Answer Happy

I NEED ANSWER ASAP!
i) Show how to obtain Eq 18.7-7 from 18.7-6
ii) Show how to obtain Eq 18.7-8 from 18.7-7
OF $18.7 Diffusion and Chemical Reaction Inside a Porous (18.7-4) dr We now define an "effective diffusivity" for species A in the porous medium by dca Nar = -DA SOS in which ca is the concentration of the gas A contained within the pores. The effective and temperature and also on the catalyst pore structure. The actual mechanism for diffusion in pores is complex, since the pore dimensions may be smaller than the mean free path of the diffusing molecules. We do not belabor the question of mechanism here but as- sume only that Eq. 18.7-4 can adequately represent the diffusion process (see $24.6). When the preceding expression is inserted into Eq. 18.7-3, we get, for constant diffusivity = A ( = dr 1 d dca (18.7-5) DA = -RA z dr dr We now consider the situation where species A disappears according to a first-order chemical reaction on the catalytic surfaces that form all or part of the "walls” of the winding passages. Let a be the available catalytic surface per unit volume (of solids + voids). Then RA - Kaca, and Eq. 18.7-5 becomes (see Eq. C.1-6). ya 1 d odca z dr = Kíaca (18.7-6) This equation is to be solved with the boundary conditions that ca = Car at r = R, and that ca is finite at r = 0. Equations containing the operator (1/r^)(d/dr)[r?(d/dr)] can frequently be solved by using a "standard trick" —namely, a change of variable calcar = (1/r)f(r). The equation for f(r) is then of (kia dr? (18.7-7) DA/ This is a standard second-order differential equation, which can be solved in terms of ex- ponentials or hyperbolic functions. When it is solved and the result divided by r we get the following solution of Eq. 18.7-6 in terms of hyperbolic functions (see SC.5): kha kia r+ sinh CAR (18.7-8 9 CA - C1 cosh = r D A r 1 0 A