Ind. Eng. Chem. Res. 1994,33, 3202-3208
3202
Parametric Sensitivity, Runaway, and Safety in Batch Reactors: Experiments and Models Pavan K. ShuMa and S. Pushpavanam' Department of Chemical Engineering, IIT Kanpur-208016,India The batch reactor is a dynamic system whose trajectory depends on various parameters. Parametric sensitivity signifies large changes in the reactor trajectory induced by small changes in parameters across threshold values. This is a form of critical behavior and can lead to runaway conditions, resulting in unsafe reactor operation. In this paper experimental results of three exothermic chemical systems are considered: (a)the homogeneous liquid phase reaction between sodium thiosulfate and hydrogen peroxide, (b) the heterogeneous (liquid-liquid) hydrolysis of acetic anhydride catalyzed by dilute sulfuric acid, and (c) t,he noncatalytic heterogeneous (liquidsolid-gas) hydrogenation of nitrobenzene using Sn and HC1. The systems are modeled using pseudohomogeneous rate expressions available in the literature. The only measured variables are the initial compositions and the reactor temperature as a function of time. Effect of several parameters, i.e., initial temperature, catalyst concentration, initial composition, etc., on the reactor trajectory are discussed. Existing criteria in the literature are used to determine the parametric sensitivity of these systems.
Introduction Batch reactors have etched a permanent place for themselves in the chemical industry. Some of the industries where batch reactors are predominantly used include drugs, pharmaceuticals, polymers, and speciality chemicals. They are preferred when better control in end product quality is desirable and when the volume processed is relatively small. Two standard variations of batch reactor are (a) semibatch reactors, which are used in polymerization reactors as they help drive the equilibrium to the right (Martin et al., 19911, and (b) fed-batch reactors, which are used extensively in the cont.extof fermentation industry (Teixeira et al., 1992). Batch systems are inherently dynamical systems. The different variables depend on time in contrast to continuous systems where the preferred mode of operation is steady, i.e., time independent. Numerous investigations have been carried out in determining optimal temperature profiles in batch systems (Westerholt et al. 1991). Lewin et al. (1990)have estimated the safe startup time in terms of design parameters for a batch reactor. These studies also have implications in determining start-up conditions. They discuss control strategies which would enable implementing a safe trajectory in a batch reactor. Open systems like CSTRs exchange mass and energy with the surroundings. The system state is determined here by a balance between transport processes and kinetics. Here the system can operate away from the state of equilibrium. These systems under suitable conditions exhibit different instabilities like multiple steady states, autonomous oscillations or limit cycles, chaotic behavior, etc. These instabilities cannot occur in batch systems. Batch reactor operation, however, can be unsafe. The trajectory of the system t o reach the terminal state depends on the operating conditions or system parameters. Typical control parameters include coolant temperature, initial reactor temperature, initial composition, etc. Under conditions of sufficiently low coolant temperature with all other variables remaining con-
* Author to whom correspondence
should be addressed.
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stant, an exothermic reaction proceeds slowly and smoothly (Eigenberger, 1989). The temperature remains almost steady and constant for a long length of time, before finally dropping down to the ambient or coolant temperature after the reaction has gone to completion. These exists a threshold value of the coolant temperature above which the reaction proceeds very fast. This is true for exothermic reactions, and the progress of the reaction now is accompanied by a sharp fast rise in temperature. At this high temperature the reaction goes to completion, and under nonadiabatic conditions the temperature decreases to the ambient value. This drastic change in system behavior across a threshold value of a parameter is called a runaway. Alternatively, the system is said t o be parametrically sensitive. This behavior can be characterized by analyzing the dynamic temperature profile of the batch system. Most investigations of parametric sensitivity have been theoretical or based on numerical simulations. Semenov's classical theory of thermal explosions is concerned with evaluating the critical temperature above which the temperature rise in the reactor is very fast (Frank-Kamenetskii,1959). He analyzed the zerothorder reaction. This approximation is applicable for reactions going to low conversion or reactions occurring slowly where the change in concentration can be neglected over the progress of the reaction or when the reactants are present in excess. The second derivative test of the temperature profile can also be used to determine if a system is parametrically sensitive. A positive second derivative of the temperature versus time profile implies that the system is parametrically sensitive. Thus the conditions for parametric sensitivity are
Barkelew (1959) divided the parameter space into parametrically sensitive and insensitive regions. The parameters he chose are a dimensionless rate of heat generation and a rate of heat removal. This condition can be used t o divide the parameter space into sensitive and insensitive regions for nth order reactions.
