Ind. Eng. Chem. Fundam.
expressed as kcal/m3. The energy storage density becomes limiting when sufficient water has been sorbed to initiate dripping from the bed of pellets, a condition which must be avoided to maintain the integrity of the bed. If one uses literature information associated with the concentration and densities of CaC1, solutions (Weast, 1971) and the measured internal void fraction of the Celite pellets (ti = 0.575), the drip molar ratio becomes xd
= 3.20/L'"O4
(6)
Equation 6 relates to the maximum water that can be retained within the pellet for any loading, L , before the solution begins to drip. By combining eq 6 with eq 1 and 5 and substituting for the bulk density of Celite P b = 550 kg/m3, the energy storage density becomes 247900 15882 950 (7) (ESD)max = L2.312
impregnated with CaC1, ( L = 0.265 g of CaC12/g of Celite). The conclusion from these experimental results and comparison with other viable means for storing thermal energy indicates a promising utilization of the proposed scheme. More specifically, these means of chemical energy storage are not associated with the presence of continuous temperature gradients which normally would exist in solar ponds or rock bed storage and problems associated with phase changes. Glossary
energy storage density, kcal/m3 energy storage density at drip conditions, kcal/m3 heat of reaction, kcal/g-mol of CaC1, loading, g of CaCl,/g of Celite g-mol of water reacted molecular weight of CaC1, molar ratio, g-mol of H20/g-mol of CaC1, drip molar ratio, g-mol of H,O/g-mol of CaCl, volume of impregnated Celite, m3
+-
Equation 7 predicts the maximum energy storage density in kcal/m3 for any loading associated with the Celite pellets. A comparison now can be made between this system and other thermal energy storage schemes. In this connection, the capability of this system is compared with a sensible heat storage by using water and a phase-change storage using Glauber's salt (Na2S04.10H20).As a basis for comparison, eq 7 shows that 1 m3 of Celite with a loading of L = 0.265 g of CaC12/g of Celite is capable of storing 226 074 kcal. For the sensible heat storage of water cooling from 66 to 20 "C, 4.91 m3 of liquid water is needed to match the chemical energy storage of the loaded Celite. Glauber's salt, utilized as a phase change storage system a t its melting point, 32.4 "C, is capable of storing 84 330 kcal/m3, thus requiring 2.68 m3 of this hydrate to match the equivalent energy storage attainable by 1m3 of Celite
771
m6,25,771-775
Greek Letters bulk density of CaC1,-free Celite, kg/m3 ti internal void fraction
Pb
Registry No. CaCl,, 10043-52-4. Literature Cited Bailer, Jr., J. C.; Emeleus, H. J.; Nyhoim, R.: Trotman-Dickenson, A. F. Comprehensive Inorganic Chemistry, 1st ed.: Pergamon: New York, 1973; VOl I. Greiner, L. Presented at the ERDA Solar Energy Storage Program Information Exchange Meeting, Cleveland, OH, Sept. 1976. Heiti, R. V. Ph.D. Dissertation, Northwestern University, Evanston, IL, 1984. Weast, R. C. Handbook of Chemistry and Physics, 51st. ed.; The Chemical Rubber Co.: Cleveland, OH, 1971.
Received for review January 28, 1985 Revised manuscript received January 2, 1986 Accepted January 28, 1986
Selection of Sampling Frequency for Digital Process Control Systems Carl P. Jeffreson' Chemical and Metallurgical Engineering Department, University of Nevada -Reno,
Reno, Nevada 89557-0047
Hallvard F. Svendsen Institut for Kjemiteknikk, N-7034 Trondheim-NTH, Norway
A simple method is proposed for estimating the performance degradation caused by repeated sampling at intervals in the digital, proportional-plus-integralcontrol of process systems. Linear, single-loop, n -lag systems with and without dead time are considered. The method is based on the use of Harriott's empirical index of performance extended from continuous to discrete systems. Experimental performance data are correlated against the index. The sharp minima reported by Roberts in plots of integral time-weighted absolute error against sampling interval are not found when the reset rate is unconstrained.
