Chisl Model Trub Seleznev 2007

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2

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3

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4

! : .., .., .. / . .. . .: , 2005. 496 . .. , .. .. , 2006 . , , , . 2007 . .. , .. .. .

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_____________________________________________________________________________

3

............8 ...........13

1 ..14

2 - 2.1. .........22 2.2. ...24 2.3. -....27

3 3.1. ....37 3.2. ..38 3.3. ....44 3.4. ....46 3.5. ....49 3.6. ...49 3.6.1. ........49 3.6.2. .....62

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_____________________________________________________________________________3.6.3. .................74 3.6.4. - .....80 3.6.5. ......86 3.6.6. ......88 3.6.7. ...93 3.6.8. ......97 3.6.9. ....101 3.7. ..107 3.7.1. ....107 3.7.2. ......116 3.7.3. ......132 3.7.3.1. ....132 3.7.3.2. ........132 3.7.4. ...135 3.8. ..138 3.8.1. ....138 3.8.2. ................................................139 3.8.3. , .........140 3.8.4. ...148

4

4 4.1. ..153 4.2. ....158 4.3. ....161 4.4. - .....163 4.4.1. - ..................................164 4.4.2. ........165

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_____________________________________________________________________________4.4.2.1. .165 4.4.2.2. - ..173 4.5. ...........185 4.5.1. .185 4.5.2. ....186 4.5.2.1. ...188 4.5.2.2. ....193 4.5.2.3. .....196 4.5.3. .....199 4.5.4. ...200 4.5.5. , .....204 4.5.6. .....217 4.5.6.1. ..218 4.5.6.2. .......224

5

5 , 5.1. ....229 5.2. .........231 5.3. .....235 5.3.1. ...235 5.3.2. .255 5.3.3. .....266 5.3.3.1. ..266 5.3.3.2. .279 5.3.3.3. ....286 5.3.4. 289 5.3.5. 290 5.4. ..292 5.4.1. ...292 5.4.2. 294

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_____________________________________________________________________________5.4.3. ........298 5.4.4. .....312 5.4.5. ...315

6

6 6.1. ....................................................................317 6.2. ..323 6.3. ...329 6.4. ...332 6.5. ...333 6.6. ..343 ......348 ...349 1. . .., .., .., .., ...367 2. . .., .. ....411 3. . .., .., .....435 4. ANSYS ABAQUS . .., .. ......459

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_____________________________________________________________________________ 5. . .., ......466 ....479

7

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8 _____________________________________________________________________________

. . . , . . , . . , 25 . , , , , . , , 1992 2001 550 [224232]. 5060 . 2003 2004 , , 18 , 15 . , . , , ( 1999 2000 ) Olympic Pipe Line Co. El Paso Natural Gas Co.. . , , , 16 . , - , 2004 Fluxys NV. , () , , Butagaz (.), 30- . ,

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9 _____________________________________________________________________________

, Butagaz. (, - ), , 15 100 . - , Euronews, . . , . . . . , . , , . , . , , , . , - . [1 4]: 1) .., .., .. / . .. . 2- ., . 2005; 2) .., .., .., .., .., .., .. / . .. . 2003; 3) .., .., .., .., .. / . .. .. . 2003; 4) .., .., .., .., .., .., .. / . .. . 2004.

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10 _____________________________________________________________________________

[5]: Ilkaev R.I., Seleznev V.E., Aleshin V.V., Klishin G.S. Numerical simulation of gas pipeline networks: theory, computational implementation, and industrial applications / Ed. by V.E. Seleznev. 2005. . , , , 23 . , , [1 5], . , . [1] . . -. ( [1]) -. , . . [1,2,4], , . , , . . , [1,3,4]. , , , , , - . , [1,3,4].

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11 _____________________________________________________________________________

. . [1,2], , . . , - .. . .. . , , , : .. 1, 2, 3 ( 3.6.4 3.6.9), 5, 6; .. 4 2.2, 5.2, 5.4.4; .. 3.6, 5.3.1, 5.3.3.2, 5.3.3.3. , . . ,

1 .. , .. , .. , .. .. . , .. , . 3 . [1,2] . 2 .. .. . . 3 .. , .. .. . , . 4 .. .. . , [1,3] ANSYS ABAQUS . 5 .. .. . , .

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12 _____________________________________________________________________________

. URSS , - .

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13 _____________________________________________________________________________

(-) ; ( ); ; ; (-) (-) (); (-) (-); (-) (-); ; (-) (-) (-); ( ) ( , ); ; (-) (-) (-); (-) (-); - - ; - - ; - - ; - - ; ; (-) (-) ; ; (-) (-); ; () ; ; ; - ; ; - ; (-); (-) ; (-) ; - ; (-) (-) (); - ; (-) (-) (-); (-) ; (-) (-) ; SCADA- Supervisory Control And Data Acquisition System. , , .

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14

_____________________________________________________________________________

1 , , .. [1,15]. .. [1,3,1522], .. [2,2329], .. [2,26,27,3034], .. [2,32,3539], .. [4,4044] .. [2,4548], .. . .. [3,4,4953] .. [3,4,51,5458], .. . - : ( ); ; ;

; . , , . : 1) ( ; ); 2) (

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15

_____________________________________________________________________________ , ; , , ); 3) ( - () ; , ); 4) ( , , ; ); 5) , ( , , ; , ). [6, 16]. , , , [6, 16]. . , , ,

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16

_____________________________________________________________________________ [1,6]. . [2, 59]. [1, 2, 34]. , , . , , , . [1, 6]: , , , ; . . , ( ). : Alfargus, ; PipEst [1, 3, 4, 10, 1522, 60]; - CorNet [1,2,4,79,26],

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17

_____________________________________________________________________________ ; CorNet [2,8,27] , . , 20002003 SPP () - CorNet, AMADEUS [27, 28, 32, 61]. , AMADEUS, , Alfargus/PipeFlow. PipEst Alfargus/ StructuralAnalysis .. [1, 3, 4, 6] .. [1,3,4,16]. () () (). , PipEst .. [1, 11, 1618, 21]. , () - () . ( ) , , . -. PipEst Alfargus/ StructuralAnalysis , - .. [1, 21]. PipEst, Alfargus, 2, 4, 5 4, 5 . - CorNet .. [1, 2, 49, 12, 59] .. [2, 2329]. (). , CorNet .. , (, ), , , [2, 23, 24]. .. , .. , .. , 2005, 2006, 2007

1

18

_____________________________________________________________________________ - CorNet Alfargus 3, 6 1, 2, 3 . PipEst, CorNet, CorNet ( AMADEUS (. )) Alfargus [15, 10, 1820, 27, 41, 45, 49, 6164]: - ; ; ;

; () ; , (); - / ; , .. : , (, , , , , ..); ; - , , (

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19

_____________________________________________________________________________ ), ; - ; - , .. , . , . , , (. 2). , . [1]: , ; , ; ; ; . [16, 16]: 1) ; 2) , ; 3) ;

4) ; 5) - .

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20

_____________________________________________________________________________ , , [1]. . , [65,66], , [67, 68]. , , [1]. , . , (., , [15]): ; ;

/ ; , . , , , , .. [79]. (., , [79, 49, 53]). , , , .. , , , . - , . .

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21

_____________________________________________________________________________ , , , [2, 45, 49, 53, 61]. , . ( ( [69])) ( ) ( () [49, 53]). . - () . , , , , , .. [7, 10, 17]. , , . [4, 35, 39, 49]. . , .. , [69, 70]. ( ) [15, 71]: ; ; ( ); .. .

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22

_____________________________________________________________________________

2 2.1. [72]. , , - (.2.1) [72].

.2.1. - 3720 ( ) - (.2.2), PII (). , -. - (.2.3). - , , [72, 73]. , . .. , .. , .. , 2005, 2006, 2007

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23

_____________________________________________________________________________

.2.2. -

.2.3. - ( PIPETRONIX ()) - - (, ) [1]. - . . . -, , (., , .2.1 2.2), . , ,

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24

_____________________________________________________________________________ -, ( ), . , -, ( -). - . . - - [1]. , . , , - -, . , 90- . - . - , , (, BJ Pipeline Inspection Services (), PII (), Linscan (), , , SPP (), GasCo ()). 2.2. [74, 75], 2.1 : ! ! ! ! rot rotA grad ve divA rot H = 0, (2.1)

(

)

(

)

(

)

! A ; = 1 ;

! ; HC -. (2.1)

1 3 ve = vii ; 3 i =1

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25

_____________________________________________________________________________ :! ! 1 2 A = rot H , ve

(2.2)

2 . . (2.1) (2.2) () [1]:! A ( x, , z ) ! = Q1 ( x , , z ) ; x [0; x0 ], ; , z = 0 2 2 ! = Q2 ( x , , z ) ; x [0; x0 ], ; , z = z0 2 2

(2.3)

! A ( x , , z )

(2.3)

AZ ( x , , z )

x [0; x0 ], =

2

, z [0; z0 ]

= 0;

(2.3)

AX ( x , , z )

x [0; x0 ], =

2

, z [0; z0 ]

= 0,

(2.3)

! Qi ( x , , z ) ,

i = 1, 2, ( x x0 ) ,

z ( z z0 ) ,

-; AX ( x , , z ) AZ ( x , , z ) ! ! A = A ( x, , z ) ( x ). ! A + - [1]: ! ! ! ! ! ! ! ! ! ! (2.3) A + = A , n A + = n A ; n + rotA + = n rotA ,

( x , , z )

(

)

(

)

! n , . : ; , -; -; , . (2.1,2.3) (2.2,2.3)

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_____________________________________________________________________________ , [7678]. , , EMAG ANSYS () [79]. ANSYS (, , ..). - ANSYS . - (-) . - () , ( , , ..). , . , . - ANSYS - (), . , , . , , - - ANSYS . . , -, [80]: ! ! ! ! ! B = rotA; H = B HC , (2.4) ! ! : B H . -. , , , [81]: ! ! (2.5) F = T nds ,S

! F

;

S

,

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27

_____________________________________________________________________________! ! 1 ! ! ; T = H B H B [82] ( 2 , ).

2.3. -

2.2 - 3720. - (. ) (. ) [1]. : - . - 3720 . (. .2.1). , . , -. , -. - . , , - . . , - - . - 3720 . , , ( ) - 3720 (. ), , . . .. , .. , .. , 2005, 2006, 2007

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28

_____________________________________________________________________________ , 2.2. - , - 3720, . - : ( , ) (.2.4,), (.2.4,), (.2.4,), (.2.4,). (2.2,2.3). .2.5 2.6. .2.5 -, , . .2.6 . , . (2.5) , - 3720 .

