K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 P
d 50c 50c ¼
50:5 Dc0:46 D0i :6 Do1:21 e0:08C V Du0:71 h0:38 Q0:45 ðqs À q1 Þ
ð2Þ
0:5
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 however show the trend that as the particle size becomes finer, the d 50c increases’’. 50c size increases’’. The The Plitt Plitt mode modell in its its curr curren entt form form as revi revise sed d by Flintoff et al. (1987) has no dependence for feed size characteristics in any of the equations and is given below: P
d 50c 50c ¼ F 1
39:7 Dc0:46 D0i :6 Do1:21 g0:5 e0:063C V
m ¼ F 2 1:94 P ¼ F 3
S ¼ F 4
Du0:71 h0:38 Q0:45 Dc2 h Q
0:15
eð
ð3Þ
qs À1 k 1:6
 à Þ
À1:58S 1þS
ð4Þ
P
1:88Q1:8 e0:0055C V Dc0:37 D0i :94 h0:28 ð Du2 þ Do2 Þ
18:62qp0:24 ð Du = Do Þ
3:31 0:54
h
ð5Þ
0:87
split, for example, that used in the Plitt model, could be made. 3.2.2 3.2.2.. Operating Operating and design design variables variables The choice choice of independent independent operating variables/facvariables/factors, tors, which which are relevant relevant for modelling modelling,, was based on phenomenolo phenomenological gical consideratio considerations. ns. A suitably suitably modified modified product of the Euler and Froude numbers, f P =ðqp gDc Þg was considered an appropriate factor that could be used to account for the centrifugal centrifugal force field generated in the cyclone. The hindered settling factor ðvH =vT Þ, k was chosen to account for the effect of the differential movement of solid particles and hence the effect of feed solids concentration centration on d 50c was assu assumed med that that the the hind hinder ered ed 50c . It was settling factor would adequately account for the changes in pulp viscosity and viscous effects due to changes in feed solids content. The obviou obviouss choice choicess for design design variab variables les were included: •
P
ð Du2 þ Do2 Þ0:36 e0:0054C V
•
Dc1:11 P 0:24
•
ð6Þ
•
Since Flintoff et al. (1987) do not include a specific feed size size term, term, but provide provide F factors factors for calibra calibratio tion n it is probably safe to assume that the model should be recalibrated whenever feed data are available, in preference to using the uncalibrated equations.
•
3.2. Nageswarar Nageswararao ao model model development development––Nag ––Nageswar eswararao arao (1978) Althoug Although, h, the basic basic model model equatio equations ns as develo developed ped and and in a modi modifie fied d form form are are publ publis ishe hed d (Lyn (Lynch ch and and Morr Morrell ell,, 1992 1992;; Na Nage geswa swara rara rao, o, 1995 1995;; Na Napi pierer-Mu Munn nn 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 considered relevant relevant to describe describe cyclone cyclone performance, performance, collectively referred to as P i , were: • • • •
q ffiffi ffi
The Euler number, EU defined as Q= Dc2 q P . p The dimensionless cut size, d 50c 50c = 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 direct comparison with other available equations for water
675
•
cyclone diameter, Dc ; reduced reduced vortex finder, Do = Dc ; reduced spigot, Du = Dc ; reduced 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 geometrically similar cyclones cyclones operating 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, tangential, etc.) were explicitly explicitly ignored. ignored. Consequentl Consequently y the effects of these variables, if significant, would introduce duce errors errors in the model. model. Anothe Anotherr signific significant ant implici implicitt assumption in the model building exercise is the fixed proper properties ties of the fluid medium. medium. This implies 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 considered as an independent independent variable. variable. The following following thought thought experiments experiments elucidate elucidate this contention. contention. If we visualise a cyclone treating two homogeneous but different materials (say, limestone and iron ore) under identical operating conditions, also assuming the particle size and shape distributions to be identical, such that the only difference is the material being treated, we would still expect that the cyclone performance characteristics (Q, d 50c 50c and Rf ) would be different. different.
672
K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
Nomenclature C cyclone water split to overflow a; b; . . .; . ; g parameters in equation for Rf d size of the particle, lm d 50c corrected classification size, lm 50c Dc , Do , Du , Di diameters of the cyclone, vortex finder, spigot and inlet Dc;std diameter of the standard cyclone EU Euler number E UC ‘corrected’ cyclone split to underflow UC E oa actual cyclone split to overflow oa F 50 medi median an size size (that (that is, 50% 50% passi passing ng)) of feed feed 50 solids f ; f 1 ; . . .; . ; f 11 functions of . . . 11 functions f i size distribution of feed solids C W per cent solids (by weight) in feed slurry g acceleration due to gravity F 1 ; F 2 ; F 3 ; F 4 calibra calibration tion parame parameter terss for Plitt’s Plitt’s equaequations H head of feed slurry (Plitt’s equation for flow split) h free vortex height K po common material dependent constant in the po genera generalise lised d model model for perfor performan mance ce charac charac-teristic, P i ( p ¼ Q, d , W and V respectively for thro throug ughp hput, ut, cut cut size size,, water water recov recover ery y and and volumetric recovery equations) 0 K po material dependent constant in the reformulate lated d gene genera ralis lised ed mode modell for for perf perform orman ance ce characteristic, P i K p1 function of K p0 p1 p0 and cyclone diameter K p2 function function of K p1 design variables variables p2 p1 and minor design ( DI , Lc and h) k hydr hydrod odyna ynamic mic expo expone nent nt,, to be esti estima mated ted from from data data,, in Plitt Plitt’s ’s equa equatio tion n (3) (3) for for d 50c 50c (default value for laminar flow 0.5)
2. the quality of separation of the products, as quantified 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 design and operating conditions. While theoretical methods for the prediction of cyclone clone perform performanc ancee based based on consid considerin ering g the physica physicall principles of motion of solid particles in a fluid medium do exist, (Barrientos and Concha, 1992; Concha et al., 1996; 1996; Monred Monredon on et al., 1992, 1992, etc.), etc.), they they have have not yet made a significant impact on the prediction of hydrocyclon cyclonee perfor performan mance ce in mineral mineralss proces processing sing industr industry y applications.
