Minerals Engineering 17 (2004) 671–687This article is also available online at:

www.elsevier.com/locate/mineng

Two empirical hydrocyclone models revisited

K. Nageswararao a, D.M. Wiseman b,*, T.J. Napier-Munn c

a NFTDC, Hyderabad 500 058, Indiab David Wiseman Pty Ltd., Adelaide 5000, Australia

c Julius Kruttschnitt Mineral Research Centre, The University of Queensland, Brisbane 4072, Australia

Received 2 December 2003; accepted 1 January 2004Available online

Abstract

There has been an abundance of literature on the modelling of hydrocyclones over the past 30 years. However, in the commi-

nution area at least, the more popular commercially available packages (e.g. JKSimMet, Limn, MODSIM) use the models developed

by Nageswararao and Plitt in the 1970s, either as published at that time, or with minor modification.

With the benefit of 30 years of hindsight, this paper discusses the assumptions and approximations used in developing these

models. Differences in model structure and the choice of dependent and independent variables are also considered. Redundancies are

highlighted and an assessment made of the general applicability of each of the models, their limitations and the sources of error in

their model predictions.

This paper provides the latest version of the Nageswararao model, based on the above analysis, in a form that can readily be

implemented in any suitable programming language, or within a spreadsheet. The Plitt model is also presented in similar form.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Hydrocyclones; Classification; Separation; Modelling; Simulation

1. Introduction

Versatile in application, the hydrocyclone is the

standard classifier used in closed circuit milling in min-

eral processing plants. This paper focuses on that spe-

cific usage and on industrial scale units.

In 1962, a small group led by Lynch started a re-

search project on control, modelling and optimisation of

mineral processing plants at the University of Queens-

land (later to become AMIRA project P9). Modelling ofindustrial cyclone classifiers was an integral part of that

project. The first ever comprehensive model for the

description of the performance of industrial hydrocy-

clones (Rao, 1966; Lynch and Rao, 1968) and its

application at Mount Isa Mines were significant out-

comes. The methodology has been successfully adopted

within the mineral industry (Lynch, 1977).

Further hydrocyclone research at JKMRC (Marlow,1973; Lynch and Rao, 1975; Nageswararao, 1978; Cas-

tro, 1990) resulted in a generalised model for hydrocy-

*Corresponding author. Address: P.O. Box 94, Blackwood, SA

5051, Australia. Tel.: +61-8-8370-2584; fax: +61-8-8370-2584.

E-mail address: david@davidwiseman.com.au (D.M. Wiseman).

0892-6875/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.mineng.2004.01.017

clones. An alternate model incorporating angle of

inclination too has been developed (Asomah, 1996;Asomah and Napier-Munn, 1996).

Combining the first industrial database on cyclones

generated at JKMRC (Rao, 1966) with his own labo-

ratory data, Plitt (1976) developed an alternative gen-

eral-purpose cyclone model.

These two models (known as the Plitt and the Nage-

swararao models) are the two general-purpose hydro-

cyclone models most widely used for industrial scalesimulation studies of comminution and classification

circuits. In commercial simulation software both the Plitt

model (MODSIM) and the Nageswararao model

(JKSimMet) have been available since the early 1980s

and have been used with very few changes since then.

2. Hydrocyclone models for industrial application––anoverview

From the point of view of a plant engineer, the per-

formance characteristics of interest are:

1. the quantity (tonnage) of slurry a cyclone can treat

and

Nomenclature

C cyclone water split to overflow

a; b; . . . ; g parameters in equation for Rfd size of the particle, lmd50c corrected classification size, lmDc, Do, Du, Di diameters of the cyclone, vortex finder,

spigot and inlet

Dc;std diameter of the standard cyclone

EU Euler number

EUC ‘corrected’ cyclone split to underflow

Eoa actual cyclone split to overflow

F50 median size (that is, 50% passing) of feedsolids

f ; f1; . . . ; f11 functions of . . .fi size distribution of feed solids

CW per cent solids (by weight) in feed slurry

g acceleration due to gravity

F1; F2; F3; F4 calibration parameters for Plitt’s equa-tions

H head of feed slurry (Plitt’s equation for flowsplit)

h free vortex height

Kpo common material dependent constant in the

generalised model for performance charac-

teristic, Pi (p ¼ Q, d, W and V respectively for

throughput, cut size, water recovery and

volumetric recovery equations)

K 0po material dependent constant in the reformu-

lated generalised model for performance

characteristic, PiKp1 function of Kp0 and cyclone diameterKp2 function of Kp1 and minor design variables

(DI, Lc and h)k hydrodynamic exponent, to be estimated

from data, in Plitt’s equation (3) for d50c(default value for laminar flow 0.5)

Lc length of the cylindrical section of the cyclone

m classification index

P cyclone feed pressurePI performance characteristics, EU, d50c=Dc, Rf ,

RVQ throughput of the cyclone, l/min

Rf recovery of water to underflow

RWf recovery of water to underflow calculated

form equation for RfRVf recovery of water to underflow calculated

form equation for RVRV volumetric recovery of feed slurry to under-

flow

S volumetric flow split (volumetric flow in

underflow/volumetric flow in overflow)

VH, VT terminal velocities––hindered and unhindered

conditions

s scale-up parameter

CPV percent solids in feed by volumeCV volumetric fraction of feed solids

a cyclone efficiency curve shape parameter

b cyclone efficiency curve shape parameter

k hindered settling factor, CV=ð1� CVÞ3, 8:05�101:82CV=ð1� CVÞ2

m1, m2 unknown/unquantifiable operating/design

variables

g liquid viscosity (in Plitt’s equation for d50c)h full cone angle, degrees

qp density of feed pulp

qs density of feed solids

ql density of feed fluid medium (water)

672 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

2. the quality of separation of the products, as quanti-

fied by the recovery of• water, Rf and• feed particles of each size to one product, that is,

the actual efficiency curve, at any given set of de-

sign and operating conditions.

While theoretical methods for the prediction of cy-

clone performance based on considering the physical

principles of motion of solid particles in a fluid mediumdo exist, (Barrientos and Concha, 1992; Concha et al.,

1996; Monredon et al., 1992, etc.), they have not yet

made a significant impact on the prediction of hydro-

cyclone performance in minerals processing industry

applications.

2.1. Theoretical/phenomenological models––possibilities

and limitations

Considerable progress is being made in the funda-

mental modelling of hydrocyclones using solutions of

the basic fluid flow equations, either directly or via

commercial Computational Fluid Dynamics codes

(Chakraborti and Miller, 1992; Rajamani and Milin,

1992; Concha et al., 1997; Dyakowski and Williams,

1997; Slack et al., 2000; Brennan et al., 2002; Brennanet al., 2003). It is likely that this approach will soon

provide useful results, particularly with regard to the

optimisation of cyclone design.

However such solutions are computationally inten-

sive; current JKMRC work on the CFD modelling of a

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 673

hydrocyclone operating under normal industrial condi-

tions using parallel processing in a super computer can

consume two weeks of CPU time for one steady state

simulation.

Invoking Moore’s Law (Moore, 1965), we would

expect solution time to halve every 112to 2 years. Existing

CFD models could therefore not be expected to beuseable in process simulators (1–2 s execution times) for

at least the next 25 years.

Robust empirical models that can easily be coded into

process simulators or spreadsheets will therefore con-

tinue to be the main basis of process simulation and

optimisation at least in the short to medium term.

Indeed, it is likely that a hybrid approach, where

computationally intensive models are used to assist inbuilding empirical models, will become more common

as development in theoretical and phenomenological

models continues.

