Dev. Chem. Eng, Mineral Process. 13(3/4). pp. 289-3 00, 2005. Electrolyte Inventory Control in a Sulphide Leach Plant M.W. Foley*, C.R. Christie and R.H. Julien' Dept. of Chemical Engineering, University of the West Indies, St. Augustine, Trinidad, West Indies K.Home Engineering Ltd., Suite 110, Bretton Hall, 16 Victoria Avenue, Port-of spain, Trinidad, West Indies ' The regulation of electrolyte inventoly in a zinc leaching plant is a challenging process control problem. This paper presents a simple plant-wide control strategy to maintain level in the Electrolytic and Melting ( E M ) feed tank at the operatorspecified upper limit. Electrolyte production is thereby maximized while ensuring that the outlet temperature of the Hot Stage Purification heat exchanger remains greater than its lower limit, and that levels in the other vessels are kept in their desired ranges. The basic principle underlying this approach is to set throughput at a value which will consume spare capacity in the E M tank over the next M hours of operation. To tune the strategy, the user must choose the prediction horizon M, as well as a maximum allowable rate-ofchange in plant throughput. The algorithm is easily programmed using standard DCS functions and does not require the purchase of third-party optimization software. Simulation results are included which demonstrate that the proposed method provides high-performance control of electrolyte inventoly in response to typical plant disturbances, such as reductions in steam header pressure and step changes in E&M feedrate. Introduction The focus of this report is throughput maximization and inventory control in the Sulphde Leach Plant (SLP) at Teck Cominco Metals Ltd. Trail Operations. Figure 1 is a simplified schematic diagram of the SLP process. The leaching section is not shown because all of its tanks and thckeners overflow, i.e. there is no surge capacity [l]. Other vessels, such as the hot-stage mix tanks, are omitted for the same reason. To slmplify the dynamic simulation model described in the following section, it has been assumed that every stream in the plant operates under perfect flow control. The worlung volumes of the tanks, each of which is a vertical cylinder, are listed in Table 1. The corresponding residence times were calculated for a steady-state throughput of 400 m3/hr. The setpoint of FC8 determines the feed rate to the E&M plant. It is entered by * Author for correspondence (mfoley@uwi.tt). 289 M.W.Foley, C.R. Christie and R.H.Julien the E&M console operator and must therefore be regarded as a disturbance variable in the SLP inventory control scheme. In principle, the setpoint of any of the remaining flow controllers could be used to fix SLP production rate. The hot-stage heat exchanger flowate F4 was selected because it provides the most immediate handle on outlet temperature, which often constrains throughput during the winter. us H..l Tank 1 (NTO Storsps) Tank 2 (CS Mh Tank) Tank 3 (Eachanpsf Pump TL.) Tank 4 Tank 5 (PS Mix Tank) Tank 6 (SLP Electrnlyle (HSFiler Pump Tk.) storwe) Tank 1 (EhM Eiedrolyte Storwe) EhM Faad Figure 1. Regulatoly control scheme for Sulphide Leach Plant. Therefore, for Tanks 1-3 upstream of the exchanger, liquid level must be controlled by manipulating inlet flowrate. Conversely, effluent flow will be adjusted to regulate level in Tanks 4-6. Tanks 2 and 5 are purification reactors and hence level is to be tightly controlled to a setpoint of 80% in order to utilize reactor capacity. The remaining vessels are surge drums for which it is not necessary to control level to any particular value. Rather, level may be allowed to float between limits, say 20 and 80% (see Table 1). The electrolyte inventory control problem can now be stated: Maintain level in the E&M storage tank (LI7) at the operatorspecified upper limit through adjustment of the heat exchanger flowrate F4, while keeping exchanger outlet temperature above its lower bound and the levels in tank 1- 7 in their desired ranges. Table 1. Surge capacity in the Sulphide Leach Plant. Tank No. Volume (m? Residence time (min) Desired operating range for level (?Aof scale) 1 2 3 4 5 6 7 250 100 30 30 100 960 2400 37.5 15 4.5 4.5 15 144 3 60 [20,801 ~75,851 [20,801 PO, 801 [75,851 [20,801 [20,80] 290 Electrolyte Inventory Control in a Sulphide Leach Plant Simulation Model Presuming that flowrates remain constant between the sampling instants, a dynamic material balance around any of the tanks in Figure 1 leads to: ,..(l) F/ and F/+, refer to the inlet and outlet volumetric flowrates in m3/hr, and hf the height of liquid in vessel i expressed in % of scale. The process is sampled every T minutes; the integer t denotes sample number. The “velocity gain” K i I 100T/(60Vj), where Vi is the volume of the tank (m3). The variables hf , F/ and F,.