0 =
initial steady-state velocity of tube-side fluid
ir = deviation in velocity from steady state x = dimensionless space variable y = distance along the heat exchanger z = 2 transform variable Greek Letters y = a constant used to specify the value of outlet state a t
the second sampling instant after the load is sensed (see eq 22) p = density of tube-side fluid v k i = random numbers in the interval [-0.5,0.5] from the set k 0 = time Superscripts tilde = refers to deviation state bar = refers to steady state
Literature Cited Koppel, L. E.,Kamman, D. T., Woodward, J. L., I d . Eng. Chem., Fundam., 9, 198 (1970). Luus, R., Jaakola, T. H. I., AEhEJ., 19, 760 (1973). Luyben, W. L., “Process Modelling. Simulation, and Control for Chemical Engineers”, p 520, McGraw-Hill, New York, N.Y.. 1973. Moore. C. F., Smith, C. L., Murill, P. W., lnst. Contr. Systems, 43, 70 (1970). Mosler. H. A,, Koppel, L. E.,Coughanowr, D. R.. AEhEJ., 13, 768 (1967). Mutharasan, R., Coughanowr. D. R.. Ind. Eng. Chem.. Process Des. Dev., 13, 168 (1974). Paraskos, J. A,, McAvoy, T. J., AlChE J., 18, 754 (1970). Tou, J. T., “Digital and SampledData Control Systems”, p 501, McGraw-Hill. New York, N.Y., 1959.
Received for review March 6, 1975 Accepted August 18, 1975 This work was performed with the asssistance of a grant from the National Research Council of Canada, Grant No. A-3515. Computations were carried out with the facilities of the University of Toronto Computer Center.
Sampled-Data Proportional Control of a Flow-Forced Tubular Reactor Rajakkannu Mutharasan’ and Donald R. Coughanowr Deparfment of Chemical Engineering, Drexel University, Philadelphia. Pennsylvania 19 104
Though continuous proportional feedback control of an isothermal tubular reactor is always stable, sampleddata proportional control can make the system output to oscillate indefinitely. Through analysis and simulations it is shown that for any feedback gain, there is a magnitude of step change in inlet concentration, which can cause the system output to oscillate indefinitely.
Introduction and Previous Literature Several published papers have appeared on continuous feedback control of distributed parameter systems. Koppel (1966) considered continuous nonlinear feedback control of tubular chemical reactors and heat exchangers. Koppel et al. (1970) reported theoretical and experimental results on two-point linear control of a flow-forced heat exchanger and extended the principle to other parametrically forced distributed parameter systems. Paraskos et al. (1970) studied experimental feedforward computer control of a flowforced heat exchanger and reported that their algorithm was far superior to conventional Ziegler-Nichols settings. Implementation of their feedforward algorithm involved solution of finite difference equations, which is rather elaborate for an industrial application. Seinfeld et al. (1970) compared continuous proportional feedback, feedforward, and optimal control of a flow-forced isothermal tubular reactor. Their paper contains useful results on offset and stability of the system under proportional control. They reported that the system is stable, irrespective of the value of the proportional gain. Oscillations in outlet concentration increased as the proportional gain is increased; however, there is an upper limit on the gain because of the physical requirement that the velocity should be greater than zero. Several continuous optimal control algorithms, for open loop and closed loop, configurations have been reported for the flow-forced reactor-heat exchanger system (Vermeychuk et al., 1973; Seinfeld et al., 1968, 1970). Review of the existing literature indicates that very few authors have worked on feedback sampled-data control of
distributed-parameter systems (Palas, 1970; Hassan et al., 1970). This topic is of great importance in an industrial environment because of an exponential increase in the use of computers as a control element. Several authors have developed methods of design of direct digital control algorithms for lumped-parameter systems (Mosler et al., 1967; Moore et al., 1970; Cox et al., 1966; Dahlin, 1968), but similar work for distributed systems is lacking. This investigation helps fill part of the gap that exists in the literature. In this paper we present the results of a study on the nature of feedback sampled-data proportional control of a flow-forced tubular reactor. Ultimate loop gain of a sampled-data control of a first-order system is lower than that of a continuous control system (Mosler et al., 1967). One conclusion we may draw is that sampling operations cause the lumped parameter system to be less stable, but a stable and responsive proportional controller can still be designed. In the present distributed parameter system, we show that sampling the output with manipulation proportional to error yields a system which sustains oscillations indefinitely (at least theoretically), thus leading to the conclusion that proportional sampled-data control is unsatisfactory for the flow-forced tubular reactor. T h e Process The process under consideration is a flow-forced tubular reactor in which an nth-order isothermal reaction, A B, is taking place. Under the assumptions of (1) plug flow of fluid, (2) uniform physical properties, and (3) negligible axial dispersion, the dynamic behavior of the tubular reac-
-
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
141
.
