢɭɫɚ r3 (ɢɥɢ ɞɢɚɦɟɬɪɚ). ɍɫɥɨɜɢɟ ɷɤɫɬɪɟɦɭɦɚ ɷɬɨɣ ɮɭɧɤɰɢɢ ɟɫɬɶ
w
R
w d 3 T ,C
w § 1 ·
¨
¸
wd3 © /C ¹
1
2O 2 d 3
1
D 2d 32
0.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ɞɢɚɦɟɬɪɚ
2O 2
D2
d 3 (5.33)
ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɧɟɲɧɟɝɨ ɞɢɚɦɟɬɪɚ d 2 ɢɡɨɥɢɪɭɟɦɨɝɨ ɬɪɭɛɨɩɪɨɜɨɞɚ, ɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɵɛɪɚɧɧɨɝɨ ɢɡɨɥɹɬɨɪɚ
O 2 ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɬɞɚɱɢ ɫ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ D 2 .
ɇɚɯɨɞɢɦ ɜɬɨɪɭɸ ɩɪɨɢɡɜɨɞɧɭɸ
1 ·
w2 § 1 · 1 § 2
¨
¸
¨
¸
w d 32 © / C ¹ d 32 © D 2d 3 2O 2 ¹
ɢɥɢ (ɫ ɭɱɟɬɨɦ (5.33))
w 2 § 1 · D 32
(5.34)
! 0.
¨
¸
w d 32 © / C ¹ 8O 32
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ d
d 3 ɩɨɥɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɦɢɧɢɦɚɥɶɧɨ.
Ɇɨɠɧɨ ɫɞɟɥɚɬɶ ɜɵɜɨɞ: ɟɫɥɢ ɞɢɚɦɟɬɪ ɢɡɨɥɢɪɭɟɦɨɣ ɬɪɭɛɵ ɛɨɥɶɲɟ
ɤɪɢɬɢɱɟɫɤɨɝɨ ɞɢɚɦɟɬɪɚ, ɧɚɣɞɟɧɧɨɝɨ ɞɥɹ ɜɵɛɪɚɧɧɨɝɨ ɢɡɨɥɹɰɢɨɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ d ! d 3 ɢ ɡɚɞɚɧɧɵɯ ɭɫɥɨɜɢɣ ɬɟɩɥɨɨɛɦɟɧɚ ɫɨ ɫɪɟɞɨɣ, ɬɨ ɩɨɤɪɵɬɢɟ ɬɪɭɛɵ ɫɥɨɟɦ ɬɚɤɨɣ ɢɡɨɥɹɰɢɢ ɭɦɟɧɶɲɚɟɬ ɬɟɩɥɨɨɬɞɚɱɭ ɱɟɪɟɡ ɰɢɥɢɧɞɪɢɱɟɫɤɭɸ ɫɬɟɧɤɭ. ȿɫɥɢ ɠɟ d d 3 , ɬɨ ɧɚɧɟɫɟɧɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ
ɬɪɭɛɵ ɜɵɛɪɚɧɧɨɝɨ ɢɡɨɥɹɬɨɪɚ ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɩɪɢɜɟɞɟɬ ɤ ɜɨɡɪɚɫɬɚɧɢɸ
ɬɟɩɥɨɨɬɞɚɱɢ. ɂ ɥɢɲɶ ɩɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɛɭɞɟɬ ɞɨɫɬɢɝɧɭɬ ɤɪɢɬɢɱɟɫɤɢɣ ɞɢɚ124
ɦɟɬɪ, ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɫɬɟɧɤɭ ɧɚɱɧɟɬ ɭɛɵɜɚɬɶ; ɡɚɬɟɦ ɨɧ ɞɨɫɬɢɝɧɟɬ
ɬɨɣ ɜɟɥɢɱɢɧɵ, ɤɨɬɨɪɚɹ ɛɵɥɚ ɛɵ ɩɪɢ ɨɬɫɭɬɫɬɜɢɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ, ɢ ɥɢɲɶ
ɡɚɬɟɦ ɫɬɚɧɟɬ ɦɟɧɶɲɟ ɟɟ.
ȼ ɩɨɫɥɟɞɧɟɦ ɫɥɭɱɚɟ ɫɥɟɞɭɟɬ ɩɨɞɨɛɪɚɬɶ ɞɪɭɝɨɣ ɢɡɨɥɹɰɢɨɧɧɵɣ ɦɚɬɟɪɢɚɥ ɢɥɢ ɫɞɟɥɚɬɶ ɦɧɨɝɨɫɥɨɣɧɭɸ ɢɡɨɥɹɰɢɸ ɫ O ɷɤɜ ! O 2 ɢ ɥɢɲɶ, ɩɨɬɨɦ
ɩɨɣɬɢ ɧɚ ɫɧɢɠɟɧɢɟ ɬɟɩɥɨɨɬɞɚɱɢ ɩɭɬɟɦ ɡɧɚɱɢɬɟɥɶɧɨɝɨ ɭɜɟɥɢɱɟɧɢɹ ɬɨɥɳɢɧɵ ɢɡɨɥɹɰɢɨɧɧɨɝɨ ɫɥɨɹ.
5.7. ɒɚɪɨɜɚɹ ɫɬɟɧɤɚ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨ ɨɞɧɨɦɟɪɧɭɸ ɫɬɚɰɢɨɧɚɪɧɭɸ ɡɚɞɚɱɭ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɲɚɪɨɜɨɣ ɫɬɟɧɤɟ ɫ ɪɚɞɢɭɫɚɦɢ ɜɧɭɬɪɟɧɧɟɣ ɢ ɜɧɟɲɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɟɣ r1 ɢ r2 ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ O .
Ɉɞɧɨɦɟɪɧɨɫɬɶ ɡɚɞɚɱɢ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɬɟɧɤɟ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɪɚɞɢɭɫɚ r , ɱɬɨ ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɲɚɪɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɬɨɝɨ,
ɱɬɨɛɵ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɲɚɪɨɜɨɣ ɫɬɟɧɤɟ, ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɫɥɟɞɭɸɳɟɟ ɢɡ (5.10)
ɞɥɹ qV 0 ,
d 2T
2 dT
0.
2
r
dr
dr
Ɋɟɲɚɟɬɫɹ ɡɚɞɚɱɚ ɚɧɚɥɨɝɢɱɧɨ ɩɪɟɞɵɞɭɳɟɦɭ.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
dT
.
u
dr
Ɍɨɝɞɚ ɜɦɟɫɬɨ (5.35) ɩɨɥɭɱɢɦ
du
u
2 .
dr
r
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ln u
(5.35)
2ln r ln C1 .
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɬɟɦɩɟɪɚɬɭɪɟ, ɧɚɣɞɟɦ
T r C1
dT
C2 ;
r
dr
C1
2
.
(5.36)
r
ȼ ɫɢɥɭ ɬɟɯ ɠɟ ɩɪɢɱɢɧ, ɱɬɨ ɢ ɜ ɫɥɭɱɚɟ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ – ɧɟɥɢɧɟɣɧɨ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ
r
r1 :
125
T
T1 ;
r
r2 :
T
T2 ,
ɧɚɣɞɟɦ ɬɟɦɩɟɪɚɬɭɪɭ
§1 1 ·
§ 1 1·
T1 ¨ ¸ T2 ¨ ¸
¨r r ¸
© r1 r ¹
2 ¹
©
§1 1·
¨ ¸
© r1 r2 ¹
T r (5.37)
ɢ ɩɨɥɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ
Q O
dT
˜ 4S r 2
dr
4SO C1 ,
ɤɨɬɨɪɵɣ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɬɟɤɭɳɟɝɨ ɪɚɞɢɭɫɚ, ɬɚɤ ɤɚɤ ɨɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɩɪɨɯɨɞɹɳɟɟ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɱɟɪɟɡ ɢɡɨɬɟɪɦɢɱɟɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ,
ɤɚɤɨɣ ɡɞɟɫɶ ɹɜɥɹɟɬɫɹ ɥɸɛɚɹ ɫɮɟɪɚ ɫ ɪɚɞɢɭɫɨɦ
r1 d r d r2 ,
ɞɨɥɠɧɨ ɛɵɬɶ ɨɞɢɧɚɤɨɜɨ.
ɂɡ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɪɚɜɟɧɫɬɜ ɧɚɯɨɞɢɦ
Q
4SO
T T .
1 1 1 2
r1 r2
(5.38)
Ⱦɥɹ ɦɧɨɝɨɫɥɨɣɧɨɣ ɲɚɪɨɜɨɣ ɫɬɟɧɤɢ ɜ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
ɩɟɪɜɨɝɨ ɪɨɞɚ ɚɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɧɚɣɞɟɦ ɪɚɜɟɧɫɬɜɨ
Q
4S T1 T2 n
1 §1
1 ·
¦O ¨r r ¸
i 1 ¹
i 1 i© i
,
ɝɞɟ Oi ɢ ri – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɜɧɭɬɪɟɧɧɢɣ ɪɚɞɢɭɫ i -ɝɨ
ɫɥɨɹ.
ɇɚɣɞɟɦ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɬɪɟɬɶɟɝɨ ɪɨɞɚ
wT
r r1 :
O
D1 Te1 T ,
(5.39)
wr
wT
r r2 :
O
D 2 T Te 2 .
wr
ɉɨɞɫɬɚɜɥɹɹ ɨɛɳɟɟ ɪɟɲɟɧɢɟ (5.36) ɜ ɜɵɩɢɫɚɧɧɵɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ,
ɩɪɢɞɟɦ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
126
D1r1 O C1 D1r12C 2
D 1r12Te1 ,
D 2r2 O C1 D 2r22C 2
D 2r22Te 2 ,
(5.40)
ɨɬɤɭɞɚ ɢɦɟɟɦ
D 1r12 Te 2 Te1 C1
D1r1 O C2
D 1r12
D 2r22
,
D 2r2 O D1r1 O T D1r12 T
D 2r2 O e2 D 2r22 e1
.
D1r1 O D1r12
D 2r2 O D 2r22
Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɪɟɲɚɟɬɫɹ ɡɚɞɚɱɚ ɨ ɧɚɯɨɠɞɟɧɢɢ ɩɨɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-ɫɨɫɬɚɜɧɵɯ ɲɚɪɨɜɵɯ ɨɛɨɥɨɱɤɚɯ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ (5.38) ɞɨɥɠɟɧ ɛɵɬɶ ɪɚɜɟɧ ɬɟɩɥɨɜɨɦɭ
ɩɨɬɨɤɭ, ɩɪɢɯɨɞɹɳɟɦɭ ɤ ɜɧɭɬɪɟɧɧɟɣ ɫɬɟɧɤɟ
4 S r1 2D1 Te1 T ɢ ɬɟɩɥɨɜɨɦɭ ɩɨɬɨɤɭ, ɩɨɤɢɞɚɸɳɟɦɭ ɜɧɟɲɧɸɸ ɫɬɟɧɤɭ
4 S r2 2D 2 T Te 2 ɉɨɷɬɨɦɭ ɜ ɫɥɭɱɚɟ ɲɚɪɨɜɨɣ ɫɬɟɧɤɢ Te1 ! Te 2 ɪɟɲɟɧɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ (5.37), ɝɞɟ
§
T1
r12D1Te1 / s ¨ Te 2
¨
©
·
T
e1 ¸¸
r22D 2
¹,
r12D1
§
r 2D ·
/ s ¨1 12 1 ¸ r12D1
¨ r D ¸
2 2¹
©
§
·
r12D 1
/ s ¨ Te 2 2 Te1 ¸
¨
¸
r2 D 2
©
¹,
2
§
r D ·
/ s ¨1 12 1 ¸ r12D 1
¨ r D ¸
2 2¹
©
r12D1Te 2
T2
127
(5.41)
/s
O
1 1
r1 r2
{
O r1r2
.
r2 r1
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɬɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɤɚɡɵɜɚɟɬɫɹ ɛɨɥɟɟ ɭɞɨɛɧɵɦ.
ȼ ɩɪɟɞɟɥɟ ɩɪɢ ɢɞɟɚɥɶɧɨɦ ɬɟɩɥɨɨɛɦɟɧɟ ɫɪɟɞ ɫ ɡɚɞɚɧɧɵɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ ɩɨɜɟɪɯɧɨɫɬɟɣ ɲɚɪɨɜɨɣ ɫɬɟɧɤɢ (ɬ.ɟ., ɩɪɢ ɛɟɫɤɨɧɟɱɧɵɯ ɡɧɚɱɟɧɢɹɯ
Di ) ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɩɟɪɟɯɨɞɢɬ ɜ
ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ.
5.8. Ɋɟɲɟɧɢɹ ɩɪɨɫɬɟɣɲɢɯ ɡɚɞɚɱ
ɜ ɛɟɡ ɪɚɡɦ ɟɪɧɨɣ ɮɨɪɦɟ
ɋɨɛɟɪɟɦ ɪɟɲɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɞɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ
ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɜɦɟɫɬɟ:
Tp
T1 T1 T2 r
;
r2
Tc
T1ln r2 r T2ln r r1 ;
ln r2 r1 Ts
§1 1 ·
§ 1 1·
T1 ¨ ¸ T2 ¨ ¸
¨r r ¸
© r1 r ¹
©
2 ¹
,
§1 1·
¨ ¸
© r1 r2 ¹
ɝɞɟ ɞɥɹ ɨɛɳɧɨɫɬɢ ɡɚɩɢɫɢ ɤɨɨɪɞɢɧɚɬɚ x ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ
ɡɚɦɟɧɟɧɚ ɧɚ r ; ɚ ɬɨɥɳɢɧɚ ɩɥɚɫɬɢɧɵ ɨɛɨɡɧɚɱɟɧɚ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɛɨɥɶɲɢɣ
ɪɚɞɢɭɫ ɫɬɟɧɤɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɢ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ
( h r2 ). ɇɢ ɨɞɧɨ ɢɡ ɷɬɢɯ ɪɟɲɟɧɢɣ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɭɫɬɚɧɨɜɢɜɲɟɦɭɫɹ
ɫɨɫɬɨɹɧɢɸ, ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
ȼ ɩɟɪɟɦɟɧɧɵɯ
T
T T2
;[
T1 T2
r
r2
ɜɵɩɢɫɚɧɧɵɟ ɩɪɨɫɬɟɣɲɢɟ ɪɟɲɟɧɢɹ ɩɪɢɧɢɦɚɸɬ ɜɢɞ
128
T p 1 [ , 0 d [ d 1,
4
ln [ 1
0,8
, H d [ d 1 , (5.42)
Tc ln H
0,6
1 [
1
2
, H d [ d 1,
Tp
1
3
0,4
[ H 1
ɝɞɟ
0,2
H
r2
H
! 1.
0,0
r1
0,0
0,2
0,4
0,6
0,8
ȼɢɞɢɦ, ɱɬɨ ɜ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ ɤɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɟɟ Ɋɢɫ. 5.8. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ
ɩɥɨɫɤɨɣ (1), ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ (2) ɢ
ɬɨɥɳɢɧɵ. Ⱥ ɜɨɬ ɜ ɰɢɥɢɧɞɪɢɱɟɲɚɪɨɜɨɣ (3) ɫɬɟɧɤɟ. ɋɩɥɨɲɧɵɟ ɥɢɧɢɢ –
ɫɤɨɣ ɢ ɲɚɪɨɜɨɣ – ɧɟɥɢɧɟɣɧɨ
H 1 0 ; ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ – H 5
ɦɟɧɹɟɬɫɹ ɫ ɪɚɞɢɭɫɨɦ, ɩɪɢɱɟɦ
ɯɚɪɚɤɬɟɪ ɪɚɫɩɪɟɞɟɥɟɧɢɹ (ɤɪɢɜɢɡɧɚ ɤɪɢɜɨɣ) ɡɚɜɢɫɢɬ ɨɬ ɫɨɨɬɧɨɲɟɧɢɹ
ɜɧɟɲɧɟɝɨ ɢ ɜɧɭɬɪɟɧɧɟɝɨ ɪɚɞɢɭɫɨɜ. Ɋɚɡɧɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪ ɫɬɟɧɨɤ ɜ ɮɨɪɦɭɥɵ ɧɟ ɜɯɨɞɢɬ.
ȼ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɪɟɲɟɧɢɹ ɩɪɨɫɬɟɣɲɢɯ
ɡɚɞɚɱ ɡɚɜɢɫɹɬ ɨɬ ɩɚɪɚɦɟɬɪɨɜ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɬɟɩɥɨɨɛɦɟɧ. ɉɨɥɚɝɚɹ,
ɱɬɨ D1 D 2 D ɩɪɟɞɫɬɚɜɢɦ ɪɟɲɟɧɢɹ ɩɪɨɫɬɟɣɲɢɯ ɡɚɞɚɱ ɞɥɹ ɩɥɨɫɤɨɣ,
ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɢ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦ ɤɨɨɪɞɢɧɚɬ ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
T Te 2
r
;[
.
r2
Te1 Te 2
ɂɦɟɟɦ ɞɥɹ ɩɥɚɫɬɢɧɵ
4
T
0,8
0,6
0,4
3
1
2
0,2
0,0
0,2
Tp
3
0,4
0,6
1
0,8
2
T1 T1 T 2 [
T1 1 [
Ɋɢɫ. 5.9. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɞɨɥɶ ɤɨɨɪɞɢɧɚɬɵ ɜ ɩɥɨɫɤɨɣ (1),
ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ (2) ɢ ɫɮɟɪɢɱɟɫɤɨɣ (3)
ɫɬɟɧɤɚɯ ɜ ɭɫɥɨɜɢɹɯ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɨɛɦɟɧɚ. ɋɩɥɨɲɧɵɟ ɥɢɧɢɢ –
B i 2 ; ɩɭɧɤɬɢɪɧɵɟ – B i 1 0
129
1
,T
2 Bi 2
(5.43)
1
,
2 Bi
ɞɥɹ ɰɢɥɢɧɞɪɚ
T1 T 2 ln [ T 2ln H
, (5.44)
Tc
ln H
H
T1 1 ,
1 H H Bi ln H
H
T2
1 H H Bi ln H
ɢ ɞɥɹ ɫɮɟɪɵ
Ts
T1
T2
1 [ T1 [H 1 T 2 ,
[ H 1
H 1 B i 1 ,
1 H H 1 B i
(5.45)
1
,
1 H H 1 Bi
D r1
.
O
ɇɚ ɪɢɫ. 5.9. ɩɪɟɞɫɬɚɜɥɟɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜɞɨɥɶ ɤɨɨɪɞɢɧɚɬɵ [ . ȼɢɞɧɨ, ɱɬɨ ɤɚɱɟɫɬɜɟɧɧɵɣ ɯɚɪɚɤɬɟɪ ɤɪɢɜɵɯ ɪɚɡɥɢɱɟɧ ɞɥɹ ɫɬɟɧɨɤ ɫ ɪɚɡɧɨɣ ɤɪɢɜɢɡɧɨɣ. ɋ ɭɜɟɥɢɱɟɧɢɟɦ ɱɢɫɥɚ B i ɬɟɦɩɟɪɚɬɭɪɵ ɫɬɟɧɨɤ
ɩɪɢɛɥɢɠɚɸɬɫɹ ɤ ɬɟɦɩɟɪɚɬɭɪɚɦ ɨɦɵɜɚɸɳɢɯ ɫɪɟɞ, ɱɬɨ ɜɢɞɧɨ ɢɡ ɫɪɚɜɧɟɧɢɹ ɫɩɥɨɲɧɵɯ ɢ ɩɭɧɤɬɢɪɧɵɯ ɤɪɢɜɵɯ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɧɨɦɟɪɚɦɢ.
ɝɞɟ B i
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. Ʉɚɤɢɟ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ ȼɚɦ ɢɡɜɟɫɬɧɵ? ɂɡɨɛɪɚɡɢɬɟ ɢɯ ɝɪɚɮɢɱɟɫɤɢ.
2. Ʉɚɤɨɜ ɨɛɳɢɣ ɜɢɞ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ?
3. Ɂɚɩɢɲɢɬɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ
ɮɨɪɦɵ (ɩɥɚɫɬɢɧɵ, ɞɥɢɧɧɨɝɨ ɰɢɥɢɧɞɪɚ ɢ ɫɮɟɪɵ)
4. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɬɥɢɱɢɟ ɪɟɲɟɧɢɣ ɩɪɨɫɬɟɣɲɢɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɞɥɹ ɪɚɡɧɵɯ ɫɢɫɬɟɦ ɤɨɨɪɞɢɧɚɬ?
5. Ɂɚɩɢɲɢɬɟ ɩɨɥɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɞɥɢɧɧɨɣ
ɬɪɭɛɵ.
6. ɂɡ ɱɟɝɨ ɫɤɥɚɞɵɜɚɟɬɫɹ ɩɨɥɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɬɪɭɛɵ
ɜ ɭɫɥɨɜɢɹɯ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ?
7. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɚɧɚɥɨɝɢɹ ɞɥɹ ɦɧɨɝɨɫɥɨɣɧɨɣ
ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɬɟɧɤɢ?
8. ɑɬɨ ɬɚɤɨɟ «ɤɪɢɬɢɱɟɫɤɢɣ ɞɢɚɦɟɬɪ ɬɟɩɥɨɢɡɨɥɹɰɢɢ»?
9. Ɂɚɩɢɲɢɬɟ ɩɨɥɧɵɣ ɩɨɬɨɤ ɬɟɩɥɚ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɫɮɟɪɢɱɟɫɤɨɣ
ɫɬɟɧɤɢ.
10. Ʉɚɤɢɟ ɞɜɚ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɚɪɚɦɟɬɪɚ ɩɨɹɜɥɹɸɬɫɹ ɜ ɡɚɞɚɱɚɯ ɨ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ ɤɨɧɜɟɤɬɢɜɧɨɦ ɨɯɥɚɠɞɟɧɢɢ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɢ ɫɮɟɪɢɱɟɫɤɨɣ ɫɬɟɧɨɤ?
130
Ɂɚɞɚɧɢɹ
1. ɉɨɥɶɡɭɹɫɶ ɪɟɡɭɥɶɬɚɬɚɦɢ ɪɚɡɞɟɥɨɜ 3.4, 5.6 ɢ 5.7, ɩɨɤɚɡɚɬɶ, ɱɬɨ ɪɟɲɟɧɢɹ ɩɪɨɫɬɟɣɲɢɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɜ ɞɟɤɚɪɬɨɜɨɣ, ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɢ ɫɮɟɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɚɯ ɤɨɨɪɞɢɧɚɬ ɜ ɛɟɡɪɚɡɦɟɪɧɨɣ ɮɨɪɦɟ ɢɦɟɸɬ ɜɢɞ (5.43) –
(5.45).
2. ɋɮɨɪɦɭɥɢɪɨɜɚɬɶ ɡɚɞɚɱɭ ɨ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɞɜɭɯɫɥɨɣɧɨɣ ɲɚɪɨɜɨɣ ɨɛɨɥɨɱɤɟ ɩɪɢ ɟɟ ɤɨɧɜɟɤɬɢɜɧɨɦ ɨɯɥɚɠɞɟɧɢɢ, ɩɨɥɶɡɭɹɫɶ
ɦɚɬɟɪɢɚɥɨɦ ɪɚɡɞɟɥɚ 5.7. Ɍɟɩɥɨɜɨɣ ɤɨɧɬɚɤɬ ɦɟɠɞɭ ɫɥɨɹɦɢ ɫɱɢɬɚɬɶ ɢɞɟɚɥɶɧɵɦ. ɉɪɢɜɟɫɬɢ ɡɚɞɚɱɭ ɤ ɛɟɡɪɚɡɦɟɪɧɨɣ ɮɨɪɦɟ. ɉɨɫɬɪɨɢɬɶ ɬɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ.
3. Ɋɚɫɫɱɢɬɚɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɜɧɭɬɪɟɧɧɟɣ ɢ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɟɣ ɲɚɪɨɜɨɣ ɨɛɨɥɨɱɤɢ ɜ ɡɚɞɚɱɟ 2, ɚ ɬɚɤɠɟ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚ ɤɨɧɬɚɤɬɟ; ɨɩɪɟɞɟɥɢɬɶ ɩɨɥɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɭɯɨɞɹɳɢɣ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɲɚɪɚ, ɩɪɢɧɢɦɚɹ,
ɱɬɨ ɬɟɦɩɟɪɚɬɭɪɵ ɫɪɟɞɵ ɜɧɭɬɪɢ ɨɛɨɥɨɱɤɢ – 175 ɨɋ, ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ – 25 ɨɋ; ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɨɬɞɚɱɢ ɨɞɢɧɚɤɨɜɵ ɢ ɪɚɜɧɵ
– 28,8 ɤɤɚɥ/(ɦ2·ɱɚɫ·ɝɪɚɞ); ɜɧɭɬɪɟɧɧɢɣ, ɢ ɜɧɟɲɧɢɣ ɪɚɞɢɭɫɵ ɨɛɨɥɨɱɤɢ – 3
ɫɦ ɢ 5 ɫɦ, ɬɨɥɳɢɧɚ ɜɧɭɬɪɟɧɧɟɣ ɨɛɨɥɨɱɤɢ – 25 ɦɦ. ȼɧɭɬɪɟɧɧɹɹ ɨɛɨɥɨɱɤɚ
ɢɡɝɨɬɨɜɥɟɧɚ ɢɡ ɦɚɬɟɪɢɚɥɚ ɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ 1,45 ɤɤɚɥ/(ɦ ɱɚɫ ɝɪɚɞ);
ɜɧɟɲɧɹɹ ɢɡ ɦɚɬɟɪɢɚɥɚ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
0,137 ɤɤɚɥ/(ɦ·ɱɚɫ·ɝɪɚɞ). Ʉɚɤ ɛɭɞɟɬ ɢɡɦɟɧɹɬɶɫɹ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɬɨɥɳɢɧɵ ɜɧɟɲɧɟɣ ɨɛɨɥɨɱɤɢ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 25 ɦɦ ɞɨ 300 ɦɦ?
131
ɑȺɋɌɖ 6
Ɂ ɚ ɞ ɚ ɱ ɢ ɫ ɨ ɛ ɴ ɟ ɦɧ ɵ ɦ ɬ ɟ ɩ ɥ ɨ ɜɵ ɞ ɟ ɥ ɟ ɧ ɢ ɟ ɦ
ȼ ɜɟɳɟɫɬɜɟ ɧɚɪɹɞɭ ɫ ɩɪɨɰɟɫɫɚɦɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɨɡɦɨɠɧɨ ɨɛɴɟɦɧɨɟ ɜɵɞɟɥɟɧɢɟ ɬɟɩɥɚ, ɫɜɹɡɚɧɧɨɟ ɫ ɤɚɤɢɦɢ-ɥɢɛɨ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɦɢ
ɹɜɥɟɧɢɹɦɢ: ɤɨɧɞɟɧɫɚɰɢɟɣ, ɞɠɨɭɥɟɜɵɦ ɧɚɝɪɟɜɚɧɢɟɦ, ɹɞɟɪɧɵɦɢ ɢ ɯɢɦɢɱɟɫɤɢɦɢ ɪɟɚɤɰɢɹɦɢ ɢ ɬ.ɞ. ɋ ɩɨɡɢɰɢɣ ɬɟɩɥɨɨɛɦɟɧɚ ɬɚɤɢɟ ɩɪɨɰɟɫɫɵ ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɤɨɥɢɱɟɫɬɜɨɦ ɬɟɩɥɚ, ɜɵɞɟɥɹɸɳɢɦɫɹ ɢɥɢ ɩɨɝɥɨɳɚɸɳɢɦɫɹ ɜ ɟɞɢɧɢɰɟ ɨɛɴɟɦɚ ɡɚ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ qV , ȼɬ/ɦ2. ɗɬɚ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɨɛɴɟɦɧɨɝɨ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ. Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɜ ɷɬɨɣ ɱɚɫɬɢ ɡɚɞɚɱɚɯ ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɨɛɴɟɦɧɨɝɨ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ – ɜɟɥɢɱɢɧɚ ɩɨɫɬɨɹɧɧɚɹ – ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɜɪɟɦɟɧɢ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɵ. Ʉɨɧɟɱɧɨ, ɞɥɹ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ ɷɬɨ – ɜɟɫɶɦɚ ɭɩɪɨɳɟɧɧɵɣ
ɫɩɨɫɨɛ ɨɩɢɫɚɧɢɹ. ɋ ɜɨɡɦɨɠɧɨɫɬɹɦɢ ɛɨɥɟɟ ɫɬɪɨɝɨɝɨ ɫɩɨɫɨɛɚ
ɨɩɢɫɚɧɢɹ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ ɦɵ ɩɨɡɧɚɤɨɦɢɦɫɹ ɩɨɡɠɟ
(ɱɚɫɬɢ 10 ɢ 11).
6.1. Ɂɚ ɞɚɱɚ ɨ ɩɥɨɫ ɤɨɣ ɫ ɬɟɧ ɤɟ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ, ɨɦɵɜɚɟɦɨɣ ɫɪɟɞɚɦɢ ɫ ɡɚɞɚɧɧɵɦɢ ɬɟɦɩɟɪɚɬɭɪɚɦɢ (ɪɢɫ. 6.1). ɉɪɢ ɧɚɥɢɱɢɢ ɨɛɴɟɦɧɨɝɨ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ (ɭɪɚɜɧɟɧɢɟ (5.11) ɩɪɢ ɭɫɥɨɜɢɢ
wT wt 0 )
d 2T
qV
0.
(6.1)
O
dx 2
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɨɱɟɜɢɞɧɨ, ɛɭɞɭɬ
ɫɥɟɜɚ – T T1 ; ɫɩɪɚɜɚ – T T2 , (6.2)
ɥɢɛɨ
dT
D1 Te1 T ;
ɫɥɟɜɚ: O
dx
dT
ɫɩɪɚɜɚ: O
D 2 T Te 2 . (6.3)
dx
ȿɫɥɢ Te1 T ! 0 , ɬɨ ɨɦɵɜɚɸɳɚɹ ɫɥɟɜɚ
Te 2
D2
T1
Te1
D1
G2
G1
T2
0
Ɋɢɫ 6.1. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
ɡɚɞɚɱɟ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ
ɫɪɟɞɚ ɹɜɥɹɟɬɫɹ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ ɞɥɹ ɩɥɚɫɬɢɧɵ. ȿɫɥɢ Te1 T 0 , ɨɦɵɜɚɸɳɚɹ ɫɪɟɞɚ ɫɩɨɫɨɛɫɬɜɭɟɬ ɨɯɥɚɠɞɟɧɢɸ ɩɥɚɫɬɢɧɵ. ȼ ɥɸɛɨɦ ɫɥɭɱɚɟ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɩɪɨɫɬɵɦ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ:
132
q
dT
V x C1 ;
dx
O
2
q x
T V
C1x C 2 .
(6.4)
O 2
ɍɪɚɜɧɟɧɢɟ (6.4) ɟɫɬɶ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.1). ɑɬɨɛɵ ɧɚɣɬɢ
ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɧɭɠɧɨ ɩɨɦɟɫɬɢɬɶ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɧɚ ɨɞɧɨɣ ɢɡ ɩɨɜɟɪɯɧɨɫɬɟɣ ɩɥɚɫɬɢɧɵ ɢ ɩɨɞɫɬɚɜɢɬɶ (6.4) ɜ ɧɭɠɧɵɟ ɝɪɚɧɢɱɧɵɟ
ɭɫɥɨɜɢɹ.
ɉɨɫɬɭɩɢɦ ɢɧɚɱɟ.
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ ɢɧɬɟɧɫɢɜɧɨɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɢ ɜɧɭɬɪɢ ɩɥɚɫɬɢɧɵ
ɬɟɩɥɨ ɦɨɠɟɬ ɨɬɞɚɜɚɬɶɫɹ ɜ ɨɦɵɜɚɸɳɢɟ ɟɟ ɫɪɟɞɵ ɫ ɞɜɭɯ ɩɨɜɟɪɯɧɨɫɬɟɣ.
Ɍ.ɟ., ɤɪɢɜɚɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɛɭɞɟɬ ɢɦɟɬɶ ɦɚɤɫɢɦɭɦ. ɉɭɫɬɶ
ɨɫɶ ɤɨɨɪɞɢɧɚɬ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɷɬɨɬ ɦɚɤɫɢɦɭɦ, ɤɨɬɨɪɵɣ ɧɚɯɨɞɢɬɫɹ ɧɚ
ɪɚɫɫɬɨɹɧɢɢ G2 ɨɬ ɩɪɚɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ. ȼɟɥɢɱɢɧɚ G2 ɧɟɢɡɜɟɫɬɧɚ ɢ ɛɭɞɟɬ ɨɩɪɟɞɟɥɟɧɚ ɩɨɡɠɟ.
ɂɡ ɭɫɥɨɜɢɹ ɷɤɫɬɪɟɦɭɦɚ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɤɪɢɜɨɣ d T d x 0 ɩɪɢ x 0
ɫɥɟɞɭɟɬ, ɱɬɨ ɜ (6.4) C1 0 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
qV x 2
C2x .
O 2
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɜ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɧɚ ɩɪɚɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ, ɬ.ɟ. ɩɪɢ x G 2 , ɧɚɣɞɟɦ
T
§
·
qV G 22
(6.5)
D 2 ¨ C2 Te 2 ¸ qV G 2 .
¨
¸
2
O
©
¹
ɉɨɫɥɟɞɧɟɟ ɫɨɨɬɧɨɲɟɧɢɟ ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɢɡ ɮɢɡɢɱɟɫɤɢɯ
ɫɨɨɛɪɚɠɟɧɢɣ. Ɍɚɤ ɤɚɤ ɩɥɨɫɤɨɫɬɶ x 0 ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ, ɬɨ
dT
q x 0 O
0
dx x 0
ɢ ɜɫɟ ɬɟɩɥɨ, ɜɵɞɟɥɢɜɲɟɟɫɹ ɜ ɩɥɚɫɬɢɧɟ ɫɩɪɚɜɚ, ɞɨɥɠɧɨ ɛɵɬɶ ɨɬɜɟɞɟɧɨ ɜ
ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ ɩɨɫɪɟɞɫɬɜɨɦ ɬɟɩɥɨɨɬɞɚɱɢ ɫ ɩɪɚɜɨɣ ɫɬɟɧɤɢ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɛɭɞɟɬ ɧɚɪɭɲɟɧɨ ɭɫɥɨɜɢɟ ɫɬɚɰɢɨɧɚɪɧɨɫɬɢ ɩɪɨɰɟɫɫɚ. ȼɟɥɢɱɢɧɚ qV G 2 ɫɩɪɚɜɚ ɜ (6.5) ɟɫɬɶ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɜɵɞɟɥɹɸɳɟɟɫɹ ɜ
ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɜ ɨɛɴɟɦɟ ɩɥɚɫɬɢɧɵ ɫ ɬɨɥɳɢɧɨɣ, ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ.
ɋɥɟɜɚ ɜ (6.5) ɫɬɨɢɬ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɨɬɨɤɚ ɬɟɩɥɨɨɬɞɚɱɢ ɫ ɟɞɢɧɢɰɵ
ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ.
Ⱥɧɚɥɨɝɢɱɧɵɟ ɪɚɫɫɭɠɞɟɧɢɹ ɞɥɹ ɥɟɜɨɝɨ ɫɥɨɹ ɩɥɚɫɬɢɧɵ ɫ ɬɨɥɳɢɧɨɣ
G1 G G 2 ɩɪɢɜɨɞɹɬ ɤ ɭɪɚɜɧɟɧɢɸ
133
2
§
·
qV G G 2 ¨
D1 C 2 Te1 ¸ qV G G 2 .
(6.6)
2O
¨
¸
©
¹
ȼɵɪɚɠɚɹ C2 ɢɡ (6.5) ɢ (6.6), ɩɪɢɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɞɪɭɝ
ɞɪɭɝɭ, ɧɚɣɞɟɦ ɩɨɥɨɠɟɧɢɟ ɦɚɤɫɢɦɭɦɚ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɤɪɢɜɨɣ
G2
2O D1D 2 Te1 Te 2 qV GD 2 GD 1 O 2qV ª¬GD1D 2 O D 1 D 2 º¼
.
ɉɨɫɬɨɹɧɧɭɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɬɟɩɟɪɶ ɥɟɝɤɨ ɧɚɣɬɢ, ɢɫɩɨɥɶɡɭɹ
ɥɢɛɨ (6.6). Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɩɪɢɧɢɦɚɟɬ ɨɫɨɛɟɧɧɨ ɩɪɨɫɬɨɣ ɜɢɞ,
D1 D 2 D ɢ Te1 Te 2 Te . ȼ ɷɬɨɦ ɫɥɭɱɚɟ, ɨɱɟɜɢɞɧɨ, G1 G 2 G 2
ɦɚɤɫɢɦɭɦ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚɯɨɞɢɬɫɹ ɜ ɰɟɧɬɪɟ ɩɥɚɫɬɢɧɵ) ɢ
qV G qV G 2
C2
Te ;
2D
8O
2
º q G
qV ª§ G ·
2
«¨ ¸ x » V Te .
T
2O «© 2 ¹
»¼ 2D
¬
Ɇɚɤɫɢɦɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɨɱɤɟ x 0
Tmax
qV G qV G 2
Te
2D
8O
(6.7)
(6.5)
ɟɫɥɢ
(ɬ.ɟ.,
(6.8)
(6.9)
ɛɭɞɟɬ ɬɟɦ ɛɨɥɶɲɟ, ɱɟɦ ɦɟɧɶɲɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɩɥɚɫɬɢɧɵ.
Ɍɟɦɩɟɪɚɬɭɪɵ ɫɬɟɧɨɤ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɞɢɧɚɤɨɜɵ
T
1
qV G
Te
2D
T2
ɢ ɪɚɫɬɭɬ ɫ ɭɯɭɞɲɟɧɢɟɦ ɬɟɩɥɨɨɬɞɚɱɢ.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ ɥɟɝɤɨ ɩɨɥɭɱɢɬɶ ɫ ɩɨɦɨɳɶɸ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ, ɫɥɟɞɭɸɳɟɣ ɢɡ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ
(6.2),
qV G 22
C 2 T2 ;
2O
2
q G G 2 V
C 2 T1 .
2O
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɦɚɤɫɢɦɭɦ ɬɟɦɩɟɪɚɬɭɪɵ ɛɭɞɟɬ ɪɚɫɩɨɥɚɝɚɬɶɫɹ ɧɚ ɪɚɫɫɬɨɹɧɢɢ G2 ɨɬ ɩɪɚɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ
134
G2
G § 2O T1 T2 ·
¨1 ¸.
2
¸
2 ¨©
qV G
¹
Ɋɟɲɟɧɢɟ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢɦɟɬ ɜɢɞ
T
2
­
½
·º
qV ° ª G §
2O
2°
T2 T T ¸ » x ¾ .
® « ¨1 2O ° « 2 ¨© qV G 2 1 2 ¸¹ »
°
¼
¯¬
¿
(6.10)
ɉɪɢ ɨɱɟɧɶ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ D ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɬɶɟɝɨ ɪɨɞɚ
ɩɟɪɟɯɨɞɹɬ ɜ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɩɟɪɜɨɝɨ ɪɨɞɚ. Ɍɨɝɞɚ, ɩɨɥɚɝɚɹ D o f , ɢɡ
ɪɟɲɟɧɢɹ ɫɢɦɦɟɬɪɢɱɧɨɣ ɡɚɞɚɱɢ (6.8) ɧɚɣɞɟɦ
2
º
qV ª§ G ·
(6.11)
T
«¨ ¸ x 2 » T s ,
2O «© 2 ¹
»¼
¬
ɝɞɟ Ts T1 T2 Te (ɬɟɦɩɟɪɚɬɭɪɚ ɫɬɟɧɨɤ ɪɚɜɧɚ ɬɟɦɩɟɪɚɬɭɪɟ ɨɦɵɜɚɸɳɢɯ ɫɪɟɞ), ɢ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɟɫɬɶ
qV G 2
Tmax
Ts .
(6.12)
8O
ɉɭɫɬɶ ɬɟɩɟɪɶ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O ɡɚɜɢɫɢɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ
(6.13)
O O 0 1 E T .
Ɍɨɝɞɚ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɨɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɚɫɬɢɧɟ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
d § dT ·
¨O
¸ qV 0 .
dx © dx ¹
ɉɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɟɫɬɶ
O 0 1 E T dT
qV x C1 0 .
dx
ɉɨɫɥɟɞɭɸɳɟɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɚɟɬ
T2
x2
(6.14)
O 0T O 0E
qV
C1x C 2 0 .
2
2
ɋ ɩɨɦɨɳɶɸ (6.14) ɢ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ (6.2) ɧɚɣɞɟɦ
ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
T2
O 0T1 O 0E 1 C 2 0 , x 0 ;
2
135
T2 2
G2
O 0T2 O 0E
qV
C1G C 2
2
2
0, x G.
ɉɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɫɬɪɚɡɭ ɞɚɟɬ C2 , ɚ ɢɡ ɜɬɨɪɨɝɨ ɧɚɯɨɞɢɦ
C1 O 0
T1 T2 O 0E T12 T2 2
G
qV .
G
G
2
2
ȼ ɪɟɡɭɥɶɬɚɬɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ (6.14) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ
ɜɢɞɟ
T2
x2
O 0T O 0E
qV
2
2
ª
§
O 0E 2
G2 º x
T12 ·
2
T1 T2 qV
«O 0 T1 T2 ¸¸ 0
» O 0 ¨¨ T1 E
2
2 »¼ G
2
«¬
©
¹
ɢɥɢ
J 2
T 1 VJ T f [ 0 ,
(6.15)
2
ɝɞɟ
M 2 M
§ J
·
f [
[ [ 1 [ ¨1 J V ¸ ,
2
2
© 2
¹
T2
x
T T2
qV G 2
J E T1 T2 , V
,M
;[
;T
,
O 0 T1 T2 G
T1 T2
T1 T2
ɬɚɤ ɱɬɨ
§ J
·
f 0 ¨1 J V ¸ ,
© 2
¹
f 1 0 .
ɂɡ (6.15) ɧɚɯɨɞɢɦ
T
1 VJ r
1 VJ 2 2J f [ .
J
ɉɨɞɤɨɪɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜɫɟɝɞɚ ɩɨɥɨɠɢɬɟɥɶɧɨɟ ɢ ɦɟɧɹɟɬɫɹ ɨɬ
1 VJ 2 2J ¨§1 ©
J
·
JV ¸ ! 0
2
¹
ɩɪɢ [ 0 , ɝɞɟ T 1 , ɞɨ
1 VJ 2 ! 0
136
ɩɪɢ [ 1 , ɝɞɟ T 0 .
ɉɨɷɬɨɦɭ ɩɟɪɟɞ ɤɨɪɧɟɦ ɜɵɛɢɪɚɟɦ ɡɧɚɤ «+».
ɉɨɥɨɠɟɧɢɟ ɷɤɫɬɪɟɦɭɦɚ ɧɚ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɤɪɢɜɨɣ ɧɚɣɞɟɦ ɢɡ ɭɫɥɨɜɢɹ
dT
0.
d[
ɂɦɟɟɦ
1 1§ J
·
¨1 JV ¸ .
2 M© 2
¹
[
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɬɨɛɵ ɷɤɫɬɪɟɦɭɦ ɧɚɛɥɸɞɚɥɫɹ ɩɪɢ [ ! 0 , ɧɟɨɛɯɨɞɢɦɨ
ɜɵɩɨɥɧɟɧɢɟ ɭɫɥɨɜɢɹ
§ J
·
M ! 2 ¨1 JV ¸ .
© 2
¹
J 4
1,00
0,75
4
M J 1,5
J J 1,2
J 0,9
0,50
J J J J 0,6
0,25
0,00
0,0
M J 0,3
0,2
0,4
0,6
0,8
0,0
0,0
[
0,2
ɚ
0,4
0,6
0,8
[
ɛ
Ɋɢɫ. 6.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɪɚɞɢɭɫɭ ɰɢɥɢɧɞɪɚ ɜ ɧɟɥɢɧɟɣɧɨɣ
ɡɚɞɚɱɟ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ. ɉɭɧɤɬɢɪ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɥɭɱɚɸ
ɩɨɫɬɨɹɧɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ: ɚ) ij=2,5; ɛ) ij=5,0
ȿɫɥɢ J 0 (ɬ.ɟ., ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɧɟ
ɡɚɜɢɫɢɬ), ɢɡ (6.15) ɢɦɟɟɦ
M
M
T [ 2 [ 1 [ .
2
2
(6.16)
ɇɚ ɪɢɫ. 6.2 ɢɡɨɛɪɚɠɟɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ T ɜ ɩɥɨɫɤɨɦ
ɫɥɨɟ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ J ɢ ɢɫɬɨɱɧɢɤɚ M . ɉɪɢɧɹɬɨ, ɱɬɨ V 0,5
T1 3T2 . ȿɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɪɚɫɬɟɬ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ,
J ! 0 , ɦɚɤɫɢɦɭɦ ɫɦɟɳɚɟɬɫɹ ɤ ɥɟɜɨɣ ɝɪɚɧɢɰɟ.
ɉɪɢɜɟɞɟɦ ɪɟɲɟɧɢɹ ɟɳɟ ɧɟɤɨɬɨɪɵɯ ɡɚɞɚɱ ɞɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ.
137
6.2. ɐɢɥɢɧɞɪ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥ ɨɜɵɞɟɥ ɟɧɢɟɦ
Ɋɚɫɫɦɨɬɪɢɦ ɛɟɫɤɨɧɟɱɧɵɣ ɫɩɥɨɲɧɨɣ ɰɢɥɢɧɞɪ, ɪɚɜɧɨɦɟɪɧɨ ɧɚɝɪɟɜɚɟɦɵɣ (ɢɥɢ ɨɯɥɚɠɞɚɟɦɵɣ) ɫ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ. ȼ ɨɛɴɟɦɟ ɰɢɥɢɧɞɪɚ ɧɚɯɨɞɢɬɫɹ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ qV . Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɞɥɹ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɪɟɠɢɦɚ.
Ɉɞɧɨɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɰɢɥɢɧɞɪɚ ɫ ɨɛɴɟɦɧɵɦ
ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ ɟɫɬɶ (ɭɪɚɜɧɟɧɢɟ (5.19) ɩɪɢ ɭɫɥɨɜɢɢ w T w t 0 )
d 2T
1 dT qV
0.
O
dr 2 r dr
ɋ ɩɨɦɨɳɶɸ ɧɨɜɨɣ ɮɭɧɤɰɢɢ u d T d r ɭɪɚɜɧɟɧɢɟ ɩɟɪɟɩɢɲɟɬɫɹ ɜ ɜɢ
ɞɟ
r
q r
du
u V
dr
O
ɢɥɢ
0.
d ru qV r
0.
dr
O
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɭɪɚɜɧɟɧɢɹ ɟɫɬɶ
qV r 2
ru C1 .
2O
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɬɟɦɩɟɪɚɬɭɪɟ, ɧɚɣɞɟɦ
q r C
dT
V 1
2O
dr
r
ɢɥɢ
qV r 2
T C1ln r C 2 .
(6.17)
4O
ɗɬɨ ɢ ɟɫɬɶ ɨɛɳɟɟ ɪɟɲɟɧɢɟ.
Ⱦɥɹ ɫɩɥɨɲɧɨɝɨ ɰɢɥɢɧɞɪɚ ɢɡ ɭɫɥɨɜɢɹ ɫɢɦɦɟɬɪɢɢ d T dr 0 ɩɪɢ r 0
ɢɦɟɟɦ C1 0 .
ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɧɚ ɜɧɟɲɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ r R :
dT
O
D T Te .
dr
ɉɨɞɫɬɚɜɥɹɹ ɫɸɞɚ ɪɟɲɟɧɢɟ (6.17), ɧɚɯɨɞɢɦ
qV R qV R 2
C2
Te .
2D
4O
138
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɪɚɞɢɭɫɭ ɰɢɥɢɧɞɪɚ
ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
qV
q R
T
R 2 r 2 V Te .
(6.18)
4O
2D
ɂɡ ɪɚɜɟɧɫɬɜɚ (6.18) ɦɨɠɟɦ ɧɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ (ɤɨɬɨɪɚɹ ɪɚɫɩɨɥɚɝɚɟɬɫɹ, ɨɱɟɜɢɞɧɨ, ɧɚ ɨɫɢ ɰɢɥɢɧɞɪɚ)
qV 2 qV R
Tmax
R Te
(6.19)
4O
2D
ɢ ɬɟɦɩɟɪɚɬɭɪɭ ɫɬɟɧɨɤ
qV R
Ts
Te .
2D
ɍɫɬɪɟɦɥɹɹ D ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɩɪɢɞɟɦ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɩɟɪɜɨɝɨ ɪɨɞɚ, ɤɨɝɞɚ Ts Te .
ɇɚɣɞɟɦ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ
qV R
q D Ts Te 2
ɢ ɩɨɥɧɵɣ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ
qV R
Q qF
2S Rl qV S R 2l .
2
ȼɵɪɚɠɟɧɢɹ (6.16) ɢ (6.19) ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɦɚɤɫɢɦɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɛɟɡɪɚɡɦɟɪɧɨɣ ɮɨɪɦɟ
T
r
, Bi
R
DR
,M
O
T Te
T Te
T max
ɝɞɟ [
M
M
1 [2 ,
4
2 Bi
Tmax Te
T Te
M
M
,
4 2B i
qV R 2
.
O T Te ɉɨɥɚɝɚɹ M 1 , ɨɩɪɟɞɟɥɢɦ ɦɚɫɲɬɚɛɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ
q
(6.20)
T Te V R 2 .
O
ɗɬɨ ɟɫɬɶ ɦɚɤɫɢɦɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ, ɞɨ ɤɨɬɨɪɨɣ ɦɨɠɟɬ «ɞɨɝɪɟɬɶɫɹ»
ɰɢɥɢɧɞɪ ɪɚɞɢɭɫɚ R ɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ O ɩɪɢ «ɦɝɧɨɜɟɧɧɨɦ» ɜɵɞɟɥɟɧɢɢ ɜ ɧɟɦ ɬɟɩɥɚ ɦɨɳɧɨɫɬɶɸ qV .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
1
1
T
1 [2 ,
4
2Bi
139
1
1
.
4 2Bi
Ɂɚɞɚɱɚ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɫ ɜɧɭɬɪɟɧɧɢɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ – ɬɢɩɢɱɧɚɹ ɡɚɞɚɱɚ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɨ ɩɪɨɜɨɞɟ, ɩɨ ɤɨɬɨɪɨɦɭ ɬɟɱɟɬ ɷɥɟɤɬɪɢɱɟɫɤɢɣ ɬɨɤ. Ɇɨɳɧɨɫɬɶ ɜɧɭɬɪɟɧɧɟɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ qV ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɮɨɪɦɭɥɨɣ
T max
I 2 Re
,
qV
V
ɝɞɟ I – ɫɢɥɚ ɬɨɤɚ, Ⱥ; Re – ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, V –ɨɛɴɟɦ ɩɪɨɜɨɥɨɤɢ (ɬ.ɟ., ɷɬɨ – ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɧɚ ɟɞɢɧɢɰɭ ɨɛɴɟɦɚ
ɩɪɨɜɨɞɚ).
ɉɪɢɦɟɪ. ɇɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɭɸ ɫɢɥɭ ɬɨɤɚ, ɤɨɬɨɪɵɣ ɦɨɠɧɨ ɩɪɨɩɭɫɤɚɬɶ ɩɨ ɚɥɸɦɢɧɢɟɜɨɣ ɩɪɨɜɨɥɨɤɟ ( O 204 ȼɬ/(ɦ·Ʉ)) ɞɢɚɦɟɬɪɨɦ 1 ɦɦ,
ɱɬɨɛɵ ɟɟ ɬɟɦɩɟɪɚɬɭɪɚ ɧɟ ɩɪɟɜɵɲɚɥɚ 200 ɨɋ. ɉɪɨɜɨɥɨɤɚ ɩɨɞɜɟɲɟɧɚ ɜ
ɜɨɡɞɭɯɟ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ 25 ɋ. Ʉɨɷɮɮɢɰɢɟɧɬ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ
ɨɬ ɩɪɨɜɨɥɨɤɢ ɤ ɜɨɡɞɭɯɭ ɪɚɜɟɧ 10 ȼɬ/(ɦ2·Ʉ). ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ ɩɪɨɜɨɥɨɤɢ ɟɫɬɶ 0,037 Ɉɦ/ɦ.
Ɋɟɲɟɧɢɟ. ȼɨɫɩɨɥɶɡɭɟɦɫɹ ɮɨɪɦɭɥɨɣ (6.19), ɢɡ ɤɨɬɨɪɨɣ ɫɥɟɞɭɟɬ
qV R ª RD º
I 2 Re ª RD º
1
1
.
Tmax Te Te 2 D «¬ 2O »¼
2 S RD l «¬ 2O »¼
ȼɟɥɢɱɢɧɚ Re l ɢ ɟɫɬɶ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɪɨɜɨɥɨɤɢ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ.
ɉɨɞɫɬɚɜɥɹɟɦ ɡɚɞɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ:
ª
10 3 2 ˜ 10 º
I2
».
200 25 ˜ 0 . 0 3 7 «1 3
«
2 ˜ 204 »
2 ˜ S ˜ 10 2 ˜ 10
«¬
»¼
ɨ
Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ ɫɢɥɭ ɬɨɤɚ I 12,2 Ⱥ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɧɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɫɨɛɨɝɨ ɬɪɭɞɚ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨ
ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɪɨɜɨɥɨɤɟ ɫ ɬɨɤɨɦ ɞɥɹ ɩɟɪɟɦɟɧɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɧɚɩɪɢɦɟɪ, ɩɨɥɚɝɚɹ O O 0 1 E T .
ɉɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɛɭɞɟɬ ɫɥɟɞɨɜɚɬɶ ɢɡ ɭɪɚɜɧɟɧɢɹ
d r u
qV r
0.
dr
O 0 1 E T ɏɨɞ ɪɟɲɟɧɢɹ ɚɧɚɥɨɝɢɱɟɧ ɡɚɞɚɱɟ ɨ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ ɫ ɩɟɪɟɦɟɧɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O O T .
140
Ɂɚɞɚɱɚ ɨɛ ɨɯɥɚɠɞɟɧɢɢ ɰɢɥɢɧɞɪɚ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ
ɩɪɟɞɫɬɚɜɥɹɟɬ, ɜ ɱɚɫɬɧɨɫɬɢ, ɢɧɬɟɪɟɫ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɤɚɬɨɞɚɯ, ɢɫɩɨɥɶɡɭɟɦɵɯ ɜ ɩɥɚɡɦɨɬɪɨɧɚɯ ɞɥɹ ɝɟɧɟɪɚɰɢɢ ɩɨɬɨɤɨɜ ɢɨɧɨɜ. ȼ ɩɪɚɤɬɢɱɟɫɤɨɦ ɩɪɢɥɨɠɟɧɢɢ ɷɬɚ ɡɚɞɚɱɚ ɦɨɠɟɬ ɛɵɬɶ ɩɟɪɟɮɨɪɦɭɥɢɪɨɜɚɧɚ ɬɚɤ: ɧɚɣɬɢ ɦɨɳɧɨɫɬɶ ɢɫɬɨɱɧɢɤɚ, ɞɨɫɬɚɬɨɱɧɭɸ ɞɥɹ ɪɚɫɩɵɥɟɧɢɹ ɤɚɬɨɞɚ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɥɹ ɷɬɨɝɨ ɧɭɠɧɨ ɞɨɫɬɢɱɶ ɬɟɦɩɟɪɚɬɭɪɵ
ɩɥɚɜɥɟɧɢɹ Tm ɦɚɬɟɪɢɚɥɚ ɤɚɬɨɞɚ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɮɨɪɦɭɥɭ (6.19), ɢɡ ɤɨɬɨɪɨɣ ɫɥɟɞɭɟɬ
Tm Te
.
qV
R2 R
4O 2D
ȼ ɭɫɥɨɜɢɹɯ ɬɟɩɥɨɨɛɦɟɧɚ ɢɡɥɭɱɟɧɢɟɦ (ɝɥɚɜɚ 9) ɡɚɞɚɱɚ ɭɫɥɨɠɧɹɟɬɫɹ ɢ
ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ ɬɨɥɶɤɨ ɱɢɫɥɟɧɧɨ.
6.3. ɉɪɨɜɨɞ ɫ ɢɡ ɨɥ ɹɰɢɟɣ
ɂɫɩɨɥɶɡɭɹ ɨɛɳɟɟ ɪɟɲɟɧɢɟ (6.17), ɦɨɠɧɨ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɬɨɥɳɢɧɟ ɫɬɟɧɤɢ ɩɨɥɨɝɨ ɰɢɥɢɧɞɪɚ ɢɥɢ ɩɨ ɬɨɥɳɢɧɟ ɰɢɥɢɧɞɪɚ, ɩɨɤɪɵɬɨɝɨ ɡɚɳɢɬɧɵɦ ɫɥɨɟɦ. ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɧɭɠɧɨ ɡɚɞɚɬɶ ɭɫɥɨɜɢɹ
ɧɚ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ. ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ (ɪɢɫ. 6.3) ɩɨɬɪɟɛɭɟɬɫɹ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɞɜɭɯ ɦɚɬɟɪɢɚɥɨɜ
ɫ ɪɚɡɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ, ɬ.ɟ. ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɱɟɬɜɟɪɬɨɝɨ ɪɨɞɚ.
Ɋɢɫ. 6.3. Ʉ ɡɚɞɚɱɟ ɨɛ
ɢɡɨɥɢɪɨɜɚɧɧɨɦ ɩɪɨɜɨɞɟ
ɋɬɪɨɝɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɷɬɨɣ
ɡɚɞɚɱɢ ɜɤɥɸɱɚɟɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɞɥɹ ɩɪɨɜɨɞɚ (ɢɧɞɟɤɫ «1») ɢ ɢɡɨɥɹɰɢɢ (ɢɧɞɟɤɫ
«2»)
d 2T1 1 d T1 qV
0,
(6.21)
2
O
r
d
r
dr
1
d 2T2
dr 2
1 dT2
r dr
0
(6.22)
ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
r
0 : d T dr 0 ;
r R : d T dr 0 , T1 T2 ;
(6.23)
dT
r R G : O 2 2 D T2 Te .
dr
ɉɟɪɜɨɟ ɭɫɥɨɜɢɟ ɟɫɬɶ ɭɫɥɨɜɢɟ ɫɢɦɦɟɬɪɢɢ; ɜɬɨɪɨɟ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ
ɬɟɩɥɨɜɨɣ ɤɨɧɬɚɤɬ ɦɟɠɞɭ ɩɪɨɜɨɞɨɦ ɢ ɢɡɨɥɹɰɢɟɣ – ɢɞɟɚɥɶɧɵɣ, ɚ ɬɪɟɬɶɟ
141
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɤɨɧɜɟɤɬɢɜɧɨɦɭ ɬɟɩɥɨɨɛɦɟɧɭ ɩɪɨɜɨɞɚ ɫ ɢɡɨɥɹɰɢɟɣ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ (6.17) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ
ɬɚɤ
qV r 2
T1 C1ln r C 2 ,
4O 1
T2 C 3ln r C 4 .
(6.24)
ɉɨɞɫɬɚɜɥɹɹ (6.24) ɜ ɩɟɪɜɨɟ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ, ɧɚɣɞɟɦ
C1 0 .
ȼɬɨɪɨɟ ɭɫɥɨɜɢɟ ɞɚɟɬ
§ q R·
C
O1 ¨ V ¸ O 2 3 ,
R
© 2O 1 ¹
ɨɬɤɭɞɚ ɢɦɟɟɦ
qV R 2
.
C3 2O 2
ɂɡ ɪɚɜɟɧɫɬɜɚ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɫɥɟɞɭɟɬ
qV R 2
qV R 2
ln R C 4 .
C2 4O 1
2O 2
ɉɨɫɥɟɞɧɟɟ ɢɡ ɭɫɥɨɜɢɣ (6.23) ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ C4 . ɂɦɟɟɦ
C 3 O 2 qV R 2
O 2
{
˜
R
R 2O 2
ª qV R 2
º
ln R G C 4 Te »
D «
«¬ 2O 2
»¼
ɢɥɢ
qV R 2
q R
ln R G V .
C 4 Te 2O 2
2D
Ɍɟɩɟɪɶ ɦɨɠɧɨ ɧɚɣɬɢ ɩɨɫɥɟɞɧɸɸ ɩɨɫɬɨɹɧɧɭɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
C2
qV R 2 § 2O 1 · qV R 2 § R G ·
1
ln ¨
Te ¸.
4O 1 ¨©
D R ¸¹ 2O 2
© R ¹
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɪɨɜɨɞɟ ɫ ɢɡɨɥɹɰɢɟɣ
ɨɩɢɫɵɜɚɟɬɫɹ ɮɨɪɦɭɥɚɦɢ
T1
qV R 2 § 2O 1 · qV R 2 § R G · qV r 2
Te ,
1
ln ¨
¸
4O 1 ©¨
D R ¹¸ 2O 2
© R ¹ 4O 1
142
(6.25)
T2
qV R 2 O 2 qV R 2 § R G ·
ln ¨
Te ¸.
2O 2 D R 2O 2
© r ¹
Ɉɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
[2
1§
Bi · K O
, 0 d [ d1
T1
l n 1 H ¨1 2
¸
4©
KO ¹ 2
4
T2
(6.26)
KO KO §1 H ·
ln ¨
¸ , 1 d [ d1 H ,
2 Bi
2
© [ ¹
ɝɞɟ
Ti Te
r
qV R 2
DR
O1
,[
, T Te , Bi
, KO
,H
Ti
R
O2
O2
T Te
O1
Ɍɚɤɚɹ ɮɨɪɦɚ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɟɲɟɧɢɹ ɭɞɨɛɧɚ ɞɥɹ ɚɧɚɥɢɡɚ.
G
.
R
Ɉɩɪɟɞɟɥɢɦ ɩɨɬɨɤ ɬɟɩɥɚ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɨɜɨɞɧɢɤɚ
q D T2 R G Te ɢ
Q S R 2 R lD T2 R G Te .
ɉɨɞɫɬɚɜɢɦ ɫɸɞɚ T2 ɢɡ (6.25) ɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɛɟɡɪɚɡɦɟɪɧɵɦɢ ɩɚɪɚɦɟɬɪɚɦɢ
KO
Q
.
(6.27)
S R 2 R lD T Te Bi
ȿɫɥɢ K O Bi 1 , ɬɨ ɢɡɨɥɹɰɢɹ ɧɟ ɨɬɜɨɞɢɬ ɬɟɩɥɨ ɨɬ ɩɪɨɜɨɞɧɢɤɚ ɫ ɬɨɤɨɦ. ȿɫɥɢ K O Bi ! 1 , ɬɨ ɜɨɡɦɨɠɧɨ ɟɝɨ ɨɫɬɵɜɚɧɢɟ ɡɚ ɫɱɟɬ ɩɨɬɟɪɶ ɬɟɩɥɚ ɜ
ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ.
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɩɨ ɞɥɢɧɧɨɣ ɚɥɸɦɢɧɢɟɜɨɣ ɩɪɨɜɨɥɨɤɟ ɞɢɚɦɟɬɪɨɦ 1 ɫɦ
ɬɟɱɟɬ ɷɥɟɤɬɪɢɱɟɫɤɢɣ ɬɨɤ ɫɢɥɨɣ ɬɨɤɚ 1000 ɨȺ. ɉɪɨɜɨɥɨɤɚ ɩɨɤɪɵɬɚ ɫɥɨɟɦ
ɪɟɡɢɧɨɜɨɣ ɢɡɨɥɹɰɢɢ ɬɨɥɳɢɧɨɣ 3 ɦɦ ( O 2 0 ,15 ȼɬ/(ɦ·Ʉ)). Ɍɟɦɩɟɪɚɬɭɪɚ
ɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɨɥɹɰɢɢ 30 ɋ. ɇɚɣɬɢ ɬɟɦɩɟɪɚɬɭɪɭ ɜɧɭɬɪɟɧɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɨɥɹɰɢɢ. Ɉɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɪɨɜɨɥɨɤɢ ɧɚ ɟɞɢɧɢɰɭ ɞɥɢɧɵ 3,7 ˜ 104 Ɉɦ/ɦ.
Ɋɟɲɟɧɢɟ. Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɜɬɨɪɨɣ ɮɨɪɦɭɥɨɣ
(6.25). ɋ ɭɱɟɬɨɦ ɬɨɝɨ, ɱɬɨ ɡɚɞɚɧɚ ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ
Re I 2
ɢɡɨɥɹɰɢɢ, ɬ.ɟ. D 2 o f , ɢ qV
, ɢɦɟɟɦ
S R 2l
143
T2 r
Re I 2
§ RG·
ln ¨
Te ˜
¸
l 2SO 2 © R ¹
R
273 30 3,7 ˜10
4
˜
10002
ª 0 ,005 0 ,003 º
ln «
»¼ | 477 ,6 Ʉ.
2 ˜ 3,14 ˜ 0 ,15 ¬
0 ,005
ɂɫɩɨɥɶɡɭɹ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɚɥɸɦɢɧɢɟɜɨɣ
ɩɪɨɜɨɥɨɤɢ O1 | 2 3 2 ȼɬ/(ɦ·Ʉ) ɢ ɩɟɪɜɭɸ ɢɡ ɮɨɪɦɭɥ (6.25), ɦɨɠɟɦ ɪɚɫɫɱɢɬɚɬɶ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɰɟɧɬɪɟ ɩɪɨɜɨɞɚ. ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɭɫɥɨɜɢɹɯ ɢɦɟɟɦ
Re I 2
Re I 2
§ R G · Re I 2
ln ¨
T1 r R Te T2 r R ¸
l 2SO 2 © R ¹ l 4SO 1
l 4 SO 1
3,7 ˜ 10 4 1000 2
477 ,6 | 477 ,7 ,
4 ˜ 3,14 ˜ 232
ɬ.ɟ. ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɰɟɧɬɪɟ ɩɪɨɜɨɞɚ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɟɝɨ
ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɝɪɚɧɢɰɟ ɫ ɢɡɨɥɹɰɢɟɣ.
6.4. ɒɚɪ ɫ ɨɛ ɴɟɦ ɧɵɦ ɬɟ ɩɥ ɨɜ ɵɞɟɥɟ ɧɢɟɦ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɯɨɠɞɟɧɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɲɚɪɟ ɫ ɨɛɴɟɦɧɵɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ. Ɉɞɧɨɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ (ɮɨɪɦɭɥɚ (5.10) ɩɪɢ ɭɫɥɨɜɢɢ w T w t 0 )
1 d 2 dT qV d 2T 2 dT qV
{
r
2 dr
2
dr
O
r
dr
O
r
dr
0.
(6.28)
Ɉɛɳɢɣ ɢɧɬɟɝɪɚɥ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɡɚɞɚɱɟ ɨ
ɰɢɥɢɧɞɪɟ
q r2 C
T V 1 C2 .
(6.29)
r1
6O
ɂɡ ɭɫɥɨɜɢɹ ɫɢɦɦɟɬɪɢɢ d T dr 0 ɩɪɢ r 0 ɞɥɹ ɫɩɥɨɲɧɨɝɨ ɲɚɪɚ
ɢɦɟɟɦ C1 0 . ɍɫɥɨɜɢɟ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ
dT
O
D T Te ɩɪɢ r R ɞɚɟɬ
dr
q
q
C 2 Te V R V R 2 .
3D
6O
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
144
2
qV
qV 2 ª § r · º
T Te R
R «1 ¨ ¸ » .
3D
6O
«¬ © R ¹ »¼
ɇɚɯɨɞɢɦ ɞɚɥɟɟ ɦɚɤɫɢɦɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ (ɜ ɰɟɧɬɪɟ ɲɚɪɚ)
Tmax
Te (6.30)
qV
q
R V R2 ,
3D
6O
ɬɟɦɩɟɪɚɬɭɪɭ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ
qV
R
3D
ɢ ɩɨɥɧɵɣ ɩɨɬɨɤ ɬɟɩɥɚ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɲɚɪɚ
Te TS
R2 § dT ·
Q S
O
4 ¨© d r ¸¹ r
ȼ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ T
R
1 3
S R qV .
3
T Te
, [
T Te
r
, ɝɞɟ, ɤɚɤ ɢ ɜ ɡɚɞɚɱɚɯ
R
q R2
, ɮɨɪɦɭɥɚ (6.30) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ɨ ɰɢɥɢɧɞɪɟ, T Te V
O
1
1 [2
Ts
3Bi
6
(6.31)
ɉɨɞɨɛɧɚɹ ɡɚɞɚɱɚ ɨ ɰɢɥɢɧɞɪɟ
4
ɞɚɟɬ ɧɚɦ ɮɨɪɦɭɥɭ
1
1,2
1
1 [2
,
Tc
1
2B i
4
0,8
2
ɫɥɟɞɭɸɳɭɸ ɢɡ (6.18).
3
Ⱦɥɹ ɩɥɚɫɬɢɧɵ ɫɢɦɦɟɬɪɢɱɧɚɹ
0,4
2
ɡɚɞɚɱɚ (ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɬɟɦɩɟɪɚ3
ɬɭɪɚɦɢ ɨɦɵɜɚɸɳɢɯ ɫɪɟɞ ɢ ɨɞɢɧɚ0,0
0,0
0,2
0,4
0,6
0,8
[
ɤɨɜɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɬɟɩɥɨɨɬɞɚɱɢ) ɩɪɢɜɨɞɢɬ ɤ ɮɨɪɦɭɥɟ (6.8),
ɨɬɤɭɞɚ ɩɪɢ ɭɫɥɨɜɢɢ R G 2 ɢɦɟ- Ɋɢɫ. 6.4. ɋɪɚɜɧɟɧɢɟ ɪɟɲɟɧɢɣ ɡɚɞɚɱ ɞɥɹ
ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ ɫ ɨɛɴɟɦɧɵɦ
ɟɦ
ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ. ɋɩɥɨɲɧɵɟ ɥɢɧɢɢ –
1 1 [2
Bi = 1,0; ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ Bi = 2,5.
.
Tp
1. – ɩɥɚɫɬɢɧɚ, 2. – ɰɢɥɢɧɞɪ, 3 – ɫɮɟɪɚ.
2
Bi
ɋɪɚɜɧɟɧɢɟ ɩɨɥɭɱɟɧɧɵɯ ɪɟɲɟɧɢɣ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.4.
145
6 . 5 . Ɂ ɚ ɞ ɚ ɱ ɚ ɨ ɛ ɨ ɲ ɢ ɛ ɤ ɚ ɯ ɬ ɟ ɪ ɦ ɨ ɩ ɚ ɪ ɵ 19
ȿɫɥɢ ɬɟɩɥɨ ɜɵɞɟɥɹɟɬɫɹ ɢɥɢ ɩɨɝɥɨɳɚɟɬɫɹ ɜ ɨɞɧɨɣ ɢɥɢ ɛɨɥɟɟ ɥɨɤɚɥɶɧɵɯ ɨɛɥɚɫɬɹɯ («ɬɨɱɤɚɯ»), ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ ɫɢɫɬɟɦɚ ɫɨɞɟɪɠɢɬ ɦɟɫɬɧɵɟ ɢɥɢ
ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ. Ʉ ɬɚɤɢɦ ɫɢɫɬɟɦɚɦ ɨɬɧɨɫɹɬɫɹ ɫɢɫɬɟɦɵ ɫ ɬɟɪɦɨɩɚɪɚɦɢ. ɇɨ ɩɪɟɠɞɟ ɱɟɦ ɩɟɪɟɣɬɢ ɤ ɡɚɞɚɱɟ ɨɛ ɨɲɢɛɤɚɯ, ɫɜɹɡɚɧɧɵɯ ɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɩɪɨɜɨɞɨɜ ɬɟɪɦɨɩɚɪɵ, ɪɚɫɫɦɨɬɪɢɦ ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɢɡɨɥɢɪɨɜɚɧɧɨɣ ɛɟɫɤɨɧɟɱɧɨɣ
ɩɥɨɫɤɨɣ ɩɥɚɫɬɢɧɟ ɫ ɩɨɦɟɳɟɧɧɵɦ ɜ ɧɟɣ ɰɢɥɢɧɞɪɢɱɟɫɤɢɦ ɢɫɬɨɱɧɢɤɨɦ
ɬɟɩɥɚ (ɪɢɫ. 6.5).
ɉɭɫɬɶ ɨɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɜɵɞɟɥɹɟɦɨɟ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɦ ɨɛɴɟɦɟ S rs2G ( rs – ɪɚɞɢɭɫ ɥɨɤɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɢɱɟɫɤɨɝɨ
ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ; G – ɬɨɥɳɢɧɚ ɩɥɚɫɬɢɧɵ), ɪɚɜɧɨ q0 ɢ ɢɫɬɨɱɧɢɤ ɜɫɸɞɭ
ɢɦɟɟɬ ɨɞɢɧɚɤɨɜɭɸ ɬɟɦɩɟɪɚɬɭɪɭ T0 .
Ɋɢɫ. 6.5. Ɇɟɫɬɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ ɜ ɧɟɚɞɢɚɛɚɬɢɱɟɫɤɨɣ ɩɥɚɫɬɢɧɟ
Ⱦɨɩɨɥɧɢɬɟɥɶɧɨ, ɤɪɨɦɟ ɬɟɩɥɚ ɨɬ ɰɟɧɬɪɚɥɶɧɨɝɨ ɢɫɬɨɱɧɢɤɚ, ɩɥɚɫɬɢɧɚ
ɩɨɥɭɱɚɟɬ ɬɟɩɥɨ ɱɟɪɟɡ ɥɢɰɟɜɭɸ ɩɨɜɟɪɯɧɨɫɬɶ 1 ɨɬ ɨɦɵɜɚɸɳɟɝɨ ɟɟ ɝɚɡɚ ɫ
ɬɟɦɩɟɪɚɬɭɪɨɣ T g1 ɢ ɨɬɞɚɟɬ ɬɟɩɥɨ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ 2 ɝɚɡɭ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ T g 2 . Ʉɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɨɬɞɚɱɢ ɧɚ ɷɬɢɯ ɩɨɜɟɪɯɧɨɫɬɹɯ ɪɚɜɧɵ D1 ɢ
D 2 . ȼɫɟ ɷɬɢ ɜɟɥɢɱɢɧɵ, ɚ ɬɚɤɠɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɩɥɚɫɬɢɧɵ ɫɱɢɬɚɸɬɫɹ
ɩɨɫɬɨɹɧɧɵɦɢ.
ȿɫɥɢ ɩɥɚɫɬɢɧɚ ɬɨɧɤɚɹ, ɬɨ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɟɟ ɧɨɪɦɚɥɢ ɧɟɬ ɝɪɚɞɢɟɧɬɚ
ɬɟɦɩɟɪɚɬɭɪɵ, ɬ.ɟ. T1 T2 . Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
ɩɨ ɪɚɞɢɭɫɭ T r ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ ɞɥɹ rs r f .
19
ɒɧɟɣɞɟɪ ɉ. ɂɧɠɟɧɟɪɧɵɟ ɩɪɨɛɥɟɦɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɇ.:ɂɅ, 1960. 480 ɫ.
146
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɨɟ ɞɜɭɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
1 w § w T · w 2T
r
0.
(6.32)
r w r ¨© w r ¸¹ w z 2
Ɍɚɤ ɤɚɤ ɩɥɚɫɬɢɧɚ ɬɨɧɤɚɹ, ɬɨ ɩɪɟɧɟɛɪɟɝɚɹ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɣ, ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ (6.32) ɩɨ z ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0 ɞɨ G :
G
1 w § wT
³ r w r ¨© r w r
0
G
w 2T
·
¸dz ³ 2 dz
¹
0 wz
0
ɢɥɢ
§ w 2T 1 w T · § w T ·
§ wT ·
G¨ 2 ¨
¨
0.
¸
¸
¸
¨ wr
¸ © wz ¹
w
w
r
r
z
©
¹
z G
z 0
©
¹
Ɍɚɤ ɤɚɤ ɩɪɢ z G ɢɦɟɟɬ ɦɟɫɬɨ ɭɫɥɨɜɢɟ
wT
O
D1 T g1 T ,
wz
ɚ ɩɪɢ z 0 - ɭɫɥɨɜɢɟ
wT
O
D 2 T Tg 2 ,
wz
ɬɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨɫɥɟ ɩɨɞɫɬɚɧɨɜɤɢ ɜ ɧɟɝɨ ɭɫɥɨɜɢɣ ɧɚ
ɩɨɜɟɪɯɧɨɫɬɹɯ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
d 2T
dr 2
D1T g1 D 2T g 2 ·
1 dT D 1 D 2 §
T
¨
¸ 0.
r dr
OG ©
D1 D 2
¹
(6.33)
ɉɪɢ ɨɬɫɭɬɫɬɜɢɢ ɢɫɬɨɱɧɢɤɚ q0 ɩɥɚɫɬɢɧɚ ɛɵɥɚ ɛɵ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ,
ɬ.ɟ. ɢɦɟɥɚ ɛɵ ɩɨɫɬɨɹɧɧɭɸ ɜɫɸɞɭ ɬɟɦɩɟɪɚɬɭɪɭ. Ɍɚɤɭɸ ɬɟɦɩɟɪɚɬɭɪɭ Tf
ɩɥɚɫɬɢɧɚ ɢɦɟɟɬ ɩɪɢ r o f , ɝɞɟ d T dr 0 :
Tr of
Tf
D 1T g1 D 2T g 2
D1 D 2
, r ! rs .
(6.34)
ɗɬɨ ɟɫɬɶ ɬɟɦɩɟɪɚɬɭɪɚ ɜɫɟɣ ɩɥɚɫɬɢɧɵ ɩɪɢ q0 0 .
ɉɟɪɟɣɞɟɦ ɤ ɩɟɪɟɦɟɧɧɨɣ ) T Tf . Ɍɨɝɞɚ ɜɦɟɫɬɨ (6.33) ɩɨɥɭɱɢɦ
ɭɪɚɜɧɟɧɢɟ ɜɢɞɚ
d 2) 1 d )
(6.35)
H 2) 0 ,
2
r dr
dr
ɝɞɟ
147
D1 D 2
.
OG
ɗɬɨ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ Ȼɟɫɫɟɥɹ ɧɭɥɟɜɨɝɨ ɩɨɪɹɞɤɚ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɢɦɟɟɬ ɜɢɞ
H
) T Tf
C1I 0 H r C 2 K 0 H r ,
ɝɞɟ I 0 ɢ K 0 – ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɵɟ ɮɭɧɤɰɢɢ Ȼɟɫɫɟɥɹ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ
ɪɨɞɚ ɧɭɥɟɜɨɝɨ ɩɨɪɹɞɤɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. ȼɢɞ ɷɬɢɯ ɮɭɧɤɰɢɣ ɜ ɧɟɛɨɥɶɲɨɣ
ɨɛɥɚɫɬɢ ɢɡɦɟɧɟɧɢɹ ɚɪɝɭɦɟɧɬɚ ɩɨɤɚɡɚɧ ɧɚ ɪɢɫ. 6.6.
Ɏɭɧɤɰɢɹ I 0 H r ɧɟɨɝɪɚɧɢɱɟɧɧɨ ɜɨɡɪɚɫɬɚɟɬ ɩɨ ɦɟɪɟ ɬɨɝɨ, ɤɚɤ ɟɟ ɚɪɝɭɦɟɧɬ H r o f . ɉɨɷɬɨɦɭ, ɱɬɨɛɵ ɬɟɦɩɟɪɚɬɭɪɚ ɨɫɬɚɜɚɥɚɫɶ ɤɨɧɟɱɧɨɣ,
ɧɭɠɧɨ ɩɨɥɨɠɢɬɶ C1 0 .
ȼɫɥɟɞɫɬɜɢɟ ɧɚɥɢɱɢɹ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ q0 ɩɪɢ r rs ɩɥɚɫɬɢɧɚ ɹɜɥɹɟɬɫɹ ɧɟɢɡɨɬɟɪɦɢɱɟɫɤɨɣ. ɉɪɢ r rs ɢɦɟɟɦ
q0
§ dT ·
OFs ¨
¸
© dr ¹ r
rs
ªd
º
2SOGrs « C2 K 0 Hr »
¬ dr
¼r
rs
2SOGrs C2 HK1 Hrs ,
ɝɞɟ K1 – ɦɨɞɢɮɢɰɢɪɨɜɚɧɧɚɹ ɮɭɧɤɰɢɹ Ȼɟɫɫɟɥɹ ɜɬɨɪɨɝɨ ɪɨɞɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ, Fs – ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ. ɗɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɞɚɟɬ ɧɚɦ ɩɨɫɬɨɹɧɧɭɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ C2 :
q0
C2
.
2 SO G rs H K 1 H rs Ɋɢɫ. 6.6. Ɇɨɞɢɮɢɰɢɪɨɜɚɧɧɵɟ ɮɭɧɤɰɢɢ Ɋɢɫ. 6.7. Ɋɚɞɢɚɥɶɧɨɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ
Ȼɟɫɫɟɥɹ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɪɨɞɚ ɧɭɥɟ- ɬɟɦɩɟɪɚɬɭɪɵ ɜɨɤɪɭɝ ɥɨɤɚɥɶɧɨɝɨ ɢɫɜɨɝɨ ɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɨɜ
ɬɨɱɧɢɤɚ ɬɟɩɥɚ ɜ ɧɟɚɞɢɚɛɚɬɢɱɟɫɤɨɣ
ɩɥɚɫɬɢɧɟ
148
ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɚɞɢɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɨɩɪɟɞɟɥɢɬɫɹ
ɮɨɪɦɭɥɨɣ
K H r q0
T Tf ˜ 0
(6.36)
2SO GH rs K 1 H rs ɢɥɢ
T
T Tf
T0 Tf
K 0 H rs [ ,[
K 1 H rs r
,
rs
ɝɞɟ ɬɟɦɩɟɪɚɬɭɪɚ T0 ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ q0
T0
Tf q0
.
2SOGH rs
Ⱦɥɹ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ H rs [ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ
ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɮɭɧɤɰɢɢ20
§ 2 ·
¸¸ 0 ,577 , Hrs [ 0,05
K 0 >Hrs [@ | ln¨¨
r
H
[
© s ¹
(6.37)
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɚɫɬɢɧɟ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ H rs
ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 6.7.
Ɍɟɩɟɪɶ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ ɡɚɞɚɱɟ ɨɛ ɨɲɢɛɤɚɯ ɬɟɪɦɨɩɚɪɵ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ. Ʉɚɤ ɢɡɜɟɫɬɧɨ, ɢɡɦɟɪɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɟɣ ɨɛɵɱɧɨɣ
ɬɟɪɦɨɩɚɪɨɣ ɫɨɩɪɹɠɟɧɨ ɫ ɪɹɞɨɦ ɨɲɢɛɨɤ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ, ɮɢɡɢɤɨɯɢɦɢɱɟɫɤɨɝɨ ɢ ɬɟɪɦɢɱɟɫɤɨɝɨ ɩɪɨɢɫɯɨɠɞɟɧɢɹ. ȼɨɡɦɨɠɧɨ, ɱɬɨ ɧɚɢɛɨɥɟɟ
ɫɟɪɶɟɡɧɨɣ ɢɡ ɨɲɢɛɨɤ, ɜɫɬɪɟɱɚɸɳɢɯɫɹ ɩɪɢ ɢɡɦɟɪɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɪɦɨɩɚɪɨɣ ɨɛɵɱɧɨɝɨ ɬɢɩɚ, ɹɜɥɹɟɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɨɲɢɛɤɚ ɡɚ ɫɱɟɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɨɜɨɞɨɜ ɬɟɪɦɨɩɚɪɵ.
ɉɪɨɢɫɯɨɠɞɟɧɢɟ ɨɲɢɛɨɤ ɩɪɢ ɢɡɦɟɪɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɪɦɨɩɚɪɨɣ
ɢɥɥɸɫɬɪɢɪɭɟɬ ɪɢɫ. 6.8. ȿɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɫɪɟɞɵ ɛɨɥɶɲɟ ɢɡɦɟɪɹɟɦɨɣ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ T g1 ! T g 2 , ɬɨ ɩɪɨɜɨɞɚ ɬɟɪɦɨɩɚɪɵ, ɧɚɯɨɞɹɳɢɟɫɹ ɩɪɢ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɛɭɞɭɬ ɩɨɞɜɨɞɢɬɶ ɬɟɩɥɨ ɤ ɩɥɚɫɬɢɧɟ ɫ ɛɨɥɟɟ ɧɢɡɤɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɱɬɨ ɜɵɡɨɜɟɬ ɦɟɫɬɧɨɟ ɩɨɜɵɲɟɧɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ ɤɚɤ ɪɚɡ ɜ ɬɨɣ ɬɨɱɤɟ, ɝɞɟ ɫɩɚɣ ɬɟɪɦɨɩɚɪɵ ɞɨɥɠɟɧ ɢɡɦɟɪɹɬɶ
ɬɟɦɩɟɪɚɬɭɪɭ ɩɨɜɟɪɯɧɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɤɚɡɚɧɢɹ ɬɟɪɦɨɩɚɪɵ ɞɚɞɭɬ ɡɚɜɵɲɟɧɧɨɟ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɢɫɬɢɧɧɵɦ ɡɧɚɱɟɧɢɟɦ
ɬɟɦɩɟɪɚɬɭɪɵ.
20
əɧɤɟ ȿ., ɗɦɞɟ Ɏ., Ʌɟɲ Ɏ. ɋɩɟɰɢɚɥɶɧɵɟ ɮɭɧɤɰɢɢ. Ɇ.: 1978.
149
ɂ ɧɚɨɛɨɪɨɬ, ɟɫɥɢ T g T g 2 , ɬɨ
1
ɬɟɪɦɨɩɚɪɚ ɛɭɞɟɬ ɪɟɝɢɫɬɪɢɪɨɜɚɬɶ
ɬɟɦɩɟɪɚɬɭɪɭ, ɦɟɧɶɲɭɸ ɮɚɤɬɢɱɟɫɤɨɣ. Ɍɚɤɢɟ ɨɲɢɛɤɢ ɩɨɥɧɨɫɬɶɸ
ɭɫɬɪɚɧɹɸɬɫɹ ɥɢɲɶ ɜ ɬɨɦ ɫɥɭɱɚɟ,
ɟɫɥɢ T g1 T g 2 , ɱɬɨ ɜɨɡɦɨɠɧɨ
ɬɨɥɶɤɨ ɜ ɢɞɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ.
ȿɫɥɢ ɩɪɢɧɹɬɶ, ɱɬɨ ɩɪɨɜɨɞɚ
ɬɟɪɦɨɩɚɪɵ ɩɢɬɚɸɬ ɢɫɬɨɱɧɢɤ Q0 ,
ɬɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɨɲɢɛɤɢ ɦɨɠɧɨ
ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɬɨɥɶɤɨ ɱɬɨ ɩɨɥɭɱɟɧɧɵɦ ɪɟɲɟɧɢɟɦ.
ȿɫɥɢ T g ! T g 2 , ɬɨ ɬɟɪɦɨɩɚɪɚ
1
Ɋɢɫ. 6.8. ɂɡɨɥɢɪɨɜɚɧɧɚɹ ɬɟɪɦɨɩɚɪɚ ɜ
ɧɟɚɞɢɚɛɚɬɢɱɟɫɤɨɣ ɩɥɚɫɬɢɧɟ
ɫɬɚɧɨɜɢɬɫɹ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ, ɢ
Q0 ɛɭɞɟɬ ɩɪɟɞɫɬɚɜɥɹɬɶ ɫɨɛɨɣ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɬɟɩɥɨɦ, ɩɨɞɜɨɞɢɦɵɦ ɩɪɨɜɨɞɚɦɢ ɤ ɫɩɚɸ Q0 1 , ɢ ɬɟɩɥɨɦ, ɪɚɫɫɟɹɧɧɵɦ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɤɪɭɝɥɨɝɨ ɢɫɬɨɱɧɢɤɚ Q02 . ɂɦɟɟɦ
/ 2 T0 T g 2 ,
Q01 / 1 T g1 T0 ;
Q02
ɝɞɟ / 1 – ɬɟɪɦɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ ɩɪɨɜɨɞɨɜ ɬɟɪɦɨɩɚɪɵ / 1 S l K T 1 ;
/ 2 – ɩɨɜɟɪɯɧɨɫɬɧɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ. ɗɬɢ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɤɚɤ ɭɠɟ ɢɡɜɟɫɬɧɨ, ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɱɟɪɟɡ ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɨɜɨɞɨɜ ɢ
ɢɡɨɥɹɬɨɪɨɜ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɨɛɦɟɧɚ ɢ ɯɚɪɚɤɬɟɪɧɵɟ ɥɢɧɟɣɧɵɟ ɪɚɡɦɟɪɵ (ɫɦ. ɪɚɡɞɟɥɵ 5.3, 3.4).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
Q0
Q01 Q02
/1T g1 / 2Tg 2 /1 / 2 T0 .
(6.38)
ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɪɟɲɟɧɢɟɦ (6.36), ɧɚɣɞɟɦ
T Tf
ɢɥɢ
T Tf
/1Tg1 / 2Tg 2 /1 / 2 T0 ˜ K 0 H rs 2SO GH rs
K 1 H rs / 1 T g1 Tf / 2 T g 2 Tf / 1 / 2 Tf T0 K 0 H rs .
˜
2SO GH rs
K 1 H rs 150
ɉɪɢɧɢɦɚɹ, ɱɬɨ T T0 ɢ ɟɫɬɶ ɬɟɦɩɟɪɚɬɭɪɚ, ɡɚɩɢɫɵɜɚɟɦɚɹ ɬɟɪɦɨɩɚɪɨɣ
(ɩɪɢ r rs ), ɩɨɥɭɱɢɦ
ª
T0 Tf «1 ¬«
K 0 H rs / 1 / 2 º
˜
»
K 1 H rs 2SO GH rs ¼»
/ 1 T g1 Tf / 2 T g 2 Tf
2SO GH rs
˜ K 0 H rs K 1 H rs ɢɥɢ
T0 Tf
/ 1 T g1 Tf / 2 T g 2 Tf
/1 / 2 ª¬ 2SO GH rs K1 H rs º¼
K 0 H rs .
(6.39)
ɂɬɚɤ, T0 – ɬɟɦɩɟɪɚɬɭɪɚ, ɡɚɩɢɫɚɧɧɚɹ ɬɟɪɦɨɩɚɪɨɣ; Tf – ɯɚɪɚɤɬɟɪɢɡɭɟɬ
ɬɟɦɩɟɪɚɬɭɪɭ ɩɥɚɫɬɢɧɵ ɜ ɨɬɫɭɬɫɬɜɢɟ ɬɟɪɦɨɩɚɪɵ ( q0 0 ) ɢɥɢ ɠɟ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɛɟɫɤɨɧɟɱɧɨ ɭɞɚɥɟɧɧɨɣ ɬɨɱɤɟ. Ɍ.ɟ., ɪɚɡɧɨɫɬɶ T0 Tf ɢ ɟɫɬɶ ɚɛɫɨɥɸɬɧɚɹ ɨɲɢɛɤɚ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ, ɢɡɦɟɪɟɧɧɨɣ ɬɟɪɦɨɩɚɪɨɣ.
ɇɚ ɩɪɚɤɬɢɤɟ, ɤɚɤ ɩɪɚɜɢɥɨ, /1 !! / 2 . ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɲɟɧɢɟ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
T
T0 Tf
T g1 Tf
1 2S O G / 1 1
.
H rs K1 H rs K 0 H rs (6.40)
Ⱦɥɹ ɪɚɫɱɟɬɨɜ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɮɭɧɤɰɢɢ K1 ɞɥɹ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ ɚɪɝɭɦɟɧɬɚ
K1 x | x 1 , x 0 ,05 .
(6.41)
ȿɫɥɢ ɬɟɪɦɨɩɚɪɵ ɧɟ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɵ, ɬɨ ɬɟɪɦɢɱɟɫɤɚɹ ɩɪɨɜɨɞɢɦɨɫɬɶ /1 ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɬɞɚɱɢ ɬɟɪɦɨɩɚɪɵ ɜ ɫɪɟɞɟ), ɬ.ɟ. / 1 S l D1rW ( l – ɞɥɢɧɚ ɩɪɨɜɨɞɨɜ ɬɟɪɦɨɩɚɪɵ, rW – ɢɯ ɪɚɞɢɭɫ).
Ɍɨɝɞɚ ɫ ɭɱɟɬɨɦ (6.41), (6.37) ɡɚɩɢɲɟɦ (6.40) ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
T
ɝɞɟ x
OG
, y H rr
S lrWD1
1
1
1 2Sx
ln2 y 0 ,577
,
D1 D 2
rs (ɫɦ. ɮɨɪɦɭɥɭ (6.35)).
OG
151
(6.42)
Ɋɢɫ. 6.9. Ɉɛɳɚɹ ɨɲɢɛɤɚ ɡɚ ɫɱɟɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɬɟɪɦɨɩɚɪɵ ɞɥɹ
ɧɟɚɞɢɛɚɬɢɱɟɫɤɨɣ ɩɥɚɫɬɢɧɵ.
ȼ ɤɚɱɟɫɬɜɟ ɪɚɞɢɭɫɚ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɜɟɥɢɱɢɧɭ
rs 1 . 41 4rW ,
ɝɞɟ rW – ɪɚɞɢɭɫ ɩɪɨɜɨɥɨɤɢ.
ɇɚ ɪɢɫ. 6.9. ɢɡɨɛɪɚɠɟɧɚ ɚɛɫɨɥɸɬɧɚɹ ɨɲɢɛɤɚ ɬɟɪɦɨɩɚɪɵ ɜ
ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ x ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɤɨɦɩɥɟɤɫɚ y .
ɑɟɦ ɬɨɥɳɟ ɩɥɚɫɬɢɧɚ, ɬɟɦ ɛɨɥɶɲɟ x , ɧɨ ɬɟɦ ɦɟɧɶɲɟ y , ɬɚɤ ɱɬɨ
ɞɨɛɢɬɶɫɹ ɧɭɥɟɜɨɣ ɨɲɢɛɤɢ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɜɨɡɦɨɠɧɨ.
6.6. Ɉɯɥɚɠɞɟɧɢɟ ɪɚɛɨɱɢɯ ɥɨɩɚɬɨɤ ɬɭɪɛɢɧɵ
ɏɨɪɨɲɨ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɵɟ ɰɢɤɥɢɱɟɫɤɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɝɚɡɨɜɵɯ ɬɭɪɛɢɧɚɯ, ɩɪɟɠɞɟ ɜɫɟɝɨ, ɨɝɪɚɧɢɱɢɜɚɸɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚɦɢ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɪɚɡɪɭɲɟɧɢɹ ɬɭɪɛɢɧɧɵɯ ɥɨɩɚɬɨɤ, ɪɚɛɨɬɚɸɳɢɯ ɩɪɢ
ɜɵɫɨɤɨɣ ɰɟɧɬɪɨɛɟɠɧɨɣ ɧɚɝɪɭɡɤɟ ɢ ɩɨɞɜɟɪɝɧɭɬɵɯ, ɤɪɨɦɟ ɬɨɝɨ, ɧɚɩɪɹɠɟɧɢɹɦ ɢɡɝɢɛɚ ɫɨ ɫɬɨɪɨɧɵ ɝɚɡɚ, ɜɢɛɪɚɰɢɨɧɧɵɦ ɢ ɬɟɪɦɢɱɟɫɤɢɦ ɧɚɩɪɹɠɟɧɢɹɦ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɩɨɜɵɲɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɵɯ ɜɢɛɪɚɰɢɨɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪ ɫ ɰɟɥɶɸ ɭɥɭɱɲɟɧɢɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɪɚɛɨɬɵ ɝɚɡɨɜɨɣ ɬɭɪɛɢɧɵ ɜɨɡɦɨɠɧɨ ɞɜɭɦɹ ɩɭɬɹɦɢ. ɉɟɪɜɵɣ ɩɭɬɶ ɫɜɹɡɚɧ ɫ ɫɨɡɞɚɧɢɟɦ ɦɚɬɟɪɢɚɥɨɜ, ɫɩɨɫɨɛɧɵɯ ɪɚɛɨɬɚɬɶ ɜ ɭɫɥɨɜɢɹɯ ɜɵɫɨɤɢɯ ɬɟɦɩɟɪɚɬɭɪ ɢ ɧɚɩɪɹɠɟɧɢɣ ɞɥɢɬɟɥɶɧɨɟ ɜɪɟɦɹ. ȼɬɨɪɨɣ – ɫɜɹɡɚɧ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɢ ɭɫɨɜɟɪɲɟɧɫɬɜɨɜɚɧɢɟɦ ɦɟɬɨɞɨɜ ɢɫɤɭɫɫɬɜɟɧɧɨɝɨ ɨɯɥɚɠɞɟɧɢɹ ɥɨɩɚɬɨɤ ɜ ɩɪɨɰɟɫɫɟ ɪɚɛɨɬɵ.
ɇɚɩɪɢɦɟɪ, ɬɭɪɛɢɧɧɵɟ ɥɨɩɚɬɤɢ ɦɨɠɧɨ ɨɯɥɚɠɞɚɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ, ɧɚɝɧɟɬɚɹ ɨɯɥɚɠɞɚɸɳɭɸ ɠɢɞɤɨɫɬɶ ɩɨ ɤɚɧɚɜɤɚɦ ɜɧɭɬɪɢ ɥɨɩɚɬɨɤ ɢɥɢ, ɨɬɜɨɞɹ ɬɟɩɥɨ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɨɬ ɥɨɩɚɬɨɤ ɤ ɬɭɪɛɢɧɧɨɦɭ ɪɨɬɨɪɭ ɫ ɜɧɭɬɪɟɧɧɢɦ ɨɯɥɚɠɞɟɧɢɟɦ. ȼɬɨɪɨɣ ɫɩɨɫɨɛ ɧɚɡɵɜɚɸɬ ɨɯɥɚɠɞɟɧɢɟɦ ɡɚ ɫɱɟɬ
ɩɪɨɜɨɞɢɦɨɫɬɢ. ɗɮɮɟɤɬɢɜɧɨɫɬɶ ɷɬɨɝɨ ɦɟɬɨɞɚ ɨɰɟɧɢɜɚɟɬɫɹ, ɧɚɩɪɢɦɟɪ, ɩɨ
ɭɫɥɨɜɢɹɦ, ɩɪɢ ɤɨɬɨɪɵɯ ɮɚɤɬɢɱɟɫɤɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɥɨɩɚɬɤɢ
ɞɨɫɬɢɝɚɟɬ ɦɚɤɫɢɦɚɥɶɧɨ ɞɨɩɭɫɬɢɦɨɣ ɞɥɹ ɞɚɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɬɟɦɩɟɪɚɬɭɪɵ.
ɉɟɪɟɧɨɫ ɬɟɩɥɚ ɜ ɥɨɩɚɬɤɟ ɬɭɪɛɢɧɵ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɥɭɱɚɣ ɧɟɩɪɟɪɵɜɧɨɝɨ ɩɪɢɬɨɤɚ ɬɟɩɥɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ. ȼɵɜɟɞɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ
ɭɪɚɜɧɟɧɢɟ, ɩɪɢɝɨɞɧɨɟ ɞɥɹ ɨɩɢɫɚɧɢɹ ɨɯɥɚɠɞɟɧɢɹ ɥɨɩɚɬɤɢ, ɢɫɩɨɥɶɡɭɹ
ɪɢɫ. 6.10.
152
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɩɨɩɟɪɟɱɧɨɟ ɫɟɱɟɧɢɟ ɥɨɩɚɬɤɢ A ɢ ɩɟɪɢɦɟɬɪ p ɨɫɬɚɸɬɫɹ
ɩɨɫɬɨɹɧɧɵɦɢ ɞɥɹ ɥɸɛɵɯ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɡɧɚɱɟɧɢɣ ɤɨɨɪɞɢɧɚɬɵ x . Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ D ɢ ɷɮɮɟɤɬɢɜɧɚɹ
ɬɟɦɩɟɪɚɬɭɪɚ ɝɚɡɚ Tg ɩɨɫɬɨɹɧɧɵ ɜɞɨɥɶ p
ɢ L ; ɝɪɚɞɢɟɧɬɚɦɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜɨ ɜɫɟɯ
ɩɨɩɟɪɟɱɧɵɯ ɫɟɱɟɧɢɹɯ ɥɨɩɚɬɤɢ ɦɨɠɧɨ
Ɋɢɫ. 6.10. Ʌɨɩɚɬɤɚ ɬɭɪɛɢɧɵ,
ɩɪɟɧɟɛɪɟɱɶ. ȼ ɷɬɢɯ ɭɫɥɨɜɢɹɯ ɪɚɫɩɪɟɞɟɨɯɥɚɠɞɚɟɦɚɹ ɡɚ ɫɱɟɬ
ɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɥɨɩɚɬɤɟ ɛɭɞɟɬ ɨɞɩɪɨɜɨɞɢɦɨɫɬɢ
ɧɨɦɟɪɧɵɦ, T T x .
Ɋɚɫɫɦɨɬɪɢɦ ɷɥɟɦɟɧɬ ɥɨɩɚɬɤɢ dx (ɪɢɫ. 6.10). ɗɥɟɦɟɧɬ ɩɪɨɜɨɞɢɬ ɬɟɩɥɨ ɜ ɩɪɨɞɨɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ q x ɢ q x d x , ɚ ɫ ɛɨɤɨɜ ɩɨɥɭɱɚɟɬ ɬɟɩɥɨ ɡɚ
ɫɱɟɬ ɜɵɧɭɠɞɟɧɧɨɣ ɤɨɧɜɟɤɰɢɢ ɝɚɡɚ ɜ ɤɨɥɢɱɟɫɬɜɟ qg (ɝɚɡ ɢɦɟɟɬ ɬɟɦɩɟɪɚɬɭɪɭ Tg ).
ȿɫɥɢ ɜ ɬɨɱɤɟ x ɬɟɦɩɟɪɚɬɭɪɚ ɟɫɬɶ T , ɬɨ ɜ ɬɨɱɤɟ x dx ɛɭɞɟɦ ɢɦɟɬɶ
ɬɟɦɩɟɪɚɬɭɪɭ
dT
T x dx T dx .
dx
Ɍɟɩɥɨɜɨɣ ɛɚɥɚɧɫ ɞɥɹ ɷɬɨɝɨ ɷɥɟɦɟɧɬɚ ɢɦɟɟɬ ɜɢɞ
q g q x dx q x 0
ɢɥɢ
ª
d §
dT · º ª
dT º
D pdx T g T « O A ¨ T dx ¸ » « O A
dx ©
dx ¹ ¼ ¬
dx ¼»
¬
0
ɢɥɢ
d 2T
dx
2
N 2 T Tg
0,
(6.43)
Dp
. Ɍ.ɟ., ɦɵ ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɨɛɴɟɦɧɵɦ
OA
ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ, ɡɚɜɢɫɹɳɢɦ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ.
ȼɜɟɞɟɦ ɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ
T Tg
x
T
,[
,
L
Tr T g
ɝɞɟ N
ɝɞɟ Tr – ɬɟɦɩɟɪɚɬɭɪɚ ɭ ɨɫɧɨɜɚɧɢɹ ɥɨɩɚɬɤɢ.
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (6.43) ɩɪɢɦɟɬ ɜɢɞ
153
d 2T
B 2T 0
d[ 2
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ ɭɪɚɜɧɟɧɢɸ (6.43) ɛɭɞɭɬ
x 0 : T Tr ;
x L : dT dx 0 ,
ɢɥɢ ɜ ɧɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ
(6.44)
[ 0 : T 1;
[ 1: d T d [ 0 .
(6.45)
ȼɬɨɪɨɟ ɭɫɥɨɜɢɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜɟɪɲɢɧɚ ɥɨɩɚɬɤɢ ɧɟɩɪɨɧɢɰɚɟɦɚ ɞɥɹ ɬɟɩɥɚ.
Ʌɟɝɤɨ ɩɨɧɹɬɶ ɫɦɵɫɥ ɩɚɪɚɦɟɬɪɚ B ,
B2
D pL
OA L
/s
.
/i
Ʉɜɚɞɪɚɬ ɷɬɨɝɨ ɩɚɪɚɦɟɬɪɚ ɪɚɜɟɧ ɨɬɧɨɲɟɧɢɸ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɩɪɨɜɨɞɢɦɨɫɬɟɣ ɥɨɩɚɬɤɢ.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (6.44) ɢɦɟɟɬ ɜɢɞ
T C1e xp B[ C 2e x p B[ .
(6.46)
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɢɦɟɟɦ ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ, ɤɨɬɨɪɚɹ ɫɥɟɞɭɟɬ ɢɡ (6.46) ɩɪɢ ɩɨɞɫɬɚɧɨɜɤɟ ɟɝɨ ɜ ɭɫɥɨɜɢɹ (6.45)
C1 C 2 1
C1B exp B C 2 B exp B 0 .
ɇɚɯɨɞɢɦ
e x p B e x p B , C2
.
2ch B 2ch B ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɥɨɩɚɬɤɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
T Tg ch B 1 [ ,
(6.47)
T
Tr Tg
ch B C1
ɬ.ɟ., ɞɥɹ ɡɚɞɚɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪ ɝɚɡɚ ɢ ɨɫɧɨɜɚɧɢɹ ɥɨɩɚɬɤɢ ɬɟɦɩɟɪɚɬɭɪɚ
ɫɚɦɨɣ ɥɨɩɚɬɤɢ ɡɚɜɢɫɢɬ ɨɬ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɩɚɪɚɦɟɬɪɚ B . Ɉɱɟɜɢɞɧɨ, ɱɬɨ
ɩɪɢ B o 0 ɢɦɟɟɦ T 1 , ɬ.ɟ. T Tr , ɚ ɩɪɢ B o f ɢɦɟɟɦ T 0 ɢɥɢ T Tg .
Ɇɚɥɵɟ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ B , ɠɟɥɚɬɟɥɶɧɵɟ ɞɥɹ ɩɨɜɵɲɟɧɢɹ ɷɮɮɟɤɬɢɜɧɨɫɬɢ ɨɯɥɚɠɞɟɧɢɹ, ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɥɢɛɨ ɫɧɢɠɟɧɢɟɦ ɩɨɜɟɪɯɧɨ154
ɫɬɧɨɣ ɩɪɨɜɨɞɢɦɨɫɬɢ ɥɨɩɚɬɤɢ, / s D p L , ɥɢɛɨ ɭɜɟɥɢɱɟɧɢɟɦ ɟɟ ɜɧɭɬɪɟɧɧɟɣ ɩɪɨɜɨɞɢɦɨɫɬɢ, / i O A L . ɉɪɚɤɬɢɱɟɫɤɨɟ ɫɧɢɠɟɧɢɟ ɪɚɛɨɱɟɝɨ ɡɧɚɱɟɧɢɹ B ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɫɜɹɡɚɧɨ ɥɢɛɨ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɢɡɨɥɹɰɢɢ, ɥɢɛɨ ɫ ɩɨɜɵɲɟɧɢɟɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɫɚɦɨɣ ɥɨɩɚɬɤɢ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɋ ɤɚɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ ɦɨɠɟɬ ɛɵɬɶ ɫɜɹɡɚɧɨ ɨɛɴɟɦɧɨɟ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟ?
2. Ɂɚɩɢɲɢɬɟ ɩɪɨɫɬɟɣɲɟɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɨɛɴɟɦɧɵɦ
ɢɫɬɨɱɧɢɤɨɦ.
3. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɤɚɱɟɫɬɜɟɧɧɨɟ ɨɬɥɢɱɢɟ ɜ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɨɫɤɨɣ ɫɬɟɧɤɟ ɛɟɡ ɨɛɴɟɦɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ ɢ ɫ ɨɛɴɟɦɧɵɦɢ
ɢɫɬɨɱɧɢɤɚɦɢ ɬɟɩɥɚ?
4. Ɉɬ ɤɚɤɢɯ ɮɢɡɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɡɚɜɢɫɢɬ ɩɨɥɨɠɟɧɢɟ ɦɚɤɫɢɦɭɦɚ
ɬɟɦɩɟɪɚɬɭɪɵ?
5. Ƚɞɟ ɧɚɯɨɞɢɬɫɹ ɦɚɤɫɢɦɭɦ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ ɬɟɩɥɨɨɛɦɟɧɟ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɫɩɥɨɲɧɵɯ ɲɚɪɚ ɢ ɰɢɥɢɧɞɪɚ, ɫɨɞɟɪɠɚɳɢɯ ɜɧɭɬɪɟɧɧɢɟ
ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ?
6. ɉɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪɵ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɝɞɟ
ɜɫɬɪɟɱɚɸɬɫɹ ɜɧɭɬɪɟɧɧɢɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ.
7. Ʉɚɤɢɟ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɮɨɪɦɭɥɢɪɭɸɬɫɹ ɜ ɡɚɞɚɱɚɯ ɨ ɬɟɩɥɨɨɛɦɟɧɟ ɫɩɥɨɲɧɵɯ ɲɚɪɚ ɢ ɰɢɥɢɧɞɪɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ?
8. Ʉɚɤɢɟ ɧɨɜɵɟ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɚɪɚɦɟɬɪɵ ɜɫɬɪɟɬɢɥɢɫɶ ɜ ɡɚɞɚɱɚɯ ɫ
ɨɛɴɟɦɧɵɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɬɟɩɥɚ?
9. ɋ ɱɟɦ ɦɨɝɭɬ ɛɵɬɶ ɫɜɹɡɚɧɵ ɨɲɢɛɤɢ ɩɪɢ ɢɡɦɟɪɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɫ
ɩɨɦɨɳɶɸ ɬɟɪɦɨɩɚɪɵ?
10. Ʉɚɤ ɜɵɝɥɹɞɢɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢ ɪɚɫɱɟɬɟ ɩɪɨɰɟɫɫɚ ɨɯɥɚɠɞɟɧɢɹ ɪɚɛɨɱɢɯ ɥɨɩɚɬɨɤ ɬɭɪɛɢɧɵ? ɋ ɱɟɦ ɫɜɹɡɚɧɨ ɨɛɴɟɦɧɨɟ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟ (ɩɨɝɥɨɳɟɧɢɟ) ɜ ɷɬɨɦ ɫɥɭɱɚɟ?
Ɂɚɞɚɧɢɹ
1. ɉɨ ɞɥɢɧɧɨɦɭ ɩɨɥɨɦɭ ɰɢɥɢɧɞɪɭ ɞɥɢɧɨɣ L ɩɪɨɩɭɫɤɚɟɬɫɹ ɜ ɨɫɟɜɨɦ
ɧɚɩɪɚɜɥɟɧɢɢ ɬɨɤ ɩɨɫɬɨɹɧɧɨɣ ɫɢɥɵ I . ȼɧɭɬɪɟɧɧɹɹ ɩɨɜɟɪɯɧɨɫɬɶ ɰɢɥɢɧɞɪɚ ( r R1 ) ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɚ, ɚ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚɪɭɠɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ
( r R 2 ) ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɪɚɜɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ Te . ɗɥɟɤɬɪɢɱɟɫɤɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɰɢɥɢɧɞɪɚ Re . ɇɚɣɬɢ ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɰɢɥɢɧɞɪɟ.
155
2. Ɍɨɤ ɫɢɥɨɣ I 200 Ⱥ ɩɪɨɩɭɫɤɚɟɬɫɹ ɱɟɪɟɡ ɩɪɨɜɨɥɨɤɭ ɢɡ ɧɟɪɠɚɜɟɸɳɟɣ ɫɬɚɥɢ ɞɢɚɦɟɬɪɨɦ 2 ɦɦ ɢ ɞɥɢɧɨɣ 1 ɦ. ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɪɨɜɨɥɨɤɚ – 0,125 Ɉɦ, ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
17 ȼɬ/(ɦ Ʉ). Ɍɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɨɜɨɥɨɤɢ 150 ɋ. Ɍɪɟɛɭɟɬɫɹ ɪɚɫɫɱɢɬɚɬɶ ɬɟɦɩɟɪɚɬɭɪɭ ɧɚ ɨɫɢ ɩɪɨɜɨɥɨɤɢ.
ɉɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɩɪɨɜɨɥɨɤɚ ɩɨɤɪɵɬɚ ɫɥɨɟɦ ɢɡɨɥɹɰɢɢ (ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢɡɨɥɹɰɢɢ 0,15 ȼɬ/(ɦ Ʉ)), ɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɨɥɹɰɢɢ ɪɚɜɟɧ 60 ȼɬ/(ɦ2Ʉ). Ʉɚɤ ɧɭɠɧɨ ɢɡɦɟɧɢɬɶ
ɫɢɥɭ ɬɨɤɚ (ɭɜɟɥɢɱɢɬɶ ɢɥɢ ɭɦɟɧɶɲɢɬɶ), ɱɬɨɛɵ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ
ɩɪɨɜɨɥɨɤɢ ɨɫɬɚɥɚɫɶ ɪɚɜɧɨɣ 150 ɋ.
3. ɉɥɨɫɤɚɹ ɩɥɚɫɬɢɧɚ ɬɨɥɳɢɧɨɣ L ɢɦɟɟɬ ɩɨɫɬɨɹɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɉɨɜɟɪɯɧɨɫɬɶ x 0 - ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɚ, ɚ ɩɨɜɟɪɯɧɨɫɬɶ x L ɨɦɵɜɚɟɬɫɹ ɠɢɞɤɨɫɬɶɸ, ɤɨɬɨɪɚɹ ɨɛɟɫɩɟɱɢɜɚɟɬ ɩɨɫɬɨɹɧɧɭɸ
ɬɟɦɩɟɪɚɬɭɪɭ T1 . ȼ ɫɬɟɧɤɟ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɵ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ ɫ
ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ qV . ɇɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɭɸ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɜ ɨɛɴɟɦɟ ɩɥɨɫɤɨɣ ɫɬɟɧɤɢ, ɩɪɢ ɤɨɬɨɪɨɣ ɬɟɦɩɟɪɚɬɭɪɚ T1 ɧɟ ɩɪɟɜɵɫɢɬ 120 ɋ, ɟɫɥɢ L 0,1 ɦ, O 0 0 ,15 ȼɬ/(ɦ Ʉ).
Ɋɚɫɫɱɢɬɚɬɶ ɬɟɦɩɟɪɚɬɭɪɭ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ.
ɇɚɣɬɢ ɬɟɦɩɟɪɚɬɭɪɭ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɧɤɢ, ɦɚɤɫɢɦɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɢ ɟɟ ɩɨɥɨɠɟɧɢɟ, ɟɫɥɢ
O O 0 ª1 0 , 0 1T 1 0 4 T 2 º .
¬
¼
4. (Ɂɚɞɚɱɚ ɧɚ ɩɨɜɬɨɪɟɧɢɟ) ȼ ɲɚɪɨɜɨɣ ɟɦɤɨɫɬɢ ɪɚɞɢɭɫɨɦ R1 ɧɚɯɨɞɢɬɫɹ ɠɢɞɤɨɫɬɶ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ T1 . ɋɬɟɧɤɢ ɟɦɤɨɫɬɢ ɬɨɥɳɢɧɨɣ G ɢɡɝɨɬɨɜɥɟɧɵ ɢɡ ɦɟɬɚɥɥɚ ɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ O1 ɢ ɩɨɤɪɵɬɵ ɫɥɨɟɦ ɬɟɩɥɨɢɡɨɥɹɰɢɢ ɬɨɥɳɢɧɨɣ ' ɫ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ O 2 . Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɟɦɩɟɪɚɬɭɪɭ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɨɥɹɰɢɢ ɩɪɢ ɭɫɥɨɜɢɢ ɤɨɧɜɟɤɬɢɜɧɨɝɨ
ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɬɞɚɱɢ D 2 .
Ɋɟɲɢɬɶ ɡɚɞɚɱɭ ɞɥɹ ɬɪɟɯ ɜɚɪɢɚɧɬɨɜ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ:
ɚ) ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɫɬɟɧɤɢ ɟɦɤɨɫɬɢ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɪɚɜɧɨɣ T1 ;
ɛ) ɜɧɭɬɪɟɧɧɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɪɚɜɟɧ D1 ;
ɜ) ɬɟɦɩɟɪɚɬɭɪɚ ɜɧɭɬɪɟɧɧɟɣ ɫɬɟɧɤɢ – T1 , ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɩɥɨɢɡɨɥɹɰɢɢ – T2 , ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɟɬɚɥɥɚ ɢ ɢɡɨɥɹɰɢɢ ɢɦɟɟɬɫɹ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɜɟɥɢɱɢɧɨɣ RT ɜɫɥɟɞɫɬɜɢɟ ɧɟɢɞɟɚɥɶɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɤɨɧɬɚɤɬɚ (ɫɦ. ɪɚɡɞɟɥ 2.10).
156
ɑȺɋɌɖ 7
ɇ ɟ ɫ ɬ ɚ ɰ ɢ ɨ ɧ ɚ ɪ ɧ ɵ ɟ ɡ ɚ ɞ ɚ ɱ ɢ ɬ ɟɩ ɥ ɨ ɩ ɪ ɨ ɜ ɨ ɞ ɧ ɨ ɫ ɬ ɢ
7.1. Ɉɛɡɨɪ ɡɚɞɚɱ ɢ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ
ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ⱦɥɹ ɞɚɥɶɧɟɣɲɟɝɨ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɧɟɤɨɬɨɪɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɦɟɬɨɞɚɯ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɋɟɱɶ ɛɭɞɟɬ ɢɞɬɢ ɨ ɬɨɱɧɵɯ ɦɟɬɨɞɚɯ. ɉɨɱɬɢ ɜɫɟ (ɢɥɢ ɜɫɟ) ɢɡ ɢɡɜɟɫɬɧɵɯ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ
ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɩɪɢɦɟɧɢɦɵ ɤ ɡɚɞɚɱɚɦ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɚ ɦɟɬɨɞɵ, ɯɨɪɨɲɨ ɪɚɡɪɚɛɨɬɚɧɧɵɟ ɞɥɹ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɜɟɫɶɦɚ ɩɨɥɟɡɧɵ ɜ ɞɪɭɝɢɯ ɨɛɥɚɫɬɹɯ. ɇɚɢɛɨɥɟɟ ɢɡɜɟɫɬɧɵɦɢ
ɹɜɥɹɸɬɫɹ ɦɟɬɨɞ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ, ɦɟɬɨɞ Ⱦɸɚɦɟɥɹ-ɇɟɣɦɚɧɚ, ɦɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ (ɦɟɬɨɞ Ɏɭɪɶɟ), ɦɟɬɨɞɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɜ ɤɨɧɟɱɧɵɯ ɢ ɛɟɫɤɨɧɟɱɧɵɯ ɩɪɟɞɟɥɚɯ. ɂɡɭɱɟɧɢɟ ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ
ɡɚɞɚɱ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ ɬɪɟɛɭɟɬ ɦɧɨɝɨ ɜɪɟɦɟɧɢ ɢ ɞɨɥɠɧɨ ɩɪɨɯɨɞɢɬɶ ɜ ɫɩɟɰɢɚɥɶɧɵɯ ɤɭɪɫɚɯ ɥɟɤɰɢɣ ɢ ɧɚ ɩɪɚɤɬɢɱɟɫɤɢɯ ɡɚɧɹɬɢɹɯ. Ɍɟɦ ɧɟ
ɦɟɧɟɟ, ɩɪɢɦɟɪɵ ɪɟɲɟɧɢɹ ɱɚɫɬɧɵɯ ɡɚɞɚɱ ɦɨɝɭɬ ɨɤɚɡɚɬɶɫɹ ɜɟɫɶɦɚ ɩɨɥɟɡɧɵɦɢ.
Ɂɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɦɨɠɧɨ ɤɥɚɫɫɢɮɢɰɢɪɨɜɚɬɶ ɩɨ ɪɚɡɧɵɦ ɩɪɢɡɧɚɤɚɦ. ȼɨɡɦɨɠɧɵɣ ɜɚɪɢɚɧɬ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɩɪɟɞɫɬɚɜɥɟɧ ɧɚ ɪɢɫ. 7.1.
Ɋɢɫ. 7.1.
Ⱦɨ ɫɢɯ ɩɨɪ ɦɵ ɡɚɧɢɦɚɥɢɫɶ ɚɧɚɥɢɡɨɦ ɬɨɥɶɤɨ ɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɜ ɤɨɬɨɪɵɯ ɩɨɥɟ ɬɟɦɩɟɪɚɬɭɪɵ ɧɟ ɡɚɜɢɫɟɥɨ ɨɬ ɜɪɟɦɟɧɢ.
ȼ ɷɬɨɦ ɢ ɫɥɟɞɭɸɳɟɦ ɪɚɡɞɟɥɚɯ ɦɵ ɛɭɞɟɦ ɢɡɭɱɚɬɶ, ɜ ɨɫɧɨɜɧɨɦ, ɧɟɫɬɚɰɢɨ157
ɧɚɪɧɵɟ ɡɚɞɚɱɢ ɪɚɡɥɢɱɧɨɣ ɫɬɟɩɟɧɢ ɫɥɨɠɧɨɫɬɢ ɢ ɞɥɹ ɬɟɥ ɪɚɡɧɨɣ ɝɟɨɦɟɬɪɢɢ, ɪɟɲɚɹ ɤɨɬɨɪɵɟ ɦɵ ɛɭɞɟɦ ɡɧɚɤɨɦɢɬɶɫɹ ɫ ɧɟɤɨɬɨɪɵɦɢ ɬɨɱɧɵɦɢ ɢ
ɩɪɢɛɥɢɠɟɧɧɵɦɢ ɚɧɚɥɢɬɢɱɟɫɤɢɦɢ ɦɟɬɨɞɚɦɢ.
ȼ ɷɬɨɦ ɪɚɡɞɟɥɟ, ɤɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɢɯ, ɢɡɥɨɠɟɧɢɟ ɜɟɞɟɬɫɹ ɞɨɫɬɚɬɨɱɧɨ
ɩɪɨɫɬɵɦ ɹɡɵɤɨɦ, ɛɟɡ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɫɩɟɰɢɚɥɶɧɵɯ ɬɟɨɪɟɦ, ɤɨɬɨɪɵɟ
ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɥɸɛɨɦ ɭɱɟɛɧɢɤɟ ɩɨ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɟ.
Ⱦɨɫɬɭɩɧɵɦ ɹɡɵɤɨɦ ɨɫɧɨɜɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɦɧɨɝɨɱɢɫɥɟɧɧɵɦɢ ɩɪɢɦɟɪɚɦɢ ɩɨɞɪɨɛɧɨ ɨɩɢɫɚɧɵ ɜ ɭɱɟɛɧɨɣ ɥɢɬɟɪɚɬɭɪɟ, ɞɚɥɟɤɨ ɧɟ ɩɨɥɧɵɣ ɫɩɢɫɨɤ ɤɨɬɨɪɨɣ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɤɨɧɰɟ ɩɨɫɨɛɢɹ.
7.2. ɉɪɨɫɬɟɣɲɢɟ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ
ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɂɬɚɤ, ɞɨ ɫɢɯ ɩɨɪ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɥɢ ɫɢɫɬɟɦɵ, ɜ ɤɨɬɨɪɵɯ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɨɰɟɫɫɵ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜɨ ɜɪɟɦɟɧɢ ɛɵɥɢ
ɡɚɜɟɪɲɟɧɵ. ɇɨ ɞɥɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɫ ɡɚɞɚɧɧɨɣ
ɬɨɱɧɨɫɬɶɸ ɬɪɟɛɭɟɬɫɹ ɧɟɤɨɬɨɪɨɟ ɜɪɟɦɹ. Ʉɪɨɦɟ ɬɨɝɨ, ɛɨɥɶɲɢɧɫɬɜɨ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ
ɢɫɬɨɱɧɢɤɚɦɢ ɹɜɥɹɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ ɧɟɫɬɚɰɢɨɧɚɪɧɵɦɢ. ɇɚ ɚɧɚɥɢɡɟ ɧɟɤɨɬɨɪɵɯ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɨɫɬɚɧɨɜɢɦɫɹ, ɧɟ ɩɪɢɛɟɝɚɹ ɤ ɢɡɭɱɟɧɢɸ
ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɨɫɧɨɜ ɬɨɱɧɵɯ ɚɧɚɥɢɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ.
ɉɭɫɬɶ ɢɦɟɟɬɫɹ ɬɟɥɨ, ɨɛɥɚɞɚɸɳɟɟ
ɛɨɥɶɲɨɣ («ɛɟɫɤɨɧɟɱɧɨɣ») ɬɟɩɥɨɩɪɨɜɨɞTe ɧɨɫɬɶɸ. ȿɫɥɢ ɩɪɢ ɷɬɨɦ ɬɟɥɨ ɨɛɥɚɞɚɟɬ ɧɢɡɤɢɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɨɛɦɟɧɚ, ɬɨ ɬɟɩV
&
n
ɥɨɜɨɣ ɩɨɬɨɤ ɤ ɬɟɥɭ ɢɥɢ ɨɬ ɬɟɥɚ ɨɩɪɟɞɟɥɹT
ɟɬɫɹ, ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ, ɤɨɧɜɟɤɬɢɜɧɵɦ ɫɨc ,U,O
ɩɪɨɬɢɜɥɟɧɢɟɦ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜ ɬɟɥɟ ɫɭɳɟA
ɫɬɜɭɸɬ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɵɟ ɝɪɚɞɢɟɧɬɵ ɬɟɦɩɟɪɚɬɭɪɵ, ɥɢɛɨ ɨɧɢ ɫɨɜɫɟɦ ɨɬɫɭɬɫɬɜɭɸɬ.
ɇɟɤɨɬɨɪɵɟ ɪɟɚɥɶɧɵɟ ɬɟɥɚ ɦɚɥɵɯ ɪɚɡɦɟɪɨɜ
Ɋɢɫ. 7.2. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
ɢ ɛɨɥɶɲɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɭɞɨɜɥɟɬɜɨɮɨɪɦɭɥɢɪɨɜɤɟ ɩɪɨɫɬɟɣɲɟɣ
ɪɹɸɬ ɬɚɤɢɦ ɭɫɥɨɜɢɹɦ. Ɍɚɤɢɦ ɭɫɥɨɜɢɹɦ,
ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɧɚɩɪɢɦɟɪ, ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɨɛɪɚɡɰɵ, ɩɨɞɜɟɪɝɚɟɦɵɟ ɜ ɩɟɱɢ ɢɡɨɬɟɪɦɢɱɟɫɤɨɦɭ ɨɬɠɢɝɭ, ɫɩɟɤɚɟɦɵɟ ɨɛɪɚɡɰɵ, ɤɨɝɞɚ
ɫɭɳɟɫɬɜɭɟɬ ɬɪɟɛɨɜɚɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɩɪɨɝɪɟɜɚ. Ⱦɥɹ ɨɩɢɫɚɧɢɹ ɷɜɨɥɸɰɢɢ
ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɛɪɚɡɰɟ ɜɦɟɫɬɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
wT
(7.1)
cU
O'T
wt
158
ɫ ɝɪɚɧɢɱɧɵɦɢ ɢ ɧɚɱɚɥɶɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜɢɞɚ
§ wT ·
(7.2)
¨
D T Te ,
¸
© wn ¹ A
t 0 : T T0 ,
(7.3)
ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ
wT
(7.4)
cU V
D A T Te ,
wt
ɤɨɬɨɪɨɟ ɩɨɥɭɱɚɟɬɫɹ ɢɡ (7.1) ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ ɩɨ ɨɛɴɟɦɭ V ɫ ɭɱɟɬɨɦ
ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ (7.2) ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ A .
Ɍɟɦɩɟɪɚɬɭɪɭ ɨɤɪɭɠɚɸɳɟɝɨ ɩɨɬɨɤɚ ɫɱɢɬɚɟɦ ɩɨɫɬɨɹɧɧɨɣ.