+’ represent deviations from initial steady-state values. As explained above, level in Tanks 1 , 3 , 4 and 6 can be allowed to surge within acceptable limits, hence the algorithm of the corresponding level controllers should be proportional-only. It is essential that surge-tank level controllers be configured as proportional, not proportional-integral, to obtain smoothing of flowrate changes [2]. Thus for Tanks 1 and 3: F: = KC,^ hf, i = 1,3 ...(2) assuming for simplicity that the level setpoints remain constant at their initial values. The proportional gain Kc,i will be given a positive value to obtain a reverse-acting or increase-decrease level controller. For Tanks 4 and 6, the effluent flowrates are manipulated and so: F;+, = - 8 K c , i h f , i = 4,6 ...(3) with Kc,i < 0 for direct action. Flows were normally limited to a maximum of 800 m 3 h ; the factor 8 converts level controller output from % to m3/hr. On the other hand, the control objective for Tanks 2 and 5 is that reactor level is regulated tightly around a setpoint of 80%. Here, the correct choice of feedback algorithm is proportional-integral [3]: Fi = Fl = -8Kc.2 [ ( l + T / r 1 , 2 ) - ~ - ’ ] V h: - 8 Kc,5 [ ( l + T / r,,5)- z-’1 ...(4) ...(5 ) h: V For convenience, it has been assumed that each controller executes at the same frequency (e.g.every 5 seconds). A discrete-time simulation of the SLP levellflow dynamics was created in MATLAB Sirnulink@based upon Equations (1)-(5). The proportional-only level controllers were tuned for a proportional band (loo/)K ~ , I); of 60% [2]. PI level controllers LC2 and LC.5 were designed using the technique of [4] for a desired peak deviation in level of 5%, and a maximum rate of change of 20 m3/hr/min in manipulated flowrate for a worst-case step flow disturbance of 100 m3/hr. This produced a proportional band of 66.7% and an integral time of 7.3 minutes. Transient behaviour of the hot-stage heat exchanger outlet temperature was modelled by discretizing the Laplace-domain expression: 291 M. W. Foley, C.R. Christie and R.H. Julien ...(6 ) This equation was adapted from Problem 16.4 of [ 5 ] and is not intended as an exact representation of the exchanger dynamics. It was included so that the performance of the proposed strategy could be evaluated in response to reductions in steam header pressure. A proportional-integral control loop was configured to regulate outlet temperature To to 84°C by manipulating the setpoint of a steam pressure controller. The dynamics of the secondary pressure control loop were neglected, i.e. perfect steam pressure control was assumed. The tuning constants of the temperature controller were selected using the IMC method [6] with e = 1 / 6 minutes, which can be interpreted as the desired closed-loop time constant. Optimal Control of Inventory in the E&M Storage Tank Having established a regulatory control structure for the plant, it remains to develop a throughput maximization strategy which maintains Tank 7 level at its upper limit while observing process constraints. The first step is the construction of a dynamic model relating the throughput handle F4 and load variable F8 to E&M electrolyte storage tank level h7. Substitution of Equation (3) into (1) yields the following closed loop expression for i = 4: F{ = Z-' G4(z-l) G4(z-l) = ...(7) Fi ...(8) -8K4 KC,4 1-(1+8K4 Kc,~)z-' Continuing in this manner, we obtain: Fj = z - ~C~(Z-')C~(Z-~)G~(Z-~)F~~ G ~ ( Z - '1 ) -8K5 Kc,5 [ ( I + T / T ~ , ~ ) - z - ' ] 1 -[ 2 + 8 K5 Kc.5 (1 + T / r 1 , 5 ) ] z - '+ (1 + 8 K.j K = , ~ ) z - * ...(9) ...( 10) ,.(12) The polynomials w ( z - ' ) and 6(z-l) represent, respectively, the numerator and denominator of the transfer function K , G4(z-' G5(2-l) c6(=-'I. Extended Horizon Control Model Predictive Control (MPC) refers to a general class of algorithms which use a dynamic model (such as Equation 12) to compute a sequence of future control moves that minimize a cost index without violating process constraints. Several MPC packages are available commercially, most or all of which are well-suited to 292 Electrolyte Inventory Control in a Sulphide Leach Plant the problem at hand. However, significant capital and engineering expenses may be involved in purchasing the vendor software and interfacing it to the plant DCS. Thus, a key objective of the current study was the development of a constrained MPC scheme for Tank 7 level control which could be implemented within the DCS using standard control functions. One of the simplest MPC strategies is Extended Horizon Control [7]. This approach minimizes the quadratic objective function: ^t+N 2 J = ( h S P -h7 1 ...