Ind. Eng. Chem. Fundam. 1980, 79, 93-98 Wankat, P. C., Chem. Eng. Sci., 3 2 , 1283 (1977). Wankat, P. C., in A. E. Rodrigues, Ed., “Percolation Processes: Theory and Applications”, NATO Advanced Study Institute, Espinho, Portugal, July 17-29, 1978, Noordhoff, Leyden, Netherlands, in press, 1979. Wilhelm, R. H., Rice, A. w., Roike, R. w., SWeed, N. H., Ind. Eng. Chem. Fundam.,7 , 337 (1968). Zhukhovitsky, A. A,, ( G a s Chromatography-1960”, R. P. w. scott, Ed., pp 213-300, Butterworths, London, 1960.
93 Receiued for reuiew April 30, 1979 Accepted August 3, 1979
Presented at the 72nd AIChE ~~~~~l Meeting, sari ~ ~ calif., Nov 1979. This research was Partially supported by NSF Grant No. ENG74-02002 A02.
Nonlinear and Nonconventional Liquid Level Controllers Tak-Fa1 Cheung EXXON Research and Engineering Company, Florham Park, New Jersey 07932
William L. Luyben” Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 180 15
The nonlinear “wide-range’’ level controller proposed by Shunta and Fehervari has been examined quantitatively. It was found to be more versatile than the proportional-only, proportional-integral, and proportional-lag (as proposed by Luyben and Buckley) controllers, versatility being defined as the abilii to reach different response specifications. However, it has drawbacks in face of large upsets or large amplitude noise. Piecewise linear approximations of the nonlinear controller have been proposed and their responses examined through digital simulation. A constraint on the rate of change in output was also proposed as a safety feature. These proposed schemes were designed to capture most of the advantages of nonlinear level control while avoiding the pitfalls.
Introduction “Averaging” liquid level control in surge vessels has two conflicting objectives. First, the changes in flow rate out of the vessel should be as smooth as possible so as not to upset downstream processes. Second, level should not be permitted to deviate too far from set point. Obviously, the desirable controller has to achieve some compromise between these two objectives. Averaging level control methods with proportional-only (P) and proportional-integral (PI) controllers have been discussed in textbooks by Buckley (1964) and Shinskey (1967). Cheung and Luyben (1979) examined quantitatively the details of these conventional control systems, particularly when applied to a sequence of process vessels. Design charts were presented that permit the control engineer to select controller settings by specifying the maximum peak height in level (MPH) and maximum rate of change in outflow (MRCO) in response to an expected inflow step disturbance. Luyben and Buckley (1977) proposed a feedforwardl feedback strategy, called a proportional-lag (PL) controller, that maintains most of the desirable flow smoothing features and also eliminates the level offset of P control. In a quantitative analysis, Cheung and Luyben (1979) showed that the PL controller gave responses in between that of P and PI. However, the attainable responses in terms of the two performance criteria (MPH and MRCO) are limited. Shunta and Fehervari (1976) proposed a nonlinear wide-range controller for level control. Such a nonlinear controller is said to fulfill the control objectives of slow response to small disturbances, but fast response to larger upsets. The purposes of the present paper are to verify the claims of the wide-range controller and t o explore a variety of nonconventional level control strategies. 0019-7874/80/1019-0093$01 .OO/O
Wide-Range Controller The Shunta and Fehervari (1976) wide-range controller is actually a PI controller with the proportional mode output given by C , = (251elK)(Kd)e
Kd is the gain at zero error and K is a rangeability factor adjusted in terms of the incremental error that would double the gain. The controller gain is given by d CP K , = - = (1 + (e(K In 25)(251‘IK)(Kd) de The reset time is changed to keep the damping coefficient constant, giving TI
=
710
(1+ le1 K In 25) (25IelK)
The zero-error gain and reset time (Kcoand q0)determine the damping coefficient. Shunta and Fehervari suggested the advantages of the wide-range controller to be: (i) the gain and reset time are changed only as much as necessary to keep level in bounds; (ii) the outflow changes are continuous; and (iii) noise filtering characteristics are good. The responses of the wide-range controller are illustrated by digital simulation. The simulated system (Cheung and Luyben, 1979) is a tank with one inflow and one outflow stream, the outflow being manipulated by the level controller. The parameters of the system are specified as follows: A = 1 ft; AH = 2 ft; Q ,, = 4 ft3/min; initial steady-state level = 50%; initial steady-state flow = 1 ft3/min. Figures 1and 2 compare the responses of a wide-range controller with a PI controller. Figure 1 shows that the wide-range controller responds much faster than the PI 0 1980 American Chemical Society
~
94
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 100
ik
1: NL: K,,
901
TIo = 2; K = 0.47
= 1,
Kc
2: PI.