)

)

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29

_____________________________________________________________________________

)

)

.2.4. : ) ; ) ; ) -. - (). [1]. 1823. 3720 1811. 0,7% [1]. , . , , 3720 [1]. - 3720 25% [1]. (, , ..). - . , (.2.7). ( ) . - 3720 (.2.8). .2.9 2.10, ,

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_____________________________________________________________________________ ( 0,72, 0,008), - 3720. ( ) BX ( ) BZ . .2.9 2.10 , 0,001 . -.

.2.5. B [] - 3720 -

.2.6. B []

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31

_____________________________________________________________________________

.2.7.

.2.8. B [], - 3720

. 2.9. B []

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32

_____________________________________________________________________________

. 2.10. B [] - 3720 () : ( ); ( ); ( ). , 0,01, 0,005 0,00135 . , 3720 , , , 10-6 10-4 [1]. . , 10-7 [1]. - 3720 10-3 [1]. , , PIPETRONIX 1998 , - 10-4. , - 3720. 3720 : (.2.11) [64]. , 0,02, 0,004 0,00075 . 0,005.

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_____________________________________________________________________________ , - 3720 , .2.122.14. .2.152.17 , . .2.14 2.17 , 0,001 . -.

.2.11. () , .2.122.17, .. [64].

.2.12. BZ [], - 3720

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34

_____________________________________________________________________________

.2.13. B [], - 3720

.2.14. BZ []

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35

_____________________________________________________________________________

.2.15. BZ [], - 3720

.2.16. B [], - 3720

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36

_____________________________________________________________________________

.2.17. B [] (. .2.14 2.17), - 3720 , , - , 10-4 . , - 3720 10-3. . , 10-9. , , , . , , , , , . , -, (. .2.1 2.3), . - [1, 5].

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37

_____________________________________________________________________________

3 3.1. , , , , . , , , . 145 [83]. 234 , 38,2 . 600 , [84]. 1260 . 21 , 3 [83]. . , , . . . , , . , . , , . : [3537]; (, ..), [7];

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38

_____________________________________________________________________________ [2,30]; [2] .. , , [1, 2, 31, 59]. [4, 8]: ; , .. , [27] .. , , [7]. : , ; ; ; ; . , . 3.2. (.3.1) () (.3.2), () (.3.3), () [85]. .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________ , , . (), , 3,57,8. , , , ( , ..).

.3.1.

.3.2.

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40

_____________________________________________________________________________

.3.3. ( ) () (.3.4). () (.3.5). () (.3.6). , , . - / () (.3.7). . [86]. , , , () (.3.8). : ( ). . 4 27. , , . . , , - .

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41

_____________________________________________________________________________ . (), 308 ( ) (.3.9). .3.10 [86].

.3.4.

.3.5.

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42

_____________________________________________________________________________

.3.6.

.3.7.

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43

_____________________________________________________________________________

.3.8.

.3.9.

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44

_____________________________________________________________________________1220

1220

1020

1020

1020

530 1020

530

1020

1020 1020

1020

720

.3.10. - [86] , , ( , (. )) [87]. , - (). 3.3. , , . : ; ( ) ; , ;

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720

1020

720

720

720

720

720

3

45

_____________________________________________________________________________ / , ; , ( ) ; , ; - ;

- SCADA (Supervisory Control And Data Acquisition) , , ; , ; , ; , , . , , [88]. [1, 2, 4, 5]: , , ..; , c ,

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46

_____________________________________________________________________________ ( , , , , / ..); - , ; ; - ; ; , , , . 3.4. , . , , [88, 91]. ( ) . , . , , . , , - 3.3 , ( )

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_____________________________________________________________________________ ( ) , - . ( ). : 1) - (. 3.6); 2) - (. 3 3.8.3); 3) (. 3.6); 4) - (. 3.7.3.2 3.8.2); 5) ( ) - ( ) (, (. 3.8.4)); 6) (. 6); 7) .. , : 1) (. 6); 2) - (. 5.3.3.2). [2]. , [7].

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_____________________________________________________________________________ : /; ; ;

; (); , .. , . 1, , (). () . () , , , , .. , () ( ) , , . - , : ( , ); , . , . . , , . ( ) - ( , , , , ..). - CorNet AMADEUS. - CorNet

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_____________________________________________________________________________ 90- . 1, - AMADEUS CorNet, SPP (). GDS-AMADEUS v.4.2, CorNet SPP () 2003 . , [88] . [88] . , , . [88] , A, , SIMONE .., , , . 3.5. . , . SCADA-, . - . . , ( ) . . . - . - . 3.6. 3.6.1.

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50

_____________________________________________________________________________ . [75]: ! ! ! d t dV = Vt dV St JdS , dt V ( ) () ()! ! - , ; ! -, ( ! ); J -, ! S ( t ) ; V ( t ) ; S ( t )

, V ( t ) ; t . - :0 Y 1 0 " " ! ! = YN S ; ; = 0 ! ! F 2 ! ! + 2 F + Q 0 ! ! 1 n " ! ! ! J = NS n , ! ! p n n ! ! ! ! ! ! NS p n n + W n + m m n m =1

! ! ! ! ; ; n = n ! ! ! n dS ; p ; n = n

, ! ! n ( ); F ; ( ) ; Q ( ! ! ) ; W = k T ( ! ! k , T ); m = Dm Ym m - ( Dm () m - , Ym m - ); N S

. , , . (. 3.2), , . [8991]. ,

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_____________________________________________________________________________ , , ( ) . , . 1 [1]. V , S . , , [75] 1: # # (3.1) t dV + n dS = 0; V S # ! ( ) ! # ! # ! # ! (3.2) t dV + n dS = p ndS + n dS + FdV ; V S S S V t + V S

# # 2 # # 2 # dV + + n dS = p n dS + 2 2 S S ! ! # ! ! # ! ! + n dS + F dV + QdV W ndS ,V V S

(3.3)

# # ; p ; ( ) # ; Q ( ) . (3.13.3) , . ( ) , (3.13.3) , (. (1.27) 1 [1]):#

( f ) ( f w) + = 0; t x2 ( f w) ( f w ) + =f t x

(3.4)z p + g 1 w w R; x 4 x

(3.4)

( p f w) w2 w2 f + + f w + = 2 x 2 t x z f T f w g 1 p + Q f + k f ( T , T ) ; x t x x

(3.4)

1 , , .

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_____________________________________________________________________________p = p (,T );

= ( p, T ) ,

(3.4)

f = f ( x, t ) ; w ; z1 , ( ); g ; R = f ; [92]; x ; . (3.4) . ( T , Toc ) , . ( ) f ( ( T , Toc ) > 0 ; T - ). , , (3.4), (. (1.40) 1 [1]): ( f ) + ( w f ) = 0; t xY ( Ym f ) + ( Ym w f ) f Dm m = 0, m = 1, N S 1 ; YN S = 1 t x x x ( w f ) + t ( w2 f ) x z p = f + g 1 w w R; x x 4 N S 1 m =1

(3.5)

Y

m

;

(3.5) (3.5)

z1 w2 w2 f + + w f + = ( p w f ) w f g 2 x 2 t x x NS Ym f T p + +Q f + k f ; ( T , Toc ) + f m Dm t x x x x m =1

(3.5)

m = m ({S }) ,p = p ({S }) ;

m = 1, N S ;

T1 = T2 = = TN S ;

(3.5) (3.5) (3.5) (3.5)

= ({S }) ;k = k ({S }) ;

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_____________________________________________________________________________ Dm = Dm ({S }) , m = 1, N S , (3.5)

{S } , , ( , ()). (3.5), , . , ( - ) : w ( x, t0 ) = w0 ( x ) ; T ( x, t0 ) = T0 ( x ) ; p ( x, t0 ) = p0 ( x ) . ( x, t0 ) ( x, t0 ) (3.5) (3.5) . :T ( xB , t ) = TB (t ); w( xB , t ) = wB (t ); p( xB , t ) = pB (t );T ( xB , t ) = jB ( t ) ; x

( w f ) B = qB ( t ) ,

xB ; jB ( t ) qB ( t ) . (3.5) , [93], ( ). , / ( ). (3.5) ( T , Toc ) , . - ( T , Toc ) . , . .3.11 . ( T , Toc ) . ( T , Toc )

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_____________________________________________________________________________ . . , , .

