Finite Quench Revisited

We looked earlier at the solution for quenching an object in a finite bath—that is, where the bath is small enough (relative to the object being quenched) that the bath temperature rises while the object’s temperature goes down.  As you’d expect, eventually the object and the bath arrive at the same equilibrium temperature.  Today, we’ll look at getting the quenched object to two specific temperatures at two different times.
We’ll assume that the initial temperature of the steel, along with all of the material properties of the steel and bath are fixed.   We’ll also assume that we want to get a specified temperature at equilibrium, and also one other specified temperature at some specified time.  The parameters that we can vary are bath volume, and bath initial temperature.
Based on the very simple expression from the earlier post for the eventual equilibrium temperature:

we can get the initial bath temperature, T_B0, that will result in a final temperature of the bath (and object) (call it T_f) as a function of ε:

Now, we also have an equation for the steel temperature which is a function of time, ε, and the initial bath temperature.  So, if we put the specified time and specified steel temperature into that equation, and substitute the equation above in for the initial bath temperature, we have a single equation in ε which can be solved by any convenient root-finding method.
This figure shows the variation in required oil volume, and required oil initial temperature as a function of the specified 20 second temperature for the case of a 1 kg steel billet quenched in oil with an initial temperature of 750 deg C and a final temperature of 450 deg C. The dotted line corresponds to the case illustrated in the next plot.  

This plot shows the temperature variation as a function of time for both the steel billet and the oil bath for the single case indicated by the dotted line in the previous figure.  The dotted line in this figure demonstrates that the temperature of the steel billet is 600 deg C at 20 seconds, as designed, and the right side of the figure shows that the long-term temperature converges to 450 deg C, as designed.