14.4. CENTRIFUGAL SEPARATION PROCESSES

14.4A. Introduction

1. Centrifugal settling or sedimentation

In Section 14.3 were discussed the processing methods of settling and sedimentation, where particles are separated from a fluid by gravitational forces acting on the particles. The particles were solid, gas, or liquid and the fluid was a liquid or a gas. In the present section we discuss settling or separation of particles from a fluid by centrifugal forces acting on the particles.

Use of centrifuges increases the forces on particles manyfold. Hence, particles that will not settle readily or at all in gravity settlers can often be separated from fluids by centrifugal force. The high settling force means that practical rates of settling can be obtained with much smaller particles than in gravity settlers. These high centrifugal forces do not change the relative settling velocities of small particles, but these forces do overcome the disturbing effects of Brownian motion and free convection currents.

Sometimes gravity separation may be too slow because of the closeness of the densities of the particles and the fluid, or because of association forces holding the components together, as in emulsions. An example in the dairy industry is the separation of cream from whole milk, giving skim milk. Gravity separation takes hours, while centrifugal separation is accomplished in minutes in a cream separator. Centrifugal settling or separation is employed in many food industries, such as breweries, vegetable-oil processing, fish-protein-concentrate processing, fruit juice processing to remove cellular materials, and so on. Centrifugal separation is also used in drying crystals and for separating emulsions into their constituent liquids or solid–liquid. The principles of centrifugal sedimentation are discussed in Sections 14.4B and 14.4C.

2. Centrifugal filtration

Centrifuges are also used in centrifugal filtration, where a centrifugal force is used instead of a pressure difference to cause the flow of slurry in a filter where a cake of solids builds up on a screen. The cake of granular solids from the slurry is deposited on a filter medium held in a rotating basket, washed, and then spun “dry.” Centrifuges and ordinary filters are competitive in most solid–liquid separation problems. The principles of centrifugal filtration are discussed in Section 14.4E.

14.4B. Forces Developed in Centrifugal Separation

1. Introduction

Centrifugal separators make use of the common principle that an object whirled about an axis or center point at a constant radial distance from the point is acted on by a force. The object being whirled about an axis is constantly changing direction and is thus accelerating, even though the rotational speed is constant. This centripetal force acts in a direction toward the center of rotation.

If the object being rotated is a cylindrical container, the contents of fluid and solids exert an equal and opposite force, called centrifugal force, outward to the walls of the container. This is the force that causes settling or sedimentation of particles through a layer of liquid or filtration of a liquid through a bed of filter cake held inside a perforated rotating chamber.

In Fig. 14.4-1a a cylindrical bowl is shown rotating, with a slurry feed of solid particles and liquid being admitted at the center. The feed enters and is immediately thrown outward to the walls of the container, as in Fig. 14.4-1b. The liquid and solids are now acted upon by the vertical gravitational force and the horizontal centrifugal force. The centrifugal force is usually so large that the force of gravity may be neglected. The liquid layer then assumes the equilibrium position, with the surface almost vertical. The particles settle horizontally outward and press against the vertical bowl wall.

Figure 14.4-1. Sketch of centrifugal separation: (a) initial slurry feed entering, (b) settling of solids from a liquid, (c) separation of two liquid fractions.


In Fig. 14.4-1c two liquids having different densities are being separated by the centrifuge. The denser fluid will occupy the outer periphery, since the centrifugal force on it is greater.

2. Equations for centrifugal force

In circular motion the acceleration due to the centrifugal force is

Equation 14.4-1


where ae is the acceleration from a centrifugal force in m/s2 (ft/s2), r is radial distance from the center of rotation in m (ft), and ω is angular velocity in rad/s.

The centrifugal force Fc in N (lbf) acting on the particle is given by

Equation 14.4-2


where gc = 32.174 lbm · ft/lbfs2.

