Математическое моделирование трубопроводных сетей и систем каналов: методы, модели и алгоритмы 978-5-317-02011-8
eBook (изначально компьютерное) В монографии приводится подробное описание базовых методов численного моделирования труб
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(01) Титульный лист.pdf......Page 0
(02) Оглавление.pdf......Page 5
(03) Предисловие.pdf......Page 11
(04) Список сокращений.pdf......Page 14
(05) Глава 1.pdf......Page 16
_06_ Глава 2_1.pdf......Page 61
_06_ Глава 2_2.pdf......Page 159
(07) Глава 3.pdf......Page 279
(08) Глава 4.pdf......Page 342
(09) Глава 5.pdf......Page 452
(10) Заключение.pdf......Page 553
(11) Список литературы.pdf......Page 554
(12) Приложение 1.pdf......Page 569
(13) Приложение 2.pdf......Page 597
(14) Приложение 3.pdf......Page 630
(15) Приложение 4.pdf......Page 638
(16) Приложение 5.pdf......Page 653
(17) Приложение 6.pdf......Page 656
(18) Приложение 7.pdf......Page 661
(19) Приложение 8.pdf......Page 664
(20) Приложение 9.pdf......Page 674
(21) Приложение 10.pdf......Page 685
(22) Об авторах.pdf......Page 695
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В.Е. Селе нев, В.В. Алешин, С.Н. П ялов ______________________________________________
МАТЕМАТИЧЕСК Е М ДЕЛИ ВАНИЕ Т П В ДН СЕТЕ И СИСТЕМ КАНАЛ В Мето ы, о ели и ал орит ы ______________________________________________
М СКВА – 2007
ДК 621.64:519.8 К 39.71-022:22.18 29
С 29
В.Е., А
и В.В., П я
а
а и и ы, ииа г и 2007. – 695 . ISBN 978-5-317-02011-8
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. ДК 621.64:519.8 К 39.71-022:22.18
ISBN 978-5-317-02011-8
© В. .
, В.В. А
, .Н.
, 2007
ВА АЕМ Е ЧИТАТЕЛИ! ВАШЕМ ВНИМАНИ П ЕДЛАГАЕТС АВТ СКА ЭЛЕКТ ННА ВЕ СИ ТЕКСТА М Н Г АФИИ: Селе нев В.Е., Алешин В.В., П ялов С.Н. Ма е а че ру р в ых е е е а ал в: е ы, ел ре . В.Е. Селе ева. – М.: МАКС Пре , 2007. – 695 .
е ел р ва е ал р ы / П
ЭТА ВЕ СИ ЕГ Л Н С П В ДАЕТС В.Е. СЕЛЕЗНЕВ М, В.В. АЛЕШИН М И С.Н. П Л В М, НАЧИНА С СЕНТ 2007 Г ДА. СН ВН Е ТЛИЧИ ТЕКСТА АВТ СК ВЕ СИИ Т ВА ИАНТА ТЕКСТА, П ЛИК ВАНН Г В ИЗДАТЕЛ СТВЕ «МАКС П е », М СКВА, ТМЕЧЕН ЕЛТ М МА КЕ М. П СЛЕДНИЕ П АВКИ В ЭЛЕКТ НН ТЕКСТ В СЕНТ Е 2009 Г ДА.
ЛИ ВНЕСЕН
ВСЕ П АВА НА П ЕДСТАВЛЕНН ЭЛЕКТ НН ВЕ СИ П ИНАДЛЕ АТ В.Е. СЕЛЕЗНЕВ , В.В. АЛЕШИН И С.Н. П Л В .
лавление П е и ловие ............................................................................................................ 9 С и о о новных и
Н ВА П
оль уе ых о
ащени ................................................. 12
ГЛАВА 1
Е АК И К Н ЕП ИИ ЧИС ЕНН М Е И ВАНИ В НЫ СЕ Е И СИС ЕМ КАНА В С К Ы ЫМ С
М
1.1. .......................................................................................... 14 1.2.
................ 16
1.3. В ........................................................................................................... 18 1.4. В ........................................................................... 33 1.5. ................................................................................................................ 36 1.6. .... 46 1.7.
............................................ 53
МА ЕМА ИЧЕСК Е М
2.1.
Е И
ГЛАВА 2
ВАНИЕ АНСП И ВАНИ П В НЫМ СИС ЕМАМ
П
К
ВП
............................................................................................ 59
2.2. В .................................................................................................................... 60 2.3.
........................................... 63
2.3.1.
......................................................................... 63
2.3.2.
........................................................................................... 77
2.3.3.
............................................................ 80
2.3.4.
................................................................ 88
2.4. © В.Е. Селе
........................................... 123 ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
4 лавле е _______________________________________________________________________________________
2.4.1.
.................................123
2.4.2.
, ...................................................................................................................134
2.5. Ч
.............138
2.5.1.
..................163
2.5.2.
.........................................................................................................168
2.5.3. К ..........................................................................183 2.5.4. Ч
...........................................................................190
2.5.5. Ч
Ч , .................................................................................225
2.5.6. Ч ..................................................................................................................233 2.6. ................................................................................................237 2.6.1.
(
)...............237
2.6.2. .............240 2.6.3. .........................................................................................252 2.6.3.1.
К ..............252
2.6.3.2.
К ..............................................253
2.6.4. К
......256
2.7. ........................................................................................................................258 2.7.1.
......259
2.7.2.
..260
2.7.3. Ч ................................................................................................260 2.7.4. А ................................263 2.8. ....266 2.8.1. К ......................................................................................267 2.8.2. ....................................................268 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лавле е 5 _______________________________________________________________________________________
2.8.3. А ..................................................................................... 271 2.8.4. .......................................................................................................... 274
ЧИС ЕННЫ
АНА ИЗ П
ГЛАВА 3 ЧН С И
П
В
НЫ СИС ЕМ
3.1.
.......................................................................................................... 277
3.2.
.............................. 281
3.3.
....................................................................................... 285
3.3.1.
-
............................................... 285
3.3.2.
......... 286
3.3.2.1. И .................................................................................. 286 3.3.2.2.
-
.................................... 294
3.4.
................................................................. 308
3.4.1. В
НД
......................................... 309
НД
3.4.2.
................................................... 310
3.4.2.1.
.............................................................. 312
3.4.2.2.
....................................................... 315
3.4.2.3.
............................................................ 318
3.4.3. А
НД
............................... 320
3.4.4. А
. 321
3.4.5. А
, ................................................................................................................ 325
МА ЕМА ИЧЕСК Е М
ГЛАВА 4
Е И П В
ВАНИЕ АВА И НЫ СИ НЫ СИС ЕМА
4.1.
А И В
................. 340
4.2. .................................................................................. 341 4.3. Ч ............................................................................................. 348 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
6 лавле е _______________________________________________________________________________________
4.3.1.
-
.......................348
4.3.2. Ч
................363
4.3.3. Ч ........................................................................................................................365 4.3.4.
ЭК ............................................................372
4.3.5. К ..................................................................................373 4.4. Ч
375
4.4.1.
....................................................................................................375
4.4.2. Ч ...............................................................................................377 4.4.3. Ч ..................................................................................................................381 4.4.4. А .................................................................................................394 4.4.5. Ч ................................................................................................................401 4.4.6. К ..................................................................................................402 4.5. ....................................................................403 4.5.1. К .....................................................................................................................403 4.5.2. Ч
.....................................................417
4.6. К .............................................................................................................................424 4.7.
, ..............................................................425
МА ЕМА ИЧЕСК Е М СИС ЕМАМ П
5.1.
ГЛАВА 5
Е И ВАНИЕ ЕННЫ КАНА
АНСП И ВАНИ ВС К Ы ЫМ С
И К С Е П М И ЕКАМ
...........................................................................................................450
5.2. ..............................................................................................................................451 5.3. ..............................................................................................................................460 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лавле е 7 _______________________________________________________________________________________
5.4. ........................................................................................................... 477 5.5.
.......... 489
5.6.
. 510
5.7. К
............. 520
5.8.
............................................ 525
5.9. (
«
»
) .................................................................. 528
5.10. ...................................................................................................................... 550 5.11.
550
За лючение .............................................................................................................. 551 С и о ли е а у ы................................................................................................. 552 П ило ение 1 Ч ле ы а ал вл я я а ч а ря е - е р р ва я я ру а р ч а рал ых ру р в в е и . .. ........................................................................................................................... 567
П ило ение 2 К в р у ра че ых е ах равл че х р вле ре я в ру р в ах Пря ов С.Н., Се е ев .Е.................................................................................................... 595
П ило ение 3 ре еле е ара е р в е ер ых еле в а е в я ру р в а ру ре ул а а ч ле ел р ва я е и . .. ........................................................................................................................... 628
П ило ение 4 Кр
ер е и
ла ч ру в ал верх е уче . .. ........................................................................................................................... 636
П ило ение 5 К в р у ч ле ел р ва е ых уча в р я е ых ру р в ых е е и . .. ........................................................................................................................... 651
П ило ение 6 ч е реше е МКЭ а ач ев е е ру р в а в ру е е и . .. ........................................................................................................................... 654 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
8 лавле е _______________________________________________________________________________________
П ило ение 7 Пре ел е и
е рав ве е в « еал ых» ру в . ..............................................................................................................................659
П ило ение 8 Кра о е
р а е о . ., Се е
в в ла ару е я ра рыв в а р в в ев .Е. ................................................................................................662
П ило ение 9 М ел р ва е авар в а ра р ых е ах, вя а ых еу ч в ра а ере ач ва е ру ва я оми аров .С., Пря ов С.Н., Се е ев .Е.......................................................................672
П ило ение 10 К в р у ра че ых е ах аче э е а Ше Пря ов С.Н., Ю и . . ..........................................................................................................683
ав о ах................................................................................................................ 693
© В.Е. Селе
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АВА 1 Новая е а ия он е ии чи ленно о о ели ования у о ово ных е е и и аналов о ы ы у ло
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ер ра че
ев, В.В. Алеш
хе ы ЧМ ,
, С.Н. Прял в, 2007–2009
р е
в ре а
ре
л
С
24 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
( )
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ер а а
ев, В.В. Алеш
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;
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:
;
я ара е р в ЧМ в ре а
, С.Н. Прял в, 2007–2009
ре
л
С
, -
лава 1 25 ______________________________________________________________________________________
К
Ч
, ,
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(
, .
1.3).
и . 1.3. Фра
(
е
а ы а
ых
а а в ЧМ в
е
ре С
ре еле
И . 1.4).
ы
в
ва
, ,
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( ).
и . 1.4. Фра
е
а ы а
ых е
е
в ЧМ в
С
:
•
;
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,
;
•
-
, , В Д Д :
; . .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
Д ( ;
. , . . , . 1.5).
,
-
26 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
и . 1.5. Пр
ер
ер
л
ва ел
( )
ре
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ыв
С
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); . .
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) . 1.6).
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, ;
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(АВ ), ,
-
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Д
Д ,
Д
, С.Н. Прял в, 2007–2009
К
. .
лава 1 27 ______________________________________________________________________________________
и . 1.6. Пр
В
ер ра че
р е
К , .
(ЦН)
хе ы КС,
А(
© В.Е. Селе
ре
л
С
Ч , К ,
,
.Д
,
А А
. 1.7).
и . 1.7. Пр
в ре а
ер а а
ев, В.В. Алеш
я ара е р в ра
, С.Н. Прял в, 2007–2009
ы
А
К
-
Д
Н в ре а
А.
ре
л
КС
-
28 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
Д
К К : А;
; К ,
. .Э
; .
АВ ;
:
К ;
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)
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:
• (
. 1.8);
и . 1.8. Пр
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ал
в
а р
а
в е а ач в
•
;
•
• •
; А
Ч
,
-
К
, . .; SCADA-
© В.Е. Селе
С
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
( . 1.9));
,
,
-
лава 1 29 ______________________________________________________________________________________
и . 1.9. Пр
ер ав
авле
луче
•
я а а а вх а а я SCADA-
ра е е ы
С
вре е
,
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•
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. .
: ( -
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); ;
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. 1.10);
и . 1.10. Пр © В.Е. Селе
ев, В.В. Алеш
ер
ал
в
, С.Н. Прял в, 2007–2009
а а р е
С
30 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
•
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[7, 39]. Д
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Ч
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. 1.11).
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. 1.12. ,
Д
К
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:
К ;
К
К ; Ч Ч
К .
К
К .Н
-
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«
.В
2
4. -
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«real time»
. 1.14.
,
,
Д , .
, (
.
).
-
, ,
Д © В.Е. Селе
. -
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 1 31 ______________________________________________________________________________________
,
. .В Ч
(
А,
,
),
, / ,
.В (
. 1.15)
.Д
Д
.
и . 1.11. Пр
и . 1.12. Пр
© В.Е. Селе
ер
ал
ер
ев, В.В. Алеш
в
ал
в ра
ав ече
р
С р а а а а в ЧМ
а в С р вы р ва я а а
, С.Н. Прял в, 2007–2009
раевых у л в
ре е ЧМ
ля ра че а
а ра че а ара е р в
-
32 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
и . 1.13. Пр
ер
ал
ра
в
р
и . 1.14. Пр ер ал в ра р р ва я а а чере © В.Е. Селе
ев, В.В. Алеш
а в С р р ве е р ва я а а ЧМ
а
ав
ра
С р ч ле р у е
, С.Н. Прял в, 2007–2009
ра че а ара е р в
а ал е ара е р в в ре е «real time»
лава 1 33 ______________________________________________________________________________________
и . 1.15. Пр
ра
ер ре авле я в С ре ул а в ч ле а ал а ара е р в р р ва я а а чере а ра р е ре р я е
-
Д
.
Д
, ,
, .
Д ( ),
,
-
А, А Д, АВ ,
-
Д
,
. (
-
.
. 1.16–1.19).
1.4. Вычи ли ельные ехноло ии анали а е иальных лучаев ун иони ования у о ово ных и анальных и е В ,
© В.Е. Селе
ев, В.В. Алеш
.В
, С.Н. Прял в, 2007–2009
-
34 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
:
[28];
[40, 41]; [42]. (
)
, .
и . 1.16. Пр
ер выв
и . 1.17. Пр
© В.Е. Селе
а ре ул а
в ра че а ара е р в ра ч ЧМ в С
ер выв а ре ул а в а ра р е в
ев, В.В. Алеш
а С (в в
, С.Н. Прял в, 2007–2009
ра е ра
р
р ва
р р ва в)
я а а
я а а
,
лава 1 35 ______________________________________________________________________________________
и . 1.18. Пр
и . 1.19. Пр
© В.Е. Селе
ер выв а ре ул а в а ра р е в
ер выв
ев, В.В. Алеш
а С (в в
е
ра р р ва а ра )
а ре ул а в ра че а ара е р в ра а ра р е в С , С.Н. Прял в, 2007–2009
р
я а а
р ва
я а а
36 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
В
,
-
,
, (
)
,
, .
-
ЭК (
)
,
,
-
,
.В (
-
)
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,
. (
) -
(
)
.
, (
) -
,
[1, 5, 6].
Д
(
.
1.5) (
.
1.6).
1.5. П и е еали а ии о новных ин и ов вы о о очно о о ели ования у о ово ных и е и ве и и а ии ое ных ешени ля а и альных у о ово ов Н
ЭК , .
-
. ,
: -
; -
; ; © В.Е. Селе
ев, В.В. Алеш
. . , С.Н. Прял в, 2007–2009
,
-
лава 1 37 ______________________________________________________________________________________
,
,
, . , .
,
, -
(
. Э
) -
. В
1.3 ,
-
. ,
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:
-
(НД )
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1.1
1.2). ,
.
ЭК
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.
В
( , ,
•
1.4). К
.
, : -
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
38 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
,
,
,
, ,
, ,
. .;
•
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)
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), , -
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1.3). , , ,
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.
И К
А
И
(КАИ) «Alfargus» (
.
1.6)
:
•
, ;
• © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 1 39 ______________________________________________________________________________________
;
•
;
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;
• . Д (
Э ,
, [7]).
.,
,
-
. , (
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1.1
-
1.2),
(
),
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: НД ( );
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1.1 ,
-
, . .К
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, , .Э © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
40 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
.В
,
-
. -
.Э
НД
.Д
НД , ( . -
В.В. А
,
[3, 6, 17, 23, 43]). Д
3
-
. Н ( . [44]):
•
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(
);
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К (
, (Н
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)); ;
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( ,
А .
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(
, [18, 19])
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[44]. В ,
-
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/ ,
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2). Э
.
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. (
)
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,
, (
.
2). Э
.В -
, , [44]. © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 1 41 ______________________________________________________________________________________
(
-
4). В
.
(
.,
.В
-
, [44])
-
. ,
. , . Ч . -
.Э -
,
,
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КАИ «Alfargus»,
( . 1.6): 1)
-
,
,
,
,
,
. .; 2)
, К , ,
, (
3) © В.Е. Селе
ев, В.В. Алеш
-
Н
, С.Н. Прял в, 2007–2009
)
. .; -
42 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
,К
Н ; НД
4) ; 5)
, ;
6)
НД );
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,
7)
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10)
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(
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) (
-
); 13)
;
14)
© В.Е. Селе
-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 1 43 ______________________________________________________________________________________
(
);
15)
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;
17)
( ,
);
18)
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19)
,
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1
:
1)
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) НД
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,
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; НД
2)
,
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;
3)
,
1
,
, .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
44 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
,
-
,
, 4)
,К
, ; НД
Н
,
, ; НД
5) : 5.1)
(
,
,
-
, . .);
5.2)
(
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. .); 5.3)
( ,
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5.4)
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, . .); НД
6) ,
;
7)
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НД
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10)
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11)
(
Н
К (
Н
,
, К
) , )
; ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
Н
) -
;
,
© В.Е. Селе
, -
,
,
лава 1 45 ______________________________________________________________________________________
12) ; 13)
;
14) (
),
;
15) ( (
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. .)
-
),
;
16)
;
17) ; 18)
,
;
19)
;
20)
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( ,
В
.
[1–7]).
,
•
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: ;
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;
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• SCADA-
(
)
, , ;
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© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
. .
46 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
1.6. П и е еали а ии о новных ин и ов вы о о очно о о ели ования у о ово ных и е и у авлении ело но ью у о ово ных е е 1.1 – 1.4 .Х КАИ «Alfargus». К «PipEst»,
-
Д .
«AMADEUS», ( . [1–6]), КАИ «Alfargus». КАИ «Alfargus» – ,
«CorNet» -
ЭК.
ЭК (
, )
-
, ,
, КАИ
. К
«Alfargus» 1)
: ,
),
(
,
;
2)
(
)
-
; 3)
;
4)
;
5)
,
-
; 6) , ; 7)
( );
8) ; 9)
;
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 1 47 ______________________________________________________________________________________
ЭК
10)
[7, 39],
И
. .
КАИ «Alfargus» . :
•
В. .
;
•
, ;
•
;
•
;
•
, ( ,
«Alfargus», •
.,
-
1.3). Д
.В
КАИ
К
(
.
1.3), -
: (
,
-
, );
•
(
-
, ,
,
,
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. .); (
-
,
, «Alfargus», ).
1)
2)
КАИ «Alfargus» «Alfargus/PipeManufacture» НД , «Alfargus/NDTest»
: , ; ;
3)
«Alfargus/PipeFlow»
;
4)
«Alfargus/StructuralAnalysis» ,
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
48 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
; «Alfargus/DebrisHazard»
5)
-
; 6)
«Alfargus/ToxicGasHazard» ;
7)
«Alfargus/ToxicLiquidHazard»
,
; 8)
«Alfargus/FireHazard» ;
9)
«Alfargus/RiverFlow»
10)
, «Alfargus/Ranking»
11)
«Alfargus/OptimFlow»
12)
; «Alfargus/Planning»
; ; , . КАИ «Alfargus»
Д
.Э
«Alfargus/NDTest», . .
«Alfargus/PipeManufacture» 1. П В
-
е а «Alfargus/NDTest». ,
, -
-
. .Н
,
-
, [47]. ,
К
. В
Х
.
Х
, © В.Е. Селе
. -
ев, В.В. Алеш
,
,
. , С.Н. Прял в, 2007–2009
-
лава 1 49 ______________________________________________________________________________________
,
«Alfargus/NDTest»,
rot
(
)
(
)
(
⋅ rotA − grad ve ⋅ divA − rot
A –
=
; ; HC – -
−1
)
:
⋅ HС = 0,
1 3 ve = ⋅ ∑ vii ; 3 i =1
;
(1.1) –
-
.Д
(1.1) (
:
∇2 –
. В
∇2 A = −
.
-
)
1 ⋅ rot HС , ve
(1.2)
.Д
. (
(1.1)
.,
, [6, 47])
(1.2)
,
-
, (
): A ( x, θ , z )
= Q1 ( x , θ , z ) ; ⎡ π π⎤ x ∈ [0; x0 ], θ ∈ ⎢ − ; ⎥ , z = 0 ⎣ 2 2⎦
(1.3 )
A ( x ,θ , z )
= Q2 ( x , θ , z ) ; ⎡ π π⎤ x ∈ [0; x0 ], θ ∈ ⎢ − ; ⎥ , z = z0 ⎣ 2 2⎦
(1.3 )
AZ ( x , θ , z )
AX ( x , θ , z ) Qi ( x , θ , z ) ,
( x − x0 ) → ∞ ,
z→∞
i = 1, 2,
AZ ( x , θ , z ) –
x ∈ [0; x0 ], θ = ±
π
x ∈ [0; x0 ], θ = ±
π
2
2
, z ∈ [0; z0 ]
= 0;
(1.3 )
, z ∈ [0; z0 ]
= 0,
(1.3 )
( z − z0 ) → ∞ –
, -
( x ,θ , z )
(
x –
A
:
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
; AX ( x , θ , z )
A = A ( x, θ , z )
«+ »
-
).
«− »
-
50 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
n×
A+ = A− , n × A+ = n × A− ;
(
+
)
⋅ rotA + = n ×
n – Д
(
−
)
⋅ rotA − ,
(1.3 )
. : –
«
»;
,
-
; -
;
-
,
Д
-
. (1.1,1.3)
( КЭ),
(1.2,1.3) -
[48]. , -
,
[49]:
B = rotA; H = ⋅ B − HC ,
(1.4)
H –
B
. -
-
. ,
,
,
, [49]:
Fмаг = ∫ T ⋅ nds ,
(1.5)
S
Fмаг –
; S –
,
1 ; T = H ⊗ B − ⋅δ⋅H⋅B – 2
-
, ⊗
(δ –
–
). «Alfargus/NDTest»
.
[1, 6,
50–53]. В
«Alfargus/NDTest»
, -
, -
«
,
-
–
»,
.
90[50, 51]. -
, «BJ Pipeline Inspection Services» (К «LinScan» ( АЭ), А « » ( ), АК « ( – – ), «GasCo» ( АЭ)). © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
( ), «PII» (В » (
, ,
), ), «SPP, a.s.»
лава 1 51 ______________________________________________________________________________________
П
е а «Alfargus/PipeFlow». К :
•
,
•
; ;
•
;
•
;
•
;
•
Д ,
«Alfargus/PipeFlow» «offline», «online» «real time»;
,
•
-
;
•
;
•
;
• ;
•
, ,
. .
«Alfargus/PipeFlow», ( .
(
. П
•
2, 4
2
) 9
[1, 2, 4–7]). е а «Alfargus/StructuralAnalysis». Э К :
-
. К
НД ,
,
-
;
•
;
•
К
•
;
,
•
;
. , «Alfargus/StructuralAnalysis»,
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
КЭ.
-
52 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
3, 4
3–7 (
.
[1, 3,
4, 6]). П
•
е а «Alfargus/DebrisHazard». : ;
•
. «Alfargus/DebrisHazard»
( Д
), ,
. 4.2 (
.
[1, 6]). П
•
е а «Alfargus/ToxicGasHazard». К :
-
;
•
, ;
•
. «Alfargus/ToxicGasHazard»
(
. П
•
.Д 4
. [1, 6, 45, 54, 55]). е а «Alfargus/ToxicLiquidHazard». :
-
;
•
. «Alfargus/ToxicLiquidHazard» . П
•
4.
е а «Alfargus/FireHazard». В :
;
•
.
, «Alfargus/FireHazard»,
,
Д (
-
.
© В.Е. Селе
. [4, 6, 56–58]). ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
. 4
лава 1 53 ______________________________________________________________________________________
П
е а «Alfargus/RiverFlow». В :
•
-
;
•
. «Alfargus/RiverFlow» -В , 5( . [5, 6, 27]). П
.
е а «Alfargus/Ranking».
-
.
5(
«Alfargus». [1]).
. П
3, 4
е а «Alfargus/OptimFlow». В :
•
,
(
-
) ;
•
, ,
. . «Alfargus/OptimFlow»
, , .
2(
.
[1, 2, 5, 6, 30, 59, 60]). П »
е а «Alfargus/Planning»
«
,
.
4(
.
-
«Alfargus». [61, 62]).
1.7. Фо
2, 3
ули ов а е во ве
ии а ши енно
он е
ии
А ( 1.5
1.1 – 1.4),
. 1.6),
(
,
.Д
1
:
1
,
« .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
»
. -
54 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
1. В
ч е е ч е х е е е а а яа е ь а ча ь е я1, а а е
е
а
я
е
е е
х .
ех
х х
а ь а я, -
е
– , (
)
(
)
. , (
), ,
. .
, -
ЭК .
) НД , -
( , . 2. Д
,
, аа е е ч е е а я а а ь я а е ч е а а е е а а яа х а ч а х а е а че х е е е х ( а а ь х) е е / че х е , е а щ х х. С е ь е я е я я я е х а е е а ь е х ех че а а е щеаще я. В а яь е е я е а е е а ще я ь е а а я ае х а щ х а а х. , . . . Д , , . ( ) . 3. В
а а а
ае
х
ч
х
,
е
)
а
щ х а а
е
.Д
х х
ь е , а а а ач ЭК
а е а е а
че
а е
х
е е
е а е я еха я е е я е е щ х а ,
1
© В.Е. Селе
«
»
ев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
(
х -
лава 1 55 ______________________________________________________________________________________
, . А
-
,
,
,
.Д
-
. 4. П
е ех е а х а а е а че е я е е е е я ь ще ще .
е а
е
еха а
а
е
а
х е а а ь х х -
х е х ,
,
-
. В
1
,
(
,
.Э
)
/
-
,
2
.
,
. В
3
5.
е
а
я,
я а
-
а
ч
е
я
я
е
х а е я, а а ч х
а е е
а а я а а
ь
е х ч х х ( а а ь х)
че а ах , е
. еа а е е
аа х ех е е а я яе я а а а я ех че х ь е х я е . Д
е1 е а ь х а ае -
х
,
-
/
(
). . -
(
),
(
),
,
-
, 1
Э
«
».
2
Э
«
».
3
.В
,
-
. © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
56 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
. .В
-
,
-
ЭК
-
, , , . . 4.
3
, . 6. е
,
х
е
е
, (
)
,
е е я ч х ь е х я , еа ь ча е ч е ь е ех . е я я я е е ще е ь х х е х , е а ь а е а я еа ь х а а х а , а а а а ь а е я е ч е а еа а а я а а ь е а , е е а я е х ех че х е а а, . .В ч е ь е ех а а а е а ь х чае а я х ( а а ь х) е е е а я е ь х е а е я а а а ь а а ч а е е е я а яа х а а х а е е а ЭК / а х а а , а а е х а е е а е а е е е е е х а ач, а а - а е а че е ече я, а ще аче е ч е а я е е я а е х а ач. В ( ) . ь
7. Э
а а я аа х е ь е х я ча е ь х ех е а е а ь е а а х а а е а ях ч е а я, ч е ь еха , ч е ь е а е а че а . ЭК / ,
х ЭК а
ч / е -
.Э
-
, , . е а
И В. . (
© В.Е. Селе
.,
ев, В.В. Алеш
-
.
-
-
[5–7].
-
, [16, 17, 22]).
, С.Н. Прял в, 2007–2009
лава 1 57 ______________________________________________________________________________________
, а
е
х
. е
е
, ч е а а
е
е
а
я
а .
а ь
х
. ,
,
•
-
: , ,
, ( Н), . .) ;
•
(
( Э ),
, (АЭ ),
-
, , ;
• ;
•
, ,
),
( ;
•
;
•
-
,
. .
И е
е .
е
а
а ч
а
е
х
а
а
е
че а
я
я
я
-
,
че а е Э а
а а е
я е 1.6 е я е
а
е ь
. я
е е
я
ь
ь а ь х а а
аче е че х
. яе е
а а а я
а ь я
,
е ее -
«Alfargus». ще е ч е ь е ех [1, 5, 6, 23, 61, 62], ЭК (
) В
е
-
-
.
© В.Е. Селе
-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
58 К е я ел р ва я ру р в ых е е е а ал в ры ы ру л _______________________________________________________________________________________
, -
. В
, . Д
,
,
-
, . ,
-
.
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
АВА 2 Ма е а иче ое о ели ование ан о и ования о у ов о и е а о
2.1.
е е
у о
ово ны
о ели ования ,
,
,
-
, .
.
-
И .
÷
, , (
) [6]. Д
Ч
, (
,
, ,
,
К А ,
К , . 3,0÷7,8МПа. , ,
. .).
.В -
.
КЦ
.Д
÷
Д
.
,
-
А -
/
А
А
. . .
,
-
,
4МВт
К А. Н Д
: ЦН
). Ц . 27МВт.
Ч К
А
.
(
, . Э
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
А
, . -
.В , 308К.
© В.Е. Селе
,
А К
,
-
-
60 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
В
, 1
,
.
К
-
, -
(
2.2. Вы о о очны о ью е ны и уля о а о ан о но о
).
а о ина иче е ия ия
и
В
, 1.7,
,
-
.
Д –
, [90]:
•
2
-
-
;
• .
Д
3
•
-
.
Д ,
, :
(
) ,
•
(
.
2.1); , ,
-
, ,
-
;
•
Ч
•
К ; ;
• В
,
. . -
, ,
-
1
« (
–
)–
( ».
)»
«
2
.
3
–
К
© В.Е. Селе
,
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 61 _______________________________________________________________________________________
( В
)
. . Д
-
: 1) 1
;
2) ; 3)
;
4)
; 2
5)
-
3
;
6)
;
7)
, SCADA-
,
-
. . ,
Д :
•
, -
(
•
)
(
.
1.5
[2]); -
;
•
. Д
К
Д Ч
К
.
.
Н
А, АВ
Д
-
-
, .Д 1
Э
(
Д ,
.
1).
2 3
. Н
,
© В.Е. Селе
, ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
62 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
,
К SCADA-
.
Д
1.3, ,
.
Д
-
.
Д
:
•
(
ЦН,
•
)
; А(
•
/
К
•
Ч
); ; -
;
•
;
•
АВ , Д
, .
К
Д
хе а ДС,
Д
1.3, .
К ,
. .
е
,
Ч
,
. . В 1
,
Д . а че ая а
а
х
, а че
Д
. -
ея
(
-
), :
•
Д ;
•
Д
-
;
•
Д , Д
. , -
Д .
1
И
Ч
© В.Е. Селе
К . ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
лава 2 63 _______________________________________________________________________________________
Д .Э
, Д
(
ЭК [1–7].
.
-
1.4).
2.3. Мо ели ование ечени в о нони очно
у о
ово е
.
-
,
Д ,
.
В
, .
-
.Н . 2.3.1. Течение о но о
онен но о а а .В .
S. К
V,
, ,
:
1
,
∂ρ
∫∫∫ ∂t dV + ∫∫ ρ ⋅υ dS = 0;
(2.1)
n
V
∫∫∫
∂ ( ρ ⋅ υ) ∂t
V
S
dV + ∫∫ρ ⋅ υ ⋅υ n dS = − ∫∫ p ⋅ n dS + ∫∫ τ n dS + ∫∫∫ ρ ⋅ FdV ; S
S
S
(2.2)
V
⎛ υ 2 ⎞⎤ υ2 ⎞ ∂ ⎡ ⎛ ρ ε ρ ε dV ⋅ + + ⋅ + ⎢ ⎥ ⎜ ⎜ ⎟ ∫∫∫ ∫∫S ⎜⎝ 2 ⎟⎠ ⋅υn dS = −∫∫S p ⋅υn dS + ⎜ 2 ⎠ ⎥⎦ V ∂t ⎢ ⎣ ⎝ + ∫∫ τ n ⋅ υdS + ∫∫∫ ρ ⋅F ⋅ υdV + ∫∫∫ Q dV − ∫∫ W ⋅ ndS , S
V
ρ –
V
; t –
(2.3)
S
; υ –
n
τn = τ ⋅ n –
; υn = υ ⋅ n –
dS ; υ = υ ; p –
υ
;
,
n (τ –
1
,
); F – , .
© В.Е. Селе
-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
64 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
;ε – (
( )
T –
(
)
; W – k p, T –
W = − k ⋅∇T ,
.
; Q –
)
,
.
∫∫∫
:
∂ ( ρ ⋅ υ) ∂t
V
–
(2.2)
(
)
dV + ∫∫ ρ ⋅ υ ⋅υn dS = ∫∫∫ ρ ⋅ F − ∇p dV + ∫∫ τ n dS . S
V
Ox . , . . f = f ( x, t ) . В
f
f1 ,
f
,
υn = υ x
(
,
υn = −υ x
. 2.1).
(2.4)
S
Δx f
f1 ,
(2.1 – 2.3),
:
и . 2.1. И ле уе ы
е
∂ρ
∫∫∫ ∂t dV + ∫∫ ρ ⋅υ df − ∫∫ ρ ⋅υ df + ∫∫ω ρ ⋅υ dω = 0; V
∫∫∫ V
∂ ( ρ ⋅ υ) ∂t
(
x
x
f1
(2.5)
n
f
dV + ∫∫ ρ ⋅ υ ⋅υ x df − ∫∫ ρ ⋅ υ ⋅υ x df + ∫∫ ρ ⋅ υ ⋅υn d ω =
)
ω
= ∫∫∫ ρ ⋅ F − ∇p dV + ∫∫ τ x df − ∫∫ τ x df + ∫∫ τ n d ω ; V
f1
f
f1
(2.6)
ω
f
⎛ ⎛ ∂ ⎡ ⎛ υ 2 ⎞⎤ υ2 ⎞ υ2 ⎞ ρ ε ρ ε υ ρ ε ⋅ + + ⋅ + ⋅ − ⋅ + dV df ⎢ ⎜ ⎟⎥ ∫∫∫ ∫∫f ⎜⎝ 2 ⎟⎠ x ∫∫f ⎜⎝ 2 ⎟⎠⋅υ x df + ∂t ⎣⎢ ⎝ 2 ⎠ ⎦⎥ V 1 ⎛ υ2 ⎞ + ∫∫ ρ ⋅ ⎜ ε + ⎟ ⋅υ n d ω = − ∫∫ p ⋅υ x df + ∫∫ p ⋅υ x df − ∫∫ p ⋅υ n d ω + 2 ⎠ ω ω f1 f ⎝
+ ∫∫ τ x ⋅υdf − ∫∫ τ x ⋅ υdf + ∫∫ τ n ⋅ υ d ω + ∫∫∫ ρ ⋅ F ⋅ υdV + ∫∫∫ Q dV − ω
− ∫∫ Wx df + ∫∫ Wx df − ∫∫ Wn d ω , f1
f1
© В.Е. Селе
ев, В.В. Алеш
f
f
ω
, С.Н. Прял в, 2007–2009
V
V
(2.7)
лава 2 65 _______________________________________________________________________________________
ω –
. ⎛
∂ρ
⎞
∂ρ
∂ρ
∫∫∫ ∂t dV = ∫ ⎜⎜ ∫∫ ∂t df ⎟⎟ dx = Δx ⋅ ∫∫ ∂t df ;
∫∫∫
V
Δx
⎝
⎠
:
∂ ( ρ ⋅ υ) ∂ ( ρ ⋅ υ) dV = Δx ⋅ ∫∫ df ; ∂t ∂t f f
f
∫∫∫ ( ρ ⋅ F − ∇p ) dV = Δx ⋅ ∫∫ ( ρ ⋅F − ∇p ) df ; V
V
∂ ⎡
⎛
∫∫∫ ∂t ⎢⎢ ρ ⋅ ⎜⎝ ε + ⎣
V
f
υ 2 ⎞⎤
υ 2 ⎞⎤ ∂ ⎡ ⎛ ⎟ ⎥ dV = Δx ⋅ ∫∫ ⎢ ρ ⋅ ⎜ ε + ⎟ ⎥ df ; 2 ⎠ ⎦⎥ 2 ⎠ ⎦⎥ ∂t ⎣⎢ ⎝ f
∫∫∫ ρ ⋅ F ⋅ υdV = Δx ⋅ ∫∫ ρ ⋅ F⋅ υdf ; V
∫∫∫ QdV = Δx ⋅ ∫∫ Qdf . f
V
Δx → 0 ,
∂ρ
∫∫ ∂t f
∫∫ f
df +
∂ ( ρ ⋅ υ) ∂t
(
∂ ⎡
⎛
df +
∫∫ ∂t ⎢⎢ ρ ⋅ ⎜⎜⎝ ε + f
⎣
=−
Δx
(2.5 – 2.7),
(2.8)
∂ 1 ρ ⋅ υ ⋅υ x df + lim ⋅ ρ ⋅ υ ⋅υn d ω = ∫∫ x Δ → 0 ∂x f Δx ∫∫ ω
)
υ 2 ⎞⎤
∂ 1 τ x df + lim ⋅ τ n dω; Δx → 0 Δx ∫∫ ∂x ∫∫f ω
(2.9)
⎛ ⎛ υ2 ⎞ 1 υ2 ⎞ ∂ ⋅ ∫∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υn d ω = ⎟ ⎥ df + ∫∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υ x df + Δlim x → 0 Δx 2 ⎠ ⎥⎦ 2 ⎠ 2 ⎠ ∂x f ω ⎝ ⎝
1 1 ∂ ∂ p ⋅ υ x df − lim ⋅ p ⋅ υ n d ω + ∫∫ τ x ⋅ υdf + lim ⋅ τ n ⋅ υd ω + Δx → 0 Δx ∫∫ Δx → 0 Δx ∫∫ ∂x ∫∫f ∂ x ω ω f
+ ∫∫ ρ ⋅ F ⋅ υ df + ∫∫ Qdf − f
f
∂ ρ ⋅ υ ⋅υ x df ∂x ∫∫f
∂ τ x df . Д ∂x ∫∫f
,
τ x = i ⋅τ x x + j ⋅τ x y + k ⋅τ x z . И
© В.Е. Селе
. 2.2). В
.
x υ
τx
,
f –
: ев, В.В. Алеш
(2.10)
1 ∂ ⋅ Wn d ω. Wx df − lim Δx → 0 Δx ∫∫ ∂x ∫∫f ω
Ox (
Ox ,
-
:
∂ 1 ρ ⋅υ x df + lim ⋅∫∫ ρ ⋅υ n d ω = 0; ∫∫ Δ → 0 x ∂x f Δx ω
= ∫∫ ρ ⋅ F − ∇p df + f
f
, С.Н. Прял в, 2007–2009
i , j, k
Ox . : υ = i ⋅ υx + j ⋅ υ y + k ⋅ υz ; ,
66 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
и . 2.2.
р ы
е ы
(
р
)
а
∂ ∂ ρ ⋅ υ ⋅υ x df = ∫∫ ρ ⋅υ x ⋅ i ⋅υ x + j ⋅ υ y + k ⋅ υ z df = ∂x ∫∫f ∂x f = i⋅ +
∂ ∂ ∂ ρ ⋅υ x ⋅υ x df + j ⋅ ∫∫ ρ ⋅υ x ⋅υ y df + k ⋅ ∫∫ ρ ⋅υ x ⋅υ z df + ∂x ∫∫f ∂x f ∂x f
∂i ∂j ∂k ⋅ ρ ⋅υ x ⋅υ x df + ⋅ ∫∫ ρ ⋅υ x ⋅υ y df + ⋅ ρ ⋅υ x ⋅υ z df ; ∂x ∫∫f ∂x f ∂x ∫∫f
(
)
∂ ∂ ∂ τ x df = ∫∫ i ⋅τ xx + j ⋅ τ xy + k ⋅ τ xz df = i ⋅ ∫∫ τ xx df + ∫∫ ∂x f ∂x f ∂x f + j⋅
∂ ∂ ∂i ∂j ∂k τ xy df + k ⋅ ∫∫ τ xz df + ⋅ ∫∫ τ xx df + ⋅ ∫∫ τ xy df + ⋅ ∫∫ τ xz df . ∫∫ ∂x f ∂x f ∂x f ∂x f ∂x f
В
∂i j = ; ∂x R1
R1 –
Ox ,
∂j i k =− + ; R1 ζ ∂x
–
ζ –
:
∂k j =− , ∂x ζ Ox ,
∂ ∂ ∂ ∂ ρ ⋅ υ ⋅υ x df = i ⋅ ∫∫ ρ ⋅υ x ⋅υ x df + j ⋅ ∫∫ ρ ⋅υ x ⋅υ y df + k ⋅ ∫∫ ρ ⋅υ x ⋅υ z df + ∂x ∫∫f ∂x f ∂x f ∂x f ⎛ i k⎞ j j + ⋅ ∫∫ ρ ⋅υ x ⋅υ x df + ⎜ − + ⎟ ⋅ ∫∫ ρ ⋅υ x ⋅υ y df − ⋅ ∫∫ ρ ⋅υ x ⋅υ z df ; R1 f ζ f ⎝ R1 ζ ⎠ f
∂ ∂ ∂ ∂ τ x df = i ⋅ ∫∫ τ xx df + j ⋅ ∫∫ τ xy df + k ⋅ ∫∫ τ xz df + ∂x ∫∫f ∂x f ∂x f ∂x f ⎛ i k⎞ j j + ⋅ ∫∫ τ xx df + ⎜ − + ⎟ ⋅ ∫∫ τ xy df − ⋅ ∫∫ τ xz df . R1 f ζ f ⎝ R1 ζ ⎠ f
В
, , . . ,
Ox .Э
15°.
d ω ≈ d χ ⋅ dx ,
, © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
(2.11)
(2.12)
-
χ –
-
лава 2 67 _______________________________________________________________________________________
lim
Δx → 0
1 1 1 ⋅ ρ ⋅υn d ω = lim ⋅ ⋅ Δx ⋅ ∫ ρ ⋅υn d χ = ∫ ρ ⋅υ n d χ , ρ ⋅υn d χ d x ≈ lim Δx → 0 Δx ∫ ∫ Δx → 0 Δx Δx ∫∫ ω χ χ Δx χ lim
Δx → 0
1 ⋅ ρ ⋅ υ ⋅υ n d ω ≈ ∫ ρ ⋅ υ ⋅υ n d χ , Δx ∫∫ ω χ
lim
Δx → 0
⎛ ⎛ 1 υ2 ⎞ υ2 ⎞ ⋅ ∫∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υn d ω ≈ ∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υ n d χ , Δx → 0 Δx 2 ⎠ 2 ⎠ ω χ ⎝ ⎝ lim
lim
Δx → 0
1 ⋅ τ n ⋅ υd ω ≈ ∫ τ n ⋅ υ d χ , Δx ∫∫ ω χ
lim
Δx → 0
1 ⋅ τ n dω ≈ ∫ τ n d χ , Δx ∫∫ ω χ
lim
Δx → 0
1 ⋅ p ⋅υn d ω ≈ ∫ p ⋅υn d χ , Δx ∫∫ ω χ
1 ⋅ Wn d ω ≈ ∫ Wn d χ . Δx ∫∫ ω χ
.
,
1 R1
-
1ζ
. .
, (2.8), (2.9), (2.10)
(2.12) ∂ρ
(2.9)
Ox ,
∫∫ ∂t df + ∂x ∫∫ ρ ⋅υ df + ∫χ ρ ⋅υ d χ = 0; ∂
x
f
∫∫
∂ ( ρ ⋅υ x ) ∂t
f
∂ ⎡
⎣
: (2.13)
n
f
⎛ ∂ ∂ p⎞ ∂ ρ ⋅υ x2 df + ∫ ρ ⋅υ x ⋅υn d χ = ∫∫ ⎜ ρ ⋅ Fx − τ xx df + ∫ τ n x d χ ; ⎟ df + ∂x ∫∫f ∂ ∂ x x ∫∫f ⎠ f ⎝ χ χ (2.14) 2 ⎤ 2 ⎛ ∂ υ ⎞ υ ⎞ + ⎟ ⎥ df + ∫∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υ x df + ∂x f 2 ⎠ ⎦⎥ 2 ⎠ ⎝
df +
⎛
∫∫ ∂t ⎢⎢ ρ ⋅ ⎜⎜⎝ ε f
(2.11),
⎛ ∂ ∂ υ2 ⎞ + ∫ ρ ⋅ ⎜⎜ ε + ⎟ ⋅υn d χ = − ∫∫ p ⋅υ x df − ∫ p ⋅υ n d χ + ∫∫ τ x ⋅ υ df + ∫ τ n ⋅ υd χ + ∂x f ∂x f 2 ⎠ χ χ χ ⎝ ∂ + ∫∫ ρ ⋅F ⋅ υdf + ∫∫ Qdf − ∫∫ Wx df − ∫ Wn d χ . ∂ x f χ f f
∂ ϕ ( x, y, z, t ) df , ∂ t ∫∫f
Д
ϕ ( x, y , z , t ) – f = f ( x, t ) ,
. ,
dχ
dt (
cos ϑ 0 –
f
; T
–
-
-
). Φ (T , Toc )
-
Д [7]. -
, Д 0,01м/ ,
1
.
, Re кр = 3000 ÷ 5000 .
, Re
. 2
Д
⎡ ⎤ w = ⎢ f −1 ⋅ ∫∫ υdf ⎥ , ⎢⎣ ⎥⎦OX f
© В.Е. Селе
[…]OX
ев, В.В. Алеш
: –
, С.Н. Прял в, 2007–2009
.
70 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Φ (T , Toc )
Φ (T , Toc )
.
.Д -
.А ,
,
Д А
И
Д
τ xx
τ. ⎛
τ xx = μd ⋅ ⎜ 2 ⋅ ⎝
∂υ x 2 ⎞ 2 − ⋅∇ ⋅ υ ⎟ = ⋅ μ d ∂x 3 ⎠ 3
τ nx = ( τ ⋅ n) ⋅ i = τ xx ⋅ nx + τ xy ⋅ n y + τ xz ⋅ nz =
,К
, 1.6). (2.17), (2.18), τ nx , [63, 108]:
«Alfargus» (
.
⎛ ∂υ ∂υ y ∂υ z ⎞ 4 ∂υ x ; ⋅⎜ 2⋅ x − − ⎟ ≈ ⋅ μd ⋅ 3 ∂ x ∂ y ∂ z ∂x ⎝ ⎠
⎛ ∂υ ∂υ y ∂υ z ⎞ ⎛ ∂υ x ∂υ y ⎞ ⋅⎜ 2⋅ x − − + ⎟ ⋅ nx + μ d ⋅ ⎜ ⎟ ⋅ ny + ∂y ∂z ⎠ ∂x ⎠ ⎝ ∂x ⎝ ∂y ∂υ ∂υ ⎛ 4 ∂υ ⎞ ⎛ ∂υ ∂υ ⎞ + μ d ⋅ ⎜ x + z ⎟ ⋅ nz ≈ μ d ⋅ ⎜ ⋅ x ⋅ n x + x ⋅ n y + x ⋅ nz ⎟ = μ d 3 z x x y z ∂ ∂ ∂ ∂ ∂ ⎝ ⎠ ⎝ ⎠
=
2 ⋅ μd 3
μd –
∂υ ⎞ ⎛ 1 ∂υ ⋅ ⎜ ⋅ x ⋅ nx + x ⎟ , 3 x ∂ ∂n ⎠ ⎝
. ,
:
∂υ ∂ ∂w ⎞ 4 ∂ 4 ∂ ⎛ τ xx df ≈ ⋅ ∫∫ μd ⋅ x df = ⋅ ⎜ μdf ⋅ f ⋅ ⎟ ; 3 ∂x f 3 ∂x ⎝ ∂x ∫∫f ∂x ∂x ⎠
(2.20)
χ ⎛ 1 χ ∂υ x ∂υ ⎞ ∂υ x ⎞ ⎛ 1 ∂υ x ⋅ nx + x ⎟ d χ = μ dχ ⋅ χ ⋅ ⎜⎜ ⋅ ⋅ nx + ⎟; ∂n ⎠ ∂x ∂n ⎟⎠ ⎝ 3 ∂x ⎝3
∫ τ nx d χ = ∫ μd ⋅ ⎜ ⋅ χ
χ
∂ ∂ τ x ⋅ υdf = ∫∫ (τ xx ⋅υ x + τ xy ⋅υ y + τ xz ⋅υ z ) df = ∂x ∫∫f ∂x f =
μdf
μdχ –
∂υ x ∂x © В.Е. Селе
∂υ ∂ ∂w ⎞ 4 ∂ 4 ∂ ⎛ τ xx ⋅υ x df ≈ ⋅ ∫∫ μd ⋅υ x ⋅ x df = ⋅ ⎜ μdf ⋅ w ⋅ f ⋅ ⎟ , 3 ∂x f 3 ∂x ⎝ ∂x ∫∫f ∂x ∂x ⎠
∂υ x . ∂n
μd ,
ев, В.В. Алеш
χ
∂υ x ∂x
. -
χ
∂υ x – ∂n
, С.Н. Прял в, 2007–2009
χ
(2.21)
(2.22)
-
лава 2 71 _______________________________________________________________________________________
μd ∼ 10−5 Н ⋅ м 2 [64]. μd ⋅ ∂w ∂x ,
Д (2.17) Д
, (2.18)
, (2.20)
(2.22)
.
χ ).
( (
(τ
,
(2.21)
χ
nn
) ( p ⋅υ ′ ⋅ χ ) ( (2.17)
⋅υn′ ⋅ χ
) , .
: τ nn = μd ⋅ G ,
w ∼ 10 м
(τ
(2.18)
−1
Oz ,
nn
⋅υn′ ⋅ χ
(r –
,
)
p ∼ 10 Па .
. .
-
τ nn ∼ 10 Па . −1
)
6
-
G – .
0,001м
∂w ∂r ∼ 104
τ nn
. (2.19)).
n
( p ⋅υ ′ ⋅ χ ) .
,
n
,
.
(
(2.16 – 2.18) (2.20), (2.22) (
.
К ,
)
,
(2.19) ′ τ nn ⋅υn ⋅ χ ,
). Н
β
,
β ≈ 1.
, ,
.
. И
,
∂(ρ ⋅ f ) ∂t
+
∂(ρ ⋅ w⋅ f )
∂ ( ρ ⋅ w⋅ f ) ∂t
∂x
+
: = 0;
∂ ( ρ ⋅ w2 ⋅ f ) ∂x
(2.23)
∂z ⎞ ⎛ ∂p = − f ⋅ ⎜ + ρ ⋅ g ⋅ 1 ⎟ + χ ⋅τ χ ; ∂x ⎠ ⎝ ∂x
⎛ ⎛ ∂( p⋅w⋅ f ) w2 ⎞ ⎤ ∂ ⎡ w2 ⎞ ⎤ ∂ ⎡ − ⎢ρ ⋅ f ⋅⎜ ε + ⎟⎥ + ⎢ ρ ⋅ w ⋅ f ⋅ ⎜ ε + ⎟⎥ = − 2 ⎠ ⎥⎦ ∂x ⎢⎣ 2 ⎠ ⎥⎦ ∂t ⎢⎣ ∂x ⎝ ⎝ ∂z ∂ ⎛ ∂T ⎞ − ρ ⋅ w ⋅ g ⋅ f ⋅ 1 − p ⋅υ n′ ⋅ χ + Q ⋅ f + ⎜ k ⋅ f ⋅ ⎟ − Φ ( T , Toc ) . ∂x ∂x ⎝ ∂x ⎠
,
, (
:
© В.Е. Селе
ев, В.В. Алеш
(2.24)
(2.25)
,
τχ ,
τχ = −
),
λ⋅ w 8
-
⋅ ρ ⋅ w,
, С.Н. Прял в, 2007–2009
(2.26)
72 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Д
λ – –В
[65, 66]. Д
λ (
.Н. ,
(
2). -
.
) .
,
(
λ
, .
–
, . .).
-
. Д
В
, : D = 2⋅
π f
υn′ =
;
p ⋅υn′ ⋅ χ = p ⋅
p = p ( ρ, T )
. 1 ∂D 1 ∂f ⋅ = ⋅ ; 2 ∂t 2 ⋅ π ⋅ f ∂t
∂f ; ∂t
χ ⋅τ χ = −
χ = π ⋅ D = 2⋅ π ⋅ f ;
λ⋅ π ⋅ w
⋅ ρ ⋅w⋅ f .
4
(2.23 – 2.26), ε = ε ( p, T ) .
,
: p ⋅V B C = 1 + + 2 + …, R0 ⋅ T V V V – ,
(2.27)
; R0 –
; B, C,…–
. .
(V → ∞ )
, . –К
.
,
,
(2.27) . .
(2.27) , –К [67]:
Д
м ль ) .
-
. . К
,
,
⎡ ⎤ a* ⎢p+ ⎥ ⋅ (υ − b* ) = R ⋅ T , T ⋅υ ⋅ (υ + b* ) ⎥⎦ ⎢⎣
a * = 0, 4278 ⋅ R 2 ⋅ TК2,5 p К ; b* = 0, 0867 ⋅ R ⋅ TК
( кг
. . ,
,
К
(2.27)
; R = R0 M – ,
pК ; TК
υ =1 ρ –
(2.28)
( Д ( кг ⋅ K ) ) ; pК –
M –
. -
:
ε ( p, T ) = h ( p, T ) − dh = h – © В.Е. Селе
ρ
p
∂h ∂h ∂h ⋅ dT + ⋅ dp = c p ⋅ dT + ⋅ dp, ∂T ∂p ∂p
; cp – ев, В.В. Алеш
(2.29)
;
(2.30) .
, С.Н. Прял в, 2007–2009
-
лава 2 73 _______________________________________________________________________________________
( ∂h
∂p )T
(
⎛ ∂h ⎞ ⎛ ∂υ ⎞ ⎜ ⎟ = −T ⋅ ⎜ ⎟ + υ. ⎝ ∂T ⎠ p ⎝ ∂p ⎠T
, μ ( p, T ) ∂ ∂ h p ( )T . [68], μ ( p, T ) = ( ∂T ∂p )h . ,
.,
, [67]): (2.31)
Д
–
. Д
– (
dh = c p ⋅ dT +
Н
-
∂h ⋅ dp = 0. ∂p
. (2.30)): (2.32)
(2.32) :
⎛ ∂T ⎞ 1 ⎟ =− cp ⎝ ∂p ⎠ h
μ ( p, T ) = ⎜
(2.34) (2.30),
.И
⎛ ∂h ⎞ ⋅⎜ ⎟ , ⎝ ∂p ⎠T
(2.33)
⎛ ∂h ⎞ ⎜ ⎟ = −μ ⋅ c p . ⎝ ∂p ⎠T
,
,
-
(2.34)
:
dh = c p ⋅ dT − μ ⋅ c p ⋅ dp.
(2.29)
, ε = ε ( p, T )
(2.35)
(2.30)
-
, –К ,
(2.30) ,
.
, (
.В -
)
.Н. [6, 7] .А
, (
Э
,
.,
, [11]).
, (
,
-
)
. В
.Н.
[6, 7]
-
, . , ,
,
. (2.28 –
2.30). , © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
74 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
, ∂(ρ ⋅ f ) ∂t
+
∂ ( ρ ⋅ f ⋅ w)
∂ ( ρ ⋅ f ⋅ w) ∂t
∂x
+
: = 0;
∂ ( ρ ⋅ f ⋅ w2 ) ∂x
(2.36 )
∂z ⎞ π ⎛ ∂p = − f ⋅ ⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x
∂ ( p ⋅ f ⋅ w) ⎛ ⎛ w2 ⎞ ⎤ ∂ ⎡ w2 ⎞ ⎤ ∂ ⎡ − ⎢ρ ⋅ f ⋅ ⎜ε + ⎟⎥ + ⎢ ρ ⋅ f ⋅ w ⋅ ⎜ ε + ⎟⎥ = − 2 2 x ∂t ⎣ ∂ ∂x ⎝ ⎠⎦ ⎝ ⎠⎦ ⎣ ∂z ∂f ∂ ⎡ ∂T ⎤ − ρ ⋅ f ⋅ w ⋅ g ⋅ 1 − p ⋅ + Q ⋅ f + ⎢ k ⋅ f ⋅ ⎥ − Φ (T , T ) ; ∂x ∂t ∂x ⎣ ∂x ⎦ p = p ( ρ,T );
ε = ε ( p, T ) ,
f π –
R=
(2.29, 2.30).
; p = p ( ρ,T ) –
. (2.28); ε = ε ( p, T ) –
.
.Н
(2.36) w ( x, t0 ) = w0 ( x ) ; T ( x, t0 ) = T0 ( x ) ;
ρ ( x, t0 ) (2.36 ).
Н
, -
p ( x, t0 ) = p0 ( x ) .
ε ( x, t0 )
-
∂T ( xB , t ) = jB ( t ) ; ∂x
: T ( x B , t ) = TB (t );
xB
.
(2.36 ), (2.36 )
(2.36 ),
Н
(2.36 )
(2.36 )
(2.36), (2.36 ),
:
(2.36 )
w( xB , t ) = wB (t );
p( xB , t ) = pB (t );
( ρ ⋅ w ⋅ f ) B = qB ( t ) ,
; jB ( t )
–
qB ( t )
–
.
(2.36)
(
,
.В
[69–72], , ) [73, 74]. , ,
, [6],
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
-
/
-
(2.36)
. .Н. -
лава 2 75 _______________________________________________________________________________________
,
(2.36) (
Д
.
[75]). (2.36)
-
:
ρ ⋅ f ⋅ w = C т = const ; C2 d ⎛ 1 dp =− т⋅ ⎜ dx f dx ⎝ ρ ⋅ f
(2.37 ) dz1 λ ⋅ C т ⋅ C т ⎞ − ; ⎟− g⋅ρ ⋅ dx 4 ⋅ R ⋅ f 2 ⋅ ρ ⎠
(2.37 )
C2 d ⎛ 1 ⎞ d ⎛ p ⎞ 1 d ⎡ dε dz Q ⋅ f dT ⎤ Φ ( T , T = − т ⋅ ⎜ 2 2 ⎟− ⎜ ⎟− g⋅ 1 + + ⋅ ⎢k ⋅ f ⋅ − 2 dx ⎝ ρ ⋅ f ⎠ dx ⎝ ρ ⎠ dx dx C т C т dx ⎣ dx ⎥⎦ Cт p = p ( ρ,T );
)
ε = ε ( p, T ) .
(2.37 ) ЭК
В
. -
(2.36).
[2]. Д
К
; (2.37 )
(КЦ) . Э
Д
[76]
(2.37) (2.37)
–В
[65].
. ,
.Э 3,0МПа
( ≤ 10000м ) ,
( . И
10м, 7,8МПа (
.
( ≤ 0, 25К 1000 м )
). Э
( . ,
,
π ⋅ D2 4
⋅ ρ1 ⋅ w1 =
π ⋅ D2 4
1 2 ; D = const – ев, В.В. Алеш
, [76, 77]).
.,
,
А)
J = J1 = J 2 =
-
,
:
-
).
К (
© В.Е. Селе
, 40м/ ,
⋅
p12 − p22 ⎛λ ⋅ ρ1 ⋅ ⎜ + ξ p1 ⋅ l ⎝D
; l – , С.Н. Прял в, 2007–2009
⎞ ⎟ , ⎠ −1
(2.38)
;
76 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ξ
= const – (2.38)
1
. -
, . . Z Z1 ≈ 1 ,
Z –
, ; Z1 –
,
[78]:
p = Z ⋅ ρ ⋅ R ⋅T.
А (2.37 )
(2.38) w = w2 ⋅ ρ 2 ρ .
ρ1 p1 ,
(2.39) Z Z1 ≈ 1
,
-
К
.
(2.38)
ρ 2 p2 .
-
.
. Д
p1 − p2 p1 ⋅100% ≤ 5%
10000м
( P1 ∈ [3, 0; 7,8] ( МПа ) , T ∈ [ 273; 313] ( К ) ) , . , ( ≤ 10м
(2.28)
)
500м.
, 0,02%. Д P1 − P2 = λ ⋅
(ρ
р
= ( ρ1 + ρ 2 ) 2 ) :
J = J1 = J 2 =
0,25% ( ≤ 0, 018МПа ) . В
,
(2.38),
И
(
–К
) К
0,41%.
π ⋅ D2 4
⋅ρ ⋅w =
–В
l ρ ⋅ w2 ⋅ D 2
(2.40)
-
π ⋅ D2 4
⋅ 2⋅
(2.41)
p1 − p2 ⎛λ ⋅ ρ р ⋅⎜ +ξ l ⎝D
⎞ ⎟ . ⎠ −1
(2.41)
10000м
,
0,20%
,
1
, [66];
( dl . © В.Е. Селе
)
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.38). 500м, dp = ξ
⋅ dl ⋅ ρ ⋅ w 2 2 –
-
лава 2 77 _______________________________________________________________________________________
,
(2.38)
(2.41),
-
0,01% . , (2.38)
(2.41)
. (2.41)
Д .
К
, -
ξ
,
ξ
,
[66].
-
. 2.3.2. Течение а ово
е и
, ,
(2.36),
-
, .
В
. ,
∑(ρ
w m = υm − υ; υm –
:
NS
m =1
m
⋅ w m ) = 0; ρ = ∑ ρ m ; ρ ⋅ υ = ∑ ( ρ m ⋅ υm ), NS
NS
m =1
m =1
m-
; ρ – m-
(
) (
NS –
;
ρm –
(2.42)
; υ –
m-
);
. wm .
