OPTIMAL CONTROL OF A BALL MILL GRINDING CIRCUIT-I. GRINDING CIRCUIT MODELING AND DYNAMIC SIMULATION Utah Generic Center Control m Comminution, International RAJ K. RAJAMANI 115 EMRO, University of Utah, Salt Lake City, UT X4112, U.S.A. JOHN A. HERBST Inc., 419 Wakara Way, Suite 101, Salt Lake City, UT 84108, U.S.A. (Firsr received 28 April 1988; acrepled in revisedform 29 June 1990) Abstract-Mineral grinding circuits can be controlled with a set of proportional-integral (PI) controllers or alternatively by specialized controllers which make use of optimal control theory. The latter control strategy is superior in the sense that feed solid and water addition rates are manipulated in concert to achieve a specitied control objective. A dynamic model is needed for optimal control, and for PI control the model can be used for ofi-line tuning. Off-line tuning circumvents the problem of on-line tuning, during which transients persist for a long time, resulting in lost production. The key elements of the full dynamic model are the population balance model of the ball mill and an empirical model of the hydrocyclone. The model development and its verification for both steady- and unsteady-state responses are shown. On-line computations with the full dynamic model require the solution of 37 differential equations at every sampling instant. In addition, optimal control calculations may overburden the control computer. Therefore, a slmplltied model usmg Just three state variables 1s shown to be adequate for dynamic predictions. In Part 11 the full dynamic model is used in off-line tuning and the simplified model is used in the optimal controller. Both model predictions and pilot scale ball mill circuit responses are shown. INl’KODUCTION Ball mill grinding circuits constitute the primary unit operation in the production of metal from mined ore. In a typical mineral processing plant for copper ore, crushers are used to break large lumps of mined ore to pieces of 2.5 cm size which are further reduced to a fineness of 100 pm in ball milling circuits. After the grinding stage, the ore is subjected to a concentration operation such as flotation. The metal is extracted from the concentrate by leaching, solvent extraction, electrowinning or smelting operations. Grinding is an inherently energy-intensive process in which less than 10% of the total electrical energy input is utilized in the size reduction of ore particles. Considering the fact that thousands of tonnes of ore are processed every day in a typical plant, the cost of operation of the grinding plant often dictates the overall cost of metal production. Hence, it is essential that such circuits be run as efficiently as possible. A ball mill draws a certain level of power just to keep the ball charge in motion and a fraction thereof to grind the ore. Therefore, the cost of operation is minimized by maintaining the ore feed rate to the mill at the maximum design capacity at all times. But, at the same time, the product from the circuit-the fine stream from the hydrocyclone classifier-must meet the size specification for metal extraction efficiency in subsequent processing. During operation the grinding circuit experiences many disturbances: ore hardness variations, ore feed rate changes and feed size variations. Ore feed rate and feed size disturbances can be easily countered by adjusting the water additions to the circuit, but ore hardness variations can cause drastic reduction in mill throughput. Hardness of the ore varies considerably, depending on the location from which it was mined. Owing to ore hardness disturbances, the mass rate of recycled stream and the fineness of the circuit product tend to fluctuate and so the objective of computer control strategy is to counteract these disturbances in the optimum sense. Grinding circuits can be controlled with a series of proportional-integral controllers. By their very own nature these controllers gradually vary process streams until setpoint is reached. In other words, the controllers hunt for steady-state conditions. As a result, circuit transients persist for a very long time, especially due to interaction between controllers. To avoid these problems, some control strategies employ a mathematical model of the circuit to predict the output of the process one sampling step ahead and apply control actions accordingly. The optimal control discussed in this manuscript uses a result known as “Pontryagin’s Maximum Principle” which provides a mathematical solution to the dynamic optimization problem. In Part I of this paper models of the subunits in the grinding circuit are developed. Then experimental verification of the dynamic model is shown. The use of dynamic models in control