Model based supervisory control of a ball mill grinding circuit

Journal of Process Control 9 (1999) 195±211

Model based supervisory control of a ball mill grinding circuit V.R. Radhakrishnan* School of Chemical Engineering, University Science Malaysia, 31750 Tronoh, Malaysia

Abstract Optimum operation of grinding mills are important for the economic recovery of the valuable minerals, for energy eciency, as well as from the point of view of pollution control. In many mineral bene®ciation operations, the economic objectives translate into the maximization of throughput with suitable constraints on the product particle size distribution. While normal PID type controllers at the regulatory level are capable of controlling the process at the desired values of the process variables, a supervisory control system based on a process model will be required for optimizing the operation. In the present paper the ball mill model, together with the hydrocyclone separation system model is used in a simulation study to generate a response surface relating the control variables with plant load variables and manipulated variables. This response surface together with a suitably formulated economic objective function is used for on-line optimization to determine the optimum setpoints for the controlled variables in the supervisory control system. The model based supervision together with the regulatory layer control has been tested using simulation. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Ball mill; Supervisory control; Selection function

H i

Nomenclature bil Bil C CH Cs d50 d50 di dinlet dspig dvf E…t†

F

breakage function of particles in size interval l which reports to interval i cumulative breakage function de®ned by Eq. (A2) volumetric concentration of the solids in the slurry fractional solid content in cyclone feed fractional solids ®lling in the mill cut size of particles of standard density cut size of particles of density particle diameter in size fraction i, discretised size variable cyclone inlet diameter cyclone spigot diameter cyclone vortex ®nder diameter „ residence time distribution, E(t)dt represents the fraction of the material which has resided in the mill from time t to t+dt feed rate

k1 ; k2

kg/m3

k3 mi M n

m m m m m m

P Q Rf si s1 sE1

t1 ; t2 tonne/h

W x

* Tel.:+60-05-3676901; fax:+60-05-3677055. E-mail address: [email protected] (V.R. Radhakrishnan). 0959-1524/99/$Ðsee front matter # 1999 Elsevier Science Ltd. All rights reserved PII: S0959 -1 524(98)00048 -1

ball mill holdup number of size fractions into which the total continuous distribution is divided penalties for under size and over size respectively unit price of product mass fraction of size fraction i, superscript: uf in under¯ow, of in over¯ow, n+1 in cyclone feed mass ¯ow rate exponent in Eq. (A3). Also cell number power draft cyclone volumetric feed rate fraction of cyclone feed water reporting to under¯ow selection function of the ith size fraction selection function for the top size, size 1 speci®c selection function tolerance allowed in undersize and oversize respectively water ¯ow rate coded variable, subscript 1=feed rate, subscript 2=water ¯ow rate, subscript 3=selection function

tonne

$/tonne

tonnes/h kW m3/h

hÿ1 hÿ1 tonne/kWh tonne/h

196

x Yi Yif Y

  



V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

orthogonal variables for x subscript 1=feed rate, 2=water ¯ow rate, 3=selection function eciency of the ith size fraction in the cyclone without considering short circuiting eciency of the ith size fraction in the cyclone considering short circuiting fraction of the product in the particular size range, `p'=desired product, `u'=undersize, `o'=oversize constant in the selection function equation constant in Eq. (A3), also scalar objective ftmction in Eq. (A29) constant in the equation relating the selection function for the top size, size 1 with the selection function for the ith size, Eq. (A5) density kg/M3

Superscript F UF OF n N N+1

feed under¯ow, over¯ow cell number mill discharge cyclone feed

Subscript F i 1 ini S

feed ith size range ®rst or top size initial value sump

1. Introduction Distributed control systems based on microprocessor based controllers have become standard technology for process control at the regulatory level. The regulatory level controllers control the process variables at their setpoints. These setpoints are normally calculated on the basis of the technical factors governing the process operations together with economic and management factors like raw material and product prices, required quality speci®cations of the product, optimum production rate based on market demand, pollution and energy considerations. The determination of the optimum setpoints for process control is known as the supervisory function which was performed traditionally by the process engineer manually using his slide rule and calculator. In such manual supervision the supervisor

provides the setpoints on the basis of the techno-economic parameters of the system and a simple technical model of the process amenable to hand calculation. With such an arrangement only very simple models can be employed and optimum operation of the plant under changing conditions of raw materials, production rate and other market and management parameters become dicult. In modern distributed process control systems, with the availability of substantial on-line computing, the supervisory function is being increasingly automated. When a computer is utilized at the supervisory level a more realistic plant model can be utilized, coupled with suitable techniques of optimization and the optimum setpoints can be evaluated as often as necessary. When the process supervision is integrated with process control in a hierarchical structure, the system is known as supervisory control. In modern Distributed Control Systems, facilities exist for implementing supervisory control. However implementation of supervisory control requires more than the hardware platform.Development of suitable plant models and objective functions are required for the successful implementation of supervisory control systems. There are considerable incentives in utilizing setpoint supervisory optimization in mineral industries due to the following reasons. 1 The high energy requirements of mineral processing operations and the consequent need to increase the energy eciency. 2 The high variability in the mineral raw material grades necessitating frequent changes in the processing conditions. 3 Requirements of tight processing controls to minimize pollution. 4 Frequent changes in the market demand and price requiring adjustment of the production rate. Setpoint supervisory control requires an appropriate process model which can be used for determining the optimum values of the manipulated variable under any processing conditions. These process models should be suciently simple to be utilized on line. Process models based on the fundamental theoretical relations of a process in many cases are too complex for use on-line for real time optimization. These fundamental models can be utilized to derive simpli®ed models based on response surface or neural networks which in turn are combined with economic parameters and operational constraints to formulate the objective function which is to be optimized. Depending on the nature of the objective function a variety of techniques are available for optimization. The solution of the optimization problem will give the best values of the controlled variables for a given set of process input variables and the corresponding values of the manipulated variables to achieve these best values. These values of the manipulated variables are the setpoints to the controllers in the lower level of process regulation. The use of the fundamental process models to derive simpli®ed response surface model of the grinding mill in a copper concentrator plant and its

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

utilization for on-line optimization and set point supervisory control is presented in this paper.

