Scrigroup - Documente si articole

Username / Parola inexistente      

Home Documente Upload Resurse Alte limbi doc  

CATEGORII DOCUMENTE





ArhitecturaAutoCasa gradinaConstructiiInstalatiiPomiculturaSilvicultura


DISTRIBUTIA TEMPERATURII INTR-O PLACA METALICA DREPTUNGHIULARA CU SURSA DE CALDURA

Instalatii

+ Font mai mare | - Font mai mic







DOCUMENTE SIMILARE

Trimite pe Messenger
CONDUCTIA TERMICA-2D STATIONAR - Metoda volumelor finite
DISTRIBUTIA TEMPERATURII INTR-O PLACA METALICA DREPTUNGHIULARA CU SURSA DE CALDURA

DISTRIBUTIA TEMPERATURII INTR-O PLACA METALICA DREPTUNGHIULARA CU SURSA DE CALDURA

Se considera oplaca metalica dreptungiulara de dimensiuni 0,05 x 0,04[m].Conductivitatea termica a placii este λ= 4[W/m/K]. Toate frontierele placii sunt mentinute la temperatura constanta de 0° C si termenul sursa este S=40∙106[W/m3].



Sa se calculeze distributia stationara de temperatura in placa folosind reteaua de dicretizare

din figura de mai jos(Δx=Δy=0,01 m).

Rezolvare :

Ecuatiaconductiei termice stationare 2D pentru conditiile enuntate este :

Ecuatiadiscretizata pentru un nod interior (de exemplu :nodul 16) este urmatoarea :

apTp= awTw+ aETE+ aSTS+ aNTN+ b

sau:

;; ; ;ap=aW+aE+ aS+ aN ;b=

Termenul sursa ,S fiind constant,nu este necesar sa fie liniarizat.Valorile coeficientilor sunt:

aw = aE = aS = aN =4;ap=16 ;b=4000

Rezulta sistemul de ecuatii:

Solutia sistemului este:


PROGRAMUL IN FORTRAN:

parameter (nnx=21,nny=26,nit=1500)

double precision temp(nnx,nny),ax(nnx,nny),ay(nnx,nny)

double precision a(nnx,nny),b(nnx,nny),c(nnx,nny),wk(nnx,nny)

double precision a1(nny),b1(nny),c1(nny),wk1(nny),temp1(nny)

double precision dif(nnx,nny),temp2(nnx,nny)

double precision dx,dy,tb,l,h,la,gz,s,err

data tb/0.0/,l/0.04/,h/0.05/,la/4.0/,gz/1.0/,err/0.001/

data s/40000000.0/

c -----------------calculul pasului dx si dy------------------------------------------------

dx=l/(nnx-1)

dy=h/(nny-1)

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

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

c-----------------initializarea temperaturii--------------------------------------------

do i=1,nnx

do j=1,nny

temp(i,j)=0.01

enddo

enddo

c-----------introducerea conditiilor de tip Dirichlet----------------------------------

c-----------------frontiera NORD-----------------------------------------------------

do i=1,nnx

temp(i,nny)=tb

enddo

c-------- frontiera SUD-----------------------------------------------------------------

do i=1,nnx

temp(i,1)=tb

enddo

c--------- frontiera WEST------------------------------------------------------- -

do j=1,nny

temp(1,j)=tb

enddo

c--------- frontiera EST------------------------------------------------------- ---------

do j=1,nny

temp(nnx,j)=tb

enddo

c==================bucla de iteratie==============================

do k=1,nit

c------------formarea diagonalelor vectorilor --------------------------------------------------

do i=2,nnx-2

if(i.eq.2)then

c-------------diagonala inferioara ‘ a’------------------------------------------------------

do j=2,nny-1

if(j.eq.2)then

a(i,j)=0.0

a1(j-1)=a(i,j)

else

a(i,j)=-la*(dx*gz)/dy

a1(j-1)=A(i,j)

endif

enddo

c-----------diagonala superioara ‘c’ -------------------------------------------------------

do j=2,nny-2

if(j.eq.2)then

c(i,j)=-la*(dx*gz)/dy

c1(j-1)=c(i,j)

else

c(i,j)=-la*(dx*gz)/dy

c1(j-1)=c(i,j)

endif

enddo

c(i,nny-1)=0.0

c1(nny-2)=c(i,nny-1)

