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Ind. Eng. Chem. Res. 1988,27, 204-206
Optimization of a Pressure Swing Adsorption Cycle A relatively simple and efficient optimization scheme based on the existing optimization techniques is presented for the optimization of a pressure swing adsorption (PSA) cycle. A black-box approach is used to convert the complex physical process (PSA cycle) into a simple mathematical problem. The method uses an interior penalty function. The simplicity and flexibility of the method are illustrated by three examples. Pressure swing adsorption (PSA) has become a major separation tool in chemical and petrochemical industries. A variety of sophisticated PSA cycles have been developed for various commercial applications. The performance of these cycles can be reasonably well predicted by mathematical models. Through the models, the parametric effects, i.e., the effects of the individual operating variables on the separations, are fairly understood (Yang, 1987). A typical PSA cycle involves three to five operating variables: high (adsorption)/low (desorption) pressures, cycle time, purge/feed ratio, and feed throughput. The performance of the cycle is expressed in terms of product purity and recovery. Optimization of the PSA cycle has been performed based on the parametric effects with the aid of a mathematical model (Doshi et al., 1971; Chihara and Kondo, 1986). This approach, although instructive, is time-consuming and inefficient. The purpose of this paper is to acquaint PSA designers and users with an efficient and relatively simple optimization scheme based on techniques well established in optimization. The proposed scheme converts the complex PSA process into a simple mathematical problem.
PSA Cycle and Separation The optimization scheme is illustrated by the separation of a 50/50 H2/C0 mixture using a four-step PSA cycle with zeolite type 5A as the sorbent. The sequential steps in each PSA cycle are (Kapoor and Yang, 1987) step I, H2 pressurization (2 min); step 11, high-pressure feed (2 min); step 111, cocurrent depressurization (2 min); and step IV, countercurrent blowdown (2 min). In step I, the bed is pressurized to the feed pressure by H,. In step 11, the high-pressure feed flows through the adsorber column, while the effluent is taken as H, product. Cocurrent depressurization (step 111) is used to recover H2 remained in the voids. The step IV operated countercurrent to the feed direction produces a CO product and provides a cleaned bed for the next cycle. Three separation results, product purity, product recovery, and throughput all at cyclic steady state, are used to compare the performance of this PSA process. The effluents from steps I1 and I11 are collected as H2product and that from step IV as CO product. The product purity is defined as the volume average concentration over the entire product and is expressed as vol %. The product recoveries are defined as H, recovery = H2 from steps I1 + I11 - H, used in step I (1) H, in feed (step 11) CO from step IV CO recovery = CO in feed (step 11)
(2)
The throughput is defined as the total feed volume treated per unit time per unit weight of sorbent and is expressed as L (STP)/h/kg. For a given gas mixture/adsorbent system, the process variables for this PSA cycle are feed pressure, end pressure
of step 111, end pressure of step IV, throughput, and cycle sequence and time. The end pressure of step IV (countercurrent blowdown) should be as low as possible, and for the purpose of simplification in this study, this pressure is fixed at 1 atm. Also, a fixed cycle sequence and time are assumed. Thus, the remaining three process variables, feed pressure (PF),end pressure of step I11 (PI), and throughput (F),are chosen for optimization study. Generation of Objective Function and Constraints. An equilbrium model was developed for this system and is discussed in detail elsewhere (Kapoor and Yang, 1987). It was shown that the model is capable of predicting all steps of the PSA process, and the comparison between the model predictions and the experimental results is good. This model is used directly here. Because of the complexity of PSA process, a simple black-box approach is used. A three-factor central-composite design of experiments (Beveridge and Schechter, 1970) is used to fit a set of black-box outputs in terms of the inputs by mathematical relationships. The three process variables, feed pressure, end pressure of cocurrent depressurization, and throughput, are the inputs for the black box, and product purities and recoveries for H2 and CO are the outputs from the black box. The following ranges of conditions of process variables are selected: feed pressure, PF, 200-400 psig; end pressure of step 111, PI, 30-100 psig; feed throughput, F , 300-800 L (STP)/h/kg. To simplify the mathematical computation, three new variables are defined:
v, =
PF - 300 100
(3)
PI - 70
v, = 30
(4)
- 550 v, = F235
(5)
The output, y, is expressed as y = a.