0 1994 American Chemical Society
Ind. Eng. Chem. Res., Vol. 33,No. 12, 1994 3203 Table 1. Kinetic Expressions from Literature, Best Fit Values Results system rate expression (moVcm3 s) conditions (mol/L) I (6.85 1 0 1 1 ) ~ - 9 1 5 0 / ~ ci, = 0.1 ci, = 0.2 C,,, = 0.4 I1 with catalyst (1.85 x 1017)e-'12MTC~Cs C, = 0.225 C, = 0.375 no catalyst (6.6 x 105)e-578WTC~ I11 (2.35 1 0 4 ) ~ - 5 ~ ~ , 20-80 mL of HCl ~
Morbidelli et al. (1988, 1989) have determined a generalized criterion for parametric sensitivity for batch and tubular reactors. They determined the variation in the maximum temperature of the nonadiabatic reactor with respect to different parameters. The maximum temperature in the batch reactor is equivalent to the hot spot temperature of a tubular reactor. The method quantifies the trajectory by evaluating the derivatives of the hot spot temperature with respect to various parameters. The parameter value a t which the derivative is a maximum is the threshold separating sensitive and insensitive regions. Eigenberger et al. (1989)have discussed the concepts of stability and safety in the context of batch and continuous systems. They explain that though the operation of a continuous system a t a state may be stable, it could be unsafe. This can be best understood in the context of multiple steady states where a system may go from a desirable stable steady state to an undesirable stable steady state due to a large amplitude perturbation. In batch systems they emphasize that it is important to ensure that the reaction occurs smoothly a t a reasonable rate. A reaction occurring slowly is inherently unsafe, as a local hot spot can trigger a runaway condition. They discuss several strategies for carrying out reactions in semibatch mode t o overcome this problem. Villermaux et al. (1991) discuss runaway criteria in terms of time constants for reactions and cooling. Rao et al. (1988) determined the kinetics of the acetic anhydride hydrolysis by measuring temperature profiles of the reactor. To date most of the work on parametric sensitivity performed has been of a theoretical nature. Consequently the various criteria existing in the literature has been seldom verified. In this work we have chosen the following three reactions: (i) The reaction between sodium thiosulfate and hydrogen peroxide (system I). 4H,O,
+ 2Na,S,03 - Na,S,O, + Na,SO, + 4H,O
(ii) The sulfuric acid-catalyzed hydrolysis of acetic anhydride (system 11). (CH,CO),O
+ H,O -2CH3COOH H2S04
(iii)The reduction of nitrobenzene using Sn and HC1 (system 111).
Sn
+ 2HC1-
SnCl,
+ H,
Here we present the results of experiments in a batch reactor. These systems are all exothermic and of varying degrees of complexity. System I is homoge-
~
parameters B , y 0.044,30 0.088,30 0.176,30
parameters fit (B,y ) 0.052,27.2 0.096,28.1 0.144,29.0
0.1,37.4 0.1, 37.4
0.13, 37.3 0.12, 37.4
0.1, 19.5 0.65, 18.15
0.15,20.9 0.17-0.24, 18-17
neous, whereas I1 and I11 are heterogeneous being L-L and G-L-S, respectively. The kinetics of these reactions are well established in the literature. This is used to simulate the behavior of these reactors. The simulations are compared with the experimental results. The system is then classified as being sensitive or otherwise according to the different criteria.
Modeling A pseudohomogeneous model is used to simulate reactor behavior. The kinetics of all these systems have been discussed in the literature (Smith, 1981; Kearns et al., 1969;Mukherjee, 1990). The reaction is assumed to occur in one phase, and its rate expression is determined by considering it to be a single step elementary reaction. The mass and energy balance equations are then given by
v-dC = Vr(C,T) dt
(la)
dT VQCp dt = V ( - W r ( C , T ) - UA(T - T,) ( l b ) For an nth order reaction, the rate is given by
r(C,T)= k,e -EIRT cn These equations can be transformed in dimensionless form to yield
& =i S k , C ~ ~ ' (-l x)ne-y'y- BO, - y,) dt
(2b)
Every term in this equation has unit of s-l. All the variables are defined in the nomenclature. Equations 2a and 2b are integrated subject to the initial conditions t = 0, x = 0, y = yin using a fourthorder Runge-Kutta method. The parameters KO,B, and y are characteristic of the chemical system. The values of these parameters reported in the literature are used in the simulations. These are shown in Table 1. The heat transfer coefficient parameter /3 has to be determined experimentally for our system. The activation energy and the heat of reaction are adjusted to obtain the best fit between theoretical prediction and experimental results.
