Page 454 - Modelling in Transport Phenomena A Conceptual Approach

434 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

The unknown coefficients Cn can be determined by using the initial condition in
Eq. (10.1-26). The result is
00
1 - < = C, sin(nn<) (10.1-41)
n=O
Since the eigenfunctions are simply orthogonal, multiplication of Eq. (10.1-41) by
sin(m?r<) and integration from E = 0 to < = 1 gives

Note that the integral on the right side of Eq. (10.1-42) is zero when n # m and
nonzero when n = m. Therefore, when n = m the summation drops out and Eq.
(10.1-42) reduces to the form

(10.1-43)

Evaluation of the integrals gives
2
c, = - (10.1-44)
n?r
The transient solution takes the form

(10.1-45)

Substitution of the steady-state and the transient solutions, Eqs. (10.1-24) and
(10.1-45), into Eq. (10.1-20) gives the solution as

(10.1-46)

The volumetric flow rate can be determined by integrating the velocity distri-
bution over the cross-sectional area of the plate, i.e.,
C2 = Lw I”
dzdy
v,
1
= WBVL @e (10.1-47)

Substitution af Eq. (10.1-46) into Eq. (10.1-47) gives

Note that when r + 00, 8 -, WBV/2 which is identical with Eq. (8.1-15).