The problem we solve is illustrated in Figure 4–29. The flow has solidified to the depth y = ym(t). We assume that molten material of uniform temperature Tm lies everywhere below the growing surface layer. The fact that the molten region does not extend infinitely far below the surface is of no consequence to the solution. We must solve the heat conduction equation (4–68) in the space 0 ≤ y ≤ ym(t) subject to the conditions T = T0 at y = 0, T = Tm at y = ym(t), and ym = 0 at t = 0. The position of the solidification boundary is an a priori unknown function of time. As in the case of the sudden heating, or cooling, of a semi-infinite half-space, there is no length scale in this problem. For this reason, we once again introduce the dimensionless coordinate η = y/2 √ κt as in Equation (4–96); it is also convenient to introduce the dimensionless temperature θ = (T − T0)/(Tm − T0) as in Equation (4–93).
The dimensionless coordinate η is obtained by scaling the depth with the thermal diffusion length √ κt because there is no other length scale in the problem. Similarly, the depth of the solidification interface m must also scale with the thermal diffusion length in such a way that ym/ √ κt is a constant.
In other words, the depth of the solidification boundary increases with time proportionately with the square root of time. We have used dimensional arguments to determine the functional form of the dependence of ym on t, a nontrivial result. Because η = y/2 √ κt and ym is proportional to √ κt, the
solidification boundary corresponds to a constant value ηm = ym/2 √ κt of the similarity coordinate η. We denote this constant value by ηm = λ1. Thus we have ym = 2λ1 √ κt. (4.136) With our definitions of θ and η, the heat conduction equation for θ(η) is clearly identical to Equation (4–100), whose solution we already know to be proportional to erf(η). This form of solution automatically satisfies the condition θ = 0(T = T0) on η = 0(y = 0). To satisfy the remaining condition that θ = 1(T = Tm) at η = ηm(y = ym) = λ1, we need simply choose the constant of proportionality appropriately.
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