ȼɜɟɞɟɦ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɟɪɟɦɟɧɧɵɟ
T Te
t
,W
,
(7.5)
T
t
T0 Te
ɝɞɟ t L2 a , a O cU , L – ɯɚɪɚɤɬɟɪɧɵɣ ɥɢɧɟɣɧɵɣ ɪɚɡɦɟɪ ɬɟɥɚ, ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɟɝɨ ɨɛɴɟɦɚ ɤ ɩɥɨɳɚɞɢ ɟɝɨ ɜɧɟɲɧɟɣ
ɩɨɜɟɪɯɧɨɫɬɢ:
Ⱦɥɹ ɬɟɥ ɤɚɧɨɧɢɱɟɫɤɨɣ ɮɨɪɦɵ ɯɚɪɚɤɬɟɪɧɵɣ ɪɚɡɦɟɪ ɥɟɝɤɨ ɜɵɱɢɫɥɹɟɬɫɹ. Ɍɚɤ, ɞɥɹ ɲɚɪɚ ɪɚɞɢɭɫɚ r0 ɢɦɟɟɦ
4 3 S r03
L
4S r02
ɞɥɹ ɰɢɥɢɧɞɪɚ ɪɚɞɢɭɫɚ r0 ɢ ɞɥɢɧɵ l !! r0 –
Sr02 l
2Sr0l
L
r0
;
3
r0
;
2
ɞɥɹ ɤɭɛɚ ɫ ɪɚɡɦɟɪɨɦ ɪɟɛɪɚ l –
l3
l
.
6l 2 6
Ȼɟɡɪɚɡɦɟɪɧɨɟ ɜɪɟɦɹ ɜ (7.5) ɟɫɬɶ ɧɟ ɱɬɨ ɢɧɨɟ, ɤɚɤ ɱɢɫɥɨ Ɏɭɪɶɟ.
ɍɫɥɨɜɢɟ ɩɪɢɦɟɧɢɦɨɫɬɢ ɩɨɫɬɚɧɨɜɤɢ ɡɚɞɚɱɢ (7.4), (7.3) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɬɚɤ
L
1 ,
Ot* cU L
159
ɝɞɟ t – ɧɟɤɨɬɨɪɨɟ ɯɚɪɚɤɬɟɪɧɨɟ ɜɪɟɦɹ (ɧɚɩɪɢɦɟɪ, ɜɪɟɦɹ ɜɵɯɨɞɚ ɧɚ ɧɭɠɧɵɣ ɪɟɠɢɦ).
ɂɫɩɨɥɶɡɭɹ (7.5), ɜɦɟɫɬɨ (7.4) ɢ (7.3) ɧɚɣɞɟɦ
dT
D At DL
T T B iT ,
dW
cU V
O
(7.6)
W 0 : T 1,
DL
– ɤɪɢɬɟɪɢɣ Ȼɢɨ.
ɝɞɟ B i
O
ɗɬɚ ɡɚɞɚɱɚ ɥɟɝɤɨ ɢɧɬɟɝɪɢɪɭɟɬɫɹ:
dT
Bi d W ɢɥɢ T exp B i ˜ F o . (7.7)
T
ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɛɟɡɪɚɡɦɟɪɧɵɯ ɤɪɢɬɟɪɢɟɜ ɩɨɡɜɨɥɹɟɬ ɩɪɟɞɫɬɚɜɢɬɶ ɪɟɡɭɥɶɬɚɬ
(7.7) ɞɥɹ ɜɫɟɯ ɬɟɥ ɫ ɛɟɫɤɨɧɟɱɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɨɞɧɢɦ ɭɧɢɜɟɪɫɚɥɶɧɵɦ Ɋɢɫ. 7.3. Ɋɟɲɟɧɢɟ ɩɪɨɫɬɟɣɝɪɚɮɢɤɨɦ ɞɥɹ ɪɚɡɧɵɯ ɤɪɢɬɟɪɢɟɜ Ȼɢɨ ɲɟɣ ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ ɡɚɞɚɱɢ
(ɪɢɫ. 7.3).
ɉɪɢɦɟɪ. ɉɪɢ ɢɡɦɟɪɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɪɦɨɦɟɬɪɨɦ ɜɚɠɧɨ ɡɧɚɬɶ, ɧɚ
ɫɤɨɥɶɤɨ ɛɵɫɬɪɨ ɬɟɪɦɨɦɟɬɪ ɪɟɚɝɢɪɭɟɬ ɧɚ ɢɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ.
ɉɨɥɭɩɟɪɢɨɞɨɦ ɧɚɡɵɜɚɸɬ ɢɧɬɟɪɜɚɥ ɜɪɟɦɟɧɢ, ɜ ɩɪɟɞɟɥɚɯ ɤɨɬɨɪɨɝɨ ɧɚɱɚɥɶɧɚɹ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɢɫɬɢɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ɢ ɩɨɤɚɡɚɧɢɹɦɢ ɬɟɪɦɨɦɟɬɪɚ ɫɨɤɪɚɳɚɟɬɫɹ ɧɚɩɨɥɨɜɢɧɭ ɩɨɫɥɟ ɜɧɟɡɚɩɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɢɫɬɢɧɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɵ. Ɉɩɪɟɞɟɥɢɦ ɷɬɨɬ ɩɨɥɭɩɟɪɢɨɞ ɞɥɹ ɪɬɭɬɧɨɝɨ ɬɟɪɦɨɦɟɬɪɚ,
ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɩɨɬɨɤɟ ɜɨɡɞɭɯɚ. ɉɭɫɬɶ ɪɬɭɬɧɵɣ ɲɚɪɢɤ ɢɦɟɟɬ ɮɨɪɦɭ ɰɢɥɢɧɞɪɚ
r 0 ɦɦ.
Ʉɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɪɬɭɬɢ
O 7,45 ɤɤɚɥ/(ɦ ɱɚɫ ɝɪɚɞ). Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ a 0,0166 ɦ2/ɱɚɫ. Ɍɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɬɨɧɤɨɣ ɫɬɟɤɥɹɧɧɨɣ
ɫɬɟɧɤɢ ɩɪɟɧɟɛɪɟɝɚɟɦ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɩɨɬɨɤɨɦ ɜɨɡɞɭɯɚ
ɩɪɢɦɟɦ ɪɚɜɧɵɦ D 50 ɤɤɚɥ/(ɦ2·ɱɚɫ·ɝɪɚɞ).
Ɋɟɲɟɧɢɟ. ɇɚɯɨɞɢɦ ɩɚɪɚɦɟɬɪ ɬɟɩɥɨɨɛɦɟɧɚ:
Bi
DL
O
50 ˜ 0 ,003
7 ,45 ˜ 2
0 ,01 .
Ɉɬɧɨɲɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪ T ɢ T0 ɛɭɞɟɬ ɪɚɜɧɨ 0,5 , ɤɨɝɞɚ ɱɢɫɥɟɧɧɨɟ
ɡɧɚɱɟɧɢɟ ɩɨɤɚɡɚɬɟɥɹ ɫɬɟɩɟɧɢ ɜ (7.7) ɫɬɚɧɟɬ ɪɚɜɧɵɦ 0,693 . Ɍɨɝɞɚ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɥɭɩɟɪɢɨɞɚ ɢɦɟɟɦ ɫɨɨɬɧɨɲɟɧɢɹ
Bi ˜ Fo
at DL
L2 O
0 ,693 ɢɥɢ
160
at0 ,5
L2
0 ,693
,
0 ,01
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
69 ,3 ˜ 9 ˜10 6
t0 ,5
33,8 ɫ.
0 ,0166 ˜ 4
Ɍ.ɟ., ɦɨɠɧɨ ɩɨɥɚɝɚɬɶ, ɱɬɨ ɩɨɤɚɡɚɧɢɹ ɬɟɪɦɨɦɟɬɪɚ ɩɪɚɜɢɥɶɧɨ ɨɬɪɚɠɚɸɬ ɢɫɬɢɧɧɨɟ ɢɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɟɫɥɢ ɷɬɨ ɢɡɦɟɧɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ
ɛɨɥɟɟ ɦɟɞɥɟɧɧɵɦ ɬɟɦɩɨɦ.
ɉɭɫɬɶ ɬɟɩɟɪɶ ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɝɨ ɩɨɬɨɤɚ – ɥɢɧɟɣɧɚɹ ɮɭɧɤɰɢɹ
ɜɪɟɦɟɧɢ:
Te Bt T0 .
Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ ɩɪɢɦɟɬ ɜɢɞ
wT
cU V
D AT D A Bt T0 ,
wt
ɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜɵɩɨɥɧɹɟɬɫɹ ɬɨ ɠɟ ɭɫɥɨɜɢɟ (7.3)
§ T T0 ·
ȼ ɩɟɪɟɦɟɧɧɵɯ T ¨
¸ ɢ W , ɝɞɟ T ɩɨɤɚ ɧɟ ɨɩɪɟɞɟɥɟɧɚ, ɢɦɟɟɦ
© T T0 ¹
ɡɚɞɚɱɭ
dT
(7.8)
B iT Bi ˜E ˜ W ,
dW
W 0: T 0,
Bt .
ɝɞɟ E
T T0
Ɉɛɳɟɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (7.8) (ɤɚɤ ɫɭɦɦɚ ɨɛɳɟɝɨ
ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ) ɢɦɟɟɬ ɜɢɞ
T E W Bi 1 C1e x p B i ˜ W .
4
3
4e
2
2,5
1
Bi=0,5
1
0
0
1
2
3
Fo
Ɋɢɫ. 7.4. Ɍɟɦɩɟɪɚɬɭɪɚ ɬɟɥɚ ɜ
ɫɪɟɞɟ, ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɣ
ɦɟɧɹɟɬɫɹ ɩɨ ɥɢɧɟɣɧɨɦɭ ɡɚɤɨɧɭ
ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ, ɧɚɯɨɞɢɦ
ɩɨɫɬɨɹɧɧɭɸ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
C1 E Bi 1 . Ɍɚɤ ɤɚɤ W { Fo , ɬɨ ɡɚɩɢɲɟɦ
ɨɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ
E
ª1 e xp B i ˜ Fo º¼ . (7.9)
T E Fo Bi ¬
ɉɪɢɧɢɦɚɹ E 1 , ɨɩɪɟɞɟɥɢɦ ɦɚɫɲɬɚɛɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ
BL2
T T0 .
a
161
ɂɡ (7.9) ɫɥɟɞɭɟɬ, ɱɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɞɟɬɚɥɢ ɜɫɟɝɞɚ ɨɬɫɬɚɟɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɫɪɟɞɵ, ɦɟɧɹɸɳɟɣɫɹ ɩɨ ɡɚɤɨɧɭ
Te T0
Te
E Fo .
T* T0
ɉɪɢ W Fo o f , ɷɬɨ ɨɬɫɬɚɜɚɧɢɟ ɫɬɚɧɨɜɢɬɫɹ ɩɨɫɬɨɹɧɧɵɦ (ɪɢɫ. 7.4).
7.3. ɉɪɨɫɬɟɣɲɢɟ ɧɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ
ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ
ɷɥɟɦɟɧɬɵ ɨɩɟɪɚɰɢɨɧɧɨɝɨ ɦɟɬɨɞɚ
7.3.1. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ
ɩɟɪɜɨɝɨ ɪɨɞɚ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɯɨɠɞɟɧɢɢ ɩɨɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɦ ɬɟɥɟ (ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɟ), ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɤɨɬɨɪɨɝɨ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɚ Ts (ɪɢɫ. 7.5). ɗɬɨ ɭɫɥɨɜɢɟ ɜɵɩɨɥɧɹɟɬɫɹ, ɟɫɥɢ ɪɚɡɦɟɪ
ɬɟɥɚ H ɦɧɨɝɨ ɛɨɥɶɲɟ ɡɨɧɵ ɩɪɨɝɪɟɜɚ, ɮɨɪɦɢɪɭɸɳɟɣɫɹ ɡɚ ɜɪɟɦɹ ɧɚɛɥɸɞɟɧɢɹ
H !! O t cU .
Ɋɢɫ. 7.5. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
ɮɨɪɦɭɥɢɪɨɜɤɟ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ
Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɟɥɚ
c,U , O ɫɱɢɬɚɸɬɫɹ ɩɨɫɬɨɹɧɧɵɦɢ.
ɇɚ ɛɟɫɤɨɧɟɱɧɨɦ (ɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ
ɫɦɵɫɥɟ) ɭɞɚɥɟɧɢɢ ɨɬ ɡɨɧɵ ɩɪɨɝɪɟɜɚ ɢɫɬɨɱɧɢɤɢ ɢ ɫɬɨɤɢ ɬɟɩɥɚ ɨɬɫɭɬɫɬɜɭɸɬ ɢɥɢ
ɬɟɦɩɟɪɚɬɭɪɚ ɧɟ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɧɚɱɚɥɶɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ. ȼ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɨɬɧɨɲɟɧɢɢ ɷɬɢ ɭɫɥɨɜɢɹ ɷɤɜɢɜɚɥɟɧɬɧɵ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɬɚɤɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ
wT
w § wT ·
(7.10)
cU
¨O
¸;
wt wx © wx ¹
x 0 : T Ts ;
x o f : T T0 ;
t 0 : T T0 .
ɗɬɨ ɢ ɟɫɬɶ ɩɟɪɜɚɹ ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɧɟ ɡɚɜɢɫɹɬ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ.
162
ɉɟɪɟɣɞɟɦ ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ
T T0
t
x
,W
,[
.
T
x
Ts T0
t
ȿɫɥɢ ɫ ɦɚɫɲɬɚɛɨɦ ɜɪɟɦɟɧɢ ɜɫɟ ɩɨɧɹɬɧɨ, ɢ ɛɟɡɪɚɡɦɟɪɧɨɟ ɜɪɟɦɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɱɢɫɥɨ Ɏɭɪɶɟ, ɬɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɣ ɦɚɫɲɬɚɛ ɩɨɤɚ ɧɟ ɨɩɪɟɞɟɥɟɧ. ɉɪɢ ɩɟɪɟɯɨɞɟ ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ ɜɦɟɫɬɨ ɭɪɚɜɧɟɧɢɹ
(7.10) ɧɚɯɨɞɢɦ
wT t O w 2T
.
wW cU x2 w[ 2
ɂɫɩɨɥɶɡɭɹ ɞɨɩɭɫɬɢɦɵɣ ɩɪɨɢɡɜɨɥ ɜ ɜɵɛɨɪɟ ɩɚɪɚɦɟɬɪɨɜ ɢ ɩɪɢɧɢɦɚɹ
x L , ɩɪɢɪɚɜɧɹɟɦ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɢ ɜɬɨɪɨɣ ɩɪɨɢɡɜɨɞɧɨɣ ɩɨ ɤɨɨɪɞɢɧɚɬɟ ɟɞɢɧɢɰɟ. ɗɬɨ ɞɚɫɬ ɧɚɦ ɯɚɪɚɤɬɟɪɧɵɣ ɬɟɩɥɨɜɨɣ ɦɚɫɲɬɚɛ
L
at .
ȼ ɪɟɡɭɥɶɬɚɬɟ ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ ɡɚɞɚɱɚ ɩɪɢɦɟɬ ɜɢɞ
wT w 2T
;
wW w[ 2
[ 0 : T 1;
[ o f : T 0;
W 0: T 0.
ɗɬɚ ɡɚɞɚɱɚ ɫɨɜɫɟɦ ɧɟ ɫɨɞɟɪɠɢɬ ɩɚɪɚɦɟɬɪɨɜ (ɤɪɨɦɟ ɱɢɫɥɚ Ɏɭɪɶɟ, ɹɜɥɹɸɳɟɝɨɫɹ ɛɟɡɪɚɡɦɟɪɧɵɦ ɜɪɟɦɟɧɟɦ). ȿɟ ɬɨɱɧɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ
§ [ ·
T e rfc ¨
(7.11)
¸,
2
W
©
¹
ɫɨɞɟɪɠɢɬ ɮɭɧɤɰɢɸ erf c z f
2
exp t 2 d t – ɢɧɬɟɝɪɚɥ ɜɟɪɨɹɬɧɨɫɬɢ,
³
Sz
ɯɨɪɨɲɨ ɡɧɚɤɨɦɭɸ ɮɭɧɤɰɢɸ ɬɟɦ, ɤɬɨ ɡɚɧɢɦɚɥɫɹ ɨɛɪɚɛɨɬɤɨɣ ɞɚɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚ. ɗɬɚ ɮɭɧɤɰɢɹ ɜɯɨɞɢɬ ɜ ɱɢɫɥɨ ɫɬɚɧɞɚɪɬɧɵɯ ɮɭɧɤɰɢɣ ɜɨ ɦɧɨɝɢɯ ɹɡɵɤɚɯ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ ɢ ɢɦɟɟɬ ɫɜɨɣɫɬɜɚ:
erfc 0 1 ; erfc f 0 .
ɉɨɥɟɡɧɨ ɡɧɚɬɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɟ ɪɚɡɥɨɠɟɧɢɹ ɷɬɨɣ ɮɭɧɤɰɢɢ:
2
z 1 : erfc z |
z;
S
z !!1:
erfc z |
163
.
exp z 2
z S
(7.12)
ɂɬɚɤ, ɞɥɹ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɜɟɳɟɫɬɜ ɪɟɲɟɧɢɟ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ (7.11). ɗɬɨ ɪɟɲɟɧɢɟ ɢɡɨɛɪɚɠɟɧɨ
ɧɚ ɪɢɫ. 7.6.
ȼ ɮɢɡɢɱɟɫɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɢɡ
4
Fo=1. - 0,1
(7.11) ɢɦɟɟɦ
0,9
2. - 0,2
3. - 0.5
§ x ·
T T0 Ts T0 erfc ¨
.
(7.13)
4. - 1.0
¸
0,6
5. - 3.0
© 2 at ¹
5
4
Ɍɚɤɢɟ ɡɚɞɚɱɢ ɭɞɨɛɧɨ ɪɟɲɚɬɶ ɨɩɟ0,3
2 3
1
ɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ ɢɥɢ ɦɟɬɨɞɨɦ ɢɧ0,0
ɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨ Ʌɚɩɥɚ0,0
1,5
3,0
4,5
ɫɭ. ɗɬɨ – ɨɞɢɧ ɢɡ ɫɚɦɵɯ ɷɮɮɟɤɬɢɜɧɵɯ
[
ɦɟɬɨɞɨɜ ɪɟɲɟɧɢɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɊɢɫ. 7.6. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
ɂɧɬɟɝɪɚɥɶɧɵɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ
ɩɨ Ʌɚɩɥɚɫɭ ɧɚɡɵɜɚɸɬ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
ɜɢɞɚ
F p
ɪɚɬɭɪɵ ɜ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɟ ɩɪɢ
ɡɚɞɚɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɧɚ
ɩɨɜɟɪɯɧɨɫɬɢ (ɪɟɲɟɧɢɟ ɩɟɪɜɨɣ
ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ)
f
³ f W e x p pW d W ,
(7.14)
0
ɩɟɪɟɜɨɞɹɳɟɟ ɮɭɧɤɰɢɸ f ɞɟɣɫɬɜɢɬɟɥɶɧɨɣ ɩɟɪɟɦɟɧɧɨɣ W ɜ ɮɭɧɤɰɢɸ F
ɤɨɦɩɥɟɤɫɧɨɣ ɩɟɪɟɦɟɧɧɨɣ p . Ƚɨɜɨɪɹɬ, ɱɬɨ ɮɭɧɤɰɢɹ f W ɹɜɥɹɟɬɫɹ ɨɪɢ-
ɝɢɧɚɥɨɦ, ɚ ɮɭɧɤɰɢɹ F p – ɟɟ ɢɡɨɛɪɚɠɟɧɢɟɦ. ȼ ɥɢɬɟɪɚɬɭɪɟ ɦɨɠɧɨ
ɜɫɬɪɟɬɢɬɶ ɪɚɡɧɵɟ ɨɛɨɡɧɚɱɟɧɢɹ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɫɢɦɜɨɥɵ. ɇɚɩɪɢɦɟɪ,
ɦɨɠɧɨ ɧɚɩɢɫɚɬɶ ɬɚɤ:
W o p , f W o F p ,
ɚ ɦɨɠɧɨ ɢ ɬɚɤ
W y p , f W y F p .
Ⱦɥɹ ɢɡɨɛɪɚɠɟɧɢɹ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɢ ɨɛɨɡɧɚɱɟɧɢɟ f p .
ɂɞɟɹ ɦɟɬɨɞɚ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ.
ɂɫɩɨɥɶɡɭɹ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ (7.14) ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɟɦɭ ɧɚɛɨɪ ɬɟɨɪɟɦ ɢ ɩɪɚɜɢɥ, ɫɜɟɫɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɜ ɱɚɫɬɧɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɤ ɪɟɲɟɧɢɸ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɡɚɞɚɱɢ, ɚ ɩɨɬɨɦ, ɢɫɩɨɥɶɡɭɹ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
f W
1
2S i
V if
³
F p e pW d p ,
V if
ɩɟɪɟɣɬɢ ɢɡ ɩɪɨɫɬɪɚɧɫɬɜɚ ɢɡɨɛɪɚɠɟɧɢɣ ɜ ɩɪɨɫɬɪɚɧɫɬɜɨ ɨɪɢɝɢɧɚɥɨɜ.
164
Ⱦɥɹ ɪɹɞɚ ɡɚɞɚɱ ɬɚɤɨɣ ɩɭɬɶ ɛɵɥ ɛɵ ɜɟɫɶɦɚ ɫɥɨɠɧɵɦ, ɟɫɥɢ ɛɵ ɧɟ
ɢɦɟɸɳɢɟɫɹ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ
ɢ ɬɟɨɪɟɦɵ, ɫɭɳɟɫɬɜɟɧɧɨ ɭɩɪɨɳɚɸɳɢɟ ɯɨɞ ɪɟɲɟɧɢɹ21. ɋɜɨɞɤɚ ɨɫɧɨɜɧɵɯ
ɩɪɚɜɢɥ ɨɩɟɪɚɰɢɨɧɧɨɝɨ ɢɫɱɢɫɥɟɧɢɹ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɩɪɢɥɨɠɟɧɢɢ 1. Ⱦɥɹ
ɛɨɥɟɟ ɩɨɞɪɨɛɧɨɝɨ ɡɧɚɤɨɦɫɬɜɚ ɫ ɨɩɟɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ ɢ ɢɧɵɦɢ ɦɟɬɨɞɚɦɢ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɭɸ ɥɢɬɟɪɚɬɭɪɭ ɢɡ ɫɩɢɫɤɚ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɝɨ ɜ ɤɨɧɰɟ ɩɨɫɨɛɢɹ.
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɩɟɪɟɣɬɢ ɤ ɨɪɢɝɢɧɚɥɚɦ ɜɫɟ ɠɟ ɧɟ ɭɞɚɟɬɫɹ. ɂ ɬɨɝɞɚ ɧɚ ɩɨɦɨɳɶ ɩɪɢɯɨɞɹɬ ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɩɪɢɛɥɢɠɟɧɧɵɟ ɦɟɬɨɞɵ ɢ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɟ ɪɚɡɥɨɠɟɧɢɹ, ɩɨɡɜɨɥɹɸɳɢɟ ɩɨɫɬɪɨɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ ɢɡɭɱɚɟɦɨɣ ɡɚɞɚɱɢ, ɱɬɨ ɨɱɟɧɶ ɜɚɠɧɨ ɞɥɹ ɢɧɠɟɧɟɪɧɵɯ ɩɪɢɥɨɠɟɧɢɣ.
ɇɟɤɨɬɨɪɵɟ ɩɪɨɫɬɵɟ ɩɪɢɦɟɪɵ ɦɵ ɞɚɥɟɟ ɪɚɫɫɦɨɬɪɢɦ.
Ɉɩɢɫɚɧɧɚɹ ɜɵɲɟ ɤɪɚɟɜɚɹ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɫɚɦɨɣ ɩɪɨɫɬɨɣ. ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ ɨɧɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
d 2T
pT T 0 , [ ;
(7.15)
2
d[
1
[ 0:
;
T
p
[of:
T 0.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (7.15) ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
T Ae k1[ Be k 2[ ,
ɝɞɟ ki , i 1, 2 – ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜɧɟɧɢɹ
(7.16)
k2 p 0;
k1 p ; k 2
p.
ɂɫɩɨɥɶɡɭɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɧɚɣɞɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
1
, B 0.
A
p
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɢɦɟɟɦ
T
1
e x p p[ .
p
21
(7.17)
Ȼɟɣɬɦɟɧ ɇ.Ƚ., ɗɪɞɟɣɢ Ⱥ. Ɍɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɜ ɞɜɭɯ ɬɨɦɚɯ.
ɋɩɪɚɜɨɱɧɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɛɢɛɥɢɨɬɟɤɚ. Ɍ.1. ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ɏɭɪɶɟ, Ʌɚɩɥɚɫɚ, Ɇɟɥɥɢɧɚ. Ɇ.1969 ɝ. Ɍ.2. ɉɪɟɨɛɪɚɡɨɜɚɧɢɹ Ȼɟɫɫɟɥɹ. ɂɧɬɟɝɪɚɥɵ ɨɬ ɫɩɟɰɢɚɥɶɧɵɯ ɮɭɧɤɰɢɣ.
Ɇ., 1970.
165
ɂɫɩɨɥɶɡɭɹ ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɩɪɢɥɨɠɟɧɢɟ 2),
ɧɚɯɨɞɢɦ ɪɟɲɟɧɢɟ (7.11).
Ⱥɧɚɥɨɝɢɱɧɵɦ ɨɛɪɚɡɨɦ ɦɨɠɧɨ ɧɚɣɬɢ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɞɪɭɝɢɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɧɚ ɫɜɨɛɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ.
7.3.2. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɜɬɨɪɨɝɨ ɪɨɞɚ
ɉɭɫɬɶ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɩɚɞɚɟɬ ɩɨɬɨɤ ɬɟɩɥɚ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɵ q0 (ɪɢɫ. 7.7). Ɍɨɝɞɚ ɜ ɡɚɞɚɱɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (7.10) ɭɫɥɨɜɢɟ ɩɨɫɬɨɹɧɫɬɜɚ
ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɦɟɧɹɟɬɫɹ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɟɦ ɜɬɨɪɨɝɨ ɪɨɞɚ
Ɋɢɫ. 7.7. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
wT
ɮɨɪɦɭɥɢɪɨɜɤɟ ɜɬɨɪɨɣ
(7.18)
O
q0 .
ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
wx
ɞɥɹ
ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ
Ⱦɪɭɝɢɟ ɭɫɥɨɜɢɹ ɢ ɭɪɚɜɧɟɧɢɟ ɧɟ ɢɡɦɟɧɹɸɬɫɹ.
ɉɟɪɟɣɞɟɦ ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ T , W , [ , ɝɞɟ ɦɚɫɲɬɚɛɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɤɚ ɧɟ ɨɩɪɟɞɟɥɟɧɚ:
T T0
.
T
T T0
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ
wT w 2T
;
wW w[ 2
q0 a t
wT
;
w[ O T T0 [ o f : T 0;
W 0: T 0.
[ 0: (7.19)
ɉɪɢɧɢɦɚɹ
q 0 at q 0 t
{
{ 1,
O T T0 cUO T T0 ɨɩɪɟɞɟɥɢɦ ɦɚɫɲɬɚɛɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɤɚɤ ɫɪɟɞɧɸɸ ɬɟɦɩɟɪɚɬɭɪɭ, ɞɨ ɤɨɬɨɪɨɣ ɧɚɝɪɟɟɬɫɹ ɫɥɨɣ ɬɨɥɳɢɧɵ x ɡɚ ɜɪɟɦɹ t ɩɨɬɨɤɨɦ ɜɟɥɢɱɢɧɵ q0 .
ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɨɩɹɬɶ ɩɨɥɭɱɢɦ ɡɚɞɚɱɭ ɛɟɡ ɩɚɪɚɦɟɬɪɨɜ, ɧɨ ɫ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɟɦ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɜɬɨɪɨɝɨ ɪɨɞɚ
166
wT
1.
w[
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ W o p ; T o T ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (7.15) ɢ ɭɫɥɨɜɢɹ
dT 1
[ 0:
;
d[ p
[ of : T 0.
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɬɚɤɠɟ ɢɦɟɟɬ ɜɢɞ (7.16).
ɂɫɩɨɥɶɡɭɹ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɧɚɯɨɞɢɦ ɩɨɫɬɨɹɧɧɵɟ A ɢ B
1
A
, B 0
p p
ɢ ɪɟɲɟɧɢɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ
1
T
(7.20)
e x p p[ .
p p
ɂɫɩɨɥɶɡɭɹ ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɉɪɢɥɨɠɟɧɢɟ 2),
ɧɚɯɨɞɢɦ ɨɤɨɧɱɚɬɟɥɶɧɨ ɪɟɲɟɧɢɟ
§ [2 ·
W
§ [ ·
§ [ ·
T 2 Wi erfc ¨
{
2
exp
(7.21)
¨¨ ¸¸ [ erfc ¨
¸
¸.
4
S
W
2
2
W
W
©
¹
©
¹
©
¹
[ 0: 4
0,8
5
3
0,4
4
Fo= 1. - 0,05
2. - 0.2
3. - 0.5
4. - 1.0
5. - 2.0
1,2
1,2
[ 1. - 0,0
2. - 0.1
3. - 0.2
4. - 0.5
5. - 1.0
2
1
3
0,8
4
4
5
0,4
2
0,0 1
0
2
4
0,0
0,0
[
ɚ
0,5
1,0
1,5
Fo
ɛ
Ɋɢɫ. 7.8. ɂɥɥɸɫɬɪɚɰɢɹ ɪɟɲɟɧɢɹ ɜɬɨɪɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɬɟɨɪɢɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ: ɚ) ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
ɜ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɟ ɜ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ; ɛ) ɡɚɜɢɫɢɦɨɫɬɶ
ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ ɧɚ ɪɚɡɥɢɱɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ
Ɋɟɲɟɧɢɟ (7.21), ɫɩɪɚɜɟɞɥɢɜɨɟ ɞɥɹ ɥɸɛɵɯ ɡɧɚɱɟɧɢɣ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ
ɫɜɨɣɫɬɜ ɢ ɥɸɛɨɣ ɜɟɥɢɱɢɧɵ ɜɧɟɲɧɟɝɨ ɩɨɬɨɤɚ ɬɟɩɥɚ, ɢɡɨɛɪɚɠɟɧɨ ɧɚ
ɪɢɫ. 7.8.
167
ȼɢɞɧɨ ɤɚɱɟɫɬɜɟɧɧɨɟ ɨɬɥɢɱɢɟ ɷɬɨɝɨ ɪɟɲɟɧɢɹ ɨɬ ɪɟɲɟɧɢɹ ɩɟɪɜɨɣ
ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (ɪɢɫ. 7.6): ɜ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɟ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ (ɩɨ ɭɫɥɨɜɢɸ), ɚ ɜɬɨɪɨɣ ɤɪɚɟɜɨɣ
ɡɚɞɚɱɟ – ɦɟɧɹɟɬɫɹ ɤɚɤ W :
W
T 0,W 2
.
S
ȼ ɮɢɡɢɱɟɫɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɢɡ (7.21) ɢɦɟɟɦ
§ x2 · x
q 0 °­ t
§ x · °½
(7.22)
T T0 2
exp
erfc
¨
¸
®
¨
¸¾ ,
¨ 4a t ¸
2
cU O °¯ S
a
a
t
©
¹ °¿
©
¹
ɬɚɤ ɱɬɨ
q0
t
T 0 ,t T0 2
.
S
cU O
ɉɭɫɬɶ ɬɟɩɟɪɶ ɩɨɬɨɤ ɬɟɩɥɚ, ɩɚɞɚɸɳɢɣ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ, ɹɜɥɹɟɬɫɹ
ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɜɪɟɦɟɧɢ. Ɍɨɝɞɚ ɜɦɟɫɬɨ (7.18) ɢɦɟɟɦ
wT
O
q0 f t (7.23)
wx
ɢɥɢ ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
wT
f W .
w[
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɚɧɚɥɨɝɢɱɧɵɣ ɩɭɬɶ ɩɪɢɜɟɞɟɬ ɤ ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ
F p
T
e x p p[ ,
(7.24)
p
ɝɞɟ ɮɭɧɤɰɢɹ ɤɨɦɩɥɟɤɫɧɨɣ ɩɟɪɟɦɟɧɧɨɣ
ɮɭɧɤɰɢɢ f W :
F p
ɟɫɬɶ ɢɡɨɛɪɚɠɟɧɢɟ
f W o F p .
ɉɟɪɟɯɨɞɹ ɜ (7.24) ɤ ɨɪɢɝɢɧɚɥɚɦ (ɫɦ. ɩɪɢɥɨɠɟɧɢɟ 1), ɧɚɣɞɟɦ
W
ª
f y
[2 º
(7.25)
e x p «
T [ , W ³
»d y
4
y
W
y
W
¬«
¼»
0
( y – ɩɟɪɟɦɟɧɧɚɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ) ɢɥɢ ɜ ɮɢɡɢɱɟɫɤɢɯ ɩɟɪɟɦɟɧɧɵɯ –
T x ,t T0 q0
t
S cU O ³0
f y
ª cU x 2 º
e x p «
»d y .
O
4
t
y
ty
¬«
¼»
168
(7.26)
Ⱦɥɹ ɨɞɢɧɨɱɧɨɝɨ ɢɦɩɭɥɶɫɚ ɢɦɟɟɦ
­1, 0 d t d t i ,
f t K t i t ®
t ! ti ,
¯0 ,
ɝɞɟ K z – ɟɞɢɧɢɱɧɚɹ ɮɭɧɤɰɢɹ ɏɟɜɢɫɚɣɞɚ, ɨɬɥɢɱɧɚɹ ɨɬ ɧɭɥɹ ɬɨɥɶɤɨ ɞɥɹ
ɩɨɥɨɠɢɬɟɥɶɧɨɝɨ ɡɧɚɱɟɧɢɹ ɚɪɝɭɦɟɧɬɚ. Ɍɨɝɞɚ
t
ª cU x 2 º
Kt i y q0
e
x
p
T x ,t T0 «
»d y .
4
O
t
y
S cU O ³0 t y
«¬
»¼
ȼɜɟɞɟɦ ɧɨɜɭɸ ɩɟɪɟɦɟɧɧɭɸ z ɩɨ ɮɨɪɦɭɥɟ
x2
2
,
z
4a t y ɝɞɟ a O cU – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ. Ɍɨɝɞɚ
y t
ȿɫɥɢ y
0 , ɬɨ z
T x ,t T0 x2
4az 2
ɢ dy
x2
2az 3
dz .
x
; ɟɫɥɢ y t , ɬɨ z o f . ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
2 at
O S
exp ª z 2 º §
·
x2
¬
¼
x
t
t
dz .
K
¨
¸
³
2
¨ i 4az 2 ¸
z
©
¹
2 at
f
q0
x
(7.27)
ɉɨɞɢɧɬɟɝɪɚɥɶɧɚɹ ɮɭɧɤɰɢɹ ɜ (7.27) ɪɚɜɧɚ ɧɭɥɸ, ɟɫɥɢ
x2
z d
,
4a t i t 2
ɱɬɨ ɢɦɟɟɬ ɫɦɵɫɥ, ɟɫɥɢ t d ti . ɉɨɷɬɨɦɭ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
T x ,t T0 q0
O S
x 2 at i
³
x 2 at
exp ª z 2 º
¬
¼ dz .
x
2
z
ɂɧɬɟɝɪɢɪɭɟɦ ɩɨ ɱɚɫɬɹɦ. ɉɨɥɚɝɚɹ u e x p z 2 , dv
du
dz z 2 , ɧɚɣɞɟɦ
2 z e x p z 2 d z , v 1 z ɢ
T x ,t º
§ x2 ·
§ x2 ·
q0 ­ 2 ª
e
x
p
e
x
p
T0 t
t
«
»
¨
¸
¨
¸
®
¨ 4a t ¸
¨ 4a t i ¸ i »
cU O ¯ S «¬
©
¹
©
¹
¼
169
ª
§ x ·º ½°
§ x ·
«erfc ¨
¸» ¾ .
¸ erfc ¨¨
¸
at
2
© 2 at ¹
«¬
i ¹ »¼ °
©
¿
ɉɪɢ ti o f ɢɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɧɚɯɨɞɢɦ (7.22).
x
a
ȿɫɥɢ ɢɦɟɟɬɫɹ ɢɦɩɭɥɶɫɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ (ɪɢɫ. 7.9), ɡɚɩɢɲɟɦ
­1 , t i t p k 1 d t t i t i t p k 1
°
,
(7.28)
f t ®
d
t
t
t
k
t
t
t
k
0
1
,
°̄
i
i
p i
p
ɝɞɟ ti – ɞɥɢɬɟɥɶɧɨɫɬɶ ɢɦɩɭɥɶɫɚ; t p – ɞɥɢɬɟɥɶɧɨɫɬɶ ɩɚɭɡɵ ɦɟɠɞɭ ɞɜɭɦɹ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ ɢɦɩɭɥɶɫɚɦɢ; k – ɧɨɦɟɪ
ɢɦɩɭɥɶɫɚ. ɉɨɞɨɛɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ ɱɚɫɬɨ
ɜɫɬɪɟɱɚɸɬɫɹ ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɩɨɜɟɪɯɧɨɫɬɟɣ
ɜɧɟɲɧɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɷɧɟɪɝɢɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɩɨɬɨɤɨɦ ɢɨɧɨɜ, ɷɥɟɤɬɪɨɧɨɜ ɢ ɬ.ɩ. ɋɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɞɥɢɬɟɥɶɧɨɫɬɹɦɢ ɢɦɩɭɥɶɫɨɜ ɢ
ɩɚɭɡ, ɚ ɬɚɤɠɟ ɱɢɫɥɨ ɢɦɩɭɥɶɫɨɜ ɦɨɝɭɬ ɛɵɬɶ ɪɚɡɥɢɱɧɵɦɢ22.
Ⱦɥɹ ɞɚɧɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ
Ɋɢɫ. 7.9. ɂɦɩɭɥɶɫɧɵɣ
ɷɧɟɪɝɢɸ, ɩɨɝɥɨɳɚɟɦɭɸ ɡɚ ɨɞɢɧ ɢɦɩɭɥɶɫ ɟɞɢɢɫɬɨɱɧɢɤ ɬɟɩɥɚ
ɧɢɰɟɣ ɩɨɜɟɪɯɧɨɫɬɢ
I 0 q 0t i , Ⱦɠ/ɫɦ2,
ɱɚɫɬɨɬɭ ɫɥɟɞɨɜɚɧɢɹ ɢɦɩɭɥɶɫɨɜ
*
1
ti t p
(7.29)
ɢ ɩɨɥɧɭɸ ɷɧɟɪɝɢɸ, ɡɚɤɚɱɚɧɧɭɸ ɜ ɟɞɢɧɢɰɭ ɩɨɜɟɪɯɧɨɫɬɢ ɜɟɳɟɫɬɜɚ
In
q 0t i n .
Ʉɚɱɟɫɬɜɟɧɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ ɜ ɬɨɱɤɚɯ, ɧɚɯɨɞɹɳɢɯɫɹ ɧɚ ɪɚɡɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɧɚɝɪɟɜɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɨɤɚɡɚɧɚ ɧɚ
ɪɢɫ. 7.10.
ɑɟɦ ɜɵɲɟ ɱɚɫɬɨɬɚ ɫɥɟɞɨɜɚɧɢɹ ɢɦɩɭɥɶɫɨɜ, ɬɟɦ ɦɟɧɟɟ ɡɚɦɟɬɟɧ ɢɦɩɭɥɶɫɧɵɣ ɯɚɪɚɤɬɟɪ ɜɧɟɲɧɟɝɨ ɢɫɬɨɱɧɢɤɚ ɞɥɹ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɨɝɨ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɟɧɢɹ ɢ ɞɥɹ ɬɨɱɟɤ, ɭɞɚɥɟɧɧɵɯ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ. ɉɨɷɬɨɦɭ ɩɪɢ
ɦɨɞɟɥɢɪɨɜɚɧɢɢ ɩɨɜɟɞɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ ɜ ɭɫɥɨɜɢɹɯ ɢɦɩɭɥɶɫɧɨɣ ɨɛɪɚɛɨɬ22
Ⱦɟɱ Ƚ. Ɋɭɤɨɜɨɞɫɬɜɨ ɤ ɩɪɚɤɬɢɱɟɫɤɨɦɭ ɩɪɢɦɟɧɟɧɢɸ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɚɩɥɚɫɚ ɢ Zɩɪɟɨɛɪɚɡɨɜɚɧɢɹ, Ɇ.: ɇɚɭɤɚ. 1971.
170
ɤɢ, ɨɫɨɛɟɧɧɨ ɜɵɫɨɤɨɱɚɫɬɨɬɧɨɣ, ɢɦɩɭɥɶɫɧɵɣ ɢɫɬɨɱɧɢɤ ɡɚɦɟɧɹɸɬ ɧɟɩɪɟɪɵɜɧɵɦ ɫ ɧɟɤɨɬɨɪɵɦ ɷɮɮɟɤɬɢɜɧɵɦ ɡɧɚɱɟɧɢɟɦ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɬɟɩɥɚ,
ɢɫɯɨɞɹ ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɟɧɫɬɜɚ ɩɨɥɧɵɯ ɷɧɟɪɝɢɣ.
T,K
T,K
T,K
1
600
1
600
2
500
400
0,0
0,3
t,c
4
400
5
6
300
0,6
1
2
3
500
4
5
400
6
300
2
3
500
3
4
5
600
6
300
0,0
0,3
ɚ
0,6
t,c
ɛ
0,0
0,3
0,6
t,c
ɜ
Ɋɢɫ. 7.10. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ ɜ ɪɚɡɧɵɯ ɬɨɱɤɚɯ
ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɩɪɢ ɢɦɩɭɥɶɫɧɨɦ ɢɫɬɨɱɧɢɤɟ ɬɟɩɥɚ: * 2 0 , 5 0 , 2 5 0 0 ɞɥɹ
ɪɢɫɭɧɤɨɜ ɚ, ɛ, ɜ; x 1 0 ; 2 – 0,025; 3 – 0,05; 4 – 0,1; 5 – 0,15; 6 – 0,25 ɫɦ;
ɢɫɩɨɥɶɡɨɜɚɧɵ ɫɜɨɣɫɬɜɚ: Ȝ = 0,92 ȼɬ/ɫɦ Ʉ ɫ, ɫ = 0,482 Ⱦɠ/ɝ Ʉ, ȡ = 8,2 ɝ/ɫɦ3.
7.3.3. Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ
Ɂɚɞɚɱɚ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɝɪɟɜɭ ɢɥɢ ɨɯɥɚɠɞɟɧɢɸ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɢɥɢ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɧɨɝɨ ɫ ɛɨɤɨɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɟɪɠɧɹ ɛɟɫɤɨɧɟɱɧɨɣ ɞɥɢɧɵ ɜ ɩɨɬɨɤɟ ɝɚɡɚ ɢɥɢ ɠɢɞɤɨɫɬɢ. ɇɚ ɝɪɚɧɢɰɟ ɡɚɞɚɟɬɫɹ ɬɟɩɥɨɨɛɦɟɧ ɫ ɩɨɬɨɤɨɦ ɩɨ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɬɚɤɨɣ ɡɚɞɚɱɢ ɜɤɥɸɱɚɟɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (7.10), ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ
x 0:
O
wT
wx
D Te T ,
ɝɞɟ Te – ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɝɨ ɩɨɬɨɤɚ, ɢ ɭɫɥɨɜɢɟ ɨɝɪɚɧɢɱɟɧɧɨɫɬɢ
ɪɟɲɟɧɢɹ ɩɪɢ ɛɟɫɤɨɧɟɱɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ.
ȼ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ T , W , [ , ɝɞɟ
T
T T0
,
Te T0
ɩɪɢɞɟɦ ɤ ɡɚɞɚɱɟ
wT
wW
w 2T
w[ 2
171
;
wT
(7.30)
B i 1 T ;
w[
[ o f : T 0;
W 0: T 0,
ɫɨɞɟɪɠɚɳɟɣ ɟɞɢɧɫɬɜɟɧɧɵɣ ɩɚɪɚɦɟɬɪ – ɱɢɫɥɨ Ȼɢɨ.
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ W o p; T o T ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (7.15) ɢ ɭɫɥɨɜɢɹ
[ 0: [ 0:
dT
d[
Bi
B iT ;
p
[of:
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ
T
Bi
p Bi p
T 0.
(7.33)
5
4
172
3
2
2,5
5,0
1
7,5
[
Ɋɢɫ. 7.11. Ɂɚɜɢɫɢɦɨɫɬɶ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ
(ɫɩɥɨɲɧɵɟ ɥɢɧɢɢ) ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɬɨɱɤɟ [ 2 (ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ) ɨɬ ɜɪɟɦɟɧɢ ɜ
ɡɚɞɚɱɟ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɟɦ
ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ III ɪɨɞɚ:
Bi = 1 – 0,1; 2 – 0,25; 3 – 0,5;
4 –1,0; 5 – 2,5
exp Bi 2W erfc Bi W .
ȼ ɮɢɡɢɱɟɫɤɢɯ ɩɟɪɟɦɟɧɧɵɯ ɢɦɟɟɦ
4
1
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɢ W 0 ɬɟɦɩɟɪɚɬɭɪɚ
ɩɨɜɟɪɯɧɨɫɬɢ – ɧɭɥɟɜɚɹ, ɚ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ
ɜɪɟɦɟɧɢ – ɚɫɢɦɩɬɨɬɢɱɟɫɤɢ ɫɬɪɟɦɢɬɫɹ ɤ
ɬɟɦɩɟɪɚɬɭɪɟ ɩɨɬɨɤɚ T o 1 (ɪɢɫ. 7.11).
ȼ ɬɨɱɤɟ, ɭɞɚɥɟɧɧɨɣ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ,
ɪɨɫɬ ɬɟɦɩɟɪɚɬɭɪɵ, ɨɱɟɜɢɞɧɨ, ɧɚɱɢɧɚɟɬɫɹ ɫ
ɡɚɩɨɡɞɚɧɢɟɦ.
ɇɚɯɨɞɢɦ ɩɨɬɨɤ ɬɟɩɥɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ
0
5
2
0,00
0,0
3
0,50
0,25
T 1 exp Bi 2W erfc Bi W . (7.34)
§ wT ·
¨ ¸
© w[ ¹ [
(7.32)
0,75
4
[ ·
§
exp B i[ B i 2W erfc ¨ Bi W ¸.
2 W¹
©
ɇɚ ɩɨɜɟɪɯɧɨɫɬɢ [ 0 ɢɦɟɟɦ
exp [ p
ɢɥɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɪɢɝɢɧɚɥɨɜ –
§ [ ·
T erfc ¨
¸
©2 W¹
(7.31)
(7.35)
T
ª
§ x
T0 Te T0 «erf c ¨
© 2 at
¬«
§ D
D2 ·
x
·
t¸u
¸ exp ¨¨
¸
c
UO
c
U
O
a
¹
©
¹
§ D
x
uerfc ¨¨
t
2 at
© cUO
·º
¸¸ »
¹ ¼»
ɢ
ª
§ D
§ D
·º
x ·
T 0 ,t T0 Te T0 «1 exp ¨¨
t ¸¸ » .
¸¸ erfc ¨¨
«¬
© cU O a ¹
© cUO ¹ »¼
ɂɧɬɟɪɟɫɧɨ ɨɬɦɟɬɢɬɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɩɨɥɭɱɟɧɧɨɝɨ ɪɟɲɟɧɢɹ.
ɉɪɢ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɨɛɦɟɧɚ D !! 1, ɩɨɥɶɡɭɹɫɶ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ ɮɭɧɤɰɢɢ ɨɲɢɛɨɤ (7.12), ɧɚɣɞɟɦ
§ D2 ·
e x p¨ t
¨ cU O ¸¸
§ D
·
©
¹ cUO ,
erfc ¨
t |
3
¨ cUO ¸¸
D St
©
¹
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
T | T0 Te T0 ª«1 O D 3 º» .
¬
¼
Ɍ.ɟ., ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɜɟɞɟɬ ɫɟɛɹ ɬɚɤ ɠɟ, ɤɚɤ ɢ ɜ ɩɟɪɜɨɣ
ɤɪɚɟɜɨɣ ɡɚɞɚɱɟ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɦɚɥɵɯ ɜɟɥɢɱɢɧ.
ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɦ ɫɥɭɱɚɟ, D 1 , ɢɦɟɟɦ
ª §
· §
·º
D2
D
2
T T0 Te T0 «1 ¨ 1 t ...¸ ˜ ¨ 1 t .. .¸ » |
¸
¸ ¨
S
U
O
c
«¬ ©¨ cUO
¹ ¼»
¹ ©
t
D
.
| T0 2 Te T0 cUO S
Ɍ.ɟ., ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɦɟɧɹɟɬɫɹ ɤɚɤ t , ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɪɟɲɟɧɢɸ ɡɚɞɚɱɢ ɨ ɧɚɝɪɟɜɟ ɩɨɬɨɤɨɦ ɜɟɥɢɱɢɧɵ q 0 | D Te T0 .
7.4. Ɇɟɬɨɞ Ⱦɸɚɦɟɥɹ
Ⱦɥɹ ɰɟɥɨɝɨ ɪɹɞɚ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɜɟɫɶɦɚ ɭɞɨɛɧɵɦ ɨɤɚɡɵɜɚɟɬɫɹ
ɦɟɬɨɞ Ⱦɸɚɦɟɥɹ. Ɋɚɫɫɦɨɬɪɢɦ ɟɝɨ ɧɚ ɩɪɢɦɟɪɟ ɡɚɞɚɱɢ
wT
w § wT ·
cU
¨O
¸
wt wx © wx ¹
173
t 0 : T T0 ; x t 0
(7.36)
t ! 0: x 0 :
T f t ;
x of:
T T0 .
ɉɨɥɚɝɚɹ T u v , ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɜɜɟɞɟɧɧɵɟ ɮɭɧɤɰɢɢ u ɢ v
ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɦɭ ɭɪɚɜɧɟɧɢɸ ɬɨɣ ɠɟ ɮɨɪɦɵ,
ɧɨ ɛɨɥɟɟ ɩɪɨɫɬɵɦ ɤɪɚɟɜɵɦ ɭɫɥɨɜɢɹɦ ɞɥɹ ɮɭɧɤɰɢɢ u
u T0 , t 0, x t 0;
(7.37)
u 0, x 0, t ! 0
ɢ ɞɥɹ ɮɭɧɤɰɢɢ v –
v 0,
t 0, x t 0;
(7.38)
v f t , x 0, t ! 0.
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɞɥɹ ɮɭɧɤɰɢɢ u ɧɚɦ ɢɡɜɟɫɬɧɨ ɢɥɢ, ɜɟɪɧɟɟ, ɥɟɝɤɨ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɢɡ ɪɟɲɟɧɢɹ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (7.11), ɡɚɩɢɫɚɧɧɨɝɨ ɜ ɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
O
§ x ·
T T0 Ts T0 erfc ¨
(7.39)
¸ , a cU .
at
2
©
¹
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɮɭɧɤɰɢɢ u ɢɦɟɟɦ
ª
§ x ·º
§ x ·
(7.40)
u T0 «1 erfc ¨
¸ » T0erf ¨
¸,
at
at
2
2
©
¹
©
¹
¬
¼
ɝɞɟ
erf z z
2
y2
e
dy .
S ³0
Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɹ ɜɬɨɪɨɣ ɡɚɞɚɱɢ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɟɨɪɟɦɨɣ
Ⱦɸɚɦɟɥɹ, ɤɨɬɨɪɚɹ ɝɥɚɫɢɬ:
ȿɫɥɢ M x ,t ɟɫɬɶ ɪɟɲɟɧɢɟ ɞɥɹ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɬɜɟɪɞɨɦ
ɬɟɥɟ, ɧɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ ɧɭɥɸ, ɚ ɩɨɜɟɪɯɧɨɫɬɶ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ, ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ, ɬɨ ɪɟɲɟɧɢɟ v x ,t ɞɥɹ
ɫɥɭɱɚɹ, ɤɨɝɞɚ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ,
ɬ.ɟ. v f t , x 0 , ɞɚɟɬɫɹ ɮɨɪɦɭɥɨɣ
v x ,t t
w
³ f y w t M x ,t y d y .
(7.41)
0
Ɍɟɨɪɟɦɚ Ⱦɸɚɦɟɥɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɢɦɟɧɟɧɚ ɢ ɤ ɫɥɭɱɚɹɦ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɤɨɝɞɚ ɬɜɟɪɞɨɟ ɬɟɥɨ, ɢɦɟɸɳɟɟ ɨɞɧɨɪɨɞɧɭɸ ɧɚɱɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ, ɜɧɟɡɚɩɧɨ ɩɨɞɜɟɪɝɚɟɬɫɹ ɜɨɡɞɟɣɫɬɜɢɸ ɨɤ174
ɪɭɠɚɸɳɟɣ ɠɢɞɤɨɫɬɢ (ɝɚɡɚ), ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɣ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ
ɩɨ ɡɚɞɚɧɧɨɦɭ ɡɚɤɨɧɭ.
ɑɬɨɛɵ ɩɪɢɦɟɧɢɬɶ ɬɟɨɪɟɦɭ Ⱦɸɚɦɟɥɹ, ɧɚɣɞɟɦ ɪɟɲɟɧɢɟ M x ,t , ɤɨɬɨɪɨɟ ɬɚɤɠɟ ɫɨɞɟɪɠɢɬɫɹ ɜ (7.39).
ª
§ x ·º
§ x ·
M «1 erf ¨
erfc
»
¸
¨
¸.
2
at
2
at
©
¹¼
©
¹
¬
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
§
·
x
¨
¸
M x ,t y erfc
¨ 2 a t y ¸
©
¹
ɢ
2
x exp x 4a t y .
2 Sa
t y 3 2
ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɟ ɞɥɹ ɮɭɧɤɰɢɢ v ɩɪɢɧɢɦɚɟɬ ɜɢɞ
w
M x ,t y wt
v x ,t x
2 Sa
t
³
f y
e x p x 2 4a t y t y 0
32
t y 3 2
.
(7.42)
x 2 4a t y , ɬɨ ɧɚɣɞɟɦ
ȿɫɥɢ ɦɵ ɜɜɟɞɟɦ ɧɨɜɭɸ ɩɟɪɟɦɟɧɧɭɸ z 2
dy
dy
4 a
dz ɢ
x
v x ,t 2
S
f
x
§
x 2 · z 2
³ f ¨¨ t 4az 2 ¸¸ e dz .
¹
2 at ©
(7.43)
ɉɨɥɧɨɟ ɪɟɲɟɧɢɟ ɞɚɟɬɫɹ ɫɭɦɦɨɣ ɜɟɥɢɱɢɧ u v .
ɇɚɣɞɟɦ ɪɟɲɟɧɢɟ ɞɥɹ ɥɢɧɟɣɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ f t T0 C t . Ɇɵ ɞɨɥɠɧɵ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɭɪɚɜɧɟɧɢɟ (7.43).
ɂɦɟɟɦ
f
§
§
2
x 2 · · z 2
v x ,t ³ ¨¨ T0 C ¨¨ t 4az 2 ¸¸ ¸¸ e dz
S
©
¹¹
x 2 at ©
2T0
S
f
³e
z 2
2C t
dz S
f
³e
z 2
Cx2
dz 2 aS
f z 2
³
e
dz , X
x
.
2 at
z2
ȼ ɬɪɟɬɶɟɦ ɫɥɚɝɚɟɦɨɦ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɩɨ ɱɚɫɬɹɦ. Ɉɛɨɡɧɚɱɢɦ
X
X
175
X
dz
2
p ez ; d q
z2
ɬɨɝɞɚ
2
2 z e z dz ; q
dp
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
f z 2
³
X
e
z
2
dz
pq
f
x
f
³ qdp
e
z 2
z
X
f
f
,
1
.
z
2
2 ³ e z dz
2
X
x
eX
X2
Ɉɛɴɟɞɢɧɹɟɦ ɪɟɡɭɥɶɬɚɬɵ
2º
2
ª
T T0 Ct « 1 2 X 2 erfc X Xe X » , X
S
¬
¼
T,K
x=1. - 0 ɫɦ
2. - 1 cɦ
3. - 2 ɫɦ
4. - 3 ɫɦ
1200
1000
1
800
2
3
4
600
400
0
25
50
75
t,c
Ɋɢɫ. 7.12. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ ɜ ɬɨɱɤɚɯ, ɪɚɜɧɨɭɞɚɥɟɧɧɵɯ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ (ɩɪɢ ɥɢɧɟɣɧɨɦ
ɢɡɦɟɧɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ);
ɋ = 1, Ɍ0 = 273 Ʉ.
2
Se rfc X .
x
.
2 at
(7.44)
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ
ɜɛɥɢɡɢ ɧɚɝɪɟɜɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ
ɪɚɡɥɢɱɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 7.12. ȼ ɪɚɫɱɟɬɚɯ ɢɫɩɨɥɶɡɨɜɚɧɵ ɫɜɨɣɫɬɜɚ, ɛɥɢɡɤɢɟ ɤ
ɫɜɨɣɫɬɜɚɦ ɠɟɥɟɡɚ
c 452 Ⱦɠ/(ɤɝ·Ʉ),
U 7870 ɤɝ/ɦ3,
O 78 ȼɬ/(ɦ·Ʉ).
Ⱦɚɠɟ ɩɪɢ ɥɢɧɟɣɧɨɦ ɢɡɦɟɧɟɧɢɢ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɬɨɱɤɚɯ, ɨɬɥɢɱɧɵɯ ɨɬ x 0 ɬɟɦɩɟɪɚɬɭɪɚ ɦɟɧɹɟɬɫɹ ɫɨ ɜɪɟɦɟɧɟɦ ɧɟɥɢɧɟɣɧɨ, ɱɬɨ, ɟɫɬɟɫɬɜɟɧɧɨ, ɫɜɹɡɚɧɨ ɫ
ɩɪɨɰɟɫɫɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
ɋɭɳɟɫɬɜɭɟɬ ɦɧɨɝɨ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɵɯ ɫɥɭɱɚɟɜ, ɜ ɤɨɬɨɪɵɯ ɦɟɬɨɞ
Ⱦɸɚɦɟɥɹ ɦɨɠɟɬ ɧɚɣɬɢ ɩɪɢɦɟɧɟɧɢɟ.
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɛɨɥɶɲɚɹ ɚɥɸɦɢɧɢɟɜɚɹ ɨɬɥɢɜɤɚ ɫ ɧɚɱɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɪɚɜɧɨɣ ɩɨɜɫɸɞɭ 26,7 ɨɋ, ɜɧɟɡɚɩɧɨ ɩɨɝɪɭɠɚɟɬɫɹ ɜ ɜɚɧɧɭ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ 260 ɨɋ ɉɨɫɥɟ ɩɨɥɭɱɚɫɨɜɨɝɨ ɧɚɝɪɟɜɚɧɢɹ ɨɬɥɢɜɤɚ ɛɵɫɬɪɨ ɜɵɧɢɦɚɟɬɫɹ ɢ ɩɨɦɟɳɚɟɬɫɹ ɜ ɜɚɧɧɭ ɫ ɥɟɞɹɧɨɣ ɜɨɞɨɣ, Ʉɚɤɨɜɚ ɛɭɞɟɬ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɨɱɤɟ, ɪɚɫɩɨɥɨɠɟɧɧɨɣ ɧɚ ɪɚɫɫɬɨɹɧɢɢ 30 ɫɦ ɧɢɠɟ ɩɨɜɟɪɯɧɨɫɬɢ ɨɬɥɢɜɤɢ, ɩɨɫɥɟ ɩɪɟɛɵɜɚɧɢɹ ɟɟ ɜɨ ɜɬɨɪɨɣ ɜɚɧɧɟ ɜ ɬɟɱɟɧɢɟ 1 ɱɚɫɚ?
176
Ɋɟɲɟɧɢɟ. ȿɫɥɢ ɩɪɟɧɟɛɪɟɱɶ ɩɨɜɟɪɯɧɨɫɬɧɵɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ, ɦɵ
ɦɨɠɟɦ ɩɪɢɧɹɬɶ, ɱɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɨɬɥɢɜɤɢ ɟɫɬɶ ɫɬɭɩɟɧɱɚɬɚɹ
ɮɭɧɤɰɢɹ ɜɪɟɦɟɧɢ, ɨɩɪɟɞɟɥɹɟɦɚɹ ɪɚɜɟɧɫɬɜɚɦɢ
T 0 ,t T1 , 0 t t1 ;
T 0 ,t T2 , t t t1 .
Ɋɚɡɛɢɜɚɹ ɜ ɢɧɬɟɝɪɚɥɟ (7.42) ɩɪɨɦɟɠɭɬɨɤ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚ ɞɜɚ ɩɪɨɦɟɠɭɬɤɚ 0 ,t1 ɢ t1 ,t , ɩɨɥɭɱɚɟɦ
v x ,t ªt1 ex p x 2 4a t y « T
dy ³ 1
32
«
2 Sa 0
t y «¬
x
ex p x 2 4a t y t
³ T2
t1
t y3 2
ɢɥɢ, ɜɜɨɞɹ ɩɟɪɟɦɟɧɧɭɸ [
x
2 a t y º
dy »
»
»¼
, ɡɚɩɢɲɟɦ
ª x 2 a t t1 º
f
2 «
»
2
2
v x ,t T
d
T
d
exp
exp
[
[
[
[
1
2
³
³
»
S«
x 2 a t t1 «¬ x 2 at
»¼
ª f
f
2 «
2
T1exp [ d [ T1exp [ 2 d [ ³
³
«
S
x 2 at t1
«¬x 2 at
º
f
»
2
³ T2exp [ d [»
x 2 at t1
»¼
ɋɤɥɚɞɵɜɚɹ ɷɬɨɬ ɪɟɡɭɥɶɬɚɬ ɫ ɪɟɡɭɥɶɬɚɬɨɦ ɞɥɹ ɮɭɧɤɰɢɢ u , ɩɨɥɭɱɚɟɦ ɪɟɲɟɧɢɟ ɞɥɹ ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ t t t1
T
§ x
T0erf ¨
© 2 at
·
§ x
¸ T1erfc ¨
¹
© 2 at
§
·
x
·
¨
¸.
¸ T2 T1 erfc ¨
¸
¹
© 2 a t t1 ¹
(7.45)
ɂɫɩɨɥɶɡɭɹ ɢɫɯɨɞɧɵɟ ɞɚɧɧɵɟ ɢ ɫɜɨɣɫɬɜɚ ɚɥɸɦɢɧɢɹ
U 2710 ɤɝ/ɦ3, O 174 ɤɚɥ/(ɦ ɱɚɫ ɝɪɚɞ), ɫ=0,181 ɤɤɚɥ/(ɤɝ ɝɪɚɞ),
ɩɨɥɭɱɚɟɦ
177
x
0 ,3
0 ,3
0 ,21 ,
0 ,26 .
1,5
2 a t t1 1
2
2
2 ,82
2 ,82
ɉɨɥɶɡɭɹɫɶ ɬɚɛɥɢɱɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɞɥɹ ɮɭɧɤɰɢɢ ɨɲɢɛɨɤ, ɧɚɯɨɞɢɦ
T 26 ,7 u 0 ,234 260 u 1 0 ,234 0 260 u 1 0 ,287 6,3 198,7 185 20 ɨɋ.
x
2 at
7.5. ɉɪɢɦɟɪɵ ɫɨɩɪɹɠɟɧɧɵɯ ɡɚɞɚɱ
ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
7.5.1. Ɍɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ ɜ ɫɪɟɞɟ
Ɉɩɟɪɚɰɢɨɧɧɵɣ ɦɟɬɨɞ ɧɟɡɚɦɟɧɢɦ ɩɪɢ ɪɟɲɟɧɢɢ ɬɚɤ ɧɚɡɵɜɚɟɦɵɯ ɫɨɩɪɹɠɟɧɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɜ ɤɨɬɨɪɵɯ ɩɪɢɫɭɬɫɬɜɭɟɬ ɝɪɚɧɢɰɚ
ɪɚɡɞɟɥɚ ɦɟɠɞɭ ɦɚɬɟɪɢɚɥɚɦɢ ɫ ɪɚɡɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ. Ɋɚɫɫɦɨɬɪɢɦ ɧɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ ɫɨɩɪɹɠɟɧɧɵɯ ɡɚɞɚɱ, ɢɦɟɸɳɢɯ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ ɨɬɧɨɲɟɧɢɟ ɤ ɨɛɪɚɛɨɬɤɟ ɦɚɬɟɪɢɚɥɨɜ ɢ ɞɟɬɚɥɟɣ ɫ ɩɨɤɪɵɬɢɹɦɢ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɬɟɪɦɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ ɬɟɩɥɨɜɵɦ ɩɨɬɨɤɨɦ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɵ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɜ ɫɪɟɞɟ (ɪɢɫ. 7.13), ɫɜɨɣɫɬɜɚ ɤɨɬɨɪɨɣ ɨɬɥɢɱɧɵ ɨɬ ɫɜɨɣɫɬɜ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ. Ɍɪɟɛɭɟɬɫɹ
ɨɰɟɧɢɬɶ ɜɥɢɹɧɢɟ ɫɪɟɞɵ ɧɚ ɜɟɥɢɱɢɧɭ ɬɟɦɩɟɪɚɬɭɪɵ ɦɚɬɟɪɢɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɟɝɨ ɨɛɪɚɛɨɬɤɨɣ ɜ ɜɚɤɭɭɦɟ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɬɚɤɨɣ ɡɚɞɚɱɢ ɛɭɞɟɬ
ɜɤɥɸɱɚɬɶ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ ɢ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ. Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɨɛɨɡɧɚɱɢɦ
ɢɧɞɟɤɫɚɦɢ 2 ɢ 1 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɍɨɝɞɚ
w T1
wt
w § w T1 ·
, x0
Ȝ1
w x ¨© w x ¸¹
w T2
wt
w § w T2 ·
, x ! 0.
O2
w x ¨©
w x ¸¹
c1ȡ 1
c 2U 2
Ɋɢɫ. 7.13. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɩɨɫɬɚɧɨɜɤɟ
ɡɚɞɚɱɢ ɨ ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɟ
ɦɚɬɟɪɢɚɥɚ ɜ ɫɪɟɞɟ
ɇɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɡɚɩɢɫɵɜɚɟɦ ɭɫɥɨɜɢɹ ɱɟɬɜɟɪɬɨɝɨ ɪɨɞɚ
178
x 0 : O1
w T1
wT
O2 2
wx
wx
q 0 , T1 T2 .
ɇɚ ɛɟɫɤɨɧɟɱɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ ɬɟɦɩɟɪɚɬɭɪɵ ɧɟɢɡɦɟɧɧɵ
x o r f : Ti
T0 , i 1, 2
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɬɟɦɩɟɪɚɬɭɪɵ ɨɞɢɧɚɤɨɜɵ
t
0 : Ti
T0 , i 1, 2
Ɉɱɟɜɢɞɧɨ, ɬɚɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɛɭɞɟɬ ɤɨɪɪɟɤɬɧɚ, ɟɫɥɢ ɬɟɥɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɛɟɫɤɨɧɟɱɧɵɦ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɪɚɡɦɟɪɨɦ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ (ɡɨɧɨɣ ɩɪɨɝɪɟɜɚ), ɮɨɪɦɢɪɭɸɳɟɝɨɫɹ ɡɚ ɜɪɟɦɹ ɧɚɛɥɸɞɟɧɢɹ, ɚ
ɩɨɬɨɤ ɬɟɩɥɚ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧ ɩɨ ɜɫɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɬɟɥɚ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ
ɫɭɳɟɫɬɜɟɧɧɨ ɭɫɥɨɠɧɹɟɬɫɹ.
Ɋɟɲɢɬɶ ɷɬɭ ɡɚɞɚɱɭ ɦɨɠɧɨ ɨɩɟɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ.
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ ɢɦɟɟɦ
d 2T1
pT1 T0 a1
; x0
dx 2
d 2T2
; x!0
pT2 T0 a 2
dx 2
dT
dT
q0
x 0 : O1 1 O 2 2
; T1 T 2 .
dx
dx
p
Ⱥɧɚɥɨɝɢɱɧɨ ɛɨɥɟɟ ɩɪɨɫɬɵɦ ɡɚɞɚɱɚɦ, ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɭɱɟɬɨɦ ɭɫɥɨɜɢɣ ɜ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɢɦɟɸɬ ɜɢɞ
T1 T0
p
§ p ·
T
A1exp ¨¨
x ¸¸ ; T2 0
p
© a1 ¹
§ p ·
A2exp ¨¨
x ¸¸ .
© a2 ¹
ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɹ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɫɪɟɞ, ɧɚɣɞɟɦ
A1
A2
A
q0
p p
1
c1U1O 1 c 2U 2O 2
qe
p p c 2U 2O 2
ɝɞɟ
qe
q0
, KH
1 KH
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
179
c1U1O 1
.
c 2U 2O 2
,
§
p ·
e x p ¨¨ x ¸¸ ,
a
p p c 2U 2O 2
2 ¹
©
§
T
qe
p ·
exp ¨¨ T1 0
x ¸¸ .
p p p c 2U 2O 2
a
1 ¹
©
ɉɟɪɟɣɬɢ ɤ ɨɪɢɝɢɧɚɥɚɦ ɞɨɫɬɚɬɨɱɧɨ ɥɟɝɤɨ, ɢɫɩɨɥɶɡɭɹ ɬɚɛɥɢɰɵ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɩɪɢɥɨɠɟɧɢɟ 2)
T2 T2
T0
p
qe
§ x · °½
§ x2 ·
x
°­ t
erfc ¨
T0 ¸
¸ . (7.46)
®2 exp ¨¨ ¸
¨ 2 a t ¸¾
4
a
t
N
c 2U 2O 2 °¯ S
2
2
2 ¹°
©
¹
©
¿
qe
ɉɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɜɧɟɲɧɟ ɧɢɱɟɦ ɧɟ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɬɨɝɨ, ɤɨɬɨɪɨɟ
ɧɚɣɞɟɧɨ ɩɪɢ ɚɧɚɥɢɡɟ ɩɪɨɫɬɟɣɲɟɣ ɜɬɨɪɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ (7.22). ȼɥɢɹɧɢɟ ɜɧɟɲɧɟɣ ɫɪɟɞɵ ɩɪɨɹɜɥɹɟɬɫɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ,
ɤɚɤ ɛɭɞɬɨ ɞɟɣɫɬɜɭɸɳɢɣ ɜɧɟɲɧɢɣ ɩɨɬɨɤ ɬɟɩɥɚ ɭɦɟɧɶɲɢɥɫɹ ɜ 1 K H ɪɚɡ. ɉɚɪɚɦɟɬɪ K H ɢɧɨɝɞɚ ɧɚɡɵɜɚɸɬ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɚɤɬɢɜɧɨɫɬɢ ɨɞɧɨɝɨ
ɦɚɬɟɪɢɚɥɚ (ɫɪɟɞɵ) ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɞɪɭɝɨɦɭ (ɤ ɨɛɪɚɛɚɬɵɜɚɟɦɨɦɭ ɦɚɬɟɪɢɚɥɭ). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ ɧɚɝɪɟɜ ɦɚɬɟɪɢɚɥɚ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɧɚ ɜɨɡɞɭɯɟ ɢ, ɫɤɚɠɟɦ ɜ ɫɪɟɞɟ ɚɪɝɨɧɚ, ɬɨ ɟɝɨ ɬɟɦɩɟɪɚɬɭɪɚ ɩɪɢ ɩɪɨɱɢɯ ɪɚɜɧɵɯ
ɭɫɥɨɜɢɹɯ ɛɭɞɟɬ ɪɚɡɥɢɱɧɨɣ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɨɛɪɚɛɚɬɵɜɚɟɦɵɣ ɦɚɬɟɪɢɚɥ – ɠɟɥɟɡɨ, ɚ ɨɤɪɭɠɚɸɳɚɹ ɫɪɟɞɚ – ɜɨɡɞɭɯ. Ɍɨɝɞɚ, ɢɫɩɨɥɶɡɭɹ ɞɚɧɧɵɟ ɬɚɛɥɢɰɵ 7.1, ɧɚɣɞɟɦ
K H | 3,19 ˜10 4 .
Ɍɚɛɥɢɰɚ 7.1
ɋɜɨɣɫɬɜɚ ɜɟɳɟɫɬɜ, ɢɫɩɨɥɶɡɨɜɚɧɧɵɟ ɜ ɪɚɫɱɟɬɚɯ
ɜɟɳɟɫɬɜɨ
c , Ⱦɠ/(ɤɝ·Ʉ)
ɠɟɥɟɡɨ
ɜɨɞɚ
ɜɨɡɞɭɯ
452
4182
1012
U , Ʉɝ/ɦ3
O , ȼɬ/(ɦ·Ʉ)
7870
998,2
1,163
81,1
0,597
0,025
ȿɫɥɢ ɨɤɪɭɠɚɸɳɚɹ ɫɪɟɞɚ ɨɛɥɚɞɚɟɬ ɫɜɨɣɫɬɜɚɦɢ ɜɨɞɵ, ɬɨ ɧɚɣɞɟɦ
K H | 0,093 .
Ɍ.ɟ., ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɜɥɢɹɧɢɟɦ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ. Ⱥ ɜɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ – ɫɪɟɞɚ ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ ɞɟɣɫɬɜɭɸɳɟɝɨ ɩɨɬɨɤɚ ɬɟɩɥɚ ɩɨɱɬɢ ɧɚ 10 ɩɪɨɰɟɧɬɨɜ.
180
Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 7.14.
Ɍɚɤ ɤɚɤ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɶ
ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɦɧɨɝɨ ɦɟɧɶɲɟ, ɱɟɦ
ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɶ
ɨɛɪɚɛɚɬɵɜɚɟɦɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɡɨɧɚ ɩɪɨɝɪɟɜɚ ɜ
ɫɪɟɞɟ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɨɣ, ɢ
ɧɚ ɦɚɥɨɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ
ɧɚɝɪɟɜɨɦ ɫɪɟɞɵ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ.
T
t2 ! t1
Ɉɤɪɭɠɚɸɳɚɹ
ɫɪɟɞɚ, 1
ɨɛɪɚɡɟɰ, 2
t1
t2
x
Ɋɢɫ. 7.14. Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ «ɦɚɬɟɪɢɚɥ – ɨɤɪɭɠɚɸɳɚɹ ɫɪɟɞɚ»
7.5.2. Ɉɛɪɚɛɨɬɤɚ ɦɚɬɟɪɢɚɥɚ ɫ ɩɨɤɪɵɬɢɟɦ
Ɋɚɫɫɦɨɬɪɢɦ ɜɨɡɦɨɠɧɵɟ ɮɨɪɦɭɥɢɪɨɜɤɢ ɡɚɞɚɱ ɨ ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɟ ɞɟɬɚɥɢ ɛɨɥɶɲɨɝɨ ɪɚɡɦɟɪɚ ɫ ɩɨɤɪɵɬɢɟɦ ɬɟɩɥɨɜɵɦ ɩɨɬɨɤɨɦ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɵ, ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɟɧɧɵɦ ɜɞɨɥɶ ɩɨɜɟɪɯɧɨɫɬɢ ɞɟɬɚɥɢ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɚɧɚɥɨɝɢɱɧɚ ɩɪɟɞɵɞɭɳɟɦɭ, ɧɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɢɧɞɟɤɫɚɦɢ «1» ɬɟɩɟɪɶ
ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɨɝɪɚɧɢɱɟɧɧɨɣ ɨɛɥɚɫɬɢ
wT
w § w T1 ·
c1U1 1
, 0 x h;
O
w t w x ©¨ 1 w x ¹¸
(7.47)
w T2 w § w T2 ·
O
, x ! h;
c 2U 2
wt
w x ¨© 2 w x ¸¹
wT
wT
x h : O 1 1 O 2 2 , T1 T2 ;
(7.48)
wx
wx
wT
(7.49)
x 0 : O 1 1 q 0 ;
wx
(7.50)
x o f : T2 T0 ;
t 0 : Ti T0 , i 1, 2 .
(7.51)
ȿɫɥɢ ɜɧɟɲɧɢɣ ɩɨɬɨɤ – ɪɚɞɢɚɰɢɨɧɧɵɣ (ɫɦ. ɪɚɡɞɟɥ 9) ɢɥɢ ɥɚɡɟɪɧɵɣ,
ɬɨ ɜ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɧɭɠɧɨ ɭɱɢɬɵɜɚɬɶ ɨɩɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɦɚɬɟɪɢɚɥɨɜ. Ɍɚɤ, ɟɫɥɢ ɩɨɤɪɵɬɢɟ ɹɜɥɹɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɩɪɨɡɪɚɱɧɵɦ, ɚ ɨɫɧɨɜɧɨɣ
ɦɚɬɟɪɢɚɥ – ɧɟɩɪɨɡɪɚɱɟɧ, ɬɨ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɜɦɟɫɬɨ (7.48) ɢɦɟɟɦ ɭɫɥɨɜɢɟ, ɚɧɚɥɨɝɢɱɧɨɟ ɡɚɞɚɱɟ, ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜɵɲɟ
wT
wT
x h : O 1 1 O 2 2 q 0 ; T1 T 2 .
(7.52)
wx
wx
181
ɍɫɥɨɜɢɟ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ ɡɚɜɢɫɢɬ ɨɬ ɯɚɪɚɤɬɟɪɚ ɬɟɩɥɨɨɛɦɟɧɚ ɦɚɬɟɪɢɚɥɚ ɫ ɩɨɤɪɵɬɢɟɦ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ (ɩɨ ɤɨɧɞɭɤɬɢɜɧɨɦɭ ɦɟɯɚɧɢɡɦɭ – ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɤɨɧɜɟɤɬɢɜɧɨɦɭ ɦɟɯɚɧɢɡɦɭ ɢɥɢ ɢɡɥɭɱɟɧɢɟɦ). Ɍɚɤ, ɜ ɚɞɢɚɛɚɬɢɱɟɫɤɢɯ ɭɫɥɨɜɢɹɯ ɜɦɟɫɬɨ (7.49) ɢɦɟɟɦ
wT
x 0 : O1 1 0 .
(7.53)
wx
T
Ts
c1 ,U1 ,O1
ɉɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɧɚ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɤɪɵɬɢɹ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ
ɩɨɫɬɨɹɧɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ (ɪɢɫ. 7.15), ɜ
ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ
ɩɨɫɬɚɧɨɜɤɟ
ɡɚɞɚɱɢ
(7.47) – (7.51) ɢɡɦɟɧɹɟɬɫɹ ɥɢɲɶ ɜɧɟɲɧɟɟ
ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ (7.49). ȼɦɟɫɬɨ ɧɟɝɨ
ɢɦɟɟɦ
x 0 : T1 Ts .
(7.54)
Ɉɫɬɚɧɨɜɢɦɫɹ ɧɚ ɚɧɚɥɢɡɟ ɩɨɫɥɟɞɧɟɣ
T0
c 2 ,U2 ,O 2
x
h
Ɋɢɫ. 7.15. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɨɛ ɨɛɪɚɛɨɬɤɟ
ɦɚɬɟɪɢɚɥɚ ɫ ɩɨɤɪɵɬɢɟɦ
ɡɚɞɚɱɢ ɞɨɫɬɚɬɨɱɧɨ ɩɨɞɪɨɛɧɨ.
ȼ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ
§ T T0 ·
t
x
Fo ; [
T ¨
¸; W
t
x
© T s T0 ¹
ɫɢɫɬɟɦɚ ɭɪɚɜɧɟɧɢɣ (7.47) ɫ ɭɫɥɨɜɢɹɦɢ (7.48), (7.50), (7.51) ɢ (7.54) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
wT1
w 2T1
; 0[G
Kc
KO
wW
w[ 2
wT 2
wW
[ 0:
[ G : KO
w 2T 2
w[
2
; [!G
T1 1 ;
wT1
w[
wT 2
; T1 T 2
w[
[ o f : T2
W 0 : Ti
(7.55)
0;
0 ; i 1, 2 ,
ɝɞɟ ɜ ɞɨɩɨɥɧɟɧɢɟ ɤ ɭɠɟ ɡɧɚɤɨɦɵɦ ɩɚɪɚɦɟɬɪɚɦ K c
ɜɢɥɫɹ ɟɳɟ ɨɞɢɧ G
h
.
x
182
c1U1
; KO
c 2U 2
O1
ɩɨɹO2
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɤɪɵɬɢɢ ɢ ɜ ɨɫɧɨɜɧɨɦ ɦɚɬɟɪɢɚɥɟ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
Ȼɭɞɟɦ ɪɟɲɚɬɶ ɷɬɭ ɡɚɞɚɱɭ ɨɩɟɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ.
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ W o p ,T i o T i ɜɦɟɫɬɨ
(7.55) ɢɦɟɟɦ
K c p T1
KO
pT 2
d 2 T1
d[2
; 0 [ G;
d 2T 2
; [ !G;
d[ 2
[ 0 : T1 1 p ;
dT
d T2
[ G : KO 1
; T1 T 2 ;
d[
d[
[ o f : T2 0 .
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
T1 A1e xp k1[ B1e xp k1[ ,
T2
(7.56)
A2ex p k 2[ B2e xp k 2[ ,
Kc
p – ɤɨɪɧɢ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ
p ; k2
KO
ɨɛɥɚɫɬɟɣ 1 ɢ 2.
ɂɡ ɭɫɥɨɜɢɹ ɤɨɧɟɱɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ [ o f ɢɦɟɟɦ B 2 0 .
ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɹ ɜ ɧɭɥɟ ɢ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɚɬɟɪɢɚɥɨɜ, ɧɚɯɨɞɢɦ
ɫɢɫɬɟɦɭ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
A1 , B1 , A2 :
1
A1 B1
;
p
ɝɞɟ k1
A1e k1G B1e k1G
K O A1k1e k1G B1k1e k1G
A2e k 2G ;
A2k 2e k 2G ,
ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɣ ɢɦɟɟɬ ɜɢɞ
A1
B1
1
e k1G
;
p e k1G H e k1G
H
e k1G
;
p e k1G H e k1G
183
1 H
1
.
p e k1G H e k1G
ȼ ɪɟɡɭɥɶɬɚɬɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɪɢɝɢɧɚɥɨɜ ɧɚɯɨɞɢɦ
A2
T1
T2
ɝɞɟ H
1 K cK O
1 KcKO
1 e
˜
p
k1 [G k [G
H e 1 k1G
k1G
He
1 1 H e k2 [ G ˜
,
p e k1G He k1G
e
;
(7.57)
(7.58)
- ɬɪɚɧɫɰɟɧɞɟɧɬɧɚɹ ɮɭɧɤɰɢɹ ɩɪɨɢɡɜɟɞɟɧɢɹ K c K O .
ɉɚɪɚɦɟɬɪ H ɜɫɟɝɞɚ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ, H 1 . ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
H ex p 2k1G 1 .
ɂɫɩɨɥɶɡɭɹ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɡɧɚɦɟɧɚɬɟɥɟɣ ɜ (7.57), (7.58) ɜ ɜɢɞɟ ɪɹɞɚ
z
1
1
e k1G
e k1G H e k1G
1 H e 2k1G
e
k1G
f
¦ H ne 2k1nG ,
n 0
ɡɚɩɢɲɟɦ
T1
ª
Kc º
1 f ­° n
exp
2
n
p» H
[
G
®
«
¦
p n 0 ¯°
K O »¼
«¬
ª
K c º ½°
p »¾ ;
H n 1ex p « G [ 2n 1 G K O »¼ °¿
«¬
T2
­° ª
½°
º
Kc
1 H f n
exp
2
1
H
G
[
G
n
p
» ¾ .
® «
¦
p n 0
K
O
°¯ ¬«
°¿
¼»
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɩɟɪɟɣɬɢ ɤ ɨɪɢɝɢɧɚɥɚɦ, ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɚɛɥɢɰɚɦɢ
ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɉɪɢɥɨɠɟɧɢɟ 2), ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦɢ
1
§ D ·
exp D p y erfc ¨
¸.
p
W
2
©
¹
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
184
­° n
ª [ 2nG K c K O º
»
¦ ®H erfc «
W
2
«
»¼
°
n 0¯
¬
f
T1
(7.59)
½
ª
º
n
K
K
2
1
G
[
G
c O °
H n 1erfc « »¾ ;
2
W
«¬
»¼ °¿
f
­° 2n 1 G K c K O [ G ½°
T 2 1 H ¦ H ne rfc ®
¾.
2
W
n 0
¯°
¿°
ɗɬɨ – ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ.
ɂɫɩɨɥɶɡɭɹ ɩɪɚɤɬɢɱɟɫɤɢ ɥɸɛɨɣ ɫɨɜɪɟɦɟɧɧɵɣ ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɩɚɤɟɬ,
ɫ ɩɨɦɨɳɶɸ ɩɨɥɭɱɟɧɧɵɯ ɮɨɪɦɭɥ ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɡɧɚɱɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɥɸɛɨɣ ɬɨɱɤɟ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫ ɥɸɛɨɣ ɡɚɞɚɧɧɨɣ
ɬɨɱɧɨɫɬɶɸ.
ɇɨ ɡɚɱɚɫɬɭɸ ɜ ɩɪɨɫɬɵɯ ɢɧɠɟɧɟɪɧɵɯ ɨɰɟɧɤɚɯ, ɤɨɝɞɚ ɩɚɤɟɬɵ ɩɪɢɤɥɚɞɧɵɯ ɩɪɨɝɪɚɦɦ ɨɤɚɡɵɜɚɸɬɫɹ ɧɟɞɨɫɬɭɩɧɵɦɢ, ɩɨɞɨɛɧɵɟ ɮɨɪɦɭɥɵ
ɨɤɚɡɵɜɚɸɬɫɹ ɦɚɥɨɩɪɢɝɨɞɧɵɦɢ ɜ ɫɢɥɭ ɫɜɨɟɣ ɝɪɨɦɨɡɞɤɨɫɬɢ. Ʉɪɨɦɟ ɬɨɝɨ,
ɜ ɩɪɚɤɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ ɱɚɫɬɨ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦ ɡɧɚɬɶ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɥɢɲɶ ɜ ɨɬɞɟɥɶɧɵɯ ɬɨɱɤɚɯ ɢ ɞɥɹ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ ɜɪɟɦɟɧɢ. Ɂɞɟɫɶ ɧɚ ɩɨɦɨɳɶ ɦɨɝɭɬ ɩɪɢɣɬɢ ɩɪɢɛɥɢɠɟɧɧɵɟ
ɦɟɬɨɞɵ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɟɲɟɧɢɹ ɤɚɤ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɪɢɝɢɧɚɥɨɜ, ɬɚɤ ɢ
ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ, ɱɬɨ ɭɞɨɛɧɨ ɢ ɞɥɹ ɛɨɥɟɟ
ɫɥɨɠɧɵɯ ɡɚɞɚɱ.
Ɍɚɤ, ɩɪɢ [ G ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɢɦɟɟɦ
T1
T2
ª
K c º ½°
1 H f ­° n
n
p »¾ .
H
G
exp
2
1
« ¦®
p n 0 °¯
K
O »¼ °
¬«
¿
(7.60)
ɢɥɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɪɢɝɢɧɚɥɨɜ
T1 T 2
­
ª
n 0°
¯
¬«
f
1 H ¦ °®H nerfc « 2n 1 G
K c K O º ½°
»¾ , [ G
2 W
¼» °¿
(7.61)
ɱɬɨ, ɪɚɡɭɦɟɟɬɫɹ, ɫɥɟɞɭɟɬ ɢ ɢɡ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ (7.59).
ɉɪɢ ɭɫɥɨɜɢɢ H 1 ɪɹɞ ɜ (7.60) ɛɵɫɬɪɨ ɫɯɨɞɢɬɫɹ. ɉɨɷɬɨɦɭ ɦɵ ɦɨɠɟɦ
ɨɝɪɚɧɢɱɢɬɶɫɹ ɧɟɫɤɨɥɶɤɢɦɢ ɱɥɟɧɚɦɢ ɪɹɞɚ ɜ (7.60):
T1
T2
½°
ª
ª
Kc º
Kc º
1 H ­°
G
H
G
exp
exp
3
p
p
...
®
«
»
«
»
¾,
p °¯
K
K
O ¼»
O »¼
°¿
¬«
¬«
185
ɱɬɨ ɬɟɦ ɛɨɥɟɟ ɤɨɪɪɟɤɬɧɨ, ɤɨɝɞɚ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɛɨɥɶɲɢɟ ɡɧɚɱɟɧɢɹ
ɤɨɦɩɥɟɤɫɧɨɣ ɩɟɪɟɦɟɧɧɨɣ p , ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɦɚɥɵɦ ɡɧɚɱɟɧɢɹɦ ɜɪɟɦɟɧɢ W .
ɉɟɪɟɯɨɞɹ ɤ ɨɪɢɝɢɧɚɥɚɦ, ɧɚɯɨɞɢɦ
­°
½°
ªG Kc KO º
ª 3G K c K O º
T1 T 2 1 H ®erfc «
H
..
.
erfc
»
«
»
¾ , (7.62)
W
W
2
2
«¬
»¼
«¬
»¼
°¯
°¿
ɱɬɨ ɨɩɹɬɶ ɠɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɱɟɬɭ ɬɨɥɶɤɨ ɞɜɭɯ ɱɥɟɧɨɜ ɪɹɞɚ ɜ (7.61).
Ȼɵɫɬɪɭɸ ɫɯɨɞɢɦɨɫɬɶ ɪɹɞɚ ɜ ɪɟɲɟɧɢɢ
ɢɥɥɸɫɬɪɢɪɭɟɬ ɪɢɫ. 7.16, ɝɞɟ ɩɪɟɞɫɬɚɜɥɟɧɵ ɡɚɜɢɫɢɦɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɝɪɚɧɢɰɟ
ɪɚɡɞɟɥɚ ɦɚɬɟɪɢɚɥɨɜ ɨɬ ɜɪɟɦɟɧɢ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɚ
4
0,75
J 0,50
0,25
J 0,00
0,0
2,5
5,0
7,5
J
W
Ɋɢɫ. 7.16. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ
ɫɪɟɞ ɨɬ ɜɪɟɦɟɧɢ. H = 0,25
G
Kc
KO
h
.
N1t ɋɩɥɨɲɧɵɟ ɥɢɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɚɫɱɟɬɭ ɩɨ ɮɨɪɦɭɥɟ (7.61) ɫ ɭɞɟɪɠɚɧɢɟɦ 25
ɱɥɟɧɨɜ ɪɹɞɚ, ɚ ɫɢɦɜɨɥɵ – ɪɚɫɱɟɬɭ ɩɨ
ɮɨɪɦɭɥɟ (7.62). ȼɢɞɧɨ, ɱɬɨ ɪɟɡɭɥɶɬɚɬɵ ɫɨɜɩɚɞɚɸɬ.
ɉɚɪɚɦɟɬɪ J , ɜɯɨɞɹɳɢɣ ɜ ɪɟɲɟɧɢɟ, ɦɨɠɧɨ ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɬɟɪɦɢɱɟɫɤɭɸ ɬɨɥɳɢɧɭ ɩɨɤɪɵɬɢɹ. ɗɬɨ ɨɬɧɨɲɟɧɢɟ ɪɟɚɥɶɧɨɣ ɬɨɥɳɢɧɵ ɩɨɤɪɵɬɢɹ ɤ
ɬɨɥɳɢɧɟ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ, ɮɨɪɦɢɪɭɸɳɟɝɨɫɹ ɜ ɧɟɦ ɡɚ ɧɟɤɨɬɨɪɨɟ ɯɚɪɚɤɬɟɪɧɨɟ ɜɪɟɦɹ.
Ɍɨɝɞɚ, ɟɫɥɢ ɩɨɤɪɵɬɢɟ ɹɜɥɹɟɬɫɹ ɬɟɪɦɢɱɟɫɤɢ ɬɨɧɤɢɦ, J 1 , ɦɵ ɦɨɠɟɦ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɚɫɢɦɩɬɨɬɢɱɟɫɤɢɦ ɩɪɟɞɫɬɚɜɥɟɧɢɟɦ ɢɡɜɟɫɬɧɨɣ ɧɚɦ
2
ɮɭɧɤɰɢɢ erfc z . Ⱦɥɹ z 1 ɢɦɟɟɦ erfc z |
z ɢ
S
ª§
º
3G K c ·
G Kc · §
T1 T 2 | 1 H «¨1 H ¨1 ...» .
¸
¸
¨
W K O ¸¹ ¨©
W K O ¸¹
¬«©
¼»
Ɇɚɥɨɫɬɶ ɩɚɪɚɦɟɬɪɚ J ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ. Ɍɨɝɞɚ ɢɡ (7.60) ɧɚɯɨɞɢɦ
º
Kc
1 H ª
T1 T 2 |
p .. .»
«1 H G 1 3H p ¬«
KO
¼»
ɢ
186
ª
G 1 3H º §
Kc
Kc ·
G
.. .» | ¨1 1 4H ¸¸ ,
¨
K
K
SW
SW
O
O ¹
¬«
¼» ©
ɝɞɟ ɢɫɩɨɥɶɡɨɜɚɧɨ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɚɜɟɧɫɬɜɨ
1
1 H |
.
1 H
ȼ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɫɢɬɭɚɰɢɢ z !! 1 ɢɦɟɟɦ ɨɰɟɧɤɭ
T1 T 2
1 H «1 H er fc z |
.
exp z 2
z S
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɚɬɟɪɢɚɥɨɜ ɞɥɢɬɟɥɶɧɨɟ ɜɪɟɦɹ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɛɭɞɟɬ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɧɚɱɚɥɶɧɨɣ.
Ɋɚɫɫɦɨɬɪɢɦ ɟɳɟ ɨɞɢɧ ɩɪɢɦɟɪ ɪɟɲɟɧɢɹ ɫɨɩɪɹɠɟɧɧɨɣ ɡɚɞɚɱɢ 23. Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɬɟɩɥɨɜɨɣ ɤɨɧɬɚɤɬ ɦɟɠɞɭ ɩɨɤɪɵɬɢɟɦ ɢ ɨɫɧɨɜɨɣ ɧɟɥɶɡɹ ɫɱɢɬɚɬɶ ɢɞɟɚɥɶɧɵɦ. ɇɟɢɞɟɚɥɶɧɨɫɬɶ ɤɨɧɬɚɤɬɚ ɢɥɢ ɧɚɥɢɱɢɟ ɞɨɩɨɥɧɢɬɟɥɶɧɨɝɨ
ɬɟɪɦɢɱɟɫɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ, ɤɚɤ ɧɚɦ ɭɠɟ ɢɡɜɟɫɬɧɨ, ɦɨɠɟɬ ɛɵɬɶ ɫɜɹɡɚɧɚ
ɫ ɲɟɪɨɯɨɜɚɬɨɫɬɶɸ ɩɨɜɟɪɯɧɨɫɬɟɣ; ɩɪɨɰɟɫɫɚɦɢ ɝɚɡɢɮɢɤɚɰɢɢ ɢ ɬ.ɩ. ɩɪɢɱɢɧɚɦɢ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɨ ɧɚɯɨɠɞɟɧɢɢ ɩɨɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɩɪɟɞɵɞɭɳɟɝɨ ɬɨɥɶɤɨ ɭɫɥɨɜɢɟɦ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɚɬɟɪɢɚɥɨɜ. Ɍɟɩɥɨɜɵɟ ɩɨɬɨɤɢ, ɩɨ-ɩɪɟɠɧɟɦɭ, ɪɚɜɧɵ ɞɪɭɝ
ɞɪɭɝɭ, ɚ ɜɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬɥɢɱɚɸɬɫɹ
T1 T2
'
§ wT ·
O2 ¨ 2 ¸
O3 © wx ¹ x
,
(7.63)
h
ɝɞɟ ' O 3 – ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɧɚ ɤɨɧɬɚɤɬɟ ɦɚɬɟɪɢɚɥɨɜ (ɪɚɡɞɟɥ 2.10).
ɉɟɪɟɯɨɞɹ ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ, ɜɦɟɫɬɨ ɪɚɜɟɧɫɬɜɚ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɝɪɚɧɢɰɟ ɜ ɡɚɞɚɱɟ (7.56) ɧɚɯɨɞɢɦ
T1 T 2
D
wT 2
, [ G,
w[
(7.64)
hO 2
- ɧɨɜɵɣ ɩɚɪɚɦɟɬɪ.
O 3 a 2t ɉɪɨɰɟɞɭɪɚ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɩɨɥɧɨɫɬɶɸ ɚɧɚɥɨɝɢɱɧɚ ɩɪɟɞɵɞɭɳɟɦɭ. ȼɵɩɢɲɟɦ ɬɨɥɶɤɨ ɧɟɤɨɬɨɪɵɟ ɤɥɸɱɟɜɵɟ ɮɨɪɦɭɥɵ.
ȼ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɨɫɧɨɜɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɜ ɬɨɱɤɟ [ G ɢɦɟɟɦ
ɝɞɟ D
23
Ʉɧɹɡɟɜɚ Ⱥ.Ƚ., Ⱦɢɤ .Ƚ. Ɂɚɠɢɝɚɧɢɟ ɝɨɪɹɱɟɣ ɩɥɚɫɬɢɧɨɣ ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɨɝɨ ɜɟɳɟɫɬɜɚ
ɫ ɢɧɟɪɬɧɵɦ ɷɤɪɚɧɨɦ ɦɟɠɞɭ ɧɢɦɢ // Ɏɢɡ. ɝɨɪ. ɢ ɜɡɪ., 1990, Ɍ. 26, ʋ 2. ɋ. 8–18
187
T2
ɝɞɟ E
ª
Kc º
e x p « G
p»
K
2K H
O ¼
¬
,
ª
º
1 KH 1 D p
Kc
p»
1 E exp « 2G
K
O
¬
¼
(7.65)
.
1 K H 1 D p 1 KH 1 D p
ɉɪɢ ɨɱɟɜɢɞɧɨɦ ɭɫɥɨɜɢɢ ɦɚɥɨɫɬɢ ɩɚɪɚɦɟɬɪɚ D 1 ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
2 DK H p
.
E|H
2
1 K H Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ ɨɫɧɨɜɵ ɦɨɠɧɨ ɛɭɞɟɬ ɩɪɟɞɫɬɚɜɢɬɶ
ɜ ɮɨɪɦɟ
K H2
D
T2 T2
2
D 0
p§ K
·
c
p K H ¸¸
¨¨ G
K
O
©
¹
ȼ ɫɥɭɱɚɟ ɬɟɪɦɢɱɟɫɤɢ ɬɨɧɤɨɝɨ ɩɨɤɪɵɬɢɹ, ɩɟɪɟɯɨɞɹ ɤ ɨɪɢɝɢɧɚɥɚɦ,
ɦɨɠɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɮɨɪɦɭɥɨɣ
D
T2 | T2 D 0 . .. ,
SW
ɬ.ɟ., ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɤɪɵɬɢɹ ɢ ɨɫɧɨɜɵ ɩɪɚɤɬɢɱɟɫɤɢ
ɜɵɪɚɜɧɢɜɚɸɬɫɹ. ȼ ɨɰɟɧɤɚɯ ɢɫɩɨɥɶɡɨɜɚɧɵ ɫɥɟɞɭɸɳɢɟ ɩɪɢɛɥɢɠɟɧɧɵɟ
ɪɚɜɟɧɫɬɜɚ
2K H
1
1 H |
, 1 H {
.
1 KH
1 H
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɉɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
2. ɉɪɢ ɤɚɤɨɦ ɭɫɥɨɜɢɢ ɫɩɪɚɜɟɞɥɢɜɵ «ɧɭɥɶɦɟɪɧɵɟ» ɩɨɫɬɚɧɨɜɤɢ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ?
3. Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɯɚɪɚɤɬɟɪɧɵɣ ɪɚɡɦɟɪ ɬɟɥɚ?
4. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɤɚɱɟɫɬɜɟɧɧɨɟ ɨɬɥɢɱɢɟ ɜ ɩɨɜɟɞɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɢ ɨɬ ɜɪɟɦɟɧɢ ɞɥɹ ɝɪɚɧɢɱɧɵɯ ɡɚɞɚɱ ɫ ɭɫɥɨɜɢɹɦɢ 1, 2 ɢ 3
ɪɨɞɚ?
5. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɨɩɟɪɚɰɢɨɧɧɨɝɨ ɦɟɬɨɞɚ?
188
6. Ʉɚɤɢɦɢ ɮɢɡɢɱɟɫɤɢɦɢ ɜɟɥɢɱɢɧɚɦɢ ɦɨɠɧɨ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɢɦɩɭɥɶɫɧɨ-ɩɟɪɢɨɞɢɱɟɫɤɢɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ?
7. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɦɟɬɨɞ Ⱦɸɚɦɟɥɹ?
8. Ʉɚɤɢɟ ɧɨɜɵɟ ɮɢɡɢɱɟɫɤɢɟ ɩɚɪɚɦɟɬɪɵ ɩɨɹɜɥɹɸɬɫɹ ɜ ɫɨɩɪɹɠɟɧɧɵɯ
ɡɚɞɚɱɚɯ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ? ȼ ɱɟɦ ɢɯ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ?
9. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɫɭɬɶ ɩɪɢɛɥɢɠɟɧɧɨɝɨ ɦɟɬɨɞɚ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ?
10. ɑɬɨ ɦɨɠɧɨ ɫɤɚɡɚɬɶ ɨ ɬɟɦɩɟɪɚɬɭɪɚɯ ɤɨɧɬɚɤɬɢɪɭɸɳɢɯ ɫɪɟɞ ɜ ɫɥɭɱɚɟ ɧɟɢɞɟɚɥɶɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɤɨɧɬɚɤɬɚ?
Ɂɚɞɚɧɢɹ
1. ɂɫɩɨɥɶɡɭɹ ɩɨɥɭɱɟɧɧɵɟ ɬɨɱɧɵɟ ɪɟɲɟɧɢɹ ɧɟɫɬɚɰɢɨɧɚɪɧɵɯ ɡɚɞɚɱ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɪɟɲɢɬɶ ɫɥɟɞɭɸɳɭɸ ɡɚɞɚɱɭ. ɒɚɪɢɤɨɩɨɞɲɢɩɧɢɤɢ ɢɯ
ɯɪɨɦɢɫɬɨɣ ɫɬɚɥɢ ( O 50 ȼɬ/(ɦ·Ʉ), N 1,3 ˜105 ɦ2/ɫ) ɩɨɞɜɟɪɝɚɸɬɫɹ ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɟ. Ɉɧɢ ɧɚɝɪɟɜɚɸɬɫɹ ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ 650 ɋ, ɚ ɡɚɬɟɦ
ɪɟɡɤɨ ɨɯɥɚɠɞɚɸɬɫɹ ɜ ɜɚɧɧɟ ɫ ɦɚɫɥɨɦ, ɢɦɟɸɳɢɦ ɬɟɦɩɟɪɚɬɭɪɭ 55 ɋ. Ⱦɢɚɦɟɬɪ ɲɚɪɢɤɨɩɨɞɲɢɩɧɢɤɚ 4.0 ɫɦ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɨɬ ɲɚɪɢɤɨɩɨɞɲɢɩɧɢɤɚ ɤ ɦɚɫɥɭ 300 ȼɬ/(ɦ2·Ʉ). Ɉɩɪɟɞɟɥɢɬɶ, ɫɤɨɥɶɤɨ ɜɪɟɦɟɧɢ ɩɨɞɲɢɩɧɢɤɢ ɞɨɥɠɧɵ ɨɫɬɚɜɚɬɶɫɹ ɜ ɦɚɫɥɟ, ɩɨɤɚ ɢɯ ɬɟɦɩɟɪɚɬɭɪɚ ɧɟ ɫɧɢɡɢɬɫɹ
ɞɨ 200 ɋ; ɨɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ, ɨɬɞɚɧɧɨɟ ɤɚɠɞɵɦ ɩɨɞɲɢɩɧɢɤɨɦ ɡɚ
ɷɬɨ ɜɪɟɦɹ; ɡɧɚɱɟɧɢɹ ɦɝɧɨɜɟɧɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɨɬ ɩɨɞɲɢɩɧɢɤɨɜ ɜ
ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ, ɤɨɝɞɚ ɨɧɢ ɩɨɝɪɭɠɚɸɬɫɹ ɜ ɦɚɫɥɨ ɢ ɤɨɝɞɚ ɢɯ ɬɟɦɩɟɪɚɬɭɪɚ ɞɨɫɬɢɝɚɟɬ 200 ɋ.
2. Ɍɟɦɩɟɪɚɬɭɪɚ ɬɨɪɰɟɜɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫɬɚɥɶɧɨɝɨ ɰɢɥɢɧɞɪɚ ɛɨɥɶɲɨɣ
ɞɥɢɧɵ ɦɝɧɨɜɟɧɧɨ ɩɨɜɵɲɚɟɬɫɹ ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ 900 ɋ ɢ ɡɚɬɟɦ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ. ɇɚɣɬɢ, ɡɚ ɤɚɤɨɟ ɜɪɟɦɹ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚ ɪɚɫɫɬɨɹɧɢɢ 1
ɫɦ ɨɬ ɧɚɝɪɟɜɚɟɦɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɜɵɫɢɬɫɹ ɧɚ 50 ɋ, ɧɚ 100 ɋ, ɟɫɥɢ ɛɨɤɨɜɵɟ ɩɨɜɟɪɯɧɨɫɬɢ ɰɢɥɢɧɞɪɚ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɵ. ɉɪɢɧɹɬɶ ɜ ɪɚɫɱɟɬɚɯ
O 70 ȼɬ/(ɦ·Ʉ), a 2,0 ˜105 ɦ2/ɫ.
3. ɂɫɩɨɥɶɡɭɹ ɨɩɟɪɚɰɢɨɧɧɵɣ ɦɟɬɨɞ, ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ
wT
w § wT ·
cU
O
w t w x ¨© w x ¸¹
T 0 ,t T0 Ts T0 cos Z t x o f : T T0
t 0 : T T0
4. ɇɚɣɬɢ ɪɟɲɟɧɢɟ ɷɬɨɣ ɠɟ ɡɚɞɚɱɢ, ɢɫɩɨɥɶɡɭɹ ɦɟɬɨɞ Ⱦɸɚɦɟɥɹ.
189
ɑȺɋɌɖ 8
Ʉ ɥ ɚ ɫ ɫ ɢ ɱ ɟ ɫ ɤ ɢ ɟ ɦɟ ɬ ɨ ɞ ɵ ɪ ɟ ɲ ɟ ɧ ɢ ɹ
ɫ ɬ ɚ ɰ ɢ ɨ ɧ ɚ ɪ ɧ ɵ ɯ ɢ ɧ ɟ ɫ ɬ ɚ ɰ ɢ ɨ ɧɚ ɪ ɧ ɵ ɯ ɡ ɚ ɞ ɚ ɱ
8.1. Ɋɟɲɟɧɢɟ ɤɪɚɟɜɵɯ ɡɚɞɚɱ
ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ
Ⱦɨɜɨɥɶɧɨ ɱɚɫɬɨ ɪɟɲɟɧɢɟ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɞɜɭɯ- ɢ
ɬɪɟɯɦɟɪɧɵɯ ɨɛɥɚɫɬɹɯ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɪɟɲɟɧɢɣ ɨɞɧɨɦɟɪɧɵɯ ɡɚɞɚɱ. Ⱦɥɹ ɷɬɨɝɨ ɧɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɞɨɥɠɧɚ ɜɵɪɚɠɚɬɶɫɹ ɜ
ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɢ ɮɭɧɤɰɢɣ, ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɞɧɨɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɩɟɪɟɦɟɧɧɨɣ, ɚ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɞɨɥɠɧɵ
ɫɥɭɠɢɬɶ ɭɫɥɨɜɢɹ ɥɢɛɨ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭɪɵ, ɥɢɛɨ ɧɭɥɟɜɨɝɨ ɩɨɬɨɤɚ, ɥɢɛɨ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɫɨ ɫɪɟɞɨɣ ɧɭɥɟɜɨɣ ɬɟɦɩɟɪɚɬɭɪɵ. ɉɨɹɫɧɢɦ ɷɬɨ ɧɚ ɩɪɢɦɟɪɟ ɪɟɲɟɧɢɹ ɫɥɟɞɭɸɳɟɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ:
w 2T
w 2T
1 w T w 2T
,
(8.1)
a w t w x1 2 w x 2 2 w x 3 2
0 x i l i , t ! 0 , i 1, 2 ,3 ;
T 0 , x1 , x 2 , x 3 ) 10 x1 ˜ ) 20 x 2 ˜ ) 30 x 3 ,
§ wT
·
E iT ¸
¨Oi
© w xi
¹ xi
0 , i 1, 2, 3 ,
(8.2)
(8.3)
0
§ wT
·
G iT ¸
0 , i 1, 2, 3 .
(8.4)
¨ Ji
x
w
©
¹ xi l i
i
Ɍɚɤ ɤɚɤ ɭ ɧɚɫ ɢɦɟɟɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɩɨ ɤɚɠɞɨɣ ɢɡ ɬɪɟɯ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ xi , ɭ ɧɚɫ ɡɚɞɚɧɨ ɲɟɫɬɶ
ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ. Ɂɞɟɫɶ ɧɟɡɚɜɢɫɢɦɵɟ ɩɟɪɟɦɟɧɧɵɟ ɨɛɨɡɧɚɱɟɧɵ ɱɟɪɟɡ
x1 , x2 , x3 , ɚ ɧɟ ɱɟɪɟɡ x , y , z .
ɉɨɤɚɠɟɦ, ɱɬɨ ɪɟɲɟɧɢɟ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ (8.1) – (8.2) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɪɟɲɟɧɢɣ ɨɞɧɨɦɟɪɧɵɯ ɡɚɞɚɱ
T t , x1 , x 2 , x 3 T1 t , x1 ˜ T2 t , x 2 ˜ T3 t , x 3 ,
(8.5)
ɟɫɥɢ Ti t , x i ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɭɫɥɨɜɢɹɦ ( i 1, 2, 3 )
w Ti
wt
a
w 2Ti
w xi 2
, 0 xi li , t ! 0 ,
190
(8.6)
Ti 0 , x i ) i 0 x i ,
(8.7)
§ wT
·
§ wT
·
(8.8)
0 , ¨ Ji
G iT ¸
0.
E iT ¸
¨Oi
x
x
w
w
©
¹ xi l i
©
¹ xi 0
i
i
ɉɨɞɫɬɚɜɥɹɹ (8.5) ɜ (8.1), ɧɚɣɞɟɦ
§ wT3
§ wT1
§ wT 2
w 2T3 ·
w 2T1 ·
w 2T2 ·
T2 ˜ T3 ¨
a 2 ¸ T1 ˜ T3 ¨
a
a
¸ 0,
¸ T1 ˜ T2 ¨
2¸
2¸
¨ wt
¸
¨ wt
¨ wt
w
x
w
x
w
x
1 ¹
2 ¹
3 ¹
©
©
©
ɬɚɤ ɤɚɤ ɜɵɪɚɠɟɧɢɹ ɜ ɫɤɨɛɤɚɯ ɪɚɜɧɵ ɧɭɥɸ ɜ ɫɢɥɭ ɭɪɚɜɧɟɧɢɣ (8.6). ɉɪɢ
ɷɬɨɦ, ɨɱɟɜɢɞɧɨ, ɭɞɨɜɥɟɬɜɨɪɹɸɬɫɹ ɧɚɱɚɥɶɧɨɟ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (8.7),
(8.8).
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜ ɩɪɹɦɨɭɝɨɥɶɧɨɦ
ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɟ ɡɚɞɚɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
§ x 2·
§ x 2·
§ x 2·
T t , x1 , x 2 , x 3 Tme xp ¨ 1 2 ¸ e xp ¨ 2 2 ¸ ex p ¨ 3 2 ¸ , (8.9)
¨ R ¸
¨ R ¸
¨ R ¸
© 1 ¹
©
©
2 ¹
3 ¹
ɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɷɬɨɦ ɩɚɪɚɥɥɟɥɟɩɢɩɟɞɟ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ
ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɛɭɞɟɬ ɫɥɟɞɨɜɚɬɶ ɢɡ ɪɟɲɟɧɢɹ ɡɚɞɚɱ (8.6)-(8.8), ɝɞɟ
§ xi 2 ·
13
(8.10)
) i 0 x i Tm e x p ¨ 2 ¸ .
¨ R ¸
i ¹
©
Ɂɚɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ ɨɬɥɢɱɧɚ ɨɬ ɧɭɥɟɜɨɣ, ɧɚɩɪɢɦɟɪ, ɪɚɜɧɚ T0 , ɬɨ ɫɞɟɥɚɜ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ T T T0 , ɡɚɞɚɱɭ
ɥɟɝɤɨ ɫɜɟɫɬɢ ɤ ɬɨɥɶɤɨ ɱɬɨ ɪɚɫɫɦɨɬɪɟɧɧɨɣ.
Ⱥɧɚɥɨɝɢɱɧɵɣ ɩɪɢɟɦ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɪɢ ɧɚɯɨɠɞɟɧɢɢ ɩɨɥɹ
ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɥɨɦ ɢɥɢ ɫɩɥɨɲɧɨɦ ɰɢɥɢɧɞɪɟ, ɟɫɥɢ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɦ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɭɧɤɰɢɣ, ɡɚɜɢɫɹɳɢɯ ɨɬ ɪɚɞɢɚɥɶɧɨɣ ɢ ɨɫɟɜɨɣ ɤɨɨɪɞɢɧɚɬ.
8.2. Ɇɟɬɨɞ ɢɫ ɬɨɱɧɢ ɤɨɜ
Ɏɢɡɢɱɟɫɤɚɹ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɢɫɬɨɱɧɢɤɨɜ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɥɸɛɨɣ
ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɬɟɩɥɚ ɜ ɬɟɥɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ ɫɨɜɨɤɭɩɧɨɫɬɶ ɩɪɨɰɟɫɫɨɜ ɜɵɪɚɜɧɢɜɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ
ɦɧɨɠɟɫɬɜɚ ɷɥɟɦɟɧɬɚɪɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ, ɪɚɫɩɪɟɞɟɥɟɧɧɵɯ ɤɚɤ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬɚɤ ɢ ɜɨ ɜɪɟɦɟɧɢ. Ɋɟɲɟɧɢɟ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨ ɷɬɨɦɭ
ɦɟɬɨɞɭ ɫɜɨɞɢɬɫɹ, ɜ ɨɫɧɨɜɧɨɦ, ɤ ɩɪɚɜɢɥɶɧɨɦɭ ɜɵɛɨɪɭ ɢɫɬɨɱɧɢɤɨɜ ɢ ɢɯ
ɪɚɫɩɪɟɞɟɥɟɧɢɸ.
Ɍɚɤ, ɞɟɣɫɬɜɢɟ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ ɬɟɥɟ ɩɪɢ
ɨɞɧɨɦɟɪɧɨɦ ɩɨɬɨɤɟ ɬɟɩɥɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɮɨɪɦɭɥɨɣ
191
§ x [ 2 ·
T
¸,
G x ,[ ,t exp ¨ (8.11)
4at ¸
¨
4S at
©
¹
ɧɚɡɵɜɚɟɦɨɣ ɮɭɧɤɰɢɟɣ ɢɫɬɨɱɧɢɤɚ (ɮɭɧɤɰɢɟɣ Ƚɪɢɧɚ) ɧɚ ɛɟɫɤɨɧɟɱɧɨɣ
ɩɪɹɦɨɣ.
Ɏɭɧɤɰɢɹ G x , [ ,t ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
wT
w 2T
a 2.
(8.12)
wt
wx
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɞɢɮɮɟɪɟɧɰɢɪɭɟɦ (8.11) ɩɨ ɜɪɟɦɟɧɢ ɢ ɞɜɚɠɞɵ ɩɨ ɤɨɨɪɞɢɧɚɬɟ
2
§ x [ 2 ·
wG
T ª x [
1º
¸;
«
» exp ¨ 2t »
4at ¸
wt
¨
4S at « 4at
¬
¼
©
¹
2
§ x [ 2 ·
w 2G
T ª x [
1º
¸,
«
» exp ¨ 2
4
at
2
t
¨
4
at
¸
4 S at «
»
wx
¬
¼
©
¹
ɬ.ɟ.
wG
w 2G
a 2.
wt
wx
Ɏɭɧɤɰɢɸ G x , [ ,t ɨɛɵɱɧɨ ɧɚɡɵɜɚɸɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɦ ɪɟɲɟɧɢɟɦ
ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɇɟɩɨɫɪɟɞɫɬɜɟɧɧɨɣ ɩɪɨɜɟɪɤɨɣ ɦɨɠɧɨ ɭɛɟɞɢɬɶɫɹ, ɱɬɨ ɮɭɧɤɰɢɹ G ɩɪɟɞɫɬɚɜɥɹɟɬ ɬɟɦɩɟɪɚɬɭɪɭ ɜ ɬɨɱɤɟ x , ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɜ ɬɨɱɤɟ [ ɜɵɞɟɥɹɟɬɫɹ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ
Q T cU .
Ʉɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ ɧɚ ɩɪɹɦɨɣ ɪɚɜɧɨ
Q
f
§ x [ 2 · dx
cU T
³ exp ¨¨ 4at ¸¸ 2 at
S f
©
¹
f
f
cU T
exp u 2 du
³
S f
cU T ,
(8.13)
2
x[
ɝɞɟ
; ³ e u d u
S.
u
2 a t f
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q ɧɟ ɦɟɧɹɟɬɫɹ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɢ ɱɢɫɥɟɧɧɨ ɪɚɜɧɨ ɩɪɨɢɡɜɟɞɟɧɢɸ ɩɥɨɳɚɞɢ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɤɪɢɜɨɣ G ɢ
ɨɫɶɸ ɚɛɫɰɢɫɫ x , ɧɚ ɨɛɴɟɦɧɭɸ ɬɟɩɥɨɟɦɤɨɫɬɶ cU . Ⱦɥɹ ɦɚɥɵɯ ɡɧɚɱɟɧɢɣ
ɜɪɟɦɟɧɢ ɩɨɱɬɢ ɜɫɟ ɬɟɩɥɨ ɫɨɫɪɟɞɨɬɨɱɟɧɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ [ .
192
Ɏɭɧɤɰɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɜɥɢɹɧɢɹ ɦɝɧɨɜɟɧɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɞɥɹ
ɬɟɥɚ ɤɨɧɟɱɧɵɯ ɪɚɡɦɟɪɨɜ ɢ ɨɞɧɨɦɟɪɧɨɝɨ ɩɨɬɨɤɚ ɬɟɩɥɚ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɬɚɤ:
§ n 2S 2 ·
2T f
§ nS x ·
§ nS[ ·
(8.14)
Gl x , [ , t ¦ sin ¨© l ¸¹ sin ¨© l ¸¹ exp ¨¨ l 2 at ¸¸ .
l n 1
©
¹
Ɏɭɧɤɰɢɹ Gl ɞɚɟɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ
ɩɥɚɫɬɢɧɟ ɤɨɧɟɱɧɨɣ ɬɨɥɳɢɧɵ 0 x l ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t , ɟɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɜɧɚ ɧɭɥɸ ɢ ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜ
ɬɨɱɤɟ [ ɦɝɧɨɜɟɧɧɨ ɜɵɞɟɥɹɟɬɫɹ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɚ Q T cU .
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɯɨɠɞɟɧɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ ɬɟɥɟ (ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox ) ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɡɚɞɚɧɨ:
T x ,0 f x .
(8.15)
Ⱦɥɹ ɥɸɛɨɝɨ x ɫɩɪɚɜɟɞɥɢɜɨ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜɢɞɚ (8.12),
ɚ ɧɚ ɛɟɫɤɨɧɟɱɧɨɦ ɭɞɚɥɟɧɢɢ ɨɬ ɧɚɱɚɥɚ ɤɨɨɪɞɢɧɚɬ ɢɫɬɨɱɧɢɤɢ ɢ ɫɬɨɤɢ ɬɟɩɥɚ ɨɬɫɭɬɫɬɜɭɸɬ
w T f , t w T f ,t 0.
wx
wx
Ɋɟɲɢɦ ɡɚɞɚɱɭ ɦɟɬɨɞɨɦ ɢɫɬɨɱɧɢɤɨɜ, ɧɟ ɧɚɥɚɝɚɸɳɢɦ ɧɢɤɚɤɢɯ ɨɝɪɚɧɢɱɟɧɢɣ ɧɚ ɮɭɧɤɰɢɸ f x .
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.12), ɤɚɤ ɩɨɤɚɡɚɧɨ ɜɵɲɟ, ɢɦɟɟɬ ɜɢɞ
§ x [ 2 ·
C
¸.
T
exp ¨ (8.16)
¨
4at ¸
4S at
©
¹
ɂɡ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɢɞɧɨ, ɱɬɨ
ɩɪɢ ɡɚɞɚɧɧɨɦ ɜɪɟɦɟɧɢ t ɤɪɢɜɚɹ
ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɚɩɪɚɜɥɟɧɢɢ x ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ, ɤɨɬɨɪɵɣ ɧɚɯɨɞɢɬɫɹ ɜ ɬɨɱɤɟ x [
(ɪɢɫ. 8.1). ɉɟɪɟɧɟɫɟɦ ɧɚɱɚɥɨ ɤɨɨɪɞɢɧɚɬ ɜ ɷɬɭ ɬɨɱɤɭ. ɉɥɨɳɚɞɶ S ɩɨɞ
ɤɪɢɜɨɣ, ɬ.ɟ. ɩɥɨɳɚɞɶ, ɨɛɪɚɡɨɜɚɧɧɚɹ
ɤɪɢɜɨɣ ɢ ɨɫɶɸ ɚɛɫɰɢɫɫ, ɟɫɬɶ ɜɟɥɢɱɢɧɚ ɤɨɧɟɱɧɚɹ ɢ ɪɚɜɧɚɹ ɢɧɬɟɝɪɚɥɭ
ɨɬ (8.16) ɜ ɩɪɟɞɟɥɚɯ ɨɬ f ɞɨ f :
193
T
[
t1
t2 ! t1
t3 ! t 2
x
Ɋɢɫ .8.1. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ
ɬɟɦɩɟɪɚɬɭɪɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ
ɮɨɪɦɭɥɟ (8.16)
f
S
³
f
§ x [ 2 ·
C
¸ d x [
exp ¨ ¨
4at ¸
4S at
©
¹
f
C
z 2
e
dz C .
³
S f
C
.
4S a t
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɨɪɞɢɧɚɬɚ ɭɦɟɧɶɲɚɟɬɫɹ, ɢ
ɤɪɢɜɚɹ ɫɬɚɧɨɜɢɬɫɹ ɛɨɥɟɟ ɩɨɥɨɝɨɣ. ɉɪɢ ɭɦɟɧɶɲɟɧɢɢ ɜɪɟɦɟɧɢ t o 0 ɩɨɥɭɱɚɟɬɫɹ ɛɟɫɤɨɧɟɱɧɨ ɭɡɤɚɹ ɩɨɥɨɫɤɚ, ɧɨ ɩɥɨɳɚɞɶ ɟɟ ɫɨɯɪɚɧɹɟɬɫɹ. Ɉɧɚ
ɪɚɜɧɚ ɩɨɫɬɨɹɧɧɨɣ C .
ɉɨɥɶɡɭɹɫɶ ɷɬɢɦ ɫɜɨɣɫɬɜɨɦ, ɦɨɠɧɨ ɡɚɞɚɧɧɨɟ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ T x , 0 f x ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ ɬɟɥɟ ɩɪɟɞɫɬɚɜɢɬɶ ɤɚɤ
Ɉɪɞɢɧɚɬɚ ɤɪɢɜɨɣ ɜ ɬɨɱɤɟ ɦɚɤɫɢɦɭɦɚ ɪɚɜɧɚ
ɫɭɦɦɭ ɨɬɞɟɥɶɧɵɯ ɱɚɫɬɧɵɯ ɪɟɲɟɧɢɣ ɜɢɞɚ (8.16), ɬ.ɟ. ɤɪɢɜɭɸ f x ɡɚɦɟɧɢɬɶ ɫɭɦɦɨɣ ɛɟɫɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɤɪɢɜɵɯ ɜɢɞɚ
ª
§ x [ 2 ·º
C
¸» .
li m «
e x p¨ ¨
4a t ¸ »
t o0 « 4 S a t
©
¹¼
¬
ɉɪɢ ɷɬɨɦ, ɧɟ ɫɦɨɬɪɹ ɧɚ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɭɸ
ɲɢɪɢɧɭ ɨɬɞɟɥɶɧɨɣ ɩɨɥɨɫɤɢ d[ , ɜɵɫɨɬɚ ɟɟ ɛɭɞɟɬ ɜɟɥɢɱɢɧɨɣ ɤɨɧɟɱɧɨɣ ɢ ɪɚɜɧɨɣ f [ . ɉɥɨɊɢɫ. 8.2. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɳɚɞɶ ɬɚɤɨɣ ɩɨɥɨɫɤɢ, ɪɚɜɧɚɹ C , ɛɭɞɟɬ ɩɨɫɬɨɹɧɧɨɣ ɜɟɥɢɱɢɧɨɣ, ɬ.ɟ. (ɪɢɫ. 8.2)
ɦɟɬɨɞɭ ɢɫɬɨɱɧɢɤɨɜ
T [ ,0 d [ f [ d [ C .
ɉɨɥɧɨɟ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɨɝɪɚɧɢɱɟɧɧɨɦ
ɬɟɥɟ ɛɭɞɟɬ ɪɚɜɧɨ
f
ª
§ x [ 2 · º
1
«
¸ d [» .
lim T x ,t lim
f [ exp ¨ ³
¨
4a t ¸ »
t o0
t o0 « 4S a t
f
©
¹ ¼
¬
Ɍɚɤɨɟ ɠɟ ɫɨɨɬɧɨɲɟɧɢɟ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɨ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɧɚɱɚɥɶɧɨɝɨ
ɦɨɦɟɧɬɚ ɜɪɟɦɟɧɢ, ɧɨ ɢ ɞɥɹ ɥɸɛɨɝɨ ɩɨɫɥɟɞɭɸɳɟɝɨ ɩɪɨɦɟɠɭɬɤɚ ɜɪɟɦɟɧɢ,
ɬ.ɟ. ɨɛɳɢɦ ɪɟɲɟɧɢɟɦ ɧɚɲɟɣ ɡɚɞɚɱɢ ɛɭɞɟɬ
f
§ x [2 ·
1
¸d[ .
(8.17)
T x ,t f [ exp ¨ ³
¨
¸
4
at
4S at f
©
¹
Ɇɨɠɧɨ ɫɞɟɥɚɬɶ ɨɛɨɛɳɟɧɢɟ ɧɚ ɫɥɭɱɚɢ ɩɥɨɫɤɨɣ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ
ɡɚɞɚɱ.
194
Ɍɚɤ, ɟɫɥɢ ɜ ɩɥɨɫɤɨɫɬɢ ɡɚɞɚɧɨ ɧɚɱɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
T 0 , x , y f x , y , ɬɨ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɛɭɞɟɦ ɢɦɟɬɶ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ
f f
§ x [ 2 y K 2 ·
1
¸ d [d K . (8.18)
T x , y ,t f [ , K e xp ¨ ¨
¸
4S at ³ ³
4at
f f
©
¹
Ⱥɧɚɥɨɝɢɱɧɨ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
ɨɩɢɫɵɜɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟɦ
1
T x , y , z u
4Sat 3
f f f
(8.19)
§ x [ 2 y K2 z ] 2 ·
¸d[dKd]
f [ ,K,] exp¨ u
¨
¸
4
at
©
¹
³³³
f f f
ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t 0 ɛɵɥɨ ɡɚɞɚɧɨ
T 0, x , y , z f x , y , z .
ɉɪɢɦɟɪ. ɉɭɫɬɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɡɚɞɚɧɚ ɬɟɦɩɟɪɚɬɭɪɚ
ɩɨɥɭɨɝɪɚɧɢɱɟɧɧɨɝɨ ɬɟɥɚ T x , 0 f x . ȼ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ (ɢɥɢ ɬɨɪɰɚ ɫɬɟɪɠɧɹ) ɩɪɢɧɢɦɚɟɬ ɬɟɦɩɟɪɚɬɭɪɭ
Ts , ɤɨɬɨɪɚɹ ɡɚɬɟɦ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɜɪɟɦɟɧɢ. Ɏɚɤɬɢɱɟɫɤɢ ɷɬɨ – ɡɚɞɚɱɚ ɨɛ ɨɫɬɵɜɚɧɢɢ ɬɟɥɚ ɫ ɡɚɞɚɧɧɨɣ ɧɚɱɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ.
