Ives (1985) classified the various forces acting on particles in a flowing suspension in three categories as

a forces related to the transport mechanisms,

b forces related to the attachment mechanisms, and

c forces related to the detachment mechanisms, and characterized them in terms of the relevant dimensionless groups.

Forces Related to the Transport Mechanisms

The important relevant quantities governing the particle behavior in a suspension can be summarized as following: d and D are particle and porous media grain diameters, respectively; p5 is the density of particles; p and \JL are the density and viscosity of the carrier liquid, respectively; va is the convective velocity; g is the gravitational acceleration coefficient; and T is the absolute temperature.

Inertia Force

The inertia of a particle forces it to maintain motion in a straight line. The inertia force can be expressed by the dimensionless group as (Ives, 1985):

Gravity Force

As a result of the density difference between the particle and the carrier liquid, particles tend to move in the gravity direction according to Stokes' law. The velocity of a spherical particle undergoing a Stokes' motion is given by:

The gravity force acts upward when particles are lighter and, therefore, buoyant. The gravity force acts downward when particles are heavier and, therefore, tend to settle. The gravity force can be expressed by a dimensionless group, which relates the Stokes and convection velocities as (Ives, 1985):

Centrifugal Forces.

The centrifugal forces are generated by external acceleration. The centrifugal force created with an angular velocity of w and a radius of R is expressed in dimensionless form by

Diffusion Force

Particles smaller than 1 mm diameter tend to move irregularly in a liquid media and disperse randomly. This phenomena is called the Brownian notion. The diffusivity of fine particles undergoing a Brownian notion is given by Einstein (McDowell-Boyer et al., 1986):

The diffusion force can be expressed by the Peclet number as the ratio of the convection velocity to the average Brownian velocity given by (Ives, 1985).

Hydrodynamic Force

Hydrodynamic forces are the fluid shearing and pressure forces (Wojtanowicz et al., 1987, 1988). Ives (1985) explains that during fluid flow econdary circulation flows can be formed around the particles, which can generate out-of-balance hydrodynamic forces acting on the particles to move them across the flow field. Ives (1985) states that a proper dimensionless group rigorously expressing the hydrodynamic force is not available. Ives (1985) points out that the Reynolds number given by:

and its other forms such as those "relating to the shear gradient, the relative velocity between particle and liquid, the angular velocity of the rotating particle, and the frequency of pulsation liquid have been suggested." Khilar and Fogler (1987) expressed the hydrodynamic lift force pulling a spherical particle off the pore surface by the following equation given by Hallow (1973):

where us is the slip velocity, K is the linearized velocity gradient near the particle, and d is the diameter of the spherical particle.

Forces Related to the Attachment Mechanisms

These forces act on the particles when they are near the grain surface less than a 1 Jim distance (Ives, 1985). These forces and the characteristic dimensionless groups are described below.

London—van der Waals Force

This is the attractive force due to the electromagnetic waves generated by the electronic characteristics of atoms and molecules. The attraction force is expressed by (Ives, 1985):

in which X is a dimensionless wavelength of the dispersion force divided by nd product and Fn is a function assuming different forms for (s - 2)/X less and greater than unity.

Friction—Drag Force and Hydrodynamic Thinning

Particles approaching the grain surfaces experience a flow resistance because they must displace the liquid at the surface radially as they attach to the grain surface (Ives, 1985; Khilar and Fogler, 1987).

Forces Related to the Detachment Mechanisms

Shearing Force.

This is the friction or drag force. When the shear stress of the liquid flowing over the deposited particles creates a shearing force greater than those attaching the particles to the grain surface, then the particles can be detached and mobilized (Ives, 1985):

Electrostatic Double-Layer Force

These forces are created due to the ionic conditions measured by pH and ionic strength. When the particle and grain surfaces carry the electrostatic charges of the same sign, they repel each other. The repulsive force is expressed by (Ives, 1985):

where s is the dimensionless separation distance expressed as the ratio of the radial separation distance divided by the particle radius (d/2), k is the Debye reciprocal double-layer thickness, and d is the particle diameter. When the ionic strength is higher, then the double-layer thickness is smaller, and hence k is larger.

Born Repulsion Force

This force is generated as a result of the overlapping of the election clouds (Wojtanowicz et al., 1987, 1988).


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