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Control of Scheibel Extraction Contactors Using Neural-Network-Based Control Algorithms Farouq S. Mjalli*,† and Nabil M. Abdel-Jabbar‡ Department of Chemical Engineering, University of Qatar, P.O. Box 2713, Doha, Qatar, and Department of Chemical Engineering, Jordan University of Science and Technology, Irbid, Jordan
Control of the liquid-liquid extraction process using advanced neural-network-based algorithms was applied to control product compositions of a Scheibel agitated extractor of type I. The nonequilibrium backflow mixing cell model was used to model the extractor hydrodynamics and mass-transfer characteristics of the column. Two neural-network-based control algorithms were used to control the extractor, namely, the model predictive control and the feedback linearization control algorithms. The performance of the feedback linearization controller was compared to that of the model predictive controller. Both algorithms were capable of solving the set-point tracking control problem with a noticeable superiority of the model predictive algorithm. Despite the good set-point direction tracking for the feedback linearization controller, its performance was slow and suffered from steady-state offsets. The application of this controller is restricted to certain processes and cannot be generalized for all processes. The model predictive control algorithm was better in terms of set-point tracking accuracy and speed of response. Introduction The complex dynamics of the liquid-liquid extraction process due to model nonlinearity and the time-dependent dynamics have drawn attention for the implementation of advanced control algorithms to solve its control problem. Conventional control algorithms, which are based on linear systems, are limited in applicability in such situations. This is because they cannot handle efficiently situations such as constrained actuator moves, model mismatch, and nonlinear dynamics. Among the advanced control algorithms are the model-based ones. These algorithms solely rely on a mathematical representation of the process; consequently, a poor model will result in a modest controller performance, and sometimes the controller fails. The implementation of these algorithms necessitates the need for an accurate process model prior to the consideration of the control system design. Neural networks have the remarkable capability of capturing system dynamics and approximating the behavior of the plant under different operating conditions. They are used to estimate the unknown nonlinear dynamics and to compensate for them.1 They are costeffective, easy to implement, and data-driven. They have been implemented successfully in different modeling areas where nonlinearity and complexity are major issues. Chouai et al.2 have modeled a pulsed liquidliquid extraction column using a multilayer artificial neural network (ANN). The model was able to approximate the extractor dynamics under the tested operating conditions. In addition to its use for modeling purposes, the neural networks are utilized for the design and training of advanced controllers such as model predictive control (MPC), nonlinear internal model control, model reference control or neural adaptive control, and linear feedback control.3 * To whom correspondence should be addressed. Tel.: (974)4852495. Fax: (974)4852101. E-mail:
[email protected]. † University of Qatar. ‡ Jordan University of Science and Technology. Current address: American University of Sharjah, Sharjah, UAE.
The neural-network-based control algorithms attempt to approximate the process model and then use a certain control strategy based on the model approximation. Dirion et al.4 developed a neural controller based on the process inverse dynamics modeling. They tested the controller on a semibatch pilot-plant reactor equipped with a monofluid heating-cooling system. Botto and Costa5 compared two different approximation schemes based on the predictive control techniques using neuralnetwork models. Irwin et al.6 studied the nonlinear model-based control to an industrial high-purity distillation tower. An improved proportional-integralderivative control scheme using linearization through a specified neural network was developed by Chen and Huang7 to control nonlinear processes. Other applications of the neural-network-based process controllers were considered by Bhat and McAvoy,8 Bjarne et al.,9 and Mujtaba and Hussain.10 Feedback linearization is a technique used to approximate process nonlinearity by a linear model that can be adopted for both conventional and advanced control schemes. Decentralized control using online function approximators for feedback-linearizable systems has proven to be a very effective way to design controllers based on an approximate knowledge of the system dynamics.11 The neural-network-based feedback linearization implements its control strategy by using two subnetworks for the model approximation. The MPC, on the other hand, uses successive recursive use of a nonlinear time series model of the process. The optimization problem in this case is much more difficult and needs an iterative search technique. A comparative study of the two control algorithms was carried out to control the exit concentrations of a Scheibel extraction column. The validated nonequilibrium backflow mixing cell extraction model was used to approximate the extractor. The study was performed using the same column operating conditions. The performance of the two neural-network-based control algorithms was compared in terms of set-point tracking capability, actuator speed, and accuracy of response.