2.2. Practical mathematical modelling of hydrocyclones

The term ‘model’ in general and ‘mathematical

model’ in particular, have context sensitive meanings(Davis and Hersh, 1981; Edwards and Hanson, 1989;

Murthy et al., 1990). In simplistic terms, we can say that

a mathematical model of a system is an ‘idealised rep-

resentation of a physical reality’, in the form of a set of

self consistent equations. In this paper ‘model’ and

‘mathematical model’ have the same meaning.

Typically, model equations predict output charac-

teristics in terms of input variables. The ease of appli-cation and the usefulness of any model is dependent on

the choice of characteristics to be predicted, the factors

or variables that are assumed to affect the physical

process and the assumptions and approximations used

in expressing these variables in the mathematical struc-

ture.

The independent variables for the model equations

are the operating regime and design parameters of thecyclone.

In view of the current limitations of the theory as

outlined above, simplified models that are based on

specific observed performance characteristics can pro-

vide a viable alternative. Since current understanding

of the mechanics of fluid flow cannot yet allow deter-

mination of the model parameters from purely theo-

retical considerations, these are determined fromexperimental data only and the models are termed

‘empirical’ models.

Specifically with regards to cyclones, the performance

characteristics that have been identified for modelling

are:

• the pressure-throughput relationship;

• the ‘corrected efficiency curve’ and the corrected cutsize, d50c;

• the reduced efficiency curve, a plot of corrected effi-

ciency versus normalised size, d=d50c;• the distribution of water into the products usually, as

recovery of water to underflow, Rf but some times asflow ratio into the products, S.

The early cyclone literature abounds with equationsfor one or more (for example, pressure––throughput,

water split, etc.) of the above characteristics (Bradley,

1965). Their applicability was naturally limited.

The initial modelling approach at the JKMRC was

toward development of site-specific models (Lynch and

Rao, 1968). This methodology proved effective and was

extended to other operating plants (Lynch, 1977). Out-

side of the JKMRC, there are other examples of modelsof this genus, for example, those due to Brookes et al.

(1984) and Vallebuona et al. (1995).

These models were based on an implicitly assumed

structure for each of the performance characteristics.

The machine and operating variables were varied as part

of the experimental design. Interpolation on such

models could be used to get a reasonable estimate of the

cyclone performance for a particular machine-materialcombination. Applicability beyond the database from

which they are derived is questionable. Such models can

be simplistically described as curve fitting to experi-

mental data.

A recent example in this category is the model due to

Firth (2003). Although characterized by the use of

dimensionless groups such as Reynolds Number, Euler

Number and Froude Number, together with dimen-sionless design variables, this model also relies on curve

fitting to arrive at a site-specific model. This is

acknowledged in Firth’s unambiguous conclusion:

‘‘given that the flow patterns will be expected to change

with change in the cyclone diameter and geometric shape,

the actual values for the empirical parameters and power

indices could be expected to change.’’

The other category includes those models in which themodel parameters are not application specific. With this

type of model it was possible to estimate the relative

changes in performance characteristics with changes in

the design and operating conditions, without resorting to

further experimental work. However, such models re-

quire material specific constants, which must be deter-

mined from experimental data. The models due to Plitt

(Plitt, 1976; Flintoff et al., 1987), Nageswararao (1978,1995), Svarovsky (1984), Asomah (1996) and Asomah

and Napier-Munn (1996) belong to this category.

3. Hydrocyclone models for industrial application––two

specific models

Where models are required to describe the perfor-mance of hydrocyclones used as classifiers in closed

674 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

grinding circuits in mineral industry, the most com-

monly reported use appears to be of those due to Plitt

(1976), Flintoff et al. (1987) and Nageswararao (1978).

Although both of these models can be reduced to a

similar form, there are distinct differences in model

formulation (that is the choice of dependent variables

and model structure) and evaluation of model parame-ters. The development of the two models and their

specific differences are elaborated below.

3.1. Plitt model development––Plitt (1976) and Flintoff

et al. (1987)

Plitt’s development methodology was relatively

straightforward.The dependent variables, chosen by Plitt were:

• cyclone throughput, Q;• cut size, d50c;• volumetric flow split, S;• sharpness of classification, m.

As design or independent variables, he chose

• diameters of the cyclone, vortex finder, spigot and in-

let, Dc, Do, Du, Di;• combinations of the above, (D2u þ D2o) and (Du=Do);• free vortex height, h.

When the inlets were not circular, the inlet size corre-

sponded to circle of the same area. To account for thelength of the cyclone, he used free vortex height, h, de-fined as the distance between the bottom of the vortex

finder to the top of spigot. His choice for pressure drop

across the cyclone in the equation for S is the head offeed slurry, H .Plitt also took into account that the feed solids con-

tent significantly affects the pulp viscosity, which in turn

influences d50c. In addition, hindered settling andcrowding were also considered as possible factors. To

account for the influence of solids content in the feed

slurry, CV1 (volumetric fraction of feed solids) was the

preferred variable, as the rheological properties are

more comparable if expressed volume basis rather than

weight basis.

His choice of functional relationship appears to have

been governed by the results of regression analysis only.The functional relationship, which was found to best

represent the effect of CV on d50c was an exponentialform. This was finally incorporated only because it

1 Plitt used the symbol b to represent volumetric fraction of feedsolids. This paper uses CV to avoid confusion with the use of b in theWhiten cyclone efficiency equation. When the volumetric solids

content in the feed is expressed as per cent, the symbol used is CPV.

provided better fit than any of the other functional

forms such as CxV, ð1þ Cx

VÞ=ð1� CxVÞ and fð1þ 0:5CVÞ=

ð1� CVÞ4gx that were tried.The data for Plitt’s regression equation(s) included

• the industrial data of Rao (123 data sets including cy-

clone diameters of 2000, 1500, 1000 and 600, treating suchdiverse materials as silica, copper ore, tailings);

• his experimental work which included 9 tests on 600

cyclone where the feed solids content was varied be-

tween 0.8% and 13% by weight;

• 28 tests on 600, 33 tests on 2.500 and 8 tests on 1.2500 cy-

clones at 5% solids (by weight) in the feed slurry;

• 80 tests with water on 600 and 20 tests also with water

on 2.500. (These of course could be used for Q and Sequations only.)

The original model (Plitt, 1976) was obtained by

using a stepwise multiple linear regression program. Plitt

repeated the linear regression procedure with different

functional forms (linear, power and exponential) and

different variable combinations. He included in the

model equations only those variables that were foundsignificant at 99% level.

It is appropriate to mention here that in proposing the

equations for pressure drop, P and flow split, S, Plitt used297 sets of data, including the tests run with water only.

As d50c values were not available for all the data sets,only 179 of the sets were used for the d50c equation. Onlythe 162 tests with sufficient data points above and below

d50c to form a complete classification curve, were usedfor the equation for m.By combining data from different feed materials, such

as silica, copper, ore, tailings and silica flour (and cy-

clones too) in developing the model equations, Plitt

implicitly assumed that the cyclone performance is

independent of feed material characteristics. He was

then able to claim that the performance could be esti-

mated with reasonable accuracy even when no experi-mental data are available. This is the most conspicuous

feature of his model.

In the original reference, Plitt (1976) offered two

forms of the d50c equation, one with and the otherwithout feed size effects.

This is the Plitt (1976) equation for d50c when feed sizeeffect is included. F50 is the weight median size of feedsolids in microns (50% passing size) 2

d50c ¼50:5D0:46c D0:6i D1:21o e0:08C

PV=F 500:52

D0:71u h0:38Q0:45ðqs � q1Þ0:5

ð1Þ

and the Plitt (1976) equation for d50c without consider-ing feed size effect

2 Note that in Eqs. (1)–(6), the units are: Dc, Di, Do, Du, h (cm); Q (l/m); P (kPa); gp (cP); CV (%); d50c (lm); qs, ql (g/cm

3).