,
В
. .
, ,
, wm ,
-
. Д [79]:
Dm –
(
ρ m ⋅ w m = − ρ ⋅ Dm ⋅∇Ym ,
) Ym =
m-
ρm , ρ
Ym –
© В.Е. Селе
(2.43)
(2.44) m,
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
;
.В
,
78 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
( [79]:
. (2.36)). Д
-
⎡ ⎛ ∂ ⎡ ⎛ υ 2 ⎞⎤ υ2 ⎞ ⎤ ⎢ ρ ⋅ ⎜ ε + ⎟ ⎥ + ∇ ⋅ ⎢ ρ ⋅ ⎜ ε + ⎟ ⋅ υ ⎥ = −∇ ⋅ ( p ⋅ υ ) + ∇ ⋅ ( τ ⋅ υ ) + ρ ⋅ F ⋅ υ + Q − 2 ⎠⎦ 2 ⎠ ⎦ ∂t ⎣ ⎝ ⎣ ⎝ − ∇ ⋅ W − ∑ ∇ ⋅ ( ε m ⋅ ρ m ⋅ w m ), NS
(2.45)
m =1
εm –
(
)
m-
. ,
,
T.Д
(
, . .
,
) -
,
. ,
(
∂ ρ ⋅ Ym ∂t
) +∇⋅
( N S − 1)
,
(2.44),
( ρ ⋅ Y ⋅ υ) + ∇ ⋅ ( ρ m
m
:
⋅ w m ) = 0, m = 1, N S − 1.
(2.46 )
(2.46 ) .К
NS
YN = 1 − S
∑Y
:
N S −1 m =1
m
.
(2.46 )
Д
, -
, k,
Д
,
,
,
{S ме и } , .
k = k ({S ме и }) ,
.И
Dm = Dm ({S ме и }) .
:
«
.Д
-
, , (2.47) »
.
Д © В.Е. Селе
. . Н
cV
.
: ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 79 _______________________________________________________________________________________
p = p ({S ме и }) ;
(
ε = ε ({S ме и }) .
(2.48)
(2.43),
∂ ρ ⋅ Ym ∂t
) +∇⋅
-
( ρ ⋅ Y ⋅ υ) − ∇ ⋅ ( ρ ⋅ D m
m
)
:
⋅ ∇Ym = 0, m = 1, N S − 1;
⎡ ⎛ ∂ ⎡ ⎛ υ 2 ⎞⎤ υ2 ⎞ ⎤ ⋅ + + ∇ ⋅ ⋅ + ρ ε ρ ε ⎢ ⎜ ⎢ ⎜ ⎟⎥ ⎟ ⋅ υ⎥ = 2 ⎠⎦ 2 ⎠ ⎦ ∂t ⎣ ⎝ ⎣ ⎝
YN = 1 − S
(
∑Y ;
N S −1 m=1
m
)
= −∇ ⋅ ( p ⋅ υ ) + ∇ ⋅ ( τ ⋅ υ ) + ρ ⋅ F ⋅ υ + Q − ∇ ⋅ W + ∑ ∇ ⋅ ε m ⋅ ρ ⋅ Dm ⋅ ∇Ym . NS
m =1
В
(
∫∫∫
(
∂ ρ ⋅ Ym
V
∂t
YN = 1 − S
) dV
∑Y
+ ∫∫ ρ ⋅ Ym ⋅υ n dS − ∫∫ ρ ⋅ Dm ⋅ S
m
(2.49 )
)
:
N S −1 m =1
–
S
∂Ym dS = 0 , ∂n
m = 1, N S − 1 ;
(2.50 )
;
⎛ ∂⎡ ⎛ υ 2 ⎞⎤ υ2 ⎞ ⋅ + + ⋅ + ρ ε ρ ε dV ⎢ ⎥ ⎜ ⎜ ⎟ ∫∫∫ ∫∫S ⎝ 2 ⎟⎠ ⋅υn dS = 2 ⎠ ⎥⎦ ∂t ⎢⎣ ⎝ V
= − ∫∫ p ⋅υ n dS + ∫∫ τ n ⋅ υ dS + ∫∫∫ ρ ⋅F ⋅ υdV + ∫∫∫ QdV − ∫∫ W ⋅ ndS + ∑ ∫∫ ε m ⋅ ρ ⋅ Dm ⋅ NS
S
S
V
V
(2.50) (2.48) .
Д
m =1 S
S
(2.1, 2.2)
(2.50) Δx
,
(
∂ ρ ⋅ Ym
f
−
∂t
ρ ⋅Y ∂x ∫∫
⋅υ x df + lim
Δx → 0
f
Δx
):
1 ⋅ ρ ⋅ Ym ⋅υn d ω − Δx ∫∫ ω
∑Y
(2.51)
N S −1 m =1
m
Д © В.Е. Селе
m
.
∂Y ∂Y ∂ 1 ⋅ ρ ⋅ Dm ⋅ m d ω = 0 , m = 1, N S − 1 ; ρ ⋅ Dm ⋅ m df − lim Δx → 0 Δx ∫∫ ∂x ∫∫f ∂x ∂n ω
YN = 1 − S
(
) df + ∂
. 2.1.
,
∫∫
∂Ym dS . ∂n (2.50 )
-
,
,
(2.49 )
.
« ев, В.В. Алеш
– » (2.51)
, С.Н. Прял в, 2007–2009
, .
80 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Д
,
,
(
«
,
»
),
∂Y ∂ ∂ ∂ ( ρ ⋅ f ⋅ Ym ) + ( ρ ⋅ w ⋅ f ⋅ Ym ) = ⎛⎜ ρ ⋅ f ⋅ Dm ⋅ m ∂t ∂x ∂x ⎝ ∂x
А
,
∑ ∫∫ ε m ⋅ ρ ⋅ Dm ⋅ NS
m =1 S
:
⎞ ⎟ , m = 1, N S − 1 ; YN S = 1 − ⎠
∂Ym ∂Y ∂ S dS = ∑ ε m ⋅ ρ ⋅ Dm ⋅ ∂xm df + ∂n ∂x m =1 ∫∫f
∑Y
N S −1 m =1
m
. (2.52)
N
N NS ∂Y ∂Y ∂ ⎛ 1 S + lim ⋅ ∑ ∫∫ ε m ⋅ ρ ⋅ Dm ⋅ m d ω = ⎜ f ⋅ ρ ⋅ ∑ ε m ⋅ Dm ⋅ m ⎜ Δx → 0 Δx ∂n ∂x ⎝ ∂x m =1 ω m =1
⎞ ⎟⎟ . ⎠
(2.53)
,
,
-
,
∂(ρ ⋅ f ) ∂t
: +
∂ ( ρ ⋅ w ⋅ f ) = 0; ∂x
(2.54 )
∂Y ∂ ∂ ∂ ⎛ ( ρ ⋅ Ym ⋅ f ) + ( ρ ⋅ Ym ⋅ w ⋅ f ) − ⎜ ρ ⋅ f ⋅ Dm ⋅ m ∂t ∂x ∂x ⎝ ∂x
YN = 1 − S
∑Y
N S −1 m =1
∂ ( ρ ⋅ w⋅ f ) ∂t
+
m
⎞ ⎟ = 0, ⎠
m = 1, N S − 1 ;
(2.54 )
;
∂ ( ρ ⋅ w2 ⋅ f ) ∂x
∂z ⎞ π ⎛ ∂p = − f ⋅ ⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x
⎛ ∂ ⎡ w2 ⎞ ⎤ ∂ ⎡ ⎢ ρ ⋅ f ⋅ ⎜⎜ ε + ⎟⎥ + ⎢ ρ ⋅ w ⋅ f ∂t ⎣⎢ 2 ⎠ ⎦⎥ ∂x ⎣⎢ ⎝
⎛ ∂z1 ∂ w2 ⎞ ⎤ ⋅ ⎜⎜ ε + − ⎟⎥ = − ( p ⋅ w ⋅ f ) − ρ ⋅ w ⋅ f ⋅ g ⋅ ∂x ∂x 2 ⎠ ⎦⎥ ⎝
NS ∂Y ∂f ∂ ⎛ ∂T ⎞ ∂ ⎛ − p⋅ +Q⋅ f + ⎜k ⋅ f ⋅ − Φ + ⋅ ⋅ T T f ρ ε m ⋅ Dm ⋅ m ( , ) ⎜ ∑ oc ⎟ ∂t ∂x ⎝ ∂x ⎠ ∂x ⎝⎜ ∂x m =1
ε m = ε m ({S ме и } ) ,
(2.54 )
m = 1, N S ; ε = ε ({S ме и } ) ; T1 = T2 = … = TN S = T ;
⎞ ⎟⎟ ; ⎠
(2.54 )
p = p ({S ме и } ) ; k = k ({S ме и } ) ; Dm = Dm ({S ме и } ) , m = 1, N S .
2.3.3. Течение
но о о
онен ных и
). В
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.54 )
е
(
, © В.Е. Селе
о
(2.54 )
, [80]. В
-
лава 2 81 _______________________________________________________________________________________
0,1 ÷ 2,0МПа , .
(
10 ÷ 160 C )1 (
-
):
ρ p1 ( t ) − ρ p 2 ( t ) ⋅100%
δρT ( t ) =
⎡⎣ ρ p1 ( t ) + ρ p 2 ( t ) ⎤⎦ / 2
t = T − 273,15 K –
; ρ p1 ( t ) , ρ p 2 ( t ) –
Ц
δρT (10 C ) ≈ 0, 09%
t = 10 C
ρ p 2 (10 C ) = 1000, 6 кг м , [80]).
(2.55)
,
.
ρ p1 (10 C ) = 999, 7 кг м , 2
3
(
3
В
. :
δρ P ( p ) =
ρ t1 ( p ) , ρt 2 ( p ) –
ρt1 ( p ) − ρt 2 ( p ) ⋅100% ⎣⎡ ρt1 ( p ) + ρt 2 ( p ) ⎦⎤ / 2
.
,
(2.56)
p = 0, 7 МПа
δρ P ( 0,7 МПа ) ≈ 9,7% 3
ρ t1 ( 0, 7 МПа ) = 1000 кг м 3 , ρ t 2 ( 0, 7 МПа ) = 907,5 кг м 3 , [80]). ,
( И
ρ = ρ ( p, T )
ρ = ρ (T ) .
.
-
∂(ρ ⋅ f ) ∂t
(2.36)
+
∂ ( ρ ⋅ f ⋅ w)
∂ ( ρ ⋅ f ⋅ w) ∂t
∂x
+
,
:
= 0;
∂ ( ρ ⋅ f ⋅ w2 ) ∂x
1
(2.57 ) ∂z ⎞ π ⎛ ∂p = − f ⋅ ⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x (
«
–
, . 2
В
[80] .
[80]
10 C
0; 10; 20 ... C ). 3
(
)
. © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
»),
(2.57 )
82 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∂ ( p ⋅ f ⋅ w) ⎞⎤ ∂ ⎡ ⎛ w2 ⎞ ⎤ − ⎟⎥ + ⎢ ρ ⋅ f ⋅ w ⋅ ⎜ ε + ⎟⎥ = − 2 ⎠⎦ ∂x ⎠ ⎦ ∂x ⎣ ⎝ ∂z ∂f ∂ ⎡ ∂T ⎤ − ρ ⋅ f ⋅ w ⋅ g ⋅ 1 − p ⋅ + Q ⋅ f + ⎢ k ⋅ f ⋅ ⎥ − Φ (T , T ) ; ∂x ∂t ∂x ⎣ ∂x ⎦
⎛ ∂ ⎡ w2 ⎢ρ ⋅ f ⋅ ⎜ε + 2 ∂t ⎣ ⎝
ρ = ρ (T ) ; Д )
(2.57 )
h = h ( p, T ) .
ε = h− p ρ;
(2.57 ) (
.
)
:
δ hP ( p ) =
ht1 ( p ) , ht 2 ( p ) –
ht1 ( p ) − ht 2 ( p ) ⋅100% ⎡⎣ ht1 ( p ) + ht 2 ( p ) ⎤⎦ / 2
hp1 ( t ) , hp 2 ( t ) –
p
t = 40 C ,
(
hp1 ( t ) − hp 2 ( t ) ⋅100% ⎡⎣ hp1 ( t ) + hp 2 ( t ) ⎤⎦ / 2
.В
. : (2.59)
,
.
40 ÷ 160 C . Н
.Н
δ hT ( t ) p
δ hT ( 40 C ) ≈ 1%
t = 40 C
(
кг , [80]), . .
-
.
h
δ hT ( 60 C ) ≈ 0, 635% (
(
кг , [80]), . .
кг , h p 2 ( 40 C ) = 169,3 кД
δ hT ( t )
δ hT (10 C ) ≈ 4, 4%
t = 10 C
кг , h p 2 (10 C ) = 44 кД
h h p1 ( 40 C ) = 167,6 кД
-
кг , [80]). И
h
t
δ hT ( t ) =
10 ÷ 160 C )
δ hP ( 0, 7 МПа ) ≈ 176, 2%
кг , ht 2 ( 0, 7 МПа ) = 675, 6 кД
,
h p1 (10 C ) = 42,1 кД
(2.58)
,
0,1 ÷ 2,0МПа ,
( p = 0, 7 МПа
ht1 ( 0, 7 МПа ) = 42, 7 кД
: 10 ÷ 40 C
t.В -
.
ΔhT ( t )
-
. Д
h (
h p1 ( 60 C ) = 251, 2 кД
. Н
,
t = 60 C
кг , h p 2 ( 60 C ) = 252,8 кД
,
– кг ,
[80]). 1
, 40 C )
(
1
© В.Е. Селе
1% .
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 83 _______________________________________________________________________________________
h = h (T ) .
1
-
40 C )
(
.В
-
4%.
И
ρ = ρ (t ) , а [t − 0,5 ⋅ Δt; t + 0,5 ⋅ Δt ] : -
.
[80]. е
Δρ ( t , Δt ) = ρ ( t + 0,5 ⋅ Δt ) − ρ ( t − 0,5 ⋅ Δt ) . ,
.А
(2.60) ,
(2.60)
δρ ( t , Δt ) = (2.60) (2.61)
⎡⎣ ρ ( t + 0,5 ⋅ Δt ) + ρ ( t − 0,5 ⋅ Δt ) ⎤⎦ 2
δρ ( t , Δt ) = Δρ ( t , Δt ) ⋅
) (2.62)
:
Δρ ( t , Δt ) .
,
,
Δρ
: ( Δρ Δt ) p ≈ ( ∂ρ ∂t ) p . ,
Э
2
( ∂ρ
p = 0, 7 МПа .
δρ )
t = 160 C ,
. Н
t < 160 C ),
(2.62)
Δt
∂t ) p
Δt
,
(2.63)
δρ ( t , Δt )
( ∂ρ
. 2.4 3
.
∂t ) p
ρ = ρ (t )
( -
1
. К
.Н
, ,
3
(2.61)
100% ∼ Δρ ( t , Δt ) . ⎡⎣ ρ ( t + 0,5 ⋅ Δt ) + ρ ( t − 0,5 ⋅ Δt ) ⎤⎦ 2
.А
(2.63),
2
⋅100%.
⎣⎡ ρ ( t + 0,5 ⋅ Δt ) + ρ ( t − 0,5 ⋅ Δt ) ⎦⎤ 2 ≈ ρ = const ,
ρ –
Δρ (
е ь е t + 0,5 ⋅ Δt ] :
«∼ »
(
(2.62)
Н
[t − 0,5 ⋅ Δt;
ρ ( t + 0,5 ⋅ Δt ) − ρ ( t − 0,5 ⋅ Δt ) ,
(2.60)
Д
© В.Е. Селе
,
-
[80]),
-
. , t = 160 C ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
( .
84 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ 1
160 C )
( t = 150 C . ρ (160 C ) = 907,523 кг м 3 ,
1%. . 150 C – ρ (150 C ) = 917,095 кг м 3 [80].
δρ (160 C , 10 C ) δρ (155 C , 10 C ) =
ρ (160 C ) − ρ (150 C ) ⋅100% :
⎡ ρ (160 C ) + ρ (150 C ) ⎤ / 2 ⎣ ⎦
,
[ кг м3 ] в ы 0,7МПа 2
л
е
ера уры [ C ] р
.
δρ ( t , Δt ) = Δt :
δρ ( t , Δt )
(2.65)
Δt
,
Н
е
е
, е а
© В.Е. Селе
100% ⋅ Δρ ( t , Δt ) . ρ (t )
ρ ( t ) ⋅ δρ
Δt ( t )
, [80]. ев, В.В. Алеш
.
, С.Н. Прял в, 2007–2009
(2.65)
100% d ρ ( t ) . ⋅ dt ρ (t )
d ρ ( t ) dt ⋅100%
).
2
авле
:
100% Δρ ( t , Δt ) ⋅ ρ (t ) Δt
Δt ( t )
. 2.5
1
=
(2.64)
δρ ( t , Δt ) (
Д (2.60)),
≈ 1%.
Δt = 10 C = 10 K .
, 1%,
и . 2.4. Зав
160 C
(2.66)
(2.67)
.
δρ = 1% . а
(
а
аче-
лава 2 85 _______________________________________________________________________________________
и . 2.5. в
е аш р р в у
[80].
d ρ dt
а а ра
а е ера ур (в ав а р ва а е ае у
ы
. 2.5
ре у
а
в ч
е
Δt ( t ) . В
1% [105 °C − Δt 2; 105 °C + Δt 2] = [99 °C; 111 °C ] . Δt ( t ) . Д
Δt ( t )
.
δρ ,
Δt ( t ) ,
10
, ,
. 2.5 (
, Δt ( t )
δρ = 0,1% . 2.5.
. (2.67)),
δρ .
(2.67), . 2.5, 35 C ),
( ⎡⎣0 C ; 35 C ⎤⎦
. -
t = 35 C )
Δt = 35 C
, . ,
,
: аче е ⎡⎣0 C ; 40 C ⎤⎦ . В
ρ ( 0 C ) = 999,800 кг м 3 , а
е
е а
ев, В.В. Алеш
, чае
я
[80] е ее 1%
40 C – ρ ( 40 C ) = 992,260 кг м 3 [80]. И
δρ ( t = 20 C , Δt = 20 C ) = © В.Е. Селе
-
,
,
(
-
,
1% . В
К
ера уры), 1%
Δt (105 °C ) ≈ 12 °C . Э
t = 105°C
,
.
,
ρ ( 40 C ) − ρ ( 0 C ) ⋅100% ⎡ ρ ( 40 C ) + ρ ( 0 C ) ⎤ / 2 ⎣ ⎦
, С.Н. Прял в, 2007–2009
я 0C
а а-
, :
≈ 0,76 %.
(2.68)
86 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Δt = Δt ( t )
, ,
,
( t = 160 C ) .
,
ρ = const ;
.
. 2.5)
Δt = 10 C
-
ρ = const
.Д :
t > 40 C (
(2.57). В
(2.69 )
w = w (t ) ;
(2.69 )
∂z λ ⋅ w ⋅ w 1 ∂p dw ; = − ⋅ −g⋅ 1 − ρ ∂x 4⋅ R ∂x dt
(2.69 )
∂z Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T w ∂p ∂ε d ⎛ w2 ⎞ ∂ε + ⎜ ⎟ + w⋅ = − ⋅ − w ⋅ g ⋅ 1 + + ⋅ ⎢k ⋅ ⎥ − ρ ∂x f ⋅ρ ∂t dt ⎝ 2 ⎠ ∂x ∂x ρ ρ ∂x ⎣ ∂x ⎦ h = h ( p, T ) .
ε = h− p ρ; ,
(2.69 ) (2.69 )
(2.69 ) ( . (2.69 )). (2.69 ) w
И
);
-
dw dt
.
:
∂z1 λ ⋅ w d ⎛ w2 ⎞ w ∂p ; − ⎜ ⎟ = − ⋅ − w⋅ g ⋅ ρ ∂x 4⋅ R dt ⎝ 2 ⎠ ∂x 3
В
(2.70)
(2.69 )
(2.70)
:
Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T ∂ε ∂ε λ ⋅ w + w⋅ = + + ⋅ ⎢k ⋅ ⎥ − ρ ρ ∂x ⎣ ∂x ⎦ 4⋅R f ⋅ρ ∂t ∂x 3
,
).
(2.71)
:
ρ = const ; w = w (t ) ;
(2.72 ) (2.72 )
∂z λ ⋅ w ⋅ w 1 ∂p dw ; = − ⋅ −g⋅ 1 − ρ ∂x 4⋅ R ∂x dt
∂ε ∂ε λ ⋅ w Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T + w⋅ = + + ⋅ ⎢k ⋅ ⎥ − ρ ρ ∂x ⎣ ∂x ⎦ ∂t ∂x 4⋅ R f ⋅ρ 3
ε = h− p ρ;
h = h ( p, T ) .
ев, В.В. Алеш
(2.72 ) (2.72 )
( ,
© В.Е. Селе
);
(2.72 )
, С.Н. Прял в, 2007–2009
. (2.72 )) ,
-
лава 2 87 _______________________________________________________________________________________ 1
:
λ ⋅ w⋅ w ⋅l p − p2 dw , ⋅l = 1 + g ⋅ ⎡⎣( z1 )1 − ( z1 )2 ⎤⎦ − ρ 4⋅ R dt ; p1 – x = 0 ); p2 –
l –
(2.73) (
x = l ); ( z1 )1 –
(
; ( z1 )2 –
. ,
ρ = const ;
:
w = w (t ) ;
(2.74 ) (2.74 )
⎡( z1 )1 − ( z1 )2 ⎤⎦ λ ⋅ w ⋅ w dw p1 − p2 ; = +g⋅⎣ − 4⋅ R ρ ⋅l dt l
∂ε ∂ε λ ⋅ w Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T + w⋅ = + + ⋅ ⎢k ⋅ ⎥ − ρ ρ ∂x ⎣ ∂x ⎦ ∂t ∂x 4⋅ R f ⋅ρ
);
3
h = h ( p, T ) .
ε = h− p ρ;
(2.74 )
(2.74 ) (2.74 ) -
.
.К
2.3.2,
-
,
, ,
.
И (2.57)),
( ( )
∂(ρ ⋅ f ) ∂t
+
∂ ( ρ ⋅ f ⋅ w) ∂x
YN = 1 − S
∑Y
(
.
∂ ( ρ ⋅ f ⋅ w) ∂t
+
∂x
⎞ ⎟ = 0, ⎠
m = 1, N S − 1 ;
∂z ⎞ π ⎛ ∂p = − f ⋅ ⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂ ∂x ⎠ 4 x ⎝
,
(2.75 )
(2.75 )
.
© В.Е. Селе
(2.75 )
;
∂ ( ρ ⋅ f ⋅ w2 ) m
(2.54)):
= 0;
N S −1 m =1
-
,
∂Y ∂ ∂ ∂ ⎛ ( ρ ⋅ Ym ⋅ f ) + ( ρ ⋅ Ym ⋅ w ⋅ f ) − ⎜ ρ ⋅ f ⋅ Dm ⋅ m ∂t ∂x ∂x ⎝ ∂x
1
.
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
88 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∂ ( p ⋅ f ⋅ w) ⎛ ⎛ w2 ⎞⎤ ∂ ⎡ w2 ⎞⎤ ∂⎡ ∂z − ρ ⋅ f ⋅w⋅ g ⋅ 1 − ⎢ρ ⋅ f ⋅ ⎜ ε + ⎟⎥ + ⎢ ρ ⋅ f ⋅ w ⋅ ⎜ ε + ⎟⎥ = − ∂t ⎣ ∂x ∂x 2 ⎠ ⎦ ∂x ⎣ 2 ⎠⎦ ⎝ ⎝ ∂f ∂ ⎡ ∂T ⎤ − p ⋅ + Q ⋅ f + ⎢k ⋅ f ⋅ ⎥ − Φ (T , T ∂t ∂x ⎣ ∂x ⎦
(
)
NS ∂ ⎛ ∂Y ⎞ ) + ⎜⎜ ρ ⋅ f ⋅ ∑ ε m ⋅ Dm ⋅ m ⎟⎟ ; ∂x ⎝ ∂x ⎠ m =1
(
(2.75 )
)
ρ = ρ T , {Ym , m = 1, N S } ; ε = h − p ρ ; h = h p, T , {Ym , m = 1, N S } . В
{Y , m = 1, N }
(2.75)
Ym , m = 1, N S .
m
(2.75 )
S
е
ае
1
.
(2.75)
:
ρ = const ;
(2.76 )
w = w (t ) ;
(2.76 )
⎡( z1 )1 − ( z1 )2 ⎦⎤ λ ⋅ w ⋅ w dw p1 − p2 = +g⋅⎣ − ; ρ ⋅l 4⋅ R dt l ∂Ym ∂Y ∂Y ⎞ ∂ ⎛ + w ⋅ m − ⎜ Dm ⋅ m ⎟ = 0, ∂t ∂x ∂x ⎝ ∂x ⎠
(2.76 )
m = 1, N S − 1 ; YN = 1 −
Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T ∂ε ∂ε λ ⋅ w + w⋅ = + + ⋅ ⎢k ⋅ ⎥ − ρ ρ ∂x ⎣ ∂x ⎦ f ⋅ρ 4⋅R ∂t ∂x
)+
3
(
)
ε = h − p ρ ; h = h p, T , {Ym , m = 1, N S } .
S
∑Y
N S −1 m =1
m
(2.76 )
;
N ∂Ym ⎞ ⎤ ∂ ⎡ S⎛ ⎢ ∑ ⎜ ε m ⋅ Dm ⋅ ⎥; ∂x ⎣⎢ m =1 ⎝ ∂x ⎟⎠ ⎦⎥
(2.76 )
(2.76 )
2.3.4. Пленочное ечение вух а но
е ы
Д -
В. .
.Н.
-
[2, 6]. .
В
[68],
-
–
.
,
– -
. Ч ,
,
,
. ,
1
.В
, ). В
© В.Е. Селе
,
ев, В.В. Алеш
, , (2.76)
( « (
, С.Н. Прял в, 2007–2009
»
-
, ,
).
лава 2 89 _______________________________________________________________________________________
, 1
,
-
2
.
.
.
, ( .Э
•
)
,
-
, : («
»
•
–
», «
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.); («
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»);
•
( ). , ,
а еа
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), е
я
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я
[68]. , .В
аа
( . Не ь я
–
,
,
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я ь –
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–
(
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,
.).
,
-
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3
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, К
,
,
(
,
. Э [68].
е
)
, ,
: ,
2.3.2). В
,
(
, ,
( N − n)
), (
) –
,
, : N
-
n [68].
,
:
[79, 81]. И -
,
1
C2H4O . . [68].
, (C2H4O)3
,
H2O
(H2O)2 ,
2
[68].
3
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.
, . И
(
-
.
[85]. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
90 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
, ,
С
,
.
е
(
,
[79])
[82]. В
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, ,
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.). Э
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[82] .
В [82]
-
:
•
а а
•
а
е
ече
ь е
(
е
я
ч
; ече
)
я
-
;
•
а е
ече
я
ч а
.
ече
я
ь
е,
е(
а я
е)
е
е-
[82].
[79],
,
20÷30%.
ó
,
, (
)
В
-
ь е
:
.
[79, 82]
В
-
е
ч
(
е
.Э
-
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) е
-
.
е
-
( а е ь
ь е
) е
) е , ече я –
(
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-
.
, .
, :
,
. [83], © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 91 _______________________________________________________________________________________
,
,
,
,
. .
А
, [83]. В »)
(«
, .
,
-
(
), , .
,
,
,
. ,
,
[83].
1 1
.
ρ1
–
ψ1
r –
V
∫∫∫ V
,
ρ2 .
[82]: :
ρ м = ψ 1 ⋅ ρ1 +ψ 2 ⋅ ρ 2 , (
(2.77)
)
( r, t )
⎧⎪1, ⎪⎩0,
[82], ∂t
, ,
ψ m ( r, t ) = ⎨
∂ ( ρ m ⋅ψ m )
,
ψ2 – ,
∫∫∫
2
,
m−
[82]: ,
,
,t –
.
-
, ,
dV + ∫∫ ρ m ⋅ψ m ⋅υmn dS = ∫∫∫ J km dV , k ≠ m, k , m = 1, 2;
∂ ( ρ m ⋅ψ m ⋅ υm ) ∂t
S
-
: (2.79 )
V
dV + ∫∫ ρ m ⋅ψ m ⋅ υm ⋅υmn dS = − ∫∫ pm ⋅ ndS +
+ ∫∫ τ nm dS + ∫∫∫ ρ m ⋅ψ m ⋅ Fm dV − ∫∫∫ ρ m ⋅ψ m ⋅ a m dV + ∫∫∫ Pkm dV , S
S
k ≠ m, k , m = 1, 2; S
(2.78)
V
V
(2.79 )
V
⎡ ⎛ υ m2 ⎞ ⎤ υ m2 ⎞ ⎤ ∂ ⎡ n ⎛ ⋅ ⋅ + + ⋅ ⋅ ⋅ + dV ρ ψ ε ρ ψ υ ε ⎢ ⎥ ⎢ ⎜ ⎟ ⎜ ⎟ ⎥ dS = m m m m m m m ∫∫∫ ∫∫S 2 ⎠⎦ 2 ⎠⎦ ∂t ⎣ ⎝ ⎝ V ⎣
(
)
= − ∫∫ pm ⋅υ mn dS + ∫∫ ( τ mn ⋅ υ m ) dS + ∫∫∫ ρ m ⋅ψ m ⋅ Fm ⋅ υ m dV −
− ∫∫∫ ρ m ⋅ψ m ⋅ ( am ⋅ υ m ) dV − ∫∫ Wmn dS + ∫∫∫ Qm dV + ∫∫∫ Ekm dV , k ≠ m, k , m = 1, 2, S
S
V
V
1
S
V
V
В
, ,
© В.Е. Селе
ев, В.В. Алеш
(2.79 )
.
, С.Н. Прял в, 2007–2009
-
92 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
V –
; υ = ( υm ⋅ n ) –
; S –
m1
V ; υm –
,
; υm –
n
(J
, ; τ = τm ⋅ n –
km
υm ; J km – m, pm –
)
k= − J mk ;
n m
m-
,
m[63,
n
( τm – 79]); Fm
,
am – k-
(P
,
m-
km
m-
)
mn ; Ekm –
,
m-
= − Pmk ; ε m –
(
(E
km
; Wm –
m-
,
)
-
; Qm –
,
k-
m,
; Pkm –
m-
-
υm
n m
)
; W = Wm ⋅ n – n m
= − Emk .
Wm Wm = − km ⋅∇Tm ,
m-
km –
m-
; Tm –
-
. Ox . К
,
, V,
f
Δx
Δx
,
. (
,
. 2.6).
-
. V,
Vm Sm –
m-
V1 + V2 = V .
m-
ω
f1 + f 2 = f .
, (2.80 )
Vm . А
, f ,
V;
f ′,
f
, fm –
-
, (2.80 ) ,
S гр –
V
.В . , . .
1
J km = 0.
В
« [89],
© В.Е. Селе
ев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
(2.81) »
-
лава 2 93 _______________________________________________________________________________________
и . 2.6. Схе а
ч
е ре ру
авле е л ев ече я вух а р в у( р л е ече е)
(2.79 )
m,
Vm .
-
∂ρ m dV + ∫∫ ρ m ⋅υ mn dS = 0, m = 1, 2. t ∂ [Vm ] [ Sm ]
∫∫∫
:
–
Sm Vm (
ω2 ≡ S гр ). К
ре ы
(2.82) f m′ , ω1 ≡ S гр ∪ ω , fm
ωm , .
[84],
ρ –
:
V
∂ ∂ρ ρ dV = ∫∫∫ dV + ∫∫ ρ ⋅ U n dS , ∂t ∫∫∫ ∂t V V S
(
(2.83)
)
V ; U n = U ⋅n –
; U –
dS . В
: ⎡1 ⎛ ⎞⎤ ∂ ρ dV = lim ⎢ ⋅ ⎜ ∫∫∫ ρ ( t + Δt ) dV − ∫∫∫ ρ ( t ) dV ⎟ ⎥ = ∫∫∫ Δt → 0 Δt ⎜ ⎟⎥ ∂t V ⎢⎣ V (t ) ⎝ V (t +Δt ) ⎠⎦ = lim
Δt → 0
∫∫∫ ρ ( t + Δt ) dV − ∫∫∫ ρ ( t ) dV + ∫∫∫ ρ ( t ) dV − ∫∫∫ ρ ( t ) dV
V ( t +Δt )
∫∫∫
V ( t + Δt )
ρ ( t ) dV
∂ρ V ( t + Δt ) −V ( t ) = ∫∫∫ dV + lim 0 t Δ → ∂t Δt V
В
© В.Е. Селе
(2.84)
ев, В.В. Алеш
Δt
∫∫∫
V ( t + Δt ) −V ( t )
V ( t +Δt )
V (t )
= (2.84)
.
ρ ( t ) dV
, С.Н. Прял в, 2007–2009
-
94 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ 1
.
V
,
(
ρ ⋅ U n ⋅ Δt ⋅ dS –
V Δt
V
∫∫∫
V ( t + Δt ) −V ( t )
dS – U n ⋅ Δt ⋅ dS –
. 2.7), Δt , Δt ,
dS .
,
:
ρ ( t ) dV = ∫∫ ρ ⋅ U n ⋅ ΔtdS .
(2.85)
S
(2.85) (2.84),
(2.83).
и . 2.7. В
А
C1 ( t ) –
а ел
ая хе а
y ( r, t ) ∈ C1 ( t ) ,
, ( ),
-
[84]:
∂ ∂y ydV = ∫∫∫ dV + ∫∫ y ⋅ U n dS . ∂t ∫∫∫ V V ∂t S
(2.86)
(2.82):
⎛ ⎞ ⎛ ∂ρ m ∂ρ dV + ∫∫ ρ m ⋅υ mn dS = ⎜ ∫∫∫ m dV + ∫∫ ρ m ⋅υ mn d ω ⎟ + ⎜ ∫∫ ρ m ⋅υ mx df − ∫∫ ρ m ⋅υ mx df ⎜ [V ] ∂t ⎟ ⎜[f′] [Vm ] ∂t [ Sm ] [ωm ] [ fm ] ⎝ m ⎠ ⎝ m m = 1, 2,
∫∫∫
υmx –
υm
∫∫ ρ
[ fm ] c 1
dS ;
[ f m′ ]
m
Ox 2.
⋅υ mx df − ∫∫ ρ m ⋅υ mx df = Δx ⋅
–
[ fm ]
,
, ∂ ρ m ⋅υ mx df , ∂x [ ∫∫ fm ]
,
В .
2
© В.Е. Селе
«x» ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
Ox.
: (2.88)
c
m-
⎞ ⎟, ⎟ (2.87) ⎠
лава 2 95 _______________________________________________________________________________________
xc ,
f ′. В
f xc , . .
. .
f
ωm . Δx = const ),
f ′ ),
Vm
ωm ( . .
,
-
Um f
(2.83),
υm
f′
(
:
∂ρ m ∂ρ dV + ∫∫ ρ m ⋅υ mn d ω = ∫∫∫ m dV + ∫∫ ρ m ⋅ U mn dS = [Vm ] ∂t [ωm ] [Vm ] ∂t [ Sm ]
∫∫∫
∂ ∂ ⎛ ρ m dV = ∫ ⎜ ∫∫ ρ m df ∫∫∫ ∂t [Vm ] ∂t Δx ⎜ [ f m ] ⎝ m = 1, 2. =
,
Δx
(2.88)
(2.89)
⎞ ∂ ⎟ dx = Δx ⋅ ∫∫ ρ m df , ⎟ ∂t [ fm ] ⎠ c
Δx ,
(2.87), :
∂ ∂ ρ m df + ∫∫ ρ m ⋅υ mx df = 0, m = 1, 2. x [ fm ] ∂t [∫∫ ∂ fm ]
А :
∫∫∫ ∂t dV + ∫∫ y ⋅υ [Vm ]
∂y
[ Sm ]
n m
dS = Δx ⋅
(2.90)
∂ ∂ ydf + Δx ⋅ y ⋅υ mx df . ∫∫ ∂t [ fm ] ∂x [ ∫∫ fm ] c
В
∂ ( ρm ⋅ f m ) ∂t
(2.92 ) ев, В.В. Алеш
, +
1 ⋅ ρ m df ; f m [∫∫ fm ]
(2.92 )
f 1 ⋅ ∫∫ψ m df = m . f f f
∂ ( ρ m ⋅ wm ⋅ f m )
f = const ,
∂x
, С.Н. Прял в, 2007–2009
(2.92 )
1 ⋅ ρ m ⋅υ mx df ; f m ⋅ ρ m [∫∫ fm ]
ψm =
© В.Е. Селе
m-
ρm = wm =
(2.91)
c
, :
-
y ( r, t ) ∈ C1 ( r, t )
,
,
(2.89)
(2.92 ) :
= 0,
m = 1, 2,
(2.93 )
96 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∂ ( ρ m ⋅ψ m ) ∂t
+
∂ ( ρ m ⋅ wm ⋅ψ m ) ∂x
(2.79 ) :
(2.93 )
m,
Vm .
-
∂ ( ρ m ⋅ υm ) dV + ∫∫ ρ m ⋅ υ m ⋅υ mn dS = − ∫∫ pm ⋅ ndS + ∂ t [Vm ] [ Sm ] [ Sm ]
∫∫∫
+
n ∫∫ τm dS + ∫∫∫ ρ m ⋅ Fm dV − ∫∫∫ ρ m ⋅ am dV .
[ Sm ]
[Vm ]
(2.94) Пе
= 0, m = 1, 2.
е
е
а ае
(2.94)
[Vm ]
.
Ox
е.
(2.91),
:
⎛ ⎞ 2 ∂ ( ρ m ⋅ υm ) ⎞ ⎛ ∂ ∂ ⎜ ∫∫∫ dV ⎟ + ⎜ ∫∫ ρ m ⋅ υ m ⋅υ mn dS ⎟ = Δx ⋅ ∫∫ ρ m ⋅υ mx df + Δx ⋅ ρ m ⋅ (υ mx ) df . ∫∫ ⎜ [V ] ⎟ ⎜ [S ] ⎟ ∂t ∂t [ fm ] ∂x [ fm ] ⎝ m ⎠ ⎝ m ⎠ c c x
x
(2.95) В
, fm :
β mV 2 = ,
(2.92 )
2 1 ⋅ ∫∫ ρ m ⋅ (υ mx ) df . 2 ρ m ⋅ wm ⋅ f m [ fm ]
(2.96),
:
⎛ ⎞ ∂ ( ρ m ⋅ υm ) ⎞ ⎛ ⎜ ∫∫∫ dV ⎟ + ⎜ ∫∫ ρ m ⋅ υ m ⋅υ mn dS ⎟ = ⎜ [V ] ⎟ ⎜ [S ] ⎟ ∂t ⎝ m ⎠ ⎝ m ⎠ ∂ ∂ = Δx ⋅ ( ρ m ⋅ wm ⋅ f m )c + Δx ⋅ ( β mV 2 ⋅ ρ m ⋅ wm2 ⋅ f m ) . c ∂t ∂x x
е ье
(2.96)
а ае
x
е.
⎛ ⎞ ⎛ ⎜ ∫∫ pm ⋅ ndS ⎟ = ⎜ ∫∫ pm df − ∫∫ pm df ⎜ [S ] ⎟ ⎜[f ′] [ fm ] ⎝ m ⎠ ⎝ m x
∫∫ p
[ f m′ ]
:
m
df − ∫∫ pm df = Δx ⋅
pm – © В.Е. Селе
[ fm ]
ев, В.В. Алеш
x
⎛ ∂ ( pm ⋅ f m ) ⎞ ∂ pm df = Δx ⋅ ⎜ ⎟ , ∫∫ ∂x [ fm ] ∂x ⎝ ⎠c c
fm
⎞ ⎛ ⎞ ⎟ + ⎜ ∫∫ pm ⋅ nd ω ⎟ . ⎟ ⎜ [ω ] ⎟ ⎠ ⎝ m ⎠
:
, С.Н. Прял в, 2007–2009
(2.97)
(2.98)
(2.99)
лава 2 97 _______________________________________________________________________________________
pm =
Д
∫∫ ndS = 0 ,
, . .
1 ⋅ pm df . f m [∫∫ fm ]
(2.100)
:
[ Sm ]
⎛ ⎞ ⎛ ⎞ ⎛ ∂f ⎞ ⎜ ∫∫ pm ⋅ nd ω ⎟ = pm ⋅ ⎜ − ∫∫ ndS + ∫∫ nd ω ⎟ = − pm ⋅ ( f m′ − f m ) = − pm ⋅ Δx ⋅ ⎜ m ⎟ , (2.101) ⎜ [ω ] ⎟ ⎜ [S ] ⎟ ⎝ ∂x ⎠ c [ωm ] ⎝ m ⎠ ⎝ m ⎠ x
x
ω
pm – ⎛ ⎞ pm = ⎜ ∫∫ pm ⋅ nd ω ⎟ ⎜ ⎟ ⎝ [ωm ] ⎠
x
,
x x ⎡⎛ ⎞ ⎤ ⎛ ⎞ 1 ⎢ ⎥ pm ⋅ nd ω ⎟ . ⋅ ⎜ ∫∫ nd ω ⎟ = ⋅⎜ ⎟ ⎥ ⎟ ⎢⎜ f m − f m′ ⎝⎜ [∫∫ ωm ] ⎠ ⎦ ⎠ ⎣⎝ [ωm ]
В
−1
(2.102)
:
β mp ω = pm pm . (2.103),
(2.103) :
⎛ ⎞ ⎛ ∂ ( pm ⋅ f m ) ⎞ ⎛ ∂f ⎞ ⎜ ∫∫ pm ⋅ ndS ⎟ = Δx ⋅ ⎜ − Δx ⋅ β mp ω ⋅ pm ⋅ ⎜ m ⎟ . ⎟ ⎜ ⎟ ∂x ⎝ ∂x ⎠ c ⎝ ⎠c ⎝ [ Sm ] ⎠ x
е
е
е
а ае
∫∫ τ
е.
[ Sm ]
Д
τ mxx
Д
n m
dS =
⎛ ⎜ ∫∫ τ mx df ⎜[f ′] ⎝ m
τ mxx –
∫∫ τ
[ f m′ ]
x m
df − ∫∫ τ mx df + [ fm ]
τ ∫∫ ω
[
m
]
(2.105)
⎞ ⎛ ⎟ − ⎜ ∫∫ τ mx df ⎟ ⎜[f ] ⎠ ⎝ m x
f m′
n m
(2.104)
d ω.
(2.105)
:
⎞ ⎟ = τ mxx ⋅ f m′ − τ mxx ⋅ f m , ⎟ ⎠ x
τ mx
fm
(2.106) Ox .
:
⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎜ ∫∫ τ 2n d ω ⎟ = ⎜ ∫∫ τ 2n dS ⎟ = ∫ ⎜ ∫ τ 2n d χ ⎟ dx = Δx ⋅ ⎜ ∫ τ 2n d χ ⎟ = Δx ⋅ ( Tp2, гр ⋅ χ гр ) , (2.107 ) ⎜ ⎟ ⎜ ⎡S ⎤ ⎜ ⎡χ ⎤ ⎟ ⎟ Δx ⎜ ⎡ χ гр ⎤ ⎟ ⎝ [ω2 ] ⎠ ⎝ ⎣ гр ⎦ ⎠ ⎝⎣ ⎦ ⎠ ⎝ ⎣ гр ⎦ ⎠ x
x
x
x
χ гр –
,
χ гр
; Tp2, гр –
⎞ 1 ⎛⎜ = ⋅ ∫ τ 2n d χ ⎟ . χ гр ⎜ ⎡ χ гр ⎤ ⎟ ⎝⎣ ⎦ ⎠
τ
n 2
Ox :
x
Tp2, гр
Д © В.Е. Селе
: ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.107 )
98 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ x ⎞ ⎛ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ∫∫ τ1n d ω ⎟ = ⎜ ∫∫ τ1n dS ⎟ + ⎜ ∫∫ τ1n d ω ⎟ = ∫ ⎜ ∫ τ1n d χ ⎟ dx + ⎜ ⎟ ⎟ ⎟ ⎠ Δx ⎝⎜ ⎡⎣ χ гр ⎤⎦ ⎝ [ω1 ] ⎠ ⎝⎜ ⎡⎣ Sгр ⎤⎦ ⎠ ⎝ω ⎠ x
x
x
⎛ ⎞ n ∫Δx ⎜⎜ ∫ τ1 d χ ⎟⎟ dx = ⎝ ⎡⎣ χ тру а ⎤⎦ ⎠ x
⎛ ⎞ ⎛ ⎞ = Δx ⋅ ⎜ ∫ τ1n d χ ⎟ + Δx ⋅ ⎜ ∫ τ1n d χ ⎟ = Δx ⋅ (Tp1, гр ⋅ χ гр ) + Δx ⋅ (Tpтру а ⋅ χ тру а ) , ⎜ ⎡χ ⎤ ⎟ ⎜ ⎡χ ⎤ ⎟ ⎝ ⎣ гр ⎦ ⎠ ⎝ ⎣ тру а ⎦ ⎠ (2.108) x
x
χ тру а – τ
χ тру а
; Tpтру а –
n 1
χ гр
Ox ; Tp1, гр –
Tpтру а =
⎛ ⎞ ⋅ ⎜ ∫ τ1n d χ ⎟ ; ⎜ ⎡ χтру а ⎤ ⎟ ⎦ ⎝⎣ ⎠ x
χ тру а 1
,
τ
n 1
Ox :
⎞ 1 ⎛⎜ = ⋅ ∫ τ1n d χ ⎟ ; χ гр ⎜ ⎡ χ гр ⎤ ⎟ ⎝⎣ ⎦ ⎠ x
Tp1, гр
Tp1, гр = −Tp2, гр .
(2.106)
(2.109 ) (2.109 )
f m′
fm
ωm , m = 1, 2 .
.В
[84, 108] (
⎡⎛ ∂ υ i
):
τ ij = μ ⋅ ⎢⎜
⎣⎝ ∂ x
τ ij –
υ i , i = 1,3,
);
j
+
∂ υ j ⎞ 2 ij ∂ υ k ⎤ ⎟ − ⋅ δ ⋅ k ⎥ , i, j , k = 1,3, ∂ xi ⎠ 3 ∂x ⎦
τ ( –
; δ ij –
m υ;
; μ –
К
(2.110) x i , i = 1,3,
– -
. В
, : Re1 =
δ
μ1 –
ρ1 ⋅ w1 ⋅ δ , μ1
(2.111) -
f1
. Re1 > 400
[79], ,
δ* f2
, .К
, .И
1
f1 ,
, , 1
© В.Е. Селе
. 2.8.
, ев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
-
лава 2 99 _______________________________________________________________________________________
и . 2.8. Схе а
ч
е ре
авле
И
е ра
ре еле ру ы
я
ω1
, ,
f m′
р
е
ереч
ω2 .
(
fm 1
.
(
у ече
τ mxx
. (2.110)),
)
τ mxx ,
-
, Tpтру а (
Tpm , гр
. (2.107 – 2.109)). ,
:
⎛ ⎞ ⎜ ∫∫ τ nm dS ⎟ = Δx ⋅ (Tpm , гр ⋅ χ гр ) + δ 1m ⋅ Δx ⋅ ( Tpтру а ⋅ χ тру а ) . ⎜ [S ] ⎟ ⎝ m ⎠ x
Пя
е
а ае
е.
⎛ ⎞ ⎛ ⎜ ∫∫∫ ρ m ⋅ Fm dV ⎟ = ∫ ⎜ ∫∫ ρ m ⋅ Fmx df ⎜ [V ] ⎟ Δx ⎜ [ f ] ⎝ m ⎠ ⎝ m x
Fmx –
Fmx = Ше
е
а ае
е.
x
1
В
⎞ ⎟dx = Δx ⋅ ∫∫ ρ m ⋅ amx df = Δx ⋅ ( ρ m ⋅ amx ⋅ f m ) , c ⎟ [ f m ]c ⎠ fm
,
,
, ,
© В.Е. Селе
,
ев, В.В. Алеш
Fmx :
(2.114)
Ox ; amx –
am
fm ,
fm
1 ⋅ ρ m ⋅ Fmx df . ρ m ⋅ f m [∫∫ fm ]
⎛ ⎞ ⎛ ⎜ ∫∫∫ ρ m ⋅ am dV ⎟ = ∫ ⎜ ∫∫ ρ m ⋅ amx df ⎜ [V ] ⎟ Δx ⎜ [ f ] ⎝ m ⎠ ⎝ m amx –
⎞ ⎟dx = Δx ⋅ ∫∫ ρ m ⋅ Fmx df = Δx ⋅ ( ρ m ⋅ Fmx ⋅ f m ) , (2.113) c ⎟ [ f m ]c ⎠
Ox ; Fmx –
Fm
(2.112)
. , С.Н. Прял в, 2007–2009
(2.115)
amx :
.
100 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
amx = Δx
∂ ( ρ m ⋅ wm ⋅ f m ) ∂t
ρm ⋅ fm 1
⋅ ∫∫ ρ m ⋅ amx df .
Δx
(2.94)
(2.116)
[ fm ]
,
-
.
∂ ( β mV 2 ⋅ ρ m ⋅ wm2 ⋅ f m )
:
+
+ δ 1m ⋅ Tpтру а ⋅ χ тру а
⎛ ∂ ( pm ⋅ f m ) ∂f ⎞ = −⎜ − β mp ω ⋅ pm ⋅ m ⎟ + Tpm , гр ⋅ χ гр + ∂x ∂x ∂x ⎠ (2.117) ⎝ x x + ρ m ⋅ Fm ⋅ f m − ρ m ⋅ am ⋅ f m .
,
,
:
Fmx = g ⋅ cos Ω = − g ⋅
∂z1 , ∂x
(2.118) ,
z1
–
); Ω – Ox ); g –
)( . x (x –
g
-
(
. [82],
p1
.Д
p2 , σ′
1 Rкрив [68, 82]:
p1 = p2 + σ ′ Rкрив ,
R ′′ –
1 Rкрив = 1 R′ + 1 R′′ , R ′ . В :
[82]
(2.119) -
p1 = p2 = p1 = p2 = p;
(2.120)
β mp ω = 1.
R –
(2.121)
,
χ тру а = 2 ⋅ π ⋅ R = 2 ⋅ π ⋅ f . ,
(R −δ ) ,
, δ –
R −δ =
.И
f2 π = ψ 2 ⋅ f π ;
χ гр = 2 ⋅ π ⋅ ( R − δ ) = 2 ⋅ π ⋅ f 2 = 2 ⋅ π ⋅ψ 2 ⋅ f . ев, В.В. Алеш
-
.В
δ = R − f2 π = R − ψ 2 ⋅ f π ;
© В.Е. Селе
(2.122)
, С.Н. Прял в, 2007–2009
,
: (2.123 ) (2.123 ) (2.123 )
лава 2 101 _______________________________________________________________________________________
,
,
-
2 V2 ∂ ( ρ m ⋅ wm ⋅ f m ) ∂ ( β m ⋅ ρ m ⋅ wm ⋅ f m ) + = ∂t ∂x (2.124 ) ∂p ∂z1 x 1m = − f m ⋅ + 2 ⋅ Tpm , гр ⋅ π ⋅ f 2 + 2 ⋅ δ ⋅ Tpтру а ⋅ π ⋅ f − ρ m ⋅ g ⋅ ⋅ f m − ρ m ⋅ am ⋅ f m ∂x ∂x
:
∂ ( ρ m ⋅ wm ⋅ψ m ) ∂t
+ 2 ⋅δ
1m
+
∂ ( β mV 2 ⋅ ρ m ⋅ wm2 ⋅ψ m )
⋅ Tpтру а ⋅
π
∂x
= −ψ m ⋅
π ⋅ψ 2 ∂p + 2 ⋅ Tpm , гр ⋅ + f ∂x
∂z − ρ m ⋅ g ⋅ 1 ⋅ψ m − ρ m ⋅ amx ⋅ψ m , f ∂x
(2.124 )
m = 1, 2 . (2.124 )
(2.124 ), -
, fm :
∂ ( ρ m ⋅ wm ⋅ f m ) ∂t
+
∂ ( ρ m ⋅ wm2 ⋅ f m )
+
∂ ( ρ m ⋅ wm2 ⋅ψ m )
∂x
=
∂z ∂p = − f m ⋅ + 2 ⋅ Tpm , гр ⋅ π ⋅ f 2 + 2 ⋅ δ 1m ⋅ Tpтру а ⋅ π ⋅ f − ρ m ⋅ g ⋅ 1 ⋅ f m ∂x ∂x ∂ ( ρ m ⋅ wm ⋅ψ m ) ∂t
∂x
=
∂z π ⋅ψ 2 ∂p π = −ψ m ⋅ + 2 ⋅ Tpm , гр ⋅ + 2 ⋅ δ 1m ⋅ Tpтру а ⋅ − ρ m ⋅ g ⋅ 1 ⋅ψ m , ∂x ∂x f f m = 1, 2 . К
(2.124 )
Tpтру а
(2.124 )
-
Tpm , гр
.
m,
(2.79 ) :
∂ ⎡
∫∫∫ ∂t ⎢ ρ [Vm ]
⎣
m
-
⎡ ⎛ ⎛ υ 2 ⎞⎤ υ 2 ⎞⎤ ⋅ ⎜ ε m + m ⎟ ⎥ dV + ∫∫ ⎢ ρ m ⋅υ mn ⋅ ⎜ ε m + m ⎟ ⎥ dS = 2 ⎠⎦ 2 ⎠⎦ ⎝ ⎝ [ Sm ] ⎣
= − ∫∫ pm ⋅υ mn dS + [ Sm ]
Vm .
∫∫ ( τ
[ Sm ]
n m
(
[Vm ]
− ∫∫∫ ρ m ⋅ ( am ⋅ υ m ) dV − ∫∫ Wmn dS + ∫∫∫ Qm dV . [Vm ]
[ Sm ]
(2.125). © В.Е. Селе
ев, В.В. Алеш
)
⋅ υ m ) dS + ∫∫∫ ρ m ⋅ Fm ⋅ υ m dV −
, С.Н. Прял в, 2007–2009
[Vm ]
(2.125)
102 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Пе
е
е
а ае
∂ ⎡
∫∫∫ ∂t ⎣⎢ ρ [Vm ]
m
е.
(2.91),
⎡ ⎛ ⎛ υ 2 ⎞⎤ υ 2 ⎞⎤ ⋅ ⎜ ε m + m ⎟ ⎥ dV + ∫∫ ⎢ ρ m ⋅υ mn ⋅ ⎜ ε m + m ⎟ ⎥ dS = 2 ⎠⎦ 2 ⎠⎦ ⎝ ⎝ [ Sm ] ⎣
⎡ ⎡ ⎛ υ m2 ⎞ ⎤ υ m2 ⎞ ⎤ ∂ x ⎛ ρ ε ρ υ ε df x ⋅ + + Δ ⋅ ⋅ ⋅ + ⎢ m ⎜ m ⎢ m m ⎜ m ⎟⎥ ⎟ ⎥ df = ∂x [ ∫∫ 2 ⎠⎦ 2 ⎠⎦ ⎝ ⎝ f ⎣ ⎣ ] m c c (2.126 ) 2 ⎞ w ∂⎛ = Δx ⋅ ⎜ ρ m ⋅ ε m ⋅ f m + β mV 2 ⋅ ρ m ⋅ m ⋅ f m ⎟ + ∂t ⎝ 2 ⎠c
= Δx ⋅
+ Δx ⋅
∂ ∂t [ ∫∫ fm ]
⎞ wm3 ∂ ⎛ VE V3 ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ fm ⎟ , w f ρ ε β ρ β ⎜ m m m m m m m ∂x ⎝ 2 ⎠c
εm =
ρm ⋅ fm
β mV 2 =
β mVE =
β mV 3 = е ье
а ае
е.
∫∫ p
[ Sm ]
∫∫ p
β
[ f m′ ] PV m
m
m
⋅υ mn dS =
∫∫
1
[ f m′ ]
:
(2.126 )
[ fm ]
(2.126 )
ρ m ⋅ ε m ⋅ wm ⋅ f m
(2.126 )
1
⋅ ∫∫ ρ m ⋅υ mx ⋅ ε m df ; [ fm ]
1 ⋅ ρ m ⋅υ mx ⋅υ m2 df . ρ m ⋅ wm3 ⋅ f m [∫∫ fm ]
m
⋅υ mx df − ∫∫ pm ⋅υ mx df +
⋅υ mx df − ∫∫ pm ⋅υ mx df = Δx ⋅ [ fm ]
⋅ ∫∫ ρ m ⋅ ε m df ;
1 ⋅ ρ m ⋅υ m2 df ; ρ m ⋅ wm2 ⋅ f m [∫∫ fm ]
∫∫ p
[ fm ]
(2.126 )
p ∫∫ ω
[
m
]
m
⋅υ mn d ω.
∂ ∂ pm ⋅υ mx df = Δx ⋅ ( β mPV ⋅ pm ⋅ wm ⋅ f m ) , (2.128) c x ∂x [ ∫∫ ∂ fm ]
–
,
pm ⋅υ mn d ω =
∫∫
⎣⎡ Sгр ⎦⎤
-
fm :
1 ⋅ pm ⋅υ mx df . pm ⋅ wm ⋅ f m [∫∫ fm ]
(2.127). . . , S гр :
pm ⋅υ mn dS = pm ⋅
⎛
n ∫∫ υm dS = pm ⋅ ∫ ⎜
⎣⎡ Sгр ⎦⎤
Δx
⎜ ⎡ χ∫гр ⎤ ⎝⎣ ⎦
(2.129) ⎞
υ mn d χ ⎟ dx = ⎟ ⎠
⎛ ⎞ = pm ⋅ Δx ⋅ ⎜ ∫ υ mn d χ ⎟ = pm ⋅ Δx ⋅ ( wmn , гр ⋅ χ гр ) = Δx ⋅ ( β mPV ω ⋅ pm ⋅ wmn , гр ⋅ χ гр ) , ⎜ ⎡χ ⎤ ⎟ ⎝ ⎣ гр ⎦c ⎠
© В.Е. Селе
ев, В.В. Алеш
(2.127)
c
β mPV =
[ωm ]
:
, С.Н. Прял в, 2007–2009
(2.130)
лава 2 103 _______________________________________________________________________________________
pm =
n ∫∫ υm dS
1
∫∫
⋅
⎡⎣ S гр ⎤⎦
( β ) = ( pp ) PV ω m c
wmn , гр =
⎣⎡ S гр ⎦⎤
m
∫
(2.131 )
(2.131 )
;
m c
⋅
χ гр 1
pm ⋅υ mn dS ;
⎡⎣ χ гр ⎤⎦
υ mn d χ ,
w1,n гр = − w2,n гр .
(2.131 )
[82].
∫∫ p
:
[ Sm ]
е
е
⋅υ mn dS = Δx ⋅
m
∫∫ ( τ
е
а ае [ Sm ]
∂ ( β mPV ⋅ pm ⋅ wm ⋅ f m )c + Δx ⋅ ( β mPV ω ⋅ pm ⋅ wmn , гр ⋅ χ гр ) . ∂x
⋅ υ m ) dS =
∫∫ ( τ
е. n m
,
[ f m′ ]
⋅ υ m ) df − ∫∫ ( τ mx ⋅ υ m ) df +
x m
[ fm ]
(τ ∫∫ ω
[
m]
n m
⋅ υ m ) d ω.
(2.133). ,
∫∫ ( τ
[ωm ]
n m
⋅ υm ) d ω =
= Δx ⋅
AmТр –
∫∫ ( τ
∫ (τ
⎡⎣ χ гр ⎤⎦ c
⋅ υ m ) dS =
AmТр =
χ гр 1
(2.134)
Тр m
∫ (τ
⎡⎣ χ гр ⎤⎦
n m
⋅ υm ) d χ ;
И
(
,
∫∫ ( τ
[ Sm ]
е
а ае
е.
∫∫∫ ρ ⋅ ( F [Vm ]
m
= Δx ⋅
© В.Е. Селе
ев, В.В. Алеш
)
⋅ υ m dV =
∫∫ ρ ⋅ ( F m
n m
A1Тр = − A2Тр .
(2.135)
[ f m ]c
m
m
(2.105).
⋅ υ m ) dS = Δx ⋅ ( AmТр ⋅ χ гр ) .
∫ ⎜⎜ ∫∫ ρ ⋅ ( F ⎛
⎝ [ fm ]
)
, С.Н. Прял в, 2007–2009
m
(2.136)
)
⎞ ⋅ υ m df ⎟ dx = ⎟ ⎠
⋅ υ m df = Δx ⋅ ( β Δx
m
:
(2.133) , ,
Пя
⎛ ⎞ n ⎜ d τ υ ⋅ χ ( ) ∫ ⎜ ∫ m m ⎟⎟ dx = Δx ⎣⎡ χ гр ⎦⎤ ⎝ ⎠
χ гр ⋅
. .
S гр :
m)
(2.133)
⋅ υ m ) d χ = Δx ⋅ ( A ⋅ χ гр ) ,
⎣⎡ Sгр ⎦⎤ n m
n m
(2.132)
gw m
⋅ ρ m ⋅ wm ⋅ F ⋅ f m ) , x m
c
(2.137)
104 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(
Ше
е
а ае
е.
∫∫∫ ρ ⋅ ( a [Vm ]
m
⎛
∫ ⎜⎜ ∫∫ ρ ⋅ ( a
⋅ υ m ) dV =
= Δx ⋅ ( β m
aw m
е
а ае
⎝ [ fm ]
⋅ ρ m ⋅ wm ⋅ amx ⋅ f m ) , Δx
е.