Since ω = ν/r, where ν is the tangential velocity of the particle in m/s (ft/s),

Equation 14.4-3


Often rotational speeds are given as N rev/min and

Equation 14.4-4


Equation 14.4-5


Substituting Eq. (14.4-4) into Eq. (14.4-2),

Equation 14.4-6


By Eq. (14.3-2), the gravitational force on a particle is

Equation 14.3-2


where g is the acceleration of gravity and is 9.80665 m/s2. In terms of gravitational force, the centrifugal force is as follows, by combining Eqs. (14.3-2), (14.4-2), and (14.4-3):

Equation 14.4-7


Hence, the force developed in a centrifuge is rω2/g or ν2/rg times as large as the gravitational force. This is often expressed as equivalent to so many g forces.

EXAMPLE 14.4-1. Force in a Centrifuge

A centrifuge having a radius of the bowl of 0.1016 m (0.333 ft) is rotating at N = 1000 rev/min.

  1. Calculate the centrifugal force developed in terms of gravity forces.

  2. Compare this force to that for a bowl with a radius of 0.2032 m rotating at the same rev/min.

Solution: For part (a), r = 0.1016 m and N = 1000. Substituting into Eq. (14.4-7),


For part (b), r = 0.2032 m. Substituting into Eq. (14.4-7),



14.4C. Equations for Rates of Settling in Centrifuges

1. General equation for settling

If a centrifuge is used for sedimentation (removal of particles by settling), a particle of a given size can be removed from the liquid in the bowl if there is sufficient residence time of the particle in the bowl for the particle to reach the wall. For a particle moving radially at its terminal settling velocity, the diameter of the smallest particle which can be removed can be calculated.

In Fig. 14.4-2 a schematic of a tubular-bowl centrifuge is shown. The feed enters at the bottom, and it is assumed that all the liquid moves upward at a uniform velocity, carrying solid particles with it. The particle is assumed to be moving radially at its terminal settling velocity νt. The trajectory or path of the particle is shown in Fig. 14.4-2. A particle of a given size is removed from the liquid if sufficient residence time is available for the particle to reach the wall of the bowl, where it is held. The length of the bowl is b m.

Figure 14.4-2. Particle settling in sedimenting tubular-bowl centrifuge.


At the end of the residence time of the particle in the fluid, the particle is at a distance rB m from the axis of rotation. If rB < r2, then the particle leaves the bowl with the fluid. If rB = r2, it is deposited on the wall of the bowl and effectively removed from the liquid.

For settling in the Stokes' law range, the terminal settling velocity at a radius r is obtained by substituting Eq. (14.4-1) for the acceleration g into Eq. (14.3-9):

Equation 14.4-8


where νt is settling velocity in the radial direction in m/s, Dp is particle diameter in m, ρp is particle density in kg/m3, ρ is liquid density in kg/m3, and μ is liquid viscosity in Pa · s. If hindered settling occurs, the right-hand side of Eq. (14.4-8) is multiplied by the factor (ε2ψp) given in Eq. (14.3-16).

Since νt = dr/dt, then Eq. (14.4-8) becomes

Equation 14.4-9


Integrating between the limits r = r1 at t = 0 and r = r2 at t = tT,

Equation 14.4-10


The residence time tT is equal to the volume of liquid V m3 in the bowl divided by the feed volumetric flow rate q in m3/s. The volume . Substituting into Eq. (14.4-10) and solving for q,

Equation 14.4-11


Particles having diameters smaller than that calculated from Eq. (14.4-11) will not reach the wall of the bowl and will go out with the exit liquid. Larger particles will reach the wall and be removed from the liquid.

A cut point or critical diameter Dpc can be defined as the diameter of a particle that reaches half the distance between r1 and r2. This particle moves a distance of half the liquid layer or (r2r1)/2 during the time this particle is in the centrifuge. The integration is then between r = (r1 + r2)/2 at t = 0 and r = r2 at t = tT. Then we obtain

Equation 14.4-12


At this flow rate qc, particles with a diameter greater than Dpc will predominantly settle to the wall and most smaller particles will remain in the liquid.