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 675

d50c ¼50:5D0:46c D0:6i D1:21o e0:08C

PV

D0:71u h0:38Q0:45ðqs � q1Þ0:5

ð2Þ

Plitt remarks that the ‘‘effect of feed size analysis is not

significant and for normal situations can be neglected’’.He comments however that the above ‘‘equation does

however show the trend that as the particle size becomes

finer, the d50c size increases’’.The Plitt model in its current form as revised by

Flintoff et al. (1987) has no dependence for feed size

characteristics in any of the equations and is given be-

low:

d50c ¼ F139:7D0:46c D0:6i D1:21o g0:5e0:063C

PV

D0:71u h0:38Q0:45 qs�11:6

� �k ð3Þ

m ¼ F21:94D2chQ

� �0:15e

�1:58S1þSð Þ ð4Þ

P ¼ F31:88Q1:8e0:0055C

PV

D0:37c D0:94i h0:28ðD2u þ D2oÞ0:87

ð5Þ

S ¼ F418:62q0:24p ðDu=DoÞ3:31h0:54ðD2u þ D2oÞ

0:36e0:0054C

PV

D1:11c P 0:24

ð6Þ

Since Flintoff et al. (1987) do not include a specific feed

size term, but provide F factors for calibration it is

probably safe to assume that the model should be re-calibrated whenever feed data are available, in prefer-

ence to using the uncalibrated equations.

3.2. Nageswararao model development––Nageswararao

(1978)

Although, the basic model equations as developed

and in a modified form are published (Lynch and

Morrell, 1992; Nageswararao, 1995; Napier-Munn

et al., 1996), the details regarding its development are

not. Accordingly, an outline of the methodology used is

presented here.

3.2.1. Dependent variables

For this generalised cyclone model, the factors con-

sidered relevant to describe cyclone performance, col-

lectively referred to as Pi, were:

• The Euler number, EU defined as Q= D2cffiffiffiffiPqp

q� �.

• The dimensionless cut size, d50c=Dc.• Recovery of water to underflow, Rf .• Volumetric recovery of feed slurry to underflow, RV.

As will be discussed later, RV is a redundant factor.However, an equation for RV is developed so that a di-rect comparison with other available equations for water

split, for example, that used in the Plitt model, could be

made.

3.2.2. Operating and design variables

The choice of independent operating variables/fac-

tors, which are relevant for modelling, was based on

phenomenological considerations. A suitably modified

product of the Euler and Froude numbers, fP=ðqpgDcÞgwas considered an appropriate factor that could be used

to account for the centrifugal force field generated in the

cyclone.The hindered settling factor ðvH=vTÞ, k was chosen to

account for the effect of the differential movement of

solid particles and hence the effect of feed solids con-

centration on d50c. It was assumed that the hinderedsettling factor would adequately account for the changes

in pulp viscosity and viscous effects due to changes in

feed solids content.

The obvious choices for design variables were in-cluded:

• cyclone diameter, Dc;• reduced vortex finder, Do=Dc;• reduced spigot, Du=Dc;• reduced inlet, Di=Dc;• reduced length of the cylindrical section, Lc=Dc;• cone angle––h.

Where the inlets were not circular, the inlet size was

assumed equivalent to a circle of the same area. Clearly

geometrically similar cyclones operating under identical

operating conditions (that is pressure gauge reading at

inlet and feed solids concentration) are not expected to

show identical performance. This necessitated inclusion

of cyclone size (diameter) as an independent variable.Other design variables such as interior wall roughness

of the liners and type of inlet entry (such as involute,

tangential, etc.) were explicitly ignored. Consequently

the effects of these variables, if significant, would intro-

duce errors in the model. Another significant implicit

assumption in the model building exercise is the fixed

properties of the fluid medium. This implies that the

model is applicable only when water is the fluid medium.To extend the range of applicability of the model, it

was felt that the ‘feed material characteristics’ should be

considered as an independent variable. The following

thought experiments elucidate this contention.

If we visualise a cyclone treating two homogeneous

but different materials (say, limestone and iron ore)

under identical operating conditions, also assumingthe particle size and shape distributions to be iden-

tical, such that the only difference is the material

being treated, we would still expect that the cyclone

performance characteristics (Q, d50c and Rf ) wouldbe different.

676 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

Similarly, we can imagine operation of a cyclone

treating a single material (say limestone) with differ-

ent feed size distributions (say 25%––270 mesh in

one case and 50%––270 mesh in the other) under

identical operating and design conditions. In this

case also, we would expect the performance charac-

teristics to be different.

This explains in a simple way the effects of the size

distribution, density of feed material characteristics on

the cyclone performance. There may be other effects.

Realising that a suitable description of the feed material

effect is complex, no simplifications to quantify any

specific material effect in terms of say, nominal product

size, density, etc. was attempted. Instead, the feedmaterial characteristics were simply combined in a single

parameter Km.The above considerations can be summarised math-

ematically as:

Pi ¼ f ðKm;Dc;Do=Dc;Du=Dc;Di=Dc; Lc=Dc; h; k;fP=ðqpgDcÞgt1; t2; . . .Þ ð7Þ

Clearly, t1; t2; . . . etc., are the unknown/unquantifiableoperating and design variables/factors and those whose

independent effect, if any, we shall not attempt to

determine quantitatively.

3.2.3. Model formulation

In formulating the model structure, it was assumed

explicitly that the effects of the independent operating

and design variables on performance characteristics, Piare separable. Eq. (7) can then be written as:

Pi ¼ f1ðKmÞ þ f2ðDcÞ þ f3ðDo=DcÞ þ f4ðDu=DcÞþ f5ðDi=DcÞ þ f6ðLc=DcÞ þ f7ðhÞ þ f8ðkÞþ f9ðfP=ðqpgDcÞgÞ þ f10ðt1Þ þ f11ðt2Þ . . . ð8Þ

It was further assumed that the influence of the design

and operating variables (that is, f1; f2; . . . ; f11, etc.) fol-lows a monomial power function relationship.

As a simplification, the influence of those knownfactors that cannot be determined and the effect of un-

known factors are clubbed together with the material

effects in the form of a material specific performance

constant, Kp0.The effect of fixed fluid properties (water) is also ab-

sorbed by Kp0.Eq. (8) becomes:

Pi ¼ Kp0ðDcÞsDuDc

� �a DoDc

� �b DiDc

� �c LcDc

� �d

he PqpgDc

!f

kg

ð9ÞFor a system where the variables are only Do, Du, feedpressure and pulp density of the feed slurry, this could

be further reduced to:

Pi ¼ Kp0DuDc

� �a DoDc

� �b PqpgDc

!f

kg ð10Þ

where

Kp2 ¼ Kp0ðDcÞsDiDc

� �c LC

Dc

� �d

he ð11Þ

For convenience, the effect of cyclone size and the

material and other effects could be combined when scaleup from one cyclone to the other is not required, as:

Kp1 ¼ Kp0ðDcÞs ð12Þ

Furthermore, it was assumed that the effect of spigot

diameter on Euler number is insignificant and can be

ignored. If the pressure throughput relationship is con-

sidered similar to fluid flow through pipes, the factor to

account for centrifugal forces need not be additionally

considered as the Euler number includes both the feed

pressure and pulp density factors. That is, the model

parameters a and f are both zero when Pi ¼ EU in Eq.(9). Data available in the literature (for example, Brad-

ley, 1965; Lynch and Rao, 1975) and a preliminary

study (Nageswararao et al., 1974) are the basis for these

additional assumptions.

