Ind. Eng. Chem. Res. 2009, 48, 7631–7636
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An Algorithm for Stabilizing Unstable Steady States for Jacketed Nonisothermal Continually Stirred Tank Reactors Farouq S. Mjalli* and N. S. Jayakumar Department of Chemical Engineering, UniVersity of Malaya, 50603, Kuala Lumpur, Malaysia
The operation of nonisothermal jacketed continually stirred tank reactors (CSTRs) involves a high degree of nonlinearity, and for certain operating conditions this nonlinearity may cause concentration multiplicity at steady state. At these conditions, certain steady states will be unstable and difficult to observe and maintain. The dynamics of the cooling jacket has a profound effect on these conditions and hence should be taken into consideration for the proper reactor design. A novel steady-state design algorithm is presented here to stabilize the CSTR at such conditions. This is achieved by replacing the single CSTR by a cascade of reactors that are capable of stabilizing and maintaining the unstable steady state. The new design approach implements an optimization of the reactors operating parameters to achieve this solution. The implementation of the solution algorithm is shown graphically and also shown is the use of numerical computational optimization solution techniques. For a first order reaction, both solutions were successful at achieving the unstable steady state; however, the numerical solution was more accurate. Simulations of the new design showed that the reactors’ cascades were capable of stabilizing the metastable temperature to a high degree of accuracy. 1. Introduction The high nonlinearity and complexity entailed in the operation of continually stirred tank reactors (CSTR’s) present a real challenge to process designers and engineers. Difficulties such as input-output steady-state multiplicities, quasi-periodicity behavior, parametric sensitivity, and sometimes chaotic behavior result in reactor stability problems and degradation of products. Multiplicity of steady states is one of these challenging phenomena. In this case the CSTR operation will have the probability of attaining more than one equilibrium condition were some of these conditions are inaccessible and even unstable even under close loop conditions. Chemical engineers have become very interested in studying the features of steady-state multiplicity of reacting systems of multiparameter space using bifurcation analysis.1-4 The objective of the bifurcation theory is to describe any sudden qualitative changes in behavior of a system as a control parameter is smoothly varied. The CSTR reactor operating at steady state might give way to periodic oscillations or multiple steady states as the feed temperature increases. The possibilities of changes are obtained in the form of bifurcation diagrams. The system exhibiting these nonlinear phenomena can be useful or harmful to the chemical process. This nonlinear phenomenon can be utilized in combustion applications to the mixing of air and fuel and thus leading to improved performance. Moreover, unsteady state operations have been the object of much attention. In reactor design and in the analysis of chemical reactors, chemical engineers frequently use reaction rate expressions which in their origin contain a number of assumptions and resulting approximations, the effects of which on a reactor design and, in particular, on the stability of a reactor are difficult to determine. One of the most thoroughly studied systems in chemical engineering is the classic problem of a first-order exothermic reaction in an adiabatic CSTR.5 The isothermal-back mixed reactor (CSTR) has a nonlinear feature of exhibiting concentration * To whom correspondence should be addressed. E-mail: farouqsm@ yahoo.com.