r
L
I
L
X,l&
INLLT
Figure 1. The control system.
tor is described by the following dimensionless hyperbolic partial differential equation
aX aX a8 all with initial and boundary conditions
-+ (1+ m(8))- = -OXn 2
3
4
5
6
TlME.8
Figure 2. Discrete proportional control response. Parameters: = 1;X l ( 8 ) = 1 0 . 3 ~ ( 8 )8,; = 1.
+
X(0,O) = Xl(0)
Ob)
X I ( 0 ) = X(0,O) = 1
(14
m ( 0 )= 0
(Id)
In the above equation, m (8) represents the manipulated variable, which is equal to the normalized deviation in the velocity of the fluid through the reactor relative to the initial steady-state velocity. In a control situation the load variables for the above system are: (1)changes in inlet concentration which changes the boundary condition, eq Ib, and (2) changes in wall temperature, which causes changes in the value of the kinetic constant with the result that the parameter of the system, P, takes on new values. The control system is illustrated in Figure 1. The output of the process, X z ( 8 ) ,is sampled every sampling period, 8,. The manipulated variable is computed using the proportional algorithm and sent to the valve. The value of the manipulated variable, m(8) is held constant between sampling instants.
Analysis Let the order of reaction be one for simplicity; that is n = 1 in eq 1. Consider the isothermal reactor, initially a t steady state, to be subjected to a step change of magnitude A in the inlet concentration. That is
aX
aX + (1 + m(8))= -PX ao X i ( 8 )= 1 + A u ( 8 ) (8 2 0 ) a8
(0 < 'I < 1)
X(o,O) = exp(-Po) m(8) = K ( X , - X(l,i8,)), i8,
(2a) (2b)
= 0,1,2,. .)
(2c)
where X , = exp(-P) = X(1,O).Let the sampling period, 8, = 1. Consider K in eq 2c to have a value such that the output is forced to the set point value by time 8 = 28,. This value of gain, denoted by K * , can be calculated as follows. Error at 8 = 1 is exp(-P) - (1 A ) exp(-P) = - A exp(-P). Therefore m(8) = -K*A exp(-P) during 1 < 8 < 2 and Xz(2) = (1 A ) exp(-@,). One can compute the resident time of the exit fluid element to be 8, = 1 K * A exp(-P). Therefore X a ( 2 ) = (1 A ) exp{-P[l K*A exp(-P)]{. Since it is required that X 2 ( 2 ) = X , = exp(-P), we have
+
+
+
+
(1
where the subscript eq 4 is
dX + m m )= -px, do X1,(8) = 1 + A
m
+ +
142
Ind. Eng. Chern., Process Des. Dev., Vol. 15. No. 1, 1976
(44
for K < K * . When K > K * , simulations reveal that the response oscillated indefinitely and no unstable response was observed. The oscillations have the same period as the oscillations for the case K = K * ; however, their magnitudes were different. The oscillations have an initial transient period and then settle down to oscillate between two limits, denoted by X u and X L , which are reached a t the sampling instants. Consider the time, i8,, when the output is at the upper limit, XU. The manipulated variable for the next sampling period is K ( X , - X u ) ; this value is such that the output reaches X L at the following sampling instant, (i 1)8,. For the period (i l)8, 6 8 < (i 2)8,, the manipulated variable is K ( X , - x ~ this ) ; value is such that the output reaches X u at (i 2)8,, and the process repeats itself. The residence time of the exit fluid element a t (i 2)8, is 1/[1 K ( X , - X L ) ] .Therefore
+
+
+
+
+
+
]
[
X u = (1 A ) exp -
(6)
l+K(X,-XL) Also, the residence time of the exit fluid a t 8 = (i + l)&, when the output is X L , is 1 + K [ X , - X I , ] - K [ X ,- X u ] 1 K [ X ,- X L ] Therefore 1+ K ( X " - X L ) X L = (1+ A ) exp (7) 1 K(X,-XL) To determine the amplitude of oscillation, one needs to solve simultaneously eq 6 and 7 for X u and X L . Consider an example with A = 0.3, P = 1, 8, = 1.0, K* = 2.37727, X,
+
+
If K = K * , the error at 8 = 2 is zero, and m(8)is zero during the time interval 2 < 8 < 3. The residence time of the exit fluid element at the next sampling instant, 8 = 3, is one,
(4)
denotes steady state. The solution of
+
< 8 < (i + l)8, (i
and one can see that the process output will oscillate indefinitely between exp(-P) and (1 A ) exp(-P). A simulation response shown in Figure 2 verifies this fact. It is of interest to study the system behavior for feedback gain K > K * and K < K*. When K is less than K * , the system settles down to a constant value with an offset as shown in Figure 2. This offset can be predicted using the steady-state solution of eq 2. Under steady-state conditions, eq 2 becomes
]
I
m
0.51
1
0.44-
1
0.28
t 0
0.35 I
1
1
I
2
3
I
TIME,
4
e
$ 4
I 30
I
I
5 y28
29
Figure 3. Response when K > K*. Parameters: p = 1; X l ( 8 ) = 1.0 + 0.3U(e); or = 1.0; K* = 2.37727.
