Ind. Eng. Chem. Res. 1993,32, 466-475
466
PROCESS DESIGN AND CONTROL Dynamics and Control of Recycle Systems. 1. Simple Open-Loop and Closed-Loop Systems William L. Luyben Department of Chemical Engineering, Zacocca Hall, Lehigh University 111, Bethlehem, Pennsylvania 18015
This paper is the first part of a series of papers that explore the challenging problems associated with the dynamics and control of recycle systems. The purpose of this paper is to set the stage for the subsequent parts. The problem will be defined and the literature surveyed. Then some of the interesting phenomena, most of which have been discussed by previous workers, will be illustrated by the use of some very simple systems. First transfer function models will be used to show the effects of various parameters (steady-state gains and time constants) of the individual units in a recycle process and the overall dynamics of the total process. Then a simple reactor/column process will be considered. No claim is made that the material in this paper provides a significant advance to the field of recycle control. Its purpose is to illustrate some well-known and some not-so-wellknown phenomena that occur in recycle systems. Generic guidelines and a general design methodology will be developed in future parts.
Introduction A. Cascaded Units. The dynamics and control of continuous process units that operate as a cascade of units (either in series or in parallel) have been extensively studied for many years. A wealth of knowledge on how to design easily controllableprocesses and how to configure effective control systems is available for a large number of unit operations when these units are run completely independently. This knowledge can be directly applied to the plant-wide control problem if a number of process units are linked together essentially as a sequenceof units. Each downstream unit simply sees disturbances coming from its upstream neighbor. The design procedure was proposed almost three decades ago (Buckley, 1964)and has been widely used in industry for many years. The first step is to lay out a logical and consistent "material balance" control structure that handles the inventory controls (liquid levels and gas pressures). This "hydraulic" structure provides gradual, smooth flow rate changes from unit to unit. This filters flow rate disturbances so that they are attenuated and not amplified as they work their way down through the cascade of units. Slow-acting, proportional-only level controllers provide the most simple and the most effective way to achieve this flow smoothing. Then "product quality" loops are closed on each of the individual units. These loops typically use fast proportional-integral controllers to hold product streams as close to specification values as possible. Since these loops are typically quite a bit faster than the slow inventory loops, interaction between the two is often not a problem. Also, since the manipulated variables used to hold product qualities are quite often streams that are internal to each individual unit, there may be little effect on downstream processes of making these changes in manipulated variables. The manipulatedvariables frequently are utility streams (cooling water, steam, refrigerant, etc.) which are provided by the plant utility system. Thus the boiler house will be disturbed, but the other process units in the plant will not see disturbances coming from other process units. This is, of course, only true when the plant 0888-588519312632-0466$04.00/0
utilities systemsthemselves have effective control systems that can respond quickly to the many disturbances that they see coming in from units all over the plant. As an example of a cascade system, let us consider a sequence of distillation columns in which the bottoms product from each column feeds a downstream column. The inventory structure is chosen to have the base level in each column controlled by manipulating its bottoms flow rate, and reflux drum levels are controlled by manipulating distillate product flow rates. We assume for purposes of this example that column pressures are controlled by manipulating cooling water flow rates to the condensers. Each column will have two degrees of freedom left (reflux and vapor boilup), so some combination of two variables can be controlled in each column: two compositions, two temperatures, or one temperature and one flow. Vapor boilup changes require changes in steam flow rate and indirectly also in cooling water flow rate (through the action of the pressure controller). Both vapor boilup and reflux changes affect the two liquid levels (and therefore the distillate and bottoms flow rates), but the proportional level controllers can usually provide effective filtering of these disturbances. Since the propagation of disturbances in such a system is sequential down the flow path, the use of feedforward control on each unit can also help to improve the product quality control. It should be noted that the inventory controls can be in the direction of flow (products come off on level control) or in the opposite direction (feed is brought in on level control). The same design procedure applies in either case. B. Recycle Systems. All of the above discussion applies to cascades of units. If recycle streams occur in the plant, the procedure for designing an effective "plantwide" control system becomes much less clear and much less studied. Processes with recycle streams are quite common, but their dynamics are poorly understood at present. This is one of the most important areas in process control that cries out for some engineering research. The typical approach in the past for plants with recycle 1993 American Chemical Society
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 467 KR
R
I
I
I
I
Figure 1. Simple open-loop process with recycle.