10.1021/ie049308f CCC: $30.25 © 2005 American Chemical Society Published on Web 03/09/2005
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Experimental Apparatus The experimental pilot-plant-scale apparatus is composed basically of a Scheibel extraction column of type I, which is one of three different column architectures. This mechanically agitated extractor is known for its high throughput and good separation efficiency. The general design of this type is given in work by Scheibel.12 In this type, the column is divided into a series of wire mesh (stainless steel-polypropylene) packed calming sections followed by mixing sections. The column is made of a QVF borosilicate pipe of 8.7 cm diameter and 185 cm length. It is divided into nine compartments, each of 14.5 cm height, with a dual coalescer wire gauze packings of 12 cm height inserted in each compartment, making a stage of a mixing zone and a coalescence zone. The mixing zone of each stage was supplied with a hole of 15 mm diameter in the column’s wall to support the single-phase sampling head probe and needle. The chemical system used is a water-acetone solution as the feed and toluene as the solvent. The feed streams are introduced countercurrently. The aqueous inlet stream is introduced at the top of the column, 1.5 cm above the ninth stage, whereas the solvent inlet stream is introduced at the bottom of the column through a stainless steel distributor of 4.5 cm diameter and 2 mm hole diameter. The process stream tubes are made of either stainless steel (organic phase stream) or glass pipes of 1.25 cm diameter (aqueous phase stream) so as to prevent any kind of corrosion or material deterioration to occur because of the presence of solvents. The feed tanks are constructed of stainless steel plates (2 mm thickness) for the same reason. They were installed on a wall-mounted support, 2.5 m above the ground, to give enough head for the feed pumps. The raffinate concentration was monitored using an online refractometer (Anacon model 47) and a data-logging system using an IBM-compatible PC. The software used in the data-logging system is a Basic language based program written to operate a DT2811-PGL type A/D board.
Figure 1. Schematic diagram of the extraction mixing-stage backflow model.
Figure 2. NN-MPC structure.
the interface and the extract exit stream at the top of the column and the volume between the solvent distributor and the raffinate exit stream at the bottom of the column are considered to comprise two perfectly mixed, single-phase stages without mass transfer. The figure shows the input streams (xf and yf), the output streams (xout and yout), and the input-output internal flow streams for a typical ith stage. The details of this model are described in the appendix. This model was applied and verified successfully for an eight-stage Scheibel column using a wateracetone-toluene chemical system.
Scheibel Column Extraction Model Neural-Network-Based MPC Algorithm During the past few decades, continuous effort was devoted to the modeling of the extraction process. This is still a research-active area, where new advances in computer technology and the related advanced control theory offer new tools for tackling the problem from a new perspective. The Scheibel extractor is categorized as a mechanically agitated column. It has three main geometrical configurations. These configurations differ in terms of ease of maintenance features and masstransfer characteristics. This contactor is well-known for its improved efficiency and high throughput. Modeling of such a contactor is usually done by considering the column as a cascade of well-mixed nonideal stages between which backflow occurs.13 This approach is termed stagewise backflow modeling. The backflow model was improved to include the column end effects,14 the effect of the drop state on the mass-transfer coefficient,15 and most recently the effect of mass transfer in the calming zones.16 The final developed model is called the nonequilibrium backflow mixing cell model. A schematic diagram of the nonequilibrium backflow mixing cell model is shown in Figure 1. The column is considered as a cascade of n stages. The volume between
MPC was successfully applied in solving process control and showed very wide acceptance.17,18 This is basically related to its ability to deal with complex situations such as systems with large delays, process variable constraints, non-minimum-phase systems, and its robust performance against model inaccuracies.19 The basic implementation of MPC was mainly for linear systems. Recently, great effort is being devoted to the development and implementation of nonlinear versions of this algorithm. One of these implementations is the use of neural networks for controller design purposes. Neural networks are capable of capturing the system nonlinear dynamics and can be used to approximate the process as well as to design the MPC. The neuralnetwork-based MPC (NN-MPC) uses a neural-network model of a nonlinear plant to predict future plant performance. The controller then calculates the predicted control input that will optimize the plant performance over a specified future time horizon. This control method is based on the receding horizon technique.20 The NN-MPC structure is shown in Figure 2. It is composed of four components in addition to the plant.
