e
=
fluid temperature
7
= time domain of process, [O,T]
6
= =
D
eigeiifuiiction of operator equation a simply connected open region in ?n-dimensional Euclidean space
SUPERSCRIPTS I = dimensional quantity * = adjoint SUBSCRIPTS b = boundary = initial time or condition 0 S = steady state
Denn, RI. RI., Gray, R. D., Ferron, J. R., IND.ENG.CHEM. FUNDAhI. 5 , 59 (1966). Katz, L., J . Electron. Contr. 16, 189 (1964). Koppel, L. B., Shih, Y. P., IKD.ENG.CHEJI. FUNDAM. 7, 414 ( 1968). Koppel, L. B., Shih, Y. P., Coughanowr, D. R., IND.ENG. CHEWFUNDAhf. 7,286 (1968). Tricomi, F. G., “Integral Equations,” pp. 3, 116, Interscience, New York, 1957. Vainberg, AI. M.,“Variational Methods for the Study of Son-Linear Operators,” p. 77, Holden-Day, San Francisco, 1964. JT7eigand, W. A., D’Souza, A. F., J . Basic Eng. 91D,161 ( 1969).
RECEIVED for review August 15, 1969
literature Cited
ACCEPTEDJuly 6, 1970
Axelband, E. I., Proceedings of Sixth Annual Joint Automatic Control Conference, Troy, N. Y., p. 374, 1965. Denn, 11. lI.,ISD EKG.CHCM.FENDAM. 7, 410 (1968). Denn, A l . AI., Int. J . Contr. 4, 167 (1966).
Work supported by Purdue University and the Xational Science Fciundation (GK4994). Computer time supplied by Purdue.
Control of Plug-Flow Tubular Reactors by Variation of Flow Rate John H. Seinfeld,‘ George R. Gavalas, and Myungkyu Hwang Department of Chemical Engineering, Calijornia Institute of Technology, Pasadena, Calif. 91 109
The control of isothermal and adiabatic plug-flow tubular reactors b y variation of flow rate was studied. Proportional feedback, feedforward, and optimal control responses were compared for the regulation of reactor conversion in the presence of inlet disturbances. The optimal control, consisting of a singular solution in each case, produces a considerably improved response over both feedforward and proportional feedback control.
T h e control of tubular reactors is a problem of considerable importance in chemical processing. Based on t h e mode of operation-e.g. , isothermal, adiabatic, etc.-control can be exercised in a variety of ways-e.g., flow rate variation, inlet condition variation, heating or cooling rate variation, etc. From the standpoint of control, a convenient method of classification is by the form of the mathematical model used t o describe the reactor. Reactor models can generally be placed in two categories: hyperbolic systems, in which axial and radial diffusion effects are neglected (plug-flow) ; and parabolic systems, in which diffusion effects are included. In the present study we consider both isothermal and adiabatic plug-flow reactors for which the control objective is t o maintain the outlet composition a t a desired value in t h e presence of inlet concentration aiid temperature fluctuations. In the isothermal case, control can be exercised by variation of the flow rate and the temperature. I n the adiabatic case, control can be exercised by variation of t h e flow rate and the inlet temperature, assuming that the inlet concentration is not available for adjustment. Ogunye aiid R a y (1970) determined the optimal temperature control policy in both t h e isothermal To whom correspondence should be sent.
and adiabatic plug-flow cases in the presence of catalyst decay. We consider the other alternative for plug-flow reactor control-namely, control of the flow rate. This mode of control is of practical importance, since flow rate is an easily manipulated variable. hIanipulation of the flow rate of an isothermal plug-flow reactor t o control the exit composition was considered by Koppel (1966a,b). Since Koppel based his feedback proportional law on a transformed variable rather than directly on the outlet concentration, his results are not generally applicable. I n fact, as a result of using a transformed variable, the control no longer has a linear relationship to the error and is not proportional a s stated. The objectives of this work are the following. \Ye wish to solve directly the nonlinear problem for the dynamic response of t h e isothermal reactor n ith proportional control of the flow rate. Then, we wish t o determine the optimal flow rate control policy for both the isothermal and adiabatic case5 that minimizes the integral square error of the outlet concentration for a given inlet disturbance. Finally, the optimal respoiise is compared t o the proportional feedback and simple feedforward responses t o determine the degree of improvement achieved by optimal control. Ind. Eng. Chern. Fundarn., Vol. 9, No.