Experimental Procedure The batch reactor used was a cylindrical glass vessel of inner diameter 7 cm and height 30 cm. The reactor was clamped and its contents were well-mixed by a magnetic stirrer. The temperature was recorded using a sheathed thermocouple. A n Indotherm temperature
3204 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
-"
45
.
c
I
I-
40.
%
35c
30t
25l 0
\
I
I
I
I
I
1
2
3
4
5
T i m e , min
Figure 1. Dimensionless heat transfer coefficient measurement.
controller and indicator was used for temperature measurement. It used a K-type thermocouple. The heat losses from the glass reactor are such that runaway behavior and safe operation can be seen for different operating conditions. The heat transfer coefficient /3 in eq 2b is a characteristic of our experimental setup. It has to be determined experimentally. All other parameters are intrinsic to the system and can be obtained from the literature (Table 1). The heat loss is measured in terms of an overall heattransfer coefficient. An accurate measurement is necessary here as the work involves simulation of the reactor temperature profile and comparing with experimental results. To determine the heat-transfer coefficient, 200 mL of water was heated to a temperature of about 80 "C and poured into the vessel. The water was stirred, and its temperature was monitored as a function of time. The experiment was repeated for reproducibility. A plot of T - T, as a function of time is depicted in Figure 1. The best exponential fit shown as a solid line yields the value of the parameter /3 as 0.001 s-l. We assume that the reactant mixture properties are the same as that of water for all systems. The experimental procedure for all three systems is similar. We will now discuss them individually. System I. The reaction is second-order in thiosulfate and zeroth-order in the concentration of peroxide. To ensure that the peroxide concentration does not vary significantly, we use peroxide in 300% excess. A 100 mL sample of a known concentration of thiosulfate was charged to the reactor. The initial reactor temperature was noted. To this was added 100 mL of peroxide (30% w/v). The stirring was commenced, and the temperature profile was determined by the thermocouple hooked to the Indotherm. The experiment was carried out for three different initial concentrations of thiosulfate 0.10 mol/cm3 0.21 mol/cm3, 0.42 mol/cm3. The ambient temperature here was between 21 "C and 23 "C. System 11. This is a heterogeneous (L-L) system. A 50 mL sample of the anhydride was taken in the
vessel and was well-stirred. To this was added 150 mL of dilute sulfuric acid (catalyst). The initial temperature was noted using the digital indicator, and the progress of the reaction was determined by maintaining the reactor temperature. The experiments were carried out for two different initial temperatures, 21 "C, 26 "C, and for three concentrations of sulfuric acid, 0%, 3%, 5%. The ambient temperature here varied from 20 to 24 "C. System 111. The three reactants here are Sn, HC1, and nitrobenzene. The hydrogen liberated by the first two reduces the last to aniline. In this experiment the total volume of the reacting mixture was maintained at 100 mL. The volume of nitrobenzene was always 18 mL. The remaining volume was made of HC1(33%)and water. Different volumes of HC1(33%),i.e., 80 mL, 60 mL, 40 mL, 20 mL, were chosen. The initial temperature was maintained at 40 "C, and the liquid reactants were well-stirred. The reaction was t.riggeredby adding a fxed amount of Sn. The amount of Sn added was also varied. Samples of 10 g, 20 g, and 30 g were added for different runs. The progress of the reaction was monitored by measuring the reactor temperature as a function of time. The ambient temperature varied from 32 to 35 "C. All the chemicals used were AR grade chemicals. Each run was tested for reproducibility by repeating the run a t least twice. Some of the runs (75%) were randomly selected and carried out a third time. The criterion used for reproducibility is a &2 "C variation in maximum temperature and a f0.5 min variation in time taken to achieve this maximum temperature for a particular run.