streams has been to install large surge tanks. This isolates sequences of units and permits the use of conventional cascade-processdesign procedures. However,this practice can be very expensive in tankage capital costs and in working capital investment. In addition and increasingly more importantly, the large inventories of chemicals can greatly increase safety and environmental hazards if dangerous or environmentally unfriendly chemicals are involved. There have been a limited number of literature papers considering recycle dynamics. One of the earliest papers by Gilliland et al. (1964) considered a reactorldistillation column process in which the dynamics of the column were neglected. They were one of the first to point out that the effect of recycle was to increase the overall time constant of the process. Verykios and Luyben (1978) studied a slightly more complex process with simplified column dynamics included. They showed the steady-state sensitivity of recycle and purge flow rates to various parameters, and they were the first to indicate that these recycle systems can exhibit underdamped behavior. Luyben and Buckley (1977) discussed some of the problems that occur in a liquid recycle system in which each of the individual units in the series uses a proportional level controller, not proportional-integral, and a liquid surge tank is used in the loop. The proportional controllers produce steady-state offsets in the levels in all of the process units, and this results in very large changes in the level of liquid in the surge tank. They proposed a combined feedforwardlfeedback level control structure that maintained the flow smoothing of proportional controllers and at the same time avoided the very large surge-tank sizes. Kapoor et al. (1986) used a simple transfer function model, similar to what is done in the first part of this paper, to show why high-purity distillation column time constants are so large. They refer to the previous work by Denn and Lavie (1982) and Rinard and Benjamin (1982) which showed that the response time of recycle processes can be substantially longer than the response time of the forward path alone. Taiwo (1986) discussed robust control of plants with recycle. Douglas (1988) gives several examples of recycle processes and studies in detail the steady-state design and economics of the HDA process. This paper attempts to illustrate some phenomena of recycle systems by studying both the open-loop and the closed-loop behavior of some simple systems. Most of these phenomena have been discussed by previous workers. Later papers will provide analysis of more realistically complex processes from which generic guidelines and a control system design methodology will be developed. Open-Loop System with Recycle The first system studied, shown in Figure 1,consists of two units. The one in the forward path has a simple linear
real
Figure 2. Roots of characteristic equation of open-loop recycle process as a function of recycle loop gain KR.
transfer function consisting of a steady-state gain, K , and a first-order lag with time constant T. The unit in the recycle path also has a simple gain and lag transfer function (KRand T R ) . Thus an extremely simple linear open-loop plant is assumed. The load disturbance into the plant is L, and the output of the plant is X. The output of the plant is also fed back into the recycle unit, producing variable R,which then recycles back into the first unit. The equation describing the overall system is
K(?RS + 1)
-X=
L
TTRS'
+ ( T + TR)S + 1- KKR
(1)
The characteristic of this open-loop system is
+ + T R ) S + 1- KKR
T T R S ~ (T
0
(2)
This is a second-order system, and the roots of its characteristic equation can be plotted as a function of various parameters. Figure 2 gives a plot of the roots of eq 2 as a function of the gain in the recycle loop KR.Only the upper half of the s-plane is shown since the lower half is simply the reflection over the real axis. Specific numerical values of the parameters in the forward path are chosen to be T = K = 1. The time constant in the recycle path T R is set equal to 1 in Figure 2, but other values give similar plots that are shifted to the right along the real axis as TR is increased. This plot is not a conventional root locus plot, which by definition is a plot of the roots of the closed-loop characteristic equation as a function of the feedback controller gain. This plot applies for an open-loop system with variable recycle loop gains. This plot provides a nice picture that helps us visualize what happens