Mathematical simulation of hydrocyclones - ScienceDirect

Mathematical simulation of hydrocyclones K. A. Pericleous CHAM Limited, Bakery House, Wimbledon, London, SW19 SAU, UK (Received February 1986; revised November 1986)

Cyclones are used widely in industry as classifying devices. Due to the complexity of the multiphase flow field in such a device, prediction methods are at best semiempirical and geared towards predicting overall performance parameters. This article presents a real alternative, using state-of-the-art numerical techniques to address the physics describing the flow and solving the Navier-Stokes equations describing the mixture velocities and the transport equations for air and particle concentrations. Turbulence is modelled in a way that takes into account the effects of swirl and also the presence of particles. An algebraic slip model (ASM) is used to represent the relative migration of particles and air in the liquid mixture. The calculation yields field values of velocity, pressure, particle and air concentrations, as well as the overall performance parameters more familiar to cyclone operators.

Keywords :

cyclones,

separation,

multi-phase

flow,

numerical

methods

Introduction

Operational principles

General

Cyclones in various forms have been with us since the end of the last century, and during the late 1940s the first large-scale applications appeared in the mining and paper industries. In these first few years the basic cyclone design was established and has since remained largely unchanged. Since the early 1960s there has been an upsurge in the scale and diversity of cyclone applications, which now include the food, chemical, oil, cement, nuclear, and metallurgical industries. With this diversification, limitations in design and prediction methods have become apparent, and as a result there has been an increase in research effort. With the advent of these new applications and a greater expectation of efficiency in modern process plants, cyclones have undergone a transformation from a low-technology device to a mediumor hightechnology one. New advances in computer modelling now make it possible to simulate the complex multiphase flow inside the cyclone from first principles, without resort to empiricism. Experimental evidence is still necessary for validation, although now the emphasis is no longer on the overall performance parameters but on internal flow details.

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A cyclone is a sedimentation device that uses the enhanced gravity principle to separate solid particles from a liquid (hydrocyclones) or air (air cyclones) medium. It can also be used to separate liquid from liquid, or air from liquid, in some applications. The centrifugal principle is used to generate the necessary g-forces by rotating the feed mixture in a conocylindrical body (Figure 1). Separation takes place in the radial direction, with coarse particles moving towards the wall and fines towards the axis. Two exits, at the top and bottom of the cyclone, yield the separation products together with the carrier liquid. Coarse particles report at the lower exit (or underflow), and fines report at the top (or overflow). The overflow ducting usually protrudes into the cyclone body to form the ‘vortex finder’. In hydrocyclones operating in air, a column of air (called the air core) forms along the axis and has an important influence on the cyclone behaviour. From the designer’s point of view the most important characteristics of cyclone performance are 1. The particle cut size 2. The pressure drop 3. The volume throughput The particle cut size is usually given by the cyclone classification efficiency curve, shown in Figure 2. This 0307-904X/87/040242-14/$03.00 0 1987 Butterworth Publishers

Mathematical PICTURE OPTION

Streanline

shows

e

th ,e

path of fine particles towards

I

OVERFLOW The dotted surface represents the air core

simulation of hydrocyclones:

K. A. Pericleous

designs. The point on the efficiency curve at which there is a 50% probability of a particle appearing at either exit is called the d,, point or, more commonly, the cyclone cut size, and is measured in microns. The slope of the curve at d,, represents the sharpness of classification, and the tail ends represent fines escaping through the underflow, and vice versa. The second parameter, the pressure drop, is important because it affects the pumping requirements: it is especially important where cyclones are operated in series. It represents the energy required to spin the incoming fluid, but it also represents the losses associated with inefficient inlet and outlet designs. The cyclone throughput is related to the pressure drop, and is important because it determines the amount of slurry that can be processed per unit cyclone at any time. Cyclone size is also directly related to unit size, which means that for classification of fine suspensions or water clarification many small cyclones have to be operated in parallel. Performance prediction The process plant designer’s task is that of selecting a suitable cyclone that possesses the best compromise set of performance characteristics for a given cost outlay. Up to now, performance prediction has relied heavily on a number of semiempirical relationships developed over the years from a combination of experimental and plant evidence and through the use of idealised mechanistic models that attempt to simulate the way the flow in a cyclone behaves by modelling a single dominant feature of it. The major models are