2. Special case for settling

For the special case where the thickness of the liquid layer is small compared to the radius, Eq. (14.4-8) can be written for a constant rr2 and Dp = Dpc as follows:

Equation 14.4-13


The time of settling tT is then as follows for the critical Dpc case:

Equation 14.4-14


Substituting Eq. (14.4-13) into (14.4-14) and rearranging,

Equation 14.4-15


The volume V can be expressed as

Equation 14.4-16


Combining Eqs. (14.4-15) and (14.4-16),

Equation 14.4-17


The analysis above is somewhat simplified. The pattern of flow of the fluid is actually more complicated. These equations can also be used for liquid–liquid systems where droplets of liquid migrate according to the equations and coalesce in the other liquid phase.

EXAMPLE 14.4-2. Settling in a Centrifuge

A viscous solution containing particles with a density ρp = 1461 kg/m3 is to be clarified by centrifugation. The solution density ρ = 801 kg/m3 and its viscosity is 100 cp. The centrifuge has a bowl with r2 = 0.02225 m, r1 = 0.00716 m, and height b = 0.1970 m. Calculate the critical particle diameter of the largest particles in the exit stream if N = 23 000 rev/min and the flow rate q = 0.002832 m3/h.

Solution: Using Eq. (14.4-4),


The bowl volume V is


Viscosity μ = 100 × 103 = 0.100 Pa · s = 0.100 kg/m · s. The flow rate qc is


Substituting into Eq. (14.4-12) and solving for Dpc,


Substituting into Eq. (14.4-13) to obtain νt and then calculating the Reynolds number, the settling is in the Stokes' law range.


3. Sigma values and scale-up of centrifuges

A useful physical characteristic of a tubular-bowl centrifuge can be derived by multiplying and dividing Eq. (14.4-12) by 2g and then substituting Eq. (14.3-9) written for Dpc into Eq. (14.4-12) to obtain

Equation 14.4-18


where νt is the terminal settling velocity of the particle in a gravitational field and

Equation 14.4-19


where Σ is a physical characteristic of the centrifuge and not of the fluid–particle system being separated. Using Eq. (14.4-17) for the special case of settling for a thin layer,

Equation 14.4-20


The value of Σ is really the area in m2 of a gravitational settler that will have the same sedimentation characteristics as the centrifuge for the same feed rate. To scale up from a laboratory test of q1 and Σ1 to q2 (for νt1 = νt2),

Equation 14.4-21


This scale-up procedure is dependable for centrifuges of similar type and geometry and if the centrifugal forces are within a factor of 2 from each other. If different configurations are involved, efficiency factors E should be used, where q11 E1 = q22 E2. These efficiencies must be determined experimentally, and values for different types of centrifuges are given elsewhere (F1, P1).

4. Separation of liquids in a centrifuge

Liquid–liquid separations in which the liquids are immiscible but finely dispersed as in an emulsion are common operations in the food and other industries. An example is the dairy industry, in which the emulsion of milk is separated into skim milk and cream. In these liquid–liquid separations, the position of the outlet overflow weir in the centrifuge is very important, not only in controlling the volumetric holdup V in the centrifuge but also in determining whether a separation is actually made.

In Fig. 14.4-3, a tubular-bowl centrifuge is shown in which the centrifuge is separating two liquid phases, one a heavy liquid with density ρH kg/m3 and the second a light liquid with density ρL. The distances shown are as follows: r1 is radius to surface of light liquid layer, r2 is radius to liquid–liquid interface, and r4 is radius to surface of heavy liquid downstream.

Figure 14.4-3. Tubular bowl centrifuge for separating two liquid phases.


To locate the interface, a balance must be made of the pressures in the two layers. The force on the fluid at distance r is, by Eq. (14.4-2),

Equation 14.4-2


The differential force across a thickness dr is

Equation 14.4-22


But,

Equation 14.4-23


where b is the height of the bowl in m and (2πrb) dr is the volume of fluid. Substituting Eq. (14.4-23) in (14.4-22) and dividing both sides by the area A = 2πrb,

Equation 14.4-24


where P is pressure in N/m2 (lbf/ft2).

Integrating Eq. (14.4-24) between r1 and r2,

Equation 14.4-25


Applying Eq. (14.4-25) to Fig. 14.4-3 and equating the pressure exerted by the light phase of thickness r2r1 to the pressure exerted by the heavy phase of thickness r2r4 at the liquid–liquid interface at r2,

Equation 14.4-26


Solving for, the interface position,

Equation 14.4-27


The interface at r2 must be located at a radius smaller than r3 in Fig. 14.4-3.

