Page 404 - Modelling in Transport Phenomena A Conceptual Approach

384 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

9.4.2.1 Macroscopic equation

Integration of the governing differential equation, Eq. (9.426), over the spherical
aggregate of bacteria gives
1'" 1" T2 dr (r2 2) sin6drded4
r2
DAB
d
= [ 6" 1" k CAT^ sin 6 drd6d4 (9.446)

Carrying out the integrations yields

(9.447)

Rate of moles of species A Rate of consumption of species A
entering into the bacteria by homogeneous chem. rxn.

Substitution of Eq. (9.445) into Eq. (9.447) gives the molar rate of consumption
of species A, jl~, as
[h~ -~?~RDAB CA~ -htanhh) 1 (9.448)
(1
=
The minus sign in Eq. (9.448) indicates that the flux is in the negative r-direction,
i.e., towards the center of the sphere.

9.5 MASS TRANSFER WITH CONVECTION

9.5.1 Laminar Forced Convection in a Pipe

Consider the laminar flow of an incompressible Newtonian liquid (23) in a circular
pipe under the action of a pressure gradient as shown in Figure 9.17. The velocity
distribution is given by Eqs. (9.1-79) and (9.1-84) as

(9.51)

Suppose that the liquid has a uniform species d concentration of CA, for z < 0. For
z > 0, species A concentration starts to change as a function of r and a as a result
of mass transfer from the walls of the pipe. We want to develop the governing
equation for species d concentration. Liquid viscosity is assumed to be unaffected
by mass transfer.