Convective diffusion equation boundary layer

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. 

Then an approximate analytical solution of the convective diffusion equation (43), which satisfies the boundary conditions, equation (44), is available under the assumption that the thickness of the diffusion layer <5, is small compared with the body radius r0 (p. 80 in [25]). As in the example of Section 4.1 (see equation (33)), the results of the derivation can be formally written in terms of the diffusion layer thickness, which now is ... 

Because the kinetic and mass-transport phenomena occur in a thin region adjacent to the electrode surface, this area is treated separately from the bulk solution region. Since kinetic effects are manifested within 100 A of the electrode surface, the resulting overpotential is invariably incorporated in the boundary conditions of the problem. Mass transport in the boundary layer is often treated by a separate solution of the convective diffusion equation in this region. Continuity of the current can then be imposed as a matching condition between the boundary layer solution and the solution in the bulk electrolyte. Frequently, Laplace s equation can be used to describe the potential distribution in the bulk electrolyte and provide the basis for determining the current distribution in the bulk electrolyte. 

The rate of deposition of Brownian particles is predicted by taking into account the effects of diffusion and convection of single particles and interaction forces between particles and collector [2.1] -[2.6]. It is demonstrated that the interaction forces can be incorporated into a boundary condition that has the form of a first order chemical reaction which takes place on the collector [2.1], and an expression is derived for the rate constant The rate of deposition is obtained by solving the convective diffusion equation subject to that boundary condition. The procedure developed for deposition is extended to the case when both deposition and desorption occur. In the latter case, the interaction potential contains the Bom repulsion, in addition to the London and double-layer interactions [2.2]-[2.7]. Paper [2.7] differs from [2.2] because it considers the deposition at both primary and secondary minima. Papers [2.8], [2.9] and [2.10] treat the deposition of cancer cells or platelets on surfaces. 

The objective of the present research is to predict the rate of deposition of Brownian particles by considering the effects of diffusion, convection, and interaction forces between particle and collector. It will be shown that, when the repulsion due to the double-layer is sufficiently large, the interaction forces can be incorporated into a boundary condition for the convective-diffusion equation. This boundary condition takes the form of a virtual first-order chemical reaction which occurs on the surface of the collector. 

In the previous section it was shown that, when double-layer repulsions are sufficiently strong, the resistance of the interaction-force boundary layer can be taken into account by imposing on the convective-diffusion equation a virtual, first-order,... 

Consider a sphere of radius a held fixed in a creeping flow field with approach velocity U. The fluid contains Brownian particles having a diffusion coefficient D. Should the Peclet number 2aUiDUj have a value much greater than unity, the diffusion boundary layer will be sufficiently thin so that curvature effects and tangential diffusion are negligible. Under these conditions the convective-diffusion equation assumes the following form ... 

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). 

To determine the net rale of adsorption of particles suspended in a fluid that is flowing over the collector, one may then solve the usual convective-diffusion equation subject to a reversible first-order reaction as the boundary condition, provided the diffusion boundary layer is much thicker than the interaction boundary layer. 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

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-... 

In this paper we combine the approach of [6], which consists in solving the equations for the electric fields in the anode, cathode and the electrolyte under steady state conditions, with our own approximation of the electrochemical reaction and the transport of reactants. We solve a 2D problem for the Laplace equation coupled with a system of the convection-diffusion equations through use of the boundary conditions. Therefore om problem becomes non-stationary. We study the time period of about one horn and observe the formation of the C02 boundary layer and the variation of the Galvani potential caused by it. 

The analysis of the second case leads to two limiting conditions which are realised also in the case of a strongly retarded surface. Under condition (8.70) the surface concentration variation along the surface is insignificant. In the opposite case at sufficiently large T / C(, the motion of the surface pushes the adsorption layer down to the bottom pole of the bubble so that everywhere condition (8.72) is fulfilled except at 7t. This enables us to use the approximate boundary condition (8.74) for solving the convective diffusion equation. 

To derive a convective diffusion equation, it is important to simplify the expression for Vj(0,x) inside the boundary layer... 

Since D plays the same role as the kinematic viscosity v, we may expect for large Schmidt numbers (v>D) that the viscous boundary layer thickness should be considerably larger than the diffusion boundary layer thickness. A consequence of this is that the velocity seen by the concentration layer at its edge is not the free stream velocity U but something much less, which is more characteristic of the velocity close to the wall (Fig. 4.2.1). We note also that since c is understood to be c, then in a multicomponent solution there may be as many distinct boundary layers as there are species, with the thickness of each defined by the appropriate diffusion coefficient. With this caveat in mind, we may write the convective diffusion equation for a two-dimensional diffusion boundary layer and estimate the diffusion layer thickness. 

In the diffusion boundary layer, we have d Cfdx d C/dy, so the equation of convective diffusion in the layer becomes ... 

A completely general solution of the governing convective diffusion equation (7.2.110), and equation (7.2.111), subject to the boundary conditions (7.2.114), (7.2.115) and (7.2.116) is not available. There are two types of solutions, similarity solutions and integral boundary layer solutions (apart from complete numerical solutions). Common to both of these solutions is the assumption that the particle concentration boundary layer is very thin compared to the membrane channel dimension normal to the axial flow further, the shear stress due to the axial velocity gradient in the particle concentration boundary layer is equal to that at the wall, namely... 

The convective diffusion equation in the diffusion boundary layer is... 

At any point within the boundary layer, the convective flux of the macromolecule solute to the membrane surface is given by the volume flux,/ of the solution multipfled by the concentration of retained solute, c. At steady state, this convective flux within the laminar boundary layer is balanced by the diffusive flux of retained solute in the opposite direction. This balance can be expressed by equation 1 ... 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

The concentration boundary layer forms because of the convective transport of solutes toward the membrane due to the viscous drag exerted by the flux. A diffusive back-transport is produced by the concentration gradient between the membranes surface and the bulk. At equiUbrium the two transport mechanisms are equal to each other. Solving the equations leads to an expression of the flux ... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

One example would be ice melting or methane hydrate dissociation when rising in seawater. Convective melting rate may be obtained by analogy to convective dissolution rate. Heat diffusivity k would play the role of mass diffusivity. The thermal Peclet number (defined as Pet = 2aw/K) would play the role of the compositional Peclet number. The Nusselt number (defined as Nu = 2u/5t, where 8t is the thermal boundary layer thickness) would play the role of Sherwood number. The thermal boundary layer (thickness 8t) would play the role of compositional boundary layer. The melting equation may be written as... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

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