Question

5.5 Write Eq. (5.4) by replacing x with n, where n is defined as dx Hence show that, just as Eq. (5.17) is the solution of Eq. (5.4) for the case of uniform Г, the solution for nonuniform Г is given by where η. 1s the value of η at x L. Note that ρ1mL is the Peclet number. If the derivation on these lines is continued, we get Eq. (5.22), where Pe must be defined as Pea(ρυ)e(6n)e. Assuming that a grid-point value of Γ prevails throughout the control volume surrounding it, we can express (δη)e in terms of the Γ's and the distance increments (shown in Fig. 4.1). Hence, we have ΓΡ ΓΕ (5.4) dx 489 = Fe (4p ф-фо -exp (Ps/L)-1 虹--φο exp (P)-1 (5.17) (5.22) exp(%) (x)e- (8x) Figure 4.1 Distances associated with the interface e.

Answer 1

Given:

dx Substituting this in Eq 5.4 we get: (Γ Above is the required modification of equation 5.4

Seeing carefully one can notice that modified version is exactly similar to original equation 5.4 if we substitute「-1. This is valid as「is assumed constant or uniform in original equation. Hence solution by equation 5.17 is valid by making following modification (ie, using original solution at「= 1): exp(pun) -1 exp(P,-M/ L)-1 -00 exp(pun)-1 OL -o exp(pun)-1 is the required solution. Hence proved