strategies is described in Part II. 861 862 RAJ K. RAJAMANI CIRCUIT MODELS Grinding circuit models have been used for the evaluation (Herbst and Mular, 1980) of alternative circuit configuration and also for the evaluation of alternative control strategies (Stewart, 1970, Herbst and Rajamani, 1979) and to obtain a quantitative measure of transients of the circuits under control. However, dynamic modeIs of the nature described here are nonexistent in the literature. In the following, mathematical models of the mill, sump and hydrocyclone are developed in detail. These models form the basis of dynamic simulations and an on-line simplified model. Ball mill model The application of the population balance concept to particle breakage processes occurring in a ball mill results in an integro-differential equation in which the particle size distribution is expressed as a function of time and particle size. A more convenient model expression results when the continuous size range d,-d, is divided into a set of n discrete intervals. The linear, size-discretized model for breakage kinetics is obtained by dividing the particulate assembly being ground into n narrow size intervals with a maximum size d, and a minimum size d,. The ith size interval is bound by di above and di+ , below, and the mass fraction of material in this size interval at time t is denoted by mi. Then, for a continuous mill, a mass balance for the material in the ith size interval at time t yields accumulation dH(t)m,MP,i(f) = input - output = dt - S;N(t)m~,,Atl M Mf m MF.i -M i-L + C b,SjH(t) j=l + generation MPm MP.i(t) m,,,,(t) JOHN A. HERBS “concentration” is the solids concentration, C,, MP, defined as the mass of solids per unit volume of slurry. Then, the hold-up mass within the mill is given by H(C) = V,C,,,P. Then the variation is described by (2) in solids concentration dCs MP Vnrk = dt in the mill QMFG,MF - QMPCS,MP (3) where VM is the volume of slurry in the mill, and Q is the volumetric flow rate of slurry. The subscripts MF and MP denote mill feed and mill discharge streams, respectively. Since for overflow ball mills V, is a constant the volumetric feed rate to the mill equals the volumetric discharge rate at all times. Utilizing this fact in eq. (3) yields dC S.MP ___ = +%s. dt M MF - C,, hfp). (4) A total of n + 1 differential equations, i.e. n equations for size distribution and one equation for solids concentration in the mill slurry is to be solved for complete dynamic description of the ball mill. Sump model The slurry in the sump is kept suspended by an impeller and hence the sump behaves as a single perfect mixer. Under the assumption that size changes do not occur in the sump, i.e. negligible attrition due to impeller blades, the following equations provide a complete description of sump behavior in a circuit employing post classification: dm,, L=M dt (11 where H(t) is the total mass hold-up in the mill, and are the mass rate of solids flow into and discharge from the mill, respectively. In eq. (1) Si, the size-discretized selection function for the ith size interval, denotes the fractional rate at which material is broken out of the ith size interval, and bij, the sizediscretized breakage function, represents the fraction of the primary breakage product material in the jth size interval. The perfect mixing assumption is implied in eq. (1). For the case in which the mill residence time distribution (RTD) corresponds to an N mixers-inseries model, eq. (1) applies to each of the N perfect mixers. Physically the N mixers-in-series model may be considered as N smaller mills connected in series such that the discharge from one mill becomes the feed to the next mill in the series. A description of holdup mass, H(t), as a function of time, t, is needed to solve eq. (I). For overflow mlHs of the type used in this study, the volume of slurry present in the mill is reasonably constant over a wide range of operating conditions, hence the slurry “concentration” must be solved to compute holdup. A convenient definition for M MF and M,, and i MPm MP.i - Mspwp,, (5) z=Q+%-Q,, ; ( YS’~CS.SP) = QM&S.MP - QS&S.SP where M,, is the mass rate of solids discharge from discharge rate of the sump, QSP is the volumetric slurry, W,, is the mass rate of water added to the sump, msp.