ABSTRACT
Note that as t → 0, Equation 12.1, the initial condition is recovered and when t → ∞, C(x, ∞) = C0: the composition is uniform.
This homogenization problem can also be considered a finite boundary problem if the values of x are restricted to 0 ≤ x ≤ L, where L is the grain size since the composition (and solution, Equation 12.4) are periodic in L (Figure 12.4). As is shown later, Equation 12.4 is also the solution of a finite boundary problem by the techniques developed for solving infinite and semi-infinite boundary condition problems. Note that as was pointed out in Chapter 8, finite boundary problem solutions typically are of the form
C x t f x g x t, ,( ) = ( ) + ( ) (12.5) where:
f(x) is an equilibrium or steady-state concentration g(x, t) is a transient term, so that g(x, ∞) = 0.
