Quasi-steady-state (QSS) and transient models, developed using the CrysMAS code, are employed to study the effects of transport mechanisms and cold finger design on the temperature distribution, melt flow field, and melt-crystal interface shape during the crystal growth of sapphire by a small-scale, modified heat exchanger method (HEM). QSS computations show the importance and effects of various heat transfer mechanisms in the crystal and melt, including conduction, internal radiation, and melt convection driven by buoyant and Marangoni forces. The design of the cold finger is demonstrated to have significant effects on growth states. Notably, transient computations on an idealized heat transfer model, supplemented with QSS calculations of a model with rigorous heat transfer representation, show that non-uniform growth conditions arise under uniform cooling of the system via a linear decrease in furnace set points. We suggest that more uniform HEM growth conditions may be achieved by using non-linear cool-down strategies. (C) 2013 Elsevier B.V. All rights reserved.