=
1; Tlo = 2
50 ...................
40t
80.
'
x =
x
3
,
,
01
,
1
,
,
,
,
,
I
,TIME
,
60
40
2,5}
20
O L .25
0
.5
.75
1.25
1.0 KC =
7,
1.5
1.75
2.0
2.25
MRCO/AQ,
Figure 3. P controller tuning chart. PI
0.5 0
1
3
2
4
5
7
6
9 1 0 1 1 1 2
8
120
TIME (MIN.1
~
Figure 1. Comparison of NL and PI controllers responding to large disturbance (AQi = 1 ft3/min). 1: NL Kco = 1, 70= 2; K = 0 . 4 7 2: PL K, = 1, 7, = 2
t
55.
::w 54 53
='
4;e
...............................................
50 49
q,
-
,
45
,
,
,
,
,
,
,
,
-
0
'-.0.3
1
20
0
.25
,5
.75
1.0 K, =
TI ME
1.25
7,
1.5
1.75
2.0
2.25
l.>5
2'.0
2:25
MRCO/AQ,
Figure 4. PI controller tuning chart.
I 1.151
140
120
I
I
,
0
1
2 3
,
8 4
, 5
. 6
1 7
. 8
, 9
1
8
,
101112
T I M E IMIN.1
Figure 2. Comparison of NL and PI controllers responding to small disturbance (AQi = 0.1 ft3/min).
controller after a 100% step increase in inflow. However, as shown in Figure 2, the two controllers give about the same response to a 10% step change. It is seen that the wide-range controller fulflls the objectives of fast response to large disturbances and slow response to small ones. Comparison with P, PI, and PL Controllers Tuning charts based on specifications of MPH and MRCO have been developed for P, PI, and PL controllers (Cheung and Luyben, 1979). These are shown in Figures 3 , 4 , and 5. A similar generalized tuning chart cannot be developed for the wide-range controller because of nonlinearity between system response and size of disturbance. However, it is possible to prepare a tuning chart for a particular system from simulation results. Figure 6 shows such a chart for the previously specified simulated system, with a damping coefficient of one and a 100% step increase disturbance in inflow. In order to compare the various controllers, five sets of specifications were chosen and the corresponding settings read off the charts. Table I lists the results. In this comparison, the P controller is able to meet only one of the five sets of specifications. The limited versatility of this controller is clearly shown. The PL controller, by
*O
t
.k
.25
Ob
.i5
1:O
1.25
1.5
T, MRCO/AQi
Figure 5. PL controller tuning chart. j5
r
J, Y\\\ o
NL
\
S t m i d a r d Taiw
100
51eo Cllallve
25;
51
O
n
I
0 5
'
.
1 0
' 1 5
, 2 0
2 5
3 0
3.5
4 0
4 5
5 0
h l r C O FT3/MIN/MlN
Figure 6. NL controller tuning chart.
adding a feedforward element, meets three of the five sets of specifications. PI and wide-range controllers can meet
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 95 Table I.
Comparison of Settings performance specifications
case 1
controllers
MRCO, MPH, f t 3 min-' % min-' 30 1
p, K ,
KC 0.5
1.6
1.0
2.8
>1
'0.8'
1.25
1.6
1
0.8
1.75
1.2
1
71
KC
I 0.64
(0.83)' 1
20
NL ( 5 = 1)
PL (KF = 1)
PI
KCO 0.36
710
5.6
K 0.25
3
0.32
6.2
0.75
1.1
0.52
3.9
0.77
1.32
1.5
0.5
0.24
8.3
1.0
TF
(0.5)"
2 (1.25)'
3
2.5
15
(1.67)" 4 5 a
10
3.5
20
(2.5)' 1.25
2.5
1.25
4.0
(1.0)' 1.23
>1
12
Satisfies only MPH specification. Case
5
MPH = 205.
