ChE 541 Homework 2 Fall 2012Mass Transfer Due 9/26/12 M. Sahimi1. Diffusion with Nonlinear Enzymatic ReactionsOne of the best known nonlinear reaction kinetics is the Michaelis-Menten kinetics (namedin honor of German biochemist Leonor Michaelis and Canadian physician Maud Menten):RA= −RmaxCAKm+ CA.Here, Rmaxis the rate of consumption when CAis large enough that it makes the avail-ability of the enzymes the limiting step, and Kmis the concentration at which the rate ishalf of the maximum. Suppose that reactants diffusive in a catalyst pellet (the availabilitystep), assumed to be like a slab, and take part in an enzymatic reaction according to theMM kinetics. Assume steady state, one-dimensional diffusion, with no convective effect.(a) Write down the governing equation for the reactant A. The concentration of thereactant is CA0at the entrance (z = 0) to the catalyst, whereas it is zero on the catalyticsurface 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 formusing, C = CA/CA0, and ζ = z/L. You may also use t he notation, κ = Km/CA0, if needby. Then, a Damk¨ohler number Da should emerge. Give an explicit expression for Da.(c) The governing equation cannot be solved analytically, but in practice one needs anexpression for the flux of the reactants, which can be obtained analytically. To do so,multiply both sides of the dimensionless governing equation by 2dC/dζ. After rearrangingit, you should obtain an expression for the derivative of (dC/dζ)2. Integrate it to obtainan 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=−kCnA.(e) In both cases, L, the thickness of the slab disappears, i.e. the it has no effect. Underwhat physical condition(s) can this be true?2. Pseudo-steady-State Analysis of Diffusion in MembranesThin membranes, whether biological or synthetic, play an important role in many branchesof science and technology, from industrial separation and purification processes to bio-logical systems. Suppose that a membrane of thickness L is inserted in the middle of asystem 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 com-ponents. 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 assumedto be a one-dimensional system, as its surface area A perpendicular to the direction ofdiffusion 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 fluxN 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 mem-brane. You should obtain two equations for the two u nknowns C1(t) and C2(t). Solve thetwo equations to derive the unknown functions C1(t) and C2(t).(d) If (c) is done correctly, a time scale tmshould emerge. Give an explicit expression forit. What is its physical meaning?(e) The diffusion time scale is, td= L2/D. Under what condition(s) is the PSS analysisexpected to be accurate? Express your result in terms of td/tm.3. Pseudo-steady-State Analysis of Evaporation of a LiquidLet us reconsider the problem of evaporation of a column of a liquid. We analyzed theproblem under the steady-state condition, but we now reanalyze it using a pseudo-steady-state (PSS) analysis, which is more realistic. Liquid A evaporates in a column of heightH, which is open, allowing the vapor to escape. Initially, the column is filled with theliquid, but as it evaporates, the vapor-liquid interface is lowered by an amount h(t) (unlikethe problem in the class where the interface was held fixed), measured from the top. Asin the problem described in the class, the origin z = 0 is at the interface, where xA= x0is 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 containh(t).(c) Next, write a mass balance at the interface, assuming that C is the molar concentrationin 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 forh(t). Solve the equation to obtain an expression for h(t).(e) What is the time tefor evap orating all the liquid?(f) The time scale for diffusion in the column is, td= H2/D. Under what condition(s)the PSS assumption is valid? Express the condition(s) in terms of the ration