length of the cylindrical section of the cyclone classification index cyclone feed pressure performance performance characteristic characteristics, s, EU, d 50c 50c = Dc , Rf , RV Q throughput of the cyclone, l/min Rf recovery of water to underflow Rf W recove recovery ry of water water to underflo underflow w calcul calculate ated d form equation for Rf V Rf recove recovery ry of water water to underflo underflow w calcul calculate ated d form equation for RV RV volumetric recovery of feed slurry to underflow S volu volumet metric ric flow flow spli splitt (vol (volume umetr tric ic flow flow in underflow/vo underflow/volumetric lumetric flow in overflow) overflow) V H , V T terminal velocities––hindered and unhindered conditions s scale-up parameter P C V percent solids in feed by volume C V volumetric fraction of feed solids a cyclone efficiency curve shape parameter b cyclone efficiency curve shape parameter 3 k hindered settling factor, C V =ð1 À C V Þ , 8:05 Â 2 101:82C V =ð1 À C V Þ m1 , m2 unknow unknown/u n/unqu nquant antifia ifiable ble operat operating ing/de /design sign variables g liquid viscosity (in Plitt’s equation for d 50c 50c ) h full cone angle, degrees qp density of feed pulp qs density of feed solids ql density of feed fluid medium (water)
Lc m P P I
2.1. Theoretical/phenomenological models––possibilities and limitations Conside Considerab rable le progre progress ss is being being made made in the fundafundamental modelling of hydrocyclon hydrocyclones es using solutions of the the basic basic fluid fluid flow flow equa equati tion ons, s, eith either er dire directl ctly y or via via comm commer ercia ciall Comp Computa utatio tiona nall Fluid Fluid Dy Dynam namic icss code codess (Chakr (Chakrabo aborti rti and Miller, Miller, 1992; 1992; Rajaman Rajamanii and Milin, Milin, 1992; 1992; Concha Concha et al., 1997; 1997; Dya Dyakow kowski ski and William Williams, s, 1997; Slack et al., 2000; Brennan et al., 2002; Brennan et al., 2003). 2003). It is likely that this approa approach ch will soon soon provid providee useful useful results results,, particu particular larly ly with regard regard to the optimisation of cyclone design. However However such solutions are computationa computationally lly intensive; current JKMRC work on the CFD modelling of a
K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
hydrocyclone operating under normal industrial conditions using parallel processing in a super computer can consume two weeks of CPU time for one steady state simulation. Invokin Invoking g Moore’s Moore’s Law (Moore (Moore,, 1965), 1965), we would would 1 expect solution time to halve every 12 to 2 years. Existing CFD CFD model modelss coul could d ther therefo efore re not not be expec expecte ted d to be useable 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 process simulators simulators or spreadsheets spreadsheets will therefore therefore continue tinue to be the main basis of process process simulati simulation on and optimisation at least in the short to medium term. Indeed, Indeed, it is likely likely that that a hybrid hybrid approach approach,, where where computationally intensive models are used to assist in building empirical models, will become more common as develo developme pment nt in theore theoretica ticall and phenom phenomeno enolog logical ical models continues. 2.2. Practical mathematical modelling of hydrocyclones The The term term ‘mode ‘model’ l’ in gene genera rall and and ‘ma ‘math them emati atica call model’ model’ in particu particular lar,, have have contex contextt sensiti sensitive ve meanin meanings gs (Davis (Davis and Hersh, Hersh, 1981; 1981; Edward Edwardss and Hanson Hanson,, 1989; 1989; Murthy et al., 1990). In simplistic terms, we can say that a mathematical model of a system is an ‘idealised representation of a physical reality’, in the form of a set of self self cons consist isten entt equa equatio tions ns.. In this this pape paperr ‘mod ‘model’ el’ and and ‘mathematical model’ have the same meaning. Typica Typically, lly, model model equatio equations ns predict predict output output charac charac-teristics in terms of input variables. The ease of application and the usefulness of any model is dependent on the choice of characteristics to be predicted, the factors or vari variab able less that that are are assu assumed med to affec affectt the the phys physic ical al process and the assumptions and approximations used in expressing these variables in the mathematical structure. The independent independent variables variables for the model equations equations are the operating regime and design parameters of the cyclone. In view of the current limitations of the theory as outline outlined d above, above, sim simplifi plified ed models models that that are based based on specific specific observed observed performance performance characteristics characteristics can provide a viable alternative. alternative. Since current understanding understanding of the mechanics of fluid flow cannot yet allow determinatio mination n of the model model paramet parameters ers from from purely purely theotheoreti retica call cons consid idera eratio tions, ns, thes thesee are are dete determi rmine ned d from from expe experim rimen ental tal data data only only and and the the mode models ls are are term termed ed ‘empirical’ models. Specifically with regards to cyclones, the performance characteristi characteristics cs that have been identified for modelling modelling are: • •
the pressure-throughput relationship; the ‘corrected efficiency curve’ and the corrected cut size, d 50c 50c ;
• •
673
the reduced efficiency curve, a plot of corrected efficiency versus normalised size, d =d 50c 50c ; the distribution of water into the products usually, as recovery of water to underflow, Rf but some times as flow ratio into the products, S .
The early cyclone literature abounds with equations for one or more more (for (for example example,, pressu pressure– re––th –throu roughp ghput, ut, 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). Outside of the JKMRC, there are other examples of models of this genus, for example, those due to Brookes et al. (1984) and Vallebuona et al. (1995). These models were based on an implicitly assumed structur structuree for each each of the perfor performan mance ce charac character teristi istics. cs. The machine and operating variables were varied as part of the the expe experim rimen enta tall desig design. n. Inte Interpo rpolat lation ion on such such models could be used to get a reasonable estimate of the cyclone performance for a particular machine-material combination. combination. Applicability beyond the database database from which they are derived is questionable. Such models can be simplist simplistica ically lly describ described ed as curve curve fitting fitting to experi experi-mental data. A recent example in this category is the model due to Firth Firth (200 (2003) 3).. Altho Although ugh char charac acter terize ized d by the the use use of dimensionless groups such as Reynolds Number, Euler Number Number and Froude Froude Number Number,, togeth together er with dimendimensionless design variables, this model also relies on curve fitti fitting ng to arri arrive ve at a site site-s -sp pecifi ecificc mode model. l. This This is ackn acknow owle ledg dged ed in Firth Firth’s ’s unam unambi bigu guou ouss conc conclu lusio sion: n: given that the flow patterns will be expected to change ‘‘ ‘‘ with change in the cyclone diameter and geometric shape, shape, the actual values for the empirical parameters and power indices could be expected to change.’’ change.’’ The other category includes those models in which the model 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 design and operat operating ing conditi conditions ons,, witho without ut resort resorting ing to further experimental experimental work. However, such models require material specific constants, which must be determined 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 Napier-Munn (1996) (1996) belong belong to this category. category.
3. Hydrocyclone Hydrocyclone models for industrial industrial application– application––two –two specific models
Where Where models models are require required d to describ describee the perfor perfor-mance mance of hydroc hydrocyclo yclones nes used used as classifi classifiers ers in closed closed
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K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
grindin grinding g circuit circuitss in minera minerall industr industry, y, the most most comcommonly reported use appears to be of those due to Plitt (1976), Flintoff et al. (1987) and Nageswararao (1978). Altho Althoug ugh h both both of thes thesee mode models ls can can be redu reduce ced d to a simila sim ilarr form, form, ther theree are are dist distin inct ct diffe differe renc nces es in mode modell formulation (that is the choice of dependent variables and model structure) and evaluation of model parameters. ters. The The deve develo lopm pmen entt of the the two mode models ls and and thei theirr specific differences are elaborated below. 3.1. Plitt model development––Plitt (1976) and Flintoff et al. (1987) Plitt Plitt’s ’s deve develo lopm pmen entt meth method odolo ology gy was was rela relativ tively ely straightforward. The dependent variables, chosen by Plitt were: • • • •
cyclone throughput, Q; cut size, d 50c 50c ; volumetric flow split, S ; sharpness of classification, m. As design or independent variables, he chose
• • •
diameters diameters of the cyclone, vortex finder, spigot and inlet, Dc , Do , Du , Di ; combinations of the above, ( Du2 þ Do2 ) and ( Du = Do ); free vortex height, h.
When the inlets were not circular, the inlet size corresponded to circle of the same area. To account for the length of the cyclone, he used free vortex height, h, defined 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 of feed slurry, H . Plitt also took into account that the feed solids content significantly affects the pulp viscosity, which in turn influences d 50c additi tion on,, hind hinder ered ed sett settli ling ng and and 50c . In addi crowding were also considered as possible factors. To account for the influence of solids content in the feed slurry, C V 1 (volumetric fraction of feed solids) was the preferr preferred ed variab variable, le, as the rheolo rheologic gical al proper properties ties 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 functio functional nal relation relationship ship,, which which was found found to best best represe represent nt the effect effect of C V on d 50c exponential 50c was an exponential form. form. This This was was final finally ly inco incorp rpor orate ated d only only beca becaus usee it
prov provide ided d bett better er fit than than any any of the the othe otherr func functio tiona nall x x x forms such as C V , ð1 þ C V Þ=ð1 À C V Þ and fð1 þ 0:5C V Þ= ð1 À C V Þ4 g x that were tried. The data for Plitt’s regression equation(s) included •
•
• •
the industrial data of Rao (123 data sets including cyclone diameters of 2000 , 1500 , 1000 and 600 , treating such diverse materials as silica, copper ore, tailings); his experimental work which included 9 tests on 6 00 cyclone where the feed solids content was varied between 0.8% and 13% by weight; 28 tests on 600 , 33 tests on 2.500 and 8 tests on 1.25 00 cyclones at 5% solids (by weight) in the feed slurry; 80 tests with water on 6 00 and 20 tests also with water on 2.500 . (These of course could be used for Q and S equations only.)
