A CONTROL STUDY ON ISOTHERMAL MIXED CRYSTALLIZERS C H A N G
D A E
H A N
Department of Chemical Engineering, Polytechnic Institute of Brooklyn, Brooklyn, N . Y . 11201
An analog simulation study on the dynamic behavior of an isothermal mixed crystallizer to disturbances in feed concentration was carried out. The nature of the stability of the system was examined, using both linear and nonlinear system equations. Feedforward controllers using both linear and nonlinear system equations are designed effecting supersaturation as a control output variable, feed concentration as a n input disturbance, and feed rate as a manipulative input. Supersaturation is significantly improved when the feedforward controllers are applied to the system which is basically stable.
R ECENTLY
analytical studies on the dynamic behavior of mixed isothermal crystallizers have been published (Randolph and Larson, 1962; Sherwin et a / . , 1967), but papers on the control of such crystallizers are very scarce. Randolph and Larson (1962) studied the dynamic behavior of a mixed crystallizer with an analog computer and simulated interactions between nuclei dissolving and production rates. They found that the net effect of nuclei dissolving is to force the growth rate to a higher level, enabling the same size distribution t o be produced a t a higher production rate. On the basis of their study of the frequency response, based on suspension area, of a mixed suspension crystallizer to period inputs in the nucleation rate, they proposed two control schemes: one, based on the measurement of suspension area, and a closed-loop feedback scheme based on the measurement of seed crystal density in suspension. However, i t would seem very difficult to adopt these two control schemes in practice, because a t present no instrument is available for measuring either suspension area or nucleus density on a continuous basis. Sherwin et al. (1967) reported that self-induced cyclic instability for well-mixed crystallizers would be expected if the nucleation rate was more sensitive than the growth rate to a small change in supersaturation, and they determined the region of stability in terms of a sensitivity parameter. However, they did not consider the stability of the same system to disturbances of input variables, such as feed rate and feed concentration. This paper shows the dynamic behavior of an isothermal crystallizer as affected by disturbances of feed conditions, and presents design of a control system to improve the dynamics of the uncontrolled process. The approach chosen for designing a control scheme was the feedforward control, which seems to have definite advantages over the conventional feedback control. By the time a feedback control system detects an unexpected upset it is too late to make an appropriate change because of the relatively large time delay of the crystallization system, in general. Furthermore, one can do nothing to change the size distribution of crystals in the system, once crystals have grown beyond a stable nucleus size. Therefore a predictive type of control would be better than the corrective type. The basic technique for designing a feedforward controller has been discussed (Bollinger and Lamb, 1962; Luyben and Lamb, 1963; Tinkler and Lamb, 1966). 150
I & E C PROCESS D E S I G N A N D DEVELOPMENT
Supersaturation was controlled using feed rate as the manipulative variable and feed concentration as the input disturbance. The feedforward controller was designed in two ways, using linearized and nonlinear system equations. The improved control by use of the feedforward controller was demonstrated by an analog simulation. The present paper is concerned with the control of isothermal crystallizers. However, there is another very important type, nonisothermal crystallizers, in which control of temperature is very sensitive to the system stability. For instance, in the vacuum crystallizer an upset in vacuum can remove an extra amount of solvent and cool the solution a few degrees, and thus create a large shower of nuclei. A control study on such nonisothermal crystallizers is under way and will be presented in a future paper. Dynamic Equations of a n Isothermal Mixed Continuous Crystallizer
Since the object of the present study was to show the improved control by the use of a feedforward control scheme, the dynamic equations followed those studied by
dC W (1- k p 3 ) - = - (CO- C)CO dt V
(P
- C)
Growth and nucleation models used are
G(C) = Ki(C - Cs)
(6)
respectively. If we choose a physical system whose feed stream contains no solid particles and if the size of nuclei is assumed to be negligibly small compared to the average particle size of the product, Equations 1 to 5 are simplified to
b=(-)
Co - C
dlnB(C)
6
dC
t-,
N -
2K2
b
Equations 13 to 17 are nonlinear dynamic equations to be used for our control study here. Linearized equations may be derived by expanding each term in Equations 13 to 17 about a steady state in a Taylor’s series and retaining only the first-order term in each series. Again by defining the following dimensionless perturbation variables as -
The dimensionless variables may be defined as: 20= poi po, zi = p 1 J i 1 , Z? = p 2 i i 2 , 2 3 = p3/;3, y = (C - Cdi ( € - c_~ )_j,( t ) _= w ( ~ ) / Wh,( t ) = C , ( t ) / G , r = t W / V , _ where p n , p l , p 2 , p 3 , (7, and W denote the steadystate values of each variable. Equations 8 to 12 may be written as
ca,
-
Zh(T) =
(PO - P O ) / PO
Zi(7) =
(p2
-
-
= 2 0- 1
-
ILL)/ p r = 22 -
1
Y’(r) = ( C - C ) / ( C - C,) = Y - 1 j’(r) =
(W- W)/W=j-I
we obtain
dZi - = YZO- jZl dr
dZ3 - = YZ? - jZ, dr
Y
1
- (1 -
-
,)YZ2[(*)g+
1-
1 - Y]
(17)
where
dY’ = - g Z ; - ( I + g ) Y ’ + g j ’ + g dr
-
(--€0
-
d h’
VOL. 8 NO. 2 APRIL 1 9 6 9
(26) 151
Figure 1. Analog circuit diagram of simulated isothermal crystallizer 0.4
I
9
0
0.4
y'.
