Eect of Process Design on the Open-loop Behavior of a. Rensselaer Polytechnic Institute, Howard P. Isermann Department of Chemical Engineering, Troy,

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1 Eect of Process Design on the Open-loop Behavior of a Jacketed Exothermic CSTR Louis P. Russo and B. Wayne Bequette Rensselaer Polytechnic nstitute, Howard P. sermann Department of Chemical Engineering, Troy, NY , USA Abstract. The classic two-state continuous stirred tank reactor (CSTR) model has been the focus of much of the previous research on exothermic reactor operation. One assumption of this model is that the cooling jacket temperature dynamics are negligible, hence the cooling jacket temperature is the manipulated input (instead of the cooling jacket owrate) for feedback control of reactor temperature. The inuence of process design parameters on the open-loop behavior of a threestate CSTR model (which incorporates an energy balance around the cooling jacket) is considered in this paper. Elementary catastrophe theory is used to study the eect of process parameters on the steady-state multiplicity of the three-state CSTR model. We demonstrate the existence of disjoint bifurcations associated with infeasible reactor operation regions. Reactor scaleup is shown to have an eect on the presence of these infeasible reactor operation regions. A multiple time-scale perturbation analysis is used to understand the eect of reactor design on the oscillation amplitude of the three-state CSTR model at a Hopf bifurcation point. Keywords. Chemical Reactors, Nonlinear Systems, Multiple steady-states, Open-loop Dynamics, Process Design. 1 NTRODUCTON Continuous stirred tank reactors (CSTR's) often present challenging operation problems due to complex open-loop behavior such as input and output multiplicities, ignition/extinction behavior, parametric sensitivity, and nonlinear oscillations. These characteristics demonstrate the need for and diculty of feedback control, particularly when the CSTR is open-loop unstable or when it exhibits nonlinear oscillations. An objective of this paper is to show the eect of process design on the open-loop multiplicity and dynamic characteristics of exothermic CSTR's. This multiplicity analysis gives practical guidance for process redesign to eliminate dicult operating regions associated with output multiplicity behavior. The open-loop dynamic analysis allows one to understand the eect of process design on the oscillation behavior of the three-state CSTR model (incorporating an energy balance around the cooling jacket). CONTNUOUS STRRED TANK REACTOR MODELS The standard two-state CSTR model describing an exothermic diabatic irreversible rst-order reaction (A! B) is a set of two nonlinear ordinary dierential equations obtained from dynamic material and energy balances (with the assumptions of constant volume, perfect mixing, negligible cooling jacket dynamics, and constant physical parameters) dx d dx 1 =?x1(x) + q(x1f? x1) (1) d = x1(x)? (q + )x + x3 + qxf () where x 1 and x are the dimensionless concentration and reactor temperature, respectively. An additional energy balance around the cooling jacket yields dx 3 = 1[qc(x3f? x3) + (x? x3)] (3) d where x 3 is the cooling jacket temperature and is the cooling jacket owrate. The other parameters (, q,, etc.) are dened in Table 1 of the Appendix. Representative values of the parameters (Cases 1-3, adapted from Ray, 19) are given in Russo and Bequette (1993). Many dynamics and control studies have been based on the two-state CSTR model (eqns. 1 - ) and consider x (reactor temperature) to be the controlled variable and x 3 (coolant temperature) to be the manipulated input. We will study a three-state CSTR model (eqns. 1-3) where x is the controlled variable and (cooling jacket owrate) is the manipulated input.

