86
Ind. Eng. Chem. Res. 1988,27,86-93
Design of the Control Scheme of a Concentration-Controlled Recycle Reactor Dong-Feng Pan, Klaus Schnitzlein, and Hanns Hofmann* Institut fur Technische Chemie der Universitat Erlangen-Nuernberg, D 8520 Erlangen, West Germany
For kinetic investigations of complex heterogeneous catalytic reactions, the concentration control concept for differential recycle reactors proposed by Lowe has proven a useful tool. The actual control scheme, however, i.e., the respective pairing of set and control variables in the control loop, and the determination of the set values seem to be purely based on empiricy, since no control engineering treatment of this type of concentration control was available at that time. Therefore, in the following, criteria are developed that enable the determination of a control scheme suitable for the reaction network under investigation. These criteria are developed on the basis of a theoretical description of the control system. Besides information about the stability for the selected control scheme, the concept allows estimation of the controller settings without requiring any great expense of control and measurement installations. The details of the theory are illustrated by examples of simulation studies.
Starting Point and Objective Reaction engineering investigations of heterogeneous, catalytic, gas-phase reactions in recycle reactors are incomplete if no account is given for the possible dynamic change of the activity and selectivity of the catalyst. The concept proposed by Lowe (1980a) for the concentration control of such reactors allows the adjustment of certain concentration levels and also allows us to maintain these a t a constant level during the deactivation of a catalyst. In this way, not only the temperature and concentration dependence of the reaction rates but also the kinetics of the deactivation process can be determined (if separability of the deactivation kinetics can be assumed). This concept has apparently been applied successfully in kinetic investigations of a simple reaction (Bilgesu, 1978) and a complex reaction (Krokoszinsky, 1983). In those examples the control scheme was presumably mainly guided by the intuitive pairing of the respective set and control variables. Attempts to obtain some information about the optimal control scheme with the aid of the relative gain array did not prove successful always according to the experience of others and ourselves (Schnitzlein, 1981; Lowe, 1980b). Furthermore, in some cases, the control scheme based on this measure evidently led to unstable control loops. This is in contradiction to common empirical control principles (Lowe, 1980~).This discrepancy about the application of the relative gain array in the case of the concentration control according to Lowe has thrown users into doubt about the validity of this measure in designing control loops (Fischer, 1983). The goal of this work is to develop criteria that can facilitate the design of the control scheme for this type of concentration control. The method is applicable without great experimental expense and allows the assessment of the stability of the control scheme for any arbitrary, complex reaction system. The Control Concept of Lowe (1980a) The control concept of Lowe is based on the following assumptions. (i) The stoichiometry of the reactions is known. (ii) The system is in a pseudo steady state; Le., one can apply steady-state mass balances for a gradientless recycle reactor. (iii) Every steady state is characterized by the mole fractions of the reaction components and the flow rates of the inert gases (state variables). (iv) Some preliminary knowledge about the reaction rates is available. 0888-5885/88/2627-0086$01.50/0
For a system of R linear (stoichiometrically), independent reactions, the following N mass balances (including inert components) for steady state are obtained:
4" - f s x s+ N r ( x , ) = 0
(1)
with N
d x , ) = wCatr*(xs)
f, = 1=1 Cf,,
[The nomenclature is chosen following the suggestions of the Working Party of Chemical Reaction Engineering (EFChE, June 1977)l. On the basis of assumption iii, the ultimate goal of concentration control for a recycle reactor is the adjustment of all other unknown variables, i.e., N inlet flow rates. Furthermore, there is no need for installing N feedback control loops, since it can be shown by splitting and rearranging system 1, fRso = fSxRs+ N R r ( x , ) = 0 fLso -
f , x ~ =, -NLNR-'(fR:
- fS-%)
(2a) (2b)
with
L=N-R that only R balance equations are linear independent. Therefore, feedback control loops for only R species are necessary to maintain all the concentrations a t their set values. For the ( N - R ) species, feedforward control loops are installed, following eq Zb given above. These R species (in the following called control components) can be identified satisfying the condition det NR # 0 where N R is the matrix of stoichiometric coefficients of the R control components. Since this condition generally applies for arbitrary combinations of species, the selection of these control components is in principle not subjected to any constraints (Aris, 1965). The selection can, therefore, be guided by practical aspects: (a) The outlet mole fractions of the control components should be detectable continuously, and the analysis should be as free as possible of any delays; an electrical output signal is desirable. If in certain cases this condition is not fulfilled, a model-supported observer can make the continuous estimation possible. (b) The concentrations of the control components should be sensitive to disturbances, particularly to changes in the 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 87
~
-
f;
I
1
c~miu'e?
p
pF
Figure 1. Schematic illustration of the concentration-controlled recycle reactor: R , control components; L , noncontrol components; I , inert components.