MRCO = 2 . 5 Ft3/Min./Min.
80
1.15
1
: IF+--==2.51
m '
2.0
PI, PL
O
1.0
LLpL_.-_L 0
1
2
3
4
5
6
TIFhlE
7
8
9
1 0 1 1 1 2
\:IN
Figure 7. Comparison of P, PI, PL, and NL controllers responding to a small disturbance (AQi = 0.1 ft3/min).
specifications in all five cases. The PI controller in case 1 uses a low gain to keep the outflow change slow and large reset action to keep the level in bounds. As a result, the system is underdamped (C; = 0.64). In contrast, the wide-range controller can always be tuned to give critical damping. Moreover, the zero-error gain of the wide-range controller is much lower than the gains of the other controllers in d cases. This is desirable in smoothing out small flow disturbances. Responses of these four controllers at case 5 specifications are illustrated in Figures 7 and 8. Figure 7 shows the responses to a 10% step increase in inflow. The wide-range controller reacts much slower than P, PI, and P L controllers. Remember, the nonlinear controller settings were determined for a design case of a 100% change in inflow. Figure 8 shows the responses to a 100% step change. The wide-range controller drives level back to set point very quickly, but a sharp outflow overpeak results. This outflow overpeaking may be a major drawback in applications where such flow behavior is highly undesirable. It also points out the problem in tuning a nonlinear controller, whose response to large disturbances is difficult to predict. Flow Noise Filtering The noise filtering (flow smoothing) characteristics of the P and PI controlled systems were shown in Bode plots (Cheung, 1978). The wide-range controller is similarly
0.5'
0
'
1
2
3
4
5
6
8
7
9
101112
TIME (MIN.)
Figure 8. Comparison of P, PI, PL, and NL controllers responding to a large disturbance (AQi = 1 ft3/min). K=0.5, K c o = O . l ,
G(ji
1. N o ~ s eBaiid
=m
3
-05 LOGlO
0
E
=1.0
100,.
2. N o # r e B a n d = 50-
P,:S!
-10
=
0 5
NaiseBLnd=
1 0
I@.
1 5
FREQUEYCY
Figure 9. Frequency response of outflow to noise in inflow for NL controller.
examined in Figure 9. Noise filtering in this case is dependent on the noise amplitude because of nonlinearity. For low amplitude noise, the wide-range controller approximates a PI controller at corresponding gain (Kd) and damping (C; = 1). Since zero-error gain is usually low, the wide-range controller should be a good filter of low-frequency, low-amplitude noise. However, as the noise amplitude increases, the filtering characteristics of the wide-range controller deteriorate very rapidly.
96
Ind. Eng. Chem. Fundam., Vol. 19, No. 1 , 1980
Other Nonconventional, Nonlinear Level Controllers In order to achieve the benefits of nonlinear level controllers while avoiding some of the problems and pitfalls, a “split-range” controller is proposed. This strategy is a generalization of conventional “auto-overrides’’ where a PI controller is used in the small error range and high gain P-only control in the large error range. Both Shunta and Buckley (1971) have discussed the traditional “auto-override’’ scheme applied to level control. It is a piecewise linear approximation of a nonlinear controller. One of its problems is that the outflow may change faster than desirable when the high gain P controller takes over control. Also, the PI controller does not allow good noise filtering in the small error range. Moreover, as pointed out by Shunta and Fehervari, the level may hang up away from the set point during override, while the PI controller slowly integrates up or down to resume control. The “split-range’’ control scheme seeks to improve on the conventional override scheme by arranging different controllers in different configurations. These will be discussed in the following sections. A. PIP (Proportional-Integral/Proportional) Controller. Instead of the conventional “auto-override’’ scheme, the controllers can be reversed so that the P controller acts within the normal error band and the PI controller comes into action when the error goes outside the normal error band. The advantage of this is that for most small disturbances, P-only control is used and allows good noise filtering. For large disturbances, the integral action forces the error to return to within the band quickly. The gain is kept constant at all times so that the outflow changes smoothly. The controller action can be summarized by the following equations. For le1 I eb CO = Bias Kce
+
For le1 > eb CO = Bias
&‘lo/ OL
2.51
0.5l 0
71
’
1
’
2
’
3
’
’
4 5 6 7 8 9 1 0 1 1 1 2 TIhlE IMIN.1
Figure 10. Response of PIP controller.