The The orig origin inal al mode modell (Plit (Plitt, t, 1976) 1976) was was obta obtain ined ed by using a stepwise multiple linear regression program. Plitt repeated the linear regression procedure with different functi functiona onall forms forms (linear (linear,, power power and expone exponenti ntial) al) and differe different nt variab variable le combin combinatio ations. ns. He includ included ed in the model equations only those variables that were found significant at 99% level. It is appropriate to mention here that in proposing the equati equations ons for pressur pressuree drop, drop, P and and flow flow split split,, S , Plit Plittt used used 297 sets of data, including the tests run with water only. As d 50c 50c values were not available for all the data sets, only 179 of the sets were used for the d 50c 50c equation. Only the 162 tests with sufficient data points above and below d 50c 50c to form a complete classification curve, were used for the equation for m. By combining data from different feed materials, such as silica, copper, ore, tailings and silica flour (and cyclones clones too) too) in develo developing ping the model model equati equations ons,, Plitt Plitt impli implicit citly ly assu assumed med that that the the cycl cyclon onee perf perform orman ance ce is indepe independe ndent nt of feed feed materia materiall charac character teristi istics. cs. He was then able to claim that the performance could be estimated with reasonable accuracy even when no experimental data are available. This is the most conspicuous feature of his model. In the the orig origin inal al refe referen rence ce,, Plit Plittt (197 (1976) 6) offer offered ed two two form formss of the the d 50c equati tion on,, one one with with and and the the othe otherr 50c equa without feed size effects. This is the Plitt (1976) equation for d 50c 50c when feed size effect is included. F 50 50 is the weight median size of feed solids in microns (50% passing size) 2 0:52
P
d 50c 50c ¼
50:5 Dc0:46 D0i :6 Do1:21 e0:08C V = F 50
0:5
Du0:71 h0:38 Q0:45 ðqs À q1 Þ
ð1Þ
and the Plitt (1976) equation for d 50c 50c without considering feed size effect 1
Plitt used the symbol b to represent volumetric fraction of feed solids. This paper uses C V to avoid confusion with the use of b in the Whiten Whiten cyclone cyclone efficiency efficiency equatio equation. n. When When the volumet volumetric ric solids solids content in the feed is expressed as per cent, symbol used is C VP . cent, the symbol
2
Note that in Eqs. (1)–(6), (1)–(6), the units are: Dc , Di , Do , Du , h (cm); Q (l/ 3 m); P (kPa); gp (cP); C V (%); d 50c 50c (lm); qs , ql (g/cm ).
K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687 P
d 50c 50c ¼
50:5 Dc0:46 D0i :6 Do1:21 e0:08C V Du0:71 h0:38 Q0:45 ðqs À q1 Þ
ð2Þ
0:5
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 however show the trend that as the particle size becomes finer, the d 50c increases’’. 50c size increases’’. The The Plitt Plitt mode modell in its its curr curren entt form form as revi revise sed d by Flintoff et al. (1987) has no dependence for feed size characteristics in any of the equations and is given below: P
d 50c 50c ¼ F 1
39:7 Dc0:46 D0i :6 Do1:21 g0:5 e0:063C V
m ¼ F 2 1:94 P ¼ F 3
S ¼ F 4
Du0:71 h0:38 Q0:45 Dc2 h Q
0:15
eð
ð3Þ
qs À1 k 1:6
 à Þ
À1:58S 1þS
ð4Þ
P
1:88Q1:8 e0:0055C V Dc0:37 D0i :94 h0:28 ð Du2 þ Do2 Þ
18:62qp0:24 ð Du = Do Þ
3:31 0:54
h
ð5Þ
0:87
split, for example, that used in the Plitt model, could be made. 3.2.2 3.2.2.. Operating Operating and design design variables variables The choice choice of independent independent operating variables/facvariables/factors, tors, which which are relevant relevant for modelling modelling,, was based on phenomenolo phenomenological gical consideratio considerations. ns. A suitably suitably modified modified product of the Euler and Froude numbers, f P =ðqp gDc Þg was considered an appropriate factor that could be used to account for the centrifugal centrifugal force field generated in the cyclone. The hindered settling factor ðvH =vT Þ, k was chosen to account for the effect of the differential movement of solid particles and hence the effect of feed solids concentration centration on d 50c was assu assumed med that that the the hind hinder ered ed 50c . It was settling factor would adequately account for the changes in pulp viscosity and viscous effects due to changes in feed solids content. The obviou obviouss choice choicess for design design variab variables les were included: •
P
ð Du2 þ Do2 Þ0:36 e0:0054C V
•
Dc1:11 P 0:24
•
ð6Þ
•
Since Flintoff et al. (1987) do not include a specific feed size size term, term, but provide provide F factors factors for calibra calibratio tion n it is probably safe to assume that the model should be recalibrated whenever feed data are available, in preference to using the uncalibrated equations.
•
3.2. Nageswarar Nageswararao ao model model development development––Nag ––Nageswar eswararao arao (1978) Althoug Although, h, the basic basic model model equatio equations ns as develo developed ped and and in a modi modifie fied d form form are are publ publis ishe hed d (Lyn (Lynch ch and and Morr Morrell ell,, 1992 1992;; Na Nage geswa swara rara rao, o, 1995 1995;; Na Napi pierer-Mu Munn nn 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 considered relevant relevant to describe describe cyclone cyclone performance, performance, collectively referred to as P i , were: • • • •
q ffiffi ffi
The Euler number, EU defined as Q= Dc2 q P . p The dimensionless cut size, d 50c 50c = 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 direct comparison with other available equations for water
675
•
cyclone diameter, Dc ; reduced reduced vortex finder, Do = Dc ; reduced spigot, Du = Dc ; reduced 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 geometrically similar cyclones cyclones operating 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, tangential, etc.) were explicitly explicitly ignored. ignored. Consequentl Consequently y the effects of these variables, if significant, would introduce duce errors errors in the model. model. Anothe Anotherr signific significant ant implici implicitt assumption in the model building exercise is the fixed proper properties ties of the fluid medium. medium. This implies 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 considered as an independent independent variable. variable. The following following thought thought experiments experiments elucidate elucidate this contention. contention. If we visualise a cyclone treating two homogeneous but different materials (say, limestone and iron ore) under identical operating conditions, also assuming the particle size and shape distributions to be identical, such that the only difference is the material being treated, we would still expect that the cyclone performance characteristics (Q, d 50c 50c and Rf ) would be different. 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 different feed size distributions (say 25%––270 mesh in one case and 50%––270 mesh in the other) under identical identical operating operating and design conditions. conditions. In this case also, we would expect the performance performance characcharacteristics to be different.
P i ¼ K p0 p0
s
K p2 p2 ¼ K p0 p0 ð Dc Þ
P i ¼ f ð K m ; Dc ; Do = Dc ; Du = Dc ; Di = Dc ; Lc = Dc ; h; k;
ð7Þ
Clearly, t1 ; t2 ; . . . etc., are the unknown/unquantifiable operating and design variables/factors and those whose inde indepe pend nden entt effect effect,, if any, any, we shal shalll not not attem attempt pt 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, P i are separable. Eq. (7) can then be written as: P i ¼ f 1 ð K m Þ þ f 2 ð Dc Þ þ f 3 ð Do = Dc Þ þ f 4 ð Du = Dc Þ
þ f 5 ð Di = Dc Þ þ f 6 ð Lc = Dc Þ þ f 7 ðhÞ þ f 8 ðkÞ þ f 9 ðf P =ðqp gDc ÞgÞ þ f 10 10 ðt1 Þ þ f 11 11 ðt2 Þ . . .