selectivity due to deactivation of the catalyst. Figure 1 is a schematic sketch of the resulting control scheme proposed by Lowe. As can be seen, the signals of the measured inlet gas feeds ( f R o ) of the R control components are fed into the process computer ( 2 )where the set values for the noncontrol components are computed by using eq 2b. If a constant, i.e., uncontrolled inert gas feed (f?) is assumed, those relationships reduce to a linear form and become completely uncoupled. In order to ensure in all cases adjustable, nonnegative feed flow rates as a result of the on-line computations (eq 2b), the inert gas flow rate has to be properly set for the desired steady state. This adjustment can be done by means of an optimization with the objective of achieving the highest possible recycle ratio. This, however, requires some preliminary knowledge of the reaction rates at steady state, which can be estimated from simple experiments. Up to now, for most applications of this type of concentration control, the selection of the control components and their respective input-output pairings seem purely based on empiricy. Attempts have been made for designing the control scheme by means of measures for the static and dynamic interactions (McAvoy, 1983) between the control components (Lowe, 1980a; Fischer, 1983), but the obtained results contradict each other (Lowe, 1980~). Furthermore, following the suggestions of the relative gain matrix (Bristol, 1966), in some cases unstable control loops result (Lowe, 1980b). This uncertainty in the design of the control scheme for a concentration-controlled recycle reactor overcomes the advantages of this experimental technique for the investigations of complex reaction networks. Therefore, it is our goal to cast away all these uncertainties by a thorough investigation of this special type of a multiinput-multioutput control scheme (coupled feedback-feedforward control).
Criteria for Designing the Control Scheme For a given complex reaction network with more than one linear, independent reaction, we are dealing with a multivariable control system, in which the concentrations of all species are not independent of each other, mainly because of the nonlinear reaction rates. However, since there is no possibility to eliminate these nonlinearities without any knowledge of the reaction rates, it is desirable to suppress their mutual influence as far as possible by choosing proper control components and selecting an appropriate control scheme. To get a measure for the coupling between the control variables, some information about the open-loop gain array DRis desirable. Generally, for the case of concentration control, this array is not available, because the determination of the open-loop gain requires detailed knowledge of the reaction rates, which are the very objective of the kinetic investigations. As a way out of this dilemma, i t is proposed to experimentally determine the open-loop gain array. For this purpose, neither the control equipment nor a process computer is necessary.
It is then proposed to extend the experiments for the measurement of the reaction rates in the course of the optimization of the inert gas flow rate. The following procedure is recommended: If the flow rates of a single component ( f R o or f L o ) are altered and the changes of the mole fractions of all N species at the reactor exit are recorded, then the open-loop gain matrix [(dxldf"),] at this particular steady state can be approximated by the difference ratio [(hx/Afo),]: lim ( A x / f o ) , = (ax/afo),= GN (3) AfO-0
After selecting the control components, one obtains the normalized open-loop gain array (DR) for the concentration-controlled system,
DR(O)= 2240On,,,GNN~-~
(4) where G is composed of R lines of GNwhich correspond to the control components. In order to guarantee that the situation at the respective steady state is in this way represented close to reality, the linearization had to be done within the allowable linear region. A required condition which justifies this approximation is det DR(0) > 0 (5) which is usually easy to achieve for sufficiently low input step changes. 1. Input-Output Pairing Problem. Even though there are some opposing opinions about the applicability of Bristol's (1966) relative gain array, it is proposed to characterize the coupling between the individual exit mole fractions by this measure. As will be shown in the following, the reasons for these discrepancies are not, as generally assumed, to be found in the definition of the measure, but rather in its erroneous interpretation. The advantage of this measure is that no knowledge of the internal structure and dynamics of the system is necessary for its determination. Even though the measure does not give any unequivocal clues about the stability of the control scheme, in many cases valuable information about the relationships between set and control variables can be extracted. A critical review and discussion of the properties of the relative gain array can be found in the work of Grosdidier et al. (1985). The elements of the relative gain array are calculated according to (Bristol, 1966) = dRi];iRij (6) where DR(0)= {dRyJ isJhe open-loop gain array of the system and DR-l(O) = {dRy)is the inverse of DR(O).The proof is given by Wang and Munro (1982). Based on that measure, criteria can be developed for choosing the control components as well as selecting the input-output pairing suitable for each reaction network under investigation. As additional information, the open-loop gain array will be called upon. The latter can be found experimentally in a straightforward way, as was shown above. The following statements are extensions of known criteria developed for general multiinput-multioutput systems applied for the concentration control concept proposed by Lowe. It will be assumed that each output variable x , is returned with a negative sign to the input variable f," via the controller (i = 1,...&). This arrangement will be called the canonical pairing. The mathematical proofs of the statements are given in the Appendices.