4, 1970 651
I
I
I
1
I
I
I
I
2
3
4
5
6
7
8
Figure 1 . Dynamic response of outlet concentration for A = 0.3, /3 = 2, and n = 1 Proportional control
1 Figure 2. and n = 2
,
I
2
I
5
6 7 9 Dynamic response of outlet concentration for A = 0.3, I
3
4
p
= 2,
Proportional and optimal control
Isothermal Case
Proportional Control. The dynamics of an isothermal plugflow, tubular reactor with a n nth-order irreversible reaction and proportional control of flow rate is described in dimensionless terms by
where the concentration sensing takes place a t the reactor outlet, 7 = 1, and the desired outlet coiicentration, z (e, l ) , is zd. We assume that for e < 0 the reactor is in a steady state for which z (1) = xd, so that the control is shut off. The inlet 652
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
concentration is assumed t o undergo a step change of magnitude A a t 0 = 0 from its steady-state value of 1,
z(e,o)
=
1
+A
e >o
(2)
The object of this section is to determine the exact dynamical response of the reactor for fixed values of p, n, A, and zd and different values of the gain, K . The numerical technique based on the method of characteristics for obtaining the solution of Equation 1 is described by Hwang (1968). The exit response z(0,l) is shown in Figure 1 for p = 2 , d = 0.3, n = 1, and K = 1, 5 , 10, 20. Similar responses are shown in Figure 2 for p = 2 , A = 0.3, n = 2, and K = 1, 5 , 10, and
20. The magnitude of the offset, defined as the difference between the asymptotic outlet coiiceiitratioii x( rn ,1)and d,can be determined from (1 - K[x(l,m)
- z d ] ]In
;',I
[
-- OPTIMAL
+
___
p
\
=
0
( n = 1)
(3)
CONTROL
-FEEDFORWARD CONTROL
&\ \
\
\
\ \
\
\
\
where CY1
=
-a? =
1
+ p(1 + A ) - K(zd - A
-K[zd(l
- 1)
(5)
+ p + PA) - (1 + A ) ]
(6)
I n each case, as K is increased the offset is decreased. With larger K , t h e system undergoes more rapid oscillations before reachiiig the asymptotic value. Gain K caiiiiot be chosen arbitrarily large, because the total velocity must be greater t h a n zero. The maximum allowable value of K can be determined from this requirement as K,,,
e@ ( n = 1) A
(7)
= -
L
0.31
0
I
I
I
.o
2 .o
I
3.c
8
since the maximum deviation z(e,l) - x d occurs when 0 = 1-for example, for n = 2! p = 2 , and A = 0.3, K,,, = 36. If a pure time delay of magnitude T exists in the coiitrol loop, z(0,l) in Equation 1 is replaced by z(8 - T , 1). The numerical technique used can be extended to include this case; however, these results are not, reported here. Feedforward Control. Xii alternative t o feedback proportional coiitrol is simple feedforward control, in which as soon as the step change in inlet concentration is sensed, t h e flow rate is changed t o t,he steady-state value corresponding t'o the new inlet concentration which n.ill produce t'he same outlet coiiceiit,rat,ion.The response of r(0,l) in this case is shown in Figure 3 for n = 2 . B y comparison t o Figure 2, we see that the speed of response has been improved considerably over proportional coiit,rol, mainly because the residence time lag in the reactor has been aroided. Optimal Control. It is of interest t o determine t h e optimal open-loop flow rate policy and respoiise and compare t o t h e closed-loop proportional and t h e simple feedforward responses. Let us rewritmeEquation 1 as (9)
X e formulate the optimal control problem as follows: I t is desired t o determine @) over the given time interval (O,Of) subject t o u* > v(0) > v*, the maximum and minimum allowable flow rates, such that the integral square error
p
=
Lo'
[z(e,i) -
PI*
de
(10)
is minimized. B y application of t h e necessary conditions for optimality for distributed parameter systems (Kate, 1964; Koppel et al., 1968; Seiiifeld and Lapidus, 1968a), the optimal policy is found t o be
Figure 3. Comparison of outlet concentration responses with feedforward and optimal control Isothermal reactor
rvhere the adjoint variable p ( 0 , q ) is governed by