.3.11. (D1 ; D2 ; H1 , ; 2 )

, , Alfargus. , Alfargus, .3.12.

.3.12. [] .. , .. , .. , 2005, 2006, 2007

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55

_____________________________________________________________________________ , . () , . , , , (., , [85, 87, 90, 9497]). . , .. .. N [1, 23, 28]. , , [23]. N ( ) ( .. ). [1]. c . - n , , : ( n ) . ( n )Ox , . ( ) . (.3.13): ( n ) f L 1, ( , (n)

(n)

xL )

(n)

fM

(n)

f L (n)

(n)

X ( (n)

xM ). , , , fL = fM ,

, ..

n = 1, N .

1

f .

() , .

, X xM xL ,

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56

_____________________________________________________________________________

M (1)

(N )

V

VL

(n) ! i

(o) ! n

M

L C ( 2)

L M( 0)

V

V

, (M) (L) ..3.13. N

, M , (n)

(n)

xM (n)

xL L .

V,

n = 1, N , (. .3.13), n M L , . (0)V , ( n )V . V = ( )V . , n n =0(n)

n

(n)

V ,

S.

(n)

! i

(n) (0)

Ox ,

(0)

! n

(n)

fL ,

V . , , (n) X = X 0 (. (3.8), (3.13), (3.26) [1]): N ( n ) + t n =1(n)

( w) = 0; x

(3.6)N (n)

( Ym ) N + t n =1(n)

(n)

( Ym w ) ( n ) x Ym ;

n =1

Ym ( n ) x Dm x = 0, m = 1, N S 1;

YN = 1 S

N S 1

m =1

(n)

(3.6)

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57

_____________________________________________________________________________(n)

( w) + t (n)

(n)

( w2 ) = x (n)

(3.6)

=

p g x (n)

1 z1 (n) ( n ) ( w w ), n = 1, N ; x 4 R(n)

( ) N + t n =1

N ( w) (n) (n) = p x n =1 N 1 1 (n) 3 + ( n ) ( n ) ( n ) w ( n ) + Q 4 n =1 R

w ( n ) + x +

+

n =1 N

N

(n)

N (n) (T , Toc ) ( n ) T ( n ) + (n) x k x f n =1 (n)

(3.6)

+ n =1 m =1 (n)

NS

Ym ( n ) x m Dm x ; ( )

( m ) = ( m ) , = ( n ) = ( ) , ( n ) p = ( ) p, ( n ) k = ( ) k , (n) ( ) (n) ( ) ( Dm ) = ( Dm ) , Ym = (Ym ) = (Ym ) , ( n )Q = ( )Q , ( n ) ( z1 ) = ( ) ( z1 ) T=( )

T,

=

(n)

=

,

(n)

( )

n, 1, N m 1, N S ;

(3.6) (3.6)

n =1 N

N

(n)

w ( n ) f ( n ) s = 0; T x (n) (n) f s = 0;

n =1 N

(n)

(3.6)

n =1(n)

(n)

Ym ( n ) ( n ) f s = 0; x

(3.6)

s=

(

(0)

! 1, ! n (n) i = 1,

)

( (

(0)

! ! n ( n ) i < 0; ! (0) ! ( n ) n i > 0;

) )

(3.6)

(n)

=

(n)

V = V

(n)

fL fL

k =1

N

,

0 0, 1,

(3.12)N (n)

N

(n)

w ( n ) f ( n ) s = 0;

n =1(n)

N

(n)

Ym ( n ) ( n ) f s = 0; x (n)

(3.12)

(

)

( (

) )

=

fL fL

k =1

N

,

0 . x 2

2 y x 2

.. . DER (derivation). , x , Kx . Kx1 Kx 2 . :Kx = Kx1 + Kx 2 + 1.

(3.34)

APPR (approximation). : APPR = Kx DER. (3.35) , , , {i Kx1, i + Kx 2} . {UPSTREAM , Kx} . yi(1)0,5 = xi

x

DER

yi yi 1 : hi{i Kx1, i + Kx 2} ( DER = )

x

( yi )

(3.36)

, :{UPSTREAM , Kx} ( DER = )

x

( yi ) ,

(3.36)

< ... > . (3.36) :{ Kx1, ( DER = )

x

Kx 2}

( y)

(3.37)

, :{UPSTREAM , Kx} ( DER = )

x

( y ). x x i

(3.37) ,

,

.. , .. , .. , 2005, 2006, 2007

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77

_____________________________________________________________________________ xi 1 , xi xi +1 , : {1,1} ( DER =1) {i 1, i +1} ( DER =1)

x

( i )

x

( ) .

Kx = 3 . (3.35),

APPR = Kx DER = 3 1 = 2 . , (3.33), . , . , [102].

hi2

( hi + hi 1 )

2

y x , 6 ( ). 1,52 . . . :

yi(1)0,5 =

yi yi 1 , hi

yi(1) = 1,5

yi 1 yi 2 , hi 1

(3.38)

, . (3.33) ( - xi ): yi(1)0,5 = h 2 y y i 2 + O ( h2 ) , x 2 x

(3.39) (3.39)

yi(1) = 1,5

y 2 hi + hi 1 2 y 2 + O ( h2 ) . x 2 x2 y , : x 2

(3.39) ,

h yi(1)0,5 2 + i 1 yi(1) 1,5 hi y = + O ( h2 ) x hi 1 1 + hi

(3.40)

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78

_____________________________________________________________________________ yi(1) 1,5 hi yi(1)0,5 2 hi + hi 1 + O ( h2 ) . hi 1 2 hi + hi 1

y = x

(3.40)

:y = x h yi(1)0,5 2 + i + 2 yi(1) + +1,5 hi +1 + O ( h2 ) hi + 2 1 + hi +1 yi(1) +1,5 hi +1 yi(1)0,5 + 2 hi +1 + hi + 2 + O ( h2 ) . hi +1 1 2 hi +1 + hi + 2

(3.41)

y = x

(3.41)

, xi 1 , xi xi +1 , (3.40) (3.41) :y = x yi(1)0,5 + yi(1)0,5 + hi +1 1 + hi yi(1)0,5 + hi +1 hi + O (h2 )

(3.42)

y = x hi + yi(1)0,5 hi +1 + O ( h2 ). hi + 1 hi +1

(3.42)

( 2 hi )

2 hi = hi + hi ,

hi +1 h hi , i = 1, N ( i , ..) hi hi +1 2 hi + hi 1 , . , . 4. yi(1)0,5 , i = 1, N , +

. i =

y (3.40), (3.41) (3.42) 3 . x , , 1,52 .

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79

_____________________________________________________________________________ , . (3.33) 2 y 3 y , yi . , x 2 x 3 yi . (3.32) - :{ j +1 Kt1, ( DER =1)

t

j +1}

(

j +1 i

f i j +1 ) +j +1}

{i Kx1, i + Kx 2} ( DER =1)j +1 j +1 i

x

(

j +1 i

f i j +1 wij +1 ) = 0;{i Kx1, i + Kx 2} ( DER =1)

(3.43)j +1

i j +1 f i j +1

{ j +1 Kt1,

( DER =1)

t

(w ) + i j +1 i

f i j +1 wij +1

x

(w ) =i j +1 1 i

= f i j +1

{i Kx1, i + Kx 2} ( DER =1)

x

(p ) f

j +1 i

g i j +1

{i Kx1, i + Kx 2} ( DER =1)

x

(( z ) )

(3.43)

4

i j +1 i j +1 wij +1 wij +1 Ri j +1 ;{ j +1 Kt1,

i j +1 f i j +1

( DER =1) ( DER =1) ( DER =1)

t

j +1}

( ) + j +1 i

j +1 i

f i j +1 wij +1

{i Kx1, i + Kx 2} ( DER =1)

x

( ) =j +1 i 3

= pij +1 pij +1 +

{i Kx1, i + Kx 2}

x

(w

j +1

i

f i j +1 )

{ j +1 Kt 1,

t

j +1}

( fi j +1 ) + Qi j +1 fi j +1 + f i j +1 ) { i Kx1, i + Kx 2} ( DER =1)

4

(3.43)

i j +1 i j +1 wij +1 Ri j +1 +j +1 i

{i Kx1, i + Kx 2} ( DER =1)

x

(k

j +1 i

x

(T ) + kj +1 i

f i j +1

{i Kx1, i + Kx 2} x Ti j +1 i j +1 ;( DER = 2 )

(

)

pij +1 = p ( i j +1 , Ti j +1 ) ;

(3.43) (3.43) (3.43)

i j +1 = ( pij +1 , Ti j +1 ) ;ki j +1 = k ( pij +1 , Ti j +1 ) ,

Kt1 , t . (3.43) , (3.43), Kx1 Kx 2 , :Kx1 + Kx 2 Kx1 + Kx 2 + 1.

(3.44)

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_____________________________________________________________________________ (3.43), {UPSTREA , Kx} . (3.43) . , , .. , , . (3.43) .. . : ; ; . . [93,102]. , . - . , - . , , , . 3.6.2 3.6.3 , . .