88 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
(A) Select the input-output pairing so that dRcl> 0 for all i = 1,...,E; i.e., each closed loop i is stable for itself. If this is not the case, perform the following procedure: For an even number of negative main diagonal elements of the matrix DR,change the sign of the controller corresponding to these elements; i.e., reverse the sign of the ith row of the matrix Dp The thus resulting system then fulfills the condition above and is not structurally unstable. This stability is independent of the adjusted controller settings and therefore is called structural (Niederlinski, 1971). If the number of negative main diagonal elements is odd, the control system is either unstable itself or each subsystem corresponding to one of those elements is unstable. The izth subsystem is created from the original system by opening the kth control loop. Then the sign reversal procedure will lead to a system not structurally unstable only if det DR’ < 0, where DR’ is the transfer matrix after sign reversal (compare Appendices A and B). (B) Select the input-output pairing so that for a given set of control components the absolute values of the main diagonal elements of the matrix are markedly greater than those of the off-diagonal elements, A,, > A, for i # j . In the opposite case, a strong interaction of the control loops must be anticipated, which usually also means that the controllability of the total system deteriorates. In this case, a strong coupling is present, which often suggests a different choice of the control components. Then, as indicated by the relative gain array the combinations with the largest number should be chosen for closed loops (Shinskey, 1979), the control scheme has to be rearranged by interchanging the set and control variables crosswise xi f,”, and the signs of the respective control loops are to be changed in order to maintain the stability of the total system (see Appendix A). In certain cases, however, this procedure can result in an unstable or at least sensitive control scheme. Therefore, before attempting cross interchanges, it is more advisable to search for different control components that could lead to a better measure. (C) Select the input-output pairing so that the main diagonal elements of the relative gain array are positive, Xi, > 0; otherwise, the closed control system will be unstable if the ith loop is opened (assumed that dRcc> 0). This situation may appear in practice, for instance, in the start up of the complete control system. Thus, an unstable control can prevent the system from reaching the desired steady state. From a practical point of view, it is, therefore, not recommended to use such a configuration. If in this case no other control components are at hand, for example, because of already installed analytical instruments, specific for given reactants, then the pairing of set and control variables has to be altered according to the following strategy: The connections of the ith control loop are to be reversed, so that (if possible) the diagonal element in the relative gain array becomes positive and, at the same time, the number of reversions of control loops (that is, change of sign in the control loop) and rearrangements (that is, pairing of set and control variables) is even. An exclusive new pairing without sign reversal results in an unstable control behavior (see Appendix A), as was shown by Lowe (1980a) and Fischer (1983). 2. Controller Settings. After pairing the input and output variables based on the experimentally guessed open-loop gain matrix, it is possible to derive some criteria for adjusting the PI-controller settings. With the conditions (ai) overall changes of the number of moles by the reactions are neglected, (aii) identical dead
-
times in analysis (7’3are assumed, and (aiii) all controller settings are identical (V, and TR),the critical gain VR,crlt and the respective reset time T N for each control loop can be calculated (Pan, 1985) with ?r
V R ,= ~p,,t(22400Tt) ~ ~ ~
(sml/s)
(74
and
TN= l / ( l / T s + d,) (s)
(7b)
(1 mol = 22440 sml) with d, as the eigenvalues of the matrix (I - xR,SNNNR-’)DR-’(0).If these are unknown, an estimate of d, can be obtained from the Gerschgorin theorem (Isaacson and Keller, 1966) R
R
d, = min [max ( 2 l d d - CldR,I), max (21d~jjI- ~ l d ~ , I ) l (8)
If the determined minimum is negative, the estimate of d, will be set to 0. However, only in a few cases are the conditions which underlie the assessment satisfied. One can estimate the deviation from condition ai by comparing I - xR,SNNN, I (9) The greater the deviation of the left-hand side of eq 9 from the unity matrix I, the smaller becomes the real critical gain. 3. Stability Considerations. After the control scheme is designed by choosing the control components and their respective input-output pairings are selected, a stability assessment is desirable. A t first, the stability condition is evaluated without explicit consideration of the analytical dead times. This can be justified since, as will be shown in the following simulation studies, the “added” dead times for analysis lead to a deteriorated dynamic behavior of the system. Furthermore, the open-loop gain array used for the steady state in the stability assessment is independent of the individual dead times. With the conditions (bi) only PI controllers are used and (bii) the flow rate of the inert gas is not controlled and, therefore, is constant, it can be stated that the control system is stable against small disturbances if all the roots of eq 10 have negative real parts (compare Appendix A): ++
det ( 4 1 - X , ~ S ~ N N ~ -+~KR(s) ) - I + DR-I(0))= 0 (10) with K&) = diag [K,(s),K,(s),...,KR(s)]/22440/n,, (loa) and K,(s) = v R i ( 1 + 1/TN,s) (lob) The roots of eq 10 can be identified with the aid of standard numerical routines (Appendix C). With the above procedure, a preliminary stability assessment of any chosen control scheme and the empirical determination of the controller settings are possible. Due to the experimental determination of the open-loop gain array, the above stability assessment is limited to the steady state under consideration. For this reason, the stability assessment should be performed for different operation conditions. According to experience from the experimental design, the maximum information is gained if the experimental conditions are located on the boundaries of the variable space to be investigated.
Simulation Studies Simulation studies are employed to verify the derived criteria for the design of the control scheme in the con-
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 89 Table I. Characteristic Data of the Control Loops Tt,co = 7.0 s dead time for CO analysis dead time for COz analysis Tt,co, = 14 8 T, = 17 s residence time at given feed rate
4
I I
I
1
I
Table 11. Controller Settings According to Lowe (1980a) VR,co = 7.9 sml/s VR,CO,crit = 7.1 sml/s for TN,co = 0.1 s T ~ , c o= 32.0 s VR,co2= 2.9 sml/s VR.CO2,crit = 3.5 sml/s for TN,co2 = 0.1 s TN,CO, = 67.0 s
I
1
08
1
1
I
1
m t Ill Figure 3. Change of mole fractions with time after step jump of set values with controller settings of Krokoszinsky (1980) and CO and COz as control components (change of set values as in Figure 3).