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These components are two neural networks [one for the plant and the other for the controller (shaded blocks)], an optimizer, and a performance function. For a selected time horizon, the controller optimizes the plant output by using the neural-network plant model for calculating controller moves and predicting the plant output. The neural-network controller is trained in order to produce the correct controller moves generated by the optimization algorithm.21 A series of simulation experiments are performed using different controller parameters in order to tune its performance. The tuning parameters are the control horizon, the prediction horizon, and the weighting matrix. The variation of these parameters is done in sequence for each parameter, and the controller performance is noticed. The best tuning parameters are then set based on the controller tracking capability, aggressiveness of the process, and controller moves responses. Basically, the NN-MPC solves for the control signal variable u′ by minimizing the following objective function: N2
J(t,U) ) Γy
[yr(t+i) - ym(t+i)]2 + ∑ i)N 1
Nu
Γu
[u′(t+i-1) - u′(t+i-2)]2 ∑ i)1
(1)
where Γu and Γy are the input and output weighting parameters. These two parameters determine the contribution that the sums of the squares of the control and output prediction increments, respectively, have on the performance index. N1 and N2 are the minimum and maximum prediction horizon. Nu is the controller moves horizon. It specifies the instant time because the output of the controller should be kept at a constant value. U is the Nu future control moves vector defined as U ) [u(t), u(t+1), ..., u(t+Nu-1)]T. At each time sample, the optimization problem is solved and results in a sequence of future controls U. Only the first component of the controller moves vector, u(t), is applied to the process. Nørgaard et al.22 reported in detail the derivation of the controller. The controller parameters can be used to tune the performance of the predicted output. This may require some exploratory experiments for determining the best controller parameters. Depending on the problem formulation, the main tuning parameters may involve one or more of the following: sampling time, control horizon, prediction horizon, and weighting matrices in the optimization formulation.23 Neural-Network-Based Feedback Linearization Algorithm Feedback linearization has been utilized successfully as an approximate process control strategy for nonlinear processes.24,25 This algorithm can be used in conjunction with ANNs. It is used to model the process and to transform the nonlinear system dynamics into linear dynamics by canceling the nonlinearities.22 This approach has been adopted by Slotine and Li26 and Nikolaou and Hanagandi.27 The neural-network-based feedback linearization control (NN-FLC) algorithm adopts this methodology based on the standard FLC. Figure 3 shows the structure of the neural feedback linearization algorithm. The control strategy of the FLC algorithm
Figure 3. NN-FLC structure.
expresses the plant model in terms of two functions, f[y,u] and g[y,u], as follows:
y(k+d) ) f[y,u] + g[y,u] u(k)
(2)
where y and u are the output and input vectors, respectively, expressed as
y ) y(k), y(k-1), ..., y(k-n+1)
(3)
u ) u(k), u(k-1), ..., u(k-m+1)
(4)
where d is the system delay and k is the current time incidence. With such a system configuration, Van Breemen and Veelenturf28 described the FLC as the one that is limited to nonlinear systems with the following dynamics:
y(n) p ) f(yp) + g(yp) u
(5)
where yp is the system state variables vector given by [yp, y°p, ..., y(n) p ]. The basic idea here is to split the control signal into two components. The first component cancels out model nonlinearities, and the second one is a linear state feedback controller. To achieve this, the input that linearizes the system dynamics can be expressed as
u)
1 [- f(yp) - kTyp + r] g(yp)
(6)
where k is the feedback gains and r is the reference input. Substituting eq 5 in eq 6 results in the following linear system: T y(n) p ) -k yp + r
(7)
Implementation of the FLC algorithm in neural networks is straightforward. The functions f and g are approximated by the neural networks Nf and Ng, and eq 6 can be expressed as
u)
1 [-Nf(yp) - kTyp + r] Ng(yp)
(8)
The objective of the controller is to force the system to follow the reference model: T y(n) m ) -k ym + r
(9)
The controlled process dynamics can be expressed as
y(n) p ) Nf(yp) +
g(yp)
[-Nf(yp) - kTyp + r] (10)
Ng(yp)
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Figure 4. Block diagram of the extraction subsystem block.