2 3 STEADY-STATE MULTPLCTY A number of researchers (Uppal et al., 197; Balakotaiah and Luss, 193; Guckenheimer, 19, among others) derive conditions for steady-state multiplicities in a two-state CSTR model. We direct the reader to the review articles by Razon and Schmitz (197) (concerning the multiplicity and dynamic behavior of chemically reacting systems) and Bequette (1991) (which deals with the nonlinear control of chemical processes). This section will consider the inuence of the manipulated variable () on the multiplicity behavior of a three-state CSTR. 3.1 nput and Output Multiplicities The necessary conditions for multiplicities in a twostate CSTR model are well-known. However, it is easy to show that the steady-state multiplicity behavior changes with the addition of cooling jacket dynamics (Russo and Bequette, 1993). Hence, the openloop stability may change with the addition of cooling jacket dynamics in the CSTR modeling equations. We use some results from elementary catastrophe theory to understand the dierent multiplicity results for the two and three-state CSTR models. Equations 1-3 can be combined at steady-state to give (x s) h(x s; s; p) = qx 1f? (q + )xs q + (x s) +qx f + qcsx3f + x s ( + s) = () where x is the controlled output, is the manipulated input, and p is a vector containing the system parameters. Upon examination of the derivatives of h(x s; s; p) with respect to the parameters we have determined that q is the only parameter that (when varied) can lead to input multiplicities or isola behavior. From elementary catastrophe theory, the appearance or disappearance of output multiplicities occurs when higher order derivatives of h with respect to x s are zero: h s = : : : h = k k+1 h = k+1 s The typical approach is to determine the highest order singularity (the largest value of k) which satises equation (5). We have determined that the highest order singularity of equation (5) is k =, since s = s < (Russo and Bequette, 1993). Since we are interested in the possible steady-state multiplicity from a process control viewpoint, the singularity described by equation (5) is no longer a point but a locus in the global parameter space. This locus is taken over the bounds on the primary bifurcation parameter ( - the manipulated variable for feedback control). We take the bounds on the cooling jacket owrate as [; 1) without loss of generality (the methodology remains the same). 3. Determination of Global Multiplicity Diagrams One particularly interesting aspect of steady-state multiplicity studies is the appearance of \so-called" disjoint bifurcations (Gray, 1991). Disjoint bifurcations result in nonfeasible operating regions that separate non-closed disconnected steady-state solution branches. This type of behavior arises when there are bounds on the values of the parameters or state variables. These bounds can be either arbitrary bounds or physical bounds (certain parameters are non-negative, for example the cooling jacket owrate). t should be made clear that disjoint bifurcations are extremely dierent from isola behavior, where a closed, isolated branch of solutions is either created or destroyed through a well-dened bifurcation. Determining regions of feasible and nonfeasible operation are critical for feedback control purposes. Balakotaiah and Luss (19) described a method to divide the global parameter space into regions with different types of multiplicity diagrams. One possible problem is that since singularity theory is by its nature local, there can be multiple highest order singularities of equal codimension (or a locus, or a series of loci of largest codimension), which complicate the analysis. The results are portrayed on the - -D cross section of the parameter space. We feel that this is most appropriate since the values of (nominal Damkohler number) and (dimensionless heat of reaction) are a direct measure of the kinetic and thermodynamic properties of the reaction. The solution of equation (5) (with k = ) gives a necessary condition for multiplicity on the value of : [a 1 + a ] crit = qx 1f ( + s) [a 1(? )? a ] where a 1 and a are given by: a 1 = ((s + q ) + qs) a = ((sx 3f + q x f ) + qsx f) (7) We want to understand the eect of on the critical value of. There is no reason to expect a monotonic relationship between crit and s, due to the quadratic s terms in the numerator of crit. Therefore, we want to determine the eect of the other parameters on the monotonicity of crit with respect to s. n the case when the cooling jacket feed temperature is equal to the reactor feed temperature (x 3f = x f crit = ( + x q x 1f (?? x f ) ( + s) > () Therefore, crit is a monotone increasing function of s, hence the minimum and maximum values of crit occur at the minimum and maximum values of s (which we have taken to be and 1). Now, let us consider the general case. t can be shown that there

