## Sedimenting Centrifuges

Ideal System

Newton and Stokes have promulgated the laws describing the movement of particles. When a force is applied to a particle it is accelerated:

As a practical matter, cake rates are difficult to measure. This can be addressed by manipulating mass balances. Recovery can be defined in terms of insoluble (suspended) solids concentrations, which may be more accurately determined than cake rates.

In a static settling tank under the influence of the Earth's gravity, the particle settles along the radius of the earth. When g is the gravitational constant:

In a centrifugal field, the acceleration, w2r, results in a force that acts normal to the axis of rotation (Figure 4):

In a sedimenting centrifuge, a continuous liquid phase moves through the rotor. In order to accomplish a useful separation, the discontinuous phase - either the insoluble solids or immiscible liquids drops (or both) - must move in a direction different from the flow of the continuous liquid. Stokes' law is usually applied to describe the relationship. The effective force accelerating the particle in a centrifugal field is then described:

Recovery (%) = (c/f )((f — e)/(c — e)) x 100 [12]

Recovery then is also a function of feed solids concentration. Effluent quality is not the sole measure of recovery. High solids concentration in the effluent may simply mean that the feed solids are high. Conversely, lack of solids in the effluent may simply mean lack of solids in the feed, not a high level of recovery.

The use of overall percentage recovery may not be adequate to compare dissimilar centrifuges, especially where mp is the mass of the particle and mr is the mass of the liquid displaced by the particle. If we define Ap = (pp — Pi), the difference in the density between the particle and the continuous liquid phase, then for a spherical particle of diameter, D:

If the diameter is small, or the viscosity is high, the particle moves at a velocity below the turbulent range and Stokes' law defines the force of the liquid phase

where :

[j is rotational velocity (rad s"1) r is radius of rotation in inches g is gravitational constant (32.2 in s"2)

Figure 4 G-level.

resisting the particle as:

If the particle settles long enough (reaches equilibrium), then Fl = Fp and, in a centrifugal field:

In the Earth's gravitational field:

The difference between the velocity in the centrifugal field and in the Earth's gravitational field is twofold. The first difference is that the velocity in the centrifugal field may be three to four orders of magnitude higher. The second is that the velocity in a centrifugal field depends on the distance from the centre of rotation, so that the velocity increases as the particle moves outward from the centre of rotation. In the Earth's gravitational field, the velocity is considered independent of position.

### Sigma Value

The most widely used method of quantifying capacity in sedimenting centrifuges is the sigma value which was introduced by Ambler in the 1950s. Sigma is used as an index of centrifuge size and typically has units of cm2.

The sigma concept attempts to isolate the process system factors effecting separation from the centrifuge factors effecting separation (Figure 1). The tubular bowl was the first centrifuge to which sigma is applied. The tubular bowl is a rotating cylinder in which feed is introduced through the bottom end cap. The continuous fluid flows through the rotor and overflows the top of the bowl. If the solid particles having a specific gravity higher than the liquid are successfully separated, they accumulate on the inside of the rotor and are removed batchwise by manually cleaning the bowl. If the distance settled (x) is small, the velocity is constant, eqn [19] then can be expanded:

If we consider an ideal system, half of the particles of diameter D would be removed when:

where vs characterizes the process system:

and £ characterizes the centrifuge:

with re and se being the effective radius and effective settling distance in the centrifuge.

The problem then is to define re and se. If the liquid layer is not thin, Ambler considered that:

Ambler maximized the approximation for the tubular bowl as:

Svarovsky and Vesilind each use slightly different approximations for the effective radius.

Records argues that a second derivation assuming that all particles start on the surface instead of uniformly distributed throughout the annular space yields:

Clearly as the depth of liquid decreases r1 p r2, the values for both estimates of £ become equal.

The equivalent area of a decanter and a gravity settling tank is shown in Figure 3 and Figure 5, respectively.

Pool surface area Solids compaction volume

Pool surface area Solids compaction volume

Figure 5 Gravity settling tank.

Sigma assumptions The assumptions can be divided as follows:

• Stokes' law: The particles or droplets are spherical and uniform in size. Settling of a particle is unhindered by the smaller particles ahead of it. The particles do not deaggregate, deflocculate, agglomerate, precipitate, dissolve, emulsify or flocculate. There is no change in viscosity or density (little or no temperature change).

• Reynolds' number: The value for the Reynolds' number, (vspD)/r\, is less than one, so that the deviation from the Stokes settling velocity is relatively small.