multiplicity under certain conditions of parameters such as feed concentration and residence time.6-11 Output concentration multiplicity and stability in an isothermal CSTR has been analyzed for many years.12,13 Reactions such as the heterogeneously catalyzed hydrogenation of ethylene to ethane,14 catalytic oxidation of CO in excess O2 with Pt-Al catalyst,15 and the substrateinhibited enzyme-catalyzed reactions16 are a few of the experimental works confirming the existence of concentration multiplicity. Bhattacharjee et al.17 have found steady-state multiplicity in product distribution and heat generation rate in their experiments on Fischer-Tropsch (FT) synthesis using supported ruthenium catalyst. Because of the existence of such unstable steady states, design engineers try to avoid such regions by identifying them and designing for reasonably safe operating condition. Russo and Bequette,18 studied the existence of such multiplicity for a jacketed CSTR with nth order kinetics. They used bifurcation theory to obtain design parameters that move hysteresis and limit points to extreme values of input variable resulting in an unstable temperature region. In their analysis, they linked the bifurcation results to the multiplicity behavior of CSTRs under certain operating conditions. However, in many situations, it is often desirable to operate CSTRs under open-loop unstable conditions because of the high reaction yield attained at these conditions, while the reactor temperature is still low enough to prevent side reactions or catalyst degradation or a rapid molecular weight and viscosity increase leading to a gel effect in polymerization reactors.19 The problem of stabilizing the unstable steady state has been studied experimentally using conventional feedback control by Chang et al.20 Ding et al.21 described a means of operating a reactor about an unstable state by cycling a parameter periodically between two levels. The theoretical aspects of the above study is given in a series of papers by Aris and Amundson.22 The stabilization of the unstable steady state is of interest in that these steady states exist and they are predicted by the kinetic model which predicts those steady states which are stable. Westeterp23 presented practical examples in which the desired state of operation is the inaccessible unstable steady state. Chen and Crynes24 presented a method for accessing the unstable steady state by use of a cascade of CSTRs under open
10.1021/ie900072g CCC: $40.75 2009 American Chemical Society Published on Web 07/24/2009
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loop conditions, employing a numerical method along with a graphical procedure using the reaction rate, R, expressed as kC/ (1 + KC)2. Mjalli and Jayakumar25 presented an algorithm for achieving stability under unstable conversions using CSTR cascades under open loop conditions. Jayakumar and Mjalli26 presented an approach for achieving unstable convergence for nonisothermal CSTRs implementing optimization of the reactors’ operating parameters by considering a two-state nonisothermal CSTR model, neglecting jacket dynamics implicitly. Jacketed process vessels are used when a close control of temperature is important, generally when a chemical reaction takes place. Unwanted temperature fluctuations can degrade product quality in productions of pharmaceuticals and special chemicals. Generally, jacketed vessels are used for control performance of reactors employing various jackets like plain jacket, half coil jacket, channel jacket, spiral jackets, and dimple and jackets. Nonlinear dynamics of an exothermic nonisothermal CSTR is greatly influenced by the dynamics of jackets, and the influence of design parameters on the multiplicity behavior of a three-state CSTR model incorporating jacket dynamics in the modeling equations was studied by Russo and Bequette.27 In the present work, the number of CSTRs in the cascade needed to access the unstable steady state is investigated for a first order chemical reaction in a jacketed nonisothermal CSTR reactor by including jacket dynamics in the modeling equations. The desired unstable middle steady-state temperature is chosen for attaining a solution by replacing the single CSTR with a cascade of reactors to achieve the unstable temperature. The effect of jacket dynamics on the stability of the reactors approach to the metastable steady state is investigated.
dC 1 ) (Cf - C) - k(T)C dt q
dT ) (FFCp(Tf - T) + (-∆H)Vk(T)C - UA(T - Tc))/VFCp dt (6) dTc ) (FcFcCpc(Tcf - Tc) + UA(T - Tc))/VcFcCpc dt
1 (C - C) - k(T)C ) 0 q f
(1)
dCj ) (FfCjf - FCj + Vrj)/V dt
(2)
(9) FcFcCpc(Tcf - Tc) + UA(T - Tc) ) 0
n
∑
dT ) ((Ffhf - Fh) + V (-∆Hj)rj + Q(T))/(VFCp + ws) dt j)1 (3) dTc ) ((Fcfhcf - Fchc) - Q(T))/(VcFcCpc + wsc) dt
Cf 1 + k(T)θ
And from eq 10 the value of Tc is Tc )
FcFcCpcTcf + UAT FcFcCpc - UA
Knowing the expressions of C and Tc above, the reactor energy balance eq 9 can be written in a more compact form as
[
]
∆Hθk(T)Cf 1 + kθ FcFcCpcTcf + UAT FFCp(T - Tf) + UA T FcFcCpc - UA
[
(
)]
)0
(11)
where the Egen denotes the heat generation term which involves the nonlinear portion of the equation and Erem is the heat removal term which is linear. The Erem can be expressed as a linear function of reactor temperature: Erem ) (n - mT) where
(4)
where Q(T) represents external heat addition or removal from the reactor and ws and wsc are the extraneous reactor and jacket wall capacitance respectively, which can be safely assumed negligible comparable to total material heat capacity in the reactor and the jacket. For a single constant volume reaction of first order rate law, and substituting for the heat removal term, eqs 1-4 can be simplified to
