Ind. Eng. Chem. Process Des. Dev. 1980, 19, 537-546
used for other than alqueous systems. It is expected that the equations developed in this study will be useful for systems exhibiting a wider range of viscosities than those of water at various temperatures and where surface tensions are high. For low surface tension liquids the hydrodynamic behavior is expected to be different, and therefore an investigation is under way using liquids of similar viscosities but exhibiting a range of surface tensions to assess this particular influence on mass transfer in packed columns. Nomenclature a = area per unit volume, L-' D = diffusion coefficient, L 2 T 1 d = packing dimension, L g = gravitational constant, L T 2 kL = liquid phase mass transfer coefficient, L T ' L = liquid flow rate, ll/T'L-2 L , = molar liquid flow rate, mol T'L? N = normality of solution, mol L-3 V = volume of solution, L3 Greek Symbols cy = slope u = surface tension, M T 2 r = peripheral liquid jlow rate, MT'L-' p = viscosity, M L - ' T 1 p = specific gravity, 0 = contact angle, deg Dimensionless a n d Related Groups
537
G a = Galileo number, p2gd3/p2 Ka = Kapitsa number, pu3/p4g R e = Reynolds number, 4 L / a p or Re' = modified Reynolds number, L / p , L-' S c = Schmidt number, p / p D Literature Cited Bemer, C. G., Zuiderweg, F. J., Chem. Eng. Sci., 33, 1637 (1978). Copp, D., Ponter, A. B., Waerme Stoffuebertrag., 5, 229 (1972). Davidson, J. F., Trans. Inst. Chem. Eng., 1, 131 (1959). Fukushima, S.,Kusaka, K., J. Chem. f n g . Jpn., 10, 491 (1977). Higbie, R., Trans. Am. Inst. Chem. Eng., 31, 365 (1935). Hikita, H., Kataoka, T., Nakanishi, K., Chem. Eng. Jpn., 24, 2 (1960). Knoedler, E. L., Bonilla, C. F., Chem. Eng. Prog., 50, 125 (1954). Koch, H. A., Stutzman, L. F., Blum, H. A,, Hutchings, L. E., Chem. Eng. Prog., 45, 677 (1949). Mangers, R. J., Ponter, A. B., Chem. Eng. J., 10, 39 (1980). Mika, V., Collect. Czech. Chem. Commun., 32, 2933 (1967). Mohunta, D., Vaidyanathan, A., Laddha, G., Indlan Chem. Eng., 11, 73 (1969). Munakata,T., Watanabe, K., Miayashtta, K., J. Chem. Eng. Jpn., 8, 440 (1975). Norman, W. S., Sammak, F. Y. Y., Trans. Inst. Chem. Eng., 41, 109 (1963). Onda, K., Takeucki, H., Okumoto, Y., J. Chem. Eng. Jpn., 1, 56 (1966). Ponter, A. B., Boyes, A. P., J. Chem. Eng., Jpn., 5(1), 80 (1972). Ponter, A. B., Davles. G. A., Ross, T. K., Thofnley, P. G., J. Heat Mass Transfer, 10, 349 (1967). Rixon, F. F., Trans. Inst. Chem. Eng., 28, 119 (1948). Sherwood, T. K., Holloway, F. A. L., Trans. Am. Inst. Chem. Eng., 38, 39 119401 - -, Shuiman, H. L., Ullrich, C. F., Prouix, A. Z., Zimmerman, J. C., J . Am. Inst. Chem. fng., 253 (1955). Stephens, E. J., Morris, G. A., Chem. Eng. Prog., 47, 232 (1951). Taylor, R. F., Roberts, F., Chem. f n g . Sci., 6, 49 (1956). VanKrevelen, D. W., Hoftijzer, P. J., Red. Trav. Chim. Pays-Bas, 66, 49 (1947). VanKreveien, D. W., Hoftijzer, P. J., Chem. Eng. Prog., 44, 529 (1948). \
Received f o r review April 27, 1979 Accepted April 28, 1980
Computer Control of a Distributed Parameter System James E. Fuhrman, Rajakkannu Mutharasan, and Donald R. Coughanowr Department of Chemical Engineering, Drexel University, Philadelphia, Pennsylvania 19 104
Control algorithms for a class of distributed-parameter systems, which were presented earlier (Mutharasan and Coughanowr, 1974), are shown experimentally to have superior transient response characteristics when compared with th'e traditional proportional-integralalgorithm. This paper discusses the modifications to the earlier algorithms to include dynamics of the control valve and the measurement device.
Background a n d Olbjectives The design of direct digital control algorithms presents interesting and challenging problems. Several authors have presented direct digital control algorithms for lumpedparameter systems (Cox et al., 1966; Mosler et al., 1967; Dahlin, 1968; Moore et al., 1970; Luyben, 1973). Several authors have studied continuous control of distributedparameter systems (Lim and Fang 1972; Seinfeld et al., 1970; Vermeychuk and Lapidus 1973; Koppel et al. 1970). Koppel et al. (1970) investigated, theoretically and experimentally, the continuous control of a flow-forced heat exchanger using proportional action at an intermediate point within the exchanger and integral action a t the exit. The results showed a significant improvement over control based on exit response only. Lee et al. (1973) studied the control of countercurrent processes by two-point linear controllers, which also provide considerable improvement in control. Mutharasan and Coughanowr (1976) studied proportional control of a flovv-forced isothermal tubular reactor. Through analysis and simulations it was shown that for
any feedback gain, there is a magnitude of step change in inlet concentration which can cause the system to oscillate indefinitely. It was also shown that the inclusion of integral action does yield a stable system. However, the response of the system to a positive step change in inlet concentration is much more oscillatory than the response to a negative step change because of the nonlinearity of the process. Paraskos and McAvoy (1970) studied feedforward sampled-data control of a flow-forced heat exchanger. They derived their feedforward algorithm using a finite difference formulation of the dynamic equation. Feedback proportional-integral control action was added to eliminate steady-state errors arising from modeling errors. Their algorithm was tested on an experimental heat exchanger using analog computer control. Proportional-integral control using Ziegler-Nichols settings, which was experimentally tested, produced a sluggish response. The feedforward-feedback algorithm proposed by Paraskos and McAvoy proved to be far superior to the conventional control.