∫∫ W
[ Sm ]
Д
n m
∫∫ W
[ fm ]
x m
[ f m′ ]
⎞ ⋅ υ m ) df ⎟ dx = Δx ⋅ ∫∫ ρ m ⋅ ( am ⋅ υ m ) df = ⎟ (2.139) [ f m ]c ⎠
dS =
∫∫ W
x m
[ f m′ ]
df − ∫∫ Wmx df + [ fm ]
df = − ∫∫ km ⋅ [ fm ]
W ∫∫ ω
n m
[ m]
(2.140)
d ω.
(2.141)
∂Tm ∂T df = − km ⋅ f m ⋅ m , ∂x ∂x
(2.141) : (2.142)
m-
fm
; Tm –
x m
m
1 ⋅ ρ m ⋅ ( am ⋅ υ m ) df . ρ m ⋅ wm ⋅ amx ⋅ f m [∫∫ fm ]
km –
∫∫ W
m
(2.138)
c
β ma w = Се ь
)
1 ⋅ ρ m ⋅ Fm ⋅ υ m df . ρ m ⋅ wm ⋅ Fmx ⋅ f m [∫∫ fm ]
β mg w =
.
fm
⎛ ⎞ ∂T ∂T ∂T ⎞ ∂ ⎛ df − ∫∫ Wmx df = − ⎜ ∫∫ km ⋅ m df − ∫∫ km ⋅ m df ⎟ = −Δx ⋅ ⎜ km ⋅ f m ⋅ m ⎟ . (2.143) ⎜ ⎟ x x x ∂ ∂ ∂ ∂x ⎠c ⎝ [ fm ] [ fm ] ⎝ [ fm′ ] ⎠
Д
(2.141) :
∫∫ω W
[ 2]
n 2
dω =
∫∫ W
n 2
⎡⎣ S гр ⎤⎦
dS =
⎞ W2n d χ ⎟ dx = Δx ⋅ ∫ W2n d χ = Δx ⋅ ( Φ 2, гр ) , ⎟ Δx ⎡ χ гр ⎤ ⎡⎣ χ гр ⎤⎦ ⎝⎣ ⎦ ⎠ c ⎛
∫ ⎜⎜ ∫
Φ 2, гр = Φ 2, гр
(2.145)
)
n ∫∫ W1 d ω =
[ω1 ]
© В.Е. Селе
⎡⎣ χ гр ⎤⎦
W2n d χ .
χ гр ( Φ 2, гр > 0 –
( Д
∫
(2.144)
= Δx ⋅
∫
∫∫
⎡⎣ Sгр ⎤⎦
⎡⎣ χ гр ⎤⎦ c
).
:
W1n dS + ∫∫ W1n d ω = [ω ]
W d χ + Δx ⋅
ев, В.В. Алеш
n 1
∫W
[ χ ]c
n 1
⎛ ⎞ n ⎜ ⎟ dx + W d χ 1 ∫⎜ ∫ ⎟ Δx ⎡⎣ χ гр ⎤⎦ ⎝ ⎠
⎛ ⎞ n W d dx = χ ⎜ ⎟ 1 ∫ ⎜∫ ⎟ Δx ⎝ χ ⎠
d χ = Δx ⋅ ( Φ1, гр ) + Δx ⋅ ( Φ oc ) .
, С.Н. Прял в, 2007–2009
(2.146)
лава 2 105 _______________________________________________________________________________________
∫
Φ1, гр =
⎡⎣ χ гр ⎤⎦
W1n d χ ;
(2.147 )
Φ oc = ∫ W1n d χ ;
(2.147 )
χ
Φ1, гр = −Φ 2, гр .
Φ1, гр
χ гр ( Φ1, гр > 0 –
(
[ Sm ]
,
n m
а ае
). ) ( Φ oc > 0 –
(
∫∫ W
е
Φ oc
)
χ
В ь
(2.147 )
dS = −Δx ⋅
е.
∫∫∫ Q [Vm ]
:
dV =
⎛
∫ ⎜⎜ ∫∫ Q
Δx
⎝ [ fm ]
Qm = Δx
).
∂T ⎞ ∂ ⎛ km ⋅ f m ⋅ m ⎟ + Δx ⋅ ( Φ m , гр ) + δ 1m ⋅ Δx ⋅ ( Φ oc ) . ∂x ⎜⎝ ∂x ⎠ c
m
-
(2.125) Δx
m
(2.148)
⎞ df ⎟ dx = Δx ⋅ ( Qm ⋅ f m )c , ⎟ ⎠
(2.149)
1 ⋅ Qm df . f m [∫∫ fm ]
(2.150) ,
.
:
⎞ ∂ ⎛ ⎞ wm2 w3 ∂⎛ V2 ⋅ f m ⎟ + ⎜ β mVE ⋅ ρ m ⋅ ε m ⋅ wm ⋅ f m + β mV 3 ⋅ ρ m ⋅ m ⋅ f m ⎟ = ⎜ ρm ⋅ ε m ⋅ f m + β m ⋅ ρm ⋅ 2 2 ∂t ⎝ ⎠ ∂x ⎝ ⎠ ∂ = − ( β mPV ⋅ pm ⋅ wm ⋅ f m ) − β mPV ω ⋅ pm ⋅ wmn , гр ⋅ χ гр + AmТр ⋅ χ гр + β mg w ⋅ ρ m ⋅ wm ⋅ Fmx ⋅ f m − (2.151) ∂x ∂T ⎞ ∂ ⎛ − β ma w ⋅ ρ m ⋅ wm ⋅ amx ⋅ f m + ⎜ km ⋅ f m ⋅ m ⎟ − Φ m , гр − δ 1m ⋅ Φ oc + Qm ⋅ f m . ∂x ⎝ ∂x ⎠
[79],
wΣ ≈ 1,1 ⋅ w1 .
(2.152 )
χ гр
, wΣ
:
wгрx = 1,1 ⋅ w1 .
, , © В.Е. Селе
:
ев, В.В. Алеш
.В
, С.Н. Прял в, 2007–2009
,
-
x(2.152 )
δ
106 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(2.123 ). Д
χ гр
-
wгрx , . . w гр = {wгрr , 0, wгрx } .
wгрr
:
T
Н
wгрr .
-
r.
(
dt (
)
dr = w ⋅ dt.
. 2.9):
r гр
и . 2.9. П
ереч ы ра ре
л
(2.153)
ев
ече
я в ру е
И f2
f1 )
(
df 2 = − df1 = χ гр ⋅ dr = 2 ⋅ π ⋅ ( R − δ ) ⋅ w ⋅ dt
.
( . 2.9):
r гр
,
(2.123),
wгрr =
1
2 ⋅ π ⋅ f2
⋅
(2.154 )
∂f 2 . ∂t
(2.154 ) ,
wгрn
(
и . 2.10. Пр © В.Е. Селе
ев, В.В. Алеш
л
ы ра ре
. 2.10).
л
ев
, С.Н. Прял в, 2007–2009
ече
-
я в ру е ( ра
е
)
лава 2 107 _______________________________________________________________________________________
δ ( x) –
Д . xOr (
.
.Н ,
. 2.10)
:
h ( x ) = ⎡⎣ R − δ ( x0 ) ⎤⎦ − δ x′ ( x0 ) ⋅ ( x − x0 ) ,
(2.155)
r ( x ) = ( R − δ ( x )) ,
x0 –
δ x′ ≡ ∂δ ∂x .
(2.155), Θ χ = {1, − δ x′} –
И
T
;
,
.
Ox,
–
-
Or.
, (
.
. 2.10): n1, χ
⎧ δ x′ 1 ⎪ , − = ⎨− 2 2 ⎪⎩ 1 + (δ x′ ) 1 + (δ x′ ) ⎧ δ x′ ⎪ , =⎨ 2 ⎪⎩ 1 + (δ x′ )
n 2, χ
В
xOr ( n m , гр
w
(
1 + (δ x′ )
wmn , гр = ( w гр ⋅ n m , χ ) = (−1) m ⋅
δ x′ =
(2.156 )
⎫ ⎪ ⎬ . ⎭⎪
2
(2.156 )
. 2.10) w гр = {wгрx , wгрr } .
.
,
T
T
1
. (2.131 ))
δ x′
⎫ ⎪ ⎬ ; ⎭⎪
wгрx ⋅ δ x′ + wгрr 1 + (δ x′ )
2
(2.156),
: (2.157)
.
(2.123 ):
∂ ⎛ ⎜R− ∂x ⎝⎜
∂f ∂ψ 2 f2 ⎞ f 1 1 ⋅ 2 =− ⋅ ⋅ . ⎟=− 2 π ⋅ψ 2 ∂x π ⎠⎟ 2 ⋅ π ⋅ f 2 ∂x
(2.158)
AmТр ,
(
τ
,
. (2.135)).
(2.135)
. В
τ nm
2.10)
n m
(
.
.
,
τ n ≈ τ nx .
:
(2.159)
. .
, ( .
V1
S гр . (2.135)):
© В.Е. Селе
, AmТр =
χ гр 1
ев, В.В. Алеш
⋅
∫
⎣⎡ χ гр ⎦⎤
τ mn ⋅ υ m d χ ≈
χ гр 1
⋅
∫
⎣⎡ χ гр ⎦⎤
, С.Н. Прял в, 2007–2009
τ mnx ⋅υ грx d χ ≈ Tpm , гр ⋅ wгрx ,
(2.160)
108 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
τ mnx –
τ nm
, (2.151)
Ox . (2.118), (2.120)
(2.123),
:
⎞ ∂ ⎛ VE ⎞ wm2 wm3 ∂⎛ V2 V3 f f w f ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ fm ⎟ = ρ ε β ρ β ρ ε β ρ ⎜ m m m ⎜ m m m m⎟ m m m m m m ∂t ⎝ 2 2 ⎠ ∂x ⎝ ⎠ x r ′ ⋅ + w w δ ∂ m гр = − ( β mPV ⋅ p ⋅ wm ⋅ f m ) − β mPV ω ⋅ 2 ⋅ π ⋅ ( R − δ ) ⋅ ( −1) ⋅ p ⋅ гр x + 2 ∂x 1 + (δ x′ ) + 2 ⋅ π ⋅ ( R − δ ) ⋅ Tpm ,гр ⋅ wгрx − β mg w ⋅ ρ m ⋅ wm ⋅ f m ⋅ g ⋅
− β ma w ⋅ ρ m ⋅ wm ⋅ amx ⋅ f m +
∂Tm ⎞ ∂ ⎛ ⎜ km ⋅ f m ⋅ ⎟− ∂x ⎝ ∂x ⎠
∂z1 − ∂x
(2.161 )
− Φ m , гр − δ 1m ⋅ Φ oc + Qm ⋅ f m
⎞ ∂ ⎛ ⎞ wm2 w3 ∂⎛ V2 ⋅ψ m ⎟ + ⎜ β mVE ⋅ ρ m ⋅ ε m ⋅ wm ⋅ψ m + β mV 3 ⋅ ρ m ⋅ m ⋅ψ m ⎟ = ⎜ ρ m ⋅ ε m ⋅ψ m + β m ⋅ ρ m ⋅ 2 2 ∂t ⎝ ⎠ ∂x ⎝ ⎠ =−
wгрx ⋅ δ x′ + wгрr ∂ PV PV ω 2 ⋅ π ⋅ ( R − δ ) ⋅ ( −1) p w p ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ + β ψ β ( m m m) m 2 f ∂x 1 + (δ ′ ) m
2 ⋅π ⋅ ( R − δ ) ∂z + ⋅ Tpm , гр ⋅ wгрx − β mg w ⋅ ρ m ⋅ wm ⋅ψ m ⋅ g ⋅ 1 − f ∂x − β ma w ⋅ ρ m ⋅ wm ⋅ amx ⋅ψ m + −
Φ m , гр f
−
m = 1, 2 .
δ 1m ⋅ Φ oc f
x
(2.161 )
∂Tm ⎞ ∂ ⎛ ⎜ km ⋅ψ m ⋅ ⎟− ∂x ⎝ ∂x ⎠
+ Qm ⋅ψ m ,
(2.161 )
(2.161 ), , ,
,
⎛ ⎛ wm2 ⎞ ⎤ ∂ ⎡ wm2 ⎞ ⎤ ∂ ⎡ + ⋅ ⋅ + w f ρ ε ⎢ ρm ⋅ fm ⋅ ⎜ ε m + ⎥ ⎢ ⎟ ⎟⎥ = m m m ⎜ m 2 ⎠ ⎥⎦ ∂x ⎢⎣ 2 ⎠ ⎥⎦ ∂t ⎢⎣ ⎝ ⎝ wгрx ⋅ δ x′ + wгрr ∂ m = − ( p ⋅ wm ⋅ f m ) − 2 ⋅ π ⋅ ( R − δ ) ⋅ ( −1) ⋅ p ⋅ + 2 ∂x 1 + (δ x′ ) + 2 ⋅ π ⋅ ( R − δ ) ⋅ Tpm , гр ⋅ wгрx − ρ m ⋅ wm ⋅ f m ⋅ g ⋅
+
∂T ∂ ⎛ km ⋅ f m ⋅ m ∂x ⎜⎝ ∂x
© В.Е. Селе
∂z1 + ∂x
⎞ 1m ⎟ − Φ m , гр − δ ⋅ Φ oc + Qm ⋅ f m ⎠
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, fm :
(2.161 )
лава 2 109 _______________________________________________________________________________________
⎛ ⎛ wm2 ⎞ ⎤ ∂ ⎡ wm2 ⎞ ⎤ ∂ ⎡ + ⋅ ⋅ + ρ ψ ε w ⎢ ρ m ⋅ψ m ⋅ ⎜ ε m + ⎥ ⎢ ⎟ ⎟⎥ = m m m⎜ m 2 ⎠ ⎦⎥ ∂x ⎣⎢ 2 ⎠ ⎦⎥ ∂t ⎣⎢ ⎝ ⎝
wгрx ⋅ δ x′ + wгрr 2 ⋅ π ⋅ ( R − δ ) ⋅ ( −1) ∂ = − ( p ⋅ wm ⋅ψ m ) − ⋅ p⋅ + 2 ∂x f 1 + (δ ′ ) m
2 ⋅π ⋅ ( R − δ )
∂z + ⋅ Tpm , гр ⋅ wгрx − ρ m ⋅ wm ⋅ψ m ⋅ g ⋅ 1 + f ∂x 1m ∂T ⎞ Φ m , гр δ ⋅ Φ oc ∂ ⎛ + ⎜ km ⋅ψ m ⋅ m ⎟ − − + Qm ⋅ψ m , f f ∂x ⎝ ∂x ⎠
Φ m , гр
m = 1, 2 . К
Д
Φ oc
, ,
.
(2.161 )
x
. ,
, . [79], ,
«
δ
»,
, .
, ,
-
ν υ L ( r ) = υ L ⋅ (1 − r R ) 1 ,
:
υL –
(2.162) ,
L ; ν1 –
.
Tp тру а ,
-
Tpm , гр
,
-
,
( )
Н
. Tp тру а = Cтру а ⋅
ρ1 ⋅ w
2 1
2
[79]: (2.163)
,
[79]. Д
Cтру а –
-
, ,
ν 1 = 1 7 [79]: [85],
Tpтру а = Cтру а 4 ⋅ C тру а ( L ) Re L = © В.Е. Селе
ев, В.В. Алеш
( L) ⋅
-
ρ1 ⋅ wL2
2 0,316 = 0,25 ; Re L
ρ1 ⋅ wL ⋅ D ; μ1
, С.Н. Прял в, 2007–2009
;
(2.164 ) (2.164 )
(2.164 )
110 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Re L , wL , Cтру а
(L)
–
,
-
; D – wL =
14 ⎛ δ ⎞ ⋅⎜ ⎟ 15 ⎝ R ⎠
Re L = 2 ⋅ Cтру а =
И
(2.165),
(2.164 ) [79]:
. −1 7
wL2 ⋅ Cтру а w12
(2.165 )
(L)
(2.165 ) (2.165 )
.
«
»
[79], Cтру а ,
⋅ w1 ;
R wL ⋅ ⋅ Re1 ; δ w1
Cтру а = 0, 0589 Re10,25 ,
В
[79]:
-
Re1 > 300 ÷ 400.
(2.166) Δp
(2.166) ΔL ,
δ
-
. А
Cтру а
Δp p 26 П ,
.А (2.178).
Д
Fr м > FrA . Э
,
ϕk = μ
1 − 0, 78 ⋅ β 2 − 0, 22 ⋅ ⎣⎡1 − exp ( −15 ⋅ ρ ) ⎦⎤ ⋅ β 2
[82]:
(
1 − β 2 + 0, 03 ⋅ exp −1,35 ⋅10 ⋅ (1 − β 2 ) 3
,
(2.178 ) ;
μ1 , (2.176 )
(2.178 )
«
3
)
-
⎛ 1 − β2 ⎞ − t ⋅ exp ⎜ − ⋅ κ 03 ⎟ . 5,5 ⎝ ⎠
–
(2.179)
»,
-
.
«
– f (μ
[82]:
»
) = 1 + 0, 03 ⋅ μ
(2.180)
,
. В
(
ϕk
[82])
We =
В
.
σ′ g ⋅ ( ρ1 − ρ 2 ) ⋅ D 2
κ0 ,
[82] (2.179)
.
(2.180)
ϕ k = (1 + 0, 03 ⋅ μ
)×
[82]:
⎡1 − 0, 78 ⋅ β − 0, 22 ⋅ ⎡1 − exp −15 ⋅ ρ ⎤ ⋅ β ⎤ (2.181) ( )⎦ 2 ⎛ 1 − β2 2 ⎣ 3 ⎞⎥ ⎢ × − t ⋅ exp ⎜ − ⋅ κ0 ⎟ . ⎢ 1 − β + 0, 03 ⋅ exp −1,35 ⋅103 ⋅ (1 − β )3 ⎝ 5,5 ⎠⎥ 2 2 ⎢⎣ ⎥⎦
В
)
[82]:
wм⋅ ρ
⎛ ρ − ρ2 ⎞ ⋅⎜ 1 s* = ⎟ 3 3,3 ⋅ 0, 25 + 0, 75 ⋅ sin α ⎝ g ⋅ σ ⎠
(
α – © В.Е. Селе
(
)
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
0,25
,
β2 = 1 , . .
(2.182) -
лава 2 115 _______________________________________________________________________________________ 1
,
,
[82]:
ρ1 0,25 ρ2 ⎛ g ⋅σ ⎞ ⋅ w= ⎜ ⎟ . 1 + 2, 75 ⋅10−3 ⋅ ( μ − 1) ⎝ ρ1 − ρ 2 ⎠ 0,86 ⋅
s* , ,
s∗k =
[82]: 3,3 ⋅ ⎡⎣1 + 2, 75 ⋅10−3 ⋅ ( μ − 1) ⎤⎦ 0,86
ψ 2 = (ψ 2 ) р
[82],
(ψ 2 ) р
(2.183)
к
(2.184)
.
:
+ Δψ 2 ,
(2.185)
; Δψ 2 –
–
к
.
,
[82]
ψ 2 = β 2 ⋅ ( 0,5 + 0,31 ⋅ exp ⎣⎡0, 067 ⋅ (1 − μ
(
⎡ + ⎢ 0,5 − 0,31 ⋅ exp ⎡⎣ 0, 067 ⋅ (1 − μ ⎢ ⎣
sk∗ ≤ s* ≤ 1;
)⎦⎤ ) ⋅ ⎢1 − exp ⎜⎜ −4, 4 ⋅ ⎡
)⎤⎦ ) ⋅
ψ 2 = β 2 ⋅ ( 0,5 + 0,31 ⋅ exp ⎡⎣0, 067 ⋅ (1 − μ
(
+ ⎡ 0,5 − 0,31 ⋅ exp ⎡⎣0, 067 ⋅ (1 − μ ⎣ s* > 1.
⎛
⎢⎣
⎝
:
Fr м FrA
⎞⎤ ⎟⎟ ⎥ + ⎠ ⎥⎦
(
)
⎤ s * − s∗k − 2 ⋅ (1 − β 2 ) ⎥ ⋅ exp −7,5 ⋅ 1 − β 2 , (2.186 ) ⎥ 1 − s∗k ⎦
)⎤⎦ ) +
)⎤⎦ ) − 2 ⋅ (1 − β 2 )⎤⎦ ⋅ exp ( −7,5 ⋅
s* ≤ s∗ k
(2.186 )
[82] (
Δψ 2 = 0 ,
),
)
1 − β2 ,
.
К
β2
-
[82]. .В
, , wкр . В
,
, .
1
–
© В.Е. Селе
, ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
[85].
-
116 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
:
• ;
• ,
.
Э В
. [82]
,
α:
0,86 ⋅ ρ1 ρ 2 ⎛ g ⋅σ ⎞ ⋅⎜ ⎟ 1 + 2, 75 ⋅10−3 ⋅ ( μ − 1) ⎝ ρ1 − ρ 2 ⎠ = , ⎡ (1 + 0,5 ⋅ sin α ) ⋅102 ⎤ ⋅ (1 − β 2 ) ⎥ exp ⎡⎣ −9 ⋅ (1 − β 2 ) ⎤⎦ − Δw0 ⋅ exp ⎢ − 1 − sin α ⎣ ⎦
-
0,25
wk ,α
0,86 ⋅ ρ1 ρ 2 ⎛ g ⋅σ ⎞ ⋅⎜ ⎟ −3 1 + 2, 75 ⋅10 ⋅ ( μ − 1) ⎝ ρ1 − ρ 2 ⎠ Δw0 = 1 − . 0,25 ρ1 ⎛ g ⋅ σ ⎞ μ1 ρ1 σ g ⋅σ ⋅⎜ 2,3 ⋅ ⎟ + 0,1 ⋅ + 0,8 ⋅ ⋅ ⋅ ρ 2 ⎝ ρ1 − ρ 2 ⎠ μ1 σ ρ 2 ρ1 − ρ 2
(2.187 )
0,25
В .
α = 900 [82], (2.187)
(2.187) ,
.В (2.187).
(T
И
тру а
Φ oc = π ⋅ D ⋅ qтру а ,
Tгр –
),
[79]:
qтру а = k1 ⋅ Nu тру а ⋅
q(r ) = k1 ⋅ ( ∂T ∂r ) ,
В
Φ oc
− Tгр ) ( Tтру а
Н
k1 –
(2.187 )
Tтру а − Tгр
δ
; Nu тру а –
(2.188)
,
Н
.
Ω = Φ1( гр ) Φ oc ( Φ1( гр ) –
),
-
,
.В
,
-
[79]: Nu тру а = © В.Е. Селе
ев, В.В. Алеш
3 . 2+Ω
, С.Н. Прял в, 2007–2009
(2.189)
лава 2 117 _______________________________________________________________________________________
, Ω,
,
.
Nu тру а
,
,
, . ,
(R − δ < r < R)
,
«
», Tгр [79].
Tтру а
(
,
δ
,
)
:
TL (r ) − Tтру а = TL − Tтру а ⋅ (1 − r R )
θ1
-
(2.190)
,
TL –
–
( L ); θ1 .
Д
qтру а = k1 ⋅ Nu тру а ( L ) ⋅ (Tтру а − TL ) D ;
[79]: (2.191 )
0,4 Nu тру а ( L ) = 0, 023 ⋅ Re0,8 L ⋅ Pr1 ,
TL
Pr1 = ν
(υ ) 1
–
ν
; μ1 Д
(T ) 1
= μ1 ⋅ cP ,1 k1 –
;ν
; ν (T ) 1
(υ ) 1
; –
–
-
cP ,1 –
ν 1 = θ1 = 1 7
.
5 ⎛δ ⎞ TL − Tтру а = ⋅ ⎜ ⎟ 6 ⎝R⎠
, (2.165) −1 7
Nu тру а = Nu тру а ( L ) ⋅
В
(2.191 )
(
. [79]):
⋅ (Tгр − Tтру а ) ;
(2.192 )
2 ⋅ R Tтру а − TL . ⋅ δ Tтру а − Tгр
Nu тру а = 0, 016 ⋅ Re10,8 ⋅ Pr10,4 ⋅ (δ R )
[79]:
−0,057
(2.192 )
(2.193)
.
, [79]. Д (2.191)
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
-
,
118 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(Д.А.
). В
, 1957
[79]
-
Nu тру а = 0, 010 ⋅ Re10,83 ⋅ Pr10,5 ,
:
( Re1 > 300 ) .
(2.194)
Н
[79]:
⎛ 3 ⎞ 0, 83 0,5 2 Nu тру а = ⎜ ⎟ + ( 0, 010 ⋅ Re1 ⋅ Pr1 ) , 2 + Ω ⎝ ⎠ 2
(2.195)
±19 % .
. Э
δш
(δ ш > δ ) .
δ .
[87]
,
-
:
В
, .
δш D (
. Re
, D –
), .
Re
(δ ш >> δ ) ,
-
. ,
,
. ,
-
,
.
, .
-
.В
, ó
-
,
(
). Re . Ч
δш D ,
-
Re р ,
.
.
В
( ).
δш D
,
, .Э
ев, В.В. Алеш
-
.
.
© В.Е. Селе
,
,
,
, С.Н. Прял в, 2007–2009
-
лава 2 119 _______________________________________________________________________________________
δш D
,
,
. . ,
. . [87], ( sш δ ш ) ).
–
[87]:
-
= 12 ÷ 14 (
sш δ ш ≥ 8 т
В.И. ,Н
, –
,
0,47 Nu Тру а ( L ) = 0, 022 ⋅ Re0,8 ⋅ ( Pr1 Pr те ка ) L ⋅ Pr1
Pr те ка –
:
s –
⎡
ε ш = exp ⎢0,85 ⋅ ⎣⎢
⎡
ε ш = exp ⎢0,85 ⋅ ⎢⎣
( sш δ ш )
sш δ ш
sш δ ш
( sш δ ш )
0,25
; εш –
т
⎤ ⎥ ⎦⎥
δш sш
⎤ ⎥ т ⎥ ⎦
⋅εш ,
(2.196)
,
-
;
(2.197 )
.
(2.197 )
⎛s ⎞ ≥⎜ ш ⎟ ⎝ δш ⎠
⎛s ⎞ 0;
n ⋅ ( n ) i < 0;
(0)
n⋅
(n)
∑
Θ < 1,
N
n =1
fL
(n)
(2.225 )
Θ = 1.
(2.225 )
. (2.225 – )
(2.225) ,
(2.225 – )
-
. , . . Д
V , n = 1, N ,
(n)
(2.225 ).
(n)
ΔX → 0
(2.2) (2.2) ,
(0)
V.
V
(0)
. (0)
,
V
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
130 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
.Д
,
,
(0)
-
∫∫
:
.
⎡⎣ S ⎤⎦
V,
ρ ⋅ υ ⋅υn dS = −
(0)
В
(
∫∫
⎡⎣ S ⎤⎦
p ⋅ ndS = −
(0)
.,
∫∫∫
(2.2)
⎡⎣ V ⎤⎦
∇pdV .
, [91])
.
.Д
V (2.226).
∫∫
⎡⎣ S ⎤⎦
, p ⋅ ndS =
(0)
,
ρ ⋅ υ ⋅υn dS = ∑ N
⎡⎣ (0) S ⎤⎦
= −∑ N
(n)
n =1
(ρ
n =1
∫∫
ρ ⋅ υ ⋅υn df = ∑
⎡⎣ ( n ) f L ⎤⎦
)
∑
(n)
n =1
−
(
w⋅
⎡⎣ V ⎤⎦
)
∇pdV = 0.
(2.227)
∫∫
⎡⎣ ( n ) f L ⎤⎦
ρ ⋅υ x ⋅ ( n ) i ⋅υ x ⋅ ( ( n ) i ⋅ (0) n ) df = (2.228)
:
N
(n)
n =1
⋅ s ⋅ f L ⋅ wL2 ⋅ i .
L
∫∫∫
-
-
(0)
N
К
(n)
,
(0)
(2.226)
∫∫
(2.226)
(0)
ρL ⋅
(n)
wL ⋅
( n)
f L ⋅ ( n ) s = 0.
(2.229)
(2.229)
i ,
∫∫
(2.228),
⎡⎣ S ⎤⎦
-
,
.
ρ ⋅ υ ⋅υ n dS .
,
,
(0)
(0)
(2.226)
.
V
, .Д
(0)
V,
,
-
, (
. 2.13). Δξ . И
(0)
V Δξ . Д
«
» Д
© В.Е. Селе
-
,
«
,
. (0)
V,
Δξ . ев, В.В. Алеш
,
(ρ ⋅w ) 2
, С.Н. Прял в, 2007–2009
»
лава 2 131 _______________________________________________________________________________________
x рав (
xлев
Δξ ,
. 2.13). В
. Δξ
p( x)
2.13).
(
.
-
,
. p( x) . В
.
,
.
и . 2.13. Схе а
ч
е ре
авле
е
p ( x)
у ру
авле
Δξ → 0 (
. 2.13
[ x1 ,
я а уча
⎡⎣ x рав , x2 ⎤⎦ , .Э
чле е
.
я вух
,
).
x лев ]
е
-
. 2.13,
(0)
ó
V
ó
,
,
,
-
.
В
,
-
(0)
V,
«
( 0,5 ⋅ ρ ⋅υ ) )
» «
, ( »
.
2
[91], ,
, . . -
,
, : (k )
Д
,
pL =
3,5 ÷ 7,8МПа ,
(n)
pL
n, k ∈ 1, N .
(2.230) ,
,
20 м / ,
50 кг / м . Д 3
10−2 МПа . И © В.Е. Селе
ев, В.В. Алеш
, , С.Н. Прял в, 2007–2009
(
-
132 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
) . ,
(2.230)
, .
И
,
(2.54)
(2.225), :
я а
∂(ρ ⋅ f )
–
∂t
а ще
,
∂ ( ρ ⋅ w ⋅ f ) = 0; ∂x
+
ч е е
(2.231 )
∂Y ∂ ∂ ∂ ⎛ ( ρ ⋅ Ym ⋅ f ) + ( ρ ⋅ Ym ⋅ w ⋅ f ) − ⎜ ρ ⋅ f ⋅ Dm ⋅ m ∂t ∂x ∂x ⎝ ∂x
YN = 1 −
∑Y
N S −1 m =1
S
∂(ρ ⋅w⋅ f ) ∂t
∂ ⎡ ⎢ρ ⋅ f ∂t ⎢⎣
m
+
я
⎞ ⎟ = 0, m = 1, N S − 1 , ⎠
;
∂ ( ρ ⋅ w2 ⋅ f )
⎛ w2 ⋅ ⎜⎜ ε + 2 ⎝
∂x
∂z ⎞ π ⎛ ∂p = − f ⋅ ⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x
⎛ ⎞⎤ ∂ ⎡ w2 ⎟ ⎥ + ⎢ ρ ⋅ w ⋅ f ⋅ ⎜⎜ ε + 2 ⎠ ⎥⎦ ∂x ⎢⎣ ⎝
я а
ч е е
∂ρ N ( n ) +∑ Θ ⋅ ∂t n =1 ∂ ( ρ ⋅ Ym ) ∂t
+∑ N
(n)
( n)
n =1
m = 1, N S − 1; (n)
⎛ ∂ ( ρ ⋅ w) ⎞ ⎜ ⎟+ ⎝ ∂t ⎠ =−
(n)
© В.Е. Селе
YN = 1 − S
(n)
∑
m =1
(n)
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ ρ ⋅ Dm ⋅ ⎥⋅ Θ = 0, x ∂ ∂x ⎟⎠ ⎦ ⎣ ⎝
(n)
ев, В.В. Алеш
(2.231 )
(n)
Ym ;
⎛ ∂ ( ρ ⋅ w2 ) ⎞ ⎜ ⎟= ⎜ ⎟ ∂x ⎝ ⎠
⎛ ∂p ⎞ ⎜ ⎟− g⋅ρ ⋅ ⎝ ∂x ⎠
(2.231 )
(2.231 )
N ⎛ ∂ ( ρ ⋅ Ym ⋅ w ) ⎞ ( n ) ⎜ ⎟⋅ Θ − ∑ ∂x n =1 ⎝ ⎠
(n)
⎞ ; ⎟⎟ ⎠
я
⎛ ∂ ( ρ ⋅ w) ⎞ ⎜ ⎟ = 0; ⎝ ∂x ⎠
N S −1
(2.231 )
⎞⎤ ∂z1 ∂ − ⎟ ⎥ = − ( p ⋅ w⋅ f ) − ρ ⋅ w⋅ f ⋅ g ⋅ ∂x ∂x ⎠ ⎥⎦
NS ∂Ym ∂f ∂ ⎛ ∂T ⎞ ∂ ⎛ − p⋅ + Q⋅ f + ⎜ k ⋅ f ⋅ ⎟ − Φ (T , Toc ) + ⎜⎜ ρ ⋅ f ⋅ ∑ ε m ⋅ Dm ⋅ ∂t ∂x ⎝ ∂x ⎠ ∂x ⎝ ∂x m =1
–
(2.231 )
(n) 1 ⎛ ∂z1 ⎞ ⎜ ⎟ − ( n ) ⋅ ( λ ⋅ ρ ⋅ w ⋅ w ) , n = 1, N ; ⎝ ∂x ⎠ 4 ⋅ R
, С.Н. Прял в, 2007–2009
(2.231 )
лава 2 133 _______________________________________________________________________________________
∂ ( ρ ⋅ε )
+∑
∂t
1 + ⋅∑ 4 n =1
n =1
N
−∑
(n)
N
n =1
N ⎛ ∂ ( ρ ⋅ ε ⋅ w) ⎞ ( n) ⎜ ⎟ ⋅ Θ = −∑ ∂x n =1 ⎝ ⎠
(n)
N
1 (n) (n) ( n) 3 ⋅ λ⋅ ρ⋅ w ⋅ (n) R
N S Φ (T , Toc ) ( n ) ⋅ Θ + ∑∑ (n) f n =1 m =1
(n)
(n)
N
(n)
p⋅
Θ+Q+
(n)
∑ N
⎛ ∂w ⎞ ( n ) ⎜ ⎟⋅ Θ+ ⎝ ∂x ⎠
(n)
n =1
⎡ ∂ ⎛ ∂ T ⎞⎤ (n) ⎢ ∂x ⎜ k ⋅ ∂x ⎟ ⎥ ⋅ Θ − ⎠⎦ ⎣ ⎝
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ ρ ⋅ ε m ⋅ Dm ⋅ ⎥⋅ Θ ; x ∂ ∂x ⎟⎠ ⎦ ⎣ ⎝
( ε m ) = ( ε m ) , ρ = ( n ) ρ = (ξ ) ρ , (n) (ξ ) (n) (ξ ) (n) (ξ ) ( Dm ) = ( Dm ) , Ym = (Ym ) = (Ym ) , ( z1 ) = ( z1 ) T=
ε=
(ξ )
(n)
T,
ε=
(n)
ε,
(ξ )
(ξ )
(n)
m ∈ 1, N S ;
∑ N
(n)
n =1
(n)
(n)
s=−
(
Θ=
V = V
(n)
n⋅
(n)
∑ N
(n)
n =1
⎛ ∂T ⎜ ⎝ ∂x
( (
)
⎧ ⎪ 1, е ли i =⎨ ⎪⎩ − 1, е ли
∑
(n)
,
0
0;
∑ N
(n)
n =1
⎛ ∂Ym ⎜ ∂x ⎝
⎞ (n) (n) ⎟ ⋅ f ⋅ s = 0; ⎠
(2.231 )
n ⋅ ( n ) i < 0;
(0)
n ⋅ (n)
∑
Θ < 1,
N
(n)
n =1
fL
е
(2.231 )
Θ = 1;
(2.231 )
я:
p = p ({S ме и } ) ; ε = ε ({S ме и } ) ; k = k ({S ме и } ) ; ε m = ε m ({S ме и } ) , Dm = Dm ({S ме и } ) ,
m = 1, N S ; T1 = T2 = … = TN S = T .
(2.231 )
. (2.231).
А
-
, , (
В
. ,
: (2.231)
( © В.Е. Селе
). ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
5). [71]
,
. -
134 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
2.4.2.
о ели овании ечени в а ве вленно о и ующе и о и
ан
у о
ово е,
И [11].
,
ρ = const.
1
:
-
( 0,5 ⋅ ρ ⋅ w )
(2.232)
2
,
p.
В
(2.76), ,
,
. ( n)
е е е
ае
, . .
pL =
(ξ )
я
,
я. В
n, ξ = 1, N .
pL
.
е
К
V
∑ N
n =1
(n)
⎛ ∂Ym ⎜ ∂x ⎝
ч е е
я
е ь
е
.
(2.50 )
( 0)
:
(2.233)
(2.225 )
⎞ (n) (n) ⎟ ⋅ f ⋅ s = 0, m = 1, N S . ⎠
(2.234)
А
V
(2.225 ). :
∂Ym N +∑ ∂t n =1
(n)
⎛ ∂ (Ym ⋅ w ) ⎞ ( n ) ⎜ ⎟⋅ Θ − ∂x ⎝ ⎠
m = 1, N S − 1;
YN = 1 −
(n)
S
-
∑
( n)
N
∑
n =1
N S −1
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ Dm ⋅ ⎥ ⋅ Θ = 0, x ∂ ∂x ⎟⎠ ⎦ ⎣ ⎝
(2.235)
(n)
Ym .
m =1
В
.И ,
Ym , m = 1, N S , . Э
,
-
, .
( 0)
«
»
-
V
.
, « Ym , m = 1, N S ,
, . . «
» .
( 0)
V
1
В
© В.Е. Селе
(2.231). ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
»
, «
-
лава 2 135 _______________________________________________________________________________________
» .
»1
«
«
∂Ym ⎞ ⎛ ⎜ ρ ⋅ Dm ⋅ ∂x ⎟ ⎝ ⎠
( ρ ⋅ Ym ⋅ w )
. . .
»
,
,
. (2.46 )
( 0)
V,
:
∑ρ⋅ N
⎣⎡(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎦⎤ = ρ ⋅ ∑ N
(n)
n =1
(n)
n =1
В
«
»
⎣⎡(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎦⎤ = 0, m = 1, N S .
«
IN
∑
n∈
Д
(n)
IN
« , . .
(n ∈
»
∑
n∈
∑
n∈
(n)
OUT
IN
IN
(n)
),
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎤⎦ = 0, m = 1, N S . ( n)
,
( n)
Ym = −
(0)
∑
n∈
( n)
Ym ⋅
(0)
IN
n∈
(n)
(n)
(n)
1
© В.Е. Селе
«
∑
(n)
OUT
fL «
) -
[ wL ⋅ f L ⋅ s ] = 0, m = 1, N S ;
(2.237): (2.238 )
n∈
OUT
( 0)
(2.238 )
,
[ wL ⋅ f L ⋅ s ]
(2.238)
,
, m = 1, N S ;
(2.239 )
.
(2.239 )
OUT
(Ym )L = (0)Ym ,
[ wL ⋅ f L ⋅ s ] = ( wL ( n)
[ wL ⋅ f L ⋅ s ] = − ( wL (n)
n∈
OUT
s, n = 1, N , (
⋅ fL )
⋅ fL )
»
ев, В.В. Алеш
(n ∈
OUT
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎤⎦
( n)
:
( n)
V.
Н ,
.В
∑
(Ym )L = (0)Ym ,
n∈
OUT
(2.237)
, . .
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎤⎦ +
m-
fL ( n)
V.К
Ym –
. К -
:
( 0)
(0)
IN
–
(2.236)
» –
OUT
,
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⋅ s ⎤⎦ +
(2.236)
»
, .
-
, , С.Н. Прял в, 2007–2009
. (2.207)), n∈
IN
n∈
OUT
(2.240 )
; .
(2.240 )
.
136 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
,
∑ [w (n)
n∈
⋅ fL ⋅ s] +
L
∑ [w (n)
∑
n∈
∑
(2.240) Ym =
(0)
L
IN
(n)
n∈
⋅ fL ⋅ s] = −
IN
n∈
∑
(w
⋅ fL ) =
L
IN
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⎤⎦
n∈
IN
∑
n∈
( n)
(w
⋅ fL )
L
OUT
( n)
=
(2.217 ) :
[ wL ⋅ f L ⋅ s ] = 0;
(n)
∑
OUT
n∈
n∈
n∈
(n)
∑
OUT
(n)
(2.241 )
[ wL ⋅ f L ⋅ s ];
(w
(2.241 )
⋅ fL ).
L
(2.241 )
OUT
∑
(2.241 ), ( n)
К
(2.240)
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⎤⎦
(2.239):
(n)
IN
∑
n∈
(Ym )L = (0)Ym ,
(n)
(w
L
⋅ fL )
, m = 1, N S ;
IN
n∈
OUT
(2.242 )
.
К
(2.242) .
,
,
,
( 0)
V
.Д
-
. Н
,
»
(2.242 )
(Ym )L
( n)
,
(n ∈ «
«
»
)
-
IN
(«
»)
. .К /
–
,
: ;
–
-
. -
( 0)
К
V
,
∑ N
(2.225 )
:
(n)
n =1
⎛ ∂T ⎞ ( n ) ( n ) ⎜ ⎟ ⋅ f ⋅ s = 0. ⎝ ∂x ⎠
(2.243)
А
-
V
(2.225 ). :
∂ε N +∑ ∂t n =1
(n)
1 N + ⋅∑ 4 n =1 −
ρ 1
⋅∑ N
n =1
⎛ ∂ (ε ⋅ w) ⎞ (n) 1 N ⎜ ⎟ ⋅ Θ = − ⋅∑ ρ n =1 ⎝ ∂x ⎠
1 (n) (n) 3 ⋅ λ⋅ w ⋅ (n) R (n)
(n)
Θ+
ρ
Q
N S Φ (T , Toc ) ( n ) ⋅ Θ + ∑∑ (n) f n =1 m =1 N
(n)
+
(n)
, . . © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
p⋅
ρ 1
(n)
⎛ ∂w ⎞ ( n ) ⎜ ⎟⋅ Θ+ ⎝ ∂x ⎠
⋅∑ N
n =1
(n)
⎡ ∂ ⎛ ∂T ⎢ ∂x ⎜ k ⋅ ∂x ⎣ ⎝
⎞⎤ (n) ⎟⎥ ⋅ Θ − ⎠⎦
(2.244)
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ ε m ⋅ Dm ⋅ ⎥⋅ Θ . x ∂ ∂x ⎟⎠ ⎦ ⎣ ⎝
-
лава 2 137 _______________________________________________________________________________________
.
ε=
(0)
,
∑
:
n∈
∑
ε –
(n)
(w
⋅ fL )
L
εL =
OUT
(n)
∑
=
n∈
⎣⎡ε L ⋅ wL ⋅ f L ⎦⎤
(n)
∑ IN
n∈
ε, n∈
( n)
(w
L
OUT
1
: (2.246 )
а ще
,
ч е е
я
(2.246 )
⎡( z1 )1 − ( z1 )2 ⎦⎤ λ ⋅ w ⋅ w dw p1 − p2 ; = +g⋅⎣ − 4⋅ R dt l ρ ⋅l ∂Ym ∂Y ∂Y ⎞ ∂ ⎛ + w ⋅ m − ⎜ Dm ⋅ m ⎟ = 0, ∂t ∂x ∂x ⎝ ∂x ⎠
(2.246 )
m = 1, N S − 1 ; YN = 1 −
Q 1 ∂ ⎡ ∂T ⎤ Φ ( T , T ∂ε ∂ε λ ⋅ w + w⋅ = + + ⋅ ⎢k ⋅ ⎥ − ρ ρ ∂x ⎣ ∂x ⎦ f ⋅ρ 4⋅R ∂t ∂x
)+
3
pL =
(ξ )
∑ [w N
( 0)
V.
,
w = w (t ) ;
я а
(n)
n =1
L
∂Ym N +∑ ∂t n =1
ч е е ⋅ f L ⋅ s ] = 0;
∂ε N +∑ ∂t n =1
(n)
1 N + ⋅∑ 4 n =1 −
ρ 1
⋅∑ N
n =1
∑ N
n =1
YN = 1 −
( n)
S
1 (n) ( n) 3 ⋅ λ⋅ w ⋅ (n) R
(n)
;
N ∂Ym ⎞ ⎤ ∂ ⎡ S⎛ ⎢ ∑ ⎜ ε m ⋅ Dm ⋅ ⎟⎥ ; ∂x ⎣⎢ m =1 ⎝ ∂x ⎠ ⎦⎥
∑
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ Dm ⋅ ⎥ ⋅ Θ = 0, x ∂ ∂x ⎠⎟ ⎦ ⎣ ⎝
N S −1
(2.246 )
(2.246 )
(n)
Θ+ N
(2.246 )
( n)
Ym ;
m =1
(n)
ρ
Q
N S Φ (T , Toc ) ( n ) ⋅ Θ + ∑∑ (n) f n =1 m =1
+
(n)
p⋅
ρ 1
(n)
⎛ ∂w ⎞ ( n ) ⎜ ⎟⋅ Θ+ ⎝ ∂x ⎠
⋅∑ N
n =1
(n)
⎡ ∂ ⎛ ∂T ⎢ ∂x ⎜ k ⋅ ∂x ⎣ ⎝
⎞⎤ (n) ⎟⎥ ⋅ Θ − ⎠⎦
⎡∂ ⎛ ∂Ym ⎞ ⎤ ( n ) ⎢ ⎜ ε m ⋅ Dm ⋅ ⎥⋅ Θ ; x ∂ ∂x ⎟⎠ ⎦ ⎣ ⎝ .
ев, В.В. Алеш
m
(2.246 )
1
© В.Е. Селе
m =1
я
⎛ ∂ (ε ⋅ w) ⎞ (n) 1 N ⎜ ⎟ ⋅ Θ = − ⋅∑ ρ n =1 ⎝ ∂x ⎠
(n)
S
(2.246 )
⎛ ∂ (Ym ⋅ w ) ⎞ ( n ) ⎜ ⎟⋅ Θ − ∂x ⎝ ⎠
m = 1, N S − 1;
∑Y
N S −1
n, ξ ∈ 1, N ;
pL
(n)
(2.245 )
(2.245 )
,
,
я а
( n)
;
)
ρ = const ;
–
⋅ fL )
IN
(0)
(
В
–
⎡⎣ε L ⋅ wL ⋅ f L ⎦⎤
IN
n∈
(0)
(n)
(2.242),
, С.Н. Прял в, 2007–2009
(2.246 )
138 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∑ N
(n)
n =1
∑ N
(n)
n =1
(n)
(n)
⎛ ∂Ym ⎜ ∂x ⎝ ⎛ ∂T ⎜ ⎝ ∂x
⎞ (n) (n) ⎟ ⋅ f ⋅ s = 0, m = 1, N S ; ⎠
⎞ (n) (n) ⎟ ⋅ f ⋅ s = 0; ⎠
s=−
(
Θ=
V = V
(0)
n⋅
(n)
)
∑
(n)
(n)
( (
⎧ 1, е ли ⎪ i =⎨ ⎪⎩ − 1, е ли fL
(k )
k =1
– УРС
0
0;
n ⋅ ( n ) i < 0;
(0)
n ⋅ (n)
Θ < 1,
(
{
е
∑ N
(n)
n =1
fL
е ь
(2.246 )
(2.246 )
Θ = 1;
(2.246 )
})
е
я:
ε = h − p ρ ; h = h p, T , Ym , m = 1, N S ; ε m = ε m ({S ме и }) , Dm = Dm ({S ме и }) , m = 1, N S ; T1 = T2 = … = TN S = T .
(2.246 ) Ym =
n∈
(0)
А
⎡⎣(Ym ) L ⋅ wL ⋅ f L ⎤⎦
«*»:
(n)
IN
(Ym )L =
( n)
.Н
(2.246 )
∑
∑
n∈
( n)
(w
⋅ fL )
L
Ym , n ∈
=
∑
n∈
(n)
IN
OUT
∑
n∈
OUT
(0)
( Dm = 0 ) ⎡⎣(Ym ) L ⋅ wL ⋅ f L ⎤⎦ (n)
(w
L
⋅ fL )
ε=
(0)
n∈
εL =
( n)
(n)
∑ IN
n∈
(n)
(w
L
ε, n∈
=
⋅ fL )
OUT
n∈
(n)
∑
:
⎣⎡ε L ⋅ wL ⋅ f L ⎦⎤
IN
n∈
OUT
(0)
∑
( n)
(w
L
⋅ fL )
(2.246 *)
;
IN
(2.246 *)
.
2.5. Чи ленны анали а е а иче а ве вленных у о ово ов В ( К
.В -
, [63, 92, 93]). А К -
© В.Е. Селе
( k = 0) ,
(2.246 *)
(2.246 )
⎡⎣ε L ⋅ wL ⋅ f L ⎦⎤
(2.246 *)
( Dm = 0 )
,
∑
, m = 1, N S ;
IN
.
(2.246 )
(2.246 )
ев, В.В. Алеш
о
о ели
(2.36), (2.54), (2.225) (2.231) ( К ). В К , ( К ) [71, 72, 92],
) ,
К ( .В ,
,
,
, .Э
, С.Н. Прял в, 2007–2009
.,
-
-
лава 2 139 _______________________________________________________________________________________
К , ,
[69, 92]
А.А.
. В-
К
, , , [71, 95]. В-
К
К ,
-
А.Н.
[70, 94],
-
,
, , . . ,
[71]. ,
,
.
В
{x , t }
-
i
tj –
xi
ра
ве
(
-вре е
.
ая е а ( ра
. 2.14),
x= x , x= x , t=t a i
)
b i
xi ≤ xib ≤ xi +1 , t j −1 ≤ t aj ≤ t j , t j ≤ t bj ≤ t j +1 , xia ≠ xib , t aj ≠ t bj ). В
hi +1 = xi +1 − xi – ; τ j = t j − t j −1 j-
© В.Е. Селе
«
» « τ j +1 = t j +1 − t j –
{x , t }
)
t=t
a j
i
b j
(
( xi −1 ≤ xia ≤ xi , j
»
«
»
» :
,
y = y ( x, t )
yij = y ( xi , t j ) ; yi = yi ( t ) = y ( xi , t ) ; y j = y j ( x ) = y ( x, t j ) . ев, В.В. Алеш
«
.
β j = τ j +1 τ j – -
В
е
ta − t ta − t tb − t x b − xi xia − xi −1 xia − xi −1 ; rib = i ; s aj = j j −1 = j j −1 ; s bj = j j , = τ j +1 t j − t j −1 τj hi +1 xi − xi −1 hi
hi = xi − xi −1 i-
αi = hi +1 hi
,
j
.
и . 2.14. Пр
ri a =
j
i
i, j ∈ Z , Z –
:
К . ( . 2.14),
.
, С.Н. Прял в, 2007–2009
: (2.247 )
140 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
y ( x, t ) = aiy − ( t ) ⋅ x 2 + biy − ( t ) ⋅ x + ciy − ( t ) , x ∈ [ xi −1 , xi ] ; y ( x, t ) = a
y+ i
(t ) ⋅ x
2
+ bi
y+
(t ) ⋅ x + c (t ) , y+ i
x ∈ [ xi , xi +1 ] ,
(2.247 ) (2.247 )
∂y = 2 ⋅ aiy − (t ) ⋅ x + biy − (t ), x ∈ [ xi −1 , xi ] ; ∂x ∂y δ y ( x, t ) = = 2 ⋅ aiy + (t ) ⋅ x + biy + (t ), x ∈ [ xi , xi +1 ] ; ∂x
:
δ y ( x, t ) =
(δ y ) a
В
j i
= δ y ( xia , t ) = 2 ⋅ aiy − ( t j ) ⋅ x a + biy − ( t j ) ;
(δ y )
j
b
i
(2.247 ) (2.247 )
= 2 ⋅ aiy + ( t j ) ⋅ x b + biy + ( t j ) . (2.247 )
: y = yij , h = hi , α = αi , τ = τ j , β = β j , x = xi , t = t j , y = yij +1 , y = yij −1 , y ( +1) = yij+1 ,
y ( −1) = yij−1 , r a = ri a , r b = rib , s a = s aj , s b = s bj , i, j ∈ Z ;
y ( ) = σ ⋅ y + (1 − σ ) ⋅ y; σ
y(
−S)
y(θ ) = θ ⋅ y + (1 − θ ) ⋅ y ( −1) ; y (
= s a ⋅ y + (1 − s a ) ⋅ y; y (
σ ,θ )
+S)
= σ ⋅ y + (1 − σ − θ ) ⋅ y + θ ⋅ y;
= s b ⋅ y + (1 − s b ) ⋅ y;
y( − R ) = r a ⋅ y + (1 − r a ) ⋅ y ( −1) ; y( + R ) = r b ⋅ y ( +1) + (1 − r b ) ⋅ y;
y( − R ) ( +1) = y( + R ) ; y( + R ) ( −1) = y( − R ) ; y (
−S)
= y(
+S)
; y(
+S)
= y(
Δt − = t − t a = (1 − s a ) ⋅τ ; Δt + = t b − t = s b ⋅τ = β ⋅ s b ⋅τ ;
−S)
;
Δx − = x − x a = (1 − r a ) ⋅ h; Δx + = x b − x = r b ⋅ h ( +1) = α ⋅ r b ⋅ h;
Δt = t b − t a = Δt + + Δt − = (1 − s a ) ⋅τ + s b ⋅τ = (1 − s a ) ⋅τ + β ⋅ s b ⋅τ ;
Δx = x b − x a = Δx + + Δx − = (1 − r a ) ⋅ h + r b ⋅ h ( +1) = (1 − r a ) ⋅ h + α ⋅ r b ⋅ h;
γ− =
γ+ =
(1 − r a ) ⋅ h Δx − 1 − ra = = ; a b Δx (1 − r ) ⋅ h + r ⋅ h ( +1) 1 − r a + α ⋅ r b
r b ⋅ h ( +1) α ⋅ rb Δx + = = ; a b Δx (1 − r ) ⋅ h + r ⋅ h ( +1) 1 − r a + α ⋅ r b
δ y a ( +1) = δ y b ; δ y b ( −1) = δ y a ; y x = y x ( −1) = y x ; yt =
Д
y ( +1) − y ; h ( +1)
(2.231)
.И
.Н.
В. . (
.,
(
.
-
(2.248). ,
© В.Е. Селе
y − y ( −1) ; y x ( +1) = y x ; h
y ( +1) − y y ( +1) − y y− y y−y ; yx = ; yx = ; yt = . (2.248) + + Δt Δx 0,5 ⋅ (1 + α ) ⋅ h 0,5 ⋅ (1 + β ) ⋅τ
, [1, 2, 6]). В , . 2.14) (2.54), Д
yx =
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, -
лава 2 141 _______________________________________________________________________________________
,
[70, 92]. Н
,
∂ (ρ ⋅ w ⋅ f ) ∂(ρ ⋅ w⋅ f ) 1 1 dtdx + dtdx = ⋅∫∫ ⋅∫ ∫ Δx ⋅ Δt Δx Δt ∂t Δx ⋅ Δt Δx Δt ∂x
(2.54 ).
:
2
∂p ∂z π 1 1 1 =− ⋅ ⋅ fdtdx − ⋅ ⋅ g ⋅ ρ ⋅ 1 ⋅ fdtdx − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ Rdtdx. Δx ⋅ Δt Δ∫x Δ∫t ∂x Δx ⋅ Δt Δ∫x Δ∫t ∂x 4 Δx ⋅ Δt Δ∫x Δ∫t
(2.249)
(2.249). Пе
е
а ае
е:
⎛ ∂ ( ρ ⋅ w⋅ f ) ⎞ ∂ ( ρ ⋅ w⋅ f ) 1 1 ⋅∫ ∫ ⋅∫⎜ ∫ dtdx = dx ⎟ dt = Δx ⋅ Δt Δx Δt ∂t Δx ⋅ Δt Δt ⎝ Δx ∂t ⎠
⎛∂ ⎡ ⎛∂ ⎡ ⎤⎞ ⎤⎞ 1 1 ⋅ ∫ ⎜ ⎢ ∫ ρ ⋅ w ⋅ fdx ⎥ ⎟ dt = ⋅ ∫ ⎜ ⎢ ρ ( t ) ⋅ w ( t ) ⋅ ∫ fdx ⎥ ⎟ dt = ⎜ ⎟ ⎜ ⎟ Δx ⋅ Δt Δt ⎝ ∂t ⎣ Δx Δx ⋅ Δt Δt ⎝ ∂t ⎣ Δx ⎦⎠ ⎦⎠ 1 ⎛∂ 1 ⎛∂ ⎞ ⎞ = ⋅ ∫ ⎜ ⎡ ρ ( t ) ⋅ w ( t ) ⋅ Fi ( t ) ⎤ ⎟ dt ≈ ⋅ ∫ ⎜ ⎣⎡ ρi ⋅ wi ⋅ Fi ( t ) ⎦⎤ ⎟ dt = ⎣ ⎦ Δt Δt ⎝ ∂t Δt Δt ⎝ ∂t ⎠ ⎠ =
=
ρi ( t bj ) ⋅ wi ( t bj ) ⋅ Fi ( t bj ) − ρi ( t aj ) ⋅ wi ( t aj ) ⋅ Fi ( t aj ) Δt
ρ (t ) ⋅ w (t ) – i-
,
Δx (
);
Fi =
1 ⋅ fdx Δx Δ∫x
(2.251)
( ρi ⋅ wi ⋅ Fi )
Δx ). В
(
(2.250 )
( ρi ⋅ Fi ⋅ wi ) t = t a = ( ρi ⋅ Fi ) t = t a ⋅ wi t = t a :
t aj
t bj
-
j j −1 = ⎡ s aj ⋅ ( ρ ⋅ F )i + (1 − s aj ) ⋅ ( ρ ⋅ F )i ⎤ × ⎣ ⎦
(− S ) × ⎡⎣ s ⋅ wi + (1 − s ) ⋅ wi ⎤⎦ = ⎡⎣ s ⋅ ( ρ ⋅ F ) + (1 − s a ) ⋅ ( ρ ⋅ F ) ⎤⎦ ⋅ ⎡⎣ s a ⋅ w + (1 − s a ) ⋅ w ⎤⎦ = ( ρ ⋅ F ) ⋅ w( − S ) ; j
a j
j
a j
j −1
j
j
a
( ρ i ⋅ Fi ⋅ wi ) t = t b = ( ρ i ⋅ Fi ) t = t b ⋅ wi t = t b
«→ » (2.249)
j
j
j
→ (ρ ⋅ F )
(+ S )
.
∂(ρ ⋅ w ⋅ f ) 1 dtdx → ⋅∫∫ Δx ⋅ Δt Δx Δt ∂t
( ρ ⋅ F )(
:
+S)
⋅ w( + S ) − ( ρ ⋅ F ) Δt
⋅ w(+ S ) .
, (− S )
⋅ w( − S )
= ⎡( ρ ⋅ F ) ⎣
(− S )
⋅ w( − S ) ⎤ . ⎦t +
(2.250 ) ⎛ ∂ ( ρ ⋅ w2 ⋅ f ) ⎞ ∂ ( ρ ⋅ w2 ⋅ f ) 1 1 dtdx = dx ⎟ dt = ⋅∫∫ ⋅∫⎜∫ ⎟ Δx ⋅ Δt Δx Δt ∂x Δx ⋅ Δt Δt ⎜⎝ Δx ∂x ⎠ В
е
© В.Е. Селе
а ае
е:
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
142 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
1 = ⋅∫ Δt Δt Δt . В
(ρ ⋅ w
(ρ ⋅ w
2
⋅f)
⋅f)
2
x = xib
− ( ρ ⋅ w2 ⋅ f )
x = xia
Δx
dt =
(ρ ⋅ w
2
⋅f)
xib
− ( ρ ⋅ w2 ⋅ f )
Δx
xia
= (ρ ⋅ w ⋅ f )
⋅ w x a = ⎡⎣ ri ⋅ ( ρ ⋅ w ⋅ f )i + (1 − ri ) ⋅ ( ρ ⋅ w ⋅ f )i −1 ⎤⎦ × i
(2.252 ):
a
2
⋅f)
= (ρ ⋅ w ⋅ f )
xib
∂ (ρ ⋅ w ⋅ f ) 1 dtdx → ⋅∫∫ Δ x ⋅ Δ t Δx Δ t ∂x 2
=
a
× ⎡⎣ ri a ⋅ wi + (1 − ri a ) ⋅ wi −1 ⎤⎦ = ( ρ ⋅ w ⋅ f )( − R ) ⋅ w( − R ) ; xia
,
(( ρ ⋅ w ⋅ f )
⋅ w x b → ( ρ ⋅ w ⋅ f )( + R ) ⋅ w( + R ) . i
( ρ ⋅ w ⋅ f )( + R) ⋅ w( + R) − ( ρ ⋅ w ⋅ f )( − R ) ⋅ w( − R)
⋅ w( − R )
(− R)
xib
(2.249)
) → (( ρ ⋅ w ⋅ f ) +
x
:
Δx
(− R)
⋅ w( − R )
)
(σ , θ )
=
(2.252 ) .
+
x
ó а ае
. (2.252 )
(2.252 )
(ρ ⋅ w
е ье
xia
.