Table 1 Mill speci®cations a. Ball mill

2. Closed circuit ball mill grinding The grinding circuit (Fig. 1) used for the present study is part of a copper concentrator plant. The system consists of a ball mill, hydrocyclones, pulp sump and associated pumps and solids feeding conveyors. The feed, copper ore (from primary crusher, size 3 in) is fed into the ball mill by vibratory conveyors. Since the system uses wet milling adequate quantities of water are also fed to the system. The principal speci®cations of the mill simulated in the present study is shown in Table 1. A ball mill consists of a cylindrical shell rotated about its axis by a motor. Heavy metallic balls called grinding media are loaded into the cylinder. The coarse ore is fed into the mill together with water in wet grinding. The tumbling action of the balls within the revolving mill crush the feed to ®ner sizes. The slurry containing the ®ne product is discharged from the mill into a sump. The slurry is pumped into a hydrocyclone. The hydrocyclone works on the principle of centrifugal separation. The feed stream to the hydrocyclone is separated into two streams. An over¯ow stream containing the ®ner particles and the under¯ow stream containing the larger particles. The over¯ow is the desired product. The under¯ow is recycled back to the ball mill for further grinding. The product size is speci®ed as 80% passing 200 mesh screen. The controlled variables in this process are the circulating load and the product size. Two manipulated variables are available, the solids feed rate

197

Diameter Length Slurry Volume Mill Power Draft Mill Type b. Hydro cyclones Number of cyclones Diameter Connection Inlet diameter Vortex ®nder diameter Spigot diameter c. Sump Base area

=4 m =5 m =5 m3 =580 kW =Over¯ow discharge =2 =75 cm =parallel =10 cm =15 cm =7.5 cm =12 m2

and water addition rate. An increase in the solids feed rate increases the particle size and the action is slow. Sump water addition rate increases the percent passing through 200 mesh screen. This action is fast. Increase in feed rate increases the circulating load. The increase in the sump water rate increases the circulating load and the action is fast. Increase in sump water at ®rst decreases the mill % solids and then increases it. This action is slow. Similarly an increase in the feed water rate at ®rst decreases the particle size and then increases it, the net result being a decrease in the particle size. This action is slow. The interrelationships between these process variables have been presented by Herbst and Rajamani [1] in a process matrix which shows whether the increase in a particular manipulated variable increases, decreases or has a complex relationship with a controlled variable and whether the action is slow or fast. The principal load variable a€ecting the system is the ore hardness.

Fig. 1. Schematic diagram of closed circuit grinding mill.

198

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3. Closed circuit ball mill control The control objective in ball mill control can be stated as maximizing the throughput subject to maintaining the product size within an allowable range. In such a control system the throughput is a controlled variable to be maximized and the product size is a controlled variable to be maintained within constraints. For these objectives the available manipulated variables are the feed rate and the water addition rate and the principal load variable is the ore hardness. The conventional feed back control schemes do not achieve this objective of operating at the optimum throughput rate. In such classical control systems the two controlled variables are paired with the manipulated variable through a control law like the Proportional-Integral-Derivative action. Since there are two controlled variables and two manipulated variables, it is possible to pair them in two alternate ways as shown in Table 2 [1]. The control schemes in both these pairing structures are regulatory in nature and can help to maintain the control variables at their setpoints. Investigations have been reported on control using Type 1 [2] and Type II [3] structures. Both these types of controls are essentially Single Input-Single Output (SISO) structures and do not take into consideration the interaction between the loops. Hence such control systems may exhibit loop interactions. Model based Multi Input±Multi Output (MIMO) schemes have also been proposed [1]. Regulatory layer control schemes require, besides the process measurements, the desired values of the controlled variables or the setpoints for implementing feedback control. The setpoints in general are obtained by the optimization of a techno-economic model of the process under the set of present operating conditions. The determination of the optimum setpoints or the supervisory function is performed at one hierarchical level above the regulatory level. In the speci®c grinding system studied, the objective function for the optimal control can be stated as, Table 2 Controlled variable-manipulated variable pairing Type

Controlled variable

Manipulated variable

I

Product size Circulating load Mill solids holdup Sump level Product size Circulating load Mill solids holdup Sump level Product size distribution Circulating load

Solids feed rate Sump water addition Feed water addition Pumping rate Sump water addition Solids feed rate Feed water addition Pumping rate Solids feed rate and total water addition Constraint on feed and water addition Feed water addition Pumping rate

II

III Proposed

Mill solids holdup Sump level

``Maximize throughput (or some other economic function) subject to the condition, the product particle size lies between a maximum and minimum size.'' The manipulated variables of the process (feed rate and water addition rate) and the load variable of the process (feed hardness) will a€ect both the throughput as well as the product particle size distribution. For a given set of values of the load variables the optimization procedure should give the values of the manipulated variable which will maximize the objective function subject to the constraints on the particle size. For the solution of this optimization problem we require a ball mill grinding model which can be solved for any given set of input conditions to give the product particle size distribution and throughput. Once the optimum operating conditions are determined as the solutions of the optimization algorithm they can be down loaded from the supervisory level to the regulatory level, where they are used as set points for the controllers. The model of the grinding system consisting of the ball mill operated in closed circuit with a hydrocyclone [4] is summarised in the Appendix. 4. Grinding mill optimization The grinding circuit like any other process has to be operated to maximize the pro®t. The function of the grinding mill is to reduce the particle size of the ore such that the valuable mineral constituent is exposed and can be recovered in the subsequent ¯otation operation. For a grinding mill we can write the pro®t function as, Profit=unit time ˆ throughput …unit price of product ÿ unit price of feed† ÿ cost of energy=unit time ÿ fixed cost=unit time

