Page 318 - Modelling in Transport Phenomena A Conceptual Approach

298 CHAPTER 8. STEADY MICROSCOPIC BALmCES WITHOUT GEN.

Since the limits of the integration are constant, the order of differentiation and
integration in the second term of E!q. (8.453) can be interchanged to get

Substitution of Eq. (8.454) into Eq. (8.453) yields

(8.455)

The use of the boundary condition given by Eq. (8.449) leads to

(8.456)

Note that Eq. (8.456) contains two dependent variables, (CA) and CA~,.=R, which
are at two different scales. It is generally assumed, although not expressed explic-
itly, that
CAlr=R E (CA) (8.457)
This approximation is valid for BiM << 1. Substitution of Eq. (8.457) into Eq.
(8.4-56) gives
(8.458)

Integration of Eqs. (8.450) and (8.451) over the cross-sectional area of the
pore gives the boundary conditions associated with Eq. (8.458) as

at z = 0 (CA) = CA, (8.459)

(8.460)

Equations (8.447) and (8.458) are at two different scales. Equation (8.458) is ob-
tained by averaging Eq. (8.447) over the cross-sectional area perpendicular to the
direction of mass flux. As a result, the boundary condition, i.e., the heterogeneous
reaction rate expression, appears in the conservation statement.
Note that the term 2/R in Eq. (8.458) is the catalyst surface area per unit
volume, i.e.,
2 - 2nRL Catalyst surface area
- --=a,= (8.461)
R nR2L Pore volume
Since heterogeneous reaction rate expression has the units of moles/(area) (time),
multiplication of this term by a, converts the units to moles/(volume)(time).
The physical significance and the order of magnitude of the terms in Q. (8.4
58) are given in Table 8.11.