• Distribution: The particles are evenly distributed in the continuous liquid phase. The feed is uniformly introduced into the full space available for its flow. The flow is streamlined. There is no displacement of flow of the continuous phase by the sedimented particle phase or the introduction of feed. There is no remixing of the continuous and discontinuous phases.

Sigma limitation: similarity of feed Since £ is the index of the size of the centrifuge, traditionally the throughput (Q1) of a centrifuge of a size (£1) has been used to determine the throughput (Q2) to a usually larger size (£2) centrifuge. In the normal course of commerce, the performance of the test centrifuge with occurs at a time and place different from that in which the centrifuge with £2 will operate. The small unit may be tested on lab batches, months or even years ahead of the construction of a full-scale plant. Eqn [24] can be restated as:

and:

It is important to remember that:

if and only if:

The process system parameters that allow vs1 must be duplicated to allow vs1.

The feed stream and process system should be properly documented to ensure that the process system does not adversely effect the following properties described in eqn [15]:

It is generally assumed that increasing the sediment-ing velocity (vs) produces a better (more complete, faster, possibly more economical) separation. Therefore increasing vs increases sedimentation capacity at constant Eqn [14] illustrates several important relationships:

• The larger the particle diameter, the greater the sedimentation rate.

Corollaries:

A. Flocculation may enhance performance by increasing particle size.

B. Care should be taken in those process steps ahead of the centrifuge to limit particle-size degradation by either mechanical or biological means.

• The greater the difference in the density between the particle and the continuous phase, the greater the sedimentation rate.

Corollaries:

A. Temperature is important. If the density differences are small, the percentage change in density of the continuous phase may be significant. The density of water is normally taken as unity, but actually changes by approximately 10% from 10°C to 30°C.

B. In certain systems, e.g. mineral oil and water, there may be no density difference at a given temperature, therefore separation would not be possible. Changing the temperature and thus the densities would make separation possible. In extreme cases, changing the temperature may invert the light and heavy immiscible phases.

• The lower the viscosity of the continuous phase, the greater the sedimentation rate. Corollaries:

A. Again, temperature is important. Warmer (not approaching the boiling point, and in the absence of significant increases in the solubility of the particles) is generally better than colder.

B. Materials such as tar, that may be solid at room temperature, may be liquid with a low enough viscosity for processing at elevated temperature.

Parameters such as the speed of the feed tank agitator, the type of feed pump impeller, and ambient cooling owing to seasonal temperature fluctuations, may adversely impact the separation. In biologically active systems, factors such as differences in pH, alkalinity or volatile solids may indicate a difference in the feed stock to the separation system.

Sigma limitation: efficiency The sedimentation that the sigma value attempts to quantify is only a portion of the task to be accomplished. By assumption, sigma allows comparison of centrifuges which are geometrically and hydrodynamically similar. In practice, an efficiency factor is often introduced to extend the use of sigma to compare dissimilar centrifuges. Therefore we can expand eqn [21]:

If the two centrifuges are geometrically and hydro-dynamically equal, the efficiency factors cancel. Axelsson has attempted to quantify the efficiency of the various types of sedimenting centrifuges and has provided the data in Table 1.

Scale-up The sigma formula for the various types of imperforate centrifuges are listed in Table 2.

When testing a new material for separability on a centrifuge, a bottle centrifuge (Figure 6) is usually used to estimate the G-level required. To estimate size from the bottle centrifuge:

By adapting eqn [34], the full-scale centrifuge (EL) for the full-scale flow (QL) can be determined:

Sedimenting centrifuge type |
Efficiency factor (e) | |

Disc stack |
45 |
73% |

Decanter |
54 |
67% |

Tubular bowl |
90 |
98% |

Centrifuge type |
Sigma formula |

Bottle centrifuge |
®2V/2g[rc/(rc # r,)] |

Imperforate bowls |
2nl(®2/g)(4r2 + ir?) |

(tubular, decanter, basket) | |

Disc stack |
(2nn®2/3g)(cot 0(r3 - r?)) |

The sizing should then be confirmed by testing the selected centrifuge type.

The sigma concept indexes the size of centrifuges based solely on sedimentation performance. Other criteria and limitations must also be considered. These limitations most often involve the ability of the centrifuge to handle solids once they are sedimented. This may require knowledge of solids residence time, G-level, solids transportability (conveyability or flowability), compressibility and recognition of the limits on torque and solids loading.

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