According to Krokoszinsky (1980) VR.CO,crit = 7.1 sml/s for TN,co= 0.1 s VR,COz,crit = 3.5 sml/s for TN,co2 = 0.1 s
VR,CO= 2.9 sml/s TN,CO = 32.0 s VR,CO,= 2.9 sml/s TN,co2= 67.0 s
0
18
100
1
I
0.6'
6
100
200
1bo
200
1
I
I
I 300
460
t
300
460
t [SI
I
Is1
I
7'
c
x
4,oJ
1 0 ~
2do
160
0
t
Figure 4. Mole fractions of control components as a function of time with controller settings according to Schnitzlein (1981) with control components Hz and HzO.
102
09
0
100
200
1 151
Figure 2. Change of mole fractions with time after step jump of set values with controller settings of Lowe (1980a) and CO and COZ as control components: (xco, xCo2, XH,, xH20, xc%)102 = (0.28,1.08, 7.39, 1.53, 9.69) - (0.2, 1.0, 5.35, 1.89, 10.4).
centration control according to Lowe. As a first case study, the following complex reaction system was considered (Lowe, 1980a, Krokoszinsky, 1980, Schnitzlein, 1981): CO + 3H2 = CHI + H2O
CO
+ H2O = COZ + Hz
The kinetic description was taken from Krokoszinsky (1980). If the species CO and COz are chosen as control components (Lowe, 1980a), the open-loop gain array and the relative gain array are obtained in the forms p.1547 DR(O)= 0.2130
I
Is]
X X
lo2 -0.2850 0.1634 10'
X X
101 10'
A = p.9773 0.02271 0.0277 0.9773
Following the suggestions proposed by Shinskey (19791, it is apparent that the choice of the control components and their pairing is correct. Table I contains the characteristic data of the control loops (Krokoszinsky, 1980). The controller settings including the determined critical values using eq 7 (see Appendices) are compiled in Table 11. The transients of the closed control loops after changing the set values are shown in Figure 2 for the controller settings of Lowe. It can be seen that both control loops are operating in the stable region. The marginally supercritical gain of the CO control loop is compensated by the relatively large reset time. However, the oscillatory characteristic of the CO control loop is more pronounced than that of the COz loop. In the latter control loop, the gain is set to a level markedly below the critical value. For the controller settings of Krokoszinsky, one also obtains a stable control, but instead
Table 111. Characteristic Data of the Control Loop for Control Components Hz and H,O dead time for Hz analysis Tt,H2= 10.0 Tt,H20= 10.0 s dead time for HzO analysis residence time T, = 17 s Table IV. Controller Settings According to Schnitzlein (1981) for Control Components H2and HzO VR,, = 1.7 sml/s V ~ . ~ , , c n= t 4.9 sml/s for TN,H2 = 0.1 s TN,H2= 33.0 s VR,H,o = 14.0 sml/s VR,HzO,crit = 4.9 sml,'s TN,H~O = 33.0 s for TN,H20 = 0.1 s
of the previous example, the CO response now approaches steady state asymptotically without great oscillatory behavior. The reason is the choice of the controller gain in the latter case, which was set below the critical gain of the first control loop (compare Figure 3). If, in contrast to the above case, Hz and H20are chosen as control components, the condition of the matrix DR(0) already indicates the strong interaction of the two control loops: DR(0)= p.4227 X 10' 0.6120 X 10'
-0.2680 X 1 0 1 0.3160 X lo2
A = [0.45 0.551 0.55 0.45
The characteristic data of the control loops are summarized in Table 111. Table IV contains the pertaining controller settings (Schnitzlein, 1981). Figure 4 contains a plot of the mole fractions as a function of time in the start up of the control system. Since the actual gain for H 2 0 is much larger than the critical gain, the control loop exhibits an increasing oscillatory behavior continuously from the start. As a consequence, the H2loop, even though stable in itself, is being forced into instability due to the strong loop interaction. By reducing the controller gain as well as increasing the reset time, one would yield a stable control; however, the loop interaction is responsible for a prolonged damping characteristic as compared to the control with the components CO and C02.
