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Ind. Eng. Chem. Res. 1998, 37, 1404-1409
Dynamic Simulation Approach to Digester Ratio Control in Alumina Production Yousry L. Sidrak† Department of Chemical Engineering, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
In the Bayer process, used for the production of alumina from the bauxite ores, the caustic-toalumina molar ratio in the digester slurry leaving the digesters train is a critical operating parameter. Overall process productivity depends on the ability to stabilize the digestion causticto-alumina molar ratio. It is desirable to operate the plant at a target molar ratio that is as low as possible to ensure that production and processing efficiences are not lost, yet high enough for complete alumina extraction from the bauxite and stable processing to precipitation. This paper describes the development and validation of a simple, but accurate, simulation model for the digestion process. The model developed is used to gain understanding of the dynamics of the digestion process, to determine an optimal value for the caustic-to-alumina molar ratio, and to test the performance of possible control schemes, thus minimizing the expenses involved in experimentation with poorly designed control schemes on the physical plant. The design, tuning, and testing of a feedforward-predictive-feedback control scheme are outlined. Sensitivity analysis has shown the dependency of the closed-loop performance on the model structure. 1. Introduction 1.1. Digester Operation in the Bayer Process. In the 100-year-old Bayer process, used for the production of alumina from the bauxite ores, the purpose of digestion is to extract alumina from the bauxite pulp slurry to the required caustic-to-alumina molar ratio in the digester slurry, DIS, leaving the digesters train and to promote the formation of desilication product in order to regulate levels of silica in liquor. In a train of digesters, the bauxite pulp slurry from the grinding area is mixed with the main preheated strong feed liquor, SFL, from the evaporation area and heated to 145 °C. The alumina is extracted from the bauxite pulp according to the following reaction: heat
Al2O3‚3H2O + 2NaOH 98 2NaAlO2 + 4H2O About 85-87% of the bauxite alumina is extracted and goes into solution as sodium aluminate (NaAlO2). The Bayer process flow sheet and detailed outline have been reported by Sidrak (1996). 1.2. Need for Molar Ratio Control. The causticto-alumina molar ratio in the digester slurry leaving the train of digesters is a critical operating parameter (Langa, 1989). Overall process productivity depends on the ability to stabilize the digestion caustic-to-alumina molar ratio. Generally, a target ratio is set by plant production demands and the digestion process is operated to keep the target ratio steady. The choice of a target ratio is balanced by two opposing factors. If the target ratio is too low, the process will suffer decreased extraction efficiency and run the risk of alumina loss due to premature precipitation. If the target ratio is too high, production and processing efficiencies are lost. Therefore, it is desirable to operate the digester at a target ratio that is as low as possible, yet high enough † Telephone: 966-3-860-2286. Fax: 966-3-860-4234. Email:
[email protected].
for complete alumina extraction from bauxite and stable processing to precipitation. Choice of the target ratio also depends on how well the target ratio can be controlled. Tight control can be accomplished only if an accurate on-line measurement of the caustic-toalumina ratio is available. In many alumina refineries, the electrical conductivity of the digesters’ downstream process liquor is measured, and the measurement is used to determine the caustic-to-alumina molar ratio. With a reliable on-line measurement, feedback control systems have been developed to control a ratio setpoint by varying the bauxite slurry flow entering the digestion process (Dalen and Ward, 1972). The existence, however, of long time delays on the digesters train may cause substantial degradation in the performance of the feedback controller. A rule-based expert system for the diagnosis of process instrument and equipment problems associated with the Bayer digestion ratio control has been reported by Langa (1989). The design of a satisfactory control system for the digester molar ratio would require performance testing prior to implementation. Clearly, experimentation with poorly designed control schemes on the physical plant can be very expensive. Such expenses are minimized by thorough testing through computerized simulation techniques. In the development of any control algorithm using simulation, it is axiomatic that a reasonable representation of the plant to be controlled is known in the form of a mathematical model. Although there have been attempts to simulate the dynamics of the Bayer process (Crama and Visser, 1994; Chapman et al., 1991; Donaldson et al., 1991; Audet and Larocque, 1989; Langa et al., 1986), the models reported either were too complicated for the purpose of control system design or did not simulate the digester molar ratio in particular. This paper describes the development and validation of a simple, but accurate, simulation model to describe the dynamics of the digester molar ratio. The model developed is then used to design, tune, and test a
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Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1405
Figure 1. Digesters train model structure.
