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CONDUCTIA TERMICA-2D STATIONAR - Metoda volumelor finite

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CONDUCTIA TERMICA-2D STATIONAR - Metoda volumelor finite

CONDUCTIA TERMICA-2D STATIONAR - Metoda volumelor finite

Se considera o placa dreptunghiulara cu dimensiunile urmatoare( 0.5 [m] x 0.4 [m] ) si grosimea de 0.01 [m]. Conductivitatea termica a materialului placii este . Frontiera vest a placii primeste o densitate de flux de valoare constanta , iar frontierele sud si est sunt izolate,frontiera nord fiind mentinuta la temperatura constanta de 100 [ºC].




Sa se determine distributia stationara de temperatura si calculul temperaturilor in nodurile retelei de discretizare(130).

∆x = ∆y = 0.1 [m]

Solutie

Ecuatia diferentiala ce guverneaza transferul termic este:

Se utilizeaza o retea de discretizare cu 30 de noduri uniform distribuite ca in figura :

Pentru un nod interior ecuatia discretizata este :

Ecuatiile discretizate pentru nodurile interioare ( 8, 9, 10, 11, 14, 15, 16, 17, 20, 21, 22, 23 ) ale domeniului de calcul sunt :

Pentru nodurile situate pe frontiera vest ( 2, 3, 4, 5 ) ecuatiile discretizate se obtin integrand ecuatia conductiei termice pe jumatatea de volum de control hasurat din figura :

Pentru nodurile situate pe frontiera nord ( 26, 27, 28, 29 ) ecuatiile discretizate se obtin integrand ecuatia conductiei termice pe jumatatea de volum de control hasurat din figura :

Pentru nodurile situate pe frontiera sud ( 7, 13 ,19 ) ecuatiile discretizate se obtin integrand ecuatia conductiei termice pe jumatatea de volum de control hasurat din figura :

Pentru nodul 1 ecuatia discretizata se obtine integrand ecuatia conductiei termice pe sfertul de volum de control hasurat din figura :

Pentru nodul 25 ecuatia discretizata se obtine integrand ecuatia conductiei termice pe sfertul de volum de control hasurat din figura :

MVF

30 noduri

MEF

31 noduri

Eroarea

MVF

546 noduri

MEF

485 noduri

Eroarea

T1

T2

T3

T4

T5

T7



T8

T9

T10

T11

T13

T14

T15

T16

T17

T19

T20

T21



T22

T23

T25

T26

T27

T28

T29

Graficul in MATCHAD 7 ( 546 noduri )este :

Graficul in QUICKFIELD 5.0 ( 343 noduri ) este :

program lucrarea 6

c=================================================================

c------------REZOLVAREA ECUATIEI DE CONDUCTIE TERMICA 2D STATIONARA--

c (metoda volumelor finite)

c------------ECUATIA REZOLVATA:(divk(gradT))=0-------- ----- ------ -------------

c------------VARIABILELE DE INTRARE-------- ----- ------ ----- ----- ----- ----- ------

c NNX : numarul de noduri pe directia x

c NNY : numarul de noduri pe directia y

c L : lungimea domeniului de calcul

c H : inaltimea domeniului de calcul

c    TB : conditia Dirichlet

c------------CONDITIILE LA LIMITA-------- ----- ------ ----- ----- --------- ----- ------

c Dirichlet la frontiera nord TB(x,H)=100

c Neumann (flux nul) la frontiera est si sud

c Flux impus (q) la frontiera vest

c------------VARIABILELE DE INTRARE-------- ----- ------ ----- ----- ----- ----- -----

c TEMP : temperatura T(x,y)

c------------AUTOR :Irimia Marius EG 11502

c=================================================================

parameter(nnx=21, nny=26, nit=25000)

double precision TEMP(nnx,nny),AX(nnx,nny),AY(nnx,nny)

double precision A(nnx,nny),B(nnx,nny),C(nnx,nny)

double precision wk(nnx,nny)

double precision A1(nny),B1(nny),C1(nny),wk1(nny),temp1(nny)

double precision DX,DY,TB,L,H,la,gz,q

data TB/100.0/,L/0.4/,H/0.5/,la/1000.0/,gz/0.01/

data q/500000.0/

c------------CALCULUL PASULUI SPATIULUI(DX)-------- ----- ------ -------------

DX=L/(nnx-1)