The error signal is defined as e ) yp - ym, and the error differential is expressed as
e(n) ) -kTe + [f(yp) - Nf(yp)] + [g(yp) - Ng(yp)]u (11) Figure 5. ith stage subsystem schematic diagram.
Prior to the implementation of the NN-FLC, the two networks Nf and Ng must be trained on the reference trajectory to ensure proper modeling of the process at hand. The error signal expressed by eq 11 should be minimized by proper training of the controller’s Nf and Ng networks. Results and Discussion Model Formulation and Solution. The nonequilibrium backflow mixing cell extraction mechanistic model is coded in a MATLAB/SIMULINK environment. The model is expressed in the form of a SIMULINK subsystem block to facilitate its use in the control system design. The internal structure of the SIMULINK extractor subsystem is shown in Figure 4. This subsystem block consists of three subsystems, namely, a main subsystem representing the n middle stages plus two mixing-stage subsystems. The two phase streams flow countercurrently throughout the three blocks. The two mixing stages represent the delay action introduced by the phase separation volumes (single phase), located between the interfaces and the contactor ends. They are perfectly mixed, single-phase stages without mass transfer. The two mixing stages are named subsystems 1 and 3 in the figure. Subsystem 2 contains the modeling equations for stages 1 - n. Each of the n middle stages is represented by a SIMULINK subsystem. These n subsystems have basically similar structures. The general layout of a typical middle-stage subsystem is shown in Figure 5. A single ith stage consists of two parts, representing the aqueous and organic phases. The model equations of these phases are included within two separate subblocks. Each of the two phase sub-blocks contains the equations needed for solving the material balance equations at that stage for the specific phase. These equations are the hydrodynamic equation, mass-transfer rate equation, physical property correlations, distribution coefficient correlation, and model parameter correlations (backmixing coefficients and mass-transfer weight factor). The inputs for any typical ith stage are the exit concentrations from the previous and next stages (xi-1, yi-1, xi+1, and yi+1). The outputs are the concentrations of the ith stage, xi and yi. The mass-transfer rate (Qxi) is calculated in the aqueous-phase block and then fed to the organic-phase block to be used in the material balance equations of the model. The model parameter
correlations are included in a separate block that takes the rotor speed (RPM), the feed flow rate (Ri), and the solvent feed flow rate (Si) as inputs and then calculates the fractional holdup coefficient (i), the mass-transfer weight factor (fi), and the continuous phase backmixing coefficient (Ri) using model correlations. These parameters are then fed to each stage in the model. The model parameter block is shown on the left side of the ith stage in Figure 5. The total number of model equations included in all subsystems is comprised of a set of 8(n + 2) + 4 equations. This set of differential-algebraic equations (DAEs) is stiff because of the variation of equation time constants and, hence, the numerical solution technique must efficiently handle this equation stiffness. A variable-order odes15 stiff ordinary differential equation solver, which is based on the numerical differentiation formulas, was used to solve the model equations and store results to the MATLAB workspace. Dynamic analysis of the model16 showed some degree of nonlinearity encountered in the column dynamics under set-point and disturbance excitations. This is more pronounced in the case of the rotor speed excitations. Additionally, it was noticed that the degree of nonlinearity is higher in the extract phase than in the aqueous one. This column behavior necessitates the need for an advanced control algorithm to be adopted to handle these difficulties. Moreover, the control system should handle the presence of dead time such as the one present in the raffinate concentration due to excitations in the feed flow rate. Neural-Network-Based Control System Design System synthesis is performed to select the best variable pairings and ensure minimum interaction between loops. Using the same system synthesis results as those adopted by Mjalli,16 the variable pairings used in this work are as follows: rotor speed-raffinate concentration and solvent flow rate-extract concentration. The system disturbances are the feed flow rate, solvent concentration, and feed concentration. This system configuration is comprised of a 2 × 2 system with the rotor speed and solvent flow rate as the process manipulated variables and the raffinate concentration and extract concentration as the process controlled variables.
