Thin concentration boundary layer approach

The form of Eq. (5-25) was suggested by noting that the first order curvature corrections to Eqs. (3-47) and (5-35) are near unity and by matching the expression to the creeping flow result, Eq. (3-49), at Re = E Equation (5-25) also represents the results of the application of the thin concentration boundary layer approach (Sc oo) through Eq. (3-46), using numerically calculated surface vorticities. Thus the Schmidt number dependence is reliable for any Sc > 0.25. 

As Re increases further and vortices are shed, the local rate of mass transfer aft of separation should oscillate. Although no measurements have been made for spheres, mass transfer oscillations at the shedding frequency have been observed for cylinders (B9, D6, SI2). At higher Re the forward portion of the sphere approaches boundary layer flow while aft of separation the flow is complex as discussed above. Figure 5.17 shows experimental values of the local Nusselt number Nuj c for heat transfer to air at high Re. The vertical lines on each curve indicate the values of the separation angle. It is clear that the transfer rate at the rear of the sphere increases more rapidly than that at the front and that even at very high Re the minimum Nuj. occurs aft of separation. Also shown in Fig. 5.17 is the thin concentration boundary layer... 

The second approach to concentration polarization, and the one used in this chapter, is to model the phenomenon by assuming that a thin layer of unmixed fluid, thickness S, exists between the membrane surface and the well-mixed bulk solution. The concentration gradients that control concentration polarization form in this layer. This boundary layer film model oversimplifies the fluid hydrodynamics occurring in membrane modules and still contains one adjustable parameter,... 

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

The first boundary condition is equivalent to the well-known Levich approach (ca=1 for according to which, it is supposed that the concentration values vary only within a very thin concentration layer while it is supposed to keep its bulk value elsewhere [9], Eq. (3b) has been proposed by Coutelieris et al. [8] in order to ensure the continuity of the concentration upon the outer boundary of the cell for any Peclet number. Furthermore, eq. (3c) and (3d) express the axial symmetry that has been assumed for the problem. The boundary condition (3e) can be considered as a significant improvement of Levich approach, where instantaneous adsorption on the solid-fluid interface cA(ri=Tia,ff)=0) is also assumed for any angular position 0. In particular, eq. (3e) describes a typical adsorption, order reaction and desorption mechanism for the component A upon the solid surface [12,16] where ks is the rate of the heterogeneous reaction upon the surface and the concentration of component A upon the solid surface, c,is, is calculated by solving the non linear equation... 

To obtain Eqs. 5-10, it was assumed that the concentration of solute within the adsorption boundary layer is related to the solute-surface interaction energy by a Boltzmann distribution. The essence of the thin-layer polarization approach is that a thin diffuse layer can still transport a significant amoimt of solute molecules so as to affect the solute transport outside the diffuse layer. For a strongly adsorbing solute (e.g., a surfactant), the dimensionless relaxation parameter fila (or Kid) can be much greater than imity. If all the adsorbed solute were stuck to the surface of the particle (the diffuseness of the adsorption layer disappears), then L = 0 and there would be no diffusiophoretic migration of the particle. In the limit of [l/a 0 (very weak adsorption), the polarization of the diffuse solute in the interfacial layer vanishes and Eq. 5 reduces to Eq. 1. 

Before proceeding to such types of analysis and computations in the sections that follow, we begin with a statement of the full problem with as much of the physics represented as possible. Our approach is to work with macroscopic models of the interface separating the fluid phases. This approach represents the interface by a sharp dynamic surface embedded in three-dimensional space, across which flow and concentration variables can jump in a manner specified by physical boundary conditions. The alternative microscopic approach seeks to describe the three-dimensional thin transition layer between the two phases using statistical or continuum mechanical methods. The reader is referred to Chapters 15-18 of the text by Edwards, Brenner and Wasan as well as the many references therein. 

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