DY=L/(nny-1)

write(*,*)'DX=',dx

write(*,*)'DY=',dy

c------------INITIALIZAREA TEMPERATURII-------- ----- ------ ----- ----- ---------

do i=1,nnx

do j=1,nny

TEMP(i,j)=0.0

enddo

enddo

c-----------INITIALIZAREA CONDITIILOR LA LIMITA DE TIP DIRICHLET----- ----- ----

c----- ----- -------------LA FRONTIERA NORD-------- ----- ------ ----- ----- ------------

do i=1,nnx

TEMP(i,nny)=TB

enddo

c----- ----- ------------BUCLA DE ITERATIE-------- ----- ------ ----- ----- ----- ----- ----

do k=1,nit

c------------FORMAREA VECTORILOR DIAGONALELOR-------- ----- ------ ---

do i=1,nnx-1

if(i.eq.1)then

c------------DIAGONALA INFERIOARA-------- ----- ------ ----- ----- ----- ----- ------

do j=1,nny-1

if(j.eq.1)then

A(i,j)=0.0

A1(j)=A(i,j)

else

A(i,j)=-la*(DX/2.0*gz)/DY

A1(j)=A(i,j)

endif



enddo

c------------DIAGONALA SUPERIOARA-------- ----- ------ ----- ----- ----- ----- ------

do j=1,nny-2

if(j.eq.1)then

C(i,j)=-la*(DX/2.0*gz)/DY

C1(j)=C(i,j)

else

C(i,j)=-la*(DX/2.0*gz)/DY

C1(j)=C(i,j)

endif

enddo

C(i,nny-1)=0.0

C1(nny-1)=C(i,nny-1)

c------------DIAGONALA PRINCIPALA-------- ----- ------ ----- ----- --------- ----- -----

do j=1,nny-1

if(j.eq.1)then

B(i,j)=la*(DY/2.0*gz)/DX+la*(DX/2.0*gz)/DY

B1(j)=B(i,j)

else

B(i,j)=la*(DY*gz)/DX+la*(DX/2.0*gz)/DY+la*(DX/2.0*gz)/DY

B1(j)=B(i,j)

endif

enddo

c------------FORMAREA INITIALA A TERMENULUI LIBER WK----- ----- --------- ----- -----

do j=1,nny-2

if(j.eq.1)then

wk(i,j)=(la*(DY/2.0*gz)/DX)*TEMP(i+1,j)+DY/2.0*gz*q

wk1(j)=wk(i,j)

else

wk(i,j)=(la*(DY*gz)/DX)*TEMP(i+1,j)+DY*gz*q

wk1(j)=wk(i,j)

endif

enddo

wk(i,nny-1)=(la*(DY*gz)/DX)*TEMP(i+1,nny-1)+DY*gz*q+

* (la*(DX/2.0*gz)/DY)*TEMP(i,nny)

wk1(nny-1)=wk(i,nny-1)

c-----------REZOLVAREA SISTEMULUI-------- ----- ------ ----- ----- ----- ----- -------

CALL TRIDAG(a1,b1,c1,wk1,temp1,nny-1)

c-FORMAREA SOLUTIEI PE TOATA LINIA VERTICALA(LINIA APLICARII TDMA)-

do j=1,nny-1

TEMP(i,j)=temp1(j)

enddo

else

c------------DIAGONALA INFERIOARA-------- ----- ------ ----- ----- --------- ----- ----

do j=1,nny-1

if(j.eq.1)then

A(i,j)=0.0

A1(j)=A(i,j)

else

A(i,j)=-la*(DX*gz)/DY

A1(j)=A(i,j)

endif

enddo

c------------DIAGONALA SUPERIOARA-------- ----- ------ ----- ----- ----- ----- -------

do j=1,nny-2

if(j.eq.1)then

C(i,j)=-la*(DX*gz)/DY

C1(j)=C(i,j)

else

C(i,j)=-la*(DX*gz)/DY

C1(j)=C(i,j)

endif

enddo

C(i,nny-1)=0.0

C1(nny-1)=C(i,nny-1)

c------------DIAGONALA PRINCIPALA-------- ----- ------ ----- ----- --------- ----- ----

do j=1,nny-1

if(j.eq.1)then

B(i,j)=la*(DY/2.0*gz)/DX+la*(DY/2.0*gz)/DX+la*(DX*gz)/DY

B1(j)=B(i,j)

else

B(i,j)=la*(DY*gz)/DX+la*(DY*gz)/DX+la*(DX*gz)/DY+la*(DX*gz)/DY

B1(j)=B(i,j)

endif

enddo

c------------FORMAREA INITIALA A TERMENULUI LIBER WK----- ----- --------- ----- -----

do j=1,nny-2

if(j.eq.1)then

wk(i,j)=(la*(DY/2.0*gz)/DX)*TEMP(i-1,j)+

(la*(DY/2.0*gz)/DX)*TEMP(i+1,j)

wk1(j)=wk(i,j)

else

wk(i,j)=(la*(DY*gz)/DX)*TEMP(i-1,j)+(la*(DY*gz)/DX)*TEMP(i+1,j)

wk1(j)=wk(i,j)

endif

enddo

wk(i,nny-1)=(la*(DY*gz)/DX)*TEMP(i-1,nny-1)+

* (la*(DY*gz)/DX)*TEMP(i+1,nny-1)+(la*(DX*gz)/DY)*TEMP(i,nny)

wk1(nny-1)=wk(i,nny-1)