to the roots of the openloop characteristic equation as K R changes. As K R increases from 0 toward +1,one of the roots gets closer and closer to the origin, so one of the time constants is getting larger and larger. This shows quantitatively why some recycle systems exhibit very large time constants. When K R is exactly equal to unity, the process contains a pure integrator. For recycle gains greater than K R = +1,one root of the characteristic equation lies in the right half of the s-plane, so the system is unstable. This is exactly what we would expect aince positive feedback with a gain greater than unity leads to instability. When the recycle gain is less than zero, the roots become complex (for 7 R = 1). This indicates that the presence of a recycle unit with a negative gain can produce underdamped behavior. This result may be somewhat less than
468 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993
faster since the roots of the characteristic equation are located farther from the origin (see Figure 2). Increasing the recycle time constant makes the overall process slower as expected (note the change in time scales in Figure 3). Increasing the recycle gain (negatively) reduces the steadystate gain of the overall process (XIL) since the steady-state gain is (from eq 1 )
Equation 2 can be used to calculate the damping coefficient fo and time constant T~ of this simple openloop process.
os 04
03
Equation 4 indicates that the time constant of the process increases as TR increases and as K R increases. The time constant becomes infinite as the recycle gain approaches +l. Equation 5 shows that the damping coefficient decreases as K R becomes more and more negative. Figure 4 shows these dependences of T~ and fo on the recycle gain K Rand on the recycle time constant TR using K = T = 1for illustration purposes. The more negative the gain K R ,the smaller the damping coefficient. It should be noted that the example considered above has two first-order lags, which gives a total system that is second-order. From feedback control theory we would expect underdamped behavior as the gain in the loop becomes larger in a negative sense, but the system will never be unstable. If the order of the totalsystem is greater than 2, we know that large gains will cause instability. For example, if we use two first-order lags in the forward element l/(s + 112,the value of the recycle loop gain at which the system becomes unstable is 4 + 2TR + 2/m.If TR = 1, this maximum gain is 8. We will discuss these maximum gains in more detail in the next section. These basic insights should be useful in analyzing more complex recycle systems. The important parameter is the total gain through the recycle loop: KKR.
02
01
0
-01
0
os
1
IS
2
2s
3
35
4
Time (minutes)
Time (minutes)
Figure 3. Step response of simple open-loop recycle process. top) T R = 0.1; (b, middle) TR = 1; (c, bottom) T R = 10.
(a,
obvious. The effect is illustrated in Figure 3 for three different values of TR (0.1, and 10) and three different values of K R(-2, -10, and -50). A unit step input in load was made at time t = 0. These time responses were very obtained by using the "lsim(num,den,u,t)n file from MATLAB. Recycle systems with negative gains are rare, but some do exist. One example is in a reforming process where the hydrogen gas produced in the reforming reaction is blended with natural gas and the mixture is burned in the furnace used to heat the feed stream to the reactor. An increase in furnace exit temperature causes more hydrogen to be produced. This lowers the heating value of the fuel gas, so the furnace exit temperature may be decreased by the effect of the recycle loop. As recycle gain is increased (in the negative sense), the process becomes more underdamped. It also becomes
Closed-Loop System with Recycle The next simple system to be studied is a slight modification of the one explored above. Now a feedback controller has been added to the first (forward) unit as sketched in Figure 5. Unit 1 has two inputs. The manipulated variable M,which is set by the feedback controller BB),enters the process through the transfer function GM(~).The load variable L and the output R of the recycle element G R ( ~in) unit 2 are summed and enter unit 1through the transfer function GL(~).Thus unit 1 now has a feedback controller that attempts to keep the output variable X near ita setpoint value X w t . The firat task is to design the feedback controller B(*). For illustration purposes, we pick identical second-order lag transfer functions for both GM and GL.
