Separation efficiency 1 Mass balance Equations

A separating unit operation recovers solid particles from a continuous liquid phase. Assuming no accumulation of particulate material then:

M o ~ M c ~ ^ M f 42

where M = mass flow rate (kg s'^)

o = initial (feed) particles c = coarse particle fraction f = fine particle fraction

If there is no change in particle size inside the separator (no disruption nor agglomeration), then the mass balance also applies to each particle size component. The particle size distribution frequency gives the fraction of particles of size x in a sample. Therefore the total mass of particles of a certain size range Ax present in a process stream is the mass flow rate multiplied by the appropriate size fraction. Hence:

M c ^ = 43

Ax Ax Ax

3.1.2 Total efficiency

The total separation efficiency of particles or mass yield (EJ, is defined as:

E, = — or Et = 44

h/to iVf o

These equations for separation efficiency can be substituted into the mass balance equation (45) in terms of the particle components:

45

3.1.3 Grade efficiency

It is very rare for a unit operation to yield a sharp separation where the process stream is divided entirely into particles above and below a specified size. More diffuse separations are achieved in practice where the separation efficiency of a particular piece of equipment changes with particle size. This changing performance is described by the grade efficiency T(x) and grade efficiency curves are obtained by plotting the separation efficiency for every particle size x:

T(x) = (M iL . 46

or in terms of each particle size component:

Ax /

47

Grade efficiency curves are typically S-shaped as shown in Figure 3.2.2 in section 3.2.1. They show the probability of particles of a certain size being recovered. For example, if many particles of size x$o enter the separator, then it is likely that 50% of them will follow trajectories which will allow them to settle out. "X5 0" is called the equiprobable cut size for separation. By contrast, x^oo or

Xjnax is the largest particle which may remain in the fine particle stream, and is called the limit of separation. Where diameter is used as a measure of the size of spherical particles, then X50 and Xjoo are usually termed dgg and dioo respectively.

In applying the grade efficiency concept to centrifugal separation, particle size analysis is usually carried out on the feed and supernatant. The mass yield is first calculated from:

where d^in and d^ax are the minimum and maximum particle sizes present in the feed suspension. It is assumed that negligible fluid is lost in the solids discharge. Equation (48) then takes the form:

T(d) = 49

^Fo(d)

The particle size distributions shown here are the percent-in-range distributions so that the sum over all the size fractions is 100%. Particle size measurement is restricted to particles with diameter 1 pm. It is therefore necessary to assume that sub-micron particles do not make a significant contribution to the total mass of particles. For this reason, particle size distributions are expressed in terms of particle volume, where the population of each size fraction is given spherical volume [(4/3)7i(d^/9)]. This tends to weight

the size distributions in favour of the larger particles.

Grade efficiency curves are constant for any given set of operating parameters such as fluid viscosity, flow rate and initial solids concentration. However, it is possible to make grade efficiency curves applicable to changes in solid-fluid density difference (Ap) and viscosity (p) by converting the particle size scale using Stoke's law so that:

X 2

\

50

When generating grade efficiency curves to describe separation efficiency in a centrifuge, the particle size axis is divided by d^ to create a dimensionless parameter “d/dg” (see section 3.2.1). This enables comparison of the grade efficiency curves for many systems at any flow rate, provided that the correct values for liquid viscosity and particle density inside the centrifuge are known constants. The conversion contains all the assumptions associated with Stoke's law (section 3.2), as well as assuming that particle trajectories in the equipment remain unaffected by changes in fluid viscosity.

3.1.4 Reduced efficiency

In some centrifuge operations where there is a significant underflow it is necessary to take into account the volumetric split in order to observe the net separation effect:

E '. = 51

1 -R f where E% = "reduced" efficiency

R f = (Q sed /Q o ) underflow to throughput ratio

In cases where contains compressible solids or is difficult to measure, then Rf can be found from a mass balance against the supernatant:

_ 7 0 sup

R j — 1 -

Qo

52

In practice, E i is usually determined from the solids concentration in the supernatant (Cgup) relative to the solids concentration in the feed (C J, (Svarovsky 1990). This form is also called the clarification number (0^):

E t , - Cn -

C o 'Cisup

C o

Reduced separation efficiency may also be found using:

E]

and:

-'sed

C o I ' R f

solids concentration in the underflow

53 54 E t , = C s e d - C isup C s e d ~ C su p

^ c

Hence there are three alternative ways of conveniently measuring the reduced efficiency.