feedforward-predictive-feedback control system to stabilize the digester molar ratio. Sensitivity analysis has shown the dependency of the closed-loop performance on the model structure. 2. Development of the Digestion Process Model Modeling of the chemical reactions, which occur continuously throughout the digesters train, can be very complex. Since the molar ratio of the output digester slurry is measured once the reactions are completed, the knowledge of the progress of the reactions throughout the digesters train is not required. To simplify the model, the following assumptions have been made: (a) The chemical reactions occur instantaneously at the time the bauxite slurry and the strong feed liquor enter the first digester in the digesters train. The rest of the time spent in the train constitutes a deadtime. (b) The digesters train can be considered as a large input/output tank, with two input flows, i.e., bauxite slurry and strong feed liquor, and one output, i.e., the digestion slurry molar ratio. (c) A mixture of plug-flow and perfect mixing characteristics exists in the digesters train. Figure 1 shows the digesters train model structure. The empirical bauxite charge formula, used in alumina refineries as an aid to manual control of the plant, can be used in the model structure to represent the instantaneous reactions occurring within the digesters train. The instantaneous molar ratio, MR, occurring from a given bauxite/strong feed liquor mix is calculated by the empirical formula:
MR ) {Bx/aBFC + bZ}-1
(1)
where
B)
1.645 100 - M A E 100 100 100
M ) bauxite moisture content (%), A ) alumina, Al2O3, content of bauxite (%), E ) extraction efficiency (%), F ) strong feed liquor flow rate (m3/h), C ) strong feed liquor caustic concentration (g/L), Z ) strong feed liquor molar ratio, Bx ) wet bauxite charge rate (kg/h), and a and b ) correction factors applied by the operators in the Central Control Room in order to account for small variations in the bauxite quality and instrument degradation, e.g., errors in the bauxite weighbelt and strong feed liquor flowmeters. During the course of this work, the values of a and b used were 0.925 and 0.95, respectively. The first term in eq 1 represents the ratio between the moles of alumina, Al2O3, which is extracted from the bauxite and those of the caustic soda, expressed as sodium oxide, Na2O, which enters the digesters train with the strong feed liquor, while the second term represents the same ratio in the strong feed liquor. The
reciprocal of the sum of these two terms gives the molar ratio of the digestion slurry leaving the digesters train, expressed as moles of sodium oxide to moles of alumina. The factor 1.645 represents the ratio between the molecular weight of Al2O3 and that of Na2O. Accordingly, the digester is divided into N sections, which may or may not have the same volume. Each section is considered as a perfectly mixed input/output tank. Since each section is a tank only in concept, it is here called a logical tank. The number of logical tanks, N, may be used as a tuning parameter to reproduce the experimentally obtained response curves, in which case the model depicted would be a single-parameter model. An N-parameter model would, however, result if the volumes of the N logical tanks were allowed to vary. For the special case N ) 1, a perfectly mixed digester model results. As N increases, plug-flow conditions are approached. 3. Validation of the Digester Model To validate the model structure proposed, a step disturbance in the strong feed liquor flow rate, from 728 to 776 m3/h, was fed at zero time while holding all other parameters constant. Data for the resulting delayed response of the digester molar ratio were collected from the digester plant. A comparison of the experimentally obtained response data and model output is shown in Figure 2. 