0196-4305/80/1119-0537$01.00/00 1980 American Chemical Society
538
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
Mutharasan and Coughanowr (1974) presented a simpler method of design of digital control algorithms for a class of distributed-parameter systems described by the following parital differential equation
The variable X denotes the state variable, and the variables 0 and 7 denote the time and position variables. The variables, p1(@and p 2 ( 8 ) ,are load or manipulated variables. Algorithms were developed for an isothermal tubular reactor, a flow-forced heat exchanger, a wall temperature-forced heat exchanger, and a by-pass heat exchanger using the method of characteristics. Simulations of the control system for these processes were presented for set point changes and load changes and showed that the performance of the algorithm is comparable to the twopoint continuous control algorithms reported in the literature. When large modeling errors are present the authors added a proportional control term to improve the transient response. The objective of this work is to experimentally verify the algorithms proposed by Mutharasan and Coughanowr (1974) for a double-pipe heat exchanger. To accomplish this objective, a double-pipe heat exchanger was designed, instrumented, and interfaced to a PDP-11/10 digital computer. Water flowing through the inner pipe is heated by condensing steam in the annular space. The outlet water temperature is the controlled variable, the water velocity is the manipulated variable, and the temperatures of the steam and inlet water are the load variables. The experimental verification includes: (1)evaluation of steady-state runs to determine process parameters, (2) development of real-time computer software to implement the control algorithms, (3) determination of open-loop dynamics of the process, (4) evaluation of closed-loop system response to set point and load changes, and ( 5 ) comparison of experimental and theoretical process responses. Process Model and the Control Algorithm As previously shown (Mutharasan and Coughanowr, 1974), the dynamics of a shell and tube heat exchanger can be represented by
ax ax + (1+ m(O))-= a0 atl
-P(1
+ m(0))bX
(2)
(i-k)9vLp e=e-e,
(i-k-I@
0
I
segments of constant slope and the particular characteristic path shown in Figure 1is for the fluid element which leaves the exchanger a t the ith sampling instant. The slope of the line segment between consecutive sampling instants is inversely proportional to the velocity computed a t the beginning of the sampling interval. The residence time is the time spent by a fluid element in the exchanger and is denoted by 8,. In terms of dimensionless variables, the residence time for a fluid element leaving the exchanger at time io, may be calculated from the following equation in which the integration of the velocity with respect to time must equal the length of the exchanger (unity for dimensionless variables). it,
lo,-o>l +
do = 1
(7)
Since the residence time is equivalent to an integral number (k)of sample periods plus a fraction (A) of a sample period, Or = k%, + AB,; hence the integral in eq 7 can be written in discrete form as k j=l
(1 + mi-j)o,
+ (1 + mi-k-l)Xd,
=1
(8)
where the index k is such that
c
where X denotes dimensionless temperature and the parameter /3 is given by U,,A,/A,C,u~.The associated initial and boundary conditions are given by X(0,O) = 1; m(0) = 0 (3)
X h O ) = exp(-Pa) (4) x(o,e) = x,(o) (5) The parameter b relates the variation of heat transfer coefficient to velocity. This relationship is
u = u, (1 + rn(0))b
DISTANCE,?
Figure 1. Characteristic path of a fluid element.
(6)
where the manipulated variable m(8) is the normalized deviation in velocity through the heat exchanger from the steady-state value. In a direct digital control structure, the velocity is manipulated by the computer to the desired value every sampling instant based on outlet temperature. The velocity remains constant between sampling instants and is given by Vi = 1+ mi during the period, io, I8 < (i + l)&. Figure 1 shows the motion of a particle of fluid in a position-time diagram. The characteristics consist of line
k + l (1
+ mi-j)O, > 1
j=l
Equation 8 can be rearranged to obtain X as k
h = [1 -
c(1+
j=1
+
? ? ~ - j ) & ] / [ ( i mi-k-l)e,]
(11)
As shown earlier (Mutharasan and Coughanowr, 1974),the manipulated velocity needed to cause the outlet temperature to reach the set point in one residence time is given by
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
539
Table I. List of Algorithms
algorithm
compen- insates cludes tuning for valve piiramlag eter
ALGl ALG2
no no
no yes
ALG3 ALG4
yes yes
no yes
refer to equations
I
f
eq9 to13 see Mutharasan and Coughanowr ( 1 9 7 4 ) e q 1 2 , 15a,b,c,d A L G B a n d e q 16
I I I
I I I
I
I
The algorithm consisting of eq 9 to 13 will be referred to as ALG1. In this paper, two additional algorithms are developed to compensate for valve delay. Table I summarizes the nature of the earlier algorithms ALGl and ALG2 (Mutharasan anid Coughanowr, 1974) together with the new algorithms piresented in this paper.