Results and Discussion Parameters y , B , and KO for each system as obtained from the literature are shown in the first and third columns of Table 1. The first two parameters were adjusted to obtain best fits of the theoretical predictions to the experimental results. The parameter values giving the best fits for each system are depicted in column four in Table 1. These were obtained using a minimization of least squares technique based on Powell's method. The best fit values of y for each system is within 5% of the value reported in the literature. The best fit value of parameter B exhibits a wider spread. This is possibly due to the large variation in the conditions in which each reacting system was studied. The experimental results for the thiosulfate experiments are shown in Figure 2. The concentration of thiosulfate in the feed solution used in the different runs was 0.21 mol/L, 0.42 mol& 0.84 mol&. The initial concentration in each run is half this amount as we mix equal volumes of the two reactants. Comparison of the third and fourth columns in Table 1 (literature values and best fit values) shows that the kinetic expression in the literature can predict the reactor performance accurately for this system. The solid curve in Figure 2 shows the results of the theoretical predictions of our model. We can see that the theoretical prediction agrees very well with the experimental data obtained for all three initial concentrations. This reaction was conducted with 300% excess peroxide concentration t o ensure that the rate is independent of the peroxide concentration. The kinetics of acetic anhydride hydrolysis has been studied under various conditions by earlier workers. Rao et al. used acetic acid as the solvent for this reaction,
Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 3206
a 60 66 50
40
d
E
I-
30
20
'
-0
60
120
180 Ti m e , sec
240
300
Figure 2. Temperature profiles for thiosulfate-peroxide system effect of initial concentration of thiosulfate, experiment (points) and theory (solid curves).
and Halder et al. studied the kinetics using sulfuric acid as a catalyst. In the present work no solvent was used. Sulfuric acid was used as the catalyst in some of the runs. The kinetics of uncatalyzed reaction is first-order (Smith, 1981) in anhydride concentration. This rate expression is reported on the basis of experiments carried out for an anhydride concentration of about mol/cm3, i.e., a large excess of water. The kinetics is hence independent of water concentration. The rate expression for the acid-catalyzed reaction was reported by Halder for an acid concentration of about mol/ cm3 and an anhydride concentration of lop3 mol/cm3. In the present work we have used an anhydride concentration of around mol/cm3 and an acid concentration of around 2.5 x mol/cm3. Figure 3 depicts the results of experiments carried out at an initial temperature of 21 "C for two different concentrations of sulhric acid (O%, 3%). The results of the simulations using the best fit parameters are shown in the figure as solid curves for each concentration of sulfuric acid. We see a fairly good agreement between the experimental and simulated results. The reactor behavior when 5% acid is used to catalyze the reaction is depicted in Figure 3b. This is compared with the temperature profile of the uncatalyzed reaction. The maximum reactor temperature of the uncatalyzed reaction of 44 "C is attained in 26 min. For 3% sulfuric acidcatalyzed reaction this value is 56 "C in 5 min. For 5% acid-catalyzed reaction the corresponding values are 56 "C in 3 min. In Figure 4a we show the effect of increasing the initial temperature from 21 "C t o 26 "C for the uncatalyzed reaction. The corresponding maximum temperature changes from 44 "C to 46 "C. However, the time taken to achieve this temperature is reduced from 26 to 17 min respectively. This indicates that the reaction has been accelerated significantly by this temperature change. Figure 4b depicts the behavior of the system for an initial temperature of 26 "C for 0% and 3% acid concentrations. This system shows remarkable sensitivity in the time taken t o attain the maximum temperature and not on the maximum temperature itself. This is to be expected as sensitivity is a measure of how fast the reaction occurs. When the reaction is accelerated, the time taken to acheive maximum temperature decreases drastically. The maximum temperature is limited by the adiabatic temperature rise of the reactor. The best fit parameters for the catalyzed and uncatalyzed are shown in Table
10
6
0
b
12 18 T i m e , min
24
30
t 101 0
'
"
6
"
'
12 16 Time .min
I
'
24
' 30
Figure 3. (a)Temperature profiles in acetic anhydride hydrolysis effect of 3% acid, experiment and theory, initial temperature 21 "C. (b)Temperature profiles in acetic anhydride hydrolysis effect of 5% acid in feed solution, experiment and theory, initial temperature 21 "C.
