## 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

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