Introduction
Although a variety of techniques are available for the design of digital compensators, most single- and multiple-loop process computer control systems use variations
on the standard proportional-plus-integral (PI) and three-term (PID) algorithms. The number of digital process control loops that can be supported by a multitasking operating system may be
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772
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 3:2
41
= i 1 - 2
-1
Figure 1. System under consideration. Note: disturbance dynamics are the same as process dynamics.
considerably increased if “slow” processes are sampled a t an appropriate frequency. While excessively frequent sampling wastes computer resources which could be used for other purposes, infrequent sampling may degrade performance unduly.
Objectives The main objective of this paper is to derive a simple criterion for choosing sampling frequency in linear process control systems under digital, single-loop, proportionalplus-intetgral control. It will be assumed that an exact model of the process is available which consists of an arbitrary number of first-order lags with or without dead time. The case where the process dynamics are unknown, or only available in the form of a reduced-order model, is more difficult but will be considered briefly later in the paper. The scope of the paper does not include a study of the tuning of digital control systems since this has been discussed extensively elsewhere. It will be necessary, nonetheless, to examine the implications of Robert’s tuning recommendation (1976) which relates sampling frequency to the reset rate. This is because the application of Robert’s tuning method may result in erroneous conclusions about the effects of sampling on closed-loop optimum performance. System Definition The generalized system under consideration is illustrated by Figure 1 and consists of a digital controller sampling the output of a process having steady-state gain K p and overall pulse transfer function KpGp(z-l), where Gp(z-’) is defined such that limz-l Gp(z-l) = 1. The zero-order hold has been included in the process transfer function which also incorporates the final control element and transmitter. The paper is concerned with “regulator” performance, and the step disturbance is applied a t the start of a sampling interval. Disturbance acd process dynamics are identical. Although deterministic (step) disturbances are applied, this is not a serious limitation since Thompson (1973) has shown that the mean-square output resulting from integrated white noise is directly proportional to the integral square error resulting from a step input. Previous Work Early work on digital controller tuning by Lopez et al. (1967) showed that, for a first-order-lag system with dead time, the optimum integral absolute error increased monotonically as sampling frequency decreased. More recently, Palmor and Shinnar (1979) considered the digital control of first- and second-order-lag systems with dead time subjected to random disturbances having various statistical properties. They also showed that optimum “performance”,as measured by mean-square error,
steadily degrades as sampling frequency is decreased. For their example of a first-order-lag system with a dead time, D , of 0.5 time unit and time constant, T , of 1 time unit, a sampling frequency of four samples per dead time unit would degrade output variance performance by roughly 5-10%. This degradation in performance depends on the noise properties assumed in the optimum controller design and is relative to the corresponding near-continuous system. Note that we define sampling frequency w, as the reciprocal of sampling interval T, in this paper so that the above frequency is equivalent to a sampling interval of one-fourth the system dead time. An alternative point of view was suggested in a paper by Roberts (1976). He showed that, provided the controller reset time T, is related to sampling interval T, by TR = 5T, (or l / T R = 0 . 2 4 (1) sharp minima in plots of optimum closed-loop integral absolute error against sampling interval resulted for a given first-order-lag system with dead time. These results suggest that, for this simplified discrete controller, and optimum sampling frequency may be chosen for discrete PI or PID controllers which is “better”, in the ITAE sense, than continuous control. As an example, Roberts’ Figure 6b plots “optimum” ITAE performance against sampling interval, T,, for regulator control of a first-order-lag system with dead time at a number of fixed values of normalized process dead time. For a normalized dead time to system time constant ratio, D / T ,of 0.25, the minimum ITAE occurs a t a normalized sampling interval, Ts/7,of about 0.125. The above sampling interval represents a sampling frequency, us,of eight samples per unit time constant. Our results, discussed later, show that optimum IAE (integral absolute error) performance a t this sampling frequency would be about 45% worse than an optimally tuned continuous controller, assuming reset control action to be unconstrained.