A proportional-integral controller is selected with reset time set equalto T . The closed-loop characteristic equation for just unit 1 is
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 469 Damping Coef.; solid KR10.2. duhtd: KR4.8. d u d : KR10.95
Dmpmg C a f , rohd KR--50. d s h e d KR=-IO. dotted KR=-2 101
d
101 10'
."
101
102
101
100
100
10'
10'
TUR
TauR
T h COllSImt; Solid: KR=-50, dashed: KR=-10. dotted: KR=-2
dam oh^ Coef : solid KR=0.2. dashed: KRd.8. dotted: KR4.95
100
10'
'30.1
101
10 '
TauR
TIUR
Figure 4. Damping coefficients and time constants of simple open-loop recycle process. (a, left) Negative recycle loop gains KR;(b, right) positive recycle loop gains KR.
and solve for the required controller gain.
UNIT 2
r--------I
K, = 0.778IK (11) Now the transfer function GcL(~)that relates the input (L + R ) of unit 1to the output X of unit 1can be found. This is the relationship for unit 1 with the feedback controller on automatic.
I
I
&- .
X
I L
m B
+
__________J
Figure 1. Closed-loop recycle process.
T'S'
[&]sz+
+ TS + 1 + KK, = 0 [*Is
+1=0
+
KTS
+
(12)
= = (75 1)(T2S2+ TS 1.778) Note that there is no steady-state offset (aswe would expect with the proportional-integral controller) because of the "s" term in the numerator. Figure 6 shows the time response of unit 1to a step change in load with no recycle. The loop tuning looks reasonable and settles out in about 10 min. Combining unit 1 (GcL(~)) with the recycle unit (unit 2 with GR(~)) gives the transfer function for the overallclosedloop process. GCL(s)
L+R
(8) (9)
This is in the form of a standard second-order underdamped lag with time constant T , and damping coefficient
The transfer function chosen for the recycle unit (unit 2) is the same as that used in the open-loop recycle process studied first.
(14)
fC.
+ 2Tc5',8+ 1 = 0
(10) We select a cloeed-loop damping coefficient f, = 0.375 7:s'
The characteristic equation of the entire system is the denominator of the transfer function in eq 13 set equal to
470 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 1
035,
Table I. Maximum and Minimum Recycle Gains TR
0.1 1 10 x ,
-01 0
2
4
8
6
10
12
14
t
18
16
-
20
Tim (nlinutcl)
Figure 6. Closed-loop response of unit 1 to a step change in load (with no recycle). 3, Y I
-KR=-8
1
-1 5
-0s
56
0
05
I
led
Figure 7. Roots of characteristic equation of closed-loop recycle process as a function of recycle loop gain KR.
zero. 1- G,,GR = 0
KTS
1(75
(15)
KR
+ 1)(72S2 + 75 + 1.778) 7Rs
+
(16)
For a general system with the structure shown in Figure 5, the characteristic equation can be expresed in terms of the process transfer functions and the feedback controller transfer function.