The two-point boundary value problem represented by Equations 1, 2, 11, 12, and 13 cannot be solved aiialytically. I n cases when the optimal control is given by a bang-bang law, the switching times can be determined most easily by the method of direct search on the performance index (Seinfeld and Lapidus, 1968a). One complication can arise in optimal bang-bang controlt h a t is, if G(0) = 0 on a finite time interval, a singular arc results and v(O) may be undefined. Previous work (Seinfeld and Lapidus, 1968b) has shown t h a t the direct search method is particularly effective for determining optimal singular controls, especially when the value of the control on the singular arc can be determined. The direct search is simply a systematic search over a number -11of preselected control values until the value of P can no longer be decreased. If the existence of a singular solution can be ruled out a priori, then JI = 2 with the two choices as u* and v*. I n the present case, however, the possibility of a singular solution cannot' be ruled out, since the two-point boundary value problem of Equations 1, 2 , 11, 12, and 13 cannot' be solved analytically. I n fact, hyperbolic opt'imal control problems of this type have been shown t o involve terminal singular arcs (Koppel, 1967; Seinfeld and Lapidus, 1968a). The terminal singular control v(0) in these cases corresponds t o the simple Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970 653
-S U B O P T I M A L
- - _OPTIMAL
Isothermal
I .(
1 -
POLICk POLICY
V(81
since the singular control value can be used directly in the search. The outlet response z(8,l) corresponding t o the optimal flow rate policy is shown in Figures 2 and 3 for comparison to the two previous modes of control. The value of the performanee index, P, in the simple feedforward case with v(8) = 0.895, 8 > 0, is 2.187 X whereas the value of P for optimal control is 6.309 X 10-j. This provides a quantitat'ive measure of the iinproveiiient gained by optimal cont'rol over simple feedfor\vard control, each of which is decidedly superior to proportional feedback control. Adiabatic Case
Feedforward and Optimal Control. The time-dependent behavior of a n adiabatic plug-flow reactor with an nt h-order irreversible reaction is described in dimensionless t e r m by
0 .I I
Figure 4.
3
2
8
Velocity policies in suboptimal and optimal cases
I
I
I
-- O P T I M A L CONTROL -FEEDFORWARD CONTROL
For e < 0 the reactor is in a steady state for which z(1) = z d . The inlet concentration is assumed t o undergo a step change of magnitude -1 a t 8 = 0 as ill Equation 2. The inlet teniperature is assumed t o undergo a step change of magnitude B a t e = 0,
T(e,o) = 1 \
\
\
I
I I
\ '.
I
1 .o
0
8
2.0
3.0
Figure 5. Comparison of outlet concentration responses with feedforward and optimal control Adiabatic case
feedforward value obtained by setting z(8,l) = xd. In the present example, v(8) = 0.895 for n = 2. The following computations were performed for n = 2, p = 2, A = 0.3, v* = 1.0, v* = 0.1, and 20 time increments. 1. *I[ 2. JI
= =
2 v 5 v
= =
0.1,1.0 0.1, 0.3, 0.6, 0.895, 1.0
Case 1 was carried out without regard t o the existence of a singular arc. Case 2 included several control values, one of which was the feedforward flow rate value of 0.895. Figure 4 presents the results from case5 1 and 2. The minimum value of P was achieved for the policy labeled optimal, indicating the existence of a terniinal singular arc for 8 > 0.45. The value of the direct search in handling singular solutions is evident, 654 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
e >o
(16)
The following values of the parameters were chosen: 4 = 3, 38, y = 40, w = 0.35, n = 2, A = 0.3, and B = -0.03. With these values, z d = 0.02962. Since it was shown in the isothermal case that proportional control compares poorly with even simple feedforward control, only feedforward and optimal control were examined in the adiabatic case. Feedforward control consists of set,ting v equal t o the new steady-state value corresponding to xd as soon as the inlet dist'urbaiices are sensed. The value of ~(8)corresponding to the parameters used is 0.5556. The response t'o feedforward control is shown in Figure 5. The value of P in this case is 5,372 X The optimal v(8) policy was determined by the direct search on the performance index (Figure 4). *Igain,there is a terminal singular solution for 8 > 0.3 with v(8) equal to the new steadystate value. The response t'o t'he optimal flow rate policy is shown in Figure 5, and the value of P in this case is 3.775 X 10-2. The advaiit,age in using optimal rather t,han siniple feedforward can be seen by comparing the respoiises in Figure 5 as )vel1 as the values of P obtained. This improvement is not as pronounced as in t,he previously examined isot,herinal case.