90 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
In order to study the influence of the time delays on the stability of the entire control system, in a second case study the control scheme for a complex system of irreversible reactions--the partial oxidation of methanol on a Fe/Mo oxide catalyst (Pan, 1985)-was investigated:
Table V. Computed Roots of Equation 10 for Control Components A A I, A 6: Characteristic Data According to Fischer (1983) roots open loops lnand 2b 1 and 3'
CH30H + 1/202 CHZO + H20 +
CHzO
+ '/202
--+
CO
2 and 3 1 2
HzO
f
Since, in this case, the total molar change caused by the reactions cannot be neglected, as
3 a
one finds a discrepancy between the theoretical critical gain and the value obtained from simulations: VR,crit(CH30H) = 6.1 sml/s (from simulations) and VR,crit(CH,OH) = 9.1 sml/s (from eq 7 ) for TN = 1.96 s. For the example, CH30H and CH,O were chosen as the control components. A drastic elevation of the gain beyond the critical value and/or a reduction of the reset time cause instability, even though the investigation of the stability with the aid of eq 10 predicts a stable control system for the present case. This is evidence for the fact that--as postulated in the derivation of the stability criteria-the delay times alone are responsible here €or the observed instability. Therefore, it is possible for every control system that is stable according to eq 10 to find controller settings which lead to stability of the control system, even when considering the dead times. The following fictitious reaction system was chosen by Liiwe (1982) to demonstrate the diverging opinions about the relative gain array in the design of the optimal control scheme: A, + A2 113 + A4 A , + A, = Az + A5 A2
+ A , = A, + Ai
-
Selecting species Al, A5,and A, as control components, the following arrays with the pairing f," x i are obtained: DR =
L
0.722 -0.133 -0.013
0.651 0.021 --0.062 0.131 -0.006 0.744
]
-1.05 2 09 A = [-0:04
2.05 -1.07 0.02
1 1
0 -0.02 1.02
For control components A2.A4,and A6,the arrays assume the form 0.004 -0.428 -0.824 DE = -4.066 0.594 -0.316 0.004 0.020 0.796
[
]
0.974 --0.065 0.095 A = -0.050 1.052 0.038 0.017 0.867
The stability consideration performed with eq 10 reveals that both systems are stable. This result was confirmed by Lowe (1980a) and Fischer (1983) in simulation computations. While the system A z ,A,, A, apparently exhibits little coupling (the form of the pertaining array is nearly ideal), the system A,, A,, A , indicates some strong interaction of the R concentration control loops. Despite of the suggestions derived from this fact, Lowe (1980b,c) stated that selecting A,, A,, and A , as control components would lead to better control-he chose an integral measure as the objective function-than in the opposite case. Furthermore, he found by performing simulation experiments that pairing the components fi x 5 and f 5 x1 as suggested by the matrix h1,5,6would lead to unstable control. It will be shown in the following that the application of the criteria derived above lead to unambiguous hints for
-
4-
-1.097 -1.517 -0.905 -1.512 -0.910 -1.453
1:
xi
f+
flO.
*2:
15
-0.386 -0.115 -0.612 4.389 -0.605 -0.424
-
f5?
'3:
-0.131 -0.106 -0.104 -0.118 -0.392 -0.104 xg
-
-0.098
0.025
-0.092 -0.098 -0.098 -0.089
+0.024 -0.093 -0.027
fs".
designing a suitable control scheme even in this case. For the combination A,, Ab, A,, two main diagonal elements of the matrix A1,5,6are negative, which is a criterion of increased sensitivity of the entire control system. From inspection of the derived criteria, it follows that, on one hand, the x, f 5 0 control loop is unstable when the other loops are open, since Xz2 < 0 and dRZZ< 0, and, on the other hand, the subsystem with open control loop x1 flo is unstable, since All < 0 and dRll> 0. Table V shows the results of computations that confirm the derived criteria. Only pairings of x1 f 5 0 and x5 f 1 0 with a simultaneous change of the sign of the feedback for the control loop x 5 flo will lead to a stable control. In this way the following arrays are obtained:
-
-
DR =
-
[
0.133 0.062 -0 130 0.722 0.651 0:021] 0.744 -0.013 -0.006
-
2.05 -1.05 2.09 A = [-LO7 0.2 -0.04
-
1
0.131 -0.02 1.02
An exclusive rearrangement as done by Lowe produces a structurally unstable system, as was confirmed by the criteria mentioned above and shown by Fischer (1983) in hybrid simulation computations.
Summary and Conclusion A concept for an a priori investigation of the control system resulting from the concentration control proposed by Lowe has been presented that allows, without any great experimental expense, testing of different combinations of species with regard to their efficiency as control components and examination of different sets of input-output pairings from a theoretical control point of view. With this knowledge, the user is able to predict the performance of this type of concentration control for any arbitrary reaction network. Further guidance is provided for the choice of the controller settings as a function of the selected control scheme and the time delays for the system. Nomenclature C = matrix defined by eq A-3
DR,DN = transfer matrix with feedforward controller, s d, = eigenvalue of (I - xR,SNNNR-')DR-'(O), l/s diag Ju,...,c ) = diagonal matrix with diagonal elements a,...,c dRy,dRL,= elements of matrices DR(O)and DR-'(O), respectively f = molar flow rates, sml/s (1 mol = 22440 sml) f = vector of molar flow rates, sml/s G , GN = transfer matrix without feedforward controller, s/sml I = unity matrix J = Jacobi matrix of r * for x = xg,l / s (=wcat/ntot)d r / d x (X,)
KR = "normalized" controller transfer matrix, l / s K , = controller transfer function, sml/s K, = I part of the matrix KR K P = P part of the matrix KR, 1/s m,l,i,j,k = integer figures and indices N = matrix of stoichiometric coefficients N = abbreviation for (I - x,SN)N N = number of reactants at,, = total number of moles in reactor, mol
Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988 91 oi = eigenvalues of the matrix JN, l / s p ( s ) = polynomial with variable s R = number of control components Ri = abbreviation for controller i r* = reaction rate based on catalyst mass, mol/s, g of catalyst S N = 1 x N summation vector (row vector), =(l, ...,N s = Laplace variable, l / s sgn a = sign of a T N , TNi = controller reset time, s T, = residence time, s (=ntot/molar exit flow rate) Tt = analytical dead time, s t = time axis, s V,, VRi= controller gain, sml/s wCat= catalyst mass, g x = vector of exit mole fractions of reactants x = mole fraction yR= "normalized" loop input variables, l / s
