Page 412 - Modelling in Transport Phenomena A Conceptual Approach

392 CHAPTER 9. STEADY NICROSCOPIC BALANCES WITH GEN.

Substitution of Eq. (3) into Eq. (4) gives

Example 9.11 When the concentration profile is fully developed, show that the
concentration gradient in the uxial direction, &A/&, remains constant for a con-
stant wall mass flux.
Solution
The molar flux of species A at the surface of the pipe is given by

NA, Ir=~ = kc (CAW - CAI) = Constant (1)
Since kc is constant for a fully developed concentration profile, Eq. (I) implies that

CAW - CAI = constant (2)

Therefore, Eq. (9.5-41) simplifies to

Since dcAbldz is constant according to Eq. (9.5-34), also remains constant,
z-e.,

9.5.1.2 Sherwood number for a fully developed concentration profile
Substitution of Eq. (9.51) into Eq. (9.513) gives
2(v,) [1- (32] 2 = - - ; (9.5-42)
(+)
v;B

It should always be kept in mind that the purpose of solving the above equation
for concentration distribution is to obtain a correlation to calculate the number of
moles of species A transferred between the phases. As shown in Chapter 4, mass
transfer correlations are expressed in terms of the Sherwood number. Therefore,
Eq. (9.5-42) will be solved €or a fully developed concentration profile for two
different types of boundary conditions, i.e., constant wall mass flux and constant
wall concentration, to determine the Sherwood number.