0.51
1
TIME,B
Figure 5. Sustained oscillations when 8, < 1. Parameters: 0 = 0.6; K* = 3.96212; X l ( 8 ) = 1.0 + 0 . 3 ~ ( 8 ) .
c
0.441
0.40
I
0.35I
0
I
1
1
I
1
1
I
2
3
4
5
6
7
TO 0 . 2 5 7 5 2
0
1 I
,,/\An-0.3v I
I
2 TIME,
e
3
I
4
5
Figure 6. Response of the process with discrete proportional integral controller to step changes in load: 0 = 1;8, = 1;X l ( 8 ) = 1.0 + A u ( 8 ) .Controller parameters: go = 3.5; gl = -0.02.
where go and gl are controller constants and e is the error, given by the difference between the set point and the measured value of the process output. Figure 6 shows typical responses of the process with the above control algorithm for both positive and negative step changes in inlet condition. The algorithm in eq 9 requires that, only when the process output has reached the set point value (Le., e[i0,] = e [ ( i - 1)0,] = 0), the manipulated value will reach a steady value (i.e., rn[iOs]= m [ ( i- I)&]. As can be seen from Figure 6, the outlet concentration settles down to the set point value. The response to positive step change is more oscillatory because of the nonlinearity of the process. Hence, there is an optimal choice of the parameters go and gl for different load conditions. No attempt was made in this investigation to optimally tune the parameters since the objective here is to point out the effect of inclusion of integral action.
Conclusions
The above equation is derived by the same procedure used to derive eq 3. Examples of simulation responses are given in Figures 4 and 5. Discrete Proportional Integral Controller The sustained oscillation exhibited by the proportional control can be eliminated by the inclusion of integral action. The discrete proportional integral controller can be expressed in the following form
From the analysis and simulations presented here one can see that for a given K and Or, there exists a magnitude of step change in load, A , which causes the system to oscillate indefinitely. Such a response is undesirable in a control situation, since the change in load is an unknown quantity in a practical situation. Therefore, the conclusion drawn is that a discrete proportional control algorithm is unsuitable for the flow-forced reactor system. However, inclusion of integral action in the controller would yield a stable and responsive control system.