Ɋɟɲɟɧɢɟ. Ɇɵ ɢɦɟɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (8.12) ɢ ɭɫɥɨɜɢɹ
T x ,0 f x ;
T 0 ,t Ts
const ;
w T f ,t wx
Ⱦɥɹ ɧɚɱɚɥɚ ɩɨɥɨɠɢɦ Ts 0 .
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɦɨɠɟɬ ɛɵɬɶ
ɩɨɥɭɱɟɧɨ ɢɡ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. Ⱦɥɹ ɷɬɨɝɨ ɩɪɨɞɨɥɠɢɦ ɫɬɟɪɠɟɧɶ ɜ ɨɬɪɢɰɚɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɨɫɢ Ox , ɬ.ɟ. ɛɭɞɟɦ ɫɱɢɬɚɬɶ
ɟɝɨ ɧɟɨɝɪɚɧɢɱɟɧɧɵɦ (ɪɢɫ. 8.3).
ɇɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɬɨɱɤɟ
x ! 0 ɟɫɬɶ f x , ɚ ɧɚɱɚɥɶɧɭɸ ɬɟɦɩɟ195
0.
T
f x ɭɫɥɨɜɢɟ ɧɚ
ɩɨɜɟɪɯɧɨɫɬɢ
ɜɵɩɨɥɧɹɟɬɫɹ
0
x
f x
Ɋɢɫ. 8.3. ɉɨɹɫɧɟɧɢɟ ɤ ɪɟɲɟɧɢɸ
ɡɚɞɚɱɢ ɞɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ
ɪɚɬɭɪɭ ɜ ɬɨɱɤɟ x ɜɵɛɢɪɚɟɦ ɪɚɜɧɨɣ f x , ɬ.ɟ. ɫɱɢɬɚɟɦ ɮɭɧɤɰɢɸ f x ɧɟɱɟɬɧɨɣ, f x f x .
Ɍɨɝɞɚ ɢɡ ɫɨɨɛɪɚɠɟɧɢɣ ɫɢɦɦɟɬɪɢɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɫɥɟɞɭɸɳɢɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ ɛɭɞɟɬ ɧɟɱɟɬɧɨɣ ɮɭɧɤɰɢɟɣ, ɚ ɞɥɹ x 0 ɟɟ
ɡɧɚɱɟɧɢɟ ɜɫɟɝɞɚ ɛɭɞɟɬ ɪɚɜɧɨ ɧɭɥɸ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɜɵɩɨɥɧɹɟɬɫɹ.
ɉɪɢ ɡɚɦɟɧɟ x ɧɚ [ ɜ ɤɪɢɜɨɣ ɧɚɱɚɥɶɧɨɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ
ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɚ ɨɫɧɨɜɚɧɢɢ ɩɪɟɞɵɞɭɳɟɝɨ ɢɦɟɟɬ ɜɢɞ
­ f
§ x [ 2 ·
1 °
¸ d[ T x ,t ® ³ f [ e x p¨ 4a t ¸
¨
4S a t ° f
©
¹
¯
f
§ x [ 2 · ½°
¸ d[¾
³ f [ e x p ¨ 4a t ¸ °
¨
f
©
¹ ¿
ɢɥɢ
f
ª
§ x [ 2 ·
§ x [ 2 ·º
1
¸ e x p¨ ¸ » d [ . (8.20)
T x ,t f [ «e x p ¨ ³
4
a
t
4
a
t
¨
¸
¨
¸»
«
2 S a t f
©
¹
©
¹¼
¬
ɗɬɨ ɢ ɟɫɬɶ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɧɚɲɟɣ ɡɚɞɚɱɢ.
ȿɫɥɢ ɧɚɱɚɥɶɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɫɬɨɹɧɧɚ ɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ x (ɧɚɩɪɢɦɟɪ,
ɬɟɦɩɟɪɚɬɭɪɚ ɫɬɟɪɠɧɹ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɨɞɢɧɚɤɨɜɚ ɢ
ɪɚɜɧɚ T0 ), ɬ.ɟ. T x ,0 T0 const , ɬɨ ɪɟɲɟɧɢɟ ɦɨɠɧɨ ɭɩɪɨɫɬɢɬɶ. ȼ ɩɟɪɜɭɸ ɱɚɫɬɶ ɩɨɞɢɧɬɟɝɪɚɥɶɧɨɣ ɮɭɧɤɰɢɢ ɩɨɞɫɬɚɜɢɦ
[ x 2u at ,
ɚ ɜɨ ɜɬɨɪɭɸ –
[ x 2u at .
Ɍɨɝɞɚ ɩɨɥɭɱɢɦ
T x ,t Ɍɚɤ ɤɚɤ ɮɭɧɤɰɢɹ e u
ɬɨ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ
T x ,t T0
2
2
S
T0
S
x 2 at
³
2
e u d u .
x 2 at
ɹɜɥɹɟɬɫɹ ɫɢɦɦɟɬɪɢɱɧɨɣ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ u ,
x 2 at
³
2
e u du
0
ɡɧɚɤɨɦɚɹ ɧɚɦ ɮɭɧɤɰɢɹ.
196
§ x ·
erf ¨
¸ –
© 2 at ¹
(8.21)
ȿɫɥɢ Ts z 0 , ɬɨ, ɞɟɥɚɹ ɡɚɦɟɧɭ ɩɟɪɟɦɟɧɧɵɯ w T Ts , ɫɜɟɞɟɦ ɡɚɞɚɱɭ
ɤ ɬɨɥɶɤɨ ɱɬɨ ɪɚɫɫɦɨɬɪɟɧɧɨɣ (ɬɚɤ ɤɚɤ w 0 , t T 0 ,t Ts 0 ).
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɜɵɝɥɹɞɢɬ ɬɚɤ
T x ,t Ts
§ x ·
erf ¨
(8.22)
¸.
T0 Ts
© 2 at ¹
ɗɬɭ ɩɪɨɫɬɭɸ ɡɚɞɚɱɭ, ɨɱɟɜɢɞɧɨ, ɦɨɠɧɨ ɪɟɲɢɬɶ ɢ ɨɩɟɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ, ɱɬɨ ɦɵ ɢ ɫɞɟɥɚɥɢ ɜ ɪɚɡɞɟɥɟ 7.3.
8.3. Ɇɟɬɨɞ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ
8.3.1. Ɂɚɞɚɱɚ ɞɥɹ ɤɪɭɝɚ
Ⱦɜɭɦɟɪɧɵɟ ɫɬɚɰɢɨɧɚɪɧɵɟ ɤɪɚɟɜɵɟ ɡɚɞɚɱɢ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɫɜɨɞɹɬɫɹ ɤ ɪɟɲɟɧɢɸ ɭɪɚɜɧɟɧɢɹ Ʌɚɩɥɚɫɚ
div O grad T qV 0
q
ɢɥɢ
'T V 0
O
ɫ ɡɚɞɚɧɧɵɦɢ ɤɪɚɟɜɵɦɢ ɭɫɥɨɜɢɹɦɢ. ȼ ɫɥɭɱɚɟ ɧɟɤɨɬɨɪɵɯ ɩɪɨɫɬɟɣɲɢɯ ɨɛɥɚɫɬɟɣ (ɤɪɭɝ, ɩɪɹɦɨɭɝɨɥɶɧɢɤ, ɲɚɪ, ɰɢɥɢɧɞɪ) ɪɟɲɟɧɢɟ ɬɚɤɢɯ ɡɚɞɚɱ ɦɨɠɟɬ
ɛɵɬɶ ɧɚɣɞɟɧɨ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. ɉɨɥɭɱɚɸɳɢɟɫɹ ɜ ɯɨɞɟ
ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɧɚ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ (ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɡɚɞɚɱɢ
ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ) ɩɪɢɜɨɞɹɬ ɤ ɪɚɡɥɢɱɧɵɦ ɤɥɚɫɫɚɦ ɫɩɟɰɢɚɥɶɧɵɯ ɮɭɧɤɰɢɣ, ɢɡɭɱɟɧɢɸ ɫɜɨɣɫɬɜ ɤɨɬɨɪɵɯ ɩɨɫɜɹɳɟɧɵ ɫɩɟɰɢɚɥɶɧɵɟ ɪɚɡɞɟɥɵ ɤɭɪɫɨɜ ɢ ɭɱɟɛɧɢɤɨɜ ɩɨ ɦɟɬɨɞɚɦ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɮɢɡɢɤɢ.
ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
ɜɧɭɬɪɢ ɤɪɭɝɚ ɫ ɡɚɞɚɧɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɟɝɨ ɝɪɚɧɢɰɟ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɦ ɧɭɠɧɨ ɪɟɲɢɬɶ ɡɚɞɚɱɭ
'T 0 ,
(8.23)
T 0 , R0 ,M f M ,
(8.24)
ɝɞɟ f M – ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ.
Ʉ ɬɚɤɨɣ ɡɚɞɚɱɟ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɢɡɭɱɟɧɢɟ ɩɨɜɟɞɟɧɢɹ ɨɛɪɚɡɰɚ ɜ ɮɨɪɦɟ ɬɨɧɤɨɝɨ ɞɢɫɤɚ ɜ ɭɫɥɨɜɢɹɯ
ɡɚɞɚɧɧɨɝɨ ɧɟɫɢɦɦɟɬɪɢɱɧɨɝɨ ɧɚɝɪɟɜɚ.
ȼɜɟɞɟɦ ɩɨɥɹɪɧɭɸ ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ (ɪɢɫ. 8.4)
ɫ ɧɚɱɚɥɨɦ ɜ ɰɟɧɬɪɟ ɤɪɭɝɚ. ȼ ɷɬɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
ɭɪɚɜɧɟɧɢɟ Ʌɚɩɥɚɫɚ ɢɦɟɟɬ ɜɢɞ
Ɋɢɫ. 8.4. Ʉ ɩɨɫɬɚ1 w § w T · 1 w 2T
ɧɨɜɤɟ ɡɚɞɚɱɢ ɞɥɹ
'T
(8.25)
0.
r
r w r ¨© w r ¸¹ r 2 wM 2
ɤɪɭɝɚ
197
Ȼɭɞɟɦ ɢɫɤɚɬɶ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.25) ɜ ɜɢɞɟ
T r ,M R r ˜ ) M z 0 .
(8.26)
ɉɨɞɫɬɚɜɢɜ (8.26) ɜ ɭɪɚɜɧɟɧɢɟ Ʌɚɩɥɚɫɚ (8.25), ɧɚɯɨɞɢɦ
1 w § wT ·
1 w § wR ·
¨r
¸ ) M
¨r
¸,
r wr © wr ¹
r wr © wr ¹
1 w 2T
1 w 2)
R
r
2 2.
2
2
r wM
r wM
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
d § dR ·
¨r
¸
)cc
dr © dr ¹
const O2 .
(8.27)
Rr
)
ȼ ɭɪɚɜɧɟɧɢɢ (8.27) ɩɟɪɟɦɟɧɧɵɟ ɪɚɡɞɟɥɟɧɵ, ɬ.ɟ. ɥɟɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ r , ɩɪɚɜɚɹ – ɬɨɥɶɤɨ ɨɬ M . ɉɨɫɤɨɥɶɤɭ r ɢ M ɧɟ ɡɚɜɢɫɹɬ ɞɪɭɝ ɨɬ ɞɪɭɝɚ, ɬɨ ɤɚɠɞɚɹ ɱɚɫɬɶ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɨɥɠɧɚ ɛɵɬɶ ɤɨɧɫɬɚɧɬɨɣ, ɤɨɬɨɪɭɸ ɨɛɨɡɧɚɱɢɥɢ ɱɟɪɟɡ O 2 . ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɢɦɟɟɦ ɞɜɚ
ɭɪɚɜɧɟɧɢɹ
w § wR ·
2R
0, R z 0,
(8.28)
¨r
¸O
r
wr © wr ¹
(8.29)
)cc O 2) 0 ,
ɪɟɲɟɧɢɹ ɤɨɬɨɪɵɯ ɦɵ ɦɨɠɟɦ ɧɚɣɬɢ ɧɟɡɚɜɢɫɢɦɨ.
ɉɪɨɰɟɞɭɪɚ ɧɚɯɨɠɞɟɧɢɹ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.29)ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɡɚɞɚɱɭ ɧɚ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɢɥɢ ɡɚɞɚɱɭ ɒɬɭɪɦɚ–Ʌɢɭɜɢɥɥɹ.
Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.29) ɢɳɟɦ ɜ ɜɢɞɟ
) A exp k M .
(8.30)
ɉɨɞɫɬɚɜɥɹɹ (8.30) ɜ (8.29), ɩɪɢɞɟɦ ɤ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɦɭ ɭɪɚɜɧɟɧɢɸ
k 2 O 2 ,
ɨɬɤɭɞɚ ɢɦɟɟɦ
k riO ,
1 – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ.
ɝɞɟ i
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (8.29) ɢɦɟɟɬ ɜɢɞ
) Ac e iO M Bc e iO M A co s O M B si n O M .
(8.31)
Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɭɝɥɚ M ɧɚ ɜɟɥɢɱɢɧɭ 2S ɨɞɧɨɡɧɚɱɧɚɹ
ɮɭɧɤɰɢɹ T r ,M ɞɨɥɠɧɚ ɜɟɪɧɭɬɶɫɹ ɤ ɢɫɯɨɞɧɨɦɭ ɡɧɚɱɟɧɢɸ, ɬ.ɟ.
T r , M T r , M 2S .
198
ɗɬɨ ɟɫɬɶ ɭɫɥɨɜɢɟ ɩɟɪɢɨɞɢɱɧɨɫɬɢ. Ɉɬɫɸɞɚ ɢɦɟɟɦ
) M ) M 2S .
Ɍ.ɟ., ) ɹɜɥɹɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɮɭɧɤɰɢɟɣ ɭɝɥɚ M ɫ ɩɟɪɢɨɞɨɦ 2S . ɗɬɨ
ɜɨɡɦɨɠɧɨ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ
O n,
ɝɞɟ n – ɰɟɥɨɟ ɱɢɫɥɨ, ɢ
) n M Ancos nM B nsin nM .
(8.32)
Ɏɭɧɤɰɢɸ R r ɛɭɞɟɦ ɢɫɤɚɬɶ ɜ ɜɢɞɟ
R rP .
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜ ɭɪɚɜɧɟɧɢɟ (8.28), ɧɚɣɞɟɦ
n 2 P 2 ɢɥɢ P r n , n ! 0 .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
R r C r n D r n ,
(8.33)
ɝɞɟ C , D - ɩɨɫɬɨɹɧɧɵɟ.
ȼɧɭɬɪɢ ɤɪɭɝɚ ɢɦɟɟɦ
R r Cr n ,
ɬɚɤ ɤɚɤ ɢɧɚɱɟ ɪɟɲɟɧɢɟ ɜ ɬɨɱɤɟ r 0 ɨɛɪɚɳɚɥɨɫɶ ɛɵ ɜ ɛɟɫɤɨɧɟɱɧɨɫɬɶ.
ȼɧɟ ɤɪɭɝɚ –
R r Dr n ,
ɬɚɤ ɤɚɤ ɪɟɲɟɧɢɟ ɩɪɢ r o f ɞɨɥɠɧɨ ɛɵɬɶ ɨɝɪɚɧɢɱɟɧɧɵɦ.
ɂɬɚɤ, ɱɚɫɬɧɵɟ ɪɟɲɟɧɢɹ ɧɚɲɟɣ ɡɚɞɚɱɢ ɧɚɣɞɟɧɵ
1
Tn r ,M r n Ancos nM B nsin nM , r d R ,
2
Tn r ,M r n Ancos nM B nsin nM , r t R .
(8.34)
ɋɭɦɦɵ ɷɬɢɯ ɪɟɲɟɧɢɣ
1
T r ,M
2
T r ,M
f
¦ Tn r , M , r d R ,
1
n 0
f
2
¦ Tn r , M , r t R
(8.35)
n 0
ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ ɯɨɪɨɲɟɣ ɫɯɨɞɢɦɨɫɬɢ ɬɚɤɠɟ ɛɭɞɭɬ ɝɚɪɦɨɧɢɱɟɫɤɢɦɢ
ɮɭɧɤɰɢɹɦɢ.
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ An , Bn ɢɫɩɨɥɶɡɭɟɦ ɝɪɚɧɢɱɧɨɟ
ɭɫɥɨɜɢɟ
199
f
¦ R n Ancos nM Bnsin nM
T R ,M f M .
(8.36)
n 0
Ɍɚɤ ɤɚɤ f M - ɡɚɞɚɧɧɚɹ ɮɭɧɤɰɢɹ ɭɝɥɚ M , ɬɨ ɦɵ ɦɨɠɟɦ ɩɨɫɬɪɨɢɬɶ ɟɟ
ɪɚɡɥɨɠɟɧɢɟ ɜ ɪɹɞ Ɏɭɪɶɟ
D0 f
f M
(8.37)
¦ D cos nM E nsin nM ,
2 n 1 n
ɝɞɟ
1
S
D0
Dn
En
S
³ f \ d\ ,
S
S
1
S
³ f \ c os n\ d \ , n = 1,2, … ,
S
S
1
S
³ f \ sin n\ d \ , n = 1,2, … .
S
ɋɪɚɜɧɢɜɚɹ (8.36) ɢ (8.37), ɧɚɣɞɟɦ
A0
D0
, An
2
Dn
n
, Bn
En
R
R
ɞɥɹ ɜɧɭɬɪɟɧɧɟɣ ɡɚɞɚɱɢ (ɬ.ɟ., ɜ ɤɪɭɝɟ ɪɚɞɢɭɫɚ R ) ɢ
A0
D0
, An
2
D n R n , Bn
n
–
E nR n –
ɞɥɹ ɜɧɟɲɧɟɣ ɡɚɞɚɱɢ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɨɥɭɱɢɥɢ ɮɨɪɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɩɟɪɜɨɣ ɜɧɭɬɪɟɧɧɟɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɞɥɹ ɤɪɭɝɚ ɜ ɜɢɞɟ ɪɹɞɚ
n
D0 f § r ·
¦
T r , M
D ncos nM E nsin nM 2 n 1¨© R ¸¹
ɚ ɪɟɲɟɧɢɟ ɜɧɟɲɧɟɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɜ ɜɢɞɟ
n
(8.38)
D0 f § R ·
T r , M
¦
(8.39)
D ncos nM E nsin nM .
2 n 1¨© r ¸¹
Ɇɨɠɧɨ ɞɨɤɚɡɚɬɶ, ɱɬɨ ɪɹɞɵ ɜ (8.38), (8.39) ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɫɯɨɞɹɬɫɹ, ɢ
ɮɭɧɤɰɢɹ T r ,M - ɧɟɩɪɟɪɵɜɧɚ ɜ ɡɚɦɤɧɭɬɨɣ ɨɛɥɚɫɬɢ r R d 1 .
200
8.3.2. ɂɧɬɟɝɪɚɥ ɉɭɚɫɫɨɧɚ
ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɤ ɛɨɥɟɟ ɩɪɨɫɬɨɦɭ ɜɢɞɭ. Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɪɚɫɫɦɨɬɪɢɦ ɬɨɥɶɤɨ ɜɧɭɬɪɟɧɧɸɸ ɡɚɞɚɱɭ. ɉɨɞɫɬɚɜɥɹɹ ɤɨɷɮɮɢɰɢɟɧɬɵ Ɏɭɪɶɟ ɮɭɧɤɰɢɢ f ɜ ɭɪɚɜɧɟɧɢɟ (8.38) ɢ ɦɟɧɹɹ ɩɨɪɹɞɨɤ ɫɭɦɦɢɪɨɜɚɧɢɹ ɢ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɧɚɣɞɟɦ
S
­° 1 f § r · n
½°
T r , M ³ f \ ® ¦ ¨ ¸ cos n\ cos nM sin n\ sin nM ¾ d \
°¯ 2 n 1© R ¹
°¿
S
S
­° 1 f § r · n
½°
f
n
(8.40)
\
M
\
c
o
s
¾ d\ ,
³ ® 2 ¦ ¨© R ¸¹
°¯
°¿
n 1
S
ɬɚɤ ɤɚɤ
1
ªcos n \ M cos n \ M º¼ ,
cos n\ cos nM
2¬
1
ªcos n \ M cos n \ M º¼ .
sin n\ sin nM
2¬
ɉɪɨɜɟɞɟɦ ɫɥɟɞɭɸɳɢɟ ɬɨɠɞɟɫɬɜɟɧɧɵɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
1 f n
1 1 f n ª i n M\ i n M\ º
¦ t c os ª¬ n M \ º¼
¦t e
e ¬«
¼»
2
2 2
n 1
f ª
1 ­°
i M\ 1
® ¦ « te
2 ¯° n 1 ¬
n 1
n
i M\ te i M\ i M\ º
te te 1ª
«1 »
2 « 1 t e i M\ 1 te i M\ »
¬
¼
»¼¾¿°
n º½
°
1
1 t 2
,
2 1 2t cos M \ t 2
r
1.
R
ɉɨɞɫɬɚɜɥɹɹ ɩɨɞɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɜ (8.40), ɩɨɥɭɱɚɟɦ
ɝɞɟ t
T r , M
1
2S
S
R2 r 2
³ f \ r 2 2r R c o s M \ R 2 d \ .
(8.41)
S
Ɏɨɪɦɭɥɚ (8.41), ɞɚɸɳɚɹ ɪɟɲɟɧɢɟ ɩɟɪɜɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ ɜɧɭɬɪɢ
ɤɪɭɝɚ, ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɨɦ ɉɭɚɫɫɨɧɚ, ɚ ɩɨɞɢɧɬɟɝɪɚɥɶɧɨɟ ɜɵɪɚɠɟɧɢɟ
R2 r 2
–
K r ,M, R ,\ r 2 2r R cos M \ R 2
ɹɞɪɨɦ ɉɭɚɫɫɨɧɚ. ɂɦɟɟɦ K ! 0 ɩɪɢ r R .
201
ɂɧɬɟɝɪɚɥ ɉɭɚɫɫɨɧɚ ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɣ ɮɭɧɤɰɢɟɣ ɜ ɡɚɦɤɧɭɬɨɣ
ɨɛɥɚɫɬɢ r d R .
Ɋɟɲɟɧɢɟ ɜɧɟɲɧɟɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ (ɩɪɢɜɨɞɢɦ ɛɟɡ ɩɨɞɪɨɛɧɨɝɨ ɨɩɢɫɚɧɢɹ)
­1 S
r2 R2
°°
³ f \ r 2 2r R cos M \ R 2 d \ , r ! R ;
T r ,M ® 2S S
°
r R.
°̄ f M ,
8.3.3. ɇɟɫɬɚɰɢɨɧɚɪɧɵɟ ɡɚɞɚɱɢ
Ɋɚɫɫɦɨɬɪɢɦ ɩɟɪɜɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɧɚ ɨɬɪɟɡɤɟ. ɉɭɫɬɶ ɬɪɟɛɭɟɬɫɹ ɨɩɪɟɞɟɥɢɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɥɚɫɬɢɧɟ ɬɨɥɳɢɧɵ L ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɟɫɥɢ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɞɚɧɨ
T T0
T m T0 M x , t
0,
ɚ ɜ ɞɚɥɶɧɟɣɲɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɜɟɪɯɧɨɫɬɟɣ ɩɨɞɞɟɪɠɢɜɚɸɬɫɹ ɪɚɜɧɵɦɢ
ɬɟɦɩɟɪɚɬɭɪɟ T0 . Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɷɬɨɣ ɡɚɞɚɱɢ ɛɭɞɟɬ ɜɤɥɸɱɚɬɶ ɨɞɧɨɦɟɪɧɨɟ ɧɟɫɬɚɰɢɨɧɚɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
wT
w 2T
N 2.
wt
wx
T T0
L2
ɉɟɪɟɣɞɟɦ ɤ ɩɟɪɟɦɟɧɧɵɦ u
, W
ɢ [
N
T m T0
ɨɛ ɨɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢɦɟɬ ɜɢɞ
w u w 2u
,
wW w[ 2
0 [ 1; 0 W f ;
u 0 ,W 0 ; u 1,W 0 ,
x
. Ɍɨɝɞɚ ɡɚɞɚɱɚ
L
u [ , 0 M [ ; 0 d [ d 1.
Ɋɟɲɟɧɢɟ ɢɳɟɦ ɜ ɜɢɞɟ
u X [ T W .
ɉɨɞɫɬɚɜɥɹɹ (8.45) ɜ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɧɚɣɞɟɦ
X [ Tc W X cc [ T W ɢɥɢ
202
(8.42)
(8.43)
(8.44)
(8.45)
Tc W X cc [ (8.46)
k.
T W X [
Ɍɚɤ ɤɚɤ [ ɢ W ɧɟ ɡɚɜɢɫɹɬ ɞɪɭɝ ɨɬ ɞɪɭɝɚ, ɬɨ ɜɦɟɫɬɨ (8.46) ɢɦɟɟɦ ɞɜɚ
ɭɪɚɜɧɟɧɢɹ
(8.47)
Tc k D 2T 0 ;
X cc kX 0 .
(8.48)
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ (8.48) ɛɭɞɭɬ
X 0 0 ; X 1 0 ,
(8.49)
ɫɥɟɞɭɸɳɢɟ ɢɡ (8.43).
Ɂɚɞɚɱɚ (8.48), (8.49) ɢ ɟɫɬɶ ɡɚɞɚɱɚ ɧɚ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ (ɡɚɞɚɱɚ
ɒɬɭɪɦɚ–Ʌɢɭɜɢɥɥɹ), ɫ ɤɨɬɨɪɨɣ ɦɵ ɜɫɬɪɟɬɢɥɢɫɶ ɜ ɪɚɡɞɟɥɟ 8.3.1.
Ɉɬɦɟɬɢɦ ɜɚɠɧɨɟ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ: k 0 , ɢɧɚɱɟ (8.48), (8.49) ɛɭɞɟɬ
ɢɦɟɬɶ ɬɨɥɶɤɨ ɬɪɢɜɢɚɥɶɧɨɟ ɪɟɲɟɧɢɟ. ɂɧɚɱɟ, ɮɭɧɤɰɢɹ T W ɞɨɥɠɧɚ ɭɛɵɜɚɬɶ ɩɪɢ W o f , ɬ.ɟ.
k O 2 z 0 .
ɍɪɚɜɧɟɧɢɹ (8.47), (8.49) ɪɟɲɚɸɬɫɹ ɨɩɢɫɚɧɧɵɦ ɜɵɲɟ ɫɩɨɫɨɛɨɦ. Ɋɟɲɟɧɢɹ ɢɦɟɸɬ ɜɢɞ
T W C e O
2
W
X [ A s i n O [ B c o s O [ ,
A , B ,C – ɩɪɨɢɡɜɨɥɶɧɵɟ ɩɨɫɬɨɹɧɧɵɟ.
ɂɡ ɜɫɟɝɨ ɦɧɨɠɟɫɬɜɚ ɪɟɲɟɧɢɣ ɡɚɞɚɱɢ (8.42) – (8.44)
u [ , W e x p O 2W ª¬ A s i n O [ B c o s O [ º¼
ɧɚɦ ɧɭɠɧɨ ɜɵɛɪɚɬɶ ɬɟ, ɤɨɬɨɪɵɟ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ.
ɉɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜ ɭɫɥɨɜɢɹ (8.43), ɧɚɣɞɟɦ
u 0 ,W u 1,W B e x p O 2W
0 , ɟɫɥɢ B 0 ;
A e xp O 2W si n O 0 , ɟɫɥɢ sin O 0
(ɬɚɤ ɤɚɤ ɧɚɫ ɢɧɬɟɪɟɫɭɸɬ ɧɟɬɪɢɜɢɚɥɶɧɵɟ ɪɟɲɟɧɢɹ). ɗɬɨ ɭɫɥɨɜɢɟ ɧɚɤɥɚɞɵɜɚɟɬ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɜɨɡɦɨɠɧɵɟ ɡɧɚɱɟɧɢɹ O , ɬ.ɟ. ɧɭɠɧɨ ɩɨɬɪɟɛɨɜɚɬɶ,
ɱɬɨɛɵ
O rS , r 2S ,... , ɢɥɢ O n
u n [ ,W r nS , n 1, 2 ,. ..
Anexp S n W sin S n[ 0 , n 1, 2 ,.. .
2
ɢ
203
¦ Anexp S nD f
u [ , W
n 1
2
t sin S nx .
ɉɨɞɫɬɚɜɥɹɹ u [,W ɜ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɢɦɟɟɦ
M [
f
¦ Ansin S n[ .
(8.50)
n 1
ɋɢɫɬɟɦɚ ɮɭɧɤɰɢɣ ^ sin Sn[ , n 1,2 , ... ` ɨɛɥɚɞɚɟɬ ɬɚɤɢɦ ɫɜɨɣɫɬɜɨɦ
ɤɚɤ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɶ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ
1
­0 , m z n ;
S[
S[
[
s
in
si
n
m
n
d
®
³
¯1 2 , m n .
(8.51)
0
ɍɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ (8.50) ɧɚ sin mS[ , ɝɞɟ m – ɩɪɨɢɡɜɨɥɶɧɨɟ ɰɟɥɨɟ ɱɢɫɥɨ, ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɟɝɨ ɡɚɬɟɦ ɨɬ ɧɭɥɹ ɞɨ ɟɞɢɧɢɰɵ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ
1
³ M [ si n S m[ d [
0
1
Am ³ sin 2 S m[ d [
0
Am
.
2
Ɉɫɬɚɥɶɧɵɟ ɫɥɚɝɚɟɦɵɟ ɨɛɪɚɬɢɥɢɫɶ ɜ ɧɭɥɶ, ɛɥɚɝɨɞɚɪɹ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ
u [ , W
¦ Amexp S m f
m 1
2
W sin S m[ ,
(8.52)
ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ Am ɜɵɱɢɫɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ
1
2 ³ M [ si n S m[ d [ .
Am
(8.53)
0
ɗɬɨ ɪɟɲɟɧɢɟ ɤɚɠɟɬɫɹ ɝɪɨɦɨɡɞɤɢɦ. ɇɨ ɡɞɟɫɶ ɫɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ
ɧɚɥɢɱɢɟ ɦɧɨɠɢɬɟɥɹ exp S m W ɞɟɥɚɟɬ ɪɹɞ ɜ (8.52) ɛɵɫɬɪɨ ɫɯɨɞɹ2
ɳɢɦɫɹ ɩɪɢ W ! 0 , ɬɚɤ ɤɚɤ ɫɥɚɝɚɟɦɵɟ ɫ ɛɨɥɶɲɢɦɢ ɧɨɦɟɪɚɦɢ ɜɧɨɫɹɬ ɧɟɫɭɳɟɫɬɜɟɧɧɵɣ ɜɤɥɚɞ ɜ ɫɭɦɦɭ ɪɹɞɚ. ɗɬɨ ɜɟɫɶɦɚ ɭɞɨɛɧɨ ɜ ɩɪɚɤɬɢɱɟɫɤɢɯ ɜɵɱɢɫɥɟɧɢɹɯ.
Ɇɵ ɭɠɟ ɡɧɚɟɦ, ɱɬɨ ɷɬɚ ɡɚɞɚɱɚ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ ɞɪɭɝɢɦ ɦɟɬɨɞɨɦ.
ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɩɨɥɭɱɚɟɦ ɞɪɭɝɭɸ ɮɨɪɦɭ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɟɲɟɧɢɹ.
ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ (8.52), ɩɨɞɫɬɚɜɢɜ ɜ ɧɟɝɨ ɜɵɪɚɠɟɧɢɹ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
204
ª1
º
2
u [ , W ¦ 2 « ³ M x sin S mx dx » exp S m W sin S m[ »¼
m 1 «¬ 0
1ª f
º
2
2
exp
sin
sin
m
m
x
m
S
W
S
S
[
«
» M[ d[
³« ¦
¼»
0¬ m 1
f
ɢɥɢ
u [ , W
1
³ G [ , x , W M x dx ,
0
ɝɞɟ
f
G [ , x , W 2 ¦ exp S m W sin S m[ sin S mx – (8.54)
m 1
2
ɮɭɧɤɰɢɹ ɦɝɧɨɜɟɧɧɨɝɨ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ, ɢɥɢ ɮɭɧɤɰɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɜɥɢɹɧɢɹ ɦɝɧɨɜɟɧɧɨɝɨ ɬɨɱɟɱɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɦɨɳɧɨɫɬɢ
Q cU .
Ɋɚɫɫɦɨɬɪɢɦ ɛɨɥɟɟ ɫɥɨɠɧɭɸ ɤɪɚɟɜɭɸ ɡɚɞɚɱɭ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜ ɩɥɚɫɬɢɧɟ ɢɦɟɸɬɫɹ ɨɛɴɟɦɧɵɟ ɢɫɬɨɱɧɢɤɢ ɬɟɩɥɚ,
ɡɚɜɢɫɹɳɢɟ ɨɬ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɯ ɤɨɨɪɞɢɧɚɬ ɢ ɜɪɟɦɟɧɢ, ɚ ɧɚ ɝɪɚɧɢɰɚɯ ɡɚɞɚɧɵ ɨɛɨɛɳɟɧɧɵɟ ɭɫɥɨɜɢɹ ɬɪɟɬɶɟɝɨ ɪɨɞɚ, ɜɤɥɸɱɚɸɳɢɟ ɩɨɬɨɤɢ ɬɟɩɥɚ,
ɡɚɜɢɫɹɳɢɟ ɨɬ ɜɪɟɦɟɧɢ
wT
w 2T
a 2 f x ,t ,
wt
wx
w T 0 ,t D1
E1T 0 ,t g1 t ,
wx
w T L ,t D2
E 2T L ,t g 2 t ,
wx
T x ,0 M x .
Ɂɚɦɟɧɨɣ ɩɟɪɟɦɟɧɧɵɯ ɷɬɭ ɡɚɞɚɱɭ ɦɨɠɧɨ ɫɜɟɫɬɢ ɤ ɞɪɭɝɨɣ ɡɚɞɚɱɟ. ȿɫɥɢ
ɷɬɚ ɧɨɜɚɹ ɡɚɞɚɱɚ ɨɤɚɠɟɬɫɹ ɨɞɧɨɪɨɞɧɨɣ, ɬɨ ɟɟ ɦɨɠɧɨ ɛɭɞɟɬ ɪɟɲɢɬɶ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. ȿɫɥɢ ɨɧɚ ɨɤɚɠɟɬɫɹ ɧɟɨɞɧɨɪɨɞɧɨɣ, ɬɨ ɜɨɡɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɟɬɨɞ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɢ ɪɚɡɥɨɠɟɧɢɟ
ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ.
Ɋɚɫɫɦɨɬɪɢɦ ɱɚɫɬɧɵɟ ɜɚɪɢɚɧɬɵ.
ɉɭɫɬɶ f x ,t 0 ɢ ɡɚɞɚɧɵ ɩɨɫɬɨɹɧɧɵɟ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɤɨɧɰɚɯ ɨɬɪɟɡɤɚ > 0 , L @ , ɬ.ɟ.
205
T 0 ,t k1 const ; T L ,t k 2
ɉɪɟɞɫɬɚɜɢɦ ɪɟɲɟɧɢɟ ɜ ɜɢɞɟ
T T1 x T2 x ,t ,
const .
(8.55)
(8.56)
ɝɞɟ T1 x – ɫɬɚɰɢɨɧɚɪɧɨɟ ɢɥɢ ɭɫɬɚɧɨɜɢɜɲɟɟɫɹ ɪɟɲɟɧɢɟ
x
(8.57)
k k .
L 2 1
ɉɨɞɫɬɚɜɥɹɹ (8.56) ɜ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɭɫɥɨɜɢɹ (8.55),
ɩɪɢɞɟɦ ɤ ɡɚɞɚɱɟ ɞɥɹ ɮɭɧɤɰɢɢ T2
T1 x k1 w T2
w 2T2
a
,
wt
wx2
T2 0 , t 0 ; T2 L , t 0 ,
x
ª
º
T2 x , 0 M x « k1 k 2 k1 » M x ; 0 d x d L
L
¬
¼
ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɢ ɧɨɜɵɦ, ɧɨ ɢɡɜɟɫɬɧɵɦ, ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ.
Ʉɚɤ ɪɟɲɚɬɶ ɬɚɤɭɸ ɡɚɞɚɱɭ, ɦɵ ɭɠɟ ɡɧɚɟɦ.
«ɂɡɛɚɜɢɬɶɫɹ» ɨɬ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɭɪɚɜɧɟɧɢɢ
L2
x
ɦɨɠɧɨ ɜɜɟɞɟɧɢɟɦ ɩɟɪɟɦɟɧɧɵɯ W
ɢ[
.
N
L
ɉɭɫɬɶ ɬɟɩɟɪɶ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɡɚɜɢɫɹɬ ɨɬ ɜɪɟɦɟɧɢ.
ɂɦɟɟɦ ɡɚɞɚɱɭ
wT
w 2T
a 2,
wt
wx
w T L ,t T 0 ,t g 1 t ;
h T L ,t g 2 t ,
(8.58)
wx
T x ,0 M x .
ɉɨɫɥɟ ɧɟɤɨɬɨɪɵɯ ɩɪɨɛ ɢ ɨɲɢɛɨɤ ɨɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɜɨɬ ɧɚ ɬɚɤɨɣ
ɮɨɪɦɟ ɪɟɲɟɧɢɹ
T x ,t A t >1 x L @ B t > x L @ T2 x ,t ,
(8.59)
ɝɞɟ A t , B t ɜɵɛɢɪɚɸɬ ɬɚɤ, ɱɬɨɛɵ «ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɚɹ» ɱɚɫɬɶ ɪɟɲɟɧɢɹ
ɭɞɨɜɥɟɬɜɨɪɹɥɚ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ ɡɚɞɚɱɢ (8.48). ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
S x ,t A t >1 x L @ B t > x L @ .
206
(8.60)
ȼ ɷɬɨɦ ɫɥɭɱɚɟ T2 ɛɭɞɟɬ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɨɞɧɨɪɨɞɧɵɦ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɹɦ.
ɉɨɞɫɬɚɜɥɹɹ S ɜ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ (8.58), ɩɪɢɞɟɦ ɤ ɞɜɭɦ ɭɪɚɜɧɟɧɢɹɦ, ɢɡ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɢɦ ɮɭɧɤɰɢɢ A t , B t :
At g1 t L g 2 t .
1 Lh
g1 t ; B t (8.61)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɲɟɧɢɟ (8.59) ɩɪɢɦɟɬ ɜɢɞ
ª x º g t L g 2 t x
(8.62)
T x ,t g 1 «1 » 1
T2 x ,t .
L
1
L
h
L
¬
¼
ɉɨɞɫɬɚɜɢɜ T x ,t (8.62) ɜ ɢɫɯɨɞɧɭɸ ɡɚɞɚɱɭ, ɧɚɣɞɟɦ ɡɚɞɚɱɭ ɞɥɹ
ɮɭɧɤɰɢɢ T2 x ,t w T2
w 2T2
a
St ,
2
wt
wx
w T2 L , t hT2 L ,t 0 ,
wx
T2 0 , t 0 ,
(8.63)
T2 x , 0 M x S x , 0 .
ɗɬɚ ɡɚɞɚɱɚ ɫ ɨɞɧɨɪɨɞɧɵɦɢ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɧɨ ɫɚɦɨ ɭɪɚɜɧɟɧɢɟ ɫɬɚɥɨ ɧɟɨɞɧɨɪɨɞɧɵɦ. Ɂɚɞɚɱɚ (8.63) ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɩɪɹɦɨ ɪɟɲɟɧɚ
ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. ɇɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɦɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ
ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ.
8.4. Ɇɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ
ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ
ɑɬɨɛɵ ɭɹɫɧɢɬɶ, ɜ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɦɟɬɨɞ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ, ɪɚɫɫɦɨɬɪɢɦ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ ɡɚɞɚɱɢ (8.63), ɤɨɬɨɪɵɣ ɜ
ɩɟɪɟɦɟɧɧɵɯ u, W , [ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
wu
wW
w 2u
w[
2
f [ , W ,
u 0 ,W 0 ; u 1,W 0 ,
u [ ,0 M [ ,
0 d [ d 1.
207
(Ʉɚɤ ɦɵ ɜɢɞɟɥɢ ɜɵɲɟ, ɡɚɞɚɱɭ ɧɚ ɨɬɪɟɡɤɟ ɨɬ 0 ɞɨ L ɜɫɟɝɞɚ ɦɨɠɧɨ ɫɜɟɫɬɢ
ɤ ɡɚɞɚɱɟ ɧɚ ɨɬɪɟɡɤɟ ɨɬ ɧɭɥɹ ɞɨ ɟɞɢɧɢɰɵ, ɜɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ ɤɨɨɪɞɢɧɚɬ).
Ɉɫɧɨɜɧɚɹ ɢɞɟɹ ɦɟɬɨɞɚ ɫɨɫɬɨɢɬ ɜ ɪɚɡɥɨɠɟɧɢɢ ɮɭɧɤɰɢɢ ɩɥɨɬɧɨɫɬɢ ɢɫɬɨɱɧɢɤɚ f [ , W ɜ ɪɹɞ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ ɮɭɧɤɰɢɹɦ
(8.64)
f [ , W f 1 W X 1 [ f 2 W X 2 [ ... f n W X n [ . ..
ɢ ɜ ɨɩɪɟɞɟɥɟɧɢɢ ɨɬɤɥɢɤɨɜ ɫɢɫɬɟɦɵ u n ɧɚ ɜɨɡɞɟɣɫɬɜɢɟ ɤɚɠɞɨɣ ɤɨɦɩɨ-
ɧɟɧɬɵ f n W X n [ . ȿɫɥɢ ɜɫɟ ɨɬɤɥɢɤɢ ɛɭɞɭɬ ɧɚɣɞɟɧɵ, ɬɨ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
u [ , W
f
¦ u n [ , W .
(8.65)
n 0
Ɉɫɧɨɜɧɚɹ ɬɪɭɞɧɨɫɬɶ ɢ ɫɨɫɬɨɢɬ ɜ ɪɚɡɥɨɠɟɧɢɢ ɩɥɨɬɧɨɫɬɢ ɢɫɬɨɱɧɢɤɚ ɧɚ
ɤɨɦɩɨɧɟɧɬɵ. Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɦɧɨɠɢɬɟɥɢ X n [ ɜ ɞɚɧɧɨɣ ɡɚɞɚɱɟ ɹɜɥɹɸɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦɢ ɜɟɤɬɨɪɚɦɢ ɫɢɫɬɟɦɵ ɒɬɭɪɦɚ–Ʌɢɭɜɢɥɥɹ, ɤɨɬɨɪɚɹ
ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɪɟɲɟɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɨɞɧɨɪɨɞɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ, ɚ ɢɦɟɧɧɨ ɡɚɞɚɱɢ:
w u w 2u
,
wW w[ 2
0 [ 1; 0 W f ;
u 0 ,W 0 ; u 1,W 0 ,
u [ , 0 M [ ; 0 d [ d 1.
Ɂɚɞɚɱɚ ɒɬɭɪɦɚ-Ʌɢɭɜɢɥɥɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɦ ɬɨɠɟ ɭɠɟ ɢɡɜɟɫɬɧɚ ɢ
ɢɦɟɟɬ ɜɢɞ (8.47), (8.48)
X cc kX 0 .
X 0 0 ; X 1 0 ,
ȿɟ ɪɟɲɟɧɢɹɦɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ
X n W sin S[n , n 1,2 , ...
(8.66)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɚɡɥɨɠɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɢɫɬɨɱɧɢɤɚ ɩɪɟɞɫɬɚɜɢɦɨ ɜ ɜɢɞɟ ɪɹɞɚ
f [ , W
f1 W sin S[ f 2 W sin 2S[ ... f n W sin nS[ .. . (8.67)
ɂ, ɧɚɤɨɧɟɰ, ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɧɚɣɬɢ ɮɭɧɤɰɢɢ f n t , ɭɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ (8.67) ɧɚ sin mS x ɢ ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɨɬ ɧɭɥɹ ɞɨ ɟɞɢɧɢɰɵ ɩɨ ɤɨɨɪɞɢɧɚɬɟ x
208
1
³
f
1
n 1
0
¦ f n W ³ sin mS x sin nS x d x
f x , W sin mS x d x
0
1
f W
2 m
ɢɥɢ (ɩɨɫɥɟ ɡɚɦɟɧɵ m ɧɚ n )
1
f n W 2 ³ f x , W sin nS x d x .
(8.68)
0
ɗɬɨ ɫɨɨɬɧɨɲɟɧɢɟ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɫɜɹɡɶ ɦɟɠɞɭ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ f n t ɢ ɩɥɨɬɧɨɫɬɶɸ ɬɟɩɥɨɜɨɝɨ ɢɫɬɨɱɧɢɤɚ
f [ , W
f
¦ f n W sin nS[ .
(8.69)
n 1
Ɍɟɩɟɪɶ ɩɨɩɵɬɚɟɦɫɹ ɧɚɣɬɢ ɢɧɞɢɜɢɞɭɚɥɶɧɵɟ ɨɬɤɥɢɤɢ, ɬ.ɟ. ɮɭɧɤɰɢɢ
Tn W ɜ ɪɚɡɥɨɠɟɧɢɢ
f
¦ T n W s i n nS[ .
u [ ,W
(8.70)
n 1
ɉɨɞɫɬɚɜɢɦ (8.70) ɜ ɢɫɯɨɞɧɭɸ ɡɚɞɚɱɭ ɫ ɭɱɟɬɨɦ (8.69). ɂɦɟɟɦ
f
f
f
2
¦ T n c Ws in S n[ ¦ S n T n W sin S n[ ¦ f n Wsi n S n[ (8.71)
n 0
n 1
n 1
f
¦ T nsi n 0 0;
f
¦ T ns i n S0 0;
n 1
n 1
(ɬ.ɟ., ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɭɞɨɜɥɟɬɜɨɪɹɸɬɫɹ ɬɨɠɞɟɫɬɜɟɧɧɨ)
f
¦ T n 0 s i n S n[ M[ .
n 1
ɍɪɚɜɧɟɧɢɹ (8.61) ɩɟɪɟɩɢɲɟɦ ɜ ɜɢɞɟ
f
¦ ª«¬T nc W S n n 0
2
T n W f n W º s in S n[ 0 .
»¼
(8.72)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Ɏɭɧɤɰɢɢ Tn W ɹɜɥɹɸɬɫɹ ɪɟɲɟɧɢɹɦɢ ɡɚɞɚɱɢ Ʉɨɲɢ
ɞɥɹ ɨɛɵɤɧɨɜɟɧɧɨɝɨ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
Tnc S n T n
2
f n W ;
1
³
Tn 0 2 M[ sinSn[ d[ an .
0
209
(8.73)
ȼɫɟ ɷɬɢ ɡɚɞɚɱɢ ɥɟɝɤɨ ɪɟɲɚɸɬɫɹ ɫɬɚɧɞɚɪɬɧɵɦɢ ɦɟɬɨɞɚɦɢ ɢ ɢɯ ɪɟɲɟɧɢɹ ɡɚɩɢɫɵɜɚɸɬɫɹ ɜ ɜɢɞɟ
W
2
2
a ne xp ª nS W º ³ e x p ª nS W z º f n z d z ,
¬«
¼»
¬«
¼»
T n W
0
ɨɬɤɭɞɚ ɧɟɬɪɭɞɧɨ ɩɨɥɭɱɢɬɶ ɩɨɥɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ:
u [ ,W
f
f
¦ T WsinnS[ ¦^a exp > nS W@sinnS[`
n
2
n
n 1
n 1
(8.74)
W
­
½
°
°
2
®sin nS[ exp nS W z f n z dz ¾.
°
n 1°
¯
¿
0
ȼ (8.74) ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ ɜ ɩɪɚɜɨɣ ɱɚɫɬɢ ɨɛɭɫɥɨɜɥɟɧɨ ɧɚɱɚɥɶɧɵɦ
ɭɫɥɨɜɢɟɦ, ɚ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ – ɨɛɭɫɥɨɜɥɟɧɨ ɢɫɬɨɱɧɢɤɨɦ ɜ ɭɪɚɜɧɟɧɢɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ.
f
³ >
¦
@
8.5. Ɂɚɞɚɱɚ ɨɛ ɨɫɬɵɜɚɧɢɢ ɛɟɫɤɨɧɟɱɧɨɣ ɩɥɚɫɬɢɧɵ
ȼɫɩɨɦɧɢɦ ɮɢɡɢɱɟɫɤɨɟ ɫɨɞɟɪɠɚɧɢɟ ɡɚɞɚɱ.
Ɂɚɞɚɱɚ (8.42)–(8.43) ɟɫɬɶ ɡɚɞɚɱɚ ɨɛ ɨɫɬɵɜɚɧɢɢ ɛɟɫɤɨɧɟɱɧɨɣ ɩɥɚɫɬɢɧɵ
ɟɞɢɧɢɱɧɨɣ ɬɨɥɳɢɧɵ ɫ ɡɚɞɚɧɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ.
ɉɭɫɬɶ ɬɟɩɟɪɶ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɥɚɫɬɢɧɚ ɩɪɨɝɪɟɬɚ ɨɞɧɨɪɨɞɧɨ. ɇɚ ɥɟɜɨɣ ɟɟ ɝɪɚɧɢɰɟ ɡɚɞɚɧɨ ɭɫɥɨɜɢɟ ɚɞɢɚɛɚɬɢɱɧɨɫɬɢ (ɬ.ɟ., ɩɥɚɫɬɢɧɚ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɚ ɩɪɢ x 0 ), ɚ ɧɚ ɩɪɚɜɨɣ ɝɪɚɧɢɰɟ – ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɟɩɥɨɨɛɦɟɧ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ, ɢɦɟɸɳɟɣ ɬɟɦɩɟɪɚɬɭɪɭ Te , ɩɨ
T Te
ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ. ȼ ɩɟɪɟɦɟɧɧɵɯ u
, W ,[ ɬɚɤɚɹ ɡɚɞɚɱɚ ɩɪɢɦɟɬ ɜɢɞ
T0 Te
w u w 2u
;
wW w[ 2
0 [ 1; 0 W f ;
w u 0 ,W w u 1, W 0; BiT ;
w[
w[
u [ , 0 T 0 1 ; 0 d [ d 1,
ɝɞɟ B i
DL
, W { Fo
O
Nt
L2
.
210
(8.75)
ɗɬɚ ɡɚɞɚɱɚ ɥɟɝɤɨ ɪɟɲɚɟɬɫɹ ɨɩɢɫɚɧɧɵɦ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ. Ɋɟɲɟɧɢɟ ɢɦɟɟɬ ɜɢɞ
f
u
¦P
n 0
2sin P n
cos P n[ exp ª P n2 Wº ,
¬
¼
n sin P n cos P n
(8.76)
ɝɞɟ Pn - ɤɨɪɧɢ ɭɪɚɜɧɟɧɢɹ
c tg P P Bi .
(8.77)
Ʉɚɠɞɨɦɭ ɡɧɚɱɟɧɢɸ ɱɢɫɥɚ Ȼɢɨ B i ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɜɨɟ ɫɟɦɟɣɫɬɜɨ ɪɟɲɟɧɢɣ.
ȿɫɥɢ B i o f , ɬɨ
P1
S
; P2
2
3
S; P 3
2
0; P 2
S; P 3
5
S; ...; P n
2
2n 1 S .
2
ȿɫɥɢ Bi o 0 , ɬɨ
P1
2S; ...; P n
n 1S .
Ⱦɥɹ ɞɪɭɝɢɯ ɡɧɚɱɟɧɢɣ B i ɤɨɪɧɢ ɢɦɟɸɬ ɩɪɨɦɟɠɭɬɨɱɧɨɟ ɡɧɚɱɟɧɢɟ.
Ʉɨɷɮɮɢɰɢɟɧɬɵ
Dn
2si n P n
P n s in P nc os P n
(8.78)
ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɬɨɥɶɤɨ ɱɢɫɥɚ B i ɢ ɥɟɝɤɨ ɦɨɝɭɬ ɛɵɬɶ ɪɚɫɫɱɢɬɚɧɵ.
Ɍɚɤ ɤɚɤ Pn - ɪɹɞɵ ɜɨɡɪɚɫɬɚɸɳɢɯ ɱɢɫɟɥ, ɬɨ ɱɟɦ ɛɨɥɶɲɟ Pn , ɬɟɦ
ɦɟɧɶɲɟ ɪɨɥɶ ɩɨɫɥɟɞɭɸɳɢɯ ɱɥɟɧɨɜ ɪɹɞɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɱɟɦ ɛɨɥɶɲɟ ɱɢɫɥɨ
Ɏɭɪɶɟ, ɬɟɦ ɱɥɟɧɵ ɪɹɞɚ ɛɭɞɭɬ ɭɛɵɜɚɬɶ ɛɵɫɬɪɟɟ ɫ ɧɨɦɟɪɨɦ n .
ɉɪɢ F o t 0 . 3 ɞɨɫɬɚɬɨɱɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɩɟɪɜɵɦ ɱɥɟɧɨɦ ɪɹɞɚ, ɬ.ɟ.
u
D1c o s P1[ e x p ª P12W º ,
¬
¼
2sin P1
.
P1 sin P1cos P1
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɡɥɢɱɧɵɟ ɱɚɫɬɧɵɟ ɫɥɭɱɚɢ.
ɝɞɟ D1
1. ɉɭɫɬɶ B i o f . ɉɪɚɤɬɢɱɟɫɤɢ ɞɨɫɬɚɬɨɱɧɨ B i ! 100 . ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɬɟɦɩɟɪɚɬɭɪɚ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ ɫɪɚɡɭ ɠɟ ɫɬɚɧɨɜɢɬɫɹ ɪɚɜɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ, ɜ ɤɨɬɨɪɭɸ ɩɨɦɟɳɟɧɚ ɩɥɚɫɬɢɧɚ, ɢ ɦɵ ɢɦɟɟɦ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɩɨɤɚɡɚɧɧɨɟ ɧɚ ɪɢɫ. 8.5, ɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɩɪɨɰɟɫɫ ɨɯɥɚɠɞɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɧɭɬɪɟɧɧɢɦ ɬɟɩɥɨɨɛɦɟɧɨɦ.
211
2. ȿɫɥɢ Bi o 0 , ɬɨ D1 ~ 1 . ɉɪɚɤɬɢɱɟɫɤɢ ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ Bi 0,1 .
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɨɯɥɚɠɞɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɧɟɲɧɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ (ɜɧɟɲɧɢɦɢ ɭɫɥɨɜɢɹɦɢ), ɢ ɦɵ ɢɦɟɟɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɩɨɤɚɡɚɧɧɨɟ ɧɚ ɪɢɫ. 8.5, ɛ. Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ
u W ,0
u W ,1
exp Bi ˜ Fo cos Bi exp Bi ˜ Fo o 1.
3. ɉɪɨɦɟɠɭɬɨɱɧɵɦ ɡɧɚɱɟɧɢɹɦ ɱɢɫɥɚ B i ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɤɚɱɟɫɬɜɟɧɧɨɟ
ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɩɨɤɚɡɚɧɧɨɟ ɧɚ ɪɢɫ. 8.5, ɜ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ
ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɩɪɨɰɟɫɫɚ ɨɯɥɚɠɞɟɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɜɧɭɬɪɟɧɧɢɦ, ɬɚɤ
ɢ ɜɧɟɲɧɢɦ ɬɟɪɦɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ.
ɚ
ɛ
ɜ
Ɋɢɫ. 8.5. Ɋɟɠɢɦɵ ɨɯɥɚɠɞɟɧɢɹ ɩɥɚɫɬɢɧɵ
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɉɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɪɟɲɟɧɢɟ ɡɚɞɚɱ ɜ ɞɜɭɯ- ɢ ɬɪɟɯɦɟɪɧɵɯ ɨɛɥɚɫɬɹɯ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɪɟɲɟɧɢɣ ɛɨɥɟɟ ɩɪɨɫɬɵɯ ɡɚɞɚɱ?
2. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɮɢɡɢɱɟɫɤɚɹ ɫɭɳɧɨɫɬɶ ɦɟɬɨɞɚ ɢɫɬɨɱɧɢɤɨɜ?
3. ɑɬɨ ɬɚɤɨɟ «ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ»?
4. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɦɟɬɨɞɚ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ?
5. Ƚɞɟ ɜɫɬɪɟɬɢɥɚɫɶ ɡɚɞɚɱɚ ɒɬɭɪɦɚ–Ʌɢɭɜɢɥɥɹ?
6. ɑɬɨ ɬɚɤɨɟ «ɢɧɬɟɝɪɚɥ ɉɭɚɫɫɨɧɚ»?
7. Ʉɚɤɨɝɨ ɜɢɞɚ ɡɚɞɚɱɢ ɭɞɨɛɧɨ ɪɟɲɚɬɶ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ?
8. Ɉɯɚɪɚɤɬɟɪɢɡɭɣɬɟ ɢɞɟɸ ɦɟɬɨɞɚ ɪɚɡɥɨɠɟɧɢɹ ɩɨ ɫɨɛɫɬɜɟɧɧɵɦ
ɮɭɧɤɰɢɹɦ.
9. Ʉɚɤɢɦɢ ɮɢɡɢɱɟɫɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶ ɨɯɥɚɠɞɟɧɢɹ ɬɟɥ?
212
Ɂɚɞɚɧɢɹ
1. ɇɚɣɬɢ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɣ ɤɪɚɟɜɨɣ ɡɚɞɚɱɢ
w u w 2u
,
wW w[ 2
0 [ 1; 0 W f ;
u 0 ,W 0 ; u 1,W 0 ,
u [ ,0 M [ ; 0 d [ d 1
ɞɥɹ M M
>1 [@ 2 [ .
2. ɇɚɣɬɢ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (8.75) ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ.
3. Ɇɟɬɚɥɥɢɱɟɫɤɚɹ ɛɨɥɜɚɧɤɚ ɡɚɞɚɧɧɨɝɨ ɥɢɧɟɣɧɨɝɨ ɪɚɡɦɟɪɚ L ɢ ɫ ɡɚɞɚɧɧɵɦɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ, ɢɦɟɸɳɚɹ ɬɟɦɩɟɪɚɬɭɪɭ T0 , ɜ
ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɩɪɢɜɨɞɢɬɫɹ ɨɞɧɢɦ ɬɨɪɰɨɦ ɜ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɟ ɫ ɬɟɥɨɦ, ɢɦɟɸɳɢɦ ɩɨɫɬɨɹɧɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ Ts . ȼɬɨɪɨɣ ɬɨɪɟɰ ɢ ɛɨɤɨɜɵɟ ɩɨɜɟɪɯɧɨɫɬɢ ɛɨɥɜɚɧɤɢ – ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɵ. ɇɚɣɬɢ ɜɪɟɦɹ, ɡɚ ɤɨɬɨɪɨɟ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɬɨɪɟɰ ɛɨɥɜɚɧɤɢ ɩɪɨɝɪɟɟɬɫɹ ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ Ts
ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ H . Ɋɟɲɢɬɶ ɡɚɞɚɱɭ ɦɟɬɨɞɨɦ ɪɚɡɞɟɥɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɢ ɨɩɟɪɚɰɢɨɧɧɵɦ ɦɟɬɨɞɨɦ. ɋɪɚɜɧɢɬɶ ɪɟɲɟɧɢɹ.
213
ɑȺɋɌɖ 9
ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ:
ɬ ɟ ɩ ɥ ɨ ɨ ɛ ɦ ɟ ɧ ɢ ɡɥ ɭ ɱ ɟ ɧ ɢ ɟ ɦ
9.1. Ɉɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɬɟɨɪɢɢ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ
Ɍɟɩɥɨɜɨɟ ɢɡɥɭɱɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ
ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢɡɥɭɱɚɸɳɟɝɨ ɬɟɥɚ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɦɢ ɤɨɥɟɛɚɧɢɹɦɢ ɢ ɮɨɬɨɧɚɦɢ. Ʌɸɛɵɟ ɬɟɥɚ, ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɵɯ ɜɵɲɟ ɚɛɫɨɥɸɬɧɨɝɨ
ɧɭɥɹ, ɢɡɥɭɱɚɸɬ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɤɨɥɟɛɚɧɢɹ. Ƚɟɧɟɪɚɬɨɪɚɦɢ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɜɨɥɧ ɹɜɥɹɸɬɫɹ ɡɚɪɹɠɟɧɧɵɟ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɟ ɱɚɫɬɢɰɵ –
ɷɥɟɤɬɪɨɧɵ ɢ ɢɨɧɵ, ɜɯɨɞɹɳɢɟ ɜ ɫɨɫɬɚɜ ɜɟɳɟɫɬɜɚ. ɋɨɝɥɚɫɧɨ ɜɨɥɧɨɜɨɣ
ɬɟɨɪɢɢ, ɢɡɥɭɱɟɧɢɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜɨɥɧɨɜɵɦɢ ɤɨɥɟɛɚɧɢɹɦɢ, ɢɦɟɸɳɢɦɢ ɱɚɫɬɨɬɭ Q ɢ ɞɥɢɧɭ ɜɨɥɧɵ O . ɉɪɨɢɡɜɟɞɟɧɢɟ ɱɚɫɬɨɬɵ ɢ ɞɥɢɧɵ ɜɨɥɧɵ ɟɫɬɶ ɫɤɨɪɨɫɬɶ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɤɨɥɟɛɚɧɢɣ, ɪɚɜɧɚɹ ɫɤɨɪɨɫɬɢ ɫɜɟɬɚ:
c Q ˜ O . ɉɨɦɢɦɨ ɜɨɥɧɨɜɵɯ ɫɜɨɣɫɬɜ, ɢɡɥɭɱɟɧɢɟ ɨɛɥɚɞɚɟɬ ɢ ɤɨɪɩɭɫɤɭɥɹɪɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ, ɬ.ɟ. ɥɭɱɢɫɬɚɹ ɷɧɟɪɝɢɹ ɢɫɩɭɫɤɚɟɬɫɹ ɢ ɩɨɝɥɨɳɚɟɬɫɹ ɜɟɳɟɫɬɜɚɦɢ ɧɟ ɧɟɩɪɟɪɵɜɧɨ, ɚ ɞɢɫɤɪɟɬɧɵɦɢ ɩɨɪɰɢɹɦɢ – ɮɨɬɨɧɚɦɢ. Ʉɚɠɞɵɣ ɮɨɬɨɧ ɞɜɢɠɟɬɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ ɫɜɟɬɚ ɢ ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɭɸ ɷɧɟɪɝɢɸ, ɡɚɞɚɧɧɭɸ ɫɨɨɬɧɨɲɟɧɢɟɦ e hQ , ɝɞɟ h – ɩɨɫɬɨɹɧɧɚɹ ɉɥɚɧɤɚ.
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ ɡɚɜɢɫɢɬ ɨɬ ɦɚɬɟɪɢɚɥɚ ɢ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ, ɞɥɢɧɵ ɜɨɥɧɵ, ɫɨɫɬɨɹɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ, ɚ ɞɥɹ ɝɚɡɨɜ – ɟɳɟ ɢ
ɨɬ ɬɨɥɳɢɧɵ ɫɥɨɹ ɢ ɞɚɜɥɟɧɢɹ. ɋ ɜɨɡɪɚɫɬɚɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɷɧɟɪɝɢɹ ɢɡɥɭɱɟɧɢɹ ɭɜɟɥɢɱɢɜɚɟɬɫɹ, ɬɚɤ ɤɚɤ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ ɬɟɥɚ.
ɉɪɢ ɜɵɫɨɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɨɫɧɨɜɧɵɦ ɜɢɞɨɦ ɩɟɪɟɧɨɫɚ ɬɟɩɥɨɬɵ ɦɨɠɟɬ
ɨɤɚɡɚɬɶɫɹ ɬɟɩɥɨɜɨɟ ɢɡɥɭɱɟɧɢɟ, ɬɚɤ ɤɚɤ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ ɡɚɜɢɫɢɬ
ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɡɧɚɱɢɬɟɥɶɧɨ ɫɢɥɶɧɟɟ, ɱɟɦ ɤɨɧɜɟɤɰɢɹ ɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ.
ȼ ɨɬɥɢɱɢɟ ɨɬ ɞɪɭɝɢɯ ɜɢɞɨɜ ɬɟɩɥɨɨɛɦɟɧɚ, ɩɨɬɨɤ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɢ
ɩɟɪɟɞɚɟɬɫɹ ɤɚɤ ɨɬ ɛɨɥɟɟ ɧɚɝɪɟɬɨɝɨ ɬɟɥɚ ɤ ɦɟɧɟɟ ɧɚɝɪɟɬɨɦɭ, ɬɚɤ ɢ ɧɚɨɛɨɪɨɬ. Ʉɨɧɟɱɧɵɦ ɪɟɡɭɥɶɬɚɬɨɦ ɬɚɤɨɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɢ ɛɭɞɭɬ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɨɬɵ, ɩɟɪɟɞɚɧɧɨɣ ɢɡɥɭɱɟɧɢɟɦ. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ ɜɨɡɦɨɠɟɧ ɢ ɜ
ɜɚɤɭɭɦɟ.
Ⱦɥɹ ɥɸɛɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɜ ɥɸɛɨɣ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɦɧɨɝɨɱɢɫɥɟɧɧɵɟ ɷɥɟɤɬɪɨɧɵ ɫɨɜɟɪɲɚɸɬ ɩɟɪɟɯɨɞɵ ɧɚ ɪɚɡɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢ ɭɪɨɜɧɢ, ɩɨɷɬɨɦɭ ɷɧɟɪɝɢɹ ɮɨɬɨɧɨɜ, ɩɨɤɢɞɚɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɶ, ɪɚɫɩɪɟɞɟɥɟɧɚ
ɩɨ ɫɩɟɤɬɪɭ ɱɚɫɬɨɬ.
ȼɫɟ ɜɢɞɵ ɢɡɥɭɱɟɧɢɹ ɪɚɡɥɢɱɚɸɬɫɹ ɞɥɢɧɨɣ ɜɨɥɧɵ. Ⱦɥɹ ɧɚɫ ɧɚɢɛɨɥɶɲɢɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɸɬ ɧɨɫɢɬɟɥɢ ɬɟɩɥɨɜɨɣ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɢ: ɜɢɞɢ-
214
ɦɵɟ (ɫɜɟɬɨɜɵɟ) ɥɭɱɢ ɫ ɞɥɢɧɨɣ ɜɨɥɧɵ 0.4–0,8 ɦɤɦ ɢ ɨɫɨɛɟɧɧɨ ɢɧɮɪɚɤɪɚɫɧɵɟ ɫ ɞɥɢɧɨɣ ɜɨɥɧɵ 0,8–800 ɦɤɦ.
ȿɫɬɶ ɞɪɭɝɢɟ ɫɩɨɫɨɛɵ, ɤɪɨɦɟ ɧɚɝɪɟɜɚɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ
ɛɵɬɶ ɩɪɢɱɢɧɨɣ ɢɫɩɭɫɤɚɧɢɹ ɬɟɥɨɦ ɮɨɬɨɧɨɜ. ɇɚ ɤɨɪɨɬɤɨɜɨɥɧɨɜɨɦ ɤɨɧɰɟ
ɫɩɟɤɬɪɚ, ɧɚɩɪɢɦɟɪ, ɧɚɯɨɞɢɬɫɹ ɪɟɧɬɝɟɧɨɜɫɤɨɟ ɢɡɥɭɱɟɧɢɟ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ
ɛɵɬɶ ɜɵɡɜɚɧɨ ɛɨɦɛɚɪɞɢɪɨɜɤɨɣ ɤɭɫɤɚ ɦɟɬɚɥɥɚ ɩɨɬɨɤɨɦ ɷɥɟɤɬɪɨɧɨɜ. ɇɚ
ɞɪɭɝɨɦ ɤɨɧɰɟ ɫɩɟɤɬɪɚ ɧɚɯɨɞɹɬɫɹ ɪɚɞɢɨɜɨɥɧɵ ɫ ɛɨɥɶɲɢɦɢ ɞɥɢɧɚɦɢ ɜɨɥɧ,
ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɝɟɧɟɪɢɪɨɜɚɬɶɫɹ ɷɥɟɤɬɪɨɧɧɵɦ ɨɛɨɪɭɞɨɜɚɧɢɟɦ ɢ ɤɪɢɫɬɚɥɥɚɦɢ.
Ȼɨɥɶɲɢɧɫɬɜɨ ɬɜɟɪɞɵɯ ɢ ɠɢɞɤɢɯ ɬɟɥ ɢɦɟɸɬ ɫɩɥɨɲɧɨɣ ɫɩɟɤɬɪ ɢɡɥɭɱɟɧɢɹ, ɬ.ɟ. ɢɡɥɭɱɚɸɬ ɷɧɟɪɝɢɸ ɜɫɟɯ ɞɥɢɧ ɜɨɥɧ 0 d O d f . ɑɢɫɬɵɟ ɦɟɬɚɥɥɵ, ɦɟɬɚɥɥɵ ɫ ɩɨɥɢɪɨɜɚɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢ ɝɚɡɵ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ
ɩɪɟɪɵɜɢɫɬɵɦ ɫɩɟɤɬɪɨɦ ɢɡɥɭɱɟɧɢɹ, ɢɦɟɸɳɢɦ ɨɝɪɚɧɢɱɟɧɧɵɣ ɞɢɚɩɚɡɨɧ
ɞɥɢɧ ɜɨɥɧ. ɂɡɥɭɱɟɧɢɟ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟ ɭɡɤɨɦɭ ɢɧɬɟɪɜɚɥɭ ɞɥɢɧ ɜɨɥɧ,
(ɨɬ O ɞɨ O d O ), ɧɚɡɵɜɚɟɬɫɹ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɦ ɢɥɢ ɨɞɧɨɪɨɞɧɵɦ.
Ɍɟɩɥɨɜɨɟ ɢɡɥɭɱɟɧɢɟ ɤɨɥɢɱɟɫɬɜɟɧɧɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɨɥɧɵɦ ɩɨɬɨɤɨɦ ɢ ɩɥɨɬɧɨɫɬɶɸ ɩɨɬɨɤɚ.
ɋɭɦɦɚɪɧɚɹ ɷɧɟɪɝɢɹ, ɢɡɥɭɱɚɟɦɚɹ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ, ɜɨ ɜɫɟɦ ɢɧɬɟɪɜɚɥɟ ɞɥɢɧ ɜɨɥɧ ɫɩɟɤɬɪɚ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɵɦ
ɢɥɢ ɩɨɥɧɵɦ ɩɨɬɨɤɨɦ ɢɡɥɭɱɟɧɢɹ. ɂɡɦɟɪɹɟɬɫɹ ɜ ɜɚɬɬɚɯ – ȼɬ:
Q
³ E dF ,
(9.1)
F
ɝɞɟ E – ɷɧɟɪɝɢɹ, ɢɡɥɭɱɚɟɦɚɹ ɫ ɟɞɢɧɢɰɵ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɜ ɟɞɢɧɢɰɭ
ɜɪɟɦɟɧɢ ɩɨ ɜɫɟɦ ɧɚɩɪɚɜɥɟɧɢɹɦ ɫɮɟɪɢɱɟɫɤɨɝɨ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ, ȼɬ/ɫɦ2:
E dQ dF .
(9.2)
ȼɟɥɢɱɢɧɚ E ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ
ɬɟɥɚ ɢ ɧɚɡɵɜɚɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ ɢɡɥɭɱɟɧɢɟɦ ɢɥɢ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɬɟɥɚ. ɗɬɚ ɜɟɥɢɱɢɧɚ ɹɜɥɹɟɬɫɹ ɩɥɨɬɧɨɫɬɶɸ ɩɨɬɨɤɚ ɢɧɬɟɝɪɚɥɶɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ
E q.
Ɉɬɧɨɲɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɢɧɬɟɝɪɚɥɶɧɨɝɨ ɥɭɱɢɫɬɨɝɨ ɩɨɬɨɤɚ, ɢɫɩɭɫɤɚɟɦɨɝɨ ɜ ɛɟɫɤɨɧɟɱɧɨ ɦɚɥɨɦ ɢɧɬɟɪɜɚɥɟ ɞɥɢɧ ɜɨɥɧ, ɤ ɜɟɥɢɱɢɧɟ ɷɬɨɝɨ ɢɧɬɟɪɜɚɥɚ ɧɚɡɵɜɚɟɬɫɹ ɫɩɟɤɬɪɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɶɸ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ
E O d E d O , ȼɬ/ɦ3.
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɥɨ, ɭɱɚɫɬɜɭɸɳɟɟ ɜ ɥɭɱɢɫɬɨɦ ɬɟɩɥɨɨɛɦɟɧɟ ɫ ɞɪɭɝɢɦɢ
ɬɟɥɚɦɢ (ɪɢɫ. 9.1). ɇɚ ɩɨɜɟɪɯɧɨɫɬɶ ɞɚɧɧɨɝɨ ɬɟɥɚ ɩɚɞɚɟɬ ɷɧɟɪɝɢɹ ɢɡɥɭɱɟɧɢɹ ɞɪɭɝɢɯ ɬɟɥ Q – ɩɚɞɚɸɳɟɟ ɢɡɥɭɱɟɧɢɟ. ɗɬɚ ɷɧɟɪɝɢɹ ɱɚɫɬɢɱɧɨ ɩɨɝɥɨɳɚɟɬɫɹ ɬɟɥɨɦ, ɱɚɫɬɢɱɧɨ ɨɬɪɚɠɚɟɬɫɹ, ɚ ɱɚɫɬɢɱɧɨ ɩɪɨɯɨɞɢɬ ɫɤɜɨɡɶ ɬɟɥɨ.
Ʉɚɠɞɚɹ ɢɡ ɷɬɢɯ ɱɚɫɬɟɣ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɩɨɬɨɤɚɦɢ –
215
ɩɨɬɨɤɨɦ ɩɨɝɥɨɳɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ Q A ; ɩɨɬɨɤɨɦ ɨɬɪɚɠɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ
Q R ; ɩɨɬɨɤɨɦ ɩɪɨɩɭɫɤɚɟɦɨɝɨ ɢɡɥɭɱɟɧɢɹ Q D . Ɇɨɠɧɨ ɡɚɩɢɫɚɬɶ ɪɚɜɟɧɫɬɜɚ
Q A AQ ; Q R RQ ; Q D DQ ,
ɝɞɟ A – ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ
ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ; R – ɨɬɪɚɠɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ;
D – ɩɪɨɩɭɫɤɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ
ɫɨɯɪɚɧɟɧɢɹ ɷɧɟɪɝɢɢ ɢɧɬɟɝɪɚɥɶɧɵɣ ɥɭɱɢɫɬɵɣ ɩɨɬɨɤ,
ɩɚɞɚɸɳɢɣ ɧɚ ɬɟɥɨ, ɪɚɜɟɧ
Ɋɢɫ. 9.1. ɋɨɫɬɚɜɥɹɸɳɢɟ ɢɧɬɟɝɪɚɥɶɧɨɝɨ
ɫɭɦɦɟ ɜɫɟɯ ɫɨɫɬɚɜɥɹɸɳɢɯ
ɥɭɱɢɫɬɨɝɨ ɩɨɬɨɤɚ
Q Q A QR QD ,
ɨɬɤɭɞɚ ɫ ɭɱɟɬɨɦ ɜɵɪɚɠɟɧɢɣ ɞɥɹ ɫɨɫɬɚɜɥɹɸɳɢɯ ɫɭɦɦɚɪɧɨɝɨ ɩɨɬɨɤɚ ɫɥɟɞɭɟɬ
A R D 1.
(9.3)
Ʉɚɠɞɵɣ ɢɡ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɦɨɠɟɬ ɦɟɧɹɬɶɫɹ ɜ ɩɪɟɞɟɥɚɯ ɨɬ 0 ɞɨ 1. ȿɫɥɢ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ A 1 , ɬɨ ɞɜɚ
ɞɪɭɝɢɯ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɜɧɵ ɧɭɥɸ ( R D 0 ). Ɍɟɥɚ, ɩɨɝɥɨɳɚɸɳɢɟ ɜɫɸ
ɩɚɞɚɸɳɭɸ ɧɚ ɧɢɯ ɷɧɟɪɝɢɸ, ɧɚɡɵɜɚɸɬɫɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦɢ. ɑɟɪɧɨɟ
ɬɟɥɨ – ɷɬɨ ɷɬɚɥɨɧ, ɫ ɤɨɬɨɪɵɦ ɦɨɠɧɨ ɫɪɚɜɧɢɜɚɬɶ ɜɫɟ ɞɪɭɝɢɟ ɢɡɥɭɱɚɬɟɥɢ.
Ʉ ɷɬɨɦɭ ɷɬɚɥɨɧɭ ɦɨɠɧɨ ɩɪɢɛɥɢɡɢɬɶɫɹ ɧɚ ɩɪɚɤɬɢɤɟ ɩɨɤɪɵɬɢɟɦ ɩɨɜɟɪɯɧɨɫɬɢ ɬɟɥɚ ɢɥɢ ɜɢɞɨɢɡɦɟɧɟɧɢɟɦ ɮɨɪɦɵ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ.
ɉɨɝɥɨɳɟɧɧɚɹ ɷɧɟɪɝɢɹ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜɧɨɜɶ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɸɸ ɷɧɟɪɝɢɸ ɬɟɥɚ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɬɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ ɫɜɹɡɚɧ ɫ ɞɜɨɣɧɵɦ ɩɪɟɜɪɚɳɟɧɢɟɦ ɷɧɟɪɝɢɢ: ɬɟɩɥɨɬɚ ɬɪɚɧɫɮɨɪɦɢɪɭɟɬɫɹ ɜ ɷɧɟɪɝɢɸ ɢɡɥɭɱɟɧɢɹ, ɤɨɬɨɪɚɹ, ɱɚɫɬɢɱɧɨ ɩɨɝɥɨɳɚɹɫɶ ɞɪɭɝɢɦ ɬɟɥɨɦ, ɜɧɨɜɶ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɸɸ ɷɧɟɪɝɢɸ ɬɟɥɚ.
ȿɫɥɢ ɩɪɟɞɦɟɬ ɩɨɝɥɨɳɚɟɬ ɜɫɟ ɥɭɱɢ, ɬɨ ɨɧ ɡɪɢɬɟɥɶɧɨ ɜɨɫɩɪɢɧɢɦɚɟɬɫɹ
ɤɚɤ ɱɟɪɧɨɟ ɬɟɥɨ. ȿɫɥɢ ɠɟ ɩɨɜɟɪɯɧɨɫɬɶ ɩɨɝɥɨɳɚɟɬ ɜɫɟ ɥɭɱɢ, ɤɪɨɦɟ ɜɢɞɢɦɵɯ, ɬɨ ɨɧɚ ɧɟ ɤɚɠɟɬɫɹ ɱɟɪɧɨɣ, ɯɨɬɹ ɩɨ ɥɭɱɢɫɬɵɦ ɫɜɨɣɫɬɜɚɦ ɦɨɠɟɬ
ɛɵɬɶ ɛɥɢɡɤɚ ɤ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɦɭ ɬɟɥɭ. ɇɚɩɪɢɦɟɪ, ɫɧɟɝ ɩɨ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ( A 0,95 y 0,98 ) ɨɬɧɨɫɢɬɫɹ ɤ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦ
ɬɟɥɚɦ, ɯɨɬɹ ɢɦɟɟɬ ɛɟɥɵɣ ɰɜɟɬ. Ⱦɟɥɨ ɜ ɬɨɦ, ɱɬɨ ɛɟɥɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɯɨɪɨɲɨ ɨɬɪɚɠɚɟɬ ɬɨɥɶɤɨ ɜɢɞɢɦɵɟ (ɫɜɟɬɨɜɵɟ) ɥɭɱɢ, ɱɬɨ ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɠɢɡɧɢ: ɛɟɥɵɟ ɤɨɫɬɸɦɵ, ɨɤɪɚɫɤɚ ɜɚɝɨɧɨɜ-ɪɟɮɪɢɠɟɪɚɬɨɪɨɜ, ɰɢɫɬɟɪɧ ɢ ɬ.ɞ., ɚ
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ɧɟɜɢɞɢɦɵɟ ɬɟɩɥɨɜɵɟ ɥɭɱɢ ɛɟɥɚɹ ɤɪɚɫɤɚ ɢ ɬɤɚɧɶ ɩɨɝɥɨɳɚɸɬ ɬɚɤɠɟ ɯɨɪɨɲɨ, ɤɚɤ ɢ ɬɟɦɧɵɟ ɩɨɜɟɪɯɧɨɫɬɢ.
Ɍɟɥɨ, ɞɥɹ ɤɨɬɨɪɨɝɨ R 1 ɢ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, A D 0 , ɨɬɪɚɠɚɟɬ ɜɫɸ
ɥɭɱɢɫɬɭɸ ɷɧɟɪɝɢɸ. ȿɫɥɢ ɷɬɨ ɨɬɪɚɠɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɡɚɤɨɧɚɦ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɨɩɬɢɤɢ, ɬɨ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɶ ɧɚɡɵɜɚɟɬɫɹ ɡɟɪɤɚɥɶɧɨɣ, ɟɫɥɢ ɠɟ
ɨɬɪɚɠɟɧɢɟ – ɪɚɫɫɟɹɧɧɨɟ, ɬɨ ɚɛɫɨɥɸɬɧɨ ɛɟɥɨɣ.
Ɍɟɥɨ, ɞɥɹ ɤɨɬɨɪɨɝɨ D 1 , ɚ A R 0 , ɩɪɨɩɭɫɤɚɟɬ ɜɫɸ ɥɭɱɢɫɬɭɸ
ɷɧɟɪɝɢɸ ɢ ɧɚɡɵɜɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɩɪɨɡɪɚɱɧɵɦ. Ɍɟɥɚ, ɞɥɹ ɤɨɬɨɪɵɯ
0 D 1 , ɧɚɡɵɜɚɸɬɫɹ ɩɨɥɭɩɪɨɡɪɚɱɧɵɦɢ. Ɇɧɨɝɢɟ ɬɜɟɪɞɵɟ ɬɟɥɚ ɢ ɠɢɞɤɨɫɬɢ ɞɥɹ ɬɟɩɥɨɜɵɯ ɥɭɱɟɣ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɩɪɨɡɪɚɱɧɵ. ɋɭɳɟɫɬɜɭɸɬ ɬɟɥɚ,
ɤɨɬɨɪɵɟ ɩɪɨɡɪɚɱɧɵ ɬɨɥɶɤɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɧɵɯ ɞɥɢɧ ɜɨɥɧ. ɇɚɩɪɢɦɟɪ,
ɨɤɨɧɧɨɟ ɫɬɟɤɥɨ ɩɪɨɡɪɚɱɧɨ ɞɥɹ ɫɜɟɬɨɜɵɯ ɥɭɱɟɣ ɢ ɧɟɩɪɨɡɪɚɱɧɨ ɞɥɹ ɭɥɶɬɪɚɮɢɨɥɟɬɨɜɵɯ, ɚ ɤɜɚɪɰ – ɩɪɨɡɪɚɱɟɧ ɞɥɹ ɫɜɟɬɨɜɵɯ ɢ ɭɥɶɬɪɚɮɢɨɥɟɬɨɜɵɯ
ɥɭɱɟɣ, ɧɨ ɧɟ ɩɪɨɡɪɚɱɟɧ ɞɥɹ ɬɟɩɥɨɜɵɯ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɰɜɟɬɨɜɵɟ ɢ ɨɩɬɢɱɟɫɤɢɟ ɨɳɭɳɟɧɢɹ ɱɟɥɨɜɟɤɚ ɧɟ ɜɫɟɝɞɚ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɫɩɨɫɨɛɧɨɫɬɹɦ ɬɟɥɚ ɨɬɪɚɠɚɬɶ, ɩɨɝɥɨɳɚɬɶ ɥɢ ɩɪɨɩɭɫɤɚɬɶ ɬɟɩɥɨɜɨɟ ɢɡɥɭɱɟɧɢɟ. Ⱦɥɹ ɩɨɝɥɨɳɟɧɢɹ ɢ ɨɬɪɚɠɟɧɢɹ ɬɟɩɥɨɜɵɯ ɥɭɱɟɣ
ɪɟɲɚɸɳɭɸ ɪɨɥɶ ɢɝɪɚɟɬ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ: ɱɟɦ ɨɧɚ ɛɨɥɶɲɟ,
ɬɟɦ ɛɨɥɶɲɟ ɷɧɟɪɝɢɢ ɩɨɝɥɨɳɚɟɬ ɢ ɢɡɥɭɱɚɟɬ ɩɨɜɟɪɯɧɨɫɬɶ. ɉɪɢɦɟɪ ɫɨ ɫɧɟɝɨɦ ɫɜɢɞɟɬɟɥɶɫɬɜɭɟɬ ɨ ɬɨɦ, ɱɬɨ ɞɥɹ ɡɚɳɢɬɵ ɚɩɩɚɪɚɬɨɜ ɨɬ ɜɨɡɞɟɣɫɬɜɢɹ
ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ ɢɯ ɩɨɜɟɪɯɧɨɫɬɶ ɞɨɥɠɧɚ ɛɵɬɶ ɧɟ ɬɨɥɶɤɨ ɛɟɥɨɣ, ɧɨ ɢ
ɨɱɟɧɶ ɝɥɚɞɤɨɣ.
Ʉɚɤ ɛɵɥɨ ɫɤɚɡɚɧɨ, ɤɚɠɞɨɟ ɬɟɥɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɨɬɨɤɨɦ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ Q p . ȿɝɨ ɫɭɦɦɚ ɫ ɩɨɬɨɤɨɦ ɨɬɪɚɠɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɫɨɫɬɚɜɥɹɟɬ ɩɨɬɨɤ ɷɮɮɟɤɬɢɜɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɬɟɥɚ (ɪɢɫ. 9.1)
Qres
Q p QR .
(9.4)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɥɭɱɢɫɬɵɦ ɬɟɩɥɨɨɛɦɟɧɨɦ ɧɚɡɵɜɚɸɬɫɹ ɫɨɜɦɟɫɬɧɵɟ
ɩɪɨɰɟɫɫ ɜɡɚɢɦɧɨɝɨ ɢɫɩɭɫɤɚɧɢɹ, ɩɨɝɥɨɳɟɧɢɹ, ɨɬɪɚɠɟɧɢɹ ɢ ɩɪɨɩɭɫɤɚɧɢɹ
ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ ɜ ɫɢɫɬɟɦɚɯ ɪɚɡɥɢɱɧɵɯ ɬɟɥ. ɉɪɢ ɷɬɨɦ ɬɟɥɚ, ɜɯɨɞɹɳɢɟ
ɜ ɫɢɫɬɟɦɭ, ɦɨɝɭɬ ɢɦɟɬɶ ɤɚɤ ɪɚɡɥɢɱɧɵɟ, ɬɚɤ ɢ ɨɞɢɧɚɤɨɜɵɟ ɬɟɦɩɟɪɚɬɭɪɵ.
Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ ɦɟɠɞɭ ɷɬɢɦɢ ɬɟɥɚɦɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɨɬɨɤɨɦ
ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ Q s u m , ɤɨɬɨɪɵɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɡɧɨɫɬɶ ɦɟɠɞɭ ɫɨɛɫɬɜɟɧɧɵɦ ɢɡɥɭɱɟɧɢɟɦ ɬɟɥɚ ɢ ɱɚɫɬɶɸ ɩɚɞɚɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ, ɤɨɬɨɪɚɹ ɩɨɝɥɨɳɚɟɬɫɹ ɞɚɧɧɵɦ ɬɟɥɨɦ:
Qs um
Q p QA.
ȼ ɩɪɢɪɨɞɟ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɯ, ɛɟɥɵɯ ɢ ɩɪɨɡɪɚɱɧɵɯ ɬɟɥ ɧɟ ɫɭɳɟɫɬɜɭɟɬ, ɚ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɪɟɚɥɶɧɵɦ ɬɟɥɚɦ ɷɬɢ ɩɨɧɹɬɢɹ ɭɫɥɨɜɧɵ.
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9.2. Ɉɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ
9.2.1. Ɂɚɤɨɧ ɉɥɚɧɤɚ
ȼ 1900 ɝɨɞɭ Ɇ. ɉɥɚɧɤ, ɢɫɯɨɞɹ ɢɡ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɩɪɢɪɨɞɵ ɢɡɥɭɱɟɧɢɹ ɢ ɪɚɡɪɚɛɨɬɚɧɧɨɣ ɢɦ ɤɜɚɧɬɨɜɨɣ ɬɟɨɪɢɢ, ɭɫɬɚɧɨɜɢɥ ɞɥɹ ɚɛɫɨɥɸɬɧɨ
ɱɟɪɧɨɝɨ ɬɟɥɚ (ɢɧɞɟɤɫ 0) ɡɚɜɢɫɢɦɨɫɬɶ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɬɟɥɚ ɨɬ ɞɥɢɧɵ ɜɨɥɧɵ ɢ ɬɟɦɩɟɪɚɬɭɪɵ
E0 O
E 0O
C1O 5
C 2 O T ,
(9.5)
1
e
ɝɞɟ O – ɞɥɢɧɚ ɜɨɥɧɵ, ɦ; T – ɚɛɫɨɥɸɬɧɚɹ
ɬɟɦɩɟɪɚɬɭɪɚ ɬɟɥɚ, Ʉ; C1 ,C 2 – ɤɨɧɫɬɚɧɬɵ:
C1 3,74 ˜10 16 ȼɬ·ɦ2; C2 0 ,0144 ɦ ˜ Ʉ .
ɇɚ ɪɢɫ. 9.2. ɡɚɤɨɧ ɉɥɚɧɤɚ ɩɪɟɞɫɬɚɜɥɟɧ
ɝɪɚɮɢɱɟɫɤɢ. ɂɡ ɝɪɚɮɢɤɚ ɜɢɞɧɨ, ɱɬɨ ɫ ɭɜɟO
ɥɢɱɟɧɢɟɦ ɞɥɢɧɵ ɜɨɥɧɵ ɩɪɢ ɥɸɛɨɣ ɬɟɦɩɟɊɢɫ. 9.2. Ƚɪɚɮɢɱɟɫɤɨɟ ɩɪɟɞɪɚɬɭɪɟ ( T1 ,T2 ,T3 ) ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ
ɫɬɚɜɥɟɧɢɟ ɡɚɤɨɧɚ ɉɥɚɧɤɚ
ɫɧɚɱɚɥɚ ɛɵɫɬɪɨ ɜɨɡɪɚɫɬɚɟɬ, ɞɨɫɬɢɝɚɹ ɦɚɤɫɢɦɭɦɚ (ɬɨɱɤɢ M 1 , M 2 , M 3 ), ɚ ɡɚɬɟɦ ɦɟɞɥɟɧɧɨ ɭɛɵɜɚɟɬ. ɋ ɩɨɜɵɲɟɧɢɟɦ
ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ ( T3 ! T2 ! T1 ) ɷɧɟɪɝɢɹ ɟɝɨ ɢɡɥɭɱɟɧɢɹ ɫɭɳɟɫɬɜɟɧɧɨ
ɜɨɡɪɚɫɬɚɟɬ (ɧɚ ɝɪɚɮɢɤɟ ɨɧɚ ɢɡɨɛɪɚɠɚɟɬɫɹ ɩɥɨɳɚɞɶɸ ɩɨɞ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɢɡɨɬɟɪɦɨɣ). Ʉɪɨɦɟ ɬɨɝɨ, ɫ ɩɨɜɵɲɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɭɜɟɥɢɱɢɜɚɟɬɫɹ
ɷɧɟɪɝɢɹ ɥɭɱɚ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɞɥɢɧɵ ɜɨɥɧɵ.
9.2.2. Ɂɚɤɨɧ ɫɦɟɳɟɧɢɹ ȼɢɧɚ
ɂɡ ɡɚɤɨɧɚ ɉɥɚɧɤɚ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɞɥɢɧɭ ɜɨɥɧɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ ɦɚɤɫɢɦɚɥɶɧɨɣ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ. Ɋɚɡɪɟɲɚɹ ɭɪɚɜɧɟɧɢɟ
d E dO 0 ,
(9.6)
ɩɨɥɭɱɢɦ ɡɚɜɢɫɢɦɨɫɬɶ
2,898 ˜103
O max
,
(9.7)
T
ɩɪɟɞɫɬɚɜɥɹɸɳɭɸ ɫɨɛɨɣ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ ɡɚɤɨɧɚ ȼɢɧɚ.
ȼɟɥɢɱɢɧɚ O max ɟɫɬɶ ɞɥɢɧɚ ɜɨɥɧɵ, ɩɪɢ ɤɨɬɨɪɨɣ ɞɨɫɬɢɝɚɟɬɫɹ ɦɚɤɫɢɦɭɦ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɱɟɪɧɨɝɨ ɬɟɥɚ ɫ
ɬɟɦɩɟɪɚɬɭɪɨɣ T .
Ɇɚɤɫɢɦɚɥɶɧɚɹ ɫɩɟɤɬɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɫ ɩɨɜɵɲɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɫɦɟɳɚɟɬɫɹ ɜ ɫɬɨɪɨɧɭ ɛɨɥɟɟ ɤɨɪɨɬɤɢɯ ɞɥɢɧ ɜɨɥɧ.
218
ɂɡ ɪɢɫɭɧɤɚ 9.2. ɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ T3 ! T2 ! T1 , ɬɨ ɢ
O 3,max ! O 2 ,max ! O1,max .
Ɇɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɝɨ ɢɡɥɭɱɟɧɢɹ ɱɟɪɧɨɝɨ ɬɟɥɚ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɨɞɫɬɚɧɨɜɤɨɣ (9.7) ɜ (9.5), ɱɬɨ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɞɚɟɬ
E0O max 1,287 ˜105T 5 ȼɬ/ɦ3.
ɋɨɥɧɰɟ ɹɜɥɹɟɬɫɹ ɩɪɢɦɟɪɨɦ ɢɫɬɨɱɧɢɤɚ ɷɧɟɪɝɢɢ ɫ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ. ȼɧɟɲɧɹɹ ɩɨɜɟɪɯɧɨɫɬɶ ɋɨɥɧɰɚ ɢɦɟɟɬ ɬɟɦɩɟɪɚɬɭɪɭ ~5800 Ʉ. ɋɨɝɥɚɫɧɨ ɡɚɤɨɧɭ ȼɢɧɚ, O max ɩɪɢ ɷɬɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɪɚɜɧɹɟɬɫɹ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ
5,2 ˜ 10 7 ɦ, ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɫɟɪɟɞɢɧɟ ɜɢɞɢɦɨɣ ɨɛɥɚɫɬɢ. Ƚɥɚɡ ɱɟɥɨɜɟɤɚ ɢɞɟɚɥɶɧɨ ɩɪɢɫɩɨɫɨɛɥɟɧ ɤ ɜɨɫɩɪɢɹɬɢɸ ɦɚɤɫɢɦɭɦɚ
ɷɧɟɪɝɢɢ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɝɨ ɢɡɥɭɱɟɧɢɹ ɋɨɥɧɰɚ.
9.2.3. Ɂɚɤɨɧ ɋɬɟɮɚɧɚ–Ȼɨɥɶɰɦɚɧɚ
Ɂɚɤɨɧ ɋɬɟɮɚɧɚ–Ȼɨɥɶɰɦɚɧɚ, ɨɬɤɪɵɬɵɣ ɜ 1879 ɝɨɞɭ ɱɟɲɫɤɢɦ ɭɱɟɧɵɦ
Ƀ. ɋɬɟɮɚɧɨɦ ɢ ɬɟɨɪɟɬɢɱɟɫɤɢ ɨɛɨɫɧɨɜɚɧɧɵɣ ɜ 1884 ɝɨɞɭ ɚɜɫɬɪɢɣɫɤɢɦ
ɭɱɟɧɵɦ Ʌ .Ȼɨɥɶɰɦɚɧɨɦ, ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɚɛɫɨɥɸɬɧɨ–ɱɟɪɧɨɝɨ ɬɟɥɚ ɨɬ ɟɝɨ ɬɟɦɩɟɪɚɬɭɪɵ:
f
E0
³ E 0O d O
V 0T 4 ,
(9.8)
0
ɝɞɟ V 0
5,77 ˜ 10 8 ȼɬ/(ɦ2Ʉ4) – ɩɨɫɬɨɹɧɧɚɹ ɋɬɟɮɚɧɚ–Ȼɨɥɶɰɦɚɧɚ,
4
§ S · C1
.
V0 ¨
¸
C
15
© 2¹
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɝɨ ɬɟɥɚ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɱɟɬɜɟɪɬɨɣ ɫɬɟɩɟɧɢ ɟɝɨ ɚɛɫɨɥɸɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ.
ɂɧɬɟɝɪɚɥɶɧɨɟ ɢɡɥɭɱɟɧɢɟ ɚɛɫɨɥɸɬɧɨ
ɱɟɪɧɨɝɨ ɬɟɥɚ ɩɪɢ ɞɚɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɜ
ɩɪɟɞɟɥɚɯ ɨɬ O 0 ɞɨ O f ɝɪɚɮɢɱɟɫɤɢ
ɢɡɨɛɪɚɠɚɟɬɫɹ ɩɥɨɳɚɞɶɸ, ɨɝɪɚɧɢɱɟɧɧɨɣ
ɤɪɢɜɨɣ T const ɢ ɨɫɶɸ ɚɛɫɰɢɫɫ
(ɪɢɫ. 9.3); C1 ɢ C 2 – ɬɟ ɠɟ ɩɨɫɬɨɹɧɧɵɟ,
ɱɬɨ ɢ ɜ ɡɚɤɨɧɟ ɉɥɚɧɤɚ.
Ɋɢɫ. 9.3. ɂɧɬɟɝɪɚɥɶɧɨɟ
ɢɡɥɭɱɟɧɢɟ ɚɛɫɨɥɸɬɧɨ
ɱɟɪɧɨɝɨ ɬɟɥɚ
ɂɡ ɡɚɤɨɧɚ ɋɬɟɮɚɧɚ-Ȼɨɥɶɰɦɚɧɚ ɫɥɟɞɭɟɬ, ɱɬɨ ɜɥɢɹɧɢɟ ɢɡɥɭɱɟɧɢɹ ɜ ɛɨɥɶɲɢɧɫɬɜɟ
219
ɫɥɭɱɚɟɜ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɩɪɢ ɧɢɡɤɢɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ɜɫɥɟɞɫɬɜɢɟ ɦɚɥɨɫɬɢ
ɤɨɷɮɮɢɰɢɟɧɬɚ V . ɉɪɢ ɤɨɦɧɚɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɢɥɢ ɨɤɨɥɨ 300 Ʉ, ɢɧɬɟɝɪɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɱɟɪɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɬɨɥɶɤɨ
~ 460 ȼɬ/ɦ2. ɗɬɚ ɜɟɥɢɱɢɧɚ ɫɨɫɬɚɜɥɹɟɬ ɨɤɨɥɨ 1/10 ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ
ɩɨɬɨɤɚ, ɩɟɪɟɞɚɜɚɟɦɨɝɨ ɤɨɧɜɟɤɰɢɟɣ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɠɢɞɤɨɫɬɢ, ɤɨɝɞɚ ɤɨɷɮɮɢɰɢɟɧɬɵ ɤɨɧɜɟɤɬɢɜɧɨɣ ɬɟɩɥɨɨɬɞɚɱɢ ɢ ɩɟɪɟɩɚɞ ɬɟɦɩɟɪɚɬɭɪ ɢɦɟɸɬ
ɞɨɫɬɚɬɨɱɧɨ ɧɢɡɤɢɟ ɡɧɚɱɟɧɢɹ ~100 ȼɬ/(ɦ2Ʉ) ɢ 50 Ʉ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ.
Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɪɚɫɱɟɬɨɜ ɜɵɪɚɠɟɧɢɟ (9.8) ɩɪɟɞɫɬɚɜɥɹɸɬ ɜ ɜɢɞɟ
E0
C 0 T 100 ,
4
(9.9)
ɝɞɟ C0 5 ,77 ȼɬ/(ɦ2·Ʉ4) – ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɥɭɱɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɝɨ
ɬɟɥɚ.
Ⱦɥɹ ɪɟɚɥɶɧɵɯ ɬɟɥ, ɬ.ɟ., ɧɟɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɯ (ɫɟɪɵɯ ɬɟɥ) ɩɥɨɬɧɨɫɬɶ
ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɜɵɪɚɠɚɟɬɫɹ ɬɚɤɨɣ ɠɟ ɮɨɪɦɭɥɨɣ
E C T 100 ,
4
ɧɨ ɜɟɥɢɱɢɧɚ C ɨɬɧɨɫɢɬɫɹ ɭɠɟ ɤ ɫɟɪɵɦ ɬɟɥɚɦ.
Ⱦɥɹ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɩɥɨɬɧɨɫɬɟɣ ɩɨɬɨɤɨɜ ɢɡɥɭɱɟɧɢɹ ɪɟɚɥɶɧɨɝɨ ɢ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɝɨ ɬɟɥ ɩɪɢ ɨɞɧɨɣ ɢ ɬɨɣ ɠɟ ɬɟɦɩɟɪɚɬɭɪɟ ɢɫɩɨɥɶɡɭɸɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤɭ ɬɟɥɚ H , ɧɚɡɵɜɚɟɦɭɸ ɫɬɟɩɟɧɶɸ ɱɟɪɧɨɬɵ ɬɟɥɚ
H
E E0
C C0
ɢɥɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ.
Ɂɚɤɨɧ ɋɬɟɮɚɧɚ–Ȼɨɥɶɰɦɚɧɚ ɞɥɹ ɫɟɪɨɝɨ ɬɟɥɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ
E H V 0T 4 H C 0 T 100 .
ȼɟɥɢɱɢɧɚ H ɞɥɹ ɫɟɪɵɯ ɬɟɥ ɜɫɟɝɞɚ ɦɟɧɶɲɟ ɟɞɢɧɢɰɵ; ɨɧɚ ɡɚɜɢɫɢɬ ɨɬ
ɩɪɢɪɨɞɵ ɬɟɥɚ, ɫɨɫɬɨɹɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɧɚɯɨɞɢɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɩɭɬɟɦ.
4
9.2.4. Ɂɚɤɨɧ Ʌɚɦɛɟɪɬɚ
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ, ɢɫɩɭɫɤɚɟɦɨɣ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦ
ɬɟɥɨɦ, ɜ ɪɚɡɥɢɱɧɵɯ ɧɚɩɪɚɜɥɟɧɢɹɯ ɩɪɨɫɬɪɚɧɫɬɜɚ ɧɟɨɞɢɧɚɤɨɜɨ. ȼ 1760
ɝɨɞɭ ɧɟɦɟɰɤɢɣ ɭɱɟɧɵɣ ɂ. Ʌɚɦɛɟɪɬ ɭɫɬɚɧɨɜɢɥ ɡɚɜɢɫɢɦɨɫɬɶ ɜɟɥɢɱɢɧɵ
ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɟɟ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ
ɡɚɩɢɫɶ ɡɚɤɨɧɚ Ʌɚɦɛɟɪɬɚ ɞɥɹ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ
m , ɫɨɫɬɚɜɥɹɸɳɟɦ ɫ ɧɨɪɦɚɥɶɸ n ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɢɡɥɭɱɟɧɢɹ ɭɝɨɥ M , ɢɦɟɟɬ ɜɢɞ (ɪɢɫ. 9.4)
220
E 0M
E 0 nc o s M ,
(9.10)
ɝɞɟ E 0M – ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ
ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɝɨ ɬɟɥɚ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ M 0 .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɚɛɫɨɥɸɬɧɨ
ɱɟɪɧɨɝɨ ɬɟɥɚ ɩɨɬɨɤ ɢɡɥɭɱɟɧɢɹ ɜ ɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɩɨɬɨɤɭ ɢɡɥɭɱɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ
Ɋɢɫ. 9.4. Ʉ ɨɛɴɹɫɧɟɧɢɸ ɡɚɤɨɧɚ
ɤ ɢɡɥɭɱɚɸɳɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɤɨɫɢɧɭɫɭ
Ʌɚɦɛɟɪɬɚ
ɭɝɥɚ ɦɟɠɞɭ ɧɢɦɢ.
ɂɡ (9.10) ɫɥɟɞɭɟɬ, ɱɬɨ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ M 0 ɢɡɥɭɱɚɟɬɫɹ ɧɚɢɛɨɥɶɲɟɟ ɤɨɥɢɱɟɫɬɜɨ ɷɧɟɪɝɢɢ. ɂɧɬɟɝɪɢɪɭɹ (9.10) ɩɨ ɭɝɥɭ
M , ɧɚɣɞɟɦ
E 0n E 0 S ,
ɬ.ɟ., ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɧɨɪɦɚɥɢ ɤ ɩɨɜɟɪɯɧɨɫɬɢ
ɜ S ɪɚɡ ɦɟɧɶɲɟ ɩɨɥɧɨɣ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ.
Ⱦɥɹ ɪɟɚɥɶɧɵɯ ɬɟɥ ɡɚɤɨɧ Ʌɚɦɛɟɪɬɚ ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ ɥɢɲɶ ɩɪɢ ɭɝɥɟ
M 0 y 60 q (ɫɦ. ɪɚɡɞɟɥ 9.6)
9.2.5. Ɂɚɤɨɧ Ʉɢɪɯɝɨɮɚ
Ɂɚɤɨɧ Ʉɢɪɯɝɨɮɚ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɜɡɚɢɦɨɫɜɹɡɶ ɦɟɠɞɭ ɫɩɨɫɨɛɧɨɫɬɹɦɢ
ɬɟɥɚ ɢɡɥɭɱɚɬɶ ɢ ɩɨɝɥɨɳɚɬɶ ɷɧɟɪɝɢɸ. ɗɬɚ ɫɜɹɡɶ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ ɢɡ
ɪɚɫɫɦɨɬɪɟɧɢɹ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɩɪɢ ɥɭɱɢɫɬɨɦ ɬɟɩɥɨɨɛɦɟɧɟ ɦɟɠɞɭ ɞɜɭɦɹ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɩɨɜɟɪɯɧɨɫɬɹɦɢ (ɫɦ. ɪɢɫ. 9.5), ɥɟɜɚɹ
ɢɡ ɤɨɬɨɪɵɯ – ɫɟɪɚɹ, ɚ ɩɪɚɜɚɹ – ɱɟɪɧɚɹ. ɋɟɪɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɦɩɟɪɚɬɭɪɨɣ T , ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ A ɢ ɩɥɨɬɧɨɫɬɶɸ
ɩɨɬɨɤɚ ɫɨɛɫɬɜɟɧɧɨɝɨ ɢɡɥɭɱɟɧɢɹ E , ɚ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɚɹ – ɜɟɥɢɱɢɧɚɦɢ
T0 , A0 ɢ E 0 . ɋɨɛɫɬɜɟɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɫɟɪɨɣ ɩɨɜɟɪɯɧɨɫɬɢ E ɩɨɝɥɨɳɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦ ɬɟɥɨɦ. Ⱥɛɫɨɥɸɬɧɨ ɱɟɪɧɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɡɚ ɬɨ ɠɟ
ɜɪɟɦɹ ɢɡɥɭɱɚɟɬ ɩɨɬɨɤ, ɤɨɬɨɪɵɣ ɱɚɫɬɢɱɧɨ ɩɨɝɥɨɳɚɟɬɫɹ (ɜ ɤɨɥɢɱɟɫɬɜɟ
AE 0 ) ɫɟɪɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɚ ɱɚɫɬɢɱɧɨ – ɜ ɤɨɥɢɱɟɫɬɜɟ 1 A E 0 ɨɬɪɚɠɚɟɬɫɹ ɨɬ ɧɟɟ. Ɉɬɪɚɠɟɧɧɚɹ ɱɚɫɬɶ ɡɚɬɟɦ ɩɨɝɥɨɳɚɟɬɫɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦ
ɬɟɥɨɦ. ɉɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɪɟɡɭɥɶɬɢɪɭɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ ɜ ɫɥɭɱɚɟ T ! T0
ɧɚɯɨɞɢɬɫɹ ɢɡ ɷɧɟɪɝɟɬɢɱɟɫɤɨɝɨ ɛɚɥɚɧɫɚ ɧɚ ɫɟɪɨɣ ɩɨɜɟɪɯɧɨɫɬɢ
Esum
E AE0 .
221
(9.11)
ɉɪɢ T T0 ɫɢɫɬɟɦɚ ɧɚɯɨɞɢɬɫɹ ɜ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɦ ɪɚɜɧɨɜɟɫɢɢ,
ɬ.ɟ. ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɢɦɟɟɬ ɦɟɫɬɨ ɥɭɱɢɫɬɵɣ ɬɟɩɥɨɨɛɦɟɧ, ɧɨ
Esum 0 , ɨɬɤɭɞɚ ɧɚɣɞɟɦ
E A E0 .
(9.12)
ɋɨɨɬɧɨɲɟɧɢɟ (9.12) ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɥɸɛɵɯ ɫɟɪɵɯ ɬɟɥ, ɩɨɷɬɨɦɭ
E1 A1
E 2 A2
... E 0 .
(9.13)
Ɂɚɜɢɫɢɦɨɫɬɶ (9.13) ɟɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ ɡɚɤɨɧɚ Ʉɢɪɯɝɨɮɚ: ɨɬɧɨɲɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɫɟɪɨɝɨ ɬɟɥɚ ɤ ɟɝɨ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɩɪɢɪɨɞɵ ɬɟɥɚ ɢ ɪɚɜɧɨ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɨɝɨ ɬɟɥɚ ɩɪɢ ɬɚɤɨɣ ɠɟ ɬɟɦɩɟɪɚɬɭɪɟ.
Ɍɚɤ ɤɚɤ E E 0 H , ɬɨ H A . ɗɬɨ – ɜɬɨɪɚɹ ɮɨɪɦɚ ɡɚɩɢɫɢ ɡɚɤɨɧɚ Ʉɢɪɯɝɨɮɚ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɨɣ ɩɪɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɦ ɪɚɜɧɨɜɟɫɢɢ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɢ ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ ɱɢɫɥɟɧɧɨ ɪɚɜɧɵ ɦɟɠɞɭ
ɫɨɛɨɣ.
ɂɡ ɡɚɤɨɧɚ Ʉɢɪɯɝɨɮɚ ɦɨɠɧɨ ɫɞɟɥɚɬɶ
ɫɥɟɞɭɸɳɢɟ ɜɵɜɨɞɵ:
1. ɑɟɦ ɛɨɥɶɲɟ ɬɟɥɨ ɫɩɨɫɨɛɧɨ ɢɡɥɭɱɚɬɶ,
ɬɟɦ ɛɨɥɶɲɟ ɟɝɨ ɜɨɡɦɨɠɧɨɫɬɶ ɩɨɝɥɨɳɚɬɶ ɥɭɱɢɫɬɭɸ ɷɧɟɪɝɢɸ.
2. ɑɟɦ ɦɟɧɶɲɟ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɥɚ, ɬɟɦ ɦɟɧɶɲɟ ɟɝɨ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ɬɟɥɚ, ɯɨɪɨɲɨ ɨɬɪɚɠɚɸɳɢɟ ɥɭɱɢɫɬɭɸ
ɷɧɟɪɝɢɸ, ɫɚɦɢ ɢɡɥɭɱɚɸɬ ɨɱɟɧɶ ɦɚɥɨ
Ɋɢɫ. 9.5. Ʉ ɜɵɜɨɞɭ ɡɚɤɨɧɚ
(ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɚɛɫɨɥɸɬɧɨ
Ʉɢɪɯɝɨɮɚ
ɛɟɥɨɝɨ ɬɟɥɚ ɪɚɜɧɚ ɧɭɥɸ). ɉɨɷɬɨɦɭ ɞɥɹ
ɭɦɟɧɶɲɟɧɢɹ ɬɟɩɥɨɜɵɯ ɩɨɬɟɪɶ ɚɩɩɚɪɚɬɚ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɶ ɞɨɥɠɧɚ ɢɦɟɬɶ
ɧɚɢɦɟɧɶɲɟɟ ɡɧɚɱɟɧɢɟ.
3. ɉɪɢ ɨɞɢɧɚɤɨɜɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɚɛɫɨɥɸɬɧɨ
ɱɟɪɧɨɝɨ ɬɟɥɚ ɜɫɟɝɞɚ ɛɨɥɶɲɟ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɫɟɪɨɝɨ ɬɟɥɚ.
Ɋɟɡɭɥɶɬɚɬɵ ɩɨɞɫɤɚɡɵɜɚɸɬ ɩɪɨɫɬɨɣ ɫɩɨɫɨɛ ɢɦɢɬɚɰɢɢ ɱɟɪɧɨɝɨ ɢɡɥɭɱɚɬɟɥɹ ɫ ɩɨɦɨɳɶɸ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɥɨɫɬɢ, ɢɦɟɸɳɟɣ ɦɚɥɟɧɶɤɨɟ ɨɬɜɟɪɫɬɢɟ ɜ ɩɨɜɟɪɯɧɨɫɬɢ (ɪɢɫ. 9.6).
ɂɡɥɭɱɟɧɢɟ, ɤɨɬɨɪɨɟ ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɨɬɜɟɪɫɬɢɟ, ɛɭɞɟɬ ɦɧɨɝɨɤɪɚɬɧɨ
ɨɬɪɚɠɚɬɶɫɹ ɨɬ ɜɧɭɬɪɟɧɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɨɫɬɢ, ɢ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɫɨɫɬɨɹɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ ɩɨɥɨɫɬɢ ɩɚɞɚɸɳɟɟ ɢɡɥɭɱɟɧɢɟ, ɜ ɨɫɧɨɜɧɨɦ,
ɩɨɝɥɨɬɢɬɫɹ.
222
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟ
ɩɚɞɚɸɳɟɟ ɧɚ ɨɬɜɟɪɫɬɢɟ ɢɡɥɭɱɟɧɢɟ ɩɨɝɥɨɬɢɬɫɹ, ɢ ɨɬɜɟɪɫɬɢɟ ɜ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɩɨɥɨɫɬɢ ɛɭɞɟɬ ɜɟɫɬɢ ɫɟɛɹ ɩɨɞɨɛɧɨ
ɱɟɪɧɨɦɭ ɬɟɥɭ. ȼ ɥɢɬɟɪɚɬɭɪɟ ɢɦɟɟɬɫɹ
ɪɹɞ ɪɟɲɟɧɧɵɯ ɡɚɞɚɱ ɨ ɧɚɯɨɠɞɟɧɢɢ ɷɮɮɟɤɬɢɜɧɨɣ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɪɹɞɚ ɬɟɥ ɢɡɜɟɫɬɧɵɯ ɮɨɪɦ, ɬɚɤɢɯ
ɤɚɤ ɫɮɟɪɢɱɟɫɤɢɟ, ɩɪɹɦɨɭɝɨɥɶɧɵɟ ɢɥɢ Ɋɢɫ. 9.6. ɂɡɨɬɟɪɦɢɱɟɫɤɚɹ ɡɚɦɤɰɢɥɢɧɞɪɢɱɟɫɤɢɟ. ɂɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɨ- ɧɭɬɚɹ ɩɨɥɨɫɬɶ, ɦɨɞɟɥɢɪɭɸɳɚɹ
ɩɨɜɟɞɟɧɢɟ ɱɟɪɧɨɝɨ ɬɟɥɚ
ɥɨɫɬɢ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɜ ɥɚɛɨɪɚɬɨɪɧɵɯ ɭɫɥɨɜɢɹɯ ɞɥɹ ɫɨɡɞɚɧɢɹ ɢɫɬɨɱɧɢɤɚ ɱɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ.
9.3. Ɇɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ
ȿɫɥɢ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɜɵɪɚɠɟɧɧɭɸ ɡɚɤɨɧɨɦ ɉɥɚɧɤɚ, ɩɨ ɜɫɟɦ ɞɥɢɧɚɦ ɜɨɥɧ ɨɬ O 0 ,
ɞɨ O O 1 , ɬɨ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɢɦ ɩɨɥɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɷɧɟɪɝɢɹ ɢɡɥɭɱɟɧɢɹ ɜ ɢɧɬɟɪɜɚɥɟ ɞɥɢɧ ɜɨɥɧ ɨɬ 0 ɞɨ O 1 , ɢɫɩɭɳɟɧɧɨɝɨ ɱɟɪɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ T . Ɋɟɡɭɥɶɬɚɬ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɬɨɥɶɤɨ ɩɪɨɢɡɜɟɞɟɧɢɹ O1T :
O1
E 0oO
³ E 0O T d O
E 0 0 o O 1 ,T E 0 O 1T .
0
Ɉɛɳɟɟ ɤɨɥɢɱɟɫɬɜɨ ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ, ɢɫɩɭɳɟɧɧɨɝɨ ɜ ɢɧɬɟɪɜɚɥɟ
ɞɥɢɧ ɜɨɥɧ O 1 y O 2 , ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɚɡɧɨɫɬɶɸ ɞɜɭɯ ɬɚɤɢɯ ɢɧɬɟɝɪɚɥɨɜ
O2
O1
0
0
³ E 0O T d O ³ E 0 O T d O
E 0 O 2T E 0 O 1T .
Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɪɚɡɧɨɫɬɢ ɢɧɬɟɝɪɚɥɨɜ ɜ ɷɬɨɦ ɭɪɚɜɧɟɧɢɢ ɩɨɤɚɡɚɧ
ɧɚ ɪɢɫ. 9.3 ɤɚɤ ɩɥɨɳɚɞɶ ɩɨɞ ɤɪɢɜɨɣ E 0O ɦɟɠɞɭ ɞɥɢɧɚɦɢ ɜɨɥɧ O 1 ɢ O 2 .
ɑɬɨɛɵ ɧɚɣɬɢ ɞɨɥɸ ɩɨɥɧɨɣ ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ ɱɟɪɧɨɝɨ ɬɟɥɚ, ɢɫɩɭɳɟɧɧɨɝɨ ɜɨ ɜɫɟɦ ɫɩɟɤɬɪɟ, ɩɪɢɯɨɞɹɳɟɦɫɹ ɧɚ ɢɧɬɟɪɜɚɥ ɞɥɢɧ ɜɨɥɧ
O 1 O O 2 , ɧɭɠɧɨ ɪɚɡɞɟɥɢɬɶ ɪɚɡɧɨɫɬɶ ɷɧɟɪɝɢɣ ɧɚ ɩɨɥɧɵɣ ɢɧɬɟɝɪɚɥ
E 0 O 2T E 0 O 1T V 0T 4
223
.
ȼɟɥɢɱɢɧɵ
E 0 O T V 0T 4
ɨɛɵɱɧɨ ɧɚɡɵɜɚɸɬ ɪɚɞɢɚɰɢɨɧɧɵɦɢ
ɮɭɧɤɰɢɹɦɢ, ɤɨɬɨɪɵɟ ɡɚɬɚɛɭɥɢɪɨɜɚɧɵ ɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɜ ɪɚɫɱɟɬɚɯ.
Ⱦɨ ɫɢɯ ɩɨɪ ɦɵ ɨɩɪɟɞɟɥɹɥɢ ɢɧɬɟɝɪɚɥɶɧɵɟ ɫɜɨɣɫɬɜɚ, ɹɜɥɹɸɳɢɟɫɹ ɜɟɥɢɱɢɧɚɦɢ, ɨɬɧɨɫɹɳɢɦɢɫɹ ɤɨ ɜɫɟɦɭ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɦɭ ɫɩɟɤɬɪɭ. Ɉɩɪɟɞɟɥɢɦ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ, ɨɬɧɨɫɹɳɢɟɫɹ ɤ ɨɬɞɟɥɶɧɵɦ ɞɥɢɧɚɦ
ɜɨɥɧ.
Ɇɨɧɨɯɪɨɦɚɬɢɱɟɫɤɚɹ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ AO ɩɪɟɞɫɬɚɜɥɹɟɬ
ɫɨɛɨɣ ɩɨɝɥɨɳɟɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɫ ɞɥɢɧɨɣ ɜɨɥɧɵ O , ɞɟɥɟɧɧɨɟ ɧɚ ɢɡɥɭɱɟɧɢɟ ɫ ɞɥɢɧɨɣ ɜɨɥɧɵ O , ɩɚɞɚɸɳɟɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɶ
QO , A
AO
.
QO
ɋɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɬɟɥɚ ɢ ɟɝɨ ɢɧɬɟɝɪɚɥɶɧɨɣ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ A ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ
f
³ AOQOd O
0
A
,
f
(9.14)
³ QO d O
0
ɝɞɟ ɢɧɞɟɤɫ " O" ɨɛɨɡɧɚɱɚɟɬ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɭɸ ɜɟɥɢɱɢɧɭ.
ɉɨɞɨɛɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɦɟɠɞɭ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɦɢ ɢ ɢɧɬɟɝɪɚɥɶɧɵɦɢ ɨɬɪɚɠɚɬɟɥɶɧɵɦɢ ɢ ɩɪɨɩɭɫɤɚɬɟɥɶɧɵɦɢ ɫɩɨɫɨɛɧɨɫɬɹɦɢ. ɉɪɢ ɫɨɫɬɚɜɥɟɧɢɢ ɛɚɥɚɧɫɚ ɷɧɟɪɝɢɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɞɥɹ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɝɨ ɢɡɥɭɱɟɧɢɹ ɭɪɚɜɧɟɧɢɟ, ɚɧɚɥɨɝɢɱɧɨɟ (9.3), ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
AO RO D O 1 .
Ɇɨɧɨɯɪɨɦɚɬɢɱɟɫɤɚɹ ɢɥɢ ɫɩɟɤɬɪɚɥɶɧɚɹ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɜ ɜɢɞɟ
E O T HO
,
(9.15)
E 0O T ɚ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɨɣ ɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɢɡɥɭɱɚɬɟɥɶɧɵɦɢ ɫɩɨɫɨɛɧɨɫɬɹɦɢ – ɜ ɜɢɞɟ
f
H
³ H O E 0O T d O
0
f
³ E 0 O T d O
0
224
f
1
H E T d O .
V 0T ³ O 0O
0
(9.16)
ȿɫɥɢ ɫɪɚɜɧɢɬɶ ɭɪɚɜɧɟɧɢɹ (9.14) ɢ (9.16), ɬɨ ɦɨɠɧɨ ɡɚɦɟɬɢɬɶ ɫɭɳɟɫɬɜɟɧɧɨɟ ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɢɧɬɟɝɪɚɥɶɧɨɣ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɢ ɢɧɬɟɝɪɚɥɶɧɨɣ
ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɹɦɢ. Ʉɚɤ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ, ɬɚɤ ɢ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɢ ɹɜɥɹɸɬɫɹ ɫɜɨɣɫɬɜɚɦɢ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɡɚɜɢɫɹɬ ɨɬ ɬɢɩɚ
ɦɚɬɟɪɢɚɥɚ, ɫɨɫɬɨɹɧɢɹ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɬɟɦɩɟɪɚɬɭɪɵ. Ʉɪɨɦɟ ɬɨɝɨ, ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ – ɮɭɧɤɰɢɹ ɫɜɨɣɫɬɜ ɜɫɟɯ ɨɤɪɭɠɚɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ, ɞɚɸɳɢɯ ɜɤɥɚɞ ɜ ɩɚɞɚɸɳɟɟ ɢɡɥɭɱɟɧɢɟ QO , ɤɚɤ ɷɬɨ ɜɢɞɧɨ ɢɡ ɭɪɚɜɧɟɧɢɹ (9.14). ɂɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟ ɡɚɜɢɫɢɬ ɨɬ
ɫɜɨɣɫɬɜ ɞɪɭɝɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ. Ɉɧɚ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɟɣ ɬɨɥɶɤɨ ɫɜɨɣɫɬɜ
ɫɨɛɫɬɜɟɧɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ɂɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɫɩɟɤɬɪɚ ɩɚɞɚɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ, ɜ ɬɨ ɜɪɟɦɹ ɤɚɤ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɡɚɜɢɫɢɬ ɨɬ QO . Ⱦɚɠɟ
ɬɨɝɞɚ, ɤɨɝɞɚ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɢ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɢ ɢɞɟɧɬɢɱɧɵ, ɢɧɬɟɝɪɚɥɶɧɵɟ ɜɟɥɢɱɢɧɵ A ɢ H ɦɨɝɭɬ ɪɚɡɥɢɱɚɬɶɫɹ.
Ɉɞɧɚɤɨ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɩɚɞɚɸɳɟɟ ɢɡɥɭɱɟɧɢɟ ɢɫɯɨɞɢɬ ɨɬ ɱɟɪɧɨɝɨ ɬɟɥɚ ɫ
ɬɟɦɩɟɪɚɬɭɪɨɣ, ɪɚɜɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɩɪɢɟɦɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ, ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɩɪɢɟɦɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɜɧɚ ɟɟ ɩɨɝɥɨɳɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ. ɗɬɨ ɫɥɟɞɭɟɬ ɢɡ ɡɚɤɨɧɚ Ʉɢɪɯɝɨɮɚ.
Ɂɚɤɨɧ Ʉɢɪɯɝɨɮɚ ɞɥɹ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɞɥɹ ɬɟɥ, ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɬɟɩɥɨɜɨɦ ɪɚɜɧɨɜɟɫɢɢ, ɢɦɟɟɬ ɜɢɞ
AO H O .
ɇɨ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɚ ɡɚɤɨɧɚ Ʉɢɪɯɝɨɮɚ ɧɟ ɨɝɪɚɧɢɱɟɧɚ ɫɥɭɱɚɟɦ ɬɟɩɥɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ. Ɇɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɢ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɢ ɬɟɥɚ ɪɚɜɧɵ ɞɪɭɝ ɞɪɭɝɭ, ɞɚɠɟ ɟɫɥɢ ɩɨɜɟɪɯɧɨɫɬɢ, ɤɨɬɨɪɵɟ ɜɧɨɫɹɬ ɜɤɥɚɞ ɜ ɩɚɞɚɸɳɟɟ ɢɡɥɭɱɟɧɢɟ, ɧɚɯɨɞɹɬɫɹ ɧɟ ɩɪɢ ɨɞɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ.
9.4. ɉɨɧɹɬɢɟ ɫɟɪɨɝɨ ɬɟɥɚ
ɉɨɧɹɬɢɟ ɫɟɪɨɝɨ ɬɟɥɚ ɫɢɥɶɧɨ ɭɩɪɨɳɚɟɬ ɪɚɫɱɟɬɵ ɢɡɥɭɱɟɧɢɹ. ɋɟɪɨɟ ɬɟɥɨ – ɷɬɨ ɩɨɜɟɪɯɧɨɫɬɶ, ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ ɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɵ
ɞɥɹ ɜɫɟɯ ɞɥɢɧ ɜɨɥɧ. ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ ɫɟɪɚɹ, ɬɨ
HO const , AO const
ɢ ɬ.ɞ. ɂɧɬɟɝɪɚɥɶɧɚɹ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɞɥɹ ɫɟɪɨɝɨ ɬɟɥɚ ɟɫɬɶ
f
³ AOQOd O
A
0
f
AO ³ QO d O
f
f
0
0
³ QO d O
0
³ QO d O
225
AO ,
ɬ.ɟ. ɢɧɬɟɝɪɚɥɶɧɚɹ ɢ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɚɹ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɢ
ɫɟɪɨɝɨ ɬɟɥɚ ɪɚɜɧɵ. Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ:
H HO , R
RO , D
DO .
Ⱦɥɹ ɫɟɪɨɝɨ ɬɟɥɚ ɫɩɪɚɜɟɞɥɢɜɵ ɪɚɜɟɧɫɬɜɚ
AO
HO ɢ A H ,
ɞɚɠɟ ɟɫɥɢ ɫɟɪɨɟ ɬɟɥɨ ɧɟ ɧɚɯɨɞɢɬɫɹ ɜ ɬɟɩɥɨɜɨɦ ɪɚɜɧɨɜɟɫɢɢ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ. ɗɬɨ – ɜɚɠɧɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɫɟɪɨɝɨ ɬɟɥɚ.