1. The heat generation parameter B contains the initial reactant concentration. The value in column three has been calculated assuming equal distribution of the reactant in both phases. The difference in the B value in columns three and four can be attributed to an unequal reactant distribution between the two phases in a heterogeneous system. The initial concentrations we have mentioned are obtained assuming that the mixture is homogeneous where as it is determined by the equilibrium relationship. The rate of reduction of nitrobenzene is first-order in nitrobenzene. Our experiments involved varying amounts of Sn and HC1, keeping the concentration of nitrobenzene constant. The best fit parameters which simulate our experiments are shown in column four Table 1. The spread in parameters B and y is due to variations in the conditions of the reaction. A comparison of the experimental results with theoretical predictions is shown in Figures 5 and 6. In Figure 5a,b we depict the effect of varying the amount of HC1, with 10 g of Sn in aniline manufacture. The various amounts used in Figure 5a are 20 and 60 mL of HCl. The maximum temperatures are 44 "C and 89 "C. The time taken to acheive these temperatures are 15 and 9 min, respectively. Similarly for 40 and 80 mL of HC1, the respective values are 60 "C (in 15 min) and 100 "C (in 3 rnin). The sensitivity in this system reflects itself in a drastic increase in maximum temperature as well as
3206 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994
a
a 60
-
89
.y
73 -
i -
r-"
-
57
o o o o
10
6
0
18 Time, min
I2
0
30
24
b
6
12
6
12
-zm--
0 0 0 0 0 0 0 0 0 0 0
18 T i m e , min
24
30
18
24
3
b 105 h
2ot t
101
0
'
6
'
I
'
12 18 Ti m e , mi n
I
'
I
26
Figure 4. (a) Effect of initial temperature on reaction progress for uncatalyzed hydrolysis of acetic anhydride. (b) Temperature profile in acetic anhydride hydrolysis, with 3% acid in feed, experiment and theory, initial temperature 26 "C.
a reduction in the time taken to attain this temperature. Figure 6 shows the effect of changing the amounts of Sn used to carry out the reaction. Two analogous criteria have been proposed in the literature to demarcate sensitive regions from insensitive regions in parameter space. The first is the Barkelew criteria. It involves determining two dimensionless parameters: N , the rate of heat removal, and S, the rate of heat generation. They are defined for an nth order reaction as
25 0
T i m e , min
Figure 6. Effect of varying amount of HCI in aniline manufacture (a) 20 and 60 mL of HC1, (b) 40 and 80 mL of HC1.
89
1 t
0
The Barkelew diagram is a plot in NIS - S space (Barkelew, 1959). A critical curve through this divides the system as being sensitive or insensitive. The values of the parameters N and S for our system and whether they are sensitive or otherwise according to the Barkelew criterion is shown in Table 2. The parameters N and S have been calculated from the best fit values. The second criterion (Villermaux, 1991) uses the concept of relative time constants. Two characteristic time constants are the reaction time ( t r ) and the heat removal time &). These are defined as
6
12
18 Time ,min
24
30
Figure 6. Effect of varying amount of Sn in aniline manufacture.