Algorithm Implementation The implementation of the algorithm by the computer requires that the memory storage requirements of the past velocities be estimated. The relationship between the maximum number of stored velocities and the expected minimum dimensionless velocity, Vmin,is 1 maximum number of stored values = - (14) Vmines
An array of size 20 is used to store past values of dimensionless velocity, V, starting with the most recent value in position number one pi:ogressing to the oldest sample value in position number 201. The array is initialized to values of 1, indicating no previous deviation in velocity. The computer calculation used to implement the control algorithm can be summarized as follows: (1)Sample outlet temperature and convert to dimensionless form. (2) Compute residence time by the following procedure: (a) set R = 0, i = 1. (b) Calculate R = R + V@,,where past dimensionless velocities are stored in an array of length N . (c) If R > 1 go to step e. (d) Increment i. If i > N , print error message. Return to step b. (e) Calculate k = i - 1. (f) Calculate A. by eq 11. (3) Calculate velocity required to reach setpoint in one residence time by eq 12. (4) Update velocity airray and store the newest velocity data. (5) Send signal to valve after necessary conversion of units.
Experimental System The flow sheet of thle experimental system is shown in Figure 2 (Fuhrman, 1978). The apparatus consists of a 7.3 m long double-pipe heat exchanger in which water flows through the inner pipe while steam condenses on the shell side. The control system consists of a PDP 11/10 computer which samples the inlet and outlet water temperatures and water flow rate and manipulates the control valve to obtain the desired cutlet water temperature. The flow rate and the inlet water temperature are not required in the control algorithm lbut are required in the preliminary experiments and are useful in the evaluation of closed-loop control runs. The exchanger is provided with a cold stream and a warm stream. A load change in the inlet temperature is introduced by the interchanging of these two inlet streams, which is achieved by using the four two-way solenoid valves. The flow and temperature transmitters provide 4-20 mA signals which are converted by means of a resistor circuit into 0 to 5-V signals to provide compatible interface signals to the computer. The current load of the digital computer and the strip chart recorder proved to be more than the
I I I I
I I
--7 _-_I_
A - AGITATOR O R - DRAIN U P - ELECTRO-PNEUMATIC TRANSDUCER F M - FLOWMETER MVP-MILLIVOLT POTE NTlOMETER P - PRESSURE GAUGE 5 - S O L E N O I D VALVE 1 - S T E A M TRAP TC-THERMOCOUPLE
Figure 2. Flow diagram of the heat exchanger control system.
transmitters could handle and operational amplifiers from an analog computer were used as buffer. The signal from the temperature transmitter was sent through a differential amplifier before interfacing with the analog/digital channel in the digital computer. The computer is interfaced to the control valve through an operational amplifier. The voltage signal is converted to an equivalent current signal by use of a resistor and subsequently into a pressure signal by a current/pressure convertor. The solenoid valves are activated manually or by the computer. The flowmeter, the two temperature transmitters, the electropneumatic transducer, the control valve, and the strip chart channels were calibrated for control studies. The exponent b in eq 6 was determined from several steady-state experiments run at different velocities. The dynamics of the control valve are determined by visual observation of the rotameter. The computer sends a signal for flow change to the valve, and the elapsed time to reach the steady-state value was determined with a stop watch. The lag was determined for several values of initial and final flow rates and the average value of the lag was found to be 1.8 s, which in dimensionless time Om is 0.200. The combined dynamics of the flowmeter measurement and the control valve were determined through the use of a computer program. A t the first sampling instant ( t = 0), a step change was given in control valve flow setting. The flowmeter response was sampled by the computer every half-second. The combined first-order time constant for the control valve and the flowmeter was found to be nearly the same for different step changes, equal to 2.2 s. Since the time constant of the flowmeter was about the same order as that of the control valve, cascade control for setting of the flow rate was not needed. The thermocouple time constant in a stream flowing with a velocity of 1 m/s was calculated to be about 0.05 s; consequently, the dynamic lag of the thermocouple was neglected.
540
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
L;..j/-ll
w a
E
(i-2)~~
Ic
SIMULATION, VALVE DYNAMICS NEGLECTED SIMULATION,VALVE DYNAMICS INCLUDED EXPERIMENTAL RESPONSE
(i-kMS DIMENSIONLESS TIME, 0
Figure 3. Outlet temperature response to flow rate changes. Flow rate changed from 0.284 to 0.221 L/s (4.5 to 3.5 gpm) at 0 = 0 and from 0.221 to 0.347 L/s (3.5 to 5.5 gpm) at 6 = 3.04. !,B-I.l
‘
b.02
I
I
l
1 5:
j 0
EXPERIMENTAL RESPONSE RESPONSE
- SIMULATED 1
2
3
4
DIMEKSIOYLESS TIME
,e
-I 0
DISTANCE,7
I
Figure 5. Characteristic path of a fluid element when valve delay is present.