UNDERFLOW Figure 1

Basic design for a hydrocyclone

Particle Figure 2

A typical efficiency

diameter,

d

curve for cyclone

(micron) classification

curve represents the percentage of particles entering the cyclone that appear at the underflow for various particle diameters. The corrected efficiency curve represents the classification due to the centrifugal effects only, and it is more useful for comparing cyclones of differing

* * * *

The The The The

equilibrium orbit’ retention-time hypothesis’ crowding theory’ boundary layer approach’

Each of the above models works quite well for particular cyclones, or applications, in predicting the cyclone performance parameters. They are, however, not general enough, and they do not tell us anything about the flow behaviour inside the cyclone. The nongenerality of these methods is evidenced by the fact that, for example, Bednarski3 compiles no less than 38 expressions for volumetric throughput, whereas Plitt’ quotes three and equations for dSo, two for sharpness of separation, two for pressure drop. It is obvious that the real situation is far more complex than any of the models mentioned represents, and can only be realistically simulated by a model that includes all the above mechanisms, each in proper interaction with each other. This can be achieved not by merely looking at the overall cyclone characteristics and inferring assumptions about the flow, but by looking at the fundamental equations governing the conservation of mass, species, and momentum in the cyclone. Solution of these equations coupled with the correct boundary conditions will yield, in addition to overall parameters, details of the flow field and particle distributions. Present contribution It is obvious from the above that an alternative approach to the prediction of cyclone behaviour is necessary, which takes into account the effects of turbu-

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243

Mathematical simulation of hydrocyclones: K. A. Pericleous lence, particle interaction, air core, etc., since these factors govern the flow behaviour and its design performance. The following sections describe a hydrocyclone mathematical model that has been developed using a general-purpose, multidimensional two-phase, fluid flow computer code, PHOENICS4 The flow is assumed to be steady, two-dimensional, and axially symmetric. Equations are solved for continuity, momentum, and species concentration. The solid particles in the hydrocyclone are represented by the equations for species concentration, each equation modelling a particular size range (density range) of solid particle. Special source terms are included in these equations to simulate the particle relative velocity resulting from action of the centrifugal and drag forces. In dilute slurries the particle and fluid equations can be treated independently. However, as the particle concentration increases, the equations become interdependent due to changes in the mixture density and effective viscosity. The present model takes into account both these effects by assuming a mixture density that depends on particle concentration, and a turbulent viscosity that in turn depends on the mixture density. In addition, the particle diffusion exchange coefficient is increased in regions of very high particle concentration to prevent particle buildup and settlement and to account for the random movement caused by particle interactions. The air core in the hydrocyclone model is treated in a similar way to the particles: i.e., the centrifugal forces on pockets of air, arising due to the density difference between air and liquid, lead to a slip velocity towards the cyclone axis. As the calculation proceeds, the air concentrates along the axis to form the air core. The amount of air entrained and the size of the air core are determined from continuity considerations during the calculation. The representation of turbulence in the hydrocyclone model is by way of a mixture-effective viscosity that varies according to local flow conditions. A simple mixing-length model has been developed and validated using experimental data for air cyclones. Higher-order turbulence models, such as the k-e modeL5 where equations are solved for k, the turbulent kinetic energy, and E, its rate of dissipation, are unsuitable in standard form due to the effects of high swirl, which lead to turbulence anisotropy. To validate the model, researcher@ are experimenting with glass hydrocyclones installed in a test facility at Warren Spring Laboratory, and are comparing the model with experiments performed by other investigators. Comparison with Kelsall’s experiments’ have been chosen for presentation here (a) because these are widely known in the cyclone community, and (b) because they represent a thorough and reliable investigation into the flow behaviour inside a hydrocyclone. A slightly simplified version of the hydrocylone has also been used for air-cyclone modelling. The version was used particularly during the development of the turbulence model and also to study the effects of geometrical parameters on air-cyclone behaviour.2 The empirical contribution to the model is still present but now kept to peripheral aspects of the flow field, such as turbulence. The only assumption regarding the flow field is that of axisymmetry, which is not in fact a