ð8Þ
It was further assumed that the influence of the design . ; f 11 and operating variables (that is, f 1 ; f 2 ; . . .; 11 , etc.) follows a monomial power function relationship. As a sim simpl plifi ificat catio ion, n, the the influ influen ence ce of thos thosee known known factors that cannot be determined and the effect of unknown known factor factorss are clubbe clubbed d togeth together er with with the materia materiall effects effects in the form of a materia materiall specifi specificc perform performanc ancee constant, K p0 p0 . The effect of fixed fluid properties (water) is also absorbed by K p0 p0 . Eq. (8) becomes: s
P i ¼ K p0 p0 ð Dc Þ
f
! Du Dc
a
Do Dc
b
Di Dc
c
Lc Dc
d
e
h
P qp gDc
a
Du Dc
b
Do Dc
P qp gDc
k g
ð10Þ
where
This explains in a simple way the effects of the size distribution, density of feed material characteristics on the cyclon cyclonee perform performanc ance. e. There There may be other other effects. effects. Realising that a suitable description of the feed material effect effect is comple complex, x, no simplific simplificatio ations ns to quanti quantify fy any specific material effect in terms of say, nominal product size, size, dens density ity,, etc. etc. was was atte attemp mpte ted. d. Inst Instead ead,, the the feed feed material characteristics were simply combined in a single parameter K m . The above considerations can be summarised mathematically ematically as:
f P =ðqp gDc Þgt1 ; t2 ; . . .Þ
f
!
k g
ð9Þ For a system where the variables are only Do , Du , feed pressure and pulp density of the feed slurry, this could be further reduced to:
c
d
Di Dc
LC Dc
he
ð11Þ
For For conv conveni enien ence ce,, the the effec effectt of cycl cyclon onee size size and and the the material and other effects could be combined when scale up from one cyclone to the other is not required, as: s
K p1 p1 ¼ K p0 p0 ð Dc Þ
ð12Þ
Furthe Furthermor rmore, e, it was assume assumed d that that the effect effect of spigot spigot diamet diameter er on Euler Euler number number is insigni insignifica ficant nt and can be ignored. If the pressure throughput relationship is considered 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 pressu pressure re and pulp density density factors. factors. That That is, the model parameters a and f are both zero when P i ¼ EU in Eq. (9). Data available in the literature (for example, Bradley, ley, 1965 1965;; Lync Lynch h and and Rao, Rao, 1975 1975)) and and a prel prelimi imina nary ry study (Nageswararao et al., 1974) are the basis for these additional assumptions. To account for the complex flow pattern in the cyclone clone (spec (specific ifical ally ly due due to high high solid solidss conc concen entr trat atio ion n normally normally encountered encountered in industrial industrial practice) and consequ sequen enti tial al effec effectt on the the rela relati tive ve move movemen ments ts of solid solid particles, the hindered settling factor was the preferred variab variable. le. The approx approximat imation ion sugges suggested ted by Steino Steinour ur (1944) (1944) that the hindered hindered settling factor k is proportional to the volumetri volumetricc fractio fraction n of feed feed solids, solids, or k ¼ C V = ð1 À C V Þ3 , is used in all numerical calculations. This is certainly a simplistic approximation and does not not take take into into acco accoun untt the the inde indepe pend nden entt effec effectt of size size distribution of feed solids in particular the clay content, which could be expected to strongly influence pulp viscosity and hence the terminal settling velocities. However, as the ‘material effect’ had already considered, it was felt that K m together together with k adequately account for the overall influence of dense slurries on the cut size. The perception is that while k encompasses the differences in cyclone behaviour due to changes in percent solids in feed slurry, K m accounts for the changes due to material characteristics. As a consequence, the material dependent performance performance constants (the K values) in the model will 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 3.2.4.. Evaluation Evaluation of model parameters The set of model equations given by Eqs. (10)–(12) can can only only be mean meaning ingfu full if the the nume numeric rical al valu values es of a; b; . . .; . ; g and the scale factor s for each of the performance characteristics, P i , are known. As the theory of
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the hydrodynamics within the cyclone is not developed enough enough to evaluat evaluatee these these direct directly, ly, experim experiment ental al data data 3 were required to calibrate the model. The extens extensive ive databa database se of Lynch Lynch and Rao (1975) (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), MEDIUM (50% passing––53 lm) and COARSE (40% passing––53 lm) were were used used for for this this purp purpos osee as deta detaile iled d below. 1. The parameters a, b, f and g in equation set (10) were were evalua evaluated ted from from the Lynch– Lynch–Rao Rao data data (34 tests) tests) treati treating ng FINE limestone limestone in a 38.1 38.1 cm hydroc hydrocycl yclone one.. After After suitabl suitably y transf transform orming ing the data, data, the regress regression ion method developed by Whiten (1977) was used. For this regression analysis, the design variables were assumed to be exac exactt and and it was was furth further er assu assumed med that erro errors rs in loge EU , loge f P =ðqp gDc Þg, loge k, were ±0.03, and were ±0.01 in loge ðd 50c 50c = Dc Þ. The special feature of the Whiten regression method is that it takes into account errors in independent variables, unlike other methods available at that time, which assumed the independent independent variable variable to be exact. The parameters thus obtained were used further to evaluate the average K p2 p2 for each of the data sets of the Lynch–Rao (1975) database. 2. The Nageswararao 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 carried out with with variat variation ionss in vortex vortex finder finder diamete diameter, r, spigot, feed pressure and solids concentration. The feed material was MEDIUM limestone (containing 50%––53 Mean K p2 values es for for each each data data set set were were then then lm). Mean p2 valu determined (using the same model parameters a, b, f and g obtained in step 1 above). These together with the K p2 p2 values from the Lynch-Rao data sets were used to estimate 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 obtained on 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 c from the earlier step, K p1 p1 values for each data set could be calculated. 4. The dependence dependence of P i on cyclone cyclone size (scale (scale up factor, s), was estimated independently from the data for each of three size distributions studied by Lynch and Rao. Rao. The The relat relativ ivee erro errors, rs, if any, any, in each each of K p1 p1 were taken into consideration and the final scale up factors reported below are those that reflect the assumed functional 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: Q Dc2
À0:10
q ffiffi ffi ffi ffi ffi P =qp
¼ K Qo f Dc
0:45
Di Dc
Lc Dc
0:20
Lc Dc
h
Lc Dc
0:22
h
Dc0:00
Lc Dc
0:15
Do Dc
0:00 Rf ¼ K W Wo Dc
RV ¼ K V 0
0:20
hÀ0:10
ð13Þ 0:52
À0:50
! È É ! È É !
Â
Â
0:68
Do g Dc
d 50c Do 50c ¼ K Do f DcÀ0:65g Dc Dc
Â
4
À0:24
Do Dc
0:22
h
À0:24
Du Dc
P qp gDc
À1:19
Du Dc
P qp gDc
À0:94
Du Dc
P qp gDc
Di Dc
0:20
À0:22
k0:93
2:40
Di Dc
ð14Þ
0:50
À0:53
k0:27
1:83
Di Dc
ð15Þ
0:25
À0:31
ð16Þ
3.3. Comparison of Plitt and Nageswararao models The following section examines the assumptions and appr approx oxima imatio tions ns in the the mode modell form formula ulatio tion n for for both both models. 3.3.1. Model structure 3.3.1.1. Nageswararao model . The most significant feature of the Nageswararao model is the a priori choice of desi design gn and and oper operat atin ing g vari variab able less and and the the expl explic icit it assumptions made in binding them to the model equations. This resulted in a model with an assumed structure that explicitly deco decoup upled led the the ma mach chin inee and and ma mater teria iall characteristics. This was the first of the models developed oped at the the JKMR JKMRC C to inco incorp rpor orate ate this this import importan antt concept and represented a clear paradigm shift to a new modelli modelling ng approa approach. ch. Later Later Whiten Whiten and his studen students ts Awachie Awachie (1983) (1983) and Nar Naraya ayanan nan (1985) (1985) extend extended ed this this notion to develop material specific breakage functions for crushe crushers rs and grinding grinding mill mills. s. NapierNapier-Mun Munn n 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 prac practic ticee in deve develo lopin ping g an empi empiri rica call mode model. l. 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 , C V (fraction); d 50c 50c (lm); qp (t/m3 ).
