Sampled-Data Control of a Distributed-Parameter Process Rajakkannu Mutharasan' and Donald R. Coughanowr Deparfment of Chemical Engineering, Drexel University, Philadelph$, Pennsylvania 19 104
Direct digital control algorithms for a wall temperature forced heat exchanger are derived considering inlet temperature as the primary load variable. The minimal prototype algorithms respond well to undesigned loads and parameter shifts. The minimal algorithms for set point change and load change are identically the same and respond in the same manner as optimal open-loop algorithms reported in the literature (Koppel, 1967; Seinfeld and Lapidus, 1968) when the residence time to tube-fluid in the exchanger is an integral multiple of the sampling period.
In an earlier paper (Mutharasan and Coughanowr, 1974), we presented a method of deriving direct digital control algorithms for a class of hyperbolic distributed parameter systems. One of the processes which was considered was the wall temperature-forced heat exchanger. The direct digital algorithm we reported is nonlinear and acted upon the estimated inlet temperature disturbance and drove the outlet state of the process to the set point value of the loop in one residence time of the tube-fluid. Continuous feedback and feedforward control algorithms have been reported for the wall temperature-forced heat exchanger process. Koppel(l967) treated the time optimal control p-roblem while Ray (1969) presented the variational approach and Seinfeld and Lapidus (1968) reported the open loop time-optimal control algorithm for the wall temperature-forced exchanger. Lim and Fang (1972) considered optimal feedback control which minimizes a performance index of the quadratic type. In the present paper we investigate the development of linear direct digital control algorithm for the wall temperature-forced exchanger process. We will also see that minimal prototype algorithms derived for this system closely resemble the open-loop algorithms reported by Koppel (1967) and Seinfeld and Lapidus (1968). The Process and the Control System Consider a wall temperature-forced heat exchanger control system, and assume plug flow, negligible axial conduction, negligible capacitance of metal wall, uniform physical properties, and perfect radial mixing. The wall temperature is manipulated to control the exchanger outlet temperature. Let the process be initially at steady state. The partial differential equation describing the dynamics of the heat exchanger in dimensionless form is
and some of the past values of error and manipulated variable are stored in the computer. The output value, m(0),is kept constant between sampling instants. The time required for sampling of process output, calculation of m(0),and transmitting the new value of m(0)to the final control element is assumed to be so small compared to the process residence time that we may consider this sequence of operations to be instantaneous. For the control system, the primary load variable is the inlet temperature; that is, X I ( @takes on nonzero values. The deviation of velocity of the tube-fluid from steady state is considered to be a secondary load variable. Here, all design procedures will be developed for the primary load variable, Xl(0), but the response of the algorithm to velocity loads will be tested also. By this consideration, for the purpose of the development of an algorithm, eq l simplifies to aX aX a0 a? Introducing deviation variables, X ( 7 , O ) = X(7,S) - [I exp(-Pa)] and ni(0) = m ( 0 ) - 1 equation (2) reduces to
-+ - = P[m(0)- XI
aX aX
-+ - = P[rn(0) - X ] a0
a7
a(0) = 0; m ( 0 ) = 1
X(a,O) = 1 - exp(-Pa)
In the equation above, m(0)is the dimensionless manipulated variable calculated using the digital control algorithm which is to be designed. The outlet temperature of the exchanger is sampled every sampling instant through one of the analogdigital channels. The value is compared with the set point, X, [which in the present analysis is the initial steady state value, (1 - exp(-P))] and if a difference exists the manipulated variable value is calculated and sent to the final control element through a digital-analog chmnel. The loop's parameters 378 Ind. Eng. Chern., Process Des. Dev., Vol. 15,No. 3, 1976
(3)
with the initial condition specified by ni(0) = 0
(4)
X(7,O) = 0
(5)
The set point value in the deviation variables, 2,is zero. Laplace transformation of the above equation leads to an ordinary differential equation, which is easily solved to obtain
+ exp[-ds + P)lX(O,s) with the following boundary and initial conditions.
