1. [Effectiveness factor]
(a) Consider the following governing equation which can describe the reaction-diffusion inside a
porous catalyst with the flat geometry:
dZCA k
C = 0 (1)
dx2 D A
where C A is the concentration of species A , k is the reaction constant, and D is the diffusion
coefficient. The corresponding boundary conditions are given by
dC
CA=CAS=latx=oand4=oatx=f (2)

dx
The schematic for the domain is illustrated in Fig. 1.
??

x ? 0 x = 1
CA ? CA5 ? 1 fl ? 0
dx
Fig. 1. Schematic for the domain.
Solve Eq. (1) with the above boundary conditions. (For your convenience, you may use the following
variable, m

m = -. (3)
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Then, you have a concentration profile as a function of x. Subsequently, calculate the effectiveness
factor, 77 based on the following definition:

kCAdx
77 ? o
?????

Plot (1) the effectiveness factor and (2) – together as a function of m in the log-log scale.

[Optional] (b) Now consider the two extreme cases in the problem (a): (i) diffusion is too fast:
D ->oo & (ii) diffusion is too slow: D ->0. Provide the effectiveness factor in these cases by
taking a corresponding limit of the solution in the problem (a). Interpret the results physically (Briefly
answer).