(10)
For certain values of feed temperature, multiplicity of three steady states exist with high conversion/temperature state or low conversion state/quenched temperature state dependent upon the values of feed temperature. The steady-state mass and energy balance eqs 8 and 9 can be calculated and plotted as function steady-state temperature for a fixed value of feed temperature. From eq 8 the value of C is
dE ) Egen - Erem )
for the jth species ) 1, 2, ..., n, where n is number or reacting species
(8)
FFCp(Tf - T) + (-∆H)Vk(T)C - UA(T - Tc) ) 0
C)
dM ) FfFf - FF dt
(7)
In the derivation of these equation from the mathematical model described above, variation of density and specific heat of the reaction mixture with respect to temperature and composition are neglected. Furthermore the reactor liquid volume is constant. So the steady-state version of eqs 5-7 becomes
2. Modeling and Analysis of Nonisothermal Jacketed CSTR Conventionally a mechanistic dynamic model for a single CSTR can be written by considering the material and energy equations for the reactor and its jacket. The general case model for a constant volume reactor in which multireactions occur is written as
(5)
n ) FFCpTf +
(12)
FcFcCpcUA T FcFcCpc - UA cf
and m ) FFCp + UA -
(UA)2 FcFcCpc - UA
The heat removal is characterized by the two parameters n and m in eq 12. This linear equation can be used to study the behavior of the CSTR as a function of the parameters n and m. For the twostate model under consideration, the reactor feed temperature and
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Table 1. Operating Conditions for the CSTR Multiple Steady-States Problem
Figure 1. Effect of changing feed concentration on the position of unstable steady state.
Figure 2. Effect of changing feed temperature on the position of unstable steady state.
concentration are the two variables that influence the existence and behavior of steady-state multiplicity. Varying the reactor feed concentration results in changing the Egen term and changing the location of the unstable steady state. In Figure 1, the two terms Egen and Erem are plotted as a function of the reactor temperature for different values in feed concentration. It is clear that increasing CAf results in moving the Egen line upward, and consequently the two steady-state conversions get closer to each others until they coincide at a concentration of 0.0077 kmol/cc. At this feed concentration the reactor converges to one stable steady-state solution. Figure 2, shows the effect of increasing the reactor feed temperature Tf on the energy removal term Erem. Starting at a feed temperature of 292.16 K, the reactor exhibits two distinctive steadystate conversions. As the feed temperature is increased, the energy removal line is lowered resulting in closer steady-state solutions. At a Tf of 296.5 K there is only one reactor steady-state conversion. From this we conclude that the design of a single CSTR is highly affected by the choice of its feed concentration and temperature. And hence, theoretically it will be possible to come up with a combination of reactors that can achieve the conversion stability by carefully selecting the reactors’ feed conditions. Of course in
condition
value
reaction activation energy; E/R [K] pre-exponential factor; ko [1/min] heat of reaction; ∆H [cal/mol] reactor volume; V [cc] overall heat transfer coefficient; UA [cal/(min K] feed concentration; Cf [mol/cc] feed temperature; Tf [K] feed flow rate; Ff [cc/min] coolant temperature; Tc [K] coolant flow rate; Fc [cc/min] specific heat; Cp [cal/(gm K)]
11243.9 2.4422 × 1014 14000 400 10 0.005 292.16 73 292.16 25 0.8
this case it will be more practical to vary the feed temperature as the feed composition is a reactor specification. To investigate more the effect of these two reactor feed variables on the location of the unstable steady-state conversion, a base case reactor design problem was considered with the nominal operating conditions as given in Table 1. A wide range of reactor feed temperatures (240-350 K) and feed concentrations (0.004-0.006 kmol/cc) were considered. For each combination of feed conditions, the upper and lower unstable conversion limits were calculated by solving the reactor model. The results were plotted as a function of the m parameter in eq 12 as shown in Figure 3. The lower plane outlines the lower inaccessibility limit, whereas the upper plane indicates its upper limit. The region of unstable conversions widens as the feed concentration increases. The increase of feed temperature has a similar effect but less pronounced than that caused by the increase of reactor feed composition. Within the space between the two planes, multiplicity of conversions exists and the metastable conversions there are not achievable and can not be maintained using conventional design techniques. For certain applications, the designer is confronted with the problem of designing a reactor to achieve a conversion that may lay within the space bounded by these two planes. In this work, we attempt to address this issue by replacing the single nonisothermal CSTR with a cascade of reactors. The two-state reactor model is used to study the multiplicity behavior and a new method is presented to tackle this problem. In the next section, the method is demonstrated both graphically as well as by using an optimiza-
Figure 3. Upper and lower planes for the unstable region.
