In this paper, the problem of estimation is considered for a class of processes involving solidifying materials. These processes have natural nonlinear infinite-dimensional representations, and measurements are only available at particular points in the caster, each corresponding to a single discrete-in-time boundary measurement in the Stefan problem partial differential equation (PDE) mathematical model. The results for two previous estimators are summarized. The first estimator is based on the Stefan problem, using continuous instead of discrete-in-time boundary measurements. The second estimator employs a process model that is more detailed than the Stefan problem, but with no output injection to reduce estimation error, other than model calibration. Both of these estimation frameworks are extended in the current paper to a more realistic sensing setting. First, an estimator is considered that uses the Stefan problem under some simplifying but practically justified assumptions on the unknowns in the process. The maximum principle for parabolic PDEs is employed to prove that online calibration using a single discrete-in-time temperature measurement can provide removal of the estimation error arising due to mismatch of a single unknown parameter in the model. Although unproven, this result is then shown in simulation to apply to the more detailed process model.