Optimal Sensor Locations and Controller Settings for Class of Countercurrent Processes H. H. Lee,l 1. B. Koppel, and H. C. Lim2 School of Chemical Engineering, Purdue L-niversity,Lafayette, Ind. 47907
Control of heat exchangers and gas absorbers using two sensors, one at the process exit and one a t an intermediate point, is investigated using as a performance criterion the integral-square-error with a bound on control variable. It is shown that sensor location is quite important, the optimal location for the intermediate sensor being beyond 0.7 from the process inlet, that the intermediate sensor improves performance to the point of giving essentially the same performance as the optimal open-loop control, and that sensor location and controller settings are sensitive to process design parameters. The optimal sensor locations and controller settings are given as functions of process design parameters.
O p t i m a l feedback control laws for distributed-parameter systems pioneered by Butkovskii (1963), have been obtained by various methods, such as dynamic programming (Kim and Erzberger, 1967; Sirazetdinov, 1965; Wang, 1964; K a n g aiid Tung, 1964), modal expansion (Gould and MurrayLasso, 1966; Wiberg, 1967), and the formal Hamiltonian approach (Koppel and Shih, 1968; Denn, 1968). While these yield optimal control, there are two major practical difficulties which prevent in many cases the use of the optimal control lam-i.e., the feedback control laws obtained by the above methods require solution of nonlinear partial differential equations and a n infinite number of measuring elements, in most cases, for feedback realization. One way of overcoming these difficulties is to use a finite number of sensors and to fix the form of control function or controller. Inherent in this approach is the trade-off between the number of sensors and the cost of implementation. The more sensors vie use, the better the performance, but the higher the cost of implementation. There are many possible control schemes. One of the possibilities was considered by JIcCann (1966) based primarily on steady-state considerations. One basic question a designer often faces is that of whether any sensor(s), in addition to one a t the process exit, is iiecessary to meet the desired dynamic performance of the process. Lee (1971) has sho\m that the most important factor is the steady-state process design parameters. He showed that when heat exchangers and gas absorbers are designed so that they have large enough mixing capacity or contact area to smooth out the disturbance, a n additional sensor may not be necessary. On the other hand, when the process is designed without excess mixing or contact capacity, a n additional sensor a t the optimal location improves significantly dynamic performances such as the integral-square-error, the maximum output deviation, and response time. I n this paper, control of heat exchangers and gas absorbers using two sensors, one a t the exit and the other a t an intermediate point, is investigated. A measure of performance that yields a reasonable response time aiid yet penalizes 1 Present address, Process Control Hesearch Lahoratory, Westvaco, S o r t h Charleston, S.C. 29405. 2 T o n-horn correspondence should be addressed.
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Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1 , 1973
heavily large deviation is the integral-square-error. This performance index along n-ith a bound on the control variable is used to prepare control system charts which give optimal sensor locations and controller settings as functions of steady-state design parameters. System
Figure 1 presents schematics of the processes studied. For the heat exchanger, the control objective is to regulate the process side exit temperature (controlled or output variable) by adjusting the utility side flow rate (manipulated variable) in the presence of disturbances in process inlet temperature or flow rate (load variables). For the absorber, the output and manipulated variables are the process side exit concentration and the utility (absorbent) flow, respectively. The load variable is the concentration of the key component in the inlet gas stream or the flow rate of gas. By assuming plug flow, constant physical properties, constant flow rate of the process stream and constant inlet quality of the utility stream, linear equilibrium and negligible heat effects in the case of packed absorber, utility side transfer coefficient is the controlling factor in the overall transfer coefficient in the case of heat exchanger, and load variables of small magnitude, Lee (1971) derived the following equations to describe a class of countercurrent heat or mass transfer processes. For constant flow rate of the process stream
bx
Q *at
-
3x2 -
az
=
PR(mx2 - x l )
+ PR(1 - g)f(z)u(t)
where
Sk)
=
for naR
=
1
rllp L[(naR - 1)/(1 - 7nReP(mR-1)11 x exp [P(7nR - l)z] for mR # 1 and g is the exponent in the power law for the transfer coefficient, K = K O(1 u)"
+
T,II,O)
Table 1. Relative Performance Types of Control
of Various
: T,
Cp,(BTU/lb
'F)
, U o ( B T U / h r f1'
Muilb/fl!
Relative performance, .l(uJ/J(u~),
'F)
HR = 2
I
1 2 3 4 5 6 J ( U i ) / J ( , U 1 )=
II
= 1.5, R = 1.O
Control t y p e (i)
{ = 3.0, R = 0.5
1.00 1.00 0.87 0.78 0.60 0.30 0.50 0.24 0.47 0.22 0.46 0.21 minimum value of the integral-square-error obtained with optimal settings for type ? control minimum value of the integral-square-error obtained with optimal settings for type 1 control
U T I L I T Y SIDE
TU10,O) 0)
HEAT
EXCHANGER
LIQUID PHASE (utility side)
GAS P H A S E side 1
I process
x (1.0)
YIl,O) 'Y,
h (Ib m o l e / f t 3 )
K o l l b mole/hr f t 3 )
The initial and boundary conditions are
LIQUID PHASE
GAS PHASE Y 10.0) Yo
x 10.0)
= 0
(4)
z*(z,O) = 0
(5)
21(2,0)
L (ID
mOle/hr f t 2 )
b ) PACKED-BED
i
V(1b mole/hr f l ' )
ABSORBER
.
