ChE 541 Homework 9 Fall 2012Mass Transfer Due 11/26/12 M. Sahimi1. Taylor-Aris Dispersion in a ChannelConsider dispersion in laminar flow through a rectangular channel of height (in the z-direction) a and width (in the y-direction) b. The flow is in the x-direction. If the aspectration b/a is large, then the velocity profile is independent of y, and is given byvx= vmax"1 −2za2#.As discussed in the class, Aris (1956) showed that the dispersion coefficient DLis givenbyDL= D + ka2v2mD, (1)where vmis the mean flow velocity, and k a geometric factor that depends on the shapeof the cross section of the channel. We wish to determine k for the channel flow.(a) Aris introduced a function χ(y, z), the normalized deviation from the mean flowvelocity,χ(y, z) =vx(y, z)vm− 1 . (2)Calculate χ for the high aspect ratio case.(b) Next, one solves the following problema2∇2φ = −χ , −a/2 < z < a/2dφ/dz = 0 , z = ±a/2Solve for φ for the case of a channel with high aspect ratio.(c) The geometrical factor k is given byk = hχφi ,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 AnalysisAs discussed in the class, in order for the analysis of dispersion in tubes by G.I. Taylor tobe valid, the time scale for radial or vertical diffusion must be much shorter than that ofthe convective time. We wish to determine this condition for the channel flow of Problem1 in the limit of high aspect ratios.(a) We must first study diffusion in the z direction, which is perpendicular to the flow di-rection. 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 byC(z, t) =∞Xn=0Ancos(2πnz/a) exp(−4π2n2Dt/a2) ,where Anis 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 FilmConsider the problem of absorption of a gas A that approaches a liquid film flowing ona vertical flat surface, which was studied in the class. Suppose that once A is absorbedinto the film, it reacts with the liquid according to first-order kinetics. The rest of theproblem is similar to what was described in the class.(a) Write down the governing equation for CAand the boundary conditions.(b) Introduce the dimensionless variables, ζ = 1 − y/δ, θ = CA/CA0, and rewrite thegoverning equations and the boundary conditions in dimensionless form, where δ is thefilm’s thickness.(c) We wish to consider the limit of large x, where x is the direction of the flow, measuredfrom the top of the wall. A smart student has argued that in this limit we can ignore theconvection term. Do you agree? Why?(d) Assuming that the student is right, solve the governing equation.(e) We define a Sherwood number bySh =kδD=(∂θ/∂ζ)ζ=1θ(ζ = 1) − θb,where θbis the average of θ 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 expressionfor Sh for the two