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tion algorithm to solve a mathematical objective function for the best configuration of the CSTR cascade. 3. Reactor Cascade Design Algorithm The idea of replacing a single CSTR by a cascade of them, was considered for the isothermal case previously.25 This was followed by a similar study for the nonisothermal case.26 In this work an extension of the previous study to include the cooling jacket dynamics is considered. The proposed method of designing multiple reactors to replace a single CSTR is aimed at achieving the unstable conversion without sacrificing stability of the temperature profiles. For the isothermal case, basically, this is an iterative procedure for finding a series of CSTR each one accomplishes fraction of the desired conversion. This is done by manipulating the linear term of the material balance equation to satisfy a target of finding the point of tangency with the nonlinear term in the equation. The newly achieved concentration will then be used as a basis for the next reactor in the cascade. This is continued until no more reactors are needed when the unstable concentration is reached. The case of nonisothermal reactors follows a similar methodology. However in this case the heat removal term is manipulated and compared to the heat generation term in the energy balance equation. In this work, the effect of incorporating the thermal effect of the cooling jacket is explored. Using the analysis described in the previous section, the energy balance equation is modified to incorporate this effect, and the rest of the solution algorithm is basically an iterative search for the minimum number of reactors that can achieve the metastable steady-state conversion. The graphical solution algorithm described in a previous study26 was coded and simulated to reduce human error that may interfere in the implementation of the graphical solution, a high accuracy numerical solution must be provided. This is also important for the automation and generalization of the solution algorithm. 4. Optimization Solution Algorithm The numerical solution to this design problem is done by considering an optimization approach. This is achieved by calculating the point of tangency between the two energy terms in eq 11 in an iterative manner. Taking the right-hand side of the energy balance eq 11 as the energy accumulation term δE, the objective here is to minimize this term in order to get the heat removal line tangent to the heat generation curve. This is done by starting at Tf and searching for the best straight line defined by n + mT in eq 12 that is tangent to the heat generation curve. In this case only the slope (m) of this line needs to be evaluated for each new reactor in the cascade. The objective function in this case is derived from the steady-state energy balance eq 11 and expressed as min (δE(m, T))
(13)
m∈R+ T∈]t1,t2[
where
[
∆HVkFCAo - (n + mT) F + Vk FcFcCpcUA T n ) FFCpTf + FcFcCpc - UA cf
δE(m, T) )
and m ) FFCp + UA -
(UA)2 FcFcCpc - UA
]
Figure 4. Flowchart of the iterative design algorithm.
