*2.4. Processing of Solute Fields at Moving Boundaries*

In order to accurately calculate the solute field during movement, a solute extrapolation method for calculating the solute field at the moving boundary is proposed.

As shown in Figure 2, the white and black grid points are the liquid and solid grid points, the solid ellipse is the position of the dendrite at the previous moment, the dotted ellipse is the current position. The gray grid points are the covered liquid nodes because of dendrite movement. The concentration *CL* of the covered liquid grid points will be distributed in a certain proportion, which is:

$$
\Delta \mathbf{C}\_L^1 = A \cdot \mathbf{C}\_L \Delta \mathbf{C}\_L^2 = B \cdot \mathbf{C}\_L \Delta \mathbf{C}\_L^3 = \mathbf{C} \cdot \mathbf{C}\_L \tag{34}
$$

**Figure 2.** A schematic diagram of solid fall.

Δ*C*<sup>1</sup> *L*,Δ*C*<sup>2</sup> *<sup>L</sup>*, and <sup>Δ</sup>*C*<sup>3</sup> *<sup>L</sup>* respectively represent the concentration assigned to three lattice points along the direction of dendrite movement. The distance between the lattice points is the grain movement distance. *A*, *B*, and *C* are the distribution coefficients and satisfy the following relations:

$$A + B + C = 1 \\ A > B > C \tag{35}$$

In this paper, the undetermined coefficient method is used, and the distribution coefficients of different proportions are used to calculate. Then the results are compared with the experimental results of Liu's single dendrite drop. The empirical values of *A*, *B*, and *C* are 0.7, 0.2, and 0.1, respectively. The assigned concentration will be added to the three lattice points along the direction of dendrite movement.

$$\mathbf{C}\_{L}^{\rm u} = \mathbf{C}\_{L}^{\rm u} + \Delta \mathbf{C}\_{L}^{\rm u} \tag{36}$$
