Solute boundary layer analysis

Analysis of Solute Boundary Layer. A closed-fonn solution of the... 

The boundary layer structure predicted by the analysis of Burton et al. (74) and by Wilson (80) is much more robust than just a description of the solute boundary layer caused by the rotational flow near a large crystal. 

Techniques from boundary layer analysis can be used to construct a series solution to equations 29 and 30 of the form... 

For a liquid droplet in a gaseous medium, the densities differ significantly. Whereas the linearized treatment can still be extended to the liquid-phase analysis, for the gas phase the conventional boundary layer analysis should be used (82). For high Reynolds number flow over a solid sphere, approximate solutions have been obtained using both the Blasius series and the momentum integral techniques (82). 

This is the appropriate correlation to use when there is heat or mass (i.e., substitute Nu by Sh) transfer from a sphere immersed in a stagnant film is studied, Nu = 2. The second term in (5.294) accounts for convective mechanisms, and the relation is derived from the solution of the boundary layer equations. For higher Re3molds numbers the Nusselt number is set equal to the relation resulting from the boundary layer analysis of a flat plate ... 

With a few exceptions, the fluid flow must be simulated before the mass-transfer simulations can be rigorously performed. Nevertheless, here are several important situations, such as that at a rotating disk electrode, where the fluid flow is known analytically or from an exact, numerical solution. Thus there exists a body of work that was done before CFD was a readily available tool (for example, see Refs. 34-37). In many of these studies, a boundary-layer analysis, based on a Lighthill transformation (Ref. 1, Chapter 17), is employed. 

The amazing feature of (10-260) is that it is obtained entirely from the inviscid flow solution - the boundary-layer analysis does play an important role in demonstrating that the volume integral of 4>, based on the inviscid velocity field, will provide a valid first approximation to the total viscous dissipation but does not enter directly. 

Shigechi et al. [112] conducted a boundary layer analysis of this problem and included momentum and convection effects in the condensate film. They obtained different solutions using as a boundary condition various inclination angles of the liquid-vapor interface at the plate edge. Their maximum average Nusselt number was found to agree well with Eq. 14.98. Chiou et al. [214] included surface tension in their model and showed that heat transfer decreases in relation to Eq. 14.98 as the surface tension of the condensate increases. 

In a flat membrane variation called flat-plate contained liquid membrane (FPCLM), Pakala et al. 24) studied and modeled SO2 separation from flue gas using liquid membranes of aqueous sodium citrate or aqueous sodium sulfite solutions. The measured SOj fluxes were predicted well by a nonequilibrium boundary layer analysis for SO2 transport. The SO2 fluxes for sodium citrate films were a few times higher than that for sodium sulfite as the reagent concentration was increased form 0.0 to 0.667 M. They also studied the same system in a HFCLM module which, unfortunately, had a large EMT. Therefore, the percentage of SO2 removal was much less than that in Majumdar et al (22). 

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

The transition to a turbulent boundary layer for a flat plate has been experimentally determined to occur at an Rcx value of between 3 x 10 and 6 x 10. For this example, the transition would occur between 15 and 30 cm after the start of the plate. Thus, the computations for a laminar boundary layer at 0.6 and 1 m are not realistic. However, the Blasius solution helps in the analysis of experimental data for a turbulent boundary layer, because it can tell us which parameters are likely to be important for this analysis, although the equations may take a different form. 

Hiemenz (in 1911) first recognized that the relatively simple analysis for the inviscid flow approaching a stagnation plane could be extended to include a viscous boundary layer [429]. An essential feature of the Hiemenz analysis is that the inviscid flow is relatively unaffected by the viscous interactions near the surface. As far as the inviscid flow is concerned, the thin viscous boundary layer changes the apparent position of the surface. Other than that, the inviscid flow is essentially unperturbed. Thus knowledge of the inviscid-flow solution, which is quite simple, provides boundary conditions for the viscous boundary layer. The inviscid and viscous behavior can be knitted together in a way that reduces the Navier-Stokes equations to a system of ordinary differential equations. 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

Use of Ficks law to describe the diffusion process requires the solute particle to be small compared with the diffusion boundary layer. The analysis presented above suggests that, for Peelet numbers greater than 100, the ratio 8o/0p is proportional to (Pe)ua/R. The solid curves in Figure 3 are truncated at the value of the Peelet number corresponding to Pe/R3 10 "2, where an inspection of the radial concentration profile revealed that the ratio 8d/Op is about ten. 

After the Burgers equation, the numerical analysis of the incompressible boundary layer equations for convection heat transfer are discussed. A few important numerical schemes are discussed. The classic solution for flow in a laminar boundary layer is then presented in the example. 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

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