The optimizing routine uses the variable (m) as the search parameter and calculates the value of the objective function at each iteration. The solution procedure is summarized in the flowchart shown in Figure 4). In the current implementation of the algorithm, the LevenbergMarquardt optimization method was used for the line search. The method requires the minimization of the least-squares norm functional and the solution of the sensitivity problem. It was selected in this algorithm because of its high accuracy and its capability of handling nonlinear parameter identification. The fact that this algorithm considers the energy balance equation where all system nonlinearity is expressed by the energy generation term while the heat removal term is always linear makes this new design procedure applicable for any nonisothermal CSTR no matter what reaction rate equation is used. Hence this method can be generalized for all nonisothermal CSTR reactors exhibiting the problem of unstable conversions. The same case study problem used for the graphical solution was used to test the proposed optimization solution algorithm. Three different optimization techniques where investigated and their search performance was recorded as a function of iteration number. These optimization methods are the Matlab software implementation of the Dogleg, Gauss-Newton (GN), and the Levenberg-Marquardt (LM) method. The convergence tolerances for the solution parameter and the objective function were set to 1 × 10-5. The approach to the optimum reactor design for the first reactor is shown in Figure 4. All algorithms converged to the desired solution however with different approach speed. The GN method was relatively the slowest in convergence followed by the Dogleg and the fastest was the
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Table 2. Results of Numerical Solution (with Unjacketed Reactors Case) iteration mi (cal/min K) 1 2 final
43.173 81.915 209.694
Tf (K)
T ′1 (K)
T1′′ (K)
Ti (K)
Vi (cm3)
292.16 295.347 300.840 298.092 542.683 298.092 305.061 308.142 306.601 316.315 306.601 312.512 129.510
Table 3. Results of Numerical Solution (Considering the Jacketed Reactors) iteration mi (cal/min K) 1 2 final
43.173 80.492 161.115
T 1′ (K)
T1′′ (K)
Ti (K)
Vi (cm3)
292.16 294.994 300.84 297.92 297.92 303.700 307.91 305.90 305.90 314.47
581.45 368.40 187.54
Tf (K)
LM method. Using the LM algorithm, the computation iterations have achieved the solution within the first 7 iterations and the solution was then refined for the remaining 4 steps. The CPU time needed to achieve the solution for the three reactors in the cascade were 61, 47, and 19 s, respectively. The main reason behind the superiority of the LM method as compared to the other two, is that the LM-based code actually implements a trust-region algorithm, in which the step length, rather than the Marquardt parameter, and the multiple of a diagonal matrix added to the Hessian approximation, is explicitly controlled. Starting with Tf ) t1 ) 292.16 K, the line intersecting the heat generation curve at Tm was found to cross the heat generation curve at T 1′ ) 300.84. The optimization algorithm was then used to find the energy removal line that is tangent to the heat generation curve with a slope m1 ) 43.173 (cal/min K). The abscissa of this line at the point of tangency was T1′′ ) 304.56 K. T1 was calculated by taking the average of T 1′ and T1′′. The next reactor feed temperature Tf was set to T1 and the iteration was repeated for the second and third reactors. With the third reactor the final temperature T3 is the same as t2, which is the unstable temperature Tm (314.47 K). The results of the no-cooling jacket case as well as those for the case with reactor jacket consideration are summarized in Tables 2 and 3, respectively. Similar conclusion can be drawn for the importance of including the coolant dynamics. The achieved solution is tested by simulating the cascade CSTRs using the dynamic model of eq (5-7). Three sets of the CSTR dynamic model were coupled and the achieved volumes of cascade (V1, V2, and V3) where substituted in the corresponding equations for each reactor. The simulation concentration, temperature, and coolant temperature profiles for the three reactors and the two considered cases are plotted in Figure 5. The transients of temperatures and concentration for the case of coolant dynamics consideration show smooth dynamics with settling times of 80, 180, and 270 min for the three reactors, respectively. On the other hand, the same process operating conditions revealed a runaway condition for the third reactor when ignoring coolant dynamics in the design. This proves that ignoring the coolant dynamics results in inaccuracies in the final design and makes the solution less robust and more sensitive to process variables excitations especially when conditions approach the metastable steady-state solution. Table 4, shows a comparison between the achieved design results for both graphical-and optimization-based methods. The percentage relative absolute error (PRAE) calculation indicates that both methods achieved comparable accuracy, although the optimization-based method is more accurate. The temperature PRAE for all reactors were less than 0.17%, and negligible for the first reactor in the series. However, the calculated reactor volumes attained a PRAE of more than 1% for all reactors and