3.6.4. -

(., , [106]). xL x xR dy = p ( x, y ( x ) ) , dx

(3.45)

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81

_____________________________________________________________________________ p ( x, y ( x ) ) . (3.45) , . , , , , ( xL ) :y ( xL ) = YL ,

(3.46)

YL . - h = {xi } , xi i - , i = 0, N 1 ; N

- ;

hi = xi xi 1 i , i = 1, N 1 (.3.20).

.3.20. -

(3.45) (3.45) - : dy = p ( xi , y ( xi ) ) . dx i

(3.47)

dy dx :{Spline , } ( DER = )

x

( y),

(3.48)

{Spline, < ... >} , < ... > . ( DER =< ... > ) , (3.36), ( 1). (3.48) (3.47) :{Spline , } ( DER =1)

x

( yi ) = p ( xi , y ( xi ) )( y ) = p ( x, y ( x ) ) .

(3.49)

, :{Spline , } ( DER =1)

x

(3.49)

xi 0,5 = [ xi 1 , xi ] , .

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82

_____________________________________________________________________________ K 2 . y ( x ) . xi 0,5 , i = 1, N 1 yi 0,5 ( x ) = ak ,( i 0,5) x k ,k =0 K

(3.50)

ak ,( i 0,5) , k = 0, K , i = 1, N 1 . (3.45) : yi(1)0,5 ( xi 1 ) = p ( xi 1 , yi 0,5 ( xi 1 ) ) , yi(1)0,5 ( xi ) = p ( xi , yi 0,5 ( xi ) ) , i = 1, N 1; m) m) yi( 0,5 ( xi ) = yi(+ 0,5 ( xi ) ,

(3.51) (3.52)

m = 0, 2,3, ..., K 1,

i = 1, N 2,

m) yi( 0,5 ( x ) m - yi 0,5 ( x ) .

(3.51,3.52)

[( N 1) ( K + 1)]

( ( N 1)

; ( K + 1) ) 2 ( N 1) + ( K 1) ( N 2 ) . ( K 1) . .. (3.51,3.52) ( K 1) - . (3.44):y0,5 ( x0 ) = YL .

(3.53)

( K 2)

: ' , , K = ( int( K / 2) 1) ( int(...) ' ) ( K K 1) :( ' y Nm)1,5 ( xN 1 ) = 0, m = ( K K + 1) , K ; (m ' y0,5) ( x0 ) = 0, m = K + 2, K .

(3.54) (3.55)

-. [105, 107], , . , ( (3.45)) . , , ( y ( x ) ),

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83

_____________________________________________________________________________. , , ( y ). () , (. [105,107]). , , ( y )

( p ) , . - (3.49) (3.45) -. , . - .. ,

{x

i + 0,5

( y )}

( xi + 0,5 hi ) - y ( x ) :

{ y } , , i + 0,5

yi + 0,5 yi 0,5 xi + 0,5 xi 0,5

dy {Spline , } = = x ( yi ) . ( DER =1) dx i

(3.56)

, {xi + 0,5 ( y )} (.3.21). (3.54) , .. y . (3.49) (3.56), :yi + 0,5 yi 0,5 xi + 0,5 xi 0,5= pi .

(3.57)

(3.57) xi 0,5 , xi + 0,5 (3.49), , - .

: ( , , , N + 1 ..), , , , .

.. , .. , .. , 2005, 2006, 2007

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84

_____________________________________________________________________________

.3.21. {xi + 0,5 ( y )} y ( x )

- . - , (3.45) xi 0,5 ( y ), xi + 0,5 ( y ) . , - . : dy d + g ( y ) = p ( x, y ( x ) ) . (3.58) dx dx - (3.58) , .. ( dy dx dg dx ) . , ( xi 0,5 ( y ), xi + 0,5 ( y ) xi 0,5 ( g ), xi + 0,5 ( g ) ) . ( )

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85

_____________________________________________________________________________ , , (3.58), (, , - ). - (3.32). Kt , : Kx . - (3.32) :

i j +1 t

{Spline , Kt} ( DER =1

( f )+ f )j +1 i

j +1 i

{Spline , Kt} ( DER =1)j +1 i

t

( )j +1 i j +1 i

+{Spline , Kx} ( DER =1)

+ f i j +1 wij +1 + i j +1 f i j +1

{Spline , Kx} ( DER =1

x

( ) + ) (w ) =j +1 i

wij +1

x

( f )+j +1 i

(3.59)

{Spline , Kx} ( DER =1)

x

0;{Spline , Kx} ( DER =1)

i j +1 f i j +1 t = f i j +1 ( DER =1


(w ) )j +1 i j +1 i

+ i j +1 f i j +1 wij +1 f i j +1 g i j +1 x

x

(w )j +1 i

= 4 i j +1 i j +1 wij +1 wij +1 Ri j +1 ;

{Spline , Kx}

x

(p ) ) ( ) )j +1 i


(( z ) )j +1 1 i

(3.59)

i j +1 f i j +1 t = pij +1 x


+ i j +1 f i j +1 wij +1


x

( )j +1 i 3

=


(wj +1

j +1

i

f i j +1 )

pij +1 +


t

(f ) )i

+ Qi j +1 f i j +1 +{Spline, Kx} x Ti j +1( DER =1

4

i j +1 i j +1 wij +1 Ri j +1 +{Spline , Kx} x Ti j +1( DER = 2 )

{Spline , Kx} ( DER =1

x

(k )

j +1 i

f i j +1 )

( )

)

+ ki j +1 f i j +1

(

)

i j +1 ;

(3.59) (3.59) (3.59) (3.59)

pij +1 = p ( i j +1 , Ti j +1 ) ;

i j +1 = ( pij +1 , Ti j +1 ) ;ki j +1 = k ( pij +1 , Ti j +1 ) .

(. 3.6.3) (3.59) , (3.59), Kx Kx , : Kx > Kx. (3.60) - (3.59) , (3.32). , .. , .. , .. , 2005, 2006, 2007

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86

_____________________________________________________________________________ . , (3.32) ( Kt 2 , Kx 2 , Kx 3 ), . , - . (, , , ..). , [93, 102, 104] -. . , , - . - . - , [93, 102, 104]. - , . - .

3.6.5.

(3.12) . [108, 109], :p V B C = 1 + + 2 + , R0 T V V

(3.61)

3 V ; R0 ; B 3 6 , C 2 , , .. , . (V ) (3.61) . ,

, .. (3.61) , , .. . (3.61) , . , ,

.. , .. , .. , 2005, 2006, 2007

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87

_____________________________________________________________________________ [108, 110, 111]: a* p+ ( b* ) = R T , * T ( + b )

(3.62)

2,5 a * = 0, 4278 R 2 T p ; b* = 0, 0867 R T p ; T p . , = 1 . [111, 112]: p ( p, T ) = h ( p, T ) ; (3.63)

dh = c p dT +

h dp, p

(3.64)

h ; c p . ( h p )T (., , [111]): h +. = T T p p T

(3.65)

= ( p, T ) . (3.64) . , (3.12), , () . . . , . , , (, ) . , . , h = c p T (3.64). , 1000. d = 1 , = 0,013 , = 7,5 105 . 8,0795 290 , 6,0795 . (3.28) .. , .. , .. , 2005, 2006, 2007

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88

_____________________________________________________________________________ : s a = s b = = = 0 , r a = r b = 0,5 . .3.22 3.23. , . . , 10 / , 10%. , , . , . , , (3.62) (3.63).

3.6.6.

[23]. , . . , , .

.3.22. ,

.. , .. , .. , 2005, 2006, 2007

3

89

_____________________________________________________________________________

.3.23. ,

. (., , [92]) , . , , (, (. )). [23]. (3.28), (3.30) (3.31). / / , . , , , / . , , ( , , ), , (3.28), (3.30) (3.31), (, ). .. [23]. , , .3.24. , iL iR ( iR = iL + 1) . iL 1 , iL , iR , iR + 1

.. , .. , .. , 2005, 2006, 2007

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90

_____________________________________________________________________________ :L = X iR + 2 X iL 2 , Z = ( z1 )iR + 2 ( z1 )iL 2 , L , 3 X iL 1 = X iL = X iL 2 + Z , 3 L , 3

X iR = X iR +1 = X iR + 2

( z1 )iL1 = ( z1 )iL = ( z1 )iL2 +Z , 3

( z1 )iR = ( z1 )iR +1 = ( z1 )iR + 2

(3.66)

X i i ,

( z1 )i

i . iL iR ( ), iL 1 iR + 1 , ( ).

iL-2

iL-1, iL

iR, iR+ 1

iR +2

.3.24.

( ), (3.30), ( iL 1 , iL ) ( iR , iR + 1 ). . , , / , . , . . , , . iR . :Q = iR wiR f iR .

(3.67)

iL 1 , iL iR + 1 Q . iL 1 iL 2 , iL 1 iL 1 , ( ) ( ) iL 1 . iR + 1 , iR + 1 . :

iL = iL 1 ,

iR = iR +1.

(3.68)

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91

_____________________________________________________________________________ . . (. (3.5)). , , / , , , (3.5). : w2 ( p f w) . f w + = x 2 x

(3.69)

(3.69) [ X iL 1 , X iR +1 ] : w2 w2 + p f w + + p = 0. f w + 2 2 X X iR +1 iL 1

(3.70)

:

iR +1 =

w2 w2 1 f w + + p + p . 2 iR +1 ( f w )iR +1 2 X X iR +1 iL 1 1

(3.71)

. iL 1 iL 2 , iL 1 iL 1 . ( ) iL 1 . iR + 1 (3.71). :

iL = iL 1 ,

iR = iR +1.