ɉɨɧɹɬɢɟ ɫɟɪɨɝɨ ɬɟɥɚ ɹɜɥɹɟɬɫɹ ɢɞɟɚɥɢɡɢɪɨɜɚɧɧɵɦ, ɩɨɫɤɨɥɶɤɭ ɫɟɪɵɟ
ɬɟɥɚ ɧɚ ɩɪɚɤɬɢɤɟ ɧɟ ɫɭɳɟɫɬɜɭɸɬ. ɇɨ ɞɚɠɟ ɟɫɥɢ ɩɨɜɟɞɟɧɢɟ ɪɟɚɥɶɧɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɥɢɲɶ ɩɪɢɛɥɢɠɟɧɧɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɜɟɞɟɧɢɸ ɫɟɪɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ, ɪɚɫɱɟɬɵ, ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɩɪɢɛɥɢɠɟɧɢɢ ɫɟɪɨɝɨ ɬɟɥɚ, ɞɚɸɬ ɭɞɨɜɥɟɬɜɨɪɢɬɟɥɶɧɵɟ ɪɟɡɭɥɶɬɚɬɵ.
9.5. Ʌɭɱɢɫɬɵɣ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɬɟɥɚɦɢ
Ʌɭɱɢɫɬɵɣ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɩɚɪɚɥɥɟɥɶɧɵɦɢ ɩɥɨɫɤɨɫɬɹɦɢ (ɪɢɫ. 9.7)
ɩɪɨɢɫɯɨɞɢɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. ɉɭɫɬɶ ɢɦɟɸɬɫɹ ɞɜɟ ɩɚɪɚɥɥɟɥɶɧɵɟ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɢɟ ɩɥɚɫɬɢɧɵ ɢɡ ɪɚɡɧɵɯ ɫɟɪɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɤɚɠɞɚɹ ɢɡ ɤɨɬɨɪɵɯ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɫɜɨɟɣ ɬɟɦɩɟɪɚɬɭɪɨɣ ( T1 ɢ T2 ) ɢ ɫɬɟɩɟɧɶɸ ɱɟɪɧɨɬɵ ( H1 ɢ H 2 ). ɉɪɢɦɟɦ T1 ! T2 ɢ ɨɩɪɟɞɟɥɢɦ ɬɟɩɥɨɬɭ, ɩɟɪɟɞɚɧɧɭɸ ɥɭɱɟɢɫɩɭɫɤɚɧɢɟɦ ɨɬ ɩɟɪɜɨɣ (ɥɟɜɨɣ) ɩɥɚɫɬɢɧɵ ɤɨ ɜɬɨɪɨɣ (ɩɪɚɜɨɣ). ɋɧɚɱɚɥɚ
ɨɩɪɟɞɟɥɢɦ ɬɟɩɥɨɬɭ, ɩɟɪɟɞɚɧɧɭɸ ɨɬ ɩɟɪɜɨɣ ɩɥɚɫɬɢɧɵ ɤɨ ɜɬɨɪɨɣ, ɢɫɤɥɸɱɢɜ ɢɡ ɪɚɫɫɦɨɬɪɟɧɢɹ ɫɨɛɫɬɜɟɧɧɨɟ ɢɡɥɭɱɟɧɢɟ ɜɬɨɪɨɣ ɩɥɚɬɢɧɵ. ɋɨɛɫɬɜɟɧɧɵɣ ɩɨɬɨɤ ɢɡɥɭɱɟɧɢɹ ɩɟɪɜɨɣ ɩɥɚɫɬɢɧɵ – E1 . ɉɪɢ ɩɨɩɚɞɚɧɢɢ ɧɚ ɜɬɨɪɭɸ
ɩɥɚɫɬɢɧɭ ɱɚɫɬɶ ɷɬɨɝɨ ɩɨɬɨɤɚ A2 E1 – ɩɨɝɥɨɳɚɟɬɫɹ ɟɸ, ɚ ɨɫɬɚɥɶɧɚɹ –
R 2 E1 – ɨɬɪɚɠɚɟɬɫɹ. Ɉɬɪɚɠɟɧɧɚɹ ɱɚɫɬɶ ɩɨɬɨɤɚ ɜ ɤɨɥɢɱɟɫɬɜɟ R 2 A1E1 ɩɨɝɥɨɳɚɟɬɫɹ ɩɟɪɜɨɣ ɩɥɚɫɬɢɧɨɣ ɢ ɱɚɫɬɢɱɧɨ (ɜ ɤɨɥɢɱɟɫɬɜɟ R 2 R1E1 ) ɨɬɪɚɠɚɟɬɫɹ ɨɬ ɧɟɟ ɢ ɬ.ɞ. ɉɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɨɛɭɫɥɨɜɥɟɧɧɨɝɨ ɫɨɛɫɬɜɟɧɧɵɦ ɢɡɥɭɱɟɧɢɟɦ ɩɟɪɜɨɣ ɩɥɚɫɬɢɧɵ ɢ ɩɨɝɥɨɳɟɧɧɨɝɨ ɜɬɨɪɨɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɬɟɩɥɨɨɛɦɟɧɚ, ɨɩɪɟɞɟɥɢɬɫɹ ɤɚɤ ɫɭɦɦɚ
q2
A2 E1 A2 R 2 R1E1 A2 R 22 R12 E1 ...
A2 ª1 R2 R1 R 2 R1 . ..º E1 .
«¬
»¼
Ɍɚɤ ɤɚɤ R1 1 ɢ R 2 1 , ɬɨ ɜɵɪɚɠɟɧɢɟ ɜ ɫɤɨɛɤɚɯ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ
ɭɛɵɜɚɸɳɭɸ ɝɟɨɦɟɬɪɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ, ɩɪɨɫɭɦɦɢɪɨɜɚɜ ɤɨɬɨɪɭɸ, ɩɨɥɭɱɢɦ
2
226
q2
A2 E1
.
1 R1R 2
Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɢɬɫɹ ɩɥɨɬɧɨɫɬɶ
ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɨɛɭɫɥɨɜɥɟɧɧɨɝɨ ɫɨɛɫɬɜɟɧɧɵɦ ɢɡɥɭɱɟɧɢɟɦ ɜɬɨɪɨɣ ɩɥɚɫɬɢɧɵ
ɢ ɩɨɝɥɨɳɟɧɧɨɝɨ ɩɟɪɜɨɣ ɜ ɪɟɡɭɥɶɬɚɬɟ
ɨɞɧɨɫɬɨɪɨɧɧɟɝɨ ɬɟɩɥɨɨɛɦɟɧɚ
A1E 2
.
q1
1 R1R 2
ɋɭɦɦɚɪɧɚɹ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ
ɩɨɬɨɤɚ ɨɬ ɩɟɪɜɨɣ ɩɥɚɫɬɢɧɵ ɤɨ ɜɬɨɪɨɣ
ɟɫɬɶ
A2 E1 A1E 2
q q 2 q1
.
1 R1R 2
ɍɱɢɬɵɜɚɹ, ɱɬɨ
Ɋɢɫ. 9.7. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ
ɦɟɠɞɭ ɩɚɪɚɥɥɟɥɶɧɵɦɢ
ɩɨɜɟɪɯɧɨɫɬɹɦɢ
E C T 100 ɢ R 1 A 1 H ,
4
ɩɨɥɭɱɢɦ
ª§ T · 4 § T · 4 º
q H 3 C 0 «¨ 1 ¸ ¨ 2 ¸ » ,
«¬© 100 ¹ © 100 ¹ »¼
(9.17)
1
– ɩɪɢɜɟɞɟɧɧɚɹ ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ ɞɜɭɯ ɬɟɥ.
1 H1 1 H 2 1
Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ ɥɭɱɟɢɫɩɭɫɤɚɧɢɟɦ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ F
ª§ T · 4 § T · 4 º
(9.18)
Q qF H 3C 0 «¨ 1 ¸ ¨ 2 ¸ » F .
100
100
¹ ©
¹ »¼
«¬©
ȼ ɯɨɥɨɞɢɥɶɧɨɣ ɬɟɯɧɢɤɟ (ɧɚɩɪɢɦɟɪ, ɜ ɫɨɫɭɞɚɯ ɞɥɹ ɯɪɚɧɟɧɢɹ ɫɠɢɠɟɧɧɵɯ ɝɚɡɨɜ, ɩɪɢ ɜɚɤɭɭɦɧɨɣ ɢɡɨɥɹɰɢɢ ɤɪɢɨɝɟɧɧɵɯ ɭɫɬɚɧɨɜɨɤ ɢ ɬ.ɞ.) ɞɥɹ
ɭɦɟɧɶɲɟɧɢɹ ɥɭɱɢɫɬɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɦɟɠɞɭ ɩɨɜɟɪɯɧɨɫɬɹɦɢ ɩɪɢɦɟɧɹɸɬ
ɷɤɪɚɧɵ. ȿɫɥɢ ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ ɷɤɪɚɧɨɜ ɢ ɬɟɩɥɨɩɟɪɟɞɚɸɳɢɯ ɫɬɟɧɨɤ ɨɞɢɧɚɤɨɜɚ, ɬɨ ɩɥɨɬɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɨɬ ɜɜɟɞɟɧɢɹ n ɷɤɪɚɧɨɜ ɭɦɟɧɶɲɢɬɫɹ ɢ ɫɨɫɬɚɜɢɬ
q ɛ .ɷ
qɷ
,
n 1
ɝɞɟ q ɷ ɢ q ɛ .ɷ – ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɟɦ ɩɪɢ ɧɚɥɢɱɢɢ
ɷɤɪɚɧɚ ɢ ɛɟɡ ɷɤɪɚɧɚ.
ɝɞɟ H 3
227
ɗɮɮɟɤɬɢɜɧɨɫɬɶ ɷɤɪɚɧɨɜ ɪɟɡɤɨ ɜɨɡɪɚɫɬɚɟɬ, ɟɫɥɢ ɦɚɬɟɪɢɚɥ, ɢɡ ɤɨɬɨɪɨɝɨ ɨɧɢ ɢɡɝɨɬɨɜɥɟɧɵ, ɢɦɟɟɬ ɦɚɥɭɸ ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ. Ʉɨɝɞɚ H1 H 2 !! H ɷ ,
ɪɚɫɱɟɬ ɜɟɞɭɬ ɩɨ ɮɨɪɦɭɥɟ
H ɷ q ɛ .ɷ
qɷ
.
H1 n 1
ȼ ɤɚɱɟɫɬɜɟ ɷɤɪɚɧɨɜ ɜ ɤɪɢɨɝɟɧɧɵɯ ɭɫɬɚɧɨɜɤɚɯ ɢɫɩɨɥɶɡɭɸɬ ɚɥɸɦɢɧɢɟɜɭɸ ɮɨɥɶɝɭ, ɦɟɬɚɥɥɢɡɢɪɨɜɚɧɧɵɟ ɩɨɥɢɦɟɪɧɵɟ ɩɥɟɧɤɢ ɢ ɬ.ɞ.
Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɬɟɩɥɨɨɛɦɟɧɚ ɢɡɥɭɱɟɧɢɟɦ ɞɥɹ ɬɟɥɚ ɩɪɨɢɡɜɨɥɶɧɨɣ
ɮɨɪɦɵ, ɡɚɦɤɧɭɬɨɝɨ ɜɧɟɲɧɢɦ ɬɟɥɨɦ ɛɨɥɶɲɟɣ ɩɨɜɟɪɯɧɨɫɬɢ (ɪɢɫ. 9.8).
Ɍɟɥɚ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɩɥɨɳɚɞɶɸ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɟɦɩɟɪɚɬɭɪɨɣ, ɫɬɟɩɟɧɶɸ ɱɟɪɧɨɬɵ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɪɢɱɟɦ T1 ! T 2 .
Ɍɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɩɟɪɟɞɚɜɚɟɦɵɣ ɨɬ ɩɟɪɜɨɝɨ ɬɟɥɚ ɤɨ ɜɬɨɪɨɦɭ, ɦɨɠɟɬ
ɛɵɬɶ ɨɩɪɟɞɟɥɟɧ ɩɨ ɮɨɪɦɭɥɟ, ɚɧɚɥɨɝɢɱɧɨɣ (9.15)
Q12
ª§ T · 4 § T · 4 º
H 3C 0 «¨ 1 ¸ ¨ 2 ¸ » F1 ,(9.19)
«¬© 100 ¹ © 100 ¹ »¼
ɝɞɟ
1
.
1 H1 F1 F2 1 H 2 1
ɂɡ ɩɨɫɥɟɞɧɟɝɨ ɭɪɚɜɧɟɧɢɹ ɫɥɟɞɭɟɬ,
ɱɬɨ ɩɪɢ F 2!! F1 ɩɪɢɜɟɞɟɧɧɚɹ ɫɬɟɩɟɧɶ
ɱɟɪɧɨɬɵ ɟɫɬɶ H 3 | H1 . ɍɪɚɜɧɟɧɢɟ (9.19)
ɫɩɪɚɜɟɞɥɢɜɨ ɬɨɥɶɤɨ ɞɥɹ ɫɥɭɱɚɹ, ɤɨɝɞɚ
ɦɟɧɶɲɟɟ ɬɟɥɨ – ɜɵɩɭɤɥɨɟ.
H3
Ɋɢɫ. 9.8. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ ɦɟɠɞɭ ɬɟɥɨɦ ɢ
ɟɝɨ ɨɛɨɥɨɱɤɨɣ
9.6. ɇɚɩɪɚɜɥɟɧɧɵɟ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ
Ⱦɨ ɫɢɯ ɩɨɪ ɦɵ ɩɨɥɚɝɚɥɢ, ɱɬɨ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ ɪɟɚɥɶɧɵɯ ɬɟɥ
ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɬɨɥɶɤɨ ɞɥɢɧɵ ɜɨɥɧɵ ɢ ɭɫɥɨɜɢɣ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ
ɩɨɜɟɪɯɧɨɫɬɢ ɩɪɢɟɦɧɢɤɚ ɢ ɢɡɥɭɱɚɬɟɥɹ. ɇɚ ɫɚɦɨɦ ɞɟɥɟ ɪɚɞɢɚɰɢɨɧɧɵɟ
ɫɜɨɣɫɬɜɚ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɹɬ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ. Ⱦɥɹ ɢɞɟɚɥɶɧɵɯ ɱɟɪɧɵɯ
ɬɟɥ ɡɚɜɢɫɢɦɨɫɬɶ ɢɡɥɭɱɚɬɟɥɶɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɡɚɤɨɧ Ʌɚɦɛɟɪɬɚ (ɫɦ. ɪɚɡɞɟɥ 9.2.4). ɋɜɨɣɫɬɜɚ, ɤɨɬɨɪɵɟ ɨɩɢɫɵɜɚɸɬ
ɭɝɥɨɜɵɟ ɢɡɦɟɧɟɧɢɹ, ɧɚɡɵɜɚɸɬɫɹ ɧɚɩɪɚɜɥɟɧɧɵɦɢ ɫɜɨɣɫɬɜɚɦɢ.
ɇɚɩɪɚɜɥɟɧɧɚɹ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɯɚɪɚɤɬɟɪɧɚ ɬɟɦ, ɱɬɨ ɞɥɹ ɛɨɥɶɲɢɯ ɭɝɥɨɜ 4 ɨɧɚ ɜɵɲɟ, ɱɟɦ ɞɥɹ
ɦɚɥɵɯ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɪɨɜɨɞɧɢɤ ɛɭɞɟɬ ɢɫɩɭɫɤɚɬɶ ɛɨɥɶɲɟ ɢɡɥɭɱɟɧɢɹ
228
ɩɨ ɤɚɫɚɬɟɥɶɧɨɣ, ɱɟɦ ɩɨ ɧɨɪɦɚɥɢ. ɂɧɚɱɟ ɜɟɞɭɬ ɫɟɛɹ ɞɢɷɥɟɤɬɪɢɤɢ. Ɉɧɢ
ɢɫɩɭɫɤɚɸɬ ɛɨɥɶɲɟ ɢɡɥɭɱɟɧɢɹ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ, ɛɥɢɡɤɢɯ ɤ ɧɨɪɦɚɥɢ, ɬɨɝɞɚ
ɤɚɤ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɭɝɥɚ 4 ɞɨ 90° ɢɯ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɩɚɞɚɟɬ
ɞɨ ɧɭɥɹ.
Ⱦɥɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɧɚɩɪɚɜɥɟɧɧɵɯ ɪɚɞɢɚɰɢɨɧɧɵɯ ɫɜɨɣɫɬɜ ɧɚɦ ɧɭɠɧɨ
ɜɜɟɫɬɢ ɧɟɤɨɬɨɪɵɟ ɩɨɧɹɬɢɹ. ɉɟɪɜɨɟ ɢ ɧɢɯ – ɷɬɨ ɬɟɥɟɫɧɵɣ ɭɝɨɥ, ɤɨɬɨɪɵɣ
ɢɫɩɨɥɶɡɭɟɬɫɹ ɜ ɫɬɟɪɟɨɦɟɬɪɢɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɷɥɟɦɟɧɬɚɪɧɭɸ ɩɥɨɳɚɞɤɭ dF , ɤɨɬɨɪɚɹ ɫɬɹɝɢɜɚɟɬ ɷɥɟɦɟɧɬɚɪɧɵɣ ɬɟɥɟɫɧɵɣ ɭɝɨɥ dZ ɜ ɬɨɱɤɟ O (ɪɢɫ. 9.9). Ɍɟɥɟɫɧɵɣ ɭɝɨɥ – ɷɬɨ
ɛɟɡɪɚɡɦɟɪɧɚɹ ɜɟɥɢɱɢɧɚ, ɨɩɪɟɞɟɥɹɟɦɚɹ ɤɚɤ ɧɨɪɦɚɥɶɧɚɹ ɩɪɨɟɤɰɢɹ dF ,
ɞɟɥɟɧɧɚɹ ɧɚ ɤɜɚɞɪɚɬ ɪɚɫɫɬɨɹɧɢɹ ɦɟɠɞɭ ɬɨɱɤɨɣ O ɢ ɩɪɨɟɤɬɢɪɭɟɦɨɣ ɩɥɨɳɚɞɤɨɣ:
d Fn d F c os 4
.
(9.20)
dZ
r2
r2
ɂɡɦɟɪɹɸɬ ɬɟɥɟɫɧɵɣ ɭɝɨɥ ɜ ɫɬɟɪɚɞɢɚɧɚɯ,
ɫɨɤɪɚɳɟɧɧɨ ɫɪ.
dF
ɋɭɳɟɫɬɜɭɟɬ ɩɨɞɨɛɢɟ ɦɟɠɞɭ ɩɥɨɫɤɢɦ ɢ ɬɟdZ
ɥɟɫɧɵɦ ɭɝɥɚɦɢ. ɉɥɨɫɤɢɣ ɭɝɨɥ ɫɬɹɝɢɜɚɟɬɫɹ
ɧɨɪɦɚɥɶɧɨɣ ɩɪɨɟɤɰɢɟɣ ɥɢɧɢɢ, ɞɟɥɟɧɧɨɣ ɧɚ
ɪɚɫɫɬɨɹɧɢɟ ɞɨ ɷɬɨɣ ɥɢɧɢɢ. ɉɥɨɫɤɢɣ ɭɝɨɥ ɛɟɡO
ɪɚɡɦɟɪɟɧ ɢ ɢɡɦɟɪɹɟɬɫɹ ɜ ɪɚɞɢɚɧɚɯ. ɍɝɨɥ ɜ 2S
ɪɚɞɢɚɧ ɫɬɹɝɢɜɚɟɬɫɹ ɡɚɦɤɧɭɬɨɣ ɥɢɧɢɟɣ, ɧɚɩɪɢ- Ɋɢɫ. 9.9. Ɍɟɥɟɫɧɵɣ ɭɝɨɥ
ɦɟɪ, ɨɤɪɭɠɧɨɫɬɶɸ. ɑɢɫɥɨ ɫɬɟɪɚɞɢɚɧ ɜ ɭɝɥɟ, ɨɝɪɚɧɢɱɟɧɧɨɦ ɡɚɦɤɧɭɬɨɣ
ɩɨɜɟɪɯɧɨɫɬɶɸ, ɧɚɩɪɢɦɟɪ ɫɮɟɪɨɣ, ɦɨɠɟɬ ɛɵɬɶ ɨɩɪɟɞɟɥɟɧɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟɦ (9.20) ɩɨ ɫɮɟɪɟ
Zs
³
2S
dZ
sphere
S
³ ³
r d sin 4 d M
M 04 0
r2
4S .
(9.21)
ȼɬɨɪɚɹ ɜɟɥɢɱɢɧɚ, ɤɨɬɨɪɚɹ ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ, ɷɬɨ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ, ɤɨɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɷɧɟɪɝɢɹ ɢɡɥɭɱɟɧɢɹ, ɢɫɩɭɫɤɚɟɦɨɝɨ ɡɚ
ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ, ɜ ɟɞɢɧɢɰɭ ɬɟɥɟɫɧɨɝɨ ɭɝɥɚ, ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ, ɩɟɪɩɟɧɞɢɤɭɥɹɪɧɨɣ ɧɚɩɪɚɜɥɟɧɢɸ ɩɟɪɟɧɨɫɚ ɢɡɥɭɱɟɧɢɹ
I 4 ,M
dq
,
c os 4 d Z
(9.22)
ɝɞɟ, ɤɚɤ ɢ ɜ ɩɪɟɞɵɞɭɳɢɯ ɝɥɚɜɚɯ q – ɷɧɟɪɝɢɹ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɧɚ ɟɞɢɧɢɰɭ ɩɥɨɳɚɞɢ, ȼɬ/ɦ2=Ⱦɠ/(ɦ2ɫ). ȿɞɢɧɢɰɟɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɫɥɭɠɢɬ
ȼɬ/(ɦ2ɫɪ). ȿɫɥɢ ɢɡɜɟɫɬɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɢɧɬɟɧɫɢɜɧɨɫɬɢ, ɬɨ ɦɨɠɧɨ ɨɩɪɟ-
229
ɞɟɥɢɬɶ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ, ɩɨɤɢɞɚɸɳɟɝɨ ɩɥɨɫɤɭɸ ɩɨɜɟɪɯɧɨɫɬɶ
E q
³
I 4 ,M cos 4 d Z
2S S 2
³ ³ I 4 ,M sin 4 cos 4 d 4 d M ,
(9.23)
M 04 0
hemisphere
ɝɞɟ ɢɫɩɨɥɶɡɨɜɚɧɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ dZ ɢɡ (9.21).
ɍɪɚɜɧɟɧɢɟ (9.23) ɦɨɠɧɨ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɩɨ ɭɝɥɚɦ.
ɋɚɦɵɣ ɩɪɨɫɬɨɣ ɜɚɪɢɚɧɬ – ɩɨɫɬɨɹɧɧɚɹ ɢɧɬɟɧɫɢɜɧɨɫɬɶ. ɉɨɜɟɪɯɧɨɫɬɶ,
ɤɨɬɨɪɚɹ ɢɡɥɭɱɚɟɬ ɫ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɶɸ ɩɨ ɜɫɟɦ ɭɝɥɚɦ, ɧɚɡɵɜɚɟɬɫɹ ɞɢɮɮɭɡɧɨɣ ɩɨɜɟɪɯɧɨɫɬɶɸ ɢɥɢ ɢɧɨɝɞɚ ɩɨɜɟɪɯɧɨɫɬɶɸ, ɩɨɞɱɢɧɹɸɳɟɣɫɹ ɡɚɤɨɧɭ ɤɨɫɢɧɭɫɨɜ Ʌɚɦɛɟɪɬɚ, ɩɨɫɤɨɥɶɤɭ ɷɧɟɪɝɢɹ ɢɡɥɭɱɟɧɢɹ, ɩɨɤɢɞɚɸɳɚɹ ɟɝɨ ɞɢɮɮɭɡɧɭɸ ɩɨɜɟɪɯɧɨɫɬɶ ɜ ɞɚɧɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ, ɢɡɦɟɧɹɟɬɫɹ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɤɨɫɢɧɭɫɭ ɭɝɥɚ ɦɟɠɞɭ ɷɬɢɦ ɧɚɩɪɚɜɥɟɧɢɟɦ ɢ ɧɨɪɦɚɥɶɸ
ɤ ɩɨɜɟɪɯɧɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɞɢɮɮɭɡɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɢɦɟɟɦ
I 4 , M const ɢ E S I .
(9.24)
Ⱥɧɚɥɨɝɢɱɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɢɦɟɸɬ ɦɟɫɬɨ ɢ ɞɥɹ ɱɟɪɧɨɝɨ ɬɟɥɚ: ɱɟɪɧɚɹ
ɩɨɜɟɪɯɧɨɫɬɶ ɹɜɥɹɟɬɫɹ ɞɢɮɮɭɡɧɨɣ,
E0
SI 0 .
ɂɬɚɤ, ɧɚɩɪɚɜɥɟɧɧɵɟ ɪɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ ɨɩɪɟɞɟɥɹɸɬɫɹ ɱɟɪɟɡ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ.
ɇɚɩɪɢɦɟɪ, ɧɚɩɪɚɜɥɟɧɧɚɹ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ H 4 ,M ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɚɤ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ, ɢɫɩɭɳɟɧɧɨɝɨ ɩɨɜɟɪɯɧɨɫɬɶɸ ɜ
ɧɚɩɪɚɜɥɟɧɢɢ, ɨɩɪɟɞɟɥɹɟɦɨɦ ɭɝɥɚɦɢ 4 , M , ɞɟɥɟɧɧɚɹ ɧɚ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɢɡɥɭɱɟɧɢɹ ɱɟɪɧɨɝɨ ɬɟɥɚ ɜ ɬɨɦ ɠɟ ɧɚɩɪɚɜɥɟɧɢɢ
H 4 ,M I 4 ,M I 0 .
(9.25)
ɂɧɬɟɝɪɚɥɶɧɚɹ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɨɬɧɨɲɟɧɢɟɦ H E E 0 (ɪɚɡɞɟɥ 9.2.3.). ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ ɧɟɞɢɮɮɭɡɧɚɹ, ɬɨ ɩɪɢɯɨɞɢɦ ɤ ɜɵɪɚɠɟɧɢɸ
2S S / 2
³ ³ I 4 ,M s in 4 c os 4 d 4 d M
H
M 04 0
,
SI 0
ɨɬɤɭɞɚ ɧɚɯɨɞɢɦ ɫɜɹɡɶ ɦɟɠɞɭ ɢɧɬɟɝɪɚɥɶɧɨɣ H ɢ ɧɚɩɪɚɜɥɟɧɧɨɣ HT, M ɢɡɥɭɱɚɬɟɥɶɧɵɦɢ ɫɩɨɫɨɛɧɨɫɬɹɦɢ
230
H
1
S
2S S / 2
³ ³ H 4 ,M s i n 4 c o s 4 d 4 d M .
(9.26)
M 04 0
ȿɫɥɢ ɧɚɩɪɚɜɥɟɧɧɚɹ ɢɡɥɭɱɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɭɝɥɚ 4 (ɱɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɞɥɹ ɦɧɨɝɢɯ ɪɟɚɥɶɧɵɯ ɬɟɥ), ɬɨ ɩɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚɯɨɞɢɦ
S/ 2
H 2
³ H 4 ,M s i n 4 c o s 4 d 4 .
4 0
ɋɭɳɟɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɦɟɬɨɞɨɜ ɪɚɫɱɟɬɚ ɬɟɩɥɨɨɛɦɟɧɚ ɢɡɥɭɱɟɧɢɟɦ, ɜ
ɬɨɦ ɱɢɫɥɟ, ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɦɟɬɨɞ, ɦɟɬɨɞ ɷɥɟɤɬɪɢɱɟɫɤɢɯ ɰɟɩɟɣ24.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɪɚɫɱɟɬɨɜ ɪɚɞɢɚɰɢɨɧɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɜɜɟɞɟɧɵ ɞɜɚ
ɬɢɩɚ ɨɬɪɚɠɚɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ. ɉɟɪɜɵɣ – ɞɢɮɮɭɡɧɨ ɨɬɪɚɠɚɸɳɚɹ ɩɨɜɟɪɯɧɨɫɬɶ. ɗɬɚ ɩɨɜɟɪɯɧɨɫɬɶ ɨɬɪɚɠɚɟɬ ɨɬɞɟɥɶɧɵɣ ɩɚɞɚɸɳɢɣ ɥɭɱ ɬɚɤ, ɱɬɨ
ɟɝɨ ɷɧɟɪɝɢɹ ɢɦɟɟɬ ɨɞɧɭ ɢ ɬɭ ɠɟ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɞɥɹ ɜɫɟɯ ɭɝɥɨɜ ɨɬɪɚɠɟɧɢɹ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɫɥɟɞɢɬɶ ɡɚ ɤɚɠɞɵɦ ɨɬɞɟɥɶɧɵɦ
ɥɭɱɨɦ.
Ⱦɪɭɝɨɣ ɬɢɩ ɨɬɪɚɠɚɸɳɟɣ ɩɨɜɟɪɯɧɨɫɬɢ – ɡɟɪɤɚɥɶɧɵɣ ɨɬɪɚɠɚɬɟɥɶ.
ɉɨɞɨɛɧɨ ɡɟɪɤɚɥɭ, ɡɟɪɤɚɥɶɧɵɣ ɨɬɪɚɠɚɬɟɥɶ ɢɡɦɟɧɹɟɬ ɧɚɩɪɚɜɥɟɧɢɟ ɩɚɞɚɸɳɟɝɨ ɥɭɱɚ ɬɚɤ, ɱɬɨ ɭɝɨɥ ɩɚɞɟɧɢɹ ɪɚɜɟɧ ɭɝɥɭ ɨɬɪɚɠɟɧɢɹ. ɉɨɜɟɪɯɧɨɫɬɶ
ɫɬɚɧɨɜɢɬɫɹ ɡɟɪɤɚɥɶɧɨɣ, ɤɨɝɞɚ ɟɟ ɲɟɪɨɯɨɜɚɬɨɫɬɶ ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɞɥɢɧɨɣ ɜɨɥɧɵ ɩɚɞɚɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ.
9.7. ɉɟɪɟɧɨɫ ɢɡɥɭɱɟɧɢɹ
ɜ ɩɨɝɥ ɨɳɚɸɳɢ ɯ ɩɪɨɩɭɫɤɚ ɸɳɢɯ ɫɪɟɞɚɯ
ɉɨɞ ɩɪɨɰɟɫɫɨɦ ɩɟɪɟɧɨɫɚ ɷɧɟɪɝɢɢ ɢɡɥɭɱɟɧɢɹ ɩɪɢɧɹɬɨ ɩɨɧɢɦɚɬɶ ɫɨɛɫɬɜɟɧɧɨɟ ɢɡɥɭɱɟɧɢɟ, ɩɨɝɥɨɳɟɧɢɟ, ɚ ɬɚɤɠɟ ɦɧɨɝɨɤɪɚɬɧɵɟ ɨɬɪɚɠɟɧɢɹ ɧɚ
ɝɪɚɧɢɰɟ ɢ ɪɚɫɫɟɹɧɢɹ ɜ ɨɛɴɟɦɟ ɫɪɟɞɵ. ɗɬɢ ɹɜɥɟɧɢɹ ɢɦɟɸɬ ɦɟɫɬɨ ɩɪɢ ɩɟɪɟɧɨɫɟ ɢɡɥɭɱɟɧɢɹ ɤɚɤ ɜ ɝɚɡɨɜɵɯ ɫɪɟɞɚɯ, ɫɨɞɟɪɠɚɳɢɯ ɜɡɜɟɲɟɧɧɵɟ ɜ ɧɢɯ
ɱɚɫɬɢɰɵ ɩɵɥɢ, ɫɚɠɢ, ɤɚɩɟɥɶɤɢ ɠɢɞɤɨɫɬɢ ɢ ɬ.ɩ., ɬɚɤ ɢ ɜ ɬɜɟɪɞɵɯ ɢɥɢ
ɠɢɞɤɢɯ ɩɨɥɭɩɪɨɡɪɚɱɧɵɯ ɬɟɥɚɯ. əɜɥɟɧɢɹ ɜɫɬɪɟɱɚɸɬɫɹ ɤɚɤ ɜ ɩɪɢɪɨɞɧɵɯ
ɭɫɥɨɜɢɹɯ, ɬɚɤ ɢ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ ɬɟɯɧɢɤɢ. Ⱦɥɹ ɪɚɞɢɚɰɢɨɧɧɵɯ
ɫɜɨɣɫɬɜ ɝɚɡɨɜ ɯɚɪɚɤɬɟɪɧɚ ɤɪɚɣɧɟ ɧɟɪɟɝɭɥɹɪɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɞɥɢɧɵ
ɜɨɥɧɵ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɝɥɨɳɟɧɢɟ ɢ ɢɫɩɭɫɤɚɧɢɟ ɢɡɥɭɱɟɧɢɹ ɝɚɡɨɦ ɫɭɳɟɫɬɜɟɧɧɵ ɬɨɥɶɤɨ ɧɚ ɧɟɤɨɬɨɪɵɯ ɭɱɚɫɬɤɚɯ ɫɩɟɤɬɪɚ. Ɋɚɞɢɚɰɢɨɧɧɵɟ ɫɜɨɣɫɬɜɚ
ɧɟɩɪɨɡɪɚɱɧɵɯ ɬɜɟɪɞɵɯ ɬɟɥ ɞɨɫɬɚɬɨɱɧɨ ɩɥɚɜɧɨ ɢɡɦɟɧɹɸɬɫɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɞɥɢɧɵ ɜɨɥɧɵ, ɯɨɬɹ ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɧɚɛɥɸɞɚɸɬɫɹ ɞɨɜɨɥɶɧɨ
ɪɟɡɤɢɟ ɢɡɦɟɧɟɧɢɹ. ɂɡɥɭɱɟɧɢɟ, ɢɫɩɭɫɤɚɟɦɨɟ ɬɜɟɪɞɵɦ ɬɟɥɨɦ, ɢɫɯɨɞɢɬ ɢɡ24
Ʉɪɟɣɬ Ɏ., Ȼɥɷɤ ɍ. Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. Ɇ.: Ɇɢɪ, 1983. 512 ɫ.
231
ɧɭɬɪɢ, ɬɚɤ ɱɬɨ ɬɜɟɪɞɨɟ ɬɟɥɨ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɩɨɝɥɨɳɚɸɳɭɸ ɢ
ɢɡɥɭɱɚɸɳɭɸ ɫɪɟɞɭ ɧɚɩɨɞɨɛɢɟ ɝɚɡɚ.
ɂɡɥɭɱɚɸɳɢɣ ɝɚɡ ɦɨɠɟɬ ɫɨɫɬɨɹɬɶ ɢɡ ɦɨɥɟɤɭɥ, ɚɬɨɦɨɜ, ɢɨɧɨɜ ɢ ɫɜɨɛɨɞɧɵɯ ɷɥɟɤɬɪɨɧɨɜ. ɗɬɢ ɱɚɫɬɢɰɵ ɢɦɟɸɬ ɪɚɡɥɢɱɧɵɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ
ɭɪɨɜɧɢ. ȼ ɦɨɥɟɤɭɥɟ, ɧɚɩɪɢɦɟɪ, ɚɬɨɦɵ ɨɛɪɚɡɭɸɬ ɞɢɧɚɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ,
ɤɨɬɨɪɚɹ ɢɦɟɟɬ ɨɩɪɟɞɟɥɟɧɧɵɟ ɤɨɥɟɛɚɬɟɥɶɧɵɟ ɢ ɜɪɚɳɚɬɟɥɶɧɵɟ ɫɨɫɬɨɹɧɢɹ
ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɭɪɨɜɧɹɦɢ.
Ɋɚɞɢɚɰɢɨɧɧɵɟ ɩɪɨɰɟɫɫɵ ɭɞɨɛɧɨ ɨɩɢɫɵɜɚɬɶ ɫ ɩɨɦɨɳɶɸ ɮɨɬɨɧɨɜ ɢɥɢ
ɤɜɚɧɬɨɜɵɯ ɩɪɟɞɫɬɚɜɥɟɧɢɣ. Ɏɨɬɨɧ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɨɣ ɟɞɢɧɢɰɟɣ ɷɧɟɪɝɢɢ
ɢɡɥɭɱɟɧɢɹ. ɂɫɩɭɫɤɚɧɢɟ ɢɡɥɭɱɟɧɢɹ – ɷɬɨ ɩɪɨɰɟɫɫ ɢɫɩɭɫɤɚɧɢɹ ɮɨɬɨɧɨɜ, ɚ
ɩɨɝɥɨɳɟɧɢɟ – ɡɚɯɜɚɬ ɮɨɬɨɧɨɜ ɱɚɫɬɢɰɟɣ. ɉɪɢ ɢɫɩɭɫɤɚɧɢɢ ɢɥɢ ɩɨɝɥɨɳɟɧɢɢ ɮɨɬɨɧɚ ɷɧɟɪɝɢɹ ɢɫɩɭɫɤɚɸɳɟɣ ɢɥɢ ɩɨɝɥɨɳɚɸɳɟɣ ɱɚɫɬɢɰɵ ɛɭɞɟɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɭɦɟɧɶɲɚɬɶɫɹ ɢɥɢ ɭɜɟɥɢɱɢɜɚɬɶɫɹ. ɉɨɦɢɦɨ ɩɪɨɰɟɫɫɨɜ ɢɫɩɭɫɤɚɧɢɹ ɢ ɩɨɝɥɨɳɟɧɢɹ ɜɨɡɦɨɠɧɵ ɩɪɨɰɟɫɫɵ ɧɟɭɩɪɭɝɨɝɨ ɪɚɫɫɟɹɧɢɹ, ɩɪɢ
ɤɨɬɨɪɵɯ ɮɨɬɨɧɵ ɩɟɪɟɞɚɸɬ ɬɨɥɶɤɨ ɱɚɫɬɶ ɫɜɨɟɣ ɷɧɟɪɝɢɢ. ɗɬɢ ɩɪɨɰɟɫɫɵ
ɦɟɧɟɟ ɫɭɳɟɫɬɜɟɧɧɵ ɩɪɢ ɪɚɫɫɦɨɬɪɟɧɢɢ ɢɧɠɟɧɟɪɧɵɯ ɡɚɞɚɱ ɪɚɞɢɚɰɢɨɧɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ.
ȿɫɥɢ ɱɟɪɟɡ ɜɟɳɟɫɬɜɨ ɩɪɨɯɨɞɢɬ ɩɨɬɨɤ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɢ, ɨɧ ɨɫɥɚɛɥɹɟɬɫɹ ɧɚ ɫɜɨɟɦ ɩɭɬɢ. Ɉɫɥɚɛɥɟɧɢɟ ɩɪɨɢɫɯɨɞɢɬ ɤɚɤ ɜɫɥɟɞɫɬɜɢɟ ɩɨɝɥɨɳɟɧɢɹ
ɤɜɚɧɬɨɜ, ɬɚɤ ɢ ɜɫɥɟɞɫɬɜɢɟ ɢɯ ɪɚɫɫɟɹɧɢɹ, ɬ.ɟ. ɨɬɤɥɨɧɟɧɢɹ ɨɬ ɩɟɪɜɨɧɚɱɚɥɶɧɨɝɨ ɧɚɩɪɚɜɥɟɧɢɹ. Ɉɬɧɨɫɢɬɟɥɶɧɨɟ ɨɫɥɚɛɥɟɧɢɟ ɩɚɪɚɥɥɟɥɶɧɨɝɨ ɩɭɱɤɚ ɧɚ
ɷɥɟɦɟɧɬɟ ɩɭɬɢ dx ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɷɬɨɦɭ ɷɥɟɦɟɧɬɭ
dIO
K O I Od x ,
(9.27)
ɝɞɟ K O – ɤɨɷɮɮɢɰɢɟɧɬ ɨɫɥɚɛɥɟɧɢɹ ɢɡɥɭɱɟɧɢɹ, ɮɢɡɢɱɟɫɤɚɹ ɯɚɪɚɤɬɟɪɢɫɬɢɤɚ ɜɟɳɟɫɬɜɚ. Ʉɨɷɮɮɢɰɢɟɧɬ ɨɫɥɚɛɥɟɧɢɹ ɢɦɟɟɬ ɪɚɡɦɟɪɧɨɫɬɶ ɨɛɪɚɬɧɨɣ
ɞɥɢɧɵ ɢ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɡɚɜɢɫɢɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ, ɞɚɜɥɟɧɢɹ, ɫɨɫɬɚɜɚ
(ɜɵɪɚɠɟɧɧɨɝɨ ɤɨɧɰɟɧɬɪɚɰɢɹɦɢ ɤɨɦɩɨɧɟɧɬɨɜ) ɢ ɞɥɢɧɵ ɜɨɥɧɵ ɢɡɥɭɱɟɧɢɹ
KO
K O O ,T , p ,C i .
ɂɡ (9.27) ɧɚɯɨɞɢɦ
ª x
º
I O I O 0exp « ³ K O y dy » .
(9.28)
«¬ 0
»¼
ɍɪɚɜɧɟɧɢɟ (9.28) ɢɡɜɟɫɬɧɨ ɤɚɤ ɡɚɤɨɧ Ȼɭɝɟɪɚ, ɩɟɪɜɨɧɚɱɚɥɶɧɨ ɭɫɬɚɧɨɜɥɟɧɧɵɣ
ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ.
ɂɧɨɝɞɚ
ɟɝɨ
ɧɚɡɵɜɚɸɬ
ɡɚɤɨɧɨɦ
Ʌɚɦɛɟɪɬɚ–Ȼɭɝɟɪɚ.
Ɂɚɤɨɧ Ȼɷɪɚ ɢɥɢ Ʌɚɦɛɟɪɬɚ–Ȼɭɝɟɪɚ–Ȼɷɪɚ ɫɜɹɡɵɜɚɟɬ ɨɫɥɚɛɥɟɧɢɟ ɫɜɟɬɚ
ɫ ɧɚɥɢɱɢɟɦ ɩɨɝɥɨɳɚɸɳɢɯ ɰɟɧɬɪɨɜ (ɢɯ ɤɨɧɰɟɧɬɪɚɰɢɟɣ) ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢ
ɫɥɟɞɭɟɬ ɢɡ ɩɪɟɞɩɨɥɨɠɟɧɢɣ, ɱɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɨɫɥɚɛɥɟɧɢɟ ɫɜɟɬɚ ɜ ɛɟɫ232
ɤɨɧɟɱɧɨ ɬɨɧɤɨɦ ɫɥɨɟ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɫɜɟɬɚ ɢ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨ ɬɨɥɳɢɧɟ ɷɬɨɝɨ ɫɥɨɹ dx ɢ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɨɝɥɨɳɚɸɳɟɝɨ
ɜɟɳɟɫɬɜɚ
dI
N 0C d x .
(9.29)
I
Ɉɞɧɚɤɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɩɨɝɥɨɳɟɧɢɹ ɫɜɟɬɚ
ɤɨɧɰɟɧɬɪɚɰɢɢ C ɢɦɟɟɬ ɥɢɲɶ ɩɪɢɛɥɢɠɟɧɧɵɣ ɯɚɪɚɤɬɟɪ. ɉɪɢ ɜɵɫɨɤɢɯ
ɡɧɚɱɟɧɢɹɯ ɤɨɧɰɟɧɬɪɚɰɢɣ ɜ ɝɚɡɚɯ ɢ ɪɚɫɬɜɨɪɚɯ ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ
ɨɛɵɱɧɨ ɧɚɱɢɧɚɟɬ ɡɚɦɟɬɧɨ ɢɡɦɟɧɹɬɶɫɹ ɜɫɥɟɞɫɬɜɢɟ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɨɥɟɤɭɥ. ɇɟɡɚɜɢɫɢɦɨɫɬɶ ɩɨɤɚɡɚɬɟɥɹ ɩɨɝɥɨɳɟɧɢɹ N 0C
ɨɬ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɫɜɟɬɚ ɜɵɩɨɥɧɹɟɬɫɹ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɜɟɳɟɫɬɜ ɜ ɲɢɪɨɤɢɯ
ɩɪɟɞɟɥɚɯ ɢɡɦɟɧɟɧɢɹ ɷɧɟɪɝɢɢ ɩɨɝɥɨɳɚɟɦɨɝɨ ɫɜɟɬɚ. ɇɨ ɜɫɥɟɞɫɬɜɢɟ ɤɜɚɧɬɨɜɨɣ ɩɪɢɪɨɞɵ ɫɜɟɬɚ ɢ ɤɨɧɟɱɧɨɣ ɞɥɢɬɟɥɶɧɨɫɬɢ ɜɨɡɛɭɠɞɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ ɦɨɥɟɤɭɥ ɡɧɚɱɢɬɟɥɶɧɚɹ ɱɚɫɬɶ ɩɨɝɥɨɳɚɸɳɢɯ ɰɟɧɬɪɨɜ ɩɪɢ ɞɨɫɬɚɬɨɱɧɨ
ɛɨɥɶɲɨɣ ɦɨɳɧɨɫɬɢ ɫɜɟɬɚ ɜɫɤɨɪɟ ɨɤɚɡɵɜɚɟɬɫɹ ɜ ɜɨɡɛɭɠɞɟɧɧɨɦ ɫɨɫɬɨɹɧɢɢ, ɢ ɩɨɝɥɨɳɟɧɢɟ ɭɦɟɧɶɲɚɟɬɫɹ. ȼɟɥɢɱɢɧɚ N 0C ɡɚɜɢɫɢɬ ɢ ɨɬ ɬɨɥɳɢɧɵ
ɩɨɝɥɨɳɚɸɳɟɝɨ ɫɥɨɹ ɩɪɢ ɩɨɝɥɨɳɟɧɢɢ ɫɜɟɬɚ ɜ ɥɸɦɢɧɟɫɰɢɪɭɸɳɟɦ ɜɟɳɟɫɬɜɟ, ɤɨɝɞɚ ɪɚɫɫɬɨɹɧɢɟ ɦɟɠɞɭ ɫɜɟɬɹɳɟɣɫɹ ɢ ɩɨɝɥɨɳɚɸɳɟɣ ɦɨɥɟɤɭɥɚɦɢ
ɦɟɧɶɲɟ ɞɥɢɧɵ ɫɜɟɬɨɜɨɣ ɜɨɥɧɵ. ɉɪɢɱɢɧɚ ɷɬɨɝɨ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɪɟɡɨɧɚɧɫɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɯ ɦɟɠɞɭ ɫɜɟɬɹɳɟɣɫɹ ɢ ɩɨɝɥɨɳɚɸɳɟɣ ɦɨɥɟɤɭɥɚɦɢ.
Ɂɚɤɨɧ Ȼɷɪɚ ɹɜɥɹɟɬɫɹ ɱɚɫɬɧɵɦ ɫɥɭɱɚɟɦ ɡɚɤɨɧɚ (9.28).
Ʉɨɷɮɮɢɰɢɟɧɬ ɨɫɥɚɛɥɟɧɢɹ K O ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɨɝɥɨɳɟɧɢɹ N a ɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɫɫɟɹɧɢɹ N s . Ɉɛɪɚɬɧɵɟ ɢɦ ɜɟɥɢɱɢɧɵ –
ɟɫɬɶ ɞɥɢɧɵ ɩɪɨɛɟɝɚ ɢɡɥɭɱɟɧɢɹ: ɩɨɥɧɚɹ l O 1 K O , ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɩɨɝɥɨɳɟɧɢɸ l a 1 N a ɢ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚɫɫɟɹɧɢɸ l s 1 N s . Ⱦɥɢɧɵ
ɩɪɨɛɟɝɚ ɯɚɪɚɤɬɟɪɢɡɭɸɬ ɨɫɥɚɛɥɟɧɢɟ ɢɡɥɭɱɟɧɢɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦɭ ɩɪɨɰɟɫɫɭ ɧɚ ɟɞɢɧɢɰɟ ɩɭɬɢ.
ɉɨɞ ɪɚɫɫɟɹɧɢɟɦ ɡɞɟɫɶ ɩɨɧɢɦɚɟɬɫɹ ɥɸɛɨɟ ɫɬɨɥɤɧɨɜɟɧɢɟ ɮɨɬɨɧɚ ɫ ɨɞɧɨɣ ɢɥɢ ɛɨɥɟɟ ɱɚɫɬɢɰɚɦɢ ɞɪɭɝɨɝɨ ɫɨɪɬɚ, ɩɪɢ ɤɨɬɨɪɨɦ ɨɧ ɧɟ ɬɟɪɹɟɬ ɜɫɸ
ɫɜɨɸ ɷɧɟɪɝɢɸ. ȼɨɡɦɨɠɧɵ ɢɡɦɟɧɟɧɢɟ ɧɚɩɪɚɜɥɟɧɢɹ ɢ ɱɚɫɬɢɱɧɚɹ ɩɨɬɟɪɹ
ɢɥɢ ɩɪɢɪɚɳɟɧɢɟ ɷɧɟɪɝɢɢ. ȼɨ ɜɫɟɯ ɷɬɢɯ ɫɥɭɱɚɹɯ ɝɨɜɨɪɹɬ, ɱɬɨ ɮɨɬɨɧ ɢɫɩɵɬɵɜɚɟɬ ɪɚɫɫɟɹɧɢɟ.
ɉɨɤɚɡɚɬɟɥɶ ɷɤɫɩɨɧɟɧɬɵ ɜ (9.28) ɱɚɫɬɨ ɡɚɩɢɫɵɜɚɸɬ ɜ ɞɪɭɝɨɦ ɜɢɞɟ,
ɜɜɨɞɹ ɛɟɡɪɚɡɦɟɪɧɭɸ ɜɟɥɢɱɢɧɭ
x
NO
³ K Od x ,
0
233
ɤɨɬɨɪɭɸ ɧɚɡɵɜɚɸɬ ɨɩɬɢɱɟɫɤɨɣ ɬɨɥɳɢɧɨɣ ɫɥɨɹ ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɡɥɭɱɟɧɢɸ ɞɥɢɧɵ ɜɨɥɧɵ O . ɂɡɥɭɱɟɧɢɟ ɨɫɥɚɛɥɹɟɬɫɹ ɜ e ɪɚɡ ɧɚ ɨɩɬɢɱɟɫɤɨɣ
ɬɨɥɳɢɧɟ, ɪɚɜɧɨɣ ɟɞɢɧɢɰɟ.
ȿɫɥɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɪɚɫɫɟɹɧɢɟɦ, ɬ.ɟ. ɩɪɢɧɹɬɶ N s 0 , ɬɨ
K O N a N O ɢ ɭɪɚɜɧɟɧɢɟ (9.28) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
ª x
º
(9.30)
I O I O 0exp « ³ N O y dy » .
«¬ 0
»¼
ȿɫɥɢ ɤ ɬɨɦɭ ɠɟ N O ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɨɨɪɞɢɧɚɬɵ (ɤɚɤ, ɧɚɩɪɢɦɟɪ, ɜ ɫɥɭɱɚɟ ɝɚɡɚ ɫ ɩɨɫɬɨɹɧɧɵɦɢ ɬɟɦɩɟɪɚɬɭɪɨɣ ɢ ɞɚɜɥɟɧɢɟɦ ɢ ɫɨɫɬɚɜɨɦ), ɬɨ
IO
I O 0exp > N O x @ .
(9.31)
4SN
, ɝɞɟ N – ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ,
O
ɫɜɹɡɚɧɧɵɣ ɫ ɦɚɝɧɢɬɧɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɶɸ, ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɢ ɞɢɷɥɟɤɬɪɢɱɟɫɤɨɣ ɩɪɨɧɢɰɚɟɦɨɫɬɶɸ ɫɪɟɞɵ, ɱɬɨ ɩɨɤɚɡɵɜɚɟɬɫɹ ɜ ɷɥɟɤɬɪɨɦɚɝɧɢɬɧɨɣ ɜɨɥɧɨɜɨɣ ɬɟɨɪɢɢ ɢɡɥɭɱɟɧɢɹ25.
Ɂɚɤɨɧ Ȼɭɝɟɪɚ ɜ ɮɨɪɦɟ (9.30) ɨɩɪɟɞɟɥɹɟɬ ɨɫɥɚɛɥɟɧɢɟ ɢɡɥɭɱɟɧɢɹ ɩɪɢ
ɩɪɨɯɨɠɞɟɧɢɢ ɱɟɪɟɡ ɨɛɴɟɦ ɧɟɢɡɥɭɱɚɸɳɟɝɨ ɢ ɧɟɪɚɫɫɟɢɜɚɸɳɟɝɨ ɝɚɡɚ ɧɚ
ɩɭɬɢ x . ȼ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɩɪɢ ɩɪɨɯɨɠɞɟɧɢɢ ɢɡɥɭɱɟɧɢɹ ɱɟɪɟɡ ɜɟɳɟɫɬɜɨ ɩɨɦɢɦɨ ɩɨɝɥɨɳɟɧɢɹ ɜɨɡɧɢɤɚɟɬ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɹɜɥɟɧɢɟ, ɡɚɤɥɸɱɚɸɳɟɟɫɹ ɜ ɬɨɦ, ɱɬɨ ɜɨɡɛɭɠɞɟɧɧɨɟ ɫɨɫɬɨɹɧɢɟ ɝɚɡɚ ɹɜɥɹɟɬɫɹ ɧɟɭɫɬɨɣɱɢɜɵɦ ɢ
ɩɪɨɢɫɯɨɞɢɬ ɫɩɨɧɬɚɧɧɵɣ (ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɵɣ) ɩɟɪɟɯɨɞ ɧɚ ɛɨɥɟɟ ɧɢɡɤɢɣ
ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɭɪɨɜɟɧɶ. ɂɡɥɭɱɟɧɢɟ, ɢɫɩɭɫɤɚɟɦɨɟ ɩɪɢ ɧɚɥɢɱɢɢ ɩɨɫɬɨɪɨɧɧɟɝɨ ɩɨɥɹ ɢɡɥɭɱɟɧɢɹ, ɧɚɡɵɜɚɟɬɫɹ ɜɵɧɭɠɞɟɧɧɵɦ, ɢɥɢ ɢɧɞɭɰɢɪɨɜɚɧɧɵɦ, ɢɡɥɭɱɟɧɢɟɦ ɢ ɢɦɟɟɬ ɫɦɵɫɥ ɨɬɪɢɰɚɬɟɥɶɧɨɝɨ ɩɨɝɥɨɳɟɧɢɹ. ɂɫɬɢɧɧɵɣ
ɤɨɷɮɮɢɰɢɟɧɬ ɩɨɝɥɨɳɟɧɢɹ ɜɫɟɝɞɚ ɛɨɥɶɲɟ ɤɨɷɮɮɢɰɢɟɧɬɚ N O , ɤɨɬɨɪɵɣ
ɦɨɠɟɬ ɛɵɬɶ ɢɡɦɟɪɟɧ ɩɨ ɞɚɧɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɚ ɧɚ ɨɫɧɨɜɚɧɢɢ ɡɚɤɨɧɚ
Ȼɭɝɟɪɚ.
ȼ ɩɪɨɜɨɞɹɳɢɯ ɫɪɟɞɚɯ N O
9.8. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ
ɩɪɢ ɧɚɥɢɱɢɢ ɞɪɭɝɢɯ ɜɢɞɨɜ ɷɧɟɪɝɢɢ
ȼ ɪɟɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɥɭɱɟɢɫɩɭɫɤɚɧɢɟɦ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɞɪɭɝɢɦɢ ɜɢɞɚɦɢ ɬɟɩɥɨɩɟɪɟɧɨɫɚ – ɤɨɧɜɟɤɰɢɟɣ ɢɥɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ. Ɍɚɤɨɣ ɫɨɜɦɟɫɬɧɵɣ ɩɪɨɰɟɫɫ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ
ɫɥɨɠɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ. ȿɫɥɢ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɜɫɟɦɢ ɬɪɟɦɹ ɜɢɞɚɦɢ ɨɞɧɨɜɪɟɦɟɧɧɨ (ɪɚɞɢɚɰɢɟɣ, ɬɟɩɥɨɩɪɨɜɨɞɧɨ25
Ɂɢɝɟɥɶ Ɋ., Ⱥɜɞɭɟɜɫɤɢɣ ȼ.ɋ. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ. Ɇ.: Ɇɢɪ, 1975. 840 ɫ.
234
ɫɬɶɸ ɢ ɤɨɧɜɟɤɰɢɟɣ), ɬɨ ɨɧ ɧɚɡɵɜɚɟɬɫɹ ɪɚɞɢɚɰɢɨɧɧɨ-ɤɨɧɜɟɤɬɢɜɧɵɦ ɬɟɩɥɨɨɛɦɟɧɨɦ.
Ɍɢɩɢɱɧɵɦ ɫɥɭɱɚɟɦ ɫɥɨɠɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɹɜɥɹɟɬɫɹ ɫɨɱɟɬɚɧɢɟ ɤɨɧɜɟɤɬɢɜɧɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɥɭɱɢɫɬɵɦ: ɩɪɢ ɜɨɡɞɭɲɧɨɦ ɨɯɥɚɠɞɟɧɢɢ ɩɪɨɞɭɤɬɨɜ ɜ ɚɩɩɚɪɚɬɚɯ ɬɭɧɧɟɥɶɧɨɝɨ ɬɢɩɚ, ɬɟɪɦɨɛɚɪɨɤɚɦɟɪɚɯ ɢ ɬ.ɩ. ȼɨ ɜɫɟɯ
ɫɥɭɱɚɹɯ ɜɚɠɧɨ ɨɰɟɧɢɬɶ ɜɤɥɚɞ ɤɚɠɞɨɝɨ ɫɨɫɬɚɜɥɹɸɳɟɝɨ ɩɪɨɰɟɫɫɚ ɜ ɬɟɩɥɨɨɛɦɟɧ. ȼ ɩɪɨɰɟɫɫɚɯ ɨɯɥɚɠɞɟɧɢɹ ɜɨɡɞɭɯɚ ɨɩɪɟɞɟɥɹɸɳɢɦ, ɤɚɤ ɩɪɚɜɢɥɨ,
ɹɜɥɹɟɬɫɹ ɤɨɧɜɟɤɬɢɜɧɵɣ ɬɟɩɥɨɨɛɦɟɧ. ȼɥɢɹɧɢɟ ɥɭɱɢɫɬɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ
ɫɭɦɦɚɪɧɵɣ ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɨɤɚɡɵɜɚɟɬɫɹ ɬɟɦ ɫɭɳɟɫɬɜɟɧɧɟɟ, ɱɟɦ ɦɟɧɶɲɟ ɤɨɧɜɟɤɬɢɜɧɚɹ ɫɨɫɬɚɜɥɹɸɳɚɹ. ɇɚɩɪɢɦɟɪ, ɜ ɬɟɪɦɨɛɚɪɨɤɚɦɟɪɚɯ ɩɥɨɬɧɨɫɬɶ ɜɨɡɞɭɯɚ ɩɪɢ ɧɢɡɤɢɯ ɞɚɜɥɟɧɢɹɯ ɦɚɥɚ, ɢ ɷɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɫɧɢɠɚɟɬ
ɨɬɜɨɞ ɬɟɩɥɨɬɵ ɤɨɧɜɟɤɰɢɟɣ. ɉɨɷɬɨɦɭ ɞɥɹ ɨɯɥɚɠɞɟɧɢɹ ɢɡɞɟɥɢɣ ɞɨ ɧɭɠɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɷɬɢɯ ɤɚɦɟɪɚɯ ɨɯɥɚɠɞɚɸɳɢɟ ɭɫɬɪɨɣɫɬɜɚ ɜɵɩɨɥɧɹɸɬ
ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɦɚɤɫɢɦɚɥɶɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɷɮɮɟɤɬ ɥɭɱɢɫɬɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ. Ⱦɨɥɹ ɥɭɱɢɫɬɨɝɨ ɬɟɩɥɨɨɛɦɟɧɚ ɦɨɠɟɬ ɛɵɬɶ ɫɭɳɟɫɬɜɟɧɧɨɣ ɢ
ɞɥɹ ɨɯɥɚɠɞɚɸɳɢɯ ɭɫɬɪɨɣɫɬɜ ɫ ɟɫɬɟɫɬɜɟɧɧɨɣ ɤɨɧɜɟɤɰɢɣ (ɩɪɢɫɬɟɧɧɵɟ ɢ
ɩɨɬɨɥɨɱɧɵɟ ɛɚɬɚɪɟɢ). ɉɪɢ ɝɥɭɛɨɤɨɦ ɜɚɤɭɭɦɟ (ɜ ɤɨɫɦɨɫɟ) ɩɟɪɟɧɨɫ ɬɟɩɥɨɬɵ ɥɭɱɟɢɫɩɭɫɤɚɧɢɟɦ ɹɜɥɹɟɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɨɫɧɨɜɧɵɦ ɫɩɨɫɨɛɨɦ ɩɟɪɟɞɚɱɢ ɬɟɩɥɨɬɵ ɜɨ ɜɧɟɲɧɟɟ ɩɪɨɫɬɪɚɧɫɬɜɨ.
Ɋɚɡɥɢɱɧɵɟ ɜɢɞɚ ɩɟɪɟɧɨɫɚ ɷɧɟɪɝɢɢ ɩɨ-ɪɚɡɧɨɦɭ ɡɚɜɢɫɹɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ. Ⱦɥɹ ɧɟɱɟɪɧɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɩɨɤɚɡɚɬɟɥɶ ɫɬɟɩɟɧɢ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ
ɦɨɠɟɬ ɧɟɫɤɨɥɶɤɨ ɨɬɥɢɱɚɬɶɫɹ ɨɬ ɱɟɬɵɪɟɯ, ɬɚɤ ɤɚɤ ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ ɦɟɧɹɟɬɫɹ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ. ɉɪɢ ɧɚɥɢɱɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɡɚɜɢɫɢɦɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ ɨɬ ɥɨɤɚɥɶɧɨɝɨ ɝɪɚɞɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɵ ɨɩɢɫɵɜɚɟɬɫɹ ɡɚɤɨɧɨɦ Ɏɭɪɶɟ, ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɩɪɨɢɡɜɨɞɧɨɣ ɨɬ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɬɟɦɩɟɪɚɬɭɪɵ
(ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɧɟ ɡɚɜɢɫɢɬ). ɉɪɢ
ɧɚɥɢɱɢɢ ɤɨɧɜɟɤɰɢɢ ɩɨɹɜɥɹɟɬɫɹ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɣ
ɩɪɢɛɥɢɡɢɬɟɥɶɧɨ ɩɟɪɜɨɣ ɫɬɟɩɟɧɢ ɪɚɡɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ. ɂɡɦɟɧɟɧɢɟ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ ɩɪɢɜɨɞɢɬ ɤ ɞɨɩɨɥɧɢɬɟɥɶɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ. ɉɨɫɤɨɥɶɤɭ ɫɥɚɝɚɟɦɵɟ ɜ ɭɪɚɜɧɟɧɢɹɯ, ɭɱɢɬɵɜɚɸɳɢɟ ɢɡɥɭɱɟɧɢɟ ɨɬ ɨɤɪɭɠɚɸɳɢɯ ɩɨɜɟɪɯɧɨɫɬɟɣ, ɨɛɵɱɧɨ ɨɩɢɫɵɜɚɸɬɫɹ
ɜ ɜɢɞɟ ɢɧɬɟɝɪɚɥɨɜ, ɚ ɫɥɚɝɚɟɦɵɟ, ɭɱɢɬɵɜɚɸɳɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ, ɫɨɞɟɪɠɚɬ ɩɪɨɢɡɜɨɞɧɵɟ, ɬɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ ɹɜɥɹɸɬɫɹ
ɢɧɬɟɝɪɨɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɦɢ, ɤɨɬɨɪɵɟ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɟɲɚɸɬɫɹ ɬɨɥɶɤɨ
ɱɢɫɥɟɧɧɨ.
ȼ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɨɣ ɮɨɪɦɟ ɭɪɚɜɧɟɧɢɟ ɷɧɟɪɝɢɢ ɩɪɢ ɧɚɥɢɱɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɢɡɥɭɱɟɧɢɹ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ (ɩɪɢɜɨɞɢɦ ɛɟɡ ɜɵɜɨɞɚ)
Uc p
wT
wt
’ ˜ q c q r qV ,
235
(9.32)
ɝɞɟ q c – ɩɨɬɨɤ ɬɟɩɥɚ ɜɫɥɟɞɫɬɜɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, q r – ɜɫɥɟɞɫɬɜɢɟ ɢɡɥɭɱɟɧɢɹ,
4S
f
ª
º
«
’ ˜ qr
N
O
O
Z
Z
4
e
,
T
I
,
d
³ O « O0 ³ O »» d O ,
O 0
Z 0
¬
¼
ɝɞɟ eO 0 – ɩɨɜɟɪɯɧɨɫɬɧɚɹ ɩɥɨɬɧɨɫɬɶ ɢɡɥɭɱɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɱɟɪɧɨɦɭ ɬɟɥɭ.
ɉɪɢɦɟɪ 1. ȼ ɤɚɱɟɫɬɜɟ ɩɟɪɜɨɝɨ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɩɚɪɚɥɥɟɥɶɧɵɟ ɛɟɫɤɨɧɟɱɧɵɟ ɱɟɪɧɵɟ ɩɥɚɫɬɢɧɵ, ɪɚɡɞɟɥɟɧɧɵɟ ɫɪɟɞɨɣ ɬɨɥɳɢɧɨɣ h ɫ
ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O , ɤɨɬɨɪɚɹ ɩɪɨɡɪɚɱɧɚ ɞɥɹ ɬɟɩɥɨɜɨɝɨ
ɢɡɥɭɱɟɧɢɹ. Ʉɚɤɨɜ ɪɟɡɭɥɶɬɢɪɭɸɳɢɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ ɦɟɠɞɭ ɩɥɚɫɬɢɧɚɦɢ,
ɟɫɥɢ ɢɯ ɬɟɦɩɟɪɚɬɭɪɵ ɪɚɜɧɵ T1 ɢ T2 ?
Ɋɟɡɭɥɶɬɢɪɭɸɳɢɣ ɩɨɬɨɤ ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɪɚɞɢɚɰɢɨɧɧɨɣ Q R ɢ ɤɨɧɞɭɤɬɢɜɧɨɣ Qc ɫɨɫɬɚɜɥɹɸɳɢɯ. Ɉɧ ɪɚɜɟɧ ɩɨɬɨɤɭ ɷɧɟɪɝɢɢ, ɤɨɬɨɪɵɣ ɧɭɠɧɨ
ɩɨɞɜɟɫɬɢ ɤ ɩɥɚɫɬɢɧɟ 1, ɱɬɨɛɵ ɩɨɞɞɟɪɠɢɜɚɬɶ ɟɟ ɩɪɢ ɡɚɞɚɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ
Q1 Q R Q c
ɉɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ, ɩɟɪɟɧɨɫɢɦɨɣ ɢɡɥɭɱɟɧɢɟɦ, ɟɫɬɶ
QR
V 0 T14 T24 ,
F
ɚ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɷɧɟɪɝɢɢ, ɩɟɪɟɧɨɫɢɦɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɫɨɫɬɚɜɥɹɟɬ
Qc O
T T .
F h 1 2
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɟɫɥɢ ɤɨɧɞɭɤɬɢɜɧɚɹ ɢ ɪɚɞɢɚɰɢɨɧɧɚɹ ɫɨɫɬɚɜɥɹɸɳɢɟ
ɧɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ (ɬ.ɟ., ɧɚɥɢɱɢɟ ɨɞɧɨɝɨ ɜɢɞɚ ɬɟɩɥɨɨɛɦɟɧɚ ɧɟ ɜɥɢɹɟɬ
ɧɚ ɞɪɭɝɨɣ), ɢɦɟɟɦ
Q1
O
(9.33)
V 0 T14 T24 T1 T2 .
F
h
ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɩɪɢɯɨɞɢɬɫɹ ɢɦɟɬɶ ɞɟɥɨ ɫ ɡɚɞɚɱɚɦɢ, ɜ ɤɨɬɨɪɵɯ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢ ɢɡɥɭɱɟɧɢɟ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ.
Ɋɚɫɫɦɨɬɪɢɦ ɷɬɨɬ ɠɟ ɩɪɢɦɟɪ, ɧɨ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɩɥɚɫɬɢɧɚ 2 ɧɚɯɨɞɢɬɫɹ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T2 , ɚ ɤ ɟɞɢɧɢɰɟ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɧɵ
1 ɩɨɞɜɨɞɢɬɫɹ ɢɡɜɟɫɬɧɵɣ ɩɨɬɨɤ ɷɧɟɪɝɢɢ Q1 F , ɤɨɬɨɪɵɣ ɡɚɬɟɦ ɨɬɜɨɞɢɬɫɹ
ɤ ɩɨɜɟɪɯɧɨɫɬɢ 2. Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɬɟɦɩɟɪɚɬɭɪɭ T1 .
ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (9.33) ɜ ɢɧɨɣ ɮɨɪɦɟ
O
O
Q
V 0T14 T1 V 0T24 T2 1 .
(9.34)
h
h
F
236
ɉɨ ɨɬɧɨɲɟɧɢɸ ɤ ɢɫɤɨɦɨɣ ɧɟɢɡɜɟɫɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ T1 ɢɡɥɭɱɟɧɢɟ ɢ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɜɡɚɢɦɨɞɟɣɫɬɜɭɸɬ ɞɪɭɝ ɫ ɞɪɭɝɨɦ. ɇɟɥɢɧɟɣɧɨɟ ɭɪɚɜɧɟɧɢɟ (9.34) ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɨ ɨɬɧɨɫɢɬɟɥɶɧɨ T1 ɦɟɬɨɞɨɦ ɢɬɟɪɚɰɢɣ.
ɉɪɢɦɟɪ 2. Ɍɨɧɤɨɟ ɤɨɥɶɰɟɜɨɟ ɪɟɛɪɨ, ɧɚɯɨɞɹɳɟɟɫɹ ɜ ɜɚɤɭɭɦɟ, ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɨ ɫ ɨɞɧɨɣ ɥɢɰɟɜɨɣ ɫɬɨɪɨɧɵ ɢ ɫɨ ɫɬɨɪɨɧɵ ɧɚɪɭɠɧɨɣ ɤɪɨɦɤɢ (ɪɢɫ. 9.10). Ⱦɢɫɤ ɢɦɟɟɬ ɬɨɥɳɢɧɭ h , ɜɧɭɬɪɟɧɧɢɣ ɢ ɧɚɪɭɠɧɵɣ ɪɚɞɢɭɫɵ R1
ɢ R 2 . Ʉ ɜɧɭɬɪɟɧɧɟɣ ɤɪɨɦɤɟ ɞɢɫɤɚ ɩɨɞɜɨɞɢɬɫɹ ɷɧɟɪɝɢɹ ɬɚɤ, ɱɬɨ ɟɟ ɬɟɦɩɟɪɚɬɭɪɚ ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɨɣ ɢ ɪɚɜɧɨɣ T1 . ɇɟɢɡɨɥɢɪɨɜɚɧɧɚɹ ɩɨɜɟɪɯɧɨɫɬɶ ɹɜɥɹɟɬɫɹ ɫɟɪɨɣ ɢ ɢɦɟɟɬ
ɫɬɟɩɟɧɶ ɱɟɪɧɨɬɵ H . ɗɬɚ ɩɨɜɟɪɯɧɨɫɬɶ ɢɡɥɭɊɢɫ. 9.10. Ʉ ɮɨɪɦɭɥɢɪɨɜɤɟ
ɱɚɟɬ ɷɧɟɪɝɢɸ ɜ ɨɤɪɭɠɚɸɳɟɟ ɩɪɨɫɬɪɚɧɫɬɜɨ
ɡɚɞɚɱɢ ɨɛ ɨɯɥɚɠɞɟɧɢɢ
ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ Te . Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɪɚɫɤɨɥɶɰɟɜɨɝɨ ɪɟɛɪɚ
ɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨ ɪɚɞɢɭɫɭ ɞɢɫɤɚ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɞɢɫɤ ɞɨɫɬɚɬɨɱɧɨ ɬɨɧɨɤ, ɬɚɤ ɱɬɨ ɥɨɤɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɩɨɫɬɨɹɧɧɨɣ ɩɨ ɬɨɥɳɢɧɟ. Ɍɨɝɞɚ ɛɚɥɚɧɫ ɷɧɟɪɝɢɢ
ɞɥɹ ɥɸɛɨɝɨ ɤɨɥɶɰɟɜɨɝɨ ɷɥɟɦɟɧɬɚ ɲɢɪɢɧɨɣ dr ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
A BC,
ɝɞɟ A ɢ C – ɩɨɞɜɨɞɢɦɵɣ ɤ ɷɥɟɦɟɧɬɭ ɢ ɨɬɜɨɞɢɦɵɣ ɨɬ ɧɟɝɨ ɩɨɬɨɤɢ ɬɟɩɥɚ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ, ɚ B – ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ ɜɫɥɟɞɫɬɜɢɟ ɢɡɥɭɱɟɧɢɹ ɫ ɩɨɜɟɪɯɧɨɫɬɢ ɷɥɟɦɟɧɬɚ:
dT
A O 2S rh
,
dr
B
HV 0 T 4 Te4 2S r d r ,
dT d §
dT ·
¨ O 2S r h
¸dr .
dr dr ©
dr ¹
ɉɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ ɫɜɨɣɫɬɜ, ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ
ɩɪɢɦɟɬ ɜɢɞ
d § dT ·
4
4
0.
Oh ¨ r
¸ H r V 0 T Te
dr © dr ¹
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ ɷɬɨɦɭ ɭɪɚɜɧɟɧɢɸ ɛɭɞɭɬ
r R1 : T T1 ;
C
O 2S r h
237
dT
0.
dr
ɍɪɚɜɧɟɧɢɟ ɹɜɥɹɟɬɫɹ ɧɟɥɢɧɟɣɧɵɦ ɢ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɨ ɱɢɫɥɟɧɧɨ.
ȼ ɫɥɭɱɚɟ ɧɟɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɫɨɫɬɨɹɧɢɹ (ɧɚɩɪɢɦɟɪ, ɦɟɧɹɸɳɟɣɫɹ ɜɨ
ɜɪɟɦɟɧɢ ɬɟɦɩɟɪɚɬɭɪɟ T1 ) ɬɟɦɩɟɪɚɬɭɪɚ ɛɭɞɟɬ ɧɟ ɬɨɥɶɤɨ ɮɭɧɤɰɢɟɣ ɪɚɞɢɭɫɚ, ɧɨ ɢ ɜɪɟɦɟɧɢ t . ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɛɭɞɟɬ ɢɦɟɬɶ ɜɢɞ
wT
w § wT ·
4
4
(9.35)
c pU hr
Oh ¨r
¸ H rV 0 T Te .
wt
wr © wr ¹
Ɂɚɞɚɱɭ ɦɨɠɧɨ ɛɵɥɨ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ, ɢɫɩɨɥɶɡɭɹ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ
ɬɨɥɳɢɧɟ ɞɢɫɤɚ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɡɚɩɢɫɚɧɧɨɝɨ ɜ ɰɢɥɢɧɞɪɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɢ ɩɨɥɚɝɚɹ, ɱɬɨ ɡɚɞɚɱɚ ɹɜɥɹɟɬɫɹ ɫɢɦɦɟɬɪɢɱɧɨɣ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɫɢ ɞɢɫɤɚ.
ɉɪɢɦɟɪ 3. Ɍɪɟɬɶɢɦ ɩɪɢɦɟɪɨɦ ɛɭɞɟɬ ɡɚɞɚɱɚ ɨ ɧɚɝɪɟɜɟ ɫɥɨɹ ɪɚɞɢɚɰɢɨɧɧɵɦ ɩɨɬɨɤɨɦ ɩɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɤɨɷɮɮɢɰɢɟɧɬ ɨɫɥɚɛɥɟɧɢɹ ɢɡɥɭɱɟɧɢɹ
(ɤɨɬɨɪɵɣ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɩɨɤɚɡɚɬɟɥɟɦ ɩɨɝɥɨɳɟɧɢɹ, ɬɚɤ ɤɚɤ N O ɫɜɹɡɚɧ ɫ
ɧɢɦ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢɦɨɫɬɶɸ) ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɤɨɨɪɞɢɧɚɬɵ. Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɬɚɤɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ
wT
w §
wT ·
(9.36)
O
c pU
¨ T
¸ q 0N Oe x p N O x ,
wt wx ©
wx ¹
wT
x 0 ,G : O
0,
wx
t 0:
T T0 .
ɉɪɢ G o f ɷɬɚ ɡɚɞɚɱɚ ɪɟɲɚɟɬɫɹ ɦɟɬɨɞɨɦ ɢɧɬɟɝɪɚɥɶɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɩɨ Ʌɚɩɥɚɫɭ ɞɨɜɨɥɶɧɨ ɩɪɨɫɬɨ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ t o p ɜɦɟɫɬɨ (9.36)
ɢɦɟɟɦ
q0
O T d 2T
pT T0
N e xp N O x c pU dx 2 pc pU O
r
R2 :
ɢɥɢ
pu
T
ɝɞɟ u T 0 , N T
p
q0
O T d 2u
N exp N O x ,
c pU dx 2 pc pU O
(9.37)
OT
.
c pU
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɟɫɬɶ ɫɭɦɦɚ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɢ
ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
238
d 2u
NT
pu
.
dx 2
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (9.37) ɢɳɟɦ ɜ ɜɢɞɟ
u1 A exp N O x .
ɩɨɞɫɬɚɜɥɹɹ (9.39) ɜ (9.37), ɧɚɣɞɟɦ
q0N O
A
.
2
p p N T N O c pU
(9.38)
(9.39)
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (9.38) ɧɚɦ ɢɡɜɟɫɬɧɨ
§
§ p
p ·
u 2 B exp ¨¨ x ¸¸ C exp ¨¨
© NT ¹
© NT
·
x ¸¸ .
¹
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɫ ɭɱɟɬɨɦ ɤɨɧɟɱɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ x o f ɡɚɩɢɲɟɦ
§
q0N O
p ·
exp N O x .
u 2 B exp ¨¨ x ¸¸ 2
N
T ¹ p p N T N O c pU
©
ɂɡ ɭɫɥɨɜɢɹ ɩɪɢ x 0 ɧɚɯɨɞɢɦ ɩɨɫɬɨɹɧɧɭɸ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ
q0
N O2
NT
B .
c pU p p N T N O2 p
Ɉɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɢɡɨɛɪɚɠɟɧɢɣ ɢɦɟɟɬ ɜɢɞ
T0
q0N O
T
e xp N O x p p p N N2 c U
q0
T O
N T N O2
p p p N T N O2
p
§
p
e x p ¨¨ c pU
© NT
·
x ¸¸ .
¹
Ⱦɥɹ ɩɟɪɟɯɨɞɚ ɤ ɨɪɢɝɢɧɚɥɚɦ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɬɚɛɥɢɰɚɦɢ ɢɧɬɟɝɪɚɥɶɧɵɯ
ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ (ɩɪɢɥɨɠɟɧɢɟ 1).
ɉɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ:
T0
y T0 .
p
ȼɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ:
1
1
y ªe D t 1º ,
¼
p p D D ¬
ɝɞɟ D N T N O2 .
Ɍɪɟɬɶɟ ɫɥɚɝɚɟɦɨɟ ɭɞɨɛɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
239
1
p p pD
2
e J
1 1 ª 1
«
2 p p ¬« p D
p
1 º J
»e
p D ¼»
p
,
ɝɞɟ D 2 N T N O2 , J x N T .
Ⱦɥɹ ɤɚɠɞɨɝɨ ɫɥɚɝɚɟɦɨɝɨ ɨɪɢɝɢɧɚɥ ɫɨɞɟɪɠɢɬɫɹ ɜ ɬɚɛɥɢɰɟ:
1
p p p D2
e J
+
p
y
2 t J 2
e
D S
4t
1 DJ
D2
§ J
erfc ¨
©2 t
·
¸
¹
§ J
·
2
e
xp
t
e
r
fc
t
D
J
D
D
¨
¸
D2
©2 t
¹
1
2 t J 2
e
D S
4t
1 DJ
D2
§ J
erfc ¨
©2 t
·
¸ ) t , x ,D ¹
ȼ ɪɟɡɭɥɶɬɚɬɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (9.36) ɞɥɹ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɝɨ ɫɥɨɹ ɩɪɢɦɟɬ ɜɢɞ
T T0 q0
q0
u
exp N O x ª«exp NT N O2 t 1º» ¬
¼
OT N O
c pUO T
ª t
§ x2 ·
§ x ·
x
1
2
2
,
,
,
,
t
x
t
x
erfc ¨
u «2 exp ¨ )
N
N
)
N
N
¸
¸
O
O
T
T
2
¨
¸
¨
¸
S
NT
© 4 NT t ¹
© 2 NT t ¹ 2 NT N O
¬«
ȼ ɱɚɫɬɧɨɫɬɢ, ɩɪɢ x 0 ɢɦɟɟɦ
T
T0 q0 ª
q0
2
º
t
N
N
u
exp
1
T O
»¼
OT NO «¬
c pUOT
ª t
1
exp NT NO2 t erfc NO NT t
u «2
NT NO
«¬ S
º
».
»¼
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɑɬɨ ɩɨɧɢɦɚɸɬ ɩɨɞ ɬɟɩɥɨɜɵɦ ɢɡɥɭɱɟɧɢɟɦ?
2. Ɉɩɪɟɞɟɥɢɬɟ ɢɧɬɟɝɪɚɥɶɧɵɟ ɜɟɥɢɱɢɧɵ: ɩɨɝɥɨɳɚɬɟɥɶɧɭɸ ɫɩɨɫɨɛɧɨɫɬɶ, ɨɬɪɚɠɚɬɟɥɶɧɭɸ ɫɩɨɫɨɛɧɨɫɬɶ, ɩɪɨɩɭɫɤɚɬɟɥɶɧɭɸ ɫɩɨɫɨɛɧɨɫɬɶ.
3. Ʉɚɤɨɟ ɬɟɥɨ ɧɚɡɵɜɚɸɬ ɚɛɫɨɥɸɬɧɨ ɱɟɪɧɵɦ?
4. Ʉɚɤɢɟ ɩɨɜɟɪɯɧɨɫɬɢ ɧɚɡɵɜɚɸɬ ɡɟɪɤɚɥɶɧɨɣ ɢ ɚɛɫɨɥɸɬɧɨ ɛɟɥɨɣ?
240
º
»
»¼
5. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɨɫɧɨɜɧɵɟ ɡɚɤɨɧɵ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ (ɡɚɤɨɧ
ɉɥɚɧɤɚ, ɡɚɤɨɧ ɫɦɟɳɟɧɢɹ ȼɢɧɚ, ɡɚɤɨɧ ɋɬɟɮɚɧɚ-Ȼɨɥɶɰɦɚɧɚ, ɡɚɤɨɧ Ʌɚɦɛɟɪɬɚ ɢ ɡɚɤɨɧ Ʉɢɪɯɝɨɮɚ).
6. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɩɪɨɫɬɨɣ ɫɩɨɫɨɛ ɢɦɢɬɚɰɢɢ ɱɟɪɧɨɝɨ ɬɟɥɚ?
7. Ʉɚɤ ɨɩɪɟɞɟɥɹɸɬɫɹ ɦɨɧɨɯɪɨɦɚɬɢɱɟɫɤɢɟ ɢɡɥɭɱɚɬɟɥɶɧɵɟ ɫɜɨɣɫɬɜɚ?
8. Ʉɚɤɨɟ ɬɟɥɨ ɧɚɡɵɜɚɸɬ ɫɟɪɵɦ?
9. ɤɚɤɢɟ ɩɨɧɹɬɢɹ ɬɪɟɛɭɸɬɫɹ ɞɥɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɧɚɩɪɚɜɥɟɧɧɵɯ ɪɚɞɢɚɰɢɨɧɧɵɯ ɫɜɨɣɫɬɜ?
10. Ʉɚɤɢɦɢ ɮɢɡɢɱɟɫɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ ɢ ɩɚɪɚɦɟɬɪɚɦɢ ɦɨɠɧɨ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɩɟɪɟɧɨɫ ɢɡɥɭɱɟɧɢɹ ɜ ɩɨɝɥɨɳɚɸɳɢɯ ɩɪɨɩɭɫɤɚɸɳɢɯ ɫɪɟɞɚɯ?
Ɂɚɞɚɧɢɹ
1. ɋɮɨɪɦɭɥɢɪɨɜɚɬɶ ɡɚɞɚɱɭ ɨ ɧɚɝɪɟɜɟ ɬɟɥɚ ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɨɪɦɵ ɫ
ɩɨɜɟɪɯɧɨɫɬɢ ɡɚ ɫɱɟɬ ɬɟɩɥɨɜɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɩɨɞɨɛɧɭɸ ɡɚɞɚɱɟ ɨɛ ɨɯɥɚɠɞɟɧɢɢ ɩɨ ɡɚɤɨɧɭ ɇɶɸɬɨɧɚ (ɪɚɡɞɟɥ 7.2). ɉɪɢɜɟɫɬɢ ɡɚɞɚɱɭ ɤ ɛɟɡɪɚɡɦɟɪɧɨɣ
ɮɨɪɦɟ ɢ ɧɚɣɬɢ ɟɟ ɪɟɲɟɧɢɟ ɚɧɚɥɢɬɢɱɟɫɤɢ ɢ ɱɢɫɥɟɧɧɨ. ɋɪɚɜɧɢɬɶ ɩɨɥɭɱɟɧɧɨɟ ɪɟɲɟɧɢɟ ɫ ɪɟɲɟɧɢɟɦ ɡɚɞɚɱɢ (7.6).
2. Ɂɚɞɚɱɚ ɨɛ ɨɯɥɚɠɞɟɧɢɢ ɤɨɧɜɟɤɰɢɟɣ ɩɨɥɭɩɪɨɡɪɚɱɧɨɣ ɩɥɚɫɬɢɧɵ, ɧɚɝɪɟɜɚɟɦɨɣ ɢɡɥɭɱɟɧɢɟɦ, ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
wT w § wT ·
c pU
¨ OT
¸ q0 NOexp NO x ,
wt wx ©
wx ¹
wT
x 0 : OT
D T T0 ,
wx
wT
x G : OT
D T T0 ,
wx
t 0 : T T0 .
ɋɱɢɬɚɹ ɩɥɚɫɬɢɧɭ ɬɨɧɤɨɣ, 1) ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɬɶ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɨ ɟɟ ɬɨɥɳɢɧɟ ɫ ɭɱɟɬɨɦ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ; 2) ɧɚɣɬɢ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɩɨɥɭɱɟɧɧɨɣ ɭɩɪɨɳɟɧɧɨɣ ɡɚɞɚɱɢ; 3) ɧɚɣɬɢ ɜɵɪɚɠɟɧɢɹ
ɞɥɹ ɪɚɞɢɚɰɢɨɧɧɨɝɨ ɩɨɬɨɤɚ, ɩɪɨɩɭɳɟɧɧɨɝɨ ɩɥɟɧɤɨɣ ɢ ɩɨɝɥɨɳɟɧɧɨɝɨ
ɩɥɟɧɤɨɣ.
3. ɇɚɣɬɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɡɚɞɚɱɟ ɨɛ ɨɯɥɚɠɞɟɧɢɢ ɞɢɫɤɚ
(ɭɪɚɜɧɟɧɢɟ (9.35)) ɫ ɩɨɦɨɳɶɸ ɱɢɫɥɟɧɧɨɝɨ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɩɨɥɚɝɚɹ, ɱɬɨ
ɧɚ ɜɧɭɬɪɟɧɧɟɣ ɟɝɨ ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɞɚɧ ɩɨɬɨɤ ɬɟɩɥɚ, ɚ ɧɚ ɜɧɟɲɧɟɣ – ɨɛɟɫɩɟɱɟɧɚ ɢɞɟɚɥɶɧɚɹ ɬɟɩɥɨɢɡɨɥɹɰɢɹ:
241
wT
q0 ,
wr
wT
r R2 :
0.
wr
ɍɤɚɡɚɧɢɟ: ɢɫɩɨɥɶɡɭɹ ɧɟɹɜɧɭɸ ɤɨɧɫɟɪɜɚɬɢɜɧɭɸ ɪɚɡɧɨɫɬɧɭɸ ɫɯɟɦɭ,
ɩɪɢɜɟɫɬɢ ɡɚɞɚɱɭ ɤ ɜɢɞɭ, ɭɞɨɛɧɨɦɭ ɞɥɹ ɩɪɢɦɟɧɟɧɢɹ ɦɟɬɨɞɚ ɩɪɨɝɨɧɤɢ.26
r
R1 : O
26
ɋɚɦɚɪɫɤɢɣ Ⱥ.Ⱥ. ȼɜɟɞɟɧɢɟ ɜ ɱɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ.
Ɇ.: ɇɚɭɤɚ, 1982.
172 ɫ.; Ʉɧɹɡɟɜɚ Ⱥ.Ƚ. Ɋɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ ɦɟɬɨɞɚ ɩɪɨɝɨɧɤɢ – ɦɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ
ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɥɚɛɨɪɚɬɨɪɧɵɯ ɪɚɛɨɬ. Ɍɨɦɫɤ: ɂɡɞɚɬɟɥɶɫɬɜɨ Ɍɉɍ, 2006. 16 ɫ.
242
ɑȺɋɌɖ 10
ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ:
ɮɚɡɨɜɵɟ ɩɪɟɜɪɚɳɟɧɢɹ
10.1. ɍɫɥɨɜɢɹ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
ɉɪɨɰɟɫɫɵ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɨɣ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɢ ɢɯ ɩɨɥɭɱɟɧɢɹ ɧɟɢɡɛɟɠɧɨ ɫɨɩɪɨɜɨɠɞɚɸɬɫɹ ɮɚɡɨɜɵɦɢ ɩɪɟɜɪɚɳɟɧɢɹɦɢ, ɜ ɬɨɦ
ɱɢɫɥɟ, ɩɥɚɜɥɟɧɢɟɦ ɢ ɤɪɢɫɬɚɥɥɢɡɚɰɢɟɣ, ɫɭɛɥɢɦɚɰɢɟɣ (ɢɫɩɚɪɟɧɢɟɦ) ɢ
ɤɨɧɞɟɧɫɚɰɢɟɣ. ɂ, ɟɫɬɟɫɬɜɟɧɧɨ, ɜɨɡɧɢɤɚɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɪɚɫɱɟɬɚ ɬɟɦɩɟɪɚɬɭɪɧɵɯ ɩɨɥɟɣ ɫ ɭɱɟɬɨɦ ɩɨɝɥɨɳɟɧɢɹ ɢɥɢ ɜɵɞɟɥɟɧɢɹ ɬɟɩɥɚ, ɫɜɹɡɚɧɧɨɝɨ
ɫ ɮɚɡɨɜɵɦɢ ɩɪɟɜɪɚɳɟɧɢɹɦɢ. Ɋɚɡɧɵɟ ɮɚɡɵ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɩɪɢ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɭɫɥɨɜɢɹɯ, ɤɨɬɨɪɵɟ ɭɫɬɚɧɚɜɥɢɜɚɸɬɫɹ ɧɚ ɨɫɧɨɜɟ
ɬɟɪɦɨɞɢɧɚɦɢɤɢ. ɑɬɨɛɵ ɡɚɩɢɫɚɬɶ ɭɫɥɨɜɢɹ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ, ɧɚɦ ɩɨɬɪɟɛɭɟɬɫɹ ɟɳɟ ɨɞɢɧ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɣ ɩɨɬɟɧɰɢɚɥ (ɫɦ. ɱɚɫɬɶ 1) – ɷɧɟɪɝɢɹ Ƚɢɛɛɫɚ, ɤɨɬɨɪɚɹ ɫɜɹɡɚɧɚ ɫ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɟɣ ɮɨɪɦɭɥɨɣ
J
g u p Ts .
(10.1)
J0
Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɜ (10.1) g ɢ u ɡɚɩɢɫɚɧɵ ɞɥɹ ɟɞɢɧɢɰɵ ɨɛɴɟɦɚ,
ɬ.ɟ. ɢɦɟɸɬ ɪɚɡɦɟɪɧɨɫɬɶ Ⱦɠ/ɦ3.
Ɍɚɤ ɤɚɤ
dg du pJ 01dJ JJ 01dp Tds sdT ,
ɬɨ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ (1.16) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɜɢɞɟ
J
dg sdT dp .
(10.2)
J0
ɗɬɢ ɠɟ ɪɚɜɟɧɫɬɜɚ, ɨɱɟɜɢɞɧɨ, ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɞɥɹ ɟɞɢɧɢɰɵ ɦɚɫɫɵ,
ɬ.ɟ. ɤɨɝɞɚ g ɢ u ɢɡɦɟɪɹɸɬɫɹ ɜ Ⱦɠ/ɤɝ:
g
dg
u pJ Ts ,
sdT U 1dp
ɢɥɢ ɜ ɪɚɫɱɟɬɟ ɧɚ ɨɞɢɧ ɦɨɥɶ
g
u pmU 1 Ts { u pv Ts ;
dg
ɝɞɟ v
sdT vdp ,
m U – ɦɨɥɶɧɵɣ ɨɛɴɟɦ, ɦ3/ɦɨɥɶ.
243
Ɉɛɨɡɧɚɱɟɧɢɹ ɞɥɹ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ, ɷɧɬɪɨɩɢɢ ɢ ɞɪ. ɜɟɥɢɱɢɧ ɦɵ ɩɪɢ
ɷɬɨɦ ɧɟ ɢɡɦɟɧɹɸɬɫɹ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɜ ɷɬɢɯ ɮɨɪɦɭɥɚɯ ɷɧɬɪɨɩɢɹ ɢɦɟɟɬ
ɪɚɡɦɟɪɧɨɫɬɶ Ⱦɠ/(ɦ3·Ʉ), Ⱦɠ/(ɤɝ·Ʉ), Ⱦɠ/(ɦɨɥɶ·Ʉ).
ȿɫɥɢ ɫɢɫɬɟɦɚ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɮɚɡ, ɬɨ ɨɫɧɨɜɧɨɟ ɭɪɚɜɧɟɧɢɟ ɬɟɪɦɨɞɢɧɚɦɢɤɢ (1.16) ɞɥɹ ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɩɪɢɦɟɬ ɜɢɞ
2
du Tds pdv ¦ g k dCk ,
(10.3)
k 1
ɝɞɟ g k – ɯɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ ɮɚɡ ɢɥɢ ɢɯ ɩɚɪɰɢɚɥɶɧɵɟ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ, Ck , k 1, 2 – ɦɚɫɫɨɜɵɟ ɞɨɥɢ ɢɥɢ ɦɚɫɫɨɜɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɮɚɡ, ɬɚɤ
ɱɬɨ
g g1C1 g 2C2 .
(10.4)
Ⱥɧɚɥɨɝɢɱɧɨ, ɜɦɟɫɬɨ (10.2) ɡɚɩɢɲɟɦ
2
dg
sdT vdp ¦ g k dCk .
k 1
ɏɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ ɢɦɟɸɬ ɬɭ ɠɟ ɪɚɡɦɟɪɧɨɫɬɶ, ɱɬɨ ɢ ɢɧɵɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ.
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɮɚɡɚɦɢ ɧɚɡɵɜɚɸɬ ɝɨɦɨɝɟɧɧɵɟ ɨɛɥɚɫɬɢ ɜ ɝɟɬɟɪɨɝɟɧɧɨɣ ɫɢɫɬɟɦɟ, ɤɨɬɨɪɵɟ ɨɬɞɟɥɟɧɵ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɩɨɜɟɪɯɧɨɫɬɶɸ ɪɚɡɞɟɥɚ. ɉɪɢ ɩɟɪɟɯɨɞɟ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ ɫɤɚɱɤɨɨɛɪɚɡɧɨ ɦɟɧɹɸɬɫɹ
ɯɢɦɢɱɟɫɤɢɣ ɫɨɫɬɚɜ (ɜ ɫɥɭɱɚɟ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ) ɢɥɢ ɮɢɡɢɱɟɫɤɢɟ
ɫɜɨɣɫɬɜɚ ɜɟɳɟɫɬɜɚ (ɜ ɫɥɭɱɚɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ)27. ȼɚɠɧɟɣɲɢɦ ɜɨɩɪɨɫɨɦ ɜ ɭɱɟɧɢɢ ɨ ɮɚɡɚɯ ɹɜɥɹɟɬɫɹ ɜɵɹɫɧɟɧɢɟ ɭɫɥɨɜɢɣ, ɩɪɢ ɤɨɬɨɪɵɯ ɫɢɫɬɟɦɚ, ɫɨɫɬɨɹɳɚɹ ɢɡ ɞɜɭɯ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɮɚɡ, ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ.
ȿɫɥɢ ɩɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ ɹɜɥɹɟɬɫɹ ɩɥɨɫɤɨɣ, ɬɨ ɟɟ ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɢ
ɢɧɵɟ ɫɜɨɣɫɬɜɚ ɧɟ ɫɤɚɡɵɜɚɸɬɫɹ ɧɚ ɭɫɥɨɜɢɹɯ ɪɚɜɧɨɜɟɫɢɹ. ɉɨɷɬɨɦɭ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ ɫɜɨɣɫɬɜɚ ɷɬɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɟɦ.
ȿɫɥɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɚɹ ɫɢɫɬɟɦɚ ɢɡɨɥɢɪɨɜɚɧɧɚɹ (ɬ.ɟ. ɫɢɫɬɟɦɚ ɧɟ
ɨɛɦɟɧɢɜɚɟɬɫɹ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɧɢ ɦɚɫɫɨɣ
C C1 C2 const ,
ɧɢ ɷɧɟɪɝɢɟɣ
u u1 u 2 const ),
ɢ ɧɟ ɫɨɜɟɪɲɚɟɬ ɧɢɤɚɤɨɣ ɪɚɛɨɬɵ, ɬ.ɟ.
v v1 v2 const ,
ɬɨ ɭɫɥɨɜɢɟɦ ɟɟ ɪɚɜɧɨɜɟɫɢɹ ɛɭɞɟɬ ɧɟɢɡɦɟɧɧɨɫɬɶ ɷɧɬɪɨɩɢɢ,
ds 0 .
(10.5)
ȼɨɨɛɳɟ ɩɨɧɹɬɢɟ «ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ» ɞɨɫɬɚɬɨɱɧɨ ɭɫɥɨɜɧɨ ɢ ɫɥɟɞɭɟɬ ɝɨɜɨɪɢɬɶ ɧɟ
ɨ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ, ɚ ɨ ɧɟɤɨɬɨɪɨɦ ɬɨɧɤɨɦ ɫɥɨɟ, ɜ ɤɨɬɨɪɨɦ ɫɜɨɣɫɬɜɚ ɜɟɳɟɫɬɜɚ
ɛɵɫɬɪɨ ɦɟɧɹɸɬɫɹ ɩɪɢ ɩɟɪɟɯɨɞɟ ɨɬ ɨɞɧɨɣ ɮɚɡɵ ɤ ɞɪɭɝɨɣ.
27
244
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɢɡ (10.3) ɫɥɟɞɭɟɬ
dCk
du
dv 2
dC
ds
p
¦ gk
{ g1 g 2 1 .
T
T k 1
T
T
(10.6)
Ɍɚɤ ɤɚɤ
dC 0 ɢ d C1 d C2 ,
du 0 ɢ d u1 d u2 ,
(10.7)
dv 0 ɢ d v1 d v2 ,
ɬɨ ɪɚɜɟɧɫɬɜɨ ɞɢɮɮɟɪɟɧɰɢɚɥɚ ɷɧɬɪɨɩɢɢ ɪɚɜɧɨ ɧɭɥɸ, ɟɫɥɢ
g1 g 2 ,
(10.8)
ɬ.ɟ. ɯɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ ɮɚɡ ɪɚɜɧɵ ɞɪɭɝ ɞɪɭɝɭ.
Ɋɚɜɟɧɫɬɜɚ (10.7) ɨɡɧɚɱɚɸɬ, ɱɬɨ ɜ ɡɚɦɤɧɭɬɨɣ ɫɢɫɬɟɦɟ ɢɡɦɟɧɟɧɢɟ
ɜɧɭɬɪɟɧɧɟɣ ɷɧɟɪɝɢɢ ɢ ɨɛɴɟɦɚ ɮɚɡɵ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ
ɞɪɭɝɨɣ ɮɚɡɵ, ɬɚɤ ɤɚɤ ɜ ɫɭɦɦɟ ɢɡɦɟɧɟɧɢɟ ɷɬɢɯ ɜɟɥɢɱɢɧ ɪɚɜɧɨ ɧɭɥɸ. ɉɟɪɜɨɟ ɢɡ ɭɫɥɨɜɢɣ (10.7) ɮɚɤɬɢɱɟɫɤɢ ɟɫɬɶ ɡɚɤɨɧ ɫɨɯɪɚɧɟɧɢɹ ɦɚɫɫɵ.
ɑɬɨɛɵ ɩɨɧɹɬɶ, ɩɪɢ ɤɚɤɢɯ ɭɫɥɨɜɢɹɯ ɷɬɨ ɜɨɡɦɨɠɧɨ, ɜɨɫɩɨɥɶɡɭɟɦɫɹ
ɫɜɨɣɫɬɜɨɦ ɚɞɞɢɬɢɜɧɨɫɬɢ ɷɧɬɪɨɩɢɢ, ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ
s
s1 s2 ,
ɝɞɟ s1 ɢ s2 – ɷɧɬɪɨɩɢɢ ɩɟɪɜɨɣ ɢ ɜɬɨɪɨɣ ɮɚɡ.
Ɍɨɝɞɚ
d s1 d s2 .
(10.9)
Ɍɚɤ ɤɚɤ ɭɪɚɜɧɟɧɢɟ Ƚɢɛɛɫɚ ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɤɚɠɞɨɣ ɩɨɞɫɢɫɬɟɦɵ ɜ
ɨɬɞɟɥɶɧɨɫɬɢ, ɬ.ɟ.
d uk
Tk dsk pk d vk g k d Ck , k 1, 2 ,
(10.10)
ɬɨ ɜɵɪɚɠɚɹ d s1 , d s2 ɢɡ (10.10) ɢ ɩɨɞɫɬɚɜɥɹɹ ɢɯ ɜ (10.9), ɧɚɣɞɟɦ
1
1
du1 p1dv1 g1dC1 @ > du2 p2dv2 g 2dC2 @ 0 .
>
T1
T2
ɍɱɢɬɵɜɚɹ ɪɚɜɟɧɫɬɜɚ (10.7), ɡɚɩɢɲɟɦ
§1 1 ·
§ p1 p2 ·
§ g1 g 2 ·
¨ ¸ d u1 ¨ ¸ d v1 ¨ ¸ dC1 0 .
T
T
T
T
T
T
2¹
2 ¹
2 ¹
© 1
© 1
© 1
(10.11)
ȼɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ u , ɨɛɴɟɦ v ɢ ɤɨɧɰɟɧɬɪɚɰɢɹ C1 ɦɨɝɭɬ ɦɟɧɹɬɶɫɹ
ɧɟɡɚɜɢɫɢɦɨ ɞɪɭɝ ɨɬ ɞɪɭɝɚ, ɬ.ɟ. ɞɢɮɮɟɪɟɧɰɢɚɥɵ du1 , dv1 ɢ d C1 ɧɟɡɚɜɢɫɢɦɵ. Ɍɨɝɞɚ, ɨɱɟɜɢɞɧɨ, ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɪɚɜɟɧɫɬɜɚ (10.11) ɧɟɨɛɯɨɞɢɦɨ,
ɱɬɨɛɵ ɛɵɥɢ ɪɚɜɧɵ ɧɭɥɸ ɦɧɨɠɢɬɟɥɢ ɩɪɢ ɷɬɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɚɯ, ɬ.ɟ.
245
T1 T2 , p1
p2 , g1
g2 .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɫɥɨɜɢɟ ɨɛɳɟɝɨ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
ɞɜɭɯɮɚɡɧɨɣ ɨɞɧɨɤɨɦɩɨɧɟɧɬɧɨɣ ɫɢɫɬɟɦɵ ɦɨɠɟɬ ɛɵɬɶ ɡɚɩɢɫɚɧɨ ɜ ɜɢɞɟ
g1 T , p g 2 T , p (10.12)
ɢ ɨɡɧɚɱɚɟɬ ɪɚɜɟɧɫɬɜɨ ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧɰɢɚɥɨɜ ɮɚɡ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɞɚɜɥɟɧɢɢ ɢ ɬɟɦɩɟɪɚɬɭɪɟ ɮɚɡ.
ɋ ɭɱɟɬɨɦ ɫɜɨɣɫɬɜ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɮɚɡ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɩɨɥɭɱɚɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ (ɜɵɜɨɞ ɷɬɨɝɨ ɭɫɥɨɜɢɹ ɦɨɠɧɨ ɧɚɣɬɢ ɩɪɚɤɬɢɱɟɫɤɢ ɜ
ɥɸɛɨɦ ɭɱɟɛɧɢɤɟ ɩɨ ɬɟɪɦɨɞɢɧɚɦɢɤɟ) ɢ ɢɦɟɟɬ ɜɢɞ
§p
g2 , ¨ 1 © T1
T1 T f , T2 T f , g1
Vf
p2 ·
dv
d6 f
¸
T2 ¹ 1 T f
0.
ɗɬɨ ɭɫɥɨɜɢɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɜ ɪɚɜɧɨɜɟɫɢɢ ɪɚɜɧɵ ɯɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ ɮɚɡ, ɪɚɜɧɵ ɢɯ ɬɟɦɩɟɪɚɬɭɪɵ, ɚ ɜɨɬ ɞɚɜɥɟɧɢɹ ɜ ɮɚɡɚɯ ɫɜɹɡɚɧɵ ɪɚɜɟɧɫɬɜɨɦ
d6 f
p1 p2 V f
,
(10.13)
dv f
ɝɞɟ 6 f – ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɮɚɡ, V f – ɩɨɜɟɪɯɧɨɫɬɧɨɟ ɧɚɬɹɠɟɧɢɟ.
ɉɪɨɢɡɜɨɞɧɚɹ ɨɬ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɮɚɡ ɩɨ ɨɛɴɟɦɭ
d 6 f dv1 d 6 f dv2
ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
d6 f
dv
1
1
,
R1 R2
ɝɞɟ R1 ɢ R2 – ɝɥɚɜɧɵɟ ɪɚɞɢɭɫɵ ɤɪɢɜɢɡɧɵ ɩɨɜɟɪɯɧɨɫɬɢ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
p1
§ 1
1 ·
p2 V f ¨ ¸.
R
R
© 1
2¹
(10.14)
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ ɫɮɟɪɢɱɟɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɢɦɟɟɦ
R1
R2
R ɢ p1
p2 2V f
.
R
ȼ ɫɥɭɱɚɟ ɩɥɨɫɤɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ – R o f ɢ p1
246
p2 .
10.2. ɍɪɚɜɧɟɧɢɟ Ʉɪɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ, ɫɨɩɪɨɜɨɠɞɚɸɳɢɯɫɹ ɩɨɝɥɨɳɟɧɢɟɦ
ɢɥɢ
ɜɵɞɟɥɟɧɢɟɦ
ɬɟɩɥɚ,
ɫɩɪɚɜɟɞɥɢɜɨ
ɭɪɚɜɧɟɧɢɟ
Ʉɥɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ
dT ph 'J ph ' J ph ˜ T ph
,
(10.15)
' s ph
dp
Q ph
ɝɞɟ ' J ph J 2 J 1 ɢ ' s ph s 2 s1 – ɪɚɡɧɨɫɬɢ ɩɚɪɰɢɚɥɶɧɵɯ ɦɨɥɶɧɵɯ
ɨɛɴɟɦɨɜ ɢ ɦɨɥɶɧɵɯ ɷɧɬɪɨɩɢɣ ɮɚɡ, ɚ
Q p h h2 h1 T p h s 2 s1 –
ɟɫɬɶ ɪɚɡɧɨɫɬɶ ɷɧɬɚɥɶɩɢɣ ɮɚɡ ɜ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ ɢɥɢ ɬɟɩɥɨɬɚ ɩɟɪɟɯɨɞɚ. ȿɫɥɢ Q ph 0 , ɚ ɨɛɴɟɦ ɮɚɡɵ 2 ɛɨɥɶɲɟ, ɱɟɦ ɨɛɴɟɦ ɮɚɡɵ 1, ɬɨ dT ph dp 0 ,
ɬ.ɟ. ɬɟɦɩɟɪɚɬɭɪɚ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɭɦɟɧɶɲɚɟɬɫɹ ɫ ɞɚɜɥɟɧɢɟɦ. ɍɪɚɜɧɟɧɢɟ (10.15) ɥɟɝɤɨ ɩɨɥɭɱɚɟɬɫɹ ɢɡ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɮɚɡ (10.12), ɡɚɩɢɫɚɧɧɨɝɨ ɞɥɹ ɦɨɥɶɧɵɯ ɜɟɥɢɱɢɧ, ɫ ɩɨɦɨɳɶɸ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ
ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧɰɢɚɥɨɜ ɮɚɡ ɩɨ ɦɚɥɵɦ ɨɬɤɥɨɧɟɧɢɹɦ T ɢ p ɨɬ ɢɯ ɪɚɜɧɨɜɟɫɧɵɯ ɡɧɚɱɟɧɢɣ.
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ ɩɪɨɢɡɨɲɥɨ ɨɬɤɥɨɧɟɧɢɟ ɨɬ ɫɨɫɬɨɹɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɞɚɜɥɟɧɢɹ ɧɚ ɦɚɥɵɟ ɜɟɥɢɱɢɧɵ ' T ɢ ' p . ɉɨɥɚɝɚɹ,
ɱɬɨ ɫɢɫɬɟɦɚ ɩɟɪɟɯɨɞɢɬ ɜ ɧɨɜɨɟ ɫɨɫɬɨɹɧɢɟ ɪɚɜɧɨɜɟɫɢɹ, ɡɚɩɢɲɟɦ
g 1 T ' T , p ' p g 2 T ' T , p ' p ɢɥɢ
§ wg ·
§ wg ·
§ wg ·
§ wg ·
g1 T , p ¨ 1 ¸ ' T ¨ 1 ¸ ' p g 2 T , p ¨ 2 ¸ ' T ¨ 2 ¸ ' p .
© wT ¹ p
© wT ¹ p
© w p ¹T
© w p ¹T
ɑɚɫɬɧɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧɰɢɚɥɨɜ ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɢ
ɞɚɜɥɟɧɢɸ ɟɫɬɶ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɦɨɥɶɧɚɹ ɷɧɬɪɨɩɢɹ ɢ ɦɨɥɶɧɵɣ ɨɛɴɟɦ
§ wg k ·
§ wgk ·
,
s
(10.16)
¸ vk ,
k ¨
¨ wT ¸
©
¹p
© wp ¹
ɱɬɨ ɫɥɟɞɭɟɬ ɢɡ ɭɪɚɜɧɟɧɢɹ Ƚɢɛɛɫɚ ɞɥɹ ɮɚɡɵ k , ɡɚɩɢɫɚɧɧɨɝɨ ɜ ɪɚɫɱɟɬɟ ɧɚ
ɨɞɢɧ ɦɨɥɶ
du k Tk dsk pk dvk g k dCk , k 1,2.
Ɍɚɤ ɤɚɤ ɫɩɪɚɜɟɞɥɢɜɨ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ (10.12), ɬɨ ɧɚɣɞɟɦ
s2 s1 'T v2 v1 'p ,
ɨɬɤɭɞɚ ɜ ɩɪɟɞɟɥɟ ɩɨɥɭɱɚɟɦ ɭɪɚɜɧɟɧɢɟ (10.15).
ɍɪɚɜɧɟɧɢɟ Ʉɥɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ, ɤɨɬɨɪɨɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ ɢ
ɞɪɭɝɢɦ ɫɩɨɫɨɛɨɦ, ɨɩɢɫɵɜɚɟɬ ɪɚɡɥɢɱɧɵɟ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ – ɩɥɚɜɥɟɧɢɟ,
247
ɩɚɪɨɨɛɪɚɡɨɜɚɧɢɟ, ɫɭɛɥɢɦɚɰɢɸ. ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɭɪɚɜɧɟɧɢɟ ɦɨɠɟɬ
ɛɵɬɶ ɥɟɝɤɨ ɩɪɨɢɧɬɟɝɪɢɪɨɜɚɧɨ.
Ɍɚɤ, ɩɪɢ ɫɭɛɥɢɦɚɰɢɢ ɜ ɨɛɥɚɫɬɢ ɧɢɡɤɢɯ ɞɚɜɥɟɧɢɣ ɩɚɪɚ, ɟɫɬɟɫɬɜɟɧɧɨ
ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ ɩɚɪɚ v2 v p ɦɧɨɝɨ ɛɨɥɶɲɟ ɨɛɴɟɦɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ
v1
vs
v p !! v s .
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɪɚɜɧɟɧɢɟɦ (10.15), ɢɦɟɟɦ
Q ph
dp
!0,
dT T ph v p vs
ɟɫɥɢ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɩɟɪɟɯɨɞ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɩɚɪ.
ɉɪɟɧɟɛɪɟɝɚɹ ɜɟɥɢɱɢɧɨɣ vs ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ v p ɢ ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɟ Ʉɥɚɩɟɣɪɨɧɚ (ɭɪɚɜɧɟɧɢɟ ɫɨɫɬɨɹɧɢɹ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ)
RT
,
vp
p
ɤɨɬɨɪɨɟ ɫ ɜɵɫɨɤɨɣ ɬɨɱɧɨɫɬɶɸ ɨɩɢɫɵɜɚɟɬ ɦɨɥɶɧɵɣ ɨɛɴɟɦ ɩɚɪɚ ɧɚ ɥɢɧɢɢ
ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ, ɩɪɢɜɟɞɟɦ ɧɚɲɟ ɭɪɚɜɧɟɧɢɟ ɤ ɜɢɞɭ
d ln p Q ph
.
dT
RT 2
ȿɫɥɢ ɢɡɜɟɫɬɧɚ ɬɟɦɩɟɪɚɬɭɪɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ ɬɟɩɥɨɬɵ ɫɭɛɥɢɦɚɰɢɢ
Q p h T , ɬɨ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɥɟɝɤɨ ɢɧɬɟɝɪɢɪɭɟɬɫɹ. ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ
Q ph
c on s t ɢɦɟɟɦ
ln p
Q ph 1
lnC .
R T
10.3. ɋɥɟɞɫɬɜɢɹ ɭɫɥɨɜɢɹ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɪɚɡɥɢɱɧɵɟ ɫɥɟɞɫɬɜɢɹ ɭɫɥɨɜɢɹ ɪɚɜɧɨɜɟɫɢɹ ɮɚɡ, ɡɚɤɥɸɱɚɸɳɟɝɨɫɹ ɜ ɪɚɜɟɧɫɬɜɟ ɢɯ ɩɨɬɟɧɰɢɚɥɨɜ Ƚɢɛɛɫɚ (ɢɥɢ, ɱɬɨ ɬɨ ɠɟ ɫɚɦɨɟ, ɢɯ
ɯɢɦɢɱɟɫɤɢɯ ɩɨɬɟɧɰɢɚɥɨɜ) (10.12) ɩɪɢ ɧɟɢɡɦɟɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɞɚɜɥɟɧɢɢ.
ɋɨɫɬɨɹɧɢɟ ɜɟɳɟɫɬɜɚ ɛɭɞɟɦ ɢɡɨɛɪɚɠɚɬɶ ɬɨɱɤɨɣ ɧɚ ɩɥɨɫɤɨɫɬɢ T , p (ɪɢɫ. 10.1). Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɪɚɫɫɦɨɬɪɢɦ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɬɢɩɚ «ɢɫɩɚɪɟɧɢɟ – ɤɨɧɞɟɧɫɚɰɢɹ». Ʉɚɠɞɚɹ ɬɨɱɤɚ ɷɬɨɣ ɩɥɨɫɤɨɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɨɞɧɨɪɨɞɧɨɦɭ (ɨɞɧɨɮɚɡɧɨɦɭ) ɫɨɫɬɨɹɧɢɸ ɜɟɳɟɫɬɜɚ – ɥɢɛɨ ɠɢɞɤɨɫɬɢ, ɥɢɛɨ ɟɟ ɩɚɪɭ. ɂɫɤɥɸɱɟɧɢɟ ɫɨɫɬɚɜɥɹɸɬ ɬɨɱɤɢ ɥɢɧɢɢ D K . ɗɬɚ ɥɢɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɪɚɜɧɟɧɢɸ (10.12). ɇɚ ɥɢɧɢɢ D K ɯɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ
248
ɠɢɞɤɨɫɬɢ ɢ ɩɚɪɚ ɨɞɢɧɚɤɨɜɵ; ɡɞɟɫɶ ɷɬɢ ɮɚɡɵ ɧɚɯɨɞɹɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ
ɞɪɭɝ ɫ ɞɪɭɝɨɦ. Ʉɚɠɞɚɹ ɬɨɱɤɚ ɥɢɧɢɢ D K ɢɡɨɛɪɚɠɚɟɬ ɥɢɛɨ ɠɢɞɤɨɫɬɶ, ɥɢɛɨ ɩɚɪ, ɥɢɛɨ ɢɯ ɫɦɟɫɶ ɜ ɥɸɛɵɯ ɩɪɨɩɨɪɰɢɹɯ. Ɋɚɡɪɟɲɚɹ ɭɪɚɜɧɟɧɢɟ (10.12)
ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɚɜɥɟɧɢɹ p , ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɭɪɚɜɧɟɧɢɟ ɤɪɢɜɨɣ ɜ ɜɢɞɟ
p p T .
(10.17)
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɞɚɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɞɚɜɥɟɧɢɹ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ, ɚ ɤɪɢɜɚɹ D K ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɜɨɣ ɪɚɜɧɨɜɟɫɢɹ ɠɢɞɤɨɫɬɢ ɢ
ɟɟ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɢɥɢ ɤɪɢɜɨɣ ɢɫɩɚɪɟɧɢɹ.
Ɍɨɱɤɚ Ʉ ɧɚ ɷɬɨɦ ɪɢɫɭɧɤɟ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɤɪɢɬɢɱɟɫɤɨɦɭ ɫɨɫɬɨɹɧɢɸ
ɜɟɳɟɫɬɜɚ ɢ ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɨɣ. ɉɪɢ T ! Tk , p ! p k ɧɟ
ɫɭɳɟɫɬɜɭɟɬ ɪɚɡɥɢɱɧɵɯ ɮɚɡ, ɢ ɬɟɥɨ ɜɫɟɝɞɚ ɨɞɧɨɪɨɞɧɨ.
ɗɬɭ ɬɨɱɤɭ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ ɬɨɱɤɭ ɩɟɪɟɝɢɛɚ ɢɡɨɬɟɪɦɵ ɪɟɚɥɶɧɨɝɨ ɝɚɡɚ (ɪɢɫ. 10.2), ɜ ɤɨɬɨɪɨɣ ɤɚɫɚɬɟɥɶɧɚɹ ɤ ɢɡɨɬɟɪɦɟ ɝɨɪɢɡɨɧɬɚɥɶɧɚ.
ȿɟ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɬɚɤɠɟ ɤɚɤ ɬɨɱɤɭ, ɜ ɤɨɬɨɪɭɸ ɜ ɩɪɟɞɟɥɟ ɩɟɪɟɯɨɞɹɬ
ɝɨɪɢɡɨɧɬɚɥɶɧɵɟ ɭɱɚɫɬɤɢ ɢɡɨɬɟɪɦ ɩɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɞɨ ɤɪɢɬɢɱɟɫɤɨɣ (ɫɦ. ɷɬɨɬ ɠɟ ɪɢɫɭɧɨɤ).
Ɋɢɫ. 10.2. Ⱦɢɚɝɪɚɦɦɚ «P – J » ɜ
ɨɤɪɟɫɬɧɨɫɬɢ ɥɢɧɢɢ ɪɚɜɧɨɜɟɫɢɹ
ɠɢɞɤɨɫɬɢ ɢ ɩɚɪɚ
Ɋɢɫ. 10.1. Ⱦɢɚɝɪɚɦɦɚ «P – T» ɜ
ɨɤɪɟɫɬɧɨɫɬɢ ɥɢɧɢɢ ɪɚɜɧɨɜɟɫɢɹ
ɠɢɞɤɨɫɬɢ ɢ ɩɚɪɚ
ɉɟɪɟɫɟɱɟɦ ɤɪɢɜɭɸ ɢɫɩɚɪɟɧɢɹ ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ ɩɪɹɦɨɣ, ɬ.ɟ. ɢɡɨɛɚɪɨɣ (ɥɢɧɢɹ Ⱥȼ ɧɚ ɪɢɫ. 10.1). ȼ ɬɨɱɤɟ Ⱥ ɜɟɳɟɫɬɜɨ ɧɚɯɨɞɢɬɫɹ ɜ ɠɢɞɤɨɦ
ɫɨɫɬɨɹɧɢɢ, ɡɞɟɫɶ ɞɚɜɥɟɧɢɟ p A ɜɵɲɟ ɞɚɜɥɟɧɢɹ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɩɪɢ
ɞɚɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ. ɉɪɢ ɧɚɝɪɟɜɚɧɢɢ ɠɢɞɤɨɫɬɢ ɜ ɭɫɥɨɜɢɹɯ ɩɨɫɬɨɹɧɧɨɝɨ
ɞɚɜɥɟɧɢɹ ɬɨɱɤɚ, ɢɡɨɛɪɚɠɚɸɳɚɹ ɫɨɫɬɨɹɧɢɟ ɜɟɳɟɫɬɜɚ, ɩɟɪɟɦɟɫɬɢɬɫɹ ɜɩɪɚɜɨ, ɢ ɜ ɬɨɱɤɟ ɋ ɧɚɱɧɟɬɫɹ ɢɫɩɚɪɟɧɢɟ ɠɢɞɤɨɫɬɢ. ȼɨ ɜɫɟ ɜɪɟɦɹ ɢɫɩɚɪɟɧɢɹ
ɬɟɦɩɟɪɚɬɭɪɵ ɠɢɞɤɨɫɬɢ ɢ ɟɟ ɧɚɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɛɭɞɭɬ ɧɟɢɡɦɟɧɧɵ. ɉɨɫɥɟ
249
ɢɫɩɚɪɟɧɢɹ ɜɫɟɣ ɠɢɞɤɨɫɬɢ ɢɡɨɛɪɚɠɚɸɳɚɹ ɬɨɱɤɚ ɜɧɨɜɶ ɩɟɪɟɦɟɫɬɢɬɫɹ
ɜɩɪɚɜɨ, ɬɚɤ ɱɬɨ ɭɱɚɫɬɨɤ ɋȼ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɝɪɟɜɚɧɢɸ ɩɚɪɚ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɞɚɜɥɟɧɢɟ ɧɚ ɢɡɨɛɚɪɟ ɜɵɲɟ ɤɪɢɬɢɱɟɫɤɨɝɨ (ɥɢɧɢɹ
c
c
A B ɧɚ ɷɬɨɦ ɠɟ ɪɢɫɭɧɤɟ). Ɍɨɝɞɚ ɩɪɢ ɢɡɨɛɚɪɢɱɟɫɤɨɦ ɧɚɝɪɟɜɚɧɢɢ ɢɥɢ ɨɯɥɚɠɞɟɧɢɢ ɧɢɤɚɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ ɠɢɞɤɨɫɬɢ ɜ ɩɚɪ ɢ ɨɛɪɚɬɧɨ ɧɟ ɩɪɨɢɫɯɨɞɢɬ. ɋɥɟɞɫɬɜɢɟɦ ɨɛɪɵɜɚ ɤɪɢɜɨɣ ɢɫɩɚɪɟɧɢɹ ɹɜɥɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨɫɬɶ
ɠɢɞɤɨɝɨ ɢ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɫɨɫɬɨɹɧɢɣ.
ɉɨɧɹɬɢɟ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɧɚ ɤɪɢɜɨɣ ɪɚɜɧɨɜɟɫɢɹ «ɠɢɞɤɨɫɬɶ-ɝɚɡ»
ɛɵɥɨ ɜɜɟɞɟɧɨ Ɇɟɧɞɟɥɟɟɜɵɦ ɜ 1860 ɝɨɞɭ ɢ ɫɜɨɞɢɬɫɹ ɤ ɫɥɟɞɭɸɳɟɦɭ.
ȿɫɥɢ ɫɠɢɦɚɬɶ ɝɚɡ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T Tk (ɪɢɫ. 10.2), ɬɨ, ɞɨɫɬɢɝɧɭɜ
ɨɩɪɟɞɟɥɟɧɧɨɣ ɩɥɨɬɧɨɫɬɢ U1 ɝɚɡ ɧɚɱɧɟɬ ɤɨɧɞɟɧɫɢɪɨɜɚɬɶɫɹ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ ɞɨ ɬɟɯ ɩɨɪ, ɩɨɤɚ ɩɥɨɬɧɨɫɬɶ ɫɢɫɬɟɦɵ ɧɟ ɫɬɚɧɟɬ ɪɚɜɧɨɣ
ɩɥɨɬɧɨɫɬɢ ɠɢɞɤɨɫɬɢ U 2 . ɋ ɪɨɫɬɨɦ ɬɟɦɩɟɪɚɬɭɪɵ ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɩɥɨɬɧɨɫɬɹɦɢ ɠɢɞɤɨɫɬɢ ɢ ɝɚɡɚ ɫɬɚɧɨɜɢɬɫɹ ɜɫɟ ɦɟɧɶɲɟ ɢ ɢɫɱɟɡɚɟɬ ɩɪɢ T Tk .
ȼɦɟɫɬɨ ɩɥɨɬɧɨɫɬɢ ɦɨɠɧɨ ɝɨɜɨɪɢɬɶ ɨɛ ɭɞɟɥɶɧɵɯ ɨɛɴɟɦɚɯ.
ɋɠɢɦɚɹ ɝɚɡ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T ! Tk , ɦɨɠɧɨ ɨɫɭɳɟɫɬɜɢɬɶ ɧɟɩɪɟɪɵɜɧɵɣ ɩɟɪɟɯɨɞ ɢɡ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɜ ɠɢɞɤɨɟ. Ʉɪɢɜɚɹ ɪɚɜɧɨɜɟɫɢɹ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɩɚɪɨɦ ɨɤɚɧɱɢɜɚɟɬɫɹ ɜ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ. ȼɵɲɟ
ɬɟɦɩɟɪɚɬɭɪɵ Tk ɧɚɛɥɸɞɚɸɬɫɹ ɚɧɨɦɚɥɢɢ ɜɫɟɯ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɜɟɳɟɫɬɜɚ. ɇɨ ɩɨɥɨɠɟɧɢɹ ɦɚɤɫɢɦɭɦɨɜ ɢ ɦɢɧɢɦɭɦɨɜ ɪɚɡɥɢɱɧɵɯ ɫɜɨɣɫɬɜ ɜ
ɤɪɢɬɢɱɟɫɤɨɣ ɨɛɥɚɫɬɢ ɧɟ ɫɨɜɩɚɞɚɸɬ.
Ʉɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ ɧɚ ɞɢɚɝɪɚɦɦɟ ɫɨɫɬɨɹɧɢɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɤɪɢɬɢɱɟɫɤɢɦɢ ɡɧɚɱɟɧɢɹɦɢ Tk , p k , J k ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɱɚɫɬɧɵɣ ɫɥɭɱɚɣ
ɬɨɱɤɢ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ. ȼ ɷɬɨɣ ɬɨɱɤɟ ɫɢɫɬɟɦɚ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɩɨɬɟɪɟɣ
ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɩɨ ɩɥɨɬɧɨɫɬɢ ɢɥɢ ɫɨɫɬɚɜɭ ɜɟɳɟɫɬɜɚ.
ɉɨ ɨɞɧɭ ɫɬɨɪɨɧɭ ɨɬ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ( T ! Tk ) ɜɟɳɟɫɬɜɨ ɨɞɧɨɪɨɞɧɨ, ɩɨ ɞɪɭɝɭɸ ( T Tk ) – ɪɚɫɫɥɚɢɜɚɟɬɫɹ ɧɚ ɮɚɡɵ. ɍ ɫɦɟɫɟɣ ɢɥɢ ɪɚɫɬɜɨɪɨɜ
ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɜɟɥɢɱɟɧɢɹ ɱɢɫɥɚ ɩɚɪɚɦɟɬɪɨɜ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɫɢɫɬɟɦɭ,
ɢɦɟɟɬɫɹ ɧɟ ɢɡɨɥɢɪɨɜɚɧɧɚɹ ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ, ɚ ɤɪɢɬɢɱɟɫɤɚɹ ɤɪɢɜɚɹ,
ɬɨɱɤɢ ɤɨɬɨɪɨɣ ɪɚɡɥɢɱɚɸɬɫɹ ɡɧɚɱɟɧɢɹɦɢ Tk , p k , J k ɢ ɤɨɧɰɟɧɬɪɚɰɢɢ.
