Ind. Eng. Chem. Res. 1993,32, 1667-1675
1667
Startup of a Distillation Column Using Nonlinear Analytical Model Predictive Control Saibal Ganguly and Deoki N. Saraf Department of Chemical Engineering, Indian Institute of Technology, Kanpur-208016, India
This paper discusses the application of nonlinear analytical model predictive control to distillation column startup between the time when the trays are hydraulically sealed and the time steady-state operation is reached. A novel startup method is proposed whereby the liquid feed is initially used as reflux to seal the trays by establishing the plate holdups. The control algorithm utilizes a nonlinear analytical process model to generate an optimal control vector trajectory, which is implemented by cascade controllers. The analytical model, based on material and energy balances, additionally utilizes the secondary process measurements resulting in improved controller performance. As the steady state is approached, a neighboring optimal control signal is superimposed on the output from the supervisory level controller to eliminate fast disturbances entering the column. In the hierarchical control structure, the adaptive online optimizer resides in the PC/AT which is networked with PC/XTs for neighboring optimal and cascade controls. A PID type controller with periodic update of set points and a manual startup have been compared with the present algorithm. Experimental results showed a significant reduction in time as well as cost in the case of the proposed scheme.
Introduction Startup of chemical processes like distillation, which involve complex heat- and mass-transfer operations, is a challenging control problem. An equivalent problem is tracking a large change in set point or an elimination of a large disturbance. These disturbances are often caused by tripping/failure of equipment such as feed or reflux pumps, boilers, etc. The control objective for such startup operations originates from certain optimal regulatory tasks like minimum off-specification products, minimum time, minimum utility consumption, etc. Distillation column startup characteristics, using manual operation, have been extensively studied in the literature by Cameron et al. (1986) and Ruiz et al. (1988). It has been established through simulation studies and experimental measurements that the startup operation comprises complex transient responses in hydraulic and thermodynamic variables resulting in a highly nonlinear behavior of the output variables of the column (i.e., product compositions). A step-by-step conventional startup procedure is given in Appendix A. It has been widely accepted by process engineers that the startup operation consists of three phases (Ruiz et al., 1988): (i) The discontinuous phase. In this phase, plates weep and heating is in progress (steps 0-4 in Appendix A). During this phase, pumps and flows are triggered on and off. Controllers on secondary variables such as levels, flows, and feed temperature are turned on. This phase provides the time for initial heating and does not provide any scope of controlling the primary variables. Hence, it is not discussed in this paper. (ii) The semicontinuous phase. This is the most important and slow phase, when the trays are hydraulically sealed and the column gets shifted from the total reflux conditions to the required reflux rate (steps 5 and 6 in Appendix A). Several disturbances tend to destabilize the column operation, e.g., fluctuations in reboiler steam pressure, feed temperature and feed composition, reflux flow rate, and pulses from the piston pumps. This is a highly sensitive phase in which the reflux composition
* To whom correspondence should be addressed.
builds up slowly to the desired specifications and tends to get disturbed very quickly. Also, this highly nonlinear transition needs to be effected optimally and conventional controllers are usually less than satisfactory. A nonlinear controller based on a more rigorous physical model is, therefore, necessary for this phase. (iii) The continuous phase. Here, the column reaches the neighborhood of the desired steady state (i.e., step 7 in Appendix A). This phase can be satisfactorily taken care of by means of linear or nonlinear model predictive controllers. The present work deals with the development of a hierarchical nonlinear analytical model predictive controller (NAMPC) and its practical implementation on a pilot scale distillation column. Experimental results showed that it is an appealing proposition for handling the semicontinuous and continuousphases during column startup. Distillation Column Example. The pilot plant facility for testing online optimization includes a 17-stage sieve tray distillation column, 22.8 cm in diameter and 10 m in height, with a vertical thermosiphon type reboiler and a water-cooled 1-2 shell and tube total condenser. The column is heated by steam from an oil-fired steam boiler. Typically, it processes 150kg/h of ethanol-water feed with 20% by weight EtOH and produces 81%I by weight EtOH in the top product and 6 7% by weight EtOH in the bottom with the feed entering at the eleventh stage from the top and the reflux ratio around 3.0. The average residence times in the accumulator and reboiler are of the order of 1.5 and 2.0 min, respectively. The various open loop time constants for the column were reported earlier (Ganguly and Saraf, 1992). The top few stages are not sensitive enough to changes in the manipulated variable (reflux flow), and hence the sixth stage temperature was used as the control variable. The reboiler steam pressure was kept constant. The secondary variables must also be controlled for steady-state operation of the column and can be accomplished using single loop controllers or microcomputers. Figure 1shows a schematic of the distribution column with data acquisition and control structure. The computer networking and the communicationsbetween computers
0 1993 American Chemical Society OSS8-5S85/93/2632-~667~Q4.oo/o
1668 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
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and with the process is shown in Figure 2, and the details are included in Appendix B.
