Ind. Eng. Chem. Res. 1990,29,89-96 University of Waterloo, Waterloo, Ontario, Canada, 1987. Parks, P. C.; Schaufelberger, W.; Schmid, C.; Unbehauen, H. Applications of adaptive control systems. In Methods and Applications in Adaptive Control; Unbehauen, H., Ed.; Springer Verlag: Berlin, 1980. Penlidis, A. Latex production technology-reactor design considerations. Ph.D. Dissertation, McMaster University, Hamilton, Ontario, Canada, 1986. Penlidis, A.; MacGregor, J. F.; Hamielec, A. E. Effect of impurities on emulsion polymerization: case I Kinetics. J. Appl. Polym. Sci. 1988, 35, 20. Peterson, B. B.; Narendra, K. S. Bounded error adaptive control. IEEE Trans. Autom. Control 1982, AC-27, 1161. Ponnuswamy, S. R. On-line measurements and control of a polymerization battch reactor. Ph.D. Dissertation, University of Alberta, Edmonton, Alberta, Canada, 1984. Ray, W. H. Polymerization reactor control. Proc. Am. Control Conf. Boston June 1985,2, 842. Samson, C. Stability analysis of adaptively controlled systems subject to bounded disturbances. Automatica 1983, 19, 81. Seborg, D. E.; Edgar, T. F.; Shah, S. L. Adaptive control strategies for process control-a survey. AIChE J. 1986,32, 881. Stein, G. Adaptive flight control-a pragmatic view. In Applications of Adaptive Control; Narendra, K. S., Monopoli, R. V., Eds.; Academic Press: New York, 1980.
89
Stromer, P. R. Adaptive for self-optimizing control systems - a bibliography. IRE Trans. 1959, AC-7,65. Takamatsu, T.; Shioya, S.; Okada, Y. Design of adaptive/inferential control system and ita application to polymerization reactors. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 821. Takamatsu, T.; Shioya, S.; Okada, Y. Molecular weight distribution control in a batch polymerization. h d . Eng. Chem. Res. 1988,27, 93. Tanner, B. M.; Adebekun, A. K.; Schork, F. J. Feedback control of molecular weight distribution during continuous polymerization. Polym. Proc. Eng. 1987,5, 75. Unbehauen, H. Theory and application of adaptive control. Preprints IFAC 7th conference, Digital Computer Applications to Process Control, Vienna, 1985. Unbehauen, H.; Schmid, C. Status and application of adaptive control systems. Autom. Control Theory Appl. 1975, 3, 1. Vogel, E. F.; Edgar, T. F. Application of an adaptive pole-zero placement controller to chemical processes with variable dead time. Proc. Am. Control Conf. 1982, 536. Wittenmark, B. Stochastic adaptive control methods: a survey. Int. J . Control 1975, 21, 705.
Received for review December 21, 1988 Revised manuscript received August 10, 1989 Accepted September 26, 1989
Designing and Implementing Trajectories in an Exothermic Batch Chemical Reactor Daniel R. Lewin* and Ram Lavie Department of Chemical Engineering, Technion, IIT, Haifa 32000, Israel
Two design issues t h a t are critical to the safe operation of jacketed exothermic batch reactors are the definition of the start-up heating time and the problem of adequately tuning the feedback controller. Of the two, the definition of the safe and feasible start-up time is by far the more important. A simple algorithm is proposed to estimate the safe start-up time in terms of key design parameters (reaction kinetic parameters, heat-transfer coefficients, and jacket space velocity). The start-up rules thus generated are tested by simulation of a typical example, confirming the analysis and indicating that minor fouling of heat-exchange surfaces could in some cases lead to catastrophic loss of control. Implementation uses a n adaptive PID controller. One major characteristic differentiating batch from continuous processing is that concerning the process start-up trajectory. This added degree of freedom a t the operating stage is both a boon (opportunity to optimize) and a curse (difficult control problem). The present work concentrates on the planning of a batch reactor trajectory accounting for constraints arising from the plant design and also considering our ability of implementation through an appropriate control scheme. There has been considerable interest in the past on the control of batch chemical reactors. However, most of the previous work has concentrated on either the derivation of the optimal temperature trajectory based on reaction kinetics alone or on the design of control systems guiding the plant along this trajectory. The work of Hugo and Steinbach (1986) generates design rules for semibatch reactors but only for conditions of fixed reaction temperature and so does not solve the start-up problem, which is potentially the most problematic stage of the trajectory. Marroquin and Luyben (1973) considered the start-up problem. On the basis of simulation results, they indicated that initial heating offers little or no advantages for consecutive reaction systems and makes the control problem more difficult. This paper will address the start-up issue including start-up heating and will use an analytical ap0888-5885/90/ 2629-0089$02.50f 0