in terms of N , S, and t r = NtdS. If tr > t,, then the time scale for the reaction to occur, i.e., the reaction induction time, is much larger than the time scale for the reactor to lose heat. Here the reaction is slow and heat loss is very fast. Consequently
Ind. Eng. Chem. Res., Vol. 33,No. 12, 1994 3207 Table 2. Sensitivity from Barkelew Criterion
C (mol/L)
system
I I1
I11
S
ci, = 0.1 ci, = 0.2
N
1.46 2.81 4.35
Ci,= 0.4 with catalyst C, = 0.225 (21-26 "C) C, = 0.375 (21-26 "C) no catalyst (21-26 "C) 20-80 mL of HC1
behavior
0.015 0.021 0.026
4.8 4.45 3.75 (3-4.4)
sensitive sensitive sensitive
(0.7-0.5) (0.5-0.25) (2.91-2.03) (2.43-0.6)
sensitive sensitive insensitive insensitive or marginally sensitive
Table 3. System Sensitivity According to Existing Criteria WUermaux) system
I I1
I11
C (mom)
ci, = 0.1 ci, = 0.2
Ci, = 0.4 with catalyst C. = 0.225 (21-26 "C) C. = 0.375 (21-26 " C ) no catalyst (21-26 %) (20-80
mL of HC1)
t, ( 8 )
teoolinp (8)
103 103 103
ATad ("C)
10 7.5 6
AT,,
13 26 53
("C) 13 23 49
sensitivity sensitive sensitive sensitive
103
(142-100)
35
34
sensitive
103
(114-52)
35
35
sensitive
103
(763-550)
29
22
sensitive
103
(835-150)
186
60
sensitive
the temperature rise is not very high. The sensitivity criteria used here is t r < tcfor sensitive and vice versa. In terms of parameters N and S, this yields N -= S for sensitive and vice-versa. The classification of our systems according to this criterion is shown in Table 3. Here we also indicate ATa&Le., the theoretical adiabatic temperature, rise and the experimentally observed maximum temperature rise. It must be noted that these criteria tacitly assume that the initial temperature equals ambient temperature. In our experiments this condition does not hold. They differ by about h5 "C. To overcome this we have also used Verma's intrinsic generalized sensitivity criterion t o classify our system. The generalized sensitivity with respect to Semenov number, q, was determined as a function of q. Verma's heat of reaction parameters is identical to Barkelew's S, and the Semenov number equals SIN. In spite of the relatively low values of S, this criterion predicts all our systems to be sensitive. This is probably due to the high value of the activation energy of our systems. The sensitivity criterion from evaluating the time constants and Verma's criterion are more conservative than Barkelews criterion. The first two criterion predict all our systems as being sensitive, whereas Barkelews criterion predicts some systems as insensitive or marginally sensitive. The best fit value of B for each system shows a spread (Table 1). This is due to a variation in the reaction conditions, i.e., initial temperaturelinitial concentration, presence of catalyst, etc. There is a significant difference in the best fit value of B from the literature for aniline. This arises possibly because our assumption that the reaction is first-order in nitrobenzene is inaccurate. This expression may hold for some values of HC1 added. The rate expression we have used does not capture the effect of the value of the acid added. It, however, can yield useful information about the macroscopic behavior of the system, i.e., maximum temperature rise, etc. We have also not considered the effect of evaporation (phase change) when the temperatures reach close to 100 "C. The value of B for anhydride experiment is influenced by acid catalyst. The variation in the value from the literature is because we
have assumed that the reactant is distributed equally in both the phases. Being a heterogeneous system, this may be incorrect and has to be found using equilibrium relationship. For all these systems the value of the activation energy parameter, y , is very close to the literature value. The various criteria are effected sharply by small changes in activation energy and ambient temperature. A 5 "C change in T, changes N and tr by about 50%due to the exponential dependence on T,. Similarly, a 10% change in activation energy can result in a drastic change in N . The design engineer must ensure that the kinetics expressions in the literature are valid under the conditions in which the reactor will operate. Failure to do so may lead to mispredictions by the different criteria. The practicing engineer must exercise caution in determining N , t,, etc. The subsequent use of the sensitivity criteria otherwise could result in mispredictions. From the point of view of the practicing engineer, the demarcation between a sensitive and an insensitive system is not sharp. There exists a region in parameter space which the engineer must avoid to prevent small changes in T, rendering our system sensitive or otherwise.
Conclusions In this paper we have presented results on experiments of three exothermic reaction systems carried out in batch mode. The kinetic expressions from the literature simulate the reactor behavior effectively. The best fit values of the parameters for the heterogeneous systems were obtained using Powell's method. The parameter B contains the reactant concentration term. Variations in B value in columns 3 and 4 of Table 1can be attributed to an unequal distribution of reactant between the various phases in a heterogeneous system as explained earlier. The problem of safety plays an important role in scale up. Proper scale up can be done only when experiments and modeling are carried on side by side. The results of experiments in the lab scale can be used to validate models, obtain better estimates of parameters, etc. Modeling and simulation can then be used to predict behavior under similar conditions and help in subsequent scale up t o industrial size reactors.