(13) where hSP denotes the desired Tank 7 level (e.g. 80%) and I;$+"' is an N-step-ahead prediction of the level. The integer N is a user-specified tuning parameter known as the prediction horizon. Although the implementation of Extended Horizon Control is straightforward in comparison to other MPC algorithms, it would require userwritten code for polynomial multiplication, digital filtering and solution of a Diophantine identity. This would be problematic in some of the older-generation DCS systems. Therefore it is of interest to consider an alternative strategy which provides similar performance with far less computational complexity. It is evident from Equation (9) that the dynamic relationshp between F4 and F 7 is fourth-order with a deadtime of three samples. Of course, at steady-state, these flowrates must be equal, which suggests the approximation F! = F:. The material balance Equation (1) around Tank 7 then becomes: ...( 14) The spare capacity in cubic meters can be expressed as (hSP - h : ) V, / 100. To consume this available volume in N steps or, equivalently, M = N T / 6 0 hours, requires a flow imbalance: F i - F i = (hSp - h j ) V 7 l(100M) 3 F i = F i +(hSp - h j ) V 7 l(100M) ...(15) It can be shown that this control law optimizes the cost index (Equation 13) for the reduced-order process model (Equation 14), when the future measured disturbances F;", ..., are presumed equal to the current value F; . A deficiency in this approach is that no account has thus far been taken of unmeasured disturbances such as bias in the flow sensors. To eliminate potential offset in Tank 7 level, it is necessary to incorporate some form of "bias update" into the algorithm. This can be accomplished by augmenting Equation (14) to include a term D' whch represents the net effect of all unmeasured disturbances: D' can be back-calculated at each time step from: 2 93 M. W.Foley, C.R. Christie and R.H. Julien To reduce the sensitivity of the disturbance estimator to sensor noise and modelplant mismatch, D‘ is passed through the first-order ‘IMC filter’ [8]: D‘ - +(, - , - T / ( ~ O M ) ) ~ I , - r / ( 6 0 ~Dr-l ) f j - ...(18) The filter time constant was arbitrarily equated to the prediction horizon M mainly to avoid the complication of an additional tuning parameter. The effect of unmeasured disturbances can then be included in calculation of the exchanger flowrate as: F: = Fi - D; + ( h ~ p- h ; ) V , l(100M) ...( 19) Figure 2 displays the results obtained using Equation (19) with M = 2 hr, when a step error of -100 m 3 h in the E&M flow measurement was introduced at time = 0.5 hr. Uncorrelated measurement noise sequences of variance 10 (m3h)2 and 0.25 (%)2were also added to the flow and level signals. The lower graph shows the filtered disturbance estimate D) converging to the sensor bias of -100 m 3 h which enabled the control scheme to return the tank level to its target value of 80%. 1 A 7 5 3 WI 8 I 0 L 2 1 2 3 1 2 3 4 5 6 7 8 4 5 6 7 8 -1w0 n m <hr> Figure 2. Performance of Equation (19)following introduction of bias in E&M flow sensor. Process Constraints It is proposed to maximize throughput in the Sulphide Leach Plant using Equation (19) to regulate E&M storage tank level at its operator-defined upper limit. Thls optimization will be carried out subject to the following constraints: i) Upper limit on exchangerflowrate (e.g. 800 m3/hr): F4t 5 294 F4,max ...(20) Electrolyte Inventory Control in a Sulphide Leach Plant ii) Lower limit on exchanger outlet temperature (e.g. To,sp- 3 = 81 "c): A steady-state energy balance around the hot-stage heat exchanger yields Q" = F T p C, (Tf - q"). If the temperature controller TC in Figure 1 is saturated (i.e. controller output CO = 100%) due to inadequate steam supply, the quantity Fl-' p c , (Td - T / - ' ) is an approximation of the steady-state rate of heat transfer which the exchanger is capable of delivering. If the TC is saturated and outlet temperature has fallen more than three degrees below setpoint, then throughput must be reduced to ensure adequate cobalt removal in the hot-stage mix tanks. The required exchanger flow limit is obtained from: Fl p C, (T,,sp - qf) I Fi-' p C, (T,' - Tf-') ..(21) This constraint may become active only when the temperature controller output = 100% and To,sp - T,' 2 3 "C. iii) Lower limits on level in Tank 1-3: To prevent Tanks 1-3 from emptying, throughput must be reduced when any of these level controllers is saturated and level has dropped below the operator-entered low limit. This implies: Fi I F,' when CO: = 100% and h: I hl,min ...