Question

1. [EfTectiveness factor] (a) Consider the following governing equation which can describe the reaction-diffusion inside a porous catalyst with the flat geometry: where C is the concentration of species A, k is the reaction constant, and D is the diffusion coefficient. The corresponding boundary conditions are given by CA = Cas = 1 at x = 0 and = 0 at x = 1 dx The schematic for the domain is illustrated in Fig. I 1-0 CA-CA-1 dC dr Fig. 1. Schematic for the domain. Solve Eq. () with the above boundary conditions. For your convenience, you may use the following variable, m Then, you have a concentration profile as a function of x. Subsequently, calculate the effectiveness factor, based on the following definition: kC Plot (1) the effectiveness factor and (2)together as a function of m in the log-log scale NOTE: You should use a computer software (including Microsoft Excel) to plot them. Optional (b)Now consider the two extreme cases in the problem (a) () diffusion is too fast: D & (ii) diffusion is too slow: D0 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 1

The plot of effectiveness factor and 1/m together as a function of m on the log-log scale is attached at the end.

2. ClelydCA here, >o therfore, use bounda conditions to de amine the unknowns and C2, da and Ca(x=0)=1 sinhk and, dA implies, ご0

By definition, เร็ The Effectiveness Fact, CAS = 1 , k is constant k CAs 2Cosh (mi) dz , m is constant 6positive) sinh (m/ Sinh (m) ご4.fsinh (m)-01- าท sinh (m) SO

so, the effectiveness factor, ) - Sinh (m therefore, limat Sinh1mUn [의 indeerminate Form Use L'Hospital's rule irm z Cosh (o)1 (ii) When D 0 ) m-> oo, therefore indeerminale fotm lim Sinh (m Agin use L' Hospital's rule m cosh (m lim cosh (m) CoS Thus, lim Sinh (m) tends to inpinitu .e. limit does not exist,

→co.

4 10 sinblm 103 10 10 10-1 -1 10 10 10