Page 320 - Modelling in Transport Phenomena A Conceptual Approach

300 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

8.4.4.1 Macroscopic equation

Integration of the microscopic level equations over the volume of the system gives
the equations at the macroscopic level. Integration of Eq. (8.458) over the volume
of the system gives
Jd"
LLL2=JdRww drdedz = JdL 1"" $ ks(cA) r drdedz (8.470)
r

Carrying out the integrations yields

Rate of moles of species A entering Rate of conversion of species A
into the pore through the surface at x=O to species B at the catalyst surface
Note that EQ. (8.471) is simply the macroscopic inventory rate equation for the
conservation of species A by considering the catalyst pore as a system. The use of
EQ. (8.469) in Eq. (8.471) gives the molar rate of conversion of species A, fajl~, as

(8.472)

8.4.4.2 Effectiveness fact or

The effectiveness factor, q, is defined as the ratio of the apparent rate of conversion
to the rate if the entire internal surface were exposed to the concentration CA,, i.e.,
L
~TR 1 (CA) dz iL(CA) dZ
ICs
-
- (8.473)
'= 2nRksc~,L CAJ
In terms of the dimensionless quantities, l3q. (8.473) becomes

(8.474)

Substitution of EQ. (8.469) into Eq. (8.474) gives the effectiveness factor as

(8.475)

Note that the effectiveness factor for a first-order irreversible reaction is exactly
identical with the fin efficiency. Therefore, Figure 8.24, which shows the variation
of q as a function of A, is also valid for this case.