ȼ ɨɤɪɟɫɬɧɨɫɬɢ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɢ ɧɚɛɥɸɞɚɟɬɫɹ ɰɟɥɵɣ ɪɹɞ ɤɪɢɬɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ. ɉɪɢ ɩɪɢɛɥɢɠɟɧɢɢ ɤ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɟ ɢɫɱɟɡɚɸɬ ɬɟɩɥɨɬɚ
ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɢ ɩɨɜɟɪɯɧɨɫɬɧɨɟ ɧɚɬɹɠɟɧɢɟ; ɫɭɳɟɫɬɜɟɧɧɨ ɜɨɡɪɚɫɬɚɸɬ ɮɥɭɤɬɭɚɰɢɢ ɩɥɨɬɧɨɫɬɢ ɢ ɫɨɫɬɚɜɚ (ɞɥɹ ɫɦɟɫɟɣ ɢ ɪɚɫɬɜɨɪɨɜ); ɫɬɚɧɨɜɢɬɫɹ ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɜɡɚɢɦɧɚɹ ɪɚɫɬɜɨɪɢɦɨɫɬɶ ɤɨɦɩɨɧɟɧɬɨɜ; ɧɚɛɥɸɞɚɟɬɫɹ
ɚɧɨɦɚɥɶɧɵɣ ɪɨɫɬ ɫɠɢɦɚɟɦɨɫɬɢ; ɜɨɡɪɚɫɬɚɟɬ ɦɚɝɧɢɬɧɚɹ ɜɨɫɩɪɢɢɦɱɢɜɨɫɬɶ
ɢ ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɶ ɢ ɬ.ɞ.
Ⱦɨɩɭɫɬɢɦ ɬɟɩɟɪɶ, ɱɬɨ ɱɢɫɥɨ ɮɚɡ ɯɢɦɢɱɟɫɤɢ ɨɞɧɨɪɨɞɧɨɝɨ ɜɟɳɟɫɬɜɚ,
ɧɚɯɨɞɹɳɢɯɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɞɪɭɝ ɫ ɞɪɭɝɨɦ, ɪɚɜɧɨ ɬɪɟɦ. ɉɪɨɫɬɵɦ ɢ ɜɫɟɦ
250
ɩɨɧɹɬɧɵɦ ɩɪɢɦɟɪɨɦ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɫɢɫɬɟɦɚ, ɫɨɫɬɨɹɳɚɹ ɢɯ ɬɜɟɪɞɨɣ ɮɚɡɵ, ɠɢɞɤɨɫɬɢ ɢ ɟɟ ɩɚɪɚ. Ⱦɥɹ ɪɚɜɧɨɜɟɫɢɹ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɟɧɢɟ ɬɪɟɯ
ɭɫɥɨɜɢɣ
g f p ,T g g p ,T ;
g s p ,T g s p ,T g f p ,T ;
(10.18)
g g p ,T .
ɉɟɪɜɨɟ ɟɫɬɶ ɭɫɥɨɜɢɟ ɪɚɜɧɨɜɟɫɢɹ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɟɟ ɩɚɪɨɦ; ɜɬɨɪɨɟ – ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɬɜɟɪɞɨɣ ɮɚɡɨɣ; ɬɪɟɬɶɟ – ɦɟɠɞɭ ɬɜɟɪɞɨɣ ɮɚɡɨɣ
ɢ ɩɚɪɨɦ. ɗɬɢ ɬɪɢ ɭɫɥɨɜɢɹ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ. Ʉɚɠɞɨɟ ɢɡ ɧɢɯ
ɟɫɬɶ ɫɥɟɞɫɬɜɢɟ ɨɫɬɚɥɶɧɵɯ.
ɉɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɢɡɨɛɪɚɠɚɟɬ ɤɪɢɜɭɸ ɢɫɩɚɪɟɧɢɹ f l g (ɪɢɫ. 10.3);
ɜɬɨɪɨɟ – ɤɪɢɜɭɸ ɩɥɚɜɥɟɧɢɹ s l f ; ɬɪɟɬɶɟ – ɤɪɢɜɭɸ ɫɭɛɥɢɦɚɰɢɢ ɢɥɢ ɜɨɡɝɨɧɤɢ s l g . Ʉɪɢɜɚɹ ɩɥɚɜɥɟɧɢɹ 2 ɩɟɪɟɫɟɤɚɟɬɫɹ ɫ ɤɪɢɜɨɣ ɢɫɩɚɪɟɧɢɹ 1 ɜ
ɬɨɤɟ Ⱥ, ɱɪɟɡ ɤɨɬɨɪɭɸ ɞɨɥɠɧɚ ɩɪɨɯɨɞɢɬɶ ɢ ɤɪɢɜɚɹ ɜɨɡɝɨɧɤɢ 3.
Ɍɪɢ ɮɚɡɵ ɦɨɝɭɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɦɟɠɞɭ ɫɨɛɨɣ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɥɢɲɶ ɜ ɨɞɧɨɣ ɬɨɱɤɟ, ɤɨɬɨɪɭɸ ɧɚɡɵɜɚɸɬ ɬɪɨɣɧɨɣ ɬɨɱɤɨɣ, ɬ.ɟ. ɩɪɢ
ɜɩɨɥɧɟ ɨɩɪɟɞɟɥɟɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɞɚɜɥɟɧɢɹ ɢ ɬɟɦɩɟɪɚɬɭɪɵ.
ȼ ɬɪɨɣɧɨɣ ɬɨɱɤɟ ɤɪɢɜɚɹ ɜɨɡɝɨɧɤɢ ɜɫɟɝɞɚ ɤɪɭɱɟ ɤɪɢɜɨɣ ɢɫɩɚɪɟɧɢɹ.
ɗɬɨ ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ ɫ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ Ʉɥɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ.
Ʉɪɢɜɵɟ ɢɫɩɚɪɟɧɢɹ, ɩɥɚɜɥɟɧɢɹ ɢ
ɜɨɡɝɨɧɤɢ ɞɟɥɹɬ ɩɥɨɫɤɨɫɬɶ T p ɧɚ
ɬɪɢ ɨɛɥɚɫɬɢ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɬɜɟɪɞɨɦɭ (s), ɠɢɞɤɨɦɭ (f) ɢ ɝɚɡɨɨɛɪɚɡɧɨɦɭ ɫɨɫɬɨɹɧɢɹɦ (g). ɉɥɨɫɤɨɫɬɶ T p ɫ
ɭɤɚɡɚɧɧɵɦɢ ɤɪɢɜɵɦɢ ɧɚɡɵɜɚɟɬɫɹ
ɞɢɚɝɪɚɦɦɨɣ ɫɨɫɬɨɹɧɢɹ. Ⱦɢɚɝɪɚɦɦɚ
ɫɨɫɬɨɹɧɢɹ ɩɨɡɜɨɥɹɟɬ ɫɭɞɢɬɶ ɨ ɬɨɦ,
Ɋɢɫ. 10.3. ɍɩɪɨɳɟɧɧɚɹ ɞɢɚɝɪɚɦɦɚ ɤɚɤɢɟ ɛɭɞɭɬ ɩɪɨɢɫɯɨɞɢɬɶ ɩɪɟɜɪɚɳɟɫɨɫɬɨɹɧɢɹ ɜɨɞɵ
ɧɢɹ ɜ ɬɨɦ ɢɥɢ ɢɧɨɦ ɩɪɨɰɟɫɫɟ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɩɪɨɢɡɜɨɞɢɬɫɹ ɧɚɝɪɟɜɚɧɢɟ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ
(ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɩɪɹɦɚɹ B C ɧɚ ɪɢɫ. 10.3). ȿɫɥɢ ɷɬɚ ɤɪɢɜɚɹ ɩɪɨɯɨɞɢɬ
ɜɵɲɟ ɬɪɨɣɧɨɣ ɬɨɱɤɢ, ɧɨ ɧɢɠɟ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɢ, ɬɨ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ
ɛɭɞɭɬ ɜɨɡɦɨɠɧɵ ɞɜɚ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɚ – ɩɥɚɜɥɟɧɢɟ, ɚ ɡɚɬɟɦ ɢɫɩɚɪɟɧɢɟ.
ȿɫɥɢ ɠɟ ɷɬɚ ɩɪɹɦɚɹ ɩɟɪɟɫɟɱɟɬ ɬɨɥɶɤɨ ɤɪɢɜɭɸ ɜɨɡɝɨɧɤɢ, ɧɚɩɪɢɦɟɪ, ɤɪɢɜɚɹ D F , ɬɨ ɜ ɷɬɨɣ ɬɨɱɤɟ ɩɪɢ ɧɚɝɪɟɜɚɧɢɢ ɩɪɨɢɡɨɣɞɟɬ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɟ
ɩɪɟɜɪɚɳɟɧɢɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɜ ɝɚɡɨɨɛɪɚɡɧɨɟ.
251
Ⱦɢɚɝɪɚɦɦɚ, ɢɡɨɛɪɚɠɟɧɧɚɹ ɧɚ ɪɢɫ. 10.3, ɤɚɱɟɫɬɜɟɧɧɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɭɩɪɨɳɟɧɧɨɣ ɞɢɚɝɪɚɦɦɟ ɫɨɫɬɨɹɧɢɹ ɜɨɞɵ. ɉɭɧɤɬɢɪɧɚɹ ɤɪɢɜɚɹ AE ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɜɧɨɜɟɫɢɸ ɦɟɠɞɭ ɩɟɪɟɨɯɥɚɠɞɟɧɧɨɣ ɜɨɞɨɣ ɢ ɩɚɪɨɦ.
10.4. Ɂɚɞɚɱɚ ɋɬɟɮɚɧɚ
ɉɪɨɫɬɟɣɲɟɣ ɡɚɞɚɱɟɣ ɨ ɮɚɡɨɜɨɦ ɩɟɪɟɯɨɞɟ, ɢɦɟɸɳɟɣ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɢɧɬɟɪɩɪɟɬɚɰɢɸ, ɹɜɥɹɟɬɫɹ
ɡɚɞɚɱɚ ɋɬɟɮɚɧɚ – ɡɚɞɚɱɚ ɨ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɝɨ ɫɥɨɹ
(ɪɢɫ. 10.4). ɗɬɭ ɡɚɞɚɱɭ ɜɩɟɪɜɵɟ
ɫɮɨɪɦɭɥɢɪɨɜɚɥ ɋɬɟɮɚɧ ɩɪɢ ɢɡɭɱɟɧɢɢ ɢɡɦɟɧɟɧɢɹ ɬɨɥɳɢɧɵ ɩɨɥɹɪɧɵɯ Ɋɢɫ. 10.4. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɮɨɪɦɭɥɢɥɶɞɨɜ, ɩɨɷɬɨɦɭ ɩɨɞɨɛɧɵɟ ɡɚɞɚɱɢ ɪɨɜɤɟ ɩɪɨɫɬɟɣɲɟɣ ɡɚɞɚɱɢ ɋɬɟɮɚɧɚ
ɧɨɫɹɬ ɟɝɨ ɢɦɹ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɠɢɞɤɨɫɬɶ, ɢɦɟɸɳɚɹ ɬɟɦɩɟɪɚɬɭɪɭ Te , ɩɪɢɜɟɞɟɧɚ ɜ ɫɨɩɪɢɤɨɫɧɨɜɟɧɢɟ ɫ ɯɨɥɨɞɧɨɣ ɫɬɟɧɤɨɣ,
ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɣ T0 ɦɟɧɶɲɟ ɬɟɦɩɟɪɚɬɭɪɵ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ T0 T .
ɉɪɢ t ! 0 ɨɬ ɯɨɥɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɫ ɧɟɤɨɬɨɪɨɣ ɫɤɨɪɨɫɬɶɸ ɧɚɱɢɧɚɟɬ
ɞɜɢɝɚɬɶɫɹ ɮɪɨɧɬ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ – ɩɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɡɚɤɪɢɫɬɚɥɥɢɡɨɜɚɜɲɢɦɫɹ ɜɟɳɟɫɬɜɨɦ. ɇɭɠɧɨ ɧɚɣɬɢ ɩɨɥɨɠɟɧɢɟ
ɷɬɨɣ ɝɪɚɧɢɰɵ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ x [ t .
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɷɬɨɣ ɡɚɞɚɱɢ ɜɤɥɸɱɚɟɬ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɨɛɥɚɫɬɢ, ɡɚɧɹɬɨɣ ɬɜɟɪɞɵɦ ɬɟɥɨɦ,
w T1
w 2T1
c1U1
O1 2 , 0 d x d [ t ;
(10.19)
wt
wx
ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɨɛɥɚɫɬɢ, ɡɚɧɹɬɨɣ ɠɢɞɤɨɫɬɶɸ,
w T2
w 2T2
O2
, x t [ t (10.20)
c 2U 2
wt
wx 2
ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
x 0 : T1 T0 ;
(10.21)
x o f : T2 Te ;
(10.22)
d[
wT
wT
x [ t : O 1 1 O 2 2 LU1
; T T2 T .
(10.23)
wx
wx
dt 1
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢɦɟɟɬɫɹ ɬɨɥɶɤɨ ɠɢɞɤɨɫɬɶ, ɬ.ɟ.
t 0: [ 0.
(10.24)
ɂɧɞɟɤɫ 1 ɨɬɧɨɫɢɬɫɹ ɤ ɬɜɟɪɞɨɣ ɮɚɡɟ, 2 – ɤ ɠɢɞɤɨɣ.
252
ȼ ɩɪɨɫɬɟɣɲɟɦ ɜɚɪɢɚɧɬɟ ɡɚɞɚɱɢ ɋɬɟɮɚɧɚ ɪɚɡɥɢɱɢɟɦ ɩɥɨɬɧɨɫɬɟɣ ɮɚɡ
ɩɪɟɧɟɛɪɟɝɚɸɬ, U1 U 2 U .
ɉɟɪɜɨɟ ɢɡ ɭɫɥɨɜɢɣ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ (10.23) ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɜ
ɢɧɨɣ ɮɨɪɦɟ. Ⱦɥɹ ɷɬɨɝɨ ɪɚɫɫɦɨɬɪɢɦ ɜ ɩɥɨɫɤɨɫɬɢ x ,t ɞɜɟ ɤɪɢɜɵɟ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ: T1 x ,t T ɢ T2 x ,t T . Ɍɚɤ ɤɚɤ T const , ɢɦɟɟɦ
w T1
wT
dx 1 dx
wx
wt
w T2
wT
dx 2 dt .
wx
wt
ɋɥɟɜɚ ɢ ɫɩɪɚɜɚ ɜ ɷɬɨɦ ɪɚɜɟɧɫɬɜɟ ɡɚɩɢɫɚɧɵ ɩɨɥɧɵɟ ɞɢɮɮɟɪɟɧɰɢɚɥɵ
ɬɟɦɩɟɪɚɬɭɪ ɮɚɡ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɚ ɩɨɞɜɢɠɧɨɣ ɝɪɚɧɢɰɟ x [ t O1
w T1
wT
O2 2
wx
wx
LU
w T1 w t
w T1 w x
LU
w T2 w t
.
w T2 w x
(10.25)
ɂɡ ɬɚɤɨɣ ɮɨɪɦɵ ɡɚɩɢɫɢ ɜɢɞɧɨ, ɱɬɨ ɡɚɞɚɱɚ ɫ ɩɨɞɜɢɠɧɨɣ ɝɪɚɧɢɰɟɣ
ɪɚɡɞɟɥɚ ɮɚɡ ɹɜɥɹɟɬɫɹ ɧɟɥɢɧɟɣɧɨɣ.
ȼ ɬɪɟɯɦɟɪɧɨɦ ɫɥɭɱɚɟ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ (10.25) ɩɪɢɧɢɦɚɟɬ ɜɢɞ
O 1 ’ T1 O 2 ’ T2
rU L
w T1 w t
’ T1
rU L
w T2 w t
.
’ T2
(10.26)
ɂɳɟɦ ɪɟɲɟɧɢɟ ɨɞɧɨɦɟɪɧɨɣ ɡɚɞɚɱɢ (10.19)–(10.24) ɜ ɜɢɞɟ
Ti
§ x ·
Ai Bi ) ¨
¸,
¨2 N t ¸
i ¹
©
(10.27)
ɝɞɟ ) – ɮɭɧɤɰɢɹ ɨɲɢɛɨɤ,
) z
z
2
exp y 2 d y { erf z 1 erfc z .
³
S0
Ƚɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɤɨɬɨɪɵɯ ɨɤɚɡɵɜɚɟɬɫɹ 5 ((10.21), (10.22) ɢ ɬɪɢ
ɭɫɥɨɜɢɹ (10.23)), ɩɨɡɜɨɥɹɬ ɨɩɪɟɞɟɥɢɬɶ ɩɨɫɬɨɹɧɧɵɟ Ai , Bi , i 1, 2 ɢ ɫɤɨɪɨɫɬɶ ɞɜɢɠɟɧɢɹ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ ɮɚɡ.
ɂɡ ɭɫɥɨɜɢɣ ɩɪɢ x 0 (10.21) ɢ x o f (10.22) ɧɚɯɨɞɢɦ
A2 B2
A1 T0 ;
Te .
(10.28)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
A2
Te B2 .
ɂɡ ɭɫɥɨɜɢɹ ɪɚɜɟɧɫɬɜɚ ɬɟɦɩɟɪɚɬɭɪ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ ɫɥɟɞɭɟɬ
253
ª
§ [ ·
§ [ ·º
T0 B1) ¨
¸ T B 2 «1 ) ¨
¸» T .
¨2 N t ¸ e
¨ 2 N t ¸»
«
1 ¹
2 ¹¼
©
©
¬
(10.29)
ɋɨɨɬɧɨɲɟɧɢɟ (10.27) ɞɨɥɠɧɨ ɜɵɩɨɥɧɹɬɶɫɹ ɩɪɢ ɥɸɛɵɯ t , ɜ ɬɨɦ ɱɢɫɥɟ, ɤɨɝɞɚ t o 0 , ɩɨɷɬɨɦɭ ɦɵ ɞɨɥɠɧɵ ɩɪɢɧɹɬɶ
[ 2P N1t ,
(10.30)
ɝɞɟ N1 O 1 c1U1 , ɚ P – ɦɧɨɠɢɬɟɥɶ, ɤɨɬɨɪɵɣ ɫɥɟɞɭɟɬ ɨɩɪɟɞɟɥɢɬɶ ɢɡ ɨɫɬɚɜɲɟɝɨɫɹ ɝɪɚɧɢɱɧɨɝɨ ɭɫɥɨɜɢɹ. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ɝɪɚɧɢɰɚ ɪɚɡɞɟɥɚ ɮɚɡ
ɞɜɢɠɟɬɫɹ ɤɚɤ t
T0 B1) P ɫɥɟɞɨɜɚɬɟɥɶɧɨ
§
Te B2 B2) ¨¨ P
©
N1
N2
·
¸¸ T ,
¹
T T0
, B2
) P
Te T
.
(10.31)
§
N1 ·
1 )¨P
¸
© N2 ¹
ɉɨɞɫɬɚɜɥɹɹ ɪɟɲɟɧɢɟ (10.27) ɜ ɭɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɚ ɩɨɬɨɤɨɜ (10.23) ɫ
ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɵɯ ɩɨɫɬɨɹɧɧɵɯ (10.31), ɧɚɣɞɟɦ
B1
O 1B1exp P
2
§N ·
O 2 B2 ¨ 1 ¸
© N2 ¹
12
§ N
·
exp ¨ 1 P 2 ¸
© N2
¹
LPN1U S
ɢɥɢ
O 2 T Te
exp P
) P 2
O 1 T T0
§ N
·
exp ¨ 1 P 2 ¸
N1
© N2
¹
N2 §
N ·
)¨ P 1 ¸ 1
© N2 ¹
PL S
. (10.32)
c1 T T0 ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɬɟɦɩɟɪɚɬɭɪɚ ɠɢɞɤɨɫɬɢ ɪɚɜɧɚ ɬɟɦɩɟɪɚɬɭɪɟ
ɩɥɚɜɥɟɧɢɹ, ɭɪɚɜɧɟɧɢɟ ɞɥɹ P ɩɪɢɧɢɦɚɟɬ ɩɪɨɫɬɨɣ ɜɢɞ
P) P exp P 2
c1 T T0 .
(10.33)
L S
Ʉɨɪɧɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɧɚɣɬɢ ɝɪɚɮɢɱɟɫɤɢ (ɪɢɫ. 10.5) ɢɥɢ
ɩɪɢɛɥɢɠɟɧɧɨ
254
P2 |
c1 T T0 .
2L
ɉɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɧɚɣɞɟɧɨ P , ɥɟɝɤɨ ɧɚɣɬɢ ɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚ4
a=0,3
a=0,5
ɬɭɪɵ
3
a=0,7
T T0 § x ·
)¨
T
T
¸ , (10.34)
1
0
2
¨
¸
)
P
N
2
t
F = Pa
1 ¹
©
1
Te T
T2 Te u
F = exp(-P )/)(P)
0
ª
º
§
·
N
P
0,0
0,2
0,4
0,6
0,8
«1 ) ¨ P 1 ¸ »
«¬
© N 2 ¹ »¼ , (10.35)
Ɋɢɫ. 10.5. Ƚɪɚɮɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ
ª
ɭɪɚɜɧɟɧɢɹ (10.33)
§ x ·º
u «1 ) ¨
¸» ,
aP e x p P 2 / ) P ¨ 2 N t ¸»
«¬
2 ¹¼
©
ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɤɚɱɟɫɬɜɟɧɧɨɦɭ ɪɢɫ. 10.4.
F1
F2
1
2
2
10.5. ɉɪɨɫɬɟ ɣɲɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ
ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ
Ʉɥɚɫɫɢɮɢɤɚɰɢɹ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɩɨ ɩɨɜɟɞɟɧɢɸ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɛɵɥɚ ɩɪɟɞɥɨɠɟɧɚ ɉ. ɗɪɟɧɮɟɫɬɨɦ ɜ 1933 ɝɨɞɭ ɢ ɫɬɚɥɚ
ɤɥɚɫɫɢɱɟɫɤɨɣ. ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɜ ɤɚɱɟɫɬɜɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɣ
ɮɭɧɤɰɢɢ ɜɵɛɢɪɚɸɬ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɣ ɩɨɬɟɧɰɢɚɥ Ƚɢɛɛɫɚ.
ɋɨɝɥɚɫɧɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɩɪɢ ɮɚɡɨɜɨɦ ɩɟɪɟɯɨɞɟ ɩɟɪɜɨɝɨ ɪɨɞɚ (Ɏɉ1)
ɩɪɟɬɟɪɩɟɜɚɸɬ ɫɤɚɱɨɤ ɩɟɪɜɵɟ ɩɪɨɢɡɜɨɞɧɵɟ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ
g
g 1C1 g 2C 2 ,
(10.36)
ɝɞɟ Ci – ɤɨɧɰɟɧɬɪɚɰɢɢ ɮɚɡ, ɚ g i – ɢɯ ɯɢɦɢɱɟɫɤɢɟ ɩɨɬɟɧɰɢɚɥɵ, ɩɨ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɞɚɜɥɟɧɢɸ, ɬ.ɟ. ɷɧɬɪɨɩɢɹ ɢ ɨɛɴɟɦ
s
§ wg ·
§ wg ·
,v ¨
.
¨
¸
¸
w
p
T
w
©
¹ p ,C1 ,C 2
©
¹ T ,C1 ,C 2
(10.37)
ȼ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɢɦɟɟɦ:
's
s 2 s1 z 0 , ' v v 2 v1 z 0 ,
255
(10.38)
ɩɪɢɱɟɦ ɫɨɜɟɪɲɟɧɧɨ ɧɟ ɨɛɹɡɚɬɟɥɶɧɨ, ɱɬɨɛɵ ɜɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɚɹ ɦɨɞɢɮɢɤɚɰɢɹ ɢɦɟɥɚ ɛɨɥɶɲɢɣ ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ, ɱɟɦ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɚɹ.
Ʉ ɮɚɡɨɜɵɦ ɩɟɪɟɯɨɞɚɦ ɩɟɪɜɨɝɨ ɪɨɞɚ (Ɏɉ1) ɩɪɢɧɚɞɥɟɠɚɬ ɢɫɩɚɪɟɧɢɟ
ɢ ɤɨɧɞɟɧɫɚɰɢɹ, ɩɥɚɜɥɟɧɢɟ ɢ ɡɚɬɜɟɪɞɟɜɚɧɢɟ, ɫɭɛɥɢɦɚɰɢɹ ɢ ɤɨɧɞɟɧɫɚɰɢɹ ɜ
ɬɜɟɪɞɭɸ ɮɚɡɭ; ɧɟɤɨɬɨɪɵɟ ɫɬɪɭɤɬɭɪɧɵɟ ɩɟɪɟɯɨɞɵ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ, ɜ ɬɨɦ
ɱɢɫɥɟ, ɦɚɪɬɟɧɫɢɬɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ. Ɉɛɵɱɧɵɟ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɩɟɪɜɨɝɨ ɪɨɞɚ ɥɟɝɤɨ ɪɟɝɢɫɬɪɢɪɭɸɬ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ. ɋɤɚɱɤɨɨɛɪɚɡɧɨɟ ɢɡɦɟɧɟɧɢɟ ɨɛɴɟɦɚ ɜ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɨɬɜɟɱɚɟɬ ɢɡɦɟɧɟɧɢɸ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɵ, ɬɚɤ ɤɚɤ ɩɥɨɬɧɨɫɬɶ ɢ ɨɛɴɟɦ ɷɥɟɦɟɧɬɚɪɧɨɣ ɹɱɟɣɤɢ ɜ
ɪɚɡɥɢɱɧɵɯ ɩɨɥɢɦɨɪɮɧɵɯ ɦɨɞɢɮɢɤɚɰɢɹɯ ɪɚɡɥɢɱɧɵ. ɂɡɦɟɧɟɧɢɟ ɨɛɴɟɦɚ
ɪɟɝɢɫɬɪɢɪɭɸɬ ɞɢɥɚɬɨɦɟɬɪɢɱɟɫɤɢ, ɚ ɢɧɨɝɞɚ ɢ ɩɭɬɟɦ ɜɢɡɭɚɥɶɧɵɯ ɧɚɛɥɸɞɟɧɢɣ. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɢɡɦɟɧɟɧɢɟ ɨɛɴɟɦɚ ɩɪɢ ɩɪɟɜɪɚɳɟɧɢɢ ɬɟɬɪɚɝɨɧɚɥɶɧɨɣ ɦɨɞɢɮɢɤɚɰɢɢ ɞɢɨɤɫɢɞɚ ɰɢɪɤɨɧɢɹ Z rO 2 ɜ ɦɨɧɨɤɥɢɧɧɭɸ ɜɵɡɵɜɚɟɬ ɪɚɫɬɪɟɫɤɢɜɚɧɢɟ ɦɚɬɟɪɢɚɥɚ. Ɏɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɛɟɥɨɟ ɨɥɨɜɨ – ɫɟɪɨɟ
ɨɥɨɜɨ ɬɚɤɠɟ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɟɦ ɫɟɬɢ ɬɪɟɳɢɧ, ɱɬɨ ɫɜɹɡɚɧɨ ɫ ɫɭɳɟɫɬɜɟɧɧɵɦ ɨɬɥɢɱɢɟɦ ɦɨɥɶɧɵɯ ɨɛɴɟɦɨɜ ɮɚɡ.
ȼ ɬɨɱɤɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɩɟɪɜɨɝɨ ɪɨɞɚ ɫɤɚɱɤɨɦ ɦɟɧɹɟɬɫɹ ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ, ɱɬɨ ɜɢɞɧɨ ɢɡ ɪɚɜɟɧɫɬɜɚ
'u T's p'v T s2 s1 p v2 v1 z 0
(10.39)
Ɉɛɵɱɧɨ ɢɡɦɟɧɟɧɢɟ ɨɛɴɟɦɚ ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɢɡɦɟɧɟɧɢɟɦ ɷɧɬɚɥɶɩɢɢ
' h ' u p' v .
(10.40)
ɂɡɦɟɧɟɧɢɟ ɷɧɬɚɥɶɩɢɢ ɦɨɠɧɨ ɨɛɧɚɪɭɠɢɬɶ ɦɟɬɨɞɨɦ ȾɌȺ: ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɧɚ ɤɪɢɜɵɯ ȾɌȺ ɜɨɡɧɢɤɚɸɬ ɷɧɞɨɬɟɪɦɢɱɟɫɤɢɟ ɢɥɢ ɷɤɡɨɬɟɪɦɢɱɟɫɤɢɟ ɩɢɤɢ.
ɉɪɹɦɨɟ ɢɡɦɟɪɟɧɢɟ ɢɡɦɟɧɟɧɢɹ ɷɧɬɪɨɩɢɢ ɩɪɨɜɨɞɢɬɶ ɦɟɧɟɟ ɭɞɨɛɧɨ.
Ɍɟɦ ɧɟ ɦɟɧɟɟ, ɨ ɫɭɳɟɫɬɜɨɜɚɧɢɢ ɫɤɚɱɤɚ ɷɧɬɪɨɩɢɢ ɦɨɠɧɨ ɡɚɤɥɸɱɢɬɶ ɩɨ
ɧɚɥɢɱɢɸ ɬɟɯ ɠɟ ɩɢɤɨɜ ɧɚ ɤɪɢɜɵɯ ȾɌȺ. Ɍɚɤ ɤɚɤ ɜ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ
'g
T ' s ' h 0 ,
ɬɨ
's
'h
T
Q ph
T ph
.
(10.41)
ɗɬɨ ɢɥɥɸɫɬɪɢɪɭɟɬ ɪɢɫ. 10.6.
ȼ ɫɥɭɱɚɟ ɠɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɩɟɪɜɨɝɨ ɪɨɞɚ ɬɢɩɚ «ɩɨɪɹɞɨɤɛɟɫɩɨɪɹɞɨɤ» ɡɚɮɢɤɫɢɪɨɜɚɬɶ ɩɪɟɜɪɚɳɟɧɢɟ ɦɨɠɧɨ ɪɟɧɬɝɟɧɨɝɪɚɮɢɱɟɫɤɢɦɢ, ɚɤɭɫɬɢɱɟɫɤɢɦɢ, ɨɩɬɢɱɟɫɤɢɦɢ ɢ ɞɪɭɝɢɦɢ (ɧɚɩɪɢɦɟɪ, ɪɚɫɫɟɹɧɢɟɦ
ɦɟɞɥɟɧɧɵɯ ɧɟɣɬɪɨɧɨɜ, ɚɧɧɢɝɢɥɹɰɢɟɣ ɩɨɡɢɬɪɨɧɨɜ) ɦɟɬɨɞɚɦɢ.
256
Ɋɢɫ. 10.6. ɉɨɜɟɞɟɧɢɟ ɷɧɬɪɨɩɢɢ
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ Ɏɉ1
Ɋɢɫ. 10.7. ɉɨɜɟɞɟɧɢɟ ɬɟɩɥɨɟɦɤɨɫɬɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ Ɏɉ1
Ⱥɧɚɥɨɝɢɱɧɨ u ɢ h (ɫɦ. ɮɨɪɦɭɥɵ (10.39), (10.40)), ɦɨɠɧɨ ɨɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɢ ɩɨɜɟɞɟɧɢɟ ɪɚɡɥɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɜ ɬɨɱɤɟ Ɏɉ1. Ɍɚɤ,
ɞɥɹ ɭɞɟɥɶɧɨɣ ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ ɢɦɟɟɦ, ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ,
§ ws ·
cp T¨
¸ ,
© wT ¹ p
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɬɨɱɤɟ Ɏɉ1 c p o f . Ʉɚɱɟɫɬɜɟɧɧɨɟ ɩɨɜɟɞɟɧɢɟ ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɨɤɚɡɚɧɨ ɝɪɚɮɢɱɟɫɤɢ (ɪɢɫ. 10.7).
Ⱥɧɚɥɨɝɢɱɧɨ, ɜ ɬɨɱɤɟ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɩɟɪɜɨɝɨ ɪɨɞɚ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ ɫɬɪɟɦɹɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɫɠɢɦɚɟɦɨɫɬɢ ET ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɪɦɢɱɟɫɤɨɝɨ ɪɚɫɲɢɪɟɧɢɹ DT , ɨɩɪɟɞɟɥɹɟɦɵɟ ɤɚɤ
ET
1 § wv ·
, DT
v0 ¨© w p ¸¹ T
1 § wv ·
.
v0 ¨© w T ¸¹ p
(10.42)
ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɩɪɢ Ɏɉ1 ɜɵɞɟɥɹɟɬɫɹ (ɢɥɢ ɩɨɝɥɨɳɚɟɬɫɹ) ɨɩɪɟɞɟɥɟɧɧɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ. ȿɫɥɢ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɪɨɢɫɯɨɞɢɬ ɩɪɢ ɩɨɜɵɲɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ (ɩɪɹɦɨɣ ɩɟɪɟɯɨɞ), ɬɨ ɜ ɩɪɨɫɬɨɣ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɟ ɬɟɩɥɨ ɜɫɟɝɞɚ ɩɨɝɥɨɳɚɟɬɫɹ.
ɗɬɨ ɩɪɚɜɢɥɨ ɹɜɥɹɟɬɫɹ ɩɪɢɧɰɢɩɨɦ Ʌɟ-ɒɚɬɟɥɶɟ, ɤɨɬɨɪɨɟ ɦɨɠɧɨ
ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɬɚɤ:
«ɇɚɝɪɟɜɚɧɢɟ ɫɬɢɦɭɥɢɪɭɟɬ ɩɪɨɰɟɫɫɵ, ɫɨɩɪɨɜɨɠɞɚɸɳɢɟɫɹ ɩɨɝɥɨɳɟɧɢɟɦ ɬɟɩɥɚ ɢ ɬɟɦ ɫɚɦɵɦ ɤɚɤ ɛɵ ɩɪɨɬɢɜɨɞɟɣɫɬɜɭɸɳɢɟ ɜɧɟɲɧɟɦɭ ɜɨɡɞɟɣɫɬɜɢɸ».
ɇɚɥɢɱɢɟ ɬɟɩɥɨɬɵ ɩɟɪɟɯɨɞɚ ɹɜɥɹɟɬɫɹ ɫɚɦɨɣ ɯɚɪɚɤɬɟɪɧɨɣ ɱɟɪɬɨɣ
Ɏɉ1, ɨɬɥɢɱɚɸɳɟɣ ɢɯ ɨɬ Ɏɉ2.
ɋ ɨɫɨɛɵɦ ɩɨɜɟɞɟɧɢɟɦ ɬɟɩɥɨɟɦɤɨɫɬɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ
ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ (ɬɟɦɩɟɪɚɬɭɪɵ ɩɥɚɜɥɟɧɢɹ) ɫɜɹɡɚɧ ɟɳɟ ɨɞɢɧ ɜɚɪɢɚɧɬ
ɡɚɞɚɱɢ ɋɬɟɮɚɧɚ. Ɍɚɤ, ɜ ɭɫɥɨɜɢɹɯ ɧɚɝɪɟɜɚ ɩɥɨɫɤɨɝɨ ɨɛɪɚɡɰɚ ɞɨɫɬɚɬɨɱɧɨ
257
ɛɨɥɶɲɢɯ ɪɚɡɦɟɪɨɜ ɜɧɟɲɧɢɦ ɢɫɬɨɱɧɢɤɨɦ ɩɨɫɬɨɹɧɧɨɣ ɢɧɬɟɧɫɢɜɧɨɫɬɢ ɫ
ɩɨɜɟɪɯɧɨɫɬɢ ɡɚɞɚɱɚ ɨɛ ɨɩɪɟɞɟɥɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɧɨɝɨ ɩɨɥɹ ɜ ɨɛɪɚɡɰɟ ɦɨɠɟɬ ɛɵɬɶ ɫɮɨɪɦɭɥɢɪɨɜɚɧɚ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
wT
w wT
O
,
(10.43)
cU ef f
wt wx wx
wT
wT
x 0: O
q0 ; x o f : O
0;
wx
wx
t 0 : T T0 ,
ɝɞɟ
­° cU s ,T T ph ;
(10.44)
c
Q
T
T
U
U
G
ef f
®
ph s
ph
°̄ cU L ,T d T ph ,
ɢɧɞɟɤɫ «s» ɨɬɧɨɫɢɬɫɹ ɤ ɬɜɟɪɞɨɣ ɮɚɡɟ, «f» ɤ ɠɢɞɤɨɣ; G – ɞɟɥɶɬɚ-ɮɭɧɤɰɢɹ
Ⱦɢɪɚɤɚ. ɉɪɢ ɱɢɫɥɟɧɧɨɦ ɪɟɲɟɧɢɢ ɡɚɞɚɱɢ ɞɟɥɶɬɚ-ɮɭɧɤɰɢɹ ɡɚɦɟɧɹɟɬɫɹ
ɞɟɥɶɬɚ – ɨɛɪɚɡɧɨɣ ɮɭɧɤɰɢɟɣ, ɧɚɩɪɢɦɟɪ,
§ § T T ·2 ·
1
ph
e x p¨ ¨
G1
¸ ¸,
¨
V
SV
¹ ¸¹
© ©
ɭɞɨɜɥɟɬɜɨɪɹɸɳɟɣ ɭɫɥɨɜɢɸ ɧɨɪɦɢɪɨɜɤɢ
f
³ G1 y d y
1.
f
ɉɚɪɚɦɟɬɪ ɫɝɥɚɠɢɜɚɧɢɹ V ɩɨɞɛɢɪɚɟɬɫɹ ɬɚɤ, ɱɬɨɛɵ ɨɩɢɫɚɬɶ ɤɚɱɟɫɬɜɟɧɧɨɟ
Ɋɢɫ. 10.8. Ʉɚɱɟɫɬɜɟɧɧɨɟ
ɩɨɜɟɞɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
ɩɨɜɟɞɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɥɚɜɥɟɧɢɹ (ɪɢɫ. 10.8).
Ɏɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɩɟɪɜɨɝɨ ɪɨɞɚ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ ɩɥɚɜɥɟɧɢɹ
ɦɨɝɭɬ ɢɞɬɢ ɤɚɤ ɩɪɢ ɫɦɟɧɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɬɚɤ ɢ ɩɪɢ ɫɦɟɧɟ ɞɚɜɥɟɧɢɹ.
Ɏɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɜɬɨɪɨɝɨ (Ɏɉ2) ɪɨɞɚ ɯɚɪɚɤɬɟɪɢɡɭɸɬɫɹ ɧɚɥɢɱɢɟɦ
ɜ ɬɨɱɤɟ ɩɪɟɜɪɚɳɟɧɢɹ ɤɨɧɟɱɧɨɝɨ ɫɤɚɱɤɚ ɜɬɨɪɵɯ ɩɪɨɢɡɜɨɞɧɵɯ ɫɜɨɛɨɞɧɨɣ
ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ – ɬɟɩɥɨɟɦɤɨɫɬɢ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ, ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɪɦɢɱɟɫɤɨɝɨ ɪɚɫɲɢɪɟɧɢɹ ɢ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɫɠɢɦɚɟɦɨɫɬɢ
§ w 2g ·
¨¨ 2 ¸¸
© w p ¹T
§ w 2g
¨¨
© w pw T
·
¸¸
¹T
§ wv ·
¨ wp ¸
© ¹T
§ wv ·
¨
¸
© wT ¹ p
258
D T v0
ET v 0 ,
§ ws ·
¨ ¸ ,
© w p ¹T
(10.45)
§ w 2g ·
cp
§ ws ·
.
¨¨
¸
¨ wT ¸
2¸
T
©
¹
w
T
p
©
¹p
Ɉɛɴɟɦ ɢ ɷɧɬɪɨɩɢɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɦɟɧɹɸɬɫɹ ɧɟɩɪɟɪɵɜɧɨ. Ɉɬɫɭɬɫɬɜɢɟ
ɫɤɚɱɤɚ ɷɧɬɪɨɩɢɢ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ ɩɪɢ Ɏɉ2 ɜ ɩɪɨɫɬɵɯ ɫɢɫɬɟɦɚɯ ɧɟ
ɜɵɞɟɥɹɟɬɫɹ ɢ ɧɟ ɩɨɝɥɨɳɚɟɬɫɹ ɬɟɩɥɨ.
Ɍɟɦɩɟɪɚɬɭɪɭ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɭɸ Ɏɉ2, ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ O – ɬɨɱɤɨɣ
ɢɥɢ ɬɨɱɤɨɣ Ʉɸɪɢ. ɇɚɡɜɚɧɢɟ « O – ɬɨɱɤɚ» ɫɜɹɡɚɧɨ ɫ ɬɟɦ, ɱɬɨ ɜɛɥɢɡɢ ɬɨɱɤɢ
Ɏɉ2 ɜ ɪɟɚɥɶɧɵɯ ɫɢɫɬɟɦɚɯ ɜɫɥɟɞɫɬɜɢɟ ɪɹɞɚ ɨɛɫɬɨɹɬɟɥɶɫɬɜ ɤɪɢɜɵɟ ɦɧɨɝɢɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ ɱɚɫɬɨ ɧɚɩɨɦɢɧɚɸɬ ɝɪɟɱɟɫɤɭɸ ɛɭɤɜɭ O .
Ɍɟɦɩɟɪɚɬɭɪɚ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɜɬɨɪɨɝɨ ɪɨɞɚ ɡɚɜɢɫɢɬ ɨɬ ɭɫɥɨɜɢɣ,
ɩɪɢ ɤɨɬɨɪɵɯ ɨɧɢ ɩɪɨɢɫɯɨɞɹɬ. ɋɚɦɚ ɬɟɦɩɟɪɚɬɭɪɚ Ɏɉ2 ɩɪɢ ɢɡɦɟɧɟɧɢɢ
ɭɫɥɨɜɢɣ ɬɚɤɠɟ ɦɟɧɹɟɬɫɹ. ɂɧɨɝɞɚ ɭɫɥɨɜɢɟ (ɜɧɟɲɧɟɟ ɞɚɜɥɟɧɢɟ, ɷɥɟɤɬɪɢɱɟɫɤɢɟ ɢ ɦɚɝɧɢɬɧɵɟ ɩɨɥɹ ɢ ɬ.ɞ.) ɦɨɝɭɬ ɬɚɤ ɫɭɳɟɫɬɜɟɧɧɨ ɜɥɢɹɬɶ ɧɚ ɩɪɨɰɟɫɫ ɮɚɡɨɜɨɝɨ ɩɪɟɜɪɚɳɟɧɢɹ, ɱɬɨ ɦɟɧɹɟɬɫɹ ɯɚɪɚɤɬɟɪ ɫɚɦɨɝɨ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞ: Ɏɉ2 ɫɬɚɧɨɜɢɬɫɹ Ɏɉ1 ɢ ɧɚɨɛɨɪɨɬ. Ɍɟɦɩɟɪɚɬɭɪɚ, ɩɪɢ ɤɨɬɨɪɨɣ
ɩɪɨɢɫɯɨɞɢɬ ɬɚɤɨɟ ɢɡɦɟɧɟɧɢɟ, ɧɚɡɵɜɚɟɬɫɹ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɨɣ Ʉɸɪɢ ɢɥɢ
ɤɪɢɬɢɱɟɫɤɨɣ O – ɬɨɱɤɨɣ. ȼɛɥɢɡɢ ɷɬɨɣ ɬɟɦɩɟɪɚɬɭɪɵ Ɏɉ1 ɢ Ɏɉ2 ɦɚɥɨ
ɨɬɥɢɱɚɸɬɫɹ.
ɉɪɢɧɰɢɩɢɚɥɶɧɨ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɢ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ ɩɨ ɫɤɚɱɤɚɦ ɩɪɨɢɡɜɨɞɧɵɯ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ ɛɨɥɟɟ ɜɵɫɨɤɢɯ
ɩɨɪɹɞɤɨɜ.
ɗɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɨɛɧɚɪɭɠɢɬɶ Ɏɉ2
ɧɟɫɤɨɥɶɤɨ ɫɥɨɠɧɟɟ, ɱɟɦ Ɏɉ1, ɩɨɬɨɦɭ ɱɬɨ
ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɡɦɟɧɟɧɢɹ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɜ
ɫɢɫɬɟɦɟ, ɜɵɪɚɠɟɧɵ ɝɨɪɚɡɞɨ ɫɥɚɛɟɟ. Ɉɞɧɢɦ ɢɡ ɭɞɨɛɧɵɯ ɦɟɬɨɞɨɜ ɹɜɥɹɟɬɫɹ ɤɚɥɨɪɢɦɟɬɪɢɱɟɫɤɢɣ ɦɟɬɨɞ ɢɡɦɟɪɟɧɢɹ ɬɟɩɥɨɟɦɤɨɫɬɢ. ɉɪɢ ɩɪɢɛɥɢɠɟɧɢɢ ɤ ɬɨɱɤɟ Ɏɉ2 ɬɟɩɥɨɟɦɤɨɫɬɶ ɨɛɵɱɧɨ ɜɨɡɪɚɫɬɚɟɬ, ɚ ɜ ɬɨɱɤɟ
Ɏɉ2 ɩɪɟɬɟɪɩɟɜɚɟɬ ɫɤɚɱɨɤ, ɤɚɤ ɷɬɨ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 10.9.
Ɋɢɫ. 10.9. ɉɨɜɟɞɟɧɢɟ ɬɟɩɥɨɌɢɩɢɱɧɵɟ Ɏɉ2 ɷɬɨ ɩɟɪɟɯɨɞ «ɩɚɪɚɟɦɤɨɫɬɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ
ɦɚɝɧɟɬɢɤ – ɮɟɪɪɨɦɚɝɧɟɬɢɤ», ɫɨɩɪɨɜɨɠɬɨɱɤɢ Ɏɉ2
ɞɚɸɳɢɣɫɹ ɩɨɹɜɥɟɧɢɟɦ ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɨɝɨ ɦɚɝɧɢɬɧɨɝɨ ɦɨɦɟɧɬɚ; ɩɟɪɟɯɨɞ «ɩɚɪɚɦɚɝɧɟɬɢɤ – ɚɧɬɢɮɟɪɪɨɦɚɝɧɟɬɢɤ»,
ɫɨɩɪɨɜɨɠɞɚɸɳɢɣɫɹ ɩɨɹɜɥɟɧɢɟɦ ɚɧɬɢɮɟɪɪɨɦɚɝɧɢɬɧɨɝɨ ɭɩɨɪɹɞɨɱɟɧɢɹ;
ɩɟɪɟɯɨɞ «ɩɚɪɚɷɥɟɤɬɪɢɤ – ɫɟɝɧɟɬɨɷɥɟɤɬɪɢɤ» ɫ ɩɨɹɜɥɟɧɢɟɦ ɫɚɦɨɩɪɨɢɡɜɨɥɶɧɨɣ ɩɨɥɹɪɢɡɚɰɢɢ ɜɟɳɟɫɬɜɚ; «ɧɨɪɦɚɥɶɧɵɣ ɩɪɨɜɨɞɧɢɤ – ɫɜɟɪɯɩɪɨɜɨɞɧɢɤ» ɫ ɢɡɦɟɧɟɧɢɟ ɬɢɩɚ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɜ ɷɥɟɤɬɪɨɧɧɨɣ ɩɨɞɫɢɫɬɟɦɟ.
259
Ɂɚɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ ɜ ɫɥɭɱɚɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɩɟɪɜɨɝɨ ɪɨɞɚ ɤɪɢɜɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ ɮɚɡ ɨɬ ɬɟɦɩɟɪɚɬɭɪ ɩɟɪɟɫɟɤɚɸɬɫɹ, ɬɨ ɜ
ɫɥɭɱɚɟ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɜɬɨɪɨɝɨ ɪɨɞɚ ɬɚɤɨɝɨ ɩɟɪɟɫɟɱɟɧɢɹ ɦɨɠɟɬ ɢ ɧɟ
ɛɵɬɶ.
10.6. Ɋɚɡɦɵɬɵɟ ɢ ɬɨɱɟɱɧɵɟ
ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ
Ɋɚɡɥɢɱɢɟ ɦɟɠɞɭ ɮɚɡɨɜɵɦɢ ɩɟɪɟɯɨɞɚɦɢ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɪɨɞɚ ɨɱɟɜɢɞɧɨ, ɧɨ ɧɚ ɩɪɚɤɬɢɤɟ ɦɧɨɝɢɟ ɩɟɪɟɯɨɞɵ ɬɪɭɞɧɨ ɨɞɧɨɡɧɚɱɧɨ ɨɬɧɟɫɬɢ ɤ
ɬɨɦɭ ɢɥɢ ɢɧɨɦɭ ɬɢɩɭ, ɩɨɫɤɨɥɶɤɭ ɨɧɢ ɧɨɫɹɬ ɝɢɛɪɢɞɧɵɣ ɯɚɪɚɤɬɟɪ.
Ƚɪɚɮɢɱɟɫɤɢ ɷɬɨ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ, ɢɡɨɛɪɚɠɟɧɧɨɦ ɧɚ
ɪɢɫ. 10.10. ɉɪɢ ɧɟɤɨɬɨɪɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɧɟɦɧɨɝɨ ɧɢɠɟ T ph ɫɭɳɟɫɬɜɭɟɬ
ɡɚɦɟɬɧɚɹ ɪɚɡɧɢɰɚ ɦɟɠɞɭ ɷɧɬɚɥɶɩɢɹɦɢ
ɦɨɞɢɮɢɤɚɰɢɣ 1 ɢ 2. (ɞɥɹ ɜɬɨɪɨɣ
ɮɚɡɵ – ɷɬɨ ɩɭɧɤɬɢɪ ɜ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɨɣ ɨɛɥɚɫɬɢ). ɉɪɢ ɧɚɝɪɟɜɚɧɢɢ ɷɧɬɚɥɶɩɢɹ ɮɚɡɵ 1 ɧɚɱɢɧɚɟɬ ɚɧɨɦɚɥɶɧɨ
ɛɵɫɬɪɨ ɩɨɜɵɲɚɬɶɫɹ, ɩɨɤɚ ɜ ɬɨɱɤɟ
T ph ɧɟ ɞɨɫɬɢɝɧɟɬ ɷɧɬɚɥɶɩɢɢ ɮɚɡɵ 2,
Ɋɢɫ. 10.10. ȼɨɡɦɨɠɧɨɟ ɩɨɜɟɞɟɧɢɟ
ɷɧɬɚɥɶɩɢɢ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɨɱɤɢ
ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ
ɜ ɪɟɚɥɶɧɨɣ ɫɢɫɬɟɦɟ
ɝɞɟ h1 h2 . Ʉɚɤɨɝɨ ɪɨɞɚ ɩɪɟɜɪɚɳɟɧɢɟ
ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ?
əɫɧɨ, ɱɬɨ ɨɧɨ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɨɬɧɟɫɟɧɨ ɤ ɩɟɪɟɯɨɞɭ 1-ɝɨ ɪɨɞɚ, ɞɥɹ
ɤɨɬɨɪɨɝɨ ɯɚɪɚɤɬɟɪɧɨ ɧɚɥɢɱɢɟ ɪɚɡɪɵɜɚ ɜ h ɜ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ: h1 h 2 z 0 .
Ɍɚɤɨɟ ɩɪɟɜɪɚɳɟɧɢɟ ɦɨɠɧɨ ɛɵɥɨ ɛɵ ɨɬɧɟɫɬɢ ɤ ɩɟɪɟɯɨɞɭ 2-ɝɨ ɪɨɞɚ,
ɟɫɥɢ ɨɛɪɚɬɢɬɶ ɜɧɢɦɚɧɢɟ ɧɚ ɢɡɦɟɧɟɧɢɟ ɬɟɩɥɨɟɦɤɨɫɬɟɣ (ɪɢɫ. 10.9). ɇɨ
ɚɧɨɦɚɥɶɧɨ ɛɨɥɶɲɨɣ ɪɨɫɬ h1 ɩɪɢ T T ph ɞɥɹ ɬɚɤɢɯ ɩɟɪɟɯɨɞɨɜ ɜɨɜɫɟ ɧɟ
ɯɚɪɚɤɬɟɪɟɧ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɧɚ ɩɪɚɤɬɢɤɟ ɮɚɡɨɜɵɦ ɩɪɟɜɪɚɳɟɧɢɹɦ ɩɪɟɞɲɟɫɬɜɭɸɬ
ɨɬɞɟɥɶɧɵɟ ɢɡɦɟɧɟɧɢɹ ɜ ɫɢɫɬɟɦɟ, ɧɚɩɪɢɦɟɪ, ɭɜɟɥɢɱɟɧɢɹ ɪɚɡɭɩɨɪɹɞɨɱɟɧɢɹ ɩɪɢ ɩɪɢɛɥɢɠɟɧɢɢ ɬɟɦɩɟɪɚɬɭɪɵ ɤ T ph . ɑɚɫɬɨ ɬɚɤɢɦɢ ɹɜɥɟɧɢɹɦɢ
ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɨɫɨɛɟɧɧɨ ɟɫɥɢ ɫɤɚɱɨɤ ɷɧɬɚɥɶɩɢɢ ɩɪɢ T
T ph ɞɨɫɬɚ-
ɬɨɱɧɨ ɜɟɥɢɤ, ɬ.ɟ. ɩɪɢ Ɏɉ1. ȼ ɫɥɭɱɚɟ ɠɟ Ɏɉ2 (ɨɫɨɛɟɧɧɨ ɩɪɢ ɩɟɪɟɯɨɞɚɯ
ɬɢɩɚ «ɩɨɪɹɞɨɤ-ɛɟɫɩɨɪɹɞɨɤ» ɢɡɦɟɧɟɧɢɹ ɫɬɪɭɤɬɭɪɵ ɢ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɯ
ɫɜɨɣɫɬɜ ɩɪɢ T ph ɫɜɹɡɚɧɵ ɫ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɦɢ ɢɡɦɟɧɟɧɢɹɦɢ ɜ ɫɢɫɬɟɦɟ
ɧɢɠɟ ɷɬɨɣ ɬɟɦɩɟɪɚɬɭɪɵ. ɉɨɷɬɨɦɭ ɩɪɟɧɟɛɪɟɱɶ ɬɚɤɢɦɢ ɢɡɦɟɧɟɧɢɹɦɢ ɧɟ260
ɜɨɡɦɨɠɧɨ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ T ph ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɫɬɨ ɬɟɦɩɟɪɚɬɭɪɭ,
ɩɪɢ ɤɨɬɨɪɨɣ (ɢɥɢ ɜɵɲɟ ɤɨɬɨɪɨɣ) ɫɬɪɭɤɬɭɪɧɵɟ ɢɡɦɟɧɟɧɢɹ ɜ ɫɢɫɬɟɦɟ ɩɪɨɫɬɨ ɡɚɤɚɧɱɢɜɚɸɬɫɹ.
ɇɚɥɢɱɢɟ ɩɪɟɞɤɪɢɬɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ (ɩɨɞɨɛɧɵɯ ɭɜɟɥɢɱɟɧɢɸ ɪɚɡɭɩɨɪɹɞɨɱɟɧɢɹ ɢɥɢ ɤɨɧɰɟɧɬɪɚɰɢɢ ɞɟɮɟɤɬɨɜ) ɩɪɢ ɩɪɢɛɥɢɠɟɧɢɢ ɤ T ph ɨɛɴɟɞɢɧɹɟɬ Ɏɉ1 ɢ Ɏɉ2.
ɋ ɩɨɦɨɳɶɸ ɬɨɥɶɤɨ ɱɬɨ ɨɩɢɫɚɧɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɜɟɫɶɦɚ ɩɨɥɟɡɧɚ
ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɍɛɟɥɥɨɞɟ. ɋɨɝɥɚɫɧɨ ɟɝɨ ɩɨɞɯɨɞɭ, ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ
ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ ɪɚɡɦɵɬɵɟ ɢ ɬɨɱɟɱɧɵɟ. Ɋɚɡɦɵɬɵɣ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɥɭɱɚɸ ɢɡɨɛɪɚɠɟɧɧɨɦɭ ɧɚ ɪɢɫ. 10.10: ɩɪɢ T T ph ɜɟɥɢɱɢɧɚ ɷɧɬɚɥɶɩɢɢ ɧɟ ɦɟɧɹɟɬɫɹ ɫɤɚɱɤɨɨɛɪɚɡɧɨ, ɚ ɢɡɦɟɧɟɧɢɹ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɫɬɪɭɤɬɭɪɵ ɩɪɨɢɫɯɨɞɹɬ ɩɥɚɜɧɨ ɢ ɧɟɩɪɟɪɵɜɧɨ ɨɬ ɫɬɪɭɤɬɭɪɵ, ɯɚɪɚɤɬɟɪɧɨɣ ɞɥɹ ɮɚɡɵ 1, ɤ ɫɬɪɭɤɬɭɪɟ, ɯɚɪɚɤɬɟɪɧɨɣ ɞɥɹ ɮɚɡɵ 2. ɉɪɟɞɤɪɢɬɢɱɟɫɤɢɟ ɹɜɥɟɧɢɹ ɹɜɥɹɸɬɫɹ ɤɚɤ ɛɵ ɧɚɱɚɥɨɦ ɪɚɡɦɵɬɨɝɨ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ.
Ɍɨɱɟɱɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɯɚɪɚɤɬɟɪɧɵ ɞɥɹ ɜɟɳɟɫɬɜ, ɤɪɢɫɬɚɥɥɢɱɟɫɤɢɟ
ɪɟɲɟɬɤɢ ɪɚɡɥɢɱɧɵɯ ɮɚɡ ɤɨɬɨɪɵɯ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɸɬɫɹ
ɞɪɭɝ ɨɬ ɞɪɭɝɚ.
Ɋɚɡɦɵɬɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɢɦɟɸɬ ɦɟɫɬɨ ɩɪɢ ɨɛɪɚɡɨɜɚɧɢɢ «ɝɢɛɪɢɞɧɵɯ» ɫɬɪɭɤɬɭɪ, ɤɨɝɞɚ ɞɨɦɟɧɵ (ɨɛɥɚɫɬɢ) ɜɨɡɧɢɤɚɸɳɟɣ ɧɨɜɨɣ ɮɚɡɵ ɪɚɫɬɭɬ ɜɧɭɬɪɢ ɤɪɢɫɬɚɥɥɚ ɢɫɯɨɞɧɨɝɨ ɜɟɳɟɫɬɜɚ ɢ ɜɨɡɧɢɤɚɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ
ɝɢɛɪɢɞɧɵɣ ɤɪɢɫɬɚɥɥ. ɇɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɦɟɠɞɭ ɢɫɯɨɞɧɨɣ ɢ ɧɨɜɨɣ ɮɚɡɚɦɢ ɜɟɳɟɫɬɜɨ ɛɭɞɟɬ ɧɚɯɨɞɢɬɶɫɹ ɜ ɫɠɚɬɨɦ (ɧɚɩɪɹɠɟɧɧɨɦ) ɫɨɫɬɨɹɧɢɢ,
ɬɚɤ ɤɚɤ ɦɚɥɨ ɜɟɪɨɹɬɧɨ, ɱɬɨɛɵ ɦɨɥɶɧɵɟ ɨɛɴɟɦɵ ɮɚɡ ɛɵɥɢ ɨɞɢɧɚɤɨɜɵ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɧɟɪɝɟɬɢɱɟɫɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɬɚɤɢɯ ɩɟɪɟɯɨɞɨɜ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɹɬ ɨɬ ɷɧɟɪɝɢɢ ɧɚɩɪɹɠɟɧɢɣ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɟɚɥɶɧɵɟ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɞɚɥɟɤɨ ɧɟ ɜɫɟɝɞɚ ɭɤɥɚɞɵɜɚɸɬɫɹ ɜ ɪɚɦɤɢ ɤɥɚɫɫɢɱɟɫɤɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɣ. ȼɨ-ɩɟɪɜɵɯ, ɬɟ ɜɟɥɢɱɢɧɵ, ɤɨɬɨɪɵɟ, ɫɨɝɥɚɫɧɨ ɤɥɚɫɫɢɱɟɫɤɨɣ ɬɟɨɪɢɢ, ɜ ɬɨɱɤɟ ɩɟɪɟɯɨɞɚ ɞɨɥɠɧɵ
ɫɬɪɟɦɢɬɶɫɹ ɤ ɛɟɫɤɨɧɟɱɧɨɫɬɢ, ɜ ɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ ɜ ɥɭɱɲɟɦ ɫɥɭɱɚɟ ɞɨɫɬɢɝɚɸɬ ɬɨɥɶɤɨ ɞɨɫɬɚɬɨɱɧɨ ɛɨɥɶɲɢɯ ɡɧɚɱɟɧɢɣ, ɨɫɬɚɜɚɹɫɶ ɤɨɧɟɱɧɵɦɢ.
Ɍɟɦɩɟɪɚɬɭɪɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ ɰɟɥɨɝɨ ɪɹɞɚ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ (ɬɟɩɥɨɟɦɤɨɫɬɢ, ɞɢɷɥɟɤɬɪɢɱɟɫɤɚɹ ɢ ɦɚɝɧɢɬɧɚɹ ɩɪɨɧɢɰɚɟɦɨɫɬɢ, ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɩɥɨɜɨɝɨ ɪɚɫɲɢɪɟɧɢɹ, ɪɚɫɲɢɪɟɧɢɹ, ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɢ ɞɪ.) ɩɪɨɯɨɞɹɬ
ɜ ɨɤɪɟɫɬɧɨɫɬɢ T ph ɱɟɪɟɡ ɦɚɤɫɢɦɭɦ ɢɥɢ ɦɢɧɢɦɭɦ. ȼɨ-ɜɬɨɪɵɯ, ɜɟɫɶɦɚ
ɱɚɫɬɨ ɮɚɡɨɜɵɣ ɩɟɪɟɯɨɞ ɩɪɨɬɟɤɚɟɬ ɧɟ ɩɪɢ ɨɞɧɨɣ ɫɬɪɨɝɨ ɮɢɤɫɢɪɨɜɚɧɧɨɣ
ɬɟɦɩɟɪɚɬɭɪɟ, ɚ ɜ ɧɟɤɨɬɨɪɨɦ ɢɧɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ, ɱɬɨ ɬɢɩɢɱɧɨ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɦɚɪɬɟɧɫɢɬɧɨɝɨ ɬɢɩɚ.
261
10.7. ɗɥɟɦɟɧɬɵ ɬɟɨɪɢɢ ɞɜɭɯɮɚɡɧɨɣ ɡɨɧɵ
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɫɢɫɬɟɦ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯɫɹ ɧɚɥɢɱɢɟɦ ɢɧɬɟɪɜɚɥɚ ɬɟɦɩɟɪɚɬɭɪ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɵɟ
ɫɢɫɬɟɦɵ, ɨɛɪɚɡɭɸɳɢɟ ɬɜɟɪɞɵɟ ɪɚɫɬɜɨɪɵ. ɋɪɟɞɢ ɫɢɫɬɟɦ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯɫɹ ɨɛɪɚɡɨɜɚɧɢɟɦ ɬɜɟɪɞɵɯ ɪɚɫɬɜɨɪɨɜ, ɫɚɦɵɣ ɩɪɨɫɬɨɣ ɜɢɞ ɢɦɟɸɬ ɬɚɤɢɟ ɮɚɡɨɜɵɟ ɞɢɚɝɪɚɦɦɵ, ɜ ɤɨɬɨɪɵɯ ɤɚɤ ɜ ɠɢɞɤɨɣ, ɬɚɤ ɢ ɜ ɬɜɟɪɞɨɣ ɮɚɡɚɯ
ɧɚɛɥɸɞɚɟɬɫɹ ɧɟɨɝɪɚɧɢɱɟɧɧɚɹ ɜɡɚɢɦɧɚɹ ɪɚɫɬɜɨɪɢɦɨɫɬɶ ɤɨɦɩɨɧɟɧɬɨɜ
(ɪɢɫ. 10.11).
Ɍɟɦɩɟɪɚɬɭɪɚ ɩɥɚɜɥɟɧɢɹ ɤɨɦɩɨɧɟɧɬɚ A ɭɦɟɧɶɲɚɟɬɫɹ ɩɪɢ ɞɨɛɚɜɥɟɧɢɢ B , ɬɟɦɩɟɪɚɬɭɪɚ ɩɥɚɜɥɟɧɢɹ ɤɨɦɩɨɧɟɧɬɚ B ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɪɢ ɞɨɛɚɜɥɟɧɢɢ A .
ɉɥɚɜɧɵɟ ɤɪɢɜɵɟ ɥɢɤɜɢɞɭɫɚ (ɜɟɪɯɧɹɹ ɝɪɚɧɢɰɚ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɬɜɟɪɞɨɣ ɮɚɡɵ) ɢ ɫɨɥɢɞɭɫɚ (ɧɢɠɧɹɹ ɝɪɚɧɢɰɚ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɠɢɞɤɨɣ ɮɚɡɵ)
ɫɨɟɞɢɧɹɸɬɫɹ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɬɨɥɶɤɨ ɜ ɬɨɱɤɚɯ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɱɢɫɬɵɦ
ɜɟɳɟɫɬɜɚɦ A ɢ B .
ɇɢɠɟ ɤɪɢɜɨɣ ɫɨɥɢɞɭɫɚ ɫɭɳɟɫɬɜɭɟɬ ɨɞɧɨɮɚɡɧɵɣ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɵɣ ɬɜɟɪɞɵɣ (ɫ ɞɜɭɦɹ ɫɬɟɩɟɧɹɦɢ
ɫɜɨɛɨɞɵ) ɪɚɫɬɜɨɪ, ɚ ɜɵɲɟ ɤɪɢɜɨɣ
ɥɢɤɜɢɞɭɫɚ – ɨɞɧɨɮɚɡɧɵɣ ɠɢɞɤɢɣ
ɪɚɫɬɜɨɪ ɫ ɬɚɤɢɦ ɠɟ ɱɢɫɥɨɦ ɫɬɟɩɟɧɟɣ
ɫɜɨɛɨɞɵ. Ɇɟɠɞɭ ɤɪɢɜɵɦɢ ɥɢɤɜɢɞɭɫɚ ɢ ɫɨɥɢɞɭɫɚ ɧɚɯɨɞɢɬɫɹ ɞɜɭɯɮɚɡɧɚɹ
ɨɛɥɚɫɬɶ, ɜ ɤɨɬɨɪɨɣ ɫɨɫɭɳɟɫɬɜɭɸɬ
ɬɜɟɪɞɵɟ ɢ ɠɢɞɤɢɟ ɪɚɫɬɜɨɪɵ.
Ɋɢɫ. 10.11. Ⱦɢɚɝɪɚɦɦɚ ɫɨɫɬɨɹɧɢɹ
ɉɟɪɟɫɟɱɟɧɢɟ ɢɡɨɬɟɪɦɵ ɫ ɤɪɢɞɥɹ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɵɯ ɫɢɫɬɟɦ ɫ
ɜɨɣ ɫɨɥɢɞɭɫɚ ɞɚɫɬ ɫɨɫɬɚɜ ɬɜɟɪɞɨɝɨ
ɧɟɨɝɪɚɧɢɱɟɧɧɨɣ ɪɚɫɬɜɨɪɢɦɨɫɬɶɸ
ɪɚɫɬɜɨɪɚ a , ɚ ɫ ɤɪɢɜɨɣ ɥɢɤɜɢɞɭɫɚ –
ɤɨɦɩɨɧɟɧɬɨɜ: a – ɫɨɫɬɚɜ ɬɜɟɪɞɨɝɨ ɫɨɫɬɚɜ ɠɢɞɤɨɝɨ ɪɚɫɬɜɨɪɚ b .
ɪɚɫɬɜɨɪɚ; b – ɠɢɞɤɨɝɨ
ȼ ɫɢɫɬɟɦɚɯ ɫ ɩɨɞɨɛɧɵɦɢ ɮɚɡɨɜɵɦɢ ɞɢɚɝɪɚɦɦɚɦɢ ɧɚɛɥɸɞɚɟɬɫɹ ɹɜɥɟɧɢɟ, ɧɚɡɵɜɚɟɦɨɟ ɥɢɤɜɚɰɢɟɣ: ɜɨɡɧɢɤɚɸɬ ɦɟɬɚɫɬɚɛɢɥɶɧɵɟ ɩɪɨɞɭɤɬɵ ɜ ɪɟɡɭɥɶɬɚɬɟ ɞɪɨɛɧɨɣ (ɮɪɚɤɰɢɨɧɧɨɣ)
ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ. Ɍɚɤɢɟ ɹɜɥɟɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɜɫɹɤɢɣ ɪɚɡ, ɤɨɝɞɚ ɩɪɢɦɟɧɹɸɬ ɧɟɞɨɫɬɚɬɨɱɧɨ ɧɢɡɤɢɟ ɫɤɨɪɨɫɬɢ ɨɯɥɚɠɞɟɧɢɹ, ɱɬɨ ɧɟ ɩɨɡɜɨɥɹɟɬ ɞɨɫɬɢɱɶ
ɪɚɜɧɨɜɟɫɢɹ ɩɪɢ ɤɚɠɞɨɣ ɬɟɦɩɟɪɚɬɭɪɟ. Ʉɪɢɫɬɚɥɥɵ, ɨɛɪɚɡɭɸɳɢɟɫɹ ɩɟɪɜɵɦɢ ɩɪɢ ɨɯɥɚɠɞɟɧɢɢ ɠɢɞɤɨɫɬɢ ɫɨɫɬɚɜɚ b , ɢɦɟɸɬ ɫɨɫɬɚɜ a . ȿɫɥɢ ɧɟɬ ɜɪɟɦɟɧɢ ɞɥɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɧɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɷɬɢɯ ɤɪɢɫɬɚɥɥɨɜ ɫ ɠɢɞɤɨɫɬɶɸ, ɬɨ ɨɧɢ ɮɚɤɬɢɱɟɫɤɢ ɞɥɹ ɫɢɫɬɟɦɵ ɹɜɥɹɸɬɫɹ ɩɨɬɟɪɹɧɧɵɦɢ. ɂɡ-ɡɚ
ɷɬɨɝɨ ɧɨɜɵɟ ɩɨɪɰɢɢ ɤɪɢɫɬɚɥɥɨɜ ɨɛɨɝɚɳɟɧɵ ɤɨɦɩɨɧɟɧɬɨɦ B . Ɉɛɪɚɡɨɜɚɧɢɟ ɬɚɤɢɯ ɫɬɪɭɤɬɭɪ, ɪɚɡɜɢɜɚɸɳɢɯɫɹ ɜɨɤɪɭɝ ɰɟɧɬɪɚɥɶɧɨɝɨ ɡɚɪɨɞɵɲɚ,
262
ɜɟɫɶɦɚ ɱɚɫɬɨ ɜɫɬɪɟɱɚɟɬɫɹ ɩɪɢ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɪɚɫɩɥɚɜɨɜ. ȼ ɪɟɡɭɥɶɬɚɬɟ
ɬɜɟɪɞɚɹ ɮɚɡɚ ɫɨɞɟɪɠɢɬ ɧɟɨɞɧɨɪɨɞɧɵɟ ɩɨ ɫɨɫɬɚɜɭ ɤɪɢɫɬɚɥɥɵ.
Ⱦɥɹ ɫɢɫɬɟɦ ɩɨɞɨɛɧɨɝɨ ɬɢɩɚ ɛɵɥɚ ɪɚɡɪɚɛɨɬɚɧɚ ɬɟɨɪɢɹ ɞɜɭɯɮɚɡɧɨɣ
ɡɨɧɵ. ȿɟ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ.
ȼ ɞɜɭɯɮɚɡɧɨɣ ɡɨɧɟ ɦɟɠɞɭ ɤɪɢɜɵɦɢ ɥɢɤɜɢɞɭɫɚ ɢ ɫɨɥɢɞɭɫɚ ɧɚɯɨɞɢɬɫɹ ɤɚɤ ɠɢɞɤɨɫɬɶ, ɬɚɤ ɢ ɤɪɢɫɬɚɥɥɵ ɬɜɟɪɞɨɣ ɮɚɡɵ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɢɯ ɨɛɴɟɦɧɵɟ ɞɨɥɢ K f ɢ K s ɫɜɹɡɚɧɵ ɫɨɨɬɧɨɲɟɧɢɟɦ
K f K s 1.
ɋɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɞɨɥɹɦɢ ɮɚɡ ɡɚɜɢɫɢɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪ ɥɢɤɜɢɞɭɫɚ
Tliq ɢ ɫɨɥɢɞɭɫɚ Tsol , ɤɨɬɨɪɵɟ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɡɚɜɢɫɹɬ ɨɬ ɫɨɫɬɚɜɚ ɞɜɭɯɤɨɦɩɨɧɟɧɬɧɨɝɨ ɪɚɫɬɜɨɪɚ:
D s Es [ J s [2 ,
Tsol
D f E f [ J f [2 ,
Tliq
ɝɞɟ [ – ɞɨɥɹ ɨɞɧɨɝɨ ɢɡ ɤɨɦɩɨɧɟɧɬɨɜ; ɤɨɧɫɬɚɧɬɵ D i ,E i , J i , i f , s , ɦɨɝɭɬ
ɛɵɬɶ ɧɚɣɞɟɧɵ ɚɩɩɪɨɤɫɢɦɚɰɢɟɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɪɢɜɵɯ ɞɢɚɝɪɚɦɦɵ
ɫɨɫɬɨɹɧɢɹ (ɪɢɫ. 10.11).
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɷɮɮɟɤɬɢɜɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ ɜ ɭɪɚɜɧɟɧɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ (10.43) ɞɨɥɠɧɚ ɜɵɱɢɫɥɹɬɶɫɹ ɧɟ ɩɨ ɮɨɪɦɭɥɟ (10.44), ɚ ɩɨ ɮɨɪɦɭɥɟ
­ cU s ,
T d Ts ,
°
wK
(10.46)
cU ef f °®cU Q p hU s L , Ts T T L , ,
w
T
°
° cU L ,
T t TL ,
¯
ɝɞɟ
cU cU s 1 K f cU f K f .
ȼ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɨɛɴɟɦɧɵɟ ɬɟɩɥɨɟɦɤɨɫɬɢ ɬɜɟɪɞɨɣ ɢ ɠɢɞɤɨɣ ɮɚɡ
ɬɚɤɠɟ ɞɨɥɠɧɵ ɡɚɜɢɫɟɬɶ ɨɬ ɫɨɫɬɚɜɚ. ȼ ɩɪɨɫɬɟɣɲɟɦ ɩɪɢɛɥɢɠɟɧɢɢ ɦɨɠɧɨ
ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɮɨɪɦɭɥɚɦɢ (ɩɪɚɜɢɥɨɦ ɫɦɟɫɢ)
cU s cU A ,s [ 1 [ cU B ,s ,
cU f cU A , f [ 1 [ cU B , f .
Ɍɟɩɥɨɟɦɤɨɫɬɢ ɢɧɞɢɜɢɞɭɚɥɶɧɵɯ ɜɟɳɟɫɬɜ, ɨɱɟɜɢɞɧɨ, ɬɚɤɠɟ ɦɨɝɭɬ ɡɚɜɢɫɟɬɶ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ.
Ɉɛɴɟɦɧɚɹ ɞɨɥɹ ɠɢɞɤɨɣ ɮɚɡɵ ɜɵɱɢɫɥɹɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɫɨɨɬɧɨɲɟɧɢɹ
263
n
§ Tl iq T ·
K f 1 ¨
¸ ,
¨ Tl i q Tso l ¸
©
¹
ɝɞɟ ɩɚɪɚɦɟɬɪ n ɜɚɪɶɢɪɭɟɬɫɹ ɞɥɹ ɪɚɡɧɵɯ ɫɩɥɚɜɨɜ.
Ⱦɨɥɹ ɬɜɟɪɞɨɣ ɮɚɡɵ, ɨɱɟɜɢɞɧɨ, ɟɫɬɶ K s 1 K f .
(10.47)
Ɍɟɨɪɢɹ ɞɜɭɯɮɚɡɧɨɣ ɡɨɧɵ ɨɛɨɛɳɚɟɬɫɹ ɢ ɧɚ ɦɧɨɝɨɤɨɦɩɨɧɟɧɬɧɵɟ
ɫɩɥɚɜɵ.
10.8. ɗɥɟɦɟɧɬɵ ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ
ɮɚɡɨɜɵɯ ɩɪɟɜɪɚ ɳɟɧɢɣ
ȼ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɞɨɥɸ ɤɪɢɫɬɚɥɥɨɜ ɬɜɟɪɞɨɣ ɮɚɡɵ ɜ ɞɜɭɯɮɚɡɧɨɣ
ɡɨɧɟ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɧɚ ɨɫɧɨɜɟ ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ, ɭɱɢɬɵɜɚɸɳɟɣ
ɹɜɥɟɧɢɹ, ɩɪɟɞɲɟɫɬɜɭɸɳɢɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɜ ɛɨɥɶɲɢɯ
ɨɛɴɟɦɚɯ.
Ȼɨɥɶɲɚɹ ɱɚɫɬɶ ɮɚɡɨɜɵɯ ɩɪɟɜɪɚɳɟɧɢɣ ɧɚɱɢɧɚɟɬɫɹ ɫ ɨɛɪɚɡɨɜɚɧɢɹ
ɮɢɡɢɱɟɫɤɢ ɪɚɡɥɢɱɢɦɵɯ ɰɟɧɬɪɨɜ (ɷɬɨɬ ɩɪɨɰɟɫɫ ɢɡɜɟɫɬɟɧ ɤɚɤ ɡɚɪɨɠɞɟɧɢɟ),
ɩɨɫɥɟ ɱɟɝɨ ɨɛɥɚɫɬɢ, ɩɪɟɬɟɪɩɟɜɲɢɟ ɩɪɟɜɪɚɳɟɧɢɟ, ɪɚɫɬɭɬ ɜ ɨɤɪɭɠɚɸɳɭɸ
ɫɪɟɞɭ.
Ɍɚɤ, ɩɪɨɰɟɫɫɵ ɡɚɬɜɟɪɞɟɜɚɧɢɹ ɢɥɢ ɤɨɧɞɟɧɫɚɰɢɢ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɧɚ
ɞɜɟ ɫɬɭɩɟɧɢ: ɩɟɪɜɨɧɚɱɚɥɶɧɨɟ ɨɛɪɚɡɨɜɚɧɢɟ ɡɚɪɨɞɵɲɟɣ ɤɪɢɫɬɚɥɥɨɜ ɢ ɩɨɫɥɟɞɭɸɳɢɣ ɪɨɫɬ ɷɬɢɯ ɡɚɪɨɞɵɲɟɣ ɩɭɬɟɦ ɩɪɢɫɨɟɞɢɧɟɧɢɹ ɦɨɥɟɤɭɥ ɢɡ ɪɚɫɩɥɚɜɚ. ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɢɡɜɟɫɬɧɵɯ ɬɟɨɪɢɣ ɩɪɢɧɹɬɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɪɨɰɟɫɫ
ɡɚɪɨɞɵɲɟɨɛɪɚɡɨɜɚɧɢɹ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɜɨɡɧɢɤɧɨɜɟɧɢɢ ɢ ɪɨɫɬɟ ɚɝɪɟɝɚɬɨɜ
ɦɨɥɟɤɭɥ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɪɨɬɟɤɚɧɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɛɢɦɨɥɟɤɭɥɹɪɧɵɯ
ɪɟɚɤɰɢɣ (ɫɦ. ɱɚɫɬɶ 11).
ȼɨɡɧɢɤɧɨɜɟɧɢɟ ɧɟɛɨɥɶɲɨɣ ɤɚɩɥɢ ɪɚɞɢɭɫɨɦ r ɩɪɢ ɞɚɜɥɟɧɢɢ ɩɚɪɚ
p , ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɝɨɫɹ ɪɚɜɧɨɜɟɫɧɵɦ ɞɚɜɥɟɧɢɟɦ p0 (ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ
T ), ɫɨɩɪɨɜɨɠɞɚɟɬɫɹ ɢɡɦɟɧɟɧɢɟɦ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ
' g ' g vo l ' g s ,
ɝɞɟ ' g v o l – ɟɫɬɶ ɢɡɦɟɧɟɧɢɟ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ ɩɪɢ ɤɨɧɞɟɧɫɚɰɢɢ; ɞɥɹ
ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɷɬɚ ɜɟɥɢɱɢɧɚ, ɨɬɧɟɫɟɧɧɚɹ ɤ ɨɞɢɧɨɱɧɨɦɭ ɚɬɨɦɭ, ɫɨɫɬɚɜɥɹɟɬ
kT ln p p 0 .
ɗɬɚ ɜɟɥɢɱɢɧɚ ɜɵɜɨɞɢɬɫɹ ɢɡ ɢɡɦɟɧɟɧɢɹ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ ɢɞɟɚɥɶɧɨɝɨ ɝɚɡɚ ɩɪɢ ɢɡɨɬɟɪɦɢɱɟɫɤɨɦ ɫɠɚɬɢɢ.
Ⱦɥɹ ɫɮɟɪɢɱɟɫɤɨɣ ɤɚɩɥɢ ɪɚɞɢɭɫɚ r ɩɨɥɭɱɚɟɦ
4
S r 3v f 1k T ln p p 0 ,
3
264
ɝɞɟ v f – ɚɬɨɦɧɵɣ ɨɛɴɟɦ ɠɢɞɤɨɫɬɢ.
ȼɤɥɚɞ ' g s ɜ ɢɡɦɟɧɟɧɢɟ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ ɟɫɬɶ
4S r 2 J f g ,
ɝɞɟ J fg – ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ «ɩɚɪ – ɠɢɞɤɨɫɬɶ».
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
4
' g 4S r 2 J f g S r 3v f 1kT ln p p 0 .
(10.48)
3
ɉɪɢ ɦɚɥɵɯ ɪɚɞɢɭɫɚɯ ɤɚɩɥɢ ɩɪɟɨɛɥɚɞɚɟɬ ɩɟɪɜɨɟ ɫɥɚɝɚɟɦɨɟ, ɱɬɨ ɞɟɥɚɟɬ ' g ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɜɟɥɢɱɢɧɨɣ. ɉɪɢ ɛɨɥɶɲɢɯ ɪɚɞɢɭɫɚɯ ɜ (10.48)
ɧɚɱɢɧɚɟɬ ɩɪɟɨɛɥɚɞɚɬɶ ɜɬɨɪɨɟ ɫɥɚɝɚɟɦɨɟ, ɢ ɬɨɝɞɚ ' g ɫɬɚɧɨɜɢɬɫɹ ɨɬɪɢɰɚɬɟɥɶɧɨɣ ɜɟɥɢɱɢɧɨɣ. Ɇɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ' g , ɤɨɬɨɪɨɟ ɨɛɨɡɧɚɱɢɦ
ɤɚɤ ' g , ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɚɤɬɢɜɚɰɢɨɧɧɵɣ ɛɚɪɶɟɪ ɞɥɹ ɡɚɪɨɠɞɟɧɢɣ.
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ' g ɩɨ ɪɚɞɢɭɫɭ ɢ ɩɪɢɪɚɜɧɢɜɚɹ ɪɟɡɭɥɶɬɚɬ ɧɭɥɸ, ɧɚɣɞɟɦ
ɤɪɢɬɢɱɟɫɤɢɣ ɪɚɞɢɭɫ ɤɚɩɥɢ
2v f J fg
r*
,
kT ln p p 0 ɫɥɟɞɨɜɚɬɟɥɶɧɨ
16v 2f J 3f g
' g*
.
2
3 ª¬ kT ln p p 0 º¼
Ʉɚɩɟɥɶɤɚ ɢɥɢ ɡɚɪɨɞɵɲ ɪɚɡɦɟɪɨɦ r r ɛɭɞɟɬ ɫɤɥɨɧɧɚ ɤ ɢɫɩɚɪɟɧɢɸ,
ɚ ɪɚɡɦɟɪɨɦ r ! r – ɤ ɪɨɫɬɭ. Ʌɸɛɨɣ ɢɡ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ ɛɭɞɟɬ ɫɨɩɪɨɜɨɠɞɚɬɶɫɹ ɭɦɟɧɶɲɟɧɢɟɦ ɷɧɟɪɝɢɢ. ȿɫɥɢ p p 0 d 1, ɫɜɨɛɨɞɧɚɹ ɷɧɟɪɝɢɹ Ƚɢɛɛɫɚ
' g ɛɭɞɟɬ ɜɨɡɪɚɫɬɚɬɶ ɩɪɢ ɥɸɛɨɦ r , ɢ ɜɫɟ ɤɚɩɥɢ ɛɭɞɭɬ ɧɟɭɫɬɨɣɱɢɜɵ ɨɬɧɨɫɢɬɟɥɶɧɨ ɢɫɩɚɪɟɧɢɹ.
ɑɬɨɛɵ ɪɚɫɫɱɢɬɚɬɶ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɡɚɪɨɞɵɲɟɣ ɩɨ ɪɚɡɦɟɪɚɦ, ɩɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɡɚɪɨɞɵɲɢ ɢɡ n ɚɬɨɦɨɜ, ɱɢɫɥɨ ɤɨɬɨɪɵɯ N n , ɨɛɪɚɡɭɸɬ «ɪɚɡɛɚɜɥɟɧɧɵɣ ɪɚɫɬɜɨɪ» ɜ ɦɨɧɨɦɨɥɟɤɭɥɹɪɧɨɦ ɩɚɪɟ. Ɍɨɝɞɚ ɷɧɬɪɨɩɢɹ ɫɦɟɲɟɧɢɹ ' sm ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɷɧɬɪɨɩɢɸ N v ɩɚɪɨɨɛɪɚɡɧɵɯ ɦɨɥɟɤɭɥ ɢɡ
ɨɛɳɟɝɨ ɱɢɫɥɚ N 0 ɱɚɫɬɢɰ ɢ ɡɚɪɨɞɵɲɟɣ
N0
N v ¦ Nn .
n
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
ª
§
§N ·
§ N · ·º
' sm k «N vln ¨ v ¸ ¦ ¨ N nln ¨ n ¸ ¸» .
© N0 ¹ n ©
© N 0 ¹ ¹¼»
¬«
265
ɂɡɦɟɧɟɧɢɟ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ ɷɬɨɣ ɫɢɫɬɟɦɵ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɨɝɨ, ɱɬɨ
ɡɚɪɨɞɵɲ ɢɡ n 1 ɦɨɥɟɤɭɥ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɡɚɪɨɞɵɲ ɢɡ n ɦɨɥɟɤɭɥ ɜ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɫɨɟɞɢɧɟɧɢɹ ɤ ɧɟɦɭ ɦɨɥɟɤɭɥɵ ɩɚɪɚ, ɫɤɥɚɞɵɜɚɟɬɫɹ ɢɡ ɷɧɟɪɝɢɣ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɯ ɢɡɦɟɧɟɧɢɹɦ N v , N n ɢ N 0 , ɚ ɬɚɤɠɟ ɢɡ ɢɡɦɟɧɟɧɢɣ
ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɢ ɨɛɴɟɦɧɨɣ ɷɧɟɪɝɢɣ.
ȿɫɥɢ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɡɚɪɨɞɵɲɟɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɚɜɧɨɜɟɫɧɨɦɭ ɫɨɫɬɨɹɧɢɸ, ɬɨ
n
§N ·
§ 'g ·
N n N 0 ¨ v ¸ exp ¨ n ¸ ,
© kT ¹
© N0 ¹
ɝɞɟ ' g n ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɭɪɚɜɧɟɧɢɸ (10.48) ɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɮɨɪɦɟ
' g n 2 3J LvK nkT ln p p0 ,
(10.49)
4 3 1
13
23
S r vL , ɢ ɦɧɨɠɢɬɟɥɶ K 36S vL ɨɬɧɨɫɢɬɫɹ ɤ ɫɮɟɪɟ.
3
ɉɨɫɤɨɥɶɤɭ, ɩɨ ɩɪɟɞɩɨɥɨɠɟɧɢɸ, ɨɛɳɟɟ ɱɢɫɥɨ ɡɚɪɨɞɵɲɟɣ ɦɚɥɨ
(ɬɨɥɶɤɨ ɩɪɢ ɬɚɤɨɦ ɭɫɥɨɜɢɢ ɦɵ ɦɨɠɟɦ ɝɨɜɨɪɢɬɶ ɨ ɪɚɡɛɚɜɥɟɧɧɨɦ ɪɚɫɬɜɨɪɟ), ɬɨ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
§ 'g ·
N n | N exp ¨ n ¸ ,
© kT ¹
ɝɞɟ N - ɨɛɳɟɟ ɱɢɫɥɨ ɦɨɥɟɤɭɥ.
ɋɤɨɪɨɫɬɶ ɡɚɪɨɠɞɟɧɢɹ l ɨɩɪɟɞɟɥɹɟɬɫɹ ɱɚɫɬɨɬɨɣ, ɫ ɤɨɬɨɪɨɣ ɨɞɢɧɨɱɧɵɟ ɦɨɥɟɤɭɥɵ ɫɨɭɞɚɪɹɸɬɫɹ ɢ ɫɨɟɞɢɧɹɸɬɫɹ ɫ ɤɪɢɬɢɱɟɫɤɢɦɢ ɡɚɪɨɞɵɲɚɦɢ
ɩɥɨɳɚɞɶɸ F . ɗɬɚ ɫɤɨɪɨɫɬɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɦɨɥɟɤɭɥ J 0 , ɩɥɨɳɚɞɢ F ɢ ɱɢɫɥɭ N n . Ɍɨɝɞɚ
§ 'g ·
I | J 0 F* N exp ¨ n ¸ .
© kT ¹
ɉɨɥɚɝɚɹ, ɱɬɨ ɫɩɪɚɜɟɞɥɢɜɨ ɫɨɨɬɧɨɲɟɧɢɟ ɢɡ ɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ ɝɚɡɨɜ
p
,
J0
2S mkT
ɢ ɢɫɩɨɥɶɡɭɹ ɜɦɟɫɬɨ ' g n ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ ' g , ɞɥɹ ɟɞɢɧɢɱɧɨɝɨ
ɨɛɴɟɦɚ ɧɚɣɞɟɦ
§
·
J 0 F*
16S vL2 J 3Lv
I
¨
¸.
exp (10.50)
|
2
3
NvL
vL
¨ 3 kT ªln p p º ¸
0 ¼ ¹
¬
©
ȼɵɪɚɠɟɧɢɟ ɞɥɹ ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ ɤɪɢɫɬɚɥɥɢɱɟɫɤɢɯ ɡɚɪɨɞɵɲɟɣ
ɢɡ ɩɟɪɟɫɵɳɟɧɧɨɝɨ ɩɚɪɚ ɥɢɲɶ ɫɥɟɝɤɚ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɜɵɪɚɠɟɧɢɹ (10.50).
ɩɪɢɱɟɦ n
266
ɉɪɢ ɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɚɝɪɟɝɚɬɵ ɢɦɟɸɬ ɫɬɪɭɤɬɭɪɭ ɤɪɢɫɬɚɥɥɢɡɭɸɳɟɝɨɫɹ ɬɜɟɪɞɨɝɨ ɜɟɳɟɫɬɜɚ. Ⱥɝɪɟɝɚɬ, ɫɨɞɟɪɠɚɳɢɣ n
ɦɨɥɟɤɭɥ, ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɡɚɪɨɞɵɲ ɤɪɢɬɢɱɟɫɤɨɝɨ ɪɚɡɦɟɪɚ, ɤɨɬɨɪɵɣ
ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɛɭɞɟɬ ɪɚɫɬɢ, ɚ ɧɟ ɞɟɝɪɚɞɢɪɨɜɚɬɶ. ȼ ɩɪɨɫɬɟɣɲɢɯ ɬɟɨɪɢɹɯ
ɩɪɢɧɢɦɚɟɬɫɹ, ɱɬɨ ɞɥɹ ɡɚɪɨɞɵɲɚ, ɫɨɞɟɪɠɚɳɟɝɨ n ɦɨɥɟɤɭɥ, ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ, ɱɬɨ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɟ ɤɨɧɫɬɚɧɬɵ ɬɚɤɨɝɨ ɦɚɥɨɝɨ ɚɝɪɟɝɚɬɚ ɢɞɟɧɬɢɱɧɵ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɢɦ ɩɚɪɚɦɟɬɪɚɦ ɦɨɧɨɤɪɢɫɬɚɥɥɚ, ɚ ɷɧɟɪɝɢɹ ɞɟɮɨɪɦɚɰɢɢ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɚ, ɫɬɚɧɞɚɪɬɧɭɸ ɫɜɨɛɨɞɧɭɸ ɷɧɟɪɝɢɸ ɨɛɪɚɡɨɜɚɧɢɹ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ, ɚɧɚɥɨɝɢɱɧɨɦ ɩɪɟɞɵɞɭɳɟɦɭ. Ɏɚɤɬɨɪ
ɮɨɪɦɵ K ɜɤɥɸɱɚɟɬɫɹ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɤɪɢɬɢɱɟɫɤɨɣ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ
' g . Ʉɪɢɫɬɚɥɥɢɱɟɫɤɢɣ ɡɚɪɨɞɵɲ ɫ ɦɢɧɢɦɚɥɶɧɨɣ ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɷɧɟɪɝɢɟɣ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɟ ɛɭɞɟɬ ɢɦɟɬɶ ɫɮɟɪɢɱɟɫɤɭɸ ɮɨɪɦɭ. ȼɦɟɫɬɨ ɱɚɫɬɨɬɵ
ɫɨɭɞɚɪɟɧɢɣ ɝɚɡɨɜɵɯ ɦɨɥɟɤɭɥ J 0 ɛɟɪɭɬ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɢɟɣ ɚɛɫɨɥɸɬɧɵɯ ɫɤɨɪɨɫɬɟɣ ɪɟɚɤɰɢɣ ɱɚɫɬɨɬɭ, ɫ ɤɨɬɨɪɨɣ ɚɬɨɦ ɩɪɟɨɞɨɥɟɜɚɟɬ ɷɧɟɪɝɟɬɢɱɟɫɤɢɣ ɛɚɪɶɟɪ ɧɚ ɝɪɚɧɢɰɟ ɦɟɠɞɭ ɠɢɞɤɨɫɬɶɸ ɢ ɤɪɢɫɬɚɥɥɢɱɟɫɤɢɦ ɡɚɪɨɞɵɲɟɦ, ɫɨɞɟɪɠɚɳɢɦ n ɚɬɨɦɨɜ. ȼ ɰɟɥɨɦ, ɢɦɟɟɬɫɹ ɜɫɟɝɨ ɥɢɲɶ ɞɜɚ ɧɚɞɟɠɧɵɯ ɩɪɢɦɟɪɚ ɝɨɦɨɝɟɧɧɨɝɨ ɡɚɪɨɞɵɲɟɨɛɪɚɡɨɜɚɧɢɹ: ɡɚɬɜɟɪɞɟɜɚɧɢɟ ɜɨɞɵ ɢ ɡɚɬɜɟɪɞɟɜɚɧɢɟ ɤɚɩɟɥɟɤ ɪɬɭɬɢ. ȼɫɟ ɨɫɬɚɥɶɧɵɟ ɫɥɭɱɚɢ ɫɥɟɞɭɟɬ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɝɟɬɟɪɨɝɟɧɧɨɟ ɡɚɪɨɞɵɲɟɨɛɪɚɡɨɜɚɧɢɟ. ɉɨɞɨɛɧɵɟ ɬɟɨɪɢɢ
ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɫɩɟɰɢɚɥɶɧɵɦ ɨɛɪɚɡɨɦ.
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. ɑɬɨ ɬɚɤɨɟ «ɮɚɡɚ»?
2. ɉɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪɵ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ.
3. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɭɫɥɨɜɢɟ ɮɚɡɨɜɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.
4. Ʉɚɤ ɫɜɹɡɚɧɵ ɞɚɜɥɟɧɢɹ ɜ ɮɚɡɚɯ ɧɚ ɢɫɤɪɢɜɥɟɧɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ
ɪɚɡɞɟɥɚ?*28
5. ɑɬɨ ɨɩɢɫɵɜɚɟɬ ɭɪɚɜɧɟɧɢɟ Ʉɥɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ?
6. ɂɡɨɛɪɚɡɢɬɟ ɩɪɨɫɬɟɣɲɭɸ ɞɢɚɝɪɚɦɦɭ ɫɨɫɬɨɹɧɢɹ ɞɥɹ ɫɢɫɬɟɦɵ
«ɩɚɪ-ɠɢɞɤɨɫɬɶ»
7. ɑɬɨ ɬɚɤɨɟ «ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ»?
8. ɑɬɨ ɬɚɤɨɟ «ɬɪɨɣɧɚɹ ɬɨɱɤɚ»?
9. ɑɬɨ ɨɩɢɫɵɜɚɟɬ ɡɚɞɚɱɚ ɋɬɟɮɚɧɚ?
10. Ⱦɚɣɬɟ ɬɟɪɦɨɞɢɧɚɦɢɱɟɫɤɭɸ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ.
11. ȼ ɱɟɦ ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɪɚɡɦɵɬɵɦɢ ɢ ɬɨɱɟɱɧɵɦɢ ɮɚɡɨɜɵɦɢ ɩɟɪɟɯɨɞɚɦɢ?
28
Ɂɧɚɤɨɦ «*» ɨɬɦɟɱɟɧɵ ɜɨɩɪɨɫɵ ɢ ɡɚɞɚɧɢɹ ɩɨɜɵɲɟɧɧɨɣ ɬɪɭɞɧɨɫɬɢ
267
12. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɬɥɢɱɢɟ ɜ ɩɨɜɟɞɟɧɢɢ ɬɟɩɥɨɟɦɤɨɫɬɟɣ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪ ɮɚɡɨɜɵɯ ɩɟɪɟɯɨɞɨɜ ɩɟɪɜɨɝɨ ɢ ɜɬɨɪɨɝɨ ɪɨɞɚ?*
13. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɬɟɨɪɢɢ ɞɜɭɯɮɚɡɧɨɣ ɡɨɧɵ?*
14. ɑɬɨ ɨɩɢɫɵɜɚɟɬ ɬɟɨɪɢɹ ɡɚɪɨɠɞɟɧɢɹ?*
Ɂɚɞɚɧɢɹ
1*. ɉɨɫɬɪɨɢɬɶ ɨɛɨɛɳɟɧɢɟ ɭɪɚɜɧɟɧɢɹ Ʉɥɚɩɟɣɪɨɧɚ–Ʉɥɚɭɡɢɭɫɚ ɞɥɹ
ɫɥɭɱɚɹ ɮɚɡɨɜɨɝɨ ɩɟɪɟɯɨɞɚ ɩɪɢ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɧɨ ɩɪɢ ɪɚɡɥɢɱɧɨɦ ɢɡɦɟɧɟɧɢɢ ɞɚɜɥɟɧɢɹ ɜ ɮɚɡɚɯ (ȼɵ ɞɨɥɠɧɵ ɩɨɥɭɱɢɬɶ ɭɪɚɜɧɟɧɢɟ
ɉɨɣɧɬɢɧɝɚ).
2*. ɉɨɥɶɡɭɹɫɶ ɩɨɧɹɬɢɟɦ ɞɟɥɶɬɚɨɛɪɚɡɧɨɣ ɮɭɧɤɰɢɢ, ɩɨɫɬɪɨɢɬɶ ɱɢɫɥɟɧɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (10.43), (10.44) ɨ ɮɚɡɨɜɨɦ ɩɟɪɟɯɨɞɟ.
3*. ɉɨɞɨɛɪɚɜ ɭɞɨɛɧɵɟ ɛɟɡɪɚɡɦɟɪɧɵɟ ɩɟɪɟɦɟɧɧɵɟ, ɢɡɨɛɪɚɡɢɬɶ ɤɚɱɟɫɬɜɟɧɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ' g n (10.49) ɩɪɢ ɜɚɪɶɢɪɨɜɚɧɢɢ ɨɬɧɨɲɟɧɢɹ
p p0 . Ʉɚɤɨɣ ɜɵɜɨɞ ɫɥɟɞɭɟɬ ɢɡ ɚɧɚɥɢɡɚ ɩɨɥɭɱɟɧɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ?
268
ɑȺɋɌɖ 11
ɋɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ:
ɞɢɮɮɭɡɢɹ ɢ ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ
11.1. ɉɪɨɫɬɟ ɣɲɢɟ ɩɨɧɹɬɢɹ ɨ ɤɢɧɟɬɢɤɟ
ɯɢɦ ɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ
ȼ ɩɪɢɪɨɞɟ ɢ ɜ ɩɪɨɦɵɲɥɟɧɧɵɯ ɭɫɥɨɜɢɹɯ ɩɪɨɬɟɤɚɟɬ ɦɧɨɠɟɫɬɜɨ ɯɢɦɢɱɟɫɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ, ɧɚɱɢɧɚɹ ɨɬ ɷɥɟɦɟɧɬɚɪɧɵɯ, ɩɨɞɪɨɛɧɨ ɢɡɭɱɟɧɧɵɯ, ɢ ɡɚɤɚɧɱɢɜɚɹ ɛɢɨɯɢɦɢɱɟɫɤɢɦɢ ɩɪɨɰɟɫɫɚɦɢ. Ʌɸɛɨɟ ɫɥɨɠɧɨɟ ɯɢɦɢɱɟɫɤɨɟ ɩɪɟɜɪɚɳɟɧɢɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɜɨɤɭɩɧɨɫɬɶ ɪɹɞɚ ɩɪɨɫɬɟɣɲɢɯ
ɪɟɚɤɰɢɣ, ɩɪɨɢɫɯɨɞɹɳɢɯ ɜ ɪɟɡɭɥɶɬɚɬɟ ɨɞɧɨɝɨ ɷɥɟɦɟɧɬɚɪɧɨɝɨ ɚɤɬɚ. Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɪɨɬɟɤɚɧɢɹ ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢɡɭɱɚɟɬ ɯɢɦɢɱɟɫɤɚɹ ɤɢɧɟɬɢɤɚ – ɧɚɭɤɚ ɨ ɫɤɨɪɨɫɬɹɯ ɯɢɦɢɱɟɫɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ. Ȼɟɡ ɧɟɟ ɧɟɜɨɡɦɨɠɧɨ ɢɡɭɱɟɧɢɟ ɦɟɯɚɧɢɡɦɨɜ ɪɟɚɤɰɢɣ ɢ ɩɨɫɬɪɨɟɧɢɟ ɝɪɚɦɨɬɧɨɣ ɦɨɞɟɥɢ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɝɨ ɩɪɨɰɟɫɫɚ, ɜɤɥɸɱɚɸɳɟɝɨ ɯɢɦɢɱɟɫɤɢɟ ɫɬɚɞɢɢ. Ɇɵ ɨɝɪɚɧɢɱɢɦɫɹ ɡɧɚɤɨɦɫɬɜɨɦ ɫ ɩɪɨɫɬɟɣɲɢɦɢ ɩɨɧɹɬɢɹɦɢ ɢ ɚɧɚɥɢɡɨɦ ɩɪɨɫɬɟɣɲɢɯ ɩɪɢɦɟɪɨɜ.
ɏɢɦɢɱɟɫɤɢ ɪɟɚɝɢɪɭɸɳɭɸ ɫɢɫɬɟɦɭ ɛɭɞɟɦ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɧɚɛɨɪɨɦ
ɤɨɧɰɟɧɬɪɚɰɢɣ ɤɨɦɩɨɧɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ-ɪɚɡɧɨɦɭ.
N
ȿɫɥɢ nk – ɱɢɫɥɨ ɦɨɥɟɣ ɤɨɦɩɨɧɟɧɬɚ k ɜ ɫɢɫɬɟɦɟ, ɚ n
¦ nk
– ɫɭɦɦɚ
k
ɦɨɥɟɣ ɜɫɟɯ ɤɨɦɩɨɧɟɧɬɨɜ ɷɬɨɣ ɫɢɫɬɟɦɵ ( N - ɱɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɤɨɦɩɨɧɟɧɬɨɜ – ɷɥɟɦɟɧɬɨɜ ɢ ɢɯ ɫɨɟɞɢɧɟɧɢɣ), ɬɨ ɩɨɞ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɦɨɥɶɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɟɣ ɛɭɞɟɦ ɩɨɧɢɦɚɬɶ ɜɟɥɢɱɢɧɭ
yk
Ɉɱɟɜɢɞɧɨ, ɱɬɨ
¦ yk
k nk
.
n
1.
ȿɫɥɢ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɣ ɨɛɴɟɦ – ɟɞɢɧɢɱɧɵɣ, ɬɨ ɪɚɡɦɟɪɧɨɫɬɶ nk –
ɦɨɥɶ/ɦ3.
ɉɪɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɦ ɦɨɞɟɥɢɪɨɜɚɧɢɢ ɯɢɦɢɱɟɫɤɢ ɪɟɚɝɢɪɭɸɳɢɯ ɫɢɫɬɟɦ ɱɚɫɬɨ ɨɤɚɡɵɜɚɸɬɫɹ ɭɞɨɛɧɵɦɢ ɦɚɫɫɨɜɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɤɨɦɩɨɧɟɧɬɨɜ
ɢɥɢ ɢɯ ɦɚɫɫɨɜɵɟ ɞɨɥɢ.
Ɍɚɤ, ɩɥɨɬɧɨɫɬɶ ɜɟɳɟɫɬɜɚ, ɧɚɯɨɞɹɳɟɝɨɫɹ ɜ ɨɛɴɟɦɟ V , ɨɱɟɜɢɞɧɨ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
269
M
,
V
U
ɝɞɟ M – ɦɚɫɫɚ ɜɟɳɟɫɬɜɚ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɨɛɴɟɦɟ.
N
ȿɫɥɢ M ɟɫɬɶ ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɚɫɫ ɪɚɡɧɵɯ ɤɨɦɩɨɧɟɧɬɨɜ, M
¦Mk ,
k 1
ɬɨ ɦɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɩɚɪɰɢɚɥɶɧɵɟ ɩɥɨɬɧɨɫɬɢ ɤɨɦɩɨɧɟɧɬɨɜ ɩɨ ɮɨɪɦɭɥɟ
Mk
, ɤɝ/ɦ3.
V
Uk
N
Ɉɱɟɜɢɞɧɨ, ɱɬɨ
¦ Uk
U.
k 1
Ɍɨɝɞɚ ɦɚɫɫɨɜɭɸ ɤɨɧɰɟɧɬɪɚɰɢɸ ɤɨɦɩɨɧɟɧɬɚ ɫ ɧɨɦɟɪɨɦ k ɨɩɪɟɞɟɥɢɦ ɩɨ ɮɨɪɦɭɥɟ
Uk N
, ¦ Ck 1 .
U k 1
Ck
Ɇɚɫɫɨɜɵɟ ɢ ɦɨɥɶɧɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɨɞɧɨɡɧɚɱɧɨ ɫɜɹɡɚɧɵ ɦɟɠɞɭ ɫɨUk
ɛɨɣ. Ɍɚɤ ɤɚɤ, nk
, ɝɞɟ mk – ɦɚɫɫɚ ɨɞɧɨɝɨ ɦɨɥɹ ɤɨɦɩɨɧɟɧɬɚ k (ɚ
mk
ɩɥɨɬɧɨɫɬɶ ɷɬɨ ɦɚɫɫɚ ɟɞɢɧɢɰɵ ɨɛɴɟɦɚ), ɬɨ ɧɚɯɨɞɢɦ
yk
Uk mk
N
¦Ui
.
mi
i 1
ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ ɨɛɵɱɧɨ ɜɵɪɚɠɚɸɬ ɫɤɨɪɨɫɬɶɸ
ɪɟɚɤɰɢɢ W , ɬ.ɟ. ɤɨɥɢɱɟɫɬɜɨɦ ɜɟɳɟɫɬɜɚ, ɪɟɚɝɢɪɭɸɳɢɦ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɜ ɞɚɧɧɨɣ ɫɢɫɬɟɦɟ. ɉɨɞ ɤɨɥɢɱɟɫɬɜɨɦ ɜɟɳɟɫɬɜɚ ɦɵ ɦɨɠɟɦ ɩɨɧɢɦɚɬɶ
ɱɢɫɥɨ ɦɨɥɟɣ ɤɨɦɩɨɧɟɧɬɚ ɢɥɢ ɥɸɛɭɸ ɞɪɭɝɭɸ ɫɜɹɡɚɧɧɭɸ ɫ ɧɢɦ ɜɟɥɢɱɢɧɭ.
Ɍɨɝɞɚ
W
dn
.
dt
ɉɨɞ ɭɞɟɥɶɧɨɣ ɫɤɨɪɨɫɬɶɸ ɪɟɚɤɰɢɢ ɩɨɧɢɦɚɸɬ ɤɨɥɢɱɟɫɬɜɨ ɜɟɳɟɫɬɜɚ,
ɪɟɚɝɢɪɭɸɳɟɝɨ ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɜ ɟɞɢɧɢɰɟ ɪɟɚɤɰɢɨɧɧɨɝɨ
ɩɪɨɫɬɪɚɧɫɬɜɚ:
270
wV
1 dn
–
V dt
ɞɥɹ ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɨɛɴɟɦɟ V (ɝɨɦɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ),
wS
1 dn
–
S dt
ɞɥɹ ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ S (ɝɟɬɟɪɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ).
ȿɫɥɢ ɨɞɧɚ ɢ ɬɚ ɠɟ ɪɟɚɤɰɢɹ ɩɪɨɬɟɤɚɟɬ ɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɢ ɜ ɨɛɴɟɦɟ,
ɬɨ ɟɟ ɫɤɨɪɨɫɬɶ ɜɵɪɚɡɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
dn
wS S wV V .
dt
ɉɟɪɜɢɱɧɵɟ ɞɚɧɧɵɟ ɤɢɧɟɬɢɱɟɫɤɢɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɧɚɛɨɪ ɤɨɧɰɟɧɬɪɚɰɢɣ ɯɢɦɢɱɟɫɤɢɯ ɤɨɦɩɨɧɟɧɬɨɜ ɢɥɢ ɧɟɤɨɬɨɪɵɯ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵɯ ɢɦ ɜɟɥɢɱɢɧ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɜɪɟɦɟɧɢ ɪɟɚɤɰɢɢ.
ɉɨ ɷɬɢɦ ɞɚɧɧɵɦ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɤɢɧɟɬɢɱɟɫɤɢɟ ɤɪɢɜɵɟ
«ɤɨɧɰɟɧɬɪɚɰɢɹ – ɜɪɟɦɹ».
Ɍɨɝɞɚ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɜ ɞɚɧɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɞɚɟɬɫɹ ɬɚɧɝɟɧɫɨɦ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɨɧɰɟɧɬɪɚɰɢɨɧɧɵɯ ɤɪɢɜɵɯ, ɩɨɥɭɱɚɟɦɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ, ɜ ɷɬɨɬ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ.
ɑɚɫɬɨ ɭɪɚɜɧɟɧɢɟ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɜɢɞɟ
Q1
M k >C1@
>C2 @Q2 ...
Q
k 3 >Ci @ i ,
i
(11.1)
ɝɞɟ Q1 ,Q 2 … – ɩɚɪɰɢɚɥɶɧɵɟ ɩɨɪɹɞɤɢ ɪɟɚɤɰɢɢ ɩɨ ɤɨɦɩɨɧɟɧɬɚɦ A,B, …, ɚ
>C1@ ,>C2 @ – ɤɨɧɰɟɧɬɪɚɰɢɢ ɷɬɢɯ ɤɨɦɩɨɧɟɧɬɨɜ (ɦɨɥɶɧɵɟ, ɦɚɫɫɨɜɵɟ…), k –
ɤɨɧɫɬɚɧɬɚ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ. ɍɪɚɜɧɟɧɢɟ (11.1) ɟɫɬɶ ɩɪɨɫɬɟɣɲɟɟ ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ.
ɗɥɟɦɟɧɬɚɪɧɵɟ ɪɟɚɤɰɢɢ ɛɵɜɚɸɬ ɪɟɚɤɰɢɹɦɢ ɧɭɥɟɜɨɝɨ, ɩɟɪɜɨɝɨ ɢ
ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ. Ȼɨɥɶɲɢɧɫɬɜɨ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ
ɤɚɤ ɬɟ ɢɥɢ ɢɧɵɟ ɤɨɦɛɢɧɚɰɢɢ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɬɚɞɢɣ.
Ɉɛɵɱɧɨ ɜɵɞɟɥɹɸɬ ɫɥɟɞɭɸɳɢɟ ɩɪɨɫɬɵɟ ɤɨɦɛɢɧɚɰɢɢ: ɩɚɪɚɥɥɟɥɶɧɵɟ
(ɤɨɧɤɭɪɢɪɭɸɳɢɟ, ɨɞɧɨɜɪɟɦɟɧɧɵɟ); ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɟ; ɨɛɪɚɬɢɦɵɟ ɪɟɚɤɰɢɢ. Ɋɟɚɤɰɢɸ, ɜ ɤɨɬɨɪɨɣ ɪɟɚɝɟɧɬ ɩɨɞɜɟɪɝɚɟɬɫɹ ɩɪɟɜɪɚɳɟɧɢɸ ɩɨ ɞɜɭɦ
ɢɥɢ ɧɟɫɤɨɥɶɤɢɦ ɩɭɬɹɦ ɨɞɧɨɜɪɟɦɟɧɧɨ, ɧɚɡɵɜɚɸɬ ɩɚɪɚɥɥɟɥɶɧɨɣ. ȼ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɣ ɪɟɚɤɰɢɢ ɩɪɨɞɭɤɬ, ɨɛɪɚɡɭɸɳɢɣɫɹ ɜ ɨɞɧɨɣ ɫɬɚɞɢɢ, ɹɜɥɹɟɬɫɹ
ɪɟɚɝɟɧɬɨɦ ɜ ɞɪɭɝɨɣ. ȿɫɥɢ ɞɜɟ ɫɬɚɞɢɢ ɪɟɚɤɰɢɢ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɝɨ ɧɚ271
ɩɪɚɜɥɟɧɢɹ ɩɪɨɢɫɯɨɞɹɬ ɫ ɫɨɢɡɦɟɪɢɦɨɣ ɜɟɪɨɹɬɧɨɫɬɶɸ, ɬɨ ɬɚɤɭɸ ɪɟɚɤɰɢɸ
ɧɚɡɵɜɚɸɬ ɨɛɪɚɬɢɦɨɣ. ɋɭɳɟɫɬɜɭɸɬ ɫɥɨɠɧɵɟ ɪɟɚɤɰɢɢ, ɫɨɱɟɬɚɸɳɢɟ ɜ ɫɟɛɟ ɞɜɚ ɢɥɢ ɜɫɟ ɬɪɢ ɧɚɡɜɚɧɧɵɯ ɬɢɩɚ. ɍɪɚɜɧɟɧɢɹ (11.1), ɩɨɥɭɱɟɧɧɵɟ ɧɚ
ɨɫɧɨɜɟ ɚɧɚɥɢɡɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ ɢɥɢ ɧɚ ɨɫɧɨɜɟ ɮɨɪɦɚɥɶɧɨɣ
ɤɢɧɟɬɢɤɢ, ɢɦɟɸɬ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ, ɟɫɥɢ ɩɨɪɹɞɤɢ ɪɟɚɤɰɢɢ – ɩɪɨɫɬɵɟ
ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ. Ⱦɪɨɛɧɵɟ ɩɨɪɹɞɤɢ ɨɤɚɡɵɜɚɸɬɫɹ «ɤɚɠɭɳɢɦɢɫɹ» ɢ
ɟɫɬɶ ɫɥɟɞɫɬɜɢɟ ɫɥɨɠɧɨɝɨ ɦɟɯɚɧɢɡɦɚ ɪɟɚɤɰɢɢ, ɧɟ ɨɩɢɫɵɜɚɟɦɨɝɨ ɩɪɨɫɬɨɣ
ɪɟɚɤɰɢɨɧɧɨɣ ɫɯɟɦɨɣ ɢ ɜɤɥɸɱɚɸɳɟɝɨ ɜ ɤɚɱɟɫɬɜɟ ɨɬɞɟɥɶɧɵɯ ɫɬɚɞɢɣ ɪɚɡɧɨɨɛɪɚɡɧɵɟ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ.
11.2. ɉɪɢɦɟɪɵ ɨɩɢɫɚɧɢɹ ɤɢɧɟɬɢɤɢ
ɝɨɦ ɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɵɟ ɩɪɢɦɟɪɵ ɤɢɧɟɬɢɱɟɫɤɢɯ ɫɯɟɦ, ɯɚɪɚɤɬɟɪɧɵɯ ɞɥɹ
ɨɩɢɫɚɧɢɹ ɝɨɦɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ, ɬ.ɟ. ɞɥɹ ɝɨɦɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɨɛɴɟɦɟ ɨɞɧɨɪɨɞɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ȼ ɫɥɭɱɚɟ ɩɪɨɫɬɨɣ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
k
A 
oB
(11.2)
ɟɟ ɫɤɨɪɨɫɬɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɤɨɧɰɟɧɬɪɚɰɢɢ A :
d > A@ dt k > A@ .
(11.3)
Ⱦɥɹ ɜɟɳɟɫɬɜɚ B – ɩɪɨɞɭɤɬɚ ɪɟɚɤɰɢɢ ɫɩɪɚɜɟɞɥɢɜɨ ɚɧɚɥɨɝɢɱɧɨɟ
ɭɪɚɜɧɟɧɢɟ
d >B@ dt k >B@ .
ȿɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ
ɹɜɥɹɟɬɫɹ 1/ɫ.
ɇɨ ɜɫɥɟɞɫɬɜɢɟ ɫɨɯɪɚɧɟɧɢɹ ɜɟɳɟɫɬɜɚ ɜ ɫɢɫɬɟɦɟ
>A@ >B @ >A@0 >B @0
ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ
const
ɧɚɦ ɞɨɫɬɚɬɨɱɧɨ ɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (11.3).
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɢ ɩɨɫɬɨɹɧɫɬɜɟ ɞɚɜɥɟɧɢɹ
p const ɢ ɬɟɦɩɟɪɚɬɭɪɵ T const ɞɚɟɬ
d > A@
d ln > A@ k T , p dt ,
A
ln > A@ kt C , C ln > A@0 ,
ɝɞɟ > A@0 – ɧɚɱɚɥɶɧɚɹ ɤɨɧɰɟɧɬɪɚɰɢɹ A .
272
> A@ > A@0 exp kt .
(11.4)
ɍɪɚɜɧɟɧɢɟ (11.4) ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɜ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɪɚɫɯɨɞɨɜɚɧɢɟ ɪɟɚɝɟɧɬɚ ɜɨ ɜɪɟɦɟɧɢ ɩɪɨɢɫɯɨɞɢɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨ.
Ɋɟɡɭɥɶɬɚɬ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɝɪɚɮɢɱɟɫɤɢ (ɪɢɫ. 11.1).
ɍɝɨɥ ɧɚɤɥɨɧɚ ɩɪɹɦɨɣ ln > A@ ɢ ɨɩɪɟɞɟɥɹɟɬ ɤɨɧɫɬɚɧɬɭ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ.
ȼɜɟɞɟɦ ɩɟɪɢɨɞ ɩɨɥɭɩɪɟɜɪɚɳɟɧɢɹ
ln2
t1 2
–
k
ɜɪɟɦɹ ɞɨɫɬɢɠɟɧɢɹ ɤɨɧɰɟɧɬɪɚɰɢɟɣ ɪɟɚɝɟɧɬɚ ɩɨɥɨɜɢɧɵ ɩɟɪɜɨɧɚɱɚɥɶɧɨɣ
ɜɟɥɢɱɢɧɵ.
Ɋɢɫ. 11.1. ɂɥɥɸɫɬɪɚɰɢɹ ɤ ɦɨɞɟɥɢ
ɪɟɚɤɰɢɢ 1-ɝɨ ɩɨɪɹɞɤɚ
Ɋɢɫ. 11.2. ɂɡɦɟɧɟɧɢɟ ɜɨ ɜɪɟɦɟɧɢ
ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɟɚɝɟɧɬɚ ɢ ɩɪɨɞɭɤɬɚ
ɞɥɹ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
ȼɜɨɞɹ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɨ ɮɨɪɦɭɥɚɦ
> A@ , z
>B@ ,
y
> A@0 >B@0
> A@0 >B@0
ɬɚɤ ɱɬɨ y z 1 , ɩɪɟɞɫɬɚɜɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɧɰɟɧɬɪɚɰɢɣ ɨɬ ɜɪɟɦɟɧɢ ɧɚ
ɪɢɫ. 11.2. ɉɟɪɟɫɟɱɟɧɢɟ ɤɪɢɜɵɯ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɥɨɜɢɧɟ ɧɚɱɚɥɶɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɟɚɝɟɧɬɚ.
ɉɪɨɫɬɭɸ ɪɟɚɤɰɢɸ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
2> A@ 
o> B@ .
ȿɞɢɧɢɰɟɣ ɢɡɦɟɪɟɧɢɹ ɤɨɧɫɬɚɧɬɵ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
ɫɥɭɠɢɬ ɜɟɥɢɱɢɧɚ
k
1/(ɪɚɡɦɟɪɧɨɫɬɶ ɤɨɧɰɟɧɬɪɚɰɢɢ * ɫ)
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ȼ ɫɥɭɱɚɟ ɩɪɨɫɬɨɣ ɪɟɚɤɰɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɭɪɚɜɧɟɧɢɟ ɫɤɨɪɨɫɬɢ
ɢɦɟɟɬ ɜɢɞ
d > A@ dt 2k > A@ .
(11.5)
Ɇɧɨɠɢɬɟɥɶ 2 ɩɨɹɜɥɹɟɬɫɹ ɩɨɬɨɦɭ, ɱɬɨ ɜ ɤɚɠɞɨɦ ɷɥɟɦɟɧɬɚɪɧɨɦ ɚɤɬɟ
ɪɚɫɯɨɞɭɸɬɫɹ ɞɜɟ ɦɨɥɟɤɭɥɵ. ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɚɟɬ
1
1
(11.6)
2kt .
> A@ > A@0
2
ɢɥɢ
> A@
> A@0
.
1 2kt > A@0
ȿɫɥɢ ɦɵ ɩɨɫɬɪɨɢɦ ɡɚɜɢɫɢɦɨɫɬɶ ɨɛɪɚɬɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɨɬ ɜɪɟɦɟɧɢ, ɬɨ ɦɵ
ɩɨɥɭɱɢɦ ɩɪɹɦɭɸ ɥɢɧɢɸ ɫ ɧɚɤɥɨɧɨɦ 2k
(ɪɢɫ. 11.3). Ⱦɥɹ ɪɚɡɧɵɯ ɧɚɱɚɥɶɧɵɯ ɤɨɧɰɟɧɬɪɚɰɢɣ ɩɨɥɭɱɢɦ ɧɚɛɨɪ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɹɦɵɯ.
ȿɫɥɢ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɪɟɚɤɰɢɹ ɢɦɟɟɬ
ɜɬɨɪɨɣ ɩɨɪɹɞɨɤ, ɬɨ ɧɚ ɨɫɧɨɜɟ ɞɚɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚ ɤɨɧɫɬɚɧɬɭ ɫɤɨɪɨɫɬɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨ ɮɨɪɦɭɥɟ
k
(11.7)
Ɋɢɫ. 11.3. ɂɥɥɸɫɬɪɚɰɢɹ ɤ
ɦɨɞɟɥɢ ɷɥɟɦɟɧɬɚɪɧɨɣ
ɪɟɚɤɰɢɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ
> A@0 > A@
.
2> A@0 > A@t
ɉɨɞɫɬɚɜɥɹɹ > A@0 2 ɜɦɟɫɬɨ > A@ , ɜ ɭɪɚɜɧɟɧɢɟ (11.6), ɧɚɣɞɟɦ ɩɟɪɢɨɞ
ɩɨɥɭɩɪɟɜɪɚɳɟɧɢɹ
1
.
t1 2
2k > A@0
ȼ ɨɬɥɢɱɢɟ ɨɬ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ, ɩɟɪɢɨɞ ɩɨɥɭɩɪɟɜɪɚɳɟɧɢɹ
ɡɚɜɢɫɢɬ ɨɬ ɧɚɱɚɥɶɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɟɚɝɟɧɬɚ.
Ⱦɪɭɝɨɣ ɪɟɚɤɰɢɟɣ, ɜ ɤɨɬɨɪɨɣ ɜ ɨɛɪɚɡɨɜɚɧɢɢ ɩɪɨɞɭɤɬɚ ɭɱɚɫɬɜɭɸɬ ɞɜɟ
ɦɨɥɟɤɭɥɵ, ɛɭɞɟɬ ɪɟɚɤɰɢɹ ɜɢɞɚ
> A@1 > A@2 o >P@ .
ɗɬɨ – ɛɢɦɨɥɟɤɭɥɹɪɧɚɹ ɪɟɚɤɰɢɹ ɫ ɫɭɦɦɚɪɧɵɦ ɜɬɨɪɵɦ ɩɨɪɹɞɤɨɦ.
ɗɬɭ ɪɟɚɤɰɢɸ ɦɨɠɧɨ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɜ ɪɚɡɧɵɯ ɫɢɬɭɚɰɢɹɯ:
274
1) ɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɢɟ ɧɚɱɚɥɶɧɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ;
2) ɨɞɢɧ ɢɡ ɪɟɚɝɟɧɬɨɜ ɛɟɪɟɬɫɹ ɜ ɢɡɛɵɬɤɟ ɬɚɤ, ɱɬɨ ɟɝɨ ɤɨɧɰɟɧɬɪɚɰɢɸ ɜ
ɯɨɞɟ ɪɟɚɤɰɢɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɢɡɦɟɧɧɨɣ.
ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ
d > A@1
d > P@
2
k > A@1 > A@2 k > A@1 > A@0 > A@1 ɢɥɢ
k > A@1 .
dt
dt
(11.8)
ɗɬɨɬ ɫɥɭɱɚɣ ɫɜɨɞɢɬɫɹ ɤ ɩɪɟɞɵɞɭɳɟɦɭ, ɧɨ ɜɦɟɫɬɨ 2k ɜ ɭɪɚɜɧɟɧɢɢ
ɫɬɨɢɬ k
1
1
(11.9)
kt .
> A@ > A@0
ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɞɨɩɭɫɬɢɦ, ɱɬɨ > A@20 !! > A@10 . Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ
ɞɥɹ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
d > A@1
(11.10)
keff > A@1 ,
dt
ɝɞɟ keff k > A@20 . Ɍ.ɟ. ɷɬɨɬ ɫɥɭɱɚɣ ɫɜɨɞɢɬɫɹ ɤ ɩɟɪɜɨɦɭ. Ƚɨɜɨɪɹɬ, ɱɬɨ ɜ
ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɚɤɰɢɹ ɩɪɨɯɨɞɢɬ ɜ ɭɫɥɨɜɢɹɯ ɩɫɟɜɞɨɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ (ɩɨ
ɨɬɧɨɲɟɧɢɸ ɤ A1 ).
ɉɪɢ ɩɪɨɬɟɤɚɧɢɢ ɞɜɭɯ ɩɚɪɚɥɥɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɫɨ
ɫɤɨɪɨɫɬɹɦɢ k1 ɢ k 2 ɪɟɚɤɰɢɨɧɧɭɸ ɫɯɟɦɭ ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
P1
k1
A
k2
P2
Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɪɟɚɝɟɧɬɨɜ ɢ ɩɪɨɞɭɤɬɨɜ ɢɦɟɸɬ
ɜɢɞ
d > A@ dt k1 k2 > A@ ,
d > P1@ dt k1> A@ ,
(11.11)
d >P2 @ dt k2 > A@ .
ɇɟɡɚɜɢɫɢɦɵɦɢ ɢɡ ɧɢɯ ɹɜɥɹɸɬɫɹ ɬɨɥɶɤɨ ɞɜɚ. ɉɟɪɜɨɟ ɭɪɚɜɧɟɧɢɟ ɢɧɬɟɝɪɢɪɭɟɬɫɹ ɫɪɚɡɭ
(11.12)
> A@ > A@0 exp k1 k2 t .
ɂɡ ɞɜɭɯ ɞɪɭɝɢɯ ɫɥɟɞɭɟɬ
>P1@ >P2 @
k1 k2 ,
ɬ.ɟ. ɨɬɧɨɲɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɣ ɩɪɨɞɭɤɬɨɜ ɡɚɜɢɫɢɬ ɬɨɥɶɤɨ ɨɬ ɨɬɧɨɲɟɧɢɹ
ɤɨɧɫɬɚɧɬ ɫɤɨɪɨɫɬɟɣ.