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K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
regression equations for the chosen performance characteristics in terms of independent variables are developed. The choice of independent variables, as well as the equati equation on structu structure, re, are govern governed ed by consid considera eration tion of best fit equations for the available database. This This diffe differe renc ncee is well well illust illustra rate ted d by the the diffe differe rent nt handling of the effect of feed solids concentration on d 50c 50c . Plitt’s final choice for the independent variable ( C VP ) 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, hand, for the Nag Nagesw eswara ararao rao model, both both the indepe independe ndent nt variabl variablee (k) and the functio functional nal relationship (power) were explicitly assumed. The only choi choice ce avai availa labl blee afte afterr thes thesee assu assump mpti tion onss was was the the approx approximat imation ion(s) (s) availa available ble for the hinder hindered ed settling settling facto factor. r. Cons Conseq eque uent ntly ly,, the the only only modi modific ficati ation on to the the model model sugges suggested ted by Castro Castro (1990) (1990) was restrict restricted ed preprecisely by these constraints. (This is discussed in further detail below.) It shou should ld be ment mentio ione ned d tha that Plit Plittt too too forc forced ed À0:5 into into the the equa equatio tion n for for d 50c assuming g ðqs À ql Þ 50c by assumin laminar flow. Despite his expressed expressed reservations 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 on simplify simplifying ing assump assumptio tions ns or hypoth hypothese esess and (usual (usually) ly) proces processs measure measuremen ments. ts. Errors Errors in any measur measured ed data data used used for evaluat evaluating ing model model parame parameter ters, s, will be carried 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 the model is for our objective––in our case, prediction of the performance characteristics, within the limits of precision of their measurement, specifically, when the cyclone used used as a clas classifi sifier er in closed closed grin grindi ding ng circ circuit uits. s. This 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 the earl early y data databa base se of Rao Rao (196 (1966) 6).. This This was was supp supple le-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 3.3.2.2. .2. Nageswarara Nageswararao o model model . For For the the Na Nage gesw swara arara rao o model the accuracy of parameters is exclusively dependent on the extensive data base of Lynch and Rao (1975) and Nageswararao Nageswararao (1978). (1978). 3.3.3 3.3.3.. Evaluation Evaluation of model parameters 3.3.3.1. Plitt model . In the Plitt model, the independent variab variables les,, the model model parame parameter terss and the functio functional nal (linear (linear power power and expone exponentia ntial) l) relatio relationsh nships ips are govgoverned purely by consideration of the best fit under the multipl multiplee linear linear regres regressio sion n method method used. used. Plitt’s Plitt’s regres 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 functional relationships relationships between 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 the spigot effect. The significance ( t -test) -test) of the coefficient for the spigot term as obtained by regression was not the consideration for its omission. A similar rationale applies to the omission of the k term in the equation equation for RV (Eq. (16)). Equally, in the equation for cut size (Eq. (14)), the criteri criteria a for inclusion inclusion of a spigot spigot term and a consta constant nt factor ( K d2 for the model fitting data set) were the choice d2 of the model structure and not the significance level of the regression coefficients in a t -test. -test. Using Using this approach approach,, if the assump assumptio tions ns were were perperfectly true and the data were precise, the number of data sets needed for evaluation of model parameters would exac exactl tly y equa equall the the numb number er of unkn unknown own para parame mete ters rs.. However, However, in practice, the assumptions assumptions are never perfect, nor the data free of errors. Regression analysis is then needed to get the best estimates of the model parameters ters,, spec specifi ifica call lly y for for the the effec effects ts of Do , Du , k and P =ðqp gDc Þ. Of the 52 tests available, those tests outside the range of intere interest st for model model applica applicatio tion n (for (for example example,, feed feed solids solids conten contentt above above 70% or where where classifi classificati cation on was poor) and those with suspected high experimental error (for (for exampl example, e, tests tests with poor poor materia materiall balanc balances) es) were were not included in the regression analysis. A set of 34 tests on 38.1 38.1-c -cm m cycl cyclon ones, es, trea treati ting ng FINE FINE lim limes esto tone ne was was consid considere ered d sufficien sufficient. t. This This data data set alone alone was used used to finally determine the model parameters. All other data sets could then be used to validate the model parameters parameters and for further further evaluatio evaluation n (effect (effect of inlet inlet etc.) etc.) where where necessary. Conceptually we cannot use the Nageswararao model to predict the absolute values of the performance char-
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K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
acteristics acteristics (Q, d 50c 50c , etc.) without any test data for the desired material. This is a major difference of the model compared to that of Plitt, where default model parameters were provided. That these defaults were not reliable, able, as acknow acknowledg ledged ed by the incorpo incorporat ration ion of the F calibration parameters in Flintoff’s modified version, is a sepa separa rate te issu issue. e. At leas leastt in prin princi ciple ple,, it is poss possibl iblee to determine the absolute values of the performance characteristics using the Plitt model. With the Nageswararao model, when predictions are required required in greenfield greenfield situations, situations, it is necessary necessary to source appropriate K values from previous surveys. JKMRC/ JKTe JKTech ch has has built built a cons consid idera erabl blee reso resour urce ce of thes thesee parameters over the course of many years of use of the model. However, even a parameter database such as this can 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 with cauti caution on,, since since they they are are typic typical ally ly not not base based d on experim experiment entally ally determi determined ned K values values for the desired 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 dist distrib ributi ution on,, impl implyi ying ng that that cycl cyclon onee perf perfor orma manc ncee is independent of the feed size distribution. In the original original model model (Plitt, (Plitt, 1976) Plitt offered offered an optional equation, with F 50 50 , (median feed size, that is 50% passing size) as a variable. Such a simple approximation for the feed size effect is however, questionable and the more recent implementation does not include that equation. It is to be expect expected ed that that cyclon cyclonee perfor performan mance ce does 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 hydrocyclone 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 equations might be appropriately adjusted. Flintoff et al. (1987) revised the model by incorporating calibration factors, F 1 À F 4 , for each of the model equatio equations, ns, presuma presumably bly taking taking into into consid considera eration tion the observation observationss of independent independent researchers (for example, example, Apling et al., 1980) that the predictions are inaccurate. Thei Theirr expe expect ctat atio ion n in intr introd oduc ucin ing g the the cali calibr brat atio ion n parameters was that calibration with experimental data would give improved predictions.
3.3.4.2. Nageswararao model . Because of the observation that that cycl cyclon onee perf perform orman ance ce is affec affecte ted d by both both feed feed material material type and size distribution, distribution, the Nageswararao Nageswararao model is structured to allow it to be ‘‘tuned’’ to particular ular feed feed materia materials ls by parame parameter ter fitting fitting to measur measured ed plant data. In fact, the ideal use of the Nageswararao model is to determi determine ne the materia materiall specific specific consta constants nts from from a test test using a geometrically similar (or the same) cyclone on the the part partic icul ular ar feed feed type type and and to use use thos thosee cons consta tants nts whenev whenever er that that materia materiall is encoun encountere tered. d. For example example,, results will certainly be more accurate in a milling circuit simulation where series cycloning is to be investigated, if two different sets of material specific constants are are deriv derived ed for for mil milll disc discha harg rgee and and prima primary ry cyclo cyclone ne overflow. 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 The impo import rtan ance ce of the the reco recove very ry of wate waterr to the the underflow, Rf , is well understood. It also represents the minimum recovery of the near zero sized particles and is the starting point on the actual efficiency curve. The manner in which this performance characteristic is modelled modelled represents represents a significant 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 following earlier researchers (for example, Stas, 1957; Moder and Dahlstrom, 1952; Bradley, 1965). Rf , which is ultimately required for subsequent calculations of the cyclone performance, can then be calculated using the equation suggested by (Hinde, 1977; Plitt et al., 1990; King, 2001).