To account for the complex flow pattern in the cy-

clone (specifically due to high solids concentration

normally encountered in industrial practice) and con-

sequential effect on the relative movements of solidparticles, the hindered settling factor was the preferred

variable. The approximation suggested by Steinour

(1944) that the hindered settling factor k is proportionalto the volumetric fraction of feed solids, or k ¼ CV=ð1� CVÞ3, is used in all numerical calculations.This is certainly a simplistic approximation and does

not take into account the independent effect of size

distribution of feed solids in particular the clay content,which could be expected to strongly influence pulp vis-

cosity and hence the terminal settling velocities. How-

ever, as the ‘material effect’ had already considered, it

was felt that Km together with k adequately account forthe overall influence of dense slurries on the cut size. The

perception is that while k encompasses the differences incyclone behaviour due to changes in percent solids in

feed slurry, Km accounts for the changes due to materialcharacteristics. As a consequence, the material depen-

dent performance constants (the K values) in the modelwill not be the same even for similar material if the size

distribution effects, in particular that of the clay content,

are significant.

3.2.4. Evaluation of model parameters

The set of model equations given by Eqs. (10)–(12)

can only be meaningful if the numerical values of

a; b; . . . ; g and the scale factor s for each of the perfor-mance characteristics, Pi, are known. As the theory of

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 677

the hydrodynamics within the cyclone is not developed

enough to evaluate these directly, experimental data

were required to calibrate 3 the model.

The extensive database of Lynch and Rao (1975)

from 2000, 1500, 1000 and 600 (38.1, 25.4, 15.2 and 10.2 cm)

cyclones, treating limestone having three different feed

size distributions, FINE (65% passing––53 lm), MED-IUM (50% passing––53 lm) and COARSE (40% pass-

ing––53 lm) were used for this purpose as detailedbelow.

1. The parameters a, b, f and g in equation set (10)were evaluated from the Lynch–Rao data (34 tests)

treating FINE limestone in a 38.1 cm hydrocyclone.

After suitably transforming the data, the regression

method developed by Whiten (1977) was used. For thisregression analysis, the design variables were assumed to

be exact and it was further assumed that errors in

loge EU , logefP=ðqpgDcÞg, loge k, were ±0.03, and were±0.01 in logeðd50c=DcÞ. The special feature of the Whitenregression method is that it takes into account errors in

independent variables, unlike other methods available at

that time, which assumed the independent variable to be

exact. The parameters thus obtained were used furtherto evaluate the average Kp2 for each of the data sets ofthe Lynch–Rao (1975) database.

2. The Nageswararao (1978) database, also from 2000,

1500, 1000 and 600 (38.1, 25.4, 15.2 and 10.2 cm) cyclones is

complementary to that of Lynch and Rao (1975) in that

only inlets were different. With each cyclone, tests were

carried out with variations in vortex finder diameter,

spigot, feed pressure and solids concentration. The feedmaterial was MEDIUM limestone (containing 50%––53

lm). Mean Kp2 values for each data set were thendetermined (using the same model parameters a, b, f andg obtained in step 1 above). These together with the Kp2values from the Lynch-Rao data sets were used to esti-

mate the parameter c, which quantifies the effect of inlet.3. The dependence of cyclone length and cone angle,

(parameters d and e), were evaluated from data obtainedon a 15.2-cm hydrocyclone, where these two variables

were changed. Feed material was MEDIUM limestone

as above. Using these d and e values, together with cfrom the earlier step, Kp1 values for each data set couldbe calculated.

4. The dependence of Pi on cyclone size (scale upfactor, s), was estimated independently from the data foreach of three size distributions studied by Lynch andRao. The relative errors, if any, in each of Kp1 weretaken into consideration and the final scale up factors

reported below are those that reflect the assumed func-

tional relationship as closely as possible.

3 The term ‘calibration’ here has a different meaning from that used

by Flintoff et al. (1987). For a detailed discussion refer Nageswararao

(1999b).

The resulting equations are: 4

Q

D2cffiffiffiffiffiffiffiffiffiffiffiP=qp

q ¼ KQofD�0:10c g Do

Dc

� �0:68 DiDc

� �0:45 LcDc

� �0:20h�0:10

ð13Þ

d50cDc

¼ KDofD�0:65c g Do

Dc

� �0:52 DuDc

� ��0:50 DiDc

� �0:20

� LcDc

� �0:20h0:15

PqpgDc

!�0:22

k0:93 ð14Þ

Rf ¼ KWo D0:00c

DoDc

� ��1:19 DuDc

� �2:40 DiDc

� �0:50

� LcDc

� �0:22h�0:24 P

qpgDc

!�0:53

k0:27 ð15Þ

RV ¼ KV 0 D0:00c

DoDc

� ��0:94 DuDc

� �1:83 DiDc

� �0:25

� LcDc

� �0:22h�0:24 P

qpgDc

!�0:31

ð16Þ

3.3. Comparison of Plitt and Nageswararao models

The following section examines the assumptions and

approximations in the model formulation for both

models.

3.3.1. Model structure

3.3.1.1. Nageswararao model. The most significant fea-

ture of the Nageswararao model is the a priori choice of

design and operating variables and the explicitassumptions made in binding them to the model equa-

tions.

This resulted in a model with an assumed structure

that explicitly decoupled the machine and material

characteristics. This was the first of the models devel-

oped at the JKMRC to incorporate this important

concept and represented a clear paradigm shift to a new

modelling approach. Later Whiten and his studentsAwachie (1983) and Narayanan (1985) extended this

notion to develop material specific breakage functions

for crushers and grinding mills. Napier-Munn et al.

(1996) emphasise that this has now become a standard

practice and all JKMRC simulation models aspire to the

goal of separating ore characteristics from those of the

processing machine.

3.3.1.2. Plitt model. The Plitt model follows the standard

practice in developing an empirical model. A set of

4 Note that in Eqs. (13)–(16), the units are: Dc, Di, Do, Du, Lc (m); h(degrees); Q (m3/h); P (kPa); g (m/s2); g (cP); RV, Rf , CV (fraction); d50c(lm); qp (t/m

3).

678 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

regression equations for the chosen performance char-

acteristics in terms of independent variables are devel-

oped. The choice of independent variables, as well as the

equation structure, are governed by consideration of

best fit equations for the available database.

This difference is well illustrated by the different

handling of the effect of feed solids concentration on d50c.

Plitt’s final choice for the independent variable (CPV)and the functional form (exponential) was driven

by considerations of the best fit regression equation

for the model fitting data set.

On the other hand, for the Nageswararao model,

both the independent variable (k) and the functionalrelationship (power) were explicitly assumed. The only

choice available after these assumptions was the

approximation(s) available for the hindered settling

factor. Consequently, the only modification to the

model suggested by Castro (1990) was restricted pre-

cisely by these constraints. (This is discussed in further

detail below.)

It should be mentioned that Plitt too forcedðqs � qlÞ

�0:5into the equation for d50c by assuming

laminar flow. Despite his expressed reservations that the

flow relative to the particles may be turbulent, this is the

only explicit assumption made in building his model.

3.3.2. Model base data sets

All models are subject to the limitation that they are

merely approximations of the physical reality, based onsimplifying assumptions or hypotheses and (usually)

process measurements. Errors in any measured data

used for evaluating model parameters, will be carried

forward into the model and hence into the simulation

results.

As a consequence, the model predictions from either

model will never be perfect.

The only yardstick for comparison is how useful themodel is for our objective––in our case, prediction of the

performance characteristics, within the limits of preci-

sion of their measurement, specifically, when the cyclone

used as a classifier in closed grinding circuits. This

clearly implies that when cyclone is used as a thickener

or as a washer or when the feed solids concentration is

low (say around 20% by weight) we are out of the range

of validity and the reliability of predictions is doubtful.