proach in order to specify feasible trajectories that guarantee stable performance. The implementation of the temperature trajectory is effected by manipulation of final control elements. The actual choice of the desired trajectory is therefore also dependent on the physical limitations of the processing unit, a fact that has not been given sufficient attention in the past. Indeed, in many cases, optimality considerations are outweighed by these limitations, leading to operation along the constraints. In practice, the operation of a chemical batch reactor is formulated in terms of a temperature trajectory because temperature is the most readily available output, to date. The trajectory is specified as a sequence of steps, consisting of three parts: (A) start-up, where the reactor contents are brought from the initial loading conditions to the desired operating level; (B) maintenance of the desired nominal operating conditions for as long as it is beneficial to do so; (C)termination of the reaction according to either optimality or product specifications considerations. The relative importance of each part may differ in different cases, but the most common situation, considered here, is that where most of the reaction occurs in part B. However, the performance in part B may be strongly dependent on how part A was carried out. The logical order 0 1990 American Chemical Society
90 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
in which the trajectory is derived always starts with the specification of the nominal operating temperature. From rate considerations alone, it will usually be advantageous to operate at a high temperature level. The actual choice of this level will usually be constrained by physical limitations (e.g., choice of coolant, sensitivity of materials, etc.) or by optimality considerations. Since this choice will be specific to the particular application, it is clearly beyond the scope of the present study, and we shall assume some given arbitrary level in the following discussion. The subsequent decisions that must be made concerning the trajectory are how fast should the nominal operating level be attained and when and how should the reaction be terminated. The former issue is the main concern of this paper and will be discussed below. No attempt will be made to obtain true optimal trajectories over the whole batch processing cycle. A suboptimal path, which accounts for the constraints on start-up rate and nominal operating level. will be our main goal. In addition, we shall discuss control design alternatives in order to ensure that the desired trajectories are feasible. Whatever tools are used in the design of the control of the plant, the use of a mathematical model is essential in developing strategies. In some extreme cases where a specific deterministic detailed mathematical model of the plant is applicable, it may be feasible to derive analytical design and control strategies a priori. A t the other extreme, important features of the process may be too vaguely known to enable full mathematical formulation of the problem. In this eventuality. some kind of mathematical model will still be needed to frame the engineer’s mind, sharpen his intuition, and help him quantify directions of action. In the following, a generalized deterministic mathematical model for a batch reactor is first formulated. Then it is further detailed to a specific example to permit the study of the particular case. On the basis of the results obtained, we will conclude with generalizations of the understanding gained from the study.
Mathematical Model Generalization of batch chemical reactors as a single category is not easy since their crucial element, the chemical reaction, may differ widely from one case to another with implications on objectives and performance criteria. Thus, the production of polymers or of other reactive materials involving highly exothermic and sometimes autocatalytic reactions pose problems that are different from those related to more benign reactions aimed at high selectivity or product purity. Nevertheless, an attempt at the formulation of a conceptual model, including sufficient elements such as to allow the representation of a wide variety of typical situations of interest as special cases, is well justified. In so doing, it is worth separating the relatively simple, preparatory and postprocessing steps (such as recipe formulation, reactant loading, and product discharging), which are characteristically process specific and mostly independent from the central issue concerning the carrying out of the chemical reaction itself, which will be the subject of the following discussion. A typical batch reactor is illustrated in Figure 1. Here, split-range temperature control is employed, with steam being used to raise the reaction temperature to the desired level after which cooling water is used to remove the heat of reaction. Luyben (1975) and Liptgk (1986) provide reviews ot alternative schemes which incorporate heat removal systems of varying complexity. The model will be based on the following assumptions: (a) The reactor content is well mixed at all times. While this is not necessarily always the case in practice (in
limed f I I Ied 1
Figure 1. Simplified batch reactor showing control and state variables.
