## Feed

Figure 11. Continuous solid-bowl centrifuge.

Because the radius of the solid discharge port is ususally less than the radius of the liquid overflow at the broader end of the bowl, part of the settled solids is submerged in the pond.

The remainder, closer to the center, is inside the free liquid interface, where they can drain before being discharged. The total length of the "settling" and "pressing out" zones depends on the dimensions of the rotor. Their relative length can be varied by changing the pond level through suitable adjustment of the liquid discharge radius. When the pond depth is lowered, the length of the pressing out zone increases with some sacrifice in the clarification effectiveness. The critical point in the transport of solids to the bowl wall is their transition across the free liquid interface, where the buoyancy effect of the continuous phase is lost. At this point, soft amorphous solids tend to flow back into the pond instead of discharging. This tendency can be overcome by raising the pond level so that its radius is equal to, or less than, that of the solids discharge port. In reality, there are no dry settled solids. The solids form a dam, which prevents the liquid from overflowing. The transfer of solids becomes possible because of the difference between the rotational speed of the screw conveyor and that of the bowl shell. The flights of the screw move through the settled solids and cause the solids to advance. To achieve this motion, it is necessary to have a high circumferential coefficient of friction on the solid particles with respect to the bowl shell and a low coefficient axially with respect to the bowl shell and across the conveyor flights. These criteria may be achieved by constructing the shell with conical grooves or ribs and by polishing the conveyor flights. The conveyor or differential speed is normally in the range of 0.8 % to 5 % of the bowl's rotational speed.

The required differential is achieved by a two-stage planetary gear box. The gear box housing carrying two ring gears is fixed to, and rotates with, the bowl shell. The first stage pinion is located on a shaft that projects outward from the housing. This arrangement provides a signal that is proportional to the torque imposed by the conveyor. If the shaft is held rotational (for example, by a torque overload release device or a shear pin), the relative conveyor speed is equivalent to the bowl rotative speed divided by the gear box ratio. Variable differential speeds can be obtained by driving the pinion shaft with an auxiliary power supply or by allowing it to slip forward against a controlled breaking action. Both arrangements are employed when processing soft solids or when maximum retention times are needed on the pressing out zone. The solids handling capacity of this type centrifuge is established by the diameter of the bowl, the conveyor's pitch and its differential speed. Feed ports should be located as far from the effluent discharge as possible to maximize the effective clarifying length. Note that the feed must be introduced into the pond to minimize disturbance and resuspension of the previously sedimented solids. As a general rule, the preferred feed location is near the intercept of the conical and cylindrical portions of the bowl shell. The angle of the sedimentation section with respect to the axis of rotation is typically in the range of 3 to 15°. A shallow angle provides a longer sedimentation area with a sacrifice in the effective length for clarification. In some designs, a portion of the conveyor flights in the sedimentation area is shrouded (as with a cone) to prevent intermixing of the sedimented solids with the free supernatant liquid in the pond through which they normally would pass. In other designs, the clarified liquid is discharged from the front end via a centrifugal pump or an adjustable skimmer that sometimes is used to control the pond level in the bowl. Some displacement of the adhering virgin liquor can be accomplished by washing the solids retained on the settled layer, particularly if the solids have a high degree of permeability. Washing efficiency ranges up to 90% displacement of virgin liquor on coarse solids. Some configurations enable the settled layer to have two angles; comparatively steep in the wetted portion (10-15°) and shallow in the dry portion (3-5°). A wash is applied at the intersection of these angles, which, in effect, forms a constantly replenished zone of pure liquid through which the solids are conveyed. The longer section of a dry shallow layer provides more time for drainage of the washed solids. In either washing system, the wash liquid that is not carried out with the solids fraction returns to the pond and eventually discharges along with the effluent virgin liquor.

Separation Rates in Tubular-and Solid-Bowl Centrifuges: To evaluate the radial velocity of a particle moving toward a centrifuge wall, the expression for particle settling in a gravitational field is applied:

where ur is the peripheral velocity at a distance, r, from the axis of rotation, and 'a' is centrifugal acceleration. Expressing ur in terms of the number of rotations, n, ur = 2itrn (2)

The centrifugal acceleration is:

Depending on the particle diameter and properties of the liquid, the radial motion of particles will be laminar, turbulent or transitional. The motion of large particles at Re > 500 is turbulent. Therefore, their settling velocity in a gravitational field may be expressed as:

Replacing g by centrifugal acceleration, a:

where yp = specific weight of the solids and y = specific weight of the liquid. Substituting for "a" into this last expression, we obtain the particle velocity in the radial direction of the wall:

Combining the above expressions, we determine the number of times the particle velocity in a centrifuge is greater than that in free particle settling:

For example, at n = 1200 rpm = 20 liter/sec and r = 0.5 m, the settling velocity in the centrifuge is almost 28 times greater than that in free settling. Note that the above expressions are applicable only for Re > 500. For small particles, Re < 2, migration toward the wall is laminar. The proper settling velocity expression for the gravitational field is u = d2(y - y)/18it (8)

Substituting in pg for y, we obtain u = d2(Pp - p)a/18fi

Replacing 'a' and with some algebraic manipulations we obtain:

For the same case of n = 1200 rpm and r = 0.5, we obtain ur/ug = 800, whereas for the turbulent regime the ratio was only 28. This example demonstrates that the centrifugal process is more effective in the separation of small particles than of large ones. Note that after the radial velocity ur is determined, it is necessary to check whether the laminar condition, Re < 2, is fulfilled. For the transition regime, 2 < Re < 500, the sedimentation velocity in the gravity field is:

The expression for particle radial velocity toward the wall is:

Let's consider the following example in order to get a better appreciation for the application of the above expressions. In this problem oil droplets (dp = 10-4 m, pp = 900 kg/m3, n = 10'3 cP) are to be separated in a sedimentation centrifuge. The machine operates at 5000 rpm (« = 2ir x 5000/60). If the distance of a single droplet from the axis of rotation is 0.1 m, determine the droplet's redial settling velocity.