3.3.2.1. Plitt model. With regard to data from industrial

units, the accuracy of the model parameters for Plitt’s

equations is almost wholly dependent on the precision of

the early database of Rao (1966). This was supple-

mented with data from testwork with small (600 or less)

diameter cyclones, the vast majority of which were from

tests at low (less than or equivalent to 5% by weight)solids, or using water only.

3.3.2.2. Nageswararao model. For the Nageswararao

model the accuracy of parameters is exclusively depen-

dent on the extensive data base of Lynch and Rao (1975)

and Nageswararao (1978).

3.3.3. Evaluation of model parameters

3.3.3.1. Plitt model. In the Plitt model, the independentvariables, the model parameters and the functional

(linear power and exponential) relationships are gov-

erned purely by consideration of the best fit under the

multiple linear regression method used. Plitt’s regres-

sions were based on all of the available data and he only

included variables in the final model equations if they

were significant at the 99% confidence level.

3.3.3.2. Nageswararao model. In contrast, the structure

of the Nageswararao model was explicitly restricted by

the assumed functional relationships between the model

variables and the classification process. For example, the

omission of a spigot term in the equation for throughput

(Eq. (13)) is based on previous experience (Lynch and

Rao, 1975) and other empirical/experimental evidence,

which suggested that there was no need to include thespigot effect. The significance (t-test) of the coefficientfor the spigot term as obtained by regression was not the

consideration for its omission. A similar rationale ap-

plies to the omission of the k term in the equation for RV(Eq. (16)).

Equally, in the equation for cut size (Eq. (14)), the

criteria for inclusion of a spigot term and a constant

factor (Kd2 for the model fitting data set) were the choiceof the model structure and not the significance level of

the regression coefficients in a t-test.Using this approach, if the assumptions were per-

fectly true and the data were precise, the number of data

sets needed for evaluation of model parameters would

exactly equal the number of unknown parameters.

However, in practice, the assumptions are never perfect,

nor the data free of errors. Regression analysis is thenneeded to get the best estimates of the model parame-

ters, specifically for the effects of Do, Du, k and

P=ðqpgDcÞ.Of the 52 tests available, those tests outside the range

of interest for model application (for example, feed

solids content above 70% or where classification was

poor) and those with suspected high experimental error

(for example, tests with poor material balances) werenot included in the regression analysis. A set of 34 tests

on 38.1-cm cyclones, treating FINE limestone was

considered sufficient. This data set alone was used to

finally determine the model parameters. All other data

sets could then be used to validate the model parameters

and for further evaluation (effect of inlet etc.) where

necessary.

Conceptually we cannot use the Nageswararao modelto predict the absolute values of the performance char-

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 679

acteristics (Q, d50c, etc.) without any test data for thedesired material. This is a major difference of the model

compared to that of Plitt, where default model param-

eters were provided. That these defaults were not reli-

able, as acknowledged by the incorporation of the Fcalibration parameters in Flintoff’s modified version, is a

separate issue. At least in principle, it is possible todetermine the absolute values of the performance char-

acteristics using the Plitt model.

With the Nageswararao model, when predictions are

required in greenfield situations, it is necessary to source

appropriate K values from previous surveys. JKMRC/

JKTech has built a considerable resource of these

parameters over the course of many years of use of the

model. However, even a parameter database such as thiscan only be used as a guide. The act of selecting K values

from a library of such parameters should automatically

warn the user that the simulation results are to be used

with caution, since they are typically not based on

experimentally determined K values for the desired

material.

3.3.4. Effect of feed material characteristics

3.3.4.1. Plitt model. A distinctive feature of the version

of the Plitt model in most common use (Flintoff et al.,

1987) Eqs. (3)–(6) above), is that the equations define

cyclone performance to be independent of feed material

type. The equations also ignore the effect of feed size

distribution, implying that cyclone performance is

independent of the feed size distribution.

In the original model (Plitt, 1976) Plitt offered anoptional equation, with F50, (median feed size, that is50% passing size) as a variable. Such a simple approxi-

mation for the feed size effect is however, questionable

and the more recent implementation does not include

that equation.

It is to be expected that cyclone performance does

depend on feed size distribution. This has been clearly

shown by Lynch and Rao (1975) and Hinde (1985).Where a regression model does not take a particular

effect into account the model parameters (the regression

coefficients), are biased accordingly. The claim for the

Plitt model that it enables the performance of a hydro-

cyclone to be calculated with reasonable accuracy, when

no experimental data are available, must therefore be

treated with care. Indeed Plitt himself noted, that with

experimental data, the constants in the model equationsmight be appropriately adjusted.

Flintoff et al. (1987) revised the model by incorpo-

rating calibration factors, F1 � F4, for each of the modelequations, presumably taking into consideration the

observations of independent researchers (for example,

Apling et al., 1980) that the predictions are inaccurate.

Their expectation in introducing the calibration

parameters was that calibration with experimental datawould give improved predictions.

3.3.4.2. Nageswararao model. Because of the observation

that cyclone performance is affected by both feed

material type and size distribution, the Nageswararao

model is structured to allow it to be ‘‘tuned’’ to partic-

ular feed materials by parameter fitting to measured

plant data.

In fact, the ideal use of the Nageswararao model is todetermine the material specific constants from a test

using a geometrically similar (or the same) cyclone on

the particular feed type and to use those constants

whenever that material is encountered. For example,

results will certainly be more accurate in a milling cir-

cuit simulation where series cycloning is to be investi-

gated, if two different sets of material specific constants

are derived for mill discharge and primary cycloneoverflow. Of course, in practice this may be difficult to

obtain and then experience with the model must be the

guide.

3.3.5. Flow split and water split

The importance of the recovery of water to the

underflow, Rf , is well understood. It also represents theminimum recovery of the near zero sized particles and isthe starting point on the actual efficiency curve.

The manner in which this performance characteristic

is modelled represents a significant difference between

the Plitt and Nageswararao models.

3.3.5.1. Plitt model. Plitt (1976) chose flow split, S as the

preferred parameter for his model, presumably follow-

ing earlier researchers (for example, Stas, 1957; Moderand Dahlstrom, 1952; Bradley, 1965).

Rf , which is ultimately required for subsequent cal-culations of the cyclone performance, can then be cal-

culated using the equation suggested by (Hinde, 1977;

Plitt et al., 1990; King, 2001).

Rf ¼S=ð1þ SÞ � CV 1�

Pn1

fie�0:6931ðd=d50cÞm

�

1� CV 1�Pn1

fie�0:6931ðd=d50cÞm

� ð17Þ

To use Plitt’s equations (6) and (17), both feed pressure,

P and throughput, Q are required. This is because the

equation for S includes pressure as an independent

variable and those for d50c and m include Q. This impliesthat for a better estimate of Rf , both P and Q need to bemeasured. When one of them is estimated from the

model, model errors are introduced. Further model er-rors arise from the errors in estimation of both d50c andm, which are required for the prediction of Rf accordingto Eq. (17).

Cilliers and Hinde (1991) also noted that Plitt’s

equation for S (Eq. (6)) does not take the effect of feedsolids concentration completely into account, even after

‘calibration’. They proposed a provisional revision with

680 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

coefficients of 1.51 for Du=Do instead of 3.31 and 0.0787instead of 0.0054 for the solids concentration term, CPV.The conclusion is that the Plitt equation overesti-

mates the effect of Du=Do and underestimates that of CPV,at least for the Cilliers and Hinde data.

Despite these reservations, the current version of

MODSIM continues to use Eqs. (6) and (17) to predictRf (King, 2001).Plitt used the industrial data of Rao (1966) for eval-

uation of his model parameters. To evaluate the equa-

tion for S, the values are calculated for the data of Rao(1966) and are shown in Fig. 1.