+ GMB-GLGR-0
(17) The roots of the characteristic equation can be plotted as a function of the recycle gain KR. Substituting the specific numerical values of K = 7 = 1 into eq 16 and rearranging give a fourth-order polynomial in s. 1
(7R)s4
+ (1 + 27R)s3 + (2 + 2.7787R)s2+
(2.778 + 1.7787R - K,)s + 1.778 = 0 (18) Figure 7 gives an s-plane plot (upper half) of the roots of eq 18 as a function of the recycle loop gain K Rwith TR set equal to 1. When K R is zero, two of the roots are complex conjugates with a0.375 damping coefficient. This is from the tuning of the feedback controller in unit 1. As K R is increased positively, two of the complex roots move to right. At K R = +3.34, the roots lie right on the imaginary axis, and for larger values of K Rthe system is unstable. Remember that the open-loop system became unstable a t K R = +l. Thus, in the closed-loop system, a larger maximum gain in the recycle element can be
Kill, 1.98 3.34 19.3
Kmin
-23.4 -8.56 -40.7
Table 11. Parameter Values and Steady-State Values of Variables for Reactor/Distillation Column Recycle Process fresh feed 239.5 lb-mol/h fresh feed composition 0.90 mole fraction A reactor holdup 1250 lb-mol 0.34h-’ specific reaction rate reactor composition 0.5 mole fraction A 500 lb-mol/h column feed rate 260.5 lb-mol/h distillate flow rate reflux flow rate 572.1 lb-mol/h bottoms flow rate 239.5 lb-mol/h 833.2 lb-mol/h vapor boilup total trays 20 feed tray 12 distillate composition 0.95 mole fraction A bottoms composition 0.0105 mole fraction A 2 relative volatility column diameter 6f t 1 in. weir height weir length 4.8 ft height over weir 0.843in. base holdup 89.4 lb-mol 69.4 lb-mol reflux drum holdup 4.342lb-mol/tray tray holdup tray hydraulic time constant 5 s (0.001388h)
tolerated. This is because the effective gain in the forward element hae been changed by the addition of the feedback controller. On the other hand, as K R is made more and more negative, the two complex roots also move to the right. At K R= -8.56 these roots lie right on the imaginary axis. For larger negative values of KR, the system is unstable. Remember that the open-loop system became unstable at K R = -8 (for T R = 1). Thus the minimum gain in the closed-loop system is almost the s m e as in the open-loop system. Thus we have the very intriguing problem of trying to design and operate a system that exhibits “conditional stability”. If K Ris too large, the system is unstable. If K R is too small, the system is also unstable. Table I gives the maximum and minimum values of K R for several values of the recycle time constant 7R. The range of stable gains (the difference between Kmaxand Kmin)is the smallest when the recycle time constant TR = 1(Le., when the time constant of unit 2 is close to the time constant of unit 1). These effects are shown in Figure 8. A unit step change in load was made at time t = 0, and the output X of the entire recycle system is shown with unit 1on closed-loop control. Three different values of T R are shown in Figure 8a-c. It can be seen that if KR is too large, the system is unstable. And if it is too small (too large negatively), the system is also unstable. For example, in Figure 8a where TR = 0.1, when K R = -30 (lower figure), the process is unstable. For K R = -20, it is stable but quite oscillatory. For K R= -10, the response is good. For K R = -2, response is slower and the error is much larger. For K R= +2.5, the system is again unstable.
A More Realistic Reactor/Distillation Column Recycle Process In order to show how the results of the very simple linear systems studied above also apply to more complex processes, a reactor/distillation column system wae in-
Ind. Eng. Chem. Res., Vol. 32, No.3, 1993 471
03
i?:"
:I: 02
x
01
0
41
2
0'
4
I
6
IO
I2
I6
11
I8
"0
20
2
4
6
I
IO
I2
I4
I6
I8
t
I
XR=.10
20
I
OJ
0
45 I
IJ
2 2
"0
4
6
IO
8
12
14
I8
16
M
h1-(
8. C l o d - l o o p rerpow of total recycle system to a step change in load. (a, left) 7R = 0.1; (b, middle) m = 1; (c, right) m = 10.
-re
m
I
xe
L-
1"
Figure 9. Rsrctor/dbthtion column proam. REACTOR r-----------1
FoI 20
I
>
IF I
'-t---------
I Figure 11. C l o d - l o o p mpow of column alone to rtap change8 in feed oompooition.