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Table 1

Integral source terms

Variable

s,

ur Y

-gV+F,

w

-

C

t (PW,.,c) +i

$ (prvre,c)V

V = cell volume wreI and v,,, = velocity of particles in z- and y-directions relative to the fluid velocity F,, v, w = wall friction force

restriction of the method calculations.

but an economy

device in the

Analysis and solution method Governing diferential equations A cylindrical polar coordinate system is assumed with 8, r, and z being the circumferential, radial, and axial coordinate directions. The variables solved for are * fluid velocities, ur, u, and w in the three coordinate directions * particle and air concentrations, ci * pressure, p The fluid motion is represented by the steady, incompressible Navier-Stokes and continuity equations, which can be expressed as follows, neglecting derivatives in the circumferential, o-direction. Continuity i

$ (pm) = 0

(pw) + i

Momentum

and concentration

div(pirO - r grad 0) = S,

(2)

where @ = ur, v, w, or c p = mixture density I- = effective diffusion coefficient (for velocities this is simply p the mixture effective viscosity; for all other variables it is given by the ratio of effective viscosity and the Prandtl-Schmidt number) S, = source of @ per unit volume. The source terms in the equations, which include the effects of pressure gradient, wall friction, and particle relative motion, are given in Table 2 in the integral form in which they appear in the finite domain equations solved by the program. Auxiliary relationships Since the solution properties are relevant

is for a fluid here.

Physical properties The density of the mixture the relation 1 -= Pin

1 -&

mixture,

mixture

varies locally according

to

cj Pl

+ j1

0

;

(3)

Mathematical

where pl, the liquid density, is taken as 1000 kg/m3, and pi can be either the solid particle or air density. Wall friction Wall friction is applied to the appropriate velocity components in mesh cells adjacent to the cyclone surface, including the inner and outer surfaces of the vortex finder. For surfaces defined by the coordinate system, the method built into PHOENICS is used to calculate wall friction by a logarithmic velocity profile at the wall. In the conical section, friction is calculated by adding a sink term to the momentum equations: Fi = 3pf ix+ A, where the friction resultant velocity sponding velocity area in the cell.

(4)

factor, f, is taken to be 0.003, v is the parallel to the wall, Q is the correcomponent, and A, the cone surface

Turbulent difluusion The turbulent viscosity of the fluid is allowed to vary locally, and is estimated using a modified Prandtl mixing-length model. The usual form of the mixinglength model is

of the flow. flows. The

of hydrocyclones:

K. A. Pericleous

kinetic energy at inlet, and A is a proportionality stant taken in this study as 1.

con-

Relative velocities of particles and air Although the PHOENICS computer code, on which the hydrocyclone model is based, has a full two-phase capability, which allows the solution of a set of six differential equations for momentum, coupled with a phase conservation equation, this capability was not exploited here. Even in the simplest hydrocyclone model there are three phases present, namely liquid, monosized particles, and air. Since particles of different diameters move with different velocity, each additional particle size represents in this sense an additional phase. An algebraic slip approach was used instead, with three momentum equations solved for the mixture, and relative movement of each species taken into account in the conservation equations in an iterative manner. The relative velocities between the air and particles and the liquid in the hydrocyclones are evaluated by consideration of the forces acting. For a particle in equilibrium the centrifugal force is equal to the radial drag force, and the vertical force due to gravity equals the vertical drag force. Hence,

(5) where ;1. is a mixing-length characteristic This equation is applicable to unidirectional analogous model for swirling flows is

simulation

v,= tP1vdz$

(P, - PI ;

CD

and

(10) (6) However, this equation would predict zero viscosity in the region of no shear, i.e., near the cyclone axis. Hence, following Bloor and Ingham,* a constant term is added, which augments the shear. In addition, the particle content is incorporated by an empirical formula of Kunitz’ appropriate for suspension concentrations up to 30% by volume. Thus

where V, = volume of particle = nd3/6 p = mixture density pp = particle density C, = drag coefficient If m is the particle specific gravity,

