More Useful Lumps (Part 2)

In our last post, we examined the case of lumped capacitance where the free-stream temperature was a function of time and presented the equation governing that situation. This time, we’ll look at some specific examples.
Suppose that the free-stream temperature were a linear function of temperature:

Now, equation [2] from the last post becomes: 

Which looks like this: 

It is clear that wherever the temperature of the body starts (T_i) it soon approaches a line parallel to, and lagging, the free stream temperature. 

As a second example, suppose that the free-stream temperature oscillated sinusoidally which would be a good approximation to the temperature in a thermostatically controlled conditioned space. Now, the free-stream temperature is: 

Inserting [7] into equation [2] from the last post and carrying out the integration yields: 

where N is defined as before, and

Equation [8] looks like this: 

Starting from the initial temperature, T_i, the temperature quickly approaches a sinusoidal fluctuation at the same phase, but delayed and attenuated from the free-stream fluctuation. Smaller values of the ratio “R” increase the delay and attenuation.