EXAMPLE 14.4-3. Location of Interface in Centrifuge

In a vegetable-oil-refining process, an aqueous phase is being separated from the oil phase in a centrifuge. The density of the oil is 919.5 kg/m3 and that of the aqueous phase is 980.3 kg/m3. The radius r1 for overflow of the light liquid has been set at 10.160 mm and the outlet for the heavy liquid at 10.414 mm. Calculate the location of the interface in the centrifuge.

Solution: The densities are ρL = 919.5 and ρH = 980.3 kg/m3. Substituting into Eq. (14.4-27) and solving for r2,



13.4D. Centrifuge Equipment for Sedimentation

1. Tubular centrifuge

A schematic of a tubular-bowl centrifuge is shown in Fig. 14.4-3. The bowl is tall and has a narrow diameter, 100–150 mm. Such centrifuges, known as supercentrifuges, develop a force about 13 000 times the force of gravity. Some narrow centrifuges, having a diameter of 75 mm and very high speeds of 60 000 or so rev/min, are known as ultracentrifuges. These supercentrifuges are often used to separate liquid–liquid emulsions.

2. Disk bowl centrifuge

The disk bowl centrifuge shown in Fig. 14.4-4 is often used in liquid–liquid separations. The feed enters the actual compartment at the bottom and travels upward through vertically spaced feed holes, filling the spaces between the disks. The holes divide the vertical assembly into an inner section, where mostly light liquid is present, and an outer section, where mainly heavy liquid is present. This dividing line is similar to an interface in a tubular centrifuge.

Figure 14.4-4. Schematic of disk bowl centrifuge.


The heavy liquid flows beneath the underside of a disk to the periphery of the bowl. The light liquid flows over the upper side of the disks and toward the inner outlet. Any small amount of heavy solids is thrown to the outer wall. Periodic cleaning is required to remove solids deposited. Disk bowl centrifuges are used in starch–gluten separation, concentration of rubber latex, and cream separation. Details are given elsewhere (P1, L1).

14.4E. Centrifugal Filtration

1. Theory for centrifugal filtration

Theoretical prediction of filtration rates in centrifugal filters have not been too successful. The filtration in centrifuges is more complicated than for ordinary filtration using pressure differences, since the area for flow and driving force increase with distance from the axis and the specific cake resistance may change markedly. Centrifuges for filtering are generally selected by scale-up from tests on a similar-type laboratory centrifuge using the slurry to be processed.

The theory of constant-pressure filtration discussed in Section 14.2E can be modified and used where centrifugal force causes the flow instead of impressed pressure difference. The equation will be derived for the case where a cake has already been deposited, as shown in Fig. 14.4-5. he inside radius of the basket is r2, ri is the inner radius of the face of the cake, and r1 is the inner radius of the liquid surface. We will assume that the cake is nearly incompressible so that an average value of α can be used for the cake. Also, the flow is laminar. If we assume a thin cake in a large-diameter centrifuge, then the area A for flow is approximately constant. The velocity of the liquid is

Equation 14.4-28


Figure 14.4-5. Physical arrangement for centrifugal filtration.


where q is the filtrate flow rate in m3/s and ν the velocity. Substituting Eq. (14.4-28) into (14.2-8),

Equation 14.4-29


where mc = cSV, mass of cake in kg deposited on the filter.

For a hydraulic head of dz m, the pressure drop is

Equation 14.4-30


In a centrifugal field, g is replaced by rω2 from Eq. (14.4-1) and dz by dr. Then,

Equation 14.4-31


Integrating between r1 and r2,

Equation 14.4-32


Combining Eqs. (14.4-29) and (14.4-32) and solving for q,

Equation 14.4-33


For the case where the flow area A varies considerably with the radius, the following has been derived (G1):

Equation 14.4-34


where A2 = 2πr2b (area of filter medium), (logarithmic cake area), and (arithmetic mean cake area). This equation holds for a cake of a given mass at a given time. It is not an integrated equation covering the whole filtration cycle.