(3.72)

: T = T ( , p ) . . (3.30) . ( iL 1 , iL ) ( iR , iR + 1 ). , ( ) . ( ), . [ iL 1 , iR + 1 ] , . , , ,

.. , .. , .. , 2005, 2006, 2007

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92

_____________________________________________________________________________ / . [1,2]. , ( 1 2), , , .. [35]. SPP () [2]. . (, ..) , (. .3.16 3.17). . 0 2 . 120 c , : . : 1 c . - 870 ( ) 0, 4 c ( ). : s a = s b = r a = r b = 0,5 , = 1 , = 0 . 102,81 c .3.25.

.3.25. .. , .. , .. , 2005, 2006, 2007

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93

_____________________________________________________________________________3.6.7.

.. .. [2,23]. , L (.3.26). M L . [113]: , (. ) . [113, 114], , , , , , . [115]. [75,116]. , . , [116]. . L [113, 117]:wLj +1 = sign cLj +1 ,

(3.73)

c ; sign = 1 . , .

M

L

.3.26.

p ( ) Toc ( ). Toc , (, , ). [113]:p 2 1 > pM , + 1

(3.74)

. [113]

.. , .. , .. , 2005, 2006, 2007

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94

_____________________________________________________________________________ wL > cL . (3.74)

(3.74) ( ). , . . , . . . (, ..) , (. .3.163.18). . 0 x = 30000 1 . : 10 c . - 170 0,08 c . (3.31): s a = s b = r a = r b = 0,5 , = 1, = 0 . 120,97 c .3.27.

.3.27.

.. , .. , .. , 2005, 2006, 2007

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95

_____________________________________________________________________________ 3.6 CorNet AMADEUS [1,2]. 1. .. (. 3.6.6) , (. ). .. pL < p ,

(3.74)

(3.74). . c=

( p )S

= R T ,

(3.75)

p ; ; S ; ; R ; T . , . , ( p ) S . , : dp = p p p T = + . S T T S

p p d + dT , T

(3.76)

(3.76)

, , TdS = d + pdV = dh Vdp = 0, (3.77) h p (3.78) = V , S S ; h ; V = 1 . h h p T (3.79) = + cp , S p T S S

.. , .. , .. , 2005, 2006, 2007

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96

_____________________________________________________________________________ h c p = . (3.78) (3.79), T p : p h T (3.80) V + cp = 0. S p T S (3.76) (3.80) p T p . , : S S S p cp p T . = S p h cp + V T p T

(3.81)

(3.65), : p p cp cp T p T , = = T p S c T p V cp + 2 p T T p T T p p cP T c= . T p cP + 2 T T p

(3.82)

(3.83)

(3.83) (3.75), p = R T , cV = c p R , = c p cV , cV . (3.75) (3.83). .3.28 . c = c ( p, T ) , :c = creal cideal 100% , creal

(3.84)

cideal , (3.75); creal , (3.83).

.. , .. , .. , 2005, 2006, 2007

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97

_____________________________________________________________________________

.3.28.

.3.28, ( ) p [ 0,1 10 ] , T [ 50C (223 K ) 50C (323 K ) ] 4%. p [1 10 ] ,T [ 27C (240 K ) 0C (273 K )] .

1%. , , , , . , ( , 0,3 ), (3.75) . (, ), (3.83).3.6.8.

- .. .. [1]. .. , .. , .. , 2005, 2006, 2007

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98

_____________________________________________________________________________ , . (. (3.4)). [2]: 1) ; 2) ; 3) , ; 4) , . - [2]: f1 ( w1 f1 ) + = 0; t x ( 2 f2 ) t + ( 2 w2 f 2 ) x = 0;

(3.85) (3.85)

2 2 Tp f ( w1 f1 ) ( w1 f1 ) f p 2 Tp1, z + = 1 + f2 + g 1 f1 ; 1 x 1 1 t x x 2 ( 2 w2 f 2 ) ( 2 w2 f 2 ) p z + = f 2 + 2 Tp2, f 2 2 g 1 f 2 ; t x x x

(3.85)

(3.85)

w12 w12 1 f1 1 + + w1 f1 1 + = ( p w1 f1 ) + 2 x 2 1 x t x r x 2 ( R ) p w x + w 2 ( R ) Tp1, w + + 2 1 1 1 + ( x ) w1 f1 g Q f z1 1 T + k1 f1 1 1, oc + 1 1 ; 1 1 x 1 x x 1

(3.85)

2 2 w2 w2 2 f2 2 + + 2 w2 f 2 2 + = ( p w2 f 2 ) 2 x 2 t x x r w + w x 2 ( R ) p x + 2 ( R ) Tp2, w 2 ) 1 + ( x

(3.85)

2 w2 f 2 g Tp1, = Tp2, ;

z1 T + k2 f 2 2 2, + Q2 f 2 ; x x x

(3.85)

.. , .. , .. , 2005, 2006, 2007

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99

_____________________________________________________________________________ 1, = 2, ;R = f2 ; 1 2 f2 f 2 ; x

(3.85) (3.85) (3.85) (3.85) f 2 ; t

x =

x w = 1,1 w1 ;

r w =

1 2 f2

(3.85) (3.85) (3.85)

p = p2 ( 2 , T2 ) ;

m = m ( p, Tm ) , m = 1, 2,

1 2 , (); t ( ); x ; f () , ; f m () f , m- , ; m f m m- ; wm f m m- ; p f ; Tp ; Tpm , , m- , ; g ; z1 , ( ); m f m m- ; R ; , ; x ; w ; x r = ( x ) ; w

; km f m m- ; Tm f m m- ; oc ( oc > 0 ); m , m- ( m , > 0 ); Qm f m , m- . (3.85) (

.. , .. , .. , 2005, 2006, 2007

3

100

_____________________________________________________________________________ , ) . [119, 120] (. [121, 122]). (3.85) 3.6.2. . ( , ) , . 10000, .3.29. 1,42, 0,0187, 8 104 .

. 3.29.

. , 7,09, 6,08 , 1%. 293,15K. . 293,15K. (.3.30). . 453,68 .3.31.

. 3.30. ()

.. , .. , .. , 2005, 2006, 2007

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101

_____________________________________________________________________________

. 3.31. -

3.6.9.

, , . 2,5. , . . , , , : , , . . . : , . , . , . .. , .. , .. , 2005, 2006, 2007

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102

_____________________________________________________________________________ . - (. 4), . PipEst [1,35] , , - , . , 2.04.08-87. , . , , . , , . (3.13.3) , . , , , . , [123] . , . . . ( ) (3.13.3). .3.32 . : 1. . 2. : p , p < p ; p ( p , p ) = p , ,

(3.86)

p ; p

, p .

.. , .. , .. , 2005, 2006, 2007

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103

_____________________________________________________________________________ 3. . 4. - , .

.3.32.

(3.13.3) . (3.86) (3.2). : , . . , , . . , . , (3.13.3) :

S

n dS = 0; n Ym dS = 0,# #N S 1 # # m = 1, N S 1 ; YN = 1 Ym ; S m =1

#

(3.87)

S

(3.87)

p , p < p , p ( p , p ) = p , ;

(3.87)

.. , .. , .. , 2005, 2006, 2007

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104

_____________________________________________________________________________

S

+

! ! # # 2 # n dS = p n dS + n dS ; 2 S S

(3.87) (3.87) (3.87) (3.87) (3.87)

m = m ({S }) ,p = p ({S }) ;

m = 1, N S ;

T1 = T2 = = TN S ;

= ({S }) ;k = k ({S }) ; Dm = Dm ({S }) ,m = 1, N S ,

(3.87)

, , . (3.87) (3.87) (. .3.32), (, ) . , (3.87) (3.87) :f

n df # #

#

f

n df = 0,# # #N S 1 m =1

#

(3.88)#m

f

n Ym df

f

n Ym df = 0, m = 1, N S 1 , YN = 1 S

Y

(3.88)

w f w f = 0, w f (Ym ) w f (Ym ) = 0, m = 1, N S 1 ; YN = 1 S N S 1 m =1

(3.89)

Y

m

,

(3.89)

f f # ; , w , Ym , x # Ym . (3.89), (3.89) :

(Ym ) (Ym ) = 0,

m = 1, N S 1 , YN = 1 S

N S 1 m =1

Y

m

.

