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looking for help w/ heat equation + surface convection

looking for help w/ heat equation + surface convection

looking for help w/ heat equation + surface convection


I have an application in which I have a semi-infinite solid subjected to surface convection at the x=0 boundary and a time varying fluid temperature. I am familiar with the heat equation (d2Tdx2 = 1/alpha dTdt) and I've been able to solve my problem by discretizing the spatial domain and time marching with an rk4 solver, but I am trying a method which does not require tracking as many state variables as I potentially need to solve for many different semi-infinite bodies at once.

I came across an approximate integral method (ref. Ozisik, Heat Conduction) in which an approximate spatial distribution is enforced over a thermal boundary distance (delta) and the heat equation is reduced to solving a differential equation describing delta.

My problem is actually integrating the equation. At time zero the surface temperature rate of change is infinite (also the thermal boundary layer has zero thickness), which I've confirmed by looking at available exact solutions for my problem which have a constant fluid temperature. This equates to a thermal boundary rate initially at infinity. At any time after t=0 there will be a positive boundary layer thickness and the equations won't blow up.

How do I handle time zero and getting the whole thing started? The text just leave it at "Equation (9-36) can be integrated numerically".

Thanks for any insight anyone can offer.

RE: looking for help w/ heat equation + surface convection

Start at time t=0.1, with a reasonable approximation for the first condition.

Heat transfer requires approximations and assumptions for everything else.

RE: looking for help w/ heat equation + surface convection

Thanks, I'm thinking I'd try approximating the time derivative at t=0 assuming a very small thermal bounday depth, 1 micrometer or so. Then compare solution against constant fluid temp exact solution to be sure it's reasonable.

RE: looking for help w/ heat equation + surface convection

Real world, the IR images of the film thickness are are in the textbooks are not "scaled" but appear much thickner than that: Try 1/2 to 1 millimeter (20 - 40 thousandth inch) as an approximation.

Your time interval for heat transfer in a real-world large application (unless it is an explosion or internal combustion engine fuel combustion) is on the order of minutes and tens of minutes not milliseconds. So I'd try to stay away from the very, very tiny increments that will distort your starting conditions.

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