The Model Design of optimal control strategies for the transient semicontinuous phase of distillation column startup calls for the development of a nonlinear physical model with the capabilityof predicting future behavior. A wide variety of transport phenomena based physical models are available in literature, (Holland and Liapis, 1983;Joseph et al., 1988; Luyben, 1990). A distillation model is said to be analytical when it is able to predict, from fist principles, the primary variable interactions and the effect of secondary variable disturbances on the process. For the present problem, the primary variables are the top and the bottom temperatures and the secondary variables include the reflux and the bottom accumulator levels, the
feed and reflux temperatures, the feed flow rate and composition, the column pressure, and the product flow rates. The disturbances in the secondary variables automatically result in a feedforward effect on the final control action while using controllers based on such analytical models. The full order analytical models, also referred to as rigorous models, are computation intensive and not robust enough to accept noisy raw data. Lumped or shortcut models, on the other hand, are an example of excessive simplification since they fail to bring out the effect of secondary variables. The semirigorous formulation retains the basic structure of the full order rigorous model, but the number of equations to be solved is significantlyreduced, which makes ita use possible in real time applications. In semirigorous modeling, any staged process can be considered to consist of two types of units-single tray and multitray, where the single tray units are the simple equilibrium stages and the multitray units are a bunch of
Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 1669 SCREU TERMINAL PANEL
110 CARD
P C / A T 80386
110 CARL DT2805
DTZ805
PILOT
DISTILLATION
SCALE
SCREU TERMINAL PANEL
COLUMN
Figure 2. Schematic of networking and communication structure.
single stages grouped in one unit. For a steady-state simulation, this was orginally proposed by Ohmura and Kasahara (1978) and was subsequently modified by Ganguly et al. (1985). For the present example, the tray on which control is to be effected, the feed tray, the the terminal trays are treated as single trays while lumping the rest in multitray units. The steady-state equations for this model are given in Appendix C. The dynamic version of the same model can be obtained by assuming first-order dynamics for each unit as follows. dxi/dt = l / r i (pixsi- x i )
(1)
where zi denotes the composition of the liquid stream from the ith unit, either singletray or multitray; xsi is the steadystate solution for the model based on the current values of measured and control vectorsincludingthe disturbances; and is an estimated time constant. The steady-state solution, xsi, can be obtained by solving the model equations in Appendix C. pi is the external efficiency factor. Both Ti and pi are estimated adaptively from past transient experimental data. Nonlinear Estimation of Model Parameters. Since the process characteristics and disturbances change with time, it is necessary to continuously update the model parameters, using the most recent measurements. Given aset of averaged or filtered observations within a transient at T(tp),T(tp-At), T(tp-2At), ...,T(tp-tAt),andprocess model equations, the model parameters are to be determined such that some measure of the model error is minimized. This is achieved by using a standard nonlinear
programming technique. The objective function for dynamic optimization is formulated as: NPOINTS NUNIT J = rnin
where pi and ~i are the efficiency and time constant of ith unit in the semirigorous model. For binary distillation a t constant pressure, temperature is uniquely related to the composition a t every stage. Eq 1,when written in terms of temperature, constitutes the following constraint equations: dTi/dt = A(piTimma- Ti); i
2, ..., NUNIT
(3)
'i
along with the variable bounds and column operating constraints. The first stage is a total condenser, and the model equations of the condenser and accumulator drum are integrated in parallel with the semirigorous model. A nonlinear programming algorithm like SOCOLL with SQP (Powell, 1978; Gill et al., 1989; Biegler, 1984, 1991) or Gauss-Newton-Marquardt (Cuthbert, 1987; Constantinides, 1988) is used for estimating the parameters. In the presence of noisy data, a noise feedback markedly improves the match between the experimental data and the predictions (Eaton and Rawlings, 1990). The noise feedback is implemented by computing a disturbance di a t time t k as the difference between the measured temperature Ti at time tk and the predicted temperature
1670 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 Plant Scheduling
S e t points
Process Variables
control layers comprising (a) neighboring optimal control and (b) cascade control. (i) Generation of Nominal Trajectory. The general problem at the supervisory online optimization level can be stated as follows. Given an objective function and available discrete measurement values, estimate the nominal trajectory u*(t) such that the objective function is optimized subject to operating constraints. Mathematically,
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Figure 3. Multilayer decomposition of the control tasks.
