Re: Heat transfer -- possible in mathematica?
- To: mathgroup at smc.vnet.net
- Subject: [mg79291] Re: Heat transfer -- possible in mathematica?
- From: Miguel <misvrne at gmail.com>
- Date: Tue, 24 Jul 2007 05:55:48 -0400 (EDT)
- References: <f7v4n2$t31$1@smc.vnet.net>
On 22 jul, 10:32, meaton01 <mike.ea... at gmail.com> wrote:
> Greetings,
>
> I'm attempting to solve a 2-dimensional, transient heat transfer calculation (rectangular slab) with uniform generation. Is this (relatively easily?) possible in mathematica, or should I simply resort to trying to program a finite element solution to the problem? Additionally, if anyone knows of a notebook already constructed, I'd love to see it.
>
> Thanks!
> Mike
For a paralepiped (a,b,c) with heat generation uniform and boundary
condition isothermic, the solution is
D[T[x,y,z,t],{x,2}]+D[T[x,y,z,t],{y,2}]+D[T[x,y,z,t],{z,2}]=(1/
alfa)D[t[x,y,z,t],{t}]-E/k
where
alfa= Difussivity
E=Heat generation in kcal/m^3 or W/m^3
k=Conductivity
Boundary conditions and initial value:
t=0; T=0; (0<=x<=a);(0<=y<=b);(0<=z<=c)
t>0; x=0,T=0; y=0,T=0; z=0,T=0
x=a, T=0; y=b, T=0; z=c, T=0
t>0; E=Cte.
The distribution of temperatures is:
T(x,y,z,t)=(8*E)/(Pi^3*k)Sum[Sum[Sum[(Sin[Nn*x]*Sin[Mm*y]*Sin[Pp*z])/
(n*m*p*s^2)(1-e^(-s^2*alfa*t)),{p,1,inf}],{m,1,inf}],{n,1,inf}]
where
n=m=p=1,3,5,.... Nn=(Pi*n)/a Mm=(Pi*m)/b Pp=(Pi*p)/
c s^2=Nn^2+Mm^2+Pp^2
When the boundary condition is convection, the solution is more
complicated.