over-all heat transfer coefficient O t < O unit step function = jlt>O = normal, steady-state velocity = velocity disturbance = 0 f i ( t ) , total velocity = wall temperature = mass flow rate, massltime = dimensionless distance along tubular process, x’/L = distance along tubular process = normalized set point disturbance for heat exchanger or first-order reactor, defined by Equation 45 = normalized gain for heat exchanger or first-order reactor, defined by Equation 45 = dimensionless magnitude of set point disturbance for heat exchanger or first-order reactor = dimensionless set point value, normally zero = dummy variable of integration = dummy argument of h = dimensionless temperature for heat exchanger, ( W - T ) / ( W - T o ) ; dimensionless concentration for chemical reactor, CA/CA, = dimensionless disturbance in inlet process variable = functional of r ( t ) , defined by Equation 9 = dummy time variable of integration = transformed process variable, defined by Equation = =
Substitution of Equation 70 into Equation 65 yields, after simplification,
+
40(1, t ) --- t [ u ( t - 1) - u ( t ) ] (71) M Equation 71 is plotted in Figure 7, while Equation 69 is plotted in Figures 8 and 9, for .the cases (x = 0.5, K = 2) and (x = 0.8, K = 5), respectively. Nomenclature = dimensionless area under pulse or impulse flow = = = = = = =
= = = = = =
= = =
= = = = = = =
rate disturbance total heat exchange area dimensionless amplitude of step flow rate disturbance dimemionless linear rate of change of over-all heat transfer coefficient with flow rate concentration of reactant normal, steady-state inlet reactant concentration specific heat dimensionless time of duration of pulse controller transfer function functional of r ( t ) , defined by Equation 8 dimensionless loop gain dimension less proportional controller gain reaction velocity constant length of tubular process dimension less amplitude of inlet disturbance, expressed in fraction of full process range order of reaction heat exchange to heat capacity ratio UAh/iwCp in the case of a heat exchanger; ratio of convective to reactive rates of transport of reactant kCAOn-l.L/a in the case of a chemical reactor; dimensionless ratio of initial slopes of set point responses, nonlinear feedback to linear feedback dimensionless flow rate disturbance, C/O dimension1 ess Laplace transform variable temperature normal, steady-state inlet temperature dimensionless time, ot’/L time
9 P
i(x, t )
= deviation in transformed process variable, de-
gl(t)
= transformed inlet disturbance
a,(x) $(x, t ;
fined by Equation 3
K)
&(l,t )
final value of +(x, t ) step response of closed loop system when proportional gain is K = exit process response when proportional control is based upon value of 6 a t location x , using gain given by Equation 31
= =
Literature Cited
(1) Kamman, D. T., Koppel, L. B., IND.ENG. CHEM.FUNDAMENTALS
5. 208 (1966).
(2) Koppe1,L. B.,~Zbid.,’l, 131 (1962). ( 3 ) Zbid., 4, 269 (1965). (4) Weber, T. W., Harriott, P., Zbid., 4,155 (1965). RECEIVED for review June 21, 1965 ACCEPTED December 17, 1965
CONTROL OF TUBULAR HEAT EXCHANGERS AND CHEIMICAL REACTORS BY PIECEWISE CONSTANT MANIPULATION OF FLOW RATE L0W ELL B
. K0PPEL,
class of chemical reactors and heat exchangers, when subject to changes in flow rate (4) shows a transfer function of the form ABROAD
(1)
where
B(x,t) == -
(2)
School of Chemical Engineering, Purdue University, Lafayette, Znd.
The dependent variable b’(x,t) representsn ormalized temperature in the case of a heat exchanger, and normalized concentration in the case of an isothermal chemical reactor. The forcing function r(s) is a Laplace-transformed, normalized disturbance in flow rate, and parameter POis the ratio of heat exchange to heat capacity for the case of a heat exchanger, or, in the case of a chemical reactor, is the ratio of reactive to convective rates of transport of the reactant. Parameter b, different from zero only in the case of a heat exchanger, is the linearized rate of change of over-all heat transfer coefficient VOL. 5
NO. 3
AUGUST 1 9 6 6
413
Control of a class of tubular heat exchangers and chemical reactors by piecewise-constant (switched) manipulation of flow rate provides efficient algorithms for driving the process from one steady-state condition to another. Switching times for minimizing various performance criteria are derived from exact and linearized dynamics, and found to compare closely. Time-optimum control of the transfer function ( 1 e-’)]/s is shown to require infinite switching.
with flow rate (7, 2). For n = 1, Equation 1 gives the transfer function for a constant-wall-temperature heat exchanger, or of a reactor with a first-order isothermal, irreversible decomposition of reactant; for other values of n, the dynamics are those of a chemical reactor with an nth-order isothermal, irreversible decomposition. The assumptions upon which the transfer function of Equation 1 is based are those of plug flow, constant physical properties, perfect radial mixing, and negligible back mixing. In addition, the true dynamics have been linearized to obtain the transfer function of Equation 1. This linearization is in terms of the auxiliary variable, B ( x , t ) , which is defined in terms of the original dependent variable through the transformation of Equation 2. Since the process is of a distributed-parameter nature, the variable $ ( x , t ) depends upon both dimensionless distance x and dimensionless time t. The processes and nomenclature have been described in detail (2-4). Rewriting the transformation as
should not be drastically different from that exhibited by many, more realistic tubular processes. Manipulations have effects both at the time of manipulation and one unit of time later (Equation 1). The problem of design of optimum, or suboptimum, control algorithms for Equation 1 thus has broad implications for process control. Although the “optimum” control laws are based upon the linearized transfer function of Equation 1, where possible the results of applying these control laws to the actually nonlinear process are studied through recourse to the exact dynamic equations previously derived (2-4).