Figure 5. Reactant concentration, temperature, and coolant temperature transients of the three CSTR cascade for the nominal design (case 1) and for the unjacketed reactors dynamics case (case 2). Table 4. Comparison of the Graphical and the Optimization Based Design Results reactor 1
reactor 2
reactor 3
graphical
T V
297.9 571.25
305.65 355.24
314.40 179.51
optimization
T V
297.92 581.45
305.90 368.400
314.47 187.540
PRAE
T V
0.0067 1.78
0.082 3.7
0.0222 5.55
the third reactor volume has the highest drift of 5.55%. This is basically related to the human limitations in conducting visual measurements as well as the error propagation due to approximation. We can say that the optimization-based solution is more reliable and accurate and can be utilized for other cases with different reaction dynamics and operating conditions. 5. Conclusion The effect of cooling jacket temperature dynamics for nonisothermal reactors exhibiting unstable conversion temperatures is considered in this work. An irreversible first order reaction was used to investigate the existence of multiplicity in continuous stirred tank reactor. The middle metastable steadystate conversion was identified and a cascade of jacketed CSTRs was designed to achieve this conversion. The solution was accomplished both graphically as well as numerically. The numerical solution comprised the solution of an optimization problem in which the objective function is formulated from the energy generation and energy removal terms, respectively. The solution to the case study under consideration was achieved by designing a cascade of three reactors with steady-state temperatures of 297.92, 305.90, and 314.47 K, respectively. The final design of the three reactors was 581.45, 368.4, and 187.54 cc,
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respectively. The simulation of the new design proved that it was capable of achieving the target metastable condition of 314.47 within approximately 270 min of simulation time. Comparing the cases of design with and without jacket dynamics consideration showed the importance of incorporating the cooling temperature dynamics on the accuracy of the solution and hence on the stability of the reactors dynamics. Adopting the suggested design method for the special case of achieving metastable conversions will help the design engineer in achieving the design goal as well as stabilizing the system and have a better control over the dynamics of the process. This technique is applicable for any similar case and with different reaction kinetics. Nomenclature hf ) cal/mol enthalpy of the feed h ) cal/mol enthalpy of the output k ) min-1 reaction rate constant ko ) min-1 pre-exponential factor m ) cal/min K slope of energy removal term n ) cal/min intercept of energy removal term rj ) mol/min rate of reaction for the jth species ti ) K ith steady-state temperature solution C ) mol/cc reactant concentration Cf ) mol/cc feed concentration Cp ) cal/(gm K) specific heat E/R ) K reaction activation energy Ff ) cc/min feed flow rate F ) cc/min reactor outlet flow rate M ) mg accumulated mass R ) cal/(mol K) gas law constant Tc ) K coolant temperature Tm ) K unstable reactor temperature Tn ) K outlet temperature of the nth reactor Tf ) K feed temperature T ′ ) K temperature corresponding to intersection of energy generation curve with line connecting reactor feed temperature and the unstable reactor temperature Tm T ′′ ) K temperature corresponding to the minimum of δE UA ) cal/(min K) overall heat transfer coefficient V ) cc reactor volume F ) gm/cc density Ff ) gm/cc density of feed F ) gm/cc density of output δE ) cal/min energy accumulation term in eq 10 θ ) min average residence time ∆H ) cal/mol heat of reaction
Literature Cited (1) Kubicek M.; Marek, M. Computational Methods in Bifurcation Theory and DissipatiVe Structures; Springer:, New York, 1983. (2) Golubitsky M. Schaffer D. A Theory for Imperfect Bifurcations Via Singularity Theory; Springer: New York, 1985. (3) Balkotaiah, V.; Luss, D. Exact steady-state multiplicity criteria for two consecutive or parallel reaction in lumped parameter systems. Chem. Eng. Sci. 1982, 37, 433.
