Quenching in a Finite Bath

In an earlier post we looked at the transient 1 dimensional temperature profiles and

temperature gradients across a piece of metal being quenched in a tank large enough that the temperature of the quench fluid never changed.  In this post, we’ll look at a different case: a piece of steel small enough that the temperature is uniform across the piece which is being quenched in a bath small enough that the temperature of the quench fluid goes up as the steel cools down.  We’ll assume that the quench fluid is well-stirred so that it can also be characterized by single temperature.
Of course, the first law of thermodynamics requires that energy is conserved in this process, so it is straightforward to apply the first law to the bath and to the steel to obtain two coupled, first order differential equations.  The solution to these equations can be written generally with the following definitions:
T=temperature
t=time
m*c=mass times heat capacity
h, A=convective heat transfer coefficient and surface area, respectively, at the interface between the steel and the quench fluid
subscript "B" refers to the “bath” or quench fluid
subscript "S" refers to the steel
subscript "0" refers to an initial condition

The solutions for the steel temperature and the bath temperature are:

and both temperatures asymptote toward:

This figure shows the dimensionless transient temperatures for two values of ε, along with early and late asymptotes.

As expected, after a long time the bath temperature and the steel temperature approach one another.  In the limit of a very large bath (ε=0) the solution for the steel temperature becomes identical to lumped capacitance cooling with constant fluid temperature.

Sometimes, it might be desirable to know how long it would take for the steel to reach a particular temperature.  
This figure shows the inverted solution where the dimensionless time is plotted as a function of the dimensionless steel temperature for various values of ε.

[1] Heat Conduction, Grigull and Sandner, Hemisphere, 1984.