275
ɂɫɩɨɥɶɡɭɹ (11.12) ɢ
> A@0 > A@ >P1@ >P2 @ , ɩɨɥɭɱɚɟɦ
ɭɪɚɜɧɟɧɢɟ
ɦɚɬɟɪɢɚɥɶɧɨɝɨ
ɛɚɥɚɧɫɚ
k1
(11.13)
> A@ ª1 exp k1 k2 t º¼ ,
k1 k2 0 ¬
k2
(11.14)
>P2 @
> A@ ª1 exp k1 k2 t º¼ .
k1 k2 0 ¬
Ʉ ɬɟɦ ɠɟ ɫɚɦɵɦ ɭɪɚɜɧɟɧɢɹɦ ɦɨɠɧɨ ɩɪɢɣɬɢ, ɩɨɞɫɬɚɜɥɹɹ (11.12) ɜɨ
ɜɬɨɪɨɟ ɢ ɬɪɟɬɶɟ ɭɪɚɜɧɟɧɢɹ (11.11) ɢ ɢɧɬɟɝɪɢɪɭɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ.
>P1@
ɋɭɬɶ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɫɥɭɱɚɹ ɫɜɨɞɢɬɫɹ ɤ ɬɨɦɭ, ɱɬɨ ɤɚɤ ɪɚɫɯɨɞɨɜɚɧɢɟ
ɪɟɚɝɟɧɬɚ, ɬɚɤ ɢ ɨɛɪɚɡɨɜɚɧɢɟ ɩɪɨɞɭɤɬɨɜ, ɩɨɞɱɢɧɹɸɬɫɹ ɡɚɤɨɧɭ ɫɤɨɪɨɫɬɢ
ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. Ɍɚɤ, ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɟɣ
ln > A@ ,
ln >P@1f >P@1 ,
ln >P@2f > P@2 ɨɬ ɜɪɟɦɟɧɢ ɞɚɸɬ ɩɪɹɦɭɸ ɥɢɧɢɸ, ɬɚɧɝɟɧɫ ɭɝɥɚ ɧɚɤɥɨɧɚ ɤɨɊɢɫ. 11.4. Ɂɚɜɢɫɢɦɨɫɬɶ ɤɨɧɰɟɧɬɨɪɨɣ ɪɚɜɟɧ k1 k2 , ɱɬɨ ɚɧɚɥɨɝɢɱɧɨ
ɬɪɚɰɢɣ ɨɬ ɜɪɟɦɟɧɢ ɞɥɹ ɞɜɭɯ
ɩɚɪɚɥɥɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ
ɪɢɫ. 11.1.
ɩɨɪɹɞɤɚ
Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɚɠɞɨɣ ɢɡ ɤɨɧɫɬɚɧɬ ɩɨ ɞɚɧɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɚ ɧɭɠɧɨ ɢɡ (11.13), (11.14) ɢɫɤɥɸɱɢɬɶ ɜɵɪɚɠɟɧɢɟ ɜ ɤɜɚɞɪɚɬɧɵɯ ɫɤɨɛɤɚɯ ɫ ɩɨɦɨɳɶɸ (11.12). ɇɚɣɞɟɦ
>P@1
k1
(11.15)
k
k
A
A
> @ > @ 1 2
ɢ
0
>P@1
> A@0 > A@
k1
k1 k2
(11.16)
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ ɤɨɧɰɟɧɬɪɚɰɢɣ ɩɪɨɞɭɤɬɨɜ ɨɬ
ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɟɚɝɟɧɬɚ > A@0 > A@ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɹɦɵɟ ɥɢɧɢɢ ɫ
ɪɚɡɧɵɦɢ ɧɚɤɥɨɧɚɦɢ.
Ⱦɥɹ ɭɞɨɛɫɬɜɚ ɜɜɟɞɟɦ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɤɨɧɰɟɧɬɪɚɰɢɢ
> A@ , z >P@1 , z >P2 @ .
y
> A@0 1 > A@0 2 > A@0
Ɍɨɝɞɚ ɜɦɟɫɬɨ (11.13), (11.14) ɧɚɣɞɟɦ
276
k1
ª1 exp k1 k2 t º¼ ,
k1 k2 ¬
k2
ª1 exp k1 k2 t º¼ .
z2
k1 k2 ¬
ɂɡɦɟɧɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɣ ɜɨ ɜɪɟɦɟɧɢ ɞɥɹ ɷɬɨɝɨ ɫɥɭɱɚɹ ɩɨɤɚɡɚɧɨ ɧɚ
ɪɢɫ. 11.4. ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɤɨɧɫɬɚɧɬ ɫɤɨɪɨɫɬɢ.
z1
ɉɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɞɜɭɯ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɨɬɜɟɱɚɟɬ ɫɯɟɦɟ
k
k
1o B 
2o P
A 
ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɩɪɨɫɬɟɣɲɢɣ ɩɪɢɦɟɪ ɦɧɨɝɨɫɬɚɞɢɣɧɨɣ ɪɟɚɤɰɢɢ.
Ɂɞɟɫɶ ɧɭɠɧɨ ɪɚɡɥɢɱɚɬɶ ɬɪɢ ɜɚɪɢɚɧɬɚ.
1. ɉɪɢ k1 !! k 2 ɩɪɨɦɟɠɭɬɨɱɧɵɣ ɩɪɨɞɭɤɬ ɝɨɪɚɡɞɨ ɦɟɧɟɟ ɪɟɚɤɰɢɨɧɧɨɫɩɨɫɨɛɟɧ, ɱɟɦ ɢɫɯɨɞɧɵɣ ɪɟɚɝɟɧɬ. ɉɟɪɜɚɹ ɫɬɚɞɢɹ ɩɪɚɤɬɢɱɟɫɤɢ ɡɚɜɟɪɲɚɟɬɫɹ ɞɨ ɬɨɝɨ, ɤɚɤ ɧɚɱɢɧɚɟɬɫɹ ɜɬɨɪɚɹ ɫɬɚɞɢɹ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɤɚɠɞɭɸ ɫɬɚɞɢɸ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɧɟɡɚɜɢɫɢɦɭɸ ɪɟɚɤɰɢɸ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ.
2. ȼ ɫɥɭɱɚɟ k1 k 2 ɜɬɨɪɚɹ ɫɬɚɞɢɹ ɛɵɫɬɪɨ ɫɥɟɞɭɟɬ ɡɚ ɩɟɪɜɨɣ, ɩɨɷɬɨɦɭ ɜ ɥɸɛɨɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɫɩɪɚɜɟɞɥɢɜɨ
>B@ >A@ ɢ > P @ .
3. ȼ ɫɥɭɱɚɟ ɧɟɡɧɚɱɢɬɟɥɶɧɨɝɨ ɪɚɡɥɢɱɢɹ ɪɟɚɤɰɢɨɧɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ
ɢɫɯɨɞɧɨɝɨ ɪɟɚɝɟɧɬɚ ɢ ɩɪɨɦɟɠɭɬɨɱɧɨɝɨ ɩɪɨɞɭɤɬɚ ɧɚɛɨɪ ɨɛɳɢɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɜɢɞ
d > A@ dt k1> A@ ,
d >B@ dt k1> A@ k2 > B@ ,
(11.17)
d > P@ dt k2 > B@ .
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t
> A@ > A@0 , >B@
0 ɢɦɟɟɦ:
0 , >P@ 0 .
ɂɡ ɩɟɪɜɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫɢɫɬɟɦɵ (11.17) ɢɦɟɟɦ
> A@ > A@0 expk1t .
(11.18)
ɉɨɞɫɬɚɜɥɹɹ ɷɬɨ ɜɵɪɚɠɟɧɢɟ ɜɨ ɜɬɨɪɨɟ ɭɪɚɜɧɟɧɢɟ ɫɢɫɬɟɦɵ (11.17),
ɧɚɣɞɟɦ
d > B@
k2 >B@ k1> A@0 e k1t .
(11.19)
dt
277
ɑɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɢɳɟɦ ɜ ɜɢɞɟ
>B@
D e k1t .
Ɍɨɝɞɚ ɢɡ (11.19) ɢɦɟɟɦ
k1D e k1t k2D e k1t k1 > A@0 e k1t ,
ɨɬɤɭɞɚ ɧɚɯɨɞɢɦ
D
k1 > A@0
.
k2 k1
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɧɟɨɞɧɨɪɨɞɧɨɦɭ ɭɪɚɜɧɟɧɢɸ (11.19), ɢɦɟɟɬ ɜɢɞ
ln > B@ k2t ln C
ɢɥɢ
>B@
C exp k2t .
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (11.19) ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ
>B@
C exp k2t k1> A@0
exp k1t k2 k1
ɢɥɢ, ɢɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɤ ɡɚɞɚɱɟ (11.17), – ɜ ɜɢɞɟ
k > A@
>B@ 1 0 ª¬expk1t expk2t º¼ .
k k
2
(11.20)
1
ɂɡ ɭɪɚɜɧɟɧɢɹ ɦɚɬɟɪɢɚɥɶɧɨɝɨ ɛɚɥɚɧɫɚ ɧɚɯɨɞɢɦ ɤɨɧɰɟɧɬɪɚɰɢɸ ɩɪɨɞɭɤɬɚ ɪɟɚɤɰɢɢ
ª
º
1
k2exp k1t k1exp k2t » .
(11.21)
>P@ > A@0 «1 ¬ k2 k1
¼
Ʉɚɤ ɜɢɞɧɨ, ɤɢɧɟɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɡɚɦɟɬɧɨ ɭɫɥɨɠɧɢɥɨɫɶ, ɢ ɞɚɥɟɟ
ɫɥɨɠɧɨɫɬɶ ɨɩɢɫɚɧɢɹ ɡɚɦɟɬɧɨ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɨ ɦɟɪɟ ɭɜɟɥɢɱɟɧɢɹ ɱɢɫɥɚ
ɫɬɚɞɢɣ. Ȼɨɥɟɟ ɬɨɝɨ, ɚɧɚɥɨɝɢɱɧɵɟ ɚɧɚɥɢɬɢɱɟɫɤɢɟ ɪɟɲɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ
ɩɨɥɭɱɟɧɵ ɥɢɲɶ ɞɥɹ ɫɢɫɬɟɦɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. Ⱦɥɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɩɨɪɹɞɤɚ ɩɨɥɭɱɚɸɬɫɹ
ɫɢɫɬɟɦɵ ɧɟɥɢɧɟɣɧɵɯ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ
ɪɟɲɢɬɶ ɚɧɚɥɢɬɢɱɟɫɤɢ ɥɢɲɶ ɜ ɨɝɪɚɧɢɱɟɧɧɨɦ ɱɢɫɥɟ ɫɥɭɱɚɟɜ.
Ȼɨɥɶɲɭɸ ɝɪɭɩɩɭ ɪɟɚɤɰɢɣ ɫɨɫɬɚɜɥɹɸɬ ɝɨɦɨɝɟɧɧɵɟ ɤɚɬɚɥɢɬɢɱɟɫɤɢɟ
ɪɟɚɤɰɢɢ, ɜ ɤɨɬɨɪɵɯ ɤɚɬɚɥɢɡɚɬɨɪ ɭɱɚɫɬɜɭɟɬ ɜ ɨɛɪɚɡɨɜɚɧɢɢ ɜɵɫɨɤɨɪɟɚɤɰɢɨɧɧɨɫɩɨɫɨɛɧɨɝɨ ɩɪɨɦɟɠɭɬɨɱɧɨɝɨ ɩɪɨɞɭɤɬɚ, ɧɟ ɦɟɧɹɹ ɫɬɟɯɢɨɦɟɬɪɢɢ ɪɟ278
ɚɤɰɢɢ. ȿɫɥɢ ɤɚɬɚɥɢɡɚɬɨɪɨɦ ɪɟɚɤɰɢɢ ɹɜɥɹɟɬɫɹ ɤɨɧɟɱɧɵɣ ɢɥɢ ɩɪɨɦɟɠɭɬɨɱɧɵɣ ɩɪɨɞɭɤɬ, ɬɨ ɬɚɤɢɟ ɪɟɚɤɰɢɢ ɧɚɡɵɜɚɸɬ ɚɜɬɨɤɚɬɚɥɢɬɢɱɟɫɤɢɦɢ.
ɋɬɪɨɝɨɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ ɩɨɞɨɛɧɵɯ ɪɟɚɤɰɢɣ ɦɨɠɟɬ ɛɵɬɶ ɜɟɫɶɦɚ
ɩɪɨɛɥɟɦɚɬɢɱɧɵɦ.
11.3. Ʉɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ
ɋɭɳɟɫɬɜɟɧɧɨɟ ɭɩɪɨɳɟɧɢɟ ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɦɨɠɟɬ ɛɵɬɶ
ɞɨɫɬɢɝɧɭɬɨ ɩɪɢ ɜɜɟɞɟɧɢɢ ɫɬɚɰɢɨɧɚɪɧɨɝɨ (ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɝɨ) ɩɪɢɛɥɢɠɟɧɢɹ.
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɜɦɟɫɬɨ ɫɢɫɬɟɦɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɪɨɰɟɫɫɚ ɦɨɝɭɬ ɛɵɬɶ ɨɩɢɫɚɧɵ ɫɢɫɬɟɦɨɣ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ
ɭɪɚɜɧɟɧɢɣ, ɱɬɨ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɛɨɥɟɟ ɢɥɢ ɦɟɧɟɟ ɩɪɨɫɬɵɟ ɪɟɲɟɧɢɹ
ɞɥɹ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɡɚɞɚɱ.
Ɋɚɫɫɦɨɬɪɢɦ ɨɫɨɛɟɧɧɨɫɬɢ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɧɚ ɩɪɢɦɟɪɟ
ɫɢɫɬɟɦɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɪɟɚɤɰɢɣ.
ɋɢɫɬɟɦɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɦɵ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɫɬɚɰɢɨɧɚɪɧɨɣ, ɟɫɥɢ ɫɤɨɪɨɫɬɢ ɜɫɟɯ ɫɬɚɞɢɣ ɷɬɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɧɟ ɢɡɦɟɧɹɸɬɫɹ
ɜɨ ɜɪɟɦɟɧɢ. ɇɟɢɡɦɟɧɧɨɫɬɢ ɫɤɨɪɨɫɬɟɣ ɫɬɚɞɢɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɨɫɬɨɹɧɫɬɜɨ
ɤɨɧɰɟɧɬɪɚɰɢɣ ɜɫɟɯ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɜɟɳɟɫɬɜ. ȼ ɧɚɲɟɦ ɩɪɨɫɬɨɦ ɫɥɭɱɚɟ
ɷɬɨ ɨɡɧɚɱɚɟɬ
d > B@
k1> A@ k2 > B@ 0
dt
(11.22)
Ɍɨɝɞɚ
>B@st
k1
> A@ .
k2
(11.23)
Ɍ.ɟ., ɩɪɢ ɫɬɚɰɢɨɧɚɪɧɨɦ ɩɪɨɬɟɤɚɧɢɢ ɩɪɨɰɟɫɫɚ ɫɤɨɪɨɫɬɢ ɨɛɟɢɯ ɫɬɚɞɢɣ
ɪɚɜɧɵ. ȼ ɩɪɨɬɢɜɧɨɦ ɫɥɭɱɚɟ ɤɨɧɰɟɧɬɪɚɰɢɹ ɩɪɨɦɟɠɭɬɨɱɧɨɝɨ ɩɪɨɞɭɤɬɚ
ɦɟɧɹɥɚɫɶ ɛɵ ɫɨ ɜɪɟɦɟɧɟɦ.
ȼ ɡɚɤɪɵɬɨɣ ɫɢɫɬɟɦɟ ɷɬɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɪɟɚɤɰɢɣ ɧɟ ɦɨɠɟɬ ɩɪɨɬɟɤɚɬɶ ɫɬɚɰɢɨɧɚɪɧɨ. ɗɬɨ ɨɛɭɫɥɨɜɥɟɧɨ ɬɟɦ, ɱɬɨ ɤɨɧɰɟɧɬɪɚɰɢɹ ɪɟɚɝɟɧɬɚ (ɢ
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɫɤɨɪɨɫɬɶ ɩɟɪɜɨɣ ɫɬɚɞɢɢ) ɜ ɡɚɤɪɵɬɨɣ ɫɢɫɬɟɦɟ ɛɭɞɟɬ
ɭɦɟɧɶɲɚɬɶɫɹ ɩɨ ɦɟɪɟ ɟɝɨ ɪɚɫɯɨɞɨɜɚɧɢɹ. Ⱦɥɹ ɞɨɫɬɢɠɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɫɬɢ ɬɪɟɛɭɟɬɫɹ ɜɜɟɫɬɢ ɢɫɬɨɱɧɢɤ ɤɨɦɩɨɧɟɧɬɚ A , ɦɨɳɧɨɫɬɶ ɤɨɬɨɪɨɝɨ ɪɚɜɧɚ
ɫɤɨɪɨɫɬɢ ɪɚɫɯɨɞɨɜɚɧɢɹ ɜɟɳɟɫɬɜɚ A . Ɍɚɤɚɹ ɫɢɬɭɚɰɢɹ ɥɟɝɤɨ ɦɨɠɟɬ ɛɵɬɶ
ɪɟɚɥɢɡɨɜɚɧɚ ɜ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ.
Ɂɚɮɢɤɫɢɪɭɟɦ ɷɬɨ ɞɨɩɨɥɧɢɬɟɥɶɧɨɟ ɭɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɨɦ > A@
279
> A@0 .
Ɍɨɝɞɚ ɫɬɚɰɢɨɧɚɪɧɨɫɬɶ ɪɟɚɤɰɢɢ ɫɬɚɧɨɜɢɬɫɹ ɨɱɟɜɢɞɧɨɣ. ɇɨ ɜɨɬ ɜɨɩɪɨɫ ɨɛ ɷɜɨɥɸɰɢɢ ɫɢɫɬɟɦɵ ɤ ɫɬɚɰɢɨɧɚɪɧɨɦɭ ɪɟɠɢɦɭ ɨɫɬɚɟɬɫɹ ɨɬɤɪɵɬɵɦ, ɟɫɥɢ ɜ ɧɚɱɚɥɟ ɧɚɛɥɸɞɟɧɢɹ ɢɦɟɟɬɫɹ ɥɢɲɶ ɤɨɦɩɨɧɟɧɬ A .
ɍɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɤɨɦɩɨɧɟɧɬɚ B ɜ ɞɨɫɬɚɰɢɨɧɚɪɧɵɣ ɩɟɪɢɨɞ ɢɦɟɟɬ
ɜɢɞ
d > B@
k1 > A@0 k2 >B@ 0
dt
(11.24)
ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ
>B@ >P@
0 ɩɪɢ t
0.
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ (11.24) ɞɚɟɬ
§
·
k1> A@0
ln ¨
k t.
¨ k1 > A@ k2 > B@ ¸¸ 2
©
¹
0
(11.25)
ɂɫɩɨɥɶɡɭɹ (11.23), ɢɡ (11.25) ɧɚɣɞɟɦ
>B@
>B@st
1 e k 2t .
(11.26)
Ɋɚɜɟɧɫɬɜɨ (11.26) ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɫɬɚɰɢɨɧɚɪɧɚɹ ɤɨɧɰɟɧɬɪɚɰɢɹ
ɩɪɨɦɟɠɭɬɨɱɧɨɝɨ ɜɟɳɟɫɬɜɚ ɭɫɬɚɧɨɜɢɬɫɹ ɜ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɫɢɫɬɟɦɟ ɱɟɪɟɡ ɛɟɫɤɨɧɟɱɧɨ ɛɨɥɶɲɨɣ ɩɪɨɦɟɠɭɬɨɤ ɜɪɟɦɟɧɢ. Ɉɞɧɚɤɨ, ɟɫɥɢ ɦɵ ɡɚɞɚɞɢɦ ɞɨɫɬɚɬɨɱɧɨ ɦɚɥɵɟ, ɧɨ ɤɨɧɟɱɧɵɟ ɨɬɤɥɨɧɟɧɢɹ >B@ ɨɬ >B@st , ɬɨ ɨɧɢ
ɨɤɚɠɭɬɫɹ ɞɨɫɬɢɠɢɦɵɦɢ ɡɚ ɜɩɨɥɧɟ ɤɨɧɟɱɧɵɟ ɜɪɟɦɟɧɚ.
Ɍɚɤ, ɟɫɥɢ ɫɱɢɬɚɬɶ ɛɥɢɡɤɨɣ ɤ ɫɬɚɰɢɨɧɚɪɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɸ
>B @ 0,95>B @st , ɬɨ ɢɡ ɮɨɪɦɭɥɵ (11.26) ɫɥɟɞɭɟɬ, ɱɬɨ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ
ɪɟɚɤɰɢɢ ɷɬɚ ɤɨɧɰɟɧɬɪɚɰɢɹ ɛɭɞɟɬ ɞɨɫɬɢɝɧɭɬɚ ɡɚ ɜɪɟɦɹ t 3 k2 .
ɉɪɨɰɟɫɫ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɨɣ ɤɨɧɰɟɧɬɪɚɰɢɢ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɜɟɳɟɫɬɜ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɪɟɥɚɤɫɚɰɢɨɧɧɵɣ ɢ ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɟɝɨ ɜɪɟɦɟɧɟɦ ɩɨɥɨɜɢɧɧɨɣ ɪɟɥɚɤɫɚɰɢɢ, ɬ.ɟ. ɜɪɟɦɟɧɟɦ, ɡɚ ɤɨɬɨɪɨɟ
ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɧɚɛɥɸɞɚɟɦɨɣ ɢ ɫɬɚɰɢɨɧɚɪɧɨɣ ɜɟɥɢɱɢɧɚɦɢ ɫɨɤɪɚɳɚɟɬɫɹ
ɜɞɜɨɟ. ɗɬɨ ɯɚɪɚɤɬɟɪɧɨɟ ɜɪɟɦɹ ɪɟɥɚɤɫɚɰɢɢ, ɹɜɥɹɸɳɟɟɫɹ ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ
ɟɫɬɟɫɬɜɟɧɧɵɦ ɜɪɟɦɟɧɧɵɦ ɦɚɫɲɬɚɛɨɦ ɫɢɫɬɟɦɵ, ɤɚɤ ɫɥɟɞɭɟɬ ɢɡ (11.26),
ɛɭɞɟɬ ɪɚɜɧɨ: t ln2 k2 .
Ⱥɧɚɥɨɝɢɱɧɵɣ ɚɧɚɥɢɡ ɦɨɠɟɬ ɛɵɬɶ ɩɪɨɜɟɞɟɧ ɞɥɹ ɞɪɭɝɢɯ ɫɢɫɬɟɦ ɪɟɚɤɰɢɣ, ɜɤɥɸɱɚɸɳɢɯ ɫɬɚɞɢɢ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. ɉɪɢɦɟɪɵ ɫɥɨɠɧɵɯ ɫɯɟɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-ɩɚɪɚɥɥɟɥɶɧɵɯ ɪɟɚɤɰɢɣ ɩɪɟɞɫɬɚɜɥɟɧɵ
ɧɚ ɪɢɫ. 11.5
280
Ɋɢɫ. 11.5. ɍɫɥɨɜɧɵɟ ɫɯɟɦɵ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-ɩɚɪɚɥɥɟɥɶɧɵɯ ɪɟɚɤɰɢɣ
Ⱦɥɹ ɪɚɡɥɢɱɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɨɜɨɤɭɩɧɨɫɬɟɣ ɢɦɟɸɬ ɦɟɫɬɨ ɛɥɢɡɤɢɟ
ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɵɯ ɪɟɠɢɦɨɜ: ɜɪɟɦɟɧɚ ɭɫɬɚɧɨɜɥɟɧɢɹ ɫɬɚɰɢɨɧɚɪɧɵɯ ɤɨɧɰɟɧɬɪɚɰɢɣ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɜɟɳɟɫɬɜ ɫ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɶɸ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɵ ɫɭɦɦɟ ɤɨɧɫɬɚɧɬ ɫɬɚɞɢɣ, ɜ
ɤɨɬɨɪɵɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɯ ɩɪɟɜɪɚɳɟɧɢɟ.
ɂɡɥɨɠɟɧɧɵɣ ɦɟɬɨɞ ɨɩɢɫɚɧɢɹ ɤɢɧɟɬɢɤɢ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ ɛɵɥ
ɩɪɟɞɥɨɠɟɧ Ȼɨɞɟɧɲɬɟɣɧɨɦ ɢ ɢɦɟɟɬ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɨɩɢɫɚɧɢɹ ɪɚɡɜɟɬɜɥɟɧɧɵɯ ɪɟɚɤɰɢɣ ɢ ɝɟɬɟɪɨɝɟɧɧɵɯ ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ.
11.4. Ɉɬ ɱɟɝɨ ɡɚɜɢɫɢɬ ɫɤɨɪɨɫɬɶ
ɯɢɦ ɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ?
Ɏɢɡɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɪɟɚɤɰɢɢ ɩɨɡɜɨɥɹɟɬ ɭɫɬɚɧɨɜɢɬɶ ɫɯɟɦɭ ɪɟɚɤɰɢɢ,
ɬ.ɟ. ɧɚɛɨɪ ɨɬɞɟɥɶɧɵɯ ɷɥɟɦɟɧɬɚɪɧɵɯ ɫɬɚɞɢɣ. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɮɭɧɞɚɦɟɧɬɚɥɶɧɨɟ ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɩɨɧɹɬɢɹɦɢ «ɫɯɟɦɚ ɪɟɚɤɰɢɢ» ɢ
«ɦɟɯɚɧɢɡɦ ɪɟɚɤɰɢɢ», ɤɨɬɨɪɵɟ ɜ ɥɢɬɟɪɚɬɭɪɟ ɱɚɫɬɨ ɭɩɨɬɪɟɛɥɹɸɬɫɹ ɤɚɤ
ɬɨɠɞɟɫɬɜɟɧɧɵɟ. ɋɯɟɦɚ ɨɫɬɚɟɬɫɹ ɮɨɪɦɚɥɶɧɵɦ ɨɩɢɫɚɧɢɟɦ ɪɟɚɤɰɢɢ (ɟɟ
ɦɚɤɪɨɫɤɨɩɢɱɟɫɤɨɝɨ ɩɪɨɹɜɥɟɧɢɹ) ɢ ɧɟ ɬɪɟɛɭɟɬ ɧɢɤɚɤɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɣ ɨ
ɫɜɨɣɫɬɜɚɯ ɤɨɦɩɨɧɟɧɬɨɜ. ɍɫɬɚɧɨɜɥɟɧɢɟ ɦɟɯɚɧɢɡɦɚ ɪɟɚɤɰɢɢ ɞɨɥɠɧɨ ɞɚɬɶ
ɨɬɜɟɬ ɧɚ ɪɹɞ ɜɨɩɪɨɫɨɜ: ɝɞɟ ɪɚɫɩɨɥɨɠɟɧɵ ɚɤɬɢɜɧɵɟ ɰɟɧɬɪɵ ɜ ɪɟɚɝɟɧɬɚɯ;
ɤɚɤɢɟ ɫɜɨɣɫɬɜɚ ɩɪɨɹɜɥɹɸɬ ɪɟɚɝɟɧɬɵ; ɤɚɤɢɟ ɫɜɹɡɢ ɪɜɭɬɫɹ ɢɥɢ ɨɛɪɚɡɭɸɬɫɹ
ɜ ɤɚɠɞɨɣ ɢɡ ɫɬɚɞɢɣ ɪɟɚɤɰɢɢ; ɱɬɨ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɨɦɟɠɭɬɨɱɧɵɟ
ɩɪɨɞɭɤɬɵ ɪɟɚɤɰɢɢ; ɤɚɤɨɜɵ ɫɨɫɬɚɜ ɢ ɫɬɪɭɤɬɭɪɚ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɤɨɦɩɥɟɤɫɨɜ? ɗɬɢ ɜɨɩɪɨɫɵ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɨɜɪɟɦɟɧɧɵɦɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦɢ ɨ
ɫɬɪɨɟɧɢɢ ɜɟɳɟɫɬɜɚ ɢ ɬɟɨɪɢɢ ɯɢɦɢɱɟɫɤɨɣ ɫɜɹɡɢ, ɬ.ɟ. ɩɪɢ ɢɡɭɱɟɧɢɢ ɦɢɤɪɨɩɪɨɰɟɫɫɨɜ. Ɍ.ɨ., ɭɫɬɚɧɨɜɥɟɧɧɵɣ ɦɟɯɚɧɢɡɦ ɪɟɚɤɰɢɢ ɦɨɠɟɬ ɛɵɬɶ ɯɨɪɨɲ
ɥɢɲɶ ɧɚɫɬɨɥɶɤɨ, ɧɚɫɤɨɥɶɤɨ ɯɨɪɨɲɢ ɷɬɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹ. ɗɬɨ ɨɡɧɚɱɚɟɬ,
ɱɬɨ ɩɪɟɞɩɨɥɚɝɚɟɦɵɣ ɦɟɯɚɧɢɡɦ ɧɢɤɨɝɞɚ ɧɟ ɛɵɜɚɟɬ ɨɤɨɧɱɚɬɟɥɶɧɵɦ, ɚ
ɦɨɠɟɬ ɛɵɬɶ ɬɨɥɶɤɨ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɦ.
281
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɭɫɥɨɜɢɣ ɟɟ ɨɫɭɳɟɫɬɜɥɟɧɢɹ, ɜ ɱɚɫɬɧɨɫɬɢ, ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɞɚɜɥɟɧɢɹ.
ɋɭɳɟɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɭɪɚɜɧɟɧɢɣ, ɩɪɢɝɨɞɧɵɯ ɞɥɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨɝɨ
ɨɩɢɫɚɧɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɤɨɧɫɬɚɧɬ ɫɤɨɪɨɫɬɢ ɷɥɟɦɟɧɬɚɪɧɵɯ
ɪɟɚɤɰɢɣ. ɋɚɦɨɟ ɢɡɜɟɫɬɧɨɟ ɢɡ ɧɢɯ – ɭɪɚɜɧɟɧɢɟ Ⱥɪɪɟɧɢɭɫɚ – ɢɦɟɟɬ ɜɢɞ
k k0e Ea
RT
,
(11.27)
ɝɞɟ k0 ɤɨɧɫɬɚɧɬɚ, ɧɚɡɵɜɚɟɦɚɹ ɱɚɫɬɨɬɧɵɦ ɮɚɤɬɨɪɨɦ, ɩɪɟɞɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɦ ɦɧɨɠɢɬɟɥɟɦ ɢ ɚɪɪɟɧɢɭɫɨɜɫɤɢɦ ɦɧɨɠɢɬɟɥɟɦ; ɜɟɥɢɱɢɧɚ Ea ɧɚɡɵɜɚɟɬɫɹ ɷɧɟɪɝɢɟɣ ɚɤɬɢɜɚɰɢɢ ɪɟɚɤɰɢɢ.
ȿɫɥɢ ɩɨɫɬɪɨɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ k T ɜ ɚɪɪɟɧɢɭɫɨɜɫɤɢɯ ɤɨɨɪɞɢɧɚɬɚɯ,
ɬ.ɟ. ln k f 1 T , ɬɨ ɩɨɥɭɱɢɬɫɹ ɩɪɹɦɚɹ ɫ ɨɬɪɢɰɚɬɟɥɶɧɵɦ ɧɚɤɥɨɧɨɦ, ɪɚɜɧɵɦ Ea R , ɢ ɨɬɫɟɤɚɸɳɚɹ ɧɚ ɨɫɢ ɨɬɪɟɡɨɤ, ɪɚɜɧɵɣ lnk0 .
Ⱦɥɹ ɨɛɴɹɫɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɫɭɳɟɫɬɜɭɟɬ ɧɟɫɤɨɥɶɤɨ ɪɚɡɥɢɱɧɵɯ ɬɟɨɪɢɣ. Ɍɚɤ, ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɷɧɟɪɝɢɢ
ɚɤɬɢɜɚɰɢɢ Ea ɨɛɴɹɫɧɹɟɬɫɹ ɜ ɬɟɨɪɢɢ ɚɤɬɢɜɢɪɨɜɚɧɧɨɝɨ ɤɨɦɩɥɟɤɫɚ ɢ ɜ
ɬɟɨɪɢɢ ɫɬɨɥɤɧɨɜɟɧɢɣ. ɉɪɟɠɞɟ ɜɫɟɝɨ, ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ ɟɟ ɤɨɪɪɟɥɹɰɢɹ ɫ
ɷɧɟɪɝɢɟɣ ɫɜɹɡɢ. ɇɨ ɛɟɡ ɨɬɜɟɬɚ ɨɫɬɚɟɬɫɹ ɜɨɩɪɨɫ, ɩɨɱɟɦɭ ɬɨɬ ɢɥɢ ɢɧɨɣ
ɩɪɨɰɟɧɬ ɷɧɟɪɝɢɢ ɪɜɭɳɢɯɫɹ ɫɜɹɡɟɣ ɧɟɨɛɯɨɞɢɦ ɞɥɹ ɚɤɬɢɜɚɰɢɢ. ȼɟɥɢɱɢɧɚ
ɷɧɟɪɝɢɢ ɚɤɬɢɜɚɰɢɢ ɦɟɧɹɟɬɫɹ ɜ ɲɢɪɨɤɢɯ ɩɪɟɞɟɥɚɯ ɞɥɹ ɪɚɡɧɵɯ ɪɟɚɤɰɢɣ.
ȿɫɥɢ ɝɨɦɨɝɟɧɧɚɹ ɪɟɚɤɰɢɹ ɩɪɨɬɟɤɚɟɬ ɜ ɧɨɪɦɚɥɶɧɵɯ ɭɫɥɨɜɢɹɯ, ɬɨ ɨɬɤɥɨɧɟɧɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɟɟ ɫɤɨɪɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɚɪɪɟɧɢɭɫɨɜɫɤɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɫɜɹɡɵɜɚɸɬ ɫ ɬɟɦ, ɱɬɨ ɢɡɦɟɪɹɟɦɚɹ ɤɨɧɫɬɚɧɬɚ ɫɤɨɪɨɫɬɢ ɹɜɥɹɟɬɫɹ
ɫɥɨɠɧɨɣ ɢ ɨɬɧɨɫɢɬɫɹ ɤ ɛɨɥɟɟ ɱɟɦ ɨɞɧɨɣ ɫɬɚɞɢɢ ɪɟɚɤɰɢɢ.
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɢɟɣ ɚɤɬɢɜɢɪɨɜɚɧɧɨɝɨ ɤɨɦɩɥɟɤɫɚ,
k BT z
(11.28)
K ,
h c
ɝɞɟ k B - ɩɨɫɬɨɹɧɧɚɹ Ȼɨɥɶɰɦɚɧɚ, h - ɩɨɫɬɨɹɧɧɚɹ ɉɥɚɧɤɚ, K cz – ɤɨɧɫɬɚɧɬɚ
ɪɚɜɧɨɜɟɫɢɹ, ɤɨɬɨɪɚɹ ɫɜɹɡɚɧɚ ɫ ɢɡɦɟɧɟɧɢɟɦ ɫɜɨɛɨɞɧɨɣ ɷɧɟɪɝɢɢ Ƚɢɛɛɫɚ
k
RT ln K cz ' G z ,
ɝɞɟ ' G z ' H z T ' S z , ' H z – ɷɧɬɚɥɶɩɢɹ ɚɤɬɢɜɚɰɢɢ (ɪɚɡɧɨɫɬɶ ɦɨɥɹɪɧɵɯ ɷɧɬɚɥɶɩɢɣ ɧɚɱɚɥɶɧɨɝɨ ɢ ɤɨɧɟɱɧɨɝɨ ɫɨɫɬɨɹɧɢɣ), ' S z – ɷɧɬɪɨɩɢɹ ɚɤɬɢɜɚɰɢɢ.
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɦɟɫɬɨ (11.28) ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
k
k BT
exp ' S z R exp ' H z RT .
h
282
(11.29)
ɗɬɨ ɟɫɬɶ ɭɪɚɜɧɟɧɢɟ ɗɣɪɢɧɝɚ. ȼ ɥɨɝɚɪɢɮɦɢɱɟɫɤɨɣ ɮɨɪɦɟ ɢɦɟɟɦ
z
'H z
§k ·
§ k · 'S
.
ln ¨ ¸ ln ¨ B ¸ RT
©T ¹
© h¹ R
ȿɫɥɢ ɩɨɫɬɪɨɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ln k T ɨɬ 1 T , ɬɨ ɧɚɤɥɨɧ ɩɪɹɦɨɣ ɛɭɞɟɬ ɪɚɜɟɧ ' H z RT . Ɂɧɚɹ ɷɬɭ ɜɟɥɢɱɢɧɭ, ɦɨɠɧɨ ɪɚɫɫɱɢɬɚɬɶ ɷɧɬɪɨɩɢɸ
ɪɟɚɤɰɢɢ.
ɋɨɝɥɚɫɧɨ ɩɪɨɫɬɨɣ ɬɟɨɪɢɢ ɫɬɨɥɤɧɨɜɟɧɢɣ, ɪɟɚɝɢɪɭɸɳɢɟ ɦɨɥɟɤɭɥɵ
ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɤɚɤ ɠɟɫɬɤɢɟ ɲɚɪɢɤɢ, ɧɟ ɨɛɥɚɞɚɸɳɢɟ ɫɢɥɚɦɢ ɩɪɢɬɹɠɟɧɢɹ ɞɪɭɝ ɤ ɞɪɭɝɭ. ɋɱɢɬɚɟɬɫɹ, ɱɬɨ ɪɟɚɤɰɢɹ ɞɜɭɯ ɦɨɥɟɤɭɥ ɩɪɨɢɫɯɨɞɢɬ
ɬɨɥɶɤɨ ɩɪɢ ɢɯ ɫɬɨɥɤɧɨɜɟɧɢɢ ɢ ɱɬɨ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɱɚɫɬɨɬɵ
ɫɬɨɥɤɧɨɜɟɧɢɣ Z . Ⱦɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɩɨɫɥɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɞɜɭɯ ɦɨɥɟɤɭɥ
ɦɨɝɥɚ ɩɪɨɢɡɨɣɬɢ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ, ɧɟɨɛɯɨɞɢɦɨ, ɱɬɨɛɵ ɷɧɟɪɝɢɹ ɞɜɭɯ
ɦɨɥɟɤɭɥ ɛɵɥɚ ɪɚɜɧɚ ɷɧɟɪɝɢɢ ɚɤɬɢɜɚɰɢɢ Ea . ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ Ȼɨɥɶɰɦɚɧɚ, ɞɨɥɹ ɦɨɥɟɤɭɥ n , ɨɛɥɚɞɚɸɳɢɯ ɧɟɨɛɯɨɞɢɦɨɣ ɷɧɟɪɝɢɟɣ, ɨɩɪɟɞɟɥɢɬɫɹ ɤɚɤ
n* n exp Ea RT .
Ⱥ ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɫɬɨɥɤɧɨɜɟɧɢɟ ɦɨɝɥɨ ɩɪɢɜɟɫɬɢ ɤ ɯɢɦɢɱɟɫɤɢɦ ɢɡɦɟɧɟɧɢɹɦ, ɦɨɥɟɤɭɥɵ ɜ ɦɨɦɟɧɬ ɫɬɨɥɤɧɨɜɟɧɢɹ ɞɨɥɠɧɵ ɛɵɬɶ ɛɥɚɝɨɩɪɢɹɬɧɵɦ
ɨɛɪɚɡɨɦ ɨɪɢɟɧɬɢɪɨɜɚɧɵ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ. Ɂɚ ɷɬɨ ɨɬɜɟɱɚɟɬ ɮɚɤɬɨɪ ɜɟɪɨɹɬɧɨɫɬɢ P , ɤɨɬɨɪɵɣ ɬɚɤɠɟ ɧɚɡɵɜɚɸɬ ɫɬɟɪɢɱɟɫɤɢɦ ɮɚɤɬɨɪɨɦ.
ɑɢɫɥɨ ɫɬɨɥɤɧɨɜɟɧɢɣ Z ɧɚ 1 ɫɦ3 ɡɚ ɫɟɤɭɧɞɭ ɦɟɠɞɭ ɪɚɡɧɵɦɢ ɦɨɥɟɤɭɥɚɦɢ A1 ɢ A2 ɦɨɠɧɨ ɜɵɜɟɫɬɢ ɢɡ ɡɚɤɨɧɨɜ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɦɟɯɚɧɢɤɢ
Na
Z
r r
1000 A1 A2
ɝɞɟ N a – ɱɢɫɥɨ Ⱥɜɨɝɚɞɪɨ, P
m A1 ,mA2
m A1 m A2
m A1 m A2
1/ 2
2 § 8k BT ·
¨ P ¸
©
¹
,
(11.30)
, rA ,rA – ɪɚɞɢɭɫɵ ɪɟɚɝɟɧɬɨɜ,
1
2
– ɢɯ ɦɨɥɹɪɧɵɟ ɦɚɫɫɵ. ɑɢɫɥɨ ɫɬɨɥɤɧɨɜɟɧɢɣ ɜ (11.30)
ɢɡɦɟɪɹɟɬɫɹ ɜ 1/(ɦ·ɫ).
Ɍɟɨɪɢɹ ɫɬɨɥɤɧɨɜɟɧɢɣ ɛɵɥɚ ɪɚɡɜɢɬɚ ɞɥɹ ɝɚɡɨɮɚɡɧɵɯ ɪɟɚɤɰɢɣ. ɇɨ ɞɥɹ
ɦɧɨɝɢɯ ɪɟɚɤɰɢɣ ɜ ɪɚɫɬɜɨɪɚɯ ɟɟ ɩɪɟɞɫɤɚɡɚɧɢɹ ɬɚɤɠɟ ɨɤɚɡɵɜɚɸɬɫɹ ɫɩɪɚɜɟɞɥɢɜɵɦɢ.
Ⱦɥɹ ɦɨɧɨɦɨɥɟɤɭɥɹɪɧɵɯ ɪɟɚɤɰɢɣ ɱɚɫɬɨɬɧɵɣ ɮɚɤɬɨɪ ɧɟ ɹɜɥɹɟɬɫɹ
ɱɚɫɬɨɬɨɣ ɫɬɨɥɤɧɨɜɟɧɢɣ, ɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ ɫɪɟɞɧɟɣ ɤɨɥɟɛɚɬɟɥɶɧɨɣ ɱɚɫɬɨɬɵ, ɤɨɬɨɪɭɸ ɜ ɧɟɤɨɬɨɪɵɯ ɭɫɥɨɜɢɹɯ ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɪɚɜɧɨɣ k BT h .
283
ɗɧɬɪɨɩɢɹ ɚɤɬɢɜɚɰɢɢ ɫɜɹɡɚɧɚ ɫɨ ɫɬɟɪɢɱɟɫɤɢɦ29 ɮɚɤɬɨɪɨɦ
e' S
z
R
P,
ɬɚɤ ɱɬɨ ɫɬɟɪɢɱɟɫɤɢɣ ɮɚɤɬɨɪ, ɪɚɜɧɵɣ ɟɞɢɧɢɰɟ, ɞɨɥɠɟɧ ɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶ
ɧɭɥɟɜɨɣ ɷɧɬɪɨɩɢɢ ɚɤɬɢɜɚɰɢɢ. ɉɨɥɨɠɢɬɟɥɶɧɵɟ ɜɟɥɢɱɢɧɵ ' S z , ɧɚɣɞɟɧɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ, ɧɟɥɶɡɹ ɨɛɴɹɫɧɢɬɶ ɬɚɤɢɦ ɫɩɨɫɨɛɨɦ.
Ȼɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɞɥɹ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɢɦɟɟɬ ɢ ɬɚɤɚɹ ɜɟɥɢɱɢɧɚ,
ɯɚɪɚɤɬɟɪɢɡɭɸɳɚɹ ɩɟɪɟɯɨɞɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɤɚɤ ɚɤɬɢɜɚɰɢɨɧɧɵɣ ɨɛɴɟɦ. ȼ
ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɢɟɣ ɩɟɪɟɯɨɞɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɧɫɬɚɧɬɵ
ɫɤɨɪɨɫɬɢ ɨɞɧɨɫɬɚɞɢɣɧɨɣ ɠɢɞɤɨɮɚɡɧɨɣ ɪɟɚɤɰɢɢ (ɷɥɟɦɟɧɬɚɪɧɨɣ ɪɟɚɤɰɢɢ)
ɨɬ ɞɚɜɥɟɧɢɹ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɭɪɚɜɧɟɧɢɟɦ
(11.31)
d ln k dp ' V z RT ET Q 1 ,
ɝɞɟ Q – ɩɨɪɹɞɨɤ ɪɟɚɤɰɢɢ, ET – ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɫɠɢɦɚɟɦɨɫɬɢ ɪɚɫɬɜɨɪɢɬɟɥɹ. ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɫɩɪɚɜɟɞɥɢɜɨ ɩɪɢ ɩɨɫɬɨɹɧɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ, ɢ ɟɦɭ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɥɢɧɟɣɧɵɟ ɝɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɢ lnk ɨɬ p .
Ɉɞɧɚɤɨ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɤɪɢɜɵɟ ɫɜɢɞɟɬɟɥɶɫɬɜɭɸɬ ɨ ɡɚɜɢɫɢɦɨɫɬɢ ɚɤɬɢɜɚɰɢɨɧɧɨɝɨ ɨɛɴɟɦɚ ɨɬ ɞɚɜɥɟɧɢɹ.
ȼ ɥɢɬɟɪɚɬɭɪɟ ɨɛɫɭɠɞɚɸɬɫɹ ɢ ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɫɥɭɱɚɢ ɡɚɜɢɫɢɦɨɫɬɢ
ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɣ ɨɬ ɞɚɜɥɟɧɢɹ, ɜ ɬɨɦ ɱɢɫɥɟ ɞɥɹ ɪɟɚɤɰɢɣ, ɤɨɧɬɪɨɥɢɪɭɟɦɵɯ ɞɢɮɮɭɡɢɟɣ.
Ʉɢɧɟɬɢɤɚ ɝɟɬɟɪɨɝɟɧɧɵɯ ɪɟɚɤɰɢɣ, ɬ.ɟ. ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɧɚ
ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ, ɛɨɥɟ ɫɥɨɠɧɚ, ɬɚɤ ɤɚɤ ɢɯ «ɷɥɟɦɟɧɬɚɪɧɵɦɢ» ɫɬɚɞɢɹɦɢ ɹɜɥɹɸɬɫɹ ɞɢɮɮɭɡɢɨɧɧɵɟ ɩɟɪɟɦɟɳɟɧɢɹ ɪɟɚɝɟɧɬɨɜ ɢ ɩɪɨɞɭɤɬɨɜ, ɱɬɨ
ɬɪɟɛɭɟɬ ɨɬɞɟɥɶɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ.
1 1 . 5 . Ɋ ɟ ɚ ɤ ɰ ɢ ɢ ɫ ɭ ɱ ɚ ɫ ɬ ɢ ɟ ɦ ɬ ɜ ɟ ɪ ɞ ɵ ɯ ɜ ɟ ɳ ɟ ɫ ɬ ɜ 30
11.5. 1. Ʉɥ ɚɫɫ ɢɮ ɢɤɚɰɢɹ ɬɜɟ ɪɞ ɨɮɚɡɧ ɵɯ ɩɪɟɜ ɪɚ ɳɟ ɧɢɣ
Ⱦɨɥɝɨɟ ɜɪɟɦɹ ɫɱɢɬɚɥɨɫɶ, ɱɬɨ ɯɢɦɢɱɟɫɤɢɟ ɢ ɮɚɡɨɜɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɜ
ɬɜɟɪɞɨɣ ɮɚɡɟ ɜɨɨɛɳɟ ɧɟ ɢɞɭɬ ɢɥɢ ɢɞɭɬ ɱɪɟɡɜɵɱɚɣɧɨ ɦɟɞɥɟɧɧɨ, ɱɬɨ ɡɚɬɪɭɞɧɹɟɬ ɢɥɢ ɞɟɥɚɟɬ ɛɟɫɩɨɥɟɡɧɵɦ ɢɯ ɩɪɚɤɬɢɱɟɫɤɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ. ɋɢɫɬɟɦɚɬɢɱɟɫɤɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɩɪɟɜɪɚɳɟɧɢɣ ɜ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜɚɯ ɛɵɥɨ ɧɚɱɚɬɨ ɥɢɲɶ ɜ 1930-ɟ ɝɝ., ɱɬɨ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɜɹɡɚɧɨ ɫ ɩɨɹɜɥɟɧɢɟɦ ɧɨ-
29
ɋɬɟɪɢɱɟɫɤɢɣ ɮɚɤɬɨɪ – ɦɧɨɠɢɬɟɥɶ, ɨɬɪɚɠɚɸɳɢɣ ɞɨɥɸ ɫɨɭɞɚɪɟɧɢɣ ɱɚɫɬɢɰ ɫ ɞɨɫɬɚɬɨɱɧɨɣ ɷɧɟɪɝɢɟɣ, ɜɟɞɭɳɢɯ ɤ ɪɟɚɤɰɢɢ.
30
Ɍɪɟɬɶɹɤɨɜ ɘ.Ⱦ., ɉɭɬɥɹɟɜ ȼ.ɂ. ȼɜɟɞɟɧɢɟ ɜ ɯɢɦɢɸ ɬɜɟɪɞɨɮɚɡɧɵɯ ɦɚɬɟɪɢɚɥɨɜ. Ɇ.:
ɇɚɭɤɚ, ɂɡɞ-ɜɨ Ɇɨɫɤɨɜɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ, 2006. 400 ɫ.
284
ɜɵɯ ɯɢɦɢɱɟɫɤɢɯ ɬɟɯɧɨɥɨɝɢɣ, ɜ ɬɨɦ ɱɢɫɥɟ ɧɟɬɪɚɞɢɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɫɢɧɬɟɡɚ ɬɜɟɪɞɨɮɚɡɧɵɯ ɫɨɟɞɢɧɟɧɢɣ.
Ʉ ɱɢɫɥɭ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɨɰɟɫɫɨɜ ɨɬɧɨɫɹɬ ɬɟ, ɜ ɤɨɬɨɪɵɯ ɨɞɧɚ ɢɡ
ɮɚɡ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɩɪɟɜɪɚɳɟɧɢɢ, ɹɜɥɹɟɬɫɹ ɬɜɟɪɞɨɣ. ɗɬɨ ɦɨɠɟɬ ɛɵɬɶ
ɨɞɢɧ ɢɡ ɪɟɚɝɟɧɬɨɜ, ɩɪɨɦɟɠɭɬɨɱɧɵɣ ɩɪɨɞɭɤɬ, ɤɚɬɚɥɢɡɚɬɨɪ ɢɥɢ ɤɨɧɟɱɧɵɟ
ɫɨɟɞɢɧɟɧɢɹ. ɋɨɛɫɬɜɟɧɧɨ ɬɜɟɪɞɨɮɚɡɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ – ɷɬɨ ɬɟ, ɜ ɤɨɬɨɪɵɯ
ɪɟɚɝɟɧɬɵ – ɬɜɟɪɞɵɟ ɜɟɳɟɫɬɜɚ.
ɉɪɨɫɬɟɣɲɢɦɢ ɩɪɢɦɟɪɚɦɢ ɮɢɡɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɝɞɟ ɩɪɢɫɭɬɫɬɜɭɟɬ
ɬɜɟɪɞɚɹ ɮɚɡɚ, ɹɜɥɹɸɬɫɹ ɩɥɚɜɥɟɧɢɟ ɢ ɫɭɛɥɢɦɚɰɢɹ, ɤɨɬɨɪɵɟ ɫɯɟɦɚɬɢɱɧɨ
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɬɚɤ:
S o L,
(11.32)
S oG,
ɝɞɟ S , L ,G – ɬɜɟɪɞɚɹ, ɠɢɞɤɚɹ ɢ ɝɚɡɨɨɛɪɚɡɧɚɹ ɮɚɡɵ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ⱦɪɭɝɨɣ ɩɪɢɦɟɪ – ɷɬɨ ɮɚɡɨɜɵɟ ɩɟɪɟɯɨɞɵ ɛɟɡ ɢɡɦɟɧɟɧɢɹ ɚɝɪɟɝɚɬɧɨɝɨ ɫɨɫɬɨɹɧɢɹ – ɩɟɪɟɯɨɞɵ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɫɬɪɭɤɬɭɪɧɵɦɢ ɦɨɞɢɮɢɤɚɰɢɹɦɢ ɨɞɧɨɝɨ ɢ ɬɨɝɨ ɠɟ ɫɨɟɞɢɧɟɧɢɹ (ɚɥɦɚɡ l ɝɪɚɮɢɬ, ɛɟɥɨɟ ɨɥɨɜɨ l ɫɟɪɨɟ ɨɥɨɜɨ, D – ɠɟɥɟɡɨ l E – ɠɟɥɟɡɨ ɢ ɞɪ.):
S1 o S 2 , S1 l S 2 .
(11.33)
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɨɣ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ ɦɨɠɧɨ
ɩɪɢɜɟɫɬɢ ɪɟɚɤɰɢɸ ɨɤɢɫɥɟɧɢɹ ɦɟɬɚɥɥɚ:
S1 G o S2
(11.34)
(ɨɤɢɫɥɟɧɢɟ ɚɥɸɦɢɧɢɹ, ɧɢɤɟɥɹ, ɠɟɥɟɡɚ ɢ ɬ.ɞ.). ɋɤɨɪɨɫɬɶ ɦɧɨɝɢɯ ɬɚɤɢɯ
ɪɟɚɤɰɢɣ ɥɢɦɢɬɢɪɭɟɬɫɹ ɩɪɨɰɟɫɫɨɦ ɞɢɮɮɭɡɢɢ, ɩɪɨɢɫɯɨɞɹɳɢɦ ɜ ɩɥɨɬɧɨɦ
ɫɥɨɟ ɨɤɢɫɥɚ, ɩɨɤɪɵɜɚɸɳɟɝɨ ɦɟɬɚɥɥ.
Ɋɟɚɤɰɢɢ ɪɚɡɥɨɠɟɧɢɹ ɤɚɪɛɨɧɚɬɚ ɤɚɥɶɰɢɹ, ɫɭɥɶɮɚɬɨɜ ɦɟɬɚɥɥɨɜ, ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɤɚɤ
S1 l S2 G ,
(11.35)
ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɩɪɢɦɟɪɵ ɪɟɚɤɰɢɣ, ɜ ɤɨɬɨɪɵɯ ɥɢɦɢɬɢɪɭɸɳɚɹ ɫɬɚɞɢɹ ɥɨɤɚɥɢɡɨɜɚɧɚ ɜɛɥɢɡɢ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɞɜɭɯ ɬɟɥ (ɮɚɡ, ɫɪɟɞ). ɗɬɨ
ɪɟɚɤɰɢɢ ɬɢɩɚ
CaCO3 l CaO CO 2 ,
BeSO 4 l BeO SO 3 .
ɋɯɟɦɨɣ (11.35) ɜ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɩɪɨɰɟɫɫɵ ɫɭɲɤɢ ɢ ɞɟɝɢɞɪɚɬɚɰɢɢ
ɤɪɢɫɬɚɥɥɨɝɢɞɪɚɬɨɜ
( CuSO 4 ˜ 5H 2O , NiSO 4 ˜ 7 H 2O ,
CaCO3 ˜ 6H 2O , MnC 2O 4 ˜ 2H 2O ɢ ɞɪ.), ɤɨɬɨɪɵɟ ɫ ɭɫɩɟɯɨɦ ɦɨɝɭɬ ɛɵɬɶ
ɨɬɧɟɫɟɧɵ ɢ ɤ ɮɢɡɢɱɟɫɤɢɦ ɩɪɨɰɟɫɫɚɦ.
əɜɥɟɧɢɹ, ɩɪɨɢɫɯɨɞɹɳɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɨɝɪɚɧɢɱɢɜɚɸɬ ɫɤɨɪɨɫɬɶ ɛɨɥɶɲɢɧɫɬɜɚ ɪɟɚɤɰɢɣ, ɨɬɧɨɫɹɳɢɯɫɹ ɤ ɫɥɟɞɭɸɳɢɦ ɞɜɭɦ ɬɢɩɚɦ:
S G1 l G2
(11.36)
285
S1 G1 l S 2 G2 .
(11.37)
Ɉɛɪɚɡɨɜɚɧɢɟ ɤɚɪɛɨɧɚɬɚ ɧɢɤɟɥɹ ɢɥɢ ɟɝɨ ɪɚɡɥɨɠɟɧɢɟ – ɩɪɢɦɟɪ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɬɢɩɚ
4CO Ni l NiCO 4 ;
ɨɛɠɢɝ ɩɢɪɢɬɚ ɢɥɢ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ɝɚɡɨɦ – ɩɪɢɦɟɪɵ
ɪɟɚɤɰɢɣ ɜɬɨɪɨɝɨ ɬɢɩɚ
Fe 3O 4 4H 2 o 3Fe 4H 2O .
ɋɪɟɞɢ ɪɟɚɤɰɢɣ ɩɪɨɫɬɵɯ ɬɢɩɨɜ ɦɨɠɧɨ ɟɳɟ ɭɩɨɦɹɧɭɬɶ ɨ ɪɟɚɤɰɢɹɯ,
ɩɪɨɬɟɤɚɸɳɢɯ ɫ ɭɱɚɫɬɢɟɦ ɬɪɟɯ ɬɜɟɪɞɵɯ ɮɚɡ
S1 S 2 o S3 .
(11.38)
Ʉ ɷɬɨɦɭ ɬɢɩɭ ɨɬɧɨɫɹɬɫɹ ɪɟɚɤɰɢɢ ɨɛɪɚɡɨɜɚɧɢɹ ɫɢɥɢɤɚɬɨɜ, ɬɢɬɚɧɚɬɨɜ,
ɦɨɥɢɛɞɚɬɨɜ ɪɚɡɥɢɱɧɵɯ ɦɟɬɚɥɥɨɜ ɢɡ ɨɤɢɫɥɨɜ ɦɟɬɚɥɥɨɜ ɢ ɨɤɢɫɥɚ ɤɪɟɦɧɢɹ.
ɗɬɢ ɪɟɚɤɰɢɢ ɥɢɦɢɬɢɪɭɸɬɫɹ ɪɚɡɥɢɱɧɵɦɢ ɞɢɮɮɭɡɢɨɧɧɵɦɢ ɩɪɨɰɟɫɫɚɦɢ.
ɇɚɢɛɨɥɟɟ ɢɡɭɱɟɧɚ ɜ ɷɬɨɣ ɝɪɭɩɩɟ ɤɢɧɟɬɢɤɚ ɪɟɚɤɰɢɣ ɨɛɪɚɡɨɜɚɧɢɹ ɲɩɢɧɟɥɟɣ (ɮɟɪɪɢɬɨɜ, ɯɪɨɦɢɬɨɜ ɢ ɬ.ɩ.), ɦɨɥɢɛɞɚɬɨɜ ɢ ɜɨɥɶɮɪɚɦɚɬɨɜ ɤɨɦɩɥɟɤɫɧɵɯ ɢɨɞɢɞɨɜ, ɧɚɩɪɢɦɟɪ,
ZnO Fe 2O3 o ZnFe 2O 4 ,
CoO WO3 o CoWO 4 .
ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɨɛɪɚɡɨɜɚɧɢɟ ɫɢɥɢɤɚɬɚ ɤɚɥɶɰɢɹ ɢ ɩɨɞɨɛɧɵɟ ɪɟɚɤɰɢɢ
ɢɞɭɬ ɱɟɪɟɡ ɨɛɪɚɡɨɜɚɧɢɟ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɩɪɨɞɭɤɬɨɜ.
Ʉ ɪɟɚɤɰɢɹɦ ɛɨɥɟɟ ɫɥɨɠɧɵɯ ɬɢɩɨɜ ɦɨɠɧɨ ɨɬɧɟɫɬɢ ɪɟɚɤɰɢɢ ɞɜɨɣɧɨɝɨ
ɪɚɡɥɨɠɟɧɢɹ ɜ ɬɜɟɪɞɨɣ ɮɚɡɟ
S1 S 2 o S3 S4 .
(11.39)
ɇɚɩɪɢɦɟɪ,
Cu AgCl o CuCl Ag,
2CuI Ag 2S o 2AgI Cu 2S,
ɢɥɢ ɪɟɚɤɰɢɢ, ɜ ɤɨɬɨɪɵɯ ɩɨɹɜɥɹɟɬɫɹ ɧɟɫɤɨɥɶɤɨ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɩɪɨɞɭɤɬɨɜ ɦɟɠɞɭ ɮɚɡɨɣ ɪɟɚɝɟɧɬɚ ɢ ɮɚɡɨɣ ɤɨɧɟɱɧɨɝɨ ɩɪɨɞɭɤɬɚ. ɗɬɨ – ɪɟɚɤɰɢɹ
ɜɨɫɫɬɚɧɨɜɥɟɧɢɹ ɨɤɢɫɥɚ ɠɟɥɟɡɚ Fe 2O3 ɨɤɢɫɶɸ ɭɝɥɟɪɨɞɚ ɩɪɢ 1000 ɨɋ, ɜ
ɯɨɞɟ ɤɨɬɨɪɨɣ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɦɟɠɞɭ ɮɚɡɨɣ ɢɫɯɨɞɧɨɝɨ ɨɤɢɫɥɚ ɢ ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɮɚɡɨɣ ɞɜɟ ɪɚɡɥɢɱɧɵɟ ɱɟɪɟɞɭɸɳɢɟɫɹ ɮɚɡɵ Fe 3O 4 ɢ FeO .
Ɋɟɚɤɰɢɢ ɞɜɨɣɧɨɝɨ ɬɜɟɪɞɨɮɚɡɧɨɝɨ ɪɚɡɥɨɠɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɫɯɟɦɚɬɢɱɟɫɤɢ ɩɪɟɞɫɬɚɜɥɟɧɵ ɧɟɦɧɨɝɨ ɢɧɚɱɟ:
ABs CDs o ADs CBs .
(11.40)
ɉɪɢɦɟɪɚɦɢ ɛɨɥɟɟ ɫɥɨɠɧɵɯ ɪɟɚɤɰɢɣ ɦɨɝɭɬ ɛɵɬɶ ɪɟɚɤɰɢɢ ɨɛɪɚɡɨɜɚɧɢɹ ɞɜɨɣɧɵɯ ɫɨɟɞɢɧɟɧɢɣ ɫ ɜɵɞɟɥɟɧɢɟɦ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɩɪɨɞɭɤɬɚ ɢɥɢ ɪɟɚɤɰɢɢ ɪɚɡɥɨɠɟɧɢɹ, ɤɚɬɚɥɢɡɢɪɭɟɦɵɟ ɬɜɟɪɞɵɦɢ ɞɨɛɚɜɤɚɦɢ.
286
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɢɫɥɨ ɩɪɟɜɪɚɳɟɧɢɣ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ
ɜɟɥɢɤɨ, ɢ ɯɚɪɚɤɬɟɪ ɢɯ ɪɚɡɧɨɨɛɪɚɡɟɧ. ɂɯ ɢɡɭɱɚɸɬ ɢ ɦɨɞɟɥɢɪɭɸɬ ɜɨ ɦɧɨɝɢɯ ɧɚɭɱɧɵɯ ɞɢɫɰɢɩɥɢɧɚɯ.
ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɦɨɞɟɥɟɣ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɟɜɪɚɳɟɧɢɣ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ɜɨ ɜɧɢɦɚɧɢɟ ɩɪɢɧɢɦɚɸɬ ɤɚɤ ɢɯ ɦɚɤɪɨɤɢɧɟɬɢɱɟɫɤɢɟ ɩɪɨɹɜɥɟɧɢɹ, ɬɚɤ ɢ ɩɪɨɰɟɫɫɵ, ɩɪɨɬɟɤɚɸɳɢɟ ɧɚ ɦɢɤɪɨɭɪɨɜɧɟ. Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɪɟɜɪɚɳɟɧɢɣ ɧɚ ɦɢɤɪɨɭɪɨɜɧɟ ɨɬɪɚɠɚɟɬ ɬɨɬ ɢɥɢ ɢɧɨɣ ɬɢɩ
ɤɢɧɟɬɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɢɥɢ ɷɜɨɥɸɰɢɨɧɧɵɯ ɭɪɚɜɧɟɧɢɣ.
ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɰɟɥɟɣ ɢ ɯɚɪɚɤɬɟɪɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɩɪɢɦɟɧɹɸɬ ɬɭ
ɢɥɢ ɢɧɭɸ ɤɥɚɫɫɢɮɢɤɚɰɢɸ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ. Ʉɚɤ ɢ ɪɟɚɤɰɢɢ ɜ
ɝɚɡɚɯ ɢ ɠɢɞɤɨɫɬɹɯ, ɬɜɟɪɞɨɮɚɡɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɛɵɜɚɸɬ ɛɵɫɬɪɵɦɢ ɢ
ɦɟɞɥɟɧɧɵɦɢ; ɷɤɡɨɬɟɪɦɢɱɟɫɤɢɦɢ (ɢɞɭɳɢɦɢ ɫ ɜɵɞɟɥɟɧɢɟɦ ɬɟɩɥɚ) ɢ ɷɧɞɨɬɟɪɦɢɱɟɫɤɢɦɢ (ɢɞɭɳɢɦɢ ɫ ɩɨɝɥɨɳɟɧɢɟɦ ɬɟɩɥɚ); ɝɨɦɨɝɟɧɧɵɦɢ (ɢɞɭɳɢɟ
ɜ ɨɛɴɟɦɟ) ɢ ɝɟɬɟɪɨɝɟɧɧɵɦɢ (ɩɪɨɬɟɤɚɸɳɢɦɢ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɜɟɳɟɫɬɜ); ɩɪɨɬɟɤɚɸɳɢɦɢ ɛɟɡ ɢɡɦɟɧɟɧɢɹ ɯɢɦɢɱɟɫɤɨɝɨ ɫɨɫɬɚɜɚ ɮɚɡ ɢ ɫ ɢɡɦɟɧɟɧɢɟɦ ɯɢɦɢɱɟɫɤɨɝɨ ɫɨɫɬɚɜɚ. ɉɟɪɜɵɟ – ɷɬɨ ɩɨɥɢɦɨɪɮɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ, ɹɜɥɟɧɢɹ ɭɩɨɪɹɞɨɱɟɧɢɹ ɢ ɪɚɡɭɩɨɪɹɞɨɱɟɧɢɹ ɬɜɟɪɞɵɯ ɪɚɫɬɜɨɪɨɜ; ɜɬɨɪɵɟ – ɷɬɨ ɫɨɛɫɬɜɟɧɧɨ ɯɢɦɢɱɟɫɤɢɟ ɩɪɟɜɪɚɳɟɧɢɹ. ɉɨɥɢɦɨɪɮɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ ɱɚɫɬɨ ɬɨɠɟ ɨɬɧɨɫɹɬ ɤ ɤɥɚɫɫɭ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ, ɬɚɤ ɤɚɤ ɨɧɢ ɡɚɱɚɫɬɭɸ ɫɜɹɡɚɧɵ ɫ ɞɟɮɨɪɦɚɰɢɟɣ ɢ ɢɡɦɟɧɟɧɢɟɦ ɬɢɩɨɜ
ɯɢɦɢɱɟɫɤɢɯ ɫɜɹɡɟɣ. ɏɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ
ɦɨɝɭɬ ɩɪɨɬɟɤɚɬɶ ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɬɜɟɪɞɨɝɨ ɪɚɫɬɜɨɪɚ ɢ ɛɟɡ ɟɝɨ ɨɛɪɚɡɨɜɚɧɢɹ, ɫ ɢɡɦɟɧɟɧɢɟɦ ɚɝɪɟɝɚɬɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɢ ɛɟɡ ɬɚɤɨɝɨ ɢɡɦɟɧɟɧɢɹ; ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɦɨɠɟɬ ɥɢɦɢɬɢɪɨɜɚɬɶɫɹ ɱɢɫɬɨ ɤɢɧɟɬɢɱɟɫɤɢɦɢ ɮɚɤɬɨɪɚɦɢ
(ɬ.ɟ. ɫɨɛɫɬɜɟɧɧɨ ɯɢɦɢɱɟɫɤɨɣ ɫɬɚɞɢɟɣ) ɢɥɢ ɪɚɡɧɨɝɨ ɪɨɞɚ ɩɪɨɰɟɫɫɚɦɢ ɩɟɪɟɧɨɫɚ. ȼɵɲɟ ɩɪɢɜɟɞɟɧɵ ɩɪɢɦɟɪɵ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɨɰɟɫɫɨɜ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɥɚɫɫɢɮɢɤɚɰɢɟɣ ɩɨ ɬɢɩɭ ɜɟɳɟɫɬɜ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɪɟɚɤɰɢɢ.
ɋɭɳɟɫɬɜɭɟɬ ɰɟɥɵɣ ɪɹɞ ɛɵɫɬɪɵɯ ɪɟɚɤɰɢɣ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ɫ ɛɨɥɶɲɢɦ ɷɤɡɨɬɟɪɦɢɱɟɫɤɢɦ ɷɮɮɟɤɬɨɦ, ɫɩɨɫɨɛɧɵɯ ɩɨɞɞɟɪɠɢɜɚɬɶ
ɫɚɦɢɯ ɫɟɛɹ (ɪɟɚɤɰɢɢ ɪɚɡɥɨɠɟɧɢɹ ɜɡɪɵɜɱɚɬɵɯ ɫɨɟɞɢɧɟɧɢɣ). ȼɵɫɨɤɨɣ
ɫɤɨɪɨɫɬɶɸ ɢ ɫɩɨɫɨɛɧɨɫɬɶɸ ɫɚɦɨɪɚɫɩɪɨɫɬɪɚɧɹɬɶɫɹ ɨɛɥɚɞɚɸɬ ɢ ɧɟɤɨɬɨɪɵɟ ɪɟɚɤɰɢɢ ɩɨɥɢɦɟɪɢɡɚɰɢɢ ɜ ɬɜɟɪɞɨɣ ɮɚɡɟ, ɫɜɹɡɚɧɧɵɟ ɫɨ ɫɬɪɭɤɬɭɪɧɵɦɢ ɩɟɪɟɯɨɞɚɦɢ.
Ɉɫɨɛɨɟ ɦɟɫɬɨ ɜ ɥɢɬɟɪɚɬɭɪɟ ɡɚɧɢɦɚɸɬ ɧɢɡɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ ɪɚɞɢɤɚɥɶɧɵɟ ɪɟɚɤɰɢɢ ɜ ɫɬɟɤɥɨɨɛɪɚɡɧɵɯ ɢ ɩɨɥɢɤɪɢɫɬɚɥɥɢɱɟɫɤɢɯ ɦɚɬɪɢɰɚɯ,
ɦɚɪɬɟɧɫɢɬɧɵɟ ɩɪɟɜɪɚɳɟɧɢɹ, ɬɜɟɪɞɨɮɚɡɧɚɹ ɞɟɬɨɧɚɰɢɹ, ɦɟɬɚɥɥɨɬɟɪɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ, ɪɟɚɤɰɢɢ ɜ ɭɞɚɪɧɵɯ ɜɨɥɧɚɯ, ɤɨɬɨɪɵɟ ɩɪɟɞɫɬɚɜɥɹɸɬ ɡɧɚɱɢɬɟɥɶɧɵɣ ɢɧɬɟɪɟɫ ɞɥɹ ɬɟɯɧɨɥɨɝɢɣ ɩɨɥɭɱɟɧɢɹ ɢ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ.
ɋɭɳɟɫɬɜɭɸɬ ɢ ɱɢɫɬɨ ɯɢɦɢɱɟɫɤɢɟ ɫɩɨɫɨɛɵ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ.
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11.5. 2. Ɉɫ ɨɛɟ ɧɧɨɫ ɬɢ ɬɜɟ ɪɞ ɨɮɚɡ ɧɵ ɯ ɩɪɟ ɜɪɚ ɳɟ ɧɢɣ
ɑɬɨ ɠɟ ɹɜɥɹɟɬɫɹ ɨɛɳɢɦ ɞɥɹ ɜɫɟɯ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɱɬɨ ɨɬɥɢɱɚɟɬ
ɢɯ ɨɬ ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɝɚɡɚɯ ɢ ɠɢɞɤɨɫɬɹɯ?
ȼɨ-ɩɟɪɜɵɯ, ɷɬɨ – ɥɨɤɚɥɢɡɚɰɢɹ ɪɟɚɤɰɢɢ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ. ɉɨ
ɷɬɨɣ ɩɪɢɱɢɧɟ ɜɫɟ ɪɟɚɤɰɢɢ, ɩɪɨɬɟɤɚɸɳɢɟ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ,
ɧɟɤɨɬɨɪɵɟ ɚɜɬɨɪɵ ɧɚɡɵɜɚɸɬ ɝɟɬɟɪɨɝɟɧɧɵɦɢ. ɉɪɢ ɧɚɥɢɱɢɢ ɯɨɬɹ ɛɵ ɨɞɧɨɣ ɬɜɟɪɞɨɣ ɮɚɡɵ ɯɢɦɢɱɟɫɤɨɟ ɩɪɟɜɪɚɳɟɧɢɟ ɧɟ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜ
ɥɸɛɨɣ ɬɨɱɤɟ ɩɪɨɫɬɪɚɧɫɬɜɚ, ɤɚɤ, ɧɚɩɪɢɦɟɪ, ɜ ɫɥɭɱɚɟ ɪɟɚɤɰɢɣ, ɩɪɨɬɟɤɚɸɳɢɯ ɜ ɝɚɡɨɜɨɣ ɮɚɡɟ, ɝɞɟ ɯɢɦɢɱɟɫɤɨɟ ɩɪɟɜɪɚɳɟɧɢɟ ɦɨɠɟɬ ɫɨɜɟɪɲɚɬɶɫɹ ɜ
ɪɟɡɭɥɶɬɚɬɟ ɫɥɭɱɚɣɧɵɯ ɫɬɨɥɤɧɨɜɟɧɢɣ ɦɟɠɞɭ ɱɚɫɬɢɰɚɦɢ. ȿɫɥɢ ɨɞɧɚ ɢɡ ɮɚɡ
– ɬɜɟɪɞɨɟ ɜɟɳɟɫɬɜɨ, ɬɨ ɪɟɱɶ ɦɨɠɟɬ ɢɞɬɢ ɥɢɲɶ ɨ ɛɨɥɟɟ ɢɥɢ ɦɟɧɟɟ ɩɪɨɬɹɠɟɧɧɨɣ ɨɛɥɚɫɬɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɜɛɥɢɡɢ ɩɨɜɟɪɯɧɨɫɬɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɜ
ɤɨɬɨɪɨɣ ɚɬɨɦɵ ɢɥɢ ɦɨɥɟɤɭɥɵ ɬɜɟɪɞɨɝɨ ɫɨɟɞɢɧɟɧɢɹ ɞɨɫɬɭɩɧɵ ɞɪɭɝɢɦ
ɪɟɚɤɰɢɨɧɧɵɦ ɱɚɫɬɢɰɚɦ. ɗɬɚ ɨɝɪɚɧɢɱɟɧɧɚɹ ɨɛɥɚɫɬɶ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɨɛɥɚɞɚɟɬ ɩɨɜɵɲɟɧɧɨɣ ɪɟɚɤɰɢɨɧɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ, ɫɩɨɫɨɛɫɬɜɭɸɳɟɣ ɥɨɤɚɥɢɡɚɰɢɢ ɢ ɚɜɬɨɥɨɤɚɥɢɡɚɰɢɢ ɩɪɨɰɟɫɫɚ. ɉɨɜɵɲɟɧɧɨɣ ɪɟɚɤɰɢɨɧɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɦɨɝɭɬ ɨɛɥɚɞɚɬɶ ɢ ɭɱɚɫɬɤɢ ɜ ɨɛɴɟɦɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ, ɧɚɯɨɞɹɳɢɟɫɹ
ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɫɬɪɭɤɬɭɪɧɵɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɟɣ. ɉɪɨɰɟɫɫ ɮɨɪɦɢɪɨɜɚɧɢɹ
ɪɟɚɤɰɢɨɧɧɨɣ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ ɫɨɫɬɨɢɬ ɢɡ ɰɟɥɨɝɨ ɪɹɞɚ ɫɨɛɵɬɢɣ. Ʉɪɚɬɤɨ
ɷɬɨ – ɨɛɪɚɡɨɜɚɧɢɟ ɥɨɤɚɥɶɧɵɯ ɡɚɪɨɞɵɲɟɣ ɩɪɨɞɭɤɬɚ, ɢɯ ɪɨɫɬ ɢ ɫɥɢɹɧɢɟ.
ɋɬɚɞɢɢ ɨɛɪɚɡɨɜɚɧɢɹ ɢ ɪɨɫɬɚ ɡɚɪɨɞɵɲɟɣ ɞɥɹ ɪɚɡɥɢɱɧɵɯ ɩɪɨɫɬɟɣɲɢɯ ɪɟɚɤɰɢɣ, ɢɯ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɚɧɚɥɨɝɢɱɧɵ ɩɨɞɨɛɧɵɦ ɫɬɚɞɢɹɦ ɮɚɡɨɜɵɯ ɩɪɟɜɪɚɳɟɧɢɣ. ȼɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɩɨɞ ɡɨɧɨɣ ɪɟɚɤɰɢɢ ɩɨɧɢɦɚɸɬ ɮɪɨɧɬ ɪɟɚɤɰɢɢ – «ɝɪɚɧɢɰɭ ɪɚɡɞɟɥɚ» ɦɟɠɞɭ ɪɟɚɝɟɧɬɚɦɢ ɢ ɩɪɨɞɭɤɬɚɦɢ. Ɉɛɪɚɡɨɜɚɧɢɟ ɮɪɨɧɬɚ ɪɟɚɤɰɢɢ ɜ ɪɟɡɭɥɶɬɚɬɟ ɫɥɢɹɧɢɹ ɡɚɪɨɞɵɲɟɣ ɫɜɹɡɚɧɨ ɫ ɛɨɥɶɲɨɣ
ɪɟɚɤɰɢɨɧɧɨɣ ɫɩɨɫɨɛɧɨɫɬɶɸ ɚɬɨɦɨɜ ɢ ɦɨɥɟɤɭɥ, ɧɚɯɨɞɹɳɢɯɫɹ ɜɛɥɢɡɢ ɝɪɚɧɢɰɵ ɪɚɡɞɟɥɚ ɮɚɡ, ɢ ɫ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɦɟɞɥɟɧɧɨɫɬɶɸ ɩɪɨɰɟɫɫɚ ɦɚɫɫɨɩɟɪɟɧɨɫɚ, ɨɛɟɫɩɟɱɢɜɚɸɳɟɝɨ ɩɨɞɜɨɞ ɪɟɚɝɟɧɬɚ (ɪɟɚɝɟɧɬɨɜ) ɤ ɷɬɨɣ ɝɪɚɧɢɰɟ.
Ɍ.ɟ. ɞɥɹ ɦɧɨɝɢɯ ɬɜɟɪɞɨɮɚɡɧɵɯ ɪɟɚɤɰɢɣ ɨɛɪɚɡɨɜɚɧɢɟ ɮɪɨɧɬɚ ɧɟ ɜɫɟɝɞɚ
ɨɛɟɫɩɟɱɢɜɚɟɬɫɹ ɫɢɥɶɧɨɣ ɡɚɜɢɫɢɦɨɫɬɶɸ ɫɤɨɪɨɫɬɢ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ (ɷɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɜ ɩɪɨɰɟɫɫɚɯ ɝɨɪɟɧɢɹ, ɜ ɬɨɦ ɱɢɫɥɟ ɬɜɟɪɞɨɮɚɡɧɨɝɨ).
Ƚɪɚɧɢɰɚ ɪɚɡɞɟɥɚ «ɢɫɯɨɞɧɨɟ ɜɟɳɟɫɬɜɨ – ɩɪɨɞɭɤɬ ɪɟɚɤɰɢɢ» ɢɦɟɟɬ ɤɚɤ
ɨɛɴɟɤɬ ɢɫɫɥɟɞɨɜɚɧɢɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɢ ɹɜɥɹɟɬɫɹ ɜ ɢɡɜɟɫɬɧɨɦ
ɫɦɵɫɥɟ ɢɞɟɚɥɢɡɢɪɨɜɚɧɧɵɦ ɩɨɧɹɬɢɟɦ, ɞɨɩɭɫɤɚɸɳɢɦ ɭɩɪɨɳɟɧɧɨɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ. ɗɬɨ ɡɚɦɟɱɚɧɢɟ ɨɬɧɨɫɢɬɫɹ ɤɚɤ ɤ ɫɨɛɫɬɜɟɧɧɨ ɯɢɦɢɱɟɫɤɢɦ ɩɪɟɜɪɚɳɟɧɢɹɦ, ɬɚɤ ɢ ɤɨ ɦɧɨɝɢɦ ɮɚɡɨɜɵɦ ɩɟɪɟɯɨɞɚɦ. ȼ ɨɤɪɟɫɬɧɨɫɬɢ ɷɬɨɣ «ɝɪɚɧɢɰɵ» ɦɨɝɭɬ ɦɟɧɹɬɶɫɹ ɨɞɧɨɜɪɟɦɟɧɧɨ ɧɟɫɤɨɥɶɤɨ ɩɚɪɚɦɟɬɪɨɜ, ɫɤɚɡɵɜɚɸɳɢɯɫɹ ɧɚ ɪɟɚɤɰɢɨɧɧɨɣ ɫɩɨɫɨɛɧɨɫɬɢ ɜɟɳɟɫɬɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ ɨɛɪɚɡɨɦ.
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ȼɬɨɪɚɹ ɨɫɨɛɟɧɧɨɫɬɶ ɪɟɚɤɰɢɣ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ – ɷɬɨ ɦɚɥɚɹ ɩɨɞɜɢɠɧɨɫɬɶ ɢ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɚɹ ɪɚɡɞɟɥɟɧɧɨɫɬɶ ɪɟɚɝɟɧɬɨɜ ɢɥɢ
ɪɟɚɝɟɧɬɚ ɢ ɩɪɨɞɭɤɬɚ. ɗɬɨ ɨɛɭɫɥɨɜɥɢɜɚɟɬ ɜɚɠɧɭɸ ɪɨɥɶ ɩɪɨɰɟɫɫɨɜ ɩɟɪɟɧɨɫɚ ɬɨɝɨ ɢɥɢ ɢɧɨɝɨ ɬɢɩɚ. ȿɫɬɟɫɬɜɟɧɧɨ, ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɢɩɚ ɪɟɚɤɰɢɢ ɢ
ɭɫɥɨɜɢɣ ɟɟ ɨɫɭɳɟɫɬɜɥɟɧɢɹ, ɯɚɪɚɤɬɟɪ ɦɚɫɫɨɩɟɪɟɧɨɫɚ ɪɚɡɥɢɱɟɧ. ȿɫɥɢ ɪɟɚɝɟɧɬɵ ɢ ɩɪɨɞɭɤɬ ɪɟɚɤɰɢɢ – ɬɜɟɪɞɵɟ ɜɟɳɟɫɬɜɚ, ɬɨ ɫɤɨɪɨɫɬɶ ɯɢɦɢɱɟɫɤɨɣ
ɪɟɚɤɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɫɚɦɨɣ ɦɟɞɥɟɧɧɨɣ ɟɟ ɫɬɚɞɢɢ (ɞɢɮɮɭɡɢɢ ɪɟɚɝɟɧɬɨɜ
ɱɟɪɟɡ ɫɥɨɣ ɩɪɨɞɭɤɬɚ) ɢ ɡɚɦɟɞɥɹɟɬɫɹ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɩɨɫɥɟɞɧɟɝɨ. ȼ ɬɨ ɠɟ
ɫɚɦɨɟ ɜɪɟɦɹ ɜ ɭɫɥɨɜɢɹɯ ɜɵɫɨɤɢɯ ɞɚɜɥɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɢ ɫɞɜɢɝɚ «ɦɟɞɥɟɧɧɚɹ ɞɢɮɮɭɡɢɹ» ɱɟɪɟɡ ɫɥɨɣ ɩɪɨɞɭɤɬɚ ɧɟ ɹɜɥɹɟɬɫɹ ɩɪɟɩɹɬɫɬɜɢɟɦ ɤ
ɨɫɭɳɟɫɬɜɥɟɧɢɸ ɪɟɚɤɰɢɢ; ɡɞɟɫɶ ɡɚ ɦɚɥɵɟ ɜɪɟɦɟɧɚ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɦɟɲɢɜɚɧɢɟ ɧɚ ɦɨɥɟɤɭɥɹɪɧɨɦ ɭɪɨɜɧɟ ɪɟɚɝɟɧɬɨɜ ɢɦɟɧɧɨ ɜ ɬɜɟɪɞɨɣ ɮɚɡɟ. ȿɫɥɢ ɨɞɢɧ ɢɡ ɪɟɚɝɟɧɬɨɜ – ɝɚɡ ɢɥɢ ɠɢɞɤɨɫɬɶ, ɬɨ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ, ɨɫɭɳɟɫɬɜɥɹɸɳɟɣɫɹ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ, ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶɸ ɧɚɢɛɨɥɟɟ ɩɨɞɜɢɠɧɨɝɨ ɪɟɚɝɟɧɬɚ ɢɥɢ ɫɤɨɪɨɫɬɶɸ ɨɬɜɨɞɚ ɩɪɨɞɭɤɬɚ ɪɟɚɤɰɢɢ. ɋɤɨɪɨɫɬɶɸ ɦɚɫɫɨɩɟɪɟɧɨɫɚ ɜ ɩɨɪɚɯ ɢ ɬɪɟɳɢɧɚɯ ɬɜɟɪɞɨɝɨ ɜɟɳɟɫɬɜɚ ɨɩɪɟɞɟɥɹɸɬɫɹ, ɧɚɩɪɢɦɟɪ, ɦɧɨɝɢɟ ɨɛɪɚɬɢɦɵɟ ɪɟɚɤɰɢɢ ɪɚɡɥɨɠɟɧɢɹ.
ɂɫɤɥɸɱɢɬɟɥɶɧɨ ɜɚɠɧɚɹ ɪɨɥɶ ɩɪɨɰɟɫɫɨɜ ɩɟɪɟɧɨɫɚ ɩɪɨɹɜɥɹɟɬɫɹ ɢ ɩɪɢ
ɩɪɨɬɟɤɚɧɢɢ ɪɟɚɤɰɢɣ ɜ ɩɨɥɢɦɟɪɧɵɯ ɫɢɫɬɟɦɚɯ, ɞɥɹ ɤɨɬɨɪɵɯ ɤɢɧɟɬɢɱɟɫɤɢɟ
ɨɫɨɛɟɧɧɨɫɬɢ ɞɢɮɮɭɡɢɢ ɫɭɳɟɫɬɜɟɧɧɨ ɡɚɜɢɫɹɬ ɨɬ ɬɨɝɨ, ɜ ɤɚɤɨɦ ɫɨɫɬɨɹɧɢɢ
ɧɚɯɨɞɢɬɫɹ ɩɨɥɢɦɟɪ - ɜɵɫɨɤɨɷɥɚɫɬɢɱɟɫɤɨɦ, ɜɹɡɤɨɦ, ɫɬɟɤɥɨɨɛɪɚɡɧɨɦ ɢɥɢ
ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɦ. ɉɪɨɰɟɫɫɵ ɩɟɪɟɧɨɫɚ ɢɝɪɚɸɬ ɨɩɪɟɞɟɥɹɸɳɭɸ ɪɨɥɶ ɩɪɢ
ɨɤɢɫɥɟɧɢɢ ɦɟɬɚɥɥɨɜ; ɩɪɢ ɬɜɟɪɞɨɮɚɡɧɨɦ ɫɢɧɬɟɡɟ ɫɥɨɠɧɵɯ ɫɨɟɞɢɧɟɧɢɣ ɢɡ
ɛɨɥɟɟ ɩɪɨɫɬɵɯ; ɩɪɢ ɡɚɪɨɞɵɲɟɨɛɪɚɡɨɜɚɧɢɢ ɜ ɧɚɱɚɥɶɧɨɣ ɫɬɚɞɢɢ ɬɜɟɪɞɨɮɚɡɧɨɣ ɪɟɚɤɰɢɢ. «Ɇɚɫɫɨɩɟɪɟɧɨɫ» – ɷɬɨ ɩɟɪɟɧɨɫ ɧɟ ɬɨɥɶɤɨ ɪɟɚɝɟɧɬɨɜ ɢɥɢ
ɩɪɨɞɭɤɬɨɜ ɪɟɚɤɰɢɢ, ɧɨ ɢ ɩɟɪɟɦɟɳɟɧɢɟ ɩɪɢɦɟɫɟɣ, ɞɜɢɠɟɧɢɟ ɞɢɫɥɨɤɚɰɢɣ,
ɜɚɤɚɧɫɢɣ ɢ ɩɨɪ. ȼɨɨɛɳɟ ɝɨɜɨɪɹ, ɢɫɤɥɸɱɟɧɢɟ ɩɪɨɰɟɫɫɨɜ ɩɟɪɟɧɨɫɚ ɢɡ ɹɜɧɨɝɨ ɪɚɫɫɦɨɬɪɟɧɢɹ ɩɪɢɜɨɞɢɬ ɢɥɢ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɤ ɧɟɜɟɪɧɨɣ ɬɪɚɤɬɨɜɤɟ
ɫɦɵɫɥɚ ɮɨɪɦɚɥɶɧɨ-ɤɢɧɟɬɢɱɟɫɤɢɯ ɩɚɪɚɦɟɬɪɨɜ ɪɟɚɤɰɢɣ ɢ ɢɫɤɚɠɟɧɢɸ ɮɢɡɢɱɟɫɤɢɯ ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ.
Ɍɪɟɬɶɟɣ ɜɚɠɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ ɹɜɥɹɟɬɫɹ ɧɚɥɢɱɢɟ ɫɥɨɠɧɵɯ ɨɛɪɚɬɧɵɯ ɫɜɹɡɟɣ ɦɟɠɞɭ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɦ
ɩɪɟɜɪɚɳɟɧɢɟɦ
ɢ
ɪɚɡɥɢɱɧɵɦɢ
ɮɢɡɢɱɟɫɤɢɦɢ
ɹɜɥɟɧɢɹɦɢ,
ɢɯ
ɫɨɩɪɨɜɨɠɞɚɸɳɢɦɢ.
Ʉɚɤɢɟ ɮɢɡɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ ɫɨɩɪɨɜɨɠɞɚɸɬ ɯɢɦɢɱɟɫɤɢɟ ɩɪɟɜɪɚɳɟɧɢɹ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ɢɥɢ ɹɜɥɹɸɬɫɹ ɩɨɥɧɨɩɪɚɜɧɵɦɢ «ɷɥɟɦɟɧɬɚɪɧɵɦɢ» ɫɬɚɞɢɹɦɢ ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ? ɗɬɨ – ɜɨɡɧɢɤɧɨɜɟɧɢɟ
ɞɟɮɟɤɬɨɜ, ɪɚɡɪɵɯɥɟɧɢɟ ɤɪɢɫɬɚɥɥɢɱɟɫɤɢɯ ɪɟɲɟɬɨɤ; ɢɯ ɩɟɪɟɫɬɪɨɣɤɚ
ɜɫɥɟɞɫɬɜɢɟ ɩɨɥɢɦɨɪɮɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ; ɨɛɪɚɡɨɜɚɧɢɟ ɢ ɪɚɫɩɚɞ ɬɜɟɪɞɵɯ
ɪɚɫɬɜɨɪɨɜ; ɫɩɟɤɚɧɢɟ, «ɨɬɞɵɯ» ɪɟɤɪɢɫɬɚɥɥɢɡɚɰɢɹ; ɩɥɚɜɥɟɧɢɟ; ɪɚɫɬɜɨɪɟ289
ɧɢɟ ɤɨɦɩɨɧɟɧɬɨɜ ɜ ɪɚɫɩɥɚɜɟ; ɤɪɢɫɬɚɥɥɢɡɚɰɢɹ ɢɡ ɠɢɞɤɨɣ ɮɚɡɵ; ɜɨɡɝɨɧɤɚ;
ɞɢɫɫɨɰɢɚɰɢɹ ɢ ɞɪ..
ɇɚ ɪɟɚɤɰɢɨɧɧɭɸ ɫɩɨɫɨɛɧɨɫɬɶ ɬɜɟɪɞɵɯ ɬɟɥ ɨɤɚɡɵɜɚɸɬ ɜɥɢɹɧɢɟ,
ɤɪɨɦɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɦɟɯɚɧɢɱɟɫɤɢɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɨɛɪɚɡɰɟ (ɜɧɟɲɧɢɟ,
ɜɧɭɬɪɟɧɧɢɟ), ɞɢɮɮɭɡɢɹ (ɜɧɟɲɧɹɹ, ɜɧɭɬɪɟɧɧɹɹ, ɩɨɜɟɪɯɧɨɫɬɧɚɹ), ɭɫɥɨɜɢɹ
ɜ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɟ, ɧɚɩɪɹɠɟɧɧɨɫɬɶ ɷɥɟɤɬɪɢɱɟɫɤɨɝɨ ɩɨɥɹ, ɫɬɪɭɤɬɭɪɧɵɟ
ɨɫɨɛɟɧɧɨɫɬɢ. ȼ ɬɜɟɪɞɨɦ ɜɟɳɟɫɬɜɟ ɜɫɟ ɷɬɢ ɮɚɤɬɨɪɵ ɜɡɚɢɦɨɫɜɹɡɚɧɵ ɢ ɫɩɨɫɨɛɧɵ ɜɥɢɹɬɶ ɧɚ ɯɢɦɢɱɟɫɤɭɸ ɪɟɚɤɰɢɸ ɤɚɤ ɩɪɹɦɨ, ɬɚɤ ɢ ɤɨɫɜɟɧɧɨ.
Ɍɚɤ, ɫ ɢɡɦɟɧɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɬɟɥɚ ɢɡɦɟɧɹɟɬɫɹ ɟɝɨ ɫɬɪɭɤɬɭɪɚ (ɜ
ɪɟɡɭɥɶɬɚɬɟ ɮɚɡɨɜɵɯ ɩɪɟɜɪɚɳɟɧɢɣ); ɦɟɧɹɟɬɫɹ ɤɨɧɰɟɧɬɪɚɰɢɹ ɢ ɩɨɞɜɢɠɧɨɫɬɶ ɞɟɮɟɤɬɨɜ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɪɟɲɟɬɤɢ (ɩɪɨɢɫɯɨɞɢɬ ɢɯ ɝɟɧɟɪɚɰɢɹ
ɢɥɢ ɨɬɠɢɝ); ɩɨɹɜɥɹɸɬɫɹ ɢɥɢ ɪɟɥɚɤɫɢɪɭɸɬ ɬɟɦɩɟɪɚɬɭɪɧɵɟ ɧɚɩɪɹɠɟɧɢɹ.
Ɇɟɯɚɧɢɱɟɫɤɢɟ ɧɚɩɪɹɠɟɧɢɹ (ɥɸɛɨɣ ɩɪɢɪɨɞɵ), ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɫɩɨɫɨɛɧɵ
ɜɥɢɹɬɶ ɧɚ ɯɢɦɢɱɟɫɤɭɸ ɪɟɚɤɰɢɸ ɩɨ ɪɚɡɥɢɱɧɵɦ ɤɚɧɚɥɚɦ: ɱɟɪɟɡ ɢɡɦɟɧɟɧɢɟ
ɫɬɪɭɤɬɭɪɵ ɜɟɳɟɫɬɜɚ; ɷɥɟɤɬɪɨɧɧɨɝɨ ɫɬɪɨɟɧɢɹ; ɩɨɞɜɢɠɧɨɫɬɢ ɞɟɮɟɤɬɨɜ
ɢɥɢ ɞɪɭɝɢɯ ɱɚɫɬɢɰ; ɬɟɦɩɟɪɚɬɭɪɵ; ɤɨɧɰɟɧɬɪɚɰɢɢ ɞɢɫɥɨɤɚɰɢɣ, ɜɚɤɚɧɫɢɣ;
ɪɚɡɦɟɪɨɜ ɪɟɚɝɢɪɭɸɳɢɯ ɱɚɫɬɢɰ; ɧɚɥɢɱɢɟ ɚɤɬɢɜɧɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ.
ɋɥɨɠɧɵɣ ɯɚɪɚɤɬɟɪ ɬɜɟɪɞɨɮɚɡɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɨɬɪɚɠɚɟɬɫɹ ɧɚ
ɤɢɧɟɬɢɱɟɫɤɢɯ ɤɪɢɜɵɯ. Ⱦɥɹ ɬɜɟɪɞɨɮɚɡɧɵɯ ɩɪɟɜɪɚɳɟɧɢɣ ɤɢɧɟɬɢɱɟɫɤɢɟ
ɤɪɢɜɵɟ ɭɞɨɛɧɨ ɩɪɟɞɫɬɚɜɥɹɬɶ ɜ ɜɢɞɟ
D
f t ,
(11.41)
ɝɞɟ D – ɫɬɟɩɟɧɶ ɩɪɟɜɪɚɳɟɧɢɹ, D n n0 , n – ɱɢɫɥɨ ɦɨɥɟɣ ɪɟɚɝɟɧɬɚ ɜ ɬɟɤɭɳɢɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, n0 – ɜ ɧɚɱɚɥɶɧɵɣ.
ȿɫɥɢ ɤɨɧɰɟɧɬɪɚɰɢɹ ɜɟɳɟɫɬɜɚ ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɵɯ ɞɚɜɥɟɧɢɢ ɢ ɬɟɦɩɟɪɚɬɭɪɟ ɨɞɧɨɡɧɚɱɧɨ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɫɢɫɬɟɦɭ, ɬɨ ɫɬɟɩɟɧɶ ɩɪɟɜɪɚɳɟɧɢɹ ɧɟ
ɹɜɥɹɟɬɫɹ ɨɞɧɨɡɧɚɱɧɨɣ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɝɟɬɟɪɨɮɚɡɧɨɣ ɫɢɫɬɟɦɵ ɞɚɠɟ ɩɪɢ
ɮɢɤɫɢɪɨɜɚɧɧɵɯ ɩɚɪɚɦɟɬɪɚɯ ɫɨɫɬɨɹɧɢɹ. Ɉɞɧɚ ɢ ɬɚ ɠɟ ɫɬɟɩɟɧɶ ɩɪɟɜɪɚɳɟɧɢɹ ɦɨɠɟɬ ɛɵɬɶ ɪɟɚɥɢɡɨɜɚɧɚ ɜ ɪɚɡɧɨɨɛɪɚɡɧɵɯ ɫɢɫɬɟɦɚɯ ɨɞɢɧɚɤɨɜɨɝɨ ɫɨɫɬɚɜɚ, ɨɬɥɢɱɚɸɳɢɯɫɹ ɫɬɪɭɤɬɭɪɨɣ ɪɟɚɤɰɢɨɧɧɨɣ ɡɨɧɵ.
Ⱦɥɹ ɬɜɟɪɞɨɮɚɡɧɵɯ ɪɟɚɤɰɢɣ ɬɢɩɢɱɟɧ ɫɢɝɦɨɢɞɧɵɣ ɯɚɪɚɤɬɟɪ ɤɪɢɜɵɯ
(11.41) (ɪɢɫ. 11.6). ȼ ɧɚɱɚɥɟ ɩɪɨɰɟɫɫɚ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɦɚɥɚ, ɧɚɛɥɸɞɚɟɬɫɹ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɢɧɞɭɤɰɢɨɧɧɵɣ ɩɟɪɢɨɞ. Ɂɚɬɟɦ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ
ɪɟɡɤɨ ɜɨɡɪɚɫɬɚɟɬ, ɩɪɨɯɨɞɢɬ ɱɟɪɟɡ ɦɚɤɫɢɦɭɦ (ɤɨɬɨɪɨɦɭ ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɩɟɪɟɝɢɛ ɧɚ ɤɪɢɜɨɣ D f t ) ɢ ɞɚɥɟɟ ɫɧɢɠɚɟɬɫɹ ɞɨ ɧɭɥɹ. Ɋɚɡɭɦɟɟɬɫɹ,
ɮɨɪɦɚ ɤɢɧɟɬɢɱɟɫɤɨɣ ɤɪɢɜɨɣ ɫɜɹɡɚɧɚ ɫ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ ɢ ɯɚɪɚɤɬɟɪɨɦ ɩɪɨɰɟɫɫɨɜ, ɫɨɫɬɚɜɥɹɸɳɢɯ ɬɜɟɪɞɨɮɚɡɧɭɸ ɪɟɚɤɰɢɸ.
290
ɇɚɛɥɸɞɚɟɦɚɹ ɤɢɧɟɬɢɱɟɫɤɚɹ ɤɪɢɜɚɹ ɫɨɞɟɪɠɢɬ ɢɧɮɨɪɦɚɰɢɸ ɨ ɫɨɜɨɤɭɩɧɨɫɬɢ ɩɪɨɰɟɫɫɨɜ ɪɚɡɥɢɱɧɨɝɨ ɬɢɩɚ, ɞɥɹ ɨɩɢɫɚɧɢɹ ɤɨɬɨɪɵɯ
ɬɪɟɛɭɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɦɨɞɟɥɢ. Ɋɹɞ ɬɚɤɢɯ ɦɨɞɟɥɟɣ ɨɛɫɭɠɞɚɥɫɹ ɢ ɨɛɫɭɠɞɚɟɬɫɹ ɜ ɦɧɨɝɨɱɢɫɥɟɧɧɨɣ ɥɢɬɟɪɚɬɭɪɟ ɩɨ ɤɢɧɟɬɢɤɟ ɬɜɟɪɞɨɮɚɡɧɵɯ
ɪɟɚɤɰɢɣ. Ɉɫɨɛɨɟ ɦɟɫɬɨ ɡɚɧɢɦɚɸɬ ɦɨɞɟɥɢ ɨɛɊɢɫ. 11.6. Ʉɢɧɟɬɢɱɟɫɤɚɹ
ɪɚɡɨɜɚɧɢɹ ɢ ɪɨɫɬɚ ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɤɪɢɜɚɹ, ɬɢɩɢɱɧɚɹ ɞɥɹ
ɞɭɤɬɚ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɱɚɫɬɨ ɝɨɜɨɪɹɬ ɨ ɬɨɩɨɯɢɬɜɟɪɞɨɮɚɡɧɵɯ ɪɟɚɤɰɢɣ
ɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɹɯ – ɪɟɚɤɰɢɹɯ, ɫɤɨɪɨɫɬɶ ɤɨɬɨɪɵɯ ɜ ɫɭɳɟɫɬɜɟɧɧɨɣ ɫɬɟɩɟɧɢ ɡɚɜɢɫɢɬ ɨɬ ɫɬɪɭɤɬɭɪɧɵɯ ɢ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɪɟɚɝɟɧɬɚ.
11.6. Ʉɢɧɟɬɢɱɟɫɤɨɟ ɨɩɢɫɚɧɢɟ
ɬ ɜ ɟ ɪ ɞ ɨ ɮ ɚ ɡ ɧ ɵ ɯ ɩ ɪ ɟ ɜ ɪ ɚ ɳ ɟ ɧ ɢ ɣ 31
Ɋɚɡɞɟɥ ɬɟɨɪɢɢ ɤɢɧɟɬɢɤɢ ɬɨɩɨɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ ɞɚɟɬ ɨɫɧɨɜɭ ɞɥɹ
ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɢ ɪɚɫɱɟɬɚ ɨɫɨɛɟɧɧɨɫɬɟɣ ɢɡɦɟɧɟɧɢɹ ɜɨ ɜɪɟɦɟɧɢ ɧɚɛɥɸɞɚɟɦɨɣ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ. Ɉɫɬɚɧɨɜɢɦɫɹ ɥɢɲɶ ɧɚ ɨɩɢɫɚɧɢɢ ɤɢɧɟɬɢɤɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɞɥɹ ɪɟɚɤɰɢɣ ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ «ɝɚɡ – ɬɜɟɪɞɨɟ ɬɟɥɨ».
ɉɪɢɧɰɢɩɵ ɩɨɫɬɪɨɟɧɢɹ ɦɨɞɟɥɟɣ ɞɥɹ ɞɪɭɝɢɯ ɫɢɫɬɟɦ ɨɫɬɚɸɬɫɹ ɬɟɦɢ ɠɟ.
ɉɭɫɬɶ ɧɚ ɢɫɯɨɞɧɨɣ ɩɨɜɟɪɯɧɨɫɬɢ ɤɪɢɫɬɚɥɥɚ S ɢɦɟɟɬɫɹ Z 0 ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ ɹɞɪɨɨɛɪɚɡɨɜɚɧɢɹ, ɤɨɧɰɟɧɬɪɚɰɢɹ ɤɨɬɨɪɵɯ ɦɟɧɹɟɬɫɹ
ɬɨɥɶɤɨ ɡɚ ɫɱɟɬ ɩɪɨɰɟɫɫɚ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ. Ɍɨɝɞɚ ɫɤɨɪɨɫɬɶ ɨɛɪɚɡɨɜɚɧɢɹ
ɹɞɟɪ ɨɩɪɟɞɟɥɢɬɫɹ ɬɚɤ
dN
N·
§
W cS ¨ Z 0 ¸ ,
dt
S¹
©
ɝɞɟ W c – ɭɞɟɥɶɧɚɹ ɫɤɨɪɨɫɬɶ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ (ɜ ɪɚɫɱɟɬɟ ɧɚ ɨɞɢɧ ɰɟɧɬɪ),
ɡɚɜɢɫɹɳɚɹ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɢ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɪɟɚɝɟɧɬɚ ɩɪɢ ɤɨɧɰɟɧɬɪɚɰɢɢ
ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ Z 0 .
ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɢɦɟɟɦ
t 0: N 0.
ɉɨɫɥɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚɣɞɟɦ
N Z 0 S ª¬1 exp W ct º¼
31
Ɋɨɡɨɜɫɤɢɣ Ⱥ.ə. Ƚɟɬɟɪɨɝɟɧɧɵɟ ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ. Ʉɢɧɟɬɢɤɚ ɢ ɦɚɤɪɨɤɢɧɟɬɢɤɚ. Ɇ.: ɇɚɭɤɚ, 1980. 324 ɫ.
291
ɢ
dN
W cZ 0 S ª¬1 exp W ct º¼ .
(11.42)
dt
ɍɞɟɥɶɧɭɸ ɫɤɨɪɨɫɬɶ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɨɛɵɱɧɨ ɧɚɡɵɜɚɸɬ ɤɨɧɫɬɚɧɬɨɣ
ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ ɡɚɪɨɞɵɲɟɣ (ɹɞɟɪ) ɢ ɩɪɢɩɢɫɵɜɚɸɬ ɟɣ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ. ɍɪɚɜɧɟɧɢɟ (11.42) ɧɚɡɵɜɚɸɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɵɦ ɡɚɤɨɧɨɦ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ. ɉɪɢ ɜɵɜɨɞɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ, ɱɬɨ W c const . Ʉɪɨɦɟ ɬɨɝɨ, ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɫɦɵɫɥ ɩɪɢ ɩɨɫɬɨɹɧɫɬɜɟ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɤɨɧɰɟɧɬɪɚɰɢɢ ɪɟɚɝɟɧɬɨɜ. Ɍ.ɟ., ɭɫɥɨɜɢɟɦ ɩɪɢɦɟɧɢɦɨɫɬɢ (11.42) ɹɜɥɹɟɬɫɹ ɩɪɨɜɟɞɟɧɢɟ ɪɟɚɤɰɢɢ ɜ ɫɢɫɬɟɦɟ, ɨɬɤɪɵɬɨɣ
ɞɥɹ ɝɚɡɨɨɛɪɚɡɧɨɝɨ ɪɟɚɝɟɧɬɚ, ɢ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ. Ʉɪɨɦɟ ɬɨɝɨ, ɬɪɟɛɭɟɬɫɹ
ɩɪɟɞɩɨɥɚɝɚɬɶ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɫɬɶ ɫɢɫɬɟɦɵ.
ɉɪɢ ɜɵɜɨɞɟ ɩɪɟɞɩɨɥɚɝɚɥɨɫɶ, ɱɬɨ ɧɟɬ ɜɨɫɩɪɨɢɡɜɨɞɫɬɜɚ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ ɹɞɪɨɨɛɪɚɡɨɜɚɧɢɹ ɜ ɯɨɞɟ ɪɟɚɤɰɢɢ. ȼ ɞɟɣɫɬɜɢɬɟɥɶɧɨɫɬɢ ɷɬɨ –
ɧɟ ɜɫɟɝɞɚ ɬɚɤ. ɇɚɩɪɢɦɟɪ, ɜ ɪɚɞɢɚɰɢɨɧɧɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɚɯ ɩɨɬɟɧɰɢɚɥɶɧɵɟ ɰɟɧɬɪɵ ɦɨɝɭɬ ɩɨɹɜɥɹɬɶɫɹ ɜ ɯɨɞɟ ɪɟɚɤɰɢɢ. ɑɬɨɛɵ ɭɱɟɫɬɶ ɷɬɨɬ
ɩɪɨɰɟɫɫ, ɦɨɠɧɨ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ 1) ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɜɨɫɩɪɨɢɡɜɨɞɫɬɜɚ ɰɟɧɬɪɨɜ ɩɨɫɬɨɹɧɧɚ
Wz W ccS ,
(11.43)
2) ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ, ɜɨɫɩɪɨɢɡɜɨɞɹɳɟɝɨ ɩɨɬɟɧɰɢɚɥɶɧɵɟ ɰɟɧɬɪɵ ɹɞɪɨɨɛɪɚɡɨɜɚɧɢɹ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɪɚɜɧɨɜɟɫɧɨɣ Z e ɢ ɬɟɤɭɳɟɣ Z ɤɨɧɰɟɧɬɪɚɰɢɹɦɢ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ
Wz W cc Ze Z S .
(11.44)
Ⱦɥɹ ɦɨɞɟɥɢ 1) ɤ ɩɪɨɢɡɜɨɥɶɧɨɦɭ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ t ɤɨɥɢɱɟɫɬɜɨ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ ɧɚ ɟɞɢɧɢɰɟ ɩɨɜɟɪɯɧɨɫɬɢ ɛɭɞɟɬ ɪɚɜɧɨ ɫɭɦɦɟ ɢɫɯɨɞɧɨɝɨ ɢɯ ɤɨɥɢɱɟɫɬɜɚ Z 0 ɢ ɤɨɥɢɱɟɫɬɜɚ ɜɧɨɜɶ ɜɨɡɧɢɤɲɢɯ ɰɟɧɬɪɨɜ W cct
ɡɚ ɜɵɱɟɬɨɦ ɰɟɧɬɪɨɜ, ɢɡɪɚɫɯɨɞɨɜɚɧɧɵɯ ɧɚ ɨɛɪɚɡɨɜɚɧɢɟ ɭɠɟ ɜɨɡɧɢɤɲɢɯ ɤ
ɷɬɨɦɭ ɦɨɦɟɧɬɭ ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ
Z
Z0 N
W cct .
S
Ɍɨɝɞɚ ɜ ɪɚɦɤɚɯ ɷɬɨɣ ɦɨɞɟɥɢ ɫɤɨɪɨɫɬɶ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɛɭɞɭɬ ɩɨɞɱɢɧɹɬɶɫɹ ɭɪɚɜɧɟɧɢɸ
dN
N
§
·
W cS ¨ Z 0 W cct ¸ .
dt
S
©
¹
ɇɚɱɚɥɶɧɨɟ ɭɫɥɨɜɢɟ ɨɫɬɚɧɟɬɫɹ ɬɟɦ ɠɟ.
292
(11.45)
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɫ ɭɱɟɬɨɦ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɢɦɟɟɬ ɜɢɞ
(ɪɟɲɟɧɢɟ ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɨ, ɧɚɩɪɢɦɟɪ, ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɨɩɟɪɚɰɢɨɧɧɨɝɨ ɦɟɬɨɞɚ)
W cc
W cc ·
§
N Z0S S W ccSt S ¨ Z 0 ¸ exp W ct .
Wc
Wc¹
©
Ⱦɢɮɮɟɪɟɧɰɢɪɭɹ ɷɬɨ ɭɪɚɜɧɟɧɢɟ ɩɨ ɜɪɟɦɟɧɢ, ɧɚɣɞɟɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ
ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ
dN
W ccS S W c Z 0 W cc exp W ct .
dt
(11.46)
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɩɨ ɫɜɨɟɦɭ ɜɧɟɲɧɟɦɭ ɜɢɞɭ ɨɬɥɢɱɚɟɬɫɹ ɨɬ (11.42)
ɬɨɥɶɤɨ ɧɚɥɢɱɢɟɦ ɫɜɨɛɨɞɧɨɝɨ ɱɥɟɧɚ, ɤɨɬɨɪɵɣ ɛɭɞɟɬ ɫɥɚɛɨ ɜɥɢɹɬɶ ɧɚ ɡɚɤɨɧɨɦɟɪɧɨɫɬɢ ɩɪɢ ɦɚɥɵɯ ɜɪɟɦɟɧɚɯ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ ɷɤɫɩɟɪɢɦɟɧɬɚɯ ɡɚɜɢɫɢɦɨɫɬɶ (11.46) ɦɨɠɟɬ ɜɨɫɩɪɢɧɢɦɚɬɶɫɹ ɤɚɤ (11.42), ɝɞɟ ɜɨɫɩɪɨɢɡɜɨɞɫɬɜɨ ɰɟɧɬɪɨɜ ɧɟ ɭɱɢɬɵɜɚɟɬɫɹ. Ɏɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜ ɷɬɢɯ
ɦɨɞɟɥɹɯ – ɪɚɡɥɢɱɟɧ. ɇɚɩɪɢɦɟɪ, ɜɟɥɢɱɢɧɚ W cc , ɜ ɨɬɥɢɱɢɟ ɨɬ W c , ɦɨɠɟɬ
ɡɚɜɢɫɟɬɶ ɨɬ ɞɨɡɵ ɨɛɥɭɱɟɧɢɹ, ɧɨ ɦɨɠɟɬ ɧɟ ɡɚɜɢɫɟɬɶ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɢ ɤɨɦɩɨɧɟɧɬɨɜ ɪɟɚɤɰɢɨɧɧɨɣ ɫɦɟɫɢ.
ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɰɟɧɬɪɵ ɪɚɫɯɨɞɭɸɬɫɹ ɧɚ ɨɛɪɚɡɨɜɚɧɢɟ ɹɞɟɪ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɡɚɤɨɧɨɦ
dN
W cSZ ,
dt
(11.47)
ɚ ɜɨɫɩɪɨɢɡɜɨɞɹɬɫɹ ɩɨ ɡɚɤɨɧɭ (11.44). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɱɢɫɥɨ ɰɟɧɬɪɨɜ ɹɞɪɨɨɛɪɚɡɨɜɚɧɢɹ ɜ ɩɪɨɢɡɜɨɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɛɭɞɟɬ ɩɨɞɱɢɧɹɬɶɫɹ
ɭɪɚɜɧɟɧɢɸ
dZ
W cc Ze Z W cZ
dt
ɫ ɧɚɱɚɥɶɧɵɦ ɭɫɥɨɜɢɟɦ t 0 : Z Ze .
ɂɧɬɟɝɪɢɪɭɹ ɩɨɫɥɟɞɧɟɟ ɭɪɚɜɧɟɧɢɟ ɢ ɩɨɞɫɬɚɜɥɹɹ ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ Z ɜ (11.47), ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɜ
ɪɚɦɤɚɯ ɦɨɞɟɥɢ 2) ɜ ɜɢɞɟ
dN
Wc
Z S ^W cexp ª¬ W c W cc t º¼ W cc` .
dt W c W cc e
(11.48)
ɇɟɫɥɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (11.48) ɜɟɞɟɬ ɫɟɛɹ ɚɧɚɥɨɝɢɱɧɨ
ɭɪɚɜɧɟɧɢɸ (11.46): ɩɪɢ ɦɚɥɵɯ t ɨɛɪɚɳɚɟɬɫɹ ɜ ɭɪɚɜɧɟɧɢɟ (11.42), ɚ ɩɪɢ
ɛɨɥɶɲɢɯ t ɞɚɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɛɨɥɶɲɢɟ ɡɧɚɱɟɧɢɹ ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ
ɹɞɟɪ, ɱɟɦ ɭɪɚɜɧɟɧɢɟ (11.42).
293
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɭɫɥɨɠɧɟɧɢɹ ɦɨɞɟɥɢ ɩɪɢɜɨɞɹɬ ɤ ɨɬɤɥɨɧɟɧɢɸ ɧɚɛɥɸɞɚɟɦɨɣ ɫɤɨɪɨɫɬɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ ɨɬ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɝɨ ɡɚɤɨɧɚ ɩɪɢ ɛɨɥɶɲɢɯ ɜɪɟɦɟɧɚɯ, ɯɨɬɹ ɧɚ ɪɚɧɧɢɯ ɫɬɚɞɢɹɯ ɪɟɚɤɰɢɢ ɩɪɨɫɬɟɣɲɟɟ ɭɪɚɜɧɟɧɢɟ ɨɫɬɚɟɬɫɹ ɯɨɪɨɲɢɦ ɩɪɢɛɥɢɠɟɧɢɟɦ. ɋɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɧɚɛɥɸɞɚɟɦɭɸ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɦɨɠɟɬ ɨɤɚɡɚɬɶ ɩɨɝɥɨɳɟɧɢɟ ɪɚɫɬɭɳɢɦɢ ɹɞɪɚɦɢ ɤɚɤ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɰɟɧɬɪɨɜ
ɡɚɪɨɞɵɲɟɨɛɪɚɡɨɜɚɧɢɹ, ɬɚɤ ɢ ɫɥɢɹɧɢɟ ɡɚɪɨɞɵɲɟɣ. ɍɱɟɬ ɩɟɪɟɤɪɵɜɚɧɢɹ
ɹɞɟɪ ɩɪɢɜɨɞɢɬ ɤ ɪɟɡɤɨɦɭ ɭɫɥɨɠɧɟɧɢɸ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɨɪɨɧɵ ɡɚɞɚɱɢ,
ɯɨɬɹ ɷɬɨɬ ɦɟɯɚɧɢɡɦ ɩɨɧɹɬɟɧ ɫ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ.
ȼɬɨɪɵɦ ɨɫɧɨɜɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɤɢɧɟɬɢɤɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɮɚɡɵ
ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ ɹɜɥɹɟɬɫɹ «ɫɬɟɩɟɧɧɨɣ ɡɚɤɨɧ», ɢɦɟɸɳɢɣ ɜɢɞ
dN
dt
at b .
(11.49)
Ɋɚɡɥɢɱɧɵɟ ɨɛɨɫɧɨɜɚɧɢɹ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɜɤɥɸɱɚɸɬ ɞɨɩɭɳɟɧɢɹ ɨɛ
ɨɛɪɚɡɨɜɚɧɢɢ ɹɞɪɚ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ ɜ ɪɟɡɭɥɶɬɚɬɟ ɧɟɫɤɨɥɶɤɢɯ ɩɚɪɚɥɥɟɥɶɧɵɯ ɢɥɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɚɤɬɨɜ ɪɟɚɤɰɢɢ. ɋɨɨɛɪɚɠɟɧɢɹ, ɚɧɚɥɨɝɢɱɧɵɟ ɬɟɦ, ɤɨɬɨɪɵɟ ɜɵɫɤɚɡɚɧɵ ɩɪɢ ɨɛɫɭɠɞɟɧɢɢ ɷɤɫɩɨɧɟɧɰɢɚɥɶɧɨɝɨ ɡɚɤɨɧɚ, ɱɚɳɟ ɜɫɟɝɨ ɩɪɢɜɨɞɹɬ ɤ ɬɨɦɭ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ (11.49) ɱɚɳɟ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɤɚɤ ɭɞɚɱɧɚɹ ɚɩɩɪɨɤɫɢɦɚɰɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɧɚɛɥɸɞɚɟɦɵɯ
ɡɚɤɨɧɨɦɟɪɧɨɫɬɟɣ.
ȼ ɰɟɥɨɦ, ɧɚɛɥɸɞɚɟɦɚɹ ɤɢɧɟɬɢɤɚ ɬɜɟɪɞɨɮɚɡɧɵɯ ɪɟɚɤɰɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɨɡɧɢɤɧɨɜɟɧɢɟɦ ɢ ɪɨɫɬɨɦ ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ. Ⱦɥɹ ɬɨɝɨ
ɱɬɨɛɵ ɜɵɜɟɫɬɢ ɫɭɦɦɚɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɤɢɧɟɬɢɤɢ ɞɥɹ ɬɨɩɨɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ, ɭɱɬɟɦ, ɱɬɨ ɡɚ ɜɪɟɦɹ ɨɬ W ɞɨ W 'W ɨɛɪɚɡɭɟɬɫɹ
'N
dN
'W
dW
ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ.
ȿɫɥɢ W c – ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɞɥɹ ɨɬɞɟɥɶɧɨɝɨ ɹɞɪɚ, ɬɨ ɤ ɦɨɦɟɧɬɭ
ɜɪɟɦɟɧɢ t ɜɪɟɦɹ ɠɢɡɧɢ ɷɬɨɝɨ ɹɞɪɚ ɫɨɫɬɚɜɢɬ t W . Ɍɨɝɞɚ ɞɥɹ ɜɫɟɣ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɮɪɚɤɰɢɢ ɹɞɟɪ ɤ ɦɨɦɟɧɬɭ t ɧɚɛɥɸɞɚɟɦɚɹ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɛɭɞɟɬ
' W W ct ,W
dN W
'W .
dW
(11.50)
Ɍɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ
ɮɚɡ, ɚ ɩɨɫɥɟɞɧɹɹ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɨ ɦɟɪɟ ɪɨɫɬɚ ɹɞɟɪ, ɧɚɛɥɸɞɚɟɦɚɹ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɞɥɹ ɤɚɠɞɨɝɨ ɹɞɪɚ ɨɩɪɟɞɟɥɢɬɫɹ ɟɝɨ ɪɚɡɦɟɪɚɦɢ ɢɥɢ, ɜ ɤɨɧɟɱɧɨɦ ɫɱɟɬɟ, ɟɝɨ ɜɪɟɦɟɧɟɦ ɠɢɡɧɢ, ɱɬɨ ɢ ɮɢɤɫɢɪɭɟɬ ɭɪɚɜɧɟɧɢɟ (11.50).
294
ɉɪɨɫɭɦɦɢɪɨɜɚɜ ɭɪɚɜɧɟɧɢɹ (11.50) ɞɥɹ ɜɫɟɯ ɹɞɟɪ ɢ ɩɟɪɟɯɨɞɹ ɤ ɩɪɟɞɟɥɭ,
ɧɚɣɞɟɦ
t
W
³W ct ,W
0
dN W
dW .
dW
(11.51)
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɢ ɹɜɥɹɟɬɫɹ ɨɫɧɨɜɧɵɦ ɭɪɚɜɧɟɧɢɟɦ ɤɢɧɟɬɢɤɢ ɬɨɩɨɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɜɵɪɚɡɢɬɶ ɧɚɛɥɸɞɚɟɦɭɸ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ ɤɚɤ
ɮɭɧɤɰɢɸ ɜɪɟɦɟɧɢ, ɧɭɠɧɨ ɡɧɚɬɶ ɢɥɢ ɩɨɫɬɭɥɢɪɨɜɚɬɶ ɜɢɞ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɜɪɟɦɟɧɢ W c ɢ dN dt . ɋɤɨɪɨɫɬɶ ɪɨɫɬɚ ɨɬɞɟɥɶɧɨɝɨ ɹɞɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ
ɨɫɨɛɟɧɧɨɫɬɹɦɢ ɫɢɫɬɟɦɵ, ɚ ɫɜɹɡɶ ɫɤɨɪɨɫɬɢ ɪɨɫɬɚ ɫ ɧɚɛɥɸɞɚɟɦɨɣ ɫɤɨɪɨɫɬɶɸ ɪɟɚɤɰɢɢ ɡɚɜɢɫɢɬ ɟɳɟ ɢ ɨɬ ɮɨɪɦɵ ɹɞɪɚ (ɬɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɪɟɚɤɰɢɢ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɩɥɨɳɚɞɢ ɩɨɜɟɪɯɧɨɫɬɢ). ɂɦɟɸɳɢɟɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɞɚɧɧɵɟ ɫɜɢɞɟɬɟɥɶɫɬɜɭɸɬ, ɱɬɨ, ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɤɪɚɬɤɨɝɨ ɧɚɱɚɥɶɧɨɝɨ
ɩɟɪɢɨɞɚ, ɫɤɨɪɨɫɬɶ ɥɢɧɟɣɧɨɝɨ ɪɨɫɬɚ ɹɞɟɪ ɩɨɫɬɨɹɧɧɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜ
ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɮɨɪɦɵ ɹɞɟɪ ɥɟɝɤɨ ɜɵɪɚɡɢɬɶ W c ɤɚɤ ɬɭ ɢɥɢ ɢɧɭɸ ɮɭɧɤɰɢɸ ɜɪɟɦɟɧɢ. Ɋɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ ɦɨɞɟɥɟɣ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɥɢɬɟɪɚɬɭɪɟ,
ɜ ɬɨɦ ɱɢɫɥɟ, ɩɪɟɞɫɬɚɜɥɟɧɧɨɣ ɜ ɤɨɧɰɟ ɩɨɫɨɛɢɹ.
ɍɪɚɜɧɟɧɢɟ, ɚɧɚɥɨɝɢɱɧɨɟ (11.51), ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɢ ɞɥɹ ɨɛɴɟɦɚ ɩɪɨɪɟɚɝɢɪɨɜɚɜɲɟɝɨ ɜɟɳɟɫɬɜɚ
V t t
³V ct ,W
0
dN W
dW .
dW
11.7. Ⱦɢɮɮɭɡɢɨɧɧɚɹ ɤɢɧɟɬɢɤɚ
11.7.1. Ɉɛɳɢɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɨ ɞɢɮɮɭɡɢɢ
ȼ ɝɟɬɟɪɨɝɟɧɧɨɣ ɫɢɫɬɟɦɟ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɥɨɤɚɥɢɡɨɜɚɧɚ ɜ ɨɞɧɨɣ ɢɥɢ ɧɟɫɤɨɥɶɤɢɯ ɮɚɡɚɯ ɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɹɯ ɪɚɞɟɥɚ ɮɚɡ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɬɢɦ ɜɨɡɧɢɤɚɸɬ ɪɚɡɧɵɟ ɡɚɞɚɱɢ ɞɢɮɮɭɡɢɨɧɧɨɣ ɤɢɧɟɬɢɤɢ,
ɨɫɨɛɟɧɧɨɫɬɢ ɤɨɬɨɪɵɯ ɨɩɪɟɞɟɥɹɸɬɫɹ ɦɟɫɬɨɦ ɥɨɤɚɥɢɡɚɰɢɢ ɪɟɚɤɰɢɢ ɢ ɦɟɯɚɧɢɡɦɚɦɢ ɞɢɮɮɭɡɢɢ ɜ ɪɚɡɥɢɱɧɵɯ ɮɚɡɚɯ.
ɇɚɢɛɨɥɟɟ ɛɵɫɬɪɨ ɞɢɮɮɭɡɢɹ ɩɪɨɬɟɤɚɟɬ ɜ ɝɚɡɚɯ. ȿɫɥɢ ɛɵ ɞɢɮɮɭɡɢɹ ɜ
ɝɚɡɚɯ ɨɩɪɟɞɟɥɹɥɚɫɶ ɬɨɥɶɤɨ ɬɟɩɥɨɜɵɦ ɞɜɢɠɟɧɢɟɦ ɦɨɥɟɤɭɥ, ɬɨ ɨɧɚ ɩɪɨɬɟɤɚɥɚ ɛɵ ɩɪɚɤɬɢɱɟɫɤɢ ɦɝɧɨɜɟɧɧɨ. ȼɫɥɟɞɫɬɜɢɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɞɪɭɝ ɫ ɞɪɭɝɨɦ ɦɨɥɟɤɭɥɵ ɫɨɜɟɪɲɚɸɬ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɢɣ ɩɭɬɶ, ɱɟɦ ɪɚɫɫɬɨɹɧɢɟ
ɦɟɠɞɭ ɞɜɭɦɹ ɬɨɱɤɚɦɢ.