equal to pp/p, , then

(11) and

The summation is over all particle sizes, but not air concentration. The expression by Kunitz is only one of many that can be used in this context. Other experimental expressions exist that reflect the rapid increase in mixture viscosity with solids content. The mixing length, 2, is taken to be a fraction of the cyclone diameter, D, , and, in the vortex finder, a fraction of D,. Parametric studies performed for an aircyclone’ indicated that a value of 2 = D/30 gave the best agreement with experimental data for the predicted swirl distribution. Strictly speaking, the turbulent mixing length varies both axially and radially inside the cyclone. The assumption of a constant ;1 is therefore an approximation, but one that seems to be justified by the closeness of predicted and experimental flow fields. The term p0 may be thought of as the turbulence convected into the cyclone: it can be estimated from & = ‘4pk”2;li

(8) where ,$ = dJ10 is the inlet length scale, appropriate for fully developed turbulent pipe flows, k is the turbulent

(12) The negative sign denotes relative movement in the negative z-direction for heavy particles. The drag coefficient is a function of the slip Reynolds number, defined as Re

=

P&.,z

+ w,,,z)1’2d

(13)

4

where p, is the medium density (liquid or air). The frictional relationship was approximated to the two expressions shown, which represent the laminar and turbulent regimes. C, = 24/Re

for Re < 64

C, = 0.42

for Re 2 64

The particle diameter d is fixed for each concentration equation; therefore, to account for a range of particle sizes, we solve several equations, each representing a characteristic size. Air, which may enter in the form of

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K. A. Pericleous

of hydrocyclones:

bubbles through the cyclone inlet as well as the overflow and underflow, is treated in a similar manner. The bubble diameter d, is assumed to be a function of the local pressure difference between the bubble and liquid, and the surface tension, r, of the air-water interface. The pressure in the bubbles is assumed constant and equal to atmospheric. Hence, d, = 4rjAp

(14)

Boundary conditions A mass inflow of liquid and concentration of solid are specified at the inlet. At the vortex finder outlet the solution domain is best extended beyond the cyclone body to preserve the rotational characteristics of the flow at the exit. The pressure is prescribed here according to the relation

u,,tz

dp,,,

F’Py-

(15)

where u,,, = 80% of velocity in the cell adjacent to the exit, and represents an extrapolation based on the decay of swirl in a pipe. Also, pext = 0 at r = 0 provides the reference pressure at the exit. At the underflow exit the pressure is assumed to behave similarly, although the factor on u is taken as 20% to reflect the expansion of the flow as it leaves the underflow. Parametric variation of these two quantities did not significantly affect the overall flow behaviour. Furthermore, as the air core develops, the density p in (15) diminishes by a factor of 1000, leading to a practically constant pressure at the boundaries.

The finite difference formulation Finite difSerence equations (FDE) The discretised form of equation (1) was solved on the axisymmetric grid shown in Figure 3 by the PHOENICS code. The standard form of the FDEs for a variable 4 residing in a cell node is 4 = Cn,s,h,l (inflo;

x in;Tng ns1 . h 31

In

Ps) + sources

(16)

ows

where inflows are the sum of convective and diffusive terms through the appropriate cell face, and sources are linearised versions of the terms in Table I and boundary conditions. In the above formulation the appropriate velocities are the mixture ones, not the velocities of the carrier fluid. The algebraic slip model (ASM) approach of Pericleous et al. lo is used to simulate the migration of particles through the mixture. This model postulates that conservation of particles can still be retained even though their velocity vector is different to that of the mixture by modifying equation (16) so that the appropriate velocities are now the particle velocities. Hence if we consider the ith particle family, we have n,s,h,e (inflows ci = c

x incoming

C”,S,h.l (outflows)

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Cyclone and solution grid

Inflows and outflows are now due to a modified involving a particle velocity F, where q = v + ye*

flux,

(18)

In (17) Ci can now exceed the value of all its neighbours. To ensure that species conservation is retained, the following two conditions need to be satisfied: ci < 1

and

i$lci

d ’

(19)

Two further steps are necessary for this: (a) A flow resistance term that prevents particles flowing into a cell that is almost fully occupied by other particles:

C’s) + sources

Ni Di

Figure 3

August

(20)