I
1
c
(i-k
9-9-9,
- - - -- - -
5
Figure 4. Outlet temperature response to change in inlet temperature.
thermocouple. The simulated response and the experimental response are reasonably close; however, the experimental response is more sluggish. From the response data shown in Figures 3 and 4 it is reasonable to assume that eq 2 describes the dynamics of the heat exchanger reasonably well. Inclusion of Control Valve and Measurement Dynamics The algorithm of Mutharasan and Coughanowr (1974) described earlier is based on negligible dynamics for the valve and the sensor. However, in the experimental system used in this investigation, valve and measurement dynamics become a significant fraction of the time constant of the process. Such dynamics can be easily incorporated to modify the algorithm of Mutharasan and Coughanowr. Referring to Figure 5, a pure delay in the control valve affects the characteristic by shifting the path-line upward by BvD time units. In the present case, eq 8 is modified as follows (1 mi-l)(e,- OVD) + (1 mi-k-l)X6,
+
+
+ z(i+ mi-j)e,= 1 (15) ;=2 k
Open-Loop Dynamics Responses of the outlet temperature of the heat exchanger to inlet temperature changes and flow rate changes were determined experimentally. The experiments were repeated three times to establish reproducibility. A computer program was designed to perform the experiments and analyze the data automatically. Typical responses and the corresponding responses predicted by the model are shown in Figures 3 and 4. The theoretical response shown in Figure 3 is obtained by solving eq 2 with step changes in flow rate from 0.284 to 0.221 L/s at 0 = 0 and from 0.221 to 0.347 L/s a t 6 = 3.04. Since the flow changes were effected through a computer program, the valve delay is included in the dynamics obtained from the experiment. Inclusion of the control valve delay of 1.8 s (i.e., OvD = 0.2) in the process model shifts the theoretical response by 0.2 time unit to the right which is shown in Figure 3. The actual temperature response is more sluggish due to the thermal capacitance of the exchanger wall. Figure 4 shows the response to inlet temperature disturbance. The disturbance is not a step change and there is a delay of 0.3 time unit between the acuation of the solenoid switch and the measured temperature change of the inlet stream. This delay corresponds to the transportation lag of the fluid between the solenoid and the
and the index k is calculated such that k+l
(1
+ mi-l)(O,- BvD) + jc(1 + mi-j)e, > 1 =2
(1
+ mi-l)(e, - )e,
k
+ jc(i+ mi-;)e,I =2
1
(15a)
(m)
and the parameter X is obtained as = [I - (1 + mi-1)(6,- 8VD) k
C(1 + mi-j)oS]/[(l+ mi-k-i)e,l ( 1 5 ~ )
j=2
The equation for calculation of the manipulated variable remains the same, eq 12, with Q defined as = (0,
-
OVD)(l
+ mi-l)b + k z(1 + mi-j)be,+ (1 + mi-k-l)bX6, (15d) j=2
Equation 15 enables the calculation of the residence time of the fluid element leaving at io, when the dynamics of the control valve are significant compared to the exchanger
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980 541 N
$045 ci
-2 045c
IT
. ..
1
EXPERIMENTAL RESPONSE
- SIUULATED RESPONSE
z
W
I
0 0 2 00
2
1
3
4
5
DIMENSIONLESS T I M E ,
8
7
6
1
Q ~0025 z
9
002
q , , 0
1
e
2
,
~,
3
4
5
,
,
,
I
6
7
8
9
DIMENSIONLESST I M E , e
Figure 6. Comparison of experimental and theoretical response for ALG1. Magnitude of step change in set point is 11.1K.
dynamics. However, the algorithm given in eq 12 still holds true, while the equatio:nsused to calculate k and X are now based upon eq 15. Equation 12 together with eq 15 a,b,c,d enable the calculation of the manipulated variable and will be referred to as ALCT3. A more accurate model of the control valve is represented by a straight-1:ine approximation of flow velocity from the past ve1ocit:y to the new desired velocity in a period, Om. The characteristic will be curved in the interval id, Id < id, dvD. The average velocity is used in the calculation of the length of the exchanger travelled during the time perio'd, dvD. For d v D < e,, eq 8 becomes
Figure 7. Effect of inclusion of valve dynamics on the set point response for ALG1. Magnitude of step change is 11.1K. x.
,B=I.I b =0.3
W
rr
1
a 20.451
+
k
C(1 + mi-j)(d, - &D) +
2
0 0 2 0 ~
j=l
i[
j=l
(1 + rni-j) ~
"I,,
+ (1 + mi-j-J 2
I
2
,
,
,
,
,
,
:
3
4
5
6
7
8
9
DIMENSIONLESS TIME ,e
]&D
(1
+
+ mi-k-l)hd, = 1
In the experimental work reported here, the valve dynamics were approximated as time-delay and eq 15 was used to estimate the residence time of the fluid element leaving the exchanger. Closed-Loop Responses The algorithms proposed by Mutharasan and Coughanowr (1974) and traditional linear discrete and continuous algorithms were used in the experimental system. Experimental closed-loop responses of the control system were determined for setpoint changes and for load changes in inlet temperature and steam pressure with and without errors in the process ]parameter, p. To study the response to set point change, a closed-loop run consisted of a series of changes of set point of +5.6 K, -11.2 K and 5.6 K, u s h g a given set of algorithm parameters. A single load change of +11.2 K overall in the inlet water temperature was used to obtain load response. Steam pressure changes were introduced by changing manually the steam regulator, with the result that the change in shell temperature was not a step change. In all the closed-loop runs, the initial velocity was 0.83 m/s, corresponding to a residence time of 9 s. The range of residence time was fro:m 7.44 s at 0.35 L/s to 11.6 s a t 0.22 L/s. The sampling period was in the range of 0.5 to 8 s. Comparison with Theory. The transient response for the algorithm, ALG1, for set point change with a sampling period of 4 s was determined experimentally and is shown in Figure 6. In the [same diagram the solution to the process model, eq 2, in which the velocity is calculated by the algorithm, ALG1, is also presented. In the simulated
Figure 8. Effect of sampling period on the set point response for ALG1.