& P & P n
S =ð1 þ S Þ À C V 1 À Rf ¼
f i e
m À0:6931ðd =d 50c 50c Þ
1
n
m
50c Þ f i eÀ0:6931ðd =d 50c
1 À C V 1 À
1
'
'
ð17Þ
To use Plitt’s equations (6) and (17), both feed pressure, P and throughput, throughput, Q are required. This is because the equati equation on for S includ includes es pressu pressure re as an indepe independe ndent nt variable and those for d 50c 50c and m include Q. This implies that for a better estimate of Rf , both P and Q need to be meas measure ured. d. When When one one of them them is esti estima mated ted from from the the model, model errors are introduced. Further model errors arise from the errors in estimation of both d 50c 50c and m, which are required for the prediction of Rf according to Eq. (17). Cillie Cilliers rs and and Hinde Hinde (199 (1991) 1) also also note noted d that that Plitt Plitt’s ’s equation for S (Eq. (6)) does not take the effect of feed solids concentration concentration completely into account, account, even after ‘calibration’. They proposed a provisional revision with
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K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
coefficients of 1.51 for Du = Do instead of 3.31 and 0.0787 instead of 0.0054 for the solids concentration term, C VP . The conclu conclusio sion n is that that the Plitt Plitt equati equation on overes overestitimates the effect of Du = Do and underestimates that of C VP , at least for the Cilliers and Hinde data. Despite Despite these these reserv reservatio ations, ns, the curren currentt version version of MODSIM continues to use Eqs. (6) and (17) to predict Rf (King, 2001). Plitt used the industrial data of Rao (1966) for evaluation of his model parameters. To evaluate the equation for S , the values are calculated for the data of Rao (1966) and are shown in Fig. 1. It can can be seen seen that that the the pred predict icted ed valu values es of S are subject to significant errors even when applied to Plitt’s model fitting database. For comparison, the predicted values 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 of error in the Plitt model. A detailed discussion on flow split and water recovery in hydrocyclones is available elsewhere (Nageswararao, 2001). The Na Nage gesw swar arar arao ao 3.3. 3.3.5. 5.2. 2. Na Nage gesw swar arar arao ao mo mode del l . The model included equations for both Rf and RV (repeated below):
È É È É
Rf ¼ K W Wo
Â
Lc Dc
RV ¼ K V 0
Â
Dc0:00
Dc
0:22
:
0:22
Do Dc
À1:19
hÀ0 24
Dc0:00
Lc Dc
Do
hÀ0:24
2:40
À0:50
! ! Du Dc
P qp gDc
À0:94
Du Dc
P qp gDc
Di
Dc
For comparison with Fig. 1, the observed versus calculated 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 of Eq. (17), different estimates of Rf will be obtained from each of Eqs. (18) and (19). However, due to the indirect calculation method, Rf V woul would d carr carry y forw forwar ard d the the erro errors rs in the the estim estimat ation ion 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 approach proach is to average them with appropriate appropriate weighting weighting to calcul calculate ate a single single predic predicted ted value value for Rf . This This is the procedure followed initially at the JKMRC and subsequen quently tly used used in the the implem implemen enta tatio tion n of the the mode modell in JKSimMet. 3.3.6. Reduced efficiency curve Both Both mode models ls rely rely on the the conc concep eptt of the the ‘redu ‘reduce ced d efficien efficiency cy curve’ curve’ and each each model model assume assumess a particu particular lar form form of that that curv curvee (Nap (Napier ier-M -Mun unn n et al., al., 1996 1996). ). The The shap shapee of the the redu reduce ced d efficie efficienc ncy y curv curvee (a plot plot of the the ‘corrected’ efficiency versus dimensionless size, d =d 50c 50c ) is a measure measure of the sharpnes sharpnesss of separa separatio tion n within 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.
À0:53
:
k0 27
1:83
Di Dc
ð18Þ
Plitt (197 (1971) 1) (as (as did did Reid Reid (197 (1971) 1))) 3.3.6 3.3.6.1. .1. Plitt Plitt model model . Plitt derived a Rosin-Rammler type function: m
À ln 2ðd i =d 50c 50c Þ E UC UCi ¼ 1 À e
À0:25
À0:31
ð19Þ
ð20Þ
and and assu assume med d that that the the redu reduce ced d effici efficien ency cy curv curvee is dependent dependent on operating operating and design conditions, conditions, developing Eq. (4) to describe describe the effect of these on parameter parameter
40 70
35
) 60 % ( t 50 i l p s w 40 o l f d 30 e t c i d 20 e r P 10
) 30 % ( f
SPlitt SNageswararao
25
R d 20 e t c i d 15 e r P
10 5 0
0 0
10 20 30 40 50 60 Observed flow split (%)
70
Fig. 1. Predict Prediction ion of flow split for model fitting data of Plitt Plitt (after Nageswararao, 2001).
0
5
10 15 15
20
25 30 3 0 35 35
40
Observed R f (%) Fig. 2. Nageswararao Nageswararao model prediction of water recovery to underflow (data ex Rao, 1966).
K. Nageswararao et al. / Minerals Engineering 17 (2004) 671–687
m in Eq. (20). However, he records the poorest correlation lation coefficient coefficient (0.75) (0.75) for this Eq. (4) among among all his model equations.
Nageswa eswarar rarao ao relied relied on 3.3.6.2. Nageswarar 3.3.6.2. Nageswararao ao model . Nag the earlier Lynch and Rao, 1975) JKMRC approach, based on regarding the reduced efficiency curve as constan stantt for for cycl cyclon ones es of diffe differe rent nt physi physica call dime dimens nsio ions ns treating the same feed material. The Whiten form of the efficiency equation (NapierMunn Munn et al., al., 1996 1996), ), was was chos chosen en by Na Nage geswa swara rara rao, o, (expre (expresse ssed d below below in terms terms of actual actual recovery recovery to overoverflow, flow, the typical typical cyclon cyclonee produc productt in comminu comminutio tion n circuits): E oa oai
ea À 1 ¼ C ad =d 50c þ ea À 2 e i 50c
ð21Þ
Implicit in the Nageswararao model is the requirement that a, the parameter describing the shape of the efficiency curve must be separately separately determined 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 been well well esta establ blish ished ed over over the the last last thre threee deca decade dess of indust industria riall experie experience nce at the JKMRC JKMRC (Napie (Napier-Mu r-Munn nn et al., 1996) and elsewhere. A detailed analysis of the reduced efficiency curve is the central central theme theme of a recent recent paper paper (Nages (Nageswara wararao rao,, 1999b). That analysis also concludes that the assumption 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 throughput initially initially incr increa ease sess with with perc percen entt solid solidss in the the feed feed slur slurry ry,, C W reaching a maximum at approximately 12–18% solids by weight. Thereafter, Q decreases with C W . This effect was quantified in later studies on a 15.2 cm cyclone (Nageswararao, 1978) when slurry was MEDIUM limestone, as: K Q2 Q2 for Water ¼ 0:80 K Q2 Q2 for Slurry
ð22Þ
The Nag Nageswa eswarar rarao ao model model parame parameter terss were were evalua evaluated ted C using using data data with with W greate greaterr than than 40% 40%.. The choice choice of Euler number as a performance characteristic explicitly assume assumess that that Q / qpÀ0:5 , whic which h is comp compat atib ible le with with empirical evidence. 3.3.7.2. Plitt model . The form of the pressure/throughput equation resulting from Plitt’s regression regression analysis analysis implies implies the functio functional nal relati relations onship hip that that pressu pressure re drop drop increases increases with solids concentration. concentration.
681
However However,, Plitt’s Plitt’s equati equation on was develo developed ped from 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 assump assumptio tion n that that pressu pressure re drop drop increa increases ses 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 inappr inappropr opriat iatee range range for the functi functiona onall relatio relationsh nship ip implici implicitly tly assume assumed d via the regres regressio sion n analys analysis. is. This This highlig highlights hts one of the proble problems ms of a regres regressio sion n based based approach. 3.3.8. Interaction of variables A sign signifi ifica cant nt diffe differen rence ce betw betwee een n the the two mode models ls concerns the interactive nature of the effects of Do and Du , especially for the prediction of Q (or P ) and S (or Rf ). This is due to the way combinations of the outlet areas––( Du2 þ Do2 ) and ( Du = Do )––appear as independent variab variables les in the Plitt model. model. This This model model will predict 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 predicts dicts a consta constant nt change change (in Rf ), irrespe irrespecti ctive ve of other other variables. For example, an increase in Du of 10% will always result in an increase of 26% in Rf in the Nageswararao model, whereas the relative change in S predicted by the Plitt model will also depend depend on the changes changes to 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 (Nageswarara swararao, o, 1978; 1978; Asomah Asomah,, 1996; 1996; Asomah Asomah and Napier Napier-Munn, 1996). Althoug Although h in both both models, models, all inlet inlet geometr geometries ies are assumed equivalent to a circle of equal area, there are indications that the flow regime could be affected by the inle inlett shap shapee and and geom geomet etry ry (Rog (Roger ers, s, 1998 1998). ). Rece Recent nt experimental work at JKMRC and by cyclone manufacturers 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 independent 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, However, a distinction between the effects of Lc and h on the the cycl cyclon onee perfo perform rman ance ce is ma made de in the the mode models ls developed at the JKMRC.
