Leveque solution

At higher Graetz numbers, the experimental data lie parallel to but below the Leveque solution. This is most likely because the screen, placed in the blood flow channel, touches the membrane. Thus the membrane surface area available for gas transfer is... 

The ability of the Leveque solution to adequately describe the variation in flux with diffusivity of the retained solute is illustrated in Figure 3.42. Here, albumin data were compared with whole serum data. The larger globulins in whole serum have a lower diffusivity D = 4x 10"7 cm2/sec29 and a lower solubility limit (Cg). The theoretical curves are 15 to 20% below the experimental data in both cases. Thus, the dependence of flux on diffusivity to the 0.67 power is confirmed. 

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

Y. P. Shih and T. D. Tsou, Extended Leveque Solutions for Heat Transfer to Power Law Fluids in Laminar Flow in a Pipe, Chem. Eng. Sci. (15) 55,1978. 

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

The usefrilness of film theory becomes apparent when the effect of an aheration of a process parameter, such as bulk flow velocity, is to be investigated [Blatt et al, 1970], Using the Leveque solution for transfer coefficient, the following equation correlates flux rate with diear rate at the wall of the membrane (yw), where L is the channel length and B is a constant ... 

If it is assumed that the velocity profile is flat as in rodlike flow, the solution is more easily obtained (SI). A third solution, called the approximate Leveque solution, has been obtained, where there is a linear velocity profile near the wall and the solute diffuses only a short distance from the wall into the fluid. This is similar to the parabolic velocity profile solution at high flow rates. Experimental desigrv equations are presented in Section 7.3D for this case. 

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. 

Newman J. Extension of the Leveque solution. Journal of Heat Transfer 1969 91 177-178. 

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. 

Shih YP, Tsou JD. Extended Leveque solutions for heat transfer to power law fluids in laminar flow in a pipe. The Chemical Engineering Journal 1978 15 55-62. 

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. 

Determine the value of the water flux through this module using the length-averaged Leveque solution. (Ans. 25.3 Uter/m -hr). 

Determine the fractional feed water removal through this module. Justify the use of the form of Leveque solution recommended for use. (Ans. 0.0235.)... 

It is known further that the Leveque solution (see Example 7.2.6) is valid here. If all other conditions in the problem are similar to those of Example 7.2.6, determine the fractional reduction in solvent flux as the BSA solution is concentrated from 2 g/100 cm to 15 g/100 cm. ... 

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