( is the fraction of solids in the ith size interval present in the sump, and C,.,, is the mass of solids per unit volume of slurry in the sump. Hydrocyclone model In general terms, the centrifugal force within the hydrocyclone carries the coarser and heavier particles to the outside wall, which results in the discharge through the spigot. Simultaneously, an inner spiral of fluid carries the finer particles to the vortex finder. A dynamic model is unnecessary because the response of the hydrocyclone is virtually instantaneous. Cohen rt ul. (1966) experimentally measured the residence time of mineral particles in a 15-cm cyclone and showed that the mean residence time is less than 5 s Optimal control of a ball mill grinding hd4,) K = 1 - exp[ R, = ~1, + a8 WOFf E;= (9) (10) - 0.693(d,/d,,)a,] WF (11) Y,(l-RR,)+R, (12) where WOF is the water flow rate in the overflow, WF is the feed water rate, d,, is the size at which 50% of the solids report to overflow and 50% reports to underflow, Qc is the volumetric rate of pulp ked,f, is the volume fraction of solids in the pulp feed, and R, is the fraction of fines reporting to underflow. Ei is the fraction of particles (in size class “i”) in the feed which is classified to the underflow, and Yi is equal to Ei corrected for particles carried off by water reporting to the underflow. The model uses eight empirical constants, a,, a2 . as, which must be determined experimentally. 0 GRINDING CIRCUIT A schematic of the major pieces of process equipment and instrumentation is shown in Fig. 1. The ball mill is a standard Denver mill, 76 cm in internal diameter and 46 cm long. In all of the tests reported a ball load of 345 kg, which corresponds to 40% millfilling, was used. The classifier is a Krebs 7.5cm hydrocyclone and the sump is 30 cm in diameter and 80 cm in height. The objective of this work is to study the behavior of industrial size mills using the pilot circuit described. This would entail using ore samples typically processed in a copper ore grinding circuit. Since the quality of the metal-ore sample, in terms of its grindability, varies within a batch of sample, a clean material such as limestone is used to validate models. The feed material processed is - 1680~pm limestone. For the range of flows encountered in the study a flow meter and a density gauge installed on the cyclone-overflow line would severely restrict the flow. To avoid this problem, the cyclone overflow was piped to a sump-pump assembly as shown in Fig. 1. The Microtrac particle size analyzer which operates in the range of 2-176 nrn was used. A pneumatically operated sampling device supplied samples to the analyzer every 2 min. Though the analyzer is capable of providing 13 size fractions, the percent passing 44 pm was used in the size control experiments. A Hewlett-Packard 2100 A minicomputer system consisting of mass storage disc, plotter, monitor, analog-digital converter and printer served as the online computer control system. A specialized on-line control software enabled configuring control loops (8) a.,Qc + asf, = a3 + 863 I EXPERIMENTAL for all sizes of particles at widely different operating conditions. This study conclusively proves that the dynamics associated with the cyclone are negligible compared to the mill or the sump. Hydrocyclone modeling has not matured to fully fluid flow based models. A number of papers have made advances into the description of swirling fluid flow in this device, but a complete description of size classification is yet to emerge. Hence, we use the empirical modeling approach proposed by Lynch and Rao (1975) with a slight modification. The model equations are WOF=a,WF+a, circuit- _“II.Ic.I*C ,,: ...................... .............. ...................... ...................... level PensOr ,e I! PA ..... ...................... ...................... ::::,:::1 ......................... . . . . . . . . . :::::::::::::::::::::: 4mn nm ........... .. ................................ ......... mo~ne+ic 13mm ................................ ...................... ::: ...................... ....... -, .............. :::::: .... ...................... ...................... . .. flomlcter .: ;- ........................ .................................. ... . ...................... ..................... ..................... ......................... ..... ...................... ...................... .......................... ................ ............... 75mm ........dlmrtnthl .......... ............... ........ \ :::::::: / C.” cymnd . . ........................ ...................... .................. ...................... &ty [“-‘“be > :.: ...... y=J==J . gucqe micro rot par*ic,e size OnOiyLBl .,: ,.>:.yg:z :y.{.c..- .......... .......... _..... _... .. ............. ....................... ......... ............ Stream : ..: .......... - b,” PrmoCt : ..... ..... o,e ........................... _, . ...... :;j: q . PROCESS EQ”lPMENT ...... . 