# TAMU MEEN 344 - HW9 (2 pages)

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## HW9

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## HW9

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School:
Texas A&M University
Course:
Meen 344 - Fluid Mechanics
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ChE 541 Mass Transfer Homework 9 Due 11 26 12 Fall 2012 M Sahimi 1 Taylor Aris Dispersion in a Channel Consider dispersion in laminar flow through a rectangular channel of height in the zdirection a and width in the y direction b The flow is in the x direction If the aspect ration b a is large then the velocity profile is independent of y and is given by vx vmax 2z 1 a 2 As discussed in the class Aris 1956 showed that the dispersion coefficient DL is given by 2 a2 v m DL D k 1 D where vm is the mean flow velocity and k a geometric factor that depends on the shape of the cross section of the channel We wish to determine k for the channel flow a Aris introduced a function y z the normalized deviation from the mean flow velocity vx y z 1 2 y z vm Calculate for the high aspect ratio case b Next one solves the following problem a2 2 a 2 z a 2 d dz 0 z a 2 Solve for for the case of a channel with high aspect ratio c The geometrical factor k is given by k h i where h i denotes an average over the cross section For channel with a high aspect ratio the average reduces to one in the z direction Determine k 2 Determining the Time Scale for the Validity of G I Taylor s Analysis As discussed in the class in order for the analysis of dispersion in tubes by G I Taylor to be valid the time scale for radial or vertical diffusion must be much shorter than that of the convective time We wish to determine this condition for the channel flow of Problem 1 in the limit of high aspect ratios a We must first study diffusion in the z direction which is perpendicular to the flow direction Write down the governing equation for unsteady state diffusion in that direction as well as the boundary conditions b Show that the solution of the governing equation is given by C z t X An cos 2 nz a exp 4 2 n2 Dt a2 n 0 where An is a constant c The diffusion time scale is obtained from the solution C z t by considering the slowest in time mode term Hence identify this time scale d Thus write down the condition for the validity of Taylor s analysis for channel flow How different is it from the corresponding condition for a circular tube 3 Reaction and Absorption in a Falling Liquid Film Consider the problem of absorption of a gas A that approaches a liquid film flowing on a vertical flat surface which was studied in the class Suppose that once A is absorbed into the film it reacts with the liquid according to first order kinetics The rest of the problem is similar to what was described in the class a Write down the governing equation for CA and the boundary conditions b Introduce the dimensionless variables 1 y CA CA0 and rewrite the governing equations and the boundary conditions in dimensionless form where is the film s thickness c We wish to consider the limit of large x where x is the direction of the flow measured from the top of the wall A smart student has argued that in this limit we can ignore the convection term Do you agree Why d Assuming that the student is right solve the governing equation e We define a Sherwood number by Sh k 1 D 1 b where b is the average of over the thickness of the film weighted by the velocity vx Derive an expression for Sh f Consider the limits of very fast and very slow reactions and simplify the expression for Sh for the two limits

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