particular in large reactors and in cases where the reaction mass is viscous or otherwise difficult to mix), this is at least an objective. (b) Jacket heating/cooling occurs, with the heating/ cooling medium being well mixed in the jacket. (cj The operating procedure along the reaction path follows a predefined recipe concerning the amounts and timing for the addition of reactants, solvents, catalysts, moderators, etc. The amounts (nLk) and timing (tk)may be defined as a preprogrammed sequence or as a control action in response to conditions developing along the reaction path (feedback). This also applies to the amount and timing for the heating/cooling rate in the jacket. This leads to the simplified material and energy balances: dCi J - = 6(t - tk)Cik+ Crij(Cj,T , P ) , dt j=1 i = 1, ..., I (species), Ci(0) = 0, j = 1, ..., J (reactions), k = 1, ...,K (shots) (1) J
T(0) = T o (2)
dTm dt
-=-
Qj-Qm
81
(4)
p = $bi,TI (5) (6) h = #l(ni, T , P ) where Ciis the amount of species i at time t , Cik is the amount of species i added to the reactor at time t k , rij is the rate of reaction j with stoichiometry for species i, T is the temperature of the reacting mixture, T k is the temperature of the shot added a t time k , AHj is the heat of reaction j, P is the pressure, and tl = one of 1 reaction extent indicators. The heat-transfer rates are given by the following equations: Qm = h,(Tm - T ) (7) Qj = h,(Tj - T,) (8) Important design parameters in the state equations are (1)the coefficients hl and h2,which are thermal conductances from the reactor to the wall and from the wall to the jacket; (2) the coefficient 6,which is the jacket space velocity; and (3) T , and T,, which are the cooling and
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 91 generation is proportional to the reaction rate, which in turn increases with the temperature and reactant concentration. Considering cases where most of the reaction is carried out at the nominal operating temperature, reactants do not have a chance to be substantially depleted within the start-up time. Then obviously, maximum heat generation occurs at the earliest peak temperature. For safe operation, it is sufficient that the cooling capacity should a t all times exceed the maximum heat generation rate (including some safety margin to account for dynamic distortion). Equations 2-4 a t steady state can be combined into an expression relating the instantaneous heat removal rate (expressed as a temperature gradient) to the heat-transfer conductances hl and h2:
Topf............
Time
Figure 2. Graphical illustration showing that AH is at a maximum at * = P. (a) Start-up temperature trajectory is assumed a ramp. (b) Plot of showing a maximum at t = T~ and contributions from temperature and concentration dependencies.
heating fluid supply temperatures. The mathematical model presented by Luyben (1973) is essentially a partial case of the one brought here, with some modification of heating and cooling mechanisms.
Feasible and Safe Start-up Times In the following, it is assumed that no composition measurements are available and that the reaction under consideration is such that the composition trajectory is implicitly represented by the batch temperature trajectory. The desired product is some species i = ip,and we generally which to prevent the wasteful formation of other species (side products). The design should determine the temperature trajectory that optimizes the production of desired product while avoiding dangerous or undesirable conditions. The nominal operating temperature is specified first. Then, we should address the specification of how to attain this desired temperature level from the initial loading conditions. The start-up trajectory can be defined by specifying a start-up time, r , at which the nominal operating temperature is attained and the path followed to this point. We are therefore making an implicit assumption that we can follow an arbitrarily chosen temperature trajectory by means of the available heating and cooling capacity and an appropriately designed control system. Evidently, the implementation of a start-up path will invariably require first heating and then cooling. The start-up time is bound on the one hand by the fastest heating rate possible and on the other by safety limitations arising from our capability of removing heat generated by the reaction. Both of these considerations will be discussed in detail below. Defining the Minimum “Feasible”Start-up Time, T ~ .The minimum feasible time, rh, taken to reach the target temperature, TOP, is the time necessary to heat the reactant mass up to the reaction temperature, with some help from the heat of the reaction. Assuming the availability of a full mathematical model, it can be determined by integrating eq 1-4 from the initial conditions and assuming maximum heat input (U, = 1 and U, = 0). In the absence of detailed reaction kinetics, a lower bound for rh is trivially evaluated by omiting the reaction terms in eq 2. Defining the “Safe” Start-up Time, T,. Figure 2 shows the expected form of the concentration response to an imposed ramp temperature trajectory taking place over a period of rC and its implications on heat generation. Heat