1 / 750x450mm boll mill Fig. 1. Schematic of the pilot scale grinding circuit. a CONTROL RAJ K. RAJAMANIand JOHN A. HERBSI 864 reported here. A total of 10 instrument readings were taken every 2 s and averaged over 12 s. Five final control elements were controlled by the computer during control experiments. DYNAMIC MODEL PARAMETERIZATION AND VERIFICATION Ball mill parameters The parameters of the mill model depend both on the material consist within the mill and the mill operating condition. This dependence, although highly complex, has been well investigated. The dependence of selection and breakage functions on mill dimensions and operating conditions including critical speed, ball loading, particle loading and lifter geometry has been investigated systematically (Herbst and Fuerstenau, 1973; Kim, 1974; Malghan and Fuerstenau, 1977; Herbst er al., 1983). Herbst and Fuerstenau (1973) proposed that all of the influences on selection function can be equated as the corresponding influence of energy input to the mill on rate of breakage. This single fact is extremely useful for simplifying the modeling effort. Milling tests done on a 25, a 3% and the 76-cm mill by Herbst et al. (1977) confirmed the following hypotheses: (9 Selection proportional mill, i.e. functions for a given material are to the specific power draft of the Si = S:(P/H) i = 1,2,3 ..n (13) where SF is the specific se!ection function values, P is the power drawn by the mill, and H is the hold-up mass in the mill. (ii) Breakage functions for a given material are approximately invariant. (iii) In wet ball milling, the linear model is applicable in the neighborhood for which parameters were estimated, i.e. a linear model is applicable for the narrow range of mill discharge fineness from which parameters were estimated. According to the first hypothesis, the selection function estimated at some feed rate can be used to predict mill performance at any other feed rate by normalizing the selection function as shown in eq. (13). This fact combined with the last hypothesis enables calculation of mill dynamics with the linear model. The selection and breakage function parameters of the linear model were estimated with a computer program called ESTIMILL (Herbst et al., 1977; Rajamani and Herbst, 1984). The breakage function of the limestone feed material used here was determined experimentally in three mills-a 25 and a 3%cm laboratory scale mill and the 76-cm pilot scale mill. To a good approximation, the breakage functions are the same in the three mills. The cumulative form of the breakage function is given as Bij = 0.31(di/dj+,)0.4s The cumulative breakage + 0.69(d,/dj+,)Z function R. (14) BZiis the fraction of the material in size interval i that reports to all size intervals below and including the interval i upon breakage. Circulating load refers to the ratio of mill feed rate to the fresh feed rate. Mills are operated at high circulating loads so that overgrinding does not occur in the mill. A description of the transient response of the mifl is needed for dynamic model calculations. For all practical purposes the RTD determined with a suitabIe tracer is adequate. It is assumed that the transport characteristic of particles of all sizes is identical. Detailed investigations of the hold-up variations and RTDs on the 70-cm mill for various feed rates using a liquid tracer (lithium chloride solution) yielded the dimensionless RTD of the form (Kinneberg and Herbst, 1984) E(o) = ev( - O/&J)2+ evi - @/I - 4) 1 (15) where 8 = r/T,, = dimensionless time, T, = mean retention time, and Q = fractlonal volume of the first mixer. This expression corresponds to two mixers of unequal volume in series. Further studies with a solid tracer (quartz particles) yielded identical RTD confirming that water and particles behaved identically within the mill. In the dynamic model calculation the RTD was taken to be a single perfect mixer since the value of 4 was small (0.01-0.07) for most feed rates. From a mathematical point of view, the feed size distribution is linearly transformed to mill discharge size distribution. The transformation involves selection or breakage rate functions, a breakage distribution function and the RTD. Therefore if the breakage distribution function and RTD are known, the selection function can be estimated (Rajamani and Herbst, 1984). Accordingly, steady-state experimental data were gathered in the feed rate range of 9(X205 kg/h. As stated earlier, the selection functions depend on the fineness of the product in the mill. However, in the feed rate range of 90-136 kg/h, the operating range for both feedback and optimal control tests, a single set of selection functions given by S”(tonne/kWh) = 10.6(Jd,d,,l/a)‘.4z7 (16) was sufficient to predict all the data. Figure 2 shows the model prediction during selection function estimation at a feed rate of 159 kg/h. Close agreement is seen between model