# What coefficients should I use when modeling non-stationary heating of a steel plate?

I am trying to make a model of non-stationary heating on a plate, where I must use an explicit numerical method to solve the model. I decided the plate will be made of steel (0,5% C) and I found that the thermal diffusivity coefficients $k/\rho c_p$ are:

$$\alpha_{steel} = 14.74 \times 10^{-6} \:\mathrm{m^2/s}$$ $$\alpha_{air} = 1.9 \times 10^{-5} \:\mathrm{m^2/s}$$

I also have to implement a heat source but I don't know what coefficient I should use for that.

As it's a 2D problem, my solving code looks like this:

def heat_source(x, y, matrica):
if (x >= 0.8 and x <= 1.2) and (y >= 0.8 and y <= 1.2):
if matrica[10,10]<300.:
return 280 + randn(1)*20
else:
return 0.
else:
return 0.0

dmx = (M0[iY,iX+1] + M0[iY,iX-1] - 2.0 * M0[iY,iX])/dx**2.0 # conduction in x-direction
dmy = (M0[iY+1,iX] + M0[iY-1,iX] - 2.0 * M0[iY,iX])/dy**2.0 # conduction in y-direction
M_new[iY,iX] = M0[iY, iX] + k_diffusion*dt*(dmx+dmy) + dt*k_heating*heat_source(X[iX],     Y[iY], M0)  - dt*k_cooling*(M0[iY,iX]-T_ok)


My questions are:

• Am I using the right coefficients? If yes, what coefficient should I use for the heat source?
• What's the difference between non-stationary and stationary source?
• A non stationary source is simply non-constant in time (and possibly in space as well). How you model it is very problem specific. Could you describe your physical problem in more detail? – Paul Apr 25 '15 at 2:53
• I will just model it as 90% constant, and 10% random. There is a flat plate and part of it is being heated. Other(non-heated part) should receive heat as a diffusive part. The thing I'm worried about is that coefficient with the source is 1(K/s) and with non heated part is smth*10^-6, so there is a big gap. It should be realistic model, and when I set the time of heating to 10 minutes, the non-heating part was still too cold in my opinion. Here is a brief intro into numerical-explicit way of solving the heat equation. ewp.rpi.edu/hartford/~wallj2/CHT/Notes/ch06.pdf – cvut Apr 25 '15 at 13:31
• Explain in more detail what you mean by 90% constant and 10% random. Write your source term explicitly as a function of time and space. – Paul Apr 25 '15 at 15:05
• Why exactly are you using a random source term? The way you coded it, the values change at every timestep. Why are you doing this? – Paul Apr 26 '15 at 3:01
• cvut, a non-stationary source doesn't have to have random fluctuations. A simple non-stationary source would be $q(t) = q_0 H(t)$, where $q_0$ is the heat produced by the source and $H(t)$ is the Heaviside step function. Since the value of $q(t)$ changes at $t=0$, it is non-stationary. – regdoug Apr 26 '15 at 3:39

This would look something like: $$k_{heating}=\frac1{dx \,dy \, \rho \, c_p \, t}$$ Where $t$ is the thickness of your metal