3208 Ind. Eng. Chem. Res., Vol. 33, No. 12, 1994 Sensitivity is normally measured in terms of the dependence of the maximum temperature on parameters (Morbidelli, 1988, 1989). In many situations the maximum temperature rise is almost equal to the adiabatic temperature rise. However, the rise may occur slowly. Changes in operating conditions for such reactors do not yield significant changes in maximum temperature. However, the reaction rate can increase drastically, and now sensitivity manifests itself in a sharp reduction in the time taken to achieve the maximum temperature as in anhydride hydrolysis. Thus the sensitivity can be defined as the point where Idt,,/dPI is a maximum. Here t,, denotes the time taken to achieve maximum temperature, and P is the parameter with respect to which the sensitivity is studied. Another instance when the t,, can be more sensitive than T,, is for reactions which have an activation energy dependent on temperature. If the ambient temperature or the initial temperature is chosen as the operating parameter, then the change in activation energy can drastically alter the time to reach Tmm. T,, itself would not be significantly changed as the heat of reaction would be unaltered, and this would fix the AT,d. The existing criteria in the literature predict our systems as being sensitive or otherwise. These criteria appear to be conservative as they predict sensitivity even for relatively insensitive systems. Thus there is a need for more rigorous, i.e., strict, criteria which can demarcate the boundaries better.
Nomenclature A = area of heat transfer, cm2 B = (-LvnCiJeCpTref, dimensionless heat of reaction C = reactant concentration, mol/cm3 Ci, = initial reactant concentration, mol/cm3 C, = specific heat, cal mol-' K-l E = activation energy, cal/mol k , = rate constant ( m ~ l / c m ~ )s-l ~-~, T = reactor temperature, K T , = ambient temperature, K Tref = reference temperature U = overall heat transfer coefficient V = reactor volume x(1 - c/Cin) = dimensionless conversion
y(TlTref)= dimensionless temperature y,(T&T,,f) = dimensionless ambient temperature G(E/RT,f) = dimensionless activation energy /3( UAN,C,) = heat transfer coefficient6 parameter, e = density of mixture
Literature Cited Barkelew, C. H. Stability of Chemical Reactors. Chem. Eng. Progr. Symp. Ser. 1959,55,37. Eigenberger, G.; Schuler, H. Reactor Stability and Safe Reaction Engineering. Zntl. Chem. Eng. 1989,29,12. Frank-Kamenetskii. Mass and Heat Transfer in Chemical Kinetics; Springer Verlag: Berlin, 1959. Halder, R.; Rao, D. P. Experimental studies on limit cycle behaviour of the sulfuric acid catalysed hydrolysis of acetic anhydride in a CSTR. Chem. Eng. Sci. 1991,46,1197. Kearns, D. L.; Manning, F. S. Model simulation of adiabatic continuous flow stirred tank reactors. AIChE J. 1969,15,660. Lewin, D.R.; Lavie, R. Designing and Implementing trajectories in an exothermic batch reactor. Znd. Eng. Chem. Res. 1990, 29,89. Martin, H. C. S.; Choi, K. Y. Two phase model for continuous finalstage melt polycondensation of polyethylene terephthalate, 2, analysis of dynamic behavior. Znd. Eng. Chem. Res. 1991,30, 1712. Morbidelli, M.; Varma, A. A generalised criterion for parametric sensitivity, application to thermal explosition theory. Chem. Eng. Sci. l988,43,91. Morbidelli, M.; Varma, A. A Generalized criterion for parametric sensitivity: application to a pseudo homogeneous tubular reactor with consecutive or parallel reactions. Chem. Eng. Sci. 1989,44,1675. Mukherjee, S. Parametric sensitivity and runaway conditions in batch reactors. M. Tech. Thesis, IIT, Kanpur, 1990. Rao, D. P.; Parey, S. K. B. Modelling and Simulation of an exothermic reaction in a batch reactor. Indian Chem. Eng. 1988,30,33. Smith, J. M. Chem. Eng. Kinet. 1981,263. Teixeira, J. A.; Sousa, M. L.; Azevedo, S. F. Motam, Monitoring and Control of Fed batch fermentation. Chem. Eng. Educ. 1992, 94. Villermaux, J.;Georgakis, C. Current problems concerning batch reactors. Znt. Chem. Eng. 1991,31,434. Westerholt, V. E.; Beard, J. N.; Melsheimer, S. S. Time optimal startup control algorithm for Batch Processes. Znd. Eng. Chem. Res. 1991,30, 1205.
Received for review December 29, 1993 Accepted August 1, 1994@
* Abstract published in Advance ACS Abstracts, October 1, 1994.