+IL =
\
I
0.4-
+B
Summary
X direct comparison of feedback, feedforward, aiid optimal flow rate control has been presented for isothermal aiid adiabatic plug-flo~reactors with a single reaction. Applications of this work would be important in the control of liquid and gas phase react,ioiis carried out i n flow reactors-e.g., nitration of aromatic compounds and pyrolysis of lower paraffins. I n both isot,hermal and adiabat,ic operation, optinial control produced a considerably better response than simple feedforward control, and both modes were far superior to feedback
cont'rol. The optinial flow rate policy in each case had a terminal singular arc corresponding t o the new steady value of
49 Nomenclature
d B
dimensionless inlet concentration change dimensionless inlet temperature change G(0) = sivit'ching function K = proportional gain 31 = number of control values in direct search n = reaction order P = performance index p ( 0 , n ) = adjoint variable T(0,n) = dimensionless temperat,ure 40) = dimensionless velocity x(6,~= ) dimensionles concentratioii = =
GREIX LETTLRS Q ~ , C Y Z = constants p = dimensionless reaction group = dimensionless activation energy (: . ) = Dirac delta function ?I = dimeiisioiiless spatial variable 0 = dimensionless time 4 = dimensionless frequency factor 4 = dimensionless constant W = dimensionless heat generat,ioii constant,
SUPERSCRIPTS d = desired * = maximum SUBSCRIPT * = minimum literature Cited
Hwang, ill., 31.S. report, Depart'ment of Chemical Engineering, California 1nst.itute of Technology, June 1968. Katz, S.,J . Elect. Conlrol 16, 189 (1964). Koppel, L. B., IXD.EXG.CHEM.FUKDAJI. 5 , 403 (1966a). kroppel, L. B., ISD. EXG.CHEM. F U S D A h f . 5, 413 (1966b). Eioppel, L. 13., ISD.ESG. CHEM.FUXDAJf. 6, 299 (1967). Koppel, L. B., Shih, Y. P., Coughanow, D . R.,ISD. ESG. CHEM.FUXDAM. 7,286 (1968). Ogunye, A. F., Ray, W, H., A.1.Ch.E. J . , in press, 1970. Seinfeld, J. H., Lapidus, L., Chem. Eng. Sci. 23 (12), 1461 (1968a). Seinfeld, J. H., Lapidus, L., Chem. Eng. Sci. 23 (12), 1486 (196813).
RI:CEIVED for review February 28, 1969 ACCEPTED July 21, 1970 Work supported in part by Kational Science Foundation Grant GK-3342.
EXPERIMENTAL TECHNIQUES
Modified Ney-Armistead Cell for Gas Diffusion Measurements Arvo Lannus' and Elihu D. Grossmann Department of Chemical Engineering, Drexel Cniversity, Philadelphia, Pa. 191 04
A two-bulb gas diffusion cell i s described which i s suitable for measurements over a wide range of temperature and pressure, and i s completely and easily accessible for cleaning and inspection. The new cell equation was tested experimentally using the carbon dioxide-nitrogen system.
N e y and iirniistead's gas diffusion apparatus ( ~ e yand Arniistead, 1947) is capable of providing more accurate diffu5ion data than any of the other devices reported in the literature. It consists of two bulbs separated by a narrow capillary fitted in the center with a stopcock or valve by which the bulbs may be isolated or connected. While the cell is usable over a wider range of temperature and pressure than the older Loschmidt (1870) apparatus, it is often difficult to avoid a distorted diffusion path because of the stopcock design. The distortion can be excessive if the stopcock has been designed for hermetic sealing over a mide range of temperature. A recent two-bulb apparatus (Van Heijiiiiigeii et al., 1968) has Pre-eiit addre-$ Department of Chemical Engineering, The Coopei Uniori, New York, X. E'.10003. To Thorn correspondence should be sent.
eliminated the stopcock altoget,her, but is usable only a t very low pressures. Our modification places the valve a t the entrance to the capillary t'uhe, thus involving no distortion of the diffusion path. The design a l l o w easy access to both chambers and the tube, and maint'aiiis simple cylindrical geometry for absolute measurenients. Diffusion Cell
The diffusion cell (Figure 1) consists of a top chamber, A , of volume TIA, bottom chamber, B , of volume V B ,and a center block, C, through which has been bored a capillary of length L and area of cross section A t . The upper end of the capillary may be closed by valve E. l'alve D controls the inlet tube, F. Sections d and B are modified 6-inch-0.d. =\S1150-pound welding neck flanges of 304 stainless steel. The faces are grooved to fit the machined lip on block C. Lead gaskets provide the best seals for both pressure arid vacuum service. The capillary sealing valve, E , is a noiirotating stem high pressure Ind. Eng. Chem. Fundom., Vol. 9, No. 4, 1970
655
