## HW5

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## HW5

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

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ChE 541 Mass Transfer Homework 5 Due 10 24 12 Fall 2012 M Sahimi 1 Modeling of Intercellular Calcium Response of Endothelial Cells Part I Formulation When a change occurs in the concentration of specific agonists such as triphosphate ATP endothelial cells respond by increasing the concentration of cytosolic Ca producing prostacyclin endothelium derived relaxing factor and ectoenzymes To study this experimentally one uses a rectangular flow chamber provided with a cell surface at one of its walls and a feed stream containing ATP flowing into the chamber The concentration of ATP near the cell surface depends on the rate of its degradation and the rate at which it 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 nonreacting while the one at y h represents the endothelial cell surface The xz surface is large so that the system is two dimensional in the x y directions Assume that we have laminar flow within the chamber in the x direction The transport of ATP in the y direction is through diffusion while convective mass transfer dominates in the x direction The velocity profile is given by vx 6vm 2 y y h h where vm is the mean velocity a Write down the governing equation for CA the ATP concentration in the flowing fluid b There is a first order reaction at the endothelial cell surface while the surface at y 0 is non reactive and non porous The feed stream at x 0 contains ATP at a constant concentration C0 Write down the boundary conditions c Introduce the dimensionless variables C CA C0 y h and xD vm h2 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 of variables to solve the governing equation Thus assume that C X Y substitute it in the governing equation and derive the governing equations for the two functions X and Y Derive the solution for X 2 Drug Delivery and Release Part I Spherical Drug Delivery Material One important problem currently under extensive research is drug delivery to a target area 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 One such material is a polymer in which the drug is inserted within its matrix Then when the polymer and its content are exposed to water that penetrates the matrix the drug gradually dissolves This gives rise to a well defined interface between the liquid solution and the undisturbed part of the polymer The best drug delivery materials are those that help establish quickly a steady rate of drug release Researcher have developed the drug delivery materials in a semi spherical shape of radious b The surface is covered by an impermeable non porous coating except for a small cavity of radius a on the flat side of the semi sphere Diffusion is radially inward into the semi sphere through the cavity The radial position of the interface is t The concentration of the drug in the polymeric matrix is Cs and that of the aqueous solution at the interface is C0 We assume that the drug concentration at r a is negligible and that 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 d dt b Solve for the concentration profile within the aqueous solution given that diffusion is pseudo steady state c Combine the results from a and b to derive an equation for 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 becomes constant after sometime 3 Stretching a Polymeric Sheet to Remove Volatile Monomers Part I Analysis of Stretching One important problem in the synthesis of an elastic polymer in sheet form is to remove a small amount of volatile unreacted monomers within the polymer One solution for the problem is to stretch the polymer continuously This decreases the thickness of the polymer hence making a shorter diffusion path for the unreacted monomers to reach the surface At the same time the stretching increases the surface hence larger mass transfer between the polymer and outside We wish to analyze this problem Suppose that the elastic polymeric sheet is stretched continuously in the x direction but the volume remains constant The surface at x 0 is fixed but that at x L t is pulled at a constant speed v The sheet s thickness at time t is 2 t with the origin of the coordinates in the middle a The strain rate t is given by t 1 L dL dt 1 d dt Show that the velocities in the elastic sheet are given by vx x t x t and vy x t y t b Derive an equation for t if t 0 0 Hence derive equations for L t and t

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