е:
⎛ ∂p ⎞ 1 1 ∂p ⋅ ∫ ∫ ⋅ fdtdx = ⋅∫⎜∫ ⋅ fdx ⎟ dt. Δx ⋅ Δt Δx Δt ∂x Δ x ⋅ Δ t Δt ⎝ Δx ∂ x ⎠ p ( x, t )
-
(2.253) Δx
-
:
⎛x ⎞ ⎛ ⎞ xi x xb pi +1 − pi 1 1 ⎜ i ∂p ∂p ∂p ⎟ 1 ⎜ pi − pi −1 ⎟ ⋅∫ ⋅ fdx = ⋅⎜ ∫ ⋅ fdx + ∫ ⋅ fdx ⎟ = ⋅⎜ ⋅ ∫ fdx + ⋅ ∫ fdx ⎟ . (2.254) x h h Δx Δx ∂x Δx ⎜ a ∂x ∂x Δ i i +1 ⎟ ⎜ ⎟ xi xa xi ⎝x ⎠ ⎝ ⎠ b
В
: 1 B = −⋅ Δx − i
∫
xi xia
1 f ( x, t ) dx; B = + ⋅ Δx + i
∫ f ( x, t ) dx.
xib
(2.255)
xi
(2.255) (2.248) (2.254) : p − pi −1 − − pi +1 − pi + + ∂p 1 ⋅∫ ⋅ fdx = i ⋅ γ i ⋅ Bi + ⋅ γ i ⋅ Bi → γ − ⋅ B − ⋅ px + γ + ⋅ B + ⋅ p x . (2.256) Δ x Δx ∂ x hi hi +1 (2.256) (2.253),
:
1 1 ⎛ p − pi −1 − − pi +1 − pi + + ⎞ ∂p ⋅∫ ∫ ⋅ fdtdx = ⋅ ∫ ⎜ i ⋅ γ i ⋅ Bi + ⋅ γ i ⋅ Bi ⎟ dt → Δx ⋅ Δt Δx Δt ∂ x Δ t Δt ⎝ hi hi +1 ⎠ → (γ − ⋅ B − ⋅ px + γ + ⋅ B + ⋅ p x )
е
е
а ае
ев, В.В. Алеш
е(
а а
.
а ае ): ⎛ ⎞ 1 1 ∂z ∂z ⋅ ∫ ∫ g ⋅ ρ ⋅ 1 ⋅ fdtdx = ⋅ ∫ ⎜ g ⋅ ρ ⋅ ∫ 1 ⋅ fdx ⎟ dt ≈ Δ x ⋅ Δ t Δx Δ t ∂x Δ x ⋅ Δ t Δt ⎝ ∂x Δx ⎠ © В.Е. Селе
е
(σ , θ )
е ь
, С.Н. Прял в, 2007–2009
(2.257)
лава 2 143 _______________________________________________________________________________________
≈
Пя
π
(σ , θ ) ⎛ ⎞ ∂z 1 ⋅ ∫ ⎜ g ⋅ ρ i ⋅ ∫ 1 ⋅ fdx ⎟ dt → g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x ) ⎤⎦ . (2.258) Δx ⋅ Δ t Δ t ⎝ ∂x Δx ⎠
е
а ае
е:
⎛ ⎞ 1 π 1 ⋅ ∫ ∫ λ ⋅ ρ ⋅ w ⋅ w ⋅ Rdtdx = ⋅ ⋅ ∫ ⎜ ∫ λ ⋅ ρ ⋅ w ⋅ w ⋅ Rdx ⎟ dt ≈ 4 Δ x ⋅ Δ t Δx Δ t 4 Δ x ⋅ Δ t Δt ⎝ Δ x ⎠ ⋅
⎡ 1 ⎤⎞ 1 ⎛ ⋅ ∫ ⎜ λi ⋅ ρ i ⋅ wi ⋅ wi ⋅ ⎢ ⋅ ∫ Rdx ⎥ ⎟ dt = ⎜ ⎟ 4 Δ t Δt ⎝ ⎣ Δ x Δx ⎦⎠ (σ , θ ) π 1 π = ⋅ ⋅ ∫ ( λi ⋅ ρ i ⋅ wi ⋅ wi ⋅ ri ) dt → ⋅ ( λ ⋅ ρ ⋅ w ⋅ r ) ⋅ w(σ , θ ) , 4 Δ t Δt 4
π
≈
⋅
ri =
1 ⋅ Rdx. Δx Δ∫x
(2.249)
(
:
⎡ ( ρ ⋅ F )( − S ) ⋅ w( − S ) ⎤ + ( ρ ⋅ w ⋅ f ) ⋅ w (− R) (− R) ⎣ ⎦t
)
− g ⋅ ⎡⎣ ρ ⋅ ( B ⋅ γ ⋅ ( z1 ) x + B ⋅ γ ⋅ ( z1 ) x )⎤⎦ +
−
−
+
+
(2.259)
(2.260)
= − ( B − ⋅ γ − ⋅ px + B + ⋅ γ + ⋅ px )
(σ , θ ) +
x
(σ , θ )
−
π
4
⋅ (λ ⋅ ρ ⋅ w ⋅ r )
(σ , θ )
⋅w
(σ , θ )
, (σ , θ )
-
−
(2.261)
.
А
(
(
:
⎡ ( ρ ⋅ F )( − S ) ⎤ + ( ρ ⋅ w ⋅ f ) (− R) ⎣ ⎦t +
(
)
(σ ,θ ) +
x
a − ⎡( ρ ⋅ f ⋅ Dm )( − R ) ⋅ δ (Ym ) ⎤ ⎣ ⎦x
= 0;
(σ , θ )
(
+
)
− g ⋅ ⎡⎣ ρ ⋅ ( B ⋅ γ ⋅ ( z1 ) x + B ⋅ γ ⋅ ( z1 ) x )⎤⎦ −
−
(
+
+
⎡ ρ ⋅ F )( − S ) ⋅ ε ( − S ) ⎤ + ( ρ ⋅ w ⋅ f ) ⋅ ε (− R) (− R) ⎣( ⎦t
(
+
= − ( p ⋅ w ⋅ f )( − R )
)
(σ , θ ) +
x
(σ , θ ) +
x
−
)
(σ , θ ) +
x
(σ , θ )
(σ , θ )
YN = 1 −
+
ев, В.В. Алеш
(σ , θ )
+
x
∑ Ym ;
N S −1
(2.262 )
m =1
= − ( B − ⋅ γ − ⋅ px + B + ⋅ γ + ⋅ px )
−
S
⋅ (λ ⋅ ρ ⋅ w ⋅ r )
π
4
(σ , θ )
⋅w
(σ , θ )
(σ , θ )
+ ⎡( k ⋅ f )( − R ) ⋅ δ T a ⎤ ⎣ ⎦x
(σ , θ ) +
, С.Н. Прял в, 2007–2009
(σ , θ )
− φ (σ , θ ) +
⋅ w (σ , θ ) −
−
(2.262 )
;
⎛ ⎛ w2 ⎞ w2 ⎞ + Kt ⎜ ρ ⋅ F ⋅ ⎟ + K x ⎜ ρ ⋅ w ⋅ f ⋅ ⎟ 2 ⎠ 2 ⎠ ⎝ ⎝
− g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x ) ⎤⎦
− p (σ , θ ) ⋅ ⎡⎣ F ( − S ) ⎤⎦ t + ( Q ⋅ F )
© В.Е. Селе
)
(2.262 )
= 0, m = 1, N S − 1;
⎡ ( ρ ⋅ F )( − S ) ⋅ w( − S ) ⎤ + ( ρ ⋅ w ⋅ f ) ⋅ w (− R) (− R) ⎣ ⎦t +
. 2.14) (2.54),
⎡( ρ ⋅ F )( − S ) ⋅ (Ym )( − S ) ⎤ + ( ρ ⋅ w ⋅ f ) ⋅ (Ym ) (− R) (− R) ⎣ ⎦t +
.
(σ , θ )
=
144 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ a + ∑ ⎡( ρ ⋅ Dm ⋅ f )( − R ) ⋅ (ε m )( − R ) ⋅ δ (Ym ) ⎤ ⎣ ⎦x m=1
(σ , θ )
NS
(2.262 )
;
ε m = ε m ({S ме и }) , m = 1, N S ; T1 = T2 = … = TN ; +
p = p ({S ме и }) ;
ε = ε ({S ме и }) ;
k = k ({S ме и }) ;
(2.262 )
S
(2.262 )
Dm = Dm ({S ме и }) , m = 1, N S , F , B+ , B−
(2.262 ) (2.2623)
r –
,
f
( 0,5 ⋅ ρ ⋅ F ⋅ w )
(
( 0,5 ⋅ ρ ⋅ w ⋅ f ⋅ w )
2
Kx –
); K t
2
(
К
(2.262) . (2.262)
[1, 2, 6], : , « »Д . . [63, 92].
,
;
; -
;
.В «
-
); φ (σ , θ ) –
Φ ( T , Toc ) .
,
Э
R
» –
,
, [70, 96], .Д
(2.262)
ri a = rib−1 ;
:
s aj = s bj −1 .
(2.263)
,
В
.
, .
,
.
(2.262)
(2.263)
-
. (2.262) (2.263)
-
( , ) [63, 92, 97].
yij = y ( xi , t j ) ,
2 2 ⎛ ∂y ⎞ h ⎛ ∂ y ⎞ ⎛ ∂y ⎞ = yi − hi ⋅ ⎜ ⎟ + i ⋅ ⎜ 2 ⎟ + O ( h 3 ) = yij − hi ⋅ ⎜ ⎟ + O ( h 2 ) ; ⎝ ∂x ⎠i 2 ⎝ ∂x ⎠i ⎝ ∂x ⎠i j
y
j i −1
-
j
j
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
ℑ = {xi , t j } :
j
(2.264 )
лава 2 145 _______________________________________________________________________________________ 2 ⎛ ∂y ⎞ (α ⋅ h ) yij+1 = yij + αi ⋅ hi ⋅ ⎜ ⎟ + i i 2 ⎝ ∂x ⎠i j
⎛ ∂2 y ⎞ ⎛ ∂y ⎞ ⋅ ⎜ 2 ⎟ + O ( h 3 ) = yij + α i ⋅ hi ⋅ ⎜ ⎟ + O ( h 2 ) ; ⎝ ∂x ⎠i ⎝ ∂x ⎠i j
j
2 ⎛ ∂y ⎞ τ ⎛ ∂ y ⎞ ⎛ ∂y ⎞ yij −1 = yij − τ j ⋅ ⎜ ⎟ + j ⋅ ⎜ 2 ⎟ + O (τ 3 ) = yij − τ j ⋅ ⎜ ⎟ + O (τ 2 ) ; ∂ ∂ t t 2 ⎝ ⎠i ⎝ ∂t ⎠i ⎝ ⎠i j
yi
j +1
y(
j ⎛ ∂y ⎞ ( β j ⋅τ j ) = yi + β j ⋅ τ j ⋅ ⎜ ⎟ + 2 ⎝ ∂t ⎠i j
σ,θ)
(B )
j
2
(2.264 )
⎛ ∂2 y ⎞ ⎛ ∂y ⎞ ⋅ ⎜ 2 ⎟ + O (τ 3 ) = yij + β j ⋅τ j ⋅ ⎜ ⎟ + O (τ 2 ) ; (2.264 ) ∂ ∂t ⎠i t ⎝ ⎝ ⎠i j
j
= σ j ⋅ yij +1 + (1 − σ j − θ j ) ⋅ yij + θ j ⋅ yij −1 = yij + O (τ ) .
(2.264 )
1 1 ⋅ f ( x, t j ) dx = ⋅ ( f i j + O ( h ) ) dx = f i j + O ( h ) ; Δx Δ∫x Δx Δ∫x
,
Fi j = +
j
2
(2.264 )
= fi j + O ( h ) ,
j
ri j = Ri j + O ( h ) ; i
(B ) −
j i
(2.264 )
= fi j + O ( h ) ;
(2.264 ) (2.264 )
⎛ ∂F ⎞ ⎛ ∂f ⎞ ⎜ ⎟ = ⎜ ⎟ + O (h ). ⎝ ∂t ⎠i ⎝ ∂t ⎠i j
j
Ра
а
Первая ра
(2.264 ) и
а ы
и (2.262а)
s bj ⋅ ρ i j +1 ⋅ Fi j +1 + (1 − s bj − s aj ) ⋅ ρ i j ⋅ Fi j − (1 − s aj ) ⋅ ρ i j −1 ⋅ Fi j −1
ть:
⎡⎣( ρ ⋅ F )( − S ) ⎤⎦ t = +
Δt j
=
j ⎛ ⎞ ⎛ ∂ (ρ ⋅ F ) ⎞ j j s bj ⋅ ⎜ ( ρ ⋅ F )i + ( β ⋅ τ ) j ⋅ ⎜ + O (τ 2 ) ⎟ + (1 − s bj − s aj ) ⋅ ( ρ ⋅ F )i ⎟ ⎜ ⎟ t ∂ ⎝ ⎠i ⎝ ⎠ = − Δt j j ⎡ ⎤ ⎛ ∂ (ρ ⋅ F ) ⎞ j 2 a s F 1 ρ τ − ⋅ ⋅ − ⋅ ⎢ ( ) ( j) ⎟ + O (τ ) ⎥ j ⎜ i ⎢⎣ ⎥⎦ ⎝ ∂t ⎠i − = Δt j
⎛ ∂ (ρ ⋅ F ) ⎞ 2 ⎡(1 − s aj ) ⋅ τ j + β j ⋅ s bj ⋅τ j ⎤ ⋅ ⎜ ⎟ + O (τ ) ⎣ ⎦ ∂ t ∂ (ρ ⋅ F ) ∂ (ρ ⋅ f ) ⎝ ⎠i = = + O (τ ) = + O (τ , h ) . a b ∂t ∂t (1 − s j ) ⋅τ j + β j ⋅ s j ⋅τ j j
(( ρ ⋅ w ⋅ f ) )
Вт рая ра
(− R)
=
ть.
(σ , θ )
=
(( ρ ⋅ w ⋅ f ) )
(2.265)
(2.264 ) (− R)
+ O (τ ) =
:
rib ⋅ ρ i j+1 ⋅ f i +j 1 ⋅ wij+1 + (1 − rib − ri a ) ⋅ ρ i j ⋅ f i j ⋅ wij − (1 − ri a ) ⋅ ρ i j−1 ⋅ f i −j1 ⋅ wij−1
© В.Е. Селе
+
x
+
x
Δxi
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
+ O (τ ) =
146 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ j ⎡ ⎤ ⎛ ∂ (ρ ⋅ w ⋅ f ) ⎞ j 2 rib ⋅ ⎢( ρ ⋅ w ⋅ f )i + (α ⋅ h )i ⋅ ⎜ ⎟ + O ( h )⎥ j b a ∂x ⎢⎣ ⎥⎦ (1 − ri − ri ) ⋅ ( ρ ⋅ w ⋅ f )i ⎝ ⎠i = + − Δxi Δxi
j ⎡ ⎤ ⎛ ∂ (ρ ⋅ w ⋅ f ) ⎞ j 2 a − ⋅ ⋅ ⋅ − ⋅ r w f h 1 ρ ⎢ ( ) ( i) ⎟ + O ( h )⎥ i ⎜ i ∂x ⎝ ⎠i ⎣⎢ ⎦⎥ + O τ = − ( ) Δxi
(2.266)
⎛ ∂ (ρ ⋅ w ⋅ f ) ⎞ 2 ⎡(1 − ri a ) ⋅ hi + α i ⋅ rib ⋅ hi ⎤ ⋅ ⎜ ⎟ + O (h ) ⎣ ⎦ ∂x ∂ (ρ ⋅ w ⋅ f ) ⎝ ⎠i = + O (τ ) = + O (τ , h ) . a b ∂x (1 − ri ) ⋅ hi + αi ⋅ ri ⋅ hi j
,
(2.262 ) -
(2.54 )
-
∂ (ρ ⋅ f )
:
∂t
А
+
∂ ( ρ ⋅ f ⋅ w) ∂x
+ O (τ , h ) = 0.
(2.262 )
(2.267) (2.54 )
, (2.54 ). Ра
а
Первая ра
ть:
и
и
(2.262 )
ия (2.262 )
⎡ s b ⋅ ρ j +1 ⋅ Fi j +1 + (1 − s bj ) ⋅ ρ i j ⋅ Fi j ⎤ ⋅ ⎡ s bj ⋅ wij +1 + (1 − s bj ) ⋅ wij ⎤ ⎦ ⎣ ⎦− ⎡ ( ρ ⋅ F )( − S ) ⋅ w( − S ) ⎤ = ⎣ j i ⎣ ⎦t Δt j ⎡ s ⋅ ρ i ⋅ Fi + (1 − s aj ) ⋅ ρ i j −1 ⋅ Fi j −1 ⎤ ⋅ ⎡ s aj ⋅ wij + (1 − s aj ) ⋅ wij −1 ⎤ ⎦ ⎣ ⎦= −⎣ Δt j +
a j
j
j
j j ⎡ ⎤ ⎡ ⎤ ⎛ ∂ (ρ ⋅ F ) ⎞ j ⎛ ∂w ⎞ j b 2 2 τ β τ O w s + ⋅ + ⋅ ⋅ ⋅ ⎢ ( ρ ⋅ F )i + s bj ⋅ ( β ⋅τ ) j ⋅ ⎜ ⎥ ( ) ( ) ⎢ ⎟ ⎟ + O (τ )⎥ i j j ⎜ ⎝ ∂t ⎠i ⎢ ⎥⎦ ⎣⎢ ⎝ ∂t ⎠i ⎦⎥ − =⎣ Δt j
j j ⎡ ⎤ ⎡ ⎤ ⎛ ∂ (ρ ⋅ F ) ⎞ j ⎛ ∂w ⎞ j a 2 2 τ τ 1 O w s + ⋅ − − ⋅ ⋅ ⎢ ( ρ ⋅ F )i − (1 − s aj ) ⋅τ j ⋅ ⎜ ⎥ ( ) ⎢ i ( ⎟ ⎟ + O (τ )⎥ j ) j ⎜ ⎝ ∂t ⎠i ⎢ ⎥⎦ ⎣⎢ ⎝ ∂t ⎠i ⎦⎥ −⎣ = Δt j
j j ⎡ ⎤ ⎡ ⎤ j j ⎛ ∂w ⎞ b j ⎛ ∂ (ρ ⋅ F ) ⎞ b 2 ⎢( ρ ⋅ w ⋅ F )i + s j ⋅ ( β ⋅τ ) j ⋅ wi ⋅ ⎜ ⎟ ⎥ + ⎢ s j ⋅ ( β ⋅ τ ) j ⋅ ( ρ ⋅ F )i ⋅ ⎜ ⎟ + O (τ ) ⎥ ⎝ ∂t ⎠i ⎢ ⎥⎦ ⎝ ∂t ⎠i ⎥⎦ ⎢⎣ =⎣ − Δt j
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 147 _______________________________________________________________________________________ j j ⎡ ⎤ ⎛ ∂ (ρ ⋅ F ) ⎞ ⎤ ⎡ j j ⎛ ∂w ⎞ a τ ρ s F 1 − − ⋅ ⋅ ⋅ ⋅ ⎢( ρ ⋅ w ⋅ F )i − (1 − s aj ) ⋅ τ j ⋅ wij ⋅ ⎜ ⎥ )i ⎜ ⎟ + O (τ 2 )⎥ ⎢( ⎟ j ) j ( ⎝ ∂t ⎠i ⎢ ⎝ ∂t ⎠i ⎦⎥ ⎣ ⎦ −⎣ = Δt j
∂ (ρ ⋅ w ⋅ f ) ⎛ ∂ (ρ ⋅ w ⋅ F ) ⎞ ⎛ ∂ (ρ ⋅ w ⋅ f ) ⎞ =⎜ + O ( h, τ ) . ⎟ + O (τ ) = ⎜ ⎟ + O ( h, τ ) = t t ∂ ∂ ∂t ⎝ ⎠i ⎝ ⎠i j
(( ρ ⋅ w ⋅ f )
Вт рая ра (− R)
j
)
ть. И
⋅ w( − R )
(σ , θ )
(
(2.264 ),
= ( ρ ⋅ w ⋅ f )( − R ) ⋅ w( − R )
)
(2.268) :
+ O (τ ) =
⎛ ⎡ rib ⋅ ρ i j+1 ⋅ f i +j1 ⋅ wij+1 + (1 − rib ) ⋅ ρ i j ⋅ f i j ⋅ wij ⎤ ⋅ ⎡ rib ⋅ wij+1 + (1 − rib ) ⋅ wij ⎤ ⎦ ⎣ ⎦− =⎜⎣ ⎜ Δxi ⎝ +
x
+
x
⎡ ri a ⋅ ρ i j ⋅ f i j ⋅ wij + (1 − ri a ) ⋅ ρ i j−1 ⋅ f i −j1 ⋅ wij−1 ⎤ ⋅ ⎡ ri a ⋅ wij + (1 − ri a ) ⋅ wij−1 ⎤ ⎞ ⎦ ⎣ ⎦ ⎟+O τ = −⎣ ( ) ⎟ Δxi ⎠
j j ⎡ ⎤ ⎡ j b ⎤ ⎛ ∂(ρ ⋅ w ⋅ f ) ⎞ j ⎛ ∂w ⎞ 2 2 b ⎢( ρ ⋅ w ⋅ f )i + ri ⋅ (α ⋅ h )i ⋅ ⎜ ⎟ + O ( h )⎥ ⎟ + O ( h ) ⎥ ⋅ ⎢ wi + ri ⋅ (α ⋅ h )i ⋅ ⎜ x x ∂ ∂ ⎝ ⎠i ⎢ ⎥⎦ ⎣ ⎝ ⎠i ⎦ =⎣ − Δxi
j j ⎡ ⎤ ⎡ j ⎤ ⎛ ∂(ρ ⋅ w ⋅ f ) ⎞ j ⎛ ∂w ⎞ a 2 2 O h w r h 1 + ⋅ − − ⋅ ⋅ ⎢( ρ ⋅ w ⋅ f )i − (1 − ri a ) ⋅ hi ⋅ ⎜ ⎥ ( ) ( ) ⎢ ⎟ + O ( h )⎥ ⎟ i i i ⎜ ∂x ⎝ ∂x ⎠i ⎢ ⎝ ⎠i ⎦ ⎦⎥ ⎣ −⎣ + Δxi
+ O (τ ) =
j j ⎡ ⎤ ⎡ ⎤ j 2 b j ⎛ ∂(ρ ⋅ w⋅ f ) ⎞ w f r h w ρ α ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ ⎢( )i i ( )i i ⎜ ∂x ⎟ ⎥ + ⎢rib ⋅ (α ⋅ h )i ⋅ ( ρ ⋅ w ⋅ f )ij ⋅ ⎝⎜⎛ ∂∂wx ⎠⎟⎞ + O ( h2 )⎥ ⎢ i ⎝ ⎠i ⎥⎦ ⎣ ⎦ − =⎣ Δxi
j j ⎡ ⎤ j ⎛ ∂ (ρ ⋅ w ⋅ f ) ⎞ ⎤ ⎡ j ⎛ ∂w ⎞ a 2 − − ⋅ ⋅ ⋅ ⋅ ⋅ 1 r h w f ρ ⎢( ρ ⋅ w2 ⋅ f )i − (1 − ri a ) ⋅ hi ⋅ wij ⋅ ⎜ ⎥ ( ) ( ) ⎢ ⎟ + O ( h )⎥ ⎟ i i i ⎜ ∂x ⎝ ∂x ⎠i ⎢ ⎝ ⎠i ⎦⎥ ⎣ ⎦ −⎣ + Δxi
⎛ ∂ ( ρ ⋅ w2 ⋅ f ) ⎞ ∂ ( ρ ⋅ w2 ⋅ f ) ⎟ + O (τ , h ) = + O (τ ) = ⎜ + O (τ , h ) . ⎜ ⎟ ∂x ∂x ⎝ ⎠i j
Третья ра
(B
(2.269)
ть. И
−
=
⋅ γ − ⋅ px + B + ⋅ γ + ⋅ px )
(2.264 ),
( B ) ⋅ (γ ) −
j
−
j
⋅
(σ , θ )
:
= B − ⋅ γ − ⋅ px + B + ⋅ γ + ⋅ px + O (τ ) =
j j j p pij − pi j−1 − pij + ( B + ) i ⋅ (γ + ) i ⋅ i +1 + O (τ ) = hi hi +1
j ⎛ ∂p ⎞ j ⎛ ∂p ⎞ ∂p = f i j ⋅ (γ − ) i ⋅ ⎜ ⎟ + f i j ⋅ (γ + ) i ⋅ ⎜ ⎟ + O (τ , h ) = f ⋅ + O (τ , h ) . ∂ ∂ ∂x x x ⎝ ⎠i ⎝ ⎠i i
i
j
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
j
(2.270)
148 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
етвертая ра
ть (
g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x ) ⎤⎦
Пятая ра
(2.270)):
ть. И
π
4
(2.264 )
⋅ (λ ⋅ ρ ⋅ w ⋅ r )
(σ , θ )
(σ , θ )
= f ⋅g⋅ρ ⋅
(2.264 ),
⋅ w (σ , θ ) =
π 4
∂z1 + O (τ , h ) . ∂x
(2.271)
:
⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R + O (τ , h ) .
,
(2.272)
(2.262 ) -
(2.54 )
∂ ( ρ ⋅ f ⋅ w) ∂ ( ρ ⋅ f ⋅ w + ∂t ∂x
:
2
-
) = − f ⋅ ⎛ ∂p + g ⋅ ρ ⋅ ∂z
π 1 ⎞ ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R + O (τ , h ) . (2.273) ∂x ⎠ 4
⎜ ⎝ ∂x
Ра
а
и э
гии (2.262г)
Первая ра
ть (
⎡( ρ ⋅ F )( − S ) ⋅ ε ( − S ) ⎤ = ∂ ( ρ ⋅ ε ⋅ f ) + O (τ , h ) . ⎣ ⎦t ∂t
(2.268)):
(2.274)
+
(( ρ ⋅ w ⋅ f )
Вт рая ра
ть (
(− R)
Третья и четвертая ра е а
⎛ w2 ⎞ ( ρi Kt ⎜ ρ ⋅ F ⋅ ⎟ = 2 ⎠ ⎝
⋅ ε (− R)
(σ , θ ) +
x
=
∂ (ρ ⋅ w ⋅ε ⋅ f ) + O (τ , h ) . ∂x
(2.275)
ти (2.262). Kx
,
,
⋅ Fi j +1 + ρ i j ⋅ Fi j ) ⋅ wij +1 ⋅ wij − ( ρ i j ⋅ Fi j + ρ i j −1 ⋅ Fi j −1 ) ⋅ wij ⋅ wij −1 Kt
j +1
)
(2.269)):
2 ⋅ (1 + β j ) ⋅τ j
: ,
j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 ⎛ w2 ⎞ ( ρ i +1 ⋅ wi +1 ⋅ f i +1 + ρ i ⋅ wi ⋅ f i ) ⋅ wi +1 ⋅ wi − Kx ⎜ ρ ⋅ w ⋅ f ⋅ ⎟ = 2 ⎠ 2 ⋅ (1 + αi ) ⋅ hi ⎝
−
(ρ
j +1
i
⋅ wij +1 ⋅ f i j +1 + ρ i j−+11 ⋅ wij−+11 ⋅ f i −j1+1 ) ⋅ wij +1 ⋅ wij−+11
(2.268)
O (τ , h )
Пятая ра
2 ⋅ (1 + αi ) ⋅ hi
(2.269)
, -
ть (
(( p ⋅ w ⋅ f ) )
(2.266)): (− R)
© В.Е. Селе
.
ев, В.В. Алеш
(σ ,θ ) +
x
=
∂( p⋅w⋅ f ) + O (τ , h ) . ∂x
, С.Н. Прял в, 2007–2009
.
(2.276)
лава 2 149 _______________________________________________________________________________________
е тая ра
ть
g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x )⎤⎦
Се ьмая ра
ть (
Девятая ра
δT
δT ri a
(σ , θ )
∂ ( z1 ) + O (τ , h ) . (2.277) ∂x
(2.264 )):
j
j
u ( x, t ) = k ( x, t ) ⋅ f ( x, t ) .
(2.280)
[ xi −1 , xi ] [ xi , xi +1 ]
T ( x, t )
Tx
(σ ,θ )
= ⎡ u( − R ) ⋅ δ T a ⎤ + O (τ ) = ⎣ ⎦x
(
Tx )
:
T j −T j T j − Ti −j1 ⎡ rib ⋅ uij+1 + (1 − rib ) ⋅ uij ⎤ ⋅ i +1 i − ⎡ ri a ⋅ uij + (1 − ri a ) ⋅ uij−1 ⎤ ⋅ i ⎣ ⎦ (α ⋅ h ) ⎣ ⎦ hi i +
(2.279)
:
b
rib .
(2.278)
= ( Q ⋅ F )i + O (τ ) = ( Q ⋅ f )i + O (τ , h ) = Q ⋅ f + O (τ , h ) .
ть. В
⎡( k ⋅ f ) ⋅ δ T a ⎤ (− R) ⎣ ⎦x =
⋅ w (σ , θ ) = f ⋅ w ⋅ g ⋅ ρ ⋅
(2.265), (2.264 )): ∂F ∂f p (σ , θ ) ⋅ ⎣⎡ F ( − S ) ⎦⎤ = p ⋅ + O (τ ) = p ⋅ + O (τ , h ) . t t ∂ ∂t +
(Q ⋅ F )
,
(σ , θ )
(2.264 ):
ть (
В ьмая ра
a
(2.271)
+
Δxi
j ⎡ j b ⎤ ⎡⎛ ∂T ⎞ j (α ⋅ h )i ⎛ ∂u ⎞ 2 u r h O h α + ⋅ ⋅ ⋅ + ( ) ( ) ⎢ i i ⎥ ⋅ ⎢⎜ ⎟ ⎟ + i ⎜ 2 ⎝ ∂x ⎠i ⎢⎣ ⎥⎦ ⎢⎣⎝ ∂x ⎠i = b a ri ⋅ (α ⋅ h )i + (1 − ri ) ⋅ hi
+ O (τ ) =
j ⎤ ⎛ ∂ 2T ⎞ ⋅ ⎜ 2 ⎟ + O ( h 2 )⎥ ⎝ ∂x ⎠i ⎦⎥
−
j j ⎤ ⎡ j ⎤ ⎡⎛ ∂T ⎞ j hi ⎛ ∂ 2T ⎞ ⎛ ∂u ⎞ 2 2 a ⎢ ui − (1 − ri ) ⋅ hi ⋅ ⎜ ⎟ + O ( h )⎥ ⋅ ⎢⎜ ⎟ − ⋅ ⎜ 2 ⎟ + O ( h )⎥ ⎝ ∂x ⎠i ⎢⎣ ⎥⎦ ⎢⎣⎝ ∂x ⎠i 2 ⎝ ∂x ⎠i ⎥⎦ − + O (τ ) = b a ri ⋅ (α ⋅ h )i + (1 − ri ) ⋅ hi
(
)
j j j j 2 2 hi ⋅ ⎡ 0,5 ⋅ (αi + 1) − αi ⋅ ri b + (1 − ri a ) ⎤ ⎛ ∂T ⎞ ⎛ ∂u ⎞ j ⎛∂ T ⎞ j ⎛∂ T ⎞ ⎣ ⎦ +O τ, h = =⎜ ( ) ⎟ ⋅ ⎜ ⎟ + ui ⋅ ⎜ 2 ⎟ + ui ⋅ ⎜ 2 ⎟ ⋅ rib ⋅ (α ⋅ h )i + (1 − ri a ) ⋅ hi ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i
=
b a j j 2 ⎡ ⎤ ∂ ⎛ ∂T ⎞ j ⎛∂ T ⎞ ⎣αi ⋅ ( 0,5 − ri ) − ( 0,5 − ri ) ⎦ + O τ , h = u u ⋅ + ⋅ ⋅ ( ) i ⎜ ⎜ ⎟ 2 ⎟ b a ∂x ⎝ ∂x ⎠i ri ⋅ αi + (1 − ri ) ⎝ ∂x ⎠i
j j 2 ⎡αi ⋅ ( 0,5 − rib ) − ( 0,5 − ri a ) ⎤ ∂ ⎛ ∂T ⎞ j j ⎛∂ T ⎞ ⎣ ⎦ +O τ, h . = ⎜k ⋅ f ⋅ ( ) ⎟ + ki ⋅ f i ⋅ ⎜ 2 ⎟ ⋅ ∂x ⎝ ∂x ⎠i rib ⋅ αi + (1 − ri a ) ⎝ ∂x ⎠i
(2.281) © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
150 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(2.281),
Ч
-
ri a = 0,5 − αi ⋅ ( 0,5 − rib ) .
: (2.282)
(2.282)
1 ri a = rib = , 2
(2.283) T ( x, t )
. , 1
⎡( k ⋅ f ) ⋅ δ T a ⎤ (− R) ⎣ ⎦x
(σ , θ )
:
= ⎡⎣u( − R ) ⋅ δ T a ⎤⎦ + O (τ ) =
u( + R ) ⋅ δ T b − u( − R ) ⋅ δ T a
⎡ rib ⋅ uij+1 + (1 − rib ) ⋅ uij ⎤ ⋅ ⎡ rib ⋅ δ Ti +j1 + (1 − rib ) ⋅ δ Ti j ⎤ ⎦ ⎣ ⎦ − =⎣ Δxi +
x
+
Δxi
+ O (τ ) =
⎡ ri a ⋅ uij + (1 − ri a ) ⋅ uij−1 ⎤ ⋅ ⎡ ri a ⋅ δ Ti j + (1 − ri a ) ⋅ δ Ti −j1 ⎤ ⎣ ⎦ ⎣ ⎦ +O τ = − ( ) Δxi
=
1 Δxi
j j ⎞ ⎡⎛ ⎞ ⎛ ⎛ ∂T ⎞ j b ⎛ ∂ 2T ⎞ ⎛ ∂u ⎞ r h α ⋅ ⎢⎜ uij + rib ⋅ (α ⋅ h )i ⋅ ⎜ ⎟ + O ( h 2 ) ⎟ ⋅ ⎜ ⎜ + ⋅ ⋅ ⋅ + O ( h2 ) ⎟ − ( ) ⎜ ⎟ i ⎟ 2 i ⎜ ⎟ ⎜ ⎟ ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎢⎣⎝ ⎠ ⎝ ⎝ ∂x ⎠i ⎠
j j ⎞⎤ ⎛ ⎞ ⎛ ⎛ ∂T ⎞ j ⎛ ∂ 2T ⎞ ⎛ ∂u ⎞ a ⎡ ⎤ r h − − ⋅ ⋅ + O ( h 2 ) ⎟ ⎥ + O (τ ) = 1 − ⎜ uij − ⎣⎡1 − ri a ⎦⎤ ⋅ hi ⋅ ⎜ ⎟ + O ( h 2 ) ⎟ ⋅ ⎜ ⎜ i ⎦ i ⎜ ⎟ 2 ⎟ ⎣ ⎜ ⎟ ⎟ ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ⎠ ⎝⎜ ⎝ ∂x ⎠i ⎠ ⎦⎥ j j j ⎛ j ⎛ ∂T ⎞ j ⎞ ⎛ ∂ 2T ⎞ ⎛ ∂T ⎞ b ⎛ ∂u ⎞ j b ⎜ ui ⋅ ⎜ u r h r h α α + ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ ⋅ + O (h2 ) ⎟ ( ) ( ) ⎟ ⎜ ⎟ ⎜ ⎟ i i i ⎜ 2 ⎟ i i ⎜ ⎟ ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎠− =⎝ ri b ⋅ (α ⋅ h )i + (1 − ri a ) ⋅ hi
j j j ⎛ j ⎛ ∂T ⎞ j ⎞ ⎛ ∂ 2T ⎞ ⎛ ∂T ⎞ ⎛ ∂u ⎞ j a ⎜ ui ⋅ ⎜ u r h 1 −⎜ ⋅ (1 − ri a ) ⋅ hi ⋅ ⎜ ⎟ + O ( h 2 ) ⎟ − ⋅ − ⋅ ⋅ ⎟ ⎟ i ( i ) i ⎜ 2 ⎟ ⎜ ⎟ ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎠ +O τ = −⎝ ( ) rib ⋅ (α ⋅ h )i + (1 − ri a ) ⋅ hi
=
∂ ⎛ ∂T ⎞ ∂ ⎛ ∂T ⎜u⋅ ⎟ + O ( h, τ ) → ⎜ k ⋅ f ⋅ ∂x ⎝ ∂x ⎠i ∂x ⎝ ∂x j
⎞ ⎟ + O ( h, τ ) . ⎠
(2.284) Де ятая ра
ть (
φ
(σ , θ )
= φi + O (τ ) = Φ ij + O (τ , h ) = Φ + O (τ , h ) .
(2.264 )):
j
,
(2.285)
,
,
,
(2.262 )
1
В
© В.Е. Селе
δ T ( x, t = const ) ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.54 ) : -
-
.
лава 2 151 _______________________________________________________________________________________
∂ ⎡ ⎢ρ ⋅ f ∂t ⎢⎣
⎛ ⎛ ∂z w2 ⎞ ⎤ ∂ ⎡ w2 ⎞ ⎤ ∂ ⋅⎜ε + + ⋅ ⋅ ⋅ + f w ρ ε ( p ⋅ f ⋅ w) − ρ ⋅ f ⋅ w ⋅ g ⋅ 1 − ⎢ ⎜ ⎟⎥ ⎟⎥ = − x x ∂ ∂ ∂x 2 2 ⎠ ⎦⎥ ⎠ ⎦⎥ ⎝ ⎝ ⎣⎢ (2.286) ∂f ∂ ⎛ ∂T ⎞ − p⋅ +Q ⋅ f + ⎜k ⋅ f ⋅ ⎟ − Φ ( T , Toc ) + O (τ , h ) . ∂t ∂x ⎝ ∂x ⎠
Н
,
-
(2.262)
(2.54) -
.
(2.262)
, .
-
.
Д
(2.225) [69, 70, 96].
( . (3.27) – еч - а
еч V⋅
(ρ
j +1 L
×∑ n =1
(
j +1 L
а
j −1 L
е
я е а
я
)
а
е
я е а
(
+ ρ Lj ) ⋅ (Ym ) L + (Ym ) L − ( ρ Lj + ρ Lj −1 ) ⋅ (Ym ) L + (Ym ) L (n)
f Lj +1 ⋅
N
n =1
j +1
( ( )ρ n
- а
(
j +1 M
(n)
2 ⋅ (1 + β j ) ⋅τ j j
ρ Mj +1 ⋅
(n)
а а
( Dm ) M
j +1
а
(n)
,
) ( ( ) (Y
е
∑ ( )ρ n =1
n
L
+ ( n ) ρ Lj +1 ⋅
)
j +1 m M
n
(n)
я
j −1
j
⋅ ( n ) wMj +1 + ( n ) ρ Lj +1 ⋅ ( n ) wLj +1 ⋅
N
( Dm ) L
j +1
а
)
⋅
) − 0,25 ×
+
(n)
(n)
(Ym ) L
j +1
(Ym ) L
j +1
(2.287 ) е
)⋅
−
а
(n)
(n)
е-
s+
(Ym ) M
2 ⋅ ( n ) ΔX
j +1
= 0;
(2.287 )
яК
х
а
n
n
n
⋅ ( ) wL ⋅ ( ) f L ⋅ ( ) s = 0
(2.287 ’)
ρL ,
V , n = 1, N ,
(n)
1
0,5 ⋅ ⎡⎣ ( n ) xL +
е
N
а а
+ 0,5 ⋅ ∑ ( n ) f Lj +1 ⋅
еч
[1])1: а а
-
−ρ ⋅V − 0,5 ⋅ ∑ ( n ) ρ Mj +1 ⋅ ( n ) wMj +1 ⋅ ( n ) f Mj +1 ⋅ ( n ) s = 0; (1 + β j ) ⋅τ j n =1
ρ
- а
N
–
,
O ( h 2 ,τ )
(2.54)
–
,
( n)
xM ⎤⎦ (
.
N
n =1
(2.217 ), © В.Е. Селе
( n)
xL
. 2.11). В .Н
V= ∪ N
V ⋅ ∂ρ ∂t − 0,5 ⋅ ∑
(n)
n-
⎡⎣( ρ ⋅ w ⋅ f ) L + ( ρ ⋅ w ⋅ f ) M ⎤⎦ ⋅ ( n ) s = 0 . (2.287 ).
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
n=0
,
(2.221 )
V
:
( n)
ρ
L
К
152 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∑ ( )w N
(
еч
–
- а
ρ Lj +1 +
(n)
+ ( ) s ⋅ ( ) f Lj +1 ⋅
(n)
V⋅
(n)
(n)
n
n
=−
(n)
(
−0,5 ⋅
– V⋅
f Lj +1 ⋅ 2 (n )
еч
(ρ
j +1 L
(
(( )p n
( z1 ) L
а а
ρ Lj ⋅
wLj +1 +
ρ Lj +1 ⋅
(n)
(w )
j +1 L
−
j +1
)(
n =1
(n)
)
n
(n)
( z1 )M
j +1
- а
ρ Lj +
(n)
n
е
ρ
(n )
4
⋅
j +1 L
(n )
а
f
е
2 ⋅ (1 + β j ) ⋅τ j
n
( ( )ρ n
N
n
π ⋅ ( n ) f Lj +1
n =1
N ⎡ − ∑ ⎢0,5 ⋅ n =1 ⎣
(
(n )
m =1 n =1
n
n
(Ym ) L
j +1
−
)
(
(n)
n
(n)
ρ Mj +1 ⋅
(Ym ) M
n
(n)
(n)
fL
И (2.225)
wLj +
(n)
)+
е
wLj −1
j +1 L
n
⋅
(n )
(n)
n
е е −
я
n
)
n
wLj +1 ⋅ ( ) ΔX − n
(2.287 )
j +1 M
е
е
а
е-
)
ε Lj +1 ⋅ ( n ) s =
+
(n)
n
3
n
TLj +1 − ( )TMj +1 ( n ) j +1 ⋅ fL + n 2 ⋅ ( ) ΔX n
j +1
е
) ( ( )ε
ρ Lj +1 ⋅ ( n ) wLj +1 ⋅
)
(n)
n
wLj +1 ⋅ ( ) ΔX + QLj +1 ⋅V −
(n )
( Dm ) M
n
s;
я
n
n
⋅ ( )λLj +1 ⋅ ( ) ρ Lj +1 ⋅ ( ) wLj +1 ⋅
+ ( ) ρ Lj +1 ⋅ n
(n)
( Dm ) L
j +1
j +1
2 ⋅ ( ) ΔX
Д (2.225 ) ,
(n)
(n)
n
− ( ) wLj +1 ⋅ ( ) s +
n
k Mj +1 + ( )k Lj +1 ⋅
× ∑∑ ( ) f Lj +1 ⋅ N
j +1 M
n
⋅ ( )λLj +1 ⋅ ( ) ρ Lj +1 ⋅
4
(n)
n
( ( )w
n
n =1
N
⋅ ( ) wMj +1 +
j +1 M
)(
ρ Lj −1 ⋅
я а
n
π ⋅ ( n ) f Lj +1
n
= 0,5 ⋅ ∑ ( ) pMj +1 ⋅ ( ) f Lj +1 ⋅
(n)
я
(2.287 ’’)
− 0, 25 ⋅ ⎡⎣ ( ) ρ Lj +1 ⋅ ( ) wLj +1 + ( ) ρ Mj +1 ⋅ ( ) wMj +1 ⎤⎦ ⋅ ⎡⎣ ( ) wLj +1 + ( ) wMj +1 ⎤⎦ =
а а
n =1
×
wLj −
(n)
n
+ ρ Lj ) ⋅ ( ε Lj +1 + ε Lj ) − ( ρ Lj + ρ Lj −1 ) ⋅ (ε Lj + ε Lj −1 ) N
NS
) (
е
2 ⋅ (1 + β j ) ⋅τ j
j +1 2 L
)⋅ g ⋅
n
L
а
− ( ) pMj +1 ⋅ ( ) s −
− 0, 25 ⋅ ∑ ( ) f Lj +1 ⋅
+∑
(n)
⋅ ( ) f L ⋅ ( ) s = 0;
n
⎤
φLj +1 ⋅ ( n ) ΔX ⎥ − 0, 25 ×
(n )
)⋅(
(n)
(ε m )M
j +1
⎦
+
(n )
(ε m )L
j +1
)×
(2.287 ’)
.
(2.287) . . 2.11),
fM (
ΔX n V = f L ⋅ ( ) ΔX . (n)
(n)
(2.225 ), (2.225 ), -
.
(n )
(2.287) ,
(2.287)
-
(2.225)
[1].
К
,
,
–
.
, .
(2.54)
,
, (
© В.Е. Селе
-
. В
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
«n »
лава 2 153 _______________________________________________________________________________________
):
∂ ( ρ ⋅ ε ) ∂ ( ρ ⋅ε ⋅ w ) 1 ∂w ∂ ⎛ ∂T ⎞ Φ (T , Toc ) 3 + = −p⋅ + ⋅λ ⋅ρ ⋅ w + Q + ⎜k ⋅ + ⎟− f ∂t ∂x ∂x 4 ⋅ R ∂x ⎝ ∂x ⎠ ∂ ⎛ NS ∂Y ⎞ + ⎜ ρ ⋅ ∑ ε m ⋅ Dm ⋅ m ⎟ . ∂x ⎝ m=1 ∂x ⎠
(n)
(2.288)
(2.288)
Θ
-
,
(2.225 ). (2.225 ). Д ,
А (2.54 ) :
(2.289), w⋅ w⋅
-
∂ρ ∂ ( ρ ⋅ w ) + = 0. ∂t ∂x
(2.225 ) ( w2 2 ) ,
(2.289)
w, :
∂ ( ρ ⋅ w ) w2 ∂ρ ∂ ⎛ w2 ⎞ − ⋅ = ⎜ ρ ⋅ ⎟; ∂t 2 ∂t ∂t ⎝ 2 ⎠
∂ ( ρ ⋅ w2 ) ∂x
−
Θ. -
(n)
(2.290 )
w2 ∂ ( ρ ⋅ w ) ∂ ⎛ w2 ⎞ ⋅ = ⎜ ρ ⋅ w⋅ ⎟, ∂x ∂x ⎝ 2 2 ⎠
(2.290 )
:
∂⎛ w ⎞ ∂ ⎛ ∂p ∂z1 1 w ⎞ 3 − ⋅λ ⋅ ρ ⋅ w . ⎜ ρ ⋅ ⎟ + ⎜ ρ ⋅ w ⋅ ⎟ = −w − g ⋅ ρ ⋅ w ⋅ ∂t ⎝ ∂x ∂x 4 ⋅ R 2 ⎠ ∂x ⎝ 2 ⎠ 2
(2.288)
2
(2.291),
(2.291) :
⎛ w2 ⎞ ⎤ ∂ ⎡ w2 ⎞ ⎤ ∂ ⎡ ⎛ ⎢ρ ⋅ ⎜ ε + ⎟⎥ + ⎢ ρ ⋅ w ⋅ ⎜ ε + ⎟⎥ = 2 ⎠ ⎦ ∂x ⎣ 2 ⎠⎦ ∂t ⎣ ⎝ ⎝
∂ ( p ⋅ w) ∂z ∂ ⎛ ∂T =− − g ⋅ ρ ⋅w⋅ 1 + Q + ⎜k ⋅ ∂x ∂x ∂x ⎝ ∂x
⎞ Φ ( T , Toc ) . ⎟− f ⎠
(2.292)
Д
,
,
, (2.292),
-
. (
( n)
V.Э
(2.231 )
1
В
(2.262 )
M = i −1 ,
© В.Е. Селе
(2.288)
, sia = sib = ria = rib = 0,5 , σ = 1 , θ = 0 , f = const . y ( ев, В.В. Алеш
⎡0,5 ⋅ ⎣
( n)
xL +
(n)
)
xM ;
( n)
,
xL ⎤ 1: ⎦
(2.293)
, x, p, w
, С.Н. Прял в, 2007–2009
(
)
. .)
yij++11 = y Lj +1 .
L=i ,
154 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ (n )
(n)
(ρ + =
+ ρ Lj ) ⋅
(ε
(n)
( ρ ⋅ ε ⋅ w)L
j +1
A − ( n ) pMj +1 ⋅
j +1 L
(n)
+ ε Lj ) −
( ρ ⋅ w )L
j +1
(n)
(ρ
2 ⋅ (1 + β j ) ⋅τ j
− 0,25 ⋅
(n)
j L
+ ρ Lj −1 ) ⋅
(( ρ ⋅ w) (n)
− 0,5 ⋅
j +1
(n )
(n)
(( ρ ⋅ w )
(n )
ΔX ⋅
(ε
j L
j +1
j +1
s
+ ε Lj −1 )
) (ε
+ ( ρ ⋅ w)M ⋅
ΔX ⋅ ( n ) s L
(n )
+ ( ρ ⋅ w )M
j +1 L
(n)
+
j +1 L
+ ε Mj +1 )
1 w − w + ( n ) j +1 ⋅ ( n )λLj +1 ⋅ ( n ) ρ Lj +1 ⋅ (n) 2 ⋅ ΔX ⋅ s 4 ⋅ RL
(n)
j +1 L (n)
(n)
j +1 M
⎡ ( n ) j +1 ( n ) ⎛ ∂T ⎞ j +1 1 ⋅ kL ⋅ ⎜ ⎢ ⎟ − 0,5 ⋅ (n) ΔX ⋅ ( n ) s ⎢⎣ ⎝ ∂x ⎠ L
( ( )k n
j +1 M
)
+ ( n ) k Lj +1 ⋅
(n)
) = 0;
=
(n)
ρ Mj +1 ⋅
(n)
( Dm ) M
j +1
+ ( n ) ρ Lj +1 ⋅
(n)
( Dm ) L
2
wLj +1 + QLj +1 + 3
TLj +1 − ( n )TMj +1 ⎤ ( n )φLj +1 , ⎥− 2 ⋅ ( n ) ΔX ⋅ ( n ) s ⎥⎦ ( n ) f Lj +1
j +1
⋅
(n)
(ε m ) L
(n)
+
j +1
(ε m ) L
j +1
2
⋅
(n)
(Ym ) L
j +1
−
(n)
аще
ая а
=
(ρ
ь
2 ⋅ (1 + β j ) ⋅τ j
j +1 L
+ ρ Lj ) ⋅
⋅w
j +1 L
⎛ ⎜⎜ ⎝
(2.293),
j +1 L
+ ρ Lj ) ⋅
© В.Е. Селе
ρ j +1 − ρ Lj −1 ( wL − L ⋅ (1 + β j ) ⋅τ j 2
(2.295 )
)
j +1 2
=
wLj +1 ⋅ wLj w j ⋅ w j −1 − ( ρ Lj + ρ Lj −1 ) ⋅ L L w j +1 − wLj −1 2 2 + ( wLj +1 − wLj ) ⋅ ( ρ Lj + ρ Lj −1 ) ⋅ L . 2 ⋅ (1 + β j ) ⋅τ j (1 + β j ) ⋅τ j
(2.287 )
(ρ
⎤ ⎥. ⎥⎦
ь
+ ρ Lj ) ⋅ ( wLj +1 + wLj ) − ( ρ Lj + ρ Lj −1 ) ⋅ ( wLj + wLj −1 )
ая а
j +1
:
w2 ⎡ (0,5) w ⋅ w ⎤ ( 0,5) ⎡ ( 0,5) ⎤ ⎡ ρ ( 0,5) ⋅ w( 0,5) ⎤ t ⋅ w − ⎡ ρ (0,5) ⎤ t ⋅ ⋅ ⋅ ⎣w ⎦ t ; t + ( w − w) ⋅ ρ ⎣ ⎦ ⎣ ⎦ 2 = ⎢⎣ ρ 2 ⎥⎦
а е
(Ym ) M
2 ⋅ ( n ) ΔX
,
j +1 L
(2.294)
(n)
j +1 (n) s N S ⎡ ( n ) j +1 ( n ) j +1 ( n ) j +1 ⎛ ∂Ym ⎞ ρ ε ⋅ ⋅ ⋅ ⋅ − D ( ) ( ) ⎢ ∑ ⎜ ⎟ L m m (n) L L ΔX m=1 ⎢⎣ ⎝ ∂x ⎠ L
Д
(ρ
(2.293)
(n)
A= −
(n)
(n)
+ (n)
j +1 L
ρ Lj +1 − ( n ) ρ Lj −1 + (1 + β j ) ⋅τ j
(2.295),
wLj +1 ⋅ wLj w j ⋅ w j −1 − ( ρ Lj + ρ Lj −1 ) ⋅ L L 2 2 + (1 + β j ) ⋅τ j ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
⎞ w ⎟ f ⋅ ΔX ⎟⎠ ⎛ ( n ) j +1 2 ⎞ ⎜ ( wL ) 2 ⎟ . ⎝ ⎠
(n)
j +1 L
w = V
(n )
(n )
(n )
j +1 L (n)
(2.295 )
:
лава 2 155 _______________________________________________________________________________________
+
ρ Lj +1 ⋅
(w )
− 0,5 ⋅ ( ρ Lj +1 ⋅ wLj +1 + ρ Mj +1 ⋅ wMj +1 ) ⋅
j +1 3 L
wLj +1 ⋅ wMj +1 2 =
ΔX ⋅ s 3 1 pLj +1 − pMj +1 j +1 =− ⋅ wL − ⋅ λLj +1 ⋅ ρ Lj +1 ⋅ wLj +1 − j +1 2 ⋅ ΔX ⋅ s 4 ⋅ RL
( −
2
( z1 ) L
−
j +1
(n)
( z1 ) M
j +1
(n)
2 ⋅ ΔX ⋅ s
(2.296) (n) V:
(ρ
)⋅g ⋅ρ
j +1 L
(2.296)
⋅ wLj +1 − ( wLj +1 − wLj ) ⋅ ( ρ Lj + ρ Lj −1 ) ⋅
wLj +1 − wLj −1 . 2 ⋅ (1 + β j ) ⋅τ j
(2.294), j +1 L
(ρ
+
j +1 L
+ ρ Lj ) ⋅
( ρ ⋅ ε ⋅ w )L
j +1
+
+
+ ρ Lj ) ⋅ ( ε Lj +1 + ε Lj ) − ( ρ Lj + ρ Lj −1 ) ⋅ (ε Lj + ε Lj −1 )
ρ Lj +1 ⋅
=−
p
(w )
j +1 3 L
2
j +1 L
2 ⋅ (1 + β j ) ⋅τ j
wLj +1 ⋅ wLj w j ⋅ w j −1 − ( ρ Lj + ρ Lj −1 ) ⋅ L L 2 2 + (1 + β j ) ⋅τ j
(
)
− 0, 25 ⋅ ( ρ ⋅ w ) L + ( ρ ⋅ w ) M ⋅ ( ε Lj +1 + ε Mj +1 ) j +1
(n)
ΔX ⋅
j +1
(n )
s
− 0,5 ⋅ ( ρ Lj +1 ⋅ wLj +1 + ρ Mj +1 ⋅ wMj +1 ) ⋅
⋅w − p ⋅w 2 ⋅ ΔX ⋅ s j +1 L
+
j +1 M
j +1 M
ΔX ⋅ s
−
( z1 ) L
− ( z1 ) M
j +1
2 ⋅ ΔX ⋅ s
j +1
+
wLj +1 ⋅ wMj +1 2 =
⋅ g ⋅ ρ Lj +1 ⋅ wLj +1 +
j +1 1 ⎡ j +1 ⎛ ∂T ⎞ TLj +1 − TMj +1 ⎤ j +1 j +1 j +1 + ⋅ ⎢k L ⋅ ⎜ ⎥ + QL − ⎟ − 0,5 ⋅ ( k M + k L ) ⋅ 2 ⋅ ΔX ⋅ s ⎦⎥ ΔX ⋅ s ⎣⎢ ⎝ ∂x ⎠ L
φLj +1
−
К
f
j +1 L
− ( wLj +1 − wLj ) ⋅ ( ρ Lj + ρ Lj −1 ) ⋅
(2.297)
wLj +1 − wLj −1 + A. 2 ⋅ (1 + β j ) ⋅τ j
(2.297),
, . . (
(2.297) ). Д
– (2.287) (2.262)
,
V⋅
(ρ
j +1 L
+ρ
j L
) ⋅ (ε
j +1 L
) − ( ρ + ρ ) ⋅ (ε 2 ⋅ (1 + β ) ⋅ τ +ε
− 0,25 ⋅ ∑ ( n ) f Lj +1 ⋅ N
n =1
j L
( ( )ρ n
j −1 L
j L
j
j +1 M
= 0,5 ⋅ ∑ ( n ) pMj +1 ⋅ ( n ) f Lj +1 ⋅ N
n =1
© В.Е. Селе
(2.287 ’)
.В
ев, В.В. Алеш
j
⋅ ( n ) wMj +1 +
( ( )w n
j +1 M
(n)
V,
j L
+ε
j −1 L
)−
(2.287 ’), :
) ( ( )ε
ρ Lj +1 ⋅ ( n ) wLj +1 ⋅
)
− ( n ) wLj +1 ⋅ ( n ) s +
ri = 0,5 , a
, С.Н. Прял в, 2007–2009
n
j +1 M
+
)
ε Lj +1 ⋅ ( n ) s =
(n)
156 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
π ⋅ ( n) f Lj +1
+∑ N
n =1
( ( )k
⋅ ( n )λLj +1 ⋅ ( n ) ρ Lj +1 ⋅
4
⎡ − ∑ ⎢0,5 ⋅ n =1 ⎣ N
n
j +1 M
×
(n)
N
m =1 n =1
(Ym ) L
j +1
2⋅
−
(n)
(n)
(
(n)
(Ym ) M
j +1
ΔX
ρ Mj +1 ⋅
+∑ N
n =1
(
3
TLj +1 − ( n )TMj +1 ( n ) j +1 ⋅ fL + 2 ⋅ ( n ) ΔX (n)
(n)
,
К
wLj +1 ⋅ ( n ) ΔX + QLj +1 ⋅V −
(n)
+ ( n ) k Lj +1 ⋅
− 0,25 ⋅ ∑∑ ( n ) f Lj +1 ⋅ NS
)
(n)
( Dm ) M
j +1
+
(n)
ρ Lj +1 ⋅
(n)
⎤
φLj +1 ⋅ ( n ) ΔX ⎥ −
(n)
( Dm ) L
j +1
)⋅(
(n)
⎦
(ε m ) M
j +1
(n)
(ε m ) L
j +1
)×
⎡ j +1 ⎤ N w j +1 − wLj −1 ⋅ f L ⋅ ΔX ⎥ + ∑ ( n ) A. ⎢( wL − wLj ) ⋅ ( ρ Lj + ρ Lj −1 ) ⋅ L 2 ⋅ (1 + β j ) ⋅τ j ⎢⎣ ⎥⎦ n =1 (2.287 ) (2.262) (2.287) -
. (2.231)).
[69], .
(
+
) ,
,
-
.В
-
[97]. Н
[97]. В :
-
(
)
Э
, . . (
⎧ ∂ρ ⎪ ∂ t + υ0 ⋅ ∇ρ + ρ 0 ⋅ ∇ ⋅ υ = 0; ⎪ 1 ⎪ ∂υ ⎨ + υ0 ⋅ ∇ ⋅ υ + ⋅ ∇p = 0; ρ0 ⎪ ∂t ⎪ ∂p 2 ⎪ + υ0 ⋅ ∇p + ρ 0 ⋅ c0 ⋅ ∇ ⋅ υ = 0, ⎩ ∂t
(
υ, p, ρ –
)
(
(2.298)
)
,
-
; c0 –
(
c0 = γ ⋅ R ⋅ T ); υ0 , ρ 0 – (2.298)
И
, . -
. (2.298),
,
⎧ ∂ρ N ∂ρ ⎞ N ∂w ⎞ ⎛ ⎛ + ∑ ⎜ w0 ⋅ ⋅ Θ ⎟ + ∑ ⎜ ρ0 ⋅ ⋅ Θ ⎟ = 0; ⎪ ∂ ∂ ∂x t x ⎝ ⎠ ⎝ ⎠ = = n n 1 1 ⎪ (n ) (n) ⎪ (n) ∂w ⎞ 1 ⎪ ∂w ⎛ ⎛ ∂p ⎞ + ⎜ w0 ⋅ ⎨ ⎟ + ⋅ ⎜ ⎟ = 0, n = 1, N ; ∂x ⎠ ρ 0 ⎝ ⎝ ∂x ⎠ ⎪ ∂t ⎪ N (n) N (n) ⎪ ∂p + ∑ ⎛⎜ w0 ⋅ ∂p ⋅ Θ ⎞⎟ + ∑ ⎛⎜ ρ 0 ⋅ c02 ⋅ ∂w ⋅ Θ ⎞⎟ = 0. ∂x ⎠ n =1 ⎝ ∂x ⎠ ⎩⎪ ∂t n =1 ⎝ (n )
© В.Е. Селе
)
ев, В.В. Алеш
(n )
, С.Н. Прял в, 2007–2009
:
(2.299)
лава 2 157 _______________________________________________________________________________________
В
q = c02 ⋅ ρ − p; a1 = w0 ; a1 = υ0 .
:
(2.298) ,
(2.300)
c02
(2.299)
:
∂q + a1 ⋅ ∇q = 0; ∂t
(∇ ⋅ a ) = 0 , 1
И ,
∂q N +∑ ∂t n =1
(2.301 )
(n )
(2.301 )
∂q ⎛ ⎞ ⋅Θ ⎟ = 0. ⎜ a1 ⋅ ∂x ⎝ ⎠
:
∂q + ∇ ⋅ ( q ⋅ a1 ) = 0. ∂t V,
(2.301 ) (2.301 )1:
qLj +1 − qLj −1 N −∑ 2 ⋅τ n =1
(n)
(2.301 )
(2.301 ) -
⎛ ⎞ qLj +1 + qMj +1 a ⋅ ⋅ s ⋅ Θ ⎟ = 0. ⎜ 1 2 ⋅ ΔX ⎝ ⎠
(2.302)
(2.302) (2.302)
:
qLj +1 − ∑ N
(n)
n =1
(2.303) y
j C
= max yi i
«
[97]. a1 ⋅τ ⎛ j +1 ⎞ j +1 ⋅ s ⋅ Θ ⎟ = qLj −1. ⎜ ( qL + q M ) ⋅ ΔX ⎝ ⎠
»
,
( j
(2.303)
y
[97]): q
j +1 C
N (n ) ⎡ a1 ⋅τ = ⎢1 + 2 ⋅ ∑ ⋅s⋅Θ ΔX n =1 ⎢⎣
,
(2.304)
(2.302)
⎤ ⎥ ⋅ q ⎥⎦ −1
j −1 C
N ⎡ 0 < ⎢1 + 2 ⋅ ∑ n =1 ⎣⎢
,
(2.304)
.