Figure 1 Schematic diagrams for countercurrent processes
sn(1,t) = 0
(7)
When the load variable is the process-side flow rate, Equation l must be modified by replacing u ( t ) by u ( t ) - 2' and t'he boundary and initial conditions, Equat'ions 5-8, are all zero. Primary attention will be given to the first kind of system dynamics described by Equations 1 t'lirough 7 since, for the same niagiiit,ude, this kind of disturbance is much more difficult to control. Types of Control Schemes
Because of the distributed nature of the process, many condifferent control schemes are possible. In this work -1s :' trol schemes were tested computationally and compared with the optimal open-loop cont'rol vhich minimizes the integral-square-error performance criterion with a bound on the control variable. I n each cont'rol scheme, a n integral mode based on the process esit variable is used to eliminate any offset. Sis different control schemes were used, types 1 through 6. Type 1 is a proportional integral ( P I ) control a t the process-side esit, t.ype 2 a PI control a t the processside esit plus a proportional (P)control a t the utility-side inlet, t'ype 3 a n I control a t the process-side esit plus a P control a t ail optimum point on the process side, type 4 a PI a t t'he process-side esit plus a P control a t an optimal point on the process side, type 5 a PI control a t process-side esit' plus two P controls a t tlyo optimum points on the process side, and type 6 is the optimal open-loop control. Optimal Sensor Location and Optimal Controller Settings
A problem of determining the optimal controller set'tings and optimal sensor locations can be put into the following parameter optimization problem: Given the system wit,h appropriate boundary and initial conditions, Equations 1 through 7 find optimal controller settings and optimal sensor locations for each feedback control of types 1 through 5 , such that a n integral of squared error /-a
J
=
J
0
x1*(1,~)d.r
G iInax, and m j > 1 or m j < 0, then Ginew = G imax and m j = 1 or m j = 0, where Gi nlax is maximum gain possible such that the constraint on the overall gain is satisfied. Calculate ds by Equation A2. Repeat the whole procedure until dJ 5 e , where e is some small positive number. For the control modes of type 1 to type 5, the forcing functions, bu/bGi and du/bmj, that appear in the auxiliary equations of the algorithm can be expressed explicitly. For example, for the type 4 control mode, the forcing functions can be written as follows:
+
+
-
Y(1,O)l
normalized utility side temperature; normalized mole fraction of key component, in the liquid phase = [T,iz,t) Tu(z,O) Il[TP(O,O) - Tp(1,O)1 or, [xu(z,t) - Z U ( Z , O ) !/[2/,(0,0) - YP(1,0)1 y = mole fraction of the key component in the gas phase z = normalized length = z’/Z or z‘/HoG Z = total length of heat’ exchange, ft f = sensor location
x2
=
SUPERSCR~PTS = design value - = deviation from the design value SCBSCRIPTS = final steady state G = gas phase (process side) L = liquid phase (utility side) p = process side, gas phase for absorber u = utility side, liquid phase for absorber
f
GREEKLETTERS = constant in equilibrium relationships, zero for heat exchanger 8 = residence time, hr p = density, lb/ft3 f = degree of approach = ( T o- T,)/(T, - TI), or [yo ?nx(1,0)l/(Yo- 2/11
p
literature Cited
Nomenclature
A A, C, Dd FR
transfer area, f t 2 cross-sect>ionalarea, ft* = specific heat capacit’y, Btu/lb O F = degree of approach, = fractional recovery, I/( G, = dimensionless controller gain G, = process gain q = exponent in the mass or heat transfer coefficient correlat’ion KGa = K o ( l u ) f or L- = C o ( l u)g E l = holdup in the liquid phase, Ib mol/ft3 IIR = heat capacity ratio or holdup ratio: = (pCpAc)u/ ( P G A , )of ~Hih HOG = total height of absorber, ft h = holduil in the gas uhase. Ib m o m 3 J = performance index, intekral-error-square ( I E S ) K = L- or k‘,. KGU = mass transfer coefficient, Ib mol/hr f t 3 K O = design value of Koa =
=
+
Y~
+
Bryson, A . E., Jr., Ilenhani, W.F., Dreyfus, d. F., A I A A J., 1, 247 (1963). Butkovskii, .4. G., Proc. ZndInt. Congr. I F A C , Preprint Paper 513 (1963). llenn, I f . M., IncZ. Eng. Chem. Fundam., 7, 410 (1968). Gould, L. A . , JIiirray-Lasso, 31. A,, IEEE Trans. Automat. Contr., AC-11. 729 (1966). ~ - Kim, l f . , Erzberger: H., ihid,, AC-12, 22 (1967). Koppel, L. B., Shih, Y. P., l n d . Eng. Chem. Fundam., 7, 414 (1968). Lee, H. H., PhD Thesis, Purdue University, Lafayette, Ind., I
~~~
11,-1
1:li I.
McCann, lf.J., Res. Rept., Systems Research Center, Cave Institute of Technology, Cleveland, Ohio, October 1966. Sirazetdinov, T. K., Automat. Remote Contr., 2 6 , 1449 (196,5). Wang, P. K . C., “Control of Distributed Parameter Systems,’’ Advances in Control Systems, Ed. by C. T. Leondes, Academic Press, New York, N. Y., Vol. 1, 1964. Wang, P. K . C., Tung, F., .I. Basic Eng., Trans. A S M E , SdD, 67 (1964). Wiberg, D. >I., ihid., 1, 37’9 (1967). RKCKIVKD for review November 13, 1971 ACCEPTED August 31, 1972 One of the authors (H. H. L.) was supported by Purdue Research Foundation. Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 1, 1973
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