It can be seen that the predicted values of S are

subject to significant errors even when applied to Plitt’s

model fitting database. For comparison, the predictedvalues of S using the Nageswararao equation for RV (Eq.(16) or (19)) are also shown.

Not surprisingly, King (2001) remarks that prediction

of the flow split, S (and hence Rf ) is the chief source oferror in the Plitt model. A detailed discussion on flow

split and water recovery in hydrocyclones is available

elsewhere (Nageswararao, 2001).

3.3.5.2. Nageswararao model. The Nageswararao

model included equations for both Rf and RV (repeatedbelow):

Rf ¼ KWo D0:00c

DoDc

� ��1:19 DuDc

� �2:40 DiDc

� ��0:50

� LcDc

� �0:22h�0:24 P

qpgDc

!�0:53

k0:27 ð18Þ

RV ¼ KV 0 D0:00c

DoDc

� ��0:94 DuDc

� �1:83 DiDc

� ��0:25

� LcDc

� �0:22h�0:24 P

qpgDc

!�0:31

ð19Þ

Observed flow split (%)0 10 20 30 40 50 60 70

Pre

dict

ed fl

ow s

plit

( %)

0

10

20

30

40

50

60

70

SPlitt

SNageswararao

Fig. 1. Prediction of flow split for model fitting data of Plitt (after

Nageswararao, 2001).

For comparison with Fig. 1, the observed versus calcu-

lated data for Eq. (15) or (18) for the Rao (1966) data

base are shown in Fig. 2.

Since Rf can be calculated from RV in the manner ofEq. (17), different estimates of Rf will be obtained fromeach of Eqs. (18) and (19).

However, due to the indirect calculation method, RVfwould carry forward the errors in the estimation of

corrected efficiency, the same problem as identified in

the Plitt method.

When determining how to apply these two different

values in a practical simulation model, a cautious ap-

proach is to average them with appropriate weighting to

calculate a single predicted value for Rf . This is theprocedure followed initially at the JKMRC and subse-quently used in the implementation of the model in

JKSimMet.

3.3.6. Reduced efficiency curve

Both models rely on the concept of the ‘reduced

efficiency curve’ and each model assumes a particular

form of that curve (Napier-Munn et al., 1996). The

shape of the reduced efficiency curve (a plot of the‘corrected’ efficiency versus dimensionless size, d=d50c) isa measure of the sharpness of separation within the

hydrocyclone.

Plitt explicitly included an expression for efficiency.

Nageswararao did not and expected the efficiency curve

shape factor to be obtained from testwork.

3.3.6.1. Plitt model. Plitt (1971) (as did Reid (1971))derived a Rosin-Rammler type function:

EUCi ¼ 1� e� ln 2ðdi=d50cÞm ð20Þ

and assumed that the reduced efficiency curve isdependent on operating and design conditions, devel-

oping Eq. (4) to describe the effect of these on parameter

Observed R f (%)

0 5 10 15 20 25 30 35 40

Pre

dict

ed R

f(%

)

0

5

10

15

20

25

30

35

40

Fig. 2. Nageswararao model prediction of water recovery to under-

flow (data ex Rao, 1966).

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 681

m in Eq. (20). However, he records the poorest corre-

lation coefficient (0.75) for this Eq. (4) among all his

model equations.

3.3.6.2. Nageswararao model. Nageswararao relied on

the earlier Lynch and Rao, 1975) JKMRC approach,

based on regarding the reduced efficiency curve as con-stant for cyclones of different physical dimensions

treating the same feed material.

The Whiten form of the efficiency equation (Napier-

Munn et al., 1996), was chosen by Nageswararao,

(expressed below in terms of actual recovery to over-

flow, the typical cyclone product in comminution cir-

cuits):

Eoai ¼ Cea � 1

eadi=d50c þ ea � 2

� �ð21Þ

Implicit in the Nageswararao model is the requirement

that a, the parameter describing the shape of the effi-ciency curve must be separately determined by test work

for each material type.The invariant nature of the reduced efficiency curve

for a given cyclone design and feed characteristics has

been well established over the last three decades of

industrial experience at the JKMRC (Napier-Munn

et al., 1996) and elsewhere.

A detailed analysis of the reduced efficiency curve is

the central theme of a recent paper (Nageswararao,

1999b). That analysis also concludes that the assump-tion of invariance of the reduced efficiency curve with

cyclone geometry is an excellent approximation.

3.3.7. Effect of solids concentration on pressure––through-

put relationship

3.3.7.1. Nageswararao model. From studies on 38.1, 25.4

and 10.2 cm cyclones, Lynch et al. (1975) observed that,

with all other variables constant, throughput initiallyincreases with percent solids in the feed slurry, CWreaching a maximum at approximately 12–18% solids by

weight. Thereafter, Q decreases with CW. This effect wasquantified in later studies on a 15.2 cm cyclone (Nage-

swararao, 1978) when slurry was MEDIUM limestone,

as:

KQ2 for WaterKQ2 for Slurry

¼ 0:80 ð22Þ

The Nageswararao model parameters were evaluated

using data with CW greater than 40%. The choice of

Euler number as a performance characteristic explicitly

assumes that Q / q�0:5p , which is compatible with

empirical evidence.

3.3.7.2. Plitt model. The form of the pressure/through-put equation resulting from Plitt’s regression analysis

implies the functional relationship that pressure drop

increases with solids concentration.

However, Plitt’s equation was developed from 297

sets of data of which 100 sets were runs with water only,

28 sets were at 5% solids (w/w), and 9 sets were between

0.8 and 13% solids.

The assumption that pressure drop increases with

solids concentrations is valid only for those datasets in

which feed solids concentration is greater than 12–18%by weight (Lynch et al., 1975).

Thus, a significant portion of the Plitt data was in an

inappropriate range for the functional relationship

implicitly assumed via the regression analysis. This

highlights one of the problems of a regression based

approach.

3.3.8. Interaction of variables

A significant difference between the two models

concerns the interactive nature of the effects of Do andDu, especially for the prediction of Q (or P ) and S (orRf ).This is due to the way combinations of the outlet

areas––(D2u þ D2o) and (Du=Do)––appear as independentvariables in the Plitt model. This model will predict

different S values for the same percentage spigot change,depending on whatever other vortex finder changes have

been made.

By contrast, the Nageswararao model equation pre-

dicts a constant change (in Rf ), irrespective of othervariables. For example, an increase in Du of 10% will

always result in an increase of 26% in Rf in the Nage-swararao model, whereas the relative change in S pre-dicted by the Plitt model will also depend on the changesto Do.

3.3.9. Effect of feed inlet

Plitt explicitly ignored the independent effect of inlet

on flow split, while this was identified as an independent

variable in model development at the JKMRC (Nage-

swararao, 1978; Asomah, 1996; Asomah and Napier-

Munn, 1996).Although in both models, all inlet geometries are

assumed equivalent to a circle of equal area, there are

indications that the flow regime could be affected by the

inlet shape and geometry (Rogers, 1998). Recent

experimental work at JKMRC and by cyclone manu-

facturers indicates that the influence of inlet design is

crucial in some cases. Future modelling efforts need to

necessarily take this factor into consideration.

3.3.10. Effect of cyclone length

Plitt considered the free vortex height, h as an inde-pendent factor in his equation, thus simplifying the effect

to be of the same magnitude whether due to change in

cone angle or the cylinder length.

However, a distinction between the effects of Lc and hon the cyclone performance is made in the modelsdeveloped at the JKMRC.

682 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

3.3.11. Effect of angle of inclination of the cyclone

The significant effect of cyclone inclination has to

date only been quantified by Asomah (Asomah and

Napier-Munn, 1997).

Neither the Plitt, nor the Nageswararao models

included the effect. However, the latter is formulated in

such a manner that the effect can easily be included.