prediction and experimental distribution. The measurement of the mill discharge root mean squared error between values and the experimental data is the fitted model 0.0057, which is well below 0.0100, the standard norm established for such estimation. The close fit assures that the selection functions stated in eq. (16) would be good enough for dynamic model calculations. Hydrocyclone A separate taken model parametrization experimental to determine cyclone test program model was under- parameters. The Optimal control of a ball mill grinding circuit-1 TEST NO. 5 IS9 UG./HR. 0 - 865 FRESH EXPERIMENTAL LINEdR MODEL FEED FIT Fig. 2. Linear model fit to the closed-circuit steady-state data. 75-mm cyclone was mounted on an experimental test stand consisting of a sump, pump and return pipes. A density gauge and a flow meter mounted on the cyclone feed line gave mass rate measurements of solids and water. The cyclone feed flow was varied over the range 15 to 30 l/min and the feed solids percentage was varied in the range 3C50. Samples of feed, underflow and overflow were collected and their size distributions determined by sieving. The parameters of the cyclone model eqs (8)-(12) were determined as follows: Water split equation. Multiple linear methods were used in correlating cyclone WF-WOF relationship was expressed as one for higher feed rates and one for lower given as WOF (kg/min) = 1.363WF (kg/min) regression data. The two lines, feed rates, - 10.75 for WF < 21.4 WOF (kg/min) = 0.837WF (kg/min) + 0.35 for WF > 21.4. The d,, (17) (18) The dependence of d,, on volumetric feed flow rate and percent solids in the feed is given by the regression relationship log,d,,(pm) equation. = 3.616 - 15.006 x 10-2Qc(1/min) + 2.3S,. (19) The sign of the coefficients of Q, andf, are in agreement with the expected performance characteristics of the cyclone: as flow rate increases the cyclone produces a finer split and a coarser split is produced as percent solids in the feed increases. Short circuiting. The fraction of fine material reporting to the underflow is proportional to the frac- tion of feed water reporting to underflow. The regression relationship between R, and fractional water split (to underflow) WS is given as R, = 0.818 - 0.7932 WS. (20) Corrected ejiciency curve. This functional form is given as x = 1 - exp[ - 0.6931(dJd,,)“]. The value be 1.6. of the parameter MODEL Steady-state m was determined (21) to VERIFICATION response The steady-state predlctions using rate parameters given by eqs (14) and (16) for 170 kg/h feed rate conditions are shown in Fig. 3. It should be noted that the steady-state predictions made with the dynamic simulator involve both the mill and hydrocyclone models. The circulating load, defined as the ratio (expressed as percent) of mass feed rate of solids in the mill feed (includes recycle) to the mass feed rate of fresh feed entering the circuit, is also closely predicted. The excellent quality of predictions confirms the appropriateness of the models chosen for the mill and cyclone and the accuracy of the parameter estimates. Dynamic response The dynamic models incorporated in the simulator can be checked for predictive accuracy only by comparing the transient responses of the circuit with model computed responses. The transient response of the circuit to changes in sump water addition rate is well suited since to this disturbance both the circulating load and product particle size react rapidly. Control loop tuning done with the simulator for feedback control relies on the ability of the simulator to predict particle size responses to changes in sump water addition rate. For these reasons, step changes in sump water addition rates were studied experimentally and compared with model predictions. Millfeedbox delay model. Dynamic modeling studies indicated that there was a lag associated with the mill to changes in feed flow. This lag was traced to the mill feedbox which surrounded the scoop-feeder. Although the slurry volume in the mill is constant, it and JOHN A. HERBST RAJ K. RAJAMANI 866 PARTICLE SIZE ( MESH ) IO iub TEST NO 5 FP,ZD ,WTE: 17, 0 - EXPERIMENTAL PREDICTED CIRCVLATING EXPERIMENT&‘ PREDICTED I 20 Fig. 3. Modet PL?