## HW4

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## HW4

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

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ChE 541 Mass Transfer Homework 4 Due 10 9 12 Fall 2012 M Sahimi 1 Transport Through a Charged Hydrogel Membrane Some membranes are made of hydrogels which are a particular type of material in which over 90 of the volume is water The membrane is held together by crosslinking of the polymer s monomers Consider such a charged membrane of thickness L in which the concentration of the fixed immobile negative charges is Cm The membrane is connected on both sides to salt solutions where at z 0 C C C0 while at z L C C CL The diffusivities D and D are given but the charge potential within the membrane is not known Assume the system to be one dimensional Recall that the net electrical current is zero a The implication of the assumptions for the fluxes is that N N Ns where Ns is the common value of N and N Why b Write down the expression for Ns Due to a one can write the expression in terms of C and D or C and D Give both expressions c The electroneutrality condition implies that within the membrane C z C z Cm Why d Use c to eliminate C from the equation in b to obtain an equation for C Then use this equation to eliminate the potential from the equation in b e Solve the final equation for C obtained in d and give the expression for Ns that does not contain The expression should contain C 0 and C L f To be practical we need to relate C 0 and C L to C0 and CL which are known as well as measurable To do so proceed as follows First write down the general expression for the flux Ni of a component i in terms of the diffusive convective and migration contributions given in the class Then use the fact that vi 0 and Ji 0 to show that zi F ln Ci constant A RT Use this to show that C C is a constant Thus show that C 0 C 0 C02 and C L C L CL2 g Now solve for C 0 and C L in terms of Cm CL and C0 h Using g derive an expression for Ns i Use Eq A to derive an equation for 0 assuming that the external potential is 0 A similar expression can be derived for L 2 Diffusion of Water and Ammonia Through a Membrane A mixture of ammonia component A and water component B vapors passes through a slab of thickness h that contains helium compoent C The slab represents a membrane and the passage of the two types of vapor is intended for their separation A and B enter the slab through one face and leave from the opposite face with mole fractions xA1 and xB1 and xA2 and xB2 respectively a To be practical one must relate the fluxes of NA and NB to the boundary conditions where the mixture s composition is known and or can be measured Assume that the three diffusivities DAm DBm and DCm are known where m represents the mixture and that the transport processes is at steady state Derive expressions for NA and NB if helium is stationary b Rework the problem if instead of the diffusivities of the gases in the mixture we only know the binary diffusivities DAB DBC and DAC Is there a difference with the results in a Why 3 Fast Reaction of Radicals in Liquid Solutions When free radicals such as NO and O 2 are in a liquid they react so fast that can be considered essentially as instantaneous Determining the reaction rate constant for such cases is not easy and is usually based on diffusion measurement We assume that molecules A and B that are spheres of radii a and b react immediately as soon as they come into contact that is to say when the distance between their centers is a b Assume steady state and use spherical coordinates a We assume that A is stationary and is surrounded by diffusing molecules of B If CB CB far from A determine the concentration profile CB r and the rate of reaction per molecule of A b One can show that if A is also mobile the effective diffusivity is DA DB Determine the rate of reaction for this case c Determine the volumetric rate of reaction RA for b If we define an effective reaction rate constant k by RA RB kCA CB derive an expression for k where CA is the bulk concentration of A

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