Lumped Capacitance Revisited

Lumped capacitance is a very useful heat transfer approximation that we mentioned originally here,  and explored in more depth with extensions here and here.  Today, we’ll look at another extension of the basic lumped capacitance approach.
As a reminder, in lumped capacitance, we have assumed that the internal thermal conductivity of the body is high enough relative to the external heat transfer coefficient that the entire body can be characterized by a single temperature.  In the traditional (elementary) lumped capacitance analysis, the body temperature is calculated as a function of time assuming constant values for the convective heat transfer coefficient, the free stream temperature, the geometry, and all properties.  Normally, we would expect the geometry and properties to remain relatively constant, but it would be quite reasonable to expect variation in the free stream temperature or the convective heat transfer coefficient. 

Earlier, we looked at a couple of examples where the free stream temperature was a function of time. 

Today, we’ll look at a case where the free stream temperature is constant, but the heat transfer coefficient varies with time.

Specifically, let’s look at a case where the heat transfer coefficient increases linearly with time from a value of zero:

This might approximate the first few moments of a situation where a body is suddenly immersed in a stream which increases in velocity, or a body in duct when a fan first starts up, or a body falling under the acceleration of gravity.

Using the same approach described here, except with h(t)=a*t, we can derive this equation for the temperature as a function of time:

This figure shows the heat transfer coefficient, h(t), for various values of the parameter “a” along with the heat transfer coefficient for the constant case.

The corresponding temperature profiles are illustrated in this figure.  Since the heat transfer coefficient starts out at zero, the temperature change with time is originally flat.  As the heat transfer speeds up, so does the change in temperature.  Eventually, for the steepest increase in heat transfer coefficient, the temperature catches up and passes the constant heat transfer coefficient case.