Greek Symbols A = relative gain array Xij = elements of the relative gan array
+
= f," - f , ~ , N ~ ( x , )
yields = ((I - x,SN)NJ - l/T,I)x
+ (I - x,SN)f"/n,t
(A-1)
From the equation of the feedforward controller (eq 2a) with an uncontrolled inert gas flow rate, the input vector f"is obtained in the form
f"= NNR-lfR"
From the above equation, the transfer function of the loop without feedforward control is derived (after applying the Laplace transform): = GN(s)f"
with
G N ( s ) = ( ( s + l / T s ) I - (I - x,SN)NJ)-l(I - x , S N ) / ~ , , (-4-2) The transfer function of the loop with feedforward control is given by x ( s ) = DN(S)YR(S) with DN(s) = %tG&)NNR-l as the "normalized" transfer matrix and yR(s)= fRo(s)/n,, as the "normalized" input function. Substituting N=
Since the only nonvanishing eigenvalue of xR~SNNNR-~ is equal to the eigenvalue of SNNNR-'xRS (scalar) (compare Pan (1985)), and because its value is almost always less than 1 in practice, C can be inverted and det C > 0. With the above manipulations, the transfer matrix of the control components is obtained as
DR(s)= MDN(s)I where ;[DN(s)] is the matrix composed of R rows of the matrix DN(s),the latter of which is associated with the control component. Assuming that det k[JT] # 0 is true (this is almost always the case in practice), the matrix factor D R ( s ) = A[((s l/T,)I - NJ)-']RNR-' (A-4)
Then it can be stated that DR(s)= DR(s)C-l= (sC + DR-l(0))-l
Appendix A 1. Stability of a Closed Concentration Control Loop Free of Dead Times. In the evaluation of the asymptotic stability of a nonlinear system, it is sufficient according to Lyaponov to merely consider the corresponding linearized system (Kailath, 1980). For simplicity, the dimension o f f " in this appendix is chosen as mol/s. The linearization of the dynamic loop equation
X(S)
+
+
Superscripts
= NLNR-lfR'
(A-3)
the matrix DN(s)can be simplified to DN(s) = ((s l/Ts)I - NJ)-167RR-1C-'
can be transformed into the form (Pan, 1985) D&) = ((s l/T,)I - NRJNNR-l)-l
T = transponation of a matrix o = reactor inlet
fLo
N E = (I - xRsSNNNR-l)NR = C-'NR
+
Subscripts
L = noncontrol component R = control component s = steady state sp = set point tot = total
X
and with the condition that
= (I - x,SN)N
With KR(s)as the "normalized" control transfer matrix, that is, substituting fRo for yR,the characteristic determinant is obtained for the feedback configuration as det (I + DR(s)KR(s)) = det (sC + KR(s)+ DR-l(0))det DR-l(s) (A-5) This means that the closed loop will become unstable if one of the determinants in the above equation vanishes with a positive real part. Since the reactor control loop remains stable for small input step changes, regardless of the residence time of the reactant mixture, as long as temperature and pressure are maintained constant, the poles of GN(s) must have negative real parts. The reaction system itself, furthermore, does not exhibit any oscillations after such step changes, as is confirmed by simulation studies. From this fact, it can be concluded that the poles are r_eal. From eq A-2, it is concluded that all eigenvalues of NJ are negative. Since, on th_eother hand, JN possesses nonvanishing eigenvalues like NJ, all eigenvalues of the matrix JN,i.e., oi (i = 1,...,R), are negative. With this result, the poles of DR(s)are derived as d, = 1/T, - 0, With eq A-4, it further follows that the poles of DR(s)are identical with the eigenvalues of C-lDR(0). With the above statement and eq A-4, det DR&) does not contain any positive roots. From eq A-5, it then follows that the closed system is only unstable if any root of det (sC + KR(s)+ DR-l(0))= 0
(A-6)
with positive real part exists. For the nonnormalized control matrix K'(s) = diag (Kl(s),...,KR(s)),it then follows that KR'(s) = 22440n,tKR(~) Similarly, from the above consideration and eq A-3, it can be concluded that det DR(0) > 0. This is therefore the necessary condition for the approximation of the experimental loop gain matrix in the linear region. 2. Coordination Problem. For the feedback pairing x, f:, for all i = 1,,..,R, which will be called in the fol-
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92 Ind. Eng. Chem. Res., Vol. 27, No. 1, 1988
lowing the canonical pairing, it follows that K&) = diag (Kl(s),...,KR(s)) /ntOt/22440 For PI controllers, the elements of the controller matrix are computed by K J s ) = VRL1 + l/T,,s) With det C > 0, which is almost always the case in practice, it can be stated that (i) each ( 2 m - 1)rearrangement, where m is an integer, of the canonical pairing renders the closed system unstable, (ii) the sign reversal of ( 2 m - 1)controllers in the canonical pairing renders the closed system unstable, and (iii) if the sum of rearrangements and sign reversals of the canonical pairing is an even number, then the closed system will be stable. Proof. For very large values of s, it is p ( s ) N det (sC) > 0. [p(s) = det (sC + KR(s)+ DR-l(0))]For very small, but positive, values of s where 1/(TN,s) is very large, it is p ( s ) N det KR(s). In this case, for the canonical pairing, one finds det K&) > 0. For i. A rearrangement corresponds to a permutation of the matrix KR(s),which in turn leads to a sign change of det KR(s). That is, p ( s ) becomes negative for small, positive values of s. Since p ( s ) > 0 for large values of s, there is a value s* > 0, such that p(s*) = 0. This is equivalent to a positive root of eq A-6 and implies the instability of the closed system. This result is also true for every ( 2 m - 1)-fold rearrangement, since the determinant of the matrix changes signs after an odd number of permutations. For ii. The sign reversal of a controller is equivalent to the sign change of the feedback, that is, the sign change of an element of K&). This, on the other hand, affects the sign change of det K&). For the same reason as in case i, the consequence is an instability of the system. For iii. A combination of the above arguments leads to the given statement.