(3.89)

, (3.87) :2 2 w w ' + + p ('xx ) w f + = + + p ('xx ) w f , 2 2 (3.90)

.. , .. , .. , 2005, 2006, 2007

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105

_____________________________________________________________________________# # , p p ; 'xx ' xx ; . [1] , 'xx . ' . , :

w f w f = 0;

(3.91)N S 1 m =1

(Ym ) (Ym ) = 0,

m = 1, N S 1 ; YN = 1 S

Y

m

;

(3.91)

p , p < p ; p ( p , p ) = p , ;

(3.91)

+

p

+

2 w p w2 + = + + ; 2 2

(3.91)T1 = T2 = = TN S ;

m = m ({S }) ,p = p ({S }) ;

m = 1, N S ;

(3.91) (3.91) (3.91) (3.91)

= ({S }) ;k = k ({S }) ;Dm = Dm ({S }) ,m = 1, N S ,

(3.91) # T;

T ' ( w f ) . =

, (3.91) , . . , , (. 3.6.5). , (. (3.633.64)):

( p, T ) = h ( p, T )

p

,

dh = cP dT +

h dp, p

.. , .. , .. , 2005, 2006, 2007

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106

_____________________________________________________________________________ (1 ) h 1 + . = T p T T p (3.92)

() :

= cV T

h = cP T ,

(3.93)

cV . (, , ) ( h p )T (. 3.6.5). . , [123] , , (3.92). , , :dT =

1 h dp = dp, cP p T

(3.94)

[112]. [112]. ( (3.94)) [112], .. p p p CQ , Q1min

(3.103)

min Q 1 , [136], ( [137] CQ = 1,1 ).

. , , , . . , [136,142,143]. , , , ( ). , , , , . , , (.3.35).

1 2 3 4 5 .3.35.

, , , . . , ,

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_____________________________________________________________________________ . , . , [142,143]. , , [1]. , . . . , , . . . . ( ) (. .3.35) . . .. .. .. .. 199597 . , [142,143]. , . (3.12) . , , . ( ), . , (. ). , . , [136]. . . ( )

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_____________________________________________________________________________ ( ) . , . . , .3.36.

n

(Q1 ) =

P2 P 1

S

P Q

K1 QK1

PK 1

P1 Q1

P2 Q2

2

PK 2

PK QK

S

QK2

.3.36. , , : ( Q1 ) = P2 P1 ( , = n, P , T1 , Z1 , Q1 , n 1 ); , L1 , S1 , L2 , S2 ( ), ; , PK 1 PK 2 ; ( ) S , S K ( K P1 K P 2 ). , : (.. , PK 1 = P1 , PK 2 = P2 ); , , . , (3.12) [1]: .. , .. , .. , 2005, 2006, 2007

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)

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_____________________________________________________________________________dQ1 PK 1 P = (Q1 ) K 2 , dt La La dPK 1 1 = QK 1 ( PK 1 ) Q1 , dt Ca 1 dPK 2 1 = dt Ca 21 Q1 [ ( Q1 ) ] n QK 2 ( PK 2 ) ,

(3.104) (3.104)

(3.104)

n ; Q ; t ; La = P1 R Z1 T1 K L1 L1 K + ( Q1 ) , S1 a 1 = C1 L1 S1 K L S , Ca 2 = C 2 2 2 , PK 1 PK 2G

P 2 QK 1 = S H K 1 R Z H TH 1 K 1 , G PH Q 2 P 2 = SK K 2 R Z K 2 T 2 K 2 G PK G 1

P K PK 1 + PK 2 , 1 K , P1 = P 2 K + ( Q1 )

G

1 K PK 1 + PK 2 K L 2 L2 S1 1 = 1 P K + (Q ) , K = K L S , 1 L1 1 2 K1 P K1 = K1 , P G

G

K 2 = 1 (Q1 )

n 1 n

PK 2 (Q ) P . 1 1

G

: PH PK ; TH ; K 1 , K 2 ; n ; Li Si ; R ; . K L KC , [142], 0,5. ( (Q1 ) ) .. . (. .3.36) . (3.104) , . [5].

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_____________________________________________________________________________ , , (3.104) [1]. [144]. .3.37 3.38. ( ) [1,5].

3.7.2.

() , (), () ( () ) [2,34].

.3.37.

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_____________________________________________________________________________

.3.38.

, . (, ) , ( ). , . , , , . () . () (.3.39) [2]. , . , , . . (. .3.39) [2]:

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_____________________________________________________________________________

n1J 11

P11

P12

J 12

11

12

n2J P T J 21

P21

P22

J 22

T P

J

21

22

n3J 31

P31

P32

J 32

31

32

.3.39.

i1 = i1 + ( )i1 Di1 i 2 = i 2 + ( )i 2 , i = 1, N ; Di 2

i1 , i- (. (3.96));

Di1 , i- ;

( )i1 , i- ;

i 2 ,

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_____________________________________________________________________________ i- () (. (3.96)); Di 2 , i- ();

( )i 2 , i- ();li1 , i- ; li 2 , i- ;

, ( , .. , ); J (); T P (). () [1]: , ; ; ( nmin n nmax ) ; ; , ; , ; , ; , ; ..

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_____________________________________________________________________________ . (. 6) ( ), [1]. , . . , () , () [2]. , , , [1]. . ( ) (. [1]): Pi1 J i1 . , J i1 (. (3.96)):J i1 ( Pi1 ) = Sri1 2 Dij 2 2 P Pi1 , P

(3.105)

Srij =

4 lij ij

,

j = 1, 2 ;

; i , i . , Pi1 (. (3.105)):2 Pi1 ( J i1 ) = P

1 J i2 P . 1 2 Sri1

(3.106)

, , [1]. . ( ) , , . , ( , ) [1]. ( P ). , T . N ( ) (

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_____________________________________________________________________________ ) [1] :J J i1 = 0;i =1 N

(3.107) J i1 > 0 i = 1, N .

P > Pi1 i = 1, N

(3.107) . , . , . . , , , , . , , , .. [1]: J i1 Pi 2 = Pi 2 ni , T , Pi1 , , i = 1, N ; i1 ( Pi1 ) J i1 Ti 2 = Ti 2 ni , T , Pi1 , , i = 1, N ; i1 ( Pi1 ) J i1 M i = M i ni , T , Pi1 , , i1 ( Pi1 ) i = 1, N ;

(3.108)

(3.108)

(3.108)

i 2 = i 2 ( Pi 2 ) = i 2 ni , T , Pi1 ,

i1 ( Pi1 )

J i1

, i = 1, N ,

(3.108)

Pi 2 i- ; ni i- ; i1 i- ( ); Ti 2 i- ; M i , i- ; i 2 i- ( .. ). , , , . [1] (. .3.40). .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________

.3.40. ( ). , [1]: J i1 2 Pi 2 ni , T , Pi1 , P 2 i1 ( Pi1 ) J i1 J i1 Sri 2 i 2 ni , T , Pi1 , = 0, i1 ( Pi1 ) J i1 Pi 2 ni , T , Pi1 , i1 ( Pi1 ) J i1 Pi 2 ni , T , Pi1 , P , i1 ( Pi1 ) i = 1, N .

(3.109)

[1]:N J i1 T Sri 2 Ti 2 ni , T , Pi1 , i1 ( Pi1 ) i i =1 N Sri 2 i = 0, i =1

(3.110) J i1 2 Pi 2 ni , T , Pi1 , P 2 i1 ( Pi1 ) J i1 i 2 ni , T , Pi1 , . i1 ( Pi1 ) J i1 Pi 2 ni , T , Pi1 , i1 ( Pi1 )

i =

, i i- ( ) , .. [135]:

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_____________________________________________________________________________ i = P P . 0,5 g ( + )

, ( ) (. .3.39). , ( ) (), . () (3.107), , , , , , . (), . [1]. , i (. (3.105), (3.106) (3.109)): J i1 2 Pi 2 ni , T , Pi1 , P 2 i1 ( Pi1 ) J i1 J i1 Sri 2 i 2 ni , T , Pi1 , + i1 ( Pi1 ) J i1 Pi 2 ni , T , Pi1 , i1 ( Pi1 ) + G ( Pi1 , P ) = 0, i = 1, N 1,

(3.111)

J N1 2 J N 1 PN 2 nN , T , PN 1 , N 1 ( PN 1 ) J N1 1 2 = PN 2 nN , T , PN 1 , , 2 N 1 ( PN 1 ) SrN 2 J N1 N 2 nN , T , PN 1 , N 1 ( PN 1 )

2 P

(3.111*)

P Pi 2 ( Pi1 ), 0, G ( Pi1 , P ) = 1 , P > Pi 2 ( Pi1 ), ( P Pi 2 ( Pi1 ) )

(3.111)

0 < < 101 ; N 1 N ; N 2 N . (3.111) (3.111)

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_____________________________________________________________________________ ( ). (3.111), , ( ). , (3.111) . (. (3.107)):J N 1 = J J i1.i =1 N 1

(3.112)

, , J N 1 , (3.106) PN 1 :2 PN 1 = P 2 1 J N 1 P . 2 SrN 1

(3.113)

(3.112) (3.113) (3.111*), :N 1 J J i1 PN 2 1 i =1 2 , = PN 2 2 N 1 SrN 2 J J i1 i =1 N 2 nN , T , PN 1 , N 1 PN 1 ( Pi1 ) 2

2 P

(3.111)

N 1 J J i1 P 1 i =1 2 PN 1 = P 2 , SrN 1 N 1 J J i1 i =1 . = PN 2 nN , T , PN 1 , N 1 PN 1 ( Pi1 )

2

PN 2

, (3.111),

( N 1)

.