the PI controller tuning chart (Cheung and Luyben, 1979). An initial estimate of the reset time can also be found from the PI chart using the MPH specification. Another reset time estimate can be obtained by using the MPH minus the band. The correct rI should fall between these two estimates and is found by trial-and-error. The width of the band deDends on how much steadv-state offset can be tolerated. B. DRIP (Dual Range Integral/ProDortional) Controller. The DRIP controllerises’ PI control both inside and outside of the band. Only the reset time is changed when the error goes from one region to the other. The controller action is summarized as follows. For le1 < eb
t + Kce + Kc - 1 ( e f eb) dt
CO = Bias
+ K,e + Kc - f
t
e dt
712 J O
0
If e > 0, the sign within the integral is negative and vice versa. The error integral is set up so that only the error outside of the band is integrated. This allows smooth outflow changes during control mode switching. The integral action is frozen during P-only control. The frozen error integral simply adds on to the old Bias to form a new Bias. The responses of this PIP controller to positive and negative inflow step changes, for the previously defined standard tank, are shown in Figure 10. The PIP controller responses are very much like that of a PI controller. The outflow changes smoothly and even though the PI settings would normally give a damping coefficient of 0.5, the responses seemed overdamped. The main drawback of the PIP controller is that the level will return to within the band, but not necessarily to the set point. Where the final steady-state level ends up depends on the magnitude of the frozen error integral. With PIP control, it is possible to use a relatively low gain to allow good noise filtering and fast reset action to keep the level in bounds. The P-only control region acts as a damper to check the oscillatory response caused by the PI control. However, the normal error band cannot be made too large because of the path dependent steadystate level offset. When tuning the PIP controller, the MRCO specification immediately fixes K,. This gain can be found from
TIME
For le1
> eb CO = Bias
+ K,e + Kc 711
r‘e dt
Jo
A large r12is used to give [ > 1 within the band and a small 711 is used to give [ C 1 outside the band. By adjusting the width of the normal error band and the two reset times, it is possible to keep the overall process response always near critical damping. Figure 11shows the simulated responses of the standard tank with DRIP control. The DRIP controller fulfills the objectives of slow action at small error and fast action at large error. No level offset would result from load changes; therefore, a wider band can be used. The disadvantage, of course, is poorer noise filtering at small error. In tuning the DRIP controller, the gain is again fixed by the MRCO specification. A reset time that meets the MPH specification and gives E = 1 is found from the PI tuning chart. Estimates of rI1(giving E < 1)and r12(giving .$ > 1) can then be made. The controller is fine tuned experimentally. A wider band can be used with this controller. C. Limited Output Change (LOC). The rate of change in the controller output can be constrained to meet the MRCO specification. This constraint can be applied to any controller as a safety feature. It can also be used to eliminate some of the uncertainties in tuning nonlinear
Ind. Eng. Chem. Fundam., Vol. 19, No. 1, 1980 97 -
LOC
MRCO
=
PI
0.75 FL3/Min./Mtn
-------
301
3
40
20
2.51
I"-;.-;--
F z , 0 _. c __._.-._.
\\
L4
0,50
1
2
3
a
4 k 7 Q 9
io
ii i z
TIME, MIN.
