ChE 541 Homework 7 Fall 2012Mass Transfer Due 11/7/12 M. Sahimi1. Effectiveness Factor Larger than 1As discussed in the class, the effectiveness factor of a catalyst pellet is typically muchsmaller t han 1. This is true so long as the catalytic system op erates under isothermalconditions. In this problem we consider a case in which the effectiveness factor can belarger than 1.Consider a spherical catalyst particle of radius R in which a first-order reaction takesplace. The heat of the reaction is −∆H, and the system is not isothermal. The reactant’sconcentration and temperature on the catalyst’s external surface are, resp ectively, CsandTs. Heat transfer occurs by conduction only, and the reaction rate coefficient k dependson the temperature,k = k(Ts) exp"−ERgTTsT− 1#,where E is an activation energy, and Rgthe gas constant.(a) Write down the governing equations for heat and mass transfer in the catalyst. Sp ecifythe boundary conditions for the two equations.(b) Let, γ = E/(RgT ). What is the physical implication of large values of γ?(c) Introduce the dimensionless variables, ξ = r/R, and ζ = CA/Cs. Rewrite the gov-erning equation for CAand its boundary conditions in dimensiolness form. Identify theexpression for the Thiele modulus. Another dimensionless group should also emerge;identify the expression for it and denote it by β. What is the physical meaning of β?What is the implication of large values of β?(d) Derive a general expression for the effectiveness factor ηA(no need to solve the equa-tion for ζ).(e) Show, by any method that you can, that if β is large and the Thiele modulus is closeto 1, ηAwill be larger than one.2. Effectiveness Factor for a General Nonlinear Reaction with Multiple Reactantsand ProductsThe discussions in the class of the effectiveness factor were limited to the case in whicha single reaction occurs inside a catalyst. In this problem we consider the case in whichreactants A, B, · · · produce X, Y , · · ·:A + bB + · · · · · · → xX + yY + · · · · · · .The rate of the reaction is given byR =kpA1 + KApA+PiKipi,where i denotes any reaction product or any reactant other than A, and piis the partialpressure of species i. Assume that the catalyst can be represented by a slab, so thatdiffusion in only the z direction is important. All the reactants are assumed to be idealgases. Let νibe the stoichiometric coefficient of any reactant i, which is negative (exceptfor A).(a) Write down the governing equation for CA. Then, rewrite it in terms of the partialpressure of A.(b) Write down the governing equation for any other species i in terms of its partialpressure.(c) The boundary conditions are, pA= pAs, pi= pisat z = 0, and dpA/dz = dpi/dz = 0at z = L. Combine the equations in (a) and (b), keep in mind the stoichiometry of thereaction, and integrate the combined equation to show that,R =kpA1 + pA[KA− DAPi(Kiνi/Di)] +PiKi[pis+ (pAsνiDA/Di)],(d) Let, ω = 1+PiKi[pis+(pAsνiDA/Di)], k′= k/ω, and K = [KA− DAPi(Kiνi/Di)]/ω.Rewrite the expression for R in terms of the new quantities. Then, define a suitable Thielemodulus (recall that the problem is formulated in terms of the partial pressures, not theconcentrations).(e) For what value of K the reaction reduces to a first-order one? Can K be negative? Ifso, what does K < 0 mean physically?(f) Assume that pA= pA0at z = L. Substitute the expression for R from (d) into theequation in (a) and integrate it once to obtain an equation for d(KpA)/d(z/L), which, asyou may recall from the discussions in the class (do you?), is all one needs to computethe effectiveness factor ηA. Hence, derive an expression for ηA. In practical terms, do yousee any problem with the expression for ηA?(g) Simplify the expression for ηA, if the pA0at z = L is very low.3. Effectiveness Factor with the Michaelis-Menten KineticsConsider a catalyst particle with slab geometry. Assume that the reaction follows theMichaelis-Menten kinetics, which is well-known for enzymatic reactions,RA= −RmaxCAKm+ CA.Here, Rmaxis the rate of consumption when CAis large enough that it makes the avail-ability of the enzymes the limiting step, and Kmis the concentration at which the rateis half of the maximum. Derive an expression for the effectiveness factor of th e