+ alVl + a2V2 + a3V3 + blV12 + b2V2, + b3V3,+ clV,V2 + c2V2V3+ c3V3Vl (6)
For three input variables, a total of 15 sets of data is needed to fit the black box equations (Beveridge and Schechter, 1970). The equilibrium model is used in this study to generate the 15 sets of data. However, experimental or pilot-plant data may also be used for this purpose. Equation 6 is used to fit four output variables (y): H, product purity, CO product purity, H2 product recovery, and CO product recovery. The values of the fitting constants are given in Table I for all four output variables. The procedure for calculating the fitting constants from the 15 sets of data is simple and can be carried out with a hand calculator (Beveridge and Schechter, 1970). The output variables calculated from eq 6 are within 1%of the values generated from the equilibrium model (Kapoor, 1987).
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Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 205 Table I. Parameters for Equation 6 output variable
a0
a1
a2
a3
H2purity, % Hzrecovery, %
95.787 96.837 94.759 57.122
3.431 2.471 3.405 2.490
0.082 -4.442 -4.246 10.130
-10.086 4.282 1.996 -25.090
CO purity, %
CO recovery, %
It should be noted that the relationships between output and input variables so developed are valid in the following ranges of input variables:
v, I -a Iv, I -a I
-a
b3
C1
CZ
c3
-4.603 -2.979 -1.763 8.216
0.055 1.385 1.018 -0.763
-0.064 3.296 1.682 -1.612
2.297 -1.073 0.135 -0.247
Table 11. Optimum H2Product Recovery at Different Hz Product Purities and the ODtimum ODeratinn Variables optimum ~~
Hz
recovery,
CY
(8)
purity, % 85.000 90.000 95.000 97.500 99.999
99.999 99.875 98.934 98.060 96.845
(9)
minimize F(x')
(10)
gi(x') I O , i = 1, 2, ..., m
(11)
subject to
and gj@ = 0, j = m+l, m+2,
... n,
n2m
(12)
where F is the objective function (if the problem is to maximize F, then change it to minimize -F), x' is a vector of design (process) variables, gi is a set of m inequality constraints, and g j is a set of n-m equality constraints. The optimization technique used here involves the use of interior penalty function (Reklaitis et al., 1983). It uses a variable metric search method to determine sequential search directions which will decrease the objective function efficiently and uses a golden section method for obtaining the minimum along a search direction. The salient features of the technique are given below. The details of the technique followed here have been given elsewhere (Afimiwala, 1973; Afimiwala and Mayne, 1974). The objective function is modified as i
n
where R is a penalty factor. The last two terms in eq 13 are penalty terms. The penalty term with inequality constraints is small when the point is away from the constraints but becomes large as the constraints are approached. And as the value of R is decreased, the value of the penalty term with equality constraints becomes much greater and hence forces the equality constraints to be satisfied to some degree. When equality and inequality constraints are satisfied P(x'*,R ) F(x'*) as R 0 and x'* is the optimum solution. In this technique, the selection of a suitable initial value of R is important for fast convergence. If R is large, the penalty function is easy to minimize. The initial value of R is chosen to be 500 in this study. The value of R is decreased in every iteration, and iterations are continued
-
-
~
HZ
(7)
I v3 I a
i
b2
0.337 -0.228 0.376 -2.747
CY