⎤ a1 ⋅τ ⋅s⋅Θ ⎥ ≤1 . ΔX ⎦⎥ −1
(n)
(2.298) (2.299)
V , n = 1, N :
-
(2.299).
(n)
(n)
(n)
∂p + ∂t
Θ. (2.299), ):
(n )
∂p ⎞ ⎛ ⎜ w0 ⋅ ⎟ + ∂x ⎠ ⎝
(n )
⎛ 2 ∂w ⎞ ⎜ ρ 0 ⋅ c0 ⋅ ⎟ = 0, ∂x ⎠ ⎝
,
1
© В.Е. Селе
(2.305) (2.305) «n »
(
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
158 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∂ ( p + ρ 0 ⋅ c0 ⋅ w ) ∂ ( p + ρ 0 ⋅ c0 ⋅ w ) + ( w0 + c0 ) ⋅ = 0; ∂t ∂x
(2.306 )
∂ ( p − ρ 0 ⋅ c0 ⋅ w ) ∂ ( p − ρ 0 ⋅ c0 ⋅ w ) + ( w0 − c0 ) ⋅ = 0. ∂t ∂x
В
(2.306 )
:
r + = c0 ⋅ ρ 0 ⋅ w + p; a2 = w0 + c0 ; r − = − c0 ⋅ ρ 0 ⋅ w + p; a3 = w0 − c0 .
(2.307)
(r )
j +1
+
L
− (r+ )
2 ⋅τ
j −1 L
(2.306)
:
∂r + ∂r + ∂r − ∂r − + a2 ⋅ = 0; + a3 ⋅ = 0. ∂t ∂x ∂t ∂x
+ a2
(r ) ⋅ +
j +1
− (r+ )
2 ⋅ ΔX ⋅ s L
(r )
, j +1 M
j +1
−
= 0;
− (r− )
2 ⋅τ
L
(2.308)
(r ) ⋅
(2.308): j −1 L
+ a3
−
j +1
− (r− )
2 ⋅ ΔX ⋅ s L
j +1 M
(2.309)
(r ) +
(2.309)
j +1 L
+
:
j +1 j −1 a2 ⋅τ ⎡ + j +1 ⋅ (r ) − (r+ ) ⎤ = (r+ ) ; ⎢ ⎥ L M L ⎦ ΔX ⋅ s ⎣
s =1,
r+
j +1
a ⋅τ ⎞ ⎛ ⋅ ⎜1 + 2 ⋅ 2 ⎟ = r + ΔX ⎠ ⎝
(r ) −
,
j +1 L
j −1
;
r−
+
j +1 C
» .
,
-
):
a ⋅τ ⎞ ⎛ ⋅ ⎜1 + 2 ⋅ 3 ⎟ = r − ΔX ⎠ ⎝
(2.311)
(2.309)
j −1 C
⋅
(2.311)
a ⋅τ ⎤ ⎡ 0 < ⎢1 + 2 ⋅ 2 ≤1 ΔX ⎥⎦ ⎣ −1
,
a ⋅τ ⎤ ⎡ 0 < ⎢1 + 2 ⋅ 3 ≤1. ΔX ⎥⎦ ⎣ , (2.302), (2.309). Э −1
, (2.299).
(2.302)
(2.299). Д
(2.309)
(
⎡ ( ρ ⋅ F )( − S ) ⎤ + ρ ⋅ w ⋅ f (− R) ( 0,5) ( 0,5) ⎣ ⎦t +
© В.Е. Селе
ев, В.В. Алеш
)
. (2.3) (σ , θ ) +
x
= 0;
, С.Н. Прял в, 2007–2009
-
,
(2.262)
(
-
j +1 j −1 a3 ⋅τ ⎡ − j +1 ⋅ (r ) − (r− ) ⎤ = (r− ) . ⎢ ⎥ L M L ⎦ ΔX ⋅ s ⎣ (2.310)
« (
C
= 0. (2.309)
[70].
(2.310)
C
(2.307)
, (2.287). [2]
[2]):
(2.312 )
лава 2 159 _______________________________________________________________________________________
(
⎡( ρ ⋅ F )( − S ) ⋅ (Ym )( − S ) ⎤ + ρ ⋅ w ⋅ f ⋅ (Ym ) (− R) ( 0,5) ( 0,5) (− R) ⎣ ⎦t +
a − ⎡( ρ ⋅ f ⋅ Dm )( − R ) ⋅ δ (Ym ) ⎤ ⎣ ⎦x
(σ ,θ ) +
(
= 0,
− g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x ) ⎤⎦
(
⎡ ( ρ ⋅ F )( − S ) ⋅ ε ( − S ) ⎤ + ρ ⋅ w ⋅ f ⋅ ε (− R) ( 0,5) ( 0,5) ( − R ) ⎣ ⎦t
(
+
= − p( − R ) ⋅ w( 0,5) ⋅ f ( 0,5) −p
(σ , θ )
⋅ ⎡⎣ F
)
(σ , θ ) +
x
⎦ t + (Q ⋅ F )
(− S ) ⎤
(σ , θ ) +
x
−
m = 1, N S − 1;
⎡ ( ρ ⋅ F )( − S ) ⋅ w( − S ) ⎤ + ρ ⋅ w ⋅ f ⋅ w (− R) ( 0,5) ( 0,5) (− R) ⎣ ⎦t +
)
)
(σ , θ ) +
x
(σ , θ )
)
(σ , θ )
−
YN = 1 − S
(σ , θ )
+
π
4
⋅ (λ ⋅ ρ ⋅ w ⋅ r )
(σ , θ )
(σ , θ )
NS
m=1
(σ , θ )
⋅ w(
σ,θ)
(σ , θ )
−
(2.312 )
;
⎛ ⎛ w2 ⎞ w2 ⎞ + Kt ⎜ ρ ⋅ F ⋅ ⎟ + K x ⎜ ρ ⋅ w ⋅ f ⋅ ⎟ 2 ⎠ 2 ⎠ ⎝ ⎝
+ ⎡ ( k ⋅ f )( − R ) ⋅ δ T a ⎤ ⎣ ⎦x
a + ∑ ⎡( ρ ⋅ Dm ⋅ f )( − R ) ⋅ ( ε m )( − R ) ⋅ δ (Ym ) ⎤ ⎣ ⎦x m =1
(2.312 )
= − ( B − ⋅ γ − ⋅ px + B + ⋅ γ + ⋅ px )
− g ⋅ ⎡⎣ ρ ⋅ ( B − ⋅ γ − ⋅ ( z1 ) x + B + ⋅ γ + ⋅ ( z1 ) x ) ⎤⎦ +
x
∑ Ym ;
N S −1
+
−φ(
σ,θ)
(σ , θ )
(σ , θ )
=
⋅ w(σ , θ ) −
+
;
+
ε m = ε m ({S ме и }) , m = 1, N S ; T1 = T2 = … = TN ; p = p ({S ме и }) ; ε = ε ({S ме и }) ;
(2.312 ) (2.312 )
S
k = k ({S ме и }) ;
(2.312 )
Dm = Dm ({S ме и }) , m = 1, N S .
(2.312 ) (2.312 ) (2.262)
(2.312)
, (2.312).
В ∂(ρ ⋅ f ) ∂t
ρ⋅ f ⋅
+
∂(ρ ⋅ w⋅ f ) ∂x
S
ρ⋅ f ⋅
∑Y
(2.54):
= 0;
(2.313 )
N S −1 m =1
m
⎞ ⎟ = 0; ⎠
m = 1, N S − 1 ;
(2.313 )
;
∂w ∂w + ρ ⋅ w⋅ f ⋅ =−f ∂t ∂x
© В.Е. Селе
. Д
,
∂ Ym ∂Y ∂Y ∂ ⎛ + ρ ⋅ w ⋅ f ⋅ m − ⎜ ρ ⋅ f ⋅ Dm ⋅ m ∂t ∂ x ∂x ⎝ ∂x
YN = 1 −
-
ев, В.В. Алеш
⎛∂p ∂z ⎞ π ⋅⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x
, С.Н. Прял в, 2007–2009
(2.313 )
160 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ρ⋅ f ⋅
∂( w ⋅ f ) ∂ε ∂ε ∂f π 3 + ρ ⋅ w⋅ f ⋅ = −p⋅ − p⋅ + Q⋅ f + ⋅λ ⋅ ρ ⋅ w ⋅ R+ 4 ∂t ∂x ∂x ∂t
NS ∂Ym ∂ ⎛ ∂T ⎞ ∂ ⎛ + ⎜k ⋅ f ⋅ ⎟ − Φ (Toc , T ) + ⎜⎜ ρ ⋅ f ⋅ ∑ ε m ⋅ Dm ⋅ ∂x ⎝ ∂x ⎠ ∂x ⎝ ∂x m =1
ε m = ε m ({S ме и } ) ,
⎞ ⎟⎟ ; ⎠
(2.313 )
m = 1, N S ; ε = ε ({S ме и } ) ; T1 = T2 = … = TN S ;
p = p ({S ме и } ) ; k = k ({S ме и } ) ; Dm = Dm ({S ме и } ) , m = 1, N S .
(2.313 ) (2.313 ) :
ρi j +1 ⋅ Fi j +1 − ρi j −1 ⋅ Fi j −1 ρi j++11 ⋅ wij++11 ⋅ f i +j 1+1 − ρi j−+11 ⋅ wij−+11 ⋅ f i −j1+1 + = 0; (1 + α i ) ⋅ hi (1 + β j ) ⋅τ j
(ρ
j +1
i
(
⋅ Fi j +1 + ρi j ⋅ Fi j ) ⋅ (Ym )i
j +1
)
(
− (Ym )i + ( ρi j ⋅ Fi j + ρi j −1 ⋅ Fi j −1 ) ⋅ (Ym )i − (Ym )i 2 ⋅ (1 + β j ) ⋅τ j j
(
)
j −1
j
⎡ ( ρ j +1 ⋅ w j +1 ⋅ f j +1 + ρ j +1 ⋅ w j +1 ⋅ f j +1 ) ⋅ (Y ) j +1 − (Y ) j +1 i +1 i +1 i +1 i i i m i +1 m i +⎢ + ⎢ 2 ⋅ (1 + α i ) ⋅ hi ⎢⎣ +
(ρ
j +1 i
(
⋅ wij +1 ⋅ f i j +1 + ρi j−+11 ⋅ wij−+11 ⋅ fi −j1+1 ) ⋅ (Ym )i 2 ⋅ (1 + α i ) ⋅ hi
(
j +1
)
j +1 − (Ym )i −1 ⎤ ⎥− ⎥ ⎥⎦
)(
)+
(2.314 )
)
⎡ ρ j +1 ⋅ f j +1 ⋅ ( D ) j +1 + ρ j +1 ⋅ f j +1 ⋅ ( D ) j +1 ⋅ (Y ) j +1 − (Y ) j +1 i +1 i +1 m i +1 i i m i m i +1 m i −⎢ − 2 ⎢ (1 + α i ) ⋅ α i ⋅ hi ⎢⎣
(ρ −
j +1 i
⋅ f i j +1 ⋅ ( Dm )i
j +1
m = 1, N S − 1 ,
(ρ
j +1 i
( ) YN
S
)(
+ ρi j−+11 ⋅ f i −j 1+1 ⋅ ( Dm )i −1 ⋅ (Ym )i
(1 + α i ) ⋅ h
j +1
2 i
j +1 i
= 1−
∑ (Y )
N S −1 m =1
j +1 m i
j +1
⋅ Fi j +1 + ρi j ⋅ Fi j ) ⋅ ( wij +1 − wij ) + ( ρi j ⋅ Fi j + ρi j −1 ⋅ Fi j −1 ) ⋅ ( wij − wij −1 )
j +1
i
2 ⋅ (1 + β j ) ⋅τ j
⋅ wij +1 ⋅ fi j +1 + ρi j−+11 ⋅ wij−+11 ⋅ fi −j1+1 ) ⋅ ( wij +1 − wij−+11 ) ⎤ ⎥= 2 ⋅ (1 + α i ) ⋅ hi ⎥⎦
© В.Е. Селе
ев, В.В. Алеш
(2.314 )
;
⎡ ( ρi j++11 ⋅ wij++11 ⋅ fi +j 1+1 + ρi j +1 ⋅ wij +1 ⋅ f i j +1 ) ⋅ ( wij++11 − wij +1 ) +⎢ + 2 ⋅ (1 + α i ) ⋅ hi ⎢⎣
(ρ +
)
j +1 − (Ym )i −1 ⎤ ⎥, ⎥ ⎥⎦
, С.Н. Прял в, 2007–2009
+
лава 2 161 _______________________________________________________________________________________
=−
(B ) −
− g ⋅ ρi
(ρ
j +1
j +1 i
⋅ ⎡⎣ pij +1 − pij−+11 ⎤⎦ + ( B + )
(B ) ⋅ −
(1 + α i ) ⋅ hi
j +1 i
j +1 i
⋅ ⎡⎣ pij++11 − pij +1 ⎤⎦
⋅ ⎡⎣ ( z1 )ij +1 − ( z1 )ij−+11 ⎤⎦ + ( B + )
(1 + αi ) ⋅ hi
j +1 i
−
π 4
⋅ λi j +1 ⋅ ρi j +1 ⋅ wij +1 ⋅ wij +1 ⋅ ri j +1 −
⋅ ⎡⎣ ( z1 )ij++11 − ( z1 )ij +1 ⎤⎦
(2.314 )
;
⋅ Fi j +1 + ρi j ⋅ Fi j ) ⋅ ( ε i j +1 − ε i j ) + ( ρi j ⋅ Fi j + ρi j −1 ⋅ Fi j −1 ) ⋅ ( ε i j − ε i j −1 )
j +1
2 ⋅ (1 + β j ) ⋅τ j
i
⎡ ( ρi j++11 ⋅ wij++11 ⋅ f i +j 1+1 + ρi j +1 ⋅ wij +1 ⋅ fi j +1 ) ⋅ ( ε i j++11 − ε i j +1 ) +⎢ + 2 ⋅ (1 + α i ) ⋅ hi ⎢⎣
(ρ +
j +1
i
+
⋅ wij +1 ⋅ f i j +1 + ρi j−+11 ⋅ wij−+11 ⋅ f i −j1+1 ) ⋅ ( ε i j +1 − ε i j−+11 ) ⎤ ⎥= 2 ⋅ (1 + α i ) ⋅ hi ⎥⎦
⎡ p j +1 ⋅ w j +1 ⋅ f j +1 − pij−+11 ⋅ wij−+11 ⋅ fi −j1+1 = − ⎢ i +1 i +1 i +1 − (1 + α i ) ⋅ hi ⎢⎣
− wi +
j +1
π
⋅
(B ) −
j +1
i
⋅ ⎣⎡ pij +1 − pij−+11 ⎦⎤ + ( B + )
(1 + α i ) ⋅ hi
j +1
i
j +1 j −1 ⋅ ⎣⎡ pij++11 − pij +1 ⎦⎤ ⎤ ⎥ − p j ⋅ Fi − Fi + i ⎥ (1 + β j ) ⋅τ j ⎦
⋅ λi j +1 ⋅ ρi j +1 ⋅ wij +1 ⋅ ri j +1 + Qi j +1 ⋅ Fi j +1 − φi j + 3
⎡ ( ki j++11 ⋅ f i +j 1+1 + ki j +1 ⋅ f i j +1 ) ⋅ (Ti +j 1+1 − Ti j +1 ) − α i ⋅ ( ki j +1 ⋅ f i j +1 + ki j−+11 ⋅ f i −j1+1 ) ⋅ (Ti j +1 − Ti −j 1+1 ) ⎤ ⎥+ +⎢ (1 + α i ) ⋅ α i ⋅ hi2 ⎥⎦ ⎣⎢ 4
(
)
⎡ NS j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 ⎢ ∑ ⎡⎣ ρi +1 ⋅ f i +1 ⋅ ( ε m )i +1 ⋅ ( Dm )i +1 + ρi ⋅ fi ⋅ ( ε m )i ⋅ ( Dm )i ⎤⎦ ⋅ (Ym )i +1 − (Ym )i m = 1 +⎢ − ⎢ (1 + α i ) ⋅ α i ⋅ hi2 ⎢ ⎣
∑ ⎡⎣ ρ
j +1
NS
− m =1
( ε m )i
j +1
i
⋅ f i j +1 ⋅ ( ε m )i
j +1
(
= ε m {S ме и }i
( = ε ({S = k ({S
pij +1 = p {S ме и }i
ε i j +1
ki j +1
© В.Е. Селе
); ); );
j +1
}
j +1 ме и i
}
j +1 ме и i
j +1
⋅ ( Dm )i
),
ев, В.В. Алеш
j +1
(
+ ρi j−+11 ⋅ f i −j 1+1 ⋅ ( ε m )i −1 ⋅ ( Dm )i −1 ⎤ ⋅ (Ym )i ⎦
(1 + α i ) ⋅ h
j +1
2 i
m = 1, N S ;
(T1 )i
j +1
= (T2 )i
j +1
j +1
( )
= … = TN S
j +1
j +1 i
)
j +1 ⎤ − (Ym )i −1 ⎥ ⎥; ⎥ ⎥ ⎦
= Ti j +1 ;
(2.314 )
(2.314 )
(2.314 ) (2.314 ) (2.314 ) , С.Н. Прял в, 2007–2009
162 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
( Dm )i
j +1
fi −j1+1 + 4 ⋅
Fi j +1 =
(B )
j +1
−
(B ) +
({S
= Dm
i j +1
i
fi −j1+1 ⋅ f i j +1 + 7 ⋅ f i j +1 ⋅ (1 + α i ) + 4 ⋅ α i ⋅ f i +j 1+1 ⋅ f i j +1 + α i ⋅ fi +j 1+1
е
е
f i j +1 ⋅ (1 + α i ) +
(
)
(0,5)
ая а
(
⋅ (Ym )t ⎤ ⎦⎥
m = 1, N S − 1 ,
(
⎡ ρ ⋅F ⎢⎣
(
)
(0,5)
(0,5)
ь:
)
(0,5)
(
S
(0,5)
⎤ = 0; ⎥⎦ x
)
+ ⎡ ρ ⋅ w⋅ f ⎣⎢
YN = 1 −
⋅ wt ⎤ ⎥⎦
f i +j 1+1 ⋅ α i
4 ⋅ (1 + α i ) ⋅ π
⎡ ρ ⋅ F )(0,5) ⎤ + ⎡ ρ ⋅ w ⋅ f ⎣( ⎦ t ⎢⎣
⎡ ρ⋅F ⎣⎢
(
∑Y
N S −1 m =1
+ ⎡ ρ ⋅ w⋅ f ⎢⎣
)
m
(0,5)
)
(0,5)
(
⋅εt ⎤ ⎦⎥
(0,5)
= − ⎜⎛ ⎡ w ⋅ p ⋅ f ⎝ ⎣⎢
+
π
F=
) )
(2.314 )
)
(
+ ⎡ ρ ⋅ w⋅ f ⎣⎢
(0,5)
(
)
)
( )
x
(
⎤ − ⎡ ρ ⋅ f ⋅ Dm ⎦⎥ (0,5) ⎣⎢
(
(0,5)
( )
⋅ Ym
x
⎤ = 0; ⎦⎥ x
(2.315 )
(0,5)
(
)
(+1) ⋅ wx ⎤ = − γ − ⋅ B − ⋅ px + γ + ⋅ B + ⋅ px − ⎥⎦ (0,5)
(0,5)
π
4
⋅λ ⋅ ρ ⋅ w⋅ w ⋅ r;
= (+1) ⋅ ε x ⎤ ⎦⎥ (0,5)
)
)
(
⋅ Tx ⎤ − φ + ∑ ⎡ ρ ⋅ f ⋅ ε m ⋅ Dm ⎥⎦ x ⎢ (0,5) m =1 ⎣ NS
f ( −1) ⋅ f + 7 ⋅ f ⋅ (1 + α ) + 4 ⋅ α ⋅ 12 ⋅ (1 + α )
({ }) ε = ε ({S }) ; k = k ({S }) ; ( D ) = D ({S }) , m = 1, N ; p = p S ме и ;
)
(2.315 )
(2.315 )
⎤ − w ⋅ γ − ⋅ B − ⋅ p + γ + ⋅ B + ⋅ p ⎞ − p ⋅ ⎡ F (0,5) ⎤ + x x ⎟ ⎣ ⎦t ⎦⎥ x ⎠
3
(
(2.314 )
.
;
⋅λ ⋅ ρ ⋅ w ⋅r + Q⋅ F + ⎡ k ⋅ f ⎢⎣ 4 f ( −1) + 4 ⋅
(2.314 )
S
)
(0,5)
f ( +1) ⋅ f + α ⋅ f ( +1)
ε m = ε m {S ме и } , m = 1, N S ; T1 = T2 = … = TN = T ;
m
;
(2.314 )
(+1) ⋅ Ym
− g ⋅ ρ ⋅ ⎡⎣γ − ⋅ B − ⋅ ( z1 ) x + γ + ⋅ B + ⋅ ( z1 ) x ⎤⎦ −
⎡ ρ⋅F ⎣⎢
(2.314 )
S
1 ⋅ fi −j1+1 + 4 ⋅ fi −j1+1 ⋅ fi j +1 + 7 ⋅ f i j +1 ; 12 1 = ⋅ f i +j 1+1 + 4 ⋅ f i +j 1+1 ⋅ f i j +1 + 7 ⋅ fi j +1 ; 12 fi −j1+1 + 3 ⋅
ri j +1 =
) , m = 1, N ;
12 ⋅ (1 + α i )
( (
=
}
j +1 ме и i
;
( )
⋅ Ym
x
⎤ ; ⎥⎦ x
(2.315 )
(2.315 ) (2.315 ) (2.315 )
ме и
(2.315 )
ме и
(2.315 )
m
© В.Е. Селе
ме и
ев, В.В. Алеш
S
, С.Н. Прял в, 2007–2009
(2.315 )
лава 2 163 _______________________________________________________________________________________
( (
) )
1 ⋅ f ( −1) + 4 ⋅ f ( −1) ⋅ f + 7 ⋅ f ; 12 1 B + = ⋅ f ( +1) + 4 ⋅ f ( +1) ⋅ f + 7 ⋅ f ; 12 B− =
f ( −1) + 3 ⋅ f ⋅ (1 + α ) +
r=
4 ⋅ (1 + α ) ⋅ π
r a = r b = 0,5.
f ( +1) ⋅ α
(2.315 ) (2.315 ) (2.315 )
;
(2.315 ) 1
(2.314)
.Н
,
, -
. В
, ,
, . 2.5.1. П и е ы а но
ных хе
овышенно о о я
В
аа
о
и а ии
, . .Д
-
(
)
-
(2.36). Н ∂ (ρ ⋅ f ) ∂t
ρ⋅ f ⋅
+
∂ (ρ ⋅ w⋅ f ) ∂x
: = 0;
∂w ∂w + ρ ⋅w⋅ f ⋅ =−f ∂t ∂x
(2.316 ) ⎛∂p ∂z ⎞ π ⋅⎜ + g ⋅ ρ ⋅ 1 ⎟ − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R; ∂x ⎠ 4 ⎝ ∂x
∂( w ⋅ f ) π ∂ε ∂ε ∂f 3 + ρ ⋅ w⋅ f ⋅ = −p⋅ − p ⋅ + Q ⋅ f + ⋅λ ⋅ ρ ⋅ w ⋅ R + ∂t ∂x ∂x ∂t 4 ∂ ( k ⋅ f ) ∂T ∂ 2T + ⋅ + k ⋅ f ⋅ 2 − Φ ( Toc , T ) ; ∂x ∂x ∂x
ρ⋅ f ⋅
p = p (ρ,T );
(2.316 )
(2.316 )
(2.316 )
ε = ε ( p, T ) ;
(2.316 )
k = k ( p, T ) .
(2.316 ) y( x)
.
i
1
,
(
. © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
. )
-
164 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ ∂y ⎞ ⎜ ⎟ ⎝ ∂x ⎠i
j≤i.
y( x)
,
(
yi −1 = yi −
∂y ∂2 y h2 ∂3 y h3 ⋅ hi + 2 ⋅ i − 3 ⋅ i + O ( h 4 ) ; ∂x ∂x 2! ∂x 3!
∂y ∂2 y ( h + h ) ∂3 y ( h + h ) = yi − ⋅ ( hi + hi −1 ) + 2 ⋅ i i −1 − 3 ⋅ i i −1 + O ( h 4 ) ; ∂x ∂x ∂x 2! 3! 2
yi − 2
i ):
3
∂y ∂2 y ( h + h + h ) ∂3 y ( h + h + h ) = yi − ⋅ ( hi + hi −1 + hi − 2 ) + 2 ⋅ i i −1 i − 2 − 3 ⋅ i i −1 i − 2 + O ( h 4 ) . ∂x ∂x ∂x 2! 3! (2.317) 2
yi − 3
O (h4 ) ,
3
(2.317)
∂ y . ∂x 3
∂y ∂ y , ∂x ∂x 2
(2.317), (2.317) . . В « DER » (derivation). К
∂2 y ∂x 2
∂y ∂x
.
-
. ,
Kx . К
x, Kx1 Kx 2
Kx = Kx1 + Kx 2 + 1.
.В
: (2.318)
« APPR » (approximation). В
-
APPR = Kx − DER.
:
(2.319)
,
, ,
{UPSTREAM , Kx} . А y ( x)
{i − Kx1, i + Kx 2}
:
xΔ
( DER = )
(2.320 )
( yi ) ,
(2.320 ) .
(2.320) : ев, В.В. Алеш
-
( yi )
{UPSTREAM , Kx}
© В.Е. Селе
DER
: xΔ
< ... > –
{i − Kx1, i + Kx 2} .
x
xi
( DER = )
,
-
3
2
, С.Н. Прял в, 2007–2009
лава 2 165 _______________________________________________________________________________________ {− Kx1,
Kx 2}
xΔ
( DER = )
,
:
{UPSTREAM , Kx}
xΔ
( DER = )
Н
( y)
(2.321 )
( y ).
∂ρ ∂x x i
,
{−1,1}
xΔ
( DER =1)
xi −1 , xi
(ρ ) .
(2.321 ) ,
xi +1 ,
В
APPR = Kx − DER = 3 − 1 = 2 . В
,
{i −1, i +1}
xΔ
( DER =1)
:
Kx = 3 .
-
( ρi )
(2.319),
(2.317),
.
-
, .
( hi + hi −1 )
[70]. hi2
∂y ∂x
6(
,
-
2
, ). Д 1,5÷2
. -
.
.Д
Δ yi(1) − 0,5 =
yi − yi −1 ; hi
Δ yi(1) −1,5 =
:
yi −1 − yi − 2 , hi −1
(2.322) .И
, (2.317),
( Δ yi(1) − 0,5 =
Δyi(1) −1,5 =
И
(2.323)
h ∂2 y ∂y − i ⋅ 2 + O ( h2 ) ; 2 ∂x ∂x
ев, В.В. Алеш
(2.323 )
∂y 2 ⋅ hi + hi −1 ∂ 2 y − ⋅ 2 + O ( h2 ) . ∂x ∂x 2
,
∂2 y , ∂x 2
⎛ hi −1 ⎞ (1) Δ yi(1) ⎟ − Δ yi −1,5 − 0,5 ⋅ ⎜ 2 + h ∂y i ⎝ ⎠ = + O ( h2 ) ∂x ⎛ hi −1 ⎞ ⎜1 + ⎟ hi ⎠ ⎝ © В.Е. Селе
xi ):
-
, С.Н. Прял в, 2007–2009
(2.323 ) :
(2.324 )
166 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∂y = ∂x
Δ yi(1) −1,5 ⋅
hi − Δ yi(1) − 0,5 2 ⋅ hi + hi −1 + O ( h2 ) . ⎛ ⎞ hi − 1⎟ ⎜ ⋅ + h h 2 i i −1 ⎝ ⎠
(2.324 )
Д
: ∂y = ∂x
∂y = ∂x
⎛ hi + 2 ⎞ (1) Δ yi(1) ⎟ − Δ yi +1,5 + 0,5 ⋅ ⎜ 2 + h i +1 ⎠ ⎝ + O ( h2 ) ⎛ hi + 2 ⎞ ⎜1 + ⎟ ⎝ hi +1 ⎠
Δ yi(1) +1,5 ⋅
hi +1 − Δ yi(1) + 0,5 2 ⋅ hi +1 + hi + 2 + O ( h2 ) . ⎛ ⎞ hi +1 − 1⎟ ⎜ h 2 ⋅ i +1 + hi + 2 ⎝ ⎠
, (2.325),
(2.324)
(2.325 )
∂y = ∂x
∂y = ∂x
( 2 ⋅ hi )
xi −1 , xi
xi +1 ,
:
(1) Δ yi(1) + 0,5 + Δ yi − 0,5 ⋅
⎛ hi +1 ⎞ ⎜1 + ⎟ hi ⎠ ⎝
hi +1 hi
+ O (h2 )
(2.326 )
Δ yi(1) + 0,5 ⋅
hi + Δ yi(1) − 0,5 hi +1 + O ( h2 ). ⎛ hi ⎞ + 1⎟ ⎜ ⎝ hi +1 ⎠
αi =
.
(2.325 )
hi +1 , i = 1, N ( hi
(2.326 )
2 ⋅ hi = hi + hi , hi hi , hi +1 2 ⋅ hi + hi −1
,
. .) -
.
,
.
1,5÷2
∂y ∂x . .
Δy
(1) i + 0,5
, i = 1, N ,
, (2.324), (2.325)
,
.В © В.Е. Селе
4.
ев, В.В. Алеш
(2.326)
,
, (2.317) , С.Н. Прял в, 2007–2009
3
лава 2 167 _______________________________________________________________________________________
∂2 y ∂3 y , ∂x 2 ∂x 3
yi .
, .
yi
Н
(2.316)
{ j +1− Kt1,
tΔ
j +1}
( DER =1)
(ρ
ρ i j +1 ⋅ f i j +1 ⋅
π 4
tΔ
( DER =1)
pi
xΔ
( DER =1)
(w ) + ρ j +1
i
{i − Kx1, i + Kx 2}
xΔ
( DER =1)
{ j +1− Kt1,
tΔ
= − pij +1 ⋅
j +1}
( DER =1)
j +1 i
(p )− f j +1
i
(ε ) + ρ
{i − Kx1, i + Kx 2}
xΔ
( DER =1)
{i − Kx1, i + Kx 2}
xΔ
( DER =1)
(k
j +1 i
= p ( ρ i , Ti
j +1
j +1}
{i − Kx1, i + Kx 2}
j +1 i
j +1
j +1
(w i
j +1
i
⋅ f i j +1 ) ⋅
j +1
j +1
⋅ g ⋅ ρ i j +1 ⋅
⋅ f i j +1 ⋅ wij +1 ⋅
⋅ f i j +1 ) − pij +1 ⋅ i
{ i − Kx1, i + Kx 2}
);
xΔ
( DER =1)
{i − Kx1, i + Kx 2}
xΔ
( DER =1)
{i − Kx1, i + Kx 2}
xΔ
( DER =1)
{i − Kx1, i + Kx 2}
{ j +1− Kt1,
j +1
tΔ
xΔ
( DER =1) j +1}
( DER =1)
(T ) i
:
⋅ f i j +1 ⋅ wij +1 ) = 0;
⋅ f i j +1 ⋅ wij +1 ⋅
j +1 i
(ρ
⋅ λi j +1 ⋅ ρ i j +1 ⋅ wij +1 ⋅ wij +1 ⋅ Ri j +1 ;
ρi j +1 ⋅ f i j +1 ⋅
+
⋅ f i j +1 ) +
{ j +1− Kt1,
= − f i j +1 ⋅ −
j +1 i
-
(2.327 )
(w ) = j +1
i
(( z ) ) − j +1 1 i
(ε ) =
(2.327 )
j +1
( f )+Q i
j +1
i
-
j +1
i
⋅ f i j +1 +
(
π
)
⋅ λi j +1 ⋅ ρ i j +1 ⋅ wij +1 ⋅ Ri j +1 + 3
4 {i − Kx1, i + Kx 2} + ki j +1 ⋅ f i j +1 ⋅ xΔ Ti j +1 − φi j +1 ; ( DER = 2 )
(2.327 ) (2.327 )
ε i j +1 = ε ( pij +1 , Ti j +1 ) ;
(2.327 )
ki j +1 = k ( pij +1 , Ti j +1 ) ,
(2.327 )
Kt1 –
, t.
(2.327) (2.327 ),
, ,
Kx 2
Kx1 + Kx 2 ≥ Kx1 + Kx 2 + 1.
Kx1
: (2.328)
{UPSTREAМ , (2.327),
К
(2.327)
Kx} .
.
, , . .
© В.Е. Селе
ев, В.В. Алеш
,
, С.Н. Прял в, 2007–2009
-
168 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
, . К
(2.327)
.И
.Н.
:
;
-
;
.
Д
.
[69, 70]. , . . А
, -
.
,
,
-
. Д
, .В
-
Н
. 2.5.2.
о оении олно ью он е ва ивных овышенно о о я а а о и а ии
ла н- хе
– (
.,
,
xL ≤ x ≤ xR
[98, 329]).
dy = p ( x, y ( x ) ) , dx
p ( x, y ( x ) ) –
(2.329)
.
(2.329)
(
)
YL –
(2.330)
. (
hi = xi − xi −1 – © В.Е. Селе
,
y ( x L ) = YL ,
:
i-
( xL )
.Д
,
-
, i = 0, N − 1 ; N –
ев, В.В. Алеш
i
[63]) , i = 1, N − 1 (
, С.Н. Прял в, 2007–2009
Σ h = {xi } ,
. 2.15).
xi –
;
лава 2 169 _______________________________________________________________________________________
и . 2.15. ча
еч
Д
е
-ра
(2.329) (2.329)
-
:
⎛ dy ⎞ ⎜ ⎟ = p ( xi , y ( xi ) ) . ⎝ dx ⎠i
(2.331)
В
dy dx
:
{Spline , }
xΔ
( DER = )
{Spline, < ... >}
( y),
(2.332) ,
< ... > ( DER =< ... > ) ,
. 2.5.1,
1).
(
(2.332)
{Spline , }
xΔ
( DER =1)
,
:
{Spline , }
xΔ
( DER =1)
(2.333 )
( y ) = p ( x, y ( x ) ) .
(2.333 )
, K ≥2.
. i = 1, N − 1 ,
:
( yi ) = p ( xi , y ( xi ) )
Δxi −0,5 = [ xi −1 , xi ]
Д
-
(2.331)
y ( x) .
Δxi −0,5 ,
:
yi − 0,5 ( x ) = ∑ ak , ( i − 0,5) ⋅ ( x − xi ) , K
k
k =0
(2.334)
, k = 0, K , i = 1, N − 1 . К
ak ,( i − 0,5) –
(2.329)
(1) yi(1) − 0,5 ( xi −1 ) = p ( xi −1 , yi − 0,5 ( xi −1 ) ) , yi − 0,5 ( xi ) = p ( xi , yi − 0,5 ( xi ) ) , i = 1, N − 1; (2.335)
:
) yi(−m0,5 ( xi ) = yi(+m0,5) ( xi ) ,
) yi(−m0,5 ( x) – m -
В © В.Е. Селе
m = 0, 2,3, ..., K − 1,
(2.335, 2.336) ев, В.В. Алеш
i = 1, N − 2,
yi − 0,5 ( x ) .
[( N − 1) ⋅ ( K + 1)]
, С.Н. Прял в, 2007–2009
(2.336) ( ( N − 1) –
-
170 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
; ( K + 1) –
⎡⎣ 2 ⋅ ( N − 1) + ( K − 1) ⋅ ( N − 2 ) ⎤⎦
. .
( K − 1) -
( K − 1) . )
.
-
(2.335, 2.336) . y0,5 ( x0 ) = YL .
( K − 2)
Д
)
(2.337) -
, K = ( int( K / 2) − 1)
:
,
(2.330):
( K − K − 1)
( int(...) – :
y N( m−)1,5 ( x N −1 ) = 0, m = ( K − K + 1) , K ;
(2.338)
(m) y0,5 ( x0 ) = 0, m = K + 2, K .
(2.339) -
[97, 99], ,
, а
е а
хе а
. .
-
. П и 1. Д (2.329),
(
(2.329)
xi + 0,5
. 2.16)
Δxi = ⎣⎡ xi − 0,5 , xi + 0,5 ⎦⎤ :
, [97, 99]. Д = 0,5 ⋅ ( xi + xi +1 )
-
yi + 0,5 − yi − 0,5 = pi ⋅ Δxi yi + 0,5 − yi − 0,5
= pi ,
Δxi
Δxi
yi + 0,5 –
(2.341) xi + 0,5 ; pi –
y
p.
pi
и . 2.16. ча
еч
Д
xi : pi = pi .
е
-ра
yi − 0,5
yi + 0,5
: © В.Е. Селе
(2.340)
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
в
а ел
е
Δxi
-
лава 2 171 _______________________________________________________________________________________
yi − 0,5 = 0,5 ⋅ ( yi + yi −1 )
,
y ( x)
yi + 0,5 =
yi ⋅ yi +1 − 0,5 ⋅ ( yi + yi −1 ) Δxi
{ yi }
O ( hk ) –
(2.343) . В
-
. (2.344 )
⎛ ⎞ h ⎛ ∂y ⎞ ⎛ ∂y ⎞ yi ⋅ ⎜ yi + hi +1 ⋅ ⎜ ⎟ + O ( h 2 ) ⎟ = yi ⋅ 1 + i +1 ⋅ ⎜ ⎟ + O ( h 2 ) = yi ⎝ ∂x ⎠i ⎝ ∂x ⎠i ⎝ ⎠
⎛ 1 h ⎛ ∂y ⎞ ⎞ 1 ⎛ ∂y ⎞ = yi ⋅ ⎜ 1 + ⋅ i +1 ⋅ ⎜ ⎟ + O ( h 2 ) ⎟ = yi + ⋅ hi +1 ⋅ ⎜ ⎟ + O ( h 2 ) , 2 ⎝ ∂x ⎠i ⎝ 2 yi ⎝ ∂x ⎠i ⎠
k-
(2.343)
(2.342)
= pi
(2.329).
⎛ ∂y ⎞ yi −1 = yi − hi ⋅ ⎜ ⎟ + O ( h 2 ) ; ⎝ ∂x ⎠i
yi ⋅ yi +1 =
yi ⋅ yi +1 .
(2.344 )
h.
(2.344)
Δxi = 0, 5 ⋅ ( hi + hi +1 ) ,
,
(2.345)
⎛ ∂y ⎞ ⎜ ⎟ = pi + O ( h ) , ⎝ ∂x ⎠i
:
(2.346)
. . Д
–
(2.342) Δxi
yi + 0,5
Δxi +1 :
« ,
δ yi + 0,5 = 0,5 ⋅ ( yi +1 + yi ) − yi ⋅ yi +1 . Д y
y,
,
δ yi + 0,5 . И
⎡⎣ xi − 0,5 , xi +1,5 ⎤⎦ y :
(y
i +1,5
» -
)
101]).
0,5 ⋅ ( yi +1 + yi )
yi ⋅ yi +1
.
(
, -
− yi − 0,5 ) ,
y ( p ( x, y ( x ) ) ,
,
.
y
dy dx
.
⎡ ⎤ ⎡ ⎤ ⎣ yi + 2 ⋅ yi +1 − 0,5 ⋅ ( yi +1 + yi ) ⎦ + ⎣ yi +1 ⋅ yi − 0,5 ⋅ ( yi + yi −1 ) ⎦ = yi +1,5 − yi −0,5 − δ yi +0,5 . © В.Е. Селе
ев, В.В. Алеш
[100,
, С.Н. Прял в, 2007–2009
(2.347)
172 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
И
⎡⎣ xi − 0,5 , xi +1,5 ⎤⎦
(2.343) 1
:
yi +1,5 − yi − 0,5 = pi ⋅ Δxi + pi +1 ⋅ Δxi +1 − δ yi + 0,5 . К
(2.348) yi − 0,5
(2.348),
yi + 0,5
,
-
(2.329)
.
Д
O ( h)
, За
. [95, 100, 101].
-
а и 1. Д dy dx ) (y) е а а
( . . е
х е
е е х
е
е
-
а х.
(2.333)
(2.329)
.Н
.
, е -
,
-
(
-
), . За
а и 2.
i + 0,5
i + 0,5
xi + 0,5
(
{y } ,
{x
,
( y )}
-
hi )
,
.
yi + 0,5 − yi − 0,5 xi + 0,5 − xi − 0,5
y( x)
⎛ dy ⎞ {Spline , } = ⎜ ⎟ = x Δ ( yi ) . ( DER =1) ⎝ dx ⎠i
(2.349)
{xi +0,5 ( y )} (
Д В
, . 2.17).
-
(2.349) , . .
-
y
[92].
(2.333) :
yi + 0,5 − yi − 0,5 xi + 0,5 − xi − 0,5
(2.349), = pi .
(2.350)
(2.350) 1
(2.340)
yi − 0,5 , yi + 0,5 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
pi .
⎣⎡ xi − 0,5 , xi + 0,5 ⎦⎤
⎣⎡ xi + 0,5 , xi +1,5 ⎦⎤
лава 2 173 _______________________________________________________________________________________
⎣⎡ xi − 0,5 , xi + 0,5 ⎦⎤
ще
(2.333), , я
а
и . 2.17. Пр
а
е а
ер ра
е
{
- хе а я яе я е а а а.
}
я xi + 0,5 ( y )
ла
е а
е р р ва
я ля
Н
е
е
у
y ( x)
: я е е я че ая а
е
ч е
е е е а ь
я яе
я
. .), а а е а ь а е а а е е е ь
е а а
е
е е а ь ь( я а ( N + 1)
, х а е
ь я (2.329)
е
а а
е я ь е е е
ь е я е ая а аь а , , а е , я Ф ье еч ч
а я а , ч е ах а че е , я е ече а е х е еч ь е е , ящ х а х.
я
а е яе я
ь я -
, . С а - хе а е а а а а я ячее ⎡⎣ xi − 0,5 ( y ), xi + 0,5 ( y ) ⎤⎦ .
,
-
-
. dy d + g ( y ) = p ( x, y ( x ) ) . dx dx
:
-
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.351) (2.351)
-
174 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
, . .
( dy dx
dg dx )
. , .
( ⎡⎣ xi − 0,5 ( y ), xi + 0,5 ( y ) ⎤⎦
-
⎡⎣ xi − 0,5 ( g ), xi + 0,5 ( g ) ⎤⎦ ),
(
-
)
,
, 1
(2.351), П и
2.
+
∂t
t –
-
∂ρ ∂ ( ρ ⋅ w ) + = 0; ∂t ∂x
∂ ( ρ ⋅ Ym )
(
∂ ( ρ ⋅ w ⋅ Ym ) ∂x
= 0,
m = 1, N S − 1;
YN S = 1 −
; ρ –
m-
ω
τj
2.3.2):
m =1
m
ω=
m(
{( x , t ) x = x
i −1
j
i
i
Д
(2.352 )
,
; w –
-
:
∑Y
N S −1
; Ym = ρ m ρ –
; ρm – ).
.
(2.352 )
; x –
; NS –
.
}
hi
+ hi , t j = t j −1 + τ j .
(2.353)
(2.352)
-
:
j j j j j j j j ρ i j +1 − ρ i j 4 ⋅ ( ρ i +1 + ρ i ) ⋅ ( wi +1 + wi ) − 4 ⋅ ( ρ i + ρ i −1 ) ⋅ ( wi + wi −1 ) ; + 0,5 ⋅ ( hi + hi +1 ) τj
1
1
ρ i j +1 ⋅ (Ym )i − ρ i j ⋅ (Ym )i + τj j +1
(2.354 )
j
(
)
(
)
1 1 j j j j ⋅ ( ρ i j+1 ⋅ wij+1 + ρ i j ⋅ wij ) ⋅ (Ym )i +1 + (Ym )i − ⋅ ( ρ i j ⋅ wij + ρ i j−1 ⋅ wij−1 ) ⋅ (Ym )i + (Ym )i −1 4 , +4 0,5 ⋅ ( hi + hi +1 )
m = 1, N S − 1;
(Y ) NS
j i
= 1−
∑ ( Y ) ; (Y )
N S −1
j m i
m =1
NS
j +1
i
= 1−
∑ (Y )
N S −1 m =1
j +1 m i
.
(2.354 ) Д (2.352) xi + 0,5 = 0, 5 ⋅ ( xi + xi +1 ) . 1
© В.Е. Селе
, ев, В.В. Алеш
-
, С.Н. Прял в, 2007–2009
,
. 2.18,
.
лава 2 175 _______________________________________________________________________________________
и . 2.18. Пр
ра
ве
-вре е
ая е а ( ра
(2.354)
е
)
,
-
.
.
-
Ym ,
-
. (2.352 ) (2.352 ), :
ρ⋅
(2.355)
∂Ym ∂Y + ρ ⋅ w ⋅ m = 0. ∂t ∂x
,
-
∂Ym ∂x = 0 , ,
(
(Ym )i .
j +1
τj
j
)+
,
-
(2.355). Д
j
(2.354 ),
ρ i j +1 ⋅ (Ym )i − (Ym )i
Ym
, ∂Ym ∂t = 0 . Д .
Ym
(2.354 )
(2.355)
,
:
(
)
(
)
1 1 j j j j ⋅ ( ρ i j+1 ⋅ wij+1 + ρ i j ⋅ wij ) ⋅ (Ym )i +1 − (Ym )i + ⋅ ( ρ i j ⋅ wij + ρ i j−1 ⋅ wij−1 ) ⋅ (Ym )i − (Ym )i −1 4 4 + + δ K = 0, 0,5 ⋅ ( hi + hi +1 )
(ρ δK = В
j i +1
− ρi j ) ⋅ ( wij+1 − wij ) − ( ρi j − ρi j−1 ) ⋅ ( wij − wij−1 )
δK (2.354)
Ym ( x) = const © В.Е. Селе
ев, В.В. Алеш
2 ⋅ (hi + hi +1 )
. Д
,
(2.356 ) ⋅ (Ym )i . j
(2.356 )
-
. (2.356) , С.Н. Прял в, 2007–2009
:
176 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(
ρ i j ⋅ (Ym )i − (Ym )i j +1
j
τj
И (2.357)
) + (ρ
j i +1
− ρ i j ) ⋅ ( wij+1 − wij ) − ( ρ i j − ρ i j−1 ) ⋅ ( wij − wij−1 ) 2 ⋅ ( hi + hi +1 )
,
⋅ (Ym )i = 0. (2.357) j
Ym Ym ,
,
.
( ρ ⋅ w ⋅ Ym ) , ( ρ ⋅ w) .
. ,
( ρ ⋅ w) :
( ρ ⋅ w )i + 0,5
Ym =
ρ ⋅ w ⋅ Ym = const . ρ ⋅w
-
-
(2.354 ) (2.354 ) ⎡ 0, 25 ⋅ ( ρ i j+1 + ρ i j ) ⋅ ( wij+1 + wij ) ⎤ ⎣ ⎦
⎡ 0,5 ⋅ ( ρ i j+1 ⋅ wij+1 + ρ i j ⋅ wij )⎤ ⎣ ⎦ ,
П и
: .И
,
-
.
3. Д
∂ρ ∂ ( ρ ⋅ w ) + = 0; ∂t ∂x
[79, 99]:
∂ ( ρ ⋅ w) ∂t
t –
; x – ; p –
∂ ( ρ ⋅ w2 )
=−
∂x
∂p + Fx , ∂x
; ρ –
; Fx –
Ox . Д
F
+
(2.358 )
(
(2.358 ) ; w – )
(2.358)
-
:
j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 ρ i j +1 − ρ i j 8 ⋅ ( ρ i +1 + ρ i ) ⋅ ( wi +1 + wi ) − 8 ⋅ ( ρ i + ρ i −1 ) ⋅ ( wi + wi −1 ) + = 0; 0,5 ⋅ ( hi + hi +1 ) τj
1
1
(2.359 )
j +1 j +1 j +1 j +1 j +1 j +1 j +1 j +1 ρ i j +1 ⋅ wij +1 − ρ i j ⋅ wij 8 ⋅ ( ρ i +1 + ρ i ) ⋅ ( wi +1 + wi ) − 8 ⋅ ( ρ i + ρ i −1 ) ⋅ ( wi + wi −1 ) + = 0,5 ⋅ ( hi + hi +1 ) τj
1
=−
2
0,5 ⋅ ( pij++11 + pij +1 ) − 0,5 ⋅ ( pij +1 + pij−+11 ) 0,5 ⋅ ( hi + hi +1 )
1
2
+ ( Fx )i . j +1
(2.359 ) К © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 177 _______________________________________________________________________________________
(2.358)
-
,
. 2.18.
w
(2.358 )
0,5 ⋅ w , 2
:
∂ ⎛ ρ ⋅w ⎞ ∂ ⎛ ρ ⋅w ⎞ ∂p ⎜ ⎟ + ⎜ w⋅ ⎟ = − w ⋅ + w ⋅ Fx . 2 ⎠ ∂t ⎝ 2 ⎠ ∂x ⎝ ∂x 2
0,5 ⋅ ( wi
ρi
j +1
)
(2.359 )
j +1 2
(2.358 ),
wi
2
. Д (2.359 ),
j +1
.
,
j +1 2
1 ⎛ ⎞ − ρ i j ⋅ wij +1 ⎜ wij − ⋅ wij +1 ⎟ 2 ⎝ ⎠+
(w ) ⋅ i
(2.360)
:
τj
2
w j +1 ⋅ w j +1 1 w j +1 ⋅ wij−+11 1 ⋅ ( ρ i j++11 + ρ i j +1 ) ⋅ ( wij++11 + wij +1 ) ⋅ i +1 i − ⋅ ( ρ i j +1 + ρ i j−+11 ) ⋅ ( wij +1 + wij−+11 ) ⋅ i 2 4 2 +4 = 0,5 ⋅ ( hi + hi +1 ) =−
0,5 ⋅ ( pij++11 + pij +1 ) − 0,5 ⋅ ( pij +1 + pij−+11 )
,
ρ i j +1 ⋅
(w )
0,5 ⋅ ( hi + hi +1 )
j +1 2
i
τj
2
− ρij ⋅
(w )
⋅ wij +1 + ( Fx )i ⋅ wij +1 j +1
(2.361 )
:
j 2
i
+
2
w j +1 ⋅ w j +1 1 w j +1 ⋅ wij−+11 1 ⋅ ( ρ i j++11 + ρ i j +1 ) ⋅ ( wij++11 + wij +1 ) ⋅ i +1 i − ⋅ ( ρ i j +1 + ρ i j−+11 ) ⋅ ( wij +1 + wij−+11 ) ⋅ i 2 4 2 +4 = 0,5 ⋅ ( hi + hi +1 ) =−
0,5 ⋅ ( pij++11 + pij +1 ) − 0,5 ⋅ ( pij +1 + pij−+11 ) 0,5 ⋅ ( hi + hi +1 )
⋅ wij +1 + ( Fx )i ⋅ wij +1 + δ K , j +1
(2.361 )
δK = −
τj
δK
В
δK
ρ ⋅ ( wij +1 − wij )
j ⎛ ⎞ ⎛ ∂w ⎞ ρ ⋅ ⎜⎜ wij + τ j ⋅ ⎜ ⎟ + O (τ 2j ) − wij ⎟⎟ ∂ t ⎝ ⎠i ⎠ =O τ . =− ⎝ ( ) 2
2
τj
(τ
,
(2.360). Д .
(2.361) (2.361)
, hi → 0 )
.
,
1
,
1
,
(2.359) -
( © В.Е. Селе
j
(2.361 )
.
ев, В.В. Алеш
(2.361 )). , С.Н. Прял в, 2007–2009
178 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
. Н
, (2.359),
. Н
,
wi , ⎡⎣0,5 ⋅ ( wi j
,
+ wi ) ⎤⎦
j +1
, . . В
j
,
, ∂ ⎛ w2 ⎞ ⎜ρ⋅ ⎟ ( 2 ⎠ ∂t ⎝
-
(2.361)).
«
»
-
w2 ⎞ ∂ ⎛ ⎜ ρ ⋅ w⋅ ⎟, 2 ⎠ ∂x ⎝
, (2.361).
,
-
(2.359), .
, . К
,
, . Ве
е я (2.36).
а
а
- хе . Kt ,
– Kx . :
( f )+ f )
{Spline , Kt}
ρ i j +1 ⋅ t Δ
j +1
( DER =1
i
+ ρ i j +1 ⋅ wij +1 ⋅
{Spline , Kx}
xΔ
( DER =1)
− fi
( DER =1
⋅ g ⋅ ρi
j +1
⋅
j +1
{Spline , Kx}
xΔ
( DER =1)
= − pij +1 ⋅ +
xΔ
( DER =1
( DER =1
xΔ
j +1
(w )
{Spline , Kx} ( DER =1
(k )
{Spline , Kx}
j +1 i
tΔ
( DER =1
j +1
i
⋅ f i j +1 ) ⋅
pij +1 = p ( ρ i j +1 , Ti j +1 ) ;
j +1
j +1
i
⋅ f i j +1 ⋅ wij +1 ⋅
(( z ) ) −
(ε ) + ρ ) i
(ρ ) + )
{Spline , Kt}
( fi j +1 ) + ρij +1 ⋅ fi j +1 ⋅
i
{Spline , Kt}
ρ i j +1 ⋅ f i j +1 ⋅ t Δ
⋅
(w ) + ρ )
{Spline , Kt}
ρ i j +1 ⋅ f i j +1 ⋅ t Δ j +1
j +1 i
-
i
π
j +1 1 i
j +1
4
⋅ f i j +1 ⋅ wij +1 ⋅
⋅ f i j +1 ) − pij +1 ⋅ i
⋅ λi
(
)
f i j +1 ⋅ wij +1 ⋅
{Spline , Kx}
xΔ
( DER =1)
xΔ
{Spline , Kx}
xΔ
{Spline , Kt} ( DER =1)
⋅ wi
j +1
⋅ wi
(ε )
j +1
i
( f )+Q j +1
i
j +1
j +1 i
⋅ Ri
j +1
i
⋅ f i j +1 +
(
)
⋅
{Spline , Kx}
xΔ
( DER =1)
j +1
=
j +1
i
(2.362 )
(p )− j +1
i
(2.362 )
;
π
⋅ λi j +1 ⋅ ρ i j +1 ⋅ wij +1 ⋅ Ri j +1 + 3
{Spline , Kx} {Spline , Kx} xΔ Ti j +1 + ki j +1 ⋅ f i j +1 ⋅ xΔ Ti j +1 − φi j +1 ; ( DER =1)
( DER = 2 )
ев, В.В. Алеш
4
(2.362 ) (2.362 )
ε i j +1 = ε ( pij +1 , Ti j +1 ) ; © В.Е. Селе
(ρ ) +
j +1
i
j +1
( DER =1)
xΔ
( DER =1)
(w ) = − f )
( DER =1
⋅ ρi
{Spline , Kx}
( wij +1 ) = 0;
{Spline , Kx}
j +1
tΔ
-
(2.36)
(2.362 ) , С.Н. Прял в, 2007–2009
лава 2 179 _______________________________________________________________________________________
К 2.5.1)
(
.
(2.362) , Kx
(2.362 ), -
,
Kx
:
Kx > Kx.
-
(2.363)
(2.362) ,
ах а че х а
я
е х а е
я.
а
я
е
(2.36). е е
а ь я
я
-
ах
е
-
а
е е
е
,
а
-
,
(2.36) : Kt ≥ 2 , Kx ≥ 2 , Kx ≥ 3 ),
а - хе а я яе я е а а е ча х х. В е е а х а е я че е е а е а е ( , . .).
(
я е е
е
.
е
а ь а
е е
е а ах
х
а ь
ах а х , -
е
, е а
, Д
ь [69, 70, 96]
а
, х
- хе
.
. ,
Э
,
-
. -
[102]. ,
-
-
, [69, 70, 97].
.В
, -
. .
В
: hi = xi − xi −1 , τ j = t j − t j −1 .
(2.364)
y ( x, t )
,
(2.334).
,
i
(x
j
yi ,
i + iH
. y –
.Д ,
y ( x, t )
y ( x, t )
): © В.Е. Селе
ев, В.В. Алеш
(x , t )
, С.Н. Прял в, 2007–2009
, tj )
-
j
(x , t i
j + jH
),
iH
jH –
( -
180 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
y ( xi +iH , t ) j = y ( xi , t j + jH ) =
∑
Kx −1 k =1
∑
Kt −1 n =1
∞ 1 ⎛ ∂k y ⎞ 1 ⎛ ∂k y ⎞ k k ⋅ ⎜ k ⎟ ⋅ ( xi +iH − xi ) + ∑ ⋅ ⎜ k ⎟ ⋅ ( xi +iH − xi ) ; k ! ⎝ ∂x ⎠i k = Kx k ! ⎝ ∂x ⎠ i j
j
∞ n n 1 ⎛ ∂n y ⎞ 1 ⎛ ∂n y ⎞ ⋅ ⎜ n ⎟ ⋅ ( t j + jH − t j ) + ∑ ⋅ ⎜ n ⎟ ⋅ ( t j + jH − t j ) . n ! ⎝ ∂t ⎠i n = Kt n ! ⎝ ∂t ⎠i j
j
Kx ( y )
Д
(2.365 )
y
Kt ( y )
( Kx ( y ) − 1) -
( Kt ( y ) − 1) -
.
(2.365 )
. .
,
-
Kx y ( xi + iH , t j ) = y ( xi + iH , t j ) + O ⎡( xi +iH − xi ) ⎤ = y ( xi +iH , tm ) + O ⎡⎣ h Kx ⎤⎦ ; ⎣ ⎦
:
Kt y ( xi , t j + jH ) = y ( xi , t j + jH ) + O ⎢⎡( t j + jH − t j ) ⎥⎤ = y ( xi , t j + jH ) + O ⎡⎣τ Kt ⎤⎦ . ⎣ ⎦
y ( xi + iH , t j + jH ) = y ( xi +iH , t j + jH ) + O ⎡⎣ h Kx , τ Kt ⎤⎦ ,
А
(2.366 ) (2.366 )
,
(2.367 )
∂ ∂ y ( xi +iH , t j ) = y ( xi +iH , t j ) + O ⎡⎣ h Kx −1 ⎤⎦ , ∂x ∂x
(2.367 )
∂2 ∂2 y ( xi +iH , t j ) = 2 y ( xi +iH , t j ) + O ⎡⎣ h Kx − 2 ⎤⎦ , 2 ∂x ∂x
(2.367 )
∂ ∂ y ( xi , tm+ mH ) = y ( xi , tm+ mH ) + O ⎡⎣τ Kt −1 ⎤⎦ . ∂t ∂t
(2.367 )
, (2.362).
,
,
, -
. У а
ρ⋅
е
е (2.362а):
∂f ∂ρ ∂f ∂ρ ∂w +f⋅ + ρ ⋅ w⋅ + w⋅ f ⋅ +ρ⋅ f ⋅ = O ( h Kx1 ,τ Kt1 ) , ∂t ∂t ∂x ∂x ∂x
Kx1 = min ⎡⎣ Kx ( ρ ) − 1; Kx ( f ) − 1; Kx ( w ) − 1⎤⎦ , Kt1 = min ⎡⎣ Kt ( ρ ) − 1; Kt ( f ) − 1; Kt ( w ) ⎤⎦ .
У а
ρ⋅ f ⋅
е
е (2.362 ):
(2.368 )
(2.368 ) (2.368 )
∂w ∂w ∂p ∂z π + ρ ⋅w⋅ f ⋅ =−f ⋅ − f ⋅ g ⋅ ρ ⋅ 1 − ⋅ λ ⋅ ρ ⋅ w ⋅ w ⋅ R + O ( h Kx2 ,τ Kt2 ) , (2.369 ) ∂t ∂x ∂x ∂x 4
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 181 _______________________________________________________________________________________
Kx2 = min ⎡⎣ Kx ( ρ ) ; Kx ( f ) ; Kx ( w ) − 1; Kx ( p ) − 1; Kx ( z1 ) − 1; Kx ( λ ) ; Kx ( R )⎤⎦ ,
Kt2 = min ⎡⎣ Kt ( w ) − 1; Kt ( ρ ) ; Kt ( f ) ; Kt ( p ) ; Kt ( z1 ) ; Kt ( λ ) ; Kt ( R )⎤⎦ . У а
ρ⋅ f ⋅
е
(2.369 ) (2.369 )
е (2.362 ):
∂ε ∂ε ∂f ∂w π 3 + ρ ⋅w⋅ f ⋅ = − p⋅w⋅ − p⋅ f ⋅ + ⋅λ ⋅ρ ⋅ w ⋅ R − ∂t ∂x ∂x ∂x 4
∂f ∂ 2T ∂T ∂k ∂T ∂f − p ⋅ + Q ⋅ f − Φ (T , Toc ) + k ⋅ f ⋅ 2 + f ⋅ ⋅ +k⋅ ⋅ + O ( h Kx3 ,τ Kt3 ) , ∂t ∂x ∂x ∂x ∂x ∂x
(2.370 )
Kx3 = min ⎡⎣ Kx ( ρ ) ; Kx ( f ) − 1; Kx (ε ) − 1; Kx ( p ) ; Kx ( w ) − 1; Kx ( λ ) ; Kx ( R ) ; Kx ( Q ) ; Kx ( Φ ) ; Kx ( k ) − 1; Kx ( T ) − 2 ⎤⎦ ,
(2.370 )
Kt3 = min ⎡⎣ Kt ( ρ ) ; Kt ( f ) − 1; Kt (ε ) − 1; Kt ( p ) ; Kt ( w ) ; Kt ( λ ) ; Kt ( R ) ; Kt ( Q ) ; Kt ( Φ ) ; Kt ( k ) ; Kt ( T )⎤⎦ .