295
ȼ ɠɢɞɤɨɫɬɹɯ ɞɢɮɮɭɡɢɹ ɢɞɟɬ ɫɭɳɟɫɬɜɟɧɧɨ ɦɟɞɥɟɧɧɟɟ. ɋɭɳɧɨɫɬɶ
ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɦɟɯɚɧɢɡɦɚ ɩɟɪɟɧɨɫɚ ɜ ɠɢɞɤɨɫɬɹɯ ɫɜɨɞɢɬɫɹ ɤ ɬɨɦɭ, ɱɬɨ
ɦɨɥɟɤɭɥɚ ɜɵɪɵɜɚɟɬɫɹ ɢɡ ɨɤɪɭɠɟɧɢɹ ɨɞɧɢɦɢ ɱɚɫɬɢɰɚɦɢ ɢ ɫɤɚɱɤɨɦ ɩɟɪɟɯɨɞɢɬ ɜ ɨɤɪɭɠɟɧɢɟ ɞɪɭɝɢɯ.
Ɉɫɨɛɟɧɧɨ ɦɟɞɥɟɧɧɨ ɞɢɮɮɭɡɢɹ ɩɪɨɬɟɤɚɟɬ ɜ ɬɜɟɪɞɵɯ ɬɟɥɚɯ.
Ʉɚɤ ɩɪɚɜɢɥɨ, ɜɵɞɟɥɹɸɬ ɞɜɚ ɨɫɧɨɜɧɵɯ ɦɟɯɚɧɢɡɦɚ ɞɢɮɮɭɡɢɢ: ɜɚɤɚɧɫɢɨɧɧɵɣ, ɤɨɞɚ ɚɬɨɦɵ ɩɟɪɟɦɟɳɚɸɬɫɹ ɬɨɥɶɤɨ ɩɨ ɜɚɤɚɧɬɧɵɦ ɭɡɥɚɦ ɤɪɢɫɬɚɥɥɢɱɟɫɤɨɣ ɪɟɲɟɬɤɢ, ɢ ɦɟɯɚɧɢɡɦ ɜɧɟɞɪɟɧɢɹ, ɤɨɝɞɚ ɚɬɨɦɵ ɩɟɪɟɦɟɳɚɸɬɫɹ ɩɨ ɦɟɠɞɨɭɡɥɢɹɦ. ɉɟɪɜɵɣ ɦɟɯɚɧɢɡɦ ɫɜɨɣɫɬɜɟɧɟɧ ɞɢɮɮɭɡɢɢ ɦɟɬɚɥɥɨɜ ɞɪɭɝ ɜ ɞɪɭɝɟ, ɪɚɡɦɟɪɵ ɚɬɨɦɨɜ ɤɨɬɨɪɵɯ ɫɪɚɜɧɢɦɵ. ȼɬɨɪɨɣ ɦɟɯɚɧɢɡɦ
ɛɨɥɟɟ ɫɜɨɣɫɬɜɟɧɟɧ ɞɢɮɮɭɡɢɢ ɝɚɡɨɜ ɜ ɦɟɬɚɥɥɚɯ.
ȼ ɰɟɥɨɦ, ɮɢɡɢɤɚ ɩɪɨɰɟɫɫɨɜ ɞɢɮɮɭɡɢɢ ɹɜɥɹɟɬɫɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɵɦ
ɪɚɡɞɟɥɨɦ ɮɢɡɢɤɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ. Ɍɟɨɪɢɹ ɞɢɮɮɭɡɢɢ ɜ ɬɜɟɪɞɵɯ ɫɪɟɞɚɯ ɨɫɧɨɜɚɧɚ ɧɚ ɮɭɧɞɚɦɟɧɬɚɥɶɧɵɯ ɩɪɟɞɫɬɚɜɥɟɧɢɹɯ ɮɢɡɢɱɟɫɤɨɣ ɤɢɧɟɬɢɤɢ ɢ
ɧɟɪɚɜɧɨɜɟɫɧɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɢ ɢ ɬɟɫɧɨ ɫɜɹɡɚɧɚ ɫ ɬɟɨɪɢɹɦɢ ɞɟɮɟɤɬɨɜ
ɫɬɪɭɤɬɭɪɵ.
ȼ ɫɥɭɱɚɟ ɩɨɥɢɤɪɢɫɬɚɥɥɢɱɟɫɤɢɯ ɢ ɦɧɨɝɨɮɚɡɧɵɯ ɦɚɬɟɪɢɚɥɨɜ, ɤɪɨɦɟ
ɨɛɴɟɦɧɨɣ ɞɢɮɮɭɡɢɢ, ɜɵɞɟɥɹɸɬ ɞɢɮɮɭɡɢɸ ɩɨ ɝɪɚɧɢɰɚɦ ɡɟɪɟɧ ɢ ɮɚɡ.
ȼɫɥɟɞɫɬɜɢɟ ɨɫɨɛɨɝɨ ɯɚɪɚɤɬɟɪɚ ɝɪɚɧɢɰ, ɢɯ ɷɧɟɪɝɟɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
ɢ ɨɫɨɛɨɣ ɫɬɪɭɤɬɭɪɵ ɜɟɳɟɫɬɜɚ ɜ ɢɯ ɨɤɪɟɫɬɧɨɫɬɢ, ɞɢɮɮɭɡɢɹ ɜɞɨɥɶ ɝɪɚɧɢɰ
ɪɚɡɞɟɥɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɚɦɨɫɬɨɹɬɟɥɶɧɵɣ ɢɧɬɟɪɟɫ.
ɇɚɢɛɨɥɟɟ ɨɛɳɢɦ ɡɚɤɨɧɨɦ ɞɢɮɮɭɡɢɢ ɬɟɥɚɯ ɪɚɡɥɢɱɧɨɣ ɩɪɢɪɨɞɵ ɹɜɥɹɟɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɣ ɡɚɤɨɧ Ɏɢɤɚ
J UD grad C ,
(11.52)
ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɤɨɬɨɪɵɦ ɩɨɬɨɤ ɦɚɫɫɵ ɤɨɦɩɨɧɟɧɬɚ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ
ɝɪɚɞɢɟɧɬɭ ɟɝɨ ɤɨɧɰɟɧɬɪɚɰɢɢ. Ɇɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɢ ɞɢɮɮɭɡɢɨɧɧɵɣ
ɩɨɬɨɤ ɤɨɦɩɨɧɟɧɬɚ
j D grad C .
ɉɟɪɜɚɹ ɜɟɥɢɱɢɧɚ ɢɡɦɟɪɹɟɬɫɹ ɜ ɤɝ/(ɦ2ɫ) ɢɥɢ ɜ ɝ/(ɫɦ2ɫ).
ȼɬɨɪɚɹ – ɜ ɫɦ/ɫ.
ȼ ɨɛɨɢɯ ɪɚɜɟɧɫɬɜɚɯ C – ɦɚɫɫɨɜɚɹ ɤɨɧɰɟɧɬɪɚɰɢɹ ɤɨɦɩɨɧɟɧɬɚ (ɫɦ.
ɪɚɡɞɟɥ 11.1), ɚ D – ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ, ɦ2/ɫ ɢɥɢ ɫɦ2/ɫ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɡɚɜɢɫɢɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ
ɡɚɤɨɧɨɦ Ⱥɪɪɟɧɢɭɫɚ
§ E ·
D D0exp ¨ D ¸ ,
© RT ¹
ɩɨɞɨɛɧɵɦ ɡɚɤɨɧɭ Ⱥɪɪɟɧɢɭɫɚ ɞɥɹ ɫɤɨɪɨɫɬɢ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ
( ED – ɷɧɟɪɝɢɹ ɚɤɬɢɜɚɰɢɢ ɞɢɮɮɭɡɢɢ, D0 – ɩɪɟɞɷɤɫɩɨɧɟɧɬ).
Ⱥɧɚɥɨɝɢɱɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɦɵ ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ, ɜɵɪɚɠɚɹ ɤɨɥɢɱɟɫɬɜɨ ɜɟɳɟɫɬɜɚ ɜ ɫɢɫɬɟɦɟ ɫ ɩɨɦɨɳɶɸ ɢɧɵɯ ɟɞɢɧɢɰ ɢɡɦɟɪɟɧɢɹ.
296
Ɂɚɤɨɧ Ɏɢɤɚ (11.52) ɜɧɟɲɧɟ ɨɱɟɧɶ ɧɚɩɨɦɢɧɚɟɬ ɡɚɤɨɧ Ɏɭɪɶɟ: ɩɪɨɰɟɫɫɵ ɞɢɮɮɭɡɢɢ ɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɨɩɢɫɵɜɚɸɬɫɹ ɩɨɞɨɛɧɵɦɢ ɡɚɤɨɧɚɦɢ.
ɉɨɞɨɛɧɵɦɢ ɠɟ ɨɤɚɡɵɜɚɸɬɫɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ, ɬɪɟɛɭɸɳɢɟɫɹ ɞɥɹ ɨɩɢɫɚɧɢɹ ɷɬɢɯ ɩɪɨɰɟɫɫɨɜ.
Ɍɚɤ, ɭɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ ɜ ɧɟɩɨɞɜɢɠɧɨɣ ɫɪɟɞɟ ɢɦɟɟɬ ɜɢɞ
wC
U
div DU grad T Vc ,
wt
ɝɞɟ Vc - ɫɭɦɦɚ ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ ɤɨɦɩɨɧɟɧɬɚ ɜ ɨɛɴɟɦɟ, ɧɚɩɪɢɦɟɪ,
ɜɫɥɟɞɫɬɜɢɟ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ.
ȿɫɥɢ ɩɥɨɬɧɨɫɬɶ ɜɟɳɟɫɬɜɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɡɚɩɢɲɟɦ
wC
Vc
div D grad T V , V
,
wt
U
ɢɥɢ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ
wC w § wC · w § wC · w § wC ·
V.
D
D
¨D
¸
w t w x © w x ¹ w y ¨© w y ¸¹ w y ¨© w y ¸¹
ɗɬɨ ɭɪɚɜɧɟɧɢɟ ɜɧɟɲɧɟ ɧɚɦ ɬɨɠɟ ɭɠɟ ɢɡɜɟɫɬɧɨ.
Ɉɞɧɨɦɟɪɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɢɮɮɭɡɢɢ ɩɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ (ɧɚɩɪɢɦɟɪ, ɜ ɢɡɨɬɟɪɦɢɱɟɫɤɢɯ ɭɫɥɨɜɢɹɯ) ɢɦɟɟɬ ɜɢɞ
wC
w 2C
(11.53)
D 2 V.
wt
wx
Ⱥɧɚɥɨɝɨɦ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ ɜ ɬɟɨɪɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ (ɪɚɡɞɟɥ 2.7.)
O
.
a
cU
Ⱦɥɹ ɝɚɡɨɜ ɜ ɨɛɵɱɧɵɯ ɭɫɥɨɜɢɹɯ ɬɢɩɢɱɧɨ
D
Le
~ 1.
a
Ⱥ ɜɨɬ ɞɥɹ ɬɜɟɪɞɵɯ ɬɟɥ ɨɬɧɨɲɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɞɢɮɮɭɡɢɢ ɢ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ Le 1 . ɗɬɚ ɜɟɥɢɱɢɧɚ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɱɢɫɥɚ
Ʌɶɸɢɫɚ.
ɉɪɢ ɨɩɢɫɚɧɢɢ ɞɢɮɮɭɡɢɨɧɧɵɯ ɩɪɨɰɟɫɫɨɜ, ɤɪɨɦɟ ɱɢɫɥɚ Ʌɶɸɢɫɚ,
ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɸɬ ɬɚɤɢɟ ɱɢɫɥɚ ɩɨɞɨɛɢɹ ɤɚɤ ɱɢɫɥɨ ɒɟɪɜɭɞɚ ɢɥɢ
ɞɢɮɮɭɡɢɨɧɧɨɟ ɱɢɫɥɨ ɇɭɫɫɟɥɶɬɚ
Ed
Sh { Nu D
,
D
ɝɞɟ d - ɯɚɪɚɤɬɟɪɧɵɣ ɥɢɧɟɣɧɵɣ ɪɚɡɦɟɪ, E - ɤɨɷɮɮɢɰɢɟɧɬ ɦɚɫɫɨɨɬɞɚɱɢ ɜ
ɫɨɨɬɧɨɲɟɧɢɢ
j E'C ,
297
ɚɧɚɥɨɝɢɱɧɨɦ ɡɚɤɨɧɭ ɬɟɩɥɨɨɛɦɟɧɚ ɇɶɸɬɨɧɚ–Ɋɢɯɦɚɧɚ (ɫɦ. ɑɚɫɬɶ 3)
' C – ɪɚɡɧɨɫɬɶ ɤɨɧɰɟɧɬɪɚɰɢɣ; ɤɪɢɬɟɪɢɣ ɒɦɢɞɬɚ (ɚɧɚɥɨɝ ɱɢɫɥɚ
ɉɪɚɧɞɬɥɹ ɞɥɹ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨɨɛɦɟɧɚ)
Q
Sc
,
D
ɝɞɟ Q – ɤɨɷɮɮɢɰɢɟɧɬ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ. ȼɢɞ ɡɚɜɢɫɢɦɨɫɬɢ
St St Re ,Sc
ɭɫɬɚɧɚɜɥɢɜɚɟɬɫɹ ɧɚ ɨɫɧɨɜɟ ɚɧɚɥɢɡɚ ɞɚɧɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢɥɢ ɪɟɲɟɧɢɹ
ɱɚɫɬɧɵɯ ɡɚɞɚɱ ɦɚɫɫɨɨɛɦɟɧɚ.
Ʉɪɨɦɟ ɡɚɤɨɧɚ (11.52), ɜ ɬɟɨɪɢɢ ɦɧɨɝɨɤɨɦɩɨɧɟɧɬɧɨɣ ɞɢɮɮɭɡɢɢ
ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɢɦɟɟɬ ɬɚɤ ɧɚɡɵɜɚɟɦɵɣ ɨɛɨɛɳɟɧɧɵɣ ɡɚɤɨɧ Ɏɢɤɚ, ɤɨɬɨɪɵɣ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
n 1
ji
¦ D grad T , i
ij
j
1,2, ...,n ,
(11.54)
j 1
ɝɞɟ n – ɱɢɫɥɨ ɤɨɦɩɨɧɟɧɬɨɜ ɦɧɨɝɨɤɨɦɩɨɧɟɧɬɧɨɣ ɫɢɫɬɟɦɵ (ɧɟɡɚɜɢɫɢɦɵ
ɢɡ ɤɨɬɨɪɵɯ ɬɨɥɶɤɨ n 1 , ɫɦ. ɪɚɡɞɟɥ 11.1), Di j – ɩɚɪɰɢɚɥɶɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɞɢɮɮɭɡɢɢ. ȼ ɬɟɨɪɢɢ ɞɢɮɮɭɡɢɢ ɩɨɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɢɡ n ɞɢɮɮɭɡɢɨɧɧɵɯ ɩɨɬɨɤɨɜ ɧɟɡɚɜɢɫɢɦɵ ɬɚɤɠɟ ɬɨɥɶɤɨ n 1 ɩɨɬɨɤ.
ɍɪɚɜɧɟɧɢɟ (11.54) ɝɨɜɨɪɢɬ, ɱɬɨ ɩɨɬɨɤ ɤɚɠɞɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɝɪɚɞɢɟɧɬɚɦ ɤɨɧɰɟɧɬɪɚɰɢɣ ɜɫɟɯ ɧɟɡɚɜɢɫɢɦɵɯ ɤɨɦɩɨɧɟɧɬɨɜ.
ɂɡɜɟɫɬɧɵ ɢ ɛɨɥɟɟ ɫɥɨɠɧɵɟ ɦɟɯɚɧɢɡɦɵ ɩɟɪɟɧɨɫɚ ɦɚɫɫɵ (ɬɟɪɦɨɞɢɮɮɭɡɢɹ, ɛɚɪɨɞɢɮɮɭɡɢɹ, ɞɢɮɮɭɡɢɹ ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɝɪɚɞɢɟɧɬɚ ɧɚɩɪɹɠɟɧɢɣ,
ɤɨɧɜɟɤɬɢɜɧɵɣ ɦɚɫɫɨɩɟɪɟɧɨɫ).
11.7. 2. Ɋ ɨɥ ɶ ɞ ɢɮɮ ɭɡɢ ɢ ɜ ɧɟɤ ɨɬɨɪ ɵɯ
ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫ ɤɢɯ ɩɪɨɰɟɫɫɚɯ
Ȼɨɥɶɲɢɧɫɬɜɨ ɩɪɨɰɟɫɫɨɜ ɜ ɤɨɧɞɟɧɫɢɪɨɜɚɧɧɵɯ ɫɢɫɬɟɦɚɯ ɹɜɥɹɸɬɫɹ
ɝɟɬɟɪɨɝɟɧɧɵɦɢ, ɬ.ɟ. ɩɪɨɢɫɯɨɞɹɬ ɜ ɪɚɡɥɢɱɧɵɯ ɫɨɩɪɢɤɚɫɚɸɳɢɯɫɹ ɮɚɡɚɯ,
ɨɬɞɟɥɟɧɧɵɯ ɞɪɭɝ ɨɬ ɞɪɭɝɚ ɩɨɜɟɪɯɧɨɫɬɶɸ ɪɚɡɞɟɥɚ. ɏɢɦɢɱɟɫɤɢɟ ɢ ɮɚɡɨɜɵɟ ɩɪɟɜɪɚɳɟɧɢɹ ɜ ɬɚɤɢɯ ɫɢɫɬɟɦɚɯ ɧɟɢɡɛɟɠɧɨ ɫɜɹɡɚɧɵ ɫ ɞɢɮɮɭɡɢɟɣ ɢɥɢ
ɢɧɵɦɢ ɦɟɯɚɧɢɡɦɚɦɢ ɦɚɫɫɨɩɟɪɟɧɨɫɚ. ɉɪɢɦɟɪɚɦɢ ɬɚɤɢɯ ɩɪɨɰɟɫɫɨɜ ɦɨɝɭɬ
ɛɵɬɶ ɜɨɫɫɬɚɧɨɜɥɟɧɢɟ ɬɜɟɪɞɵɯ ɨɤɢɫɥɨɜ ɭɝɥɟɪɨɞɨɦ, ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɟ
ɷɥɟɦɟɧɬɨɜ ɦɟɠɞɭ ɦɟɬɚɥɥɨɦ ɢ ɲɥɚɤɨɦ, ɪɚɫɬɜɨɪɟɧɢɟ ɢ ɜɵɞɟɥɟɧɢɟ ɝɚɡɨɜ ɢɡ
ɦɟɬɚɥɥɚ ɢ ɲɥɚɤɚ, ɪɚɫɬɜɨɪɟɧɢɟ ɬɜɟɪɞɵɯ ɬɟɥ ɜ ɠɢɞɤɨɫɬɹɯ, ɩɪɟɜɪɚɳɟɧɢɹ ɜ
ɦɟɬɚɥɥɚɯ ɢ ɫɩɥɚɜɚɯ, ɫɨɩɪɨɜɨɠɞɚɸɳɢɟɫɹ ɞɢɮɮɭɡɢɨɧɧɵɦ ɩɟɪɟɧɨɫɨɦ ɜɟɳɟɫɬɜɚ, ɩɪɨɰɟɫɫɵ ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ, ɪɟɤɪɢɫɬɚɥɥɢɡɚɰɢɢ ɢ
ɬ.ɞ. ȼɫɸɞɭ ɞɢɮɮɭɡɢɹ ɹɜɥɹɟɬɫɹ ɨɞɧɨɣ ɢɡ ɫɬɚɞɢɣ ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ.
298
Ɍɚɤ, ɦɟɬɚɥɥɢɱɟɫɤɢɣ ɥɨɦ ɜ ɧɚɫɬɨɹɳɟɟ ɜɪɟɦɹ ɫɨɫɬɚɜɥɹɟɬ ɛɨɥɶɲɭɸ
ɱɚɫɬɶ ɦɟɬɚɥɥɢɱɟɫɤɨɣ ɲɢɯɬɵ, ɢɫɩɨɥɶɡɭɟɦɨɣ ɜ ɫɬɚɥɟɩɥɚɜɢɥɶɧɨɦ ɩɪɨɢɡɜɨɞɫɬɜɟ. ɉɪɨɰɟɫɫ ɩɥɚɜɥɟɧɢɹ ɫɤɪɚɩɚ ɫɨɫɬɚɜɥɹɟɬ ɡɧɚɱɢɬɟɥɶɧɭɸ ɱɚɫɬɶ ɨɛɳɟɝɨ ɜɪɟɦɟɧɢ ɩɥɚɜɤɢ. Ɇɟɯɚɧɢɡɦ ɩɪɨɰɟɫɫɚ ɞɨɫɬɚɬɨɱɧɨ ɫɥɨɠɟɧ, ɬɚɤ ɤɚɤ
ɬɟɱɟɧɢɟ ɟɝɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶɸ ɬɟɩɥɨɜɵɯ ɢ ɞɢɮɮɭɡɢɨɧɧɵɯ
ɹɜɥɟɧɢɣ ɢ ɡɚɜɢɫɢɬ ɨɬ ɛɨɥɶɲɨɝɨ ɱɢɫɥɚ ɩɚɪɚɦɟɬɪɨɜ. Ɇɨɠɧɨ ɩɪɢɧɹɬɶ, ɱɬɨ
ɩɥɚɜɥɟɧɢɟ ɫɬɚɥɢ ɜ ɪɚɫɩɥɚɜɥɟɧɧɨɦ ɱɭɝɭɧɟ, ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɝɨ ɧɢɠɟ
ɬɟɦɩɟɪɚɬɭɪɵ ɩɥɚɜɥɟɧɢɹ ɫɬɚɥɢ, ɜɤɥɸɱɚɟɬ ɬɪɢ ɫɬɚɞɢɢ: ɞɢɮɮɭɡɢɨɧɧɵɣ ɩɟɪɟɧɨɫ ɭɝɥɟɪɨɞɚ ɱɟɪɟɡ ɩɨɝɪɚɧɢɱɧɵɣ ɫɥɨɣ ɢɡ ɪɚɫɩɥɚɜɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɮɚɡ ɱɭɝɭɧ – ɫɬɚɥɶ; ɧɚɫɵɳɟɧɢɟ ɭɝɥɟɪɨɞɨɦ ɩɨɜɟɪɯɧɨɫɬɧɨɝɨ ɫɥɨɹ ɫɬɚɥɢ ɢ ɬɟɩɥɨɨɛɦɟɧ ɦɟɠɞɭ ɪɚɫɩɥɚɜɨɦ ɢ ɬɜɟɪɞɨɣ ɮɚɡɨɣ. Ⱦɢɮɮɭɡɢɹ ɭɝɥɟɪɨɞɚ
ɹɜɥɹɟɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦ ɡɜɟɧɨɦ ɩɪɨɰɟɫɫɚ, ɬɚɤ ɤɚɤ ɭɜɟɥɢɱɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɭɝɥɟɪɨɞɚ ɜ ɫɬɚɥɢ ɩɨɧɢɠɚɟɬ ɬɟɦɩɟɪɚɬɭɪɭ ɟɟ ɩɥɚɜɥɟɧɢɹ, ɢ ɫɤɨɪɨɫɬɶ
ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɟɪɟɧɨɫɚ ɜ ɨɫɧɨɜɧɨɦ ɨɩɪɟɞɟɥɹɟɬ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ.
ɋɜɨɣɫɬɜɚ ɦɟɬɚɥɥɨɜ ɩɪɢ ɩɨɜɵɲɟɧɧɵɯ ɬɟɦɩɟɪɚɬɭɪɚɯ ( T t Ts , Ts –
ɬɟɦɩɟɪɚɬɭɪɚ ɩɥɚɜɥɟɧɢɹ) ɫɭɳɟɫɬɜɟɧɧɨ ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɬɚɛɢɥɶɧɨɫɬɶɸ ɢɯ
ɫɬɪɭɤɬɭɪɵ. ɉɨɫɥɟɞɧɹɹ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɡɚɜɢɫɢɬ ɨɬ ɫɤɨɪɨɫɬɢ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɟɪɟɦɟɳɟɧɢɹ ɚɬɨɦɨɜ: ɫ ɩɨɜɵɲɟɧɢɟɦ ɬɟɦɩɟɪɚɬɭɪɵ ɪɨɥɶ ɞɢɮɮɭɡɢɨɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɜɨɡɪɚɫɬɚɟɬ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫɨ ɜɫɟɦɢ ɞɪɭɝɢɦɢ ɦɟɯɚɧɢɡɦɚɦɢ ɫɬɪɭɤɬɭɪɨɨɛɪɚɡɨɜɚɧɢɹ ɢ ɩɥɚɫɬɢɱɟɫɤɨɣ ɞɟɮɨɪɦɚɰɢɢ.
ɉɪɨɰɟɫɫɵ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɪɨɫɬɚ ɱɚɫɬɢɰ ɨɩɪɟɞɟɥɹɸɬ ɤɢɧɟɬɢɤɭ ɪɚɫɩɚɞɚ ɩɟɪɟɫɵɳɟɧɧɵɯ ɬɜɟɪɞɵɯ ɪɚɫɬɜɨɪɨɜ ɧɚ ɩɟɪɜɨɣ ɫɬɚɞɢɢ, ɤɨɝɞɚ ɩɪɟɫɵɳɟɧɢɟ ɜɟɥɢɤɨ, ɢ, ɧɚɪɹɞɭ ɫ ɪɨɫɬɨɦ ɪɚɧɟɟ ɜɨɡɧɢɤɲɢɯ ɡɚɪɨɞɵɲɟɣ, ɩɪɨɞɨɥɠɚɟɬɫɹ ɮɥɭɤɬɭɚɰɢɨɧɧɨɟ ɨɛɪɚɡɨɜɚɧɢɟ ɡɚɪɨɞɵɲɟɣ ɱɚɫɬɢɰ ɧɨɜɨɣ ɮɚɡɵ
ɜ ɪɚɡɥɢɱɧɵɯ ɬɨɱɤɚɯ ɪɚɫɬɜɨɪɚ. ɇɚ ɫɬɚɞɢɢ, ɤɨɝɞɚ ɱɚɫɬɢɰɵ ɜɨɡɧɢɤɲɟɣ ɮɚɡɵ ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɢ, ɩɪɟɫɵɳɟɧɢɟ ɦɚɥɨ, ɧɨɜɵɟ ɱɚɫɬɢɰɵ ɭɠɟ ɧɟ ɨɛɪɚɡɭɸɬɫɹ, ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɪɚɫɩɚɞɚ ɪɚɫɬɜɨɪɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɚɝɭɥɹɰɢɟɣ
ɱɚɫɬɢɰ ɧɨɜɨɣ ɮɚɡɵ, ɬɚɤɠɟ ɥɢɦɢɬɢɪɭɟɦɨɣ ɞɢɮɮɭɡɢɟɣ.
Ⱦɢɮɮɭɡɢɹ ɨɩɪɟɞɟɥɹɟɬ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɩɨɥɡɭɱɟɫɬɢ ɧɚ ɟɟ ɭɫɬɚɧɨɜɢɜɲɟɣɫɹ ɫɬɚɞɢɢ, ɤɨɝɞɚ ɷɧɟɪɝɢɹ ɚɤɬɢɜɚɰɢɢ ɩɨɥɡɭɱɟɫɬɢ ɛɥɢɡɤɚ ɤ ɷɧɟɪɝɢɢ
ɚɤɬɢɜɚɰɢɢ ɞɢɮɮɭɡɢɢ.
ɉɪɨɰɟɫɫ ɫɩɟɤɚɧɢɹ (ɭɩɥɨɬɧɟɧɢɹ ɩɨɪɢɫɬɨɝɨ ɬɟɥɚ ɜ ɨɛɥɚɫɬɢ ɜɵɫɨɤɢɯ
ɬɟɦɩɟɪɚɬɭɪ) ɹɜɥɹɟɬɫɹ ɫɥɨɠɧɵɦ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɦ ɩɪɨɰɟɫɫɨɦ, ɜɤɥɸɱɚɸɳɢɦ ɜ ɫɟɛɹ ɪɚɡɥɢɱɧɵɟ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɟ ɫɬɚɞɢɢ, ɧɟɦɚɥɭɸ ɪɨɥɶ ɜ ɤɨɬɨɪɵɯ ɢɝɪɚɟɬ ɞɢɮɮɭɡɢɹ ɢ ɩɪɨɰɟɫɫɵ ɧɚ ɝɪɚɧɢɰɚɯ ɪɚɡɞɟɥɚ.
ɉɪɨɰɟɫɫɨɦ ɞɢɮɮɭɡɢɢ ɢ ɫɨɩɭɬɫɬɜɭɸɳɢɦɢ ɹɜɥɟɧɢɹɦɢ (ɚɞɫɨɪɛɰɢɟɣ,
ɨɛɴɟɦɧɵɦɢ ɪɟɚɤɰɢɹɦɢ ɢ ɞɪ.) ɜ ɡɧɚɱɢɬɟɥɶɧɨɣ ɫɬɟɩɟɧɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɪɟɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɷɥɟɦɟɧɬɨɜ, ɯɢɦɢɱɟɫɤɢɯ ɫɨɟɞɢɧɟɧɢɣ ɢ ɮɚɡ, ɨɛɟɫ299
ɩɟɱɢɜɚɸɳɢɯ ɬɨ ɢɥɢ ɢɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɫɥɨɠɧɵɯ
ɦɚɬɟɪɢɚɥɨɜ, ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɩɨɜɟɪɯɧɨɫɬɟɣ ɢ ɧɚɧɟɫɟɧɢɢ ɩɨɤɪɵɬɢɣ.
Ɋɨɥɶ ɞɢɮɮɭɡɢɢ ɢ ɟɟ ɦɟɯɚɧɢɡɦɵ ɜ ɭɫɥɨɜɢɹɯ ɨɛɪɚɛɨɬɤɢ ɢ ɩɨɥɭɱɟɧɢɹ
ɦɚɬɟɪɢɚɥɨɜ ɜɟɫɶɦɚ ɪɚɡɧɨɨɛɪɚɡɧɵ.
ɇɟɫɦɨɬɪɹ ɧɚ ɫɥɨɠɧɨɫɬɶ ɢ ɜɡɚɢɦɨɨɛɭɫɥɨɜɥɟɧɧɨɫɬɶ ɪɚɡɥɢɱɧɵɯ ɮɢɡɢɤɨ-ɯɢɦɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ, ɧɚɛɥɸɞɚɸɳɢɯɫɹ ɩɪɢ ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɨɣ
ɨɛɪɚɛɨɬɤɟ ɩɨɜɟɪɯɧɨɫɬɟɣ ɢ ɧɚɧɟɫɟɧɢɢ ɩɨɤɪɵɬɢɣ, ɤɚɱɟɫɬɜɟɧɧɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨ ɪɨɥɢ ɞɢɮɮɭɡɢɨɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɜ ɩɨɜɟɪɯɧɨɫɬɧɨɦ ɫɥɨɟ ɦɨɠɧɨ
ɩɨɥɭɱɢɬɶ ɧɚ ɨɫɧɨɜɟ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɵɯ ɦɨɞɟɥɟɣ, ɜ ɤɨɬɨɪɵɯ ɚɧɚɥɢɡɢɪɭɸɬɫɹ ɪɚɡɥɢɱɧɵɟ ɮɢɡɢɱɟɫɤɢɟ ɫɢɬɭɚɰɢɢ.
11.7.3. Ɇɚɤɪɨɤɢɧɟɬɢɱɟɫɤɢɟ ɨɛɥɚɫɬɢ
ɩɪɨɬɟɤɚ ɧɢɹ ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ
ɏɚɪɚɤɬɟɪɧɨɣ ɨɫɨɛɟɧɧɨɫɬɶɸ ɝɟɬɟɪɨɝɟɧɧɵɯ ɩɪɨɰɟɫɫɨɜ ɹɜɥɹɟɬɫɹ ɢɯ
ɦɧɨɝɨɫɬɚɞɢɣɧɨɫɬɶ. ɇɚɢɛɨɥɟɟ ɬɢɩɢɱɧɚ ɫɢɬɭɚɰɢɹ, ɤɨɝɞɚ ɝɟɬɟɪɨɝɟɧɧɵɣ
ɩɪɨɰɟɫɫ ɜ ɞɜɭɯ ɫɨɩɪɢɤɚɫɚɸɳɢɯɫɹ ɮɚɡɚɯ ɜɤɥɸɱɚɟɬ ɬɪɢ ɫɬɚɞɢɢ: ɞɢɮɮɭɡɢɨɧɧɵɣ ɩɨɞɜɨɞ ɜɟɳɟɫɬɜɚ ɤ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɮɚɡ, ɯɢɦɢɱɟɫɤɭɸ ɪɟɚɤɰɢɸ
(ɢɥɢ ɚɞɫɨɪɛɰɢɸ) ɧɚ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ ɢ ɨɬɜɨɞ ɩɪɨɞɭɤɬɨɜ ɩɪɟɜɪɚɳɟɧɢɹ
(ɬɨɠɟ, ɤɚɤ ɩɪɚɜɢɥɨ, ɞɢɮɮɭɡɢɨɧɧɵɣ).
ȿɫɥɢ ɩɪɨɰɟɫɫ ɩɪɨɬɟɤɚɟɬ ɜ ɧɟɫɤɨɥɶɤɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɫɬɚɞɢɣ, ɬɨ
ɫɭɦɦɚɪɧɚɹ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɟɝɨ ɦɟɞɥɟɧɧɨɣ ɫɬɚɞɢɟɣ, ɤɨɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɤɨɧɬɪɨɥɢɪɭɸɳɟɣ.
ɉɭɫɬɶ ɜ ɨɛɥɚɫɬɢ x 0 (ɜɧɟɲɧɟɣ ɨɛɥɚɫɬɢ) ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɞɢɮɮɭɡɢɨɧɧɵɣ ɩɨɞɜɨɞ ɜɟɳɟɫɬɜɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ x 0 , ɧɚ ɤɨɬɨɪɨɣ ɩɪɨɢɫɯɨɞɢɬ
ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ. Ɍɨɝɞɚ ɩɪɢ x 0 ɫɩɪɚɜɟɞɥɢɜɨ
ɭɪɚɜɧɟɧɢɟ
wC
w 2C
(11.55)
D1 2 ,
wt
wx
ɝɞɟ D1 – ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɜɨ ɜɧɟɲɧɟɣ ɨɛɥɚɫɬɢ.
ɉɪɢɦɟɦ, ɱɬɨ ɜ ɫɥɨɟ ɬɨɥɳɢɧɨɣ ' , ɩɪɢɥɟɝɚɸɳɟɦ ɤ ɩɨɜɟɪɯɧɨɫɬɢ
x 0 , ɭɫɬɚɧɨɜɢɥɨɫɶ ɫɬɚɰɢɨɧɚɪɧɨɟ ɫɨɫɬɨɹɧɢɟ, ɩɪɢɱɟɦ
C ' ,t C0 ,
C 0 ,t C c .
ɇɚ ɩɨɜɟɪɯɧɨɫɬɢ x 0 ɩɪɨɢɫɯɨɞɢɬ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɫɨ ɫɤɨɪɨɫɬɶɸ M kC c .
ȼ ɫɬɚɰɢɨɧɚɪɧɨɦ ɫɨɫɬɨɹɧɢɢ ɜɫɟ ɤɨɥɢɱɟɫɬɜɨ ɜɟɳɟɫɬɜɚ, ɩɪɢɯɨɞɹɳɟɟ ɜ
ɟɞɢɧɢɰɭ ɨɛɴɟɦɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ x 0 ɜ ɟɞɢɧɢɰɭ ɜɪɟɦɟɧɢ ɩɨɥɧɨɫɬɶɸ ɬɪɚɬɢɬɫɹ ɜ ɪɟɚɤɰɢɢ, ɱɬɨ ɨɬɪɚɠɚɟɬ ɪɚɜɟɧɫɬɜɨ
300
kC c E C0 C c ,
(11.56)
ɝɞɟ E C 0 ,t C c ɤɨɷɮɮɢɰɢɟɧɬ ɦɚɫɫɨɨɬɞɚɱɢ, ɜɟɥɢɱɢɧɚ ɤɨɬɨɪɨɝɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɢɮɮɭɡɢɢ D 1 . ɂɡ ɭɪɚɜɧɟɧɢɹ (11.56)
ɫɥɟɞɭɟɬ ɡɧɚɱɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ
E C0
.
k E
Cc
(11.57)
Ɍɨɝɞɚ ɫɤɨɪɨɫɬɶ ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɟɫɬɶ
E kC0
.
k E
M
(11.58)
ɉɪɨɰɟɫɫ ɦɨɠɟɬ ɩɪɨɬɟɤɚɬɶ ɩɨ-ɪɚɡɧɨɦɭ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɤɚɤɚɹ
ɫɬɚɞɢɹ ɹɜɥɹɟɬɫɹ ɦɟɞɥɟɧɧɨɣ.
ȿɫɥɢ ɦɟɞɥɟɧɧɵɦ ɹɜɥɹɟɬɫɹ ɩɨɞɜɨɞ ɜɟɳɟɫɬɜɚ ɤ ɩɨɜɟɪɯɧɨɫɬɢ x 0 , ɬɨ
E k , C c C0 ɢ M E C0 ,
ɬ.ɟ. ɫɤɨɪɨɫɬɶ ɫɭɦɦɚɪɧɨɝɨ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɦɚɫɫɨɨɩɟɪɟɞɚɱɢ ɢɥɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɜɧɟɲɧɟɣ ɞɢɮɮɭɡɢɢ. Ƚɨɜɨɪɹɬ, ɱɬɨ ɩɪɨɰɟɫɫ ɤɨɧɬɪɨɥɢɪɭɟɬɫɹ ɜɧɟɲɧɟɣ ɞɢɮɮɭɡɢɟɣ ɢɥɢ ɧɚɯɨɞɢɬɫɹ ɜɨ ɜɧɟɲɧɟɞɢɮɮɭɡɢɨɧɧɨɣ ɨɛɥɚɫɬɢ.
ȿɫɥɢ ɦɟɞɥɟɧɧɨɣ ɹɜɥɹɟɬɫɹ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ, ɬɨ
E !! k , C c | C0 ɢ M kC0 ,
ɬ.ɟ. ɫɤɨɪɨɫɬɶ ɫɭɦɦɚɪɧɨɝɨ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɤɨɪɨɫɬɶɸ ɯɢɦɢɱɟɫɤɨɣ
ɪɟɚɤɰɢɢ. Ƚɨɜɨɪɹɬ, ɱɬɨ ɩɪɨɰɟɫɫ ɧɚɯɨɞɢɬɫɹ ɜ ɤɢɧɟɬɢɱɟɫɤɨɣ ɨɛɥɚɫɬɢ.
ɂɡɦɟɧɢɦ ɡɚɞɚɱɭ, ɞɨɛɚɜɢɜ ɞɢɮɮɭɡɢɸ ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ x ! 0
ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɞɢɮɮɭɡɢɢ D , ɩɪɢɱɟɦ ɩɪɢɦɟɦ, ɱɬɨ ɪɟɚɤɰɢɹ ɦɨɠɟɬ
ɩɪɨɢɫɯɨɞɢɬɶ ɜɨ ɜɫɟɣ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ, ɚ ɧɟ ɬɨɥɶɤɨ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ
ɪɚɡɞɟɥɚ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɪɟɚɤɰɢɹ ɢ ɞɢɮɮɭɡɢɹ ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ
ɩɪɨɢɫɯɨɞɹɬ ɩɚɪɚɥɥɟɥɶɧɨ, ɚ ɜ ɫɥɭɱɚɟ ɩɚɪɚɥɥɟɥɶɧɵɯ ɩɪɨɰɟɫɫɨɜ ɫɭɦɦɚɪɧɚɹ
ɫɤɨɪɨɫɬɶ ɨɩɪɟɞɟɥɹɟɬɫɹ ɛɵɫɬɪɨɣ ɫɬɚɞɢɟɣ. ɉɨ ɨɬɧɨɲɟɧɢɸ ɤ ɜɧɟɲɧɟɣ
ɞɢɮɮɭɡɢɢ ɨɛɚ ɩɪɨɰɟɫɫɚ ɹɜɥɹɸɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ.
ɋ ɭɱɟɬɨɦ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ ɭɪɚɜɧɟɧɢɟ, ɨɩɢɫɵɜɚɸɳɟɟ ɢɡɦɟɧɟɧɢɟ
ɤɨɧɰɟɧɬɪɚɰɢɢ ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ, ɢɦɟɟɬ ɜɢɞ
wC
wt
D
w 2C
w x2
301
kC .
§ wC
·
Ɋɟɲɟɧɢɟ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ¨
0¸ ɫ
© wt
¹
ɭɱɟɬɨɦ ɭɫɥɨɜɢɣ C f ,t 0 ɢ C 0 ,t C c ɟɫɬɶ
§
k ·
C C cexp ¨ x¸ .
(11.59)
D
©
¹
Ɉɛɵɱɧɨ ɡɚ ɝɥɭɛɢɧɭ ɞɢɮɮɭɡɢɢ G ɩɪɢɧɢɦɚɸɬ ɪɚɫɫɬɨɹɧɢɟ, ɧɚ ɤɨɬɨɪɨɦ ɤɨɧɰɟɧɬɪɚɰɢɹ ɭɦɟɧɶɲɚɟɬɫɹ ɜ e ɪɚɡ. ɉɨɷɬɨɦɭ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ
ɩɪɨɬɟɤɚɧɢɢ ɞɢɮɮɭɡɢɢ ɢ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ ɢɦɟɟɦ
G
D k.
ɋɭɦɦɚɪɧɚɹ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨɬɨɤɨɦ ɱɟɪɟɡ ɟɞɢɧɢɰɭ
ɩɨɜɟɪɯɧɨɫɬɢ
wC
D
(11.60)
jx 0 D
C c Dk C c .
wx x 0
G
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɷɮɮɟɤɬɢɜɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ keff , ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ
ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɩɨ ɭɪɚɜɧɟɧɢɸ M kC c , ɟɫɬɶ
keff
Dk
D G,
(11.61)
ɚ ɷɮɮɟɤɬɢɜɧɚɹ ɷɧɟɪɝɢɹ ɚɤɬɢɜɚɰɢɢ ɪɚɜɧɚ ɩɨɥɭɫɭɦɦɟ ɷɧɟɪɝɢɣ ɚɤɬɢɜɚɰɢɢ
ɞɢɮɮɭɡɢɢ ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ ED ɢ ɷɧɟɪɝɢɢ ɚɤɬɢɜɚɰɢɢ ɯɢɦɢɱɟɫɤɨɣ
ɪɟɚɤɰɢɢ Ea .
ɉɪɢɪɚɜɧɢɜɚɹ ɩɨɬɨɤ (11.60) ɩɨɬɨɤɭ ɢɡ ɜɧɟɲɧɟɣ ɨɛɥɚɫɬɢ (ɬ.ɟ. ɩɨɥɚɝɚɹ, ɱɬɨ ɫɤɨɥɶɤɨ ɜɟɳɟɫɬɜɚ ɩɨɞɜɨɞɢɬɫɹ ɤ ɩɨɜɟɪɯɧɨɫɬɢ x 0 , ɫɬɨɥɶɤɨ ɠɟ
ɩɟɪɟɪɚɛɚɬɵɜɚɟɬɫɹ ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ), ɧɚɣɞɟɦ
E ' C0 C c C c Dk .
Ɍɨɝɞɚ, ɤɨɧɰɟɧɬɪɚɰɢɹ
ɫɨɨɬɧɨɲɟɧɢɹ
ɧɚ
Cc
ɩɨɜɟɪɯɧɨɫɬɢ
E ' C0
E ' Dk
ɪɚɡɞɟɥɚ
,
ɫɥɟɞɭɟɬ
ɢɡ
(11.62)
ɚ ɩɨɬɨɤ ɱɟɪɟɡ ɩɨɜɟɪɯɧɨɫɬɶ ɪɚɡɞɟɥɚ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɣ ɫɭɦɦɚɪɧɭɸ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɫɥɟɞɭɟɬ ɢɡ (11.60)
jx
0
C c Dk
E ' C0 Dk
.
E ' Dk
Ɉɬɫɸɞɚ ɧɚɯɨɞɢɦ ɷɮɮɟɤɬɢɜɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ
302
(11.63)
Deff
DkE '
DE 'G
{
.
Dk E ' D E 'G
(11.64)
Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ, ɯɚɪɚɤɬɟɪɧɨɟ ɞɥɹ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ, ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 11.7.
ɉɪɨɰɟɫɫ ɨɩɹɬɶ ɩɪɨɬɟɤɚɟɬ ɩɨɪɚɡɧɨɦɭ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ, ɤɚɤɚɹ
ɫɬɚɞɢɹ ɹɜɥɹɟɬɫɹ ɦɟɞɥɟɧɧɨɣ.
ȿɫɥɢ ɦɟɞɥɟɧɧɵɦ ɹɜɥɹɟɬɫɹ ɩɨɞɜɨɞ
ɜɟɳɟɫɬɜɚ ɤ ɝɪɚɧɢɰɟ ɪɚɡɞɟɥɚ, ɬɨ ɩɪɨɰɟɫɫ
ɤɨɧɬɪɨɥɢɪɭɟɬ ɜɧɟɲɧɹɹ ɞɢɮɮɭɡɢɹ. ȼ
ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ
E' Dk , C c C0 ,
jx
0
Ɋɢɫ. 11.7. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɤɨɧɰɟɧɬɪɚɰɢɢ ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ
ɝɟɬɟɪɨɝɟɧɧɨɦ ɩɪɨɰɟɫɫɟ
E ' C0 .
ȿɫɥɢ ɦɟɞɥɟɧɧɵɦ ɹɜɥɹɟɬɫɹ ɩɪɨɰɟɫɫ
ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ, ɬɨ ɫɩɪɚɜɟɞɥɢɜɚ ɨɰɟɧɤɚ
E' !! Dk ɢɥɢ E'G !! D ,
ɬɨɝɞɚ
C c | C0 , jx 0 C0 Dk C0 D G .
Ɂɞɟɫɶ ɜɚɠɧɨ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɤɨɧɫɬɚɧɬɚɦɢ ɜɧɭɬɪɟɧɧɟɣ ɞɢɮɮɭɡɢɢ ɢ ɪɟɚɤɰɢɢ. ȼɜɟɞɟɦ ɯɚɪɚɤɬɟɪɧɵɣ ɝɟɨɦɟɬɪɢɱɟɫɤɢɣ ɪɚɡɦɟɪ ɞɥɹ ɜɧɭɬɪɟɧɧɟɣ ɨɛɥɚɫɬɢ. ɗɬɨ ɦɨɠɟɬ ɛɵɬɶ, ɧɚɩɪɢɦɟɪ, ɪɚɡɦɟɪ ɡɟɪɧɚ d . ȿɫɥɢ
k D d 2 , ɬ.ɟ. ɦɟɞɥɟɧɧɨɣ ɹɜɥɹɟɬɫɹ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ, ɬɨ G ~ d ɢ ɜ
ɡɟɪɧɟ ɩɪɚɤɬɢɱɟɫɤɢ ɨɬɫɭɬɫɬɜɭɟɬ ɝɪɚɞɢɟɧɬ ɤɨɧɰɟɧɬɪɚɰɢɢ. ɇɚɩɪɨɬɢɜ, ɟɫɥɢ
k !! D d 2 , ɬɨ G d ɢ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ «ɩɨɟɞɚɟɬ» ɜɫɟ ɜɟɳɟɫɬɜɨ
ɩɪɚɤɬɢɱɟɫɤɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɩɪɨɬɟɤɚɧɢɢ ɝɟɬɟɪɨɝɟɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɜ ɧɟɫɤɨɥɶɤɨ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɫɬɚɞɢɣ ɞɢɮɮɭɡɢɹ ɹɜɥɹɟɬɫɹ ɤɨɧɬɪɨɥɢɪɭɸɳɟɣ ɫɬɚɞɢɟɣ, ɟɫɥɢ ɟɟ ɫɤɨɪɨɫɬɶ ɦɚɥɚ, ɚ ɜ ɧɟɫɤɨɥɶɤɨ ɩɚɪɚɥɥɟɥɶɧɵɯ ɫɬɚɞɢɣ, – ɟɫɥɢ
ɟɟ ɫɤɨɪɨɫɬɶ ɜɟɥɢɤɚ.
ɉɪɢ ɫɨɩɨɫɬɚɜɢɦɵɯ ɫɤɨɪɨɫɬɹɯ ɨɬɞɟɥɶɧɵɯ ɫɬɚɞɢɣ ɫɭɦɦɚɪɧɚɹ ɫɤɨɪɨɫɬɶ ɩɪɨɰɟɫɫɚ ɡɚɜɢɫɢɬ ɨɬ ɜɫɟɯ ɫɨɫɬɚɜɥɹɸɳɢɯ.
303
11.8. ɏɢɦɢɱɟɫɤɢɟ ɢɫɬɨɱɧɢɤɢ ɷɧɟɪɝɢɢ
ɜ ɭ ɪ ɚ ɜ ɧ ɟ ɧ ɢ ɢ ɬ ɟ ɩ ɥ ɨ ɩ ɪ ɨ ɜ ɨ ɞ ɧ ɨ ɫ ɬ ɢ 32
ɍɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɯɢɦɢɱɟɫɤɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ ɬɟɩɥɚ
ɜɵɜɨɞɢɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɪɚɡɞɟɥɚɦ 2.6. ɢ 3.2 ɢ ɞɥɹ ɧɟɜɹɡɤɨɝɨ ɧɟɫɠɢɦɚɟɦɨɝɨ ɝɚɡɚ ɢɥɢ ɧɟɫɠɢɦɚɟɦɨɣ ɧɟɜɹɡɤɨɣ ɠɢɞɤɨɫɬɢ ɢɦɟɟɬ ɜɢɞ
dT
UcJ
dt
n
¦
’ ˜ J T U
hk
k 1
dC k
,
dt
(11.65)
ɝɞɟ
w ...
d ... w ...
w ...
w ...
,
w
v
u
wy
dt
wx
wz
wt
ɚ hk – ɩɚɪɰɢɚɥɶɧɵɟ ɷɧɬɚɥɶɩɢɢ ɤɨɦɩɨɧɟɧɬɨɜ, ɭɱɚɫɬɜɭɸɳɢɯ ɜ ɪɟɚɤɰɢɹɯ
h
h
¦h C
k
k
-
k 1
ɷɧɬɚɥɶɩɢɹ ɫɦɟɫɢ.
Ⱦɥɹ ɤɚɠɞɨɝɨ ɤɨɦɩɨɧɟɧɬɚ ɫɦɟɫɢ ɢɦɟɟɬ ɦɟɫɬɨ ɭɪɚɜɧɟɧɢɟ
dC
(11.66)
U k ’ ˜ J k V k ,
dt
ɝɞɟ J k – ɞɢɮɮɭɡɢɨɧɧɵɟ ɩɨɬɨɤɢ ɤɨɦɩɨɧɟɧɬɨɜ, ɨɩɪɟɞɟɥɟɧɧɵɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ ɡɚɤɨɧɚɦɢ (11.51) ɢɥɢ (11.54) ɢɥɢ ɫ ɩɨɦɨɳɶɸ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɩɪɟɞɫɬɚɜɥɟɧɢɣ, ɨɫɧɨɜɚɧɧɵɯ, ɧɚɩɪɢɦɟɪ, ɧɚ ɧɟɪɚɜɧɨɜɟɫɧɨɣ ɬɟɪɦɨɞɢɧɚɦɢɤɟ, ɚ V k – ɫɭɦɦɚ ɢɫɬɨɱɧɢɤɨɜ ɢ ɫɬɨɤɨɜ ɤɨɦɩɨɧɟɧɬɚ
k ɜ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɹɯ.
ɉɨɞɫɬɚɜɥɹɹ (11.66) ɜ (11.65), ɧɚɣɞɟɦ
n
n
ª
º
dT
UcJ
’ ˜ «J T hk J k » hk V k
«
»
dt
k 1
¬
¼ k 1
ȼɨ ɦɧɨɝɢɯ ɩɪɚɤɬɢɱɟɫɤɢ ɜɚɠɧɵɯ ɫɥɭɱɚɹɯ ɜɵɪɚɠɟɧɢɟ, ɫɬɨɹɳɟɟ ɜ
ɤɜɚɞɪɚɬɧɵɯ ɫɤɨɛɤɚɯ, ɨɩɪɟɞɟɥɹɸɬ ɤɚɤ ɷɮɮɟɤɬɢɜɧɵɣ ɩɨɬɨɤ ɬɟɩɥɚ, ɜɤɥɸɱɚɸɳɢɣ ɜ ɫɟɛɹ ɩɟɪɟɧɨɫ ɬɟɩɥɚ ɞɢɮɮɭɡɢɟɣ, ɢ ɭɞɨɜɥɟɬɜɨɪɹɸɳɢɣ ɡɚɤɨɧɭ
Ɏɭɪɶɟ:
¦
32
¦
Ȼɨɥɟɟ ɨɫɧɨɜɚɬɟɥɶɧɨ ɜɨɩɪɨɫɵ, ɡɚɬɪɚɝɢɜɚɟɦɵɟ ɜ ɷɬɨɦ ɩɚɪɚɝɪɚɮɟ, ɢɡɥɚɝɚɸɬɫɹ ɜ ɫɩɟɰɢɚɥɶɧɨɣ ɥɢɬɟɪɚɬɭɪɟ. ɋɫɵɥɤɢ ɧɟ ɧɟɤɨɬɨɪɵɟ ɭɱɟɛɧɢɤɢ ɩɪɢɜɟɞɟɧɵ ɜ ɫɩɢɫɤɟ ɪɟɤɨɦɟɧɞɨɜɚɧɧɨɣ ɥɢɬɟɪɚɬɭɪɵ.
304
n
JT ¦
JT c
hk J k
O T ’T .
k 1
ȼ ɪɟɡɭɥɶɬɚɬɟ ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɢɫɬɟɦɟ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
dT
Uc J
dt
’ ˜ J T c n
¦h V
(11.67)
k k
k 1
ɢ ɞɢɮɮɭɡɢɢ (11.67).
ɉɪɢ ɭɫɥɨɜɢɢ, ɱɬɨ ɞɢɮɮɭɡɢɟɣ ɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ
ɤɨɧɜɟɤɬɢɜɧɵɦ ɩɟɪɟɧɨɫɨɦ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, ɨɬ (11.66) ɨɫɬɚɟɬɫɹ ɭɪɚɜɧɟɧɢɟ ɩɟɪɟɧɨɫɚ
§ wC
wC
wC
wC
U¨¨ k u k v k w k
wz
wy
wx
© wt
·
¸¸
¹
Vk ,
ɚ ɜɦɟɫɬɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ – ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɜɨɝɨ ɛɚɥɚɧɫɚ
ª wT
wT
wT
wT º
Uc J « u
v
w »
wx
wy
wz ¼
¬ wt
h
¦h V
k k
.
k 1
Ⱦɥɹ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɬɨɠɟ ɭɩɪɨɳɚɟɬɫɹ.
ȼ ɫɥɭɱɚɟ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɤɨɧɜɟɤɬɢɜɧɵɦɢ ɫɥɚɝɚɟɦɵɦɢ
ɜ ɭɪɚɜɧɟɧɢɹɯ (11.66) ɢ (11.67):
wC
U k ’ ˜ J k V k ,
wt
wT
Uc J
wt
’ ˜ J T c n
¦h V
k k
.
k 1
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɬɟɨɪɟɬɢɱɟɫɤɢɦɢ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦɢ, ɜɵɪɚɠɟɧɢɟ ɞɥɹ
ɫɭɦɦɚɪɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɤɨɦɩɨɧɟɧɬɚ k ɜ ɪɟɚɤɰɢɹɯ, ɱɢɫɥɨ ɤɨɬɨɪɵɯ r ,
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
r
Vk
¦Q
ki mk Mi
(11.68)
l 1
ɝɞɟ Q ki – ɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɦɩɨɧɟɧɬɚ k ɜ ɪɟɚɤɰɢɢ i ,
mk – ɦɨɥɹɪɧɚɹ ɦɚɫɫɚ k -ɝɨ ɤɨɦɩɨɧɟɧɬɚ, Mi – ɫɤɨɪɨɫɬɶ i -ɣ ɪɟɚɤɰɢɢ. Ⱦɥɹ
ɤɚɠɞɨɣ ɪɟɚɤɰɢɢ ɫɩɪɚɜɟɞɥɢɜɨ ɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɨɟ ɫɨɨɬɧɨɲɟɧɢɟ
305
n
¦Q m 0 , i 1,2, ...,r ,
ki k
(11.69)
k 1
ɹɜɥɹɸɳɟɟɫɹ ɫɥɟɞɫɬɜɢɟɦ ɭɪɚɜɧɟɧɢɹ ɪɟɚɤɰɢɢ.
ɉɨ ɨɩɪɟɞɟɥɟɧɢɸ, ɫɬɟɯɢɨɦɟɬɪɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɱɢɬɚɸɬɫɹ
ɩɨɥɨɠɢɬɟɥɶɧɵɦɢ (ɞɥɹ ɤɚɠɞɨɣ ɞɚɧɧɨɣ ɪɟɚɤɰɢɢ), ɟɫɥɢ ɨɧɢ ɨɬɧɨɫɹɬɫɹ ɤ
ɩɪɨɞɭɤɬɚɦ, ɢ ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ, ɟɫɥɢ ɨɧɢ ɨɬɧɨɫɹɬɫɹ ɤ ɪɟɚɝɟɧɬɚɦ.
ȿɫɥɢ ɤɨɦɩɨɧɟɧɬ ɭɱɚɫɬɜɭɟɬ ɬɨɥɶɤɨ ɜ ɨɞɧɨɣ ɪɟɚɤɰɢɢ, ɬɨ ɜɦɟɫɬɨ
(11.68), (11.69) ɧɚɣɞɟɦ
n
Vk
Q k mk M ,
¦Q m .
k
(11.70)
k
k 1
ɇɚɩɪɢɦɟɪ, ɞɥɹ ɟɞɢɧɫɬɜɟɧɧɨɣ ɪɟɚɤɰɢɢ
2 A B C , ɝɞɟ C { A2 B ,
ɢɦɟɟɦ
Q A 2; Q B 1; QC 1 .
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɦ
VA
2 m A M , V B
mB M , VC
mC M .
Ɍɟɩɟɪɶ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ ɜ ɭɪɚɜɧɟɧɢɢ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
3
¦h V
k k
2m A h A mB hB hC mC M
QM ,
k 1
ɝɞɟ Q – ɢ ɟɫɬɶ ɬɟɩɥɨɜɨɣ ɷɮɮɟɤɬ ɷɬɨɣ ɪɟɚɤɰɢɢ.
ȿɫɥɢ ɪɟɚɤɰɢɹ ɩɪɨɬɟɤɚɟɬ ɜ ɝɚɡɨɜɨɣ ɢɥɢ ɠɢɞɤɨɣ ɮɚɡɟ, ɬɨ ɞɥɹ ɨɩɢɫɚɧɢɹ ɟɟ ɫɤɨɪɨɫɬɢ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹɦɢ ɮɨɪɦɚɥɶɧɨɤɢɧɟɬɢɱɟɫɤɨɣ ɬɟɨɪɢɢ, ɨɩɢɫɚɧɧɨɣ ɜ ɪɚɡɞɟɥɚɯ 11.1–11.3.
Ⱦɥɹ ɪɚɫɫɦɨɬɪɟɧɧɨɝɨ ɩɪɢɦɟɪɚ ɢɦɟɟɦ
M
k >A@2 >B @.
Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɪɟɚɤɰɢɣ ɫ ɭɱɚɫɬɢɟɦ ɬɜɟɪɞɵɯ ɜɟɳɟɫɬɜ ɦɧɨɝɨ ɫɥɨɠɧɟɟ, ɢ ɫɩɨɫɨɛɵ ɢɯ ɦɚɤɪɨɤɢɧɟɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɬɪɟɛɭɸɬ ɨɬɞɟɥɶɧɨɝɨ
ɢɡɭɱɟɧɢɹ. ɇɚɩɪɢɦɟɪ, ɩɪɢ ɨɩɢɫɚɧɢɢ ɪɟɚɤɰɢɣ ɜ ɬɜɟɪɞɨɣ ɮɚɡɟ ɱɚɫɬɨ ɩɪɟɧɟɛɪɟɝɚɸɬ ɹɜɥɟɧɢɟɦ ɞɢɮɮɭɡɢɢ ɜ ɭɪɚɜɧɟɧɢɢ ɞɥɹ ɤɨɦɩɨɧɟɧɬɚ ɢ ɩɨɥɚɝɚɸɬ,
ɱɬɨ ɞɢɮɮɭɡɢɹ ɢ ɪɚɡɥɢɱɧɵɟ ɞɪɭɝɢɟ ɫɨɩɭɬɫɬɜɭɸɳɢɟ ɹɜɥɟɧɢɹ ɩɪɢɜɨɞɹɬ
ɤ ɢɡɦɟɧɟɧɢɸ ɡɚɤɨɧɚ ɞɥɹ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ: ɩɨɹɜɥɹɸɬɫɹ ɞɪɨɛɧɵɟ
ɩɨɪɹɞɤɢ ɪɟɚɤɰɢɢ, ɡɚɜɢɫɢɦɨɫɬɶ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɨɬ ɬɨɥɳɢɧɵ ɫɥɨɹ
ɩɪɨɞɭɤɬɚ ɢ ɬ.ɞ.
306
ȼɨɩɪɨɫɵ ɞɥɹ ɫɚɦɨɤɨɧɬɪɨɥɹ
1. Ʉɚɤɢɟ ɬɢɩɵ ɤɨɧɰɟɧɬɪɚɰɢɹ ȼɵ ɡɧɚɟɬɟ? Ʉɚɤ ɨɧɢ ɦɟɠɞɭ ɫɨɛɨɣ ɫɜɹɡɚɧɵ?
2. Ʉɚɤɢɟ ɪɟɚɤɰɢɢ ɧɚɡɵɜɚɸɬ ɝɨɦɨɝɟɧɧɵɦɢ, ɚ ɤɚɤɢɟ ɝɟɬɟɪɨɝɟɧɧɵɦɢ?
3. ɩɪɢɜɟɞɢɬɟ ɩɪɢɦɟɪɵ ɩɪɨɫɬɟɣɲɢɯ ɪɟɚɤɰɢɨɧɧɵɯ ɫɯɟɦ ɞɥɹ ɝɨɦɨɝɟɧɧɵɯ
ɪɟɚɤɰɢɣ.
4. Ʉɚɤ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɨɧɫɬɚɧɬɭ ɫɤɨɪɨɫɬɢ ɪɟɚɤɰɢɢ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ?
5. Ʉɚɤɢɟ ɪɟɚɤɰɢɢ ɧɚɡɵɜɚɸɬ ɤɚɬɚɥɢɬɢɱɟɫɤɢɦɢ, ɚ ɤɚɤɢɟ - ɚɜɬɨɤɚɬɚɥɢɬɢɱɟɫɤɢɦɢ?
6. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɟ ɩɪɢɛɥɢɠɟɧɢɟ?
7. Ɉɬ ɱɟɝɨ ɡɚɜɢɫɢɬ ɫɤɨɪɨɫɬɶ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ?
8. Ɂɚɩɢɲɢɬɟ ɡɚɤɨɧ Ⱥɪɪɟɧɢɭɫɚ ɢ ɩɨɹɫɧɢɬɟ ɟɝɨ ɮɢɡɢɱɟɫɤɢɣ ɫɦɵɫɥ.
9. ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɪɚɡɥɢɱɢɟ ɦɟɠɞɭ ɬɟɨɪɢɟɣ ɫɬɨɥɤɧɨɜɟɧɢɣ ɢ ɬɟɨɪɢɟɣ
ɚɤɬɢɜɢɪɨɜɚɧɧɨɝɨ ɤɨɦɩɥɟɤɫɚ?
10. Ʉɚɤɢɟ ɪɟɚɤɰɢɢ ɧɚɡɵɜɚɸɬ ɬɜɟɪɞɨɮɚɡɧɵɦɢ? ȼ ɱɟɦ ɡɚɤɥɸɱɚɸɬɫɹ ɢɯ
ɨɫɨɛɟɧɧɨɫɬɢ?
11. Ɉɯɚɪɚɤɬɟɪɢɡɭɟɬɟ ɨɫɨɛɟɧɧɨɫɬɢ ɤɢɧɟɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɬɨɩɨɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ.
12. ɑɬɨ ɬɚɤɨɟ «ɡɚɪɨɞɵɲ»?
13. Ɂɚɩɢɲɢɬɟ 2 ɨɫɧɨɜɧɵɯ ɡɚɤɨɧɚ ɤɢɧɟɬɢɤɢ ɨɛɪɚɡɨɜɚɧɢɹ ɹɞɟɪ ɮɚɡɵ ɬɜɟɪɞɨɝɨ ɩɪɨɞɭɤɬɚ.
14. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɬɥɢɱɢɟ ɜ ɦɟɯɚɧɢɡɦɚɯ ɞɢɮɮɭɡɢɢ ɜ ɝɚɡɚɯ, ɠɢɞɤɨɫɬɹɯ ɢ
ɬɜɟɪɞɵɯ ɬɟɥɚɯ?
15. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɡɚɤɨɧ Ɏɢɤɚ.
16. Ʉɚɤɢɟ ɱɢɫɥɚ ɩɨɞɨɛɢɹ ɯɚɪɚɤɬɟɪɧɵ ɞɥɹ ɦɚɫɫɨɩɟɪɟɧɨɫɚ?
17. Ⱦɥɹ ɤɚɤɢɯ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɜɚɠɧɚ ɞɢɮɮɭɡɢɹ?
18. ɑɬɨ ɬɚɤɨɟ «ɤɨɧɬɪɨɥɢɪɭɸɳɚɹ ɫɬɚɞɢɹ ɩɪɨɰɟɫɫɚ»?
19. ȼ ɱɟɦ ɫɨɫɬɨɢɬ ɨɬɥɢɱɢɟ ɡɚɞɚɱ ɜɧɟɲɧɟɣ ɢ ɜɧɭɬɪɟɧɧɟɣ ɞɢɮɮɭɡɢɢ?
20. Ʉɚɤ ȼɵ ɩɨɧɢɦɚɟɬɟ ɬɟɪɦɢɧɵ «ɤɢɧɟɬɢɱɟɫɤɚɹ ɢ ɞɢɮɮɭɡɢɨɧɧɚɹ ɨɛɥɚɫɬɢ
ɩɪɨɬɟɤɚɧɢɹ ɝɟɬɟɪɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ?»
Ɂɚɞɚɧɢɹ
1. Ɂɚɩɢɲɢɬɟ ɫɢɫɬɟɦɭ ɤɢɧɟɬɢɱɟɫɤɢɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɫɯɟɦɵ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
ɉɨɩɵɬɚɣɬɟɫɶ ɧɚɣɬɢ ɪɟɲɟɧɢɟ ɩɨɥɭɱɟɧɧɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɚɧɚɥɢɬɢɱɟɫɤɢ. ɉɟɪɟɣɞɹ ɤ ɛɟɡɪɚɡɦɟɪɧɵɦ ɩɟɪɟɦɟɧɧɵɦ, ɩɪɨɢɥɥɸɫɬɪɢɪɭɣɬɟ
ɪɟɲɟɧɢɟ.
307
2. ȼɵɩɨɥɧɢɬɟ ɚɧɚɥɨɝɢɱɧɨɟ ɡɚɞɚɧɢɟ ɞɥɹ ɫɢɫɬɟɦɵ ɪɟɚɤɰɢɣ ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ
3. Ɉɫɧɨɜɵɜɚɹɫɶ ɧɚ ɚɧɚɥɨɝɢɢ ɦɟɠɞɭ ɩɪɨɰɟɫɫɚɦɢ ɩɟɪɟɧɨɫɚ ɬɟɩɥɚ ɢ ɦɚɫɫɵ,
ɫɮɨɪɦɭɥɢɪɭɣɬɟ ɡɚɞɚɱɭ ɨ ɧɚɫɵɳɟɧɢɢ ɩɥɨɫɤɨɝɨ ɨɛɪɚɡɰɚ ɩɪɢɦɟɫɶɸ ɢɯ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɵ. ɋɱɢɬɚɹ, ɱɬɨ ɤɨɧɰɟɧɬɪɚɰɢɹ ɩɪɢɦɟɫɢ ɜ ɫɪɟɞɟ ɧɟɢɡɦɟɧɧɚ, ɚ ɨɛɪɚɡɟɰ ɢɦɟɟɬ ɛɨɥɶɲɢɟ ɪɚɡɦɟɪɵ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɜɨɡɦɨɠɧɨɣ ɲɢɪɢɧɨɣ ɞɢɮɮɭɡɢɨɧɧɨɣ ɡɨɧɵ, ɧɚɣɞɢɬɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ.
4. ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɚɧɚɥɨɝɢɱɧɭɸ ɡɚɞɚɱɭ ɨ ɩɟɪɟɪɚɫɩɪɟɞɟɥɟɧɢɢ ɩɪɢɦɟɫɢ
ɦɟɠɞɭ ɱɚɫɬɢɰɟɣ ɪɚɞɢɭɫɚ R0 ɢ ɨɤɪɭɠɚɸɳɟɣ ɟɟ ɮɚɡɨɣ.
ɉɨɥɚɝɚɹ, ɱɬɨ ɜ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ ɡɚɞɚɧɵ ɭɫɥɨɜɢɹ
t 0 : C C0 , r d R0 ɢ C 0 ,r ! R0 ,
ɧɚɣɞɢɬɟ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ, ɩɨɥɶɡɭɹɫɶ ɨɞɧɢɦ ɢɡ ɢɡɜɟɫɬɧɵɯ
ȼɚɦ ɦɟɬɨɞɨɜ.
308
ɑȺɋɌɖ 12
Ɋɚɡɧɵɟ ɡɚɞɚɱɢ
1 2 . 1 . Ɉ ɯ ɥ ɚ ɠ ɞ ɟ ɧ ɢ ɟ ɩ ɨ ɪ ɢ ɫ ɬ ɨ ɣ ɩ ɥ ɚ ɫ ɬ ɢ ɧ ɵ 33
Ɋɚɫɫɦɨɬɪɢɦ ɩɥɨɫɤɭɸ ɧɟ ɜɵɞɟɥɹɸɳɭɸ ɬɟɩɥɨ ɩɥɚɫɬɢɧɭ ɫ ɩɨɪɢɫɬɨɫɬɶɸ m ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O w . ɉɭɫɬɶ ɨɞɧɚ ɢɡ ɥɢɰɟɜɵɯ ɩɨɜɟɪɯɧɨɫɬɟɣ ɩɥɚɫɬɢɧɵ ɧɚɝɪɟɬɚ ɞɨ ɬɟɦɩɟɪɚɬɭɪɵ T2 , ɚ ɨɯɥɚɠɞɚɸɳɚɹ
ɠɢɞɤɨɫɬɶ, ɧɚɝɧɟɬɚɟɦɚɹ ɫɤɜɨɡɶ ɩɥɚɫɬɢɧɭ ɜ ɩɨɥɨɠɢɬɟɥɶɧɨɦ ɧɚɩɪɚɜɥɟɧɢɢ
ɨɫɢ Ox , ɢɦɟɟɬ ɩɪɢ x o f ɬɟɦɩɟɪɚɬɭɪɭ T0 (ɪɢɫ. 12.1). Ɋɚɫɯɨɞ ɠɢɞɤɨɫɬɢ
(ɩɪɨɢɡɜɟɞɟɧɢɟ ɟɟ ɩɥɨɬɧɨɫɬɢ ɧɚ ɥɢɧɟɣɧɭɸ ɫɤɨɪɨɫɬɶ) ɟɫɬɶ G U L wL ,
ɤɝ/(ɦ2ɫ) ɢ ɫɱɢɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ. ɀɢɞɤɨɫɬɶ ɢɦɟɟɬ ɬɟɩɥɨɟɦɤɨɫɬɶ cL ɢ
ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O L . ȿɫɥɢ ɪɚɡɦɟɪɵ ɩɥɚɫɬɢɧɵ ɜ ɧɚɩɪɚɜɥɟɧɢɹɯ ɨɫɟɣ Oy ,Oz ɜɟɥɢɤɢ, ɬɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ ɜɧɭɬɪɢ ɧɟɟ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɨɞɧɨɦɟɪɧɵɦ. ȼ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɢɦɟɟɦ
T T x .
ɉɨɪɢɫɬɨɫɬɶ ɩɥɚɫɬɢɧɵ ɨɩɪɟɞɟɥɹɸɬ
ɤɚɤ ɨɬɧɨɲɟɧɢɟ ɨɛɴɟɦɚ, ɡɚɧɹɬɨɝɨ ɩɨɪɚɦɢ, ɤɨ ɜɫɟɦɭ ɨɛɴɟɦɭ ɦɚɬɟɪɢɚɥɚ,
m V p V . ɀɢɞɤɨɫɬɶ, ɩɪɨɬɟɤɚɸɳɭɸ
ɱɟɪɟɡ ɫɟɱɟɧɢɟ ɬɚɤɨɝɨ ɦɚɬɟɪɢɚɥɚ, ɦɨɠɧɨ
ɯɚɪɚɤɬɟɪɢɡɨɜɚɬɶ ɞɜɭɦɹ ɫɤɨɪɨɫɬɹɦɢ –
ɢɫɬɢɧɧɨɣ ɫɤɨɪɨɫɬɶɸ w ɢ ɫɤɨɪɨɫɬɶɸ
ɮɢɥɶɬɪɚɰɢɢ u , ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬ
Ɋɢɫ. 12.1. Ʉ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɨɛ ɩɨɬɨɤ ɠɢɞɤɨɫɬɢ ɱɟɪɟɡ ɩɪɨɫɬɪɚɧɫɬɜɨ,
ɨɯɥɚɠɞɟɧɢɢ ɩɨɪɢɫɬɨɣ ɩɥɚɫɬɢɧɵ
ɡɚɧɹɬɨɟ ɩɨɪɚɦɢ, ɢ ɮɢɤɬɢɜɧɵɣ ɩɨɬɨɤ,
Sp
ɤɚɤ ɛɵ ɪɚɡɦɚɡɚɧɧɵɣ ɩɨ ɫɟɱɟɧɢɸ, ɬɚɤ ɱɬɨ u
w sw , ɝɞɟ S p – ɩɥɨS
ɳɚɞɶ, ɡɚɧɹɬɚɹ ɜ ɫɟɱɟɧɢɢ S ɩɨɪɚɦɢ. ȼ ɢɡɨɬɪɨɩɧɨɦ ɦɚɬɟɪɢɚɥɟ (ɬ.ɟ. ɜ ɦɚɬɟɪɢɚɥɟ ɫ ɨɞɢɧɚɤɨɜɵɦɢ ɫɜɨɣɫɬɜɚɦɢ ɩɨ ɜɫɟɦ ɧɚɩɪɚɜɥɟɧɢɹɦ) s m , ɱɬɨ
ɦɵ ɞɚɥɟɟ ɢ ɛɭɞɟɦ ɩɪɟɞɩɨɥɚɝɚɬɶ.
ɉɟɪɟɧɨɫ ɬɟɩɥɚ ɜ ɬɚɤɨɦ ɬɟɥɟ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤ ɫɭɦɦɭ ɞɜɭɯ ɫɨɫɬɚɜɥɹɸɳɢɯ – ɩɟɪɟɧɨɫ ɬɟɩɥɚ ɜɧɭɬɪɢ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ
ɢ ɬɟɩɥɨɨɛɦɟɧ «ɬɜɟɪɞɨɟ ɬɟɥɨ – ɨɯɥɚɠɞɚɸɳɚɹ ɠɢɞɤɨɫɬɶ». ɀɢɞɤɨɫɬɶɸ ɬɟɩɥɨ ɩɟɪɟɧɨɫɢɬɫɹ ɡɚ ɫɱɟɬ ɞɜɭɯ ɦɟɯɚɧɢɡɦɨɜ – ɡɚ ɫɱɟɬ ɬɟɱɟɧɢɹ ɠɢɞɤɨɫɬɢ ɫɨ
33
ɇɟɫɤɨɥɶɤɨ ɢɧɨɣ ɩɨɞɯɨɞ ɤ ɷɬɨɣ ɩɪɨɛɥɟɦɟ ɢɫɩɨɥɶɡɭɟɬɫɹ ɤ ɤɧɢɝɟ ɒɧɟɣɞɟɪɚ ɉ. ɂɧɠɟɧɟɪɧɵɟ ɩɪɨɛɥɟɦɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɇ.: ɂɡɞ-ɜɨ ɢɧɨɫɬɪɚɧɧɨɣ ɥɢɬɟɪɚɬɭɪɵ,
1960. 480 c.
309
ɫɤɨɪɨɫɬɶɸ w ɢ ɡɚ ɫɱɟɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɠɢɞɤɨɫɬɢ. Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɞɥɹ ɩɨɪɢɫɬɨɝɨ ɬɟɥɚ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɪɟɠɢɦɟ ɦɨɠɧɨ
ɡɚɩɢɫɚɬɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
d 2T
dT
O eff
mc
G
0,
L
dx
dx 2
ɝɞɟ O eff O L m O w 1 m , ɢɥɢ
d 2T
dx
ɝɞɟ [ w
mcLG
O eff
2
[w
dT
dx
0,
(12.1)
C
.
O eff
Ƚɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɤ ɭɪɚɜɧɟɧɢɸ (12.1) ɛɭɞɭɬ
x G : T T2 , x 0 : T T1 .
ȿɫɥɢ ɬɟɦɩɟɪɚɬɭɪɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ x G ɡɚɞɚɧɚ, ɬɨ ɜɟɥɢɱɢɧɚ ɬɟɦɩɟɪɚɬɭɪɵ T1 ɡɚɜɢɫɢɬ ɨɬ ɭɫɥɨɜɢɣ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ x 0 . ȼ ɩɪɨɫɬɟɣɲɟɦ ɫɥɭɱɚɟ
T1 T0 .
Ȼɨɥɟɟ ɤɨɪɪɟɤɬɧɵɦ ɫ ɮɢɡɢɱɟɫɤɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ ɛɭɞɟɬ ɭɫɥɨɜɢɟ, ɭɱɢɬɵɜɚɸɳɟɟ ɧɚɝɪɟɜ ɠɢɞɤɨɫɬɢ ɧɚ ɜɯɨɞɟ ɡɚ ɫɱɟɬ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɜ ɩɨɪɢɫɬɨɣ ɩɥɚɫɬɢɧɟ ɝɪɚɞɢɟɧɬɚ ɬɟɦɩɟɪɚɬɭɪɵ, ɬ.ɟ.
wLcLU L T0 T O ef f
dT
.
dx
(12.2)
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (12.1) ɢɦɟɟɬ ɜɢɞ
T
C1e[w x C2
ɢɥɢ ɩɨɫɥɟ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ
ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɩɟɪɜɨɝɨ ɪɨɞɚ –
T
T0 T2 T0 e[ w x 1
[ wG
.
(12.3)
e
1
ȼ ɫɥɭɱɚɟ ɨɯɥɚɠɞɟɧɢɹ ɩɥɚɫɬɢɧɵ ɫ ɜɧɭɬɪɟɧɧɢɦ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ,
ɪɚɫɩɪɟɞɟɥɟɧɧɵɦ ɪɚɜɧɨɦɟɪɧɨ, ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɩɪɢɧɢɦɚɟɬ
ɜɢɞ
d 2T
dT qw
(12.4)
[
0,
w
dx O eff
dx 2
ɝɞɟ qw – ɭɞɟɥɶɧɚɹ ɦɨɳɧɨɫɬɶ ɜɧɭɬɪɟɧɧɢɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ, Ⱦɠ/(ɦ3ɫ).
310
Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (12.4) ɟɫɬɶ ɫɭɦɦɚ ɨɛɳɟɝɨ ɪɟɲɟɧɢɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (12.1) ɢ ɱɚɫɬɧɨɝɨ ɪɟɲɟɧɢɹ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ
(12.4). ɉɨɫɥɟɞɧɟɟ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
Tp
qw
x.
O eff [ w
ȼ ɬɨɦ, ɱɬɨ ɨɧɨ ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɭɪɚɜɧɟɧɢɸ (12.4), ɦɨɠɧɨ ɭɛɟɞɢɬɶɫɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɣ ɩɨɞɫɬɚɧɨɜɤɨɣ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
T
ɝɞɟ A
qw
O eff [ w
C1e[ w x C2 Ax ,
qw
.
C
T,K
Ɉɩɪɟɞɟɥɹɹ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɫ ɩɨɦɨɳɶɸ ɝɪɚɧɢɱɧɵɯ
ɭɫɥɨɜɢɣ, ɧɚɣɞɟɦ
T
T0 Ax T2 T0 AG (12.5)
1500
1
1000
2
3
4
500
e [ w x 1 . (12.6)
e [ wG 1
ɇɚ ɪɢɫ. 12.2 ɩɪɟɞɫɬɚɜɥɟɧɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɪɢɫɬɨɣ
ɩɥɚɫɬɢɧɟ ɞɥɹ qw 0 (ɫɩɥɨɲɧɵɟ ɥɢ-
0,00
0,08
0,16
0,24 X,M
Ɋɢɫ. 12.2. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɩɨɪɢɫɬɨɣ ɩɥɚɫɬɢɧɟ. 1 –
C 1, 2. – C 30 ; 3 – C 75 ; 4 –
2
C 150 ȼɬ/(ɫɦ Ʉ)
ɧɢɢ) ɢ qw 3 ˜ 105 ȼɬ/ɦ3 (ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ). ȼ ɪɚɫɱɟɬɚɯ ɩɪɢɧɹɬɨ
O eff 10 ȼɬ/(ɦ Ʉ), T2 1800 , T0 300 Ʉ, G 0,3 ɦ.
1 2 . 2 . Ⱦ ɜ ɢ ɠ ɭ ɳ ɢ ɟ ɫ ɹ ɢ ɫ ɬ ɨ ɱ ɧ ɢ ɤ ɢ ɬ ɟ ɩ ɥ ɚ 34
Ɂɚɞɚɱɢ ɧɟɫɬɚɰɢɨɧɚɪɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɤɨɝɞɚ ɩɨɬɨɤɢ ɬɟɩɥɚ ɩɨɪɨɠɞɚɸɬɫɹ ɞɜɢɠɭɳɢɦɢɫɹ ɢɫɬɨɱɧɢɤɚɦɢ, ɢɦɟɸɬ ɛɨɥɶɲɨɟ ɡɧɚɱɟɧɢɟ ɩɪɢ
ɢɡɭɱɟɧɢɢ ɬɪɟɧɢɹ-ɫɤɨɥɶɠɟɧɢɹ, ɜɨ ɜɧɭɬɪɟɧɧɟɣ ɛɚɥɥɢɫɬɢɤɟ, ɩɪɢ ɦɟɯɚɧɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɟ ɞɟɬɚɥɟɣ, ɜ ɪɹɞɟ ɩɪɨɰɟɫɫɨɜ ɨɛɪɚɛɨɬɤɢ ɦɟɬɚɥɥɨɜ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ, ɧɚɩɪɢɦɟɪ, ɜ ɩɪɨɰɟɫɫɚɯ ɤɢɫɥɨɪɨɞɧɨɣ ɢ
ɥɚɡɟɪɧɨɣ ɪɟɡɤɢ, ɫɜɚɪɤɢ, ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɡɚɤɚɥɤɢ ɢ ɞɪ.
ȿɳɟ ɜ ɩɟɪɜɨɣ ɩɨɥɨɜɢɧɟ ɩɪɨɲɥɨɝɨ ɜɟɤɚ ɩɪɢ ɢɡɭɱɟɧɢɢ ɩɨɞɨɛɧɵɯ
ɩɪɨɰɟɫɫɨɜ ɛɵɥɨ ɡɚɦɟɱɟɧɨ, ɱɬɨ ɜɫɟɝɞɚ ɧɚɫɬɭɩɚɟɬ ɬɚɤɨɟ ɫɨɫɬɨɹɧɢɟ, ɤɨɝɞɚ
34
ɒɧɟɣɞɟɪ ɉ. ɂɧɠɟɧɟɪɧɵɟ ɩɪɨɛɥɟɦɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɩɟɪ. ɫ ɚɧɝɥ. Ɇ.: ɂɡɞ-ɜɨ
ɢɧɨɫɬɪɚɧɧɨɣ ɥɢɬɟɪɚɬɭɪɵ, 1960. 480 c.
311
ɧɚɛɥɸɞɚɬɟɥɶ, ɧɚɯɨɞɹɳɢɣɫɹ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɡɚɦɟɱɚɟɬ
ɢɡɦɟɧɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɜɵɡɜɚɧɧɨɟ ɞɜɢɠɭɳɢɦɫɹ ɢɫɬɨɱɧɢɤɨɦ, ɧɨ ɜ ɬɨ
ɠɟ ɫɚɦɨɟ ɜɪɟɦɹ ɧɚɛɥɸɞɚɬɟɥɶ, ɧɚɯɨɞɹɳɢɣɫɹ ɜ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɨɣ ɫ ɞɜɢɠɭɳɢɦɫɹ ɢɫɬɨɱɧɢɤɨɦ, ɧɟ ɜɢɞɢɬ ɧɢɤɚɤɢɯ ɢɡɦɟɧɟɧɢɣ ɬɟɦɩɟɪɚɬɭɪɵ. Ɍɚɤɨɟ ɫɨɫɬɨɹɧɢɟ ɧɚɡɵɜɚɟɬɫɹ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɵɦ.
ȿɝɨ ɭɞɨɛɧɨ ɢɡɭɱɚɬɶ, ɩɟɪɟɯɨɞɹ ɜ
ɫɢɫɬɟɦɭ ɤɨɨɪɞɢɧɚɬ, ɫɜɹɡɚɧɧɭɸ ɫ ɢɫɬɨɱɧɢɤɨɦ. ɍɫɥɨɜɢɟɦ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɫɬɢ ɜ ɷɬɨɣ ɧɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɛɭɞɟɬ
wT
0.
wt
Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɟɫɥɢ ɩɨɫɬɨɹɧɧɵɣ
Ɋɢɫ. 12.3. ɇɟɩɨɞɜɢɠɧɚɹ ɢ ɩɨɞɩɨɜɟɪɯɧɨɫɬɧɵɣ ɢɫɬɨɱɧɢɤ ɬɟɩɥɚ ɞɜɢɜɢɠɧɚɹ ɫɢɫɬɟɦɵ ɤɨɨɪɞɢɧɚɬ
ɠɟɬɫɹ ɫ ɧɟɢɡɦɟɧɧɨɣ ɫɤɨɪɨɫɬɶɸ ɜ ɧɚɩɪɚɜɥɟɧɢɢ, ɫɨɜɩɚɞɚɸɳɢɦ ɫ ɧɚɩɪɚɜɥɟɧɢɟɦ ɨɫɢ Ox (ɪɢɫ. 12.3), ɬɨ ɬɟɦɩɟɪɚɬɭɪɚ ɜ ɥɚɛɨɪɚɬɨɪɧɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɞɨɥɠɧɚ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɸ
§ w 2T w 2T w 2T ·
wT
N¨ 2 2 2 ¸ .
(12.7)
¨ wx
¸
wt
w
w
y
z
©
¹
ȼɜɟɞɟɦ ɧɨɜɵɟ ɩɟɪɟɦɟɧɧɵɟ
[
ɬɚɤ ɱɬɨ
w[
1,
wx
w[
wt
x Vt , tc t ,
V ,
wt c
wx
0,
wt c
1
wt
ɢ
wT w[ wT wTc
w[ wx wt c wx
wT
wx
wT
wt
wT w[ wT wTc
w[ wt wt c wt
wT w 2T
,
w[ wx 2
V
w 2T
w[2
wT wT
.
w[ wt c
ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɦɟɫɬɨ ɭɪɚɜɧɟɧɢɹ (12.7) ɧɚɣɞɟɦ
wT
wT
V
wt c
w[
§ w 2T w 2T w 2T
N¨ 2 2 2
¨ w[
wy
wz
©
ɢɥɢ ɞɥɹ ɤɜɚɡɢɫɬɚɰɢɨɧɚɪɧɨɝɨ ɪɟɠɢɦɚ
312
·
¸¸
¹
,
§ w 2T w 2T w 2T
N¨ 2 2 2
¨ w[
wy
wz
©
wT
V
w[
·
¸¸ .
¹
(12.8)
ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɩɪɢɦɟɧɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (12.8) ɪɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɭɸ ɡɚɞɚɱɭ.
ɉɭɫɬɶ ɱɟɪɟɡ ɧɟɨɝɪɚɧɢɱɟɧɧɨɟ ɬɜɟɪɞɨɟ ɬɟɥɨ ɫ ɧɚɱɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɨɣ T0 ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɥɨɫɤɢɣ ɢɫɬɨɱɧɢɤ ɡɚɞɚɧɧɨɣ ɩɨɫɬɨɹɧɧɨɣ ɩɥɨɬɧɨɫɬɢ
q0 . ɉɪɢ ɬɚɤɢɯ ɭɫɥɨɜɢɹɯ ɭɪɚɜɧɟɧɢɟ (12.8) ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɨɛɵɤɧɨɜɟɧɧɨɟ
ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ.
V
dT
d[
N
d 2T
d [2
,
(12.9)
ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɤɨɬɨɪɨɝɨ ɯɨɪɨɲɨ ɢɡɜɟɫɬɧɨ
ª V º
C1exp « [ » C2 .
¬ N ¼
T
(12.10)
ȼ ɤɚɱɟɫɬɜɟ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ ɡɚɩɢɲɟɦ
dT
d[
0 ɩɪɢ [ o rf ,
dT
d[
O
q0 , [
0.
ɉɟɪɜɨɟ ɭɫɥɨɜɢɟ ɝɨɜɨɪɢɬ ɨ ɬɨɦ, ɱɬɨ ɧɚ ɛɨɥɶɲɢɯ ɪɚɫɫɬɨɹɧɢɹɯ ɨɬ ɢɫɬɨɱɧɢɤɚ ɬɟɦɩɟɪɚɬɭɪɚ ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɚ ɜɬɨɪɨɟ ɭɫɥɨɜɢɟ ɩɪɨɫɬɨ ɭɫɬɚɧɚɜɥɢɜɚɟɬ ɪɚɜɟɧɫɬɜɨ ɤɨɥɢɱɟɫɬɜɚ ɬɟɩɥɚ, ɜɵɞɟɥɟɧɧɨɝɨ ɢɫɬɨɱɧɢɤɨɦ, ɢ ɬɟɩɥɚ,
ɜɨɫɩɪɢɧɹɬɨɝɨ ɬɜɟɪɞɵɦ ɬɟɥɨɦ ɨɬ ɢɫɬɨɱɧɢɤɚ ɩɪɢ [ 0 (ɢɥɢ x Vt ). ɋɥɟɜɚ
ɢ ɫɩɪɚɜɚ ɨɬ ɢɫɬɨɱɧɢɤɚ ɦɵ ɞɨɥɠɧɵ ɡɚɩɢɫɚɬɶ
dTL
d[
C1L
V
ª V º
exp « [ » ,
N
¬ N ¼
dTR
d[
C1R
V
ª V º
exp « [ » ,
N
¬ N ¼
ɝɞɟ ɞɥɹ ɭɞɨɛɫɬɜɚ ɜɜɟɞɟɧɵ ɢɧɞɟɤɫɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɪɟɲɟɧɢɸ ɫɥɟɜɚ
(L) ɢ ɫɩɪɚɜɚ R . Ɉɬɫɸɞɚ ɜɢɞɧɨ, ɱɬɨ
TL
C2 L , T
ª V º
C1R exp « [ » C2 R .
¬ N ¼
313
Ɍɚɤ ɤɚɤ ɫɩɪɚɜɚ ɩɪɢ [ o f ɬɟɦɩɟɪɚɬɭɪɚ ɪɚɜɧɚ ɧɚɱɚɥɶɧɨɣ, ɬɨ
C2 R T0 . ɍɫɥɨɜɢɟ ɧɟɩɪɟɪɵɜɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ ɩɪɢ [ 0 ɞɚɟɬ
C2 L
C1R T0 .
ɋɨɝɥɚɫɧɨ ɜɬɨɪɨɦɭ ɝɪɚɧɢɱɧɨɦɭ ɭɫɥɨɜɢɸ, ɢɦɟɟɦ
Vº
ª
O « C1R »
N¼
¬
q0 ,
Nq0
.
OV
Ɉɤɨɧɱɚɬɟɥɶɧɨ ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
Ɋɢɫ. 12.4. Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫNq0
ɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɤ, [d 0;
(12.11)
T
T
L
0
ɪɟɫɬɧɨɫɬɢ ɩɥɨɫɤɨɝɨ ɞɜɢɠɭɳɟOV
ɝɨɫɹ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ
Nq
§ V ·
TR T0 0 exp ¨ [ ¸ , [ ! 0 .
OV
© N ¹
Ʉɚɱɟɫɬɜɟɧɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɩɨɥɭɱɟɧɧɨɦɭ ɪɟɲɟɧɢɸ, ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 12.4.
ɨɬɤɭɞɚ C1R
12.3. ɗɥɟɤɬɪɢɱɟɫɤɨɟ ɧɚɝɪɟɜɚɧɢɟ ɩɪɨɜɨɥɨɤɢ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɢɡ ɪɚɡɞɟɥɚ 6.2 ɨ ɧɚɝɪɟɜɚɧɢɢ ɩɪɨɜɨɥɨɤɢ ɪɚɞɢɭɫɚ
R ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɬɨɤɨɦ. ɂɫɬɨɱɧɢɤ ɨɛɴɟɦɧɨɝɨ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
I 2 Re I 2
,
qV
V
Oe
ɝɞɟ O e – ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ ɩɪɨɜɨɥɨɤɢ, 1/(Ɉɦ ɫɦ). ɉɥɨɬɧɨɫɬɶ ɬɨɤɚ ɫɜɹɡɚɧɚ ɫ ɧɚɩɪɹɠɟɧɢɟɦ U (ɪɚɡɧɨɫɬɶɸ ɩɨɬɟɧɰɢɚɥɨɜ) ɧɚ ɤɨɧɰɚɯ ɩɪɨɜɨɞɚ ɞɥɢɧɵ L ɫɨɨɬɧɨɲɟɧɢɟɦ
U
I Oe ,
L
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
U2
qV O e 2 .
L
ɉɨɜɟɪɯɧɨɫɬɶ ɩɪɨɜɨɥɨɤɢ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɪɢ ɬɟɦɩɟɪɚɬɭɪɟ T0 , ɱɬɨ
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɢɞɟɚɥɶɧɨɦɭ ɬɟɩɥɨɨɛɦɟɧɭ ɦɟɠɞɭ ɩɪɨɜɨɥɨɤɨɣ ɢ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ.
314
ɂɡɜɟɫɬɧɨ, ɱɬɨ ɞɥɹ ɛɨɥɶɲɢɧɫɬɜɚ ɦɟɬɚɥɥɨɜ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɜɢɫɢɬ
ɧɟ ɬɨɥɶɤɨ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ɧɨ ɢ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɶ.
Ⱦɥɹ ɧɟɛɨɥɶɲɢɯ ɩɟɪɟɩɚɞɨɜ ɬɟɦɩɟɪɚɬɭɪɵ ɷɬɢ ɡɚɜɢɫɢɦɨɫɬɢ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
O
1 D1T D 2 T 2 ...;
O0
(12.12)
Oe
1 E1T E 2 T 2 ... ,
O e0
T T0
ɝɞɟ T
, ɚ ɜɟɥɢɱɢɧɵ ɫ ɢɧɞɟɤɫɨɦ «0» ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɚɱɚɥɶɧɨɣ
T0
ɬɟɦɩɟɪɚɬɭɪɟ T0 . Ɍɚɤ ɤɚɤ ɬɟɦɩɟɪɚɬɭɪɧɨɟ ɩɨɥɟ ɧɟɨɞɧɨɪɨɞɧɨ, ɬɨ ɜ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ (ɜ ɫɬɚɰɢɨɧɚɪɧɨɦ) ɫɨɫɬɨɹɧɢɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɢ ɦɟɧɹɸɬɫɹ ɨɬ ɬɨɱɤɢ ɤ ɬɨɱɤɟ. ɉɨɷɬɨɦɭ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ (12.12) ɡɚɦɟɧɢɦ ɡɚɜɢɫɢɦɨɫɬɹɦɢ ɨɬ ɤɨɨɪɞɢɧɚɬɵ
O
1 D1[ D 2 [ 2 ... ;
O0
Oe
1 E1[ E 2 [ 2 ...,
O e0
ɝɞɟ [ r R .
Ɂɚɞɚɱɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
1 d §
dT ·
U2
r
O
r
O
r
¸ e 2 ,
r dr ¨©
dr ¹
L
dT
r 0:
0,
dr
r R : T T0
ɢɥɢ ɜ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɟɪɟɦɟɧɧɵɯ T , [ - ɜɢɞ
1 d § O dT ·
¨[
¸
[ d [ © O0 d [ ¹
B
dT
d[
0,
[ 0:
[ 1: T 0,
ɝɞɟ
B
O e0 R 2U 2
O 0 L2T0
315
.
Oe
,
O e0
(12.13)
ȿɫɥɢ Di 0 ,Ei 0 , ɪɟɲɟɧɢɟ ɷɬɨɣ ɡɚɞɚɱɢ ɫɥɟɞɭɟɬ ɢɡ ɪɟɲɟɧɢɹ, ɩɨɥɭɱɟɧɧɨɝɨ ɜ ɪɚɡɞɟɥɟ 6.2, ɢ ɜ ɧɨɜɵɯ ɩɟɪɟɦɟɧɧɵɯ ɢɦɟɟɬ ɜɢɞ
T
B
1 [2 .
4
(12.14)
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ (12.13) ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
T
B
1 [ 2 1 BT1 B 2 T 2 ... ,
4
(12.15)
ɝɞɟ Tn ɧɟ ɡɚɜɢɫɹɬ ɨɬ B ɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɹɦɢ ɬɨɥɶɤɨ ɤɨɨɪɞɢɧɚɬɵ [ .
ɉɨɞɫɬɚɜɢɜ (12.15) ɜ (12.13) ɢ ɩɪɢɪɚɜɧɹɜ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢ ɨɞɢɧɚɤɨɜɵɯ ɫɬɟɩɟɧɹɯ B , ɧɚɣɞɟɦ ɡɚɞɚɱɢ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ T1 , T2 ,... . Ɋɟɲɟɧɢɹ
ɞɨɥɠɧɵ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɞɨɩɨɥɧɢɬɟɥɶɧɵɦ ɭɫɥɨɜɢɹɦ: Tn ɞɨɥɠɧɵ ɛɵɬɶ ɤɨɧɟɱɧɵ ɩɪɢ [ 0 ɢ Tn 0 ɩɪɢ [ 1 .
Ɋɟɡɭɥɶɬɚɬ ɢɦɟɟɬ ɜɢɞ
T
B
1 [2
4
­®¯1 B «¬ª D81 1 [2 16E1 3 [2 »¼º O B2 ...½¾¿ .
(12.16)
ɂɡ ɪɢɫ. 12.5 ɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɛɵɫɬɪɟɟ ɭɦɟɧɶɲɚɟɬɫɹ ɫ ɬɟɦɩɟɪɚɬɭɪɨɣ, ɱɟɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɩɪɨɜɨɥɨɤɢ, ɬɨ
ɬɟɦɩɟɪɚɬɭɪɚ (ɤɪɢɜɚɹ 2) ɩɪɟɜɵɲɚɟɬ ɬɭ, ɤɨɬɨɪɚɹ ɦɨɝɥɚ ɛɵ ɛɵɬɶ ɜ ɩɪɨɬɢɜɨɩɨɥɨɠɧɨɣ ɫɢɬɭɚɰɢɢ (ɤɪɢɜɚɹ 3).
T
0,25
2
0,20 1
3
0,15
0,10
0,05
0,00
0,0
0,2
0,4
0,6
0,8
[
Ɋɢɫ. 12.5. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ
ɜ ɩɪɨɜɨɥɨɤɟ, ɧɚɝɪɟɜɚɟɦɨɣ ɷɥɟɤɬɪɢɱɟɫɤɢɦ ɬɨɤɨɦ (ɮɨɪɦɭɥɚ (12.16)); B 1 ;
1 – Į1 = ȕ1=0,1; 2 – Į1 = 0,5; ȕ1=0,1;
3 – Į1 = 0,1; ȕ1=0,5.
316
Ɋɢɫ. 12.6. Ɍɟɩɥɨɜɵɞɟɥɹɸɳɢɣ
ɷɥɟɦɟɧɬ
1 2 . 4 . ə ɞ ɟ ɪ ɧ ɵ ɣ ɬ ɟ ɩ ɥ ɨ ɜ ɵ ɞ ɟ ɥ ɹ ɸ ɳ ɢ ɣ ɷ ɥ ɟ ɦ ɟ ɧ ɬ 35
Ɋɚɫɫɦɨɬɪɢɦ ɹɞɟɪɧɵɣ ɬɟɩɥɨɜɵɞɟɥɹɸɳɢɣ ɷɥɟɦɟɧɬ ɫɮɟɪɢɱɟɫɤɨɣ
ɮɨɪɦɵ (ɪɢɫ. 12.6). ɗɬɨɬ ɷɥɟɦɟɧɬ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɲɚɪɨɨɛɪɚɡɧɵɣ
ɫɩɥɨɲɧɨɣ ɤɭɫɨɤ ɞɟɥɹɳɟɝɨɫɹ ɦɚɬɟɪɢɚɥɚ ɪɚɞɢɭɫɨɦ R f , ɨɤɪɭɠɟɧɧɵɣ ɫɮɟɪɢɱɟɫɤɨɣ ɨɛɨɥɨɱɤɨɣ ɢɡ ɚɥɸɦɢɧɢɹ ɪɚɞɢɭɫɚ RC . ȼɧɭɬɪɢ ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ ɨɛɪɚɡɭɸɬɫɹ ɨɫɤɨɥɤɢ ɞɟɥɟɧɢɹ, ɨɛɥɚɞɚɸɳɢɟ ɨɱɟɧɶ ɛɨɥɶɲɨɣ ɤɢɧɟɬɢɱɟɫɤɨɣ ɷɧɟɪɝɢɟɣ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɬɨɥɤɧɨɜɟɧɢɹ ɦɟɠɞɭ ɷɬɢɦɢ
ɨɫɤɨɥɤɚɦɢ ɢ ɚɬɨɦɚɦɢ ɞɟɥɹɳɟɝɨɫɹ ɦɚɬɟɪɢɚɥɚ ɜ ɪɟɚɤɬɨɪɟ ɜɵɞɟɥɹɟɬɫɹ
ɨɱɟɧɶ ɛɨɥɶɲɨɟ ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɜɨɣ ɷɧɟɪɝɢɢ. ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɢɫɬɨɱɧɢɤɚ
ɬɟɩɥɚ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɨɛɴɟɦɭ ɫɮɟɪɵ ɧɟɪɚɜɧɨɦɟɪɧɨ. Ⱦɥɹ ɩɪɨɫɬɨɬɵ ɩɪɢɦɟɦ, ɱɬɨ qV ɹɜɥɹɟɬɫɹ ɪɚɞɢɚɥɶɧɨɣ ɮɭɧɤɰɢɟɣ ɤɨɨɪɞɢɧɚɬɵ
2
ª
§ r · º
(12.17)
qV qV 0 «1 b ¨
¸ »,
¨ Rf ¸ »
«
©
¹ »¼
«¬
ɝɞɟ qV 0 - ɨɛɴɟɦɧɚɹ ɫɤɨɪɨɫɬɶ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɜ ɰɟɧɬɪɟ ɫɮɟɪɵ, b - ɛɟɡɪɚɡɦɟɪɧɵɣ ɩɚɪɚɦɟɬɪ, 0 d b 1.
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ ɫɬɚɰɢɨɧɚɪɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɫɢɫɬɟɦɟ ɩɪɢ ɭɫɥɨɜɢɢ ɢɞɟɚɥɶɧɨɝɨ ɬɟɩɥɨɜɨɝɨ ɤɨɧɬɚɤɬɚ ɦɟɠɞɭ ɷɥɟɦɟɧɬɨɦ ɢ
ɨɛɨɥɨɱɤɨɣ ɢ ɩɪɢ ɡɚɞɚɧɧɨɣ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɨɥɨɱɤɢ ɬɟɦɩɟɪɚɬɭɪɟ T0 .
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ
1 d 2
r qrf qV r 0 , 0 r d R f ;
2 dr
r
1 d 2
r qrC
2 dr
r
0 , R f d r d RC ;
qrf 0 , r
qrf
qrC , T f
T
ɝɞɟ qrf
O f
dT f
, qrC
OC
dr
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɞɚɟɬ
qrf
35
T0 , r
0;
TC , r
Rf ;
(12.18)
Rc .
dTC
.
dr
ª r b r 3 º C1 f
qV 0 « 2 » 2 ,
«¬ 3 R f 5 »¼ r
Ȼɟɪɞ Ɋ., ɋɬɶɸɚɪɬ ȼ., Ʌɚɣɬɮɭɬ ȿ. əɜɥɟɧɢɹ ɩɟɪɟɧɨɫɚ. Ɇ.: ɏɢɦɢɹ, 1974. 688 ɫ.
317
C1C
.
r2
ɂɫɩɨɥɶɡɭɹ ɭɫɥɨɜɢɹ ɞɥɹ ɩɨɬɨɤɨɜ, ɧɚɣɞɟɦ
ª r b r3 º
dT f
qrf O f
qV 0 « 2 » ,
dr
«¬ 3 R f 5 »¼
qrC
qrC
OC
dTC
dr
§1 b· 1
qV 0 R3f ¨ ¸ 2 .
©3 5¹r
ɗɬɢ ɜɵɪɚɠɟɧɢɹ ɨɩɢɫɵɜɚɸɬ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɩɥɨɜɵɯ ɩɨɬɨɤɨɜ ɜɧɭɬɪɢ
ɫɮɟɪɢɱɟɫɤɨɝɨ ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ ɢ ɜ ɨɤɪɭɠɚɸɳɟɣ ɷɬɨɬ ɷɥɟɦɟɧɬ ɨɛɨɥɨɱɤɟ. ɉɪɢ ɩɨɫɬɨɹɧɧɵɯ ɡɧɚɱɟɧɢɹɯ O f ɢ OC ɩɨɫɥɟɞɧɢɟ ɭɪɚɜɧɟɧɢɹ ɥɟɝɤɨ ɢɧɬɟɝɪɢɪɭɸɬɫɹ.
qV 0 ª r 2
b r4 º
«
» C2 f ,
Tf O f « 6 R 2f 20 »
¬
¼
qV 0 3 § 1 b · 1
TC
R
C2C .
O C f ¨© 3 5 ¸¹ r
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ ɧɚɯɨɞɢɦ ɢɡ ɭɫɥɨɜɢɣ ɞɥɹ ɬɟɦɩɟɪɚɬɭɪɵ.
Ɉɤɨɧɱɚɬɟɥɶɧɨɟ ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
2 ·
4 ·º q R2
§
qV 0 R 2f ª§
3
3 § R ·
r
b
r
«¨ 1 ¸ ¨1 ¸ » V 0 f §¨1 b ·¸ ¨1 f ¸ ,
T f T0
6O f «¨ R 2f ¸ 10 ¨ R 4f ¸ »
3OC © 5 ¹ © RC ¹
¹
©
¹¼
©
TC T0
qV 0 R 2f § 3 · § R f R f ·
1 b ¨
¸.
RC ¹
3OC ¨© 5 ¸¹ © r
ɉɪɢ r 0 ɧɚɣɞɟɦ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɰɟɧɬɪɟ
ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ:
Tmax T0
qV 0 R 2f § 3b · qV 0 R 2f
1
6O f ¨© 10 ¸¹
3OC
§ 3 ·§ R f ·
¨1 5 b ¸ ¨1 R ¸ .
©
¹©
C ¹
ɗɬɭ ɮɨɪɦɭɥɭ ɦɨɠɟɦ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ
Tmax
Tmax T0
qV 0 R 2f 6O f
O f § 3 ·§ R f ·
§ 3b ·
1
2
1 b ¨1 ¸.
¨ 10 ¸
OC ¨© 5 ¸¹ © RC ¹
©
¹
318
(12.19)
ȼɟɥɢɱɢɧɚ ɦɚɤɫɢɦɚɥɶɧɨɣ ɬɟɦɩɟɪɚɬɭɪɵ ɡɚɜɢɫɢɬ ɨɬ ɬɪɟɯ ɛɟɡɪɚɡɦɟɪɧɵɯ ɩɚɪɚɦɟɬɪɨɜ: ɨɬɧɨɲɟɧɢɹ ɪɚɞɢɭɫɨɜ ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ ɢ
ɟɝɨ ɨɛɨɥɨɱɤɢ; ɨɬɧɨɲɟɧɢɹ ɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɩɚɪɚɦɟɬɪɚ b , ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɝɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɩɥɨɬɧɨɫɬɢ ɨɛɴɟɦɧɨɝɨ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ.
12.5. Ɍɟɩɥɨɨɛɦɟɧ ɩɪɢ ɧɚɥɢɱɢɢ
ɜ ɹ ɡ ɤ ɨ ɝ ɨ ɢ ɫ ɬ ɨ ɱ ɧ ɢ ɤ ɚ ɬ ɟ ɩ ɥ ɚ 36
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɱɟɧɢɟ ɧɟɫɠɢɦɚɟɦɨɣ ɠɢɞɤɨɫɬɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɦɟɠɞɭ
ɞɜɭɦɹ ɤɨɚɤɫɢɚɥɶɧɵɦɢ ɰɢɥɢɧɞɪɚɦɢ (ɪɢɫ. 12.7). ɉɪɢ ɫɨɨɛɳɟɧɢɢ ɜɧɟɲɧɟɦɭ ɰɢɥɢɧɞɪɭ ɜɪɚɳɚɬɟɥɶɧɨɝɨ ɞɜɢɠɟɧɢɹ ɜ ɠɢɞɤɨɫɬɢ ɜɨɡɧɢɤɚɸɬ ɫɢɥɵ ɬɪɟɧɢɹ, ɤɨɬɨɪɵɟ ɞɟɣɫɬɜɭɸɬ ɦɟɠɞɭ ɤɚɠɞɨɣ ɩɚɪɨɣ ɫɦɟɠɧɵɯ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ
ɫɥɨɟɜ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɞɟɣɫɬɜɢɹ ɷɬɢɯ ɫɢɥ ɦɟɯɚɧɢɱɟɫɤɚɹ ɷɧɟɪɝɢɹ ɠɢɞɤɨɫɬɢ
ɧɟɩɪɟɪɵɜɧɨ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɬɟɩɥɨɜɭɸ, ɢ ɠɢɞɤɨɫɬɶ ɧɚɝɪɟɜɚɟɬɫɹ. ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɨɛɴɟɦɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ, ɜɨɡɧɢɤɚɸɳɟɝɨ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ
ɦɟɠɞɭ ɰɢɥɢɧɞɪɚɦɢ ɜɫɥɟɞɫɬɜɢɟ «ɜɹɡɤɨɣ ɞɢɫɫɢɩɚɰɢɢ», ɡɚɜɢɫɢɬ ɨɬ ɥɨɤɚɥɶɧɨɝɨ ɝɪɚɞɢɟɧɬɚ ɫɤɨɪɨɫɬɟɣ: ɱɟɦ ɛɵɫɬɪɟɟ ɞɜɢɠɭɬɫɹ ɞɪɭɝ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɪɭɝɚ ɫɦɟɠɧɵɟ ɫɥɨɢ ɠɢɞɤɨɫɬɢ, ɬɟɦ ɢɧɬɟɧɫɢɜɧɟɟ «ɜɹɡɤɚɹ ɞɢɫɫɢɩɚɰɢɹ» ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɬɟɦ ɛɵɫɬɪɟɟ ɧɚɝɪɟɜɚɟɬɫɹ ɠɢɞɤɨɫɬɶ.
ɚ
ɛ
Ɋɢɫ. 12.7. Ʉ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱɢ ɨ ɬɟɩɥɨɨɛɦɟɧɟ ɦɟɠɞɭ ɞɜɭɦɹ ɤɨɚɤɫɢɚɥɶɧɵɦɢ
ɰɢɥɢɧɞɪɚɦɢ; 1 – ɧɟɩɨɞɜɢɠɧɚɹ ɩɨɜɟɪɯɧɨɫɬɶ; 2 – ɩɨɜɟɪɯɧɨɫɬɶ ɜɧɟɲɧɟɝɨ
ɰɢɥɢɧɞɪɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V R:
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɧɚ ɜɧɭɬɪɟɧɧɟɣ ɢ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɹɯ ɰɢɥɢɧɞɪɚ ɩɨɞɞɟɪɠɢɜɚɸɬɫɹ ɬɟɦɩɟɪɚɬɭɪɵ T0 ɢ Tb . ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɟɦɩɟɪɚɬɭɪɚ
ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɨɥɶɤɨ ɪɚɞɢɚɥɶɧɨɣ ɤɨɨɪɞɢɧɚɬɨɣ.
ȿɫɥɢ ɲɢɪɢɧɚ ɤɨɥɶɰɟɜɨɝɨ ɡɚɡɨɪɚ b ɦɚɥɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɪɚɞɢɭɫɨɦ
R , ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨ, ɚɧɚɥɢɡɢɪɭɹ ɭɩɪɨɳɟɧɧɭɸ ɫɢɬɭɚɰɢɸ, ɬ.ɟ., ɩɪɟɧɟɛɪɟɝɚɹ ɷɮɮɟɤɬɨɦ ɤɪɢɜɢɡɧɵ ɢ ɢɫɩɨɥɶɡɭɹ
36
Ȼɟɪɞ Ɋ., ɋɬɶɸɚɪɬ ȼ., Ʌɚɣɬɮɭɬ ȿ. əɜɥɟɧɢɹ ɩɟɪɟɧɨɫɚ. Ɇ.: ɏɢɦɢɹ, 1974. 688 ɫ
319
ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ. ɇɚ ɪɢɫ. 12.7 ɜɜɟɞɟɧɵ ɨɛɨɡɧɚɱɟɧɢɹ: 1 – ɷɬɨ ɧɟɩɨɞɜɢɠɧɚɹ ɩɨɜɟɪɯɧɨɫɬɶ, ɚ 2 – ɩɨɜɟɪɯɧɨɫɬɶ ɜɧɟɲɧɟɝɨ ɰɢɥɢɧɞɪɚ, ɞɜɢɠɭɳɟɝɨɫɹ ɫɨ ɫɤɨɪɨɫɬɶɸ V R: , ɝɞɟ : – ɭɝɥɨɜɚɹ ɫɤɨɪɨɫɬɶ.
ȼ ɫɥɭɱɚɟ ɬɚɤ ɧɚɡɵɜɚɟɦɨɣ ɇɶɸɬɨɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ ɢɧɬɟɧɫɢɜɧɨɫɬɶ
ɜɹɡɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
W xz
qV
2
§ dv ·
PW xz ¨ z ¸ ,
© dx ¹
dv z
dx
(12.20)
ɝɞɟ W xz – ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ, vz – ɤɨɦɩɨɧɟɧɬɚ ɜɟɤɬɨɪɚ ɫɤɨɪɨɫɬɢ,
P – ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ. ȿɞɢɧɢɰɟɣ ɟɝɨ ɢɡɦɟɪɟɧɢɹ ɫɥɭɠɢɬ 1 ɉɭɚɡ = ɝ/(ɫɦ ɫ). ɋɨɨɬɧɨɲɟɧɢɟ (12.20) ɟɫɬɶ ɫɥɟɞɫɬɜɢɟ ɡɚɤɨɧɚ
ɜɹɡɤɨɫɬɢ ɇɶɸɬɨɧɚ
dv
(12.21)
W xz P z .
dx
ɗɬɨɦɭ ɡɚɤɨɧɭ ɩɨɞɱɢɧɹɸɬɫɹ, ɧɚɩɪɢɦɟɪ, ɩɚɫɬɵ, ɫɭɫɩɟɧɡɢɢ, ɜɵɫɨɤɨɦɨɥɟɤɭɥɹɪɧɵɟ ɫɨɟɞɢɧɟɧɢɹ.
Ⱦɥɹ ɩɨɫɬɨɹɧɧɨɣ ɫɤɨɪɨɫɬɢ ɜɧɟɲɧɟɝɨ ɰɢɥɢɧɞɪɚ V const ɢ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɬɟɱɟɧɢɹ ɦɨɠɧɨ ɩɪɢɧɹɬɶ
§x·
v z ¨ ¸V ,
©b¹
ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɞɥɹ ɫɤɨɪɨɫɬɢ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɢɦɟɟɦ
2
§V ·
qV P ¨ ¸ .
©b¹
ɉɟɪɜɵɣ ɢɧɬɟɝɪɚɥ ɨɞɧɨɦɟɪɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɜ ɞɟɤɚɪɬɨɜɨɣ ɫɢɫɬɟɦɟ ɤɨɨɪɞɢɧɚɬ ɩɪɢ ɭɫɥɨɜɢɢ ɩɨɫɬɨɹɧɫɬɜɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O ɢ ɜɹɡɤɨɫɬɢ P ɟɫɬɶ
dT
O
dx
2
§V ·
P ¨ ¸ x C1 .
©b¹
ɉɨɫɥɟɞɭɸɳɟɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ (ɫɦ. ɪɚɡɞɟɥ 6.) ɞɚɟɬ
2
T
1 P § V · 2 C1
x x C2 .
2 O ¨© b ¸¹
O
ɉɨɥɶɡɭɹɫɶ ɭɫɥɨɜɢɹɦɢ
x 0:
x b:
T
T
ɧɚɣɞɟɦ
320
T0 ,
Tb ,
T
T T0
Tb T0
x 1
x ª § x ·º
Br «1 ¨ ¸ » ,
b 2 b ¬ © b ¹¼
(12.22)
PV 2
ɝɞɟ Br
- ɱɢɫɥɨ Ȼɪɢɧɝɦɚɧɚ, ɧɚɡɜɚɧɧɨɟ ɬɚɤ ɩɨ ɢɦɟɧɢ ɭɱɟɧɨɝɨ,
O Tb T0 ɜɩɟɪɜɵɟ ɪɟɲɢɜɲɟɝɨ ɡɚɞɚɱɭ ɨ ɬɟɱɟɧɢɢ ɠɢɞɤɨɫɬɢ ɜ ɤɪɭɝɥɨɣ ɬɪɭɛɟ ɫ ɭɱɟɬɨɦ ɷɮɮɟɤɬɨɜ ɬɟɩɥɨɜɵɞɟɥɟɧɢɹ ɜɫɥɟɞɫɬɜɢɟ ɜɹɡɤɨɣ ɞɢɫɫɢɩɚɰɢɢ. ɗɬɨ ɱɢɫɥɨ ɹɜɥɹɟɬɫɹ ɦɟɪɨɣ ɨɬɧɨɫɢɬɟɥɶɧɨɝɨ ɜɤɥɚɞɚ ɞɜɭɯ ɩɨɬɨɤɨɜ: ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɨɛɭɫɥɨɜɥɟɧɧɨɝɨ ɜɹɡɤɨɣ ɞɢɫɫɢɩɚɰɢɟɣ, ɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ɜɵɡɜɚɧɧɨɝɨ ɪɚɡɧɨɫɬɶɸ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɫɬɟɧɤɚɯ ɤɚɧɚɥɚ. ȼ ɫɥɭɱɚɟ Br ! 2 ɩɪɨɮɢɥɶ ɬɟɦɩɟɪɚɬɭɪɵ ɢɦɟɟɬ ɦɚɤɫɢɦɭɦ.
Ʉɚɱɟɫɬɜɟɧɧɨɟ ɩɨɜɟɞɟɧɢɟ ɬɟɦɩɟɪɚɬɭɪɵ ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 12.8.
ȼɨ ɦɧɨɝɢɯ ɡɚɞɚɱɚɯ ɷɮɮɟɤɬ ɜɹɡɤɨɣ
T
ɞɢɫɫɢɩɚɰɢɢ ɧɟ ɢɝɪɚɟɬ ɨɫɨɛɨɣ ɪɨɥɢ. ɇɨ
1,0
1
2 3
ɫɭɳɟɫɬɜɭɟɬ ɪɹɞ, ɝɥɚɜɧɵɦ ɨɛɪɚɡɨɦ, ɢɧ4
0,8
5
ɠɟɧɟɪɧɵɯ ɩɪɨɛɥɟɦ, ɤɨɝɞɚ ɷɬɨɬ ɷɮɮɟɤɬ
0,6
ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ. ɇɚɩɪɢɦɟɪ, ɫ
0,4
ɜɹɡɤɢɦ ɬɟɩɥɨɜɵɞɟɥɟɧɢɟɦ ɩɪɢɯɨɞɢɬɫɹ
0,2
ɫɬɚɥɤɢɜɚɬɶɫɹ ɩɪɢ ɬɟɱɟɧɢɢ ɫɦɚɡɨɱɧɵɯ
0,0
ɦɚɬɟɪɢɚɥɨɜ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɦɟɠɞɭ ɛɵ0,0
0,2
0,4
0,6
0,8 x/b
ɫɬɪɨ ɞɜɢɠɭɳɢɦɢɫɹ ɞɟɬɚɥɹɦɢ; ɩɪɢ ɬɟɊɢɫ. 12.8. Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɬɟɦɩɟ- ɱɟɧɢɢ ɩɥɚɫɬɢɱɟɫɤɢɯ ɦɚɫɫ ɱɟɪɟɡ ɦɚɬɪɢɪɚɬɭɪɵ ɦɟɠɞɭ ɤɨɚɤɫɢɚɥɶɧɵɦɢ
ɰɵ ɩɪɢ ɜɵɫɨɤɨɫɤɨɪɨɫɬɧɨɣ ɷɤɫɬɪɭɡɢɢ;
ɰɢɥɢɧɞɪɚɦɢ; Br 1 – 4,0; 2 –3,0; ɩɪɢ ɞɜɢɠɟɧɢɢ ɜɨɡɞɭɯɚ ɜ ɩɨɝɪɚɧɢɱɧɨɦ
3 – 2,0; 4 – 1,0; 5 – 0,5.
ɫɥɨɟ ɜɛɥɢɡɢ ɩɨɜɟɪɯɧɨɫɬɢ ɫɩɭɬɧɢɤɚ
Ɂɟɦɥɢ ɢɥɢ ɪɚɤɟɬɵ ɢ ɞɪ. ɉɟɪɜɵɟ ɞɜɚ ɬɢɩɚ ɬɟɤɭɱɢɯ ɫɪɟɞ ɨɬɧɨɫɹɬɫɹ ɤ ɇɶɸɬɨɧɨɜɫɤɢɦ ɠɢɞɤɨɫɬɹɦ, ɩɨɞɱɢɧɹɸɳɢɦɫɹ ɡɚɤɨɧɭ (12.18). Ɋɟɨɥɨɝɢɱɟɫɤɨɟ
ɩɨɜɟɞɟɧɢɟ ɛɨɥɶɲɢɧɫɬɜɚ ɠɢɞɤɨɫɬɟɣ ɩɨɞɱɢɧɹɟɬɫɹ ɛɨɥɟɟ ɫɥɨɠɧɵɦ ɡɚɤɨɧɨɦɟɪɧɨɫɬɹɦ.
12.6. ɇɚɝɪɟɜ ɬɟɥ ɢɡɥɭɱɟɧɢɟɦ ɈɄȽ
Ɂɚɞɚɱɢ ɨ ɧɚɝɪɟɜɟ ɬɟɥ ɢɡɥɭɱɟɧɢɟɦ ɈɄȽ (ɨɩɬɢɱɟɫɤɨɝɨ ɤɜɚɧɬɨɜɨɝɨ ɝɟɧɟɪɚɬɨɪɚ) ɜɫɬɪɟɱɚɸɬɫɹ ɞɨɜɨɥɶɧɨ ɱɚɫɬɨ.37 ɋɜɹɡɚɧɵ ɨɧɢ ɫ ɬɟɯɧɨɥɨɝɢɱɟ37
Ɋɵɤɚɥɢɧ ɇ.ɇ., ɍɝɥɨɜ Ⱥ.Ⱥ. Ɉ ɧɚɝɪɟɜɟ ɪɚɡɧɨɪɨɞɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɩɪɢ ɫɜɚɪɤɟ ɜɫɬɵɤ
ɩɨɜɟɪɯɧɨɫɬɧɵɦ ɢɫɬɨɱɧɢɤɨɦ ɬɟɩɥɚ // ɎɏɈɆ, 1970, ʋ 5. ɋ. 23–28; ɍɝɥɨɜ Ⱥ.Ⱥ.,
ɂɫɚɟɜɚ Ɉ.ɂ. Ɉ ɪɚɫɱɟɬɟ ɫɤɨɪɨɫɬɢ ɧɚɝɪɟɜɚ ɦɟɬɚɥɥɨɜ ɩɪɢ ɜɨɡɞɟɣɫɬɜɢɢ ɢɡɥɭɱɟɧɢɹ ɈɄȽ
// ɎɏɈɆ, 1976, ʋ 2. ɋ. 23–28; ɍɝɥɨɜ Ⱥ.Ⱥ., Ʉɨɧɫɬɚɧɬɢɧɨɜ ɋ.Ƚ. ɑɢɫɥɟɧɧɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɩɥɨɜɵɯ ɩɪɨɰɟɫɫɨɜ ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɤɨɧɰɟɧɬɪɢɪɨɜɚɧɧɵɦɢ ɩɨɬɨɤɚɦɢ ɷɧɟɪɝɢɢ
ɩɨɤɪɵɬɢɣ ɢ ɫɨɫɬɵɤɨɜɚɧɧɵɯ ɦɚɬɟɪɢɚɥɨɜ // ɎɏɈɆ, 1995, ʋ 3. ɋ. 34–39 ɢ ɞɪ.
321
ɫɤɢɦɢ ɩɪɢɥɨɠɟɧɢɹɦɢ: ɩɨɜɟɪɯɧɨɫɬɧɨɣ ɡɚɤɚɥɤɨɣ ɥɨɤɚɥɶɧɵɯ ɨɛɥɚɫɬɟɣ,
ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɨɣ, ɢɫɫɥɟɞɨɜɚɧɢɟɦ ɬɟɪɦɨɷɦɢɫɫɢɨɧɧɵɯ
ɫɩɨɫɨɛɧɨɫɬɟɣ ɦɚɬɟɪɢɚɥɨɜ, ɩɪɨɰɟɫɫɚɦɢ ɪɟɡɤɢ ɢ ɫɜɚɪɤɢ. ȼ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɛɥɟɦɚɯ ɡɧɚɱɢɬɟɥɶɧɵɣ ɢɧɬɟɪɟɫ ɩɪɟɞɫɬɚɜɥɹɟɬ ɨɰɟɧɤɚ ɫɤɨɪɨɫɬɟɣ
ɧɚɝɪɟɜɚ ɢ ɨɯɥɚɠɞɟɧɢɹ ɦɚɬɟɪɢɚɥɨɜ, ɬɟɦɩɟɪɚɬɭɪɧɵɯ ɝɪɚɞɢɟɧɬɨɜ ɜ ɧɟɫɬɚɰɢɨɧɚɪɧɨɦ ɢ ɭɫɬɚɧɨɜɢɜɲɟɦɫɹ ɫɨɫɬɨɹɧɢɹɯ. ȼ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɬɚɤɢɟ
ɡɚɞɚɱɢ ɨɤɚɡɵɜɚɸɬɫɹ ɫɭɳɟɫɬɜɟɧɧɨ ɧɟɥɢɧɟɣɧɵɦɢ ɢ ɬɪɟɛɭɸɬ ɩɪɢɜɥɟɱɟɧɢɹ
ɱɢɫɥɟɧɧɵɯ ɦɟɬɨɞɨɜ ɢɫɫɥɟɞɨɜɚɧɢɹ. ɇɨ ɪɹɞ ɢɧɬɟɪɟɫɧɵɯ ɷɮɮɟɤɬɨɜ ɦɨɠɟɬ
ɛɵɬɶ ɨɩɢɫɚɧ ɫ ɩɨɦɨɳɶɸ ɞɨɫɬɚɬɨɱɧɨ ɩɪɨɫɬɵɯ ɦɨɞɟɥɟɣ.
