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hw2_che150b_solutions_KS_3_03_pM_16sept

hw2_che150b_solutions_KS_3_03_pM_16sept - Chemical...

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1 Chemical Engineering 150B- Fall 2005 Problem Set #2 Solutions Problem 1. (25 Points) Dilute concentrations of toxic organic solutes in aqueous solution can be often degraded by a “biofilm” attached to an inert, nonporous solid surface. A biofilm consists of living cells immobilized in a gelatinous matrix. A toxic organic solute (species A) diffusing into the biofilm and is degraded to harmless products, hopefully CO 2 and water, by the biofilm. For engineering applications, the biofilm is considered a homogenous substance (i.e., species B). The rate of degradation of the toxic solute per unit volume of the biofilm is described by a kinetic rate equation of the form: R A = -R A,max c A / (K A +c A ) where R A,max is the maximum possible degradation of species A in the biofilm, and K A is the half- saturation constant for the degradation of species A within the biofilm. In the present process, a biofilm of 0.5 cm thickness is immobilized around an inert, nonporous spherical pellet of 1 cm diameter. The concentration of the toxic contaminant in the bulk aqueous phase is a constant, 0.002 mol/m 3 . Furthermore, there are no resistances to convective mass transfer across the fluid boundary layer between the bulk fluid and the biofilm surface. The contaminant is equally soluble in both water and the biofilm, and the density difference between the biofilm and water can be neglected. (a) Draw a picture of the physical system, with an appropriate coordinate system. State at least two reasonable assumptions for the mass-transfer aspects of this process. (b) What is the general differential equation for mass transfer in terms of concentration c A ? Propose two boundary conditions for species A needed to solve the resulting differential equation. 0 A A A dc N R dt ∇⋅ + = We start with the overall flux expression: ( ) A Az AB A Az Bz dx N cD x N N dz = − + + , which can be simplified due to constant c and D AB as well dilute concentrations of A: 3 pt 4 pt for any 2 assumptions
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2 A Az AB dc N D dr = − 2 2 0 AB A A D d dc r R r dr dr + = (1.1) (c) Plot the concentration profile of A within the biofilm where D AB = 2 x 10 -10 m 2 /s; K A = 1.3 mol/m 3 ; and, R A,max =1.9 x 10 -5 mol.m -3 .s -1 . Use Mathcad to solve the ODE. As shown in the numerical handout, transform the second order ODE into a system of first order ODEs by reduction of order: Let 1 A y C = and A o dC y dr = . Then Equation (1.1) becomes 1 1 2 A,max R o o AB A dy y y dr r D K y = − + + 1 o dy y dr = with boundary conditions: y o =0 at R1 and y 1 =0.002 mol/m 3 at R2. The Mathcad worksheet is below, where I've redefined R A,max as k1, just to east notation. Note: Credit will be given if one assumes K A >>C A , although the profile must look similar (within 5%) to the one below. 5 pt 5 pt for Mathcad setup 3 pt for B.C.s
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3 The units on the above plot are as follows: c A in mol/m 3 and r in meters.
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