More Lumped Capacitance

Last post we looked at another extension of lumped capacitance beyond our original exploration here and here.  In the last post, we looked at a case where the heat transfer coefficient increased linearly with time, which might be a good approximation for the initial moments of the heat transfer for an object in a duct with the fan just starting up.  Today we’ll look at a case where the heat transfer coefficient is decreasing.

We’ll consider the case where the heat transfer coefficient starts at some initial value, call it h₀, and decreases in a "1/x" way:

With this equation, when t=0, h=h₀, and as t gets very large, h will approach zero moderated by the constant parameter "a".  This variation might be a rough approximation to a case where a body was quenched in a still pool, or a case where there was rapid fouling that interfered with the heat transfer coefficient.  Following the procedure from here, we arrive at this expression for the temperature as a function of time:

This figure demonstrates h(t) for the constant case, along with two cases where h₀ is equal to the constant h, and two cases where h₀ is larger.  Each value of h₀ has two values of the parameter “a”.

This figure shows the temperature profiles corresponding to the various heat transfer coefficient profiles.  For the two cases where h₀ was the same as the constant case (orange and green lines) the temperature never catches up to the temperature of the constant case.  For the other two cases, the temperature profiles originally drop faster than the constant case, but eventually cross over and fall behind.