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Figure 6. Block diagram for control of the extraction process.
Figure 7. Layer structure of NN-MPC and NN-FLC.
The two neural-network-based algorithms (MPC and FLC) were adopted for the previously described model by setting up the control block diagram shown in Figure 6 for each case. The block diagram consisted of two decentralized controllers in two feedback loops. The aim of the upper controller is to track set-point changes in the raffinate concentration, whereas the lower one tracks these in the extract concentration. After the model and controller block subsystems are set up, the neural-network controller structures were first selected. This was achieved based on testing of different networks that vary in terms of structure and simulation parameters. Network simplicity and accuracy of the model prediction and performance were the main criteria of structure selection. The finally selected network contained 1 hidden layer with 12 neurons for the NN-MPC case and 10 neurons for the NN-FLC. The structures are 1-12-2 and 1-10-2 for the two controllers, respectively. These two networks are shown in Figure 7. Two delayed plant inputs and outputs were used in the network to account for the time variation of variables in the system. To ensure the accuracy and performance of the controllers, the neural network should be trained to model the process and capture its dynamics efficiently. This was done by selecting a proper network structure and specifying its structural parameters as well as the simulation parameters. To facilitate the generalization of the trained neural network and confirm the acceptance of the network performance over a wide range of process operating conditions, the network needs to be trained with data that cover the entire range of possible network inputs. For the process under consideration, practical experience was utilized to select a practical range of 120-600 rpm for the rotor speed and the range 200-600 cm3/min for the solvent flow rate. The training proceeded by randomly selecting values for the input variables within the specified ranges at a sampling time of 0.5 min. The Simulink simulated process model was used for generating the training and
validating and testing the data. The inputs to the model are the rotor speed and solvent flow rate, and the outputs are the extractor raffinate and extract concentrations. The 15 000 generated input/output pairs were then divided into a ratio of 2:1:1. This gives 7500 pairs for the training subset, 3750 for the validating subset, and a similar subset for the testing data. To make the neural-network training more efficient, the inputs and targets were normalized so that they have zero mean deviation and unity standard deviation. Network training is then performed on the normalized input/output data set. This is a crucial step in the development of any ANN. It involves the process of preparation of the network to emulate the actual process. A neural network is trained to perform a particular function by adjusting the values of the connections (weights) between elements. Commonly, neural networks are adjusted, or trained, so that a particular input leads to a specific target output. Such a situation is shown below. There, the network is adjusted, based on a comparison of the network output (ynet) and the process target (yactual), until the network output matches the target. Typically, many such input/ target pairs are used, in this supervised learning, to train a network. During training, the weights and biases of the network are iteratively adjusted to minimize the network performance. The error in the validation set is monitored during the training process. The validation error will normally decrease during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error in the validation set will typically begin to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned; this is called early stopping. The testing subset is used after training is complete. The test subset error is plotted during the training process. If the error in the test set reaches a minimum at a significantly different iteration number than the validation set error, this may indicate a poor division of the data set. The basic training algorithm is termed the backpropagation algorithm. The simplest implementation of back-propagation learning updates the network weights and biases in the direction in which the performance function decreases most rapidly (the negative of the gradient). The weights and biases of the network are updated only after the entire training set has been applied to the network. The gradients calculated at each training example are added together to determine the change in the weights and biases. This is referred to as batch training, in contrast to incremental training in which the weights and biases of the network are updated each time an input is presented to the network. The training is accomplished using an optimization algorithm that searches for network parameters that minimize the performance function. The performance function is based on the mean square errors between actual plant output yi and network prediction yˆ ii for n sample points. This is expressed as
1 n performance index ) [ (yˆ i - yi)2] n i)1
∑
(12)
The optimization algorithm used is the Levenberg-
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Figure 8. Tuning of the NN-MPC.