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3.3.11. Effect of angle of inclination of the cyclone The signific significant ant effect effect of cyclon cyclonee inclina inclination tion has to date date only only been quantifie quantified d by Asomah Asomah (Asoma (Asomah h and Napier-Munn, 1997). Neith Neither er the the Plitt, Plitt, nor nor the the Na Nage geswa swara rara rao o mode models ls 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 recalculation lation of the model paramete parameters rs and some times times even even modification of the model form. If this model is to be used used furth further er,, atte attent ntio ion n to seve severa rall area areass woul would d seem seem worthwhile: •
•
•
Plitt observed that the equation for classification index, m, is poorly poorly correla correlated ted.. A detaile detailed d analysi analysiss of this issue (Nageswararao, 1999b) concluded that the equation equation for m (Eq. (4)) is of little value. King (2001) observed that the chief source of uncertainty is in the prediction of the flow split, S . Further in estimating Rf (the parameter actually required for further calculations) from S by an indirect procedure (Hinde, (Hinde, 1977; Plitt et al., al., 1990; 1990; King, King, 2001), 2001), additional error propagation is inevitable. In the throughput equation, (Eq. (5)), the functional relationship relationship chosen for dependence of P on C W is clearly clearly inconsist inconsistent ent with the low solids solids portion portion of the data used for regression.
So far, no serious efforts to remedy these shortcomings 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 With hind hindsig sight ht,, it 3.4.2 3.4.2.1.1 .1.1.. Throughput Throughput equation equation.. With À0:10 could could be argu argued ed that that the the scal scalee fact factor or,, Dc in the the equation for throughput (Eq. (13)) is an example of an attempt to arrive at the best possible fit to the available experimental data! Removing this term from the equation would introduce an error of the order of only 7% for a 2 times scaling scaling.. This This error error could could well be within within the range range of precisi precision on of the experi experimen mental tal measure measuremen ments ts during during original original data collection. collection. If further further modelling modelling attempts attempts were carried out with a different data set using similar functional relationships, the results could well show that Euler Euler numb number er is inde indepe pend nden entt of cycl cyclon onee diam diamet eter. er. Alternatively, Alternatively, an a priori assumption assumption that Euler number is indepe independe ndent nt of cyclon cyclonee diamet diameter er and a consequ consequent ent
disc discar ardi ding ng of the the scal scalee up term term in the the mode model, l, mig might ht prov provee equa equally lly as accu accura rate te as the the orig origin inal al (Eq. (Eq. (13)) (13)).. Recently Tavares et al. (2002) examined discarding this scale scale factor factor.. They They report reported ed good good agreem agreement ent betwee between n 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. /hr. The issue issue certain certainly ly merits merits furthe furtherr investigation. 3.4.2.1.2. Water split equation. equation. The equations for Rf and RV are already already indepe independe ndent nt of cyclon cyclonee diamete diameter. r. This is the result of observations made during the original modelling work that the K v1 v1 and K w1 w1 values, while not the same for all the cyclones treating the same feed, did not follow follow a monoto monotonic nic relatio relationsh nship ip with cyclone cyclone diameter. 3.4.2 3.4.2.1.3 .1.3.. Cut size equation––di equation––dimensi mensional onal inhomogeinhomogeneity. neity. The appearance of the awkward scale up factor DcÀ0:65 in an otherwi otherwise se dimensi dimensiona onally lly homoge homogeneo neous us 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 describe the performance characteristics characteristics relative to a standard standard cyclone, say, Dc;std . For a cyclone of any size, Dc , the variable 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 unquantified variables now reflect the model constants for 0 the standa standard rd cyclone cyclone,, say. say. K p0 . In this this case case,, Eq. Eq. (23) (23) would result in place of Eq. (12). 0
0
K p1 ¼ K p0
s
Dc Dc;std
ð23Þ
0 The relation relation betwee between n the new materia materiall consta constants, nts, K p0 and the current K p0 p0 can be expressed as: s
0 K p0 ¼ K p0 p0 ð Dc;std Þ
ð24Þ
Back-3.4.2 3.4.2.1.4 .1.4.. Model Model enhancement enhancement opportuniti opportunities es.. Back ward ward comp compat atib ibili ility ty impo impose sess cert certain ain cons constr train aints ts to develo developme pment. nt. In view view of the extens extensive ive databa database se accuaccumulated at JKMRC/JKTech and the adjustments that woul would d need need to be ma made de to inco incorp rpor orat atee the the abov abovee improv improveme ements, nts, such such change changess are unlike unlikely ly to be incorincorporated into the current JKSimMet implementation of the Nageswararao Nageswararao model. model. Similarly, the effort involved in recalculating model parameters probably prohibits their being included in an update updated d model model impleme implemente nted d in Limn or othe otherr such such packages. The above modifications may however, be incorporated in a future hydrocyclone model where the combined ined ben benefits efits of thes thesee and and as yet yet unid uniden enti tifie fied d improvements are sufficient to warrant a move to a new model regime.
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3.4.2.2. Flow split and water split. split. The concept that Rf can be calculated either directly or from RV has been introduced previously. In the following discussion Rf W is used to denote the resu result lt of Eq. Eq. (18) (18),, and and Rf V the the indir indirec ectt resu result lt of the the applying Eqs. (17) and (19). If we examine the options available as a result of two equations for estimation of Rf : Case Case 1: Where Where Rf W is more accurate than Rf V . There is no need to use Rf V for averaging. averaging. W V Case 2: Rf and Rf are equally accurate. There is no need to do extra computation. We can use Rf W only, without loss of accuracy. Case 3: Rf V is more accurate than Rf W . This is most unlikely, since calculation of Rf from RV involves a procedure where errors would accumulate, as show shown n abov above. e. Such Such a case case impl implies ies a subsub-op optim timum um equation equation for Rf , and we should attempt to develop a more accurate equation, rather than using an indirect method. We may therefore conclude that the direct method for prediction prediction of Rf is at least as good and probably better than the indirect method. We are more likely to introduce errors in the estimation of Rf if it is calculated using the equation for RV and the corrected efficiencies. During During the origin original al model model develo developmen pment, t, all of the available data including those of Lynch and Rao (1975) were were test tested ed.. The The resu result ltss confi confirm rmed ed that that the the dire direct ct method, that is using the Nageswararao equation (15) for Rf , was prefera preferable ble (Nages (Nageswara wararao rao,, 1978). 1978). In fact, fact, it was only when the model was implemented in compute puterr soft softwa ware re that that the the indi indire rect ct meth method od was was conconR ceiv ceived ed.. The The equa equati tion on for for V coul could d be cons consid ider ered ed superfluous. The more recent recent cyclone cyclone model model from from the JKMRC (Asomah and Napier-Munn, 1997) does not include an equatio equation n for RV , impl implyi ying ng that that incl inclusi usion on of the the RV equation does not add significantly to the accuracy of prediction prediction of Rf . The validity of the equation for Rf alone is illustrated illustrated with the data of Rao (1966). The results of prediction are shown in Fig. 2, where the excellent agreement between the observed and predicted values can be seen. These Rao data were not used in building the Nageswar swarar arao ao mode modell (that (that is, eval evalua uatio tion n of any any mode modell parameters); parameters); they represent represent a completely completely independent independent data set. Further, when the results are viewed in comparison with the predictions of S using Plitt’s equation (Fig. 1), the the adva advant ntag agee of Eq. Eq. (15) (15) comp compar ared ed to the the Plitt Plitt apapproach is obvious. In view view of the the abov above, e, the the implem implemen entat tatio ion n of the the Nageswa Nag eswarar rarao ao model model in Limn does does not not inclu include de the the equation equation for RV .