response the delay in the valve is taken into account. From Figure 6, it is apparent that there is reasonable agreement between the theoretical and the experimental responses. The overshoot in the response is caused by the presence of valve delay. Furthermore, the experimental reponse is slightly more sluggish than the simulated response because of the thermal capacitance, the dynamic effects of which are not included in the model equations. Effect of Valve Delay. Figure 7 compares the set point response for the algorithm ALGl (eq 12 and eq 9 to 11) in which valve dynamics are not included with the response of the modified algorithm ALG3 (eq 12 and eq 15a to 15d). The overshoot present in the AlGl response is reduced from 30% to 20% and settling time from 4.3 to 2.8 time units when the valve dynamics are included in the computation of the manipulated variable. The earlier algorithm, ALG1, assumes that the flow rate settings are made instantaneously. In the actual process, the valve lag delays the flow rate settings which results in an inaccurate computation of residence time and the occurrence of overshoot in the ALGl response. Effect of Sampling Period. Figure 8 shows the effect of sampling period on the set point response for ALG1. As sampling period increases, the settling time increases, the overshoot decreases, and the response becomes more oscillatory. In the earlier paper, Mutharasan and Coughanowr (1974) reported that sampling period has no effect on the set point response if the parameters and dynamics of the process are known accurately. In an experimental system, the parameters and the dynamics are not known accurately. Furthermore, in the responses shown in Figure
Ind. Eng. Chem. Process Des.
542
57-z:r--2-__
w ~
z$
Dev., Vol. 19, No. 4, 1980
7
L
L
I
--- np.2o.. -
_ - -A@-0 ng -20%
CK OacI
1
W
I
c 043
I % 040
8-1
_--es- o 4 4 _ - e5-o 6 6
&6=0
b.02
I
W J
z
S0
I
"So
, I
, 2
,
,
,
,
,
3
4
5
6
7
C I F S E ~ S I O Y L E S TIME S
, I 8
2
9
e
Figure 9. Effect of errors in 0on the set point response for ALG3. hl
I"-
x
w-
I ..,'3
~-
~
&-244 .ls-22a.
, 7 +
!Y 2345
3
4
5
6
7
8
9
DIMENSIONLESS T I M E , 0
Figure 11. Effect of sampling period and errors in @ on the load response for ALG3. Magnitude of load change is +11.1 K.
-7,
-~
N
ax
2J W2 0 4 0 6+ I
B=I I b-02 e5-044 Ag=+20%] '
i=o (ALi3) '
_ _ K-l
5
.__ K-2 5
DIMENSIONLESS TIME.0
Figure 12. Effect of the parameter K on the load response for ALG4 when errors in are present. Magnitude of load change is 11.1
2
K-I
K.
5
025-
z
L
r
7:s;-!03.;5
8, the dynamics due to the valve are not included in the algorithm. For this reason, the responses shown in Figure 8 are affected by sampling period. In general, decreasing the sampling period improves the transient response because the control algorithm is able to adjust the manipulated variable and calculate the estimated load condition more frequently. Effect of Errors in Process Parameters. Figure 9 shows the set point response of the control algorithm ALG3 when modeling errors are present. The errors in p were introduced into the control algorithm. Figure 9 shows that the proposed algorithm performs well even when f20% error is present in the knowledge of the process parameter,
----by
1
z
0
-e5- o 4 4 --- es=o 6 6 _ _ es-o 88
ci/
8030
z W 2
n0250
i
I
2
3
4
5
6
7
8
9
DIMENSIONLESS TIME,^
Figure 13. Effect of sampling period and errors in @ on the response for ALG3 when steam pressure changes occur.
P.
p. The settling time of the response is 5 time units when Ap = -20% and is increased to 5.5 time units when A@ =
and ALG3 are compared in Figure 10 for values of K = 0.5 and 1.5. A low value of K improves the transient response when sampling period is large, and the improvement is marginal for smaller sampling periods. For large values of K , overshoot increases and the response becomes oscillatory. Load Response. Figure 11 shows the response for algorithm ALG3 to changes in inlet temperature. The load change is not a step change but one similar to the change shown in Figure 4. The top part of Figure 11 shows the effect of errors in the knowledge of the process parameter,
+20%. The maximum deviation does not change significantly for either positive or negative errors in p. In the bottom part of Figure 11 the effect of sampling period on the load response is shown. The increase in sampling period causes a small increase in settling period and only a slight increase in maximum deviation. The use of ALG4 with a A@of +20% and a proportional constant greater than 1 decreases the settling time with little significant change in maximum deviation compared to ALG3. Figure 12 shows this improvement of response for constants of 1.5 and 2.5. The response for the value of 2.5 is somewhat better than the value of 1.5; however, higher values (greater than 5) yield an oscillatory response. Load Changes in Steam Pressure. The response for ALG3 and ALG4 to changes in steam pressure was determined by manually changing the steam regulator as quickly as possible. The ability of the ALG3 to respond well to steam pressure changes is shown in the upper part
Effect of the Parameter K. The control algorithm ALG3 was modified by adding a proportional corrective action as shown in eq 16. Set point responses for ALG4 manipulated variable value calcd by ALG4 = manipulated variable value calcd by ALG3 + K(X8- X2(ios)) (16)
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
543
Table 11. Ziegler-Nichols Controller Constants continuous
.-
. -- . . . . . . . .