…1†

For an installed grinding mill the ®xed cost does not change and can be omitted from the optimization. Further for the small changes from the normal operating conditions involved in the optimization, the energy consumption does not change much. The unit price of feed may be considered constant. Hence the pro®t function can be simpli®ed to, Profit=unit time ˆ throughput …unit price of product ÿ unit price of feed† or

…2†

Profit=unit time ˆ throughput X value addition The product of the grinding mill is the feed to the next stage in the bene®ciation process, that is the ¯otation

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

unit. The purpose of the mineral bene®ciation process is the recovery of the valuable mineral constituent of the ore. The value addition associated with the grinding circuit will be the increased recovery of the valuable constituent in the ¯otation unit. In an ore body the valuable constituent is dispersed in a matrix of gangue material. In the ¯otation process surface active agents modify the exposed mineral surface in such a way that the air which is bubbled in the ¯otation cell attaches to these particles and makes them ¯oat to the surface. Hence the valuable material comes up as the ¯oat and the gangue material goes out as the tailings. Since the ¯otation operation depends on the surface properties of the exposed mineral particles, only such particles which are exposed will be recovered. Mineral particles covered by gangue material will not be a€ected by the surface active agents and will be lost in the tailings. Hence it is necessary to crush the run of mine ore to such a size that the imbedded mineral particles are exposed to the action of the ¯otation process. This is called the liberation size. Grinding to a size much smaller than the liberation size in turn brings some disadvantages. Besides the higher power consumed in grinding, very ®ne particles or slimes reduce the eciency of recovery in the ¯otation process. Slimes also add to the pollution control problems. Higher concentration of slimes increases the cost of sedimentation in the ®nal tailing discharge. With high slime concentrations it may be dicult to meet pollution standards for the ®nal discharge from the mill. Therefore depending on the liberation size of the mineral there will be a range of sizes which maximize the value recovery in the grinding mill. Oversize and undersize reduce the value recovery of the process. Hence maximization of the pro®t function given by Eq. (2) can be written in an equivalent form as, Profit=Unit time ˆ throughput X unit price of product X ‰fraction of product in the desired size range ÿ penalty for under size

…3†

199

of the product. Yp ; Yu and Yo are the fractions of the total product which are in the required size range, in the under size and in the oversize respectively (Fig. 2). The objective function of Eq. (4) has been formulated with particular reference to the recovery process and economics of the mill investigated. For example in the mill investigated oversize and undersize up to tolerances of t1 and t2 respectively could be allowed without any deleterious e€ects on the ¯otation and pollution control units. Further due the particular shape of the ore inclusions, oversize up to the tolerance limit t2 reported to the ¯oat and could be recovered as product. As illustrated here the ®nal form of the objective function must be formulated taking into consideration the technical aspects of the ¯otation process and the economics and speci®cation of the product. In Eq. (4), F…Yp ‡ t2 † represents the production rate per unit time of the useful saleable product from the concentrator, Yp being the product in the required size range and t2 the allowable oversize in the product. If this product had no undersize and and no oversize above the value t2 associated with it its selling price would be k3 . However, the presence of undersize and oversize above the tolerance level decreases the unit sales price. This reduction in income has been accounted by using a pseudo production rate which subtracts from the actual production rate penalties for the undersize and oversize produced. Another way of doing would have been to associate the reduction with the unit sales price k3 . In the present work the former method was used since the commercial agreement of the plant with its customer unit used this method for the penalties associated with the undersize and the oversize. It may be noted that the two penalty terms in Eq. (4) can take only positive values or zero. If the values of the oversize and undersize are less than their allowable tolerances then the penalties take the value zero. 5. Variables a€ecting the size fractions Yp,Yu,Y0 The principal variables which a€ect the size fractions in the desired product range as well as in the oversize

ÿ penalty for over sizeŠ In the particular copper concentrator plant investigated in this paper the above objective function takes the speci®c form given by, Objective function ˆ k3 F ‰…Yp ‡ t2 † ÿ k1 …Yu ÿ t1 † ÿ k2 …Y0 ÿ t2 †Š:

…4†

Yu > t1 ; Y0 > t2 The parameters k1 and k2 represent the penalties associated with the production of the under size and oversize respectively. The parameter k3 is the unit price

Fig. 2. Size distribution of product showing the regions of desired product, over size and under size.

200

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

and the undersize are the ore feed rate, water addition rate and the ore hardness. A fourth variable, the power draft of the mill has not been considered since in an operating mill the power draft is essentially constant. Of these three variables the ore feed rate and the water addition rate are the manipulated variables while the ore hardness is the load variable. The relationships between YP ; Yu ; Y0 and the manipulated and the load variables represent the static characteristics of the system. These static characteristics may be expressed in the form of the functional relationships, Yp ˆ f1 …F; W; sE1 † f1 …f; W; sE1 †

