Page 460 - Modelling in Transport Phenomena A Conceptual Approach

440 CHAPTER IO. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

where A and B are constants. Since the problem is symmetric around the z-axis,
then 0, and hence G, must be even functions2 of E. Therefore, A = 0. Application
of the boundary condition defined by Eq. (10.222) gives
BcosX = 0 (10.225)

For a nontrivial solution, the eigenvalues are given by

COSX=O 3 An= T n=0,1,2, ... (10.2-26)

Therefore, the general solution is

(10.227)

The unknown coefficients C, can be determined by using the initial condition in
Eq. (10.2-15). The result is
1 = c c, cos [ (n + ;) 4 (10.2-28)
00

n=O
Since the eigenfunctions are simply orthogonal, multiplication of Eq. (10.228) by
cos [(m + i) 7r<] and integration from < = 0 to < = 1 gives

(10.2-29)
Note that the integral on the right side of Eq. (10.2-29) is zero when n # m and
nonzero when n = m. Therefore, when n = m the summation drops out and Eq.
(10.2-29) reduces to the form

Evaluation of the integrals gives

sin [(n + +) 7r3
Cn = 2 (10.2-31)
(n++)a
Since sin (n + a) ?r = (- l)", the solution becomes

(10.232)

2A function f(s) is said to be an odd function if f(-s) = -f(z) and an even function if
f(-z) = fb).