................ (I
. . . . . . . .
discrete (I
6
0
0
5.4 7.2
0.7 0.421
0 0.105
0 0
-6 -5.4 -34.5
6
P PI PID
5.8 21.8
13.7
For fast sampling, e, = 0.0555.
Table 111. Predicted and Experimental Offsets for Proportional Control4 offset response
expe ri mental
predicted
positive change in set point negative change in set point load
0.022 -0.019 0.028
0.035 -0.023 0.027
See Figure 15,
LI w
I
'
8-1.0 b - 0 2 0,=0.055 KpC=61
3
2 0.45-
SET P O I N T S -
t
Figure 14, Strip chart xecording of response for ALG4 to steam pressure changes.
of Figure 13 for a change from 170.3 to 128.9 KN/m2 corresponding to steam temperature change from 385 to 378 K. The lower part of Figure 13 shows the effect of sampling period on the response of ALG3 to a sudden change in steam prelesure from 135.8 to 170.3 KN/m2 which corresponds to steam temperature change from 381.5 to 385 K. It is clear from the figure that at lower sampling periods the undershoot and the settling time of the response is decreased. Figure 14 shows the response of ALG4 to three pressure changes when an error of -20% in p is present and the propclrtional constant is 0.5. It should be pointed out that the iresponse has no oscillations and no undershoot. Conventional Control Studies Several experimental responses were determined using conventional proportional (P), proportional-integral (PI), and proportional-integral-derivative(PID) algorithms. The velocity form of the discrete PID controller is (17) m i= mi-l + goei + g1ei-,+ g2ei-, The discrete constants go, g,, and g , are related to the continuous constants of gain, K,, integral time, q, and derivative time, 7 D , as follows
g1
=: -Kc(l
+ 27D/eS)
(19)
g2 = KcrD/Oa (20) When 4 is small, the Zegler-Nichols procedure of choosing the controller paramleters is applicable. From the Ziegler-Nichols settings, the discrete controller parameters, go, g,, and g2 can be computed using eq 18-20. The ultimate gain and period of the exchanger control system under fast sampling (period of 0.5 s or 0.0555 dimensionless time units) were determined experimentally to be 12.0 and 0.841 dimensionless time units, respectively. The transients were introduced by introducing a 20% decrease in set point. The Ziegler-Nichols settings are
I-
.-I
.---/ -------
..--A/----
.._
z
--_
0
3 025-
- -10'F
z W
__
+lODF SET POINT CHANGE
SET POINT CHANGE
LOAD CHANGE I N INLET TEMPERATLRE
2
0020~
1 ~
2
3
4
5
6
7
6
I
9
DIMENSIOhiESS T I M E , @
Figure 15. Proportional control response with Ziegler-Nichols settings.
given in Table I1 for both the continuous and the discrete cases. Proportional Control. For discrete proportional control, the relationships between discrete and continuous control parameters as given in eq 18-20 reduce to go = Kc
(184
g , = -Kc
(19a)
g, = 0 (204 Figure 15 shows the Ziegler-Nichols proportional control responses to changes in set point of f20% and a load change in inlet temperature, X I . A small sampling period of 0.055 time unit is used in this response. The response has choppy oscillatory behavior and a large offset. The offset can be predicted from the steady-state solution of process model, eq 2, which is
where (1 + mm)= 1 + KC(X2,- Xzm) Solution of eq 21 results in an algebraic equation which must be solved by trial and error. The solution to eq 21 is
Ind. Eng. Chem. Process Des. Dev., Vol. 19, No. 4, 1980
544
,
0.
x-
- I 0 b-02 85-0055 g0=56gj-541
C W
2 045B
SET POINTS 1
-
/
2
+I
g
wZ
____
0.25
% I2oi
2
3
b - 0 2 OS-044
-
1
4
, 2
I
5
6
7
8
9
0 025i
I
DIMENSIONLESS TIME , e
/p=lC b-02 es-0055 go=219,--345 g,-1371
2
3
4
5
6
7
8
9
DIMENSIONLESS TIME,^
Figure 16. Response of on-line tuned proportional-integral algorithm.
5
SET W I N T S
ALG 3
SETPOINT CHANGE
LOAD CHANGE I N INLET TEMPERATURE I
~
I
t I O ' F SETPOINT CHANGE
- -1O'F __
/-I
% 035W
'
Figure 18. Comparison of responses for ALGB and PI algorithms for set point and inlet temperature changes. N X
C
I
8-1 I SET POINT
a
i
32
5P
2c030
L' \--
"?
0 0.25
z
0
3
z
W
r
025-
- _ _ +iO'F SETPOINT CHANGE - -IO'F SETPOINT CHANSE _ _ LOAD CHANGE Ih INLET TEMPERATURE
DIMENSIONLESS TIME.@
Figure 19. Response for proposed algorithms for small sampling periods.