Marquardt back-propagation (LMBP) algorithm.3 It was adopted for the training of the MPC as well as the adaptive FLC. A total of 42 epochs (training steps) was used to achieve a performance index of 2.55 × 10-8 within a search time of 124 s. This is a very good performance because it ensures the efficiency of the trained network for predicting process dynamics. The network was validated and tested using the data previously generated from the model. The results showed that the extract and raffinate network predictions were very close to the model-generated ones with a regression coefficient R > 0.99 for both phases. This training accuracy offers a good foundation for confidently using the neural network as a representation of the process model. The model and controller training data were generated using the following settings: rotor speed ) 120600 rpm; solvent flow rate ) 200-600 cm3/min; sampling time ) 0.5 min; total number of sampling points ) 15 000. After the controller training process was accomplished, the two controller (NN-MPC and NN-FLC) implementations proceeded. The controllers were trained on the same process data as those described earlier. To compare the performance of the two controllers, the simulation and optimization parameters were unified for the two controller algorithms. For the NN-MPC, the selected numerical values of the tuning parameters are as follows: prediction horizon (N2) ) 10; control horizon (Nu) ) 5; search
Figure 9. NN-MPC and NN-FLC response to random raffinate concentration set points and the corresponding controller moves.
parameter (R) ) 0.01; control weighting factor (Γu) ) 10-7; output weighting factor (Γy) ) 1. These tuning parameter numerical values were selected as described below. The prediction and control horizons were set at their best values of 10 and 5, respectively, after attempting different values. These values showed moderate aggressiveness and good stability of the controller response. Figure 8 shows the effect of changing the next two controller parameters, namely, the control weighting factor (Γu) and the search parameter (R). The figure indicates that the controller performance is very sensitive to the selected value of the control weighting factor Γu. The Γu value determines the contribution of the sum of the squares of the control increments on the performance index. A value of 10-3 results in an overdamped response and low controller sensitivity. Decreasing this value by a factor of 10 improved the control performance. An optimum value of 10-7 was selected for this parameter. This value gave very good tracking with fast and low oscillatory behavior. The output weighting parameter Γy was fixed at a value of unity in all experiments. The last tuning parameter is the search parameter R. This parameter is used to control the optimization speed and performance. It determines the termination of the search process. The Levenberg-Marquardt optimization algorithm uses this scale factor to minimize the performance training function along the search direction. After several trials, a value of 0.01 was selected. This value ensures reasonable performance criteria as well as optimization speed.
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properties to that for the first loop. The only difference in this case is that the final offset was present in both positive and negative directions. The controller moves of the NN-FLC showed low sensitivity to set-point excitations and did not reach the actuation targets, as was the case for the NN-MPC. Conclusion
Figure 10. NN-MPC and NN-FLC response to random extract concentration set points and the corresponding controller moves.
