Mathematical model based optimization of the compositional uniformity of a thin BiMn/Bi bar grown in a strictly zero-gravity environment by Bridgman-Stockbarger method

Abstract

The objective of this paper is to show that in the framework of the mathematical model conceived by Zheng, Larson and Zhang [1] (to explain the Mn distribution in a thin BiMn/Bi cylindrical bar, grown in the Shuttle mission 51-G experiment MRS77F075 [2]) can be proved that there exists an adequate initially nonuniform Mn distribution for which the deviation from a prescribed value of the Mn concentration in the first 70% of a thin BiMn/Bi bar, grown in a Bridgman- Stockbarger system, in zero gravity, is of order $10^{-3}$. In the theoretical part, the presentation of the mathematical model is followed by a general theorem of existence and uniqueness of the solution of the heat and mass transport equations in the melt. In the numerical part the Mn distribution in the melt and in the BiMn/Bi bar is computed for an initially uniform and for an adequate initially nonuniform Mn distribution, respectively. The deviation from a prescribed value of Mn concentration is estimated.