(2.370 )
O ( h KxCXEMA , τ KtCXEMA ) ,
,
-
KxCXEMA = min ⎡⎣ Kx ( ρ ) − 1; Kx ( f ) − 1; Kx ( w ) − 1; Kx ( p ) − 1; Kx ( z1 ) − 1; Kx ( λ ) ;
(2.371 )
KtCXEMA = min ⎡⎣ Kt ( ρ ) − 1; Kt ( f ) − 1; Kt ( w ) − 1; Kt ( p ) ; Kt ( z1 ) ; Kt ( λ ) ; Kt ( R ) ;
(2.371 )
Kx ( R ) ; Kx ( Q ) ; Kx (ε ) − 1; Kx ( Φ ) ; Kx ( k ) − 1; Kx ( T ) − 2 ⎤⎦ ,
Kt ( ε ) − 1; Kt ( Q ) ; Kt ( Φ ) ; Kt ( k ) ; Kt (T ) ⎤⎦ . y ( x, t )
(
-
( Kx( y ) + 1) ( Kt ( y ) + 1) .
Ox )
(
Ot )
-
;
-
.
Ox
(
.
y ( x, t = const )
)
,
-
.В :
.
.Д
Ot j-
( xi , t j ) , © В.Е. Селе
ев, В.В. Алеш
. Д
, С.Н. Прял в, 2007–2009
. ( :
, . . (2.36 )),
182 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ ∂ρ ⎞ ⎜ ⎟ ⎝ ∂t ⎠i
j +1
И Д ⎛ ∂ρ ⎞ ⎜ ⎟ ⎝ ∂t ⎠ i
=−
∂f ∂ρ ∂w ⎞ ⎛ ∂f ⋅ ρ ⋅ + ρ ⋅ w⋅ + w⋅ f ⋅ +ρ⋅ f ⋅ ⎟ . j +1 ⎜ fi ∂x ∂x ∂x ⎠i ⎝ ∂t
n (n- ) , ∂ρ ∂t
( n +1)
=−
,
1 fi
j +1
j +1
1
(n-
yi
( n + 1)
)
.
j +1
:
j +1 j +1 (n) (n) ⎡ ⎤ ⎛ ∂f ⎞ ⎛ ∂f ⎞ ⎛ ∂ρ ⎞ (n) j +1 ⎛ ∂w ⎞ ⋅ ⎢ ρ i( n ) ⋅ ⎜ ⎟ + ρ i( n ) ⋅ wi( n ) ⋅ ⎜ ⎟ + wi( n ) ⋅ f i j +1 ⋅ ⎜ ⎟ + ρi ⋅ fi ⋅ ⎜ ⎟ ⎥ t x x x ∂ ∂ ∂ ∂ ⎝ ⎠i ⎝ ⎠i ⎝ ⎠i ⎝ ⎠i ⎦⎥ ⎣⎢ (2.373 ) ,
( n +1)
=−
1
( )
( )
f t j ≤ t ≤ t j +1 (
« ( n ) ».
« j +1»
, Д
( n + 1) -
-
ρi ( t )
Kt ( ρ ) )
. К
( Kt ( ρ ) − 1) )
,
( ,
t j −1 ≤ t ≤ t j .
-
-
,
»
ρi ( t )
( n +1) i
А ( n + 1) -
,
-
)
ρ
(2.373 )
,
(
«
yi . ( n)
.
{Spline , Kx ( f )} ⎡ ( n ) {Spline , Kt ( f )} j +1 (n) (n) t f w xΔ ( fi j +1 ) + ρ ρ ⋅ ⋅ Δ + ⋅ ⋅ ( ) ⎢ i i i i ( DER =1) ( DER =1) f i j +1 ⎣ {Spline , Kt ( ρ )} {Spline , Kx ( w)} n ⎤ + wi( n ) ⋅ fi j +1 ⋅ xΔ wi( ) ⎥ . ρi( n ) + ρi( n ) ⋅ f i j +1 ⋅ xΔ ( DER =1) ( DER =1) ⎦
⎛ ∂ρ ⎞ ⎜ ⎟ ⎝ ∂t ⎠i
tj .
(2.372)
. t j ≤ t ≤ t j +1
(2.373).
ρi( n +1) = ρi ( t j +1 ) . :
.
(2.374)
yi( n ) ⎯⎯⎯ → yij +1 , n →∞
y
. . -
. В (
.,
, « , [103, 104]). .В ,
-
» .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 183 _______________________________________________________________________________________
-
. В хе а
,
».
[103, 104]
« а
а
-
, .
,
ь
(
е а
а
- хе а
-
-
. Д ),
-
.
2.5.3. К во о у о о оении не авно е ных о лине и и ованных а но ных е о
у о
ово ов
Д .
H .Д
-
,
h− /
,
h+
.
(
Q ).
.К
: 1≤ q ≤ Q ,
/
q –
. Д
.М (
-
е
) .В .
-
1
е
,
-
. Н :
.В
; ( :
•
«
•
»;
«
е
В
, -
)
». а ь
Σ
) е а ,
( . H , h− , h+ -
N
, ,
. В .
, ,
,
(
е
Q
),
. 1
Д
, (
© В.Е. Селе
)
,
ев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
184 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
П ,
е
е е
е яче
Σ
е
« ще :
е
е
(
. яче Σ,
).
а
». Σ+ .
яя яче
.
-
а е
: N = N + N −1 .
Σ
Н «0». Н
,
h N − −1 , ( )
−
+
« + », : N−
« ще
« − », , Σ−
hk− = Q ⋅ hk−−1 = Q k ⋅ h0− ;
И
N+ .
Σ+ . -
h , h , h , …, − 0
– h0+ , h1+ , h2+ , …, h +N + −1 . ( )
,
е
е
».
К
Q
Σ
е яче
−
е
,
е
–
−
е
,
H.Н H а ае
е
е
− 1
,
− 2
:
hk+ = Q ⋅ hk+−1 = Q k ⋅ h0+ . ,
( ) ( ) = h ⋅ (1 + Q + Q + ... + Q ) = h ⋅ 11−−QQ )
:
H − = h0− + h1− + h2− + ... + h −N − −1 = h0− ⋅ 1 + Q + Q 2 + ... + Q ( )
H + = h0+ + h1+ + h2+ + ... + h +N + −1 (
+ 0
(N
−
)
−1
N + −1
2
= h0− ⋅
1− QN ; 1− Q
+ 0
N
, :
Q
(N
h N − −1 = h N + −1 ; ( ) ( ) −
)
−1
⋅ h0− = Q
(N
+
)
−1
H = h0− ⋅
h−
, :
−
H, h , h (2.377). Д
+
h0− = h − ;
⋅ h0+ ;
(2.375 )
)
+
h+
h0+ = h + .
(2.376 ) (2.376 ) Σ
, (2.377)
, Q
, ( © В.Е. Селе
.
(2.375 )
1− QN 1− QN ( N + −1) + + h0+ ⋅ −Q ⋅ h0 . 1− Q 1− Q −
+
-
H = H − + H + − h +N + −1 ;
(
−
): ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
(2.375 ), (2.376 ), , -
лава 2 185 _______________________________________________________________________________________
f1 ( N + ) = h − ⋅
(N )
1− Q 1− Q N
−
+
+ h+ ⋅
1− QN ( N + −1) + −Q ⋅ h − H = 0, 1− Q +
(2.378 )
+ ⎛ h+ ⎞ N − ( N + ) = 1 + logQ ⎜ − ⋅ Q N −1 ⎟ . ⎝h ⎠
Н
(2.378) а е е ь
. , е аче
е
N− ,
я ό
е
N−, H.
N+, аче Σ
,
Σ
Σ
, ,
N− е е 5)
(
(2.378 )
я (
, , N+ а
N− H. ,
,
: 0, logQ ⎡⎣1 − H ⋅ (1 − Q ) h + ⎤⎦ . ;
– -
1− QN , 1− Q +
. В
N
+
а
х ó ь
х е х аче N+ N−
» .
q−
N−
( q )( −
H =h ⋅ −
« -
q−
(2.375), (2.376)):
1 − ( q− ) 1− (q
)
N − −1
−
)
N
−
q+ .
⋅ h− = ( q+ )
+h ⋅ +
(
(N
1 − ( q+ ) 1− (q
+
)
−1
+
)
N
⋅ h+ ;
− ( q+ )
+
Д
(N
+
)
⋅ h+ .
−1
, : f2 ( q+ ) = h− ⋅
1
1 − ⎡⎣ q − ( q + ) ⎤⎦ 1 − ⎣⎡ q
−
( q )⎦⎤ +
N
−
+ h+ ⋅
1 − ( q+ ) 1− (q ,
. © В.Е. Селе
N−
. .Н
Q.
»
N+
1
q+ ,
В «
ев, В.В. Алеш
-
.В H = h+ ⋅
е
(2.378) N+
)
. Д
5,32 Σ ь е
, С.Н. Прял в, 2007–2009
+
)
N
+
− ( q+ )
-
(N
ó
+
)
−1
⋅ h + − H = 0,
-
186 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ h+ ( N + −1) ⎞ N − −1 q ( q ) = ⎜ − ⋅ ( q+ ) . ⎟ ⎝h ⎠ −
1
+
( 0, Q ] .
q+
1,
H , h− , h+
-
, .
Q
.Н
, N−
, , q+
q−
-
N+
.
Q,
,
,
-
. П
е
е е
Σ
,
е
Σ
е
е
« а е : .
+
е
« а е
е яче а е
. а
а е х ячее ».
Σ
Н
Σ
е
−
: N =N +N . −
+
х ячее ». Σ+ .
, е е яче
.Д
– К
а
« + », : N−
« − », , Σ− , Σ+ . -
N+ .
«0». Н : h0− , h1− , h2− , …, h −N − −1 , ( )
: h0+ , h1+ , h2+ , …, h +N + −1 . ( )
Σ− е е -
,
, ,
Q
:
hk− = Q ⋅ hk−−1 = Q k ⋅ h0− ; hk+ = Q ⋅ hk+−1 = Q k ⋅ h0+ .
И
,
(
:
H = h + h + h + ... + h N − −1 = h ⋅ 1 + Q + Q + ... + Q −
− 0
− 1
− 2
−
(
− 0
)
2
(
H = h + h + h + ... + h N + −1 = h ⋅ 1 + Q + Q + ... + Q ( ) +
+ 0
+ 1
+ 2
+
+ 0
2
( N −1) −
( N −1) +
)
)
1− QN ; =h ⋅ 1− Q −
− 0
1− QN =h ⋅ . 1− Q +
+ 0
, :
© В.Е. Селе
ев, В.В. Алеш
h N − −1 = h N + −1 ;
(
)
(
, С.Н. Прял в, 2007–2009
)
(2.379 )
лава 2 187 _______________________________________________________________________________________
Q
(N
−
)
−1
⋅ h0− = Q
(N
+
H =H +H ; −
H = h0− ⋅
H , h− , h+ (2.381). Д
⋅ h0+ ;
+
(2.379 ) (2.380 )
1− Q 1− Q . + h0+ ⋅ 1− Q 1− Q −
N
N
h−
,
+
(2.380 ) Σ
h+
h0− = h − ;
:
)
−1
,
h0+ = h + .
(2.381)
, (2.379 ), (2.380 ),
Q
,
(N )
1− Q f3 ( N ) = h ⋅ 1− Q
:
+
N
−
−
+
+ h+ ⋅
1− QN − H = 0; 1− Q +
(2.382 )
+ ⎛ h+ ⎞ N − ( N + ) = 1 + logQ ⎜ − ⋅ Q N −1 ⎟ . h ⎝ ⎠
(
Н
+
. Д
)
: 0, logQ ⎡⎣1 − H ⋅ (1 − Q ) h ⎤⎦ . +
(2.382 )
N ,
N−
-
N+
ó
N
N
q−
»
+
(
(2.379), (2.380)):
( q )( −
H =h ⋅ −
Д
)
N − −1
⋅ h− = ( q+ )
1 − ( q− )
1 − ( q− ) N
−
q+ .
q+ ,
(N
+h ⋅ +
+
)
−1
-
⋅ h+ ;
1 − ( q+ )
1 − ( q+ ) N
+
.
, :
f4 ( q
© В.Е. Селе
q−
. N+ -
Q.
В −
N−
» .
«
-
,
«
,
(2.382)
+
ев, В.В. Алеш
)=h
−
⋅
1 − ⎡⎣ q − ( q + ) ⎤⎦
1 − ⎣⎡ q − ( q + )⎦⎤ N
−
+h ⋅
, С.Н. Прял в, 2007–2009
+
1 − ( q+ )
1 − ( q+ ) N
+
− H = 0;
188 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ h+ ( N + −1) ⎞ N − −1 q ( q ) = ⎜ − ⋅ ( q+ ) . ⎟ ⎝h ⎠ −
1
+
( 0, Q ] .
q+
-
1,
H , h− , h+
-
, .
Q
.В -
,
.В . ,
, ,
/ .
П
В
е
е
е
: h− ≤ h+ .
,
.
.
Н h . H = H − h− − h+ . H +
H. Д h0 , h1 , h2 , ..., hN −1 , H.Д
h−
H
: -
N –
: N = N + 2.
H
:
hk = q ⋅ hk −1 = q k ⋅ h0 ;
H = h0 + h1 + h2 + ... + hN −1 = h0 ⋅
1 − qN , 1− q
q –
.
В
1≤ q ≤ Q .
q
К
,
, , , , H,
В
: . q
–
{h
−
, h0 }
{h
N −1
, h+ } .
:
h0 = q ⋅ h − ; hN −1 = q N −1 ⋅ h0 = H = h0 ⋅
(2.383 ) 1 + ⋅h ; q
1 − qN . 1− q
, © В.Е. Селе
ев, В.В. Алеш
(2.383 )
(2.383 ) .
, С.Н. Прял в, 2007–2009
-
лава 2 189 _______________________________________________________________________________________
h0
(2.383 ) (2.383 ),
:
N +1
H = q ⋅ h− ⋅
N = N (q)
(2.384 ),
N
⋅ h− .
(2.384 )
(2.383 ) (2.383 ):
h0
В
h =q +
1 − qN . 1− q
(2.384 )
,
f5 ( q ) = q ⋅ h − ⋅
q
:
1− q ( ) − H = 0, 1− q N q
N ( q ) = −1 + log q
(2.385) , а
( а
(2.385 )
h+ . h−
(2.385 ) Q,
q
е
). .
/
-
,
, (
),
(
).
−
h , h
+
H, Q.
В
.
N
, Q.
{h
N −1
-
q. Д
,
, h+ }
H
{h
,
qin ,
, h0 }
-
qbound . В h0 = qbound ⋅ h − ;
:
(2.386 )
h + = qbound ⋅ hN −1 = qbound ⋅ qinN −1 ⋅ h0 ;
H = h0 ⋅ h0
(2.386 )
: f 6 ( qin ) = qbound ( qin ) ⋅ h − ⋅ qin © В.Е. Селе
−
ев, В.В. Алеш
(2.386 )
1 − qinN . 1 − qin
(2.386 )
1 − qinN − H = 0, 1 − qin
qbound
, С.Н. Прял в, 2007–2009
(2.386 ) (2.386 ), qbound ( qin ) =
h+ . h ⋅ qinN −1 −
190 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
qin ∈ [1/ Q , Q ] ;
,
. (2.387) , . .
, qin
qbound ∈ [1/ Q , Q ] ,
qbound
( [1/ Q , 1)
, +
[1,
hN −1 h . Д
hN − 2 hN −1
(2.387) , [1/ Q, 1) . h − h0
Q ] ). А
-
h1 h0
,
H.
H)
H (
». 1/ Q , Q] , [
H « а
, H
, . . 2.5.4. Чи ленны анали
-
, .
а о ы
ана
В
,
.Д .
. Н
, .
[66],
,
,
:
Δp ì = ζ 0 ⋅
ρ0 ⋅ w02 2
(2.388)
,
«0 »
. (2.388)
К
ζ
, ,
, . , Δp ì ζi = , ρi ⋅ wi2 2
, К
( fi ) ,
(
Δp ì = ζ 0 ⋅
© В.Е. Селе
.
ев, В.В. Алеш
2
= ζi ⋅
i-
, f0 )
[66]:
Δpм ρ w2 = ζ i ⋅ i ⋅ i2 , 2 ρ 0 ⋅ w0 2 ρ 0 w0
ρ 0 ⋅ w02
ζ wi
)
ζ0 =
-
,
(
ρi
.
ρi ⋅ wi2 2
, С.Н. Прял в, 2007–2009
.
(2.389)
лава 2 191 _______________________________________________________________________________________
( ρ0 ⋅ w0 ⋅ f 0 = ρi ⋅ wi ⋅ fi ) ,
:
ρ ⎛f ⎞ ζ 0 = ζi ⋅ 0 ⋅⎜ 0 ⎟ . ρi ⎝ f i ⎠ 2
ρ0 = ρi = ρ
:
(2.390)
⎛ f0 ⎞ ⎟ . ⎝ fi ⎠
ζ 0 = ζi ⋅⎜
2
(2.391)
[66], (
.Д
), ⋅ζ ⋅
Δp м = k
ρ в ⋅ w0,2 в 2
-
: (2.392)
,
w0,в –
p0 ; ρ в –
, k
,
–
p0 p1 p0 = 1 − Δp м p0 .
:
≈ 1,0
k
k
( p1
=
[66]: Δp м ⋅ p0
p0 )крит < p1 p0 < 0, 9
1 − ( p1 p0 )крит = 0, 47 [66]. К
=
p0 )крит –
; γ –
,
, [66], .
p1 p0 > 0,9
γ −1
γ ⋅ ⎢( p1 p0 ) ⎡ ⎣
2
Δp м < 0,1 ⋅ p0 ;
− ( p1 p0 )
[1 − 0, 46 ⋅ Δp м 1
p0 ]
γ +1 γ
⎤ ⎥⎦
(2.394)
( p1
.Д
[66], k ,
(2.395)
,
p0 )крит = 0,53 Δp м .
(2.392–2.395) k Δp м , p1 p0
[105].
-
(2.392–2.395) ,
ζ, © В.Е. Селе
γ
(2.393)
1 − ( p1 p0 )крит > Δ p м p0 > 0,1 k
( p1
p1
. В ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
192 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ζ (
1
Д . , ⎡⎣ xiL −1,5 , xiR +1,5 ⎤⎦ ,
. (2.388))
, . 2.19. е ь а а xiL−1,5 = 0,5 ⋅ ( xiL−1 + xiL−2 ) xiR +1,5 = 0,5 ⋅ ( xiR +1 + xiR +2 ) .
и . 2.19. Схе а
ч
е ре
авле
:
ΔZ = ( z1 )iR + 2 − ( z1 )iL − 2 ;
( z1 )iL −1 = ( z1 )iL = ( z1 )iR = ( z1 )iR +1 = ( z1 )iL − 2 +
ΔL = xiR + 2 − xiL − 2 ;
xi –
xiL −1 = xiL = xiR = xiR +1 = xiL − 2 + i , ( z1 )i –
i. Н . 2.19 iL ( iL − 1) ( iR + 1)
),
( iL − 1, iL ) ,
( iR,
,
, . xiL = xiL −1
( [ xiL − 2 , xiR + 2 ] )2,
xiR = xiR +1
[ xiL , xiR ]
( xiR+2 − xiL−2 ) ,
[ xiL , xiR ]
2
,
2
© В.Е. Селе
, С.Н. Прял в, 2007–2009
-
-
(2.397 )
,
. ев, В.В. Алеш
-
,
.
[ xiL , xiR ]
(2.396 )
,
Δx[iL , iR] = k[iL , iR] ⋅
1
ΔZ ; 2
(
iR
.
k[iL , iR] ∈ ( 0,1) –
iR -
-
iR + 1)
, :
ΔL , 2
iL
.
( xiL−1 − xiL−2 ) + ( xiR+2 − xiR+1 )
( . . xiL ≠ xiR ),
е ра а
( iL − 1) , iL , iR , ( iR + 1)
Д , iR = iL + 1 . ( )
Д
.
k
лава 2 193 _______________________________________________________________________________________
, k[iL , iR] = 0, 2 ). В
(
-
⎡( z1 )iR +2 − ( z1 )iL−2 ⎤⎦ . Δz[iL , iR ] = k[iL , iR] ⋅ ⎣ 2
В
(2.396 )
:
ΔZ = ( z1 )iR + 2 − ( z1 )iL − 2 ;
( z1 )iL −1 = ( z1 )iL = ( z1 )iL − 2 +
xiL −1 = xiL = xiL − 2 +
ΔZ − Δz[iL , iR] 2
(2.397 )
ΔL = xiR +2 − xiL−2 ;
( z1 )iR = ( z1 )iR +1 = ( z1 )iL + Δz[iL, iR] ;
;
ΔL − Δx[iL , iR] 2
; xiR = xiR +1 = xiL + Δx[iL , iR] .
,
xiL − 2 < xiL −1 = xiL < xiR = xiR +1 < xiR + 2 .
Да ее а е ь а а. (2.398 ),
ь
а
а
а
е
ае
а
х
е
а а е е
(2.396 )
(2.398 )
ече я а а че е е е я ь я а а е
–
а е
е а
:
xiR + 2 < xiR +1 = xiR < xiL = xiL −1 < xiL − 2 .
(2.398 )
К
(
)
, (2.392) (
В
-
(2.388)).
(2.392),
( iL − 1) 1.
,
(
),
. iR , ( iR + 1) ),
(
( iL − 1)
( iR + 1) .
Д : ( iL − 1) , iL , 2
.
,
.В
-
( )3:
ρiL −1 ⋅ wiL −1 = ρiR +1 ⋅ wiR +1 .
(2.399) -
1
,
2
. 2.19
.
, .
3
, ,
© В.Е. Селе
, ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
194 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ρ,
ε
(
w,
p,
T
). ( iL − 1) ,
,
. 2.1. Та ли а 2.1
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ .
1. 2.
(2.399)
-
.
piL −1 = p ( ρiL −1 , TiL −1 ) .
3. 4.
TiL −1 = T ( ε iL −1 , piL −1 ) .
К
5.
Д
ρiL −1 ;
. – piL −1 ;
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ .
:
-
wiL −1 ;
–
–
ε iL −1 ;
–
–
TiL −1 .
,
-
.К
, .
«
.К –
» «
»1 -
.
xi ± 0,5 = 0, 5 ⋅ ( xi + xi ±1 ) .
. К ⎡⎣ xi − 0,5 , xi + 0,5 ⎤⎦ ,
» « ( (
i + 1 = iL ).
К
© В.Е. Селе
i)
,
( iL − 1) ) ⎣⎡ xiL −1,5 , xiL −1 ⎦⎤ ( », . . xiL −1,5 = 0,5 ⋅ ( xiL −1 + xiL − 2 ) .
,
1
. (
.Н
« «
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ , -
⎡⎣ xi − 0,5 , xi + 0,5 ⎤⎦ . « » » . ), ( i + 1) ,
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
«
».
-
, -
,
i = iL − 1 .
.В
.
лава 2 195 _______________________________________________________________________________________
yi +1 = yi , .В
yk – yk
«
»
«
⎡⎣ xi − 0,5 , xi ⎤⎦ = ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ . , ( i − 1)
»
А
⎡⎣ xi − 0,5 , xi + 0,5 ⎤⎦
«
⎣⎡ xi − 0,5 , xi + 0,5 ⎦⎤
xk , f k , ρ k , wk
k-
. .
«
»
⎡⎣ xi , xi + 0,5 ⎤⎦ .
»
( iR + 1) ,
В
.
,
. 2.2. Та ли а 2.2
1.
(2.392)
2.
⎡⎣ xiR +1 , xiR +1,5 ⎤⎦ .
ρiR +1 = ρ ( piR +1 , TiR +1 ) .
3. 4.
(2.400) , (
5.
(2.388).
(
.
).
)
TiR +1 = T ( ε iR +1 , piR +1 ) .
К
⎡ ⎤ ⎛ wiR2 +1 ⎞ ρ ⋅ ε + ⎢ iR +1 ⎜ iR +1 ⎟ + piR +1 ⎥ ⋅ wiR +1 ⋅ f iR +1 = 2 ⎠ ⎝ ⎣⎢ ⎦⎥
⎡ ⎤ Φ ⋅ ( iR − iL ) ⎛ w2 ⎞ , = ⎢ ρiL −1 ⋅ ⎜ ε iL −1 + iL −1 ⎟ + piL −1 ⎥ ⋅ wiL −1 ⋅ f iL −1 − 2 ⎠ f iL −1 ⎢⎣ ⎥⎦ ⎝
Φ –
(Φ > 0 –
), Вт . В
(2.400) (
.
,
(
Φ
–
. .
)).
-
, :
ε iR +1 ;
wiR +1 ;
.К
, С.Н. Прял в, 2007–2009
– –
–
, ев, В.В. Алеш
iL
iR (
( iR + 1)
ρ iR +1 ;
-
2.7.3). В ( iR − iL )
–
.
© В.Е. Селе
(2.400)
piR +1 ;
–
TiR +1 .
-
196 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ ,
–
⎡⎣ xiR +1 , xiR+1,5 ⎤⎦ . Н
,
, .А
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ .
(2.399)
: wiL −1 =
Qvalve –
-
Qvalve , ρ iL −1 ⋅ f iL−1
(2.401 )
:
Qvalve = wiR +1 ⋅ ρiR +1 ⋅ f iR +1 . 1
2.1)
( iL − 1)
Q = Qvalve ( t ) (
(2.401 ) (
.
Q –
. ). -
, 2
.
y (
ρ , p, w
. .)
yi → yi +1
(2.402) . ( iL − 2, iL − 1)
(2.402) , ,
.
.
. 2.1,
,
.И «
» ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦
, :
ρiL −1 ⋅ wiL −1 ⋅ f iL −1 − 0,5 ⋅ ( ρiL −1 ⋅ wiL −1 ⋅ f iL −1 + ρiL − 2 ⋅ wiL − 2 ⋅ f iL − 2 ) 0,5 ⋅ ( xiL −1 − xiL − 2 )
= 0,
(2.403)
,
ρiL −1 = ρiL − 2 ⋅
wiL − 2 . wiL −1
(2.404) -
(2.404)
, ,
1
В
(2.403),
.
2
© В.Е. Селе
,
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 197 _______________________________________________________________________________________
( iL − 2 )
. .
( iL − 1)
1
,
(
)
.
-
(2.401) ,
, )
ρ , p, w
, )
. ),
y (
. .)
:
yiL − 2 → yiL −1 .
(2.405)
( iL − 2 )
(
«
»
е
, я
е
-
. Д
, . . (2.399) ( ( .
е
е
а
х -
ех е е ( iL − 2 ) ). Э
а ь
а е
х
2
.
е е а ь е ( iL − 1)
) х а
,
х а е
,
я ( . . ( iL − 1)
е
-
а а а
а
( а ь-
.
( iL − 2, iL − 1)
, (2.402)
3
.
4
(
)
(2.36)
d ( ρ ⋅ w2 ) dx
-
.Д :
= 0.
(2.406) , «
( iL − 2 ) )
» (
:
1 ⋅ ⎡0, 25 ⋅ ( ρiL −1 ⋅ wiL −1 + ρiL − 2 ⋅ wiL − 2 ) ⋅ ( wiL −1 + wiL − 2 ) − 0,5 ⋅ ( xiL −1 − xiL −3 ) ⎣ −0, 25 ⋅ ( ρiL − 2 ⋅ wiL − 2 + ρiL −3 ⋅ wiL −3 ) ⋅ ( wiL − 2 + wiL − 3 ) ⎤⎦ = 0.
(2.407)
. .
(
.
), yiL −3 → yiL − 2 . 1
Н
2
,
3 4
⎡⎣ xiL−2,5 , xiL−1,5 ⎤⎦
,
(2.408)
.
⎡⎣ xiL−1,5 , xiL−1 ⎤⎦ .
,
.
Д
© В.Е. Селе
, ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
198 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(
)
K = ρiL − 2 ⋅ wiL2 − 2 = ρiL −3 ⋅ wiL2 −3 .
(2.409)
(2.407), K ( ⎣⎡0,5 ⋅ ( xiL −1 − xiL −3 )⎦⎤ :
2.409), (2.407)
,
(2.408, ).
0, 25 ⋅ ( ρiL −1 ⋅ wiL −1 + ρiL − 2 ⋅ wiL − 2 ) ⋅ ( wiL −1 + wiL − 2 ) = K .
(2.410)
(2.404) (2.410):
(2.401)
ч
, е е е
е е Д
е
0,5 ⋅ ρ iL − 2 ⋅ wiL − 2 ⋅ ( wiL −1 + wiL − 2 ) = K = ρ iL − 2 ⋅ wiL2 − 2 .
(2.411)
wiL −1 = wiL − 2 .
(2.412)
ρiL −1 = ρiL − 2 .
:
К
е а ач я яе я а ρ = ρ ( x) w = w( x) е
е е
, е а
(2.413)
е
х
а
. Д
, ае
.Ч
а
. .И
, «
»
, ( » .Э (
. .
( iL − 1)
. В , « [95]). «
. » -
( iL − 2 )
(2.404)) .
Q = Q (t ) (
. К
,
, )
-
.
,
, , .Н
, .
. -
⎣⎡ xiR +1 , xiR +1,5 ⎦⎤ , ,
.Д ( Qi −0,5 = Qi + 0,5 ,
1
Qi ± 0,5 = 0,5 ⋅ [ ρi ⋅ wi ⋅ f i + ρi ±1 ⋅ wi ±1 ⋅ f i ±1 ]) ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ ,
1
Н
© В.Е. Селе
1
.
, ев, В.В. Алеш
Д
.
, С.Н. Прял в, 2007–2009
2.1,
:
QiL −1,5 = QiL −1 , .
-
лава 2 199 _______________________________________________________________________________________
QiL −1,5 = 0, 5 ⋅ [ ρiL −1 ⋅ wiL −1 ⋅ f iL −1 + ρiL − 2 ⋅ wiL − 2 ⋅ f iL − 2 ] ;
( iL − 1)
QiL −1 = ρiL −1 ⋅ wiL −1 ⋅ f iL −1 .
( iR + 1) ,
(
2.19), QiR +1 = ρiR +1 ⋅ wiR +1 ⋅ f iR +1 .
( iR + 1)
Д
(
.
. (2.399)): QiL −1 = QiR +1 ,
.
, xiR +1 ).
( . .
-
,
. А
,
xiR +1,5 = 0,5 ⋅ ( xiR +1 + xiR + 2 )
,
( iR + 1) )
. xiR +1
1
-
xiR +1,5 (
, . . (
.
. 2.2). ,
«
⎡⎣ xiR +1 , x iR +1,5 ⎤⎦ .
» , ,
. Д .
-
. В , ,
.В (
-
),
. .
-
.
Д
(
)
, .
5
-
. . , 2
.
-
: 3
4
.
Д . 1
Д
«
( iR + 2 ) )
«
» »
⎡⎣ xiR+1 ; x iR+1,5 ⎤⎦ (
⎡⎣ xiR+1,5 ; x iR+2,5 ⎤⎦ (
( iR + 1) ),
. 2
,
.
3 4
. Д
( .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
[69])
-
200 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
В
.Д (
iR ,
iL
.
. 2.19)
f valve = kvalve ⋅ f ,
; kvalve ∈ [ 0,1] –
f –
.
( iL − 1) ,
kvalve = 0
(2.414) ,
iL , iR , ( iR + 1)
(
kvalve = 1
. В
)
,
«
».
.Д
,
( iL − 1)
,
.
iL .
,
iL
,
. В
. В
, (
и . 2.20. Схе а ече
В ( iL − 1)
-
. 2.20).
я а а чере
у е
е
ереч
ече
я
, iL (
.
. 2.19)
( .В
), ,
,
.В (2.36).
: 1)
;
2)
;
3) © В.Е. Селе
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 201 _______________________________________________________________________________________
Э
. Э
, [66].
,
И
, ,
.
, ),
(
∂ ( ρ ⋅ f ⋅ w2 )
:
∂x
(2.36 ) +f⋅
∂p = 0. ∂x
(2.415)
Δx0,1 = [ x0 , x1 ] (
ρ1 ⋅ f1 ⋅ w12 − ρ 0 ⋅ f 0 ⋅ w02 + f ⋅ ( p1 − p0 ) = 0, f –
f =
Δx0,1
. 2.20). (2.416 )
Δx0,1 .
f
:
.
-
f 0 + f1 . 2
(2.36 ) ( . . 2.20)
( ρ ⋅ f ⋅ w)1 − ( ρ
(2.417) (2.36 ) :
-
⋅ f ⋅ w )0 = 0;
(2.416 )
⎡ ⎛ ⎛ w2 p ⎞ ⎤ ⎡ w2 p ⎞ ⎤ ρ ⋅ ⋅ ⋅ ε + + − ρ ⋅ ⋅ ⋅ ε + + ⎟ ⎥ = 0. f w f w ⎢ ⎜ ⎟⎥ ⎢ ⎜ 2 ρ ⎠ ⎦1 ⎣ 2 ρ ⎠⎦0 ⎝ ⎝ ⎣ Φ
, , (2.416 )
(2.416 )
.
-
. (2.416 )
(2.416 )
(2.416 ), ,
x1 (
. w1 =
-
. 2.20):
ρ 0 ⋅ f 0 ⋅ w0 ; f1 ⋅ ρ1
p1 = p0 − ⎛
ε1 = ⎜ ε 0 + ⎝
(2.418 )
ρ1 ⋅ f1 ⋅ w12 − ρ 0 ⋅ f 0 ⋅ w02 f
= p0 − ρ 0 ⋅ f 0 ⋅ w0 ⋅
w02 p0 ⎞ ⎛ w12 p1 ⎞ + ⎟−⎜ + ⎟; 2 ρ 0 ⎠ ⎝ 2 ρ1 ⎠
w1 − w0 ; f
(2.418 ) (2.418 )
T1 = T ( ε1 , p1 ) ; ρ1 = ρ ( p1 , T1 ) ;
(2.418 )
f = 0,5 ⋅ ( f 0 + f1 ) .
(2.418 )
x1 © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
, (2.418). В )
-
202 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
,
-
.
x0 –
(
)
,
,
. , (
), )
wiL =
(2.419 )
piL = piL −1 − ρ iL −1 ⋅ f iL −1 ⋅ wiL −1 ⋅ ⎛
⎝
(
:
ρiL −1 ⋅ wiL −1 ; kvalve ⋅ ρiL
ε iL = ⎜ ε iL −1 +
-
.Д
wiL − wiL −1 ; f iL −0,5
( 2.419 )
wiL2 −1 piL −1 ⎞ ⎛ wiL2 piL ⎞ + + ⎟−⎜ ⎟; ρiL −1 ⎠ ⎝ 2 ρiL ⎠ 2
(2.419 )
TiL = T ( ε iL , piL ) ; ρiL = ρ ( piL , TiL ) ;
(2.419 )
f iL − 0,5 = 0,5 ⋅ fiL −1 ⋅ (1 + kvalve ) .
Д
wiR =
(
(2.419 )
)
ρiR +1 ⋅ wiR +1 ; kvalve ⋅ ρiR
: (2.420 )
piR = piR +1 − ρ iR +1 ⋅ f iR +1 ⋅ wiR +1 ⋅ ⎛
ε iR = ⎜ ε iR +1 + ⎝
wiR − wiR +1 ; f iR + 0,5
(2.420 )
wiR2 +1 piR +1 ⎞ ⎛ wiR2 piR ⎞ + + ⎟−⎜ ⎟; ρiR +1 ⎠ ⎝ 2 ρiR ⎠ 2
(2.420 )
TiR = T ( ε iR , piR ) ; ρiR = ρ ( piR , TiR ) ;
(2.420 )
f iR + 0,5 = 0,5 ⋅ f iR +1 ⋅ (1 + kvalve ) .
А
(2.420 )
iL
-
iR
.
. ( ,
. ( .,
К 1
( 2 ,
)
.В
-
. , [84]), [84]
p0 а е
,
-
е ),
.Д ,
. ,
1
Д
2
К
© В.Е. Селе
. Н
, ,
( ,
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
). .
лава 2 203 _______________________________________________________________________________________
,
Ч «
,
,
.
, »
,
.
е
p0
В
. ,
p0
( ).
Н
. , (
iR ). 1
, (
,
)
iR
γ –
:
c0 = γ ⋅ R ⋅ T ,
(2.421)
; R –
. . ,
,
« ,
.
»
-
, . . , . -
Н ( . . ).
). -
(
И
,
( iL − 1)
, .
iR iR (
iL
.
, . .
yL = yR ,
:
( ,
-
(2.422)
y –
Д
. 2.19)
,
,
. .). . 2.20. В iL ( iL − 1) ,
, ,
-
wiL = wiR = ( iR − iL ) ⋅ ( c0 )iR ;
(2.423 )
wiL −1 =
(2.423 )
(
.
(2.419), (2.421)):
wiL ⋅ kvalve ⋅ ρiL
ρiL −1
;
piL = piL −1 − ρ iL −1 ⋅ f iL −1 ⋅ wiL −1 ⋅
1
© В.Е. Селе
, ( ев, В.В. Алеш
wiL − wiL −1 ; f iL −0,5
(2.423 )
.
. 2.1
2.2,
, С.Н. Прял в, 2007–2009
(2.419), (2.420)).
204 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛
ε iL = ⎜ ε iL −1 + ⎝
wiL2 −1 piL −1 ⎞ ⎛ wiL2 piL ⎞ + + ⎟−⎜ ⎟; ρiL −1 ⎠ ⎝ 2 ρiL ⎠ 2
(2.423 )
TiL = T ( ε iL , piL ) ; ρiL = ρ ( piL , TiL ) ;
(2.423 )
f iL −0,5 = 0,5 ⋅ f iL −1 ⋅ (1 + kvalve ) ;
(2.423 )
( c0 )iR = γ iR ⋅ R ⋅ TiR . В
(2.423 )
( iR − iL )
(2.423 )
Д
, Д
iL ,
,
( iL − 1) ),
.В
iL > iR (
, ( iR − iL )
,
iR .
( Q = Qvalve ( t ) . Д
, -
( iL − 1) ). ,
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ (
-
iR − iL = −1 ).
,
. 2.3
«
»
Та ли а 2.3
( iL − 1) .
1.
,
2.
piL −1 = p ( ρiL −1 , TiL −1 ) .
3. 4.
TiL −1 = T ( ε iL −1 , piL −1 ) .
К
5.
,
iL
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ . ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ .
.Д -
. В
iR
: Qvalve = ρiR ⋅ wiR ⋅ f iR = ρiR ⋅ wiR ⋅ f iL −1 ⋅ kvalve . В :
( iL − 1) ,
(2.423 )). ,
-
( iL − 1)
iL , iR
( iR + 1)
, QiL −1 = Qvalve . Э
,
Qvalve
(
.,
,
-
. 2.3. , © В.Е. Селе
ев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
лава 2 205 _______________________________________________________________________________________
.Д
piL (2.423 )
( iL − 1)
-
ε iL
.
, (2.423)
iL
(2.423 ). ( iL − 1) .
,
»
«
( iR + 1)
. 2.3.
( iR + 1) ,
) , 1
(
iR .
, -
,
. ,
( iR + 1) )
( iR + 1)
, . = Qvalve .
: QiR +1
,
-
, ( ,
. 2.4. Та ли а 2.4
( iR + 1) .
1.
,
2.
iR
⎡⎣ xiR +1 , xiR +1,5 ⎤⎦ .
piR +1 = p ( ρiR +1 , TiR +1 ) .
3. 4. iR
( iR + 1)
К
5.
(2.424) (
.
).
TiR +1 = T ( ε iR +1 , piR +1 ) .
2 2 ⎛ ⎞ ⎛ ⎞ wiR ) ( ( iR − iL ) ⋅ Φ piR ⎟ ⎜ ( wiR +1 ) p ⎜ . ε iR +1 = ε iR + + − + iR +1 ⎟ − ⎜ 2 ρ iR ⎟ ⎜ 2 ρ iR +1 ⎟ ρ iR +1 ⋅ wiR +1 ⋅ f iR +1 ⎝ ⎠ ⎝ ⎠
(2.424)
(2.416 ).
.
Д
,
-
.
, «
–
».
2
,
1
(2.424)
Н
.
2
Д . © В.Е. Селе
-
ев, В.В. Алеш
,
, С.Н. Прял в, 2007–2009
,
206 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
.И (
.
. 2.19). ;
, (
iL ; iR – ( iL − 1) ( iR + 1) –
)
(
-
) (
iR )
iL
,
. -
-
, -
(
). -
. ,
-
. Н y (
,
k. Д ,
,
. .)
lim yik = yij +1 (
k →∞
( j + 1) -
j-
1. ДЛ
Д
Р Т ЧЕС
ы
1.1.
1.1.1.
Г
я
ρiLk +−11 − ρiLj −1 t j +1 − t j
+
,
ρiLk +−11 − ρiLj −1 t j +1 − t j
ТЕЧЕН
.
ГАЗА ЧЕРЕЗ РАН.
( iL − 1) .
ия га а
1
ρiLk −1 ⋅ wiLk −1 − 0,5 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 )
-
,
0,5 ⋅ ( xiL −1 − xiL − 2 )
=0
(2.425 *)
:
+
ρiLk −1 ⋅ wiLk −1 − ρiLk − 2 ⋅ wiLk − 2 xiL −1 − xiL − 2
(
= 0.
(2.425 )
. (2.399)) ( wiLk +−11 =
ρiRk +1 ⋅ wiRk +1 . ρiLk −1 ρiLk +−11 , . .
Д .И
wiLk +−11 ):
(2.425 )
( iL − 1)
( k + 1) -
. . © В.Е. Селе
ó
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ (
1.1.2.
1
-
).
и а а
ρiLk +−11 ). Н
,
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 207 _______________________________________________________________________________________
( iL − 1)
1.1.4.
(
ε iLk +−11 ). Н
,
t j +1 − t j
ρiLk −1 ⋅ ( wiLk −1 ) − ρiLj −1 ⋅ ( wiLj −1 ) 2 ⋅ ( t j +1 − t j )
+
2
3
=−
0,5 ⋅ ( xiL −1 − xiL − 2 )
+
0,5 ⋅ ( xiL −1 − xiL − 2 )
=
( z1 )iL −1 − ( z1 )iL − 2 piLk −1 ⋅ wiLk −1 − piLk − 2 ⋅ wiLk − 2 − ρiLk −1 ⋅ wiLk −1 ⋅ g ⋅ + xiL −1 − xiL − 2 xiL −1 − xiL − 2 k
+QiLj +−11 −
Φ (TiLk−1 , Toc ) f iL−1
(2.425 )
( iL − 1)
TiLk −+11 = T ( ε iLk −1 , piLk −1 ) .
ы
и а а
(
ия га а
(2.425 )
( iR + 1) .
): =p
Kk =
k iL −1
−K
k
⋅ζ ( w
piLk −1 − piRk +1 ⋅ piLk −1
γ
ρ вk ⋅ ⎡⎣ w0,k в ⎤⎦ ; )⋅ 2
k 0, в
2
γ iL −1 +1 ⎤ 2 ⎡ k ⋅ ⎢( piRk +1 piLk −1 ) γ iL−1 − ( piRk +1 piLk −1 ) γ iLk −1 ⎥ ⎣⎢ ⎦⎥ k
k iL −1
( iL − 1) ), ев, В.В. Алеш
(2.426 )
γ iLk −1 − 1
ζ
© В.Е. Селе
TiLk −+11 ):
(2.392) (
k +1 iR +1
p
я
k
.
1.1.5. К
p
+
0,5 ⋅ ρiLk −1 ⋅ ( wiLk −1 ) − 0, 25 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ wiLk −1 ⋅ wiLk − 2
+
k +1 iR +1
2
ρiLk −1 ⋅ wiLk −1 ⋅ ε iLk −1 − 0, 25 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ ( ε iLk −1 + ε iLk − 2 )
+
1.2.
(2.425 ) ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ (
ρiLk −1 ⋅ ε iLk +−11 − ρiLj −1 ⋅ ε iLj −1
1.2.1.
piLk +−11 ):
piLk +−11 = p ( ρiLk −1 , TiLk −1 ) .
1.1.3.
(2.426 )
.
(
, С.Н. Прял в, 2007–2009
-
208 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ζ ( w0,k в
В
ρ вk = ρ iLk −1 ,
)
Kk =1.
piLk −1 − piRk +1 0 – 1.2.2.
w0,k в = wiLk −1 .
+
0, 25 ⋅ ( ρiRk +1 ⋅ wiRk +1 + ρiRk + 2 ⋅ wiRk + 2 ) ⋅ ( wiRk +1 + wiRk + 2 ) − ρiRk +1 ⋅ ( wiRk +1 ) 0,5 ⋅ ( xiR + 2 − xiR +1 )
2
( z1 )iR + 2 − ( z1 )iR +1 p k − piRk +1 = − iR + 2 − ρiRk +1 ⋅ g ⋅ − xiR + 2 − xiR +1 xiR + 2 − xiR +1 k
−
λ ( wiRk +1 ) ⋅ ρ iRk +1 ⋅ wiRk +1 ⋅ wiRk +1 ⋅ π 4 ⋅ f iR +1
k
(2.426 )
.
( iR + 1)
ρiRk ++11 ):
ρiRk ++11 = ρ ( piRk +1 , TiRk +1 ) .
1.2.3.
=
(
(2.426 )
1.2.4.
, ( k +1 ε iR +1 ) ( . (2.400)):
ε
k +1 iR +1
=ε
k iL −1
(w ) − (w ) + 2 k iL −1
2 k iR +1
2
Φ –
( iR + 1)
(Φ > 0 –
я
и а а
(
ия га а
1.3.1. © В.Е. Селе
ев, В.В. Алеш
(2.426 ) ), Вт . В
.
TiRk ++11 = T ( ε iRk +1 , piRk +1 ) .
ы
-
(
Φ ⋅ ( iR − iL ) ⎛ pk pk ⎞ + ⎜ iLk −1 − iRk +1 ⎟ − k , k ⎝ ρiL −1 ρiR +1 ⎠ wiR +1 ⋅ f iR +1 ⋅ ρiR +1
Φ
1.2.5. К
1.3.
)
, С.Н. Прял в, 2007–2009
TiRk ++11 ):
(2.426 )
iL .
( iL − 1)
iL (
.
лава 2 209 _______________________________________________________________________________________
wiLk +1 ):
(2.419 )) (
ρiLk −1 ⋅ wiLk −1 . kvalve ⋅ ρiLk
wiLk +1 =
1.3.2.
k +1 iL
p
(2.419 )) (
(2.427 )
( iL − 1)
):
piLk +1 = piLk −1 − ρiLk −1 ⋅ f iL −1 ⋅ wiLk −1 ⋅
wiLk − wiLk −1 , f iL − 0,5
f iL − 0,5 ≈ 0,5 ⋅ fiL −1 ⋅ (1 + kvalve ) .
1.3.3.
ε
(2.419 )) (
ε 1.3.4.
ρ
k +1 iL
T
k +1 iL
ы
1.4.
k +1 iL
k +1 iL
я
1.4.1. (2.420 )) (
(2.420 )) (
(2.427 )
( iL − 1)
и а а
ия га а
w
k +1 iR
p
(2.427 )
iR .
( iR + 1)
(
.
):
ρiRk +1 ⋅ wiRk +1 . kvalve ⋅ ρiRk
(2.428 )
iR
):
wiRk − wiRk +1 , f iR + 0,5
1.4.3.
(
.
(2.428 )
iR , С.Н. Прял в, 2007–2009
( iR + 1)
(2.428 )
f iR + 0,5 ≈ 0,5 ⋅ f iR +1 ⋅ (1 + kvalve ) .
ев, В.В. Алеш
.
. (2.419 )) (
iR
piRk +1 = piRk +1 − ρiRk +1 ⋅ fiR +1 ⋅ wiRk +1 ⋅
© В.Е. Селе
iL (
(2.427 )
TiLk +1 = T ( ε iLk , piLk ) ; ρiLk +1 = ρ ( piLk , TiLk ) .
wiRk +1 = 1.4.2.
(2.427 )
2 ⎛ ⎞ ⎛ k 2 ⎞ wiLk −1 ) ( piLk −1 ⎟ ⎜ ( wiL ) pk k ⎜ = ε iL −1 + + k − + iLk ⎟ . ⎜ ρiL −1 ⎟ ⎜ 2 ρiL ⎟ 2 ⎝ ⎠ ⎝ ⎠
k +1 iR
.
):
iL (
):
iL (
( iR + 1)
(
.
210 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ε iRk +1 ):
(2.420 )) (
ε
1.4.4.
2. ДЛ
k +1 iR
ρiRk +1 ):
TiRk +1
СВЕРХ Р Т ЧЕС
2 ⎛ ⎞ ⎛ k 2 ⎞ wiRk +1 ) ( piRk +1 ⎟ ⎜ ( wiR ) pk k ⎜ = ε iR +1 + + k − + iRk ⎟ . ⎜ ρiR +1 ⎟ ⎜ 2 ρiR ⎟ 2 ⎝ ⎠ ⎝ ⎠
iR (
Г
ТЕЧЕН
ГАЗА ЧЕРЕЗ
k +1 2 iR
2.1.1.
2.1.2.
2.1.3.
я
k +1 2 iR
и а а
wiRk +1 = ( iR − iL ) ⋅ γ iRk ⋅ R ⋅ TiRk .
iR (
) -
а .
:
= γ iRk +1 ⋅ R ⋅ TiRk +1 .
(2.429) iR .
wiRk +1 ):
. (2.423)) (
piRk +1 ):
. (2.422)) (
ρiRk +1 ):
. (2.422)) (
iR (
. (2.422)) (
-
ε iRk +1 = ε iLk . 2.1.5.
2.2.1.
(2.430 ) iR (
TiRk +1 = TiLk .
ы
2.2.
я
wiLk +1 = wiRk .
TiRk +1 ):
. (2.422)) (
(2.430 ) и а а
ия га а iL (
. (2.422)) (
iL .
wiLk +1 ): (2.431 )
2.2.2. © В.Е. Селе
iL ев, В.В. Алеш
(2.430 )
(2.430 )
ε iRk +1 ):
2.1.4.
,
(2.430 )
iR (
ρiRk +1 = ρiLk .
(2.428 )
Н
ия га а
iR (
piRk +1 = piLk .
РАН.
( ия га а
г
( w ) ≥ (c ) ы
. (2.420 )) (
TiRk +1 = T ( ε iRk , piRk ) ; ρiRk +1 = ρ ( piRk , TiRk ) .
и и
2.1.
(2.428 )
, С.Н. Прял в, 2007–2009
( iL − 1)
(
.
лава 2 211 _______________________________________________________________________________________
piLk +1 ):
(2.423 )) ( piLk +1 = piLk −1 − ρ iLk −1 ⋅ f iL −1 ⋅ wiLk −1 ⋅
2.2.3.
ε
(
ε 2.2.4.
k +1 iL
k +1 iL
wiLk − wiLk −1 . f iL − 0,5
(2.431 ) iL
( iL − 1)
ρiLk +1 ):
(2.431 )
iL (
. (2.423 )) (
TiLk +1 = T ( ε iLk , piLk ) ; ρ iLk +1 = ρ ( piLk , TiLk ) .
ы
2.3.
2.3.1.
я
( iL − 1) ( wiLk +−11 =
2.3.2.
ρ
и а а
k +1 iL −1
(2.431 )
ρiLk −1
, wiLk +−11 )1:
(2.432 ) ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ (
). Н
ε iLk +−11 ). Н
,
t j +1 − t j
+
1
© В.Е. Селе
+
(2.432 ) ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ (
ρiLk −1 ⋅ ( wiLk −1 ) − ρiLj −1 ⋅ ( wiLj −1 ) 2 ⋅ ( t j +1 − t j ) 2
2
+
ρiLk −1 ⋅ wiLk −1 ⋅ ε iLk −1 − 0, 25 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ ( ε iLk −1 + ε iLk − 2 ) 0,5 ⋅ ( xiL −1 − xiL − 2 )
+
0,5 ⋅ ( xiL −1 − xiL − 2 )
=
0,5 ⋅ ρiLk −1 ⋅ ( wiLk −1 ) − 0, 25 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ wiLk −1 ⋅ wiLk − 2
Д ,
-
,
ρiLk −1 ⋅ ε iLk +−11 − ρiLj −1 ⋅ ε iLj −1 +
iL
.
ρiLk +−11 − ρiLj −1 ρiLk −1 ⋅ wiLk −1 − ρiLk − 2 ⋅ wiLk − 2 + = 0. t j +1 − t j xiL −1 − xiL − 2 2.3.3.
( iL − 1) .
ия га а
. (2.423 )) (
wiLk ⋅ kvalve ⋅ ρ iLk
. (2.423 ))
):
2 ⎛ ⎞ ⎛ k 2 ⎞ wiLk −1 ) ( piLk −1 ⎟ ⎜ ( wiL ) pk k ⎜ = ε iL −1 + + k − + iLk ⎟ . ⎜ 2 ρ iL −1 ⎟ ⎜ 2 ρ iL ⎟ ⎝ ⎠ ⎝ ⎠
TiLk +1
(
3
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
212 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
=−
( z1 )iL −1 − ( z1 )iL − 2 piLk −1 ⋅ wiLk −1 − piLk − 2 ⋅ wiLk − 2 − ρiLk −1 ⋅ wiLk −1 ⋅ g ⋅ + xiL −1 − xiL − 2 xiL −1 − xiL − 2 k
+QiLj +−11 −
2.3.4.
Φ (TiLk−1 , Toc ) f iL−1
k
(2.432 )
.
( iL − 1)
piLk +−11 ):
TiLk −+11
(
TiLk −+11 = T ( ε iLk −1 , piLk −1 ) ; piLk +−11 = p ( ρiLk −1 , TiLk −1 ) .
ы
2.4.
( iR + 1)
2.4.1.
wiRk ++11 =
я
и а а
ρiRk +1
( iR + 1) .
ия га а
,
wiRk ++11 ):
(
wiRk ⋅ kvalve ⋅ ρ iRk
(2.432 )
iR
.
(2.433 )
ρiRk ++11 ).
⎡⎣ xiR +1 , xiR +1,5 ⎤⎦ (
2.4.2. Н
,
ρiRk ++11 − ρiRj +1 ρiRk +1 ⋅ wiRk +1 − ρiRk + 2 ⋅ wiRk + 2 + = 0. t j +1 − t j xiR +1 − xiR + 2
2.4.3.
ε
(
ε
2.4.4.
k +1 iR +1
k +1 iR +1
iR
). Д
:
(
. (2.424))
):
( iR + 1)
TiRk ++11 = T ( ε iRk +1 , piRk +1 ) ;
(
piRk ++11 = p ( ρiRk +1 , TiRk +1 ) .
(2.433 ) TiRk ++11
(2.433 )
, «1 »
(2.54 ) « 2 ». Д
( ρ ⋅ w ⋅ f ⋅ Ym )2 − ( ρ ⋅ w ⋅ f ⋅ Ym )1 = 0, © В.Е. Селе
( iR + 1)
2 ⎛ ⎞ ⎛ k 2 ⎞ wiRk ) ( ( iR − iL ) ⋅ Φ piRk ⎟ ⎜ ( wiR +1 ) pk k ⎜ = ε iR + + k − + iRk +1 ⎟ − k . ⎜ ⎟ ⎜ ⎟ ρ iR ρ iR +1 ρ iR +1 ⋅ wiRk +1 ⋅ f iR +1 2 2 ⎝ ⎠ ⎝ ⎠
piRk ++11 )
(
(2.433 )
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
:
m = 1, N S ,
(2.434 )
лава 2 213 _______________________________________________________________________________________
⎡⎣( ρ ⋅ w ⋅ f )2 = ( ρ ⋅ w ⋅ f )1 ⎤⎦ :
,
(Ym )2 = (Ym )1 ,
Н
( iL − 1) ,
m = 1, N S .
(2.434 )
,
iL , iR , ( iR + 1)
-
, :
(Ym )iL −1 = (Ym )iL = (Ym )iR = (Ym )iR +1 ,
m = 1, N S .
(2.434 )
Д
,
.
,
p = p ( ρ,T )
{S ме и }
–
. В ε = ε ( p, T ) (2.54)
: p = p ({S ме и })
,
(
)
(
,
ε = ε ({S ме и }) , .
)
: p = p ρ , T , Y0 , Y1 , ..., YN S ; ε = ε p, T , Y0 , Y1 , ..., YN S . Н
( «*». Н
-
),
.Д
.
1*. ДЛ
Д Р Т ЧЕС Г ТЕЧЕН Н Г РАН (Д П Н НИ И М ДИФИКАЦИИ).
ы
1.1*. (
1.1.3*.
е
я
и
я
а а а
(
(
( iL − 1)
(
k
( iL − 1)
(
TiLk −+11 = T ε iLk −1 , piLk −1 , ( Y0 )iL −1 , (Y1 )iL −1 , ...,
1.1.6.
© В.Е. Селе
k
(Ym )iL −1 , m = 1, N S ). Н k +1
ев, В.В. Алеш
С ЕС
и
ЧЕРЕЗ
( iL − 1)
).
k
k
ГАЗ В
ия га
piLk +−11 = p ρiLk −1 , TiLk −1 , ( Y0 )iL −1 , (Y1 )iL −1 , ...,
1.1.5*. К
П НЕНТН
, С.Н. Прял в, 2007–2009
(Y ) NS
(Y ) NS
k iL −1
k iL −1
,
).
).
piLk +−11 ): (2.425 *) TiLk −+11 ):
(2.425 *) ⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ ( -
214 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ρiLk −1 ⋅ (Ym )iL −1 − ρiLj −1 ⋅ (Ym )iL −1 k +1
j
t j +1 − t j
+
k
ы
1.2*. е
(
(Y ) NS
я
k +1
iL −1
и
я
= 1−
∑ (Y )
N S −1
k +1 m iL −1
m =1
(Ym )iR +1 , k +1
( iR + 1)
(
k
( iR + 1)
k
(
k +1
1.3*.
k
е
(
ы
k
я
я
и а
( = ρ ( p , T , (Y ) k iL
(Ym )iL
k +1
(Ym )iL
1.4*. (
k iR +1
(Y ) NS
k iR +1
).
(2.426 *)
).
е
TiRk ++11 ):
( iR + 1)
(2.426 *) (
k
а а ).
k
k 0 iL
k iL
(2.426 )
, (Y1 )iL , ..., k
ия га
(Y )
(Y ) NS
NS
и
k iL
); ).
k iL
iL TiLk +1
iL (
(
= (Ym )iL −1 , m = 1, N S . k
ы
я
я
и а
а а ).
(2.427 ) ия га
iR ( ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
ρiLk +1 ):
(2.427 *) iL
1.4.4*. © В.Е. Селе
)
ρiRk ++11 ):
, m = 1, N S ):
k +1
( iR + 1)
и
m = 1, N S .
TiLk +1 = T ε iLk , piLk , ( Y0 )iL , (Y1 )iL , ...,
ρiLk +1
S
m = 1, N S ):
(Ym )iR +1 = (Ym )iL −1 ,
1.3.5.
(2.425 )
ия га
TiRk ++11 = T ε iRk +1 , piRk +1 , ( Y0 )iR +1 , (Y1 )iR +1 , ...,
1.3.4*.
) = 0,
k
).
k
(
k
.
а а
а
(
0,5 ⋅ ( xiL −1 − xiL − 2 )
ρiRk ++11 = ρ piRk +1 , TiRk +1 , ( Y0 )iR +1 , (Y1 )iR +1 , ..., (YN
1.2.5*. К
1.2.6.
(
ρiLk −1 ⋅ wiLk −1 ⋅ (Ym )iL −1 − 0, 25 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ (Ym )iL −1 + (Ym )iL − 2
m = 1, N S − 1;
1.2.3*.
+
и
iR TiRk +1
ρiRk +1 ):
); ).
лава 2 215 _______________________________________________________________________________________
( = ρ ( p , T , (Y )
TiRk +1 = T ε iRk , piRk , ( Y0 )iR , (Y1 )iR , ...,
ρiRk +1
k iR
(Ym )iR
1.4.5.
k +1
(Ym )iR
k
k
k 0 iR
k iR
, (Y1 )iR , ..., k
(Y )
(Y )
k
NS
iR
NS
iR
k
iR
, m = 1, N S ): k
(2.428 )
2*. ДЛ
СВЕРХ Р Т ЧЕС Г ТЕЧЕН Н Г ЧЕРЕЗ РАН (Д П Н НИ И М ДИФИКАЦИИ).
е
(
(Ym )iR
2.1.6.
k +1
(Ym )iR
ы
2.2.4*.
я
и
а а
ия га
ы
(
(Ym )iL
(Ym )iL
я
я
и а
(
а а ).
ия га
k
k
S
я
(2.431 *)
iL
(
я
(
(2.431 )
а а
а
ия га
).
( iL − 1)
TiLk −+11 = T ε iLk −1 , piLk −1 , ( Y0 )iL −1 , (Y1 )iL −1 , ..., © В.Е. Селе
iL
iL
и
ев, В.В. Алеш
k
ρiLk +1 ):
k
k
k
iL TiLk +1
iL (
k
е
piLk +−11 ):
k
); ) ).
(Y ) NS
и
= (Ym )iL −1 , m = 1, N S .
ы
2.3*.
2.3.4*.
(
(2.430 )
, m = 1, N S ):
k +1
(
iR
k
TiLk +1 = T ε iLk , piLk , ( Y0 )iL , (Y1 )iL , ...,
k +1
С ЕС
и iR
ρiLk +1 = ρ piLk , TiLk , ( Y0 )iL , (Y1 )iL , ..., (YN
2.2.5.
ГАЗ В
= (Ym )iL −1 , m = 1, N S .
е
(
я).
П НЕНТН
, m = 1, N S ):
k +1
2.2*.
(
= (Ym )iL −1 , m = 1, N S .
k +1
2.1*.