&LE’“&E predicrion of closed-circuit was detected that the mill feedbox tends to accumulate some slurry whenever transients occurred in the circuit. The transient response of the mill was determined by introducing a step change of 3.8 I/min in the feed to the mill and measuring the mill discharge volumetric rate. The mill discharge flow response was modeled as a second-order lag. The resulting input-output relationship expressed in the Laplace transfer function form is given as 1 m Dynamic = (0.45s + 1)(0.41s + 1)’ circuit (22) simulation The computation embodied in the models for the mill, hydrocyclone and sump are done in a simulation program called DYNAMILL. The program simulates the transient response of the circuit, and with the proportional-integral controller subroutine it can also simulate the response of the circuit under control. Dynamic prediction A unique feature of the product particle size response obtained under step changes in sump water addition rate is that, upon a step increase (decrease), the product particle size rapidly gets finer (coarser) Pig. 4. /“R 326 x WAD 354 % 1 0 QdS) KG %?CRCJNS 500 I 1000 2000 steady state at 170 kg/h fresh feed. and then slowly gets coarser (finer) until it reaches a steady-state value that is finer (coarser) than the product size at start. Under a step increase in sump water addition rate, the cyclone feed slurry becomes dilute and also the volumetric rate to the cyclone increases as a result of sump level control. Then, the cyclone cut size decreases producing a finer product, and simultaneously the amount of coarse solids returned to the mill increases. Due to the increase in solids feed to the mill, the mill discharge stream gets coarser. The coarse material now begins to appear in the cyclone feed, resulting in a coarser product. Clearly then, the initial part of the response is due to the dynamic elements sump and cyclone, and the final part is due to the mill. Similar reasoning applies for the inverse response in percent solids in the sump slurry, observed experimentally. The particle size response observed experimentally and the dynamic model prediction for step changes of 2.3 kg/min in the sump water rate at feed solids rates of 136 kg/h are shown in Fig. 4. For these step tests the only active control loop is the sump level control loop. The dynamic prediction is excellent both in the time response and magnitude. Such a dynamic predictive capability is imperative for classical feedback control loop tuning. Model prediction of product particle size transients for step changes in sump waler addition at 136 kg/h fresh feed. Optimal SIMPLIFIED control of a ball mill grinding MODEL The optimal control approach described in Part II requires a state space model of the grinding circuit. The state space model couId be linear or nonlinear and in either case Pontryagin’s Maximum Principle used in this study could be applied. Recourse to linearization and model simplification is taken owing to the slow speed of the HP 2100 operating system. One approach to the development of simplified dynamic models for control is to use “black box” models, i.e. a form of the model is assumed and the parameters are determined by forcing the model to fit plant data. For example, the linear difference equation form used by Borrison and Syding (1976) for crushers and the Laplace transform blocks determined by Hulbert (1977) for ball mills takes the “black box” approach. The principal drawback is that the model is not physically meaningful and hence it may not extrapolate well. On the other hand, control schemes developed with phenomenological models can be much more easily extended to a different system, i.e. an industrial installation. An alternative approach is to simplify the general phenomenological models of the subsystems by suitable assumptions while retaining the accuracy of predictions. Fortunately, such simplifications are readily available for the grinding circuit case. The parameters of this model are then readily obtained from steady-state data. Simplified model formulation The use of the detailed model for control would entail the solution of 37 nonlinear differential equations and 38 algebraic equations to provide a simulation of the grinding circuit at every control interval. Such a detailed model is not appropriate for real-time calculations on a minicomputer owing to the large number of arithmetic operations involved and the storage required. Added to this burden would be the additional real-time computations involved in optimal control problem solution and on-line model parameter estimation. However, in recent years on-line computing power has grown exponentially, and so the accuracy of predictions possible with the full set of model equations can be taken advantage of with a powerful on-line computer. In this work models had to be simplified due to the slower CPU speed of the HP 2100 computer. The models described below are simple enough for on-line use while retaining a reasonable level of predictive accuracy. 867 circui!--I hold-up mass, R,,,, is the fraction of solids retained above the size interval k, and Bkj is the cumulative breakage function given by k-l R MP,k = c mhfp, j (24) j=l (25) Equation (23) is equivalent to eq. (1) algebraically but as the independent variable. uses R,,,, Under the assumption, BkjSj = F,, where Fk is a constant with respect to the parent size j, eq. (23) can be written as d”R,, p1 k dt = MMMFRMMF.