Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15785/doc.pdf Дата изменения: Mon Oct 29 13:12:10 2007 Дата индексирования: Mon Oct 1 20:41:08 2012 Кодировка:

.. , .. , .. , .. (, ) . , , . , . , . . . . NUMERICAL MODELLING OF MAGNETOHYDRODYNAMICAL PROCESSES IN ALUMINIUM ELECTROLYZER Alatortsev A.V., Kuzmin R.N., Provorova O.G., Savenkova N.P. (Moscow, Krasnoyarsk) The problem of modeling of aluminium electrolysis in concrete electrolytic baths is examined. The mathematical modeling physical process, which is taking into account features of a design of a concrete bath, including an accretion condition at the bottom of a bath, will be carried out. The mathematical model uses information oppor243


tunities of Navier-Stokes equation that has been written down in environments of metal and electrolyte. The system of the equations multivariate, allows taking into account magnetohydrodynamical processes in two environments and interaction of environments. The problem is solved by numerical method. As a result of calculation dynamics of an interface of two environments and distribution of speeds and electric currents on border of the unit of environments is modeled. Comparison of results with results of physical experiment will be carried out. , , . . , - ( 200 ), , -, , . . , . -, . . ( ) ( ) . .
244


.. . -- -10, 2002, .243-245

. 1 .

, , . :
1 + t 1u1 t 1v1 t p1 = z 1 1 t r div ( 1V1 ) = Q r p + div ( 1u1V1 ) = - 1 + div ( 1T gradu1 ) + f1x x r p1 + div( 1v1V1 ) = - + div( 1T gradv1 ) + f1 y y - 1 g + f
1z

r v + div ( 1 1V1 ) = - p1 div (V1 ) + div (k1 gradT1 ) + div (k1T grad 1 ) + f

T1

:

245


2 + t 2 u 2 t 2 v 2 t p 2 = z 2 2 t

r div ( 2V2 ) = -Q r p + div( 2 u 2V2 ) = - 2 + div( 2T gradu 2 ) + f 2 x x r p 2 + div ( 2 v 2V2 ) = - + div( 2T gradv 2 ) + f 2 y y -2 g + f
2z

r v + div( 2 2V2 ) = - p 2 div(V2 ) + div(k 2 gradT2 ) + div (k 2T grad 2 ) + f

T2

: r rr r H = rot[V * H ] - rot ( m rotH ) t v divH = 0 : r

f 1 = ( f 1 x , f 1 y , f 1 z ) - ,

, :
r 1 f1, 2 = rotH * H; 4

f

T1

,2

=



m

. : :

4

(rotH ) 2 ;

246


.. . -- -10, 2002, .243-247

1 h1 1u1 h1 1v1 h1 + + = 2Q t x y
1u1 h1 1u1 h1 1u1v1 h1 1 [(h1 - h1 (t 0 )) * ( 1 g - f1z ) - p1 (t 0 )] + + + = 2 x t x y + FDu1 + FxH1 + Fxz0 + f1x h1
2

1v1 h1 1u1v1 h1 1v1 h1 1 [(h1 - h1 (t 0 )) * ( 1 g - f1z ) - p1 (t 0 )] + = + + 2 y y x t + FDv1 + F
1cv1T1h1 t + 1u1cv1T1h1 x + y
yH
1

2

+F

yz

0

+ f1 y h1

1v1cv1T1h1

= FDT1 + FH1 - Fz0 + f T1 h1

p1 (t 0 ) :

2 p1 2 p1 + f1x + f 1 y , p1 = p1 = x y x 2 y 2 p p 1 = f 1x , 1 = f1 y x x =l0 y y =l0 x= x y= y f1x, f1y rrr f = J в B . :

247


FDu1 = FDv1 FDT1
F
xH
1

u1 u1 T + T x y y v v = T 1 + T 1 x x y y T T = (k1T + k1 ) 1 + (k1T + k1 ) 1 x x y y x

,F

yH

1

: + FxH =- H 1 2 1 + FyH =- H 2
1 1 1 1





2

(u1 - u2 ) (u1 - u2 ) 2 + (v1 - v2 ) (v1 - v2 ) (u1 - u2 ) 2 + (v1 - v2 )

2

2

2

:

Fxz0 , F
F

yz

0

Fxz0 = - 0 1u1 u1 + v1
yz
0

2

2

= - 0 1v1 u1 + v1

2

2

FH1 ­ :
FH1 = 1 (T2 - T1 )

,
T2 > T1

Fz0 ­ ,

:
Fz0 = 0 (T1 - T0 )

:
t

H1 H1 + (V1 , )

H1 1 = , V1 - rot ( m rotH 1 ),
248

:


.. . -- -10, 2002, .243-249

2u2 h2 2u2 2 h2 2u2 v2 h2 1 + + = [(h2 - h2 (t0 )) * ( 2 g - f 2 z ) - p2 (t0 )] + t x y 2 x + FDu2 + FxH 2 + FxH1 + f 2 x h2
2 h2 2 u 2 h2 2 v 2 h2 + + = - 2Q t x y

2v2 h2 2u2v2 h2 2v2 2 h2 1 + + = [(h2 - h2 (t0 )) * ( 2 g - f 2 z ) - p2 (t0 )] + t x y 2 y + FDv2 + FyH 2 + FyH1 + f 2 y h2
2 cv2 T2 h2 t + 2u2cv2 T2 h2 x + 2v2 cv2 T2 h2 y = FDT2 + FH 2 - FH1 + fT2 h2

p 2 (t 0 ) , :

p 2 = x f 2 x + y f 2 y p2 H1 + H 2 ) y =l0 = 2 g ( H 3 - y= 2 x =0y x =l x
H1, H2, H3, - :
u2 u2 T T + x y y v2 v2 FDv 2 = T T + x x y y T2 T2 = ( k 2T + k 2 ) ( k 2T + k 2 ) + x x y y F
Du
2

=

x

F

DT

2

, , .
249


: , :
FxH 2 , FyH
2

FxH 2 = - FyH 2 = -

H H

2

2u 2v

2

u2 2 + v u2 + v
2

2 2 2

,

2

2

2

.
FH 2 :
FH 2 = 2 (T2 - T )

:
H H H2 1 + (V2 , ) 2 = 2 , V2 - rot ( m rotH 2 ), t

. , . : min(hx , h y ) t < C max(V1,max ,V2, max , H 1,max , H 2, max )



A

k, max

r = max i , j (| Ak ,ij |) A = V , H ; k = 1,2

: .
250


.. . -- -10, 2002, .243-251

. 2. .

. 3. 1000

. 4. 10000

. 5. 100000

, , , . , . . , , . 10 30 ­ , . , . , . . 1. .., .., .., ..,
251


.., .. " ". (-2000). , 2001, .96-102. 2. .., .., .. " ". ". . ". , 2002, . II, . 393-397.

252