Mathematical

and m > 1, taken as 4 in this investigation. (17) can be rewritten as

ci =

Ni Ni

+ (I - C*)Di

(b) Equation

(21)

where CT refers to the previous-sweep value of Ci. When the solution is converged, equation (21) reduces to equation (17); in the meantime, however, since CT cannot be negative, Ci can never exceed 1. Equation (21) was derived by suitably modifying the standard equation (16) of PHOENICS, through an access subroutine GROUND. Solution procedure

Equations (16) and (21) are solved for three velocity components, pressure and particle and air concentrations in an implicit iterative scheme,” for all cells in the solution domain as defined by the axisymmetric cylindrical Dolar grid shown in Figure 3. The-methoh of solution has the following sequence : The variable fields in the calculation domain are initially guessed. The mixture density and viscosity are then computed. The velocity fields are computed. The scalar fields containing particle concentrations are computed. The resulting errors in the continuity equations are then computed. These errors are used as quantitative indicators of corrections that should be made to the pressures and of the corresponding velocity corrections. The solution then iterates until the corrections required become sufficiently small to be considered negligible.

Computed results The experiment

simulated

Kelsall’ used an optical method to measure the tangential and vertical velocity components of fine aluminium particles suspended in a transparent 3-inch hydrocyclone; radial velocity components were estimated from continuity. The cyclone had interchangeable underflow and overflow sections, so the underflow and overflow diameters could be controlled. Three configurations were tested: one with practically no overflow; one with no underflow; and one with equal underflow and overflow flow rates. Each cyclone was operated at 10, 20, 30, and 40 psi indicated feed pressure. The 40 psi case with equal underflow and overflow diameters was selected for simulation, as more representative of classifying cyclones in practice. Computational

details

Calculations were performed for the above cyclone using solution grids which varied between 15 x 20 and 30 x 30 computational cells in the radial and axial directions, respectively. A 30 x 24 grid was chosen for the bulk of the runs presented here, being a good compromise between accuracy and computer cost. An average of 500 computational sweeps were needed for each point on the efficiency curve, each point requiring 50 minutes of CPU time, on a Perkin-Elmer 3250 minicomputer.

simulation

of h ydroc yclones:

K. A. Pericleous

Runs were performed for a series of monosized particles, ranging from 1 to 15 p, with particle loadings of 1% and 10% of the feed by weight. A run was also performed with three coexisting particle sizes, to investigate the effect of interaction between them. The latter case represents a run with five distinct phases present, namely air, water, and particles of three different diameters. Each phase is distinguished here as possessing a set of velocities which is in general different to that of the mean or pulp velocity. The limit to the number of particle sizes that can be accommodated in any run is determined not by the calculation procedure but by the computational cost. In dilute load cases (10% or less), runs with only a few sizes present are adequate to establish an efficiency curve. An increasing number of particle equations need to be incorporated as the loading increases in order to resolve interactions. Discussion of results Figures 4 to 14 show the results for the cases described above, in the form of computer generated streamlines, particle concentration contours, and radial profiles of velocity, particle concentration and pressure. Figure 4 shows a two-dimensional streamline representation of the mixture flow field. Solid lines indicate streamlines that exit through the underflow, while dotted lines indicate streamlines exiting through the overflow. There is a recirculating region adjacent to the vortex-finder wall, and the column along the axis indicates the extent of the air core. The streamlines indicate that a vortex reversal occurs very deep inside the cyclone cone, close to the region where the wall boundary layer meets the air core. Figures 5 to 7 show contour maps of particle mass fraction, in the range 0.01 (inlet value) to 0.03, for calculations with monosized particle loadings, of 5, 10, and 15 ~1,respectively. The 5-p case shows a gradual buildup of particles towards the underflow; at any axial location a higher concentration of particles occurs at the wall and also at a radius close to that of the overflow. The rate of particle buildup towards the underflow increases as the diameter of the particles increase, as shown by the closer spacing of the contour lines (Figures 6 and 7). At the same time, the inner buildup radius increases and moves further and further towards the outer vortex region. It is suggested that this region of secondary particle buildup corresponds to the particle equilibrium radius, the existence of which has been postulated by other investigators.’ The closer this radius is to the inner vortex the higher the proportion of particles separated to the overflow. The eficiency curve of Figure 14 shows that 5 /J lies close to dSOccJ, while 10 and 15 p lie close to the top of the curve and hence report to the underflow. Figures 8 to 10 show profiles of particle mass fraction at three axial locations in the cyclone, for particles of 1, 2.5, 3.5, 5, and 7 p. Figure 8 corresponds to the top of the cyclone. All particles enter at a mass fraction of 0.01, and at 1 p almost no classification takes place. There is a slight reduction in concentration at the vortex-finder wall, and a significant amount of particles exist in the vortex finder itself. Only the air-core region is devoid of 1 p particles. As the particle diameter increases, the particle concentration develops a maximum before it reaches the vortex finder wall. This maximum increases