Empirical Performance Index of Harriott The product P = w&L,,r of critical frequency o, and maximum allowable (proportional control action) loop gain KL,,., has been used by various authors as a rough “figure of merit” for control systems. See, for example, Harriott (1964) and Niederlinski (1971). The inverse, P’(or 1-l where I = w,(l + KLmax/2)), is approximately proportional to the optimum integral absolute error following a step disturbance. The significance of the result, investigated in some detail by Jeffreson (1976), is that an approximate figure of merit or performance index may be assigned not only to the usual firstorder-lag system with dead time but also to systems consisting of any combination of lag time constants with or without dead time subject to certain limitations outlined in the paper. It follows that if the process dynamics of the plant are known in detail, the optimal performance achievable under feedback control may be estimated assuming proportional integral control. A simple extension of this approach from continuous to discrete or digital proportional-plus-integral controllers should be possible if an equivalent dead time, equal to half the sampling interval, is added to the system dead time. This approach is very commonly used for estimating discrete system critical frequency and maximum loop gain. Knowing these two parameters, the performance index of the discrete or digital control system could then be estimated for any sampling frequency and compared with the continuous system performance index. Choice of sampling
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
frequency would be based on a specified maximum degradation in performance compared to that of the continuous system. We have tested the validity of this approach in this paper.
I ay
1
4.0
“Experimental” Work-Optimum IAE vs. Sampling Frequency In order to determine the effects of sampling frequency on performance, a PDP11/23 computer with a real-time clock and a scheduler was connected to an analog computer through i2-bit analog-to-digital and digital-to-analog converters. The integration of absolute error was performed a t a sampling frequency of 100 Hz throughout so that performance relating to the continuous “process” was measured instead of computing the sum of absolute errors a t the sampling instants. Although these “experiments” could have been performed more simply and accurately by using digital simulation, the real-time system was used to determine whether there were in fact any practical effects associated with the use of P I action a t high sampling frequency. The same system has also been connected to a variable-speed blower system with flow transmitter, and good results were obtained. The digital controller algorithm used is described by the usual difference equation pn=Kce,+S, with
s, = s , - 1 +
n = 0 , 1 , 2,...
(2)
[yen
(3)
where controller output is p n a t time t = nT, after changeover from manual to automatic control, error is e,, controller reset time is TR,and controller gain is K,. Loop gain KLis K,&p. In eq 2 and 3, the zero-time value of the “integral sumn So is initialized to equal the output of the controller’s manual regulator a t time t = 0 of manual to automatic transfer. In order to find the optimum IAE at each sampling frequency, the controller gain and reset times were varied under control of the Nelder and Mead algorithm (1965) until convergence of the normalized IAE occurred to sufficient accuracy. Amplitude normalization of the resultant IAE values was performed relative to the product of disturbance step size and disturbance open-loop gain KDwhen comparing IAE values with the earlier continuous system results of Jeffreson (1976). The time scale was also normalized relative to the first central moment of the impulse response of the open-loop plant when comparing with earlier work, e.g., relative to D + r1 + T~ + r3 for a third-order system with dead time. A t a sampling frequency of 100 Hz (sampling interval 0.01 s) computations of the control algorithm and summations consumed a negligible proportion of the sampling interval. Each run (i.e., a particular transient step test under supervision of the Nelder and Mead optimization algorithm) was terminated automatically when the change in computed IAE over a total of ten integration intervals was less than 0.005% /s.
