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
        return 0.
    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?
  • $\begingroup$ 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? $\endgroup$ – Paul Apr 25 '15 at 2:53
  • $\begingroup$ 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 $\endgroup$ – cvut Apr 25 '15 at 13:31
  • $\begingroup$ 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. $\endgroup$ – Paul Apr 25 '15 at 15:05
  • $\begingroup$ 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? $\endgroup$ – Paul Apr 26 '15 at 3:01
  • 1
    $\begingroup$ 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. $\endgroup$ – regdoug Apr 26 '15 at 3:39

The coefficient that you need to use depends on what kind of heat source you're using.

What units is that 280 in? Watts? Celsius?

If they are in watts then you need a coefficient relating joules (watts*dt) to temperature change.

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

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