(22) Fi I Fi when COi = 100% and hi I h2,min ...(23) Fi I Fi when CO; = 100% and hi I h3,,,," ...(24) iv) Upper limits on level in Tanks 4-6: To prevent Tanks 4-6 from overflowing, throughput must be reduced when any of these level controllers is saturated and level has risen above the operator-entered high limit. Thus: and hi 2 h4,mar ...(25) Fi I F; when CO; = 100% and hi 2 h5,mar ...(26) Fi I F: when CO; = 100% and hi 1 h6,mnr ...(27) Fi I F{ when CO: = 100% It is worth noting that constraints of a similar nature could be applied even if, for example, accurate flow measurement of stream 5 was not available and LC4 manipulated the effluent control valve position directly. To keep Tank 4 from overflowing, we could impose an upper limit on F4 which decreases linearly from F4,max to 0 as level rises from its upper limit to loo%, that is: v) Maximum move in exchanger jlowrate (e.g. 120 m3/hr/hr): An operatoradjustable maximum rate-of-change (maxmove) must be configured to prevent the controller from making large throughput changes which might adversely affect filter operation or electrolyte purity. 295 M.W. Foley, C.R.Christie and R.H.Julien + maxmove*Tl60 Fi I Fi-' ...(28) ...(29) Fi 2 Fi-' - maxmove* T 160 To summarize, the hot-stage heat exchanger flowrate is set to the value calculated in Equation (19) provided that none of the process constraints is thereby violated. In the event that Equation ( 1 9) is larger than any of the upper bounds imposed by Equations (20)-(27), then F4 is provisionally fixed to the lowest of these. The constrained flowrate is then clamped if required by Equation (28) or (29) to ensure that the implemented change in F4 does not exceed maxmove. However, it is recommended that the lower bound (29) be overridden by (22)-(27) in the event of a conflict, as it is of primary importance that Tanks 1-6 do not overflow or empty. Simulation Results Figure 3 illustrates the simulated behaviour of the plantwide control system when commissioned at time zero. At the initial steady-state, levels in the surge vessels (Tanks 1, 3, 4 and 6 ) were at 50%, Mix Tanks 2 and 5 were 80% full, and throughput was fixed at 400 m3/hr. All of the results presented in this section were generated using a samplingkontrol interval T = 5/60 minute, a horizon M of two hours and maximum rate-of-change maxmove = 120 m31hr/hr; sensor noise has been omitted for clarity. The E&M storage tank level in Figure 3a was observed to increase smoothly from 50% to its high limit of 80% in approximately five hours. It is evident from the ramp response of F4 in the lower graphs that constraints Equations (28) or (29) were active for much of this period. The unconstrained solution (Equation 19) was then implemented for the final 6.6 hours of operation. Figures 3a and 3b indicate that at no time did any of the vessel levels violate the limits listed in Table 1. Flowrates F3 and F5 closely follow the optimization handle I I 3 30- ---I 200 I 2 4 6 6 10 11 n m <hr> Figure 3a. Transient behaviour of level infinal four tanks. 296 Electrolyte Inventory Control in a Sulphide Leach Plant 0 I I 2 4 I 6 8 10 12 Figure 36. Transient behaviour of level infirst three tanks. F4 because Pump Tanks 3 and 4 have a residence time of only 4.5 min. Flows F2 and F6 are nearly identical to F3 and F5 since level is tightly controlled to a setpoint of 80% in Tanks 2 and 5 . Movements in Fl and F7 are attenuated somewhat due to the larger residence times of Tanks 1 and 6 as well as the flowsmoothing characteristic of their propohonal-only level controllers. Figure 4 displays the results obtained following a decrease in steam header pressure caused by a roaster shutdown. (All of the steam consumed in the SLP is generated from waste heat in the Lead Smelter and RoasterIAcid plants.) As can be seen in the lower plot of Figure 4a, the pressure dropped suddenly from 200 to 100 Wa at time = 0.5 hr. It stayed at this value for three hours until the roaster was restarted, after which it ramped back to 200 during the following 30 minutes. This large reduction in header pressure caused the temperature controller to saturate and the hot-stage heat exchanger exit temperature to initially drop to 76.6 "C. At this point, constraint (21) came into effect, and F4 was decreased until the temperature rose to its low limit of 81°C (see Figure 4b). This in turn caused Tank 7 level to fall because the E&M flowrate remained at its steady-state value of 400 m3/hr. The temperature controller was implemented with reset windup protection, so once the roaster startup began it was able to quickly shft the exchanger outlet temperature back to its setpoint of 84°C. The optimizer then gradually consumed the spare capacity in Tank 7 and the process attained its original steady-state around six hours later. The other tank levels are not shown in Figure 4b because they were kept within their desired ranges for the duration of the simulation. 