Hydrodynamic retardation

The coefficients a(p, c) and tj(p, c) describe chemical and physical effects on the kinetics of deposition. The transport of particles from the bulk of the flowing fluid to the surface of a collector or media grain by physical processes such as Brownian diffusion, fluid flow (direct interception), and gravity are incorporated into theoretical formulations for fj(p, c), together with corrections to account for hydrodynamic retardation or the lubrication effect as the two solids come into close proximity. Chemical effects are usually considered in evaluating a(p, c). These include interparticle forces arising from electrostatic interactions and steric effects originating from interactions between adsorbed layers of polymers and polyelectrolytes on the solid surfaces. 

Rajagopalan and Tien (1976) have reported the results of a theoretical study of the effects of diffusion, fluid flow, gravity, hydrodynamic retardation, electro-... 

The third group of terms on the right-hand side describes particle transport to the collector by gravity forces acting on the suspended particle. Hydrodynamic retardation is included. Here, Nc is a gravity number, that is, the ratio of the Stokes settling velocity of a suspended particle to the superficial or approach velocity of flow. 

The terms oc(i,j)s and A(i,/)s collectively describe a kinetic coefficient for the coagulation or aggregation of suspended particles of sizes i and j. They have analogies with but are not identical to the terms a(p, c) and tj(p, c) used previously in describing the kinetics of particle deposition processes in porous media. Like q p, c), the term l i,j)s incorporates information about various processes of particle transport, although as used here hydrodynamic retardation is not considered. Unlike t/(p, c), X(iJ)s is not a ratio of fluxes. It is a rate coefficient that includes most physical aspects the second-order coagulation reaction. Like a(p, c), the term a(i, j)s incorporates chemical aspects of the interactions between two colliding solids however, as used here, the effects of hydrodynamic retardation are subsumed in ot(iJ)s. The term a(i,j)s is a ratio defined here as follows ... 

Hydrodynamic Retardation. Smoluchowski assumed in the derivation of his equations that )pair = Z)1+Z)2, but this is not true if the diffusing particles are relatively close to each other. When two particles come close, the liquid between them has to flow out of the gap, and this means that (a) the local velocity gradient is increased and (b) the flow type becomes biaxial elongation rather than simple shear (see Section 5.1). Both factors cause the effective viscosity to be increased, which means in turn that the mutual diffusion coefficient of the particles is decreased, the more so as the particle separation (h) is smaller. The phenomenon is called the Spielman-Honig effect. 

The Fuchs treatment, if the colloidal interaction curve is known, and as modified by hydrodynamic retardation, i.e., use of Eq. (13.7). This equation can be considered as a solution of the problem discussed in Section 4.3.5. 

For polymer-stabilized particles (Section 12.3.1), calculation of W from theory is often not possible (and this may also be the case if some other interactions are involved see Section 12.4). Moreover, hydrodynamic retardation would not be according to the theory outlined above. In these cases, it is often very difficult to predict capture efficiencies. 

Since this book is dedicated to the dynamic properties of surfactant adsorption layers it would be useful to give a overview of their typical properties. Subsequent chapters will give a more detailed description of the structure of a surfactant adsorption layer and its formation, models and experiments of adsorption kinetics, the composition of the electrical double layer, and the effect of dynamic adsorption layers on different flow processes. We will show that the kinetics of adsorption/desorption is not only determined by the diffusion law, but in selected cases also by other mechanisms, electrostatic repulsion for example. This mechanism has been studied intensively by Dukhin (1980). Moreover, electrostatic retardation can effect hydrodynamic retardation of systems with moving bubbles and droplets carrying adsorption layers (Dukhin 1993). Before starting with the theoretical foundation of the complicated relationships of nonequilibrium adsorption layers, this introduction presents only the basic principles of the chemistry of surfactants and their actions on the properties of adsorption layers. 