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɝɪɟɜɟ ɬɜɟɪɞɨɝɨ ɨɛɪɚɡɰɚ ɢɦɟɸɳɟɝɨ ɮɨɪɦɭ
ɰɢɥɢɧɞɪɚ, ɥɚɡɟɪɧɵɦ ɢɡɥɭɱɟɧɢɟɦ (ɪɢɫ. 12.9, ɚ). Ȼɨɤɨɜɵɟ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɡɰɚ ɬɟɩɥɨɢɡɨɥɢɪɨɜɚɧɵ.
q0
r
0
h
z
ɚ
R1
ɛ
Ɋɢɫ. 12.9. Ʉ ɩɨɫɬɚɧɨɜɤɟ ɡɚɞɚɱ ɨ ɧɚɝɪɟɜɟ ɬɟɥ ɥɚɡɟɪɧɵɦ ɢɡɥɭɱɟɧɢɟɦ
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɦɚɤɫɢɦɚɥɶɧɚɹ ɩɥɨɬɧɨɫɬɶ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ ɧɟ
ɩɪɟɜɨɫɯɨɞɢɬ ɧɟɤɨɬɨɪɨɟ ɤɪɢɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ, ɧɟɨɛɯɨɞɢɦɨɟ ɞɥɹ ɧɚɱɚɥɚ
ɩɥɚɜɥɟɧɢɹ ɦɚɬɟɪɢɚɥɚ. Ɉɞɧɚɤɨ ɨɩɬɢɱɟɫɤɢɟ ɢ ɬɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ
ɦɨɝɭɬ ɡɚɜɢɫɟɬɶ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ. Ȼɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ, ɩɨɫɬɭɩɚɸɳɢɣ ɨɬ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɪɚɫɩɪɟɞɟɥɟɧ ɩɨ ɡɚɤɨɧɭ Ƚɚɭɫɫɚ
(ɪɢɫ. 12.9 ,ɛ)
(12.23)
q r , z Aq0V e x p ª r 2k V z º ,
¬
¼
ɝɞɟ A – ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɦɚɬɟɪɢɚɥɚ, V – ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ ɜ ɡɚɤɨɧɟ Ȼɭɝɟɪɚ, k – ɤɨɷɮɮɢɰɢɟɧɬ ɫɨɫɪɟɞɨɬɨɱɟɧɧɨɫɬɢ ɥɚɡɟɪɧɨɝɨ
ɢɡɥɭɱɟɧɢɹ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɬɚɤɨɣ ɡɚɞɚɱɢ ɢɦɟɟɬ ɜɢɞ
wT 1 w §
wT · w § wT ·
2
c pU
¨ OT r
¸ ¨ OT
¸ Aq0Vexp r k NO z , (12.24)
wt r wr ©
wr ¹ wz ©
wz ¹
wT
wT
z 0 : OT
D T Te ; z G : OT
D T Te ,
wz
wz
322
r
0:
wT
wr
t
0; r
Rc :
0: T
T0 .
wT
wr
0,
ɑɚɫɬɧɵɣ ɜɚɪɢɚɧɬ ɡɚɞɚɱɢ ( R1 1 k !! Rc , G o f ) ɧɚɦ ɭɠɟ ɢɡɜɟɫɬɟɧ
(ɪɚɡɞɟɥ 9.) ɢ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɝɪɟɜɚɧɢɸ ɢɡɥɭɱɟɧɢɟɦ ɬɨɥɫɬɨɝɨ ɨɛɪɚɡɰɚ ɫ
ɩɹɬɧɨɦ ɧɚɝɪɟɜɚ, ɩɪɟɜɵɲɚɸɳɢɦ ɟɝɨ ɩɨɩɟɪɱɟɧɧɵɟ ɪɚɡɦɟɪɵ.
Ɋɚɫɫɦɨɬɪɢɦ ɞɪɭɝɢɟ ɱɚɫɬɧɵɟ ɜɚɪɢɚɧɬɵ.
ɉɭɫɬɶ ɨɛɪɚɡɟɰ ɢɦɟɟɬ ɦɚɥɭɸ ɬɨɥɳɢɧɭ, ɬɚɤɭɸ, ɱɬɨ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ
ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɧɟɦ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ, G O T t* c p U . Ɍɨɝɞɚ ɢɧɬɟɝɪɢɪɭɹ (12.24) ɩɨ ɬɨɥɳɢɧɟ ɩɥɟɧɤɢ, ɩɪɢɞɟɦ ɤ ɭɪɚɜɧɟɧɢɸ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
c p UG
wT
wt
G w§
wT ·
¨ OT r
¸
wr ¹
r wr ©
(12.25)
A 1 exp GV q0 exp - r 2 k 2DT Te .
ȿɫɥɢ ɤ ɬɨɦɭ ɠɟ ɩɹɬɧɨ ɧɚɝɪɟɜɚ ɦɧɨɝɨ ɛɨɥɶɲɟ ɪɚɞɢɭɫɚ ɨɛɪɚɡɰɚ, ɡɚɞɚɱɚ ɨ ɧɚɝɪɟɜɟ ɢɡɥɭɱɟɧɢɟɦ ɈɄȽ ɩɪɢɦɟɬ ɫɨɜɫɟɦ ɩɪɨɫɬɨɣ ɜɢɞ
c pU G
dT
dt
A 1 exp GV q0 2D T Te .
(12.26)
ɉɨɬɨɤ, ɩɨɝɥɨɳɟɧɧɵɣ ɩɥɟɧɤɨɣ, ɟɫɬɶ
Aq0 ª¬1 e x p VG º¼ ,
qa
ɚ ɩɨɬɨɤ, ɩɪɨɩɭɳɟɧɧɵɣ ɩɥɟɧɤɨɣ –
)
Aq0e x p VG .
ȼ ɡɚɞɚɱɟ ɜɨɡɦɨɠɧɵ ɜɚɪɢɚɧɬɵ.
1. ȼ ɭɫɥɨɜɢɹɯ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɜɨɡɦɨɠɧɨ ɭɫɬɚɧɨɜɥɟɧɢɟ ɫɬɚɰɢɨɧɚɪɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɤɨɝɞɚ
A 1 ex p GV q0
2D T Te .
(12.27)
2. ɉɪɟɧɟɛɪɟɝɚɹ ɩɨɬɟɪɹɦɢ ɬɟɩɥɚ ɜ ɭɪɚɜɧɟɧɢɢ ɜ (12.26), ɩɨɥɭɱɢɦ, ɱɬɨ
ɬɟɦɩɟɪɚɬɭɪɚ ɬɨɧɤɨɝɨ ɨɛɪɚɡɰɚ (ɩɥɟɧɤɢ) ɪɚɫɬɟɬ ɫɨ ɜɪɟɦɟɧɟɦ ɥɢɧɟɣɧɨ
(ɪɢɫ. 12.10)
T
T0 Bt ,
ɝɞɟ
323
(12.28)
B
A
q0 ª
1 e VG º .
¼
c pUG ¬
3. ɍɱɟɬ ɬɟɩɥɨɨɛɦɟɧɚ ɫ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɨɣ ɩɪɢɜɨɞɢɬ ɤ ɢɧɨɦɭ ɡɚɤɨɧɭ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ
§ 2D t · º
qa ª
T T0
(12.29)
1
e
x
p
«
¨ cU G ¸ » .
2D ¬
©
¹¼
4. ɉɭɫɬɶ ɜ (12.11) D 0 , ɧɨ ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɡɚɜɢɫɢɬ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ. Ⱦɥɹ ɛɨɥɶɲɢɧɫɬɜɚ ɦɟɬɚɥɥɨɜ ɫɩɪɚɜɟɞɥɢɜɚ ɥɢɧɟɣɧɚɹ ɡɚɜɢɫɢɦɨɫɬɶ
A A0 A1T .
ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɧɚɦ ɧɭɠɧɨ ɧɚɣɬɢ ɪɟɲɟɧɢɟ
ɡɚɞɚɱɢ
dT
B0 1 J T ,
dt
t 0 : T T0 ,
Ɋɢɫ. 12.10. Ɂɚɜɢɫɢɦɨɫɬɶ
ɝɞɟ B0
ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ; 1 – ɫɨɨɬɜɟɬɫɬɜɭɟɬ
ɮɨɪɦɭɥɟ (12.30);
2 – ɮɨɪɦɭɥɟ (12.28)
A0
q0 ª
1 eV G º , J
¼
c pUG ¬
Ɍɨɱɧɨɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨ ɧɚɝɪɟɜɟ ɬɨɧɤɨɣ
ɩɥɟɧɤɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɢɦɟɟɬ ɜɢɞ (ɪɢɫ. 12.10)
1
T0e x p J B0t ª¬e x p J B0t 1º¼ ,
J
T
A1
.
A0
(12.30)
ɱɬɨ ɫɭɳɟɫɬɜɟɧɧɨ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɥɢɧɟɣɧɨɝɨ ɡɚɤɨɧɚ (12.28).
ɍɫɬɪɟɦɥɹɹ J ɤ ɧɭɥɸ, ɩɨɥɭɱɢɦ ɥɢɧɟɣɧɵɣ ɡɚɤɨɧ ɪɨɫɬɚ ɬɟɦɩɟɪɚɬɭɪɵ.
Ɉɬɥɢɱɚɟɬɫɹ ɢ ɜɟɥɢɱɢɧɚ ɩɨɬɨɤɚ, ɩɨɝɥɨɳɟɧɧɨɝɨ ɩɥɟɧɤɨɣ
c pU G B0 1 J T0 e xp J B0t .
Qa
5. ɉɭɫɬɶ ɦɚɬɟɪɢɚɥ ɩɨɝɥɨɳɚɟɬ ɢɡɥɭɱɟɧɢɟ ɈȽɄ ɬɨɥɶɤɨ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɢ D 0 . Ɍɨɝɞɚ ɭɪɚɜɧɟɧɢɟ (12.25) ɩɪɢɦɟɬ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɜɢɞ
c pU
wT
wt
1 w §
wT · w § wT ·
¨ OT r
¸ ¨ OT
¸,
r wr ©
wr ¹ wz ©
wz ¹
ɧɨ ɢɡɦɟɧɢɬɫɹ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ z
OT
wT
wz
Aq0e xp ª r 2k º
¬
¼
324
0:
Ɉɫɬɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɨɫɬɚɧɭɬɫɹ ɩɪɟɠɧɢɦɢ.
ɉɪɢɦɟɦ ɥɢɧɟɣɧɭɸ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ
O a bT .
Ⱦɥɹ ɩɨɥɭɩɪɨɫɬɪɚɧɫɬɜɚ (ɩɨɥɭɛɟɫɤɨɧɟɱɧɨɝɨ ɨɛɪɚɡɰɚ) ɜ ɩɟɪɟɦɟɧɧɵɯ
T T0
r
z
t
,[
,]
,W
,
T
t*
x*
x*
q0 x*
ɝɞɟ
1
, t*
k
x*
cU
,O
kO 0 0
a bT0 ,
ɡɚɞɚɱɚ ɩɪɢɧɢɦɚɟɬ ɜɢɞ
wT 1 w ª
wT º w ª
wT º
1 hT [ » «1 hT » ,
«
wW [ w[ ¬
w[ ¼ w] ¬
w] ¼
wT
] 0 : 1 hT (12.31)
e xp [2 ,
w]
wT
wT
wT
] of:
0; [ 0:
0; [ o f :
0,
w]
w[
w[
W 0: T 0,
Aq0b
.
h
ɝɞɟ
O 02 k
ɉɪɟɨɛɪɚɡɭɟɦ ɡɚɞɚɱɭ (12.31), ɢɫɩɨɥɶɡɭɹ ɩɨɞɫɬɚɧɨɜɤɭ Ʉɢɪɯɝɨɮɚ38
T
u
³ 1 h y d y
0
ɬɨɝɞɚ T
h
T T2 ,
2
1 2hu 1
. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɪɢɞɟɦ ɤ ɛɨɥɟɟ ɩɪɨɫɬɨɣ ɡɚɞɚɱɟ
h
1
w u 1 w ª w u º w 2u
«[ w[ » 2 ,
wW
[
w[
1
2
hu
¬
¼ w]
] 0:
] of:
wu
w]
wu
w]
e x p [ 2 ,
wu
wu
0; [of:
w[
w[
W 0: u 0,
0; [ 0:
38
(12.32)
0,
Ʉɨɡɞɨɛɚ Ʌ.ɂ. Ɇɟɬɨɞɵ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. Ɇ.: ɇɚɭɤɚ,
1975. 227 ɫ.
325
ɤɨɬɨɪɚɹ ɜɫɟ ɠɟ ɨɫɬɚɟɬɫɹ ɧɟɥɢɧɟɣɧɨɣ. Ⱦɥɹ ɟɟ ɪɟɲɟɧɢɹ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɟɬɨɞ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɯ ɩɪɢɛɥɢɠɟɧɢɣ, ɤɨɬɨɪɵɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ
ɫɥɟɞɭɸɳɟɦ.
1
Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ D ɫɪɟɞɧɟɟ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ
ɧɚ ɢɧ1 2h u ɬɟɪɜɚɥɟ ɬɟɦɩɟɪɚɬɭɪ >U1 ,U 2 @ , ɩɪɟɞɫɬɚɜɥɹɸɳɟɦ ɢɧɬɟɪɟɫ ɞɥɹ ɩɪɚɤɬɢɤɢ,
ɧɚɩɪɢɦɟɪ
1
D
U 2 U1
U2
du
³ 1 2hu .
U1
ȼ ɤɚɱɟɫɬɜɟ ɧɭɥɟɜɨɝɨ ɩɪɢɛɥɢɠɟɧɢɹ ɢɦɟɟɦ
u0 0 .
ɉɟɪɜɨɟ ɩɪɢɛɥɢɠɟɧɢɟ U1 - ɟɫɬɶ ɪɟɲɟɧɢɟ ɥɢɧɟɣɧɨɣ ɡɚɞɚɱɢ ɞɥɹ ɭɪɚɜɧɟɧɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
wU
D 1
wW
1 w ª w U1 º w 2U1
[
.
[ w[ ¬« w[ ¼» w] 2
ɫ ɬɟɦɢ ɠɟ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ, ɱɬɨ ɢ ɜ (12.32).
Ɋɟɲɟɧɢɟ ɥɢɧɟɣɧɨɣ ɡɚɞɚɱɢ ɦɨɠɟɬ ɛɵɬɶ ɧɚɣɞɟɧɨ, ɧɚɩɪɢɦɟɪ, ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɚ ɢɫɬɨɱɧɢɤɨɜ (ɦɟɬɨɞɚ ɮɭɧɤɰɢɣ Ƚɪɢɧɚ, ɪɚɡɞɟɥ 8.2) ɢ ɢɦɟɟɬ
ɜɢɞ
u1
2
S
WD
³
0
ª § [2
] 2 · º dy
2 ¸»
e x p « ¨
2
4 y ¸¹ »¼ 1 4 y 2
«¬ ¨© 1 4 y
ȼ ɰɟɧɬɪɟ ɩɹɬɧɚ ɧɚɝɪɟɜɚ ɢɦɟɟɦ
u1 0 , 0 , W 1
§ 2
·
W¸.
arct g ¨
S
© D
¹
(12.33)
ɉɪɢɦɟɧɹɹ ɨɛɪɚɬɧɨɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ
T1
1 2hu1 1
,
h
(12.34)
ɧɚɣɞɟɦ, ɤɚɤ ɦɟɧɹɟɬɫɹ ɢɫɯɨɞɧɚɹ ɮɭɧɤɰɢɹ T ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ.
Ⱦɥɹ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɣ D , h ɡɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɰɟɧɬɪɟ
ɩɹɬɧɚ ɧɚɝɪɟɜɚ ɨɬ ɜɪɟɦɟɧɢ ɩɨɤɚɡɚɧɚ ɧɚ ɪɢɫ. 12.11.
326
T
1
2
0,6
3
4
0,4
0,2
0,0
0
2
4
6
8
W
Ɋɢɫ. 12.11. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɰɟɧɬɪɟ ɩɹɬɧɚ ɧɚɝɪɟɜɚ
ɨɬ ɜɪɟɦɟɧɢ;1 – D=1 ( h 0 );
h 2 – 1; 3 – 2; 4 – 3.
ɋɤɨɪɨɫɬɶ ɧɚɝɪɟɜɚ ɬɜɟɪɞɨɝɨ ɬɟɥɚ ɦɨɠɧɨ ɧɚɣɬɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɦ
ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɪɟɲɟɧɢɹ ɞɥɹ u1
wT1
wu1
1
wW
1 2 hu1 wW
ª § [2
] 2 ·¸º
1
2
1
1
¨
.
exp «
»
¸
¨
1 2 hu1 S
«¬ © 1 4 W D 4W D ¹»¼ 1 4W D 2 W D
ȼ ɰɟɧɬɪɟ ɩɹɬɧɚ ɧɚɝɪɟɜɚ ɢɦɟɟɦ ɭɪɚɜɧɟɧɢɟ
wT1
wW
1
1 2hu1 0 ,0 , W D 1
1
.
S D 4W W
Ⱦɥɹ ɩɨɫɥɟɞɭɸɳɢɯ ɩɪɢɛɥɢɠɟɧɢɣ ɢɦɟɟɦ ɡɚɞɚɱɢ
w un ª
w un 1 1 w ª w un º w 2un
1 2
º
1 h un 1 D
[
D
¼» wW
wW ¬«
[ w[ «¬ w[ »¼ w] 2
wu
] 0 : n e x p [ 2 ,
w]
w un
w un
w un
] of:
0; [ 0:
0; [ o f :
0,
w]
w[
w[
W 0 : un 0 ,
ɪɟɲɟɧɢɟ ɤɨɬɨɪɵɯ ɫɜɨɞɢɦ ɤ ɧɚɯɨɠɞɟɧɢɸ ɨɬɤɥɨɧɟɧɢɣ ɨɬ ɩɪɟɞɵɞɭɳɟɝɨ
ɩɪɢɛɥɢɠɟɧɢɹ, ɩɨɥɚɝɚɹ un un 1 w .
327
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɞɥɹ ɩɨɫɬɨɹɧɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
O O 0 ɫɥɟɞɭɟɬ ɢɡ ɜɵɩɢɫɚɧɧɵɯ ɮɨɪɦɭɥ. Ⱦɥɹ ɬɟɦɩɟɪɚɬɭɪɵ T ɢɦɟɟɦ ɮɨɪɦɭɥɭ (12.33), ɝɞɟ D 1 :
1
T 0,0 ,W arct g 2 W .
S
12.7. Ɉɛɨɥɨɱɤɚ, ɨɯɥɚɠɞɚɟɦɚɹ ɢɡɥɭɱɟɧɢɟɦ, ɫ ɪɟɡɤɢɦ
ɩɟɪɟɩɚɞɨɦ ɪɚɜɧɨɜɟɫɧɵɯ ɬɟɦɩɟɪɚɬɭɪ
Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨ ɧɚɯɨɠɞɟɧɢɢ ɭɫɬɚɧɨɜɢɜɲɟɝɨɫɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ
ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɫɬɵɤɚ ɞɜɭɯ ɦɚɬɟɪɢɚɥɨɜ (ɪɢɫ. 12.12), ɤɨɬɨɪɚɹ
ɜɨɡɧɢɤɚɟɬ ɩɪɢ ɢɡɭɱɟɧɢɢ ɬɟɩɥɨɨɛɦɟɧɚ ɨɛɨɥɨɱɤɢ ɥɟɬɚɬɟɥɶɧɨɝɨ ɚɩɩɚɪɚɬɚ39.
ȿɫɥɢ ɭɫɥɨɜɢɹ ɬɟɩɥɨɨɛɦɟɧɚ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɬɨɧɤɨɫɬɟɧɧɨɣ ɨɛɨɥɨɱɤɢ
ɢɡɦɟɧɹɸɬɫɹ ɦɟɞɥɟɧɧɨ, ɬɨ ɩɟɪɟɞɚɱɭ ɬɟɩɥɚ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶɸ ɜɞɨɥɶ ɧɟɟ
ɦɨɠɧɨ ɧɟ ɭɱɢɬɵɜɚɬɶ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɭɫɬɚɧɨɜɢɜɲɢɟɫɹ ɬɟɦɩɟɪɚɬɭɪɵ ɤɚɠɞɨɣ ɱɚɫɬɢ ɬɚɤɨɣ ɨɛɨɥɨɱɤɢ ɦɨɠɧɨ ɪɚɫɫɱɢɬɵɜɚɬɶ ɧɟɡɚɜɢɫɢɦɨ.
Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɞɥɹ ɞɜɭɯ ɱɚɫɬɟɣ
ɨɛɨɥɨɱɤɢ, ɢɞɟɚɥɶɧɨ ɢɡɨɥɢɪɨɜɚɧɧɨɣ
ɫ ɜɧɭɬɪɟɧɧɟɣ ɫɬɨɪɨɧɵ, ɪɚɜɧɨɜɟɫɧɵɟ
ɬɟɦɩɟɪɚɬɭɪɵ ɢɡɜɟɫɬɧɵ (ɪɢɫ. 12.12):
T1 ɢ T2 , ɩɪɢɱɟɦ T1 ! T2 . ȼ ɨɤɪɟɫɬɧɨɊɢɫ. 12.12. Ɉɛɨɥɨɱɤɚ ɫ ɪɟɡɤɢɦ
ɫɬɢ ɫɬɵɤɚ ɢɦɟɟɦ T2 T T1 , ɩɪɢɱɟɦ
ɩɟɪɟɩɚɞɨɦ ɪɚɜɧɨɜɟɫɧɵɯ
ɬɟɦɩɟɪɚɬɭɪɚ ɦɟɧɹɟɬɫɹ ɧɟɩɪɟɪɵɜɧɨ.
ɬɟɦɩɟɪɚɬɭɪ
ɂɫɩɨɥɶɡɭɟɦ ɞɥɹ ɤɚɠɞɨɣ ɱɚɫɬɢ ɨɛɨɥɨɱɤɢ ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
ª w 2Ti w 2Ti º
(12.35)
Oi « 2 2 » 0
w y »¼
«¬ w x
ɫ ɭɫɥɨɜɢɹɦɢ
w Ti
y 0:
0,
wy
wT
y hi : O i i Hi V0 ªTi4 Tie4 º .
¬
¼
wy
Ɍɚɤ ɤɚɤ ɨɛɨɥɨɱɤɢ – ɬɨɧɤɢɟ, ɩɪɨɢɧɬɟɝɪɢɪɭɟɦ ɭɪɚɜɧɟɧɢɹ (12.35) ɩɨ
ɬɨɥɳɢɧɟ ɫ ɭɱɟɬɨɦ ɝɪɚɧɢɱɧɵɯ ɭɫɥɨɜɢɣ. ȼ ɪɟɡɭɥɶɬɚɬɟ ɧɚɣɞɟɦ
39
Ɂɚɪɭɛɢɧ ȼ.ɋ. Ɍɟɦɩɟɪɚɬɭɪɧɵɟ ɩɨɥɹ ɜ ɤɨɧɫɬɪɭɤɰɢɹɯ ɥɟɬɚɬɟɥɶɧɵɯ ɚɩɩɚɪɚɬɨɜ (ɦɟɬɨɞɵ ɪɚɫɱɟɬɚ). Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1978. 184 ɫ.
328
hi
w 2Ti
wT
³ Oi w x2 dy Oi w yi
hi
d 2Ti
(12.36)
Hi V0 ªTi4 Tie4 º 0 .
¬
¼
dx
0
0
ɍɫɥɨɜɢɹɦɢ ɤ ɞɜɭɦ ɭɪɚɜɧɟɧɢɹɦ (12.36) ɛɭɞɭɬ
x o f : T1 T1e ɢɥɢ d T1 d x 0 ;
(12.37)
x o f : T2 T2e ɢɥɢ d T2 d x 0 ;
dT
dT
x 0 : T1 T2 , O1h1 1 O 2 h2 2 .
dx
dx
ɗɬɚ ɡɚɞɚɱɚ, ɧɟ ɫɦɨɬɪɹ ɧɚ ɧɟɥɢɧɟɣɧɵɣ ɯɚɪɚɤɬɟɪ, ɥɟɝɤɨ ɢɧɬɟɝɪɢɪɭɟɬɫɹ.
ȼɜɟɞɟɦ ɨɛɨɡɧɚɱɟɧɢɟ
dTi
pi .
dx
Ɍɨɝɞɚ ɜɦɟɫɬɨ ɮɨɪɦɭɥɵ (12.36) ɡɚɩɢɲɟɦ
dp dT
Oi hi i i Hi V0 ªTi4 Ti4e º ,
¬
¼
dTi dx
ɢɥɢ
Oi hi pi d pi Hi V0 ªTi4 Tie4 º d Ti .
¬
¼
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
1
ª1
º
(12.38)
Oi hi pi 2 Hi V0 « Ti5 Tie4Ti Di » , i 1, 2 .
2
¬5
¼
ɉɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ D1 ɢ D2 ɧɚɣɞɟɦ ɢɡ ɭɫɥɨɜɢɣ (12.37).
ɂɦɟɟɦ:
4 5
4 5
D1
T1e , D2
T2e .
5
5
ɂɡ ɭɫɥɨɜɢɹ ɜ ɧɭɥɟ ɩɨɥɭɱɢɦ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɤɨɧɬɚɤɬɟ ɦɚɬɟɪɢɚɥɨɜ T1 T2 T0 :
Ɍɚɤ ɤɚɤ p1
O 2h2
p2 , p12
O1h1
Oi hi
2
2
§ O 2 h2 ·
2
¨
¸ p2 , ɬɨ ɧɚɣɞɟɦ
© O1h1 ¹
2
H V ª1
4 º § O h · H V ª1
4
º
2 1 0 « T05 T14eT0 T15e » ¨ 2 2 ¸ 2 2 0 « T05 T24eT0 T25e »
5 ¼ © O1h1 ¹ O 2 h2 ¬ 5
5
O1h1 ¬ 5
¼
ɢɥɢ
T05 > K H 1@ 5T0 ª K HT214 1º 4 ª K HT2 15 1º 0 ,
¬
¼
¬
¼
ɝɞɟ
329
(12.39)
KH
O 2h2H 2
, T0
O1h1H1
T0
,T
T1e 21
T2e
.
T1e
ȼ ɱɚɫɬɧɨɦ ɫɥɭɱɚɟ K H 1 ɢɦɟɟɦ
T0
0 ,8
1 T521
1 T 421
.
ɉɨɫɥɟ ɬɨɝɨ, ɤɚɤ ɬɟɦɩɟɪɚɬɭɪɚ T0 ɨɩɪɟɞɟɥɟɧɚ, ɤɚɠɞɨɟ ɢɡ ɭɪɚɜɧɟɧɢɣ
(12.38) ɢɥɢ (12.36) ɪɟɲɚɟɬɫɹ ɧɟɡɚɜɢɫɢɦɨ.
Ɇɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɩɪɢɛɥɢɠɟɧɧɨɟ ɪɟɲɟɧɢɟ, ɩɪɨɜɨɞɹ ɥɢɧɟɚɪɢɡɚɰɢɸ
ɭɪɚɜɧɟɧɢɣ (12.36) ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɬɟɦɩɟɪɚɬɭɪɵ T0 :
d 2Ti
dx 2
>
>
H i V0 4
Ti Tie4
O i hi
@
@
Hi V 0
Ti Tie Ti3 Ti2Tie TiTie2 Tie3 | Ni 0 Ti Tie ,
hi
O i hi
ɝɞɟ
Hi V0hi ª 3
T0 T02Ti e T0Ti2e Tie3 º .
¬
¼
Oi
Ɋɟɲɟɧɢɟɦ ɩɨɥɭɱɟɧɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɛɭɞɟɬ
ª N xº
Ti Ti e
e x p « i0 » .
(12.40)
T0 Tie
¬ hi ¼
Ɋɟɲɟɧɢɟ ɡɚɞɚɱɢ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ ɭɪɚɜɧɟɧɢɣ (12.38). ɇɨ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɪɟɲɟɧɢɟ ɧɟ ɜɵɪɚɠɚɟɬɫɹ ɱɟɪɟɡ ɷɥɟɦɟɧɬɚɪɧɵɟ ɮɭɧɤɰɢɢ.
N i20
12.8. Ɇɨɞɟɥɶ ɩɪɨɰɟɫɫɚ ɰɟɦɟɧɬɚɰɢɢ
ɤɨɦ ɩɚɤɬɧɨɝɨ ɦɚɬɟɪɢɚɥɚ
ɐɟɦɟɧɬɚɰɢɹ – ɨɞɢɧ ɢɡ ɫɩɨɫɨɛɨɜ ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ ɞɟɬɚɥɟɣ, ɩɪɟɞɧɚɡɧɚɱɟɧɧɵɣ ɞɥɹ ɢɯ ɭɩɪɨɱɧɟɧɢɹ. ɋɭɳɧɨɫɬɶ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ
ɫɨɫɬɨɢɬ ɜ ɧɚɫɵɳɟɧɢɢ ɭɝɥɟɪɨɞɨɦ ɩɨɜɟɪɯɧɨɫɬɧɵɯ ɫɥɨɟɜ ɡɚɝɨɬɨɜɤɢ, ɩɨɦɟɳɚɟɦɨɣ ɩɪɢ ɜɵɫɨɤɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɜ ɫɪɟɞɭ, ɫɨɞɟɪɠɚɳɭɸ ɫɨɟɞɢɧɟɧɢɹ ɭɝɥɟɪɨɞɚ. ɉɨɫɥɟɞɧɢɟ ɫɩɨɫɨɛɧɵ ɚɞɫɨɪɛɢɪɨɜɚɬɶɫɹ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɢ ɞɢɫɫɨɰɢɢɪɨɜɚɬɶ ɫ ɜɵɞɟɥɟɧɢɟɦ ɚɬɨɦɚɪɧɨɝɨ ɭɝɥɟɪɨɞɚ, ɞɢɮɮɭɧɞɢɪɭɸɳɟɝɨ ɜ
ɨɛɴɟɦ. ȼ ɬɟɯɧɨɥɨɝɢɹɯ, ɨɫɧɨɜɚɧɧɵɯ ɧɚ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɝɚɡɨɜɵɯ ɫɪɟɞ,
ɤɨɧɰɟɧɬɪɚɰɢɹ ɭɝɥɟɪɨɞɚ ɥɟɝɤɨ ɪɟɝɭɥɢɪɭɟɬɫɹ. ɉɪɨɫɬɚɹ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ
330
ɩɨɫɬɚɧɨɜɤɚ ɡɚɞɚɱɢ ɞɥɹ ɷɬɨɝɨ ɩɪɨɰɟɫɫɚ ɨɫɧɨɜɚɧɚ ɧɚ ɫɥɟɞɭɸɳɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɹɯ40.
1. ɍɱɢɬɵɜɚɟɬɫɹ, ɱɬɨ ɝɥɭɛɢɧɚ ɰɟɦɟɧɬɢɪɭɟɦɨɝɨ ɫɥɨɹ ɦɧɨɝɨ ɦɟɧɶɲɟ
ɪɚɡɦɟɪɨɜ ɞɟɬɚɥɢ, ɬɚɤ ɱɬɨ ɞɥɹ ɨɩɢɫɚɧɢɹ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɦɨɠɧɨ
ɨɝɪɚɧɢɱɢɬɶɫɹ ɨɞɧɨɦɟɪɧɨɣ ɩɨɫɬɚɧɨɜɤɨɣ ɡɚɞɚɱɢ ɜ ɬɨɧɤɨɦ ɩɪɢɩɨɜɟɪɯɧɨɫɬɧɨɦ ɫɥɨɟ. Ɋɚɡɦɟɪ ɞɟɬɚɥɢ ɞɥɹ ɩɪɨɰɟɫɫɚ ɞɢɮɮɭɡɢɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɛɟɫɤɨɧɟɱɧɵɦ.
2. Ɍɚɤ ɤɚɤ ɫɤɨɪɨɫɬɶ ɩɟɪɟɞɚɱɢ ɬɟɩɥɚ ɡɧɚɱɢɬɟɥɶɧɨ ɛɨɥɶɲɟ ɫɤɨɪɨɫɬɢ
ɞɢɮɮɭɡɢɢ ɜ ɚɧɚɥɨɝɢɱɧɨɣ ɫɪɟɞɟ ɢ ɝɥɭɛɢɧɚ ɩɪɨɝɪɟɬɨɝɨ ɫɥɨɹ ɡɧɚɱɢɬɟɥɶɧɨ
ɩɪɟɜɵɲɚɟɬ ɝɥɭɛɢɧɭ ɯɢɦɢɤɨ-ɬɟɪɦɢɱɟɫɤɨɣ ɨɛɪɚɛɨɬɤɢ, ɬɨ ɩɪɢ ɡɚɞɚɧɧɨɦ
ɡɚɤɨɧɟ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɫɪɟɞɵ ɢɥɢ ɩɨɜɟɪɯɧɨɫɬɢ ɨɛɪɚɛɚɬɵɜɚɟɦɨɣ
ɞɟɬɚɥɢ ɩɪɨɰɟɫɫ ɞɢɮɮɭɡɢɢ ɦɨɠɧɨ ɢɡɭɱɚɬɶ ɧɟɡɚɜɢɫɢɦɨ.
3. ɂɧɬɟɧɫɢɜɧɨɫɬɶ ɞɢɮɮɭɡɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɡɚɜɢɫɢɬ ɧɟ ɬɨɥɶɤɨ ɨɬ
ɬɟɦɩɟɪɚɬɭɪɵ, ɧɨ ɢ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɣ, ɩɪɢɱɟɦ ɡɚɜɢɫɢɦɨɫɬɶ ɤɨɷɮɮɢɰɢɟɧɬɚ
ɞɢɮɮɭɡɢɢ ɨɬ ɤɨɧɰɟɧɬɪɚɰɢɢ – ɧɟɥɢɧɟɣɧɚɹ.
ɉɪɢ ɷɬɢɯ ɩɪɟɞɩɨɥɨɠɟɧɢɹɯ ɦɚɬɟɦɚɬɢɱɟɫɤɚɹ ɮɨɪɦɭɥɢɪɨɜɤɚ ɡɚɞɚɱɢ
ɢɦɟɟɬ ɜɢɞ
wC w ª
wC º
D
C
,
T
,
(12.41)
w t w x «¬
w x »¼
C 0 , x C0 ,
x 0: D
wC
wx
E C ,T >C Ce @ ,
wC
0,
wx
ɝɞɟ D C – ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɭɝɥɟɪɨɞɚ ɜ ɦɟɬɚɥɥɟ, E – ɤɢɧɟɬɢɱɟɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɦɚɫɫɨɨɛɦɟɧɚ ɝɚɡɨɜɨɣ ɫɪɟɞɨɣ; Ce – ɭɝɥɟɪɨɞɧɵɣ ɩɨ-
xoL: D
ɬɟɧɰɢɚɥ ɫɪɟɞɵ (ɤɨɧɰɟɧɬɪɚɰɢɹ ɭɝɥɟɪɨɞɚ ɜ ɨɤɪɭɠɚɸɳɟɣ ɫɪɟɞɟ), C0 – ɧɚɱɚɥɶɧɨɟ ɫɨɞɟɪɠɚɧɢɟ ɭɝɥɟɪɨɞɚ ɜ ɨɛɪɚɛɚɬɵɜɚɟɦɨɦ ɫɥɨɟ (ɤɨɧɰɟɧɬɪɚɰɢɹ
ɢɡɦɟɪɹɟɬɫɹ ɜ ɩɪɨɰɟɧɬɚɯ).
Ⱦɥɹ ɪɹɞɚ ɤɨɦɩɚɤɬɧɵɯ ɦɚɬɟɪɢɚɥɨɜ ɢɡɜɟɫɬɧɵ ɮɨɪɦɭɥɵ, ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɢɯ ɮɢɡɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɤɨɧɰɟɧɬɪɚɰɢɣ. ɇɚɩɪɢɦɟɪ, ɞɥɹ J F e ɬɚɤɢɟ ɡɚɜɢɫɢɦɨɫɬɢ ɢɦɟɸɬ ɜɢɞ
40
Ɍɢɯɨɧɨɜ Ⱥ.ɇ., Ʉɚɥɶɧɟɪ ȼ.Ⱦ., Ƚɥɚɫɤɨ ȼ.Ȼ. Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɦɟɬɨɞ ɨɛɪɚɬɧɵɯ ɡɚɞɚɱ ɜ ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ. Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1990. 264 ɫ.
331
D
0,04 0,08C exp§¨ 31350 ·
2
¸ , ɫɦ /ɫ
© 1,987 T ¹
§ 11100 ·
E 1,36 ˜ 10 3 exp¨ ¸ , ɫɦ/ɫ,
© 1,987 T ¹
(12.42)
ɝɞɟ T – ɬɟɦɩɟɪɚɬɭɪɚ ɜ Ʉ.
ȼ ɫɥɭɱɚɟ D D0 const , E E0 const ɡɚɞɚɱɚ ɩɨɥɧɨɫɬɶɸ ɚɧɚɥɨɝɢɱɧɚ ɡɚɞɚɱɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɝɪɚɧɢɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɬɪɟɬɶɟɝɨ ɪɨɞɚ,
ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜ ɪɚɡɞɟɥɟ 7.3.3.
ɗɬɚ ɧɟɥɢɧɟɣɧɚɹ ɡɚɞɚɱɚ ɦɨɠɟɬ ɛɵɬɶ ɪɟɲɟɧɚ ɱɢɫɥɟɧɧɨ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɧɟɹɜɧɨɣ ɪɚɡɧɨɫɬɧɨɣ ɫɯɟɦɵ ɢ ɦɟɬɨɞɚ ɩɪɨɝɨɧɤɢ.
Ɉɞɢɧ ɢɡ ɜɨɡɦɨɠɧɵɯ ɚɥɝɨɪɢɬɦɨɜ ɱɢɫɥɟɧɧɨɝɨ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ
ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɫɥɟɞɭɸɳɢɟ ɷɬɚɩɵ.
ɋɧɚɱɚɥɚ ɞɥɹ ɪɟɲɟɧɢɹ ɭɪɚɜɧɟɧɢɹ (12.41) ɫɨɫɬɚɜɥɹɟɦ ɪɚɡɧɨɫɬɧɭɸ
ɫɯɟɦɭ ɧɚ ɱɟɬɵɪɟɯɬɨɱɟɱɧɨɦ ɲɚɛɥɨɧɟ (ɪɢɫ. 12.13)
Ci Ci
't
1 ª Di 1 Di Ci 1 Ci
˜
«
2
'x ¬
'x
Di Di 1 Ci Ci 1 º
˜
»,
2
'x ¼
ɝɞɟ
ɢɫɩɨɥɶɡɭɟɬɫɹ
(12.43)
ɨɛɨɡɧɚɱɟɧɢɟ
j 1
Ci { Ci
ɞɥɹ ɤɨɧɰɟɧɬɪɚɰɢɢ ɜ ɬɨɱɤɟ
xi i ˜ ' x
ɜ
ɦɨɦɟɧɬ
ɜɪɟɦɟɧɢ
t j 1 j 1 ˜ ' t (ɬ.ɟ. ɧɚ ɫɥɨɟ ɫ ɧɨɦɟɪɨɦ
j ), ' x – ɲɚɝ ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ ɤɨɨɪɞɢɧɚɬɟ, ' t – ɲɚɝ ɩɨ ɜɪɟɦɟɧɢ. Ⱦɥɹ
Ɋɢɫ 12.13. Ɋɢɫɭɧɨɤ ɤ ɩɨɫɬɪɨɟɧɢɸ ɜɟɥɢɱɢɧ ɫɨ ɫɥɨɹ j ɢɫɩɨɥɶɡɭɸɬɫɹ ɨɛɨɪɚɡɧɨɫɬɧɨɣ ɫɯɟɦɵ
ɡɧɚɱɟɧɢɹ ɜɢɞɚ Ci { Ci j , Di 1 ɢ ɬ.ɞ. ȼ
ɪɟɡɭɥɶɬɚɬɟ ɧɚ ɤɚɠɞɨɦ ɫɥɨɟ ɢɦɟɟɦ ɫɢɫɬɟɦɭ n 1 ɭɪɚɜɧɟɧɢɹ ɜɢɞɚ
aiCi 1 diCi biCi 1 fi
ɝɞɟ
0 , i 1,2,...,n 1 ,
Di 1 Di
' t Di Di 1
, bi
,
˜
˜
ai
2
2
2
'
x
' x2
' t Di 1 2 Di Di 1
, fi Ci .
di 1 2 ˜
2
'x
't
332
(12.44)
Ɋɟɲɢɬɶ ɫɢɫɬɟɦɭ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ (12.44) ɦɨɠɧɨ ɫɥɟɞɭɸɳɢɦ
ɨɛɪɚɡɨɦ. ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɢɦɟɟɬ ɦɟɫɬɨ ɫɜɹɡɶ
Ci Di 1Ci 1 J i 1 ,
(12.45)
ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬɵ Di 1 , J i 1 - ɩɨɤɚ ɧɟ ɨɩɪɟɞɟɥɟɧɵ.
Ⱥɧɚɥɨɝɢɱɧɨ (12.45) ɦɨɠɟɦ ɡɚɩɢɫɚɬɶ
Ci 1 DiCi J i .
(12.46)
ɉɨɞɫɬɚɜɥɹɹ (12.46) ɜ ɭɪɚɜɧɟɧɢɟ (12.43) ɢ ɫɨɛɢɪɚɹ ɩɨɞɨɛɧɵɟ ɫɥɚɝɚɟɦɵɟ, ɧɚɣɞɟɦ
Di ai di Ci biCi 1 fi Ji ai .
ɋɪɚɜɧɢɜɚɹ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɫ (12.44), ɧɚɣɞɟɦ ɪɟɤɭɪɪɟɧɬɧɵɟ
ɮɨɪɦɭɥɵ ɞɥɹ ɜɵɱɢɫɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ Di 1 , J i 1
bi
f J i ai
, J i 1 i
,
(12.47)
Di 1 Di ai di
Di ai di
i 1,2,...,n .
Ɉɩɢɫɚɧɧɵɣ ɦɟɬɨɞ ɪɟɲɟɧɢɹ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ ɟɫɬɶ ɨɞɢɧ
ɢɡ ɜɚɪɢɚɧɬɨɜ ɦɟɬɨɞɚ ɩɪɨɝɨɧɤɢ, ɚ ɤɨɷɮɮɢɰɢɟɧɬɵ Di 1 , J i 1 ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɩɪɨɝɨɧɨɱɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ.
ɍɪɚɜɧɟɧɢɟ (12.43) ɟɫɬɶ ɪɚɡɧɨɫɬɧɚɹ ɮɨɪɦɚ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɝɨ
ɭɪɚɜɧɟɧɢɹ. ɂɫɩɨɥɶɡɨɜɚɧɧɚɹ ɡɞɟɫɶ ɪɚɡɧɨɫɬɧɚɹ ɫɯɟɦɚ ɟɫɬɶ ɫɯɟɦɚ ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɩɨ ɜɪɟɦɟɧɢ ɢ ɜɬɨɪɨɝɨ – ɩɨ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɨɣ
ɤɨɨɪɞɢɧɚɬɟ. ɑɬɨɛɵ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɩɨɥɭɱɟɧɧɵɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ,
ɧɭɠɧɨ ɡɚɩɢɫɚɬɶ ɜ ɪɚɡɧɨɫɬɧɨɦ ɜɢɞɟ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ, ɠɟɥɚɬɟɥɶɧɨ
ɬɚɤɠɟ ɫɨ ɜɬɨɪɵɦ ɩɨɪɹɞɤɨɦ ɚɩɩɪɨɤɫɢɦɚɰɢɢ.
ɉɪɟɞɫɬɚɜɢɦ ɤɨɧɰɟɧɬɪɚɰɢɸ ɜ ɬɨɱɤɟ ɫ ɧɨɦɟɪɨɦ i 1 ɜ ɜɢɞɟ ɪɚɡɥɨɠɟɧɢɹ ɜ ɪɹɞ Ɍɟɣɥɨɪɚ ɩɨ ' x ɨɬɧɨɫɢɬɟɥɶɧɨ ɬɨɱɤɢ ɫ ɧɨɦɟɪɨɦ i 0 ɞɨ ɫɥɚɝɚɟɦɵɯ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɦɚɥɨɫɬɢ. ɂɦɟɟɦ
1 §¨ w 2C ·¸
§ wC ·
2
C1 C0 ¨
¸ 'x ¨ 2 ¸ 'x ...
2 © wx ¹
© wx ¹ 0
0
ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
C1 C0 1 §¨ w 2C ·¸
§ wC ·
(12.48)
'x ....
¨
¸ |
2 ¨© wx 2 ¸¹
'x
© wx ¹0
0
ȼɬɨɪɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɜɵɪɚɡɢɦ ɢɡ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ (12.41)
ª§ w C · § w D · § w C · º
§ w 2C ·
1
Ǭ
¸ ¨ wx ¸ ¨ wx ¸ » .
¨¨ w x 2 ¸¸
D
C
,
T
w
t
©
¹0 ©
¹0 ¼»
¹0 ©
0
©
¹0
¬«
ȼ ɪɟɡɭɥɶɬɚɬɟ ɢɡ (12.48) ɧɚɯɨɞɢɦ
333
§ wC · ª 'x § wD · 1 º
¨ w x ¸ «1 2 ¨ w x ¸ D »
©
¹0 «¬
©
¹0 0 »¼
C1 C0
1 'x § wC ·
.
'x
D0 2 ¨© w t ¸¹0
(12.49)
ȼɵɪɚɡɢɦ ɩɟɪɜɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɩɨ ɤɨɨɪɞɢɧɚɬɟ ɢɡ (12.49) ɢ ɩɨɞɫɬɚɜɢɦ ɟɟ ɜ ɝɪɚɧɢɱɧɨɟ ɭɫɥɨɜɢɟ ɩɪɢ x 0 . ɂɦɟɟɦ
ª 'x § wD · 1 º
ª C1 C0 1 ' x C0 C0 º D0 «
» E0 C ,T > C0 Ce @ ˜ «1 ¨
¸ »
D0 2
't ¼
2 © w x ¹0 D0 »¼
«¬
¬ 'x
ɢɥɢ
ª
º
't 1
't § 'x § wD · 1 · C0 «1 2 2 2
C
,
T
˜E
»
¨1 ¸
¨
¸
2 © w x ¹0 D0 ¸¹ 0
' x ¨©
' x D0
«¬
»¼
ª
º
§ 'x § wD · 1 · 't 1
C
T
2 2 C1 «C0 Ce ¨1 E
».
,
¸
¨
¸
0
¨
¸
x
D
w
2
' x D0
«¬
»¼
©
¹0 0 ¹
©
ɋɪɚɜɧɢɜɚɹ ɩɨɫɥɟɞɧɟɟ ɜɵɪɚɠɟɧɢɟ ɫ ɭɪɚɜɧɟɧɢɟɦ (12.45), ɡɚɩɢɫɚɧɧɵɦ
ɜ ɬɨɱɤɟ i 0 ,
C0
D1C1 J1 ,
ɧɚɯɨɞɢɦ
2
D1
J1
't 1
' x 2 D0
,
§
·
§
·
1
't 1
't
'x wD
¸ ˜ E0 C ,T 1 2 2 2
¨¨1 ¨
¸
2 © w x ¹0 D0 ¸¹
'x ©
' x D0
§ 'x § wD · 1 · C0 Ce ¨1 ¨ w x ¸ D ¸¸ E0 C ,T ¨
2
©
¹0 0 ¹
©
.
§
·
't 1
't
'x § wD · 1
1 2 2 2
¨1 ¨
¸ ¸ ˜E C ,T 2 © w x ¹0 D0 ¸¹ 0
' x ¨©
' x D0
(12.50)
ɍɫɥɨɜɢɟ ɧɚ ɜɧɟɲɧɟɣ ɝɪɚɧɢɰɟ x o L ɚɩɩɪɨɤɫɢɦɢɪɭɟɦ ɚɧɚɥɨɝɢɱɧɨ
ɩɪɟɞɵɞɭɳɟɦɭ. ȼ ɬɨɱɤɟ ɫ ɧɨɦɟɪɨɦ i n 1 ɢɦɟɟɦ
1 §¨ w 2C ·¸
§ wC ·
'x 2 ...
x
'
Cn 1 Cn ¨
¸
2
¨
¸
2 © wx ¹
© wx ¹ n
n
ɢɥɢ
334
Cn Cn 1 1 §¨ w 2C ·¸
§ wC ·
'x ....
¨
¸ |
2 ¨© wx 2 ¸¹
'x
© wx ¹ n
n
ɋ ɩɨɦɨɳɶɸ ɭɪɚɜɧɟɧɢɹ ɞɢɮɮɭɡɢɢ ɧɚɯɨɞɢɦ
ª§ w C · § w D · § w C · º
§ w 2C ·
1
Ǭ
¸ ¨ wx ¸ ¨ wx ¸ »
¨¨ w x 2 ¸¸
D
C
,
T
t
w
©
¹
¹n ¼»
¹n ©
n
n ©
©
¹n
¬«
Cn Cn 1 1 ' x ª§ w C · § w D · § w C · º
§ wC ·
ɢ
Ǭ
¨ wx ¸ |
¸ ¨ wx ¸ ¨ wx ¸ »
x
D
t
2
'
w
©
¹n
¹n ©
¹ n »¼
¹n ©
«¬©
n
ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɝɪɚɧɢɱɧɵɦ ɭɫɥɨɜɢɟɦ ɢɦɟɟɦ ɪɚɜɟɧɫɬɜɨ
C C
2 ' t Dn n 2n 1 Cn Cn 0 ,
'x
ɚ ɢɡ (12.45) – ɪɚɜɟɧɫɬɜɨ
Cn1 D nCn J n .
ȼ ɪɟɡɭɥɶɬɚɬɟ ɧɚɯɨɞɢɦ Cn :
2' t Cn J n 2 Dn
'x
.
(12.51)
Cn
2' t 1 1 D n 2 Dn
'x
§ wD ·
ɇɚɦ ɨɫɬɚɥɨɫɶ ɧɚɣɬɢ ɩɪɨɢɡɜɨɞɧɭɸ ¨
¸ , ɜɯɨɞɹɳɭɸ ɜ ɩɪɨɝɨɧɨɱ© w x ¹0
ɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ (12.50). ɗɬɚ ɩɪɨɢɡɜɨɞɧɚɹ ɜɵɱɢɫɥɹɟɬɫɹ ɧɚ ɫɥɨɟ j ,
ɤɨɧɰɟɧɬɪɚɰɢɹ ɢ ɜɫɟ ɫɜɹɡɚɧɧɵɟ ɫ ɧɟɣ ɜɟɥɢɱɢɧɵ ɧɚ ɤɨɬɨɪɨɦ ɢɡɜɟɫɬɧɵ.
ɉɨɷɬɨɦɭ ɩɨɫɬɭɩɢɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. Ⱥɩɩɪɨɤɫɢɦɢɪɭɟɦ ɤɪɢɜɭɸ
D C ,T D x ɩɨɥɢɧɨɦɨɦ ɜɬɨɪɨɣ ɫɬɟɩɟɧɢ D a x 2 b x d ɩɨ ɬɪɟɦ
ɬɨɱɤɚɦ x0 , x1 , x2 , ɝɞɟ ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɮɮɭɡɢɢ ɪɚɜɟɧ D0 , D1 , D2 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ. Ɍɨɝɞɚ ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɬɪɟɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ a ,b , d ɩɨɥɭɱɢɦ
ɫɢɫɬɟɦɭ ɬɪɟɯ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ
D0 d ,
D1 a' x 2 b' x d ,
D2 4a' x 2 2b' x d .
Ɋɟɲɟɧɢɟ ɷɬɨɣ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ ɢɦɟɟɬ ɜɢɞ
335
d
D0 , b
4 D1 3D0 D2
,a
2' x
2 D1 D0 D2
2' x
2
.
ɇɭɠɧɭɸ ɩɪɨɢɡɜɨɞɧɭɸ ɧɚɯɨɞɢɦ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɵɦ ɞɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟɦ
§ wD ·
4D1 3D0 D2
.
(12.52)
¨
¸ b
2
x
x
w
'
©
¹0
Ɉɤɨɧɱɚɬɟɥɶɧɨ, ɧɚ ɤɚɠɞɨɦ ɧɨɜɨɦ ɫɥɨɟ j 1 ɩɨɪɹɞɨɤ ɜɵɱɢɫɥɟɧɢɣ
ɫɥɟɞɭɸɳɢɣ. ɇɚɯɨɞɢɦ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ ɞɥɹ ɩɪɨɝɨɧɨɱɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨ ɮɨɪɦɭɥɚɦ (12.50) ɫ ɭɱɟɬɨɦ (12.52). Ɂɚɬɟɦ ɩɨ ɮɨɪɦɭɥɚɦ (12.47)
ɨɩɪɟɞɟɥɹɟɦ ɜɫɟ ɨɫɬɚɥɶɧɵɟ ɩɪɨɝɨɧɨɱɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ. ɉɨɫɥɟ ɷɬɨɝɨ ɧɚɯɨɞɢɦ ɤɨɧɰɟɧɬɪɚɰɢɸ ɜ ɬɨɱɤɟ x L ɩɨ ɮɨɪɦɭɥɟ (12.51). ɂ, ɧɚɤɨɧɟɰ, ɩɨ
ɮɨɪɦɭɥɚɦ (12.45) ɧɚɯɨɞɢɦ ɤɨɧɰɟɧɬɪɚɰɢɸ ɜɨ ɜɫɟɯ ɨɫɬɚɥɶɧɵɯ ɬɨɱɤɚɯ.
Ɋɚɫɱɟɬ ɩɨɜɬɨɪɹɟɦ ɞɨ ɨɤɨɧɱɚɧɢɹ ɜɪɟɦɟɧɢ ɨɛɪɚɛɨɬɤɢ. ȼ ɧɚɱɚɥɶɧɵɣ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɬ.ɟ. ɧɚ ɫɥɨɟ j 0 , ɤɨɧɰɟɧɬɪɚɰɢɹ ɡɚɞɚɧɚ.
ɗɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɭɫɬɚɧɨɜɥɟɧɨ, ɱɬɨ ɜ ɩɨɪɢɫɬɵɯ ɦɚɬɟɪɢɚɥɚɯ ɝɥɭɛɢɧɚ ɩɪɨɧɢɤɧɨɜɟɧɢɹ ɜɟɳɟɫɬɜɚ ɩɪɢ ɞɢɮɮɭɡɢɢ ɡɧɚɱɢɬɟɥɶɧɨ ɜɵɲɟ, ɱɟɦ ɜ
ɦɨɧɨɥɢɬɧɵɯ ɩɨɥɢɤɪɢɫɬɚɥɥɢɱɟɫɤɢɯ ɦɚɬɟɪɢɚɥɚɯ ɡɚ ɬɨ ɠɟ ɜɪɟɦɹ. ɗɬɨ ɫɜɹɡɚɧɨ ɫ ɪɚɡɥɢɱɢɟɦ ɦɟɯɚɧɢɡɦɨɜ ɞɢɮɮɭɡɢɢ ɜ ɤɨɦɩɚɤɬɧɵɯ ɢ ɩɨɪɢɫɬɵɯ ɦɚɬɟɪɢɚɥɚɯ. Ɍɟɦ ɧɟ ɦɟɧɟɟ, ɩɪɨɰɟɫɫ ɧɚɫɵɳɟɧɢɹ ɩɨɪɢɫɬɨɝɨ ɦɚɬɟɪɢɚɥɚ ɦɨɠɧɨ ɨɩɢɫɚɬɶ ɧɚ ɨɫɧɨɜɟ ɬɨɣ ɠɟ ɦɨɞɟɥɢ, ɧɨ ɡɚɜɢɫɢɦɨɫɬɢ ɤɢɧɟɬɢɱɟɫɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ ɢ ɤɨɧɰɟɧɬɪɚɰɢɣ ɛɭɞɭɬ ɢɧɵɦɢ. ȼɟɥɢɱɢɧɚ
ɩɨɪɢɫɬɨɫɬɢ ɬɚɤɠɟ ɨɤɚɡɵɜɚɟɬ ɫɭɳɟɫɬɜɟɧɧɨɟ ɜɥɢɹɧɢɟ ɧɚ ɞɢɮɮɭɡɢɨɧɧɵɣ
ɤɨɷɮɮɢɰɢɟɧɬ.
12.9. ɇɚɝɪɟɜ ɢɡɥɭɱɟ ɧɢɟɦ ɈȽɄ
ɪ ɚ ɡ ɥ ɚ ɝ ɚ ɸ ɳ ɟ ɣ ɫ ɹ ɩ ɨ ɥ ɢ ɦ ɟ ɪ ɧ ɨ ɣ ɩ ɥ ɟ ɧ ɤ ɢ 41
ɂɧɬɟɪɟɫ ɤ ɩɪɨɰɟɫɫɚɦ ɜɨɡɞɟɣɫɬɜɢɹ ɫɨɫɪɟɞɨɬɨɱɟɧɧɵɯ ɢɫɬɨɱɧɢɤɨɜ
ɷɧɟɪɝɢɢ ɧɚ ɬɜɟɪɞɵɟ ɦɚɬɟɪɢɚɥɵ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɜɹɡɚɧ ɫ ɡɚɞɚɱɚɦɢ ɥɚɡɟɪɧɨɣ ɬɟɯɧɨɥɨɝɢɢ (ɧɚɩɪɢɦɟɪ, ɫ ɡɚɞɚɱɚɦɢ ɪɟɡɤɢ ɢɥɢ ɪɚɫɤɪɨɹ ɬɤɚɧɢ).
ȼɨɡɞɟɣɫɬɜɢɟ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɧɚ ɯɢɦɢɱɟɫɤɢ ɚɤɬɢɜɧɵɣ ɦɚɬɟɪɢɚɥ
ɦɨɠɟɬ ɢɦɟɬɶ ɪɹɞ ɨɫɨɛɟɧɧɨɫɬɟɣ. ȼɨ-ɩɟɪɜɵɯ, ɫ ɩɪɨɬɟɤɚɧɢɟɦ ɯɢɦɢɱɟɫɤɢɯ
ɪɟɚɤɰɢɣ ɜ ɨɛɴɟɦɟ ɢ ɧɚ ɩɨɜɟɪɯɧɨɫɬɢ ɫɜɹɡɚɧɵ ɞɨɩɨɥɧɢɬɟɥɶɧɵɟ ɢɫɬɨɱɧɢɤɢ
ɢ ɫɬɨɤɢ ɬɟɩɥɚ. ȼɨ-ɜɬɨɪɵɯ, ɜ ɯɨɞɟ ɯɢɦɢɱɟɫɤɢɯ ɪɟɚɤɰɢɣ ɫɜɨɣɫɬɜɚ ɜɟɳɟɫɬɜ
41
Ⱦɢɤ ɂ.Ƚ., Ʉɧɹɡɟɜɚ Ⱥ.Ƚ. Ɂɚɠɢɝɚɧɢɟ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɟɣ ɬɨɧɤɨɣ ɩɥɟɧɤɢ ɫ ɦɟɧɹɸɳɢɦɢɫɹ ɜ ɯɨɞɟ ɪɟɚɤɰɢɢ ɨɩɬɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ // ɎȽȼ, 1990, ʋ 3. ɋ. 3–7; ɉɨɞɠɢɝɚɧɢɟ ɬɨɧɤɨɣ ɩɥɟɧɤɢ ɩɭɱɤɨɦ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɢ // ɎȽȼ, 1991, ʋ 6. ɋ. 3–7
336
ɢɡɦɟɧɹɸɬɫɹ, ɱɬɨ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ, ɧɚɩɪɢɦɟɪ, ɤ ɢɡɦɟɧɟɧɢɸ ɷɮɮɟɤɬɢɜɧɨɣ
ɷɧɟɪɝɢɢ, ɩɨɝɥɨɳɚɟɦɨɣ ɬɟɥɨɦ, ɢ ɤ ɪɚɡɥɢɱɧɵɦ ɧɟɥɢɧɟɣɧɵɦ ɮɢɡɢɱɟɫɤɢɦ
ɷɮɮɟɤɬɚɦ.
Ɍɚɤ, ɜ ɩɪɨɫɬɟɣɲɟɦ ɩɪɢɛɥɢɠɟɧɢɢ ɡɚɞɚɱɚ ɨ ɧɚɝɪɟɜɟ ɩɨɬɨɤɨɦ ɥɭɱɢɫɬɨɣ ɷɧɟɪɝɢɢ ɬɨɧɤɨɣ ɩɥɟɧɤɢ, ɜ ɨɛɴɟɦɟ ɤɨɬɨɪɨɣ ɦɨɠɟɬ ɩɪɨɬɟɤɚɬɶ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ, ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ
cU h
dT
dt
dK
dt
qa Qchh
dK
2D T T0 ,
dt
§ E ·
z 1 K e xp ¨ ¸ )1 ,
© RT ¹
K T T0
0, t
0,
(12.53)
(12.54)
(12.55)
ɝɞɟ T - ɬɟɦɩɟɪɚɬɭɪɚ, c,U - ɬɟɩɥɨɟɦɤɨɫɬɶ ɢ ɩɥɨɬɧɨɫɬɶ ɩɥɟɧɤɢ, h - ɬɨɥɳɢɧɚ ɩɥɟɧɤɢ, K - ɫɬɟɩɟɧɶ ɩɪɟɜɪɚɳɟɧɢɹ ɢɥɢ ɦɚɫɫɨɜɚɹ ɞɨɥɹ ɫɭɦɦɚɪɧɨɝɨ
ɩɪɨɞɭɤɬɚ ɪɟɚɤɰɢɢ, Qch - ɬɟɩɥɨɬɚ ɯɢɦɢɱɟɫɤɨɣ ɪɟɚɤɰɢɢ, E - ɷɧɟɪɝɢɹ ɚɤɬɢɜɚɰɢɢ, z - ɩɪɟɞɷɤɫɩɨɧɟɧɬ (ɫɦ. ɪɚɡɞɟɥ 11.1); qa - ɩɨɬɨɤ ɬɟɩɥɚ, ɩɨɝɥɨɳɟɧɧɵɣ ɩɥɟɧɤɨɣ, ɤɨɬɨɪɵɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɚɧɚɥɨɝɢɱɧɨ ɡɚɞɚɱɟ, ɪɚɫɫɦɨɬɪɟɧɧɨɣ ɜ ɪɚɡɞɟɥɟ 12.6
qa
q0 ^1 exp > V h@`>1 f @
ɝɞɟ q0 – ɩɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ ɩɚɞɚɸɳɟɝɨ ɩɨɬɨɤɚ ɢɡɥɭɱɟɧɢɹ, V – ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ, f – ɤɨɷɮɮɢɰɢɟɧɬ ɨɬɪɚɠɟɧɢɹ (ɫɦ. ɪɚɡɞɟɥ 9).
Ɂɞɟɫɶ ɩɪɟɞɩɨɥɚɝɚɟɬɫɹ, ɱɬɨ ɩɥɨɳɚɞɶ ɜɨɡɞɟɣɫɬɜɢɹ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɩɟɪɟɤɪɵɜɚɟɬ ɩɨɩɟɪɟɱɧɵɣ ɪɚɡɦɟɪ ɩɥɟɧɤɢ (ɫɦ. ɪɚɡɞɟɥ 12.6), ɚ ɯɢɦɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɹɜɥɹɟɬɫɹ ɷɤɡɨɬɟɪɦɢɱɟɫɤɨɣ ɢ ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧɚ ɩɪɨɫɬɟɣɲɟɣ ɪɟɚɤɰɢɨɧɧɨɣ ɫɯɟɦɨɣ
«ɪɟɚɝɟɧɬ – ɩɪɨɞɭɤɬ».
ɉɨ ɡɚɤɨɧɭ Ȼɷɪɚ ɞɥɹ ɫɥɚɛɵɯ ɪɚɫɬɜɨɪɨɜ, ɝɚɡɨɜ ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ
ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɤɨɥɢɱɟɫɬɜɭ ɩɨɝɥɨɳɚɸɳɢɯ ɰɟɧɬɪɨɜ. ȿɫɥɢ ɜ ɯɨɞɟ ɪɚɡɥɨɠɟɧɢɹ ɩɥɟɧɤɢ ɨɛɪɚɡɭɟɬɫɹ ɭɝɥɟɪɨɞ, ɫɩɨɫɨɛɫɬɜɭɸɳɢɣ ɩɨɝɥɨɳɟɧɢɸ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ, ɬɨ ɦɨɠɧɨ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɩɨɤɚɡɚɬɟɥɶ ɩɨɝɥɨɳɟɧɢɹ
ɟɫɬɶ ɥɢɧɟɣɧɚɹ ɮɭɧɤɰɢɹ ɫɬɟɩɟɧɢ ɩɪɟɜɪɚɳɟɧɢɹ K. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ɞɥɹ ɱɚɫɬɢɱɧɨ ɪɚɡɥɨɠɢɜɲɢɯɫɹ ɩɨɥɢɦɟɪɧɵɯ
ɩɥɟɧɨɤ ɫɭɳɟɫɬɜɭɟɬ ɥɢɧɟɣɧɚɹ ɤɨɪɪɟɥɹɰɢɹ ɦɟɠɞɭ ɩɨɤɚɡɚɬɟɥɟɦ ɩɨɝɥɨɳɟɧɢɹ V ɢ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɨɬɪɚɠɟɧɢɹ f . Ʌɢɧɟɣɧɚɹ ɫɜɹɡɶ ɦɟɠɞɭ V , f ɢ K
ɜɵɬɟɤɚɟɬ ɢ ɢɡ ɚɧɚɥɢɡɚ ɩɪɨɫɬɟɣɲɟɣ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɩɨɝɥɨɳɚɸɳɟɣ
ɫɪɟɞɵ ɬɢɩɚ ɰɟɩɨɱɤɢ ɨɪɢɟɧɬɢɪɨɜɚɧɧɵɯ ɜɞɨɥɶ ɥɭɱɚ ɫɤɨɩɥɟɧɢɣ ɩɨɝɥɨ337
ɳɚɸɳɢɯ ɰɟɧɬɪɨɜ. ɉɨɥɚɝɚɹ, ɱɬɨ ɡɚɤɨɧ Ȼɷɪɚ ɫɩɪɚɜɟɞɥɢɜ ɢ ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ, ɡɚɩɢɲɟɦ
V V0
V f V0
f f0
f f f0
Ș,
(12.56)
ɝɞɟ ɢɧɞɟɤɫ « f » ɨɬɧɨɫɢɬɫɹ ɤ ɤɨɧɟɱɧɨɦɭ ɫɨɫɬɨɹɧɢɸ, ɚ ɢɧɞɟɤɫ «0» – ɤ ɧɚɱɚɥɶɧɨɦɭ. Ɍɨɝɞɚ ɩɨɬɨɤ, ɩɨɝɥɨɳɟɧɧɵɣ ɩɥɟɧɤɨɣ, ɛɭɞɟɬ ɮɭɧɤɰɢɟɣ ɫɬɟɩɟɧɢ
ɩɪɟɜɪɚɳɟɧɢɹ
qa
^
`
q0 1 exp ª¬ V K h º¼ ª¬1 f K º¼ .
ɉɨɬɨɤ ɢɡɥɭɱɟɧɢɹ, ɩɪɨɩɭɳɟɧɧɵɣ ɩɥɟɧɤɨɣ, ɟɫɬɶ
) q0exp ª¬V K h º¼ ª¬1 f K º¼ .
ɉɪɢ ɱɢɫɥɟɧɧɨɦ ɢɫɫɥɟɞɨɜɚɧɢɢ ɷɬɨɣ ɡɚɞɚɱɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ ɨɬ ɜɪɟɦɟɧɢ ɢ ɜɪɟɦɹ ɩɨɥɧɨɝɨ ɪɚɡɥɨɠɟɧɢɹ ɩɥɟɧɤɢ ti , ɤɨɝɞɚ K o 0,95 . Ⱦɢɧɚɦɢɤɚ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ, ɫɬɟɩɟɧɢ ɩɪɟɜɪɚɳɟɧɢɹ ɢ ɩɨɬɨɤɚ, ɩɪɨɩɭɳɟɧɧɨɝɨ ɩɥɟɧɤɨɣ, ɩɪɨɢɥɥɸɫɬɪɢɪɨɜɚɧɵ ɧɚ
ɪɢɫ. 12.14 ɞɥɹ ɩɨɫɬɨɹɧɧɵɯ ɨɩɬɢɱɟɫɤɢɯ ɫɜɨɣɫɬɜ ɢ ɫɜɨɣɫɬɜ, ɦɟɧɹɸɳɢɯɫɹ ɜ
ɯɨɞɟ ɪɟɚɤɰɢɢ. ȼɪɟɦɹ ɩɨɥɧɨɝɨ ɪɚɡɥɨɠɟɧɢɹ ɞɥɹ ɩɨɫɬɨɹɧɧɵɯ ɢ ɩɟɪɟɦɟɧɧɵɯ ɫɜɨɣɫɬɜ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɩɥɨɬɧɨɫɬɢ ɩɨɬɨɤɚ ɩɚɞɚɸɳɟɝɨ ɢɡɥɭɱɟɧɢɹ
q0 ɩɨɤɚɡɚɧɨ ɧɚ ɪɢɫ. 12.15.
Ɋɢɫ. 12.14. Ɂɚɜɢɫɢɦɨɫɬɶ ɬɟɦɩɟɪɚɬɭɪɵ (ɤɪɢɜɵɟ 1,4), ɫɬɟɩɟɧɢ ɩɪɟɜɪɚɳɟɧɢɹ (3,6) ɢ ɩɨɬɨɤɚ, ɩɪɨɩɭɳɟɧɧɨɝɨ
ɩɥɟɧɤɨɣ (2,5) ɫ ɩɨɫɬɨɹɧɧɵɦɢ (4-6) ɢ
ɩɟɪɟɦɟɧɧɵɦɢ (1-3) ɨɩɬɢɱɟɫɤɢɦɢ
ɫɜɨɣɫɬɜɚɦɢ
Ɋɢɫ. 12.15. Ɂɚɜɢɫɢɦɨɫɬɶ ɨɬ ɜɟɥɢɱɢɧɵ
ɩɨɬɨɤɚ ɜɪɟɦɟɧɢ ɪɚɡɥɨɠɟɧɢɹ ɩɥɟɧɤɢ ɫ
ɩɟɪɟɦɟɧɧɵɦɢ (ɤɪɢɜɚɹ 1) ɢ ɩɨɫɬɨɹɧɧɵɦɢ ɨɩɬɢɱɟɫɤɢɦɢ ɫɜɨɣɫɬɜɚɦɢ (2)
338
Ȼɨɥɟɟ ɢɧɬɟɪɟɫɧɵɟ ɪɟɡɭɥɶɬɚɬɵ ɩɨɥɭɱɚɸɬɫɹ ɩɪɢ ɭɱɟɬɟ ɤɨɧɟɱɧɨɫɬɢ
ɲɢɪɢɧɵ ɩɚɞɚɸɳɟɝɨ ɩɨɬɨɤɚ ɥɚɡɟɪɧɨɝɨ ɢɡɥɭɱɟɧɢɹ ɢ ɫɬɚɞɢɣɧɨɫɬɢ ɪɟɚɤɰɢɢ
ɪɚɡɥɨɠɟɧɢɹ.
ɂɡɜɟɫɬɧɨ, ɱɬɨ ɩɨɥɢɦɟɪɵ ɪɚɡɥɚɝɚɸɬɫɹ ɜ ɧɟɫɤɨɥɶɤɨ ɫɬɚɞɢɣ. ɉɟɪɜɚɹ ɢɡ
ɧɢɯ – ɷɧɞɨɬɟɪɦɢɱɟɫɤɚɹ, ɩɪɨɬɟɤɚɟɬ ɜ ɨɛɴɟɦɟ ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɩɨɝɥɨɳɚɸɳɢɯ ɰɟɧɬɪɨɜ. ɋɥɟɞɭɸɳɚɹ ɫɬɚɞɢɹ – ɝɟɬɟɪɨɝɟɧɧɚɹ ɪɟɚɤɰɢɹ, ɩɪɢɜɨɞɹɳɚɹ ɤ
ɭɦɟɧɶɲɟɧɢɸ ɬɨɥɳɢɧɵ ɩɥɟɧɤɢ, ɹɜɥɹɟɬɫɹ ɷɤɡɨɬɟɪɦɢɱɟɫɤɨɣ. Ⱦɨɝɨɪɚɧɢɟ
ɩɪɨɞɭɤɬɨɜ ɪɚɡɥɨɠɟɧɢɹ ɩɪɨɢɫɯɨɞɢɬ ɜ ɝɚɡɨɜɨɣ ɮɚɡɟ ɢ ɞɥɹ ɧɚɫ ɢɧɬɟɪɟɫɚ ɧɟ
ɩɪɟɞɫɬɚɜɥɹɟɬ.
Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɦɨɞɟɥɶ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɭɫɥɨɠɧɹɟɬɫɹ ɢ ɜɤɥɸɱɚɟɬ
ɭɪɚɜɧɟɧɢɟ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
cU h
wT
wt
h
O w § wT
¨r
r wr © wr
·
¸ qa Q1h)1 2Q2) 2 2D T T0 ,
¹
(12.57)
ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɪɟɚɤɰɢɢ ɜ ɨɛɴɟɦɟ, ɚɧɚɥɨɝɢɱɧɨɟ (12.53),
dK
dt
k1 T 1 K )1
ɤɢɧɟɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ ɝɟɬɟɪɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ
dh
dt
) 2
­ k2 T K
, h z 0;
°
1
k
T
K
E
®
2
°
h 0,
¯0 ,
(12.58)
ɧɚɱɚɥɶɧɵɟ ɢ ɝɪɚɧɢɱɧɵɟ ɭɫɥɨɜɢɹ
K T T0
wT
wr
r 0
0 , h h0 , t
wT
wr
0;
0,
r of
§ E ·
– ɫɤɨɪɨɫɬɶ ɝɨɦɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ,
k1 T k10exp ¨ 1 ¸
© RT ¹
§ E ·
k2 T k20exp ¨ 2 ¸ – ɫɤɨɪɨɫɬɶ ɝɟɬɟɪɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ ɜ ɤɢɧɟɬɢɱɟ© RT ¹
ɫɤɨɦ ɪɟɠɢɦɟ, E - ɫɤɨɪɨɫɬɶ ɝɟɬɟɪɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ ɜ ɞɢɮɮɭɡɢɨɧɧɨɦ ɪɟɠɢɦɟ.
Ɋɚɞɢɚɰɢɨɧɧɵɣ ɩɨɬɨɤ, ɩɨɝɥɨɳɟɧɧɵɣ ɢ ɩɪɨɩɭɳɟɧɧɵɣ ɩɥɟɧɤɨɣ ɜ
ɷɬɨɦ ɫɥɭɱɚɟ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ
ɝɞɟ
339
qa
^
ª § r ·2 º
q0exp « ¨ ¸ » u
«¬ © a ¹ »¼
`
u 1 exp ª¬ V K h t º¼ ª¬1 f K º¼ ,
ª § r ·2 º
) q0exp « ¨ ¸ » u
«¬ © a ¹ »¼
.
uexp ª¬V K h t º¼ ª¬1 f K º¼ .
ɗɬɚ ɡɚɞɚɱɚ ɬɚɤɠɟ ɪɟɲɚɟɬɫɹ ɱɢɫɥɟɧɧɨ.
Ɋɢɫ. 12.16. ɂɡɦɟɧɟɧɢɟ ɜɨ ɜɪɟɦɟɧɢ
ɂɡ ɪɢɫ. 12.16. ɜɢɞɧɨ, ɱɬɨ ɢɡɦɟɩɨɬɨɤɚ, ɩɪɨɩɭɳɟɧɧɨɝɨ ɩɥɟɧɤɨɣ
ɧɟɧɢɟ ɨɩɬɢɱɟɫɤɢɯ ɯɚɪɚɤɬɟɪɢɫɬɢɤ
(1), ɬɟɦɩɟɪɚɬɭɪɵ (2), ɫɬɟɩɟɧɢ
ɩɪɟɜɪɚɳɟɧɢɹ ɜ ɨɛɴɟɦɧɨɣ ɪɟɚɤɰɢɢ ɩɪɨɢɫɯɨɞɢɬ ɧɚ ɫɬɚɞɢɢ ɯɢɦɢɱɟɫɤɢɯ
ɪɟɚɤɰɢɣ. ɇɚ ɫɬɚɞɢɢ ɢɧɟɪɬɧɨɝɨ ɧɚɝɪɟ(3), ɬɨɥɳɢɧɵ ɩɥɟɧɤɢ (4) ɢ ɩɨɝɥɨɜɚ ɢɡɦɟɧɟɧɢɣ ɧɟɬ. Ʉɪɢɜɵɟ, ɫɨɨɬɜɟɬɫɬɳɟɧɧɨɝɨ ɩɨɬɨɤɚ (5) ɜ ɰɟɧɬɪɟ
ɩɹɬɧɚ ɧɚɝɪɟɜɚ. q0 1,5 ˜103 ȼɬ/ɫɦ2, ɜɭɸɳɢɟ ɩɪɨɩɭɳɟɧɧɨɦɭ ɩɨɬɨɤɭ (1) ɢ
ɬɟɦɩɟɪɚɬɭɪɟ (2) ɤɚɱɟɫɬɜɟɧɧɨ ɫɨɨɬɜɟɬa 3,1 ˜10 2 ɫɦ
ɫɬɜɭɸɬ ɨɫɰɢɥɥɨɝɪɚɦɦɚɦ ɩɪɨɰɟɫɫɚ.
ɏɨɪɨɲɨ ɨɬɫɥɟɠɢɜɚɟɬ ɪɚɡɞɟɥɟɧɢɟ ɯɢɦɢɱɟɫɤɢɯ ɫɬɚɞɢɣ ɤɪɢɜɚɹ 5, ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɩɨɬɨɤɭ, ɩɨɝɥɨɳɟɧɧɨɦɭ ɩɥɟɧɤɨɣ.
ɑɢɫɥɟɧɧɨɟ ɢɫɫɥɟɞɨɜɚɧɢɟ ɦɨɞɟɥɢ ɩɨɡɜɨɥɢɥɨ ɜɵɹɜɢɬɶ ɪɹɞ ɢɧɬɟɪɟɫɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɷɮɮɟɤɬɨɜ, ɤɨɬɨɪɵɟ ɭɞɨɛɧɨ ɨɩɢɫɵɜɚɬɶ ɫ ɩɨɦɨɳɶɸ ɛɟɡɪɚɡɦɟɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ
Ot f f
G
,
cU a 2
ɝɞɟ t f – ɜɪɟɦɹ ɪɟɡɤɨɝɨ ɪɨɫɬɚ ɬɟɦɩɟɪɚɬɭɪɵ ɩɥɟɧɤɢ ɢɡ-ɡɚ «ɩɨɞɤɥɸɱɟɧɢɹ»
f
ɝɟɬɟɪɨɝɟɧɧɨɣ ɪɟɚɤɰɢɢ ɩɪɢ ɟɟ ɧɚɝɪɟɜɟ ɩɨɬɨɤɨɦ «ɛɟɫɤɨɧɟɱɧɨɝɨ» ɪɚɞɢɭɫɚ.
ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɪɚɫɱɟɬɨɜ ɭɫɬɚɧɨɜɥɟɧɚ ɡɚɜɢɫɢɦɨɫɬɶ
­e x p G C , G ! C ;
tf
°
®
t f f °1,
G d C,
¯
ɝɞɟ C | 0 . 22 .
ɉɪɢ G 1 ɜ ɨɛɥɚɫɬɢ r a ɮɨɪɦɢɪɭɸɬɫɹ ɢ ɪɚɫɩɪɨɫɬɪɚɧɹɸɬɫɹ ɤ ɝɪɚɧɢɰɚɦ ɩɭɱɤɚ ɬɟɦɩɟɪɚɬɭɪɧɨ-ɤɨɧɰɟɧɬɪɚɰɢɨɧɧɵɣ ɮɪɨɧɬ (ɪɢɫ. 12.17, ɚ), ɡɚɬɭɯɚɸɳɚɹ ɜɨɥɧɚ ɩɨɝɥɨɳɟɧɢɹ (ɪɢɫ. 12.17, ɛ), ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɟɪɜɨɣ
ɫɬɚɞɢɢ ɪɟɚɤɰɢɢ, ɢ ɜɨɥɧɚ ɩɪɨɫɜɟɬɥɟɧɢɹ (ɪɢɫ. 12.17, ɜ), ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ
340
ɜɬɨɪɨɣ ɫɬɚɞɢɢ ɪɟɚɤɰɢɢ, ɩɪɢɜɨɞɹɳɟɣ ɤ ɢɡɦɟɧɟɧɢɸ ɬɨɥɳɢɧɵ ɩɥɟɧɤɢ ɢ
ɨɛɪɚɡɨɜɚɧɢɸ «ɞɵɪɤɢ».
)
qa
Ɋɢɫ. 12.17. ɉɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ (ɚ, ɫɩɥɨɲɧɵɟ
ɥɢɧɢɢ), ɫɬɟɩɟɧɢ ɩɪɟɜɪɚɳɟɧɢɹ (ɚ, ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ), ɩɨɝɥɨɳɟɧɧɨɝɨ ɩɨɬɨɤɚ
(ɛ), ɬɨɥɳɢɧɵ ɩɥɟɧɤɢ (ɜ, ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ), ɩɪɨɩɭɳɟɧɧɨɝɨ ɩɨɬɨɤɚ
(ɜ, ɩɭɧɤɬɢɪɧɵɟ ɥɢɧɢɢ) ɜ ɪɚɡɥɢɱɧɵɟ ɦɨɦɟɧɬɵ ɜɪɟɦɟɧɢ; q0 = 1,5·103 ȼɬ/ɫɦ2,
į = 0,087 ɫɦ. t, c = 1 – 0,05; 2 – 0,058; 3 – 0,06; 4 –0,062; 5 – 0,065; 6– 0,073
ȿɳɟ ɨɞɢɧ ɢɧɬɟɪɟɫɧɵɣ ɷɮɮɟɤɬ – ɹɜɥɟɧɢɟ ɷɧɟɪɝɟɬɢɱɟɫɤɨɣ ɫɚɦɨɮɨɤɭɫɢɪɨɜɤɢ ɩɹɬɧɚ ɧɚɝɪɟɜɚ, ɤɨɬɨɪɵɣ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ.
Ɉɩɪɟɞɟɥɢɦ ɷɮɮɟɤɬɢɜɧɵɣ ɪɚɞɢɭɫ ɩɭɱɤɚ reff ɬɚɤ, ɱɬɨɛɵ ɩɪɢ r reff
ɩɨɝɥɨɳɚɟɦɵɣ ɩɨɬɨɤ qa ɛɵɥ ɜ e ɪɚɡ ɦɟɧɶɲɟ qa r 0 . Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ
ɜɫɥɟɞɫɬɜɢɟ ɧɟɪɚɜɧɨɦɟɪɧɨɫɬɢ ɧɚɝɪɟɜɚ ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɧɟɪɚɜɧɨɦɟɪɧɨɫɬɢ ɩɪɟɜɪɚɳɟɧɢɹ, ɨɩɬɢɱɟɫɤɢɟ ɫɜɨɣɫɬɜɚ (ɜ ɱɚɫɬɧɨɫɬɢ, ɩɨɝɥɨɳɚɬɟɥɶɧɚɹ
ɫɩɨɫɨɛɧɨɫɬɶ) ɜ ɩɪɢɨɫɟɜɨɣ ɨɛɥɚɫɬɢ r 0 ɦɟɧɹɟɬɫɹ ɫɢɥɶɧɟɟ, ɱɟɦ ɧɚ ɤɪɚɸ
ɩɭɱɤɚ. ɗɬɨ ɩɪɢɜɨɞɢɬ ɤ ɬɨɦɭ, ɱɬɨ ɨɛɥɚɫɬɶ ɧɚɢɛɨɥɟɟ ɫɢɥɶɧɨɝɨ ɩɨɝɥɨɳɟɧɢɹ
ɥɨɤɚɥɢɡɭɟɬɫɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ r 0 . ɗɬɨ ɹɜɥɟɧɢɟ ɫɥɚɛɨ ɜɵɪɚɠɟɧɨ ɞɥɹ
ɨɱɟɧɶ ɭɡɤɢɯ ɩɭɱɤɨɜ ɢɡ-ɡɚ ɭɫɢɥɟɧɧɨɣ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɫ ɝɪɚɧɢɰ ɩɹɬɧɚ
ɧɚɝɪɟɜɚ ɢ ɯɨɪɨɲɨ ɩɪɨɹɜɥɹɟɬɫɹ ɩɪɢ G d 1 .
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɪɚɡɥɢɱɧɵɯ ɮɢɡɢɱɟɫɤɢɯ ɹɜɥɟɧɢɣ
ɢɝɪɚɟɬ ɜɚɠɧɭɸ ɪɨɥɶ ɜ ɩɪɨɰɟɫɫɚɯ ɨɛɪɚɛɨɬɤɢ ɦɚɬɟɪɢɚɥɨɜ ɜɵɫɨɤɨɷɧɟɪɝɟɬɢɱɟɫɤɢɦɢ ɢɫɬɨɱɧɢɤɚɦɢ.
Ɂɚɞɚɧɢɹ
1. ɇɚɣɞɢɬɟ ɪɟɲɟɧɢɟ ɡɚɞɚɱɢ ɨɛ ɨɯɥɚɠɞɟɧɢɢ ɩɨɪɢɫɬɨɣ ɩɥɚɫɬɢɧɵ, ɩɨɥɚɝɚɹ, ɱɬɨ ɨɯɥɚɠɞɚɸɳɚɹ ɠɢɞɤɨɫɬɶ x 0 ɩɨɫɬɭɩɚɟɬ ɧɚɝɪɟɬɨɣ, ɬ.ɟ., ɢɫ341
ɩɨɥɶɡɭɹ ɜɦɟɫɬɨ ɩɨɫɬɨɹɧɫɬɜɚ ɬɟɦɩɟɪɚɬɭɪɵ ɧɚ ɝɪɚɧɢɰɟ T1 T0 ɭɫɥɨɜɢɟ
(12.2).
2. Ɇɟɞɧɵɣ ɩɪɨɜɨɞ ɞɢɚɦɟɬɪɨɦ 1 ɦɦ ɪɚɜɧɨɦɟɪɧɨ ɩɨɤɪɵɬ ɢɡɨɥɹɰɢɟɣ
ɢɡ ɩɥɚɫɬɢɤɚ. ȼɧɟɲɧɢɣ ɞɢɚɦɟɬɪ ɢɡɨɥɹɰɢɨɧɧɨɝɨ ɩɨɤɪɵɬɢɹ 3ɦɦ. ɋɪɟɞɚ, ɜ
ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɩɪɨɜɨɞ, ɢɦɟɟɬ ɬɟɦɩɟɪɚɬɭɪɭ T 37 . 8 ɋ. Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɨɬ ɜɧɟɲɧɟɣ ɩɨɜɟɪɯɧɨɫɬɢ ɩɥɚɫɬɢɤɚ ɜ ɨɤɪɭɠɚɸɳɭɸ ɫɪɟɞɭ - D 2 . 26 ɤɤɚɥ/(ɦ2ɱɚɫ Ʉ). Ʉɚɤɨɣ ɦɚɤɫɢɦɚɥɶɧɵɣ ɬɨɤ ɦɨɠɧɨ ɩɪɨɩɭɫɤɚɬɶ ɱɟɪɟɡ ɷɬɨɬ ɩɪɨɜɨɞ, ɧɟ ɨɩɚɫɚɹɫɶ ɩɟɪɟɝɪɟɜɚ ɢɡɨɥɹɰɢɢ, ɪɚɛɨɱɚɹ ɬɟɦɩɟɪɚɬɭɪɚ ɤɨɬɨɪɨɣ T 93 . 3 ɋ. Ⱦɥɹ ɪɚɫɱɟɬɚ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɚɧɧɵɦɢ: ɤɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ O ɞɥɹ ɩɪɨɜɨɥɨɤɢ 100 ɢ ɩɥɚɫɬɢɤɚ 0.092
ɤɤɚɥ/(ɦ ɱɚɫ Ʉ) ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɷɥɟɤɬɪɨɩɪɨɜɨɞɧɨɫɬɢ Oe
ɞɥɹ ɩɪɨɜɨɞɚ ɢ ɩɥɚɫɬɢɤɚ – 5 . 1 ˜ 1 05 ɢ 0.
3. Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɥɨɜɵɞɟɥɹɸɳɢɣ ɷɥɟɦɟɧɬ ɜ ɜɢɞɟ ɞɥɢɧɧɨɝɨ ɫɬɟɪɠɧɹ, ɡɚɤɥɸɱɟɧɧɨɝɨ ɜ ɚɥɸɦɢɧɢɟɜɭɸ ɨɛɨɥɨɱɤɭ. ȼɧɭɬɪɢ ɫɬɟɪɠɧɹ ɜɫɥɟɞɫɬɜɢɟ ɞɟɥɟɧɢɹ ɹɞɟɪ ɜɵɪɚɛɚɬɵɜɚɟɬɫɹ ɬɟɩɥɨ, ɩɪɢɱɟɦ ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɹɞɟɪɧɨɝɨ ɢɫɬɨɱɧɢɤɚ ɬɟɩɥɚ ɪɚɫɩɪɟɞɟɥɟɧɚ ɩɨ ɡɚɤɨɧɭ (12.15). ȼɵɱɢɫɥɢɬɶ ɦɚɤɫɢɦɚɥɶɧɭɸ ɬɟɦɩɟɪɚɬɭɪɭ ɜɧɭɬɪɢ ɬɟɩɥɨɜɵɞɟɥɹɸɳɟɝɨ ɷɥɟɦɟɧɬɚ, ɟɫɥɢ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɜɧɟɲɧɹɹ ɩɨɜɟɪɯɧɨɫɬɶ ɚɥɸɦɢɧɢɟɜɨɣ ɨɛɨɥɨɱɤɢ, ɢɦɟɸɳɟɣ ɪɚɞɢɭɫ Rc , ɧɚɯɨɞɢɬɫɹ ɜ ɤɨɧɬɚɤɬɟ ɫ ɠɢɞɤɢɦ ɬɟɩɥɨɧɨɫɢɬɟɥɟɦ ɬɟɦɩɟɪɚɬɭɪɵ
Te . Ɋɚɫɫɦɨɬɪɟɬɶ ɜɚɪɢɚɧɬɵ ɚ) ɤɨɧɬɚɤɬ – ɢɞɟɚɥɶɧɵɣ; ɛ) ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ ɟɫɬɶ De .
4. ɋɮɨɪɦɭɥɢɪɨɜɚɬɶ ɢ ɪɟɲɢɬɶ ɡɚɞɚɱɭ ɨ ɬɟɩɥɨɨɛɦɟɧɟ ɦɟɠɞɭ ɞɜɭɦɹ
ɤɨɚɤɫɢɚɥɶɧɵɦɢ ɰɢɥɢɧɞɪɚɦɢ ɞɥɹ ɧɟɧɶɸɬɨɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ, ɨɩɢɫɵɜɚɟɦɨɣ ɦɨɞɟɥɶɸ Ɉɫɬɜɚɥɶɞɚ–ȼɟɣɥɹ
n 1
dv
d vz
W xz m z
.
dx
dx
ɉɨɤɚɡɚɬɶ, ɱɬɨ ɩɪɢ ɷɬɨɦ ɞɥɹ ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɛɭɞɟɬ ɫɩɪɚɜɟɞɥɢɜɚ ɬɚ ɠɟ ɮɨɪɦɭɥɚ, ɧɨ ɜ ɧɟɣ ɱɢɫɥɨ Br ɫɥɟɞɭɟɬ ɡɚɦɟɧɢɬɶ ɧɚ ɱɢɫɥɨ
mV n 1
Br
.
O b n 1 Tb T0 5. ɉɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɷɬɭ ɠɟ ɡɚɞɚɱɭ ɞɥɹ ɧɶɸɬɨɧɨɜɫɤɨɣ ɠɢɞɤɨɫɬɢ,
ɩɨɥɚɝɚɹ, ɱɬɨ ɧɚ ɨɛɟɢɯ ɩɨɜɟɪɯɧɨɫɬɹɯ ɩɨɞɞɟɪɠɢɜɚɟɬɫɹ ɩɨɫɬɨɹɧɧɚɹ ɬɟɦɩɟɪɚɬɭɪɚ T0 , ɚ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɢ ɜɹɡɤɨɫɬɶ ɡɚɜɢɫɹɬ ɨɬ ɬɟɦɩɟɪɚɬɭɪɵ
O
1 D1T D 2T2 ... ;
O0
342
P
1 E1T E2T2 .. .
P0
ɍɤɚɡɚɧɢɟ: ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɩɪɢɟɦɨɦ, ɨɩɢɫɚɧɧɵɦ ɜ ɪɚɡɞɟɥɟ 12.3.
6. Ɉɫɧɨɜɵɜɚɹɫɶ ɧɚ ɨɩɢɫɚɧɧɨɦ ɚɥɝɨɪɢɬɦɟ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ (12.27) ɫ
ɧɟɥɢɧɟɣɧɵɦɢ ɤɢɧɟɬɢɱɟɫɤɢɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ, ɫɨɫɬɚɜɢɬɶ ɢ ɨɬɥɚɞɢɬɶ
ɩɪɨɝɪɚɦɦɭ ɟɟ ɱɢɫɥɟɧɧɨɝɨ ɢɫɫɥɟɞɨɜɚɧɢɹ. Ɂɚɞɚɬɶ ɡɚɤɨɧ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ ɜ ɜɢɞɟ
T T0 a t .
ɉɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɩɨɥɭɱɟɧɧɵɣ ɪɟɡɭɥɶɬɚɬ, ɜɚɪɶɢɪɭɹ ɫɤɨɪɨɫɬɶ ɪɨɫɬɚ ɬɟɦɩɟɪɚɬɭɪɵ. ɋɪɚɜɧɢɬɶ ɪɟɡɭɥɶɬɚɬɵ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɨɣ ɡɚɞɚɱɢ ɫ ɡɚɞɚɱɟɣ, ɜ
ɤɨɬɨɪɨɣ ɜɫɟ ɤɢɧɟɬɢɱɟɫɤɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɫɬɨɹɧɧɵ.
7. ɂɫɫɥɟɞɨɜɚɬɶ ɡɚɞɚɱɭ (12.27) ɫ ɡɚɤɨɧɨɦ ɢɡɦɟɧɟɧɢɹ ɬɟɦɩɟɪɚɬɭɪɵ
T T0 at bt 2 .
ɑɬɨ ɢɡɦɟɧɹɟɬɫɹ ɜ ɪɟɲɟɧɢɢ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɟɣ?
343
ɉɊɂɅɈɀȿɇɂə
ɉɪɢɥɨɠɟɧɢɟ 1
Ɉɫɧɨɜɧɵɟ ɬɟɨɪɟɦɵ ɢ ɩɪɚɜɢɥɚ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ Ʌɚɩɥɚɫɚ
ʋ
ɨɩɟɪɚɰɢɹ
1
Ɉɩɪɟɞɟɥɟɧɢɟ
2
3
4
ɋɥɨɠɟɧɢɟ
ɢ
ɭɦɧɨɠɟɧɢɟ
ɧɚ ɱɢɫɥɨ
Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ
ɨɪɢɝɢɧɚɥɚ
f W
8
9
10
³
F pe dp
pW
V if
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɨɪɢɝɢɧɚɥɚ
ɂɡɦɟɧɟɧɢɟ ɦɚɫɲɬɚɛɚ
ɋɞɜɢɝ
ɚɪɝɭɦɟɧɬɚ ɭ
ɨɪɢɝɢɧɚɥɚ*
ɋɞɜɢɝ
ɚɪɝɭɦɟɧɬɚ ɭ
ɢɡɨɛɪɚɠɟɧɢɹ
F p
f
³ f W e x p pW d W
0
Af W B g W AF p BG p f c W
pF p f 0 n
f W
p n F p p n 1 f 0 n 1
p n 2 f c 0 .. . f 0 1
F p
p
³ f t d t
0
WT
³³
00
7
V if
1
2S i
ɢɡɨɛɪɚɠɟɧɢɟ
W
5
6
ɨɪɢɝɢɧɚɥ
f t dt d T
f aW ,
1
p2
F p
a c on s t
1 § p·
F¨ ¸
a ©a¹
f W b , b ! 0
e bp F p e aW f W F p a
1 bW § W ·
e f¨ ¸
a
©a¹
F ap b 344
ɉɪɨɞɨɥɠɟɧɢɟ ɩɪɢɥɨɠɟɧɢɹ 1
11
12
13
14
15
Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ
ɢɡɨɛɪɚɠɟɧɢɹ
ɂɧɬɟɝɪɢɪɨɜɚɧɢɟ ɢɡɨɛɪɚɠɟɧɢɹ
ɋɨɨɬɜɟɬɫɬɜɢɟ
ɩɪɟɞɟɥɨɜ**
ɋɜɟɪɬɤɚ
ɨɪɢɝɢɧɚɥɨɜ
W f W 1
Fc p W f W
f
³ F p dp
1
f W
W
p
li m f W , a Doa
³ f1 t f 2 W t dt
f1 f 2
0
Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ ɢ
ɢɧɬɟɝɪɢɪɨɜɚɧɢɟ ɩɨ
ɩɚɪɚɦɟɬɪɭ
w
f t ,D wD
17
18
f
³
0
F1 p F2 p w
F p, D wD
a2
a2
a1
a1
³ f t,D d D
1
St
lim F p , a Doa
W
f 2 f1
16
r
F p
r r
§ y2 ·
f ye x p¨ dy
¨ 4t ¸¸
©
¹
345
³ F p,D d D
1
F
p
p
Ɉɤɨɧɱɚɧɢɟ ɩɪɢɥɨɠɟɧɢɹ 1
f
19
sy
³e f
F ª¬M p º¼ ) s ydy ,
0
e
ɝɞɟ
f
f
f
0
0
y ³ f W \ W, y d W
20
n
) p
³e
¦ \c pk e p k W
k 1
WM p k
\ p
py
) p
\ W , y dy
) p
,
\ p
ɝɞɟ
p p1 p p 2 ...
.. . p p n 21
ɋɥɭɱɚɣ
ɤɪɚɬɧɵɯ
ɤɨɪɧɟɣ
­° d k 1
1
lim
¦ k 1 ! p o p ® dp k 1 u
k °̄
k 1
n
ª ) p p p k
º ½°
W
m
p
e »¾
u«
\ p
«
»°
¬
¼¿
346
\ p
) p
,
\ p
ɝɞɟ
p p1 p p 2 ...
. .. p p m k
p p m1 .. . p p n ɉɪɢɥɨɠɟɧɢɟ 2
ɂɡɨɛɪɚɠɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɮɭɧɤɰɢɣ
1
2
ɂɡɨɛɪɚɠɟɧɢɟ
1
p
1
Ɉɪɢɝɢɧɚɥ
1
W
p2
3
W n1
n 1 !
1
SW
1
p
, n 1, 2 ,3,.. .
n
4
1
p
5
p
6
2
3 2
1
pa
1
pa
1
7
8
e aW
e -aW
W e aW
p a2
9
1
p a
10
1
W n 1e aW
n 1 !
, n 1, 2 ,3,.. .
n
1
1
e aW e bW
ab
p a p b 11
k
12
p2 k2
p
13
p2 k2
k
14
p2 k2
p
W
S
sin kW cos kW sh kW ch kW p2 k2
347
ɉɪɨɞɨɥɠɟɧɢɟ ɩɪɢɥɨɠɟɧɢɹ 2
15
p a pb
16
17
p
e bW e aW SW 3
1
2
2
1
k e k We rfc k W
SW
1
p k
1
k 2W erfc k W
p k
18
k
e k p , k t 0
1 k p
, k t0
e
p
19
20
1 k
e
p
p
2 SW 3
p p
e k
, k t0
p
, k t0
22
p b
23
25
1
p b
24
p
p
e k
e k
p
, k t0
p
k 2 4W § k ·
2W i erfc ¨
¸
©2 W¹
2
b
§ k ·
erfc ¨
¸
©2 W¹
k k 2 4W
e
SW
21
1
e
, k t0
1 k p
e
p
1
e k p
p p
W k 2 4W
§ k ·
k erfc ¨
e
¸
S
©2 W¹
§ k · b k b 2W
erfc ¨
¸e e u
©2 W¹
k ·
§
uerfc ¨ b W ¸
2 W¹
©
2
k ·
§
e bk e b W u erfc ¨ b W ¸
2 W¹
©
1
co s 2 kW
SW
1
s in 2 k W
Sk
348
ɉɪɢɥɨɠɟɧɢɟ 3
Ɍɚɛɥɢɰɚ 3.1
Ɍɟɩɥɨɟɦɤɨɫɬɢ ɢ ɩɥɨɬɧɨɫɬɢ ɧɟɤɨɬɨɪɵɯ ɜɟɳɟɫɬɜ
ɩɪɢ ɤɨɦɧɚɬɧɨɣ ɬɟɦɩɟɪɚɬɭɪɟ ɢ ɚɬɦɨɫɮɟɪɧɨɦ ɞɚɜɥɟɧɢɢ
ȼɟɳɟɫɬɜɨ
Ⱥɥɸɦɢɧɢɣ
ȼɨɥɶɮɪɚɦ
ɀɟɥɟɡɨ
Ɇɟɞɶ
ɇɢɤɟɥɶ
ɉɥɚɬɢɧɚ
Ɍɚɧɬɚɥ
ɏɪɨɦ
ɐɢɪɤɨɧɢɣ
Ⱦɸɪɚɥɸɦɢɧɢɣ
Ⱥɥɸɦɢɧɢɟɜɚɹ ɛɪɨɧɡɚ
Ⱥɫɛɟɫɬ
Ȼɟɬɨɧ (ɫɭɯɨɣ)
Ƚɪɚɧɢɬ
Ⱦɭɛ
ɉɢɯɬɚ
Ʉɢɪɩɢɱ ɫɬɪɨɢɬɟɥɶɧɵɣ
Ʉɨɪɭɧɞ
Ɍɟɮɥɨɧ
(ɩɨɥɢɬɟɬɪɚɮɬɨɪɷɬɢɥɟɧ)
ɉɨɥɢɚɦɢɞ (ɧɟɣɥɨɧ)
ɉɪɨɛɤɚ
ɋɬɟɤɥɨ ɨɤɨɧɧɨɟ
ɍɝɨɥɶ, ɚɧɬɪɚɰɢɬ
Ⱥɥɦɚɡ
Ƚɪɚɮɢɬ
ȼɨɡɞɭɯ
Ⱥɡɨɬ
Ʉɢɫɥɨɪɨɞ
Ɇɟɬɚɧ
ȼɨɞɚ
Ƚɥɢɰɟɪɢɧ
Ɍɟɩɥɨɟɦɤɨɫɬɶ,
c p , Ⱦɠ/(ɤɝ.Ʉ)
ɉɥɨɬɧɨɫɬɶ,
U , ɤɝ/ɦ3
896
134
452
383
446
133
138
440
272
833
410
816
837
820
2390
2720
837
779
1050
2702
19300
7870
8933
8900
21450
16600
7160
6570
2787
8666
383
500
2640
609-801
600
1700
3900
2200
1060-1120
1880
800
1260
510
710
1012
910,9
1041
2636
4182
1261
1470–1680
150
2800
1370
3250
1730
1,164
1,301
1,138
0,717
998,2
2350
349
Ɍɚɛɥɢɰɚ 3.2
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ ɧɟɤɨɬɨɪɵɯ ɜɟɳɟɫɬɜ,
Ȝ ȼɬ/(ɦ·Ʉ), Ɍ=273 Ʉ 42
Ⱥɥɸɦɢɧɢɣ
ȼɨɥɶɮɪɚɦ
ɀɟɥɟɡɨ
236
179
81,1
Ɇɚɝɧɢɣ
Ɇɚɪɝɚɧɟɰ
Ɇɟɞɶ
156
7,78
399
Ɇɨɥɢɛɞɟɧ
138
ɇɢɤɟɥɶ
Ɍɢɬɚɧ
ɏɪɨɦ
ɐɢɧɤ
ȼɨɡɞɭɯ
Ʌɟɞ
91
22
91,4
121
0,024–0,075
2,5
Ⱥɫɛɟɫɬ
Ȼɟɬɨɧ
Ƚɥɢɧɚ
(48,7 % ɜɥ)
Ⱦɟɪɟɜɨ
Ʉɚɪɬɨɧ
Ʉɢɪɩɢɱ ɫɢɥɢɤɚɬɧɵɣ
Ʉɢɪɩɢɱ ɲɚɦɨɬɧɵɣ
Ʉɢɪɩɢɱ ɤɪɚɫɧɵɣ
Ɉɩɢɥɤɢ
Ʌɢɧɨɥɟɭɦ
ɋɬɟɤɥɨ ɨɤɨɧɧɨɟ
ɒɬɭɤɚɬɭɪɤɚ
ȼɨɞɚ
42
0,09–0,19
0,9–1,4
1,26
0,11–0,17
0,14–0,35
1,07
1–1,4
0,55–0,8
0,071
0,081
0,81
0,814
0,55–0,7
ȼɫɟ ɞɚɧɧɵɟ ɜɡɹɬɵ ɢɡ ɤɧɢɝ: Ʌɚɪɢɤɨɜ ɇ.ɂ. Ɍɟɩɥɨɬɟɯɧɢɤɚ. Ɇ.: ɋɬɪɨɣɢɡɞɚɬ. 1985.
432 ɫ.; Ʉɪɟɣɬ Ɏ, Ȼɥɷɤ ɍ Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. Ɇ.: Ɇɢɪ, 1983. – 512 ɫ.
350
Ɍɚɛɥɢɰɚ 3.3
Ɋɚɡɦɟɪɧɨɫɬɢ ɧɟɤɨɬɨɪɵɯ ɮɢɡɢɱɟɫɤɢɯ ɜɟɥɢɱɢɧ
Ɏɢɡɢɱɟɫɤɚɹ
ɜɟɥɢɱɢɧɚ
Ⱦɥɢɧɚ
Ɇɚɫɫɚ
ȼɪɟɦɹ
Ɉɛɴɟɦ
ɉɥɨɬɧɨɫɬɶ
ɋɤɨɪɨɫɬɶ
Ⱦɚɜɥɟɧɢɟ
ɗɧɟɪɝɢɹ
Ɇɨɳɧɨɫɬɶ
ɍɞɟɥɶɧɚɹ ɷɧɟɪɝɢɹ
Ɉɛɴɟɦɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ
ɉɥɨɬɧɨɫɬɶ ɦɨɳɧɨɫɬɢ = ɩɨɬɨɤ
Ʉɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ
Ɋɚɡɦɟɪɧɨɫɬɢ
1 ɦ=100 ɫɦ=1000 ɦɦ
1 ɤɝ=1000 ɝ
1 ɱɚɫ= 60 ɦɢɧ= 3600 ɫ
1 ɦ3 = 106 ɫɦ3
1 ɤɝ/ɦ3= ɝ/ɫɦ3
1 ɦ/ɫ=100 ɫɦ/ɫ= 3,6 ɤɦ/ɱɚɫ
1 ɉɚ=1 ɇ/ɦ2= 10-5 ɛɚɪ = 7,5·10-3 ɦɦ.ɪɬ.ɫɬ.
1 Ⱦɠ= 107 ɷɪɝ =0,102 ɤɝ·ɦ = 2,78·10-4 ȼɬ·ɱɚɫ =
0,239 ɤɚɥ
1 Ⱦɠ/ɫ =1 ȼɬ
Ⱦɠ/ɤɝ
Ⱦɠ/(ɦ3·Ʉ)
Ⱦɠ/(ɦ2·ɫ)
ȼɬ/(ɦ·Ʉ)
351
ɉɊɂɇəɌɕȿ ɈȻɈɁɇȺɑȿɇɂə
a – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɦɩɟɪɚɬɭɪɨɩɪɨɜɨɞɧɨɫɬɢ
A , w – ɪɚɛɨɬɚ, ɫɨɜɟɪɲɚɟɦɚɹ ɫɢɫɬɟɦɨɣ
A – ɚɪɯɢɦɟɞɨɜɚ ɫɢɥɚ
c – ɭɞɟɥɶɧɚɹ ɦɚɫɫɨɜɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ, Ⱦɠ/(ɤɝ·Ʉ);
cc – ɭɞɟɥɶɧɚɹ ɨɛɴɟɦɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ, Ⱦɠ/(ɦ3·Ʉ);
c p – ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɞɚɜɥɟɧɢɢ, Ⱦɠ/(ɤɝ·ɫ) ɢɥɢ Ⱦɠ/(ɦ3·ɫ)
cJ – ɬɟɩɥɨɟɦɤɨɫɬɶ ɩɪɢ ɩɨɫɬɨɹɧɧɨɦ ɨɛɴɟɦɟ
d – ɞɢɚɦɟɬɪ
f – ɷɧɟɪɝɢɹ Ƚɟɥɶɦɝɨɥɶɰɚ
F – ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ, ɫɦ2 ɢɥɢ ɦ2
Fi – ɫɨɨɬɧɨɲɟɧɢɟ ɫɢɥ ɢɧɟɪɰɢɢ
Fp – ɫɢɥɚ ɞɚɜɥɟɧɢɹ
F – ɜɧɟɲɧɹɹ ɫɢɥɚ
g – ɷɧɟɪɝɢɹ Ƚɢɛɛɫɚ
g – ɭɫɤɨɪɟɧɢɟ ɫɜɨɛɨɞɧɨɝɨ ɩɚɞɟɧɢɹ
G – ɫɢɥɚ ɬɹɠɟɫɬɢ,
H , h – ɷɧɬɚɥɶɩɢɹ, Ⱦɠ ɢɥɢ Ⱦɠ/ɤɝ; ɩɨɫɬɨɹɧɧɚɹ ɉɥɚɧɤɚ
K H – ɤɨɷɮɮɢɰɢɟɧɬ ɚɤɬɢɜɧɨɫɬɢ ɨɞɧɨɝɨ ɦɚɬɟɪɢɚɥɚ (ɫɪɟɞɵ) ɩɨ ɨɬɧɨɲɟɧɢɸ
ɤ ɞɪɭɝɨɦɭ (ɤ ɨɛɪɚɛɚɬɵɜɚɟɦɨɦɭ ɦɚɬɟɪɢɚɥɭ)
L – ɬɨɥɳɢɧɚ, ɦ, ɫɦ
M – ɦɚɫɫɚ ɜɟɳɟɫɬɜɚ, ɤɝ, ɝ
m – ɦɨɥɹɪɧɚɹ ɦɚɫɫɚ
n0 – ɟɞɢɧɢɱɧɵɣ ɜɟɤɬɨɪ ɧɨɪɦɚɥɢ
p – ɞɚɜɥɟɧɢɟ
q – ɜɟɤɬɨɪ ɩɥɨɬɧɨɫɬɢ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ, ȼɬ/ɫɦ2 ɢɥɢ Ⱦɠ/(ɫɦ2·ɫ) ɢɥɢ
Ⱦɠ/(ɦ2·ɫ)
qV – ɩɥɨɬɧɨɫɬɶ ɨɛɴɟɦɧɵɯ ɢɫɬɨɱɧɢɤɨɜ ɬɟɩɥɚ
Q – ɦɨɳɧɨɫɬɶ ɬɟɩɥɨɜɨɝɨ ɩɨɬɨɤɚ (ɢɥɢ ɬɟɩɥɨɜɨɣ ɩɨɬɨɤ), Ⱦɠ/ɫ ɢɥɢ ȼɬ
QW – ɤɨɥɢɱɟɫɬɜɨ ɬɟɩɥɨɬɵ, Ⱦɠ
rT – ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɟɞɢɧɢɰɵ ɩɥɨɳɚɞɢ
R = 8,314472 – ɭɧɢɜɟɪɫɚɥɶɧɚɹ ɝɚɡɨɜɚɹ ɩɨɫɬɨɹɧɧɚɹ,
Re – ɷɥɟɤɬɪɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ,
RT , RT ,k , – ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, ɤɨɧɬɚɤɬɧɨɟ ɬɟɪɦɢɱɟɫɤɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ, (ɦ2·Ʉ)/ȼɬ
s – ɷɧɬɪɨɩɢɹ
352
t ,W – ɜɪɟɦɹ, ɫ
T – ɬɟɦɩɟɪɚɬɭɪɚ
U , u – ɜɧɭɬɪɟɧɧɹɹ ɷɧɟɪɝɢɹ ɫɢɫɬɟɦɵ
u – ɜɟɤɬɨɪ ɫɤɨɪɨɫɬɢ ɫ ɤɨɦɩɨɧɟɧɬɚɦɢ u , v , w ɢɥɢ u1 ,u2 ,u3
V – ɨɛɴɟɦ, ɫɦ3
J – ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ,
v – ɦɨɥɶɧɵɣ ɨɛɴɟɦ
x , y , z – ɞɟɤɚɪɬɨɜɵ ɤɨɨɪɞɢɧɚɬɵ
D – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɨɬɞɚɱɢ, ȼɬ/(ɦ2·Ʉ)
E – ɤɨɷɮɮɢɰɢɟɧɬ ɦɚɫɫɨɨɬɞɚɱɢ
J – ɭɞɟɥɶɧɵɣ ɨɛɴɟɦ
JT – ɢɡɨɯɨɪɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɪɦɢɱɟɫɤɨɝɨ ɭɜɟɥɢɱɟɧɢɹ ɞɚɜɥɟɧɢɹ;
U – ɩɥɨɬɧɨɫɬɶ, ɤɝ/ɦ3
O , O eq – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ, ȼɬ/(ɦ·Ʉ)
P – ɤɨɷɮɮɢɰɢɟɧɬ ɞɢɧɚɦɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ
P c – ɭɞɟɥɶɧɚɹ ɦɨɥɶɧɚɹ ɬɟɩɥɨɟɦɤɨɫɬɶ, Ⱦɠ/(ɤɦɨɥɶ·Ʉ)
G – ɬɨɥɳɢɧɚ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ
GT – ɬɨɥɳɢɧɚ ɬɟɩɥɨɜɨɝɨ ɩɨɝɪɚɧɢɱɧɨɝɨ ɫɥɨɹ
Q – ɤɨɷɮɮɢɰɢɟɧɬ ɤɢɧɟɦɚɬɢɱɟɫɤɨɣ ɜɹɡɤɨɫɬɢ
[T – ɢɡɨɬɟɪɦɢɱɟɫɤɢɣ ɦɨɞɭɥɶ ɭɩɪɭɝɨɫɬɢ;
[T 1 – ɤɨɷɮɮɢɰɢɟɧɬ ɢɡɨɬɟɪɦɢɱɟɫɤɨɣ ɫɠɢɦɚɟɦɨɫɬɢ
W – ɬɟɧɡɨɪ ɜɹɡɤɢɯ ɧɚɩɪɹɠɟɧɢɣ
W x – ɤɨɷɮɮɢɰɢɟɧɬ ɬɪɟɧɢɹ
\ – ɮɭɧɤɰɢɹ ɬɨɤɚ
K – ɤɨɷɮɮɢɰɢɟɧɬ ɬɟɩɥɨɩɟɪɟɞɚɱɢ, ȼɬ/(ɦ2·Ʉ)
6 f – ɩɥɨɳɚɞɶ ɩɨɜɟɪɯɧɨɫɬɢ ɪɚɡɞɟɥɚ ɮɚɡ
K – ɨɛɴɟɦɧɚɹ ɞɨɥɹ ɜɟɳɟɫɬɜɚ
ɂ ɧɞ ɟ ɤɫ ɵ
f – ɠɢɞɤɨɫɬɶ
e – ɨɤɪɭɠɚɸɳɚɹ ɫɪɟɞɚ
s – ɩɨɜɟɪɯɧɨɫɬɶ
353
p – ɩɥɨɫɤɨɫɬɶ
c – ɰɢɥɢɧɞɪ
i n – ɜɧɭɬɪɟɧɧɢɣ ɪɚɡɦɟɪ
ext – ɜɧɟɲɧɢɣ ɪɚɡɦɟɪ
eq – ɷɤɜɢɜɚɥɟɧɬɧɨɟ ɡɧɚɱɟɧɢɟ
Ȼɟɡɪɚɡɦɟɪɧɵɟ ɤɪɢɬɟɪɢɢ
B i – ɱɢɫɥɨ Ȼɢɨ
E u – ɤɪɢɬɟɪɢɣ ɗɣɥɟɪɚ
Fm – ɫɨɨɬɧɨɲɟɧɢɟ ɦɚɫɫɨɜɵɯ ɫɢɥ
Fr – ɤɪɢɬɟɪɢɣ Ɏɪɭɞɚ
Gr – ɱɢɫɥɨ Ƚɪɚɫɝɨɮɚ
Ho – ɤɪɢɬɟɪɢɣ ɝɢɞɪɨɞɢɧɚɦɢɱɟɫɤɨɣ ɝɨɦɨɯɪɨɧɧɨɫɬɢ
Nu – ɱɢɫɥɨ ɇɭɫɟɥɶɞɚ
Pe – ɱɢɫɥɨ ɉɟɤɥɟ
P r – ɱɢɫɥɨ ɉɪɚɧɞɬɥɹ
Re – ɱɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ
R e x – ɥɨɤɚɥɶɧɨɟ ɱɢɫɥɨ Ɋɟɣɧɨɥɶɞɫɚ
354
ɋɉɂɋɈɄ ɅɂɌȿɊȺɌɍɊɕ
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2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
Kalyan Annamalai, Iswar K.Puri Advanced thermodynamics engineering / 2001.
Lienhard J.H., Lienhard J.H. A heat transfer textbook / 2003.
Ȼɚɪɪɟ ɉ. Ʉɢɧɟɬɢɤɚ ɝɟɬɟɪɨɝɟɧɧɵɯ ɩɪɨɰɟɫɫɨɜ. – Ɇ.: Ɇɢɪ, 1976. –
400 c.
Ȼɟɪɞ Ɋ., ɋɬɶɸɚɪɬ ȼ, Ʌɚɣɬɮɭɬ ȿ. əɜɥɟɧɢɹ ɩɟɪɟɧɨɫɚ. – Ɇ.: ɏɢɦɢɹ,
1974. – 688 c.
Ȼɨɤɲɬɟɣɧ Ȼ.ɋ., Ⱦɢɮɮɭɡɢɹ ɜ ɦɟɬɚɥɥɚɯ. – Ɇ.: Ɇɟɬɚɥɥɭɪɝɢɹ,
1978. – 248 ɫ.
Ƚɭɯɦɚɧ Ⱥ.Ⱥ. ɉɪɢɦɟɧɟɧɢɟ ɬɟɨɪɢɢ ɩɨɞɨɛɢɹ ɤ ɢɫɫɥɟɞɨɜɚɧɢɸ ɩɪɨɰɟɫɫɨɜ ɬɟɩɥɨɦɚɫɫɨɨɛɦɟɧɚ. – Ɇ: ȼɵɫɲɚɹ ɲɤɨɥɚ, 1974. – 328 ɫ.
Ɂɚɪɭɛɢɧ ȼ.ɋ. ɂɧɠɟɧɟɪɧɵɟ ɦɟɬɨɞɵ ɪɟɲɟɧɢɹ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. – Ɇ.: ɗɧɟɪɝɨɚɬɨɦɢɡɞɚɬ, 1983. – 328 ɫ.
Ɂɢɝɟɥɶ Ɋ., Ⱥɜɞɭɟɜɫɤɢɣ ȼ.ɋ. Ɍɟɩɥɨɨɛɦɟɧ ɢɡɥɭɱɟɧɢɟɦ. – Ɇ.: Ɇɢɪ,
1975. – 840 ɫ.
ɂɫɚɱɟɧɤɨ ȼ.ɉ., Ɉɫɢɩɨɜɚ ȼ.Ⱥ., ɋɭɤɨɦɟɥ Ⱥ.ɋ. Ɍɟɩɥɨɩɟɪɟɞɚɱɚ. – Ɇ.:
ɗɧɟɪɝɢɹ, 1975. – 488 ɫ.
Ʉɚɪɫɥɨɭ Ƚ., ȿɝɟɪ Ⱦ. Ɍɟɩɥɨɩɪɨɜɨɞɧɨɫɬɶ ɬɜɟɪɞɵɯ ɬɟɥ. – Ɇ.: ɇɚɭɤɚ,
1964. – 488 ɫ.
Ʉɧɹɡɟɜɚ Ⱥ.Ƚ. Ɋɚɡɥɢɱɧɵɟ ɜɚɪɢɚɧɬɵ ɦɟɬɨɞɚ ɩɪɨɝɨɧɤɢ. Ɇɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɥɚɛɨɪɚɬɨɪɧɵɯ ɪɚɛɨɬ. – Ɍɨɦɫɤ.:
ɂɡɞ-ɜɨ Ɍɉɍ, 2006.
Ʉɧɹɡɟɜɚ Ⱥ.Ƚ. ɗɥɟɦɟɧɬɚɪɧɵɟ ɩɨɧɹɬɢɹ ɨ ɪɚɡɧɨɫɬɧɵɯ ɫɯɟɦɚɯ. Ɇɟɬɨɞɢɱɟɫɤɨɟ ɩɨɫɨɛɢɟ ɞɥɹ ɜɵɩɨɥɧɟɧɢɹ ɥɚɛɨɪɚɬɨɪɧɵɯ ɪɚɛɨɬ. – Ɍɨɦɫɤ.:
ɂɡɞ-ɜɨ Ɍɉɍ, 2007.
Ʉɨɡɞɨɛɚ Ʌ.Ⱥ. Ɇɟɬɨɞɵ ɪɟɲɟɧɢɹ ɧɟɥɢɧɟɣɧɵɯ ɡɚɞɚɱ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. – Ʉɢɟɜ: ɇɚɭɤɨɜɚ ɞɭɦɤɚ, 1975.
Ʉɪɟɣɬ Ɏ., Ȼɥɷɤ ɍ. Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ. – Ɇ.: Ɇɢɪ, 1983. –
512 ɫ.
Ʉɭɬɚɬɟɥɚɞɡɟ C.ɋ. Ɉɫɧɨɜɵ ɬɟɨɪɢɢ ɬɟɩɥɨɨɛɦɟɧɚ. – ɇɨɜɨɫɢɛɢɪɫɤ:
ɇɚɭɤɚ, 1970. – 670 ɫ.
Ʌɵɤɨɜ Ⱥ.ȼ. Ɍɟɨɪɢɹ ɬɟɩɥɨɩɪɨɜɨɞɧɨɫɬɢ. – Ɇ.: ȼɵɫɲɚɹ ɲɤɨɥɚ,
1967. – 600 ɫ.
Ɋɵɤɚɥɢɧ ɇ.ɇ., ɍɝɥɨɜ Ⱥ.Ⱥ., Ⱥɧɢɳɟɧɤɨ Ʌ.Ɇ. ȼɵɫɨɤɨɬɟɦɩɟɪɚɬɭɪɧɵɟ
ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɩɪɨɰɟɫɫɵ. Ɍɟɩɥɨɮɢɡɢɱɟɫɤɢɟ ɨɫɧɨɜɵ. – Ɇ.: ɇɚɭɤɚ, 1986. – 174 ɫ.
Ɉɫɧɨɜɵ ɬɟɩɥɨɩɟɪɟɞɚɱɢ ɜ ɚɜɢɚɰɢɨɧɧɨɣ ɢ ɪɚɤɟɬɧɨ-ɤɨɫɦɢɱɟɫɤɨɣ
ɬɟɯɧɢɤɟ. – Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1975. – 624 ɫ.
355
19.
20.
21.
22.
23.
24.
25.
26.
ɋɚɦɚɪɫɤɢɣ Ⱥ.Ⱥ., ȼɚɛɢɳɟɜɢɱ ɉ.ɇ. ȼɵɱɢɫɥɢɬɟɥɶɧɚɹ ɬɟɩɥɨɩɟɪɟɞɚɱɚ.
– Ɇ.: ȿɞɢɬɨɪɢɚɥ ɍɊɋɋ, 2003. – 784 ɫ.
Ɋɨɡɨɜɫɤɢɣ Ⱥ.ə. Ƚɟɬɟɪɨɝɟɧɧɵɟ ɯɢɦɢɱɟɫɤɢɟ ɪɟɚɤɰɢɢ. Ʉɢɧɟɬɢɤɚ ɢ
ɦɚɤɪɨɤɢɧɟɬɢɤɚ. – Ɇ.: ɇɚɭɤɚ, 1980. – 324 ɫ.
ɂɫɚɟɜ ɋ.ɂ., Ʉɨɠɢɧɨɜ ɂ.Ⱥ., Ʉɨɮɚɧɨɜ ȼ.ɂ. ɢ ɞɪ. Ɍɟɨɪɢɹ ɬɟɩɥɨɨɛɦɟɧɚ / ɩɨɞ ɪɟɞ. Ⱥ.ɂ. Ʌɟɨɧɬɶɟɜɚ. – Ɇ.: ȼɵɫɲɚɹ ɲɤɨɥɚ,
1979. – 495 ɫ.
Ɍɢɯɨɧɨɜ Ⱥ.ɇ., Ʉɚɥɶɧɟɪ ȼ.Ⱦ., Ƚɥɚɫɤɨ ȼ.Ȼ. Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɯ ɩɪɨɰɟɫɫɨɜ ɢ ɦɟɬɨɞ ɨɛɪɚɬɧɵɯ ɡɚɞɚɱ ɜ
ɦɚɲɢɧɨɫɬɪɨɟɧɢɢ. – Ɇ.: Ɇɚɲɢɧɨɫɬɪɨɟɧɢɟ, 1990. – 264 ɫ.
Ɏɚɪɥɨɭ ɋ. ɍɪɚɜɧɟɧɢɹ ɫ ɱɚɫɬɧɵɦɢ ɩɪɨɢɡɜɨɞɧɵɦɢ ɞɥɹ ɧɚɭɱɧɵɯ ɪɚɛɨɬɧɢɤɨɜ ɢ ɢɧɠɟɧɟɪɨɜ. – Ɇ.: Ɇɢɪ, 1985. – 384 ɫ.
Ɏɪɚɧɤ-Ʉɚɦɟɧɟɰɤɢɣ Ⱦ.Ⱥ. Ⱦɢɮɮɭɡɢɹ ɢ ɬɟɩɥɨɩɟɪɟɞɚɱɚ ɜ ɯɢɦɢɱɟɫɤɨɣ
ɤɢɧɟɬɢɤɟ. – Ɇ.: ɇɚɭɤɚ, 1987. – 502 ɫ.
ɒɢ Ⱦ. ɑɢɫɥɟɧɧɵɟ ɦɟɬɨɞɵ ɜ ɡɚɞɚɱɚɯ ɬɟɩɥɨɨɛɦɟɧɚ. – Ɇ.: Ɇɢɪ,
1988. – 544 ɫ.
ɒɦɢɞɬ Ɋ., ɋɚɩɭɧɨɜ ȼ.ɇ. ɇɟɮɨɪɦɚɥɶɧɚɹ ɤɢɧɟɬɢɤɚ. – Ɇ.: Ɇɢɪ,
1985. – 264 ɫ.
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Ⱥ.Ƚ. Ʉɧɹɡɟɜɚ
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Ɉ.ɘ. Ⱥɪɲɢɧɨɜɚ
ɉɭɪɠɲɛɭɛɨɩ ɝ Ƀɢɟɛɭɠɦɷɬɭɝɠ ɍɊɎ ɝ ɪɩɦɨɩɧ ɬɩɩɭɝɠɭɬɭɝɣɣ
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ɉɨɞɩɢɫɚɧɨ ɤ ɩɟɱɚɬɢ 25.12.2009. Ɏɨɪɦɚɬ 60ɯ84/16. Ȼɭɦɚɝɚ «ɋɧɟɝɭɪɨɱɤɚ».
ɉɟɱɚɬɶ XEROX. ɍɫɥ.ɩɟɱ.ɥ. 20,76. ɍɱ.-ɢɡɞ.ɥ. 18,77.
Ɂɚɤɚɡ 187-11. Ɍɢɪɚɠ 35 ɷɤɡ.
Ɍɨɦɫɤɢɣ ɩɨɥɢɬɟɯɧɢɱɟɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ
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NATIONAL QUALITY ASSURANCE ɩɨ ɫɬɚɧɞɚɪɬɭ ISO 9001:2008
. 634050, ɝ. Ɍɨɦɫɤ, ɩɪ. Ʌɟɧɢɧɚ, 30
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