3.1. Results for Equal-Volume Logical Tanks. Figure 2 demonstrates the effects of changing the number of logical tanks used to construct the model. The following observations can be made: (a) The experimental transfer curve shows the expected mix of plug-flow and perfect mixing characteristics. This validates assumption c in section 2. The digester molar ratio shows no evidence of change for over 0.1 h, clearly a transportation lag. From 0.1 h onward, the exponential type change has a perfect mixing appearance. (b) The one-tank model is, by definition, a perfect mixing model which simulates the experimental behavior with acceptable accuracy but only after an initial period of about 0.3 h. (c) As the number of logical tanks increases, the response characteristics change from perfect mixing to plug flow. (d) Between the two extremes of plug-flow and perfect mixing characteristics, the choice of a five-tank model structure seems to be appropriate. Other experiments, employing different strong feed liquor flow rates and magnitudes of the step changes, confirm that a five-tank model structure is sufficiently accurate for the purpose of control system design and testing. In addition, rigorous accuracy of the model is not required as the proposed control schemes will have a feedback loop to correct for minor discrepancies between the plant output and the model’s predictions. 3.2. Results for Unequal-Volume Logical Tanks. Improved agreement between the experimental step response curves and the model response curves is obtained by employing logical tanks of unequal volumes. A number of different-volume logical tank configurations were tried, with only the most successful ones shown in Figure 2. This approach requires further parametrization of the model and, given that rigorous accuracy of the model is not required, probably only adds unwarranted complex-
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Figure 2. Digester molar ratio response curves to a step change in strong feed liquor flow rate from 728 to 776 m3/h.
ity. Although this approach was not pursued further, it does provide a degree of freedom for a higher level of model accuracy. Consequently, the model adopted was an equalvolume, five-logical-tank model. 4. Control System Design 4.1. Initial Control System Design. Deviations of the digester molar ratio from its target setpoint are to be controlled by manipulating the flow rate of the strong feed liquor entering the digesters train. In principle, there are three terms to the control system: (a) A feedforward term, FF, which sets the gross strong feed liquor flow rates for a given target digester molar ratio. This term is, in fact, another form of the bauxite charge formula (eq 1):
FF ) f(Bx,B,MRsp,Z,C)
(2)
where MRsp is the digester molar ratio setpoint. (b) An adaptation of the standard Smith deadtime controller (Smith, 1957) to compensate for the long transportation lags within the digesters train. (c) A feedback term, FB, which is used to adjust for deviations of the digester molar ratio from its target setpoint. For that purpose, a standard PI controller has been adopted:
FB ) Kc∆ +
∫-∞t ∆ dt
1 τI
(3)
where Kc ) controller gain, τI ) reset time constant, and ∆ ) error term, i.e., the instantaneous deviation of the digester slurry molar ratio from the target setpoint. Figure 3 shows a diagram of the initial control system design with the plant model developed in section 2 used to represent the physical plant when the control system is being simulated. The plant model is also incorporated with the deadtime compensator of the control system. The deadtime compensator is, therefore, split into a chemical reaction part and a deadtime delay part using the same modeling concepts as in section 2.
Figure 3. Initial control system design.
4.2. Final Control System Design. The initial design of Figure 3 was tested through simulation and an attempt was made to determine the optimal values for feedback controller parameters Kc and τI. Extremely large variations of these parameters failed to produce the numerical instability expected. To reduce the insensitivity of the control system shown in Figure 3, an equivalent control system has been designed, Figure 4, in which the loop ABCD corresponds to a standard feedback loop without deadtime. Figure 5 shows an equivalent circuit for this section of the control system. The feedback equivalence, as shown in Figure 4, shows that iterating around the loop ABCD will cause flac to converge to a value such that flac ) . Since the
Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1407
Figure 4. Initial control system design redrawn.
Figure 6. Final control system design.