Yu ˆ Yo ˆ 1 ÿ Yp ÿ Yu

…5†

where the sux `max' and `min' signi®es the maximum and minimum values of the variable. Similarly the variable W is coded to x2 and sEI to x3 . It is obvious that the coded variables have upper and lower bounds of +1 and ÿ1 respectively. The fractional concentrations in the two size ranges Yp and Yu are correlated to the three coded variables. The variable Yo . is obtained by di€erence. Second order regression equations of the following form are chosen for the response surface for the two variables Yp and Yu , Y ˆ b0 ‡ b1 x1 ‡ b2 x2 ‡ b3 x3 ‡ b12 x1 x2 ‡ b13 x1 x3 ‡ b23 x2 x3 ‡ b11 x21 ‡

b22 x22

‡

…7†

b33 x23

where f1 and f2 represent the functional relationships. The static characteristics obtained by solving the system model equations Eqs. (A1)-(A15) are illustrated in Figs. 3±5. The response surface de®ning these relationships can be obtained by performing simulation experiments and standard statistical techniques. We use the system model and conduct a series of statistically designed experiments to determine the relationship between the independent variables and the dependent variables. The results are then used to obtain the surface by multiple regression techniques. In the present study a second order orthogonal design has been used for constructing the response surface. The variables F,W and sE1 are normalised so that their values lie between ÿ1 and +1. For example for the variable F the coded variable ÿ1 < x1 < ‡1 is obtained by these transformations. …6†

Fig. 4. Static characteristic, showing the variation of the (a) product, (b) undersize and (c) oversize fractions with variation in water ¯ow rate at constant feed rate and selection function.

Fig. 3. Static characteristic, showing the variation of fractions in the (a) product, (b) undersize and (c) over size fractions with variation in feed rate at constant water addition rate and selection function.

Fig. 5. Static characteristic, showing variation of the (a) product, (b) under size and (c) over size fractions with variation in the selection function at constant feed rate and water addition rate.

2F ÿ Fmax ÿ Fmin x1 ˆ Fmax ÿ Fmin

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

In Eq. (7) the coecients b are the regression coecients. With the number of independent variables R ˆ 3 and with no replications the number of observations will be 15. The corresponding star arm will have a value of 1.525. The upper and lower values of the coded variables become +1 and ÿ1. The additional star arm values are chosen at +1.525 and ÿ1.525 as recommended [5]. The quadratic terms x21 ; x22 and x23 can be converted to orthogonal form by the transformation, x 2j ˆ x2J ÿ

N 1X x2 N Iˆ1 JI

J ˆ 1; 2; 3

…8†

where n the number of observations equal to 15. The regression coecients can be directly obtained from the design matrix and are shown in Table 3 6. Optimum operating conditions With the objective function given by Eq. (4) and the functional forms for the response surfaces for the fractional recovery given by Eq. (7) and Table 3 we can proceed to determine the optimum conditions for the operation. In the case study presented here the ore hardness represented by the selection function is subject to change and hence is the load variable. The objective of the optimization is to determine the values of the manipulated variables F and W which will maximize the objective function. The desired size range is speci®ed as 100 mesh to 325 mesh. This corresponds to the size range 149 to 44mm. Sizes larger than 149mm constitute the oversize and sizes below 44mm constitute the undersize. The tolerance allowed for the over size and undersize, t1 and t2 where both ®xed at 0.2. Two sets of numerical values were used for the penalties k1 , and k2 , (2.5,1.5) and (4.5,1.5). The objective function was maximized by a Hooke and Jeeves search routine. The result of the optimization at the two sets of values of k1 and k2 are presented in Figs. 6 and 7. Table 3 Coecients of the regression equation Coecient b0 bl b2 b3 b12 b13 b23 b11 b22 b33

Yp

Yu

0.651 0.031 0.030 0.012 0.038 0.083 ÿ0.004 ÿ0.082 ÿ0.020 ÿ0.054

0.332 ÿ0.095 0.020 0.077 ÿ0.009 ÿ0.029 0.012 0.044 ÿ0.003 ÿ0.008

201

These plots show the value of the feed rate and water ¯ow rate which should be maintained at any given value of the selection function to maximize the objective function. The values of k1 and k2 determine the shape of the objective function. By altering the values of these parameters we can drive the system to produce less of the under size or over size as required. As an illustration, by increasing k1 from 2.5 to 4.5 the undersize is reduced at any value of the selection function. However, this reduction is at the cost of decreased throughput and consequent reduction in the value of the objective function. This illustrates how a careful choice of these penalties can implement the management objective of maximizing the pro®t function. 7. Regulatory/ supervisory control The schematic diagram for the supervisory computer control of the copper concentrator plant is illustrated in Fig. 8. The feedback control of the process variables are performed in the regulatory level. The setpoints required by the regulatory level controllers are determined by the optimiser which down loads the setpoint values. 7.1. Supervisory function The ore hardness is periodically determined by analysing the product particle size distribution. When the particle size distribution undergoes any change this is detected and the ore hardness is estimated by the parameter estimation method presented by Rajamani and Herbst [6]. The method is based on an error minimisation algorithm. Starting from an initial value of sE1 the product size distribution is calculated from the size discretised model of the Appendix. The calculated and measured values of the distributions are compared and the error utilised to update the selection function in a hill climbing algorithm. The new selection function calculated is used to determine the new values of the grinding mill operating conditions by the method discussed in Section 6. To illustrate the method the ball mill grinding system was simulated corresponding to a selection function of 0.025 tonnes/kW h. Corresponding to this value of the selection function, the optimum value of the feed rate is 6000 kg/min and total water addition rate is 10,000 kg/ min (Fig. 6). The operating conditions are shown in column 1, Table 4. The simulated product particle size distribution is given in Table 5 and as Case 1 of Fig. 9. This particle size distribution gives a value for the objective function equal to 6916. (Table 4, Case 1). Suppose now the hardness of the ore becomes less with the selection function taking a value of 0.028. When the system is simulated with identical operating conditions

202

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

Fig. 6. Optimum operating conditions at values of (1) ki =2.5 and k2 =1.5 and (2) k1 =4.5 and k2 =1.5: values of (a) optimum feed rate and (b) optimum water rate to maximise the objective function at di€erent values of selection function and (c) the corresponding maximized function values.