aL (1987). Ae sketched in Figure 9, the p r o c 8 ~is a continuous stirred-tankrqactor followed by a distillation veatigated. The process ie that studied by Price (1992) which wm baeed on a procese studied by Papadourakhet
-
COlrUnn. A fuetorder, isothermal, irreversible miction A B occura in the liquid phase in the reactor. Froah f e d with flow rate 3'0 (239.5lb-mol/hr)and compositionro(0.9 mole
472 Ind. Eng. Chem. Res., Vol. 32, No.3, 1993
Figure 12. Closed-loop response of reactor/column system (VR= 1250). Table 111. Effects of Changing Fresh Feed Concentration on Steady-State Values of Variables 10 z F D V %D Fo=B 0.967 239.5 440.2 200.7 773.4 0.8 0.447 833.2 0.950 239.5 500.0 260.5 0.9 0.50 926 0.924 239.5 593.7 354.2 1 0.555
fraction A) and a recycle stream from the top of the distillation column are fed to the reactor. The recycle flow rate is D (260.51 lb-mol/h) and its composition is XD (0.95 mole fraction A). The specific reaction rate is 0.34 h-1, and the reactor volume is constant at 1250 lb-mol. The column has 20 ideal trays with feed introduced onto tray 12. A partial reboiler and a total condenser are aseumed. Reflux and feed are saturated liquids. Equimolal overflow and constant relative volatility (CYA/B= 2) are assumed. Table I1gives details of the design and operating parameters and the base-case steady-state values of all variables. The reflux ratio in the column is 572.7/260.5 = 2.20). The steady-state vapor boilup is 833.21Ib-mol/h. Note that the holdup in the reactor (1250 ib-mol) is about 5 times larger than the total holdup in the column (20 trays at 4.342 lb-mol/tray, 89.4 lb-mol in the reboiler, and 69.4 lb-mol in the reflux drum give a total of 245.6 mol). The purity of the bottoms stream leaving the distillation column is 0.0105 mole fraction A. This is the only place where material leaves the process, and the fresh feed is the only place where material enters the process. The control system in this process maintains the bottoms product composition ZB near ita setpoint by adjusting the vapor boilup V. A 3-min dead time is assumed to exist in the composition analyzer loop. Perfect level controllers
were assumed for the column base (changingB),the reador (changing and the reflux drum (changing D). Figure 10 gives a multivariable block diagram of this system. It is essentially the same as that given for the scalar case (Figure 5) except for the locations where the load disturbances (Foand EO) enter the process. Unit 1 is considered to be the distillation column. Unit 2 is the reactor and is in the recycle path. A. Considering Only the Column. The standard equations (Luyben, 1990) were used to describe the dynamic response of the distillation column. A linearized version of the Francis weir formula was used (B = tray liquid hy-raulic time constant = 5 8 ) . A linear model of the column was developed and solved in the frequency domain to give various steady-state gains and transfer functions. The transfer function between XB and V was found to be a first-order lag with a 1-h time constant and a steady-state gain of -0.001 04 mole fraction/ (lb-mol/h). Adding the analyzerdead time gave the procees transfer function that was used to tune the X B / Vfeedback controller.
e,
(19)
The ultimate gain is 30 810 and the ultimate frequency is 32.03 rad/h. The Ziegler-Nichols settings are Kc= 14 OOO and 71 = 0.163 h. A nonlinear simulation of the column by itaelf showed that these Ziegler-Nichols settings were much too oscillatory. The gain was halved and the reset time was doubled, and a much more reasonable response was obtained as illustrated in Figure 11. Both step increases ( E goes from 0.5 to 0.6) and decreases (E goes
0.58
15 : . %
: -,
14-
/
i,
8
-
1.3
.,,/-
0%-
4
......_......._......
..........................
%'
I G I
.
, I
,
054-
\,
052-/
VR=125. b3 4 VR=lZS, k=3 4
looo.
400
I
...--.-.... 950
-
I
,,-+--"."
%.
-._
/
350-
..............
%.. ..................