18 and 19 results in controller constants of 8.8 and -5.4. for go and g,, respectively. However, these values produce a greatly oscillatory response, because the value of ultimate gain for the larger sampling period is smaller. The response using values of controller constants shown in Table I has a slowly decaying oscillatory behavior. However, use of the on-line tuned values of go = 5.6 and g, = -5.4 produces a slightly faster decaying response, which is shown in Figure 18. The responses of ALGB are also shown in the same figure for comparison. As can be readily seen, the overshoot, oscillatory behavior, and the settling time are improved considerably by the use of ALG3. Figure 19 shows the load and set point responses for ALGl and the set point response for ALG3 a t small sampling periods. A sampling period smaller than Os = 0.2 could not be used for ALG3 because the valve delay would be greater than the sampling period. Comparison with Figure 16 shows improvement of ALGl and ALG3 over PI control at the lower sampling periods. It should be noted that the PI control requires sensitive tuning to get a good response, which is a severe disadvantage. Also, the P I response tuning is sensitive to direction and type of change. Figure 20 shows the strip chart recording of the set point response for ALG3 for three consecutive set point changes, which demonstrates that the response is very similar for the three different set point changes and that the algorithm ALG3 gives consistently good responses. Several responses to changes in steam pressure were determined for P I control. The response, using values of Ziegler-Nichols controller constants shown in Table I, is very sluggish and the outlet temperature does not return to its set point in 55 time units. However, controller constants, go = 6 and g, = -5.4, yield a response that recovers slowly from the steam pressure change. The advantages of the algorithms derived by Mutharasan and Coughanowr (1974) are as follows: (1)Compared
Ind. Eng. Chern. Process Des. Dev., Vol. 19, No. 4, 1980 545
.
-
. .
.
.
.
Figure 20. Strip chart re’cordingof ALG3 to a series of step changes in set point.
with linear algorithms, the responses for the proposed algorithms show considerable improvement in overshoot, rise time, settling time, and oscillatory behavior. (2) The response for algorithms ALGS or ALG4 is relatively insensitive to modeling errors as compared to the response of on-line tuned conventional linear algorithms. (3) The response is nearly independent of direction and type of change, whereas a good response of the conventional control requires tuning which is dependent upon the direction and type of change. Therefore, for an adequate response, the conventional control is not practical for the heat exchanger system or for distributed parameter systems in general.
Conclusions The conclusions of this experimental work are summarized as follows. The process model (eq 2 and 6) describes the dynamics of the double-pipe heat exchanger adequately. The response of the experimental process is slightly more sluggish than that predicted by the model due to the thermal capacitance of the metal wall and due to the dynamics of the steam regulator. The inclusion of the valve lag in the control algorithm is important to minimize the overshoot. The overshoot results from an incoarect calculation of residence time caused by the valve lag. The set point response for ALG3 has a smaller overshoot. The effect of the sampling period is greater when the process parameters or the instrument dynamics are not well known. The algorithm ALG3 adequately handles set point changes with +20% error in parameter p. A low value of K (0.1 < K < 0.5) used in ALG4 marginally improves the set point response. The algorithm ALG3 adequately handles load changes in inlet water temperature, even when large errors are present in @. The algorithm ALG4, using values of K between 1.5 and 3, improves the inlet temperature change response when errors are present in p. Continuous oscillations are produced with ALG4 when the value of K is greater than five.
There exists an optimum value of K for use with ALG4 which is a function of the type and magnitude of change, the accuracy and direction of error in process parameters, and the sampling period. The algorithm ALGS adequately handles large changes in steam pressure, although the algorithm is developed for constant steam temperature. The algorithms proposed in this paper have superior transient response characteristics when compared with the linear algorithms and require essentially no tuning. Furthermore, they perform well even when modeling errors are present. In the application of the proposed algorithms to real systems, sampling periods should be from 1/4 to l/z of the normal residence time and it is recommended that the parameter K be chosen on-line. Nomenclature b = exponent of variation of overall heat transfer coefficient with velocity k = integral number of sampling periods in residence time gotag,,g, = controller constants of the discrete proportionalintegral-derivative control algorithm m(8) = deviation from initial velocity, C / u 0 , dimensionless p = load or manipulated variable t = time fi = deviation from steady-state velocity, u - uo u = velocity of tube fluid uo = initial velocity of tube fluid ui = velocity at the ith sampling instant z ,= distance along the exchanger A, = cross-sectional area of flow A, = heat transfer surface of exchanger C = heat capacity of tube fluid €?= proportional constant for the modified algorithm in ALG4 K , = proportional gain for the proportional control algorithm L = length of heat exchanger T = fluid temperature T, = steam temperature TBsi= initial inlet fluid temperature U = overall heat transfer coefficient Uo = overall heat transfer coefficient at velocity uo V = dimensionless velocity, u / u o Vmin= dimensionless minimum expected velocity X = state variable, dimensionless temperature, (T, - T)/(T, - Tsai) XI = state variable at entrance, dimensionless inlet temperature Xz= state variable at exit, dimensionless outlet temperature X,= outlet temperature set point, dimensionless Greek Letters /3 = dimensionless heat transfer parameter, U ~ , / A c C , , u o p 7 = position variable, dimensionless distance along the exchanger, z / L 8 = time variable, dimensionless parameter, tuo/L = dimensionless residence time of exit element in the exchanger 8, = dimensionless sampling period 8vD = dimensionless valve delay X = fraction of sampling period required to yield the residence time of the exit element in the exchanger, Or = k8, + AB, p = density u = expression within the brackets in eq 13 and 15d Literature Cited Cox, J. B., Heliums, L. J., Williams, T. J., Bank, R. S., Kirk, G. R., Jr., ISA J., 13, 65, (1966). Dahlin, E. B., Instrum. Control Syst., 41(6), 77 (1966). Fuhrman, J., M.S. Thesis, Drexel University, 1978. Lee, H. H., Koppel, L. B., Lim, H. C., Ind. Eng. Chem. Process Des. Dev., 12, 36 (1973). Lim, H. C., Fang, R. J., AIChE J., 18, 282 (1972). K o D D ~L.~ B.. . Kamman. D. T.. Woodward. J. L.. Ind. E m . Chem. Fundam.. ‘9, 198 (1970). Luyben, W. L., “Process Modeling, Simulation, and Control for Chemical Engineers”, Chapter 15, p 493, McGraw-Hill, New York, 1973.