(2.428 *)
k
, С.Н. Прял в, 2007–2009
(Y ) NS
k iL −1
);
и
(
( iL − 1) TiLk −+11
).
216 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(
piLk +−11 = p ρiLk −1 , TiLk −1 , ( Y0 )iL −1 , (Y1 )iL −1 , ..., k
2.3.5.
k
(Y ) NS
k iL −1
⎡⎣ xiL −1,5 , xiL −1 ⎤⎦ ( -
(Ym )iL −1 , m = 1, N S ). Н k +1
ρiLk −1 ⋅ (Ym )iL −1 − ρiLj −1 ⋅ (Ym )iL −1 k +1
j
t j +1 − t j
+
,
+
k
ы
2.4*. е
(
(Y ) NS
я
(
= 1−
k +1
iL −1
и
я
piRk ++11 ):
0,5 ⋅ ( xiL −1 − xiL − 2 )
∑ (Y )
N S −1
k +1 m iL −1
m =1
ия га
).
(
k
piRk ++11 = p ρiRk +1 , TiRk +1 , ( Y0 )iR +1 , (Y1 )iR +1 , ...,
(Ym )iR +1 , k +1
(Y ) NS
k iR +1
(Y ) NS
k iR +1
m = 1, N S ):
(Ym )iR +1 = (Ym )iL −1 , k +1
а а е а. Д
k
k
а
че
( iR + 1)
); ).
TiRk ++11
(
(2.433 *)
( iR + 1)
(
m = 1, N S .
я
я
) = 0,
и
( iR + 1) k
k
k
(2.432 )
а а
а
k
.
TiRk ++11 = T ε iRk +1 , piRk +1 , ( Y0 )iR +1 , (Y1 )iR +1 , ...,
2.4.5.
(
ρiLk −1 ⋅ ( wiLk −1 ) ⋅ (Ym )iL −1 − 0,5 ⋅ ( ρiLk −1 ⋅ wiLk −1 + ρiLk − 2 ⋅ wiLk − 2 ) ⋅ (Ym )iL −1 + (Ym )iL − 2
m = 1, N S − 1;
2.4.4*.
(2.432 *)
(2.433 )
е
а а
щь
а а, ча
ζ.В
а
е
а е е е
, , ,
,
е
-
. ,
(
.
. 2.19).
. f1 ,
− f0 . К : © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
k
-
лава 2 217 _______________________________________________________________________________________
k=
, .
f1 . f0
(2.435) -
k
В
ζ
.К (
В (
)
. , f2
f1 , . В (
f0
-
f3 (
.
. 2.19))
,
)
-
. 1
.
К
,
,
,
-
: Δp =
ζ0 –
1 ⋅ ζ 0 ⋅ ρ ⋅ w02 , 2
; ρ –
ζ0
. ,
(2.436) ; w0 – . 2.19. -
«0»
w0
.В :
Δp =
λ1 –
l ⋅ λ1 ⋅ ρ ⋅ w12 , 2 ⋅ D1
(2.437) (
; D1 – . И
λ1
( ρ ⋅ w1 ⋅ f1 = ρ ⋅ w0 ⋅ f 0 ) , w1
.
2); w1 – ; l – «1» (2.435), «0»
w1 = w0 ⋅
И (2.435)
,
D0 –
f 0 w0 = . f1 k
, l =
k 2,5 ⋅ ζ 0 ( Re0 , k )
λ1 ( Re1 , Δ )
(2.439)
«0» ( . . 2.19). (2.436) (2.437), : =
k 2,5 ⋅ ζ 0 ( Re 0 , k )
λ1 ( Re0 k , Δ )
1
(2.440)
,
( .
© В.Е. Селе
ев, В.В. Алеш
«1»: (2.438)
D1 = k ⋅ D0 ,
,
. 2.19.
, С.Н. Прял в, 2007–2009
)
-
218 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
l = l D0 –
(
D0 )
); Δ = Δ D1 – ; Δ –
( -
. ,
.К
,
,
Re ,
k,
и . 2.21. Схе а
Н
. ,
l )».
(
. 2.21
ч
-
е ре
авле
[66] . 2.21.
е « ра а в е
(
л
«
р че
)»1
fh
, k = f h f0 . [66] δ ( .
. Д
, Re > 104 . . 2.5,
[66], ζ,
, ,
k.В . 2.5. Э
. 2.21)
[66]
, Re . Э
,
. Та ли а 2.5
δ°
5
10
20
30
40
50
55
67
k
0,93
0,85
0,69
0,52
0,35
0,19
0,11
0
0,05
0,31
1,84
6,15
20,7
95,3
275
ζ И
λ = λ ( Re, Δ )
-
, Re0
1
© В.Е. Селе
l = l ( Re, k )
2.
,
[66]. ев, В.В. Алеш
l
k.
. 2.22 2.23. К Re . Д
, С.Н. Прял в, 2007–2009
∞
Δ = 0, 05 l
лава 2 219 _______________________________________________________________________________________
Δ. Д
Δ
ζ
( Δ
е
ра а k
l = l (k ) р
и . 2.23. Зав
К
. 2.22
2.23,
рых
ев, В.В. Алеш
аче
ля Δ = 0, 05
е
рых
аче
ях
ях ч
э
е
ла Re
а
ры
ля Δ = 0, 05
l
. © В.Е. Селе
Re (
.
l = l ( Re ) р
и . 2.22. Зав
). Re > 104 ),
k , С.Н. Прял в, 2007–2009
220 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
( 0, 3 < k ≤ 1)
( l .
k)
,
.В
-
,
. , ,
.
, ( . .
В
1
k ).
λ = λ ( Re, Δ )
[66], :
(
)
,
. -
Δ
, .
А
« Δ = 0, 05
)». И .Д 2.23
Δ
(
-
. 2.22 .
Re
. Д -
ó
Re .
,
( Δ,
) -
l = l ( Re, k )
.
Δ
Re ,
.
, Д
.
k .
Д ,
, .Э
(2.440) . 2.22 l ( Re )
ó
., ζ
(
, [66]
2),
Δ,
l ( Re ) (
(
., k)
2
« ев, В.В. Алеш
-
(
-
, [66]
-
Re ,
.
© В.Е. Селе
Re ,
k
1
Э
,
0, 05 ,
2.23
.
2
. . ,
. )
2).
, С.Н. Прял в, 2007–2009
(
)».
лава 2 221 _______________________________________________________________________________________
l ( Re )
В
. 2.24 – 2.29. Δ : 0, 01 ; 0, 001 ; 0, 0001 .
Re
и . 2.24. Зав
l = l ( Re ) р
е
ра а k
и . 2.25. Зав © В.Е. Селе
ев, В.В. Алеш
l (k )
k,
l = l (k ) р
е
рых
аче
ля Δ = 0, 01
рых
, С.Н. Прял в, 2007–2009
аче
ях
ях ч
э
е
ла Re
-
а
ры
ля Δ = 0, 01
222 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
и . 2.26. Зав
l = l ( Re ) р
е
и . 2.28. Зав
ев, В.В. Алеш
ях
l = l (k ) р
е
рых
аче
ях ч
l = l ( Re ) р
е
рых
аче
ях
ра а k © В.Е. Селе
аче
ля Δ = 0, 001
ра а k
и . 2.27. Зав
рых
ля Δ = 0, 0001
, С.Н. Прял в, 2007–2009
э
е
ры
ля Δ = 0, 001
ла Re
э
а
е
а
ры
лава 2 223 _______________________________________________________________________________________
l = l (k ) р
и . 2.29. Зав
( ,
.
е
рых
аче
ях ч
ла Re
ля Δ = 0, 0001
. 2.24 – 2.29)
l ( Re )
,
10 < Re < 10 4
. В 5
,
,
ζ (k ) (
,
-
, .
. 2.5),
Δ
. , . ,
,
,
-
,
-
.
, (
)
,
-
.
. 1
( iL − 1 , iL )
( iR , iR + 1 ) (
( . .
), -
, .
. 2.19). .
(
) . -
,
.Д ),
(
. В
.Э ,
-
iR
.
-
: 1
Н
© В.Е. Селе
( ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
).
224 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
( iL − 1) ,
В
iL
( iL − 1)
Q = ρ iR ⋅ wiR ⋅ f iR .
( iR + 1)
(2.441) » ( iL − 2 ) ,
Q.
В
( iL − 1)
( iL − 1) ,
( iL − 1) 1. А (
« (
ρ iL = ρ iL −1 ,
-
( iR + 1) ,
)
( iR + 1) .
)
:
ρ iR = ρ iR +1.
(2.442)
А
.Д .
Д ( ,
. (2.54 )).
,
/ ,
,
,
(2.54 ).
:
⎛ ∂ ⎡ ∂ w2 ⎞ ⎤ ( p ⋅ f ⋅ w) . ⎢ρ ⋅ f ⋅ w ⋅ ⎜ ε + ⎟⎥ = − ∂ x ⎣⎢ ∂ 2 x ⎠ ⎦⎥ ⎝
(2.443)
[ xiL−1 , xiR +1 ] :
(2.443)
⎡ ⎡ ⎛ ⎛ ⎞⎤ ⎞⎤ w2 w2 + p ⎟⎥ − ⎢ f ⋅ w ⋅ ⎜ ρ ⋅ ε + ρ ⋅ + p ⎟ ⎥ = 0. ⎢ f ⋅w ⋅ ⎜ ρ ⋅ε + ρ ⋅ 2 2 ⎠ ⎦⎥ iR +1 ⎣⎢ ⎠ ⎦⎥iL −1 ⎝ ⎝ ⎣⎢
:
ε iR +1 =
⎧⎪ ⎡ ⎛ ⎞⎤ ⎛ ρ ⋅ w2 ⎞ ⎫⎪ w2 1 ⋅ ⎨⎢ f ⋅ w ⋅ ⎜ ρ ⋅ ε + ρ ⋅ + p ⎟⎥ ⋅ −⎜ + p ⎟ ⎬. 2 ρiR +1 ⎩⎪ ⎢⎣ ⎠ ⎥⎦iL −1 ( f ⋅ w )iR +1 ⎝ 2 ⎠iR +1 ⎭⎪ ⎝ 1
( iL − 1)
».
( iL − 2 ) , ( iL − 1)
, ( ( iL − 1) .
)
ε iL = ε iL −1 ,
«
»
«
ε iR = ε iR +1. »
. © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(2.445)
« ,
(2.445). В
1
(2.444)
– -
( iL − 1) . В
-
( iR + 1)
:
(2.446)
лава 2 225 _______________________________________________________________________________________
(T = T (ε , p ) ) . К
(
(2.225))
. (2.225) ( iL − 1 , iL )
( iR , iR + 1 ). (
.Д
.
-
)
-
, ), [ iL − 1 , iR + 1 ] -
( . , . ,
,
. «
/
»
, . 2.5.5. Чи ленны анали а о ы ановых лоща о ЛЧМГ, о о у ованных е ни очны и е е ыч а и
•
Д
Д
: -
;
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.
Ч
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,
). Н
Д 1200,
Д 1400 -
,
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100÷150км.
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,
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.Н.
(
.
226 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
).
. 2.30.
и . 2.30. Схе а
и . 2.31. М
е
ел
е
че
ра
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в
ра
л
в
л
. 2.31.
а
а
. .Н
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2, 4
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-
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и . 2.32. П
р
ая
ел
е
20÷50м,
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ра
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
в
л
а
0,02÷0,10 . В 10000 . ,
,
,
. 2.32).
, Ч
, -
лава 2 227 _______________________________________________________________________________________
-
И
К
«CorNet»
А
«Alfargus»,
,
. 1
,
-
, Ч
. В 2001 .Ц
Ч
. . 2.33. Н К 04. В ( . 2.33
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) К 04. 12\2-1) . 2.34, 0,6атм, SCADA, 20÷25км)
Д
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. Д
10 –
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К
К 03 К 12
12\2-1,
и . 2.33. Схе а ЧМ ( ече
К
-
А. . Ч .В
. К 10, К 11, К 13 №1 (
№2 ( ,
. 2.33) –
-
. 2.35. , 0
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, . 3÷4ми
А. . ев, В.В. Алеш
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-
, .Н.
, С.Н. Прял в, 2007–2009
( . В. .
, [6].
228 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
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, .
400км
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,
и . 2.34. Невя а
авле
ля ра л ч ых ра
вых л
а
ал
№1
и . 2.35. Невя а
авле
ля ра л ч ых ра
вых л
а
ал
№2
, .
,
-
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-
. © В.Е. Селе
-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 229 _______________________________________________________________________________________
В
, ,
«
, (
К Ы » .
): «
(
». (
« АК Ы ». Д ).
.В
.
),
, -
Д
. Ч
-
Д , .
(
-
):
•
(
);
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Ч
( );
К
,
( ,
В
). В
. .Н.
. ,
Н
. 2.30.
,
. 2.31. Н
(
и . 2.36. а
л
е
е
Н ,
«
. 2.36).
ел
ых ра
вв
ел
че
ра
в
л
а
С (
: ев, В.В. Алеш
е
». В А
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,
. Д
, С.Н. Прял в, 2007–2009
.
. 2.30). Э
. Н
, ,
-
230 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
= А + С,
(2.447) -
.А
«+» :
L=А+D
3;
(2.448)
M=C+B
2;
(2.449)
N=B+D
4.
(2.450)
(2.447 – 2.450),
,
-
. ,
,
2.36),
, –
,
,
А
).
С,
, , . .
А
( С( С(
А
. . .В . 2.30). -
. ,
Н
.Н А,
,
№1
,
(2.447)
№3,
L (2.448).
.Д
,
-
А. .
-
,
, .
Д . . Ва иа
1.
Д
,
. 2.30 (
),
-
[6]: №1;
1) 2)
№1
3) 4)
№3
A
5) 6)
№4
D
B
9) © В.Е. Селе
№4 №1;
M ев, В.В. Алеш
№3 №1;
N
№2
K
№1; L
7) 8)
№1,
, №1;
№2
, С.Н. Прял в, 2007–2009
, A
L
C (2.447);
№3 (2.448),
D (2.448); N
№4 (2.450),
B (2.450); M C (2.449);
№2 (2.449),
лава 2 231 _______________________________________________________________________________________
10)
C
№1
№1 (2.447),
K
№1,
-
M
,
; ;
11)
,
N
; ;
12)
,
L
; ;
13)
K
,
№1
;
;
14)
– . . №1, №2, №3
, Ва иа
№4 (
.
. 2.31).
2.
К
C
(
D
и . 2.37. На
. 2.37).
е
че
ра
в
л
а
е ра ы С
D – а ры ы
[6]: №1;
1) 2)
№1
3) 4)
№1,
, №1;
A
,
D
A
C (2.447);
№3 (2.448),
L
№3
L
6)
K
№1;
5)
,
№3
D (2.448);
,
L
; ; 7)
С
,
K
,
№1
, ;
№2;
8) 9)
,
© В.Е. Селе
ев, В.В. Алеш
№1, , С.Н. Прял в, 2007–2009
-
232 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
№2,
,
№2;
,
№2
10) 11)
B
№3
N
C (2.449);
№4 (2.450),
№4
D (2.448);
,
D
B N
№2;
12) 13)
M
,
N
; ; С
14)
,
M
,
№2
, 15)
-
; –
. . :
•
•
№1,
№1
№3;
№2,
№2
№4;
. 2.38.
и . 2.38. М
Ва иа
ел
ра
в
л
а
р
ере ры ых ра ах С
D
3. C
B(
. 2.39). №1, №3 №1.
, C
B (
-
№4, №2 M
,
,
. (2.449)). Н
а
е ере ры ы ра ы С
№2
. . 2.40.
и . 2.39. На © В.Е. Селе
ев, В.В. Алеш
е
че
ра
в
л
, С.Н. Прял в, 2007–2009
B
лава 2 233 _______________________________________________________________________________________
и . 2.40. М
ел
ра
в
л
а
р
а ры ых ра ах С
В
-
Д
[6]: ,
,
.
, -
, . ,
.Э ,
«
». 2.5.6. Чи ленны анали и ечения а а и авления в а о е у
у о
ово а вы о о о
Д (
,
В. .
) .Н. [2, 6]. (
–
. 2.41). Н
L
L.
M
и . 2.41. Схе а
[106]: К
-
,
ч
е ре
,
авле
( ,
.
е ра рыва ру
)
р в
а
.
,
(
)
, ,
–
, . [107]. [108].
-
, . [108].
L © В.Е. Селе
, .В
[85, 106]: ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
234 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
wLj +1 = sign ⋅ cLj +1 ,
(2.451)
; sign = ±1
c –
. «
pатм (
, )
».
Toc (
-
). ,
Toc
– (
,
-
,
). [106]: γ
pатм
γ –
⎡ 2 ⎤ γ −1 > pM ⋅ ⎢ ⎥ , ⎣ γ + 1⎦
.
(2.452 )
[106]
wL > cL .
(2.452 )
(2.452 ) (
Н
). , . Н
-
. В ,
. В
-
. .Н.
Д
,
,
(
.
). .Н.
pL < pатм ,
(2.452 )
(2.452 ).
-
. Д c=
; ρ –
p –
; T –
( ∂p ∂ρ )S
= γ ⋅ R ⋅T ,
; S –
;γ –
.
© В.Е. Селе
ев, В.В. Алеш
,
, С.Н. Прял в, 2007–2009
; R – Д .В
,
( ∂p ∂ρ )S .
(2.453)
-
лава 2 235 _______________________________________________________________________________________
И
dp =
,
∂p ∂p ⋅dρ + ⋅ dT , ∂ρ ∂T
: (2.454 )
⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂T ⎞ ⎜ ⎟ =⎜ ⎟ +⎜ ⎟ . ⎟ ⋅⎜ ∂ ρ ⎝ ⎠ S ⎝ ∂ρ ⎠T ⎝ ∂T ⎠ ρ ⎝ ∂ρ ⎠ S
Д
,
TdS = d ε + pdV = dh − Vdp = 0, ⎛ ∂h ⎞ ⎛ ∂p ⎞ ⎜ ⎟ = V ⋅⎜ ⎟ , ⎝ ∂ρ ⎠ S ⎝ ∂ρ ⎠ S
ε –
; h –
⎛ ∂h ⎞ cp = ⎜ ⎟ – ⎝ ∂T ⎠ p :
⎛ ∂h ⎞ ⎛ ∂h ⎞ ⎜ ⎟ =⎜ ⎟ ∂ ρ ⎝ ⎠ S ⎝ ∂p ⎠T
(2.454 ) ,
(2.456) ; V =1 ρ –
.
⎛ ∂p ⎞ ⎜ ⎟ , ⎝ ∂ρ ⎠ S
(2.456)
(2.457),
(2.458)
:
(2.459)
:
⎛ ∂p ⎞ ⎛ ∂p ⎞ cp ⋅ ⎜ cp ⋅ ⎜ ⎟ ⎟ ∂ ρ ⎛ ∂p ⎞ ⎝ ⎠T ⎝ ∂ρ ⎠T = = , ⎜ ⎟ T ⎛ ∂p ⎞ ⎛ ∂ρ ⎞ ⎝ ∂ρ ⎠ S c − T ⋅ ⎛ ∂p ⎞ ⋅ ⎛ ∂V ⎞ + ⋅ ⋅ c ⎜ ⎟ ⎜ ⎟ p p ρ 2 ⎜⎝ ∂T ⎟⎠ ρ ⎜⎝ ∂T ⎟⎠ p ⎝ ∂T ⎠ ρ ⎝ ∂T ⎠ p ⎛ ∂p ⎞ cP ⋅ ⎜ ⎟ ⎝ ∂ρ ⎠T c= . T ⎛ ∂p ⎞ ⎛ ∂ρ ⎞ ⋅ cP + 2 ⋅ ⎜ ρ ⎝ ∂T ⎟⎠ ρ ⎜⎝ ∂T ⎟⎠ p
ев, В.В. Алеш
(2.457)
⎞ ⎛ ∂p ⎞ ⎛ ⎛ ∂h ⎞ ⎛ ∂T ⎞ ⎜ ⎟ ⋅ ⎜⎜ ⎜ ⎟ − V ⎟⎟ + c p ⋅ ⎜ ⎟ = 0. ρ ∂ ∂ p ⎝ ⎠ S ⎝ ⎝ ⎠T ⎝ ∂ρ ⎠ S ⎠
⎛ ∂p ⎞ cp ⋅ ⎜ ⎟ ⎛ ∂p ⎞ ⎝ ∂ρ ⎠T = . ⎜ ⎟ ⎞ ⎝ ∂ρ ⎠ S ⎛ ∂p ⎞ ⎛ ⎛ ∂h ⎞ cp + ⎜ ⎟ ⋅ ⎜ ⎜ ⎟ − V ⎟⎟ ⎝ ∂T ⎠ ρ ⎜⎝ ⎝ ∂p ⎠T ⎠
© В.Е. Селе
.
⎛ ∂p ⎞ ⎛ ∂T ⎞ ⋅⎜ ⎟ + cp ⋅ ⎜ ⎟ , ∂ ρ ⎝ ⎠S ⎝ ∂ρ ⎠ S
(2.454 ) (2.458) ⎛ ∂p ⎞ ⎛ ∂T ⎞ ⎜ ⎟ ⎜ ⎟ . ⎝ ∂ρ ⎠ S ⎝ ∂ρ ⎠ S
(2.31),
(2.455)
, С.Н. Прял в, 2007–2009
(2.460)
(2.461)
236 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
(2.461)
(2.453), p = ρ ⋅ R ⋅T ,
cV = c p − R ,
γ = c p cV ,
cV
–
. В
[6]
(2.461). Н
(2.453)
2.42 Δ c = Δ c ( p, T ) ,
:
Δc = cideal –
creal − cideal ⋅100%, creal
(2.462)
, (2.461).
,
и . 2.42.
л ч е ре ул а
К
в ра че а реал
(2.453); creal –
р а
в
ву а
р уле
(
,
,
, -
,
.
, ( ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
)
4%. p ∈ [1 МПа ÷ 10 МПа ] ,
T ∈ [ −27°C (246 K ) ÷ 0°C (273 K )] . В
1%.
-
еал
. 2.42,
p ∈ [ 0,1 МПа ÷ 10 МПа ] , T ∈ [ −30°C (243 K ) ÷ 90°C (363 K )]
© В.Е. Селе
.
.
лава 2 237 _______________________________________________________________________________________
,
0,3 МПа ),
а ),
х а е че я а еа ь
е а
е
.
е ь ч х е а а (2.461).
(2.453) е ае е х е а че а а а( а еа ь аа е е е я е ех ь
2.6. Мо ели ование ан о о е о ны ех и о
ь а че а е , я а ае я е я
и ования а ов че е е о ную ан ию
И . 2.6.1. Ма е а иче ( ехов)
ие
о ели е
ен ов о
е
о ных
ан и
-
К 2.3 – 2.5.
, К
. В -
1
,
К
.
,
.
АВ , (
[109]
. Э , [109]):
.,
TK = T
С
+ ( TH − T
-
АВ
С
В. . Ш
⎡ k⋅Fр ⎤ ⎥, ⎢⎣ J ⋅ c p ⎥⎦
) ⋅ exp ⎢ −
(2.463)
TH , TK –
АВ ; T
С
–
; k –
,
F р ); c p –
(
К АВ , 1
-
; J –
АВ . [110],
,
Д
© В.Е. Селе
-
: .
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
238 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ 1 ϕ ⋅ d0 ϕ ⋅ d0 ϕ ⋅ d0 ϕ ⋅ d0 d d 1 ⎞ k =⎜ ⋅ + ⋅ ln + Rк ⋅ + ⋅ ln 0 + ⎟ , 2 2 α λ λ α d d d d ⋅ ⋅ т к 1 1 2 ⎠ ⎝ 1 −1
α1 –
(2.464) ; ϕ –
-
( ,
); d 0 –
; d , dк – (
; d1 – , d = d к ); λ т – ; λ – ; Rк –
α1
,
α2
, ; α2 –
. -
,
.Д
,
-
[110]. : m ⎡ ⎛ d 2 − d 02i F р = π ⋅ ∑ ⎢ d 0i ⋅ l + n ⋅ δ рi ⋅ ( d pi − d 0i ) + n ⋅ ⎜ pi ⎜ 2 i =1 ⎣ ⎢ ⎝
АВ ; i – i-
m – АВ ; d рi –
; n – ; δ рi –
Д
(2.465)
i-
; l – ; d 0i –
-
[110].
-
. В
i-
АВ АВ
⎞⎤ ⎟⎟ ⎥, ⎠ ⎦⎥
[8, 110]. ,
.
К
ЦН
,
А
ЦН
P1
. Д ЦН, ε = P2 / P1 ≈ 1,1 ÷ 1,7 [111].
P2
,
: ЦН
ЦН
.
-
ЦН
,
,
,
[112, 113]: ,
. .Х
ЦН . ЦН. Д
© В.Е. Селе
ев, В.В. Алеш
ЦН [112−114]. Э
, С.Н. Прял в, 2007–2009
ЦН
,
-
ЦН
ЦН
-
лава 2 239 _______________________________________________________________________________________
А.И. ,
.
. 1)
ρ
ЦН р
(
1
, Т1 ) ,
: -
; ЦН (
2) )(
.,
: n , Nм (
,
[111]),
ЦН : P1 , Т1 , Z , R ) ЦН (
,
(
.,
,
)
[111]));
ЦН
4) [111]).
( Z1 )
( Z2 )
ЦН,
(
( P = Z ⋅ ρ ⋅ R ⋅T ) ,
ЦН, ( -
–
; Nм –
ЦН); 3)
n
-
(
,
, . . Z1 / Z 2 ≈ 1 .
, ЦН
Z
,
-
,
[1]. П
Да
• •
Т1
:
P1
а
ρ
а а ач
ЦН; р
ЦН n ,
;
: Q1 (
)
ЦН;
•
ЦН);
Т
я
Qк ( P2
ЦН.
и ь:
Т2 ,
P2 (
)
-
ЦН. А
е е
Ша 1.
я R [78].
Ша 2.
-
Z1
ЦН [78]. Ша 3. [112]: © В.Е. Селе
n0
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
240 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
n0 =
Ша 4.
n ⋅ A, n
А=
Z р ⋅ R р ⋅Т Z1 ⋅ R ⋅ T1
р
(2.466)
.
Qр,
ηр
ε = P2 / P1
, ЦН (
[112]
,
[111]). Ша 5. Д P2 = ε ⋅ P1 .
ЦН (
P2
Ша 6. Z1 Z 2 ≈ 1 ,
Т2
) ЦН
ЦН Т 2 = Т1 ⋅ ε
ηp
:
(γ −1) / (γ ⋅η p )
[112]: (2.467)
, Q1
ЦН . Д 1
M
рив
ЦН
,
[111, 112]:
⎛n⎞ P1 ⎛M ⎞ =⎜ i ⎟ ⋅ ⋅ ⎜ ⎟ + M 'м , ⎝ ρ ⎠ р Z1 ⋅ Т 1 ⋅ R ⎝ n ⎠ 3
M
рив
; ( Mi ρ )
M 'м
(2.468) -
р
ЦН. 2.6.2. Ма е а иче ое о ели ование у ановивших я е и ов ан о и ования и о но о а а че е о е о ны ех и о е о ную ан ию
В В.В. К
В. .
, [2, 6, 115].
КЦ (К ) ,
КЦ (К ), (
( НА Н) -
( НА ) ).
К
4.7). (
»(
,
НА ).
,
. 1
В
© В.Е. Селе
(
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
. -
ЦН)
,
НА Н (
,
ЦН –
«
-
.,
, [111]).
лава 2 241 _______________________________________________________________________________________
,
,
,
КЦ –
КЦ (К )
КЦ (К ) КЦ ( . 2.43). . Н
КЦ (
•
А,
• •
КЦ ,
, ЦН –
(ξ
i2
.
. 2.43)
)i 2 ) ,
= λi 2 Di 2 + (ξ
(ξ
: i = 1, N ;
и . 2.43. Схе а
•
е
i1
че
= λi1 Di1 + (ξ
)i 1 ) 1
Д
i-
, Di1 –
ЦН (
i-
,
.
-
–В 2);
ЦН;
–
, ЦН;
i-
λi 2 –
,
1
,
[66].
© В.Е. Селе
. Н -
К
λi1 –
( ξ )i 1
-
.
В
•
. .И
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
242 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ЦН
i-
•
( ξ )i 2
КЦ (К ) (
Di 2 –
•
i-
•
•
•
i-
,
.
2);
ЦН
КЦ (К );
–
,
ЦН
КЦ (К );
li1 –
,
li 2 –
,
i-
ЦН;
ЦН;
iЦН,
•
КЦ (К );
Jв
•
ЦН1;
Tв
КЦ (К ).
Pв
КЦ
ЦН:
(К )
•
ЦН,
А;
•
ЦН ( nmin ≤ n ≤ nmax ) ;
•
•
-
ЦН; А;
• •
Ч
•
;
ЦН,
-
; ЦН,
ЦН, Ч
•
; А;
,
•
ЦН
,
. . АВ КЦ
КЦ
.В А (КЦ
)
К ), АВ
, « К .
Х
© В.Е. Селе
( ».
, ЦН
) К ,
. Э
: 1
КЦ
.
. .
.
Д
КЦ
(
А (КЦ Pi1
) ,
К )
, . .
ев, В.В. Алеш
(
,
, С.Н. Прял в, 2007–2009
J i1
ЦН
ЦН
.В
ЦН.
-
лава 2 243 _______________________________________________________________________________________
. ,
J i1
(
Srij = π ⋅ Dij2
(4 ⋅
J i1 ( Pi1 ) = Sri1 ⋅
)
lij ⋅ ξij ,
. (2.38)):
Pв2 − Pi12 ⋅ ρв , Pв
(2.469)
ρв
j = 1, 2 ;
КЦ
; i – i-
,
-
ЦН.
ЦН
, . (2.469)):
( Pi1 ( J i1 ) = Pв2 −
-
Pi1
1 J i21 ⋅ Pв . ⋅ Sri12 ρ в
(2.470) А,
,
.
ЦН
А (КЦ
.
К )
,
,
.
(
).
-
( Pв ).
И
, Tв .
N J в − ∑ J i1 = 0; N
i =1
Н
А (КЦ :
)
К )(
Pв > Pi1 ∀ i = 1, N
(2.471)
К
J i1 > 0 ∀ i = 1, N .
. А.
,
ев, В.В. Алеш
КЦ
ЦН.
КЦ,
-
-
,
, ЦН, А.И. :
(2.471)
.
,
© В.Е. Селе
-
. ЦН
,
⎛ J i1 ⎞ Pi 2 = Pi 2 ⎜⎜ ni , Tв , Pi1 , ⎟ , i = 1, N ; ρ i1 ( Pi1 ) ⎟⎠ ⎝ , С.Н. Прял в, 2007–2009
,
, -
(2.472 )
244 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎛ J i1 ⎞ Ti 2 = Ti 2 ⎜⎜ ni , Tв , Pi1 , ⎟ , i = 1, N ; ρ i1 ( Pi1 ) ⎟⎠ ⎝ ⎛ J i1 ⎞ M i = M i ⎜⎜ ni , Tв , Pi1 , ⎟, ρ i1 ( Pi1 ) ⎟⎠ ⎝ ⎛
ρ i 2 = ρ i 2 ( Pi 2 ) = ρ i 2 ⎜⎜ ni , Tв , Pi1 , ⎝
Pi 2 –
ii-
ЦН; ρ i1 – ); Ti 2 – i-
(2.472 )
i = 1, N ;
(2.472 )
J i1 ⎞ ⎟ , i = 1, N , ρ i1 ( Pi1 ) ⎟⎠
ЦН; ni – ЦН ( i- ЦН; M i –
ЦН i-
ЦН1.
i; ρi 2 –
ЦН
,
(2.472 )
,
-
,
-
, КЦ (
. -
КЦ
. 2.44).
и . 2.44. Схе а у ла
еше
я
в а а 2
. -
, :
⎛ J i1 ⎞ 2 Pi 22 ⎜ ni , Tв , Pi1 , ⎟ − Pвы P ρ ⎛ ( ) J i1 ⎞ i1 i1 ⎠ ⎝ J i1 − Sri 2 ⋅ ⋅ ρ i 2 ⎜⎜ ni , Tв , Pi1 , ⎟ = 0, ρ i1 ( Pi1 ) ⎟⎠ ⎛ J i1 ⎞ ⎝ Pi 2 ⎜ ni , Tв , Pi1 , ⎟ ρ i1 ( Pi1 ) ⎠ ⎝
⎛ J i1 ⎞ Pi 2 ⎜⎜ ni , Tв , Pi1 , ⎟⎟ ≥ Pвы , ρ i1 ( Pi 1 ) ⎠ ⎝
i = 1, N .
ЦН А.И.
1 2
Д
© В.Е. Селе
. ев, В.В. Алеш
(2.473)
, С.Н. Прял в, 2007–2009
.
лава 2 245 _______________________________________________________________________________________
∑( J
КЦ
N
Tвы =
i =1
:
⋅ Ti 2 )
∑ J i1 i1
(2.474)
.
N
i =1
, А (КЦ
Нi К ) КЦ, . . [112]:
Нi = И
, К
А (КЦ ,
i-
-
Pвы − Pв . 0,5 ⋅ g ⋅ ( ρ вы + ρ в ) -
К )
КЦ (К ), (2.471),
(
.
А (КЦ
. 2.43). -
К )
НА Н НА НА Н ( НА )
. Д
-
,
-
,
, , .
КЦ (К ), В. . , i - ЦН (2.470)
,
[1]. В
,
(
. (2.469),
) = 0,
i = 1, N − 1,
(2.473)): ⎛ J i1 ⎞ 2 Pi 22 ⎜ ni , Tв , Pi1 , ⎟−P ρ i1 ( Pi1 ) ⎠ вы ⎛ J i1 ⎞ ⎝ ⋅ ρ i 2 ⎜⎜ ni , Tв , Pi1 , ⎟ + G ( Pi1 , Pвы ρ i1 ( Pi1 ) ⎟⎠ ⎛ J i1 ⎞ ⎝ Pi 2 ⎜ ni , Tв , Pi1 , ⎟ ρ i1 ( Pi1 ) ⎠ ⎝
J i1 − Sri 2 ⋅
(2.475 )
Pвы2
0 Pi 2 ( Pi1 ),
;
« N1 »
е
(2.475 )
246 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ЦН
N;
ЦН
« N2»
N.
(2.475 )
(2.475 ) А (КЦ
К ).
,
(2.475 ), 1
К
. ,
(2.475 – )
.В
(
J N 1 = J в − ∑ J i1 .
. (2.471)):
N −1
(2.476)
i =1
,
J N1 ,
,
PN 1 = Pв2 −
(2.476)
(2.470)
PN 1 :
1 J N2 1 ⋅ Pв ⋅ . SrN21 ρв
(2.477) (2.475* ),
(2.477)
:
N −1 ⎛ ⎞ J в − ∑ J i1 ⎟ ⋅ PN 2 ⎜ 1 i =1 ⎝ ⎠ , = PN22 − 2 ⋅ N −1 SrN 2 ⎛ ⎛ ⎞⎞ ⎜ ⎜ J в − ∑ J i1 ⎟ ⎟ i =1 ⎠⎟ ρ N 2 ⎜ nN , Tв , PN 1 , ⎝ ⎜ ρ N 1 ⎡⎣ PN 1 ( Pi1 ) ⎤⎦ ⎟ ⎜ ⎟ ⎝ ⎠ 2
Pвы2
(2.475 )
N −1 ⎛ ⎞ J в − ∑ J i1 ⎟ ⋅ Pв ⎜ 1 i =1 ⎠ , PN 1 = Pв2 − 2 ⋅ ⎝ SrN 1 ρв 2
N −1 ⎛ ⎛ ⎞⎞ − J J i1 ⎟ ⎟ ∑ ⎜ в ⎜ i =1 ⎠ ⎟. PN 2 = PN 2 ⎜ nN , Tв , PN 1 , ⎝ ⎜ ρ N 1 ⎡⎣ PN 1 ( Pi1 )⎤⎦ ⎟ ⎜ ⎟ ⎝ ⎠
И
НА
, Pi1
1
( N − 1)
(2.475 – ),
, i = 1, N − 1 . (2.471), , НА (2.475 – ) :
, i = 1, N − 1 ,
Э .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.В
J i1
лава 2 247 _______________________________________________________________________________________
Pi ,min < Pi1 < Pi ,max ,
i = 1, N − 1,
(2.475 )
J i ,min < J i1 < J i ,max ,
i = 1, N − 1.
(2.475 )
(2.475 ) (2.469)
(2.470). , ,
(2.475 ) -
,
. Д
К ),
А (КЦ
А
-
.
J i ,min
J i ,max
,
, А (КЦ
К ):
J N ,min + ∑ J i ,max < J в < J N ,max + ∑ J i ,min N −1
N −1
i =1
i =1
< J N ,min < J в − ∑ J i ,max , N −1
J N ,min ( те
, эк
,к
J i ,max < J i ,max ( те
тр )
i =1
, эк
,к
тр )
, i = 1, N − 1;
J в − ∑ J i ,min < J N ,max < J N ,max ( те
(2.475 )
N −1 i =1
J i ,min ( те J i ,min ( те
, эк
,к
, эк
J i ,max ( те
тр )
,к
, эк
тр )
,к
,к
< J i ,min , i = 1, N − 1,
тр )
,
, эк
тр )
,
, i = 1, N , –
ЦН
,
А
,
. .В
-
, (2.476)
. (2.477)
(2.475 ), (2.475 ) (2.475 )
,
ЦН Pi ,min Pi ,max (
N,
J i ,min
.
J i ,max )
N. (2.475 )
В
-
J N ,min < J в − ∑ J i ,max ,
:
N −1
(2.478)
i =1
J N ,max > J в − ∑ J i ,min . N −1
(2.479)
i =1
(2.479) © В.Е. Селе
ев, В.В. Алеш
«-1»
, С.Н. Прял в, 2007–2009
(2.478),
:
248 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∑J N −1 i =1
− J N ,max < ∑ J i ,min − J N ,min
∑J
N −1
i ,max
N −1
i =1
i =1
, Jт ч .
J i,min = (1 − x ) ⋅ J т ч ;
: 0 < x J N ,min( те
N тр ) .
J N ,max( те
J N,min = J N ,min( те J N ,max = J N ,max( те
, эк
,к
, эк
,к
тр )
, эк
⋅ 1, 01;
тр )
,к
тр ) − J т ч > J т ч − J N ,min( те
J N,max = 2 ⋅ J т ч − J N,min ,
⋅ 0,99; J N,min = 2 ⋅ J т ч − J N ,max . J ттарт . ч А (КЦ
= Jв N . : J ттарт ч
: © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
А, J i,min
К ) J ттарт ч J i,max
, эк
, эк ,к
,к тр ) ,
тр )
лава 2 249 _______________________________________________________________________________________
J i,max
J ттарт ч ч
2 ⋅ ( N − 1)
J N ,max − J N ,min 2 ⋅ ( N − 1)
.
4. (2.470). НА
,
(2.475)
.
,
(2.475 )
ё -
:
⎛ J i1 ⎞ 2 Pi 22 ⎜ ni , Tв , Pi1 , ⎟ − Pвы ρ P ⎛ J i1 ⎞ i1 ( i1 ) ⎠ ⋅ ρ i 2 ⎜⎜ ni , Tв , Pi1 , J i21 − Sri 22 ⋅ ⎝ ⎟ = 0, ρ i1 ( Pi1 ) ⎟⎠ ⎛ J i1 ⎞ ⎝ Pi 2 ⎜ ni , Tв , Pi1 , ⎟ ρ i1 ( Pi1 ) ⎠ ⎝ Pi ,min < Pi1 < Pi ,max ,
i = 1, N − 1,
J i ,min < J i1 < J i ,max ,
< J N ,min < J в − ∑ J i ,max , N −1
J N ,min ( те
, эк
,к
тр )
J в − ∑ J i ,min < J N ,max < J N ,max ( те N −1 i =1
, эк
,к
тр )
(2.483 )
i = 1, N − 1,
J i ,max < J i ,max ( те
i =1
i = 1, N − 1, (2.483 )
, J i ,min ( те
, эк
,к
(2.483 )
, эк
тр )
,к
тр )
, i = 1, N − 1;
< J i ,min ,
N −1 ⎛ ⎞ J в − ∑ J i1 ⎟ ⋅ PN 2 ⎜ 1 i =1 ⎝ ⎠ , = PN22 − 2 ⋅ N −1 SrN 2 ⎛ ⎛ ⎞⎞ ⎜ ⎜ J в − ∑ J i1 ⎟ ⎟ i =1 ⎠⎟ ρ N 2 ⎜ nN , Tв , PN 1 , ⎝ ⎜ ρ N 1 ( PN 1 ) ⎟ ⎜ ⎟ ⎝ ⎠
i = 1, N − 1,
(2.483 )
2
Pвы2
N −1 ⎛ ⎞ J J i1 ⎟ ⋅ Pв − ∑ в 1 ⎜⎝ i =1 2 ⎠ PN 1 = Pв − 2 ⋅ , ρв SrN 1
(2.483 )
N −1 ⎛ ⎛ ⎞⎞ J − ⎜ ⎜ в ∑ J i1 ⎟ ⎟ i =1 ⎠ ⎟. PN 2 = PN 2 ⎜ nN , Tв , PN 1 , ⎝ ⎜ ρ N 1 ( PN 1 ) ⎟ ⎜ ⎟ ⎝ ⎠
2
,
,
•
. .
В
© В.Е. Селе
,
(2.483 )
: -
К ев, В.В. Алеш
; , С.Н. Прял в, 2007–2009
250 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
•
А (КЦ
К ).
,
⎛ J i1 ⎞ 2 Pi 22 ⎜⎜ ni , Tв , Pi1 , ⎟⎟ < Pвы ρ P ( ) 1 1 i i ⎝ ⎠ ⎛ J i1 ⎞ 2 Pi 22 ⎜ ni , Tв , Pi1 , ⎟ − Pвы ρ P ⎛ ( ) J i1 ⎞ i1 i1 ⎠ − Sri 22 ⋅ ⎝ ⋅ ρi 2 ⎜⎜ ni , Tв , Pi1 , ⎟ > 0. ρ i1 ( Pi1 ) ⎟⎠ ⎛ J i1 ⎞ ⎝ Pi 2 ⎜ ni , Tв , Pi1 , ⎟ ρ i1 ( Pi1 ) ⎠ ⎝
К
, . (2.483 ) ,
, ,
(2.483 )
. (2.483 )
ЦН.
,
(2.483 )
НА
,
КЦ,
(2.483)
, .
( , А) НА (2.483) КЦ (К ),
В.В. К А (КЦ)
(
. 2.45).
и . 2.45. Схе а ра
Н © В.Е. Селе
р
р в
ЦН ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
р р
а а чере К
Pв ..
НА Н. [2]. В
-
лава 2 251 _______________________________________________________________________________________
ЦН Д
.Н
ЦН
Pi 2 ,
ЦН
, .
ЦН,
ЦН
ЦН.
, ,
-
Pвы Pi 3 ,
iИ
.
, Pi 3 – . (2.470) (2.473)).
, КЦ) (
(
КЦ. КЦ
, . .
Pi 3
i-
Pi1 . -
ЦН
-
, , . . [2]
Pi 3
⎛ J (i +1)1 ⎞ ⎛ J i1 ⎞ ⎟ = 0, i = 1, N − 1, (2.484 ) Pi 3 ⎜⎜ ni , Tв , Pi1 , ⎟⎟ − P(i +1)3 ⎜ ni +1 , Tв , P(i +1)1 , ⎜ ρ i1 ( Pi ) ⎠ ρ Pi +1 ) ⎟ ( i 1 1 + ( ) ⎝ ⎝ ⎠
⎛ J i1 ⎞ J i21 ⋅ Pi 2 ⎜ ni , Tв , Pi1 , ⎟ ρ ⎛ J i1 ⎞ 1 i1 ( Pi1 ) ⎠ ⎝ Pi 3 = Pi 22 ⎜⎜ ni , Tв , Pi1 , − ⋅ . ⎟ ρ i1 ( Pi1 ) ⎟⎠ Sri22 ⎛ J i1 ⎞ ⎝ ρ i 2 ⎜ ni , Tв , Pi1 , ⎟ ρ i1 ( Pi1 ) ⎠ ⎝
Д И
–В ,
(2.484 ) А (КЦ К ), . НА (2.484),
(2.484 )
, -
( N − 1)
.Д (2.484)
N
НА
(2.473). В N Jв
А (КЦ
( N − 1) . :
N
J N = J в − ∑ J i1 .
К ), ЦН
N −1
(2.484 )
i =1
,
(2.484 )
К
НА (2.484) (2.483 – ).
. (2.484)
А (КЦ
-
К ). (2.475 – )
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
252 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
ЦН (КЦ
К ),
,
ЦН. Д А (КЦ
В
НА
КЦ (2.474). ь а е ч я ч е
еа
а
а
е
К ). (2.483)
-
(2.484)
НА (2.483) , ч а е , е е я. ,
е х
. е я е я а
, -
В
А
Д
КЦ
ЦН А
ЦН
К
i-
,
ЦН
, ,
ЦН:
ЦН , Ti1 – Ti1 = Tв ). К , КЦ
ЦН (
ЦН
Д
-
,
p ⎛ ⎞ J i1 S = ∑ M i ⎜ ni ( J i1 ) , Ti1 , Pi1 ( J i1 ) , ⎟, ⎜ ρ i1 ( Pi1 ( J i1 ) ) ⎟⎠ i =1 ⎝
p –
-
(
, (2.472 ). В ЦН
,
ЦН
А(
А)
А. ЦН.
– . -
А.И. ЦН.
.
)
[2].
А -
(2.485)
А
[1, 2, 6]. 2.6.3. Ма е а иче ое о ели ование неу ановивших я е и ов ан о и ования и о но о а а че е о е о ны ех и о е о ную ан ию
К
:
К 2.6.3.1. Ме о
о ле ова ельно
ены
К
К . а иона ных о
-
ояни КС
[109]. -
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 253 _______________________________________________________________________________________
( ,
) [1].
НА (2.469) НА Н.
Jв ,
НА
(2.470) (
. -
(2.483)
(2.484)) -
К .
Tв
КЦ (К )
Pв
. -
[1, 2]. 2.6.3.2. Ме о анали а ина иче
их е и ов КС
Д
.Н. (2.469)
(2.470) (
(2.483)
.
. 2.43).
КЦ (
КЦ:
)
В ,
( . ). В В ( . 2.46).
А
и . 2.46. Схе а ве в
е
у
ч а
А
-
-
В
: PA
ЦН.
.Д
1.
.
J2
, КЦ
TA
КЦ (
-
(2.484)), (2.231). А,
А (
[2].
, ЦН, -
:
. 2.46): J2 ,
TA
-
PA
. P21
2.
ЦН
J2 , P21
© В.Е. Селе
T21 .
ев, В.В. Алеш
T21 , С.Н. Прял в, 2007–2009
ЦН
А.И.
254 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
P22
ЦН.
T22
3.
J2
P22
T22
. КЦ (
.
. 2.46)
ЦН
–а
,
а
е ь ЦН 1 [2].
ЦН
-
КЦ (К )
.Д
(
) .
КЦ (К ),
, ЦН
, е
Д
е я [2, 6]. ,
. Э
-
. а
я а че х е ЦН (или краще
а
е
е
.
а а а че е КЦ (КС) а а я а че х е ЦН ,
.Н. (
е
КС2) N
.
-
, -
). -
.
(2.231 ) (
)
К ),
.Д
(2.231 ) ( ,
,
, «
) А» (КЦ
–
3
.
(2.231 ), (
(
)
-
,
(
)).
,
. .
1
В
,
.
2
В ,
. А,
3
.
, .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, -
лава 2 255 _______________________________________________________________________________________
( k ⋅ ∂T
∂x ∼ ρ ⋅ w ⋅ ε ) ,
(ρ ⋅ w ⋅ε
.
>> k ⋅ ∂T ∂x ) ,
: ∂T ∂x = 0 .
КЦ (К ) (
-
ЦН КЦ ( .
)
. 2.45
2.46):
A
B.
,
, .Э PB
,
:
PA
(
).
TA
,
,
-
ЦН J 2ЦН = J 2ЦН (t ) (
. 2.43
. 2.46):
-
Ψ 2 ( J ) = 0,
(2.486 )
Ψ 2 ( J ) = P22ТГ ( J ) − P22ЦН ( J ) ,
P22ТГ ( J ) = P22ТГ ( J , T22ЦН ( J ) ) –
(2.486 ) , T22ЦН ; T22ЦН –
J
ЦН,
ЦН (
–
.
(J )
ТГ 21
); P
ТГ 21
T
,
ε2 ( P
TГ 21
(J ), T (J ), J ) ТГ 21
ЦН,
КЦ
А.И.
.
КЦ (
.
ЦН
НА
. 2.43) .В ЦН. В
. i = 1, N , N –
, КЦ. Э
-
КЦ ЦН
(
ев, В.В. Алеш
-
А (КЦ).
ЦН ) .
2.43): © В.Е. Селе
,
.
. J iЦН = J iЦН ( t ) (
;
,
,
,
ЦН.
,
(2.486) , ЦН
-
J ; P22ЦН ( J ) = ε 2 ( P21TГ ( J ) , T21ТГ ( J ) , J ) ⋅ P21TГ ( J ) –
–
,
(J )
-
, С.Н. Прял в, 2007–2009
256 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Ψ i ( J i ) = 0, i = 1, N ,
(2.487 )
ЦН Ψ i ( J i ) = Pi ТГ 2 ( J i ) − Pi 2 ( J i ) .
Д
,
(2.487 )
,
(2.487).
А, (
.
, (2.487)
Ч .В [116]
).
-
. Д
q-
r-
2.6.4. К во о у о у у ы
-
(2.487)
,
2
1
2N
о ели овании о
[116]. е
о ных
ан и
ло но
Д
К 2.6.1 – 2.6.3, КС е ь ае я КС е ь х КЦ а
,
.П че ая хе а е ПА. В
ая
В.В. К
е
А
я
а
. 2.47). Д
А
КЦ, . 2.47
А,
ЦН (
).
К ,
-
, -
К , -
, , С
а е а че ( е СНАРН ( е). В НА Н
ч СНАУ
. е я, К ( .
а ая е ь КС 2.6.1 – 2.6.3)) е а яе а че я а а яе е е е е -
К (
Н ,
© В.Е. Селе
ч е а -
.
[6]. И (
-
ев, В.В. Алеш
,
, С.Н. Прял в, 2007–2009
К , ,
). , .
лава 2 257 _______________________________________________________________________________________
,
К ,
А,
КЦ.
-
НА Н ( НА ). В
,
КЦ,
,
,
-
В.В. К
-
. Э К )
(
.
и . 2.47. Схе а а а я ех л че
Н
. 2.47
К К
© В.Е. Селе
ых е ав ых ере е ых ля а ал а ра р а а а чере КС
ев, В.В. Алеш
ера
в
( ). Д
X
. 2.47 :
, С.Н. Прял в, 2007–2009
-
258 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
⎧ P111 ( J в , Pв , Tв , J 1 ( X 1 ) , J 11 ( X 2 ) , ε111 ( X 6 ) ) − P121 ( J в , Pв , Tв , J 1 ( X 1 ) , J 12 ( X 3 ) , ε121 ( X 4 ) ) = 0; ⎪ ⎪ P121 ( J в , Pв , Tв , J 1 ( X 1 ) , J 12 ( X 3 ) , ε121 ( X 4 ) ) − P131 ( J в , Pв , Tв , J 1 ( X 1 ) , J 13 , ε131 ( X 8 ) , ε132 ( X 7 ) ) = 0; ⎪ 1 ⎪ P131 ( J в , Pв , Tв , J 1 ( X 1 ) , J 13 , ε131 ( X 8 ) , ε132 ( X 7 ) ) − P211 ( J в , Pв , Tв , J 2 , J 21 , ε 21 ( X 9 ) , ε 212 ( X 5 ) ) = 0; ⎪ ⎪ J 11 ( X 2 ) − J 111 ( ε111 ( X 6 ) ) = 0; ⎪ ⎪ J 12 ( X 3 ) − J 121 (ε121 ( X 4 ) ) = 0; ⎪ ⎪⎪ J 13 − J 131 ( ε131 ( X 8 ) ) = 0; ⎨ ⎪ J 131 (ε131 ( X 8 ) ) − J 132 (ε131 ( X 8 ) , ε132 ( X 7 ) ) = 0; ⎪ 1 ⎪ J 21 − J 21 (ε 212 ( X 5 ) ) − J 212 (ε 211 ( X 9 ) ) = 0; ⎪ 1 ⎪ J 212 (ε 21 ( X 9 ) ) − J 213 (ε 211 ( X 9 ) , ε 212 ( X 5 ) ) = 0; ⎪ ⎪0 < X i < 1, i = 1, 3; ⎪ 1 це ⎪ε max < X j < ε max , j = 4, 7; ⎪ 1 1 ⎪⎩ε min < X k < ε max , k = 8,9,
(2.488) К , J в , Pв , Tв –
P111 , P121 , P131 , P211 –
,
К , J1 , J 2 – I», J 11 , J 12 , J 13 , J 21 –
«
II», ε , ε , ε , ε 212 – 1 11
«
А А
1 12
2 13
це , ε max –
1 КЦ, ε min –
А
1 КЦ, ε131 , ε 21 – 1 КЦ, ε max –
.
К Д
, . Э
Д .
В
2.6
К [1, 2, 5 − 7].
-
.
2.7. Мо ели ование ан о и ования о у ов че е е ия ие у о ово но о ан о а И
. -
К . © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
лава 2 259 _______________________________________________________________________________________
(
), . К
(
.
)
-
. . 2.7.1. Мо ели ование у о у ов
ановивших я е и ов
ан
о
и ования
Д .
К
(
2.6.3.2). В
.
-
. а
А а е
а а е е е е
а
а
а х
а
х ече ече )
е
а
е
я(
, , [69, 77, 96]).
(
ь -
.,
-
К .Д -
(
)
Ме а а а а че е а
а а а е а е
ЦН ( . [2]. В
.Н.
е
а
я
е
. х я е а
а е я
КС К В. . :
2.6.3.2). Д Jв , ЦН. Д .
Д
а
-
Tв ,
Pв
-
: 1. В К
Ч
1,
1.
2.
Jв ,
Ч
Tв
Ч
3.
Ч
. P12
К
T12 .
ЦН
Jв , P12
-
T12
К
К (
2.6) К .
P22
4. Н
© В.Е. Селе
-
Pв
Ч
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
T22
К
.
-
260 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Ч
К
(Tв
К
5. В
= T22 , Pв = P22 ) .
, 2.
5.
Jв ,
Ч
Tв
-
Pв
Ч
.Н
P12
Ч .
T12
-
6.
.
.
2.7.2. Мо ели ование неу о у ов
ановивших я е и ов
ан
о
и ования
Д .Н.
В. .
-
а
Да е , я а че х е я ЦН ( . е я я. ЦН
[2, 117]. е , я яе я е а 2.6.3.2) а (2.486) К , (2.486)
я
ще е е я а а че е КС е ь а (2.487) К .К а
-
-
их
. , ).
А Д, ,
Э ,
е
Ч (2.487).
2.7.3. Чи ленная о ен а а а е ов а о ы ав о а иче е уля о ов авления в а о ово ных е ях
В , (
а а а
.А Д
А Д
,
2,5МПа.
А Д ,
,
,
© В.Е. Селе
.
, :
Д
А Д
.
ев, В.В. Алеш
А Д( , С.Н. Прял в, 2007–2009
.Н
. А Д -
)
лава 2 261 _______________________________________________________________________________________
А Д.
. 2.48
и . 2.48.
вая хе а а
в
а в ав
а
(2.1, 2.2, 2.50 , 2.50 ). Н
че
ре уля
ре авле
я
А Д
-
: А Д
1.
-
. А Д
2.
pвы
(p
а вы
:
а а ⎪⎧ p , е ли pвы < pв ; , pв ) = ⎨ вы ⎪⎩ pв , в р тив м лучае,
pвыа –
(2.489)
А Д; pв –
А Д; pвы –
.
3. В
А Д
.
А Д
4.
-
,
-
. (2.1, 2.2, 2.50 , 2.50 ) « .
(2.489)
-
Д
.
А Д ,
∫∫
. Д
-
, ,
. (2.1, 2.2, 2.50 , 2.50 )
ρ ⋅υ n dS = 0;
S
ев, В.В. Алеш
,
-
.А
© В.Е. Селе
(2.2).
: А Д
,
. Ч
:
»
, С.Н. Прял в, 2007–2009
(2.490 )
262 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
∫∫ S
pвы
∫∫
ρ ⋅υ n ⋅ Ym dS = 0, m = 1, N S − 1 ; YN = 1 − S
∑Y
N S −1 m =1
m
(2.490 )
;
⎧⎪ pвыа , е ли pвыа < pв , а = , p p ( вы в ) ⎨ p , в р тив м лучае; ⎪⎩ в ⎛
ρ ⋅⎜ ε + ⎝
S
(2.490 )
υ2 ⎞
⎟ ⋅υ n dS = − ∫∫ p ⋅υ n dS + ∫∫ τ n ⋅ υdS − ∫∫ W ⋅ ndS ; 2⎠ S S S
(2.490 )
T1 = T2 = … = TN S ;
(2.490 )
p = p ({S ме и }) ;
(2.490 )
ε = ε ({S ме и }) , ,
« «
(2.490 )
»
,
»– (2.490 )
« (2.490 )
2.48),
». (
∫∫
ρ ⋅υ n df −
∫∫
ρ ⋅υ n ⋅ Ym df −
:
f вы
∫∫ fв
f вы
,
А Д (
) А Д, (2.490 )
(2.490 )
ρ ⋅υ n df = 0;
∫∫ fв
(2.491 )
ρ ⋅υ n ⋅ Ym df = 0, m = 1, N S − 1 ; YN S = 1 −
∑Y
N S −1 m =1
m
(2.491 )
,
ρ вы ⋅ wвы ⋅ f вы − ρ в ⋅ wв ⋅ f в = 0;
(2.492 )
ρ вы ⋅ wвы ⋅ f вы ⋅ (Ym )вы − ρ в ⋅ wв ⋅ f в ⋅ (Ym )в = 0, m = 1, N S , fв
f вы
А
⎛ ⎜ ρ вы ⎝
(2.492 )
(Ym )вы − (Ym )в
,
(2.490 )
ε, p – . , ев, В.В. Алеш
τ xx
, С.Н. Прял в, 2007–2009
Ym .
(2.492 ) :
⎛ w2 ⋅ ⎜εв + в 2 ⎝
τ xx ; Φ –
А Д; ρ ,
:
= 0, m = 1, N S .
⎞ ⎛ ⎛ w2 ⎞ ⋅ ⎜ ε вы + вы ⎟ + pвы − (τ xx )вы ⎟ ⋅ wвы ⋅ f вы + Φ = ⎜ ρ в 2 ⎠ ⎝ ⎠ ⎝
© В.Е. Селе
(2.492 )
ρ , υx
w , Ym – (2.492 ),
. . А Д. -
⎞ ⎞ ⎟ + pв − (τ xx )в ⎟ ⋅ wв ⋅ f в , ⎠ ⎠ (2.493)
ε
p ; τ xx –
-
2.3.1
-
лава 2 263 _______________________________________________________________________________________
Φ
.В
.
,
А Д
-
:
ρ вы ⋅ wвы ⋅ f вы − ρ в ⋅ wв ⋅ f в = 0;
(Ym )вы − (Ym )в pвы
(p
а вы
ε вы +
(2.494 )
= 0, m = 1, N S ;
(2.494 )
а а ⎪⎧ p , е ли pвы < pв ; , pв ) = ⎨ вы ⎪⎩ pв , в р тив м лучае;
ρ вы
pвы
+
(2.494 )
2 Φ wвы p w2 + = εв + в + в ; ρ вы ⋅ wвы ⋅ f вы ρв 2 2
(2.494 )
T1 = T2 = … = TN S ;
(2.494 )
p = p ({S ме и }) ;
(2.494 )
ε = ε ({S ме и }) ,
(2.494 )
T –
. А Д [6]. В
А. .
.Н. ,
В.В. -
(2.494)
, .
[6]. Х
,
-
, –К
, (
.
-
2.3.1).
2.7.4. Ав о а иче ая на о а вы о о очных о ью е ных и уля о ов на еальные а а е ы он е но у о ово но и е ы
Д
, Ч
. И
( ,
. .К
)
,
-
,
-
. ,
Д
, ( ) SCADA© В.Е. Селе
, ,
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
-
264 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
Д
,
-
( ÷
,
,
). К
Д ,
-
Д
.
-
. Д В. . [4, 6]. Э
-
А. .
-
. Н
, (
-
) . ,
-
.Э
(
Д
.
).
:
•
;
•
К ;
•
(
•
);
. Э
,
,
SCADA(
).
-
Д
-
n
). Д
(
, ,
, .
В
,
k
,
-
. , (
),
Н
Д , . . , В
.В
. -
. .В
© В.Е. Селе
Д
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 2 265 _______________________________________________________________________________________
.
.
Д
Д
. Д
.К
,
, .
, SCADA-
, ,
. ,
Д , ,
l .
Д
.
(
-
)
-
,
.
(Yi = J i )
(
(Yi = Pi )
,
-
):
( )
f i X = ⎡⎣Yi ра чет
е
(X ) − Y
и мере
е
i
( X )⎤⎦
2
, i = 1, m, m = s × k × l ,
X ∈ Ω ⊂ Rn –
(2.495)
nR
n
-
( ), Ω –
, n
R (
n-
); s – В Д ,
(
Ч
), . , , (2.495),
. f i : R n → R, i = 1, m, , n f i : R → R, i = 1, m,
. . Д , .
ев, В.В. Алеш
, -
[33, 34]: © В.Е. Селе
:
,
l =1.
