# TAMU MEEN 344 - HW7 (2 pages)

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## HW7

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## HW7

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School:
Texas A&M University
Course:
Meen 344 - Fluid Mechanics
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ChE 541 Mass Transfer Homework 7 Due 11 7 12 Fall 2012 M Sahimi 1 Effectiveness Factor Larger than 1 As discussed in the class the effectiveness factor of a catalyst pellet is typically much smaller than 1 This is true so long as the catalytic system operates under isothermal conditions In this problem we consider a case in which the effectiveness factor can be larger than 1 Consider a spherical catalyst particle of radius R in which a first order reaction takes place The heat of the reaction is H and the system is not isothermal The reactant s concentration and temperature on the catalyst s external surface are respectively Cs and Ts Heat transfer occurs by conduction only and the reaction rate coefficient k depends on the temperature E Ts k k Ts exp 1 Rg T T where E is an activation energy and Rg the gas constant a Write down the governing equations for heat and mass transfer in the catalyst Specify the boundary conditions for the two equations b Let E Rg T What is the physical implication of large values of c Introduce the dimensionless variables r R and CA Cs Rewrite the governing equation for CA and its boundary conditions in dimensiolness form Identify the expression for the Thiele modulus Another dimensionless group should also emerge identify the expression for it and denote it by What is the physical meaning of What is the implication of large values of d Derive a general expression for the effectiveness factor A no need to solve the equation for e Show by any method that you can that if is large and the Thiele modulus is close to 1 A will be larger than one 2 Effectiveness Factor for a General Nonlinear Reaction with Multiple Reactants and Products The discussions in the class of the effectiveness factor were limited to the case in which a single reaction occurs inside a catalyst In this problem we consider the case in which reactants A B produce X Y A bB xX yY The rate of the reaction is given by R kpA P 1 KA p A i Ki p i where i denotes any reaction product or any reactant other than A and pi is the partial pressure of species i Assume that the catalyst can be represented by a slab so that diffusion in only the z direction is important All the reactants are assumed to be ideal gases Let i be the stoichiometric coefficient of any reactant i which is negative except for A a Write down the governing equation for CA Then rewrite it in terms of the partial pressure of A b Write down the governing equation for any other species i in terms of its partial pressure c The boundary conditions are pA pAs pi pis at z 0 and dpA dz dpi dz 0 at z L Combine the equations in a and b keep in mind the stoichiometry of the reaction and integrate the combined equation to show that R 1 pA KA DA P kpA P i Ki i Di i Ki pis pAs i DA Di d Let 1 i Ki pis pAs i DA Di k k and K KA DA i Ki i Di Rewrite the expression for R in terms of the new quantities Then define a suitable Thiele modulus recall that the problem is formulated in terms of the partial pressures not the concentrations P P e For what value of K the reaction reduces to a first order one Can K be negative If so what does K 0 mean physically f Assume that pA pA0 at z L Substitute the expression for R from d into the equation in a and integrate it once to obtain an equation for d KpA d z L which as you may recall from the discussions in the class do you is all one needs to compute the effectiveness factor A Hence derive an expression for A In practical terms do you see any problem with the expression for A g Simplify the expression for A if the pA0 at z L is very low 3 Effectiveness Factor with the Michaelis Menten Kinetics Consider a catalyst particle with slab geometry Assume that the reaction follows the Michaelis Menten kinetics which is well known for enzymatic reactions RA Rmax CA Km CA Here Rmax is the rate of consumption when CA is large enough that it makes the availability of the enzymes the limiting step and Km is the concentration at which the rate is half of the maximum Derive an expression for the effectiveness factor of the catalyst

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