c-----------REZOLVAREA SISTEMULUI-------- ----- ------ ----- ----- -----

CALL TRIDAG(a1,b1,c1,wk1,temp1,nny-1)

c----- ----- ----FORMAREA SOLUTIEI PE TOATA LINIA VERTICALA----- ----- ------------

do j=1,nny-1

TEMP(i,j)=temp1(j)

enddo

endif

enddo

c===========================PENTRU i=nnx==========================

c------------DIAGONALA INFERIOARA-------- ----- ------ ----- ----- ----- ----- -------

do j=1,nny-1

if(j.eq.1)then

A(nnx,j)=0.0

A1(j)=A(nnx,j)

else

A(nnx,j)=-la*(DX/2.0*gz)/DY

A1(j)=A(nnx,j)

endif

enddo

c------------DIAGONALA SUPERIOARA-------- ----- ------ ----- ----- ----- ----- -------

do j=1,nny-2

if(j.eq.1)then

C(nnx,j)=-la*(DX/2.0*gz)/DY

C1(j)=C(nnx,j)

else

C(nnx,j)=-la*(DX/2.0*gz)/DY

C1(j)=C(nnx,j)

endif

enddo

C(nnx,nny-1)=0.0

C1(nny-1)=C(nnx,nny-1)

c------------DIAGONALA PRINCIPALA-------- ----- ------ ----- ----- --------- ----- ----

do j=1,nny-1

if(j.eq.1)then

B(nnx,j)=la*(DY/2.0*gz)/DX+la*(DX/2.0*gz)/DY

B1(j)=B(nnx,j)

else

B(nnx,j)=la*(DY*gz)/DX+la*(DX/2.0*gz)/DY+la*(DX/2.0*gz)/DY

B1(j)=B(nnx,j)

endif

enddo

c------------FORMAREA INITIALA A TERMENULUI LIBER WK----- ----- --------- ----- -----

do j=1,nny-2

if(j.eq.1)then

wk(nnx,j)=(la*(DY/2.0*gz)/DX)*TEMP(nnx-1,j)+DY/2.0*gz*q

wk1(j)=wk(nnx,j)

else

wk(nnx,j)=(la*(DY*gz)/DX)*TEMP(nnx-1,j)

wk1(j)=wk(nnx,j)

endif

enddo

wk(nnx,nny-1)=(la*(DY*gz)/DX)*TEMP(nnx-1,nny-1)+DY*gz*q+

(la*(DY/2.0*gz)/DY)*TEMP(nnx,nny)

wk1(nny-1)=wk(nnx,nny-1)

c-----------REZOLVAREA SISTEMULUI-------- ----- ------ ----- ----- ----- ----- -------

CALL TRIDAG(a1,b1,c1,wk1,temp1,nny-1)

c----- ----- ----FORMAREA SOLUTIEI PE TOATA LINIA VERTICALA----- ----- -------------

do j=1,nny-1

TEMP(i,j)=temp1(j)

enddo

enddo

c----FORMAREA VECTORULUI PENTRU PUNCTELE DE CALCUL PE X SI PE Y-----

AX(1,1)=0.0

AY(1,1)=0.0

do i=1,nnx

do j=2,nny

AY(i,j)=AY(i,j-1)+DY

enddo

enddo

do j=1,nny

do i=2,nnx

AX(i,j)=AX(i-1,j)+DX

enddo

enddo

c-----------SCRIEREA SOLUTIEI-------- ----- ------ -------- ----- ------ ----

open(20,file='apl5.prn')

do i=1,nnx

write(20,101)(TEMP(i,j),j=1,nny)

enddo

close(20)

101 format(26(1x,f6.2))

stop

end

subroutine tridag(a,b,c,r,u,n)

parameter (NMAX=200000)

integer j

double precision a(n),b(n),c(n),r(n),u(n)

double precision bet, gam(NMAX)

c----------a=diagonala inferioara-------- ----- ------ -------- ----- ------ -------

c----------b=diagonala principala-------- ----- ------ -------- ----- ------ -------

c----------c=diagonala superioara-------- ----- ------ -------- ----- ------ ------

c----------r=termenul din partea dreapta-------- ----- ------ ----- ----- --------- ----- -------

c----------u=termenul necunoscut-------- ----- ------ -------- ----- ------ ------

c----------n=dimensiunea sistemului-------- ----- ------ -------- ----- ------ ---

c----------NMAX=numarul maxim al ecuatiei-------- ----- ------ ----- ----- ----- ----- ----

if (b(1).eq.0) pause 'tridag:rewrite equations !'

bet=b(1)

u(1)=r(1)/bet

do j=2,n

gam(j)=c(j-1)/bet

bet=b(j)-a(j)*gam(j)

if(bet.eq.0) pause 'tridag failed'

u(j)=(r(j)-a(j)*u(j-1))/bet

enddo

do j=n-1,1,-1

u(j)=u(j)-gam(j+1)*u(j+1)

enddo

return

end



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