297 M.W. Foley, C.R. Christie and R.H.Julien As a final test, the regulatory behaviour of the proposed plantwide control system was simulated in response to variation in the E&M flowrate. The F8 trend in the lower plot of Figure 5 comprises four days of actual plant data. The control scheme is clearly capable of tracking such changes with minimal disruption to electrolyte level in the E&M storage tank (or any of the other vessels in the SLP). 8 @I _ - i 4f lM) 5oo I Time * 10 12 10 12 Figure 4a. Heat exchanger temperature control with temporary restriction in steam header pressure. - B m2 r- 70- it F 8050 Time ~r Figure 46. Inventory control with temporary restriction in steam header pressure. 298 Electrolyte Inventory Control in a Sulphide Leach Plant \:k---pj loo I-80 50 10 20 30 w 50 40 70 8 0 9 0 1 P F4 F4 FB I 10 20 30 50 40 80 70 8 0 9 0 Time <hr, Figure 5. Response of inventory control scheme to changes in EMjIowrate. Conclusions This paper has presented a simple electrolyte inventory control strategy for a Sulphide Leach Plant. Production rate is maximized by regulating level in the E&M electrolyte storage tank at the operator-specified upper limit; all other vessels are to be operated under conventional P or PI level control. The hot-stage heat exchanger flowate was identified as the most appropriate optimization handle. Adjustments to this flowate are implemented subject to upper limits on this variable and its rate-ofchange, plus a lower bound on exchanger outlet temperature. Additional constraints were included to prevent vessels upstream of the exchanger from emptying and downstream tanks from overflowing. The algorithm is readily configured using standard DCS functions and will not contribute significantly to the loading on the processor. Nomenclature Abbreviations CO Controller output CS Cold Stage DCS Distributed Control System E&M Electrolytic and Melting FC Flow controller HS Hot Stage IMC Internal Model Control LC MPC NTO PI PS SLP SP Level controller Model Predictive Control Neutral Thickener Overflow Proportional-Integral Polish Stage Sulphide Leach Plant Setpoint Symbols CP D Heat capacity of electrolyte Unmeasured disturbance 299 M. W.Foley, C.R. Christie and R.H.Julien F G(z-') h K Kc M mamove N P S T Ti To V Z- Volumetric flowrate Discrete transfer function Liquid level Velocity gain in Equation (1) Proportional gain Prediction horizon Maximum move in exchanger flowrate Prediction horizon Steam pressure Laplace transform variable Sampling period Heat exchanger inlet temperature Heat exchanger outlet temperature Vessel volume Backshift operator % % hr/m3 %I% hr (m3/hr)/hr no.of samples Wa min OC "C m3 Greek letters S(z-') V P 71 o(2-l) Denominator polynomial in Equation (12) Differencing operator (1 - z-') in Equation (1) Electrolyte density kg/m3 Integral time min Numerator polynomial in Equation (12) Superscripts t Sample number Subscripts f Filtered value i Vessel number Predicted value max min Operator-specified upper limit Operator-specified lower limit Acknowledgments The cooperation of Teck Cominco Metals Ltd., especially Mr. Wannes Luppens, is gratefully acknowledged. References 1. 2. 3. 4. 5. 6. 7. 8. Belland, G.M.; Pressacco, R.G.,and Van Beek, W.A. 1995. Leaching plant flow and level control optimization at Cominco's Trail Operations. Zinc Producers Meeting, Lead-Zinc '95 Conference. Luyben, W.L., and Luyben, M.L. 1997. Essentials of Process Control. McGraw-Hill, New York. Stephanopoulos, G. 1984. Chemical Process Control. Prentice-Hall, Englewood Cliffs, USA. Cheung, T.F., and Luyben, W.L. 1979. Liquid level control in single tanks and cascades of tanks with proportional-only and proportional-integral feedback controllers. IEC Fund., 18, 15-21. Ogunnaike, B.A., and Ray, W.H. 1994. Process Dynamics, Modelling and Control. Oxford University Press, New York. Rivera, D.E.; Morari, M., and Skogestad, S. 1986. Internal model control. 4. PID controller design. Ind. Eng. Chem. Process Des. Dev.. 25,252-265. Ydstie, B.E.; Kershenbaum, L.S., and Sargent, R.W.H. 1985. Theory and application of an extended horizon self-tuning controller. AIChE J., 31, 1771-1 780. Morari, M., and Zafiriou, E. 1989. Robust Process Control. Prentice-Hall, Englewood Cliffs, USA. Received: 12 January 2004; Accepted after revision: 17 May 2004. 300

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