  Most heat transfer models for bulk crystal growth rely on the classical
Stefan formulation to evaluate interface motion during phase change. However,
when the interface is non-smooth the use of the classical Stefan formulation
may lead to singularities. To address this problem, we propose a simulation
model based on the weak formulation of the Stefan problem. Numerical solutions
of the weak Stefan formulation are obtained using the finite volume method.
This approach provides an energy conserving discretization scheme that
accurately evaluates heat transfer around non-smooth interfaces. We apply the
weak formulation to numerically simulate the solidification of silicon in the
horizontal ribbon growth process. Results exhibit a limitation on the ribbon's
pull speed, which previous classical Stefan models have failed to demonstrate.
A comparison of heat transfer between radiation and gas cooling shows that gas
cooling increases the pull speed limit for the same amount of heat removed.
