TAMU MEEN 344 - HW2 (2 pages)

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HW2



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HW2

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Texas A&M University
Course:
Meen 344 - Fluid Mechanics
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ChE 541 Mass Transfer Homework 2 Due 9 26 12 Fall 2012 M Sahimi 1 Diffusion with Nonlinear Enzymatic Reactions One of the best known nonlinear reaction kinetics is the Michaelis Menten kinetics named in honor of German biochemist Leonor Michaelis and Canadian physician Maud Menten 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 Suppose that reactants diffusive in a catalyst pellet the availability step assumed to be like a slab and take part in an enzymatic reaction according to the MM kinetics Assume steady state one dimensional diffusion with no convective effect a Write down the governing equation for the reactant A The concentration of the reactant is CA0 at the entrance z 0 to the catalyst whereas it is zero on the catalytic surface at z L which is where it is minimum Write down the boundary conditions b Rewrite the the governing equation and its boundary conditions in dimensionless form using C CA CA0 and z L You may also use the notation Km CA0 if need by Then a Damko hler number Da should emerge Give an explicit expression for Da c The governing equation cannot be solved analytically but in practice one needs an expression for the flux of the reactants which can be obtained analytically To do so multiply both sides of the dimensionless governing equation by 2dC d After rearranging it you should obtain an expression for the derivative of dC d 2 Integrate it to obtain an expression for the flux and then rewrite it in dimensional form d Repeat the analysis for the case in which the reaction is nth order n 1 RA kCAn e In both cases L the thickness of the slab disappears i e the it has no effect Under what physical condition s can this be true 2 Pseudo steady State Analysis of Diffusion in Membranes Thin membranes whether biological or synthetic play an important role in many branches of science and technology from industrial separation and purification processes to biological systems Suppose that a membrane of thickness L is inserted in the middle of a system between two compartments of volume V each that contain well mixed mixtures By passing the mixture through the membrane it is separated into its constituent components We wish to analyze this problem using a pseudo steady state PSS assumption The mixture passes through the membrane by diffusion only The membrane is assumed to be a one dimensional system as its surface area A perpendicular to the direction of diffusion is large The concentrations of the external solutions on each side are C1 t and C2 t a Write down the governing equation for the concentration C z t within the membrane b Next assume PSS Solve the governing equation and derive expressions for the flux N on each side of the membrane C1 t and C2 t are still unknown c Next write an overall mass balance for each compartment on the sides of the membrane You should obtain two equations for the two unknowns C1 t and C2 t Solve the two equations to derive the unknown functions C1 t and C2 t d If c is done correctly a time scale tm should emerge Give an explicit expression for it What is its physical meaning e The diffusion time scale is td L2 D Under what condition s is the PSS analysis expected to be accurate Express your result in terms of td tm 3 Pseudo steady State Analysis of Evaporation of a Liquid Let us reconsider the problem of evaporation of a column of a liquid We analyzed the problem under the steady state condition but we now reanalyze it using a pseudo steadystate PSS analysis which is more realistic Liquid A evaporates in a column of height H which is open allowing the vapor to escape Initially the column is filled with the liquid but as it evaporates the vapor liquid interface is lowered by an amount h t unlike the problem in the class where the interface was held fixed measured from the top As in the problem described in the class the origin z 0 is at the interface where xA x0 is known We simplify the problem by assuming that air is a single gas a Assuming that air does not dissolve in the liquid and that the PSS condition prevails write down an expression for NA b Integrate the equation in a to derive an expression for NA The result must contain h t c Next write a mass balance at the interface assuming that C is the molar concentration in the liquid and is fixed in order to obtain an expression for NA d The results in b and c must be the same which gives a differential equation for h t Solve the equation to obtain an expression for h t e What is the time te for evaporating all the liquid f The time scale for diffusion in the column is td H 2 D Under what condition s the PSS assumption is valid Express the condition s in terms of the ration td te


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