-
(6)
where d ( 7 , s ) and f i ( s ) are Laplace transforms of d(7,0)and ni (e), respectively. Substitution of 7 = 1 in the above equation will give a relationship between outlet state and the inlet and manipulated variables. Therefore, a block diagram in the conventional sense can be drawn as shown in Figure 1. The problem in the present form lends itself to linear analysis and design. Linear Digital Algorithms A major objective of this study is to derive the controller function, D ( t ) , for a given performance criterion. Both the set point and load minimal prototype algorithms (Tou, 1959) are considered. In the present context, a set point minimal prototype algorithm is one which enables the variable, X,(0), to settle a t its new set point value in one sampling period. A load
0.10
-
0.08-
0.06-
,
8 , =1.0: NO I N T E R S A M P L E RIPPLE
8,
=2.0: NO I N T E R S A M P L E
RIPPLE
?*(el 0.040.0200-
Figure 1.
prototype response, the output is required to be ;0.35 IINTERSAMPLE RIPPLE)
-
/
Substituting the appropriate variables into the following equation
0
0.2
0.4
0.6
0.8
1.0
I
I
1.2
1.4
Time. B
Figure 2.
minimal prototype algorithm is one which enables the variable, X2(8),to return to and settle at the set point value in one sampling period after the effect of the load is sensed at I) = 1; that is, if a step change in load occurs a t 8 = 0, the load will be sensed by the algorithm a t the time 8 = k8,, if k8, = 1 and a t 8 = ( k 1)8, if k8, < 1 6 ( k l)&. In this paper, settling time will be based on the response of the system at sampling instants only, and intersample ripple will be treated separately. Consider a unit-step change in set point; then, for a minimal prototype response, the output is required to be
+
+
22d(Z)
=0
+ 2 - l + Z-’ + . . . 2 - J
(7)
where the subscript d refers to the desired response. Let the set point change be a unit step function; hence 2 X,(z) = -
2-1
The controller D ( z ) is calculated from
substitution yields
D(z) go + g1z-I
1
+ hoz-’ + h k - l z - k + hkZ-(kfl) + hk+lZ-(k+2)
(10)
where the coefficients are defined in the nomenclature section. The above minimal algorithm has integral action to ensure zero steady-state error when untested inputs occur because 1
+ ho -k hk-1 + hk + h k + l = 0
This relation follows from noting that successive values of the error are zero and successive values of the manipulated variable are constant when a zero error steady-state condition is reached. For deriving the load minimal prototype algorithm, consider a unit step change in inlet state. Let 120, = 1. For a minimal
We get a transfer function which is exactly the same as the algorithm obtained for set point minimal prototype, eq 10, for q = 0 (Le., X = 0). For the case where ll0, is not an integer (i.e., X = 0), the load minimal prototype algorithm also reduces to the set point minimal prototype algorithm. The above analysis shows that the same algorithm can respond to a set point change and a load change in a minimal prototype manner. The authors are not aware of any process for which this is true. The response of the minimal prototype algorithm to a step change in set point is shown in Figure 2; response to a step change in load is shown in Figure 3. Note the intersample ripple in the response for the case 8, = 0.35. From a physical argument, one can say that an intersample ripple which decays will exist when 1/8, (with 8, < 1)is not an integer. The maximum ripple magnitude is a function of sampling rate and the value of A. The manipulated variable, m(8),for the minimal prototype algorithm response oscillates even when the process output has settled down. The oscillations decrease with time and m(8) reaches the steady state after several switches. For the case 8, = 0.2, the manipulated variable oscillates long after the output of the system settles with no intersample ripple. In contrast to this, for a linear lumped-parameter system, an oscillation in the manipulated variable implies intersample ripple (Mosler et al., 1967). Koppel (1967) derived an open-loop time-optimal continuous control algorithm for the system under consideration. It is interesting to note that the minimal algorithm derived in this section responds in the same manner as Koppel’s (1967) algorithm, when ll8, is an integer. Since Koppel posed the problem as a continuous open-loop control problem, changes in control action can be effected at any time, 8; in contrast to this, the changes in control action for the digital control situation can occur only at sampling instants. This is essentially the cause for intersample ripple. Furthermore, Koppel’s observation that “feedback realization of m based on process output is impossible, because m undergoes an infinite switching cycle even after the system output is a t rest in the final state,” is true for the continuous feedback control. As we have seen here, realization of the time-optimal algorithm in the sampled-data control is very easily implemented. Consider the design of a minimal prototype algorithm when the control variable is constrained as follows m* < m(0)d m*
(13)
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
379
T,HE
2
l,ol ,
-
SAME FOR ALL Bs>l
THE SAME FOR ALL
e,>,
0.64
SET POINT
I
/ 0.5705.6-
0.ov
0
I
1
I
2
I
3
I
4
I
I
1
I
I
I
I
5
TIME, B
Figure 4.
