Leveque solution, mass transfer

Of considerable interest is the use of small isolated electrodes, in the form of strips or disks embedded in the wall, to measure local mass-transfer rates or rate fluctuations. Mass-transfer to spot electrodes on a rotating disk is represented by Eqs. (lOg-i) of Table VII. Analytical solutions in this case have to take account of curved streamlines. Despic et al. (Dlld) have proposed twin spot electrodes as a tool for kinetic studies, similar to the ring-disk electrode applications of disk and ring-disk electrodes for kinetic studies are discussed in several monographs (A3b, P4b). In fully developed channel or pipe flow, mass transfer to such electrodes is given by the following equation based on the Leveque model ... 

Laminar Flow. The Graetz or Leveque solutions25 26 for convective heat transfer in laminar flow channels, suitably modified for mass transfer, may be used to evaluate the mass transfer coefficient where the laminar parabolic velocity profile is assumed to be established at the channel entrance but where the concentration profile is under development down the full length of the channel. For all thin-channel lengths of practical interest, this solution is valid. Leveque s solution26 gives ... 

S. M. Richardson, Leveque Solution for Flow in an Elliptical Duct, Letters in Heat and Mass Transfer, (7) 353-362,1980. 

S. M. Richardson, Extended Leveque Solutions for Flows of Power Law Fluids in Pipes and Channels, Int. J. Heat Mass Transfer (22) 1417,1979. 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

Zero-order kinetics attracts special attention, due to its analytical simplicity and particular characteristics, especially when annulment of concentration at the solid surface is involved. Sellars et al. [61] and Siegel et al. [76] gave Graetz-type solutions for uniform axial heat flux, using the eigenfunction expansion method. Compton and Unwin [77] presented the Laplace s domain analytical solution of the mass transfer problem in a channel cell-crystal-electrode system under Leveque s assumptions. Rosner [78] wrote the solutions for the wall concentration profile as c att 1 -z/zb (z < Zo)> for several classes of boundary layer problems. 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

Gottifredi JC. Flores AF. Extended Leveque solution for heat transfer to non-Newtonian fluids in pipes and flat ducts. International Journal of Heat and Mass Transfer 1985 28 903-908. 

The solution of such an equation for an actual membrane device for ultrafiltration is difficult to obtain (see Zeman and Zydney (1996) for background information). One therefore usually falls back on the stagnant film model for determining the relation between the solvent flux and the concentration profile (see result (6.3.142b)). To use this result, we need to estimate the mass-transfer coefficient kit = Dit/dt), for the protein/macromolecule. One can focus on the entrance region of the concentration boundary layer, assume to be constant for a dilute solution, V = V, Vj, = 0 in the thin boundary layer, v = y ,y (where is the wall shear rate of magnitude AVz/Ay ) and obtain the result known as the Leveque solution at any location z in terms of the Sherwood number ... 

Example 7.2.6 Calculate the value of Wj(0) in Example 7.2.5 using the following additional information. Hollow fiber I.D., 750 pm length, 50 cm. Feed solution of BSA density 1 g/cm viscosity, 0.9 cp. Diffusivity of BSA Duo = Duk = 5.94 X 10" cm /s (Table 3.A.5). Velocity of feed solution through fiber bore, 80 cm/s. For hollow fibers, the Leveque solution (equation (3.1.145)) should be employed to determine the mass-transfer coefficient for the hollow fiber UF membrane. 

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