More Useful Lumps

All beginning heat transfer classes cover the topic of lumped capacitance calculations: the case where heat conduction within a body is fast enough that the entire body can be considered to be at a single temperature. 

This is a very useful tool for estimating heat transfer in some situations.   However, with just a little work, we can extend the tool to a broader application.

First, let’s quickly review of the idea of lumped capacitance.  The physical situation is that a body with relatively high thermal conductivity (say, a piece of metal to be quenched) is initially at a uniform temperature.  It is suddenly exposed to a fluid at a constant different temperature (say, the quenching fluid) and it is desirable to know the temperature of the body as a function of time.  The solution is:

Now, it is quite reasonable to assume that the density, surface area, volume, and heat capacity of a solid would remain relatively steady with time.  However, it is very possible that the convective heat transfer coefficient might change, and even more likely that the free-stream temperature would change with time.  Let’s consider the case where the convective heat transfer coefficient stays constant, and the free stream temperature changes as a function of time: T_∞(t).  Starting with an energy balance on the body, the following expression for T(t) can be derived:

where

The relationship with the constant free-stream temperature case is easy to see if that expression [1] is re-arranged to look like this:

If T_∞(t) were set equal to a constant in [2] and taken outside the integral, the integration would promptly yield [4].

Next post we’ll show some specific examples of this solution.