Pi1 , i = 1, N 1 , J i1 , i = 1, N 1 . - (3.107), , (3.111)

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_____________________________________________________________________________ :Pi ,min < Pi1 < Pi ,max , i = 1, N 1, i = 1, N 1,

(3.111) (3.111)

J i ,min < J i1 < J i ,max ,

(3.111) (3.111) (3.105) (3.106). , , . ( ), . , J i ,min J i ,max , , ( ):J N ,min + J i ,max < J < J N ,max + J i ,mini =1 i =1 N 1 N 1

J N ,min ( , , ) < J N ,min < J J i ,max , J i ,max < J i ,max ( , , ) , i = 1, N 1;i =1 N 1

J J i ,min < J N ,max < J N ,max ( , , ) , J i ,min ( , , ) < J i ,min , i = 1, N 1,i =1

N 1

(3.111)

J i ,min ( , , )

J i ,max ( , , ) , i = 1, N ,

, , , , .. (3.112) . (3.113) (3.111), , . (3.111) (3.111) N , Pi ,min Pi ,max ( J i ,min J i ,max ) N . (3.111) :J N ,min < J J i ,max ,i =1 N 1

(3.114) (3.115)

J N ,max > J J i ,min .i =1

N 1

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_____________________________________________________________________________ (3.115) -1 (3.114), :

Ji =1

N 1

i ,max

J N ,max < J i ,min J N ,mini =1

N 1

(3.116)

Ji =1

N 1

i ,max

J i,min < J N,max J N ,min .i =1

N 1

, J . ( N 1) :J i ,max = (1 + x ) J , J i,min = (1 x ) J ,

x > 0 , J . :J i,max J i,min = 2 x J ,

(3.117)

Ji =1

N 1

i,max

= ( N 1) (1 + x ) J , = ( N 1) (1 x ) J .

Ji =1

N 1

i,min

(3.116) (3.117) :2 ( N 1) J x < J N ,max J N,min .

(3.118)

(3.117) (3.118), , N ( N 1) . ( N 1) ( N 1) . J i,min J i,max . 1. , , J i ,min( , , ) J i ,max( , , ) , i = 1, N . 2. J i ,min( , , ) ; J i ,max( , , ) . , N .

3. N J N ,min > J N ,min( , , ) J N ,max < J N ,max( , , ) . J N ,max( , , ) J > J J N ,min( , , ) ,

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_____________________________________________________________________________ :J N,min = J N ,min( , , ) 1,01, J N,max = 2 J J N,min ,

J N ,max = J N ,max( , , ) 0,99, J N,min = 2 J J N ,max . J . () , J : J =

J . J i,min J i,max N

: J i,max < J N ,max J N ,min 2 ( N 1) + J , J i,min > J

J N ,max J N ,min 2 ( N 1)

.

4. (3.106). , (3.111) . , (3.111) : J i1 2 Pi 2 ni , T , Pi1 , P 2 i1 ( Pi1 ) J i1 J i2 Sri 2 i 2 ni , T , Pi1 , = 0, 1 2 i1 ( Pi1 ) (3.119) J i1 Pi 2 ni , T , Pi1 , i1 ( Pi1 ) i = 1, N 1, Pi ,min < Pi1 < Pi ,max , i = 1, N 1,

(3.119)

J i ,min < J i1 < J i ,max , i = 1, N 1,

(3.119)

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_____________________________________________________________________________J N ,min ( , , ) < J N ,min < J J i ,max ,i =1 N 1

J i ,max < J i ,max ( , , ) , i = 1, N 1;

(3.119)J J i ,min < J N ,max < J N ,max ( , , ) ,i =1 N 1

J i ,min ( , , ) < J i ,min , i = 1, N 1,

N 1 J J i1 PN 2 1 i =1 2 , = PN 2 2 N 1 SrN 2 J J i1 i =1 N 2 nN , T , PN 1 , N 1 ( PN 1 ) 2 2

2 P

(3.119)

N 1 N 1 J J i1 P J J i1 1 i =1 i =1 2 . PN 1 = P 2 , PN 2 = PN 2 nN , T , PN 1 , SrN 1 N 1 ( PN 1 )

, , . , . (3.119) : ; ( ). , J i1 2 Pi 2 ni , T , Pi1 , < P 2 Pi1 ) i1 (

J i1 2 Pi 2 ni , T , Pi1 , P 2 i1 ( Pi1 ) J i1 Sri 2 i 2 ni , T , Pi1 , >0. 2 i1 ( Pi1 ) J i1 Pi 2 ni , T , Pi1 , i1 ( Pi1 )

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_____________________________________________________________________________ , . , (3.119) , , (3.119) . (3.119) (3.119) , , . , (3.119) , . (, ) (3.119) . (), .. [2]. () (.3.41).

n1J1 P 11 P 12 J1 P13

11n2J T P J2 P21 P 22

12

J2

P23

21n3J3 P31 P 32

22

J3

P33

31

32

.3.41. .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________ P .. Pi1 . . Pi 2 , , , . , . Pi 3 , .. . , P Pi 3 , , i- . .., Pi 3 i- ( ) (. (3.106) (3.109)). , Pi 3 , .. [2] J (i +1)1 J i1 = 0, i = 1, N 1, Pi 3 ni , T , Pi1 , P(i +1)3 ni +1 , T , P(i +1)1 , i1 ( Pi ) (i +1)1 ( Pi +1 ) J i1 J i2 Pi 2 ni , T , Pi1 , 1 i1 ( Pi1 ) J i1 1 2 Pi 3 = Pi 2 ni , T , Pi1 , . i1 ( Pi1 ) Sri 2 J i1 2 i 2 ni , T , Pi1 , i1 ( Pi1 )

(3.120)

(3.120)

(3.120) , ( ), . , (3.120), ( N 1) N . (3.120) (3.109). N J ( ), N :J N = J J i1.i =1 N 1

( N 1) .

(3.120)

, (3.120)

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_____________________________________________________________________________ . (3.120) (3.119). (3.120) ( ) [2]. (3.111) ( ), , . ( ). (3.119) (3.120) [1,2]. (3.119) (3.110). , , , [1,2]. [2]. .. . . . ( ) . ( ) , . , i - , (3.108). , , , :p J i1 S = M i ni ( J i1 ) , Ti1 , Pi1 ( J i1 ) , , i1 ( Pi1 ( J i1 ) ) i =1

(3.121)

p , Ti1 ( Ti1 = T ). , [1,2,5].

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_____________________________________________________________________________

3.7.3.

- CorNet AMADEUS [2]: .3.7.3.1.

[132]. ( ) [1]. . , (3.119) (3.120) . . J , T P () . [1,2].3.7.3.2.

.. .. [2,23]. (3.105) (3.106), - (3.12). , (. .3.39). - [1]. () : (.3.42). : J 2 , PA TA , . : , , .

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_____________________________________________________________________________

.3.42. (. .3.42) [2]: 1. J 2 , TA PA - . P21 T21 . 2. J 2 , P21 T21 .. P22 T22 . 3. J 2 P22 T22 - . (. .3.42) , [2]. () . ( ) . (), , . , . () ( ) [2,23]. , . , , ..

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_____________________________________________________________________________ N (. 3.6.1). . (3.12) ( ) (3.12) ( , , , ) . , 1. (3.12), ( () , ( )). , . . ( k T x w ), . ( w >> k T x ), : T x = 0 . () () (. .3.42): A B. , , . , : PA TA PB ( , , ). J 2 = J 2 (t ) ( .3.39 .3.42): 2 ( J ) = 0,

(3.122) (3.122)

2 ( J ) = P22 ( J ) P22 ( J ) ,

P22 ( J ) ,

J ; P21 ( J ) T21 ( J ) , T T J ; P22 ( J ) = 2 ( P21 ( J ) , T21 ( J ) , J ) P21 ( J ) ,1

, , . , .

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_____________________________________________________________________________T .. ; 2 ( P21 ( J ) , T21 ( J ) , J ) ,

. , , . , (3.122) , . (). (. .3.39) , . . . [2]. , , . J i = J i ( t ) ( i = 1, N , N ) ( .3.39): (3.123) (3.123)

i ( J i ) = 0, i = 1, N ,

i ( J i ) = Pi ( J i ) Pi 2 ( J i ) . 2

, (3.123), , (3.122). , (3.123). (3.122) (3.123) , Techno&Optim [1]. .3.7.4.

, 3.7.13.7.3, [2]. . . .. [30]. , , (.3.43) [145].