Experimental Results For “continuous” control (Le., a sampling frequency of 100 Hz) we found close agreement between the observed normalized integral absolute errors and the predicted values from the correlations of I-l against IAE in the earlier paper by Jeffreson. In extending the correlation to discrete P I control a t lower sampling frequencies, we added a dead time equal to half the sampling interval to the system dead time and computed the “equivalent” continuous system
1
I I Fin1 Order Lag Dead Timc Splema: Demd time, Time conslmt:
773
I
I
0 0.8. 4.0 Av
0.4, 4.0 0.2,
Second Order Dead Time Syslema: Dead t i m 3 i m e constmta: 0 0.8 1.23 3.89 A 0.8 0.21 2.99 V 6 . 8 2.0 1.0
O
h
/
Fourlb Order 4,
.
A ’
h u r l Time Con.CloLa
hue,
/
.’
’
.”
J
..I
Continuous S atem Pcrronnpnce Indcr. I Appror. Diacrci System l’crlormance Index
-
Figure 2. Experimental correlation of optimum discrete closed-loop PI action regulator performance.
performance index for the discrete system, This approach gave results which compared well with the more exact discrete values of critical frequency and maximum allowable loop gain from discrete system stability analysis. In Figure 2, the ratio Icont/I~isc has been plotted against the ratio IAEdisc/IAECon, of discrete system normalized optimum IAE to continuous system IAE for the following systems: solid circles and triangles, pure dead timefirst-order-lag systems; open circles and triangles, second-order lags with dead time; open squares, pure fourth-order lag and no dead time. These results indicate that the discrete system performance may be predicted from the continuous system performance index and the known sampling frequency to sufficient accuracy for engineering purposes. For very low sampling frequencies, the observed degradation in performance tends to be about 15-20% less than that predicted from the simple approximate performance indices.
Use of Equivalent Dead Time The foregoing results allow estimates of discrete system peformance to be made easily, provided the system time constants and the dead time are known. A commonly used alternative approach is to replace the exact model by an “equivalent” first-order-lag-dead-time, reduced-order model. A linear regression of the data of Figure 2 yields the following relationship with a correlation coefficient of 0.995
IAEcont - 0.125 + 0.883 IAEdisc
provided
De, + Ts/2
[
De, S e L 2
]
2
(4)
‘
Teq
In eq 4, the equivalent reduced-order model dead time, D,, is obtained by matching first moments of the exact and reduced-order models, i.e.
De, = D
+
NLAG
C
i=l
~i
-
T,,
(5)
774
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 100
where NLAG is the number of lags in the exact system, 7 , are the time constants, D is the exact system dead time, and the reduced-order model equivalent time constant T , ~ is obtained by matching second moments NLAG T , ~=
[
T,’]~
CRITICALFREQUENCY OF OSCILLATION vemw NORMALIZEDSAMPLKNG INTERVAL Solid Lines: h r e RPpoNonal Action
\$ \
Drshedhnes:
I
+
*
1-1
An alternative correlation using the “reaction curve” approach yields similar results although in this case the equivalent dead time is consistently lower than that predicted by moment matching and the correlation coefficient i s lower.
‘
1
\~ PI Action w i t h Roberta’ controller setting:
I I
10.