2. Equipment for centrifugal filtration

In a centrifugal filter, slurry is fed continuously to a rotating basket which has a perforated wall and is covered with a filter cloth. The cake builds up on the surface of the filter medium to the desired thickness. hen, at the end of the filtration cycle, feed is stopped, and wash liquid is added or sprayed onto the cake. hen the wash liquid is stopped and the cake is spun as dry as possible. he motor is then shut off or slowed and the basked allowed to rotate while the solids are discharged by a scraper knife, so that the solids drop through an opening in the basket floor. Finally, the filter medium is rinsed clean to complete the cycle. Usually, the batch cycle is completely automated. Automatic batch centrifugals have basket sizes up to about 1.2 m in diameter and usually rotate below 4000 rpm.

Continuous centrifugal filters are available with capacities up to about 25 000 kg solids/h. Intermittently, the cake deposited on the filter medium is removed by being pushed toward the discharge end by a pusher, which then retreats, allowing the cake to build up once more. As the cake is being pushed, it passes through a wash region. The filtrate and wash liquid are kept separate by partitions in the collector. Details of different types of centrifugal filters are available (P1).

14.4F. Gas–Solid Cyclone Separators

1. Introduction and equipment

For separation of small solid particles or mist from gases, the most widely used type of equipment is the cyclone separator, shown in Fig. 14.4-6. The cyclone consists of a vertical cylinder with a conical bottom. The gas–solid particle mixture enters in a tangential inlet near the top. This gas–solid mixture enters in a rotating motion, and the vortex formed develops centrifugal force, which throws the particles radially toward the wall.

Figure 14.4-6. Gas–solid cyclone separator: (a) side view, (b) top view.


On entering, the air in the cyclone flows downward in a spiral or vortex adjacent to the wall. When the air reaches near the bottom of the cone, it spirals upward in a smaller spiral in the center of the cone and cylinder. Hence, a double vortex is present. The downward and upward spirals are in the same direction.

The particles are thrown toward the wall and fall downward, leaving out the bottom of the cone. A cyclone is a settling device in which the outward force on the particles at high tangential velocities is many times the force of gravity. Hence, cyclones accomplish much more effective separation than gravity settling chambers.

The centrifugal force in a cyclone ranges from about 5 times gravity in large, low-velocity units to 2500 times gravity in small, high-resistance units. These devices are used often in many applications, such as in spray-drying of foods, where the dried particles are removed by cyclones; in cleaning dust-laden air; and in removing mist droplets from gases. Cyclones offer one of the least expensive means of gas–particle separation. They are generally applicable in removing particles over 5 μm in diameter from gases. For particles over 200 μm in size, gravity settling chambers are often used. Wet-scrubber cyclones are sometimes used, where water is sprayed inside, helping to remove the solids.

2. Theory for cyclone separators

It is assumed that particles on entering a cyclone quickly reach their terminal settling velocities. Particle sizes are usually so small that Stokes' law is considered valid. For centrifugal motion, the terminal radial velocity νtR is given by Eq. (14.4-8), with νtR being used for νt:

Equation 14.4-35


Since ω = νtan/r, where νtan is tangential velocity of the particle at radius r, Eq. (14.4-35) becomes

Equation 14.4-36


where νt is the gravitational terminal settling velocity νt in Eq. (14.3-9).

The higher the terminal velocity νt, the greater the radial velocity νtR and the easier it should be to “settle” the particle at the walls. However, the evaluation of the radial velocity is difficult, since it is a function of gravitational terminal velocity, tangential velocity, and position radially and axially in the cyclone. Hence, the following empirical equation is often used (S2):

Equation 14.4-37


where b1 and n are empirical constants.

3. Efficiency of collection of cyclones

Smaller particles have smaller settling velocities according to Eq. (14.4-37) and do not have time to reach the wall to be collected. Hence, they leave with the exit air in a cyclone. Larger particles are more readily collected. The efficiency of separation for a given particle diameter is defined as the mass fraction of the size particles that are collected.

A typical collection-efficiency plot for a cyclone shows that the efficiency rises rapidly with particle size. The cut diameter Dpc is the diameter for which one-half of the mass of the entering particles is retained.

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