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_____________________________________________________________________________

J , P ,T

J2

J1 ( X1 )

J11( X 2 )

J12 ( X 3 )1 J 13

J131 13 ( X 8 )

J 21

1 J11

B/

1 J12

A

D

A/B/C

J1 21

2 J 21 3 J 21

D

1 11( X 6 )

1 12 ( X 4 )

2 J13

2 13 ( X 7 )

2 21( X 5 )

1 (X9 ) 21

P1 11

P1 12

P1 13

1 P21

.3.43. .3.43 , , ( ). , , , . ( (. 3.7.13.7.3)) ( ). ( , ). ,

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_____________________________________________________________________________ , , , . , (). , , , , . .. [145]. ( ) [30]. .3.43 ( X ). .3.43 :1 1 P1 ( J , P , T , J 1 ( X 1 ) , J 11 ( X 2 ) , 11 ( X 6 ) ) P1 ( J , P , T , J 1 ( X 1 ) , J 12 ( X 3 ) , 12 ( X 4 ) ) = 0; 11 12 1 1 2 P1 ( J , P , T , J 1 ( X 1 ) , J 12 ( X 3 ) , 12 ( X 4 ) ) P1 ( J , P , T , J 1 ( X 1 ) , J 13 , 13 ( X 8 ) , 13 ( X 7 ) ) = 0; 12 13 1 2 1 2 1 P1 ( J , P , T , J 1 ( X 1 ) , J 13 , 13 ( X 8 ) , 13 ( X 7 ) ) P21 ( J , P , T , J 2 , J 21 , 21 ( X 9 ) , 21 ( X 5 ) ) = 0; 13 1 1 J 11 ( X 2 ) J 11 ( 11 ( X 6 ) ) = 0; 1 1 J 12 ( X 3 ) J 12 (12 ( X 4 ) ) = 0; 1 1 J 13 J 13 ( 13 ( X 8 ) ) = 0; 1 1 2 1 2 J 13 (13 ( X 8 ) ) J 13 (13 ( X 8 ) , 13 ( X 7 ) ) = 0; 1 2 2 1 J 21 J 21 ( 21 ( X 5 ) ) J 21 ( 21 ( X 9 ) ) = 0; 2 1 3 1 2 J 21 ( 21 ( X 9 ) ) J 21 ( 21 ( X 9 ) , 21 ( X 5 ) ) = 0; 0 < X i < 1, i = 1, 3; 1 max < X j < max , j = 4, 7; 1 1 min < X k < max , k = 8,9, (3.124) 1 1 1 P1 , P12 , P13 , P21 , J , P , T , 11

, J 1 , J 2 I, J 11 , J 12 , J 13 , J 21 1 1 2 2 1 1 II, 11 , 12 , 13 , 21 , 13 , 21 1 , max 1 , max , max

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_____________________________________________________________________________ . , . . 3.7 . [1,2].3.8.

3.4, (. .3.1). , (. 3.6). ..3.8.1.

[2]. . . , - CorNet AMADEUS [2]. SPP. ( ) , (., , [93, 146]). . ( ) .

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_____________________________________________________________________________ . .. .. [2]. : J , P T , . . : 1. 1, 1. 2. J , T P - . P12 T12 . 3. J , P T12 12 (. 3.7) P22 T22 . 4. . ( T = T22 , P = P22 ) . , 5. 2. 5. J , T P - . P 12 T12 . 6. . .3.8.2.

.. .. [2]. , , .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________ (. 3.7.3.2) . (3.122) (3.123) . , , (3.122) (3.123). , Techno&Optim [2]. 3.8.3. ,

: ; , . , , , . , , , , . 2030 . [86,87]. . 1860 100. ( 12 , ). , 1400 1200, . , 100150. , , . , .. (. 3.6.1). .3.44. .3.45. . , .. , .. , .. , 2005, 2006, 2007

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141

_____________________________________________________________________________ 35 110 , ( ), . , 450 3040% (.. 1 3540). Pentium 4 (1.6 , 512 ).

. 3.44.

. 3.45. . , D , 1, 3 2, 4 , , .3.45, . , . . . (.3.46). , 2050, 0,020,10. 10000 . , , , . , . , .. , .. , .. , 2005, 2006, 2007

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142

_____________________________________________________________________________ - CorNet, , [1,2].

. 3.46. , , . .. .. [35]. .. . . . , , .3.47. 03 04. 12\2-1, 12 ( .3.47 ).

. 3.47. ( )

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_____________________________________________________________________________ 2 ( ) 10 . 70 . 10, 11, 13 04. 1 ( 12\2-1) .3.48 3.49. , , 0 120, 0,6, , SCADA-.

.3.48. 1

.3.49. 2 .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________ , ( 2025) 34 . , , ( 400 ). . , , , , . , , . . , . , , (. 3.6.6), (): . . ( ). , . . .. , . ( ): ( ); ( ); , ( , ). . .. (. 3.6.1). , .3.44. , . , .3.45.

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_____________________________________________________________________________ (.3.50).

. 3.50. , . , . , (. .3.44). : = + , (3.125) + . : L = + D 3; (3.126)

M = C + B 2; N = B + D 4.

(3.127) (3.128)

(3.1253.128), , . , , , (. .3.50), . ( .3.44). , ( ). , , , .. . , , . , 1 3, , (3.125) L (3.126). , . , , . .. . .

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_____________________________________________________________________________

1. , .3.44 ( ), [35]: 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 1; , 1, , 1; 1 K A C (3.125); A L 3 (3.126), 3 1; L 3 D (3.126); D N 4 (3.128), 4 1; N 4 B (3.128); B M 2 (3.127), 2 1; M 2 C (3.127); C K 1 (3.125), 1 1, M , ; ; N , ; ; L , ; ; K , ; 1 ; .

11) 12) 13) 14)

. , 1, 2, 3 4 (. .3.45).

2. C D ( 3.51). [35]: 1) 2) 1; , 1, , 1;

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_____________________________________________________________________________ 3) 4) 5) 6) 1 K A C (3.125); A L 3 (3.126), 3 1; L 3 D (3.126); D , L , ; ;

.3.51. D 7) 8) 9) , K , , 1 ; 2; , 1, , 2, , 2;

10) 2 M B C (3.127); 11) B N 4 (3.128), 4 2; 12) N 3 D (3.126); 13) D , N , ; ; 14) , M , , 2 ;

15) . . : 1, 1 3; 2, 2 4; .3.52.

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_____________________________________________________________________________

.3.52. D 3. C B (.3.53). 1, 3 4, , 1. 2 , C B ( M , . (3.127)). 2 . .3.54.

. 3.53. B

. 3.54. [35]: , , , . - , . , .3.8.4.

, ()

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_____________________________________________________________________________ , .. , . , . , , (, , ) , SCADA-. , ( , , ). [1,2,27], . . .. .. [4,35]. . , () . , . (. ). , , ( ) . , , SCADA ( ) [4]. n ( ). , , , . k , , . , ( ), .. , .. , .. , 2005, 2006, 2007

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_____________________________________________________________________________ . , .. , . , [4]. . . . . . , , . , , SCADA . , , l , . . ( ) , . ( , , (Yi = Pi ) (Yi = J i ) ):! ! ! 2 f i X = Yi X Yi X , i = 1, m, m = s k l ,

( )

( )

( )

(3.129)

! X R n n- R n ( , ), n- R n ( ); s ( ), . , , , (3.129) . (3.129), ,

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_____________________________________________________________________________ : l =1. f i : R n R, i = 1, m , . , f i : R n R, i = 1, m , . , . , [69,70]: ! ! max f i X min, X R n . (3.130)i =1, m

( )

(. ). (3.130) : ! max f i X < C , (3.131)i =1, m

( )

C > 0 , ( : ). (3.130) . (3.130) , [147]. (3.130) xn +1 :xn +1 min ! ! n +1 f i X xn +1 0, i = 1, m; X R .

( )

(3.132)

(3.132) , Techno&Optim [1]. - CorNet [2]. [2]. AMADEUS SPP [2]. 2002 . SPP . AMADEUS (, , )

.. , .. , .. , 2005, 2006, 2007

3

152

_____________________________________________________________________________ SPP () (., , [1,2]). 35 ( 2002 2004 , 24 ) , . . 2,8% ( 0,19 ). 93% 1,7% (0,12). , , SCADA-, 0,05. 1,1% ( 3,2 ). 90% 0,7% (1,9). , , SCADA-, 0,7. 6,3% ( 94,0/ ). 1. , , 3. (), .

.. , .. , .. , 2005, 2006, 2007

4

153

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4 4.1. , , . , , 1960 1980 . . , 2030 . 1997 [83] 30% 33 , 20% . . 810 , , [83]. , , , , [83, 154]: , ; ; . , , . , , . , - (, [225232]), , , , , , . , . .. , .. , .. , 2005, 2006, 2007

4

154

_____________________________________________________________________________ , , , , , , ( ) , , , . . . , , : ; ; ; . , , , . , , - () ( , ) ( ) . [1, 3, 83, 155, 156], , . , , . , , [1, 156] . , [1], , , , . , , [1] . , [157], 19922002. , (57%), (30%). , , , [155, 156] (, ..) . , ,

.. , .. , .. , 2005, 2006, 2007

4

155

_____________________________________________________________________________ , . , , . , , , ( , , ..) , , . [72, 156]. , , , , . [158160]. - , (, ) ( ) (.4.1).

.4.1. , , , , , . - - , [160]. , , ,

.. , .. , .. , 2005, 2006, 2007

4

156

_____________________________________________________________________________ , [161167 .], . , , , , . [1, 3, 16, 21]. , . , . , , , - . . , , , , , . [155, 173182 .] [166], [167172 .], . [1, 3, 16, 21]. . , [166], (, [168]) . , ANSI/ASME B31G [166]: 1) , ; 2) ;

3) ( , () ); 4) , ;

.. , .. , .. , 2005, 2006, 2007

4

157

_____________________________________________________________________________ 5) () , ; 6) ( ), , , ; 7) , , , ; 8) . [1,3,16,21]. . , RSTRENG ( Technical Toolboxes, Inc., ), [168] , , , ( ) . , 1,5 . , [168], - . , , , , , , . . , . , , , [166,168], , . (.4.1) . - , , , ( ) [183]. , ( ) . , , 90- .. [1, 35, 11, 13, 1521, 60, 184187].

.. , .. , .. , 2005, 2006, 2007

4

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_____________________________________________________________________________ .. , .. .. [1, 35, 17, 43, 4955, 148, 183, 188, 189]. 4.2. , , , , . , , , , , .. , , , , , , , . , , , . [190], , ,

Documents

Chisl Model Trub Seleznev 2007