Mx,(t), u,(t), P, t ) = 0
pi based on measurements up to tk-I,
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Ti(tk-1).
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ulow5 ui 5 Uhigh Hierarchical Control and Communication Structure The hierarchical scheme proposed for regulatory control (Gangulyand Saraf, 1992)has been extended to distillation column startup. The multilayer decomposition of control tasks is shown in Figure 3. The supervisory online optimizer receives the process output data at a slow sampling rate (at 30-120-s intervals) and estimates parameters for the semirigorousplant model using a nonlinear optimization scheme. The identified model is then used to generate a nominal control trajectory which is communicated to the regulatory level. The regulatory level samples the plant output a t a faster rate (at 5-15-s intervals) and uses a linearized process model along with the nominal control trajectory to generate a neighboring optimal control action. This control action is implemented through a cascade controller which works at an even faster rate (at 1-24 intervals) to minimize auxilliary effects (such as boiler output pressure fluctuations, piston pump pulses, valve nonidealities, etc.). The stationary disturbances ( d l in Figure 3) get eliminated at the regulatory level, and only the nonstationary ones (da in Figure 3) reach the online optimizer because of filtering and slow sampling. At the supervisory level, the continuous time model parameters take most of the nonstationary disturbances into account. Any unmodeled disturbance present is taken care of by the noise feedback. Nonlinear Analytical Model Predictive Control (NAMPC). The nonlinear model predictive control (NMPC) problem has been studied for processes like reactors and crystallizers by Eaton and Rawlings (1990) and Patwardhan et al. (1990). Later, Patwardhan and Edgar (1991) studied the application of NMPC to packed distillation columns. Moore and Corripio (1990) used shortcut models for online optimization of distillation columns in series. Another class of control mechanisms has been proposed by Riggs (1990),who used steady-state distillation models with the generic model control law of Lee and Sullivan (1988) to generate the control action. For practical implementation of supervisory online optimization and control, the following are required: (i) A procedure for generation of nominal trajectory using a model predictive control algorithm. (ii) Implementation of the nominal control vector through the regulatory
where ri is the residual function obtained by orthogonal collocationof the dynamicstate equations, x(t)is the state vector, u(t) is the manipulated control vector and p is the vector of model parameters. The above nonlinear programming problem is solved using SOCOLL with SQP. (iia) Method of Neighboring Optimal Solution. The optimality of the nominal trajectory occasionally gets disturbed because of small process perturbations, controller offsets, or minor changes in set point. However, since all these changes are occurring fast, it is seldom possible to revert back to the supervisory level to recompute a new optimal trajectory. An alternate approach is the computation of other optimal trajectories in the neighborhood of the nominal trajectory for small variations. For this purpose, a locally linearized dynamic model has been used in conjunction with a quadratic cost function. The neighboring optimal solution is then approximated as the s u m of the nominal plus the linear optimal solution as follows: u(t) = u*(t) + Au*(t) (7) where u*(t) is the nominal control vector and Au*(t) is the linear optimal solution. The latter is obtained from a standard LQ formulation except that in place of state and control variables, one uses deviations of these from the nominal values. The optimal perturbation vector is given by Au*(t) = 4%)Ax(t) (8) where C(t) is the time varying (mxn)feedback gain matrix, which is calculated using the solution of a Riccati equation. Further details about its derivation can be found in Ganguly and Saraf (1992). For practical implementation, the discrete version of the model equation (eq 3) is used. Equation 3 can be written in discrete form as ~ ( k + i=) c b ~ ( k+) rT,(k)
