Lecture 14 February 4, 2004 2D Numerical Steady Conduction Recall the heat diffusion equation.

Or with constant conductivity and no generation,

Finite difference methods simply replace derivatives with numerical approximations, without any regard for the physics. This is dangerous, because if you do not have enough resolution, then your solution will not conserve energy and will be useless. Finite volume on the other hand takes a step back to the step before we integrated the energy equation to derive the heat diffusion equation, and strictly enforces conservation on each and every control volume. It forces conservation even with very coarse resolution. The finite volume method is thus strongly preferred in heat transfer and fluid mechanics. Consider our single control volume from a finite volume analysis, and integrate the fluxes over each face. In order

to do that, we need an approximation of the temperature gradient at each face. Start by using a first order approximation at each face,

We also need the face areas in order to calculate heat rates, and we can express these in terms of dxp, dxm, dym,dyp

Now we can calculate the heat rates through each face, and integrate them (i.e. apply conservation of energy). If the conductivity was variable, either because the material changed, or it was a function of temperature, we could

simply evaluate k locally at each face, and use that to determine the flux there. The heat rates at each face are then,

Anyhow, when we assemble the terms using conservation of energy, we find,

If there was volumetric generation, the above would not be equal to zero, but to –qdot (Volume) All we have left to do is to apply the boundary conditions, and this is done exactly as above except one the the faces will have a different heat flux through it (convection, or specified flux etc.)