Д
(2.495)
, С.Н. Прял в, 2007–2009
266 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
( )
max f i X → min, X ∈ Ω ⊂ R n . i =1, m
(2.496)
В (
.
). :
Cаа
е
(2.496)
( )
max f i X < C а а i =1, m
>0 –
е
-
(2.497)
,
,
:
( ). (2.496) Д
. (2.496) ,
[118]. В
(2.496) xn +1 :
xn +1 → min
Д ,
-
( )
f i X − xn +1 ≤ 0, i = 1, m; X ∈ Ω ⊂ R
n +1
⎫⎪ ⎬ . ⎪⎭
(2.498)
(2.498)
«Techno&Optim» [1].
2.8. Ма е а иче ие е о ы ни ения а а на ан о и ование о у ов о у о ово ны е я о ощью вы о о очных о ью е ных и уля о ов И
. , , (
и . 2.49. Схе а а
ра
р
-
, . 2.49).
е
ле
К ЦН.
(
ва ел
ы ра
л
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
е КС
А. )
-
ЦН, Pcontract
© В.Е. Селе
е
/
лава 2 267 _______________________________________________________________________________________
: Pcontract ( t )
(
Qcontract
,
Ч
К
.Д
, ( ,
,
Qcontract ( t ) )
, -
)
,
,
,
, , .
, -
.Д , .Э
Д (
.
).
2.8.1. К и е и о и и а ии неу ановивших я е и ов ан о и ования и о но о а а че е ГТС а
е
а
а еа а я ,
е
а
а
е е
е
че
аа
а
е
я а
е
,
я
-
. Д я е е я а е х че е е а я щ х е щ е а е а я е а е е е а е ях а а е а е е я а а х а че а С.
е е
х а ач е х ч ь а а а е С, е а ь х е е че х а а а я а а че е С а а е е е ече я а а х ачеаа ь х ч ах С е а ч -
, , . .
, ,
В
,
. . Д .Д ЦН
.
(
А .
,
Д
-
Д
-
2.2). . .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
268 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
,
,
,
.
( /
)
[6].
(
-
Д .
)
, ,
( ,
,
,
)
, -
,
Д .И-
SCADA-
-
.Э ,
-
. В ( )
,
(
-
)
.
2.8.2. По анов а а ачи о и и а ии неу ановивших я е и ов ан о и ования и о но о а а че е ГТС
А
Д
.Н (
, -
)
.
,
.
, ,
,
ЦН
. К
-
ЦН,
-
.
ЦН К (
К
К
u (t )
(
ев, В.В. Алеш
),
) -
,
∫ Z ( t, u ( t ) ) dt . T
0
© В.Е. Селе
. -
) [6, 119].
t
[0,T ]
Ч
(
, .Э Z ( t, u ( t ) ) .
А
А,
, С.Н. Прял в, 2007–2009
лава 2 269 _______________________________________________________________________________________
,
Φ,
(
и . 2.50. И
-
. 2.50).
е е
е
у
а ра а ра вре е
а а
р
р ва е а а р ервала
р
е ля
: J ( u ) = ∫ Z ( t , u ( t ) ) dt → min ,
t ∈ [0, T ] ,
T
{
u ( t ) ∈U ( t ) ⊂ R m ,
(2.499 )
0
specified specified W ( t , u ( t ) ) ∈ Ω = W ∈ R k : ( wmin ( t ) ) ≤ w j ( t , u ( t ) ) ≤ ( wmax ( t ) ) , j = 1, l;
j
U (t ) –
,
(
(
)
T
ев, В.В. Алеш
(2.499 )
-
–
; k –
© В.Е. Селе
j
,
; W ( t , u ( t ) ) = w1 ( t , u ( t ) ) ,..., wk ( t , u ( t ) ) ); m –
}
specified w j ( t , u ( t ) ) ≤ ( wmax ( t ) ) , j = l + 1, k , j
;
, С.Н. Прял в, 2007–2009
270 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________ specified w min (t ) ∈ Rl
specified w max (t ) ∈ Rk –
; l –
Rm –
m; Rl –
l-
-
l
,
; Rk –
W ( t, u ( t ) )
; -
k-
. ,
-
, .
-
.
. ,
. , specified w j ( t , u ( t ) ) ≤ ( wmax ( t ) ) , j = l + 1, k ,
(2.499)
j
. ,
,
-
. (
-
А
,
[0,T ]
.
,
:
)
( ti , ti +1 ) , i = 0, N − 1, u ( t ) = ui = const , t ∈ ( ti , ti +1 ) ,
( Δt = ti +1 − ti , i = 0,..., N − 1, .
( t0 , t1 ,… , tN )
,
-
ui ∈ U ( t ) . Д
)
,
, (
. 2.51):
⎧ t − t0 t ∈ [ 0, δ ) ; ⋅ ( u 0j − u j ( 0 ) ) + u j ( 0 ) , ⎪ δ ⎪ ⎪ u j (t ) = ⎨ u ij , t ∈ [ti + δ , ti +1 − δ ) ; ⎪ ⎪ t − ( ti +1 − δ ) ⋅ ( u i +1 − u i ) + u i , t ∈ [t − δ , t + δ ) , j j j i +1 i +1 ⎪⎩ 2 ⋅δ
i = 0,..., N − 1 ; j = 1,..., m ; u j ( 0 ) –
, u =u N j
N −1 j
;
(2.499 )
δ
. ЦН © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
ЦН.
-
лава 2 271 _______________________________________________________________________________________
и . 2.51. Пре
авле
е
у
ру
е е я у равля ва е С
хв
е
в
а
2.8.3. Ал о и о и и а ии неу ановивших я е и ов ан о и ования и о но о а а че е ГТС
(2.499 ) (2.499)
,
.Ш , . .(
. 2.52):
J ( u ) = ∫ Z ( t , u ( t ) ) dt ≈ ∑
N −1
T
i =0
Z i ( u ( ti ) ) + Z i +1 ( u ( ti +1 ) ) 2
⋅ Δt =
= 0,5 ⋅ Δt ⋅ Z0 ⎡⎣ u ( t0 ) ⎤⎦ + Δt ⋅ ∑ Zi ⎡⎣ u ( t0 + i ⋅ Δt ) ⎤⎦ + 0,5 ⋅ Δt ⋅ Z N ⎡⎣ u ( tN ) ⎤⎦ . 0
N −1 i =1
( u , u ,… , u ) J ( u , u ,… , u ) .
, (2.500) ,
( u , u ,… , u ) 0
1
N −1 T
0
1
N −1 T
0
1
N −1
(2.499)
N −1 i =1
© В.Е. Селе
(2.499 )):
ев, В.В. Алеш
-
:
J ( u0 , u1 ,… , u N −1 ) = 0,5 ⋅ Z0 ⎡⎣ u0 ⎤⎦ + ∑ Zi ⎡⎣ ui ⎤⎦ + 0,5 ⋅ Z N ⎡⎣ u N −1 ⎤⎦ → min,
(
(2.500)
, С.Н. Прял в, 2007–2009
(2.501 *)
272 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
{
}
ui ∈ U =
specified i i −1 specified = u ∈ R m : max ⎣⎡ u i −1 − ηconstr , umin ⎦⎤ j ≤ u j ≤ min ⎣⎡ u + ηconstr , umax ⎦⎤ j , j = 1, m , (2.501 )
i = 1, N − 1;
{
specified specified Wi ( u 0 , u1 ,… , u N −1 ) ∈ Ω = Wi ∈ R k : ( wmin ) ≤ wij ( u0 , u1 ,…, u N −1 ) ≤ ( wmax ),
j = 1, l ; w ( u , u ,… , u i j
specified u min ∈ Rm
0
1
) ≤ (w j
N −1
specified u max ∈ Rm –
specified max
)
j
}
j
, j = l + 1, k , i = 1, N ,
-
); ηconstr ∈ R m –
(
, -
j-
(2.501 )
Д .
и . 2.52.
е
ая а р
. -
Z i ( ui )
,
р
а я у е ля а а
. -
а ра вре е
а ра р р ва ервала1
, 1
,
. 2.50, 2.51
. © В.Е. Селе
ев, В.В. Алеш
(2.501 )
, С.Н. Прял в, 2007–2009
2.52
е а а р
-
лава 2 273 _______________________________________________________________________________________
, :
(2.501 *)
J ( u1 ,… , u N −1 ) = ∑ Zi ( ui ) → min,
u N = u N −1.
N
i =1
Д
(2.501 )
(2.501)
[118]. В (2.501 ) )
Э
-
( (2.501 ). (2.501 )
[32] [120],
-
. ,
-
,
.В
-
. К
А. Z i ( ui )
(2.501 ) ,
(2.501) (
-
-
) :
{
J ( u ) = Z ( u ) → min,
}
:
specified specified ⎤⎦ ≤ u j ≤ ⎡⎣ umax ⎤⎦ , u ∈ U = u ∈ R m : ⎡⎣umin j j
{
(2.502 )
j = 1, m ;
(2.502 )
specified specified W ( u ) ∈ Ω = W ∈ R k : ( wmin ) ≤ w j ( u ) ≤ ( wmax ) , j = 1, l;
}
specified w j ( u ) ≤ ( wmax ) , j = l + 1, k , j
j
u ∈ Rm -
j
(2.502 )
. . , . .
-
« » (2.502) © В.Е. Селе
ев, В.В. Алеш
(2.501). В , С.Н. Прял в, 2007–2009
(2.502)
274 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
. В. .
[2, 64, 120]. В.В. К -
– А. . К
[6, 26, 30, 121]. о и и а ии у ановивших я е и ов и о но о а а че е ГТС
2.8.4.
Д
о
и ования
(2.502)
Ка е е е е
ан
ь а а
я а а С а
. е е я а я щ х
е я х аче а а е я а а х С. а
В. . В.В. К
.Н
.К
е а ач е х е а а е С, а ь х е е че х а а а а а че е С ях е е е а е аа а че е а ч
ч е а
ь а че щ е я е ечех ч ах -
ь
[2, 59, 115]. [1, 2, 5, 6]. Н.Н.
(Д ) [1, 5])
В.В. К ,
[32] ( (
.Н
)
(Н )
) .В -
[118]
.Н.
3
,
(
.
)
К
А. Д , 1)
-
(
[122]. Н
-
К [6]: е
В.В. К ае К
Д К
( 2)
); ае
( ; © В.Е. Селе
-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
2.8.1), К
лава 2 275 _______________________________________________________________________________________
е ье
3)
ае
1 че
е
ае
я
ае
К
К ); Н ,
ЦН,
( . . 1, 2 К
К ;
3 5)
,
2, А
/ 4)
(
К )
К
(
.
4)
ЦН –
« »( ае
е
6)
.
3,
4.7); К
) ЦН
( ЦН
К (
.
3
4),
К
, (
.
5). .
В КЦ»,
а
КС.
( . ., «
А).
/
Д
К »(
«
А») ») , А. И .
«
А
К (
КЦ, К
-
К ,
А ).
В
«
-
( , ,
Н.Н. К (
[32] ( А
1
Д [1, 5]).
В.В. К К )
-
.
, . А
: ЦН,
;
1)
А
, -
; К . . [1, 5, 6]:
К ; А,
2)
,
ЦН
ЦН; К
3) , 1
© В.Е. Селе
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
276 М ел р ва е ра р р ва я р у в ру р в ы е а _______________________________________________________________________________________
А; К
4) ЦН;
,
3,
,
К
,
.
Н , А. Э •
А:
,
А
,
. К
,
А -
, А;
•
( );
•
;
•
А ;
•
А
ЦН ( В
.
, -
,
«Alfargus/OptimFlow»
© В.Е. Селе
, , 4.7)).
ев, В.В. Алеш
КАИ «Alfargus» (
, С.Н. Прял в, 2007–2009
(
ЦН -
.
1.6).
АВА 3 Чи ленны анали и е 3.1. По
очно
и
у о
ово ных
анов а а ачи , 1.7. ,
-
,
,
. -
, ,
НД , ,
,
,
-
. . ,
, ,
,
,
,
,
Н
-
. ,
В
Д
.В
, К
(
-
.
НД
[123],
,
НД ,
Н
( ,
,
-В
, . Д
,
, Н
– –
К
[123]:
σ ij , j + Fi = 0; ε ij = ⋅ ( ui , j + u j ,i ) ; 1 2
1
Д
© В.Е. Селе
( x, y , z ) . ев, В.В. Алеш
( x1 x2 x3 ) 1. В
), ), . -
(3.1)
(3.2)
-
, С.Н. Прял в, 2007–2009
278 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
–
-В
ε ki , jl + ε lj ,ik − ε li , jk − ε kj ,il = 0,
σ ij –
(3.3)
; ε ij –
; ui –
; Fi – ∂ ∂x j ; i, j , k , l = 1, 2, 3 . В
; ,j –
-
(3.1) .
Д
(3.1 – 3.3)
,
-
. ,
-
-
.В
-
,
(
)
σ ij = Eijkl ε kl ,
[123]: (3.4 )
.Д
Eijkl –
81
-
[84]. [124], -
К
.
,
,
,
.
,
,
НД
, -
,
. В
ν (
E;
:
G ). Э
[123]:
G=
Д
E . 2 ⋅ (1 + ν )
(3.5)
, Д
–Н
,
(3.4 )
, [123]:
-
σ ij = Eijkl ( ε kl − α kl ⋅ ΔT ) ,
α kl –
; ΔT –
. Д ,
α kl ≠ 0 ,
k =l ,
150°C ,
α kk = α ,
. 150°C
( k = 1, 2, 3) .
-
α
.
ев, В.В. Алеш
-
-
T
© В.Е. Селе
(3.4 )
, С.Н. Прял в, 2007–2009
лава 3 279 _______________________________________________________________________________________
И
,
(3.4 )
-
[292]:
σ ij =
E 1 +ν
ε 0 = ( ε11 + ε 22 + ε 33 ) 3 –
⎡ 1 +ν ⎤ ⎛ 3 ⋅ν ⎞ ⋅ ⎢ε ij + ⎜ ⋅ ε0 − ⋅ α ⋅ ΔT ⎟ ⋅ δ ij ⎥ , 1 − 2 ⋅ν ⎝ 1 − 2 ⋅ν ⎠ ⎣ ⎦
( δ ij = 1
К
К
( 3 ⋅ ε0 –
i = j ; δ ij = 0
); δ ij –
i ≠ j ).
,
-
. В
, ,
, (
(3.4 )
), ,
,
( (
–
.
1), [125,
) 1
126]. -
В
(
). 9
[84].
(3.4 )
.В
Gxy ; G yz ; Gxz ; α x ;α y ;α z . В
ν ij ≠ ν ji Eijkl [84]
:
ν xy Ex
В
:
=
12
i≠ j.
,
ν yx ,ν zy ,ν zx
,
ν yx Ey
ν yz
;
Ey
=
ν zy Ez
(3.6)
;
ν xz Ex
=
ν zx Ez
ν xy ;ν yz ;ν xz ;
E x ; E y ; Ez ;
(3.6)
.
Eijkl
(3.4 )
,
σx =
-
:
Ex Ψ
⎛ ⎞ ⎞ E ⎛ E E ⋅ ⎜ 1 − z ⋅ν yz2 ⎟ ⋅ ( ε x − α x ⋅ ΔT ) + y ⋅ ⎜ν xy + z ⋅ν yz ⋅ν xz ⎟ ⋅ (ε y − α y ⋅ ΔT ) + ⎜ E ⎟ ⎟ Ey Ψ ⎜⎝ y ⎝ ⎠ ⎠
Ez ⋅ (ν xz + ν xy ⋅ν yz ) ⋅ (ε z − α z ⋅ ΔT ); Ψ ⎞ E ⎛ E ⎛ E ⎞ E σ y = y ⋅ ⎜ν xz + z ⋅ν yz ⋅ν xz ⎟ ⋅ (ε x − α x ⋅ ΔT ) + y ⋅ ⎜ 1 − z ⋅ν xz2 ⎟ ⋅ (ε y − α y ⋅ ΔT ) + ⎜ ⎟ Ey Ψ ⎝ Ψ ⎝ Ex ⎠ ⎠ E ⎞ E ⎛ + z ⋅ ⎜ν yz + y ⋅ν xy ⋅ν xz ⎟ ⋅ ( ε z − α z ⋅ ΔT ); Ex Ψ ⎝ ⎠ +
1
К
,
,
.К ,
, .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
280 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
Ey ⎞ Ez E ⎛ ⋅ (ν xz + ν xy ⋅ν yz ) ⋅ (ε x − α x ⋅ ΔT ) + z ⋅ ⎜ν yz + ⋅ν xy ⋅ν xz ⎟ ⋅ (ε y − α y ⋅ ΔT ) + Ψ Ψ ⎝ Ex ⎠ ⎞ E ⎛ E + z ⋅ ⎜ 1 − y ⋅ν xy2 ⎟ ⋅ ( ε z − α z ⋅ ΔT ) ; Ψ ⎝ Ex ⎠ σ xy = Gxy ⋅ ε xy ; σ yz = G yz ⋅ ε yz ; σ xz = Gxz ⋅ ε xz ,
σz =
(3.4 )
E E E Ψ = 1− ⋅ν xy2 − z ⋅ν yz2 − z ⋅ν xz2 − 2 ⋅ z ⋅ν xy ⋅ν yz ⋅ν xz . Ex Ey Ex Ex Ey
(α
В -
,
[125]. В , ,
,
Oy
Gxy = G yz .
-
, .
,
(
».
= α y = αz = α ) ,
x
)«
» Ox, Oz
,
« : Ex = Ez ; ν xy = ν yz ;
, , Gxz
(3.5),
-
: E x ; E y ; ν xy ; ν xz ; Gxy .
, ,
E = E x ; ν = ν xz . (3.4 )
, 6
-
. )
(
.Д
,
-
, .
Д
, ,
.К
,
-
, ,
НД .
В
, ,
-
-
.В
-
f (σ ij , ε ijp , T , χ i ) = 0,
[84]:
ε ijp –
. К
, , К
© В.Е. Селе
; χi –
; T – ,
(3.4 ) -
. -
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 3 281 _______________________________________________________________________________________
. Д
К
(3.2),
, ,
ε ij = ⋅ ( ui , j + u j ,i + uk ,i ⋅ uk , j ) .
[123]:
1 2
, (3.1 – 3.4) НД
(3.7) (3.1), (3.3). ,
. Д ui = ui* ,
:
,
x ∈ S1 ;
(3.8)
σ ij ⋅ n j = ti , x ∈ S2 , ti –
(3.9)
; ni –
; S1 ∪ S 2 = S –
; x –
. (3.1 – 3.4) ,
,
.В [123].
,
В
, -
,
-
)
.
( ,
(3.2) -
(3.4), (3.1).
, (3.3)
-
. -
[89]. Н
,
, [123]:
3 ⋅ ( λ + μ ) ⋅ ε 0,i + μ ⋅ Δui + Fi = 0, i = 1, 3,
λ, μ –
( λ = 2 ⋅ν ⋅ G (1 − 2 ⋅ν ) , ,
,
μ = G) ; Δ –
(3.10)
(3.9)
(3.10)
-
[89]. (3.8)
НД
-
.
3.2. Ме о и е
ешения у авнени
авнове ия
у о
ово ных
К , , © В.Е. Селе
. ., ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, -
282 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
(
,
. .). Н
, (
, (3.10))
-
.
НД
(3.1 – 3.4). КЭ [48, 127 − 129].
,
Д
КЭ
.
КЭ – ,
, -
,
,
КЭ. (
К
,
. К КЭ – К )
, -
, , ó .
,
-
,
,
,
,
КЭ. -
(«ANSYS» [130], «LS-DYNA» [131], «MSC.NASTRAN» [132], «ABAQUS» [133], «MSC.MARC» [134], «ALGOR», «COSMOSM» .) КЭ. КЭ НД ( , ). (КЭ) Д : ( ), ( ) , , , [48]. В .Д . Д , , . 1 ( [123]) Д (3.1), (3.2), (3.8), (3.9) :
∫ (σ
V
δ ui – (3.11)
© В.Е. Селе
+ Fi ) ⋅ δ ui dv − ∫ (σ ij ⋅ n j − ti ) ⋅ δ ui ds = 0,
(3.11)
S2
(3.8). И
V,
1
ij , j
–
,
: ,
ев, В.В. Алеш
(3.2) , С.Н. Прял в, 2007–2009
(3.8).
лава 3 283 _______________________________________________________________________________________
∫σ
ij
V
⋅ δ ui , j dv − ∫ Fi ⋅ δ ui dv − ∫ ti ⋅ δ ui ds = 0. V
(3.12)
S2
(3.12) . (3.12)
:
ui ( x ) = ∑ N j (ξ ,η , ς ) ⋅ uij , k
(3.13)
j =1
(ξ , η , ς ) ;
1
Nj –
КЭ; ui – j
,
j(3.13)
. ,
i-
(3.12),
-
:
n
m =1
КЭ,
n –
КЭ;
–
КЭ; {ε th } –
b e
e
th e
m
T
pr
nd e
e
{F } = ∫ [ B ] [ D ] { ε } dv th e
T
Vm
{F } = ∫ [ N ] {F }dv [ D]
th
; { Fe pr } =
) КЭ; { Fend } – –
Vm
–
; [ B] –
КЭ;
b e
m
= 0,
(3.14)
; [ Ke ] =
∫ [ B ] [ D ] [ B ] dv T
Vm
–
∫ [ N ] {P}ds T
n
–
(
.
-
Sp
{ um }
(
КЭ;
)
КЭ;
–
КЭ; [ N ] –
; [ Nn ] – Sp ,
КЭ,
,
[]
T
–
,
(3.14).
КЭ
.
,
{u} = [ N ] {um } ; {ε } = [ B ] {um } ; {σ } = [ D ] {ε
{ε } = {ε } − {ε } – el
.В
th
N j (ξ , η , ς )
© В.Е. Селе
ев, В.В. Алеш
«
el
},
: (3.15)
(3.15)
-
1
-
-
∑ ( [ K ] {u } − {F } − {F } − {F } − {F } )
,
k –
.
».
, С.Н. Прял в, 2007–2009
НД (3.7). Д
284 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
,
V, .
t
F
S,
ε ij = ⋅ ( ui , j + u j ,i + uk ,i ⋅ uk , j ) = 0 1 2
( −σ ) [123]:
,
∫ F ⋅ δ u dv + ∫ t ⋅ δ u ds − 2 ⋅ ∫ σ
ij
1
i
i
i
V
i
S
ij
V
⋅ δ ( ui , j + u j ,i + uk ,i ⋅ uk , j ) = 0.
∫ σ ⋅ (δ ij
ki
(3.17)
σ ij ,
(3.17)
V
(3.16)
:
+ uk ,i ) ⋅ δ uk , j dv − ∫ Fi ⋅ δ ui dv − ∫ ti ⋅ δ ui ds = 0. V
(3.18)
S
(3.18) (
)
.
(3.18) .
К
,
(3.18)
−
, ,
, ⎡σ ij ⋅ (δ ki + uk ,i ) ⎤ = Fk , ⎣ ⎦, j
(3.19)
⎡σ ij ⋅ (δ ki + uk ,i ) ⎤ ⋅ n j = tk . ⎣ ⎦ S
Д
(3.19)
(3.20)
(3.20),
uk
,
(
),
ε ki – ,
[123]:
uk ,i = ε ki + ω ki ,
(3.21)
(3.7); ω ki –
ω ki = 1 2 ⋅ ( uk ,i − ui , k ) .
Д
,
, , .В , КЭ [48, 127 − 129]. , НД
© В.Е. Селе
КЭ
КЭ, Э
ев, В.В. Алеш
,
КЭ [48, 89]. И
, С.Н. Прял в, 2007–2009
-
. Д ,
КЭ
-
лава 3 285 _______________________________________________________________________________________
,
, .
3.3. Ма е а иче ие о ели ля анали а у у о- ла иче о о ове ения у о он у и
ово ных
-
-
.
( )
,
-
.В [123]. ,
.Э
,
-
. В , ,
-
, .
.К
,
-
, ,
, .
3.3.1.
у о- ла
Д
иче
ое ове ение
у ных
але
– ,
. В ,
-
[135]. ,
,
,
Ч
, . В –
σi =
σ1, σ 2 , σ 3 – 1
В
1 2
⋅
(σ 1 − σ 2 )
,
-
[136].
-
1
2
+ (σ 2 − σ 3 ) + (σ 3 − σ 1 ) , 2
2
: (3.22)
. –
-
. © В.Е. Селе
-
.
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
286 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
, (σ i = σ Т )
–
(σ i = σ в )
,
НД
-
–
.
(3.4 ) 2 ⋅σ Т
Н
,
,
3.
–
-
(
,
)
, .
, . 3.3.2. Мо ели ование в аи о е ун а
вия
у о
ово а и
иле ающе о
, -
. Н
, 90% [137, 138].
НД
-
.
, 1
, . ,
-
,
.В
В.В. А
[1, 3, 6, 139]. ) [137, 138, 140 –
( 144]. В -
-
. .
3.3.2.1. Ин ене ные о ели в аи о е о у ающи ун о
В
вия о
е но о
[137, 138, 143, 144]
у о
ово а
[140 – 142]
-
, ,
1
Д
. Э ,
, © В.Е. Селе
.
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
лава 3 287 _______________________________________________________________________________________
. [138],
-
(
,
) 1
, [1, 137, 138, 143, 144]. НД [1, 3, 6].
. . 3.1. -
, .
и . 3.1. Схе а а р р вле е ру а – р
«
а л
л е е ере е е
а ра е ру
ел е р в а»,
tg β = π ⋅ D ⋅ cx 0
ав е
,
-
,
А (ASCE) [140],
[133, 142].
,
, (
)
-
, В ,
. [138]: ( cx 0 , c y 0
, ),
-
β
( (
, ;
1
-
t
Д
р
. 3.1) )
,
-
.
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
288 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
,
qр,
(
(
.
. 3.1),
. В -
) [140, 141]
:
,
xu , yu , zu (
( . . t р , q р ).
3.1)); (
tu , pu , qu
cx 0 =
t р = tu ; tu ; π ⋅ D ⋅ xu
= pu ;
q грри
таль
c yг 0ри
таль
D –
=
:
q вертикаль = qu ; р
c вертикаль = y0
pu ; D ⋅ yu
qu , D ⋅ zu
(3.23)
. ,
.
П
ь
е
еще
е
а [137, 138]
-
t р = qтр ⋅ tgϕ гр + 2 ⋅ γ гр ⋅ qтр –
;
h
е. В
:
h
⋅ π ⋅ D 2 ⋅ tgϕ гр + 0, 6 ⋅ π ⋅ D ⋅
; ϕ гр –
–
(3.24 )
,
; γ гр – ,
h/ D ; h –
-
( ); cгр –
А
гр
.
(3.24 )
, , [1]. Д (3.24 ) -
, ,
, ,
(
, [1, 3, 5])
,
. ,
(
,
[144]),
-
. (
cx 0 )
-
[138] ( ,
.
. 3.1), ,
[137, 138] .
В
ASCE [140]
1
:
1
Д
3.3.2.1 ,
© В.Е. Селе
, .
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
лава 3 289 _______________________________________________________________________________________
–
tu =
π 2
⋅ D ⋅ γ ⋅ H ⋅ (1 + k0 ) ⋅ tg ( k ⋅ ϕ гр ) ;
–
(3.24 )
tu = π ⋅ D ⋅ α ⋅ Su ,
γ – (
(3.24 )
; H – ); k0 –
,
0,35
0,47; k –
,
(
)
; α –
,
Su
,
: 2,54 ÷ 5, 08мм
[133, 140, 142]
-
0,7; Su –
0,5
-
[140].
Д
xu
; 5, 08 ÷ 10,16мм
. , , 1984
ASCE [140]
, k0 = 1, 0 ,
.
,
,
,
.В
,
Su cгр ,
cгр = α ⋅ Su ,
[133] -
α
[140]. ,
k,
, П
, -
[142]
,
tg ( k ⋅ ϕ гр )
, (3.24 )
-
[137]. е еч
е, е
а ь
[138]:
cy 0 =
е х,
еще
е
2⋅ H 0,12 ⋅ Eгр ⋅ηгр ⎛ − ⎞ D − 1 e ⎜ ⎟; 2 (1 −ν гр ) ⋅ l0 ⋅ D ⎝ ⎠
q р = γ гр ⋅ D ⋅ ( H − 0,39 ⋅ D ) + γ гр ⋅ H 2 ⋅ tg ( 0, 7 ⋅ ϕ гр ) + Eгр –
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
а
е. В
-
(3.25 ) 0, 7 ⋅ cгр ⋅ H
cos ( 0, 7 ⋅ ϕ гр )
,
(3.26 )
, [ МПа / м ] ; ηгр –
-
290 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
; ν гр –
( l0 = 100 м ) ;
[ м] .
; l0 –
D, H –
,
(3.23), (3.24 ),
, ,
.
(3.25 ) .
cy 0
(
.Н
гр
,
= 0)
2
D
ASCE [140]
) −π
: 8⎤ . ⎦
: qu = γ ⋅ H ⋅ N qv ⋅ D,
–
(3.26 )
zu = ( 0,01 ÷ 0,015) ⋅ H ;
(3.25 )
qu = Su ⋅ N cv ⋅ D,
–
(3.26 )
zu = ( 0,1 ÷ 0, 2 ) ⋅ H ,
(3.25 )
N qv , N cv –
, , [140].
, П
е еч
е, е
а ь
-
(3.25 ), (3.26 ) -
[144]
q р = γ гр ⋅ D 2 ⋅ ⎡ H D + ( H D ) ⋅ ( tgϕ гр + 5 ⋅ e −1,7⋅H ⎣
В
,
,
-
H /D еще
,
е
а
е.
В
–Ц
-
( D)
,
[138]
-
q = cy0 ⋅ D ⋅ uy ,
u y ≤ Rгр / c y 0 ;
q = Rгр ⋅ D, q –
:
u y > Rгр / c y 0 ,
(3.27 )
; uy –
;
Rгр –
. : cy 0 =
В © В.Е. Селе
(3.28 ) ASCE [140] ев, В.В. Алеш
,
0,12 ⋅ Eгр
(1 −ν ) ⋅ 2 гр
l0 ⋅ D
(3.28 )
.
(3.25 ).
, С.Н. Прял в, 2007–2009
:
лава 3 291 _______________________________________________________________________________________
–
1 qu = γ ⋅ H ⋅ N q ⋅ D + ⋅ γ гр ⋅ D 2 ⋅ N y , 2
(3.27 )
zu = ( 0,10 ÷ 0,15) ⋅ H ;
(3.28 )
qu = Su ⋅ N c ⋅ D,
(3.27 )
–
zu = ( 0,10 ÷ 0,15) ⋅ H , Nq , N y
(3.28 )
Nc –
, ,
П
е еч е [138]
а ь
е
еще
е
[140].
а
е. В
cy 0 =
ηг р –
2⋅ H 0,12 ⋅ Eгр ⋅ηг р ⎛ − ⎞ D e 1 − ⎜ ⎟, 2 (1 −ν гр ) ⋅ l0 ⋅ D ⎝ ⎠
(3.29 )
,
a/H ,
a –
.
К
-
:
(3.29 )
,
,
:
q р.г р = γ гр ⋅ H ⋅ D ⋅ k p ,
ϕ гр ⎞ 2 ⋅ гр ϕ гр ⎞ ⎛ ⎛ k p = tg 2 ⎜ 45 + ⋅ tg ⎜ 45 + ⎟+ ⎟ – 2 ⎠ γ гр ⋅ H 2 ⎠ ⎝ ⎝ . , q р.г р < Rгр / D ,
.
-
(3.25 ).
(3.30 ) Rгр
,
Rгр / D . , . q р.г р =
В –
,
[144]
γ гр ⋅ ( H + D ) 2
2
ϕ гр ⎞ ⎛ ⋅ tg 2 ⎜ 45 + ⎟ + 2⋅ 2 ⎠ ⎝
ASCE [140] yu = Ch ⋅ ( H + D / 2 ) ,
– ев, В.В. Алеш
гр
ϕ гр ⎞ ⎛ ⋅ D ⋅ tg ⎜ 45 + ⎟. 2 ⎠ ⎝ :
pu = γ ⋅ H ⋅ N qh ⋅ D;
© В.Е. Селе
(3.30 ):
, С.Н. Прял в, 2007–2009
(3.29 ) (3.30 )
292 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
yu = ( 0, 03 ÷ 0, 05 ) ⋅ ( H + D / 2 ) ,
(3.29 )
pu = Su ⋅ N ch ⋅ D,
(3.30 )
N qh , N ch – ,
, 0,05
H /D
[140];
Ch
: ;
В
0,07 0,02
0,10 0,03
;
0,03
. [142]
,
А
[140] –
, xu , tu ( . ( 150мм
).
D = 150 мм
2%
[142] [141] -
610мм ).
, (
ASCE -
[141].
[140] D = 610 мм ).
13%
, ,
, [141],
, [140].
К
, . ,
(3.24 )
-
, , -
,
, ( ∼ 1м )
(3.24 – ).
–
-
. .
В -
,
, . , , (
,
, [137, 138] ASCE [140], x0
).
,
, , . [140, 141]
К
,
-
,
.
.В
,
«
,
,
»
,
-
[138], .
Н
,
[138]
,
,
, © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 3 293 _______________________________________________________________________________________
НД
.
, , 1
НД
,
В
.
. -
,
, -
[138]
И
. -
, 2
,
, «
. В
» .
Н
,
,
-
. В НД
. [1, 3 − 5].
3,
Н
,
-
, , ,
.
,
(
,
. Н НД -
, .) .
-
. ( Д
-
В.В. А ,
[1, 3 − 5, 145]. Д [138]
1
,
-
,
.
2
© В.Е. Селе
,
. .).
, ев, В.В. Алеш
,
, С.Н. Прял в, 2007–2009
.
294 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
,
-
. 3.3.2.2. Т ех е ная у
у о- ла
иче
ая
о ель
ун а
,
, -
, Д
НД
.
:
(
-
) . , ( . .)
,
.
,
-
,
-
.
Д
,
,
.
И
НД
,
-
[137, 138, 140 – 144, 146 − 148], ,
1
,
–
.Э
.Д
, ,
. Э
-
. Д
-
–
-
. ,
, ,
2
,
.Н
,
, .
В (
), [144, 146 –
149], -
.В ,
.В
, 1
И
,
-
. 2
«
» .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 3 295 _______________________________________________________________________________________
τ n = Φ (σ n ) ,
[148]:
τn
σn –
(3.31)
n ; Φ (σ n ) –
(
)
Φ (σ n )
В К
В
. ,
(3.32)
,
, [148, 150]:
τ n = c + tgϕ ⋅ σ n ,
;ϕ – c ϕ
c –
-
(3.32) . -
.И
,
.
К
[148], (
)
σ n < σ φ ≈ 0,5 ÷ 0, 7 МПа . Д
)
( ,
-
. , (
-
. (3.32)).
НД ,
σ 1 − σ 2 = ( 2 ⋅ c ⋅ ctgϕ − σ 1 − σ 2 ) ⋅ sin ϕ ;
К [1]:
(3.32)
σ 2 − σ 3 = ( 2 ⋅ c ⋅ ctgϕ − σ 2 − σ 3 ) ⋅ sin ϕ ;
-
(3.33)
σ 3 − σ 1 = ( 2 ⋅ c ⋅ ctgϕ − σ 3 − σ 1 ) ⋅ sin ϕ .
(3.33)
-
σ1 = σ 2 = σ 3 ( ,
-К , 4).
.
,
σi = σ j .
), -
, (3.33)
-
:
( c ≠ 0, ϕ ≠ 0 )
c
(
(
. 3.2). В
© В.Е. Селе
ев, В.В. Алеш
ϕ
σ 1* = σ 2* = σ 3* = c ⋅ ctgϕ . ,
(3.34)
(3.34) ,
–К
( c = 0, ϕ ≠ 0 ) , С.Н. Прял в, 2007–2009
. К
,
296 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
( c ≠ 0, ϕ = 0 )
(3.33)
(3.34),
, –К
), , c = σТ 2 ) [136].
(
c (
и . 3.2. П верх
е уче
. 3.2 .
М ра – Кул
а
–К
НД
-
, (
),
σ1 = σ 2 = σ 3 )
,
.
(
.В . .) 1
(
,
,
,
( ).
, -
(
,
. .) .В
,
-
, .
В
1
[144, 146 − 149]. , -
И
-
.
(
). , .
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 3 297 _______________________________________________________________________________________
(3.33) . Д
,
,
(ρ ) ,
В
-
(E) ,
ν ( ),
(ϕ )
:
(c) .
,
,
-
1
,
-
.И .
Н
, ,
. В ,
, ,
-
,
-
– К
(3.33). В -
,
-
,
[123, 151] . Э
[48, 127, 128]
, –К
[48, 123, 128]:
f ( I1 , J 2 , J 3 ) = 0,
; sij = σ ij − δ ij I1 3 – НД
α
. Д. Д
k –
[150]: (3.36)
Д
.
-
В.
α ⋅ I1 + J 2 = k ,
tg β = 6 ⋅ α (
(3.35)
; J 2 = sij sij 2 –
; J 3 = sij s jk ski 3 –
2
–К
. –
(3.36) ,
, ,
–К
1
Н
,
II-9-78 «И
». 2
© В.Е. Селе
(3.36) [150]. А
ев, В.В. Алеш
, [150] -
, С.Н. Прял в, 2007–2009
-
,
k k ⎫ ⎧ k ; ; ⎨ ⎬, ⎩ 3 ⋅α 3 ⋅α 3 ⋅α ⎭ 4).
,
[292],
-
I1 = σ ii –
В
Д
-
[48, 127 − 129]. ,
В
-
НД
В. (3.36) .
298 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
Д
ϕ, c
–
,
-
. В ,
,
α, k .
–
Д
-
.В
-
:
; (
,
4).
[150] (
,
КЭ-
, (
.,
[130, 133]),
–
. ,
4 1
– К
Д
α= α= ,
– :
sin ϕ
3 ⋅ 3 + sin ϕ 2 ⋅ sin ϕ
3 ⋅ ( 3 − sin ϕ )
Д
k=
;
2
k=
;
,
3 ⋅ c ⋅ cos ϕ 3 + sin 2 ϕ
(3.37)
,
–
6 ⋅ c ⋅ cos ϕ
:
3 ⋅ ( 3 − sin ϕ )
–
(3.38)
.
(3.38).
, ,
НД (3.37), (3.38),
, НД
-
-
Д
2
[150] -
, .В
)
НД
(3.36) , ϕ, c
(3.33). α, k
.
В
.
НД
, -
. Д
–
[1], а
1
В
2
Н
В
а ь х [1, 145]: е ч а а х е е х ь а ь
© В.Е. Селе
В.В. А е е я щ х
. а а е
е я е , я е че .
Д
а
е х че х е а–П ае а
В.В. А
. 4.
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
е че а е а М а–К
е Д х а,
– а
яа
лава 3 299 _______________________________________________________________________________________
А
( 4)
Д
α
k
В
–
т
т
-
(3.39)
⎧ 6⋅ 2 c ⋅ cos ϕ , 0 < ϕ ≤ 27,65°; ⋅ ⎪ = ⎨ 3 + 3 3 + sin 2 ϕ ⎪ ⎩c ⋅ cos ϕ , ϕ ≥ 27, 65°.
,
(3.40)
-
Д (3.39), (3.40), НД ,
–
-
, . -
. , 1952
[150]
-
:
⎧ 2⋅ 2 sin ϕ ⋅ , 0 < ϕ ≤ 27, 65°; ⎪ ⎪ 3 + 3 3 + sin 2 ϕ =⎨ ⎪ sin ϕ ⎪⎩ 3 , ϕ ≥ 27,65°;
[1, 3, 5]
.
Д. Д
1
НД . А
В.
-
. Н
-
, ,
Д
,
[48].
НД
,
.Н
-
-
, -
, –К
-
,
,
.И
, -
.
И
,
, . К
(3.36) [128]
α=
1
2 ⋅ sin ϕ
3 ⋅ ( 3 + sin ϕ )
,
k=
В
© В.Е. Селе
6 ⋅ c ⋅ cos ϕ
3 ⋅ ( 3 + sin ϕ ) –
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(3.36)
(3.41)
[147].
300 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
,
– К (
(3.38), ( . .
В
(3.36) –
Д
(3.33) (
-
, 4.4)). -
[146] ,
– 4).
Д
(3.33), ,
, ,
,
.
f ( I1 , J 2 ) = 0 ). Д
(3.35) (
-
НД [152]: p=−
σ1 + σ 2 + σ 3 3
q=
;
⎛9 ⎞3 r = ⎜ ⋅ sij s jk ski ⎟ , ⎝2 ⎠
3 ⋅ sij sij ; 2
1
(3.42)
(3.35): I p=− 1; 3
К
q = 3⋅ J2 ;
⎛ J ⎞3 r = 3⋅⎜ 3 ⎟ . ⎝ 2⎠ 1
(3.42), p – .
В
; q – . (
r , [48, 128, 133, 152])
( )
(3.43)
Д
( ),
–
-
. (3.42)
t=
:
t − p ⋅ tg β '− d = 0,
(3.44)
3 ⎡ 1 ⎛ 1 1⎞ ⎛r⎞ ⎤ ⋅ q ⋅ ⎢1 + − ⎜1 − ⎟ ⋅ ⎜ ⎟ ⎥ ; β ', d – 2 ⎢⎣ K ⎝ K ⎠ ⎝ q ⎠ ⎥⎦
, ; K –
»
Д -
Д (
–
.
(3.44)
β'
)
4) Д
– t.
4 , ев, В.В. Алеш
-
,
(
© В.Е. Селе
« .
, С.Н. Прял в, 2007–2009
« p − t ». ) , –К ,
Д
(3.44), d (
-
–
:
лава 3 301 _______________________________________________________________________________________
tg β ' =
(3.44) (3.38), (3.43), (3.36)
d=
K =1
K.
,
6 ⋅ c ⋅ cos ϕ . 3 − sin ϕ
(3.45)
–
(3.45)
(3.44) (3.44) (3.38).
,
Д
0 < K < 1,
6 ⋅ sin ϕ ; 3 − sin ϕ
-
– К
,
. -
[1], [148]. (3.44) (3.33).
,
-
β ', d
Ш
,
-
(3.44) (3.45). В
-
K
, K=
. 3.3
-
–
Н
(3.44). .
. К (3.45)
3 − sin ϕ . 3 + sin ϕ
. 3.3, (3.46),
(3.44) -
–К
(3.33): (3.46)
(3.44) (3.33). Н
. 3.4 , ϕ = 20°, c = 15 кПа .
и . 3.3. Сече е ев а р л верх е е уче : 1 – М ра – Кул а (3.33); 2 – ру ера – Пра ера (3.36), (3.38); 3 – е р ер я ру ера – Пра ера (3.44 – 3.46). © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, -
302 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
. и . 3.4. В
верх
е уче
е
р
ер я ру ера – Пра ера (3.44)
,
(3.44)
,
, -
θ ( :
-
.
, –
.
. 3.3),
,
[48, 123, 127, 128, 133, 152] -
, . . ,
,
, ,
(
). [133, 152] . 3.3
В (
θ [152, 291]:
– .В
)
⎛r⎞ cos ( 3 ⋅θ ) = ⎜ ⎟ , ⎝q⎠ 3
cos ( 3 ⋅θ ) =
(3.47)
3 ⋅ 3 ⋅ J3 . 2 ⋅ J 23 / 2
(3.48)
В
-
:
sin ( 3 ⋅θ ' ) =
arccos x = π 2 + arcsin ( − x ) ,
И
© В.Е. Селе
−3 ⋅ 3 ⋅J 3 . 2 ⋅ J 23 / 2
θ'
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(3.49) ,
θ = θ '+ π 6 , -
лава 3 303 _______________________________________________________________________________________
.
Д И
(3.47)
(3.48),
.
, (3.44) p*
2 ⋅ 2 ⋅ ( p* ⋅ tg β '+ d )
R (θ ) =
1⎞ ⎡ 1 ⎛ ⎤ 3 ⋅ ⎢1 + − ⎜1 − ⎟ ⋅ cos ( 3 ⋅θ ) ⎥ K K ⎝ ⎠ ⎣ ⎦
R (θ ) –
, C
. 3.3.
(1)
К
–К
: 0 ≤ θ ≤ 2π .
(3.50)
,
R (θ )
(3.33) :
-
Rmc (θ ) ⋅ q − p ⋅ tgϕ − c = 0,
(3.33*)
π⎞ 1 π⎞ ⎛ ⎛ ⋅ sin ⎜θ + ⎟ + ⋅ cos ⎜θ + ⎟ ⋅ tgϕ , 3⎠ 3 3⎠ 3 ⋅ cos ϕ ⎝ ⎝ . ,
Rmc (θ ) =
-
:
1
К
, 0 < K < 1. Д
(3.44), ( ),
).
,
.В
(
1
.
(3.44) ) k2 = 0 .
[89],
2
( ,
-
[123]. И
,
(
) k1 .
[89],
(3.50). Э
. 3.3 (3.44). (3.44)
(3.50) k [89]:
(
)
-
1
(
k2 (
k1
),
– ) [89]. « p−q »
2
(r q)
3
= const
© В.Е. Селе
(3.44). ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
304 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________ 2 2 k = ⎡ ρ 2 + 2 ⋅ ( d ρ dθ ) − ρ ⋅ d 2 ρ dθ 2 ⎤ ⎡ ρ 2 + ( d ρ dθ ) ⎤ 2 , ⎣ ⎦ ⎣ ⎦ 3
ρ, θ –
ρ = R (θ )
(3.51) k =
. (3.50),
v 2 + 9 ⋅ v ⋅ (1 − 1 K ) ⋅ cos ( 3 ⋅ θ )
,
(3.51)
:
2 2 v ⎡ u 2 ⎛ −3 ⋅ u ⋅ (1 − 1 K ) ⋅ sin ( 3 ⋅ θ ) ⎞ ⎤ ⋅⎢ +⎜ ⎟ ⎥ u2 ⎢ v2 ⎝ v2 ⎠ ⎥⎦ ⎣ 3
(3.52)
,
4
u = 2 ⋅ 2 ⋅ ( p* ⋅ tg β '+ d )
3 ; v = 1 + 1 K − (1 − 1 K ) ⋅ cos ( 3 ⋅ θ ) .
[89],
Oy k=
ϑ, s –
dϑ , ds
k
В
: (3.53)
,
.
k≤0
.
k ≥0,
, (3.50)
k 1
(3.44) (3.52). Н
. 0 ≤θ ≤π 3
(3.50)
(3.44)
(
-
( 3 ⋅ v )⎤⎦ , k [89] (3.52),
β '' = arctg ⎡ 2 ⋅ 2 ⋅ tg β ' ⎣ К (3.52),
(3.44) k1
)
.
sign [ k1 ] = sign [ k
И
(3.44) (3.52). ,
:
] = sign ⎣⎡8 ⋅ cos ( 3 ⋅θ ) ⋅ (1 − 1 K ) + 1 K + 1⎦⎤ .
(3.52)
[0;
, (3.50) k θ =0 ( . . 3.3). 0 ≤θ ≤π 3 k1
1
,
© В.Е. Селе
ев, В.В. Алеш
π 3]
-
( 0 < K < 1; 0 ≤ θ ≤ π 3) , С.Н. Прял в, 2007–2009
,
: (
.
(3.54)
(3.54)
(3.44)
2
. 2
.
. 3.3),
.
v
(3.52)
-
лава 3 305 _______________________________________________________________________________________
7 K≥ . 9
,
,
(3.44), (3.55)1.
K >7 9
,
K =7 9
,
,
θ = 0; 2π 3; 4 π 3 . В , (3.55) (3.55) , НД ,
И (3.46)
(3.55)
, -
,
[89]. (3.44). ϕ ≤ arcsin ( 3 8 ) ≈ 22° .
Н 22° [137, 138]. (3.44), – ,
Д , (3.44)
-
.Д
[48, 133]. (3.33),
.К , ,
-
[48, 133]: l 2 + q 2 − p ⋅ tg β '− d = 0,
l = d − pt ⋅ tg β ' ; pt – . Н . 3.5
(3.56) « p − q »,
(3.56). К (3.56),
,
.К
, -
,
pt ,
Д
pt = d / tg β '
Н
–
,
-
. 3.5 (3.56) Д
. 3.6
–
.
(3.36). (3.56)
ϕ = 20°, c = 15 кПа
pt = 30кПа . Д Д –
,
(3.36), -
1
В
K ≥ 0, 778 .
. Н
K
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
КЭ[133]
-
(3.55)
306 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
.
,
,
(3.56), ( . 3.6).
-
и . 3.5. И ра е е ер л че ру ера – Пра ера (3.56) в л
и . 3.6. П верх ла че
К
верх е уче е е ара е р че
е уче (3.36) р
р
е ер л че ер ев ру ера – Пра ера
е
р ер я а « p−q »
(3.56)
,
.
XX Н
х
, 1
,
,
. -
[154]. Д
,
1
, . [131].
© В.Е. Селе
90-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
, -
лава 3 307 _______________________________________________________________________________________
–В
ϕ, c . И
[154]
,
-
НД (3.42)
θ,
(3.47) –В
Rmw (θ ) =
1
(ε ⋅ c ⋅ tgϕ )
-
:
2
+ ( Rmw (θ ) ⋅ q ) − p ⋅ tgϕ − c = 0, 2
4 ⋅ (1 − e 2 ) ⋅ cos 2 θ + ( 2 ⋅ e − 1)
(3.57)
⎛π ⎞ ⋅ Rmc ⎜ ⎟ ; ε – ⎝3⎠ 2 ⋅ (1 − e ) ⋅ cos θ + ( 2 ⋅ e − 1) ⋅ 4 ⋅ (1 − e ) ⋅ cos θ + 5 ⋅ e − 4 ⋅ e 2
2
2
2
-
2
, ; Rmc (θ ) –
, ⎛ π ⎞ 3 − sin ϕ . Rmc ⎜ ⎟ = ⎝ 3 ⎠ 6 ⋅ cos ϕ
( . (3.50)), 0 ≤ θ ≤ 2π .
–
,
-
(3.33*),
(3.57) , , Rmw (θ )
–В
Д
; e –
0 ≤θ ≤ π 3. -
(3.57) –В
2
, , . (3.57)
–В (
) e
e=
И
–К
(3.33), (3.33*)
. (3.46)):
(
(3.57),
3 − sin ϕ . 3 + sin ϕ
(3.58)
,
1 2 ≤ e ≤1, , 0° ≤ ϕ ≤ 90° ,
( e (3.58)),
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.
[137, 138]. (3.57)
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1
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В Н
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,
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ев, В.В. Алеш
Rmw (θ )
.
, С.Н. Прял в, 2007–2009
C (1) .
-
308 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
– К θ = 0; π 3; 2 π 3, (3.57)
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НД
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.
«Alfargus/StructuralAnalysis» .
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К.И. Д
-
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1985
, С.Н. Прял в, 2007–2009
(
. [131]).
(3.57). В
лава 3 309 _______________________________________________________________________________________
и . 3.7. П верх
и . 3.8. Сече е ев а В л я а (3.57),(3.58); 2 –
3.4.1. Вы о
е
В
р
в
е уче
Ме е рэ – В л я а
л верх е е уче : 1 – Ме е рэ – е р ер я ру ера – Пра ера (3.44 – 3.46); 3 – М ра – Кул а (3.33)
о ели ования НДС
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3.2)
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,
,
ово ов
КЭ. К
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ев, В.В. Алеш
2002 ,« «ANSYS» ( . 3.9) [3, 6].
, С.Н. Прял в, 2007–2009
»
310 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
и . 3.9. И
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л
Н
ва еля выч л ел р ра ы «ANSYS»
ех
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,
НД
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«ABAQUS»
лава 3 311 _______________________________________________________________________________________
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-
312 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
(
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[1, 3].
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-
лава 3 313 _______________________________________________________________________________________
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314 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
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6). (
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ев, В.В. Алеш
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лава 3 315 _______________________________________________________________________________________
НД
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и . 3.12. а
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е ру
е
В
рал
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-
«Alfargus/StructuralAnalysis»,
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Н
е а
НД
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4]. В [1, 3–5] «ANSYS». – , 3.4.2.2.
в а ря е [Па] а уча р в а ( ал ч ая КЭел )
КЭ-
о ели
у о
[6, 7].
ово ов
НД
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,
КЭ. -
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3.3.2.2.
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. 3.13.
и . 3.13. а че
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, С.Н. Прял в, 2007–2009
ру ( ра
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316 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
КЭ-
КЭА
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, ев, В.В. Алеш
. Н
3.4.2.2, , С.Н. Прял в, 2007–2009
,
.
лава 3 317 _______________________________________________________________________________________
. ,
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2
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КЭ-
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, С.Н. Прял в, 2007–2009
, .
-
318 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
и . 3.16. Зав
лы
е ные
3.4.2.3.
р
о ели
вле
у о
я ру а
ев
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я ру ы
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. 3.17). КЭ-
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1
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НД . В , .
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
, ,
-
лава 3 319 _______________________________________________________________________________________
В
,
НД
, [1,
3–5]. [6, 7].
В
-
НД .
НД
-
КЭ-
. 3.18.
и . 3.17. а че
и . 3.18. а
ая
ре еле
ел уча
е
е
рр
в
а ру
ы
р в
рр
,
ев, В.В. Алеш
ы
а ря е [МПа] а уча е е а
Н
© В.Е. Селе
а
, С.Н. Прял в, 2007–2009
е е а
е ру
[1, 3, 6]. Н
р в
а
,
320 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
«Alfargus/StructuralAnalysis» [3].
,
НД .
3.4.3. Анали НДС и о ен а
очно
и
у о
ово ных и
е
,
НД «PipEst»
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1).
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
. , , -
лава 3 321 _______________________________________________________________________________________
.
.
3.4.4. Анали очно у о ово ов
.
и
иволине ных уча
ов
а и
альных
,
,
, [44, 137] ,
(
. 3.19). НД
.К
,
,
,
-
, .
, .
НД
, В.В. А
НД , К.И. Д
-
.
В.В. К
-
НД
[4].
и . 3.19. ру
,
ч ы
КЭ-
НД
а
-
. К
, .
НД
-
. ( © В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
КЭ-
)
322 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________ 1
, ,
-
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. НД
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δ = 13,5мм , σ Т = 488МПа ,
D = 1220 мм ,
:
σ в = 604 МПа ),
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Х70 (
. 3.21. К
. 3.21,
-
, 328МПа .
и . 3.20. а
ре еле е е ла че
в
а ря е е а ру
[Па] в е ах ру ы р у ру ч а е
,
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НД (
, .).
1
К
© В.Е. Селе
КЭев, В.В. Алеш
. , С.Н. Прял в, 2007–2009
,
лава 3 323 _______________________________________________________________________________________
и . 3.21. а
А ,
ре еле
в
НД .Н
е
е е
в я
ру
а ря е ч
КЭ-
[Па] в е ах ру ы а а
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.
и . 3.22. а ре еле е х л у ы
К
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[Па] а уча е М , луа а ых а ру
. 3.22
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( 488МПа ) .
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.
. 3.22), .
© В.Е. Селе
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ле
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
324 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
А
НД
КЭ-
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[4]. .
НД
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и . 3.24. Схе а
НД
В
в
а
в
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а
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Ч
1
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а в у ре
а в еш е
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и . 3.25. И
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в
а ря е [Па] а уча е х л а в у ре е р е
у
1
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В
© В.Е. Селе
К.И. Д ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
в
а
е е
лава 3 325 _______________________________________________________________________________________
и . 3.26. И
Н
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в
а ря е
[Па] а уча е х л а в еш е р е
у
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3.26
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а и
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ово ов,
«PipEst» «Alfargus/StructuralAnalysis» , . [4]. К
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В.В. А В.В.
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-
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
326 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
«
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© В.Е. Селе
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
лава 3 327 _______________________________________________________________________________________
и . 3.28. а че
и . 3.29. Вер
ал
ая
ые
ел уча
е е
а ру
я уча э
р в
а ру ава
а,
р в
вер ше
а
яэ
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ава
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ру а [м] р
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1
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:
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h,
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a 1
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3
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
,
328 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
( h ≥ 0,8 м; b ≥ 1м; a ≥ 0, 7 м ) .
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ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
(
. 3.31).
лава 3 329 _______________________________________________________________________________________
(
)
γ = 17000 Н м [137, 138]. В
( h90 ≈ 3 м )
3
c = 25кПа , ( 7.6), -
, [44].
и . 3.30. За ы а ра ше
и . 3.31. За ы а ра ше
ы уч
вя а
И
( ух
е ча ы ) ру
ы ( вер ы
л
ы ) ру
, ( а
а
а
.
. 3.30, 3.31),
х
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.
: ч [149]
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а -
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В.В.
[148] 50,
, ,
© В.Е. Селе
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
XX ,
.
330 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
(
.,
В.В. ,
, [148, 149]) .
,
, [143].
,
z ( x) = m ⋅ (1,57 − e − x m ) + x ⋅ tgϕ ,
[143], m=
-
2 ⋅ c ⋅ (1 + sin ϕ )
γ ⋅ (1 − sin ϕ )
; x –
; z ( x) –
(3.59)
, ,
.
и . 3.32. За ы а ра ше реал
ы
(3.59), z ( 0 ) ≠ 0 . Э
К
-
:
ру
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,
. В
,
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[148]:
q0 =
,
2 ⋅ c ⋅ cos ϕ . 1 − sin ϕ
(3.60) ,
h0 = q0 γ .
:
(3.61) .
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( © В.Е. Селе
.
ев, В.В. Алеш
, . 3.33). , С.Н. Прял в, 2007–2009
M
-
лава 3 331 _______________________________________________________________________________________
и . 3.33. Схе а а ы
ра ше
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, A)
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, С.Н. Прял в, 2007–2009
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332 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________ 1
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M.
(3.59)). . 3.33
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)
(
x 2 + y 2 − R 2 = 0,
. 3.33): R –
.
(3.62)
. (3.62):
0 ⎤ ⎡1 0 ⎢ C = ⎢0 1 0 ⎥⎥ . 2 ⎣⎢ 0 0 − R ⎦⎥
[89],
, (3.62).
[C ]
,
A
M:
⎧⎪ xM ⋅ x A + y M ⋅ y A − R 2 = 0; ⎨ 2 2 2 ⎪⎩ xM + y M − R = 0, ,
:
xM = R 2 − yM2 ;
yM =
, (3.64)
«
(3.63)
y A ⋅ R 2 ± x A2 ⋅ R 2 ⋅ ( x A2 − R 2 + y A2 )
x A = R + b + ( a + 2 ⋅ R + h ) ⋅ tgψ , y A = R + h –
M, , -
x A2 + y A2
.
(3.64) -
A
. 3.33. Д ».
НД -
.
, 2
-
. 1
Д
:
; (
2
© В.Е. Селе
, ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
.
3.3.2.1). .
лава 3 333 _______________________________________________________________________________________
Д
1
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, И
-
. ,
, -
. ( M)
-
, (
A ).
. 3.34.
и . 3.34. Схе а
л, е
ву
х а верх
ча
ру а а ы
НД
1
, © В.Е. Селе
.
ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
334 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
И
, R0 + R1 + R 2 − W = 0,
3.34: W = γ ⋅V –
(3.65)
1
;V –
–
; R 0, R1, R 2 ,
Oy
В
.
,
Oy
. (3.65) ,
(
Oxy )
R0 =
∫
xB
,
q ( x ) dx, R1 =
q ( x ) , p1 ( y ) , p2 (ζ ) – xM
∫
p1 ( y ) ⋅
yA
yB
-
cos ξ dy, R 2 = cosψ
∫ R ⋅ p (ζ ) ⋅ cos ξ (ζ ) dζ ,
ζM
2
-
; ξ –
Oy ; ζ –
(3.66)
0
,
-
q ( x ) , p1 ( y ) , p2 (ζ )
Oy .
2
. В ,
Д
-
. А
. , R 0, R1, R 2 . Н R0 . Н
-
. 3.35
3
MB (
. 3.34). К
.
, ,
MB
(
MD (
.
. 3.34)),
. ,
(
)
q
, q0 ,
DB
(3.60).
R 0 = q ⋅ S DB ,
, S DB –
(3.67) .
DB
-
Oy
R1 . Д
, AB (
1
Ш
2
К
Oz )
(
, . . 3.34). В [143, 148, 149].
,
. ,
-
. 3
Д
© В.Е. Селе
, ев, В.В. Алеш
-
, С.Н. Прял в, 2007–2009
. 3.35.
лава 3 335 _______________________________________________________________________________________
и . 3.35. Э
ра авле
я [Па] верх е ча
а
ча
а
ва ру а а ы
В «
»
(
)
(
«
и . 3.36. а че
ая хе а а
в
авле
я ру а а а л
у
е у
1
θ
, AB ,
.Н
. -
AOB . Д
-
AOB
:
W
Ea ,
Ea (
W,
)
Rg .
θ.
В
© В.Е. Селе
.Н
,
. 3.36. Н .
1
»
)
,
. ев, В.В. Алеш
, С.Н. Прял в, 2007–2009
-
336 Ч ле ы а ал р ч ру р в ых е _______________________________________________________________________________________
И
θ
, , :
k=
Ea =
(1 + z )
cos 2 (ϕ − ψ ) 2
γ ⋅ h2 2
⋅k −
⋅ cos 2 ψ ⋅ cos (ψ + δ )
В
c⋅h ⎛ k ⋅ cosψ ⎞ ⋅ ⎜1 − ⎟, ⎜ tgϕ ⎝ cos (α − ψ ) ⎠⎟
sin (ϕ + δ ) ⋅ sin (ϕ − α )
z=
;
dEa / dθ = 0 ,
cos (ψ + δ ) ⋅ cos (ψ − α )
,
(α > ϕ )
(3.68) (ψ > 65° ) [149]. В
,
.
δ
Ea 0