Appendix B: Significance of the Relative Gain Array Let y,(i = 1,...,12) and x, (i = 1,...,k ) be the input and output variables, respectively, of an arbitrary control loop, G = { g J , the open-loop gain array, and G-' = {g,]],the pertaining inverse. As a first assumption, x, is returned with a negative sign to y, via a controller R, (i = 1,...,k ) . For A,, < 0, one can discriminate two cases, which can be treated with the theorem of Koppel (1985). The simplified version of this theorem poses the following condition for the system stability: sgn (det G det (-KN)) = (-ilk (B-1) where KN = diag (Il,...,Zk), with I , as the Z contribution of the controller R,. 1. g,, < 0. In this case, the single control loop y, x, is unstable: if the remaining loops are opened, then the resulting system can be viewed as a single-variable system with positive feedback in the vicinity of a steady state. This system is obviously unstable. After a sign reversal of the controller R, (this corresponds to a sign change in the ith row of G ) ,the control loop y, x, becomes stable; however, it gives rise to a problem that is subject to case 2. 2. &, < 0. Either the system itself is unstable, or the subsystem-that is, the system with the control loop y l x, open-is unstable. If the system is stable, then according to Koppel, eq B-1 applies. If the control loop y, x, is opened, the resulting subsystem is characterized by the matrices Gs and K,,
-
-
-
-
which are obtained from G and KN by eliminating the ith row and ith column, respectively. Since Zi > 0 and gii = det G,/det G , it follows that sgn (det G , det (-KNs)) = sgn (det GZii det (-KN)(-l/Zi)) = (-l)k(+l) # (-l)k-l According to Koppel's theorem, this (sub)system is unstable. If Aii > 0 for all i = 1,...,h, then again the following two cases can be encountered. 3. g,, < 0 and gii < 0 for Certain Values of i. The conditions of cases 1 and 2 are imposed. If the signs of controller Ri are reversed, the following subcases are to be discriminated. 3.1. If the system is stable, for 2 m controllers the signs should be reversed; otherwise, the corresponding subsystems would be unstable according to Koppel. 3.2. For det G < 0 (which implies an unstable system), the signs of ( 2 m - 1) controllers are to be reversed. 4. gii > 0 and gii > 0 for All i = 1,...,k. The system is structurally unstable for det G < 0. The system is not structurally unstable for det G > 0. This case study is an extension of the theorem of Grosdidier et al. (1984) and of Gagnepain and Seborg (1982) about the negative diagonal elements of the relative gain array. Besides, it is demonstrated here how the individual cases are related to each other.
Appendix C: A Computation Method for Equation 10 If P I controllers are used, one can write KR(s) = Kp + K ~ / S With this, eq 10 is equivalent to det (s2C+ (Kp + DR-l(O))s + K,) = 0 Let us write
+
p ( s ) = det (s2C+ (Kp DR-l(O))s + K,) p ( s ) = aZRszR+ a2R-1s2R-1 + ... + a,
It then follows that a,, = det polynomial coefficients al, ..., the system of equations
KI
(C-1)
and aZR= det C. The can be determined from
The values of p ( s i )(i = 1,...,2R-1) are obtained by computing the determinant in eq C-1 with the respective values of s = si. The si can be chosen as -(R - 1),...,-1,1,..., R. The nonvanishing roots of the polynomial p ( s ) are then the roots of eq 10 and can be computed by the aid of standard numerical routines. For the cross arrangement x i f j o , only the permutations of the ith and j t h rows of the matrix K&) are necessary. However, the diagonal form of KR(s)is lost by this manipulation.
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Literature Cited Aris, R. Introduction to the Analysis of Chemical Reactors; Prentice Hall: Englewood Cliffs, NJ, 1965; Chapter 2.7. Bilgesu, E. Ph.D. Dissertation, Technische Universitat, Braunschweig, 1978. Bristol, E. IEE Trans. Autom. Control 1966, 11, 133-134. Fischer, B.; Studienarbeit, Institut fuer Technische Chemie I, Universitat Erlangen-Nuernberg, 1983. Gagnepain, J. P.; Seborg, D. E. Ind. Eng. Chem. Process Des. Deu. 1982,21, 5-11.