900-
/
300-
VR=125, k=3 4
> 850-j
n
VR1125, k=3 4
700
125).
from 0.5to 0.4)were made in the feed composition to the column. The control of the bottom composition was quite acceptable, with the column lining out in about 1 h and having a peak error of about 0.008 mole fraction. In these simulations, reflux flow rate is fixed. Distillate composition X D is not controlled. The steady-state gain between the distillate composition X D and the feed composition z was found to be 1.448 mole fraction/mole fraction. We will use this below to calculate the total recycle loop gain.
[3] = 1.448 Z column
(20)
B. Combined Reactor and Column System with Recycle. Adding the reactor to the system only required the addition of one equation to the column model, a reactor component balance on A. vR
dz = Fg0 + DxD- FZ - V& dt
(21)
where V R= reactor holdup (lb-mol), z = reactor concentration (mole fraction A), and k = specific reaction rate (h-9. To get the steady-state gains of the reactor, we must linearize eq 21. Only changes in fresh feed composition zo were considered. vR
dz = (Fo)Z, + (D)XD- (E + v ~ k ) Z+ (2, - z ) D dt
(22)
All variables are now in perturbation form. The steady-state gain between z and XD (which is the major element in the recycle loop along with the gain in
the column between
XD
and z ) can be calculated.
Substituting the numerical values gives 260 -Z= XD 500 + (1250)(0.34)
0.281 mole fraction/mole fraction (24) The total loop gain for the reactor/column system is
[53 [41 Z
column X D
= (1.448)(0.281)= +0.407
We would predict that the recycle system will be stable since the loop gain is less than +l. We would expect that ita dynamics would be quite slow since the gain is close to +l. These predictions are seen to be true in Figure 12 where both positive and negative step changes in fresh feed concentration (20 goes from 0.9 to 0.8 and 0.9 to 1.0)were made. The simulation is for the rigorous nonlinear model of the combined reactor/column system. The bottoms composition controller k on automatic and uaes the same tuning as was used for the column operating by itself without any recycle. Comparing Figure 11 (no recycle) with Figure 12 (coupled system) shows that the system with recycle is stable but haa much longer time constants. The column alone settles out in about 1 h. The coupled system takes almost 10 h to come to a new steady state. It is interesting to see the new steady-state conditions that occur when fresh feed concentration is changed. Some
474 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 RcmorIColumn. sohd zc=O 8, dashed
RcmorlColumn: solid: zo=Q.8: dashed: r o i l
20.1
11
109
LOB
'0
1
2
3
4
5
6
7
8
9
10
Time (hours)
RcactorIColumn: solid: rc=0.8:dashed:ro=l
Relcfor/Column: solid: z M . 8 duhcd 2-1
C
280 -
(I'
Time (hours)
Time olours)
Figure 14. Closed-loop response using Ziegler-Nichols settings (VR= 1250).
are shown in Figure 12, and Table I11 gives more information. As zo increases, reactor concentration z increases. This must occur because more reactant A is being fed into the system, and therefore more must react in order to keep the bottoms product purity constant. Remember fresh feed flow rate is constant, so bottoms flow rate must also be constant. Since the volume of the reactor and the specific reaction rate k are constant, the only way to increase the rate of reaction is to increase t. This increase in z is accomplished by reducing the residence time in the reactor: D increases, F increases, and V increases. With the same reflux flow rate and the same number of trays in the column, the increase in feed flow rate to the distillation column results in a decrease in distillate purity XD despite the fact that the feed composition to the column increases. Figure 13showsthe effect of changingthe reactor volume while keeping the distillation holdups the same. The reactor volume is reduced by a factor of 10. Note that the specific reaction rate has also been increased by the same factor of 10 so that the steady-state values of all concentrations and flow rates throughout the system remain the same as the base case. The response of the system is quite a bit faster as we would expect. However, the peak errors in XB are larger since the disturbances that the column sees (Fand z ) occur more quickly. Note that the same controller tuning was used for the distillation column, and it still gives good performance. Figure 14 shows what happens when the tuning of the bottoms composition controller is changed with the original reactor size. The Ziegler-Nichols settings, which were found to give very oscillatory behavior for just the column
by itself, were used in this simulation. The response of the system is only slightly faster than with the detuned Ziegler-Nichols settings, and the loop is still quite oscillatory.