In these expressions, me, Vc, Fd, F, X, a, H, t, 17, p, R, DLVO force, the hydrodynamic force, the scaled distance, the linear radius of the cells, the dimensionless closest half surface-to-surface distance between two cells, the scaled half separation distance between two cells, the time, the viscosity of liquid phase, the hydrodynamic retardation factor, the density of the cells, the gas constant, the scaled DLVO potential, the scaled van der Waals potential, and the Hamaker constant of the system, and i, , and I are dummy variables. Note that the fourth and fifth integral terms on the right-hand side of (25.141) represent, respectively, the contribution to the electrostatic repulsion force when the fixed positive and negative charge in the membrane phase of a cell appears. Equation (25.135) can be rewritten to become... 

In addition to hydrodynamic retardation, the countercharge has another influence on the velocity of a charged particle in an electric field. This influence is of electric nature it is known as the relaxation effect. The underlying cause is explained in Figure 10.6. When the particle is at rest, the centers of the particle surface charge and the countercharge coincide in the center of the particle (Figure 10.6a). However,... 

When in motion, the diffnse electrical donble-layer aronnd the particle is no longer symmetrical and this canses a rednction in the speed of the particle compared with that of an imaginary charged particle with no donble-layer. This rednction in speed is cansed by both the electric dipole field set np which acts in opposition to the applied field (the relaxation effect) and an increased viscons drag dne to the motion of the ions in the donble-layer which drag liqnid with them (the electrophoretic retardation effect). The resnlting combination of electrostatic and hydrodynamic forces leads to rather complicated eqnations which, nntil recently, conld only be solved approximately. In 1978, White and O Brien developed a clever method of nnmerical solntion and obtained detailed cnrves over the fnll range of Ka valnes (0 °°)... 

Retardation. A mobilized particle inherently falls behind the moving fluid since the drag force experienced by the particle is proportional to the relative velocity. Hydrodynamic conditions around the tortuous geometry of the pore space, gravity, inertial effects, high flow velocity and collisions enhance retardation. Retardation increases the local concentration of particles near pore throats. 

Furthermore, one would not expect to observe conditions where a —l due to hydrodynamic influences, where water must be squeezed out of the way as two particles or aggregates approach one another. As shown by Han and Lawler [3], under Brownian motion equal sized spheres should have a maximum collision efficiency somewhat less than one (interpolating from their Fig. 4, 100 nm spheres would have a maximum a = 0.7, 5 pirn. spheres would have a maximum a = 0.45). If all retarding influences are considered, then, as is the case when measurements are conducted in the laboratory, it is unlikely under the majority of common conditions for collision efficiencies to approach unity. 

Hydrodynamic boundary layer — is the region of fluid flow at or near a solid surface where the shear stresses are significantly different to those observed in bulk. The interaction between fluid and solid results in a retardation of the fluid flow which gives rise to a boundary layer of slower moving material. As the distance from the surface increases the fluid becomes less affected by these forces and the fluid velocity approaches the freestream velocity. The thickness of the boundary layer is commonly defined as the distance from the surface where the velocity is 99% of the freestream velocity. The hydrodynamic boundary layer is significant in electrochemical measurements whether the convection is forced or natural the effect of the size of the boundary layer has been studied using hydrodynamic measurements such as the rotating disk electrode [i] and - flow-cells [ii]. 

Filtration Model. A model based on deep-bed filtration principles was proposed by Soo and Radke (12), who suggested that the emulsion droplets are not only retarded, but they are also captured in the pore constrictions. These droplets are captured in the porous medium by two types of capture mechanisms straining and interception. These were discussed earlier and are shown schematically in Figure 22. Straining capture occurs when an emulsion droplet gets trapped in a pore constriction of size smaller than its own diameter. Emulsion droplets can also attach themselves onto the rock surface and pore walls due to van der Waals, electrical, gravitational, and hydrodynamic forces. This mode of capture is denoted as interception. Capture of emulsion droplets reduces the effective pore diameter, diverts flow to the larger pores, and thereby effectively reduces permeability. 

---