(d) High accuracy is not required in the bauxite charge formula used in the compensator reactor term as the feedback term of the circuit will adjust for any possible deviations of the controlled variable from its target setpoint. (e) The sensitivity of the feedback loop may be changed by inclusion of a proportionality factor, k, in the error term ∆. Accordingly, the feedback portion of the circuit will “see” the molar ratio error at any instance as
k∆ ) k(MRsp - flbp)
Figure 5. Loop ABCD of Figure 4 redrawn.
(4)
where flbp is the measured digestion slurry molar ratio. 4.3. Testing Final Control System. 4.3.1. Performance Measures. The following measures were adopted for testing the usefulness of the final control system: (a) The squared error integral:
value of is known before the iterative procedure around ABCD is begun, the final value of flac is also known in advance, and the iterative procedure is, therefore, not needed. This resulted in the design of the final control system shown in Figure 6. Advantages of the final control system over the initial one are as follows: (a) The final design is simpler than the initial one. It is cheaper and easier to implement. (b) The feedforward term has been incorporated in the deadtime compensator reactor term. (c) The need to determine the two feedback parameters of the PI controller has been eliminated.
Es
∫-∞∞∆2 dt
(5)
As the testing simply involved feeding an undetected step change of 48 tons/h in bauxite charge rate and observing the digester slurry molar ratio response, Es
Table 1. Sensitivity Analysis of the Control Systema no. of logical tanks compensator
plant
1 5 30 5 5
5 5 5 1 30
a
k ) 2.0 Es
k ) 1.2 Sp
unstable 0.0024 1.0 0.0025 1.1 0.0010 0.8 unstable
Es
k ) 1.0 Sp
0.0032 1.4 0.0036 0.9 0.0039 1.3 0.0022 1.1 unstable
k ) 0.8
k ) 0.5
k ) 0.1
Es
Sp
Es
Sp
Es
Sp
Es
Sp
0.0037 0.0043 0.0046 0.0029 0.0128
1.5 1.0 1.0 1.3 >20.3
0.0046 0.0055 0.0058 0.0039 0.0080
1.6 1.2 1.2 1.5 1.4
0.0078 0.0089 0.0093 0.0071 0.0110
2.3 1.9 1.9 2.1 1.9
0.0449 0.0465 0.0469 0.0442 0.0483
9.6 9.3 9.3 9.3 9.3
Es is the error squared integral. Sp is the settling period (h) until the digester slurry molar ratio is in the range 1.3 ( 0.01.
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Figure 7. Effect of varying k on the closed-loop response of a 5,5-logical-tank control scheme.
Figure 8. Effect of varying the number of logical tanks in the deadtime compensator, Nc, on closed-loop response.
may be written as
Es )
∫tt ∆2 dt e
s
(6)
where ts is the time of the step introduction and te is the end of the simulation.
(b) The settling period, Sp, which is the time takensafter the introduction of the step change in bauxite charge ratesfor the digester slurry molar ratio to return permanently to the target setpoint ( 0.01. This indicates the length of time needed for system oscillations to die out. Due to its arbitrary nature, no great significance should be placed on this measure.
