A mathematical model of dendrite solidification in multi-component alloys has been developed to predict microsegregation and to obtain a better understanding of the relation between the solid fraction and temperature in the solid/liquid zone. An analytical solution to the solute- and heat-balance equations in the model has been obtained by assuming that equilibrium exists at the solid/liquid interface and that the solidus and liquidus move in proportion to the square root of time.
Effective partition ratios predicted by the model for the continuous casting of steel are in good agreement with generally accepted values. Model predictions of temperature distribution in the mushy zone fit well with measured temperatures. However predictions of micro-segregation of Cr and Ni in a unidirectionally solidified stainless steel were less satisfactory, presumably because the model neglects segregation between adjacent secondary dendrite arms. The model also has provided a theoretical basis for the relationship between temperature and solid fraction in the mushy zone. It is possible to assume a linear relationship between temperature and solid fraction in cases, like plain-carbon steel, where one of the solutes has a dominant effect on depression of the liquidus temperature with respect to its concentration. This assumption simplifies the computation of solidification behaviour without introducing severe errors.