~ - F,“R.w,, This assumption was identified empirically in conjunction with a study of the zero-order production of fines (Herbst and Fuerstenau, 1968) and has been exploited to estimate breakage function values from batch bail mill data. In the present study the form of eq. (26) for closed-circuit operation becomes HM% = M,,R,, -CM,, + M,,R,, - k,HR,,., + Mu,)&,, Sump. The sump is considered to be perfectly mixed. The level control of the sump is assumed to involve a well-tuned controller which maintains the sump level steady at all times and so the slurry volume in the sump is taken to be a constant. Then, the following two mass balance equations are applicable for the sump: V,~=(M,,+M,,)-(Q,+~)C, Ball mill model. A form of the simplified model can be deduced from the more general population balance model as follows. Consider a ball mill operating in open circuit at a feed rate of M,,, then the size reduction occurring in the mill is described by d”R,, L=M kL dt MF R M~,lr - 1 ,Fl &SjHmj - MMFRMF,, (23) where M,, is the fresh solids feed rate, H is the mill (27) where R, is defined as the fraction of material above 44 pm in the mill, M is the solids feed rate, and H, mill hold-up mass. The subscript M refers to mill, FF refers to fresh feed, and CJF refers to underflow. The hold-up of solids in the mill was determined experimentally to be approximately constant for all feed rates. (29) where Vs is the volume of slurry in the sump, W,, is the rate of addition of water to the sump, QM is the total volume of slurry discharging from the mill, C, is the concentration of solids in the sump expressed as mass of solids per unit volume of slurry, and Rs is the fraction of solids above 44 pm in the sump. Hydrocyclone model. The size description in the circuit has now been reduced to two parts: the fraction of material retained above 44 pm and the fraction 868 RAJ K. RAJAMANIand JOHN A. HERBST passing 44 pm size. Accordingly, the classification action of the cyclone must be represented by two constants C, and C,, where C, is the fraction of plus 44 pm material in the cyclone feed that reports to underflow, and C, is the fraction of minus 44 pm material in the cyclone feed that reports to underflow. The variations in C, and C, with cyclone feed conditions are expressed through d,,. Owing to the widespread use of the variable d,,, both in design and modeling of cyclones, it was decided in this study to use d,, as a basis to specify classification. It is apparent from the detailed model that C, and C, are really a function of cyclone operating conditions and the size distribution of the cyclone feed stream. Several values of C, and C, were computed with the detailed dynamic model simulation program for fresh feed conditions ranging from 90 to 160 kg/h and sump water addition rates of 11.4-16 kg/mitt. C, and C, each show a linear dependence on d,, in the range of interest. Therefore the dependence of both C, and C, on d,, were expressed as c, = Ul,& + a12 (30) C, = a,,&, + ~22. (31) + where Qc is the volumetric k2Qc + k&b MM (kg/h) MU, Wmin) 159 136 125 496.3 449.4 450.3 113 355.2 model (33) C, = O.OO%d,, k, in data. parameters determined R.U RO, 0.883 0.888 0.880 0.832 0.790 0.768 0.760 0.715 0.42 0.32 0.30 0.28 40 Simplified model prediction of particle (35) from experimental data 55 50 46 42 SlMPLlFlED 0.0345 0.0354 0.0344 0.0343 [MI size response 136 kg/h. C, C* 0.8727 4.903 I 0.9205 0.905 1 0.2975 0.3184 0.4130 0.3894 MODEL 60 TIME Fig. 5. (34) C’, = O.Wl(Od,, - 2.029. k, (min-‘) R MF 20 + 1.181 The excellent dynamic model prediction obtained with eqs (34) and (35) for sump water addition tests at a fresh feed rate of 136 kg/h is shown in Fig. 5. A similar procedure resulted in excellent prediction for the 114-kg/h experimental data but the numerical coefticients in the classifier expression were different. Two different sets of classifier description equations are expected because in such a simplified model the classification constants C, and C, represent the combined effect of the size distribution of solids in the feed aud the classifier performance, i.e. C, and C, are determined by both the mass of material and the classification occurring in each of the size intervals. Variations in fresh solids feed rate cause variations in cyclone feed size distribution. Therefore, regression feed rate to the cyclone. Table 1. Simplified MM, RM, - & H, R, Similarly the values of the cyclone classification parameters C, and C, can be obtained from steady-state relations. Table 1 