Issues Issues of conflic conflictt with with the existin existing g paramet parameter er datadatabase base,, mean mean that that this this modi modific ficati ation on is unlik unlikely ely to be implemented in the JKSimMet version of the model at present. 3.4.2.3. ‘‘Fish hook’’ in efficiency curves. curves . It is logical to expe expect ct that that with with incr increa ease se in size size,, the the reco recove very ry to underflow also increases as the terminal settling velocity incr increas eases es.. This This was was the the cons consen ensu suss am amon ong g cycl cyclon onee 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 efficienc ciency y curv curve, e, this this phen phenom omen enon on gain gained ed wides widespr prea ead d accepta acceptance nce resulti resulting ng in ardent ardent suppor supportt (Kelly, (Kelly, 1991). 1991). Reports on new observations and new theories to explain 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 modifie modified d efficienc efficiency y curve curve equati equation on with with an additio additional nal parameter (b) to allow for the effect: a ð1 þ bbà d i =d 50c 50c Þðe À 1Þ E oai C ¼ à oai 50c þ ea À 2 eab d i =d 50c
ð25Þ
The value bà was introduced to preserve the definition of d 50c 50c . ie. d ¼ d 50c 50c when E oa oa ¼ 1=2C . It can be computed iteratively during evaluation of Eq. (25) by use of this definition. The current version of JKSimMet continues to use the modified Whiten function, function, which incorporates 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 significa significant nt propor proportion tion of all hydroc hydrocyclo yclone ne model model fits benefit from inclusion of the Whiten beta parameter in the fit parameter set. Whether all of these cases are genuine examples of a fish fish hook hook in the the data data,, or simp simply ly inst instan ance cess wher wheree a slightly different shape to the efficiency curve allows an improved fit, is at present undetermined. It is certainly an area area worth worth further further investi investigat gation ion (Nages (Nageswar wararao arao,, 1999a,b, 2000; Coelho and Medronoho, 2001). Until a clearer understanding is available it is likely that JKTech/JKMRC modellers will continue to apply the fish hook when the data seems to warrant it. The simple sim plerr (non (non fish fish hook hook)) form form of the the mode modell is alway alwayss availab available le by suitab suitable le choice choice of parame parameter ters. s. Limn also provides both model forms. 3.4. 3.4.2. 2.4. 4. Effect Effect of soli solids ds conc concen entra trati tion on on cut cut size size–– –– . The Nagesw Nag eswara ararao rao model mod el accoun acc ounts ts equation equation for d 50 50c for the effect of feed solids concentration through the use of the hinder hindered ed settlin settling g factor factor,, k. Steinour Steinour (1944) (1944) sugges suggested ted the simplify simplifying ing approx approximat imation ion that that k is
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proportional proportional to C V =ð1 À C V Þ . This approximation was considered satisfactory when the model was originally developed. Further studies by Castro, 1990), however, indicated that at low solids concentration, the complete Steinour 2 function for k, that is, 101:82C V =ð1 À C V Þ yielded better estimates estimates of d 50c 50c and this modification has been included in the the JKSi JKSimMe mMett and and Limn impleme implementa ntatio tions ns of the model. In the JKSimMet implementation, a scaling factor of 8.05 in the denominator was introduced to preserve the magnitu mag nitude de of the K values values to allow allow compar compariso ison n with those obtained during the early usage of the model. For sim simplic plicity ity and transp transpare arency ncy,, inclusio inclusion n of this this scaling factor was not regarded as necessary necessary for the base implementat ntation ion of the model, model, since since no compre compre-Limn impleme hensive database of previous results are available outside the JKMRC. The scaling factor is also not likely to be necess necessary ary in other other non JKMRC JKMRC impleme implementa ntation tions. s. However, potential users of the model should be aware that the K d0 d0 values obtained using the unscaled version, will be different from those obtained using JKSimMet. The impact of this change, on the model parameters for the d 50c 50c Eq. (14) will be small, since the feed solids content in the model fitting data set varied between 41% and and 70% 70% (by (by weigh weight), t), the the rang rangee in which which both both expr expres essio sions ns yield the same value for k. A comparison of the observed and predicted d 50c 50c values is shown in Fig. 3. It illustrates the small difference between the two estimates. Also shown (Fig. 4) is a comparison of the results of calculation using the original approximation for k, and the 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 d 50c 50c at d given% solids (by weight) to 50c 50c at 40% solids is plotted against% solids concentration.
d50c 0c for the Fig. 3. A comparison of observed and predicted predicted values of d 5 model fitting data set with the hindered settling factor used initially (Nageswararao, 1978) and with the modification (Castro, 1990).
Fig. 4. Relation between between the relative d50c (d 50c 50c at desired solids content/d 50c when solids content is 40% by weight) versus percent solids in 50c feed slurry.
From this graph, it can be seen that if the test data for the evalua evaluation tion of the materia materiall consta constants nts,, (K values) values),, cover the range 40–70%, and predictions are desired in the same same range, range, the estimat estimates es for for d 50c 50c using either method will 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 d 50c 50c 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 proven useful useful in contro controll and design design applica applicatio tions ns (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 the product of interest to plant engineers and accordingly, usage of efficiency curve to overflow (complementary to the conventional conventional ‘actual ‘actual efficiency efficiency curve’ to underflow 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 (Wiseman and Richardson, 1991; Napier-Munn et al., 1996, etc.) provided an avenue for more wide spread use of the Nageswararao model. Furth Further er expe experie rienc ncee led led to the the modi modific ficati ation on in the the equation for cut size, as has been discussed.
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Experience Experience with the Nageswararao 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 will require a re-evaluation of the database. While potential improvements improvements have been identified, the improvements 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 implementation,, two form formss of the the mode modell are are prov provid ided ed,, one one clos closee to the the JKMRC model, and a simpler version for general use when there are no compatibility issues.
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6. The complete Steinour approximation for hindered settl settlin ing g facto factor, r, prop propos osed ed by Cast Castro ro,, is consid considere ered d worthwhile. 7. The extensive industrial database and experience gained gained using using the Nag Nagesw eswara ararao rao model model in JKSimMe JKSimMett that that is now now avai availab lable le at the the JKMR JKMRC C and and else elsewh wher eree presen presents ts an intere interestin sting g dilemma. dilemma. Any change changess to the model 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 nece necess ssar ary y to ma main inta tain in comp compat atib ibil ilit ity y with with the the JKMRC database or to transfer parameters from another source of parameters using the JKSimMet implement mentati ation on of the the Na Nage geswa swarar rarao ao mode model, l, the the sim simpl pler er implementation implementation model using just Rf and the unscale unscaled d version version of the comple complete te Steino Steinour ur approx approximat imation ion,, is appropriate.
4. Summary and conclusions
1. Fundamental fluid flow models of hydrocyclones are improving all the time and are now beginning to be useful, especially in design. However unresolved problems in managing the fundamental fluid flow equations and and the the comp comput utati ation onal al inte intensi nsity ty requ requir ired ed for for CFD simul simulat atio ions ns ensu ensure re that that for for the the fore foresee seeab able le futu future re empirical models will continue to be the main simulation environment environment for mineral mineral processing processing engineers. engineers. 2. In the development of the Nageswararao model, dimensionless design variables and operating variables chosen on phenomenological considerations are bound together in a structure based on explicit assumptions to obta obtain in equa equatio tions ns for for perf perform orman ance ce chara characte cteri risti stics cs.. Observation of both laboratory and industrial cyclones provided the basis for these assumptions. 3. The domain of application of the Nageswararao cycl cyclon onee model model and and its succ success essfu full use use in the the mine mineral ral indu industr stry y over over the the last last 25 year yearss both both at JKMRC JKMRC and and elsewhere 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 widespread spread early use, particula particularly rly as a teachi teaching ng tool. tool. Following the addition of the Flintoff corrections, the Plitt model has also seen industrial use. With the proviso (as in the Nageswar Nageswarara arao o case) case) that, that, if at all feasible, feasible, the model should be fitted to data obtained under conditions 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 sufficiently accurate. The separate separate equation for RV can be considered redundant at least as far the estimate of Rf is concerned. concerned.
Acknowledgements
Nageswarar Nageswa rarao ao (Nagu) (Nagu) Kar Karri ri wishes wishes to thank 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 Australia. Tim Napier-Munn Napier-Munn acknowledge acknowledgess useful discussions with his colleagues at the JKMRC. Davee Wiseman Dav Wiseman thanks thanks friends friends,, colleag colleagues ues and cuscustomers, for advice, encouragement and feedback.
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