3 is at most one physically meaningful minima = ) in the plot of crit versus s (there are no maxima). The location of this minima along the s axis is a monotone decreasing function of, and that this minima occurs at s = when =, where is the physically meaningful root of equation (?? x f? x 3f ) + (?x f? x 3f )?x f x 3f = (9) n the case when x f =, is given by q =? x 3f + ( + x ) (1) 3f For example, when x 3f =?1, = 3 + p 5 5:379. For values of above, the relationship between crit and s is monotone increasing, hence the minimum value of crit occurs when is at its lowest bound ( = ) and the maximum value of crit occurs when is at its upper bound (! 1). n the limit as = (adiabatic reactor) crit is crit = x 1f ( + x f ) (?? x f ) (11) A necessary condition for steady-state multiplicity can be derived on the dimensionless activation energy (), since crit must be nonnegative. > = + p (1 + x f) (1) n the limit as s! 1, crit is crit = [(q + ) + (x 3f + qx f )] qx 1f [(q + )(? )? (x 3f + qx f )] (13) This is the same crit as the two-state model, since as! 1, x 3s! x 3f (Russo and Bequette, 1993). Therefore, the three-state model may be open-loop unstable when the two-state model is stable. This means that the three-state crit (for < 1) is less than the two-state crit, therefore the limit point instability region for the three-state model is larger than the two-state model (assuming that > ). When the feed temperatures of the cooling jacket and reactor are equal (x 3f = x f ) (as is often assumed), equation 13 becomes crit = (q + ) qx 1f ( + x f ) (?? x f ) (1) q+ q, We can see that crit changes by a factor of therefore a larger (dimensionless heat transfer coef- cient) results in a larger multiplicity area in the parameter space. This result is counter-intuitive, since it has been shown (Russo and Bequette, 1993) that multiplicity could be eliminated in the two-state CSTR model by increasing. A codimension 1 fold bifurcates from each of the parameter bounds ( [; 1)) which separates the global multiplicity characteristics in the - parameter space. h s s = or s! 1 (15) Equation 15 is satised along the \-disjoint" or \1- disjoint" loci, respectively. The codimension (cusp) hysteresis locus and codimension 1 (fold) \-disjoint" and \1-disjoint" loci divide the - parameter space into ve dierent regions, as shown in gure 1 for Case conditions (table ), which correspond to ve dierent x - bifurcation diagrams (shown in gures - ). We have determined the equations for x s, crit, and crit along the disjoint loci and the hysteresis locus (connecting = and! 1) (Russo and Bequette, 1993). Region in the - parameter space does not exhibit output multiplicities, whereas region is the standard inverse S shaped curve. Region is an example of \-disjoint" bifurcation (low temperature infeasibility region), while region is an example of \1-disjoint" bifurcation (high temperature infeasibility region). Region V exhibits both \-disjoint" and \1-disjoint" bifurcations (where the codimension 1 disjoint loci intersect), resulting in very severe operation and control problems due to the multiple infeasible operating regions. 3.3 nuence of CSTR parameters on the Multiplicity Behavior We have seen that including an energy balance around the cooling jacket changes the multiplicity results. We want to explore the eect of a process design change (reactor sizing) on the bifurcation behavior of the three-state CSTR model. The reactor parameters used in this example were obtained from Devia and Luyben (197); the dimensionless parameters (Case 5) are given in table. The CSTR studied has a height to diameter ratio of (H r = D r). As pointed out by Devia and Luyben (197), increasing the reactor size causes a reduction in the ratio of heat transfer area to reactor volume, since it is assumed that the reactor and jacket residence times are held constant with scaleup. One therefore sees a reduction in the dimensionless heat transfer coecient () when the reactor size is increased, since in this case 1 D r. Figure 7 shows the - plot when the reactor volume is 1, gallons ( = :793). The reactor operation falls in region, hence the input-output curve is inverse S shaped. However, if the reactor volume is scaled up to 15, gallons, the dimensionless heat transfer coecient decreases ( = 1:991). The reactor operation shifts to region (shown in gure ), which is associated with \1-disjoint" bifurcation. This input-output curve is shown in gure 9. Using this analysis, one can see that if the residence time for the reactor is held constant upon scaleup, then infeasible reactor operation regions may occur. n certain cases, one may even shift into a high temperature operation region (with no multiplicities), which is consistent with the loss in heat transfer capacity due to scaleup. OPEN-LOOP DYNAMCS The stability and dynamics of the two-state (stan-

4 dard) CSTR model have been thoroughly investigated (Uppal et al., 197, among others). n particular, the presence of oscillatory behavior (generally through Hopf bifurcation) inuences process performance and control. Douglas and Rippin (19) demonstrated that in certain cases oscillatory response is desirable. Dynamical systems theory (Wiggins, 199) is the basis for much of the dynamic analysis performed during the past decade, particularly the analysis of degenerate Hopf bifurcations (Golubitsky and Langford, 191). One drawback to a dynamical systems approach is that the resulting normal form coecients may be a nonlinear function of the system parameters, hence it can be dicult to understand the inuence of a process parameter on the system behavior. n order to circumvent these diculties, we use a multiple time-scale perturbation analysis (Kevorkian and Cole, 191; Kapila, 193) to determine the eect of a process parameter on the Hopf bifurcation behavior of the three-state CSTR model. This focus of this section is to demonstrate the effect of process design on the oscillation amplitude at a Hopf bifurcation point. n this work we are interested in how the reactor to cooling jacket volume ratio ( 1) changes the dynamic behavior of the dimensionless reactor temperature (x ) ( 1 does not inuence the steady-state behavior). The reader is directed to Kevorkian and Cole (191) and Kapila(193) for the details of the multiple time-scale method. The dimensionless reactor temperature (x ) is expanded as: x (t 1; t ) = x s + y 1(t 1; t ) + y (t 1; t ) + : : : (1) where x s is the steady-state dimensionless reactor temperature at the Hopf bifurcation point. The multiple time scales are t 1 = t and t = t, where = 1? 