where CY = 1.215. This value of a was chosen to make the design of experiments orthogonal (Beveridge and Schechter, 1970). Thus, eq 7-9 become constraints in the optimization. Depending on the problem, a relationship between any of the four output variables and inputs can be an objective function or a constraint. (This point will become clear in the later discussion.) Optimization Technique. The optimization problem can be formulated as
m
bl -2.847 -0.815 -1.086 -0.053
%
Vl
1.288 X 5.100 X 1.0516 0.9265 1.313 X
VZ -1.215 -0.620 1.952 X -3.591 X lo-' 2.500 X
v3
0.8133 0.557 0.1495 -0.1339 1.313 X
Table 111. Maximum CO Product Purity at Different Hz Product Purities and Outimum InDut Variables HZ max CO purity, % purity, % Vl VZ v3 80.0 96.874 0.874 -0.113 X lo-' 1.215 85.0 95.764 0.771 X 0.740 X 0.788 90.0 95.309 -0.273 X -0.273 X 0.472 95.0 94.759 -0.288 X -0.280 X lo4 0.000 -0.182 -0.401 94.586 0.303 X 99.0
until the accuracy criterion is satisfied. The method suggested here thus reduces the problem of optimization of a complex PSA process to a purely mathematical problem. The simplicity and flexibility of the method are illustrated by the following examples. Maximize H2 Recovery for a Given H2 Purity. The problem here is to find conditions for the maximum Hz product recovery at a specified H2 product purity. The objective function is
+
F(H2recovery, %) = -[96.837 + 2.471V1 - 4.442Vz 4.282v3 - 0.815V: - 0.228V: - 2.979v3' + 1.385VlV2 + 3.296VzV3 - 1.073V3V11 (14)
with inequality constraints eq 7-9 and equality constraints % = 95.787 + 3.431V1 + 0.082Vz 10.086V3 - 2.847VI2 + O.337vz2 - 4.603V: + o.055v1v2 - 0.064V2V3+ 2.297v3v1 = constant (15)
Hzpurity,
Table I1 lists the values of the optimum Hz product recoveries at different H2 product purities and the corresponding optimum process conditions. The other output results can be calculated from eq 6. The maximum Hz recovery decreases with an increase in Hz purity, in agreement with our earlier experimental observations (Kapoor and Yang, 1987). Maximize CO Purity for Given H, Purity. The optimization here is for a given H2product purity to calculate the maximum CO purity in the other product stream. The objective function is F ( C 0 purity, % ) = -[94.759 + 3.405v1 - 4.246Vz + 1.996v3 - 1.O86Vl2 + o.376vz2 - 1.763v3' + l.018V,Vz + l.682V2V3+ 0.135V,Vl] (16) The equality and inequality constraints are the same as in the preceding example. The optimum CO product purity at various H, product purities and the optimum conditions are given in Table 111, showing a decrease in maximum CO purity with an increase in Hz purity.
Ind. Eng. Chem. Res. 1988, 27, 206-208
206
Table IV. Maximum C O Product Recovery at Different CO Product Purities a n d Optimum Input Variables max CO recovery, CO purity, % % VI v2 v3 -1.215 -0.524 1.215 80.0 99.999 0.171 -1.215 -0.850 85.0 99.411 90.0 99.215 0.200 x -0.043 -1.215 -0.557 -1.215 95.0 95.980 1.215 -1.086 -1.215 87.688 1.215 98.0
Maximize CO Recovery for a Given CO Purity. The objective function is F ( C 0 recovery, 70)= -[57.122 + 2.490V1 + 10.130V2 25.090v3 - 0.O526vI2 - 2.747V2 + 8.216V32 0.763V1V2 - 1.612V,V3 - 0.247V3V1] (17) with inequality constraints eq 7-9 and equality constraints CO purity, % =
[rhs expression from eq 161 = constant Table IV shows the maximum CO recovery for different values of CO purities and optimum conditions. The CO recovery decreases with an increase in CO purity. The foregoing demonstrates that for a given separation requirement, the optimization technique can be used efficiently to locate the optimum PSA cycle conditions. The technique can be expanded to optimize more than three input variables and can be used for other P S A cycles. Moreover, total cost can also be optimized by expressing
it as a function of the input variables.