(9) where T ( k ) is the sixth stage temperature at the kth sampling instant and T,(k)is the steady-state solution of the nonlinear model based on the current values of measured ( T ( k ) )and control vectors (u(k)). 4 and r are the coefficientsof transformation. T,(k) is linearizedusing
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where T,* is a function of u*,the nominal trajectory. The values of 4, r, Tee*,and dT,$au are calculated a t the supervisory level and communicated to PC/XT 1 a t a slow sampling rate (30-120 8). It is important to note that in the hierarchical scheme, the Au*(t)of continuous domain (eq 8) gets replaced by ATee*(k) in the discrete form where AT,*(k) represents the difference between T,(k) and the last available Tee*from the supervisory level. The discrete Riccati equation is solved to obtain AT,*(k) which is used to calculate Au*(k)from eq 10. The discrete form of eq 7 provides the set point for the cascade controller residing in PC/XT 2, which acts at a fast sampling rate of 1-2 s. (iib) PI Type Cascade Controllers. PI type cascade controllers are required to eliminate boiler output pressure fluctuations, piston pump pulses, valve nonidealities, etc. The cascade controllers work at a fast sampling rate of 1-2 s/sample. Since the measured signal may show large fluctuations and spikes, (e.g., the piston pump pulses and disturbances picked up by the flow transmitter stationed after the piston pump) a filter is often required on the measured signal. Figure 4a shows the effect of a firstorder filter (constant = 0.8) on the raw data from the abovementioned flow transmitter. This filtered signal is then used for generating the cascade control action. Figures 4b and 5 present the cascade controller performances of reflux flow and steam pressure, respectively.
Results and Discussion When the conventional startup procedure under total reflux constraint was followed, the pilot scale distillation column was brought to the desired steady state using three different control schemes, namely, NAMPC, PI control, and manual control. Subsequently, results obtained with a modified startup procedure under hierarchical control are presented.
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Figure 4. (a) Filtering of the reflux flow signal. (b) Performance of the reflux flow cascade controller.
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Figure 6. Distillation column startup under manual control: (a) sixth stage temperatureversus time, (b)manipulatedvariableversus time.
Startup under Total Reflux Constraint. The total reflux constraint puts an upper limit on the output of the controller. The reflux flow rate should not exceed the amount of condensate coming into the reflux accumulator. (a) Startup Using Manual Control. The conventional procedure for startup, as listed in Appendix A, was followed. After the initial heating period, the reflux flow rate was manually adjusted to bring the sixth stage temperature to the desired set point. Any disturbance observed during the transition period was also eliminated using the same manual control. Since actions are taken only when perceptible changes show up in the plate temperature, the startup operation consumed about 2-2.5 h on the average. The resulting startup plot of the column is given in Figure 6. It showed fluctuations because of poor disturbance rejection during the entire startup period. ( b ) Startup Using PI Control. A single-loop PI controller was tuned to control the sixth stage temperature in the operating range of the column. With a large set point change (e.g., 97.5-84 "C), it failed to achieve any meaningful control and most often led to saturation or unstable operation. Therefore, the set point was gradually
1672 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993 .
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Figure 8. Distillation column startup under NAMPC: (a) sixth stage temperature versus time, (b)manipulated variable versus time.
