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TAMU MEEN 344 - HW3

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ChE 541 Homework 3 Fall 2012Mass Transfer Due 10/3/12 M. Sahimi1. Absorption from a Bubble: The Effect of ReactionConsider a spherical gas bubble that is rising slowly in a stagnant liquid. The bubble’sradius 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-liquidinterface 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 thata pseudo-steady-state assumption for the concentration profile of A is valid. DetermineCA(r, t) in the liquid.(b) Derive an expression for the flux NAat the interface. Compare it with the case inwhich there is no reaction.(c) If the concentration of A in the bubble is Cb, derive a condition in terms of C0andCbthat 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 ReceptorWhen a particular type of ligand binds to a receptor, it triggers the differentiation ofa target cell in a multicellular organism. Such a ligand is called morphogen. To modelthis phenomenon, we formulate the problem as follows. Consider two layers of cells withindefinite extent, separated by a layer of fluid with thickness h, such that one layer is atz = 0, and the second on e at z = h. The ligand is selected from a circular area of radiusR in the layer at z = 0. Its flux N0into the liquid is constant, while the rest of the lowerlayer does not release any ligand. The receptors are on the layer at z = h. The bindinghas a first-order kinetics, and the flux there is kC(r, h). Assuming steady state, we wishto deter mine the local concentration C(r, z) of the ligand in the fluid between the twolayers.(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 aone-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), thez−averaged concentration, and derive the solution for it.3. Fast Second-Order Reaction in a L iquid FilmConsider a liquid film between x = 0 and x = L. Species A is supplied into the film atx = 0 with a concentration CA0, while species B is supplied at x = L with a concentrationCBL. Within the film A and B react and produce C, with a rate kCACB. We assumethat the reaction is fast enough that A and B cannot coexist in the film. If k → ∞, thereaction occurs only at a plane xR, where on one side of it A but not B exists, while theopposite is true on the other side. Assume that DA= DB, and that this is a steady-statephenomenon.(a) Determine xR, CA(x), CB(x), and CC(x). CCis 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, arethey continuous - have the same value - at xR?


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