In , the authors introduced a novel control law for the single-phase Stefan problem, a nonlinear partial differential equation, with the eventual goal of applying the result to control of cooling in continuous steel casting. In this paper, the previously published method of controlling the Stefan problem is extended in two key ways that improve the fidelity of the nonlinear PDE as a model of the temperature and solidification of continuous steel casters. First, a non-symmetric temperature distribution is allowed, in which separate Neumann boundary control is applied at either surface. Two convergent control laws are compared in simulation, with one converging significantly faster. Second, saturation of the Neumann boundary control input is considered. Saturation is a significant concern in the actual process. The enthalpy-based algorithm still allows for convergence under saturation. However, as one would expect, severity of the constraints can affect the convergence rate.