Acknowledgment This work was supported by the U S . Department of Energy under Grant DE-AC21-85MC22060. Literature Cited Afimiwala, K. A. “Computer Programs for Optimization Including Applications”, M.S. Thesis, SUNY at Buffalo, Buffalo NY, 1973. Afimiwala, K. A.; Mayne, R. W. J. Eng. Ind. ASME 1974, 96(1), 1. Beveridge, G. S. G.; Schechter, R. S. Optimization: Theory and Practice; McGraw-Hill: New York, 1970; Chapter 3, p 41. Chihara, K.; Kondo, A. Paper presented at the Engineering Foundation Conference on Adsorption, Santa Barbara, CA, May 1986; to be published by Engineering Foundation, New York. Doshi, K. J.; Katira, C. H.; Stewart, H. A. AZChE Symp. Ser. 1971, 67 (1171, 90. Kapoor, A., SUNY at Buffalo, Buffalo, NY, unpublished results, 1987. Kapoor, A.; Yang, R. T. Sep. Sci. Technol. 1987, in press. Reklaitis, G. V.; Ravidran, A.; Ragsdell, K. M. Engineering Optimization: Methods and Applicacions; Wiley: New York, 1983; Chapter 6, p 216. Yang, R. T . Gas Separation by Adsorption Processes; Butterworth: Boston, 1987; Chapter 7, p 237.
A. Kapoor, R. T. Yang* Department of Chemical Engineering S t a t e University of New York a t Buffalo Buffalo, New York 14260 Received for review February 2, 1987 Revised manuscript received September 29, 1987 Accepted October 16, 1987
The Concept of “Eigenstructure”in Process Control Much of t h e work in multivariable process control has been directed at finding control structures that minimize interaction among loops and decouple the system. This paper claims that this approach is flawed. What is really important in the vast majority of chemical and petroleum processes is a structure t h a t does t h e best job in rejecting load disturbances. This inherent or intrinsic “eigenstructure” (choice of controlled and manipulated variables and their pairing) is that configuration which yields a system that is n a t u r d y self-regulating for load disturbances and self-optimizing. Eigenstructure is a unifying concept that links several previously published approaches to the process control problem. The literature in the area of multivariable control is quite extensive. Much of it has been directed at the problem of control system structure. The structure must be specified before the subsequent steps of controller tuning and hardware implementation can proceed. Therefore, the initial problem in process control is structure. Questions to be answered are as follows: (1)What variables should be controlled? (2) What variables should be manipulated? This means not only what control valves or flow rates to manipulate, but also consideration of the possibility of using flow ratios, sums or differences of flow rates, heat removal or addition rates, etc. (3) How should these controlled and manipulated variables be linked together? Should we use a diagonal (multiloop SISO) controller or a full multivariable controller? Many workers have concentrated on choosing the structure such that interaction among the control loops is minimized. The use of decouplers assumes coupling is undesirable. The much-used (and abused) Relative Gain Array (RGA) method and the Inverse Nyquist Array (INA) method are based on the assumption that control loop interaction is bad. This may be true in systems where 0888-5885/88/2627-0206$01.50/0
set-point changes are the principal disturbance. But set-point disturbances are usually much less frequent in chemical process control than load disturbances. Most industrial applications require a control system that can hold the process at desired values of performance (composition, yield, etc.) in the face of load disturbances such as variations in feed composition and throughput. In fact, as Niederlinski (1971) pointed out over a decade ago, designing control systems such that they are noninteracting can degrade the performance of the system in rejecting load disturbances. The predictive methods (such as DMC and IMC) assume that the controlled and manipulated variables have already been chosen and that a multivariable controller is desired. Thus these methods are essentially tuning procedures. The structure is specified, and the job is simply to calculate controller tuning parameters that give a reasonable compromise between performance and robustness. The purpose of this paper is to put forward the notion that each process has an intrinsically self-regulatingcontrol structure which makes the system as insensitive as possible to load disturbances and is self-optimizing. Thus a control system design philosophy is proposed that contains as its 0 1988 American Chemical Society
