Surface Force Boundary Layer Approximation

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

He solved this equation, using three different boundary conditions, two of which are also used in the field of particle deposition on collectors the Perfect Sink (SINK) model, the Surface Force Boundary Layer Approximation (SFBLA) and the Electrode-Ion-Particle-Electron Transfer (EIPET) model. 

Surface Force Boundary Layer Approximation (SFBLA) 210, 215... 

This relation for tangential forces with the correctness accepted for the boundary layer approximation (7) is represented. As boundary condition (30) represents a partial case of surface force r, discussed above in formulas (15-16), it is possible to use the basic model equations (16) for hydrodynamic part of the problem under consideration. We need now to discuss boundary value problem to obtain model equation for a. We follow here the recent investigations( Velarde et al., 2000). 

These relationships have been used by Spalding in the dimensionless presentation both of theoretical values obtained in his approximate solution of the boundary layer equations (58) and of the experimental data (51, 55, 60). Emmons (3), who has solved the problem of forced convection past a burning liquid plane surface in a more rigorous fashion, shows graphically the rather close correspondence between values obtained from his exact solution and that of Spalding, and between the calculated values for flat plates and the experimental values for spheres. 

The approximate integral equation method that was discussed in Chapters 2 and 3 can also be applied to the boundary layer flows on surfaces in a porous medium. As discussed in Chapters 2 and 3, this integral equation method has largely been superceded by purely numerical methods of the type discussed above. However, integral equation methods are still sometimes used and it therefore appears to be appropriate to briefly discuss the use of the method here. Attention will continue to be restricted to two-dimensional constant fluid property forced flow. 

Consider a vertical hot flat plate immersed in a quiescent fluid body. We assume the natural convection flow to be steady, laminar, and two-dimensional, and the fluid to be Newtonian with constant properties, including density, with one exception the density difference p — is to be considered since it is this density difference between the inside and the outside of the boundary layer that gives rise to buoyancy force and sustains flow. (This is known as the Boussines.q approximation.) We take the upward direction along the plate to be X, and the direction normal to surface to be y, as shown in Fig. 9-6. Therefore, gravelly acts in the —.t-direclion. Noting that the flow is steady and two-dimensional, the.t- andy-compoijents of velocity within boundary layer are II - u(x, y) and v — t/(.Y, y), respectively. 

It is not only possible to force the mobile phase through a capillary or column by means of a pump but also by electroendosmosis. Thereby one utilizes the fact that an electric double layer occurs on all boundary layers. Silica or quartz glass are surfaces covered with fixed negative excess charges and a solution in contact with it forms positive boundary charges. If a potential gradient of approximately 50 kV m is applied the solution flows in the direction of the negative electrode. 

For the heated vertical plate and horizontal cylinder, the flow results from natural convection. The stagnation configuration is a forced flow. In each case the flow is of the boimdai7 Kiyer type. Simple analytical solutions can be obtained when the thickness of the du.st-free space is much smaller than that of the boundary layer. In this case the gas velocity distribution can be approximated by the first term in an expansion in the distance norroal to the surface. Expressions for the thickness of the dust-free space for a heated vertical surface and a plane stagnation flow are derived below. 

In spite of this, we shall see that potential-flow theory plays an important role in the development of asymptotic solutions for Re i>> 1. Indeed, if we compare the assumptions and analysis leading to (10-9) and then to (10-12) with the early steps in analysis of heat transfer at high Peclet number, it is clear that the solution to = 0 is a valid first approximation lor Re y> 1 everywhere except in the immediate vicinity of the body surface. There the body dimension, a, that was used to nondimensionalize (10-1) is not a relevant characteristic length scale. In this region, we shall see that the flow develops a boundary layer in which viscous forces remain important even as Re i>> 1, and this allows the no-shp condition to be satisfied. 

Recall that within the small double layer approximation, the velocity slips at the wall going from m, to zero discontinuously. Now irrotational flow defined by the Laplace equation is a solution for the velocity field that admits of a slip condition at the surface but also satisfies the full Navier—Stokes equations, although not of course the usual no-slip boundary condition. Such an irrotational flow exerts no force or moment on the particle and the velocity is derivable from a potential, that is, u = where 0 is the velocity potential. The velocity must also satisfy the boundary conditions of no normal flow through the surface, whence n-V

velocity potential approaches the value corresponding to the uniform velocity U. 

An important parameter is the Reynolds number. At Re 1 the viscous term in (5.107) is small in comparison with the inertial one. Neglecting it, one obtains the equations of motion of an ideal liquid (Euler s equations). These equations describe flow of liquid in a volume, with the exception of small regions, adjoining the surface of an immersed body. Near such surfaces, the viscosity force can be comparable with inertial force, which results in formation of a viscous boundary layer with thickness S I/(Re), where L is the characteristic size of the body. Approximation Re 1 leads to an inertialess flow described by Stokes equations. These equations follow from (5.107), in which the inertial terms are omitted. Such equations describe the problems of micro-hydrodynamics, for example, problems of the small particles motion in a liquid. 

Leng and Quarderer (1982) reasoned that dispersion occurred in the boundary layer adjacent to the loop impeller surfaces and that the impeller vertical elements could be approximated by cylinders moving through the suspension at the relative impeller tip speed. When laminar shear forces predominated, it was shown that... 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

To calculate the double-layer force, the nonlinear Poisson-Boltzmann equation was solved for the case of two plane parallel plates, subject to boundary conditions which arise from consideration of the simultaneous dissociation equilibria of multiple ionizable groups on each surface. Deijaguin s approximation is then used to extend these results to calculate the force between a sphere and a plane. Details of the method can be found in Ref. (6). 

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