h
E ¶
0-
r= 10
.-
-
e ~~
1)iscussion-Example of Application Consider a system well described by four first-order lags with time constants of 4 , 3 , 2 , and 1 min and a dead time of 1 min. Then a continuous system critical frequency of 0.364 min-’ results, with maximum allowable loop gain of 3.445. Hence, the index of performance, I , would be about 0.991. For a sampling frequency of 0.833 samples per minute, the equivalent continuous system has a predicted index of performance of 0.821, which is about 20% worse than that of the limiting continuous system. For comparison, eq 4 predicts a reduction in performance of about 18.6% compared with the continuous system. An experiment on the time-scaled version of this system resulted in a performance degradation of about 2070, which is in agreement with both estimates to sufficient accuracy. An alternative method was proposed by Isermann (1982),who suggested that the sampling interval (for parameter-adaptive algorithms) should satisfy the inequality tg5/15 5 T , 5 tg5/4,where tgbis the 95% step-response settling time of the process. For this example, the 95% response time, tg5,is about 20.4 min. Therefore, according to Isermann’s recommendation the sampling interval should lie in the range 1.43-5.33 min. The lower value of the sampling interval (i.e., the higher sampling frequency) would result in a performance degradation of about 25% according to the correlations in this paper. The main disadvantage of the Isermann empirical recommendation is that ar. infinite number of systems with different control properties may have the same 95% response time, tg5. For example, a system consisting of a single lag of 6.65 min and a pure dead time of 0.5 min has the same tgjtime as the fifth-order-lag-dead-time system. For the single-lag system however, a sampling interval of 1.33 min will result in a much larger relative degradation in performance. Equation 4 predicts that discrete system performance would be only about 27% of the optimum continuous system performance in this case. Comparison with Roberts’ Recommendation As already noted, Roberts‘ tuning relation expressed by eq 1 causes a minimum to occur in plots of integral performance criterion against the sampling interval, T,. Stability analysis has been carried out for the simple first-order-lag systems with dead time considered by Roberts, using the discrete frequency response and Nyquist criterion (Saucedo and Schiring, 1968). Figures 3 and 4 show how the maximum allowable loop gain and closed-loop critical frequeiicy are predicted to vary with the ratio TJT of sampling interval to process time constant, T J r , for various values of the ratio of process dead time to time constant, D I T . Two cases are plotted: solid liines, purely proportional control; dashed lines, proportional-plus-integral control with the fixed ratio of reset rate to sampling frequency recommended by f?0btif‘ts i P q 1)
I.
Parameter: Ratio: Dead time Process Time constant
0.:
--
0.2 0.4 0.6 08 10 Normalised sampling interval Ts/T
Figure 3. Discrete first-order-lag-dead-time system under P or PI action. Maximum allowable loop gain is plotted vb. normalized sampling interval
\
Equahoo(4)
l u x I y U Y ALLOWABLE LOOP CAM versus NORMALIZED SAMPLING LVTERYAL
\
Parameter
’b-,I
.
Dead tune Procesa Time constant
Solid L n e s Pure Pmport~onalAchon Dashed h u e s . PI Achon n t h R o b e d ’ contmller sethn, -R
c
1
0.2
= iT,
I
1
0.4
0.8
0 8
Sormalised sampling interval Ts/T
3
Figure 4. Discrete first-order-lag-dead-time system under P or PI action. Critical Irequency of oscillation is plotted vs. normalized sampling interval.
Both the maximum allowable loop gain and the critical frequency decrease as the sampling interval approaches zero when Roberts‘ simplified tuning relationship of eq 1 is used. Application of Harriott’s performance index to these plots explains (qualitatively) why minima in plots of ITAE against sampling frequency occur with Roberts‘ tuning method. Since both w, and KL,,, exhibit maxima when plotted against the sampling interval, it is clear that the reciprocal of their product must result in a minimum when the reset rate is constrained according to eq 1. Hence, the optimum ITAE plots will also exhibit minima.
Discussion-Purely First-Order-Lag Systems When dead time is absent and the analng computer “process” consists simply of a first-order lag, it was found that the Nelder and Mead controller tuning algorithm
Ind. Eng. Chem. Fundam. 1986, 2 5 , 775-782
eventually chooses the following controller settings to a high degree of precision CY
KL = 1-a!
(7)
where cy
=
~-‘ITS
and
Ts _ -1 TR a!
