## HW3

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## HW3

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- School:
- Texas A&M University
- Course:
- Meen 344 - Fluid Mechanics

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ChE 541 Mass Transfer Homework 3 Due 10 3 12 Fall 2012 M Sahimi 1 Absorption from a Bubble The Effect of Reaction Consider a spherical gas bubble that is rising slowly in a stagnant liquid The bubble s radius at time t is R t It contains a pure species A that diffuses into the stagnant liquid where it is consumed by a first order reaction The concentration of A at the gas liquid interface is C0 Due to the reaction there is no A far from the bubble a Neglect convective mass transport and assume that the absorption is slow enough that a pseudo steady state assumption for the concentration profile of A is valid Determine CA r t in the liquid b Derive an expression for the flux NA at the interface Compare it with the case in which there is no reaction c If the concentration of A in the bubble is Cb derive a condition in terms of C0 and Cb that justifies the assumption of negligible convective mass transfer d Given c what is the condition for the validity of the pseudo steady state analysis 2 Diffusion of a Ligand and its Binding to a Receptor When a particular type of ligand binds to a receptor it triggers the differentiation of a target cell in a multicellular organism Such a ligand is called morphogen To model this phenomenon we formulate the problem as follows Consider two layers of cells with indefinite extent separated by a layer of fluid with thickness h such that one layer is at z 0 and the second one at z h The ligand is selected from a circular area of radius R in the layer at z 0 Its flux N0 into the liquid is constant while the rest of the lower layer does not release any ligand The receptors are on the layer at z h The binding has a first order kinetics and the flux there is kC r h Assuming steady state we wish to determine the local concentration C r z of the ligand in the fluid between the two layers a Write down the governing equation for C r z Specify the boundary conditions b As stated the problem is a two dimensional one but we wish to simplify it to a one dimensional problem by averaging the concentration in the z direction and hence eliminating z as an independent variable Under what conditions can we do so c Assuming that the conditions in b are satisfied formulate the problem for C r the z averaged concentration and derive the solution for it 3 Fast Second Order Reaction in a Liquid Film Consider a liquid film between x 0 and x L Species A is supplied into the film at x 0 with a concentration CA0 while species B is supplied at x L with a concentration CBL Within the film A and B react and produce C with a rate kCA CB We assume that the reaction is fast enough that A and B cannot coexist in the film If k the reaction occurs only at a plane xR where on one side of it A but not B exists while the opposite is true on the other side Assume that DA DB and that this is a steady state phenomenon a Determine xR CA x CB x and CC x CC is zero at both x 0 and x L b If we compute the concentration gradients of A B and C on both sides of xR are they continuous have the same value at xR Why

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