There were no tuning parameters to be set for the NN-FLC, and thus some limitations are imposed on the ability to influence its behavior. After training and tuning of the controllers, the process operating conditions were set to the following practical values: rotor speed ) 300 rpm; solvent flow rate ) 250 cm3/min; raffinate flow rate ) 250 cm3/min; solvent feed concentration ) 0; feed concentration ) 0.02 weight fraction. A series of random set points in the output variables were introduced in the process control loop, and the controller response was recorded. Figures 9 and 10 show the resulting raffinate and extract transients, respectively, as well as the corresponding manipulated variable moves. Figure 9 shows that the rotor speedraffinate concentration loop response for the NN-FLC attained good set-point direction tracking behavior; however, it is slow in response and resulted in a negative final steady-state offset. The NN-MPC response was very good in terms of set-point tracking, rise time, and settling speed. In terms of controller moves, the NN-MPC configuration produced smooth and nonaggressive changes in the rotor speed, whereas NN-FLC gave sudden actuator moves at the starting points of each set-point change and slowed down for the rest of the transients. This behavior was due to restricted actuator moves of the manipulated variable during the first few minutes. The solvent flow rate-extract concentration loop response (Figure 10) indicates the efficiency of the NNMPC with excellent set-point tracking properties. On the other hand, the NN-FLC behavior is similar in
The trained neural network was capable of capturing the extraction process dynamics with high prediction efficiency and thus can be used in control applications where the process exhibits high nonlinear dynamics such as the extraction process. The performance of the NN-MPC for the set-point tracking case was excellent in forcing the process output variables to their target values smoothly and within reasonable speed. The controller showed stable behavior for the whole spectrum of excitations in the output variable. The NN-FLC algorithm was capable of tracking the direction of set-point changes successfully, but it achieved that with a slow speed and with the presence of a final steady-state offset. This little deficiency of the feedback linearization technique is attributed to the fact that this algorithm is limited to systems described by eq 2 and, hence, it is applicable to a restricted class of systems. Comparing this performance to that of the NN-MPC algorithm, we recommend the use of NN-MPC in the control of the extractor because of its high set-point tracking precision, fast response, and stability under process variable excitations. The modeling errors resulting from the approximations involved in these methods are difficult to quantify beforehand. Although the two neural-network-based control algorithms showed that a stable controller behavior was obtained, it is still difficult to theoretically calculate the closed-loop stability and robustness. One reason for the deficiency of the feedback linearization algorithm is due to the first-order approximation involved in modeling of the process. This modeling error results in a drawback of the overall performance of the controller. This is not the case for the NN-MPC, where the process is approximated nonlinearly. Furthermore, integral action needs to be incorporated in the NN-FLC law to eliminate the set-point offset. This as well as testing of other neural-network-based control algorithms such as adaptive control algorithms will be investigated in future work. Appendix: Description of the Nonequilibrium Backflow Mixing Cell Model Using the nonequilibrium backflow mixing cell model, the extractor is described as a cascade of n stages with two mixing stages at the two ends of the column. Each combined mixing and calming zone is represented by a single stage in the model. The solvent is fed to the bottom of the column to stage 1 at composition yf and exits at the top of the column through the organic mixing stage with a composition of yout. The feed flows countercurrently from stage n at composition xf and exits from the aqueous mixing stage at composition xout. At each stage i, component material balance equations can be written for the solute transfer between the two phases (eqs A1 and A2). Also, solute-free material balance equations are used for calculating exiting flow
