## Separation Theory in Multipass Operation

Eqn (1) can be derived by taking the mass balance within the moving zone ABCD or A'B'C'D' as shown in Figure 1. It is based on the following assumptions: (a) constant distribution coefficient; (b) uniform composition and no diffusion in the molten zone; (c) no change in density during melting and freezing; (d) a constant cross-sectional area for the ingot.

and the boundary conditions of eqns (1) and (2) become:

in which:

Refreezing interface

A Ks

Melting interface

Refreezing interface

A Ks

Melting interface

 Refreezed solid 1 Molten ' Unmelted solid 1 Figure 1 Schematic diagram of zone-refining operation. The initial solute concentration of the ingot, C0, is uniform, the amount of solute, Wn>i, either transporting away from (k < 1), or into (k > 1) the ith section after the nth pass, can be calculated by: Since the original amount of solute in the ingot is W0 = ALC0, where A is the cross-sectional area of the ingot, the fraction of solute removed, f„^i(Y„,i(Z)), in each section after n passes, is expressed as: Once the values of fn>i( Yn>i(Z)), i = 1,2,...,1 for each section are known, the overall amount of solute removed Fn(Yn(Z)) can be obtained by summing up these values as: i = l i = l 2i = l where Xj is the place where Cn(x) = C0 or Cn(Z) = 1. Normal Freezing The optimal zone length Y[ for maximum Fq(Y1) is obtained by solving the equation, dF1(Y1)/dY1 = 0, with the use of eqns (9) and (10), since dF1(Y1)/dY1 is less than zero for k < 1 and dF1( Y1)/dY1 is larger than zero for k > 1. In order to meet both requirements, therefore, Y1 must be as large as possible, i.e. normal freezing for all values of the distribution coefficient. Accordingly, the optimal zone length for the first pass is:

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