(4) Gilmore, R. Catastrophe Theory for Scientists and Engineers; Wiley: New York, 1981. (5) Poore, B. On the theory and application of the Hopf-Friedrichs bifurcation theory. Arch. Ration. Mech. 1976, 60, 371–393. (6) Bukur, D. B. Multiplicity of steady states in an isothermal batch fluidized bed reactor. Chem. Eng. Sci. 1978, 33, 1055–1060. (7) Dudukovic, M. P. Micromixing effects on multiple steady states in isothermal chemical reactors. Chem. Eng. Sci. 1977, 32, 985–994. (8) Levenspiel O. Chemical Reaction Engineering, 2nd ed.; John Wiley: New York, 1972; p 141. (9) Luss, D. Uniqueness criteria for lumped and distributed parameter chemically reacting systems. Chem. Eng. Sci. 1971, 26, 1713–1721. (10) Perlmutter D. D. Chemical Reactor Stability, Prentice-Hall: Englewood Cliffs, NJ, 1972. (11) Knorr, R. S.; O’Driscoll, K. F. Multiplicity steady states, viscosity, and high conversion in continuous free radical polymerization. J. Appl. Polym. Sci. 1970, 14, 2683. (12) Nashaie, S. E.; Yates, J. G. Multiplicity of the steady-state in fluidized bed reactors. Chem. Eng. Sci. 1973, 28, 515–520. (13) Do, D. D.; Weiland, R. H. Substrate-inhibited kinetics with catalyst deactivation in an isothermal CSTR. AIChE. J. 1980, 26, 1020–1028. (14) Sussman, M. V.; Potter, C. Kinetics of propylene and ethylene hydrogenation. Ind. Eng. Chem. 1954, 46, 457–465. (15) Hegedus, L.; Oh, L. S. H.; Baron, K. Multiplicity steady states in an isothermal, integral reactor: The catalyst oxidation of carbon monoxide over platinum-alumina. AIChE J. 1977, 23, 632–642. (16) O’Neill, S. P.; Lilly, M. D.; Rowe, P. N. Multiple steady states in continuous flow stirred tank enzyme reactors. Chem. Eng. Sci. 1971, 26, 173–175. (17) Bhattacharjee, S.; Tierney, J. W.; Shah, Y. T. Thermal behavior of a slurry reactor: Application to synthesis gas conversion. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 117–126. (18) Russo, L. P.; Bequette, B. W. Operability of chemical reactors: Multiplicity behaviour of a jacketed styrene polymerization reactor. Chem. Eng. Sci. 1998, 53, 27. (19) Russo, L. P.; Bequette, B. W. Impact of process design on the multiplicity behavior of a jacketed exothermic CSTR. AIChE J., 1995, 41, 135–147. (20) Chang, M.; Schmitz, R. A. Feedback control of unstable states in a laboratory reactor. Chem. Eng. Sci. 1975, 30, 837–846. (21) Ding, J. S.; Sharma, Y. S.; Luss, D. Steady-state multiplicity and control of the chlorination of liquid n-decane in an adiabatic continuously stirred tank reactor. Ind. Eng. Chem. Fundam. 1974, 13, 76–82. (22) Aris, R.; Amundson, N. R. An analysis of chemical reactor stability and control. Part I. Chem. Eng. Sci. 1958, 7, 121–132. (23) Westerterp, K. R. Maximum allowable temperatures in chemical reactors. Chem. Eng. Sci. 1962, 17, 423–433. (24) Chen, C. T.; Crynes, B. L. A method for achieving inaccessible conversions by use of cascade CSTR’s. Chem. Eng. Sci. 1987, 42, 1229– 1232. (25) Mjalli, F. S.; Jayakumar, N. S. Achieving stability under inaccessible conversions using CSTR cascades. Int. J. Chem. React. Eng. 2008, 6, A77. (26) Jayakumar, N. S.; Farouq, S.; Mjalli, F. S. An approach for achieving unstable convergence for non-isothermal CSTRs. Chem. Eng. Technol. 2009, 32, 564–571. (27) Russo L. P. Bequette, B. W. CSTR performance limitations due to cooling jacket dynamics. In Dynamics and Control of Chemical Reactors, Distillation Columns, Batch Processes; Balchen, J. G. Ed.; Pergamon Press: New York, 1993; pp 149-154.
ReceiVed for reView January 16, 2009 ReVised manuscript receiVed June 27, 2009 Accepted July 7, 2009 IE900072G
