How to Solve 1-D Transient Heat Transfer for Constant Temperature Surfaces?

[Link-]

I am a little confuse here. I am trying to solve the 1-D Transient Heat Transfer for a Constant Temp. Surfaces "wall".
The PDE is
$$\frac{\partial T}{\partial t}=\alpha \frac{\partial ^2 T}{\partial x^2}$$
T(to)=200*x
I am not so sure about how I'm suppose to get the Temperature profile

any help?
-link

 

[siddharth]

I don't understand what you've written, what do you mean by "T(to)=200*x"? Is that the temperature profile at time t=0?

To solve the PDE, have you attempted the separation of variables technique?

 

[bilgealp]

This is a very simple problem in the area of heat conduction.
There are several well written books in this area. However I have followed one of those best in the field. I will suggest
Özışık (Ozisik for non-UTF viewers), M. Necati, "Heat Conduction", John-Wiley, 1993

and inquirer meant the initial condition by "T(to)=200*x" here. Btw, boundary conditions were not supplied. Most probably solution is sought for an identical Dirichlet type boundary conditions.

I'm sure the reference will help.

 

[Link-]

I don't understand what you've written, what do you mean by "T(to)=200*x"? Is that the temperature profile at time t=0?

To solve the PDE, have you attempted the separation of variables technique?

Yes, T=T(x,t) T(x,0)=200*x for 0<=x<=.5L (symetric at 0.5L) T(0,t)=0 T(L,t)=0
No, not separation of variables..
The problem was that I was using the finite difference method to find the temperature profile in the wall at different times and I had two equations for the problem but I didn't knew which one was useful.

Here are the equations
$$T_m^{p+1}= Fo(T_{m+1}^p+T_{m-1}^p)+(1-2Fo)T_m^p$$
Subscript are for spatial nodes.
Superscript are for the time nodes.

$$T_m^{n+1}-T_m^n=\frac{Fo}{2}(T_{m-1}^{n+1}-2T_m^{n+1}+T_{m+1}^{n+1}+T_{m-1}^{n}-2T_m^{n}+T_{m+1}^{n})$$

I found that both equations are good for solving the pde using finite difference.

Thanks siddharth

PS This how should look, (axis are not labeled...:P)

 

[bilgealp]

OK. Both are different FD schemes of the problem.
The first one is explicit in time* so you shoul be aware of the CFL condition that ensures stability.

The other is the Crank-Nicholson type discretization which is stable but needs system solver. In this one you have to collect unknown values (i.e. the values for the new time step which are indexed as n+1 here) to one side and solve the resulting Ax=b type linear system. This is an implicit form(*)


* Explicit in time. Values for the new time step can be obtained by direct substitution of previously known values. Spatial derivatives are obtained from the n-th time step.

* Implicit form. Spatial derivatives involves unknown values of the n+1st time step.

 

[Link-]

OK. Both are different FD schemes of the problem.
The first one is explicit in time* so you shoul be aware of the CFL condition that ensures stability.

The other is the Crank-Nicholson type discretization which is stable but needs system solver. In this one you have to collect unknown values (i.e. the values for the new time step which are indexed as n+1 here) to one side and solve the resulting Ax=b type linear system. This is an implicit form(*)


* Explicit in time. Values for the new time step can be obtained by direct substitution of previously known values. Spatial derivatives are obtained from the n-th time step.

* Implicit form. Spatial derivatives involves unknown values of the n+1st time step.

Thanks a lot for the help and info.

 

[bilgealp]

My pleasure.