In sedimentation, particles are falling from rest under the force of gravity. Therefore in sedimentation, eqn. (10.1) takes the familiar form of Stokes' Law:
Note that eqn.(10.2) is not dimensionless and so consistent units must be employed throughout. For example, in the SI system D would be m, g in m s-2, r in kg m-3 and m in N s m-2, and then vm would be in m s-1. Particle diameters are usually very small and are often measured in microns (micro-metres) = 10-6 m with the symbol mm.
Stoke's Law applies only in streamline flow and strictly only to spherical particles. In the case of spheres the criterion for streamline flow is that (Re) = 2, and many practical cases occur in the region of streamline flow, or at least where streamline flow is a reasonable approximation. Where higher values of the Reynolds number are encountered, more detailed references should be sought, such as Henderson and Perry (1955), Perry (1997) and Coulson and Richardson (1978).
For 10 mm particles since vm is proportional to the squares of the diameters,
Checking the Reynolds number for the 60 mm particles,
Stokes' Law applies only to cases in which settling is free, that is where the motion of one particle is unaffected by the motion of other particles. Where particles are in concentrated suspensions, an appreciable upward motion of the fluid accompanies the motion of particles downward. So the particles interfere with the flow patterns round one another as they fall. Stokes' Law predicts velocities proportional to the square of the particle diameters. In concentrated suspensions, it is found that all particles appear to settle at a uniform velocity once a sufficiently high level of concentration has been reached. Where the size range of the particles is not much greater than 10:1, all the particles tend to settle at the same rate. This rate lies between the rates that would be expected from Stokes' Law for the largest and for the smallest particles. In practical cases, in which Stoke's Law or simple extensions of it cannot be applied, probably the only satisfactory method of obtaining settling rates is by experiment.
In a cylinder in which a uniform suspension is allowed to settle, various quite well-defined zones appear as the settling proceeds. At the top is a zone of clear liquid. Below this is a zone of more or less constant composition, constant because of the uniform settling velocity of all sizes of particles. At the bottom of the cylinder is a zone of sediment, with the larger particles lower down. If the size range of the particles is wide, the zone of constant composition near the top will not occur and an extended zone of variable composition will replace it.
In a continuous thickener, with settling proceeding as the material flows through, and in which clarified liquid is being taken from the top and sludge from the bottom, these same zones occur. The minimum area necessary for a continuous thickener can be calculated by equating the rate of sedimentation in a particular zone to the counter-flow velocity of the rising fluid. In this case we have:
vu is the upward velocity of the flow of the liquid,
F is the mass ratio of liquid to solid in the feed, L is
the mass ratio of liquid to solid in the underflow liquid, dw/dt
is the mass rate of feed of the solids, r
is the density of the liquid and A is the settling area in the
From eqn. (10.2), vm = D2g(rp - rf)/18m
Sedimentation Equipment for separation of solid particles from liquids by gravitational sedimentation is designed to provide sufficient time for the sedimentation to occur and to permit the overflow and the sediment to be removed without disturbing the separation. Continuous flow through the equipment is generally desired, so the flow velocities have to be low enough to avoid disturbing the sediment. Various shaped vessels are used, with a sufficient cross-section to keep the velocities down and fitted with slow-speed scraper-conveyors and pumps to remove the settled solids. When vertical cylindrical tanks are used, the scrapers generally rotate about an axis in the centre of the tank and the overflow may be over a weir round the periphery of the tank, as shown diagrammatically in Fig. 10.1.
In some cases, where it is not practicable to settle out fine particles, these can sometimes be floated to the surface by the use of air bubbles. This technique is known as flotation and it depends upon the relative tendency of air and water to adhere to the particle surface. The water at the particle surface must be displaced by air, after which the buoyancy of the air is sufficient to carry both the particle and the air bubble up through the liquid.
Because it depends for its action upon surface forces, and surface forces can be greatly changed by the presence of even minute traces of surface active agents, flotation may be promoted by the use of suitable additives. In some instances, the air bubbles remain round the solid particles and cause froths. These are produced in vessels fitted with mechanical agitators, the agitators whip up the air-liquid mixture and overflow the froth into collecting troughs.
