Heat equation given constant surface heat flux

[user4417]

The finite difference equation for the flux that you wrote in post #29 is a one-sided difference approximation, and is only first order accurate in Δx. The central difference equation that I had written in an earlier post is a central difference approximation, and is second order accurate in Δx (i.e., much more accurate). For the central difference approximation, the elements are (1+2λ) and -2λ.

For the record, the correct difference equation at x = 0 for the central difference approximation to the heat flux should be:

$$(1+2λ)T_0^n-2λT_1^n=T_0^{n-1}+2λΔx\frac{Q}{k}$$

The numerical solution you presented in your previous post at the end of the first time step looks to me like the results of applying forward euler in time, not backward euler. I know this because all the temperatures should have changed significantly with backward euler, not just the temperature at the boundary.

Chet

Thanks a lot for walking through this whole thing with me, I applaud your patience and keenness in helping people out with physics problems. I figured out what my problem was. I was using alpha as Cp * ro / k, when it is k / Cp / ro. I used backward Euler for both Neuman boundaries (and backward Euler for the equation itself) and results were very close, though not in line with experiemntal data, namely temp only rises to 175C degrees at the surface after 10-20s of constant rubbing, while it should be over 326.8C. The experiement conditions were weird though. Second boundary actually had a heat sink (metal case of valve), and the rubbing was periodic and not constant (by-hand closing).