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Mathematical simulation of hydrocyclones: K. A. Pericleous in magnitude with particle diameter, and moves progressively to a larger radius. At the same time, the concentration inside the vortex finder diminishes. Figure 9 shows the behaviour of the particles at the level of the vortex-finder exit. The same comments as above apply, although there is now an additional buildup of particles on the conical wall. Figure 10 is at a level well inside the conical region. The particle build-up close to the wall is now dominant for the 5- and 7-p particles. There is however still a high concentration. at a radius close to that of the vortex finder, for the 2.5-5-p particles. In operation, the 7-p particles are classified to the underflow while smaller

Figure 5 (20).

Fig1 we 4

248

Two-dimensional

Appl.

streamline

Math. Modelling,

concentration

contours:

d = 5 /I;

levels:

0.01,

sizes are classified more and more to the overflow. The 1-p size behaves as part of the liquid medium, and it reports at the two exits in the same proportion as the medium. Figure 11 shows the behaviour of three particle sizes 3, 4.5, and 6 p, this time entering the cyclone together, at a mass fraction of 1% each. This figure can be compared with Figure 6 (for monosized particle runs), although the particle sizes considered do not corre-

pattern

1987,

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0.03

Vol. 11, August

Mathematical

simulation of hydrocyclones:

K. A. Pericleous

orbits will interfere with each other, since the preferred orbit will be partially occupied by the other particles. In high enough concentrations one would expect a smearing of the classification process to occur, and consequently an increase in d,, . This was not observed in the present low concentration calculations, as shown in Figure 14, where the points denoted by “ x ” belong to a single run, containing three particle sizes. Figure 22 shows the tangential velocity distribution

figufe 6 Particle concentration contours, d = 10 /I; levels: 0.01, (20), 0.03

spond exactly. Figure 11 shows a larger influence of particle size on radius of maximum concentration than Figure 9; hence while the 3-p radius is smaller than the corresponding 2.5-p radius, the 6-p radius is larger than the 7-p radius of the monosized run. This difference in behaviour can be attributed to the interaction between different particles. Hence, in monosized runs, each particle will settle in orbit in its own equilibrium envelope. Once a particle of a different size is introduced, the two

Figure 7 Particle concentration contours, d = 15 p; levels: 0.01, (20). 0.03

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CPRRT

I

8.886

8.885

8.015

B.Bl8

I

0.838

9.825

1

I

1

1

8.820

8.835

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Figure 8

Effect of diameter d on particle concentration (Z = 20, top of cyclone)

at various levels in the cyclone. The expected forced-free vortex distribution persists uniformly throughout the cyclone, with small deviations at the vortex-finder boundary, where there is a decay of velocity, and close to the apex, where there is a slight velocity increase as the cone surface is approached. These effects were also reported by Kelsall.’ The maximum swirl velocity at 16 m/s occurs close to the air core boundary at a radius of 0.005 m. Again agreement with Kelsall is quite good, with U,,, = 18 m/s at 0.2 inch. Kelsall’s experimental result is shown dotted on the same graph for comparison. Figure 13 shows the pressure distribution in the cyclone at three horizontal positions. The pressure is constant in the air core and is close to atmospheric. It first rises rapidly in the forced vortex region and then levels off, reaching a maximum of 2.7 x lo5 N/m2 gauge at the cylindrical wall. This value is very close to the experimental 40 psi measured by Kelsall. Axial variation of pressure is very small compared to radial variation. Figure 14 shows the efficiency curve of the cyclone