The greatest application of froth flotation is in the concentration of minerals, but one use in the food industry is in the separation of small particles of fat from water. Dissolving the air in water under pressure provides the froth. On the pressure being suddenly released, the air comes out of solution in the form of fine bubbles which rise and carry the fat with them to surface scrapers.
An important application, in the food industry, of sedimentation of solid particles occurs in spray dryers. In a spray dryer, the material to be dried is broken up into small droplets of about 100 mm diameter and these fall through heated air, drying as they do so. The necessary area so that the particles will settle can be calculated in the same way as for sedimentation. Two disadvantages arise from the slow rates of sedimentation: the large chamber areas required and the long contact times between particles and the heated air which may lead to deterioration of heat-sensitive products.
It is sometimes convenient to combine more than one force to effect a mechanical separation. In consequence of the low velocities, especially of very small particles, obtained when gravity is the only external force acting on the system, it is well worthwhile to also employ centrifugal forces. Probably the most common application of this is the cyclone separator. Combined forces are also used in some powder classifiers such as the rotary mechanical classifier and in ring dryers.
Cyclones are often used for the removal from air streams of particles of about 10 mm or more diameter. They are also used for separating particles from liquids and for separating liquid droplets from gases. The cyclone is a settling chamber in the form of a vertical cylinder, so arranged that the particle-laden air spirals round the cylinder to create centrifugal forces which throw the particles to the outside walls. Added to the gravitational forces, the centrifugal action provides reasonably rapid settlement rates. The spiral path, through the cyclone, provides sufficient separation time. A cyclone is illustrated in Fig. 10.2(a).
where Fc is the centrifugal force acting on the particle, m is the mass of the particle, v is the tangential velocity of the particle and r is the radius of the cyclone.
This equation shows that the force on the particle increases as the radius decreases, for a fixed velocity. Thus, the most efficient cyclones for removing small particles are those of smallest diameter. The limitations on the smallness of the diameter are the capital costs of small diameter cyclones to provide sufficient output, and the pressure drops.
The optimum shape for a cyclone has been evolved mainly from experience and proportions similar to those indicated in Fig. 10.2(a) have been found effective. The efficient operation of a cyclone depends very much on a smooth double helical flow being produced and anything which creates a flow disturbance or tends to make the flow depart from this pattern will have considerable and adverse effects upon efficiency. For example, it is important that the air enters tangentially at the top. Constricting baffles or lids should be avoided at the outlet for the air.
The efficiency of collection of dust in a cyclone is illustrated in Fig. 10.2(b). Because of the complex flow, the size cut of particles is not sharp and it can be seen that the percentage of entering particles which are retained in the cyclone falls off for particles below about 10 mm diameter. Cyclones can be used for separating particles from liquids as well as from gases and also for separating liquid droplets from gases.
Other mechanical flow separators for particles in a gas use the principal of impingement in which deflector plates or rods, normal to the direction of flow of the stream, abruptly change the direction of flow. The gas recovers its direction of motion more rapidly than the particles because of its lower inertia. Suitably placed collectors can then be arranged to collect the particles as they are thrown out of the stream. This is the principle of operation of mesh and fibrous air filters. Various adaptations of impingement and settling separators can be adapted to remove particles from gases, but where the particle diameters fall below about 5 mm, cloth filters and packed tubular filters are about the only satisfactory equipment.
Classification implies the sorting of particulate material into size ranges. Use can be made of the different rates of movement of particles of different sizes and densities suspended in a fluid and differentially affected by imposed forces such as gravity and centrifugal fields, by making suitable arrangements to collect the different fractions as they move to different regions.
Rotary mechanical classifiers, combining differential settling with centrifugal action to augment the force of gravity and to channel the size fractions so that they can be collected, have come into increasing use in flour milling. One result of this is that because of small differences in sizes, shapes and densities between starch and protein-rich material after crushing, the flour can be classified into protein-rich and starch-rich fractions. Rotary mechanical classifiers can be used for other large particle separation in gases.
Classification is also employed in direct air dryers, in which use is made of the density decrease of material on drying. Dry material can be sorted out as a product and wet material returned for further drying. One such dryer uses a scroll casing through which the mixed material is passed, the wet particles pass to the outside of the casing and are recycled while the material in the centre is removed as dry product.
Mechanical separations > CENTRIFUGAL SEPARATIONS
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