In such a situation, the design involves choosing the sampling rate that does not violate the above constraint. Noting that the maximum effort and minimum effort are made during the first and the second sampling periods, and calculating the manipulated variable value using eq 10, a t 8 = 0 and 6’ = Os, one can get (for a set point change) X,/(l
- exp(-p8,) d m* X, 2 m*
(15)
where the constants are given in the nomenclature section. The response of the system with the above algorithm is shown in Figure 4.It is interesting to note that the response reaches the set point value a t 0 = 1, even though the algorithm was designed to get the output to the set point value a t the first sampling period, which is greater than one. Furthermore, the responses are identically the same for all sampling periods greater than or equal to one, even though the algorithms are numerically different. A close examination of the algorithm indicates that go simplifies to 1/(1- exp(-P)) and therefore remains independent of 0,; however, g1, ho,and kl are functions of 8,. This explains why the responses are all alike in Figure 5 . A t this point it is interesting to note that Lim and Fang (1972) obtained an optimal control which minimized the quadratic performance index for this problem, which is quite different in nature from the time-optimal control problem. It is also of interest to note Lim’s observation that major control effort is made during the first residence time. Lim and Fang observe that their algorithm requires removal of energy from exiting fluid. The responses of minimal prototype algorithm to velocity-load and variations in parameter are shown in Figures 5 and 6. For the case 8, = 0.2, the output of the system oscillates within a narrow band of the set point for a longer period of Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
SET POINT
(14)
From the above the design sampling period can be determined. In this paper, analysis and design have been confined to minimal prototype algorithms. Nonminimal algorithms could easily be designed, but no attempt is made in this investigation. Design of algorithms when sampling is done a t 8, > 1 is of academic interest, but the study of the behavior of the system a t such low sampling rates gives insight into the response of the minimal prototype algorithm to the undesigned velocity-load. Since a velocity-load changes the residence time of a fluid element, changes in velocity can be visualized as changes in sampling rate. Consider the sampling rate to be such that AB, = 1 with X < 1. By the method outlined earlier, the minimal prototype algorithm takes the following form
380
c
16
Figure 6.
time compared with the responses for Os = 0.5 and 1.0. The reason for this behavior is that the algorithm makes large control effort, which is reflected in the algorithm by constants {hi1and ( g i )that are numerically larger for Os = 0.2 than for 8, = 0.5 or 1.0. Simulations show that velocity-loads do not make the system unstable, but a large velocity-load makes the system response very oscillatory. Direct digital control algorithms for a wall temperatureforced heat exchanger are derived considering inlet temperature as the primary load variable. The minimal algorithms for set-point change and load change are identically the same and respond in the same manner as optimal open-loop algorithms reported in the literature. Furthermore, the response is time-optimal without intersample ripple when ll8, is an integer. When l/8, is (and Os < 1)not an integer, the response contains intersample ripples which decay. The algorithm responds well to undesigned loads and parameter shifts.