Ind. Eng. Chem. Res., Vol. 37, No. 4, 1998 1409
Nevertheless, it provides a more understandable measure than the more definitive Es. 4.3.2. Sensitivity Analysis. Table 1 presents a summary of the sensitivity analysis results of the control design adopted. The following observations can be made: (a) As expected, the closed-loop system becomes more unstable as the value of the proportionality constant, k, increases. Conversely, the system response time increases as k decreases. Figure 7 shows the effect of varying k on the closed-loop system response of a 5,5 logical tank control scheme. For the 5,5 logical tank model, a minimum value of Es of 0.0024 is obtained for k ) 2.0 (values of k higher than 2.0 were tested). (b) Figure 8 shows the effect of varying the number of logical tanks in the deadtime compensator, Nc, on system response to a step change of magnitude of 48 tons/h in bauxite charge rate. For a particular value of k, Nc appears to have little effect on system responses. Use of a simple time delay in the compensator circuit appears, therefore, to be quite acceptable. This is demonstrated in the results in Table 1 for Nc ) 30 which approximates a plug-flow type compensator. (c) Calculations, not shown in Table 1, indicate that for values of k higher than 2.0 the integration step length has a significant effect on performance measures. For example, for the 5,5 tank case with k ) 3.0, an integration step length of 0.01 h yields an Es value of 0.0081 and Sp ) 10.0 h. For an integration step length of 0.001 h, the corresponding values are Es ) 0.0017 and Sp ) 1.8 h. Although this effect is not apparent at levels of k likely to be employed, further work is needed on the required sampling frequency for the controller. 4.4. Control System Implementation. When implementing such a model-based feedforward-predictive-feedback control system, it would be advisable to start with a low value for k. This would greatly reduce the chance of uncontrolled oscillations occurring due to erroneous models being employed. The strategy would then be to slowly increase the value of k until signs of instability occur and then settle to a “safe” value for k. 5. Conclusions To minimize the expenses involved in experimentation with the physical plant, the development of a simple, but accurate, simulation model for the digestion process in alumina production is presented in this paper. The simulation model developed, which provides understanding of both open- and closed-loop dynamics of the digester molar ratio, has been used to develop, tune, and test a feedforward-predictive-feedback control system to stabilize the digestion slurry molar ratio.
From the practical point of view, the development and testing of the proposed control system could not have been achieved in the absence of a reliable simulation model of the physical plant. This stresses the importance of simulation models during control system design, tuning, and testing phases. Sensitivity analysis has demonstrated the dependency of the closed-loop performance on the model structure. Acknowledgment The author wishes to acknowledge the contributions made by plant personnel, without which this work could not have been completed. The author gratefully acknowledges the assistance of King Fahd University of Petroleum & Minerals in the preparation of this work. Literature Cited Audet, D. R.; Larocque, J. E. Development of a Model for Prediction of Productivity of Alumina Hydrate Precipitation. Light Metals: Proceedings of Sessions, AIME 118th Annual Meeting, Warrendale, PA, 1989; pp 21-26. Chapman, J. A.; Winter, P.; Barton, G. W. Dynamic Simulation of the Bayer Process. Proceedings of the 120th TMS Annual Meeting, New Orleans, LA, 1991; pp 91-96. Crama, W. J.; Visser, J. Modelling and Computer Simulation of Alumina Trihydrate Precipitation. Proceedings of the 123rd Annual Meeting on Light Metals, San Francisco, CA, 1994; pp 73-82. Dalen, V.; Ward, L. G. Semiautomatic Thermometric Titration for the Control of the Bayer Process. Proceedings of the Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Cleveland, OH 1972. Donaldson, D. J.; Mulloy, J. W.; Oesterle, E. V.; Beaver, D. Flexible Computer Spreadsheet Model of the Bayer Process. Proceedings of the 120th TMS Annual Meeting, Warrendale, PA, 1991; pp 73-77. Langa, J. M. A Diagnostic Expert System for Bayer Digestion Ratio Control. Light Metals: Proceedings of Sessions, AIME Annual Meeting, Las Vegas, NV, 1989; pp 9-19. Langa, J. M.; Russell, T. G.; O’Neil, G. A.; Snyder, J. G.; Gacka, P.; Shah, V. B.; Stephenson, J. L. Aspen Modelling of the Bayer Process. Proceedings of the Technical Sessions at the 115th TMS Annual Meeting, New Orleans, LA, 1986; pp 169-178. Sidrak, Y. L. Control of Counter-Current Washing Operation in Alumina Production; Classical and Optimal Solutions. Control Eng. Practice 1996, 4 (3), 331. Smith, O. J. M. Closer Control of Loops with Deadtime. Chem. Eng. Prog. 1957, 53 (217), 28.
Received for review June 9, 1997 Revised manuscript received December 12, 1997 Accepted December 15, 1997 IE9704206