3.4. Current and potential improvements to the Plitt and

Nageswararao models––with hindsight

3.4.1. Plitt model

Flintoff et al. (1987) recorded that due to the structure

of the model, serious modelling efforts require recalcu-

lation of the model parameters and some times evenmodification of the model form. If this model is to be

used further, attention to several areas would seem

worthwhile:

• Plitt observed that the equation for classification in-

dex, m, is poorly correlated. A detailed analysis of

this issue (Nageswararao, 1999b) concluded that the

equation for m (Eq. (4)) is of little value.• King (2001) observed that the chief source of uncer-

tainty is in the prediction of the flow split, S. Furtherin estimating Rf (the parameter actually required forfurther calculations) from S by an indirect procedure(Hinde, 1977; Plitt et al., 1990; King, 2001), addi-

tional error propagation is inevitable.

• In the throughput equation, (Eq. (5)), the functional

relationship chosen for dependence of P on CW isclearly inconsistent with the low solids portion of

the data used for regression.

So far, no serious efforts to remedy these shortcom-

ings appear to have been attempted, apart from those of

Cilliers and Hinde (1991).

3.4.2. Nageswararao model

3.4.2.1. Cyclone diameter scaling

3.4.2.1.1. Throughput equation. With hindsight, it

could be argued that the scale factor, D�0:10c in the

equation for throughput (Eq. (13)) is an example of anattempt to arrive at the best possible fit to the available

experimental data!

Removing this term from the equation would intro-

duce an error of the order of only 7% for a 2 times

scaling. This error could well be within the range of

precision of the experimental measurements during

original data collection. If further modelling attempts

were carried out with a different data set using similarfunctional relationships, the results could well show that

Euler number is independent of cyclone diameter.

Alternatively, an a priori assumption that Euler number

is independent of cyclone diameter and a consequent

discarding of the scale up term in the model, might

prove equally as accurate as the original (Eq. (13)).

Recently Tavares et al. (2002) examined discarding this

scale factor. They reported good agreement between

measured and predicted values, although their data were

limited to 25 and 50 mm cyclones and Q varied only

from 1 to 5 m3/hr. The issue certainly merits furtherinvestigation.

3.4.2.1.2. Water split equation. The equations for Rfand RV are already independent of cyclone diameter.This is the result of observations made during the ori-

ginal modelling work that the Kv1 and Kw1 values, whilenot the same for all the cyclones treating the same feed,

did not follow a monotonic relationship with cyclone

diameter.3.4.2.1.3. Cut size equation––dimensional inhomoge-

neity. The appearance of the awkward scale up factor

D�0:65c in an otherwise dimensionally homogeneous

equation is due to the fact that the cyclone size itself is

taken as an independent variable.

An alternate model formulation would eliminate this

infelicity.

The cyclone model may be reformulated to describethe performance characteristics relative to a standard

cyclone, say, Dc;std. For a cyclone of any size, Dc, thevariable to be considered would be the scale ratio (say,

Dc=Dc;std). This factor would replace Dc in Eqs. (9), (11)and (12). The combined effect of feed material and un-

quantified variables now reflect the model constants for

the standard cyclone, say. K 0p0. In this case, Eq. (23)

would result in place of Eq. (12).

K 0p1 ¼ K 0

p0

DcDc;std

� �s

ð23Þ

The relation between the new material constants, K 0p0

and the current Kp0 can be expressed as:

K 0p0 ¼ Kp0ðDc;stdÞs ð24Þ

3.4.2.1.4. Model enhancement opportunities. Back-

ward compatibility imposes certain constraints to

development. In view of the extensive database accu-mulated at JKMRC/JKTech and the adjustments that

would need to be made to incorporate the above

improvements, such changes are unlikely to be incor-

porated into the current JKSimMet implementation of

the Nageswararao model.

Similarly, the effort involved in recalculating model

parameters probably prohibits their being included in an

updated model implemented in Limn or other suchpackages.

The above modifications may however, be incorpo-

rated in a future hydrocyclone model where the com-

bined benefits of these and as yet unidentified

improvements are sufficient to warrant a move to a new

model regime.

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 683

3.4.2.2. Flow split and water split. The concept that Rfcan be calculated either directly or from RV has beenintroduced previously.

In the following discussion RWf is used to denote theresult of Eq. (18), and RVf the indirect result of theapplying Eqs. (17) and (19).

If we examine the options available as a result of twoequations for estimation of Rf :

Case 1: Where RWf is more accurate than RVf .There is no need to use RVf for averaging.

Case 2: RWf and RVf are equally accurate.There is no need to do extra computation. We can use

RWf only, without loss of accuracy.Case 3: RVf is more accurate than RWf .

This is most unlikely, since calculation of Rf from RVinvolves a procedure where errors would accumulate, as

shown above. Such a case implies a sub-optimum

equation for Rf , and we should attempt to develop amore accurate equation, rather than using an indirect

method.

We may therefore conclude that the direct method forprediction of Rf is at least as good and probably betterthan the indirect method. We are more likely to intro-

duce errors in the estimation of Rf if it is calculated usingthe equation for RV and the corrected efficiencies.During the original model development, all of the

available data including those of Lynch and Rao (1975)

were tested. The results confirmed that the direct

method, that is using the Nageswararao equation (15)for Rf , was preferable (Nageswararao, 1978). In fact,it was only when the model was implemented in com-

puter software that the indirect method was con-

ceived. The equation for RV could be considered

superfluous.

The more recent cyclone model from the JKMRC

(Asomah and Napier-Munn, 1997) does not include an

equation for RV, implying that inclusion of the RVequation does not add significantly to the accuracy of

prediction of Rf .The validity of the equation for Rf alone is illustrated

with the data of Rao (1966). The results of prediction

are shown in Fig. 2, where the excellent agreement be-

tween the observed and predicted values can be seen.

These Rao data were not used in building the Nage-

swararao model (that is, evaluation of any modelparameters); they represent a completely independent

data set.

Further, when the results are viewed in comparison

with the predictions of S using Plitt’s equation (Fig. 1),the advantage of Eq. (15) compared to the Plitt ap-

proach is obvious.

In view of the above, the implementation of the

Nageswararao model in Limn does not include theequation for RV.

Issues of conflict with the existing parameter data-

base, mean that this modification is unlikely to be

implemented in the JKSimMet version of the model at

present.

3.4.2.3. ‘‘Fish hook’’ in efficiency curves. It is logical to

expect that with increase in size, the recovery tounderflow also increases as the terminal settling velocity

increases. This was the consensus among cyclone

researchers until the late seventies and observations to

the contrary were attributed to experimental errors.

However, since Finch (Finch and Matwijenko, 1977;

Finch, 1983) postulated a possible fish hook in the effi-

ciency curve, this phenomenon gained widespread

acceptance resulting in ardent support (Kelly, 1991).Reports on new observations and new theories to ex-

plain the effect are many (Del Villar and Finch, 1992;

Roldan-Villasana et al., 1993; Heiskanen, 1993; Brookes

et al., 1984; Rouse et al., 1987; Frachon and Cilliers,

1999; Chen et al., 2000; Kraipech et al., 2002, etc.).

In the early 1980s, Whiten at the JKMRC produced a

modified efficiency curve equation with an additional

parameter (b) to allow for the effect:

Eoai ¼ Cð1þ bb�di=d50cÞ ea � 1ð Þeab�di=d50c þ ea � 2

� �ð25Þ

The value b� was introduced to preserve the definition of

d50c. ie. d ¼ d50c when Eoa ¼ 1=2C. It can be computediteratively during evaluation of Eq. (25) by use of thisdefinition.

The current version of JKSimMet continues to use

the modified Whiten function, which incorporates a fish

hook as an option (Napier-Munn et al., 1996).