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studied, derived from a series of monosized particle runs. The dotted line represents the actual efficiency curve, y, i.e., the percentage of feed particles appearing at the underflow. The origin of the curve is close to 50%, which corresponds to the percentage of water appearing at the underflow, since particles of size below 1 p tend to behave as part of the medium. The points on the y, curve represent the corrected percentage of particles appearing at the underflow, derived from Y-

” = -1

_

R, R,

x 100%

(22)

where R, is the fraction of the feed liquid appearing at the underflow. Each point represents a calculation for a single monosized particle, and lies close to the curve defined by y, = 1 - exp[-0.6693(d/d,,)“] as given by Plitt’ sharp classification

(23)

with m = 3. This represents a fairly curve. The corrected cut size of the

Mathematical

simulation

of hydrocyclones:

K. A. Pericleous

0.015 ortex lndsr

CPbRt

0.ee Figure 9

Effect of diameter

d on particle concentration

cyclone, dSOcwas found to be 4.5 p. Points “ x ” represent the 10% particle case.

(Z = 15, vortex finder unit)

denoted

by

Comparison with experimental data and correlations Comparison of the present results with the experiment of Kelsall, and also with experimental correlations by other investigators, is summarised in Tables 2A-2C, which compare volume feed rate, flow split, and rate size d,, . Table 2A shows the volumetric flow rate through the cyclone as predicted by various investigators, assuming a feed pressure of 2.76 bar (40 psi), and as measured by

Table 28

Flow split, S

Investigator

Method

S = (underflow/overflow),

Present authors Kelsall Plitt Dahlstrom

Numerical Experiment Correlation Correlation

1.22 1.24 0.50 1.82

Kelsall. present 2.7 bar metric

The experimental value was used as input in the calculations, and the indicated pressure drop of was an outcome of the calculation. The voluflow rate through the cyclone can be predicted

Table 2C

Table 2A

Mass flow rate, Q

Investigator Present Authors Kelsall Plitt Herkenhoff3 Bednarski

Method Numerical Experiment Correlation Correlation Correlation

P (bar)

Q W/s) 0.001 0.001 0.0014 0.0009 0.0014

(input) (result) (result) (result) (resultj

2.7 2.76 2.76 2.76 2.76

(result) (input) (input) (inout) (inputj

Cut size, d,,

Investigator

Method

Present authors Kelsall Plitt Lynch” Yushioka 81 Hotta13 Mular & JuII’~ Lilge’4

Numerical Experiment

Appl.

flow rate

Math.

d 50

1987

4.5 -

1.17 9.5

8.0 9.1

-

Correlation Correlation Correlation Correlation Cone force Equation

Modelling,

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251

Mathematical simulation of hydrocyclones: K. A. Pericleous C+.esle

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Figure 70 Effect of diameter d on particle concentration (Z = 7, in apex region)

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Figure 7 7 Three-particle run; effect of diameter d on particle concentration

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Mathematical

simulation of hydrocyclones: K. A. Pericleous

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Mathematical

simulation

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K. A. Pericleous

Corrected

I

e Figure 74

2

(microns)

Classification efficiency curve

quite accurately by using existing correlations, as shown in the table. Table 2B shows a comparison of the flow split, as measured and predicted. The present method agrees well with experiment, while the empirical correlations presented can be up to 100% in error. Table 2C shows estimates of the cyclone cut size, actual or corrected. There were no experimental measurements of particle split, although Kelsall estimated a cut size of about 10 p for his cyclone. The present investigation gives a dSOc of 4.5 p, while Plitt’ and Mular and Jull (cited in reference 14) give values closer to 10 p. The reason for the discrepancy is probably due to the very long vortex finder used in Kelsall’s cyclone, which is at variance with conventional design practices and the mainstream of cyclone experiments on which all correlations are based. As a consequence, very little short-circuiting of the vortex finder by larger particles occurs, and the cut size remains low.