Fig. 7. The fractions reporting to the (a) desired product, (b) under size and (c) over size at optimum operating conditions with ki =2.5 and k2 =1.5.

as in Case 1 the particle distribution as in Case 2 of Fig. 9 is obtained. The distribution shifts (Fig. 9 Case 2, Table 5) towards the undersize since the ore is softer. Correspondingly the objective function value is considerably less at 3172 as shown in Table 4, Column II. This is because of the large penalty which has to be paid for the production of large fraction of undersize. Corresponding to the new selection function value of 0.028 the optimum value of feed rate is 6600 kg/min and total water rate 9100 kg/min. By increasing the feed rate and decreasing the water rate the optimiser attempts to shift the distribtuion to higher sizes. When we simulate the system under the new conditions we get the distribution of Case 3, Fig. 9. The corresponding function value is 5452 which is an improvement over the Case 11 value of 3192. Of course we cannot expect to get back the orginal Case 1 value of 6916 for the pro®t function since the maximum corresponding to the new hardness is di€erent from the maximum correspoding to the earlier hardness.

7.2. Regulatory Level Control The optimum setpoints calculated from the optimising strategy in the supervisory level is implemented in the regulatory level of control. The ore feed rate, feed water rate and the sump water rate are under normal PID control (Fig. 8). The setpoints for these controllers are available from the supervisory level. The ore feed rate setpoint corresponding to the new hardness is calculated by the optimiser and down loaded to the controller. The optimser also provides the value for the total water ¯ow rate. The total water ¯owrate is apportioned between the feed water and the sump water. The feed water is under ratio control with respect to the ore feed rate. The set ratio is controlled by the feed back loop from the mill discharge density. The di€erence between the total water given by the optimiser and the feed water is the sump water setpoint. The sump level is controlled by the control loop using the pumping rate as

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

203

Fig. 8. Regulatory/supervisory control system for the grinding circuit. Table 4 Supervisory controlÐchanges in operating conditions Case sE1 ,

Tonnes/kWh Feed rate, kg/min Water rate, kg/min Fraction under size Fraction right size Fraction oversize Function value

Fig. 9. Di€erential particle size distributions corresponding to cases 13 of Table 4.

1

2

3

0.025 6000 10 000 0.22 0.57 0.21 6916

0.028 6000 10 000 0.29 0.52 0.19 3172

0.028 6600 9100 0.26 0.56 0.18 5452

the manipulated variable. This controller is a simple proportional controller with a low value of proportional gain such that the sump level varies widely, but sudden and large changes in pumping rate are prevented. The control system presented above di€ers from the earlier Type I and Type II strategies (Table 2). In the earlier strategies there is a one to one pairing between particle size and circulating load as the controlled variables and

204

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Table 5 Particle size distribution at di€erent operating conditions Size, microns

18.5 40.5 48.5 58.0 68.5 81.0 96.5 115.0 137.0 163.0 213.5 273.3 P Average particle Size

Fraction Retained Case I

Case II

Case III

0.07 0.07 0.08 0.09 0.10 0.13 0.15 0.10 0.08 0.05 0.04 0.04 1.00 95.16

0.09 0.10 0.10 0.11 0.12 0.14 0.08 0.07 0.06 0.05 0.05 0.03 1.00 78.08

0.07 0.09 0.10 0.11 0.12 0.17 0.09 0.07 0.06 0.05 0.04 0.03 1.00 87.94

the feed rate and water rate as the manipulated variables. In the current strategy both the manipulated variables are used to shift the particle size distribution in a desired direction. The circulating load is allowed to change within its constraints. When the circulating load reaches it upper constraint then further change in the mill loading will not be allowed. By this process the mill is always operated at its maximum pro®tability. The grinding system model of the Appendix was used to generate step response data relating each of the outputs with each of the inputs. The inputs are hardness represented by sE1 , feed rate, feed water rate and sump water rate. The output variables are the particle size distribution, circulating load, mill holdup expressed as percentage of solids, and sump level. Since particle size distribution requires a number of points to be realistically represented, for the purpose of the control study it was converted to a mass average particle diameter. There are 4 input variables and 4 output variables and hence a total of 16 step responses can be generated. In these simulations the steady state values of the feed rate was maintained at 6000 kg/min, feed water ¯ow rate at 3000 kg/min, sump water ¯ow rate at 7000 kg/min and the speci®c selection function at 0.025 tonne/kW h. The step change in the feed rate had a a value of +600 kg/ min, feed water+1000 kg/min, sump waterÿ900 kg/min and the speci®c selection function+0.003 tonne/kW h. These values were chosen based on typical operating data from the concentrator plant studied.These reponses are presented in Figs. 10±13. Since the sump has a pumped discharge its reponse is essentially that of an integrator. The hardness of the ore within the range of study was found to have a negligible e€ect on the sump level. The feed water increase at ®rst decreased the particle size and then increased it, the net e€ect being a decrease from the intitial value. Similarly the sump water decrease at ®rst increased the mill hold up and

