A vertical Bridgman furnace, through which an ampoule containing the melt of a certain dilute binary alloy is pulled at a fixed, and predetermined speed, provides a means of improving certain alloys. Radial segregation in the finished crystalline material can make it unusable. In this paper, we construct an asymptotic theory for the flow and solidification in the ampoule, for large Rayleigh numbers but at a small Biot number. We find large-Rayleigh-number solutions to be, to leading order, completely insensitive to the character of the side-wall boundary condition on vertical velocity. Two-dimensional equivalents of optimization conditions found by Tanveer are recovered for his two limiting cases—large thermal Rayleigh number, and large negative solutal Rayleigh number. Moderate surface tension at the crystal–melt interface is found to have no effect on the optimization conditions for the two limit cases, but it does somewhat reduce the magnitude of the segregation in both limits. In addition, we present new results for the case for which the two Rayleigh numbers are of comparable magnitude and show that there is an optimization possible for this case too. Conversion of the results of this paper to an axisymmetric geometry is shown to be trivial. Keeping careful track of the ordering, we indicate how to proceed to first effects of non-linearity at small Biot and/or Prandtl number. ©1997 American Institute of Physics.