The instantanous heat generation, expressed as an exothermic temperature rise, ATgen,computed a t time rC, represents the greatest load on the cooling system: J
C m ; ( c ; ( ~ ‘T)o, p P, )
ATg& =
;=l
(10) PCP
Rigorous evaluation of eq 10 requires integration of the mass balances given by eq 1, along specified T ’(t)from t = 0 to t = rC:
and T ‘ is the imposed temperature trajectory up to t = 7,. Typically, T ’(t)could take the form of a ramp:
T‘(t) =
1
+
T(0)
(
~ c T ( o ) ) t0, 5 t < T C
(12)
Alternatively, an upper bound on the heat generation can be evaluated by taking Cj(7,) N Cj(0),but this would not help in formulating a lower bound on 7,. In any event, ATremI ATE, will ensure avoidance of temperature runaway. When only a small fraction of the reaction takes place during the start-up, there is little incentive to guide T’(t)in great detail and the ramp (eq 12) is as good a policy as any other. Then, eq 9-12 can be combined to evaluate the fastest safe start-up time. Numerical Example. For the example process given by Luyben (1973), the reaction system is the two consecutive first-order reactions: kl
A-B-C
k2
Thus, the generalized mass balance (eq 1)is reduced to that of two independent species: dCA/dt = -k,CA (14) dCB/dt = k1CA - k 2 c ~
(15)
The temperature dependence of the rate of production of B and A and loss of B to C can be expressed through Arrhenius temperature dependencies of the rate constants k , and k,: kl =
al@alIRT
k2 =
a2@a2/RT
(16)
92 Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 (a)
(bl Contour V.1uar
hl
o7
Figure 3. Contour diagrams showing the minimum heating times, rh,as a function of heat-transfer conductances, hl and h2,for (a) 4 = 1.058 (the nominal value) and (b) 4 = 0.264. Marked on the figures are the values of T~ determined for the nominal values of heat-transfer conductances. (b)
(a)
Figure 4. Contour diagrams showing the minimum safe start-up times, rC,as a function of heat-transfer conductances, h, and h2,for (a) @ = 1.058 (the nominal value) and (b) 4 = 0.264. Marked on the figures are the values of T~ determined for the nominal values of heat-transfer conductances. Table I. Initial Loading Conditions variable value (example) variable T 80 "F CB Tin 80 O F Tc 80 O F Th Tj 0.8 mol A/ft3 CA
Table 11. Design Parameters value (example) 0.0 mol B/ft3 80 O F 260 O F
For this reaction system, the heat of reaction is then given by 2
CAHjCCj, T , PI = ki(T)CAXi + k2(T)CBX,
j=1
(17)
The initial conditions are taken from Luyben (1973) and are listed in Table I. The nominal values of the design parameters, which appear in Table 11,were computed from data supplied by the same source. In this example, the reaction heat generation is large in comparison with the external heat supply capacity. Let us assume that the desired nominal operating temperature is constrained to be T O P = 200 OF. The fastest safe start-up times were computed for various values of the design parameters. It is of interest to investigate what effect changes in heat-transfer coefficients (for example, due to fouling of heat-exchange surfaces) have on the feasible and safe start-up operating policy. Figure 3 shows how the con-
variable 81 02
hl h2 4
formulation (VmPmCm)/(VpCp) (VjPjCj)/ VPC,) (hiAi)/(VpCp) (howAO)/( VpCP) J'mlVj
nominal value (examde) 0.272 0.552 0.071 min-' 0.177 min-' 1.058 min-'
ductances influence the minimum heat-up time for two values of the jacket space velocity as calculated by eq 2-4, 14, and 15. Evidently, with most of the heat supplied by the heat of the reaction, the value of T~ is not particularly sensitive to the heat-transfer conductances. It is also noted that the fastest feasible start-up time hardly changes for a 4-fold decrease in the flow of the heating medium in the jacket. On the other hand, in a reactive situation as described by this example, a safe start-up time will be quite sensitive to heat-transfer limitations. Figure 4 depicts rc as a function of hl and h2 for the same two values of the jacket space velocity, as calculated by eq 9 and 10. Here, the required value of rc may change very significantly (a 30-fold increase over a token change in the parameters). Also marked on the figures are the coordinates representing the nominal values of h, and h2in both cases. We
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990 93 Table 111. Kinetic Parameters variable nominal value variable a1 729.55 min-' Eaz aZ 6567.6 min-' X1 Ea, -15000 Btu mol-' A2
.D
m
am
15
u
6m
.a
75
1-
iaa
ias
ism
time b i n )
Figure 5. Simulation results showing temperature trajectories for a batch reactor system with nominal parameters except for 6 = 0.264. The simulations have different start-up times specified.