then decreased it with a net increase from the initial value. The step responses were approximated by suitable transfer functions. First order and second order with delay time models were found suitable for all the reponses except the sump level and the two increasingdecreasing responses discussed. The sump level was modeled as an integrator. The increasing-decreasing responses of particle size to feed water and holdup to sumpwater were modeled by two ®rst order systems with delay time connected in parallel. All the transfer function parameters were estimated by normal parameter estimation techniques. The matrix of transfer functions are presented in Table 6. The total system consisting of the transfer functions of Table 6 was simulated using the Matlab Software. The Matlab simulation model is presented in Fig. 14. The values of the optimum feed rate F is 6000 kg/min, feedwater WF 3000 kg/min and sump water Ws 7000 kg/ min. Under these conditions the particle size distribution is as shown in Case 1 of Fig. 9. This corresponds to a mass average particle size of 95.1m. The circulating load is 18814 kg/min, mill % solids 0.59% and sump level 2.5 m. The pumping rate is 16,000 kg/min. Corresponding to these operating conditions the function value is 6916. The individual regulatory level control loops for feed rate, feed water and sump water are omitted from the diagram for simplicity. As presented the diagram simulates a change in speci®c selection function from 0.025 to 0.028 at time t=0. It is assumed that there is a delay of 1000 s for the detection of this change in hardness and the consequent determination of the new selection function and new optimised values of the inputs by the supervisory system. The value has been arbitrarily assigned on the basis of experience with the operating system. This value is not critical for the control except that after a change in hardness the longer the delay, the longer will be the time period when the system is working at less than optimal conditions. When the hardness deceases as given by the higher value of the selection function and the feed and water rates are not changed then the production of the undersize is much higher and consequently the distribution shifts towards smaller sizes as shown in Fig. 9 Case II. This brings about a reduction in the function value to 3172 (Table 4). At 1000 s the supervisory system acts to increase the feed rate to 6600 kg/min and to reduce the water rate to 9100 kg/min which are the optimum values corresponding to the speci®c slection function of 0.028. This brings about the shift of the distribution to Case III of Fig. 9.The distribution curve is shifted to higher values and the function value increases to 5452. At the regulatory level the new setpoints are implemented. Since the feed rate has increased the ratio controller increases the feed water. The reduction in the total water rate

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

205

Fig. 10. Response of average particle size to step disturbance in the input variables:(1) selection function disturbance (2) feed rate disturbance (3) feed water ¯ow rate disturbance (4) sump water ¯ow rate disturbance.

Fig. 11. Response of circulating load to step disturbance in the input variables:(1) selection function disturbance (2) feed rate disturbance (3) feed water ¯ow rate disturbance (4) sump water ¯ow rate disturbance.

206

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

Fig. 12. Response of mill solid holdup to step disturbance in the input variables: (1) selection function disturbance (2) feed rate disturbance (3) feed water ¯ow rate disturbance (4) sump water ¯ow rate disturbance.

Fig. 13. Response of sump level to step disturbance in the input variables: (1) Sump water rate disturbance (2) Feed rate disturbance.

Table 6 Transfer function matrix F

WF

WS

SE1

Average particle size

0:005exp…ÿ180s† ÿ0:048exp…ÿ400s†‰1 ÿ 0:8exp…ÿ300s†Š …220s ‡ 1†…150s ‡ 1† …150s ‡ 10†…400s ‡ 10†

ÿ0:0072 …75s ‡ 1†…50s ‡ 1†

exp…ÿ240s† …400s ‡ 1†…300s ‡ 1†

Circulating load

1:6exp…ÿ200s† …500s ‡ 1†…300s ‡ 1†

0:2exp…ÿ180s …500s ‡ 1†…400s ‡ 1†

0:2 …60s ‡ 1†…40s ‡ 1†

2:97E ‡ 5exp…ÿ220s† …500s ‡ 1†…300s ‡ 1†

Mill solids

1E ÿ 4 …150s ‡ 1†

5E ÿ 5 …150s ‡ 1†

ÿ5E ÿ 5exp…ÿ20s†‰1 ÿ 0:6exp…180s†Š …150s ‡ 1†…150s ‡ 1†

ÿ25exp…220s† …150s ‡ 1†…220s ‡ 1†

Sump level

5:9E ÿ 7 s…1200s ‡ 1†

1:14E ÿ 6 s…15s ‡ 1†

1:14E ÿ 6 s

Negligible e€ect

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

Fig. 14. MATLAB simulation model of the grinding mill.

207

208

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

demanded by the optimiser and the demand for higher feed water by the ratio controller are both given by the reduction in the sump water. The change in the average particle size after these changes is shown in Fig. 15. The complex interaction of the feed rate increase, feed water increase and the sump water decrease are re¯ected in the average particle size change. However at ®nal steady state the average particle size is lower than the value it had before the hardness change but higher than what it would have been if no supervisory commands were implemented.

Fig. 15. Closed loop response of average particle size after step change in hardness (a) with supervisory control (b) without supervisory control.

The changes in the circulating load shown in Fig. 16(1) similarly relects these changes. The circulating load is within the constraints. The percent solids holdup [Fig. 16(2)] also re¯ects the competing e€ects of changes in the solid and water rates but stabilises at the set value. The sump level shown in Fig. 16(3) decreases from the initial value of 2.5 to 1.9 since with proportional control an o€set is required to maintain the new value of the manipulated variable. Since sump level is not an important variable but our aim is to prevent drastic and oscillating changes in the pumping rate, a proportional control was found to be superior to PI or PID control. The pumping rate change shown in Fig. 16(4) re¯ects the new material balance in the system. No particular e€ort was made to tune these loops and they could be made to respond faster by changing the control parameters. The proposed scheme may be called the Type III control (Table 2). The control scheme presented in this paper is free from interaction between the particle size loop and the circulating load loop which is present in the Type I and Type II control schemes proposed by earlier workers. Further, the regulatory control layer is insensitive to modeling errors since none of the control parameters on this layer is based on the process model. The e€ect of modeling error will be on the pro®t function. If there is modeling error the improvement in the pro®t function will be less. Even then the pro®tability will be far

Fig. 16. Closed loop response of mill process variables after step change in hardness: (1) circulating load (2) mill solid holdup (3) sump level (4) pumping rate.