Acknowledgment Our thanks go to Louis Haas, who assisted in preparation of the drawings. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
143
7 = dimensionless distance along the system
Nomenclature
A = magnitude of step change in inlet state A , = area of cross-section of the reactor C A = concentration of component A C A ~= initial steady-state inlet concentration A e = error, difference between set point and output of process go, g1 = discrete proportional-integral controller parameters kA = rate constant for the reaction A B K = controller gain K* = feedback gain setting a t which oscillations in the output occurs L = length of the system mi = manipulated variable value between the i t h and (i 1)st sampling instants m(0) = is the manipulated variable defined as C/o n = order of the reaction A B t = time u ( 0 ) = unit-step function, u(e) = 1 , O 2 0; u(O) = 0,O < 0 u = velocity through the system ui = velocity through the system during the time between i t h and (i 1)st sampling instants X = dimensionless state variable, equal to C A / C A ~ X L = lower limit of the output X a t the sampling instant Xu = upper limit of the output X a t the sampling instant X l ( 0 ) = inlet state of the process X*(O) = outlet state of the process X, = set point value of the loop z = distance along the reactor/exchanger
-
+
-
+
0 = dimensionless time 8, = residence time of the exit element O,i = the computed residence time a t time Os = sampling period
io,
Subscript m
= refers to ultimate steady-state value
Others - = refers to initial steady state L i t e r a t u r e Cited COX, J. B.. Hellums, L. J., Williams, T. J., Banks, R. S., Kirk, G. R., Jr., /SA J.,
13,65 (1966). Dahlin, E. B.. Instrum. ControlSyst., 41(6),77 (1968). Hassan, M. A.. Solberg, K. O., Automatica, 6, 409 (1970). Koppei. L. B.. lnd. Eng. Cbem., fundam., 5, 403 (1966). Koppel. L. B., Kamman, D. T., Woodward, J. L., Ind. Eng. Cbem., fundam., 9,
198 (1970). Moore, C. F.. Smith, C. L., Murill, P. W , Instrum. Control Syst., 43(1),70
(1970). Mosler, H. A., Koppel, L. B., Coughanowr, D. R., Ind. fng. Cbem., Process Des. Dev., 5, 297 (1966). Mosier, H. A.. Koppel, L. B., Coughanowr, D. R., AlCbEJ., 13(4),768 (1967). Mutharasan, R.. PhD. Thesis, Drexel University, 1973. Palas. R. F., Ph.D. Thesis, University of Minnesota, 1970. Paraskos, J. A,. McAvoy, T. J., AlChEJ., 16(5), 754 (1970). .Se&.!&l. J H.. Lapidus, L., Cbem. Eng. Sci., 23, 1461 (1968). Seinfeld, J. H.. Gavalas, G. R., Hwang, M., lnd. Eng. Chem., Fundam., 9, 651
(1970). Vermeychuk, J. G..Lapidus, L., AlChEJ., 19(1),123 (1973).
Greek Letters fl = process parameter, defined as LkAc).i"-l/fi
Receiued for reuiew March 17,1975 Accepted July 30, 1975
Coal Liquefaction in Coiled Tube Reactors R. E. Wood' and W. H. Wiser Department of Mining, Metallurgicaland Fuels Engineering, University of Utah, Salt Lake City, Utah 84 112
Dry, powdered coal impregnated or mixed with 5 % ZnCI, can be converted to liquid and gaseous products in small diameter coiled tubes. Yields to 70% (50% liquid, 20% primarily CH4 gas) can be obtained at 5OO0C and 1800 psi hydrogen pressures in 3/16-in. i.d. stainless steel tubes 60 to 120 ft in length. The liquid portion is a complex mixture, principally aromatic, which could serve as a synthetic crude petroleum for refinery feed stock. Procedures for catalyst recovery and or recycle are indicated.
Coal Liquefaction in Coiled T u b e Reactors Since coal represents the great majority of the fossil fuel reserves of the United States it is expected to carry a significant portion of the burden of synthetic natural gas and synthetic petroleum requirements beginning before the 1980's (Hottle, 1971). Many problems relating to mining, preparation, and conversion to liquids and gases remain to be solved. Still, the needs are such that authorization has been given for construction and operation of several large pilot plants for production of synthetic natural gas from coal and lignite. Three of these plants, those of Consolidation Coal Co. (COz-acceptor process), Bituminous Coal Research Corp. (Bi-Gas process), and Institute of Gas Technology (Hi-Gas process), have been supported in large measure by the Office of Coal Research of the Department of the Interior. A fourth process is that of the U S . Bureau of 144
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
Mines (Synthane process). The only coal liquefaction process currently funded for large pilot plant testing is the multi-stage coal pyrolysis process of FMC Corp. (COED process), This process also has had considerable Office of Coal Research support. Selection of candidate processes for large scale testing of coal hydrogenation to produce synthetic liquid fuels is yet to be made. One of the coal hydrogenation projects supported by the Office of Coal Research is that a t the University of Utah. This process has been through several bench scale phases, all of which have been directed toward a short contact time and high ratio of hydrogen to coal under moderate reaction conditions (Haddadin, 1968; Qader, 1969). The current reactor configuration (block diagram shown in Figure 1) is a series of coiled ?&-in. i.d. X ?&in, 0.d. no. 316 stainless steel tubes. The length of this tube system can