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changed in steps to reach the final desired state. The resultant startup plot is shown in Figure 7. Even though the overall startup time is reduced to about 85 min, the profile is not monotonically decreasing, which is what it ideally should be. This can be expected because a PI controller tuned at one operating condition need not stay tuned at a distant state and because the magnitude of the disturbances during startup is much larger than the capacity of such a controller. Also, operational problems like weeping, slugging, or channeling may occur during the startup phase throwing the controller into unstable/ oscillatory actions. (c) Startup Using NAMPC. As the NAMPC controller is known to perform better in the presence of large process model mismatch or large disturbances (Ganguly and Saraf, 19921,this controller was used for the startup of the pilot scale distillation column. Data from five samples, at intervals of 30 s, were used for the estimation of the model parameters. The control horizon spanned three sampling instants at intervals for 60 s each. Only the control vector calculated for the first instant was implemented. The schematic diagram for this control structure is shown in Figure 2 except that in this experiment, neighboring optional control action was not activated. The control action from the supervisory level online optimizer (on the PC/AT 80386) was directly imposed on the cascade controller. After the initial discontinuous phase heating, the online optimizer was put into operation. The desired set point of 84 "C was provided from the very beginning. Figure 8a shows the sixth stage temperature as a function of time during startup. As seen in this figure, NAMPC was able to take the process smoothly to ita desired set point. In order to test the disturbance rejection capability of the NAMPC controller during the transition stage, a load disturbance was provided by means of a feed flow rate change from 150 to 120 Llh a t 58 min. As seen in the figure, the disturbance was also eliminated successfully.
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Figure 9. Comparison of profit indices for column startup under manual control and NAMPC.
A first-order filter was required between the NAMPC and the cascade controller. The filter constant was experimentally found to be 0.6. A plot of the control action, which is the reflux flow rate, is provided in Figure 8b. The startup took about 65 min to reach the final steady state. A comparison between Figures 6,7,and 8a shows that there is a general improvement as a controller is used in place of manual startup. Also, there is a marked improvement in the startup performance when a nonlinear model predictive controller is used instead of PI. The impact of the startup performance on the profit was evaluated using a normalized profit function (Joseph et al., 1988) of the type profit index = [PPT+ PBFB - P$Q] (11) whereFT, FB,andFQare the flow rates of distillate, bottom product, and steam in kg/h and PT,PB,and PQare the cost per kg, respectively. The product costa PT and PB are assumed to be first-order functions of respective product compositions. Figure 9 shows a comparison of profitlh, in dimensionless units, for startup under manual control and using NAMPC. The manual startup results in a highly
Ind. Eng. Chem. Res., Vol. 32, No. 8,1993 1673 oscillatory profit index whereas NAMPC provides a continuous and fast increasing index reaching the maximum in a relatively short time. The profit for startup under PI control occupied an intermediate position but has not been included in the figure to avoid loss of clarity. Operational Problems for NAMPC Controllers with Total Reflux Constraint. It has been demonstrated that the startup performance of the column depends on the type of controller being used, and of all the controllers tried, NAMPC was found to be the best. However, the total reflux constraint of conventional startup strategies often leads to additional disturbances. The condensatefrom the condenser often enters the reflux drum in slugs and because of this constraint, the reflux flow controller output gets reduced which, in turn, causes oscillations in the column performance. Also, during the semicontinuous phase of column startup, reduced reflux flow rate disturbs the liquid holdup on trays resulting in increased weeping and channeling of the vapors. This tends to increase the time needed to reach steady-state operation of the column. The almost flat temperature profile observed in the initial stages (9-36 min) in Figure 8a supports the proposition. While