c-------------diagonala principala ‘b’--------------------------------------------------------

do j=2,nny-1

if(j.eq.2)then

b(i,j)=2.0*la*(dy*gz)/dx+2.0*la*(dx*gz)/dy

b1(j-1)=b(i,j)

else

b(i,j)=2.0*la*(dy*gz)/dx+2.0*la*(dx*gz)/dy

b1(j-1)=b(i,j)

endif

enddo

c---------formarea initiala a vectorului termenului liber (wk)----

do j=2,nny-2

if (j .eq. 2) then

wk(i,j) = (la*(dx*gz)/dy)*temp(i,j-1) +

* (la*(dy*gz)/dx)*temp(i-1,j) +

*(la*(dy*gz)/dx)*temp(i+1,j) + s*dx*dy*1.0

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

else

wk(i,j) = (la*(dy*gz)/dx)*temp(i-1,j) +

* (la*(dy*gz)/dx)*temp(i+1,j) + s*dx*dy*1.0

wk1(j-1) = 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) +s*dx*dy*1.0+

*(la*(dx*gz)/dy)*temp(i,nny)

wk1(nny-2) = wk(i,nny-1)

c------------- solutia sistemului ------------------------------------------------------------

CALL TRIDAG (a1,b1,c1,wk1,temp1,nny-2)

c-------formarea solutiei finale pentru toate liniile verticale ----------------------

do j = 2,nny-1

temp(i,j)=temp1(j-1)

enddo

else

c---------diagonala inferioara ‘a’ ----------------------------------------------------

do j = 2,nny-1

if (j .eq. 2) then

a(i,j) = 0.0



a1(j-1) = a(i,j)

else

a(i,j) = - la*(dx*gz)/dy

a1(j-1) = a(i,j)

endif

enddo

c---------diagonala superioara ‘c’---------------------------------------------------------

do j = 2,nny-2

if (j .eq. 2) then

c(i,j) = - la*(dx*gz)/dy

c1(j-1) = c(i,j)

else

c(i,j) = - la*(dx*gz)/dy

c1(j-1) = c(i,j)

endif

enddo

c(i,nny-1) = 0.0

C1(nny-2) = C(i,nny-1)

c----------diagonala principala ‘b’ ------------------------------------

do j = 2,nny-1

if (j.eq.2)then

b(i,j) = 2.0*la*(dy*gz)/dx+2.0*la*(dx*gz)/dy

b1(j-1) = b(i,j)

else

b(i,j) = 2.0*la*(dy*gz)/dx + 2.0*la*(dx*gz)/dy

b1(j-1) = b(i,j)

endif

enddo

c-----formarea initiala a vectorului termenului liber (wk)-----------------

do j = 2,nny-2

if (j .eq. 2) then

wk(i,j) = (la*(dx*gz)/dy)*temp(i,j-1) + (la*(dy*gz)/dx)*temp(i-1,j) +

* (la*(dy*gz)/dx)*temp(i+1,j) + s*dx*dy*1.0

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

else

wk(i,j) = (la*(dy*gz)/dx)*temp(i-1,j) +

* (la*(dy*gz)/dx)*temp(i+1,j) + s*dx*dy*1.0

wk1(j-1) = 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) +s*dx*dy*1.0+

* (la*(dx*gz)/dy)*temp(i,nny)

wk1(nny-2) = wk(i,nny-1)

c------------- solutia sistemului ------------------------------------

CALL TRIDAG (a1, b1, c1, wk1, temp1, nny-2)

c-------formarea solutiei pe toate liniile verticale--------------------------------

do j = 2,nny-1

temp(i,j)=temp1(j-1)

enddo

endif

enddo

c==================pentru ‘ i = nnx-1’ =========================

c---------diagonala inferioara’a’ -------------------------------------

do j = 2,nny-1

if (j .eq. 2) then

a(nnx-1,j) = 0.0

a1(j-1) = a(nnx-1,j)

else

a(nnx-1,j) = - la*(dx*gz)/dy

a1(j-1) = a(nnx-1,j)

endif

enddo

c---------diagonala superioara ‘c’ -------------------------------------

do j=2,nny-2

if (j .eq. 2) then

c(nnx-1,j) = - la*(dx*gz)/dy

c1(j-1) = c(nnx-1,j)

