## 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

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