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

superior compared to the case when no model based supervisory control is present. 8. Conclusions A simulation study of the ball mill grinding system of a copper concentrator plant showed that the important variables a€ecting throughput and output particle size distribution where (1). Feed rate (2). Water addition rate and (3). Ore hardness. The simulation model of a ball mill presented by earlier workers [4] is too complex for use for on-line implementation in control. An alternate approach is presented in the present paper where in the Herbst-Rajamani model is used o€-line in an experimental design to determine the response surface for the plant variables. This response surface in the form of a second order polynomial together with the economic objective function is optimized in real time supervisory control. A typical economic objective function for the particular concentrator plant has been derived as a model for such systems. The proposed supervisory control scheme is used with a regulatory level scheme di€erent from the control variablemanipulated variable pairing proposed by earlier workers. The new control scheme has been simulated and found to work well. The proposed new scheme may be termed the Type III control pairing. Appendix The ball mill grinding circuit studied consists of the ball mill, the slurry sump and hydrocyclones. By combining the models of each of these equipment a total grinding system model can be developed. A complete set of models has been presented by Herbst et al.,(1977) [4]. A1. Ball mill model The classical approach to the modeling of ball mills has been by the so called energy-size relationships (Bond, 1962) [7]. These relationships can however only be used to predict an average product particle size and cannot give the size distribution. To account for the complex breakage phenomena occurring in a size reduction equipment in which a feed material breaks into wide range of particle sizes and undergoes repeated breakages, kinetics have been formulated based on parameters de®ned as selection ftinction and breakage function. The size discretised selection function si is de®ned as the fractional rate at which particles break out of the ith size interval. The selection function is a function of the mill operating conditions like power

209

input P, mill holdup H and mill fractional solids ®lling Cs , and material properties like hardness and the particle size. The size discretised breakage function bi,l represents the fraction of the primary breakage products breaking out of size fraction l which appears in size fraction i. It is obvious that l>i. Hence by knowing the selection and breakage functions for the range of sizes of interest in a particular mill one can model the breakage kinetics. The breakage process taking place in the ball mill is distributed along the entire length of the ball mill. Particles undergo repeated breakage till they appear in the product. For describing such a distributed parameter process a commonly used method is the Nmixers in series model. In this the distributed process is modeled by N perfectly mixed cells. Using the N cells in series together with the selection function-breakage function kinetics the mass balance equation for the ith size fraction in the nth cell can be written as, d…Hn mni † ˆ Mnÿ1 mnÿ1 ÿ Mn mni i dt iÿ1 X ‡ bil sl Hn mnl ÿ si Hn mni lˆ1

…A1†

i ˆ1; 2; 3; 4 . . . . . . . . . . . . . . . I n ˆ1; 2; 3; 4 . . . . . . . . . . . . . . . N In Eq. (A1) mi represents the mass fraction of the ith size fraction and M the mass ¯ow rate. The superscripts identify the cell number. The right hand side of the equation represents the accumulation of the ith size fraction in the nth cell. The ®rst and second terms on the right hand side are the ith size fraction material entering and leaving the nth cell due to ¯ow. The third term represents the breakage of the particles of sizes higher than i, (sizes l=1 to iÿ1) and which appears in the ith size fraction, while the fourth term represents the material of size i breaking out of the ith size fraction. We can write one such equation for each component j in the ore body as well as for each cell in the nÿcell model. The cumulative breakage function is adequately represented by a three parameter equation Bi;l ˆ

I X

bk;l

kˆi



Bi;l ˆ 1

di dl‡1

2



di ‡…1 ÿ 1 † dl‡1

3

…A2†

where 1 ; 2 ; 3 are constants for a particular ore j and di and dl‡1 are the diameters of the particles corresponding to the ith and l+lth size fraction. The breakage function has been found essentially independent of the operating parameters of the ball mill except the solid

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fraction loading in the ball mill holdup [4]. The relationship with solid concentration can be accounted [4] by using a variable 1 related to the solids ®lling in the mill, 1 ˆ 1 ‡ 2 Cns

…A3†

where 1 ; 2 ; n are constants. For specifying the selection function the following approach has been adopted [4]. A parameter called energy speci®c selection function sE1 is de®ned for the largest size, size 1. This energy speci®c selection function for the top size is a constant characteristic property of the particular material, and size. The sux one denotes that it represents the selection function for the top size, the size 1. This property has to be experimentally determined for each type of material. The speci®c selection function is independent of mill operating conditions like mill power draft and mill holdup and depends only on the material type and size. The energy speci®c selection function is made mill speci®c and operating condition speci®c by the following sE P s1 ˆ 1 HCs

…A4†

The selection function for size i; si , is related to the selection function for the top size s1 , by a second order log polynomial. 0

s s2 1 d d di di‡1 A i i‡1 ‡ &2 ln si ˆ s1 exp@&1 ln d1 d2 d1 d2

…A5†

In Eq. (A5),1 &2 are constants and di ; di‡1 ; d1 ; d2 refer to the particles diameters corresponding to size ranges i; i ‡ 1, 1and 2 respectively. Eqs. (A1)±(A5) completely de®ne the model for the ball mill. Given the feed size distribution, mass ¯ow, selection and breakage function parameters and the geometric constants of the mill like holdup, the mill output size distribution can be determined by integrating the equations. For simplicity the equations have been written only for one cell and where an n-cell model is used one set of equations have to be written for each cell. Similarly if the ore contains more than one species having di€erent hardness then equations will have to be written for each species.

