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.