1 ( 1 = 1). The equation for y 1(t 1; t ) is obtained from balancing the O() terms in the expansion y 1(t 1; t ) = A(t )cos(t 1 + (t )) (17) where A(t ) and (t ) are chosen to remove the secular terms which arise from the O( ) problem. n order to illustrate the use of this approach, we consider Case 3 conditions (Table ). Case 3 has a Hopf bifurcation point when x 1s = :33, x s = 3:573, x 3s = :55, and s = :913. The reactor to cooling jacket volume ratio ( 1) is changed to 1 = 1: ( = :) in order to understand how it aects the oscillation amplitude. Figure 1 shows the plot of dimensionless reactor temperature (x ) versus dimensionless time () for the numerical solution and the () expansion. One can see that the () expansion does a good job of predicting the oscillation amplitude. 5 CONCLUSONS We have shown that adding a third state to the classic two-state exothermic CSTR to account for cooling jacket temperature dynamics has an important eect on the steady-state multiplicity behavior. The global input-output multiplicity behavior was characterized in terms of a nominal Damkohler number and a dimensionless heat of reaction. We have demonstrated that a detailed steady-state multiplicity analysis provides important information concerning the inuence of process design and operation parameters on CSTR performance. Reactor scaleup has an important effect on the presence of infeasible operation regions. A multiple time-scale perturbation analysis was used to examine the eect of the dimensionless reactor to jacket volume ratio on the oscillation amplitude at a Hopf bifurcation point. Acknowledgments Financial support from the Howard P. sermann Department of Chemical Engineering is gratefully acknowledged. REFERENCES Balakotaiah, V. and D. Luss (193). Multiplicity features of reacting systems. Dependence of the steadystates of a CSTR on the residence time. Chem. Eng. Sci., 3, Balakotaiah, V. and D. Luss (19). Global analysis of the multiplicity features of multi-reaction lumpedparameter systems. Chem. Eng. Sci., 39, 5-1. Bequette, B.W. (1991). Nonlinear control of chemical processes: A review. nd. Eng. Chem. Res., 3, Devia, N. and W.L. Luyben (197). Reactors: Size versus Stability. Hydrocarbon Process., Douglas, J.M. and D.W.T. Rippin (19). Unsteady state process operation. Chem. Eng. Sci., 1, Golubitsky, M. and W.F. Langford (191). Classication and unfoldings of degenerate Hopf bifurcations. J. Dierential Eqns., 1, Gray, B.F., J.H. Merkin, and G.C. Wake (1991). Disjoint bifurcation diagrams in combustion systems. Mathl. Comput. Modelling, 15, Guckenheimer, J. (19). Multiple bifurcation problems for chemical reactors. Physica, D, 1-. Kapila, A.K. (193). Asymptotic treatment of chemically reacting systems. Pitman, Boston. Kevorkian, J. and J.D. Cole (191). Perturbation methods in applied mathematics. Springer-Verlag, New York. Ray, W.H. (19). New approaches to the dynamics of nonlinear systems with implications for process and control system design. n Chemical Process Control, D.E. Seborg and T.F. Edgar, eds., United Engineering Trustees, New York, 5-7.

5 Razon, L.F. and R.A. Schmitz (197). Multiplicities and instabilities in chemically reacting systems - a review. Chem. Eng. Sci.,, Russo, L.P. and B.W. Bequette (1993). mpact of process design on the multiplicity behavior of a jacketed exothermic CSTR. Submitted to AChE J. Uppal, A., W.H. Ray, and A.B. Poore (197). On the dynamic behavior of continuous stirred tank reactors. Chem. Eng. Sci., 9, Wiggins, S. (199). ntroduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, New York. APPENDX φ = 5 7 "low-temperature operation" 9 1 β "high-temperature operation" 11 1 Hysteresis locus "-disjoint" loci " -disjoint" loci Figure 1: Global multiplicity behavior for Case conditions V 1 Table 1. Dimensionless variables and parameters for the two and three-state CSTR models. x 1 = Ca C af x = T?T f T f x x 3 = Tc?T f q T c f = Qc Q = E a RT f (x ) = exp( x 1+ x ) = (?H)C af UA CpT f = CpQ = V Q k e? q = Q Q 1 3 Figure : Case, = :11, = 7. 5 = Q V t 1 = V V c = Cp ccpc x 1f = C af C af x f = T f?t f T f x 3f = T cf?t f T f x Table. Three-state model parameter values. Param. Case 3 Case Case : : : : : : : : :. q x 1f x f... x 3f x Figure 3: Case, = :, = Figure : Case, = :, =

6 .1.1. Hysteresis locus "-disjoint" loci " -disjoint" loci x φ. 15, gal. reactor.. Figure 5: Case, = :, = β Figure : Global multiplicity behavior showing reactor operation in region when the reactor size is scaled up ( = 1:991). V x x Figure : Case, = :3, = Figure 9: Case 5 output multiplicity characteristics for the three-state CSTR model when V = 15; gallons. φ , gal. reactor Hysteresis locus "-disjoint" loci " -disjoint" loci β Figure 7: Global multiplicity behavior showing nominal reactor operation in region for Devia and Luyben's example ( = :793). x τ 15 numerical O(ε) perturb. Figure 1: Dynamic response of x for a small perturbation in 1 near a Hopf bifurcation point.