The experience at JKTech/JKMRC, where simulation

of hydrocyclone performance is done routinely, is that a

significant proportion of all hydrocyclone model fits

benefit from inclusion of the Whiten beta parameter inthe fit parameter set.

Whether all of these cases are genuine examples of a

fish hook in the data, or simply instances where a

slightly different shape to the efficiency curve allows an

improved fit, is at present undetermined. It is certainly

an area worth further investigation (Nageswararao,

1999a,b, 2000; Coelho and Medronoho, 2001).

Until a clearer understanding is available it is likelythat JKTech/JKMRC modellers will continue to apply

the fish hook when the data seems to warrant it. The

simpler (non fish hook) form of the model is always

available by suitable choice of parameters. Limn also

provides both model forms.

3.4.2.4. Effect of solids concentration on cut size––

equation for d50c. The Nageswararao model accountsfor the effect of feed solids concentration through the

use of the hindered settling factor, k. Steinour (1944)suggested the simplifying approximation that k is

Fig. 4. Relation between the relative d50c (d50c at desired solids con-tent/d50c when solids content is 40% by weight) versus percent solids infeed slurry.

684 K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687

proportional to CV=ð1� CVÞ3. This approximation wasconsidered satisfactory when the model was originally

developed.

Further studies by Castro, 1990), however, indicated

that at low solids concentration, the complete Steinour

function for k, that is, 101:82CV=ð1� CVÞ2 yielded betterestimates of d50c and this modification has been includedin the JKSimMet and Limn implementations of the

model.

In the JKSimMet implementation, a scaling factor of

8.05 in the denominator was introduced to preserve the

magnitude of the K values to allow comparison with

those obtained during the early usage of the model.

For simplicity and transparency, inclusion of this

scaling factor was not regarded as necessary for the baseLimn implementation of the model, since no compre-

hensive database of previous results are available out-

side the JKMRC. The scaling factor is also not likely to

be necessary in other non JKMRC implementations.

However, potential users of the model should be aware

that the Kd0 values obtained using the unscaled version,will be different from those obtained using JKSimMet.

The impact of this change, on the model parametersfor the d50c Eq. (14) will be small, since the feed solidscontent in the model fitting data set varied between 41%

and 70% (by weight), the range in which both expressions

yield the same value for k. A comparison of the observedand predicted d50c values is shown in Fig. 3. It illustratesthe small difference between the two estimates.

Also shown (Fig. 4) is a comparison of the results of

calculation using the original approximation for k, andthe the results of calculation using the complete Steinour

expression as suggested by Castro (1990). The density of

feed solids is assumed to be 2.7 and the ratio of d50c atgiven% solids (by weight) to d50c at 40% solids is plottedagainst% solids concentration.

Fig. 3. A comparison of observed and predicted values of d50c for themodel fitting data set with the hindered settling factor used initially

(Nageswararao, 1978) and with the modification (Castro, 1990).

From this graph, it can be seen that if the test data for

the evaluation of the material constants, (K values),

cover the range 40–70%, and predictions are desired in

the same range, the estimates for d50c using either methodwill be similar. If however, the K values are obtained

from data in the same feed solids range, but predictions

are required at lower feed concentrations, say, less than

30%, then the d50c predicted with the Castro correction,as implemented in the current JKSimMet version of the

model, will be higher.

3.5. Experience with the Nageswararao model

There can be no doubt that the Nageswararao model

has proven useful in control and design applications

(Napier-Munn et al., 1996).This model has been in continuous use at JKMRC

since its development in 1978. Others have also applied

it extensively via the JKSimMet simulator.

Reports (Finch and Matwijenko, 1977; Finch, 1983,

etc.) that the actual efficiency curve is not monotonic,

that is, the possibility that a dip or ‘‘fish hook’’ could

exist, have appeared in the literature.

It was also felt that the over flow product is really theproduct of interest to plant engineers and accordingly,

usage of efficiency curve to overflow (complementary to

the conventional ‘actual efficiency curve’ to underflow

used by most schools) became common at the JKMRC

and remains so (Napier-Munn et al., 1996).

The emergence of JKSimMet in the mid 1980s (Wi-

seman and Richardson, 1991; Napier-Munn et al., 1996,

etc.) provided an avenue for more wide spread use of theNageswararao model.

Further experience led to the modification in the

equation for cut size, as has been discussed.

K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 685

Experience with the Nageswararao model indicates

that it provides results with the same order of accuracy

as the data that is typically obtained from surveys in

minerals processing operations. This, coupled with the

existence of a large data and experience base in the use

of the model within JKTech/JKMRC has given rise to

an interesting dilemma. Any changes to the model willrequire a re-evaluation of the database. While potential

improvements have been identified, the improvements

are relatively minor, making the effort difficult to justify.

This dilemma extends to the use of the model outside

JKSimMet. Potential users of the published equations

may wish to maintain compatibility with the JKMRC

database. In the case of the Limn implementation, two

forms of the model are provided, one close to theJKMRC model, and a simpler version for general use

when there are no compatibility issues.

4. Summary and conclusions

1. Fundamental fluid flow models of hydrocyclones

are improving all the time and are now beginning to beuseful, especially in design. However unresolved prob-

lems in managing the fundamental fluid flow equations

and the computational intensity required for CFD

simulations ensure that for the foreseeable future

empirical models will continue to be the main simulation

environment for mineral processing engineers.

2. In the development of the Nageswararao model,

dimensionless design variables and operating variableschosen on phenomenological considerations are bound

together in a structure based on explicit assumptions to

obtain equations for performance characteristics.

Observation of both laboratory and industrial cyclones

provided the basis for these assumptions.

3. The domain of application of the Nageswararao

cyclone model and its successful use in the mineral

industry over the last 25 years both at JKMRC andelsewhere indicate that the original assumptions made in

formulating the model are realistic and are reasonable

representations of the actual separation processes taking

place in the cyclone.

4. Because the model was published in full and thus

enterered the public domain, the Plitt model saw wide-

spread early use, particularly as a teaching tool. Fol-

lowing the addition of the Flintoff corrections, the Plittmodel has also seen industrial use. With the proviso (as

in the Nageswararao case) that, if at all feasible, the

model should be fitted to data obtained under condi-

tions as close as possible to those to be simulated, the

model can be expected to give useful results.

5. For the estimation of water recovery to underflow,

the equation for Rf is sufficiently accurate. The separateequation for RV can be considered redundant at least asfar the estimate of Rf is concerned.

6. The complete Steinour approximation for hindered

settling factor, proposed by Castro, is considered

worthwhile.

7. The extensive industrial database and experience

gained using the Nageswararao model in JKSimMet

that is now available at the JKMRC and elsewhere

presents an interesting dilemma. Any changes to themodel will require reinterpretation of the database and

validation against experience. While the existing model

is seen as sufficiently accurate, it is difficult to justify

such effort.

8. For non JKMRC applications of the model, unless

it is necessary to maintain compatibility with the

JKMRC database or to transfer parameters from an-

other source of parameters using the JKSimMet imple-mentation of the Nageswararao model, the simpler

implementation model using just Rf and the unscaledversion of the complete Steinour approximation, is

appropriate.

Acknowledgements

Nageswararao (Nagu) Karri wishes to thank Drs.

A.J. Lynch, AO and T.C. Rao, from whom he learnt the

‘ABC’ of hydrocyclones, Dr. Bill Whiten from whom he

learnt ‘computer arithmetic’ and Dr. Lutz Elber from

whom he learnt how to survive during his stay in Aus-

tralia.

Tim Napier-Munn acknowledges useful discussions

with his colleagues at the JKMRC.Dave Wiseman thanks friends, colleagues and cus-

tomers, for advice, encouragement and feedback.

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