The solution to this problems is as follows:

Arc = CDRe2 = {4 x 10"12 x 1000 x 100(271 X 5000/60)2 x 0.1}/(3 x 10"6) = 3650

Note that the absolute value of pp - p has been assumed. The negative value of this difference indicates that the droplet displacement is centripetal. The value of the Reynolds number corresponding to Arc = 3650 from Figure 10 is 45. Hence we can determine the radial settling velocity from the definition of the Reynolds number:

45 = 10"4 x 1000 ur/10"3 ur = 0.45 m/sec.

If the separation were to occur in a gravitational field only, the droplet velocity would be ur = 5.45 X 10"4 m/sec. For laminar droplet motion, this corresponds to a separation number of 2800. However, in this case the flow is transitional. Particle settling velocity in a centrifuge depends on the particle location. For the laminar regime, the particle velocity is proportional to the centrifuge radius; however, for the turbulent regime, it is related to i/r. In an actual operation, a particle may blow over different regimes depending on its location. Particles of different sizes and densities may be located at the same point with the same velocity because the larger particle diameter tends to compensate for lower density. Two particles with identical settling velocities in a gravitational field will settle in the same manner in a centrifugal field, provided the regimes of motion remain unchanged. For example, if the gravitational sedimentation is laminar and centrifugal sedimentation is transitional or turbulent, then particles will have different velocities within the centrifuge. Under certain conditions of centrifugal sedimentation (characterized by r and w), the settling velocity for a gravitational field can be applied. This is exactly correct when gravitational settling occurs in the turbulent regime.. However, when a particle passes from one regime to the other, or when it is in the transitional regime, one either must calculate the velocities or extrapolate from experimental results. Extrapolation is valid only in cases in which there are no changes in the flow regime. Consider a particle (d = 5 x 10"4 in) located at a distance of 0.2 m from the axis of a centrifuge operating at 4000 rpm has a velocity of 1 m/sec through water. In this case, Re = 1 X 1000 x 0.0005/0.001 = 500 and, hence, the regime is turbulent. It is possible to extrapolate to conditions of greater radii and higher rotations. If the centrifuge operates at 5000 rpm and the distance is 0.25 m from axis of rotation, then the settling velocity is:

If the centrifuge operates at 4000 rpm and the particle's distance from the axis of rotation is 0.25 m, the settling velocity is only 0.01 m/sec, which corresponds to:

In this latter case the flow is laminar and extrapolation can be only to much smaller rotation velocities and radii. If the particle now settles in the centrifuge operating at 3000 rpm and the distance of the particle from the axis of rotation is 0.20 m, the settling velocity is:

ur = 0.01(30002 X 0.20)/(40002 X 0.25) = 0.0045 m/sec

Note that the above calculations are based on nonhindered sedimentation and, therefore, should be modified for a hindered "fall."

Estimating Capacities of Tubular- and Solid-Bowl Centrifuges: When a rotating centrifuge is filled with suspension, the internal surface of liquid acquires a cylindrical geometry of radius R,, as shown in Figure 11. The free surface is normal at any point to the resultant force acting on a liquid particle. If the liquid is lighter than the solid particles, the liquid moves toward the axis of rotation while the solids flow toward the bowl walls. The flow of the continuous liquid phase is effectively axial. A simplified model of centrifuge operation is that of a cylinder of fluid rotating about its axis. The flow forms a layer bound outwardly by a cylinder, R2, and inwardly by a free cylindrical surface, R, (Figurel3). This surface is, at any point, normal to a resulting force (centrifugal and gravity) acting on the solid particle in the liquid. The gravity force is, in general, negligible compared to the centrifugal force, and the surface of liquid is perpendicular to the direction of centrifugal force. Consider a solid particle located at distance R from the axis of rotation. The particle moves centrifugally with a settling velocity, u, while liquid particles move in the opposite direction centripetally with a velocity Uf where V - volume (m3), t = time (sec), and C = height of the bowl (m).

The resulting velocity will be centrifugal, and the solid particles will be separated; provided that:

The capacity of the centrifuge will be: dV/di = 2tiR?us

The settling capacity for a given size of particles is a function of R, (and us, which itself is proportional to R. In general, for the sedimentation of heavy particles in a suspension it is sufficient that the radial component of % be less than us at a radius greater than R2.

Because of turbulence effects, it is generally good practice to limit the settling capacity so that us again exceeds uf near Rj. The same situation occurs when the particles are lighter than the continuous liquid. The relation between us and R depends on the regime. In the laminar regime, us is proportional to R, whereas in the turbulent regime, it is related to ^R. Most industrial sedimentation centrifuges operate in the transition regime.

Disk-Bowl Centrifuges - Disk-bowl centrifuges are used widely for separating emulsions, clarifying fine suspensions and separating immiscible liquid mixtures. Although these machines are generally not applied to wastewater applications, and are more usually found in food processing, they can find niche applications in water treatment. More sophisticated designs can separate immiscible liquid mixtures of different specific gravities while simultaneously removing solids. Figure 15 illustrates the physical separation of two liquid components within a stack of disks. The light liquid phase builds up in the inner section, and the heavy phase concentrates in the outer section. The dividing line between the two is referred to as the "separating zone." For the most efficient separation this is located along the line of the rising channels, which are a series of holes in each disk, arranged so that the holes provide vertical channels through the entire disk set. These channels also provide access for the liquid mixture into the spaces between the disks. Centrifugal force causes the two liquids to separate, and the solids move outward to the sediment-holding space.

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