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Table 1. Model Physical Properties Correlations Used in the Rigorous Model property density diffusion coefficients interfacial tension viscosity distribution coeffecient Sauter mean drop diameter fractional holdup
equation
ref
Fc ) FAxi + FW(1 - xi) Fd ) FAyi + FT(1 - yi) Dc ) [R/(1 - )][-0.171 + 0.02[dRn(1 - )/Vc] Dd ) dR2n{1.3 × 10-8WeR1.54[Fc/(Fc - Fd)]4.18ReR0.61}-1 σi ) 33.4793 - 95.0502xi + 275.9171xi2 ln ηd ) xT ln ηT + xA ln ηA + xTxAGTA ln ηc ) A + B/T + CT + DT2 mi ) 0.86852 + 0.08681xi-0.483142 d32i ) dr(1.763 + 16.117i)We-0.907 i ) a0 + a1Na2 + a3Fia4 + a5(NFi)a6
Bibaud and Treybal30 Mjalli16 Grunberg and Nissan31 Weast et al.32 Mjalli16 Bonnet and Jeffreys33 Mjalli16
Table 2. Constants of the Model Parameter Correlations fractional holdup mass-transfer factor continuous backmixing
a0
a1
a2
a3
a4
a5
a6
0.0559 0.2 -2.859
5.07 × 10-9 3.6131 -2.463
8.2835 -2.2565 -0.8
0 0.2798 0.156
0 0.5526 2.0
0.000828 -0.2582 4.031
1.7736 0.2775 -0.1
rates (eqs A3 and A4). These equations can be written as
dxi ) [(1 + R)Ri+1xi+1 + RRi-1xi-1 dt (1 + 2R)Rixi - Qxi]/V(1 - i) (A1) dyi ) [βSi+1yi+1 + (1 + β)Si-1yi-1 dt (1 + 2β)Siyi + Qxi]/Vi (A2)
[
[
]
dyi / dt [(1 + β)(1 - yi)] (A3)
Si ) βSi+1(1 - yi+1) + (1 + β)Si(1 - yi) + Vi
Ri ) (1 + R)Ri+1(1 - xi+1) + RRi-1(1 - xi-1) +
]
dxi V(1 - i) /[(1 + R)(1 - xi)] (A4) dt
In the above equations, Qxi is the mass-transfer rate for the aqueous phase in both the calming and mixing sections of stage i. Its can be expressed as
Qxi ) (1 + f)Qm xi
ψi(N,Fi) ) a0 + a1Na2 + a3(Fi)a4 + a5(NFi)a6
(A6)
where Kxi is the overall mass-transfer coefficient at stage i. The equilibrium concentration is calculated using the distribution coefficient mi ) y/i /x/i . The value of the distribution coefficient for each stage is correlated as a function of the solute concentration in the raffinate phase. This correlation is given in Table 1. Hydrodynamics calculations are based on a correlated fractional holdup coefficient (). It is correlated as a function of the rotor speed (N) and the phase flow ratio at each stage Fi.16 This correlation is given in Table 1. It enables the prediction of the transient holdup behav-
(A7)
where ψi represents a value of the model parameters at stage i. The quasi-Newton optimization algorithm29 was used to minimize an objective function of the form
(A5)
where Qxmi is the mass-transfer rate in the mixing section. The parameter f is the mass-transfer weight factor. It is utilized to account for the mass transfer in the calming zones. The mass transfer in the mixing section is calculated using the following expression: / Qm xi ) KxiaiV(xi - xi )
ior at any stage. The drop state, expressed as stagnant, circulating, or oscillating, is incorporated in the calculation of the overall mass-transfer coefficient.15 For the Scheibel column under investigation and the range of operating conditions considered, it was noticed from the experimental runs that the stagnant and circulating drops comprise a much smaller proportion than the oscillating type out of the total population, so it was assumed that only oscillating drops were effectively present. The model parameters (backmixing coefficients and mass-transfer weight factor) and the fractional holdup coefficient are estimated using empirical correlations as a function of the operating variables over a wide range of column operation conditions. For any of these model parameters, the general form of the correlations as a function of rotor speed N and phase flow ratio Fi can be written as
min [f(x)] ) min
[x
N
(xei - xpi )2 ∑ i)1 N
× 100
]
(A8)
where f(x) is the objective function of the percentage deviation of the concentration profile and xei and xpi are the ith experimental and model predicted values of the acetone concentration, respectively. The optimization is performed by reconciling the model predictions with the measured experimental data until the termination tolerance was satisfied. The values for the calculated model parameters are given in Table 2. Furthermore, physical property calculations are performed throughout the column. The correlations used for physical properties and other model variables are listed in Table 2. The total number of model equations comprise a set of 8(n + 2) + 4 equations.16 This set of DAEs is stiff because of the variation of equation time constants, and hence the numerical solution technique must efficiently handle this equation stiffness.
Ind. Eng. Chem. Res., Vol. 44, No. 7, 2005 2133
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Received for review August 3, 2004 Revised manuscript received February 2, 2005 Accepted February 3, 2005 IE049308F