Nomenclature a ( @ = velocity-load, defined as 0; u(8) = 0,O < 0 u = velocity of tube fluid; equals d 6 ( t )= ~ ( a l (8)) W = wall temperature of the exchanger X = dimensionless state variable, [T - Tsi]/[Wsi - Tsi] X l ( 8 ) = inlet state of the process X 2 ( 8 ) = outlet state of the process y = distance along the exchanger
+
+
Greek Letters = pfocess parameter, defined as U O A , / ~ A , C , ~ = dimensionless distance, y / L O = dimensionless time, at/L 8, = dimensionless sampling period
6
X = a fraction defined in k8, p =
density of the fluid
+ X8,
=1
Subscripts d = desired output or state si = initial steady-state Others bar = denotes a steady-state value tilde = denotes the deviation state from the steady state Literature Cited Koppel. L. B.. lnd. Eng. Chem., Fundam., 6, 299 (1967). Lim, H. C.. Fang, R. J., AlChEJ., 18 (2).282 (1972). Mosler, H. A.. KoDDel. L. B.. Coughanowr, D. R.. lnd. €no. Chem., Process Des. Dev., 5, 297 (i966). Mosler, H. A., Koppel, L. B., Coughanowr, D. R., AlChEJ., 13, 768 (1967). Mutharasan, R., Coughanowr, D. R., lnd. Eng. Chem.. Process Des. Dev., 13, 168 119741 Ray,-W: H., Chem. Eng. Sci.. 24, 209 (1969). Seinfeld, J. H.. Lapidus, L., Chem. Eng. Sci, 23, 1461 (1968). Tou. J. T.. "Digital and Sampled-Data Control Systems," McGraw-Hill, New York, N.Y., 1959.
Receiued for reuieu, March 19,1975 Accepted March 6,1976
Axial Mixing of Grains in a Motionless Sulzer (Koch) Mixer Ruey H. Wang and L. T. Fan* Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506
Discrete deterministic and random walk models are presented which describe the axial mixing of grains in a motionless Sulzer (Koch) mixer. An experimental system was devised for determining the axial mixing of grains in this mixer. Wheat and sorghum were used. The progress of mixing was followed using a sampling technique which incorporated a radioactive tracer. The radioactive tracer, the portion of grain which was irradiated with neutrons, allows a fairly accurate determination of the concentration of the tracer particles without disturbing the mixture. The results show that the rate of mixing of sorghum was higher than that of wheat. The agreement between each of the models and the data is excellent.
Introduction The purpose of this study was to evaluate axial mixing characteristics of grains in a relatively new motionless mixer, Sulzer (Koch) mixer, by means of neutron activated tracer technique (Wolf e t al., 1974; Fan et al., 1971). The mixing of particulate solids may be broadly defined as any process that tends to eliminate existing inhomogeneity, or to reduce existing gradients. By this process, two or more solid materials are scattered in a mixer by regular and/or irregular movement of particles. The movement of particles may be caused either by motion of the mixer or motion of the particles through the mixer when it is stationary. Recently, blending of solid mixtures has been carried out by the flow of solid materials through a geometric pattern of motionless mixer components or elements mounted in a tubular barrel (Fan et al., 1971,1973). Blending of high-quality grains with low-quality grains is one of the effective methods of upgrading the latter (Kuprits, 1967). The effectiveness of blending depends on various factors: (1) physicochemical factors such as moisture content, density, shape, and surface friction; (2) characteristics of the mixing devices and their operating conditions. Continuous
mixing is the system most likely to be used for grain and feed mills in the future; it is predicted that continuous mixing will replace batch mixing probably within ten years (Christy, 1972). For a continuous mixing system, an in-line mixer, e.g., a motionless mixer, is the obvious choice. I t can be operated in a gravity flow model and can take advantage of the freeflowing property of grains. A preferred method of the radioactive tracer technique in studying the mixing and transport characteristics of particulate solids is the radioactive isotope labeling implemented by neutron activation of a portion of the material itself for use as a tracer (Wolf et al., 1974; Fan et al., 1971). This neutron activation tracer technique begins with the exposure of tracers to a neutron flux, thus creating radioisotopes from trace elements present in the material. Isotopes formed in this matter are characteristic, in numbers and types, of the compositions of the original tracer and the irradiation conditions. Assuming that the radiation source and detection system are uniform, the only variable in radioisotope production for elemental analysis is the concentration of the original tracer. The technique is possible when the material being mixed contains certain elements which can be readily activated to produce radioisotopes with suitable decay schemes and half-lives. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 3, 1976
381