I
I 1
I
I
I
I
I
1
I
I
I 36
4Q
I
1
20.4
-
-
c E-c, O
-0.8 0
I
0.4
I
0.8
I
1.2
1
1.6
I
2.0
I
I
2.8
2.4
8.
V/w,
3.2
I
1
4.4
1
4.8
1
5.2
HOUR
Figure 2. Effect of sensitivity parameter, b/g, on response of supersaturation to 10'30 step increase of feed concentration linearized system equation without controller
Analog Simulation of Process
Equations 22 to 26 were simulated on an analog computer. Figure 1 shows the simulation diagram with controller. The coefficients of linear system Equations 22 to 26 have to be determined from the steady-state equations, which may be derived by equating the left-hand 152
l 6 E C PROCESS D E S I G N A N D D E V E L O P M I N T
side of Equations 8 to 12 to zero and solving the resulting algebraic equations with rate Equations 6 and 7. Figure 2 shows the effect of sensitivity parameter, b/g, on supersaturation to a step change of feed concentration (10% increase of the steady-state feed concentration). AS the value of b/g increases, the supersaturation gives rise
I .o
I
I
1
I
I
I
I
I
I
I
1
I
I
r r )
N
-0.0
0
I
I
I
I
0.4
0.0
1.2
1.6
I
2 .o
I
I
I
I
I
I
I
2.4
2.0
3.2
3.6
4.0
4.4
4.0
,g
8
5.2
v/W, HOUR
Figure 3. Response of system variables to looh step increase of feed Concentration at critical sensitivity parameter,
b / g = 22
to an oscillatory behavior with higher frequencies and with b/g larger than 22, the supersaturation becomes oscillatory unstable-Le., the amplitude of oscillation increases and never dies out. Therefore b/g = 22 mav be called “the critical value of sensitivity parameter” which exhibits the sustained oscillation. Figure 3 shows this sustained oscillatory behavior of other system outputs-namely, 20, 2 1 , 22, and 2 3 a t b/g = 22. In Table I the steady-state operating conditions and in Table I1 the coefficients of Equations 22 to 26 are given. The nonlinear system Equations 13 to 17 were simulated on an analog computer (Figures 4 and 5). I n Figure 4 effects of the sensitivity parameter, b i g , on supersaturation are shown for the case when input is not disturbed. The steady-state operating conditions for this particular simulation are listed in Table 111. The critical value of b / g was 21.6 a t which the sustained oscillatory behavior occurred. Self-induced cyclic behavior starts to appear when blg becomes larger than about 3. I n Figure 5, with no control action provided, effects of the size of feed concentration perturbation on the supersaturation are shown a t a fixed value of the sensitivity parameter, big = 16. The size of disturbances does affect the type of responses as predicted by the theory of nonlinear control systems. Furthermore, the response to a 30% step decrease of feed concentration moves almost in the opposite direction from the response t o a 10% step increase, leading to instability. This tendency was observed where step
Table I. Steady-State Operating Conditions for Linear System Equations 22-26
Feed concentration, c,.0.3 g./cc. Void volume fraction,:, 0.926 Density of a crystal, p. 2.8 g./cc. Supersaturation, C.0.101g./ cc. Equilibrium concentration, C.. 0.1 g./cc. Shape factor of crystal, k. 4.184 Table II. Coeffkients of Linear System Equutionr 22 to 26
(F)
= 0.0798
1 + g = 216.0
blg. 20.4, 22, 32 (as shown in Figure 2)
V O l . 8 NO. 2 APRIL 1969
153
-
18
Y
=
cc-cs -cs
7 1.0
t
2
0.4
154
I
I
I
I
I
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
I
I
I
I
I
I
I
FEEDFORWARD
Table 111. Parameters Used for Simulation of Nonlinear System Equations 13 to 17
- -(
SET POINT
CONTROLLER
Steady-state feed concentration, Co.0.20 Steady-state supersaturation, C. 0.12 Steady-state void volume fraction,
