Ind. Eng. Chem. Res. 1990,29, 924-927
924
are added. This last result can be taken as further evidence of the utility of ultrasound effects in the preparative organic chemistry, including those cases where the reactions are performed under heterophase conditions and in fixed-bed reactors (Ragaini et al.. 1986, 1988). Nomenclature PTC = phase-transfer catalysis or phase-transfer catalyst(s) PAN = phenylacetonitrile BuBr = butyl bromide TEBA = triethylbenzylammonium chloride (soluble catalyst) TBBA-TEG-PB = tributylbenzylammonium chloride-benzyltetraethylene glycol-polymer bonded (insoluble catalyst) TBA = tetrabutylammonium chloride (soluble catalyst) C5 = no preconditioning of the catalyst, Le., the catalyst is the last component added to the reaction mixture C2 = preconditioning of the catalyst overnight at the reaction temperature with a mixture of PAN + NaOH(,,, (BuBr is added as the last component at the beginning of kinetic run) C, = preconditioning of the catalyst overnight at the reaction temperature with a mixture of BuBr + NaOH,q, (PAN is added as the last component at the beginning of kinetic run) UM = ultrasonic mixer (20 kHz) MS = magnetic stirring (about 1000 rpm) FB = fixed bed Registry N o . TEBA, 56-37-1; TBBA, 23616-79-7; TEG, 86259-87-2; C6H5CH2CN,140-29-4; BuBr, 109-65-9; N a O H , 1310-77-2: C,H,CH(Bu)CN. 5798-79-8
L i t e r a t u r e Cited Buzzi Ferraris, G. Quad. Ing. Chim. Ital. 1968,4, 171; 1968,4, 180. Dehmlow, E. V.; Dehmlow, S. S. Phase Transfer Catalysis; Verlag Chemie: Weinheim, 1980. Ley, S. V.; Loa, C. M. R. Ultrasound i n Synthesis; Springer-Veriag: Berlin, 1989. Mason, T. J.; Lorimer, J. P. Sonochemistry; Ellis Horwood: ChiChester, U.K. 1988. Ragaini, V.; Verzella, G.; Ghignone, A.; Colombo, G. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 878. Ragaini, V.; Chiellini, E.; D’Antone, S.; Colombo, G.; Barzaghi, P. Ind. Eng. Chem. Res. 1988, 27, 1382. Riaz, S.A. Aldrichimica Acta 1988, 21, 31. Solaro, R.; D‘Antone, S.;Chiellini, E. J . Org. Chem. 1980,45, 4179. Suslick, K. S. Mod. Synth. Methods 1986, 4, 1. Suslick, K. S. Ultrasound. Its Chemical, Physical, and Biological Effects;VCH: New York, 1988.
* To whom correspondence should be addressed. Vittorio Ragaini,* Giovanni Colombo, Paolo Barzaghi Dipartimento di Chimica Fisica ed Elettrochimica Universitci di Milano, via Golgi 19 20133 Milano, I t a l y
Emo Chiellini, Salvatore D’Antone Dipartimento di Chimica e Chimica Industriale Universitci di Pisa, via Risorgimento 35 56100 Pisa, I t a l y Received for review April 5 , 1989 Accepted February 7 , 1990
A Method for the Determination of Dahlin’s Algorithm Parameters Equations for use in tuning the first-order Dahlin control algorithm are presented in this paper. T h e algorithm contains only one adjustable or tuning parameter (A) t h a t must be specified once the sampling interval and process parameters are determined. To obtain a n optimum value of A, tuning equations relating A t o t h e process time constant and sampling rate were developed. Over the years, several digital algorithms have been developed and are becoming popular in control systems. One of these control algorithms is the one developed by Dahlin (1968). This algorithm was developed to be used in processes with significant deadtime. Chiu et al. (1973) have studied the first- and second-order Dahlin algorithms by using a model of a jacketed backmix reactor. Condon and Smith (1977a) compared the Dahlin algorithm to continuous and discrete PID controllers. In another study, Condon and Smith (1977b) performed a sensitivity analysis on the algorithm. Development of t h e Control Algorithm A basic digital control loop is shown in Figure 1. Figure 2 is a simplified version of this control loop, which was used in the study presented here. The control element and the transmitter are incorporated in the process-transfer function. The process block diagram shown in Figure 2 can be analyzed (using z transforms) following a procedure analogous to the one used to study continuous systems. The analysis starts by relating Y ( z )and R ( z ) through the closed-loop pulse-transfer function:
-Y (-z ) R(z)
1
D(z) HGk) + D(z)HG(z)
(1)
where Y ( z )is the z transform of the sampled function of the controlled variable, R ( z ) is the z transform of the sampled function of the controller’s set point, D ( z ) is the 0888-5885/90/2629-0924$02.50/0
controller’s pulse-transfer function, and HG(z) is the pulse-transfer function of the zero-order hold and process. Solving eq 1 for D ( z ) results in 1 Y(z)/R(z) (z) = D(z) = M E ( z ) HG(z) 1 - Y ( z ) / R ( z )
(2)
