Ind. Eng. Chem. Res. 1987,26, 2161-2162 the program for the column model. Note the extreme stiffness of the high-pressure C3 splitter. This is due to the very small pressure drops through the trays that result from the small vapor velocities. This produces very small time constants in the energy-balance equation.
Conclusions Vapor holdup should be included in dynamic models of columns where the vapor density becomes significant compared to the liquid density. For most common materials, this corresponds to pressures greater than 5-10 atm. A rigorous dynamic vapor hydraulic model should be used for columns where changes in pressures and in pressure drops are significant. The normal application is in vacuum columns. Nomenclature E-E = ethylene-ethane system F = feed flow rate h, = liquid enthalpy on tray n H, = vapor enthalpy on tray n L, = condensation rate in condenser L, = liquid flow rate from tray n M = molecular weight MnL= liquid holdup on tray n MnV= vapor holdup on tray n M-W = methanol-water system PD= pressure in reflux drum P, = pressure on tray n q = feed thermal condition R = perfect gas law constant t = time T,, = average column temperature T, = temperature on tray n T-X = toluene-xylene system
2161
V , = vapor flow rate from reboiler V,, = vapor flow rate from tray n VOL, = total volume of tray n VOLToT = total free vapor volume in column x,j = liquid composition on tray n ynj = vapor composition on tray n 22, = total molar holdup on tray n in both liquid and vapor phases zl,, = total holdup of component j on tray n in both liquid and vapor phases Greek Symbol p = density Superscripts L = liquid phase V = vapor phase Subscripts n = tray number j = component number
Literature Cited Choe, Y. S. M.S. Thesis, Lehigh University, Bethlehem, PA, 1985. Fuentes, C.; Luyben, W. L. Znd. Eng. Chem. Fundam. 1982,21,323. Schiesser, W. E. Lehigh University, Bethlehem, PA, personal communication, 1985. Hindmarsh, A. C.; Sherman, A. H. LSODES, Lawrence Livermore National Laboratory, Livermore, CA, 1983.
Young-Soon Choe, William L. Luyben* Process Modeling and Control Center Department of Chemical Engineering
Lehigh University Bethlehem, Pennsylvania 18015 Received f o r review December 26, 1985 Revised manuscript received February 9, 1987 Accepted June 26,1987
CORRESPONDENCE Comments on “Internal Model Control. 4. PID Controller Design” Sir: In a recent paper, Rivera et al. (1986) proposed tuning rules for Proportional-Integral-Derivative (PID) type controllers applied to first-order plus dead-time processes. They demonstrated that the Integrated Squared Error for set-point changes was a t most 10% larger than the theoretical best achievable performance. We feel that these perforinance claims are misleading since they require excessive controller action. To derive the PID controller settings, Rivera et al. (1986) developed an Internal Model Controller (IMC) for the first-order lag process and substituted a Pade’ approximation for the dead time. The resulting controller had a PID structure, with an additional optional lag. Following this idea for a process modeled as we derived a PID controller with the form u(s) =
(
K , 1 + - (1 + q , s ) e ( s ) 7
3
0888-5885/87/2626-2161$01.50/0
We also derived a “practical” PID controller with the form
where e(s) is the difference between the set pointy&) and the process variable y(s). Controller 3 is termed “practical” by Rivera et al. (1986) since the order of the numerator does not exceed that of the denominator. The tuning parameters for the controllers are given in Table I. We have parameterized the controllers according to eq 2 and 3, as opposed to what was done in Rivera et al. (1986))to enable a comparison of the tuning parameters with those from more conventional methods such as Ziegler and Nichols (1942) and to allow direct application of the parameters to most commercial controllers. The Internal Model Controller for a first-order plus dead time process is commonly known as Dahlin’s algorithm (Dahlin, 1968). It has been the subject of considerable study. The use of Dahlin’s controller as an aid for tuning PID controllers has been examined for first- and second0 1987 American Chemical Society
2162 Ind. Eng. Chem. Res., Vol. 26, No. 10, 1987
Table I. Tuning Rules for PID Controllers‘ PID
“practical“ PID
+ t)
KKC
~/(0.58
TI
T
Tn
812 0.0
TF
T/(B T
t : .
6)
@I 2 TDt/(@
c
+ e)
“ e is the single parameter. Recommended values for c for the PID controller are t Z maximum (0.80, 17) and for the “practical” PID are t 5 maximum (0.458, 0.17).
7: I-
0. 3 4 5 6 7 8 -lME (MIYCITES)
! 0 0
i G 1
i
‘
1 2
,
3 4 - ‘ M E
,
,
,
.
,
.
,
.
,
5 6 7 8 9 ( M NUTES)
.
,
1
.
surement. 2
0+----
-
-
J0 0 L
Figure 1. Set point response: practical PID derivative on error.
10
Figure 3. Set point response: practical PID derivative on mea-
3
0
9
1 I 1
9 .u4,0
1
2
,
3
,
4
7 M E
,
5
_-
. , , , 6 7 8 9 ,
i
o 1
0
(MINUTES)
Figure 4. Set point response: PID derivative on measurement.
measurement signal only. The transfer function for such a controller is
k
2 1.c: CL
i
‘/E
( ‘ ~ 4
‘,,TES:
Figure 2. Load response: practical PID derivative on error.
order processes in Martin et al. (1976). Controller settings, identical with those for the practical PID controller, have been derived in Smith and Corripio (1985). The latter authors did not directly consider the effect of the tuning parameter on robustness. Rivera et al. (1986) studied set point changes for a PID controller and a practical PID controller and compared their performances to the theoretically achievable performance of a Smith Predictor with infinite gain. They showed that the Integrated Squared Error for the PID controller was a t most 10% larger than the theoretically achievable performance. However, they did not stress that infinite control action was required for the PID controller, and in some cases, “large” control action was required for the practical PID controller. In many cases, therefore, the claimed controller performance is practically unattainable. To illustrate this, consider a process with K = 1, r = 4 min, 8 = 1 min, and a controller with a sampling interval of 0.1 s. The response of y to a unit set point change and to a unit load change (a step passing through the process) is shown in Figures 1 and 2 using the practical PID controller with e = 0.45. In both cases, the process response is excellent. The manipulated variable, u, is also shown in Figures 1 and 2. For the load change, u is acceptable, but for the set point change, u is totally unacceptable since it overshoots its steady-state value by 880%. Typically in the process industries, manipulative variable overshoot is limited to 100-200%. Large overshoots are undesirable due to modeling inaccuracies, saturation of final control elements, wear on final control elements, interactions with other process variables, and operator distress. Most industrial controllers use derivative action on the
T F is usually expressed as 017~with cy typically
