Page 406 - Modelling in Transport Phenomena A Conceptual Approach

386 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

For a cylindrical differential volume element of thickness AT and length Az, as
shown in Figure 9.17, Eq. (9.58) is expressed as

I
I
( NA, ,. 27rr AZ + NA, 27rr AT)
+
- [ NA, 2 4 ~AT) AZ + NA, lz+Az 27r~ AT] = 0 (9.59)
Dividing Eq. (9.59) by 27rArAz and taking the limit as AT 4 0 and Az 4 0
gives

(9.511)
Substitution of Eqs. (9.56) and (9.57) into Eq. (9.511) yields
DAB a
(9.512)

Note that in the z-direction mass of species A is transported both by convection
and diffusion. As stated by Eq. (2.48), diffusion can be considered negligible with
respect to convection when Pa >> 1. Under these circumstances, EQ. (9.512)
reduces to
(9.513)

As an engineer, we are interested in the variation of the bulk concentration
of species d, CA~, rather than the local concentration, CA. For forced convection
mass transfer in a circular pipe of radius R, the bulk concentration defined by Eq.
(4.1-3) takes the form
LzZ 1" V,CA T drd6
CAb = (9.514)
I'" 1" vz r drd9

In general, the concentration of species d, CA, may depend on both the radial and
axial coordinates. However, the bulk concentration of species d, CA~, depends only
on the axial direction.
To determine the governing equation for the bulk concentration of species A, it
is necessary to integrate Eq. (9.513) over the cross-sectional area of the tube, i.e.,