where M ( z ) is the z transform of the discrete controller’s output function and E ( z ) is the z transform of sampled error function. To completely develop the above control algorithm, the function HG(z) must be known and the functions Y(z)and R ( z ) must be specified. Equation 2 is referred to as the synthesis equation for a digital controller design. The basis of the Dahlin algorithm is that the closed-loop response to a step change in set point should be that of a first-order-plus deadtime (FOPDT). This closed-loop response in the Laplace domain is given by e-w 1 Y ( s ) = -(3) Xs+ls
where to, is the deadtime of the closed-loop system and X is the time constant of the closed-loop system. In Dahlin’s article, to, is assumed to be an integer number of the sampling time (T), which is the approach used in this presentation. T o obtain HG(z), the process-transfer function G(s) must be determined; the form of G(s) used is that of a FOPDT. This transfer may be obtained from the openloop step response of the process (Smith, 1972). The step 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 5 , 1990 925 Substituting eqs 6 and 9 into eq 2 yields the controller's transfer function:
where 6 , = 1 - e-Tlh
K1 = 61e-T/A
Vlil
K2 = 62e-Tlh
K,
= 6,6,
From eq 10 the following discrete controller equation can be derived after cross-multiplication: r
I
I
x'
"IO
Figure 2. Simplified digital control loop used in this simulation.
response technique consists of manually altering the manipulated variable during steady-state operation and recording the process response as detected by the transmitter. The resulting process curve is then used (see example for details) to determine the parameters of the following transfer function: G(s) =
Kpe-topS
7s + 1
(4)
where Kp is the process gain, 7 is the process time constant, and to, is the process deadtime. The transfer function of a zero-order hold is
H(s) =
1 - esT S
where T is the sampling time interval. Using eqs 4 and 5 and a table of z transforms, we obtain
where n refers to the present sampling period and n - 1 refers to the previous sampling period. Equation 11 is the first-order Dahlin algorithm. Notice that most of the parameters of eq 10, K,, T , N , and 7,are known. Incorporated within the constants K1,K2,and K 3 is the parameter A, which, as previously mentioned, is the time constant of the closed-loop system. X is the only unspecified value in the control algorithm; therefore, it is used as the tuning parameter for this algorithm.
Determining the Controller a. Performance Criteria. In this study a FOPDT process was simulated, and its response, using the controller given by eq 11, to set-point changes was obtained. The sampling time and the parameters of the first-order process (time constant, gain, and deadtime) were varied, and an optimum value of h was obtained for each variation by using the Fibonacci search technique. These values were used to perform a linear regression to obtain the tuning equations. The criterion of performance for optimization in the Fibonacci search technique was the minimization of the integral of the absolute error (IAE). b. Tuning Equations. IAE as the Performance Criterion. The equations obtained for the IAE performance criterion are log h = -0.522 + 0.918 log 7 (f0.0022) (f0.0028)
(12)
Equation 12 holds only in the range T I T = 0.001-0.25 and
where 82 = 1 -
N = to,/T
The z transform of eq 3 after setting to, = to, (for details on this requirement see example) is z-N-l(l - e-T/X) (7) (')' = (1 - -1 1 - e-T/X -1 )(
2
)
As mentioned prior to eq 3, the set point is changed in a step fashion: R(s) = l / s or
R(z) =
1 -
1 - 2-1 Then, dividing eq 7 by eq 8 yields
log ( T / h ) = -1.503 + 10.813(T/~)- 28.019(T/~)'+ 29.677(T/7I3 (13) (fo.5762) (f1.030) (f2.819) (f2.495)
Equation 13 holds only in the range T / r = 0.25-0.5. The process gain, Kp, was not found to be a determining factor in obtaining an optimum tuning value. Values in parentheses are the 95% confidence limits for the parameter coefficients. c. Example. The stirred tank reactor shown in Figure 3 was to be controlled by using the first-order Dahlin controller. For this system, the level control loop was assumed to be perfect. The design and steady-state values for this process are Vr = 0.3755 m3, ko = 8.66 X lo8 cm3/(mol.s), Vc = 4.418 X m3, E = 64648 J/mol, R = 8.314 J/(mol.K), p = 0.8814 g/cm3, Cp = 3.684 J/(g.K), AHr = -27885 J/mol, Ac = 3.344 m2, Cpc = 4.186 J / ( g K ) , U = 425.85 W/(m2.K),pc = 1 g/cm3, Cai(t) = 9.579 X mol/cm3, Ti(t) = 352.78 K, Tci(t) = 300 K, Ca(t) = 3.315 X mol/cm3, T ( t )= 377.17 K, Tc(t) = 334.8 K, Fr(t) = 6.3 X m3/s, and Fc(t) = 4.1 X m3/s.