Concluding remarks A user-oriented mathematical model of the hydrocyclone classifier has been presented. The model analyses the flowfield and particle behaviour inside the cyclone by applying numerical techniques normally

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associated with high-technology industries, (e.g., nuclear, aerospace), to an essentially empirical science. The method of solution is general, and applicable to any type of hydrocyclone and also to air-cyclones. The model has been applied to the classical experiment of Kelsall with considerable success. In addition to the usual integral parameters associated with hydrocyclone performance, i.e., pressure drop, flow split, and separation efficiency, the model is able to predict mixture velocity profiles and is unique in calculating the extent of the air core and the detailed distribution of particles. Hence, particle-concentration contour maps and profiles help to explain the separation process by showing all its essential elements; the existence of an equilibrium cylinder which varies with particle diameter, the deposition of particles onto the cyclone wall, the accumulation of particles near the underflow, and re-entrainment into the upflowing vortex surrounding the air core. The goal of this work is to establish a reliable method of hydrocyclone prediction which will eventually replace reliance on empiricism. The model possesses both the flexibility and to a large extent the physics for such a general application. However, a great amount of validation work is necessary before all the limitations are revealed and removed. The value of the method as a

Mathematical

design tool lies in the fact that since it is not empirical it can be applied with confidence to radically different cyclone configurations. Nomenclature

4 C, ci

CD d

d 50 DC Di Do

$’ U,“,W 9

k m n P Pex*

Q

Re 4 ; S, u U ext

V VP V V ret W W rel

Y YC Z

wall surface area concentration of air concentration of itch particle size drag coefficient particle diameter (microns) cyclone cut size hydrocyclone diameter inlet diameter overflow diameter underflow diameter friction factor friction force in u, u or w-direction acceleration due to gravity turbulent kinetic energy specific gravity no. of particle families pressure external pressure flow rate Reynolds number fraction of feed appearing at underflow radial coordinate direction flow split source of @ in finite domain equations circumferential velocity external circumferential velocity volume volume of particle radial mixture velocity radial velocity of particle relative to liquid axial mixture velocity axial mixture velocity separation efficiency corrected separation efficiency axial coordinate direction

E

0 I p L4 P

PP PI Q,

simulation

of hydrocyclones:

K. A. Pericleous

rate of dissipation of turbulence energy circumferential coordinate direction mixing length scale effective mixture turbulent viscosity laminar viscosity mixture density particle density liquid density general variable

Note: All quantities otherwise.

in SI units except where stated

References

4 5

10

11 12 13 14

Plitt, L. R. ‘A mathematical model of the hydrocyclone classifier’, CIM Bull., 1976, 116 Jal, E. N., Pericleous, K. A., and Rhodes, N. ‘Modelling of hydrocyclone systems’, CHAM Report No. 3250/3,1984 Bednarski, S., and Wiechowski, A. ‘A review of a hydrocyclone performance correlation’, 2nd Int. Conf on Hydrocyclones, England, 1984, pp. 335-349 Sualding, D. Brian. ‘Mathematics and commuters in simulation’. vol. 13,~i981, pp. 261-276 Launder, B. E., and Spalding, D. B. ‘The numerical computation of turbulent flows’, Comp. Methods App. Mech. Eng., 1914, 3, 269 Pericleous, K. A., Rhodes, N., and Cutting, G. W. 2nd Int. Conf: on Hydrocyclones, England, 1984, pp. 27-40 Kelsall, D. F. ‘A study of the motion of solid particles in a hydraulic cyclone’, Trans. Inst. Chem. Eng., 1952,30,87 Bloor, M., and Ingham, D. B. ‘Turbulent spin in a cyclone’, Trans. Inst. Engrs., 1975, 53 Kunitz, M. ‘An empirical formula for the relation between viscosity of solution and volume of solute’, Gen. Physiol., 9, 1926, 715-725 Pericleous, K. A., and Drake, S. N. ‘An algebraic slip model of PHOENICS for multi-phase applications’, 1st Int. PHOENICS User Conf Dartford, 1985 Patankar, S. V., and Spalding, D. B. Inc. J Heat and Mass Transfer, 1972,15, 1787-1806 Lynch, A. J. ‘Mineral crushing and grinding circuits’, Elsevier, 1976 Yoshioka, N., and Hotta, Y. ‘Liquid cyclone as a hydraulic classifier’, Chem. Eng. Japan, 1955,19(12), 632 Lilge, E. 0. ‘Hydrocyclone fundamentals’, Trans. IMM, 1962, 71,285

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