I n d . Eng. Chem. Res. 1988,27,93-99 Grosdidier, P.; Morari, M.; Holt, B. R. Proc. IEE Am. Control Conf. 1984, 3, 1290-1295. Grosdidier, P.; Morari, M.; Holt, B. R. Ind. Eng. Chem. Fundam. 1985, 24, 221-235. Isaacson, E.; Keller, H. B. Analysis of Numerical Methods; Wiley: Zurich and Frankfurt/Main, 1966. Kailath, T. Linear Systems; Prentice Hall: Englewood Cliffs, NJ, 1980; p 180. Koppel, L. B. AIChE J. 1985, 31, 70-75. Krokoszinsky, R. Ph.D. Dissertation, Technische Universitat Braunschweig, 1980. Lowe, A. Ind. Eng. Chem. Fundam. 1980a, 19, 160-166. Lowe, A. MBFSt Research Report, 1980b; MBFSt, Frankfurt/Main. Lowe, A. MBFSt Final Report, 198Oc; MBFSt, Frankfurt/Main.
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Lowe, A., private communication, 1982. McAvoy, T. J. Ind. Eng. Chem. Process Des. Dev. 1983,22, 42-48. Niederlinski, A. Automatica 1971, 7, 691-701. Pan, Diploma Thesis, Universitat Erlangen-Nurnberg, 1985. Schnitzlein, K. Diploma Thesis, Universitat Erlangen-Nurnberg, 1981. Shinskey, F. G. Process Control Systems; McGraw Hill: New York, 1979; Chapter 7. Wang, S.; Munro, N. Trans. Inst. Meas. Control (London) 1982,4, 53-56.
Received for reuiew November 17, 1986 Revised manuscript received August 24, 1987 Accepted September 9, 1987
Molecular Weight Distribution Control in a Batch Polymerization Reactor Takeichiro Takamatsu,+Suteaki Shioya,t and Yoshiki Okada*+ Department of Chemical Engineering, Kyoto University, Kyoto 606, Japan, and Department of Fermentation Technology, Osaka University, Suita, Osaka 565, Japan
This paper proposes a new method for getting the reactor temperature and initiator concentration in order t o obtain a final product of polymer which has a prescribed molecular weight distribution (MWD) in a free-radical polymerization batch reactor. At first, profiles of instantaneous average chain length and polydispersity are obtained so as t o get the desired MWD. Next, time profiles of reactor temperature and initiator concentration are determined from the profiles of instantaneous average chain length and polydispersity based on the mathematical model of the reactor. As the final step, the time profile of reactor temperature is realized by the Adaptive Internal Model Controller (AIMC) developed by us. I t is shown that AIMC can be utilized successfully for this highly nonlinear and time-dependent tracking control problem. The properties of the polymer product, such as the mechanical properties and the characteristics in molding, have a strong correlation with the molecular weight distribution (MWD) of the polymer. Martin et al. (1982) found that thermal properties, stress-strain properties, impact resistance, strength, and hardness of films of poly(methy1 methacrylate) and polystyrene were all improved by narrowing MWD. It is also generally said that the polymer of long chain length has superior mechanical properties for the polymer product but has insufficient molding characteristics for processing. Then the molding characteristics will be improved by blending short-chain polymer into this long-chain polymer, while the good mechanical characteristics will be kept. However, the adjustment by the blending always causes loss of energy, time, cost, etc. Therefore, the development of the methodology for adjusting MWD of the polymer during the reaction to a suitable one for its use is also desired in many industries, especially in producing high-quality polymers. Several papers by Hoffman et al. (1964), Tadmor and Biesenberger (1966), Nishimura and Yokoyama (1968), Hicks et al. (1969), Osakada and Fan (1970), Sachs et al. (19731, and Louie and Soong (1985) have reported the control of MWD in batch free-radical polymerization reactors. These studies have mainly focused on the optimization problem to obtain the narrowest MWD. When this optimization problem is solved by using optimization techniques such as the calculus of variations and maximum principle, it requires a great amount of iterative calculation on nonlinear differential equations for solving the twopoint boundary value problems. And there is no established method of solving the two-point boundary value problem of which convergence is assured. It is necessary Kyoto University.
* Osaka University. 0888-5885/88/2621-0093$01.50/0
to change the calculation technique according to the problem. Therefore, it is difficult to get solutions while coping with changes of reaction conditions. Thus, a simple calculation method is expected. On the other hand, there has been almost no work that developed a control strategy for broadening MWD in free-radical polymerization reaction. Arnold et al. (1980) and Couso et al. (1985) proposed the method to produce polymers of prescribed MWD in a living anionic polymerization batch reactor by perturbing the ratio of monomer to initiator concentration. This method uses a property that the instantaneous MWD in living anionic polymerization is Poisson-distributed. Unfortunately, this method cannot be used in free-radical polymerization, because in this reaction the instantaneous MWD does not have Poisson distribution. The two-step calculation method proposed here based on the instantaneous MWD can determine the time profiles of reactor temperature and initiator concentration, in a general free-radical polymerization batch reactor. By keeping the temperature and the initiator concentration along the calculated profiles, the final polymer product with prescribed MWD, that is, the average chain length and the polydispersity, will be obtained at the preestablished monomer conversion rate. Then, the required time profile of the reactor temperature should be practically realized. The adaptive controller, Adaptive Internal Model Controller (AIMC), developed by us (Takamatsu et al., 1985) is applied to a temperature control problem so that the reactor temperature tracks the desired trajectory decided above. AIMC is a combined control system of Internal Model Controller (IMC) and Model Reference Adaptive System (MRAS). IMC was developed by Garcia and Morari (1982). Parameters of the internal model used in AIMC are recursively estimated, utilizing an explicit identification 0 1988 American Chemical Society
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