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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: [email protected] (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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