Figure 11. Response of DRIP controller.
controllers. Application of the LOC feature to conventional P and PI controllers allows these to be tuned quite differently, enhancing their flexibilities. The constraint action can be accomplished in digital control by the following scheme. In each sampling period: (i) Store old value of controller output (Coold). (ii) Go through control calculations to get CO,,. (iii) Calculate change ACO = CO, - Cool+ (iv) If ACO does not exceed the specified limit ACO, ( = MRCO * TJ, CO = CO,,. with (v) If ACO exceeds the limit, CO = COold f ACO, the sign depending on the sign of the error. (vi) Update cool*= co. When the LOC feature is used with PI control, care must be taken to prevent reset windup. In digital control, an anti-windup feature can be added by recalculating the error integral (ERINT) from the constrained controller output a t the end of each sampling period; Le. ERINT = CO - Bias - K,e Figure 12 shows the behavior of the PI controller with LOC. I t is seen that constraining the output would effectively reduce damping. The best response seems to be obtained when a high gain and a large reset time (Le., 5 > 1 for no constraint case) are used. Behavior of other controllers with the LOC feature and experimental verification of results can be found in the dissertation of Cheung (1978). Conclusions The nonlinear wide-range controller fulfils the objectives of providing fast control action for large disturbances and slow action for small disturbances. On the whole, it is more versatile than the P, PI, and PL controllers, versatility being defined as the ability to reach different response specifications (MPH and MRCO). It also shows good filtering characteristics of low-frequency, low-amplitude noises. However, because of its nonlinear nature, this controller is difficult to tune and its responses hard to predict. Moreover, the noise filtering characteristics deteriorate very rapidly as noise amplitude increases. This leads to the major drawback of sharp outflow overpeaking resulting from large upsets. The PIP and DRIP controllers have been proposed as piecewise linear approximations of the nonlinear controller. They utilize conventional P and PI controllers in "splitrange" schemes, the overall effect being to have nonlinear
1.0
0.5j' 0
1
2
'
"
3
4
5
'
"
6
7
a
"
9
i
o
"
i
i
'
i
z
TIME IMIN.1
Figure 12. Response of LOC-PI controller.
control whose behavior is easily predicted. The predictability factor is extremely important in critical applications. The LOC feature forces the outflow to follow the maximum change path more closely. It allows the conventional P and PI controllers to be tuned quite differently, and ach as a safety feature on nonlinear controllers. It is highly recommended, especially in digital control applications where the implementation is relatively simple. Digital simulation results have been presented in this paper. All these controllers were tested experimentallyand simulation results were verified (Cheung, 1978). Nomenclature A = cross-sectional area CO = controller output C = proportional mode output DkIP = dual range proportional-integral controller e = error eb = error band width ERINT = error integral in PI controller K = nonlinearity factor K, = controller gain Kco = zero-error gain of wide-range controller LOC = limited output change feature AH = level range MPH = maximum peak height MRCO = maximum rate of change in outflow P = proportional-only PI = proportional-integral PIP = proportional-integral/proportional PL = proportional-lag Qi = inflow AQi = magnitude of inflow step disturbance Qo = outflow QO$mr= maximum outflow t = time T,= sampling period X = Qo.mdAQi Greek Letters
6 = damping coefficient 71 = reset time 710 = zero-error integral time constant for wide-range controller 711, 712 = integral time constants for DRIP controller 7v = effective holdup time = V/Q,,, Literature Cited Buckley, P. S., "Techniques of Process Control", p 167, Wlley, New York, N.Y., 1064.
98
Ind. Eng. Chem. Fundam. 1980, 79, 98-103
Buckley, P. S., Control Eng., 82 (Oct 1971). Cheung, T. F., Ph.D. Dissertation, Lehigh University, 1978. Cheung, T. F., Luyben, W. L., ISA Trans., 18(2), 73 (1979). Cheung, T. F., Luyben. W. L., Ind. Eng. Chem. Fundam., 18 15 (1979). Luyben, W. L., Buckley, P. S., Instrum. Techno/.,24, 65 (Dec 1977). Shinskey, F. G.,“Process Control Systems”, p 147, McGraw-Hill, New York,
N.Y., 1967. Shunta, J. P., Fehervari, W., Instrum. Techno/.,23, 43 (Jan 1976).