926 Ind. Eng. Chem. Res., Vol. 29, No. 5, 1990
1
E,-
-
- - ---
cT
Figure 3. Stirred-tank reactor with control loops 70.0
I
I
55 0
1
,
,
20
40
I
t
1
-2 1
'n
, 10
Figure 5. Response of Dahlin controller to set-point change.
1
.
65.0
55.0
1
80
!,me (n1nu:es)
I
.
I 60
73 0
1
675
-
1 65.0
I
1
J
1
process response
I
1
I
!
20
30
u1
50
time (minutes)
Figure 4. Open-loop response to 5.0% step change in controller
5 7 5 1
output. 550
The transmitter has time constants of 0.5,0.25 min and a deadtime of 10.0 min. The process transfer function, G(s), will be approximated as FOPDT or
T o determine the transfer function, G(s), the concentration controller (CIC) was placed on manual mode, and a step change of 5% in controller output was introduced. The response of the process to this change is given by Figure 4. The parameters to, and T of eq 14 are obtained from two points in the process reaction curve (Smith and Corripio, 1985), Figure 4. One of these points is at to, + 7/3, where the process attains 28.3% of its total change. The other point is a t time to,, + T , where 63.2% of the total change is attained. Thus, t0.263
I
I
= top + T / 3
t0.632
=
top
-t
(15) (16)
From Figure 4, = 16.67 min and t0.632 = 22.33 min. Solving eqs 15 and 16 simultaneously results in to, = 13.8 min and T = 8.5 min. The gain of the process (Kp) is determined as the ratio of the change in process output to the change in manipulated input, or A0 (64.375 - 56.875)%TO %TO Kp=-= = 1.5AI 5.O%CO %CO As shown earlier, the deadtime of the closed-loop system (tol) is generally set equal to the process deadtime (to ). If to, > top, then deadtime is essentially added to tke closed loop, which impedes the control loop. On the other
I1
I
i ,
20
point - set optimized ...
tuning parameters quarter decay ratio tuning
ii i
I
I
I
I
1
u)
W
W
1CX
time (minutes)
Figure 6. Response of PID controller to set-point change.
