Sedimentation force balance

The velocity of sedimentation is a result of a force balance shown in Figure 11.1 for a sphere of radius R moving at a terminal settling velocity, Vf The force of gravity, TrR p g), is a result of the... 

Figure 4-35. Forces acting on a sedimenting particle. The sedimentation and frictional forces balance each other, and the particle moves with constant velocity v.

In this type of experiment, particles are allowed to move until they arrive at their equilibrium positions. This means that sedimentation proceeds until sedimentation forces are balanced by diffusion. There are several approaches to this technique. One of these is to scan the cell, to obtain the absorbance profile at different times this allows calculation of concentration as a function of distance from the rotation axis (dc/dr). The following expression is then used to calculate the particle mass ... 

Sedimentation or creaming, depending on the relative densities of the oil and water phases, results from the action of gravity on the droplets. Under a gravitational field a spherical droplet will accelerate until it reaches a velocity for which the friction force balances the gravitational force. At this point the particle will move at a constant velocity v predicted by Stokes law ... 

When these two forces balance out (which is fairly quick), the particle sediments at a constant velocity. The rate of sedimentation (v) is then given by... 

The Stokes equation governs creaming and sedimentation of colloids in dilute dispersions. It is useful for rule-of-thumb estimations. The Stokes equation for the sedimentation velocity, u, can be then derived based on Equation 8.3 and the force balance of Figure 8.3 ... 

In combination with a general force balance, Einstein s diffiision law results in Equation 8.5b, which permits the estimation of the mass of each particle. Thus, upon combining diffiision experiments (for obtaining D) and sedimentation (gravitational) experiments (for obtaining ), we can estimate the mass of colloidal particles without any assumption about their shape. Finally, due to Einstein s diffiision law Df= kgT), the ratio f/fo is equal to DqID, where Do is the diffusion coefficient of a system containing the equivalent unsolvated spheres. 

Ni is Avogadro s number, / the particle friction coefficient, u the velocity and v the partial specific volume of the solute [79,80]. Inertial forces are negligible and the balancing of the above forces yields the sedimentation coefficient ... 

In a sedimenting suspension, however, the gravitational force on any individual particle is balanced by a combination of buoyancy and fluid friction forces, both of which are influenced by the flow field and pressure distribution in the vicinity of the particle. It is a somewhat academic exercise to apportion the total force between its two constituent parts, both of which are influenced by the presence of neighbouring particles, the concentration of which will determine the magnitude of their effect. 

Field flow fractionation (FFF), as a gentle size fractionation coupled to ICP-MS, offers the capability to determine trace metals bound to various size fractions of colloidial and particulate materials.112 On line coupling of FFF with ICP-MS was first proposed by Beckett in 1991 -113 Separation is achieved by the balance between the field force and macromolecular diffusion in the FFF channel. Depending on the field force used, FFF is classified into different techniques such as sedimentation, gravitational, electrical, thermal and flow FFF.112... 

In the sedimenter or gravity settler, the particles in the feed suspension settle due to difference in densities between the particles and the fluid. The settling particle velocity reaches a constant value - the terminal velocity - shortly after the start of sedimentation. The terminal velocity is defined by the following balance of forces acting on the particle ... 

Positive DEP drives particles towards field maxima and negative DEP moves them to field minima. It is only possible to create field maxima at the surfaces of the electrodes, no maximum can exist elsewhere in an isotropic solution. Trapping or positioning of particles by positive DEP therefore requires the dielectrophoretic force to be balanced by some other force (e.g. sedimentation, buoyancy or flow) and to be subjected to feedback control. This was first done with a two electrode system by Jones and Kaler [33-37]. 

In a sedimentation equilibrium experiment the cell is rotated at a relatively low speed (typically 5000-10000 rpm) until an equilibrium is attained whereby the centrifugal force just balances the tendency of the molecules to diffuse back against the concentration gradient developed. Measurements are made of the equilibrium concentration profiles for a series of solutions with different initial polymer concentrations so that the results can be extrapolated to c = 0. A rigorous thermodynamic treatment is possible and enables absolute values ot Mwand Mz, to be determined. The principal restriction to the use of sedimentation equilibrium measurements is the long time required to reach equilibrium, since this is at least a few hours and more usually is a few days. 

We can use the same record of the trajectory of a particle to determine at the same time its hydrodynamic radius and its density. For sedimentation at low Reynolds number, the average vertical velocity is given by the balance between the body force and the friction force ... 

Creaming or sedimentation occurs when the dispersed droplets or floccules separate under the influence of gravity to form a layer of more concentrated emulsion, the cream. Generally a creamed emulsion can be restored to its original state by gentle agitation. This process, which inevitably occurs in any dilute emulsion if there is a density difference between the phases as a consequence of Stokes law, should not be confused with flocculation which is due to particle interactions resulting from the balance of attractive and repulsive forces. Most oils are less dense than... 

---