g(x,t) =
Jf
t
Afx-f-
= -7
wheref(f) is the kinetic rate expression based upon the concentration of a particular chemical species, will extend the utility of the transfer function of Equation 1 to more complicated kinetics than simple irreversible decompositions (3); the restriction when Equation 2a is used is that only one reaction, whose rate may be expressed through the stoichiometry in terms of a single reactant, may occur. The transfer function of Equation 1 will also represent the dynamics for control of a heat exchanger of negligible wall capacitance by manipulation of the (uniform) wall heat flux (5). Since this transfer function describes the dynamics of a reasonably broad class of distributed parameter processes, it is of interest to study its control. Regulatory proportional feedback and feedforward control by flow rate manipulation have recently been studied ( 4 ) . I n the work reported here, problems of the servomechanism type were studied-that is, we assume that it is desired to drive the process operating condition to some new steady state. With the advent of steadystate optimization by digital computer control, such problems occur with increasing frequency. Once the steady-state computer optimization has indicated a desired change to a new set point, it is the objective of the present study to determine how to move the process to the new steady state, by piecewiseconstant flow rate manipulations, in a manner to optimize some desired transient performance criterion. Several such criteria are examined, including minimum area, minimum absolute area, minimum settling time, and minimum time. The piecewise constant, or switched, manipulation of flow rate offers simplicity of implementation, and ease of calculation (particularly important for on-line action) of optimum control laws. The dynamics of the class of systems described by Equations 1 and 2 is by no means general to all tubular processes; nor is the choice of flow rate as. the manipulated variable. However, the systems show a type of history-dependence which 414
ILEC FUNDAMENTALS
f
0
r(r)dr
-
r(r1d.r
r(r)dr
(4)
(5)
I t is shown that the control laws based upon the linearized transfer function give satisfactory performance. Single-Switch Control
We consider laws of the following form: The flow rate is first moved the maximum allowable amount in the desired direction and then, at a time t l chosen to provide “optimum” performance, is switched to the steady-state value which will give the desired steady-state value of the response variable 4(l,t)/(l - 6). If the new desired value of $(l,t)/(l - b ) is lower than its present value, this control law for r ( t ) will take the form shown in Figure l a ; if higher, the form of Figure 16, will be utilized. U and L represent the upper and lower bounds on the flow rate manipulation, respectively, and the current value of r ( t ) is taken to be zero in accord ,with the deviation principle on which Equation 1 is based. For later convenience, the sign convention is adopted so that both U and L will be positive. The steady-state values of the flow rate which are necessary to achieve the new set point are designated as a and a’, for the case of a decrease and an increase in 4(1,t)/ (1 - b ) , respectively. I t is clear from the definitions that the ranges of values for U and L are 0 5 U < a ,and 0 5 L < 1. Response for Single-Switch Control. To emphasize the similarities between the cases of an increase and a decrease in the set point value of & ( l , t ) ,we adopt the following nomenclatures: For an increase
c =L a = a’ k = kl’ = -a’/L
(6)
For a decrease
c = -lJ a = a k = kl =
(7)
a/U
-Ltb
0
Figure 1 .
Control function r ( t ) for single-switch control a. To achieve decrease in b. To achieve increase in
r;c;
6 (1,t)
4 (1,t)
I n both cases a = -kC, and the equation for r ( t ) is r(t) =
t < O
0< t tl
1 - k ++a-k
The reduction over a zero switching time, in settling time tp, is given in the first case of Equation 21 by P and in the second case by (1 - k ) ( l P)/(l CY). Figure 6 gives for various percentage specifications, as indicated by the values of P shown on the responses, the responses yielding minimum
-
418
'12
I&EC FUNDAMENTALS
+
k tl,* = ___ 2 (1 - k) Equation 23 is also the limit of Equation 22 as C is allowed to approach zero. To get a more direct comparison between Equation 22 and Equation 23, it is convenient to expand Equation 22 in a power series in kC. Retaining terms up to first order only yields the approximation
Figure 6. of P
2(1
Minimum settling time responses for I =
- k)
4(1
- k)
and for reasonable values of kC the linear and exact times will be close. As an example, for k = C = 1/2, tl* = 0.58, while tl,* = 0.5 and Equation 24 estimates tl* = 0.56. Comparison of Equations 14 and 23 shows that the optimum switching time to achieve minimum algebraic area is always greater than that neceijsary to achieve minimum absolute area. The disparity between these two switching times increases with increasing k. This fact may form the basis for choice between the criteria of minimum algebraic area and minimum absolute area. Double-Switch Control
T h e results of the studies 011 single-switch control showed that in all cases one could predict optimum switching times accurately by using the linearized response. While use of this approximation was not necessary in the case of the singleswitch response, derivation of the exact response when more than one switch is used becomes excessively complex. Hence, the results of the present section are all based upon the linearized transfer function of Equation 1. Adopting the nomenclature
81
= L/U
for a decrease in $ ( l , t ) , and PI’
=
U/L
for an increase in $ ( l , t ) , and letting 8 represent PI for a decrease and PI’ for an increase, the normalized control function for both cases may conveniently be written
(
0;
t < O
\--k;
tz