the use of NAMPC results in the shortest startup time with the highest profit, it has some inherent difficulties. Because of the large amount of CPU time required for computations, short-term disturbances cannot be attended to by such a controller using the available hardware. This leads to fluctuations in the column temperature (of the order of A 1 "C) particularly around the set point. An Improved Startup Procedure. In view of the operational difficulties observed with the conventional startup scheme under totalreflux constraint, an improved startup procedure is being proposed here. The difference from the earlier scheme is in the semicontinuous phase. The new startup procedure has the following operation sequence which replaces the earlier steps: Step 4. The reflux accumulatordrum is filled with feed solution, and the online optimizer is started when the condensate starts flowing into the reflux drum. Step 5. The plate liquid holdups and the vapor flows are established from the very beginning, and vapor leakage through the downcomer is sealed. Weeping also stops as the vapor boilup in the reboiler increases. Step 6. The online optimizer with the hierarchical control structure automatically tracks the set point to the desired final value, and disturbances are eliminated. The above scheme was implemented online using the three-level hierarchical control system shown in Figure 3. Amarked improvement in the startup time was observed. Figure 10a shows a plot of the sixth stage temperature with time. To eliminate short-term disturbances particularly near the final steady state, a faster acting neighboring optimal controller, resident in PC/XT No. 1, was included in the control scheme. This controller's action was implemented only when the sixth stage temperature was within A2 OC of the set point. A comparison between Figures 8a and 10a clearly brings out the superiority of the proposed startup procedure using hierarchical NAMPC. Since the total reflux constraint no longer exists in the new scheme, the disturbances and delay caused by it are eliminated. The sixth stage temperature monotonically decreases to the set point in about 35 min. The neighboring optimal control significantly reduces the temperature fluctuations in the vicinity of the setpoint. A glance at Figure 10b shows that the modified startup procedure calls for a large reflux flow rate in the beginning
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which decreases significantly as the steady state is approached. This is quite different from the former scheme where the reflux flow rate is limited by the total reflux constraint in the earlier stages but must increase subsequently in order to bring the temperature to steady state (Figure 8b). The alcohol concentration of the reflux increases as the final steady state is approached. The proposed scheme which calls for less reflux in the latter stages is, therefore, more profitable. Figure 11shows this result.
Conclusion Startup of a distillation column is a highly nonlinear process with time-varying characteristics, and the smooth monotonic transition to the final steady state gets disturbed easily. Manual startup, which is still the existing practice in many industrialunits, shows oscillatory product composition, and therefore, it is wasteful in terms of profit,
1674 Ind. Eng. Chem. Res., Vol. 32, No. 8, 1993
utilities, and time. Performance of simple controllers like PI has been found to be superior to the manual control, but it is still not satisfactory. Nonlinear analytical model predictive control (NAMPC) showed good promise. The total reflux constraint of the conventional startup procedure was found to cause operational hindrances, when used with any of the controllers. A modified startup strategy was evolved using hierarchical NAMPC in conjunction with a fast acting neighboring optimal control. Marked improvement in profit and reduction in time were observed when this new scheme was tested on the pilot plant facility. If a faster machine is available for recursive online optimization, it should be possible to do away with the neighboring optimal control layer. Moreover, some of the simplifying assumptions made in the semirigorous model may also be removed. However, generalization of the present algorithm to nonbinary systems should be made with caution. While the model is equally valid for multicomponent systems, temperature is no longer uniquely related to composition and an appropriate sensor must be found for the measurement of the state vector. This remark is valid for all model predictive controllers and calls for extensive future research.