(8)
In this case, the effect of the disturbance was eliminated in two sampling intervals. In other words, the system exhibited “dead beat” control behavior. The resulting normalized integral absolute error decreased with increasing sampling frequency such that IAE = (1- a ) / w , as predicted by theory. In the limit, continuous system “performance” should result with an optimum IAE of zero. Such a high performance is not achievable in practice because of saturation. The method of predicting the effects of sampling on performance developed in this paper is clearly not applicable to purely first-order systems. Conclusions
We have confirmed that integral “performance” of a number of representative systems under digital PI control continuously improves as sampling frequency increases. We concluded the Roberts’ observed degradation in performance at high sampling frequencies is simply due to the
775
choice of a fixed ratio of reset rate to sampling frequency. It was shown that the performance index originally proposed by Harriott for continuous systems may be applied t o the prediction of closed-loop P I controller performance under digital control. Provided that the sum of reduced-order-system equivalent dead time and the semi sampling interval, T,/2, is less than the reduced-ordersystem equivalent time constant T ~a simplified , equation (4) was found to correlate the data well. For example, a 10% reduction in performance of the discrete system (compared with the continuous system) results when the ratio of sampling interval to equivalent dead time is about Literature Cited Harriott, P. Process Control; McGraw-Hill: New York, 1964; p 103. Isermann, R. Automatica 1982, 18, 513. Jeffreson, C. P. Ind. Eng. Chem. Fundam. 1976, 15, 171. Lopez, A. M.; Smith, C. L.; Murrill, P. W. Instrum. ControlSyst. 1969, 42(2), 89. Nelder, J. A.; Mead, R . Comput. J . 1965, 7 , 308. Niederlinski, A. Automatica 1971, 6 , 691. Palmor, Z.J.; Shinnar, R. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 8. Roberts, P. D. Meas. Control 1976, 9 , 227. Saucedo, R.; Schiring, E. E. Introduction to Continuous and Digital Control Systems; Macmillan: New York, 1968; Chapter 10. Thompson, A. G. Proc.-Inst. Mech. Eng. 1973, 187, 129.
Received for review February 27, 1985 Revised manuscript received January 10, 1986 Accepted February 13, 1986
Salting-Out of Oxygen from Aqueous Electrolyte Solutions: Prediction and Measurement Werner Lang’ and Rolf Zander Physiologisches Institut der Johannes-Gutenberg Universitat Mainz, 0-6500Mainz, FRG
On the basis of an independent single-ion parameter model, the parameter k,,,, representing the salting-out of oxygen by an electrolyte, can be calculated from the individual ion parameter Hi, ionic charge zi, and stoichiometric number x, of cations and anions of the same type in the electrolyte. From extensive experimental oxygen solubility data in 68 aqueous electrolyte solutions (20 chlorides, 22 nitrates, 18 sulfates, 3 hydrogen sulfates, and 5 hydroxides) at several concentrations and measured by a photometric method at 310.2 )< with pure oxygen under atmospheric pressure, a set of relative individual ion parameters, Hi, for all of the different cations and anions of the studied electrolytes could be derived, if sodium ion is chosen as a reference ion, HNa+ = 0. Good agreement was found between predicted and experimental k,,, values of the order of f3.5%, and good agreement was also found between predicted and experimental oxygen solubilities.
I n t r o d u c t i o n and T h e o r y
The influence of dissolved electrolytes on oxygen solubility in aqueous solutions is a well-known phenomenon and is called the salting-out effect, if the solubility of oxygen in the electrolyte solution is decreased compared to that in pure water under the same experimental conditions of temperature and pressure. A quantitative treatment of this effect was first given by Setchenov (1889) by an empirical relation for carbon dioxide, which also holds for other gases for a great number of inorganic and organic substances over a wide concentration range. In its logarithmic form t o base 10, this relation can be written as log
b 0 / 4= k,,,cs
(la)
where cyo and CY are the Bunsen coefficients of oxygen solubility in pure water and in the salt solution at the same temperature, cs is the molar concentration of the salt in the solution, and k,,, is the salting-out parameter or Setchenov’s constant, which is specific with respect to the gas, electrolyte, and temperature and depends upon the solubility unit of the gas as well on the units of the salt concentration (Clever, 1983). Later, van Krevelen and Hoftijzer (1948), instead of using the molar concentration in eq la, used the ionic strength of the electrolyte, I = 1/2C,c,z,2. c, is the molar concentration of the ith ion with charge number zi. Furthermore, they split up the resulting salting-out parameter into a sum of three independent contributions which are
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