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Ind. Eng. Chem. Process Des. Dev. 1980, 79, 546-550
kbwe, C. F., Smh, C. L., Murill, P. W., Instrum. &ntrolSyst., 43(1), 70 (1970). Mosler, H. A., Koppel, L. B., Coughanowr, D. R., AIChf J., 13, 766 (1967). Mvtharasan, R., Coughanowr, D. R., Ind. Eng. Chem. Process Des. Dev., 15, 141 (1976). Mutharasan, R., Coughanowr, D.R., Ind. f n g . Chem. Process Des. Dev., 13, 168 (1974). Paraskos, J. A., McAvoy, T. J., AIChEJ., 16(5), 754 (1970).
Seinfeld, J. H., Gavalas, G. R., Hwang, M., Ind. Eng. Chem. Fundam. 9, 651 (1970). Vermeychuck, J. G., Lapidus, L., AIChE J., 19, 123 (1973).
Received for review July 23, 1979 Accepted June 12, 1980
Implementation of a Feedback Direct-Digital Control Algorithm for a Heat Exchanger Gerhard K. Giger, Donald R. Coughanowr,' and Werner Rlcharz Department of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland
The control algorithms developed by Mutharasan and Coughanowr are verified experimentally for a double-pipe heat exchanger. A method to modify these algorithms for inclusion of dynamics and hysteresis of the control valve is presented in this paper. Closed-loop responses of the actual heat exchanger for set point changes in outlet temperature showed these modifications to be important for good transient behavior of the control system. Compared with the traditional proportional-integralcontrol these algorithms have superior transient response characteristics, especially when shifting of set point over a wide range is required.
Background a n d Objectives During the past few years, several authors have studied design and implementation of direct-digital control (DDC) algorithms for distributed-parameter systems. The present work is based on this background which is reported in more detail by Fuhrman et al. (1980). The goal of this work is to verify experimentally the control algorithms, ALGl and ALG2, developed by Mutharasan and Coughanowr (1974) for a double-pipe heat exchanger. For this purpose an experimental system was designed which was similar to that used by Fuhrman et al. (1980). In this case the inner tube of the heat exchanger is filled with Sulzer static mixing elements (Tauscher and Schutz, 1973), for which the mean residence time of the heated water is much shorter than in the experiments mentioned above. The outlet water temperature represents the controlled variable, and the flow velocity is the manipulated variable. The performance of the algorithms was experimentally tested by the closed-loop system response to set point change. Algorithm Implementation by the Computer The experimental studies presented here are based on the same process model and the same control algorithms as described by Fuhrman et al. (1980). The model and algorithms are not presented here. The storage of the past velocities required by the implementation of the algorithm is organized here in a different manner than in the work by Fuhrman et al. (1980). The past 24 values of dimensionless deviation in velocity, m, are stored sequentially in a loop array with a pointer which shifts one position at the end of each sampling period. This kind of storage enables the saving of some computing time. For the same reason, the procedure for the implementation of the control algorithm is organized in a different order and can be summarized as follows: (1)
Sample outlet temperature and convert to dimensionless form. (2) Calculate velocity required to reach the set point in one residence time. (3) Send signal to valve after necessary conversion of units. (4) Store the newest value of velocity and shift the time pointer one position. ( 5 ) Compute the residence time for the following sampling period. By means of this procedure, the time between sampling of the controlled temperature and the control action can be shortened considerably. Inclusion of Static a n d Dynamic Behavior of Final Control Element In the experimental system used in this investigation, sensor dynamics contribute very little to the lag of the entire process and can be neglected. In contrast, the dynamic response of the final control element, which is a significant part of the control system dynamics, can be represented by a first-order delay. Preliminary experiments proved that this model is more accurate than the straight-line approximation used by Fuhrman et al. (1980). In this formulation of the algorithm, the valve response is considered to be slow compared to the process response and the sampling period is of the same order of magnitude as the residence time. As a result of these conditions, there is not sufficient time in one sampling period for the flow rate to reach the value requested by the computer a t the beginning of the sampling period. The variation in flow rate is shown in Figure 1. The flow rate reached at the ith sampling instant is given the symbol mfi-l; ideally, if the valve response were instantaneous rnfi-l = mi-l. As shown in Figure 1for several successive sampling intervals, the variation of flow rate between the value at the beginning of the sampling interval (mfi-2)and the value reached a t the end of the interval (mfi-J varies according to a first-order response. With this background, the flow rate m(0)during the period (i - l)0, I 0 < io, can be expressed by
* Department of Chemical Engineering, Drexel University, Philadelphia, PA 19104. 0196-4305/80/1119-0546$01.00/0
0 1980 American Chemical Society