as a simple integrator. This leads to a total material balance and component material balance for each of the size ranges. The hydrocyclone is modeled by the set of empirical equations presented by Lynch and Rao [8]. This model is based on a cut size, d50 , de®ned for a standard particle density. The d50 cut size is correlated with various geometric and operating parameters of the hydrocyclone by the equation, log10 …d50 † ˆ 0:1016dvf ÿ 0:1463dspig ‡ 0:0930dinlet ‡ 0:02999CH ÿ 0:0018927Q ‡ 0:0806 In Eq. (A6) dvf ; dspig and dinlet are the diameters of the vortex ®nder, spigot and the inlet respectively. CH is the fractional solids content in the cyclone feed and Q the volumetric feed rate to the cyclone. The correlation of Eq. (A6) is for material of a standard density. For material of any other density  a correction is made by the following equation.   4:4144 d50 ˆ d50 ÿ0:3293 ‡ g

The total grinding system consists of the ball mill, the slurry sump and the hydrocyclones. The sump is modeled

…A7†

The separation eciency without considering the short circuiting yi for any size di is related to the cut size d50 . "

di yi ˆ 1 ÿ exp ÿ0:693… †2;9 d50

# …A8†

The average size di , is de®ned as the geometric mean of diameters di and di‡1 . di ˆ

p di di‡1

…A9†

The fraction of the water fed to the cyclone which reports to the under¯ow, Rf is given by the equation, Rf ˆ 540:37

dspig 299:38 ‡ ÿ 1:617 W…n‡1† W…n‡1†

…A10†

where W…n‡1† is the water feed rate to hydrocyclone. The separation efficiency including the short-circuiting yif is given yif ˆ Rf ‡ …1 ÿ Rf †yi

A2. Total closed circuit ball mill system

…A6†

…A11†

The ¯ow split between the under¯ow and the over¯ow in the hydrocyclone can be written in terms of the material balance equations.

V.R. Radhakrishnan/Journal of Process Control 9 (1999) 195±211

MUF mUF ˆ yif QCN‡1 mN‡1 i i

…A12†

MOF mOF ˆ …1 ÿ y†if QCN‡1 mN‡1

…A13†

WUF ˆ Rf W…N‡1†

…A14†

WOF ˆ …1 ÿ Rf †W…N‡1†

…A15†

In Eqs. (A12)±(A15) M is the mass ¯ow rate of solids and mi is the mass fraction of particle in the ith size range and W is the water ¯ow rate. The subscripts/ superscript UF, OF and …N ‡ 1†identify the streams as the under¯ow, over¯ow or the feed to the cyclone. The set of Eqs. (A1)±(A15) constitute the total grinding system model. The solution of the set of equations for a given set of conditions gives information regarding the output particle size distribution. When the output particle size distribution curve like the one illustrated in Fig. 2 is available it will be possible to determine the fraction of the material in the desired size range and the fraction in the under size and over size respectively. For implementing the supervisory control it should be possible to determine the new values of the selection function from the plant operating data. There are three methods available for this purpose. Graphical methods [9] are based on the grinding of narrow size fraction materials. Tracer methods [10] involve the introduction of a tracer into one of the size intervals in the feed and analysing the product size fractions for the appearance of the tracer. The most appropriate method for continuous systems is the method based on a size discretised- n cell grinding model together with a nonlinear search technique for parameter identi®cation [6]. They used the plant input-output data together with a GaussNewton optimisation algorithm to extract the selection and breakage parameters. The method essentially involved the assumption of a set of starting values of the parameters, calculation of the output size distribution

211

using the size discretised model, comparison of the calculated size distribution with the actual measured distribution and use the error to update the breakage parameters in a hill climbing sense to minimise the error between the two. The estimation scheme for the breakage parameters can be stated as, `given a set of feed and product mass fractions determine the value of the transformation matrix and hence the values of the selection function'. Rajamani and Herbst [6] have presented a solution to this problem in the form of a linear weighted least square formulation which they solve by a Gauss±Newton optimisation algorithm. References [1] J.A. Herbst, K. Rajamani, Control of grinding circuits, in A.Weiss,(Ed.), Computer Methods for the 80s', SME Press, NewYork,1979, pp. 728±730. [2] J.H. Fewings, Digital computer control of a wet mineral grinding circuit,Proc. Symp.on Automatic Control Systems in the Mineral Processing Plants', Aus.Inst.Min. and Met., Brisbane, 1971, pp. 333±357. [3] J.L. Bolles, A.J. Broderik, H.R. Wampler, Jr., Morenci concentrator process systems, Trans.Soc.Min.Engr. (1977) 262-271. [4] J.A. Herbst, D.J. Kinnenberg, K. Rajamani, Estimill-a program for grinding simulation and parameter estimation with linear models, Met.Eng.Dept.,Univ of Utah,Saltlake City, Utah, 1977. [5] S. Akhanzarova, V. Kafarov, Experimental Optimisation in Chemistry and Chemical Engineering, Mir Publishers, Moscow, 1982, pp. 151±226. [6] K. Rajamani, J.A. Herbst, Simultaneous Estimation of Selection and Breakage Functions from Batch and Continuous Grinding Data, Trans.lnst.Min.Met.93, 1984, pp. C74±C85. [7] R.C. Bond, Crushing and grinding calculations, Allis.Chalmer publication, Milwaukee, Wisconsin, 1962. [8] A.J. Lynch, T.C. Rao, Modelling and scale up of hydrocyclones, Proc.,Int.Min.Pro.Cong., Cagliari, 1975, pp. 85±95. [9] J.A. Herbst, D.W. Fuerstenau, The zero order production of ®ne sizes in comminution and its implications in simulation, Trans.A.Inst.Min.Engrs., 241(1968) 538±548. [10] R.P. Gardner, L.G. Austin, A Chemical Engineering Treatment of batch grinding', in H. Rumpf, D. Behrens (Eds.), Symp. Zerkleinern, Weinheim; Verlag Cheme, 1962, pp. 217±2480.