Received for review M a y 4, 1979 Accepted October 1, 1979
Physical Interpretation of the Feasibility Region in the Combustion of Char by Use of Single and Double Film Theories Chrlstos Georgakis, John Congalidis, and Yam-Yee Lee Department of Chemical Engineering and Energy Laboratory, Massachusetts Institute of Technology, Cambridge. Massachusetts 02 139
Three single- and double-film models are formulated for the combustion of char particles by assuming an infinitely slow or infinitely fast reaction rate for the homogeneous oxidation of CO. Their solution provides a physical interpretation of the boundary curves of the feasibility region of a more general reaction and diffusion model in which the reaction rate of CO oxidation is finite. Upper and lower bounds are obtained not only for the char particle surface temperature and species concentrations, but also for the species fluxes. The double-film model is a generalization of a previous one in which the heterogeneous C-COP reaction was assumed as infinitely fast. At high ambient temperatures the char particle temperature rise is equal to the adiabatic,temperature rise of the 2C -k O2 -,2CO reaction and is independent of the rate of the homogeneous CO oxidation.
Introduction The phenomena associated with the combustion of coal have been the subject of extensive investigation because of their significance in a variety of industrial applications. Even after the bulk of volatile evolution has taken place, the burning of the residual char is a complicated process involving several homogeneous and heterogeneous elementary reactions. Among the mechanisms postulated for the combustion of carbon particles are: (i) a two-film mechanism in which it is assumed that carbon is consumed due to reaction with C02, while the CO thus formed diffuses away from the particle and reacts with the O2 diffusing towards it at a reaction front to form C02 (Avedesian and Davidson, 1973; Van der Held, 1961) and (ii) a single-film mechanism (Nusselt, 1924; Hougen and Watson, 1947) in which it is assumed that carbon particles react directly with O2 to produce varying amounts of CO and COPin the boundary layer of the particle. Recently, Caram and Amundson (1977) proposed a more general model for the combustion of carbon in which the homogeneous oxidation of CO to C02 is assumed to take place throughout a finite boundary layer at a finite reaction rate. Furthermore, they allowed for the heterogeneous reaction of carbon with both C02 and 02. After some mathematical manipulation of the reaction and diffusion equations, they reduced the model to a single second order nonlinear differential equation in the mass fraction of the COz (eq 33 of Caram and Amundson (1977) with boundary conditions at the surface of the char particle and at the edge of the boundary layer. Analytical expressions were also provided to relate the COPprofiles in the boundary layer to those of the CO, 02,and the temperature profile. Because of the severe numerical problem encountered, Caram and Amundson (1977) introduced the concept of a feasibility region in the plane of char surface tempera-
ture, T,,vs. bulk temperature, Tb. This feasibility region is obtained by calculating three bounding curves I, 11, and 111 (Figure 1) which correspond to zero mass fraction of C02, 02, and CO at the char particle, respectively, and provides a lower and an upper bound for the surface temperature T , for any given ambient temperature, Tb. Such bounds of the surface temperature, Caram and Amundson argued, provide the means for a better guess of the surface temperature which, they argued, facilitates the solution of the complete problem. It is obvious that there is a need to relate the general formulation of Caram and Amundson (1977) to the specific and more restrictive single and double film theories and examine under which conditions the generality of the former reduces to the specificity of the latter. As a first important goal in achieving this, the present paper presents a physical interpretation of the bounds of the feasibility region. It is shown that the three curves I, 11,and I11 that bound the feasibility region of Caram and Amundson (1977) provide relationships between T, and Tb that can be obtained by three single or double film models. In the following three sections, these film models will be postulated and solved. It will be shown that one can thus obtain eq 35-38 of Caram and Amundson (1977) when the bulk concentration of C02 and CO is zero. Although both Caram and Amundson (1977) and subsequently Mon and Amundson (1978) have looked at more general situations, we will focus our attention here on the case of a flat plate geometry and we will neglect bulk diffusion terms as was also assumed by Caram and Amundson (1977). These assumptions are made in order to limit the mathematical complexity of the equations and thus enable the physical picture to emerge quite clearly. The Feasibility Region In Table I a slightly different notation than that of Caram and Amundson (1977) is detailed. This notation
0019-7874/80/1019-0098$01.00100 1980 American Chemical Society