else

c(nnx-1,j) = - la*(dx*gz)/dy

c1(j-1) = c(nnx-1,j)

endif

enddo

c(nnx-1,nny-1) = 0.0

c1(nny-2) = c(nnx-1,nny-1)

c----------diagonala principala ‘b’ -----------------------------------------------

do j = 2,nny-1

if (j.eq.2)then

b(nnx-1,j) = 2.0*la*(dy*gz)/dx + 2.0*la*(dx*gz)/dy

b1(j-1) = b(nnx-1,j)

else

b(nnx-1,j) = 2.0*la*(dy*gz)/dx + 2.0*la*(dx*gz)/dy

b1(j-1) = b(nnx-1,j)

endif

enddo

c-----formarea initiala a vectorului termenului liber (wk)-----------------

do j = 2,nny-2

if (j .eq. 2) then

wk(nnx-1,j) = (la*(dx*gz)/dy)*temp(nnx-1,j-1) +

* (la*(dy*gz)/dx)*temp(nnx-2,j) +

* (la*(dy*gz)/dx)*temp(nnx,j) + s*dx*dy*1.0

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

else

wk(nnx-1,j) = (la*(dy*gz)/dx)*temp(nnx-2,j) +

* (la*(dy*gz)/dx)*temp(nnx,j) + s*dx*dy*1.0

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

endif

enddo

wk(nnx-1,nny-1) = (la*(dy*gz)/dx)*temp(nnx-2,nny-1) +

*(la*(dy*gz)/dx)*temp(nnx,nny-1) + s*dx*dy*1.0+

*(la*(dx*gz)/dy)*temp(nnx-1,nny)

wk1(nny-2) = wk(nnx-1,nny-1)

c------------- solutia sistemului ------------------------------------

CALL TRIDAG (a1, b1, c1, wk1, temp1, nny-2)

c-------formarea solutiei pe toate liniile verticale--------

do j = 2,nny-1

temp(i,j)=temp1(nnx-1)

enddo

c--------verificarea cu criteriul convergentei ----------------------------

do i=2,nnx-1

do j = 2,nny-1

dif(i,j) = abs(100.0*(temp(i,j) - temp2(i,j))/temp(i,j))

enddo

enddo

difmax = dif(2,2)

do i=2,nnx-1

do j=2,nny-1

if(dif(i,j) .gt. difmax) then

difmax = dif(i,j)

else

endif

enddo

enddo

write(*,*)'iter, difmax, err', k, difmax, err

write(*,*)'iter, difmax, err', difmax, err

if (difmax .le. err) then

write(*,*)'sortie par critere de convergeance, k=',k

goto 100

else

endif

c =========actualizarea temperaturii de la pasul precedent==================

do i=2,nnx-1

do j=2,nny-1

temp2(i,j) = temp(i,j)

enddo

enddo

c-------bucla fina de iteratie ‘nit’ ----------------------------------------------------------------

enddo

100continue

c------------------ scrierea solutiei ------------------------------------------------------------

OPEN(20,file='2Ds2.prn')

do i=1,nnx

write(20,101)(temp(i,j),j=1,nny,2)

enddo

CLOSE(20)

101format(26(1x,F7.2))

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)

doubleprecision bet,gam (nmax)

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

Solutia in Mathcad:


Erorile :

x=0

y[mm]

Solutie analitica

T[K]

Solutie numerica:

T[K]

Eroare

[%]

1

1430

1428.2

0.12

3

1416

1414.2

0.12

5

1388

1385.9

0.14

7

1345

1342.7

0.17

9

1286

1283.4

0.19

11

1209

1206.9

0.16

13

1113

1111.4

0.14

15

996.55

994.9

0.16

17

856.3

854.9

0.16

19

689.7

688.6

0.16

21

493.7

493

0.16

23

256

264.6

0.15

Distributia temperaturii in QuickField:








Politica de confidentialitate

DISTRIBUIE DOCUMENTUL

Comentarii


Vizualizari: 661
Importanta: rank

Comenteaza documentul:

Te rugam sa te autentifici sau sa iti faci cont pentru a putea comenta

Creaza cont nou

Termeni si conditii de utilizare | Contact
© SCRIGROUP 2019 . All rights reserved

Distribuie URL

Adauga cod HTML in site