Nomenclature C = total number of components d; = values of disturbances Fi = total feed flow rate into the ith tray, kmol/h hR = molar enthalpy of feed on ith stage, kJ/mol hi = molar enthalpy of liquid mixture on ith tray, kJ/kmol Hi = molar enthalpy of vapor mixture on ith tray, kJ/mol K = equilibrium constant L;= total liquid flow (sum of componentwise flows, 1) M ;= number of trays inside the multitray model
Greek Letters r) = Murphree vapor efficiency 7 = time constant 4, r = coefficients of discrete transformation
Appendix A A Stepby-step Conventional Startup Procedure for a Distillation Column Step 0: t = 0; column is empty. A certain liquid level is maintained in the reboiler, either feed or water, using the bottom level controller. Step 1: t = tl; reboiler heating is started with steam at a constant pressure using a single-loop controller. Vapor starts moving up, gets internally condensed, and weeps from the plates through the holes instead of the downcomer. Step 2: t = tz; as plate temperatures rise, vapor reaches the condenser and the reflux accumulator starts collecting condensate. Step 3: t = t3; feed is started into the heated column at a fixed rate and temperature using SISO feed flow and feed temperature controllers. Step 4: t = t4; total reflux is introduced into the column, so as to maintain a constant level in the reflux drum. Step 5: t = t 5 ; as the vapor boilup rate increases, the upcoming vapor seals the liquid weeping through the holes. The liquid holdup increases on plates, which causes downcomer overflow. This seals vapor leakage through downcomers. Step 6: t = t 6 ; column operation is slowly changed from total reflux to the operating reflux, and distillate is taken out. This transition has to be effected slowly and carefully, otherwise steady state is disturbed. Step 7: t = t7; column is allowed to run without changing operating conditions until steady state is reached. Here:
Ni,No = prediction horizon for input and output, respectively NT = total number of stages in distribution column NUNIT = total number of units in the column NPOINTS = total of points in time P = total pressure, mmHg Qi = heat input into the ith unit, kJ/h S = stripping factor S, = effective stripping factor S ; k j = stripping factors for jth component on kth tray inside the ith multitray unit S;V, SiL = fractional sidestream draws from ith unit for vapor and liquid TE= experimentally measured temperatures TS= model predictions for temperatures t p = present time u(t) = process input at time t Vi = total vapor flow (sum of componentwise flows, u ) x = mole fraction of component y = mole fraction of component y ( t ) = process output at time t Subscripts i = unit number j = component number k = tray number inside ith multitray unit Superscripts L = liquid state V = vapor state
Appendix B: Networking and Communication Structure Establishment of an efficient and fail-safe communication network is essential for the successful application of hierarchical online optimization and control. The present structure consists of three types of interfaces, namely, I/O cards, direct serial RS232C ports, and master/ slave configuration through communication controllers. All primary loop variables and transmitter signals for data acquisition only are connected through I/O cards for analog to digital conversion, amplification, and recording. The primary signals are processed by the software-based cascade controllers, and the resultant signals are converted to analog and imposed on the pneumatic control valves through the I/P converters. The PC/XT No. 2 (in Figure 2) constitutes the cascade control layer. The PC/AT 80386 with the peripherals acts as a supervisory level online optimizer and generates the nominal control vector trajectories. This nominal trajectory is passed on to PC/ XT No. 1 through the RS232C interface for calculating neighboring optimal control action at quick succession,to be communicated to the cascade level for effecting the requisite control action. The bidirectional data communication between the supervisory level PC/AT and the regulatory level PC/XT Nos. 1and 2 is through the serial RS232C interface of the computers. The computers tend to become slow for control purposes when used with multitasking softwares. Hence a digital triggering facility has been provided through I/O cards between the PC/AT
Ind. Eng. Chem.Res., Vol. 32,No. 8,1993 1675 and the PC/XTs to trigger on and off the desired transmission. The connection between the single-loopcontrollers and PC/XT No. 1 is through a communication controller. It supports up to 16 single-loop controllers and serves as an interface to the computer, thereby enabling the computer to use higher level language compatibility during communication, different protocol interfacing (e.g., RS422 in SISO controllers to the standard RS232C in the host), and multiple controller handling capacity through a single serial port of the host. The overall networking and communication structure is schematically shown in Figure 2.
Appendix C: The Steady-State Semirigorous Model of a Distillation Column Component Material Balance Equations (for single and multitray units) M i j = li-lj - (1+ S,L)l,, - (1+ s;luij + Ui+lj +
t:i + f i j v j = 1, ...,c (c.1) Enthalpy Balance Equations (for single and multitray units) E , = z [ l i - l j h i - l-i (1+ S:)lijhiJ - (1+ s;luipij + I
+ fij"hiJ + fij"HiJ1 + Qi
((2.2) Componentwise Equilibrium Relation (for single tray units) Vi+$i+lj
Equilibrium Relation (for multitray units)
8i.i =
where
(SUM)= (1+ S,L)l,j - Li-lj + (1+ S ? ) ~ i j Seij= (Si,,j(l+ SiNj) + 0.25)1'2 - 0.5
and
(C.4)
The Murphree vapor efficiency,q i j , has been used for each unit.
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