Heat Diffusion Equation

Friday January 16, 2004 Last day, we derived the heat diffusion equation in Cartesian coordinates,

Which is simply a specific statement of Conservation of Energy considering, heat conduction, volumetric heat generation, and sensible energy storage. So far the basic assumption is that Fourier’s law is valid, which we will not question in this course. Note that the above equation is written in conservative form, which basically means that each term is identifiable as a complete term in our statement of Conservation of Energy. Now, we will make additional assumptions, which will result in nonconservative forms that may come in handy. If k is constant, then we can take it out from the gradient term, resulting in

And we can introduce a new property,

which is called the thermal diffusivity. Thermal diffusivity can be thought of as a ratio of energy transport to energy storage.

Or, in convenient vector form,

When α is large, the material is, relatively speaking, a good conductor of thermal energy, and hence the importance of the unsteady term is diminished. You can expect then that a material with a large thermal diffusivity will reach a steady state more quickly than one

having a lower thermal diffusivity. Given enough time, a system should reach a steady state, at which time the time derivative will be zero.

Note that at steady state volumetric generation is still perfectly possible, and while the above equations are not in conservation form we can still clearly interpret the terms having derived the heat diffusion equation ourselves. We can clearly interpret from the above, that if the sum of the fluxes through our control faces is not zero, then there must be generation (or conversion) inside the volume. Otherwise conservation of energy is not satisfied. If there is no generation, then we have,

It should be clear by this point that if we know the temperature distribution, we know everything about our heat diffusion problem. Given a temperature distribution we can calculate energy fluxes anywhere along our distribution using Fourier’s Law, and by applying Conservation of Energy we should be able to interpret anything of interest in the problem domain. Let us explore some temperature distributions in this optic. MATLAB is a perfect tool to do this. Let’s begin with a one dimensional temperature distribution defined over a range (0 < x < 1). We will plot both discrete points on this range, and the midpoints between these points.

Let’s define the temperature distribution as a polynomial for this example,

where a = -50 K/m2, b = -300 K/m and c = 900 K, and plot the results on our domain

The temperature gradient, for our one dimensional case can be approximated from our temperature distribution,

This gradient calculated this way is clearly valid somewhere between the two points from which it was calculated, but we are free to choose where to represent it. It is intuitively pleasing to locate it at our midpoints as it does represent an average gradient between those two points. We can easily multiply our gradient by –k A to determine the heat rate (Fourier’s law). Let’s take k = 40 W/mK, and our A to be 1 cm x 1cm.

But now, the heat rate is not defined at the end points because of our choice to represent the gradients at the midpoints of our discrete representation. Unfortunately, we are usually interested in the heat rate at the end points as when we want to know the total power dissipated by a fin (it all had to enter at the base of the fin). One relatively easy thing to do here is to extrapolate our function to the point of interest. Before we do this, note that if we had stuck with our first representation of the gradient, then the heat rate now shown at x=0.05 m (q = 1.22 W) would have been located at x=0 m. Clearly there is some room for interpretation here. If we extrapolate the above curve, then we find that q(0) = 1.20 W. In this case, since we have an analytic expression for the temperature distribution, and can easily differentiate it to obtain the exact value at x=0.

It was obviously better in this case to interpret the gradient at the midpoints.

It should also be pointed out at this point that if this is a real 1D temperature distribution in some material, then there is clearly something interesting going on inside the material, for somehow there is increasingly more power as we move along the material. If energy is to be conserved there must be energy generation occurring inside the material. We can easily calculate this.