ChE 541 Homework 5 Fall 2012Mass Transfer Due 10/24/12 M. Sahimi1. Modeling of Intercellular Calcium Response of Endothelial Cells. Part I: For-mulationWhen a change occurs in the concentration of specific agonists, such as triphosphate(ATP), endothelial cells respond by increasing the concentration of cytosolic Ca+ +, pro-ducing prostacyclin, endothelium-derived relaxing factor and ectoenzymes. To stud y thisexperimentally, one uses a rectangular flow chamber, provided with a cell surface at one ofits walls and a feed stream containing ATP flowing into the chamb er. The concentrationof ATP near the cell surface depends on the rate of its degradation, and the rate at whichit reaches from upstream. We would like to model the experiments.Thus, consider a rectangular flow chamber of height h. The surface at y = 0 is non-reacting, while the one at y = h represents th e endothelial cell surface. The xz surfaceis large, so that the system is two dimensional in the (x, y) directions. Assume that wehave laminar flow within the chamber in the x direction. The transport of ATP in the ydirection is through diffusion, while convective mass transfer dominates in the x direction.The velocity profile is given byvx= 6vm"yh−yh2#,where vmis the mean velocity.(a) Wr ite down the governing equation for CA, the ATP concentration, in the flowingfluid.(b) There is a first-order reaction at the endothelial cell surface, while the surface at y = 0is non-reactive and non-porous. The feed stream at x = 0 contains ATP at a constantconcentration C0. Write down the boundary conditions.(c) Introduce the dimensionless variables, C = CA/C0, γ = y/h, and ξ = xD/(vmh2).Rewrite the governing equation and the boundary conditions in dimensionless forms.(d) Since the dimensions of the system are finite, we use the method of separation ofvariables to solve the governing equation. Thus, assume that, C = X(ξ)Y (γ), substituteit in the governing equation, and derive the governing equations for the two functionsX(ξ) and Y (γ). Derive the solution for X(ξ).2. Drug Delivery and Release. Part I: Spherical Drug Delivery MaterialOne important problem currently under extensive research is drug delivery to a targetarea of one’s body, so that the disease and cause of it can be directly targeted. Thus,various types of materials have been fabricated that can be used as the drug carrier. Onesuch material is a polymer in which the drug is inserted within its matrix. Then, whenthe polymer and its content are exposed to water that penetrates the matrix, the druggradually dissolves. This gives rise to a well-defined interface between the liquid solutionand the undisturbed part of the polymer. The best drug delivery materials are those thathelp establish quickly a steady rate of drug release.Researcher have developed the drug delivery materials in a semi-spherical shape of radiousb. The surface is covered by an impermeable (non-porous) coating, except for a smallcavity of radius a on the flat side of the semi-sphere. Diffusion is radially inward intothe semi-sphere through the cavity. The radial position of the interface is δ(t). Theconcentration of the drug in the polymeric matrix is Cs, and that of the aqueous solutionat the interface is C0. We assume that the drug concentration at r = a is negligible, andthat, a/b ≪ 1 and C0/Cs≪ 1. We assume that the entire process is slow and, hence,pseudo-steady state.(a) Write down a mass balance at the interface, in order to obtain an equation for dδ/dt.(b) Solve for the concentration profile within the aqueous solution, given that diffusion ispseudo-steady state.(c) Combine the results from (a) and (b) to derive an equation for δ(t).(d) Use the result to justify the assumption of pseudo-steady state.(e) Derive an equation for the rate of drug release, and show that it indeed becomesconstant after sometime.3. Stretching a Polymeric Sheet to Remove Volatile Monomers. Part I: Analysisof StretchingOne important problem in the synthesis of an elastic polymer in sheet form is to removea small amount of volatile unreacted monomers within the polymer. One solution forthe problem is to stretch the polymer continuously. This decreases the thickness of thepolymer, hence making a shorter diffusion path for the unreacted monomers to reach thesurface. At the same time, the stretching increases the surface, hence larger mass transferbetween the polymer and outside. We wish to analyze this problem.Suppose that the elastic polymeric sheet is stretched continuously in the x direction, butthe volume remains constant. The surface at x = 0 is fixed, but that at x = L(t) is pulledat a constant speed v. The sheet’s thickness at time t is 2δ(t), with the origin of th ecoordinates in the middle.(a) The strain rate ǫ(t) is given by, ǫ(t) = (1/L)(dL/dt) = −(1/δ)(dδ/dt). Show that, thevelocities in the elastic sheet are given by, vx(x, t) = xǫ(t) and, vy(x, t) = −yǫ(t).(b) Derive an equation for δ(t), if δ(t = 0) = δ0. Hence, derive equations for L(t)