gives the steady-state experimental data and model parameters. From the steady-state predictions made previously, the values of C,, C, and predictions, are d 50, which give rise to accurate known, The unknown coefficients aI 1, aI2 and azz in eqs (30) and (31) were estimated by linear regression. The resulting classification model equations that gave the best fit are (32) Model parameters. The kinetic rate parameter eq. (27) can be determined from steady-state k,, is given by k, = ~ The dependence of d,, on the cyclone feed volumetric rate and percent solids was assumed to be of the form d,, = exp(k, From the steady state the mass balance, loo NUT&o for step changes in sump water addition rate at Optimal Table 2. Comparison control of a ball mill grinding of simplified model prediction and experimental Mill feed Fresh (mic&s) (W-4 (kg/h) Experimental Predicted 136 136 136 125 117 113 113 113 684 816 954 684 816 684 816 954 585.6 592.7 610.9 636.6 484.3 457.3 412.1 415.9 426.8 563.8 468.5 equations including these two variables were developed. The simplified cyclone model equation applicable in the entire feed rate range of 114-136 kg/h is given by C, = 0.0042M,,d,o - O.O704M,, C, = O.O502M,,d,, - 2.9354M,, - 0.01546,, + 1.3412 50.0 46.0 43.5 Predicted Experimental Predicted 48.8 46.7 45.0 46.1 40.9 43.6 41.6 40.2 68.0 70.0 71.0 70.0 75.0 72.0 74.0 76.0 67.75 69.75 71.24 70.55 74.97 72.33 74.48 76.09 a2 . . a8 (37) Bij bij C CONCLUSIONS Dynamic modeling of the grinding circuit involves modeling the mill, sump, hydrocyclone and mill feedbox, and suitably linking the models together. The population balance model of the milling process is most convenient for this purpose. In the mill model, the rate parameter dependence on operating conditions can be accounted for using the specific selection function hypothesis. The sump model involves a description of the concentration solids and size distribution in the discharge of the sump. A set of empirical algebraic equations is sufficient to model the hydrocyclone since the action of this device is instantaneous. The predictive capability of the dynamic model was demonstrated by predicting the response of the circuit to changes in sump water addition rate. A simplified dynamic model is needed especially if the CPU speed of the on-line computer cannot accommodate both model and control calculations at each sampling instant. Therefore, a model of the circuit using three state variables was developed. The mill model is a limiting case of the population balance model. Correspondingly, models of the sump and hydrocyclone were simplified. In spite of these drastic simplifications, the predictive capability of the simplified model is adequate and so it can be employed in control algorithms. In Part II of this paper computer control of the grinding circuit using these dynamic data NOTATION a,, The comparison of steady-state simplified model predictions and experimental data are tabulated in Table 2. The predictions are accurate enough for the simplified dynamic model to be used in the synthesis of an optimal control strategy. steady-state models is discussed. In particular, the detailed model is used in the off-line tuning of proportional-integral controllers and the simplified model is used in the optimal control algorithm. Both model predictions and experimental responses are shown. (34) - 0.0660d,, + 4.642. Experimental 869 Overflow produce size % passing 44 pm d (Wmin) Sump water feed circuit-I E(o) Fk H(t), H ko 4, k,, k, M 4 n P Q R MP.k Rls si, s;E t vhf, VF WF WOF ws Y, cyclone model constants cumulative breakage function individual breakage function concentration of solids (units = mass of solids per unit volume of slurry) defined in eqs (30) and (31), respectively mesh opening of size interval 2 cut size fraction of solids (in the feed) reporting to underflow in size interval i residence time distribution a constant ( Bkj Sj) total mass hold-up in the mill kinetic rate parameter for milling [see eq. (33)l simplified cyclone model constants mass rate of solids flow mass fraction of material in size interval i number of size intervals net mill power draft volumetric flow rate of slurry fraction of solids retained above size interval k in mill product fraction of fines (in the feed) reporting to cyclone underflow Laplace variable selection function and specific selection function, respectively time volume of mill and sump, respectively water flow rate in the cyclone feed stream water flow rate in the cyclone overflow fractional water split to underflow fraction of solids (in the feed) reporting to cyclone underflow after correcting for entrainment RAJ K. RAJAMANI and JOIIN A. 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