Conclusions This preliminary study of some quite simple systems has illustrated several interesting and important results. The behavior of a recycle system depends strongly on the recycle loop gain and somewhat less strongly on the dynamics of the individual units in the recycle loop. Future papers will study other, more complex recycle processes with more complex reaction and separation systems. Generic rules, guidelines, and design procedures will be developed and tested for a variety of reaction (different types of reactions) and separation systems (different relative volatilities). The first-order reaction systems will contain one recycle stream. the second-order reaction systems may contain two recycle streams. A vital part of the study will be the development of measures to prevent the "snowballn effect (i.e., a small disturbance results in a very large increase in the recycle flow rate). The trade-offs between steady-state design and control in these recycle systems will also be investigated. Nomenclature B = bottoms flow rate (lb-mol/h) B(@,= feedback controller transfer function D = distillate flow rate (lb-mol/h) F = column feed flow rate = reactor effluent (lb-mol/h) Fo = fresh feed rate to reactor (lb-mol/h)
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 475
GCL= closed-loop transfer function for unit 1 CL = load transfer function CM = manipulated variable transfer function GR = transfer function of recycle unit k = specific reaction rate (h-l) K = gain in unit 1 transfer function K, = feedback controller gain K R = gain in recycle (unit 2) transfer function L = load variable R = recycle variable in Figure 1 R = reflux flow rate (lb-mol/h) s = Laplace transform variable V = vapor boilup (lb-mol/h) V R= reactor holdup (lb-mol) X = process output XB = bottoms composition (mole fraction A) XD = distillate composition (mole fraction A) z = column feed composition = reactor composition (mole fraction A) zo = fresh feed composition to reactor (mole fraction A) 0 = liquid hyraulic time constant (h) r = time constant 7c = closed-loop time constant rr = reset time of controller T~ = open-loop time constant T R = time constant in recycle unit lc= closed-loop damping coefficient lo= open-loop damping coefficient
Literature Cited Buckley, P. S. Techniques of Process Control; Wiley: New York, 1964.
Denn, M. M.; Lavie, R. Dynamics of Plants with Recycle. Chem.
Eng. J. 1982,24, 55. Douglas, J. M. Conceptual Design of Chemical Processes; McGraw-Hill: New York, 1988. Gilliland, E. R.; Gould, L. A.; Boyle, T. J. Joint Automatic Control Conference; 1964; p 140. Kapoor, N.; McAvoy, T. J.; Marlin, T. E. Effect of Recycle Structure on Distillation Tower Time Constants. AZChE J. 1986,32,411418.
Luyben, W. L. Process Modeling, Simulation and Control For Chemical Engineers; 2nd ed.; McGraw-Hill: New York, 1991; 129. Luyben, W. L.; Buckley, P. S. A Proportional-Only Level Controller Without Steadystate Offset. Znst. Tech. 1977, 24, 65. Papadourakis, A.;Doherty, M. F.; Douglas, J. M. Relative Gain Array for Units in Plants with Recycle. Znd. Eng. Chem. Res. 1987,26, 1259-1262.
Price, R. Ph.D. Thesis. Lehigh University, Bethlehem, PA. 1993. Riflard, I.; Benjamin, B. Control of Recycle Systems. 1: Continuous Control. Proc. ACC 1982, Paper WA5. Taiwo, 0. The Design of Robust Control Systems for Plants with Recycle. Znt. J. Control 1986, 43, 671-678. Verykios, X.;Luyben, W. L. Steadystate Sensitivity and Dynamics of a Reactor/Distillation Column System withRecycle. ZSA Trans. 1978, 17, 49.
Received for review January 6 , 1992 Revised manuscript received May 5, 1992 Accepted November 28, 1992
