Approximate vs. Exact Solutions in Heat Transfer (part 2)

In the last post, we showed an approximate integral solution and an exact analytical solution to the same heat transfer problem: an isothermal semi-infinite solid subjected to a step change in temperature at the surface.

Exact solutions are very useful to have because they provide the true solution to the given problem.  Unfortunately, out of all possible problems, exact solutions tend to be relatively rare and also tend be limited to fairly simple situations, like this one.

So, one advantage of approximate solutions is that they can be applied to a much wider variety of problems—they are much more flexible.  In fact, numerical solutions can be determined for almost any geometry and boundary conditions describable for conduction problems.  Returning to this example, it is relatively straightforward to extend the integral solution to solve for the case of heat generation in the semi-infinite solid, but there is no corresponding exact solution for that case.

So why do we care about exact solutions at all? 

For one thing, it is always useful to validate any approximate technique (including numerical solutions) by performing it on a problem where an exact solution is known.  For example, in the case that we are considering, a simple sign error in the approximate solution produces this result.  Such an extreme difference in size and shape would immediately make us suspect that there is something wrong with the solution.

Also, it is very useful to use an exact solution to get a sense of “how approximate” the approximate solution might be.  After looking at the comparison plot in the last post, we would have some sense of the size of the error that we might expect if we were to use a second order polynomial in an approximate solution for the case that included heat generation.  Similarly, some approximate solutions, including numerical solutions, can get “arbitrarily close” to the exact answer.  But if you don’t know the answer beforehand, how can you know when you are “close enough”?  Comparison of an approximate solution with an exact solution to a problem that is similar to the case in question is one way to get a sense of that.

Finally, approximate solutions are very sensitive to the assumptions that went into their creation.  Notice in the example in the last post, that the approximate solution veered sharply upward in an unrealistic way around η=1.7.  This is because the approximate solution was only defined between 0 < x < δ.  For η>√3,  x > δ, and the approximate solution (6) is no longer applicable.  Of course, in the approximate solution, when x > δ, T=T_i so it is trivial to find the answer and to use the solution.  However, the point is that blindly using equation (6) without being aware of the assumptions that went into the formulation can lead to erroneous results.  This is something to be particularly careful about in using numerical solution packages in which the assumptions may have been buried deep inside the program out of sight and out of awareness of the end user.