note that an accidental drop in coolant flow rate may have serious consequences in what concerns safe start-up. The estimate for T~ as calculated above is confirmed by rigorous intergration of the model as demonstrated in Figure 5. Here, we see a limiting safe start-up time of somewhat below 50 min, while the estimate based on eq 9 and 10 indicates a limit at 55 min. Thus, it seems that the equations provide a slightly conservative estimate of T ~ . It is also of interest to verify in our example how sensitive are the start-up time limits to the rate of flow in the jacket. This is illustrated in Figure 6. Delineated on the plots are regions in the 7-4 plane that are feasible and safe ( T ~ 79, feasible but unsafe ( T ~ T ~ ) and , infeasible (7 < T ~ ) .Marked on the diagrams as A are the operating conditions fixed by defining a start-up time of r = 30 min a t nominal jacket space velocity. Point B in the diagrams represent the new operating conditions caused by an upset in cooling water supply pressure, leading to a reduction in 4 by 50%. The plots illustrate an unfortunate scenario. In Figure 6a, for nominal values of h, and h2 and with the start-up time set at 30 min, a temporary loss of cooling water pressure leading to a reduction in 4 of 50% poses no problem. However, in Figure 6b, which has identical conditions except for a mere 10% reduction in hl and h2 due to fouling, the same upset can no longer be tolerated. These results are confirmed by the simulations shown in Figure 7. An even more frightening scenario is depicted in Figure 8. For the nominal values of the parameters, the fastest
88
,
68
-!
start-up time is dictated by T~ (see Figures 3a and 4a, indicating T = max (7c, T ~ =) T~ = 18.6 min). Thus, the response for a start-up time set at T = 20 min should be satisfactory, as indeed is shown in Figure 8. However, when the conductances drop by 30%, T~ soars and the reactor becomes uncontrollable. Such a reduction is well within what might be expected in an operating reactor due to fouling. This example points out dangers and opportunities in the design and start-up operation of batch reactors carrying out reactive reactions (generating substantial heat). Proper consideration of the effective cooling capacity (which can be quite sensitive to minor fouling in the heat-exchange surfaces and coolant flow) can loosen operability constraints and sometimes reduce the need for safety margins.
Reaction Kinetics and the Constrained Optimal Temperature Trajectory In the preceding discussion, we have defined start-up temperature bounds. For a ramp start-up temperature trajectory, this is expressed as the larger of T~ and T ~ We . can now proceed to define the total batch temperature trajectory as bound by this constraint. The temperature trajectory can either be optimized such as to either maximize the production or to minimize the time taken to make a product of a specified composition. This is demonstrated through the same example of the previous section. As first stated by Bilous and Amundson (1956), if Eal I Ea2, the optimal operating temperature is a constant, equal to the highest level tolerated by the reacting materials or their envelope. This ensures the highest rate of reaction and thus the shortest batch time. However, if Eal Ea2, then the optimal temperature will decrease during the batch time in order to reduce the formation of C. Table I11 gives the nominal values of the parameters for an example in which the latter case holds. The numerical parameters are taken from Luyben (1973). The maximum
(a)
I
~
S A F E
I
+A
8.
1
8.25
nominal value -20000 Btu mol-' -40000 Btu mol-' -50 000 Btu mol-'
40
{
S A F E
UNFEASIBLE
, , ,
,j,
0.5
, , , ,
, 0.15
, , , ,
,
, , , , 1.25
1
5
6
Figure 6. Sensitivity of 7Cand ihto 6 for fixed heat-transfer conductances. The two plots show results for (a) nominal values of hl and h2 and (b) values 10% lower (simulating fouling).
Ind. Eng. Chem. Res., Vol. 29, No. 1, 1990
94
(a) aeo
e
+
a-
= 0.529
zm
e
c
f
160 120
eo
0"
8.6
{
3*
8.4
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g 8.5 4
(b)
5
aao
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-
-
z m 160
k
1
..........\
// .
0 =
0,:,-7
l.m
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i
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!
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'
'
,
IOQ
'
I
I
158
, " " 200
I
250
3a
Tine ( a m )
E
Figure 9. Sensitivity of constrained optimal reaction trajectories to temperature dependence of the reaction rates. (a, top) Reaction temperature trajectories. (b, bottom) Trajectories for the concentration of B.
120
eo
is
o
30
4s
60
7s
90
loa
Lao
sa5
1so
time ( m i d
Figure 7. Resilence of the controlled reactor to an upset in cooling water supply pressure, as simulated by a 50% reduction in 4. The two figures indicate the effect of fouling. (a) Heat-transfer conductances a t their nominal values. (b) Heat-transfer conductances a t 90% of the nominal values, simulating fouling. 9..
E
4
' 8 , l -3.: A>,'
a-
a
8.3 j
1
/ 30% reduction
a-
in
b, and b,
am nominal values of h, and b, 160
iao EO
a
15
aa
u
60
7s
90
ioa
lam
13s
is0
t i m e (mid
Figure 8. Simulation showing the sensitivity of safe start-up time, T , to the heat-transfer conductances, h, and h2. Table IV. Sensitivity of Safe and Feasible Start-up Times to Activation Energy Values Ea,, Ea2, case Btu/mol Btu/mol ih, min T ~ min , tsariteh , min a -14000 -20000 14.6 111 198 18.6