hand, if to, < to,, then the control loop is faster than the process, which results in a physically unrealizable controller. In this process, a sampling time ( T )of 1.15 min was used, so for this case the ratio T / T = 0.135, using to, = NT as specified before, then N = 12. Equation 12 is used to calculate A. By use of the value determined for 7,eq 12 results in X = 2.144 min. With the process parameters and X available, all the constants for eq 11 can be calculated, and for this case, 6, = 0.4152, 6, = 0.1265, Kp = 1.50, k, = 0.3626, K2 = 0.0740, and K 3 = 0.0525. The response of the closed-loop system to a change in set point is given in Figure 5. It can be seen from this response that the Dahlin algorithm provides a fast response without excessive oscillations for this system. In general, this is the kind of response that can be expected when using the tuning equations presented here, unless the ratio of T I T is close to 0.5. For comparison purposes, the response of a PID controller tuned by using quarter decay ratio formulas (Smith and Corripio, 1985) is shown in Figure 6. For this system, Dahlin's algorithm provides a better response (IAE = 27) compared to the PID (optimum IAE = 30, quarter decay tuning IAE = 42). In general, as the process deadtime increases, Dahlin's algorithm will outperform conventional controllers, providing also deadtime compensation. d. Limitations. The equations listed above are only good for the ranges specified. However, as the T I T ratio approaches 0.5, the process responses become more oscillatory. This is due to the sampling interval approaching the time constant value, resulting in a controller unable
Ind. Eng. C h e m . Res. 1990,29, 927-928 to sample as frequently as needed to give good performance. Therefore, the ratio T/T should be kept below 0.5 for proper sampling. Conclusions The first-order Dahlin control algorithm contains the parameter A, which is used for tuning. This paper presented a method to obtain a first estimate to an optimum value of A. Tuning equations that relate A to the process and sampling parameters ( T , T ) were given. The validity of these tuning equations holds only in the range of sampling time to time constant ratio ( T I T )of 0.001-0.5. The T/7values greater than 0.5 have an insufficient sampling rate for the control algorithm to provide good control. Acknowledgment Financial support for M. Medina from the National Science Foundation through Grant RCD-8854860 is gratefully acknowledged. Nomenclature E,, = present error for discrete controller E,,-l = previous error for discrete controller FOPDT = first-order-plus deadtime IAE = integral of the absolute error ISE = integral of the squared error Kp = process gain, %ACO/%ATO M,, = present output from discrete controller M,-l = previous output from discrete controller
927
s = Laplace transform variable tol = closed-loop deadtime, min to, = process deadtime, min T = sampling time, min z = z-transform variable U = change in process input A 0 = change in process output A’= closed-loop time constant, min 7 = process time constant, min
L i t e r a t u r e Cited Chiu, K.; et al. Digital Control Algorithms, Part I: The Dahlin Algorithm. Instrum. Control Syst. 1973,46, 10. Condon, B. T.; Smith, C. A. A Comparison of Controller Algorithms as Applied to a Stirred Tank Reactor. Presented a t the ISA Meeting, Anaheim, CA, 1977a. Condon, B. T.; Smith, C. A. A Sensitivity Analysis on Dahlin’s Control Algorithm. I S A Trans. 1977b, 16 (4), 23-30. Dahlin, E. B. Designing and Tuning Digital Controllers. Int. Cem. Semin. 1968, 41, 6. Smith, C. L. Digital Computer Process Control; Intext Educational Publishers: New York, 1972. Smith, C. A.; Corripio, A. B. Principles and Practice of Automatic Process Control; John Wiley and Sons: New York, 1985.
Sylvia Bray, Maximino Medina, Carlos A. Smith* Chemical Engineering Department University of South Florida T a m p a , Florida 33620 Received for review July 17, 1989 Revised manuscript received February 1, 1990 Accepted February 26, 1990
CORRESPONDENCE Comments on “Studies on Gas Holdup in a Bubble Column Operated at Elevated Temperatures” Sir: In a recent paper, Zou et al. (1988) propose a gas hold-up correlation for bubble columns operated a t high temperatures: -0.1544 po + p, 1.6105 0.5897 tg = 0.11285(
5)
u (+)
(7)
(1)
In our opinion, the large increase in gas holdup predicted by this gas hold-up correlation with an increase in vapor pressure (P,) is caused by a misinterpretation of their experimental gas hold-up measurements. Zou et al. only measured the gas velocity a t the feed, and they did not presaturate the air feed. Therefore, the true average superficial gas velocity for their experiments increases with an increase in temperature and height due to evaporation. It can be calculated that the superficial gas velocity used by Zou et al. increases by a factor of
due to evaporation in the extreme case of a complete saturation of the air a t the exit of the bubble column. In 0888-5885/90/2629-0927$02.50/0
the relatively low bubble column ( L = 1.05 m) used by Zou et al., the air will not be completely saturated a t the exit of the column, and therefore, the increase in the superficial gas velocity due to evaporation will be less (by an unknown quantity) than predicted by the above eq 2. The fact, however, remains that the true average superficial gas velocity for their experiments increases with an increase in temperature, and because the increase in gas velocity due to evaporation is unknown, no conclusions can be drawn as to the influence of temperature on gas holdup. The correlation of Zou et al. should therefore, in our opinion, not be used for the design of bubble columns a t elevated temperatures. Nomenclature g = gravitational constant, m/sz H = dispersion height, m Po = total pressure of the system, Pa P, = saturated vapor pressure of the liquid, Pa U , = superficial gas velocity, cm/s Greek Letters cG = gas holdup c12 = liquid holdup
0 1990 American Chemical Society
