Approximate vs. Exact Solutions in Heat Transfer

In analytical heat transfer, there are a variety of techniques to determine both exact and approximate solutions to problems.  To one way of thinking, all numerical solutions are approximate, since the exact differential equations have been discretized to enable an approximate solution via a system of algebraic equations.

Both exact and approximate solutions can be useful.  Let’s look at an example, then talk about some useful features of both types of solutions.

We will consider the case of heating of a deep solid body.  This might be used to approximate the heating of the ground after a sudden temperature change at the surface, or to approximate the first few seconds of the quenching of a thick billet of metal.  If the solid is initially at a uniform temperature throughout, and a sudden change in the surface temperature is imposed, the differential equation and boundary conditions are:

T(x,0)=T_i;  T(0,t)=T_s;  T(x->∞, t)=T_i

The temperature distribution in the solid develops as shown in this figure: 

There is an exact solution for this particular differential equation and boundary conditions.  In this case, the solution is achieved through using a similarity variable defined as:

The exact solution then becomes:

where the error function, erf, is defined as:

It is also possible to solve this equation with an approximate integral method.  To use this method, the differential equation is integrated to become:

where δ is defined as the depth at which the temperature remains at T_i. At this point, the integro-differential equation (5) is still exact.  The approximate nature of this technique comes in the next step which is to assume a temperature profile that fits the boundary conditions:

It still remains to determine an expression for the function δ(t).  This is done by substituting the expression for temperature, (6), back into the integro-differential equation (5) and solving for δ:

Now, (6) in conjunction with (7) gives an approximate solution to the problem (1) while (3) along with (4) provides an exact solution to the same problem. 

This figure shows the exact solution plotted as a function of η with a blue line, and the approximate solution plotted with a red line.  (To make the vertical axes identical, the quantity 1-erf(η) is plotted for the exact solution, and to make (6) a function of η, note that x/δ=η/√3).

The two solutions are fairly close up to η≈1.7 considering that we only used a second order polynomial for the approximate solution.  (It is possible to get a slightly better approximate solution using a fourth order polynomial.)  This figure shows the difference between the two.  

In the next post we’ll talk about strengths and weaknesses of each approach.