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Modeling the effect of wall capacitance on the dynamics of an exothermic reaction system in a batch reactor
International Journal of Heat and Mass Transfer 54 (2011) 439–446
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Modeling the effect of wall capacitance on the dynamics of an exothermic reaction system in a batch reactor N.S. Jayakumar a,⇑, A. Agrawal b, M.A. Hashim a, J.N. Sahu a,c a
Department of Chemical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur 50603, Malaysia Department of Chemical Engineering, Dharmsinh Desai Institute of Technology, Nadiad 387001, Gujarat, India c Department of Chemical Engineering, Indian Institute of Technology (IIT), Kharagpur, P.O. Kharagpur Technology, West Bengal 721302, India b
a r t i c l e
i n f o
Article history: Received 28 April 2010 Received in revised form 18 August 2010 Accepted 18 August 2010 Available online 8 October 2010 Keywords: Runaway reactions Parametric sensitivity Wall capacitance Input parameters
a b s t r a c t The chemical reactors in certain range of operating conditions may exhibit parametric sensitivity where small changes in one or more of the input parameters lead to large changes in the output variable. This is a form of critical behavior that leads to runaway conditions, resulting in hazardous reactor operation. In the present work, hydrolysis of acetic anhydride reaction was used to investigate the existence and effect of parametric sensitivity with respect to the input parameters, namely cooling water flow rate, cooling water feed temperature and wall capacitance. Parametric sensitivity was observed for a small change in coolant water flow rate and feed temperature. It is found experimentally that with the introduction of extraneous wall capacitance, the batch reactor showed non-sensitivity under parametric sensitivity conditions. A mathematical model for the reactor was developed by incorporating both mass and energy balance with ordinary coupled differential equations. This dynamic model was solved numerically using ODE15s (gear method) of Matlab software, and the numerically simulated results are in satisfactory agreement with the experimental data. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Batch reactors have etched a permanent and important place for themselves in the chemical industry. Some of the industries where batch reactors are predominately used include drugs, pharmaceuticals, polymers and speciality chemicals [1]. Batch reactors are popular because of their flexibility in operation, and hence widely used in the manufacturing of products of low volume but of high added value. They are operated in closed batch system. The reactor may produce side reactions if it is not properly controlled. In an out of control system, the reactor may exhibit runaway conditions, thus causing high temperature and pressure beyond its endurance. It is known that batch reactors, in certain regions of operating conditions may exhibit a parametric sensitive behavior where small changes in one or more of the reactor input parameters lead to much larger changes in output variables [2–8]. 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 the optimal temperature profiles in batch systems [9]. Lewin and Lavie [10] have estimated the safe start-up time in terms of design ⇑ Corresponding author. Fax: +60 3 79675319. E-mail address: [email protected] (N.S. Jayakumar). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.09.025
parameters for 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. An excessive temperature excursion may adversely affect conversion of exothermic equilibrium reaction and the selectivity of the reaction process and, in the case of catalytic reaction, catalyst activity and durability and even reactor safety [11]. Knowledge of the circumstances leading to thermal runway and knowledge of the process chemistry and of the stored reagents are essential factors for the safe operation of chemical reactors and the storage of explosive chemicals. A parametric sensitivity and runaway are in fact related terms since one is the consequence of the other. A parametrically sensitive chemical reaction involves high risk of unleashing a runaway reaction by accidental causes [8,12,13]. Semonov [7] proposed a criterion to evaluate the thermal instability of a reagent by assuming the reaction is zero order and that the fluid is well mixed. The priori criterion for runaway condition was proposed by Barkelew [2] based on the examination of a number of reaction kinetics in which heat and mass balance equations were taken into account. The technique of early detection by hazardous states in chemical reactors has been developed by Gilles and Schuler [13] who predicted a safe operation detection system by applying one order and second order differentiation of reaction temperature with time. Hugo et al. [14] concluded that as far as safe operation problem in reaction is concerned, reactors should
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Nomenclature a CA CA0 Cp Cpc Cs E F1 F2 F3 J k qc rA Sj/i t T TA Tcin Ua Ua1 V Vc
heat transfer area (m2) concentration of acetic anhydride (mol/m3) initial concentration of acetic anhydride (mol/m3) specific heat of reaction mixture (J/(kg °C)) specific heat of coolant water (J/(kg °C)) initial concentration of sulphuric acid (mol/m3) activation energy (J/mol) function defined in Eq. (10) function defined in Eq. (11) function defined in Eq. (12) Jacobian matrix of the system rate constant (s1) coolant water flow rate (m3/s) reaction rate (s1) first order sensitivity coefficient matrix time (min) temperature of the batch reactor (°C) air temperature (°C) coolant water feed temperature (°C) over all heat transfer coefficient on the cooling side (W/°C) over all heat transfer coefficient due to natural convection heat loss from the reactor (W/°C) volume of the batch reactor (m3) volume of coolant (m3)
not reach high temperatures. Chemburkar et al. [15] proposed the generalized parametric criterion to determine the safe operation in a continuous stirred tank reactor while Morbidelli and Varma [6] applied the same method to determine the safe operating region for batch reactor operation. Unlike the theoretical development, experimental studies addressing parametric sensitivity of exothermic reaction in bath reactor are scarce, hence the present study on the phenomena of parametric sensitivity using exothermic reaction system is under taken. In this work, the dynamics of a batch reactor using the acid catalyzed hydrolysis of acetic anhydride was carried out. 2. Reaction pathway The overall hydrolysis of acetic anhydride reaction can be represented as: H2 SO4 Catalyst ðCH3 COÞ2 O þH2 O ! acetic acid solvent Acetic anhydride
2CH3 COOH
ð1Þ
Acetic acid
The mechanisms of the hydrolysis of acetic anhydride are well documented. It is well known that the reaction proceeds via a substitution, in which an attacking nucleophile replaces a substituent group on the central carbon. The reaction is of first order with respect to acetic anhydride. The kinetic equation for as given by Haldar and Rao [16] is:
r A ¼ ð1:85 1010 C s expð11243:9=TÞÞC A ; ðDHÞ ¼ 58520 J=mol
ð2Þ
The kinetic parameters of the above reaction were used to simulate the batch reactor experimental runs and the simulations were compared with the experimental results. 3. Modeling The quantum of investigations required for confirming parametric sensitivity in a reaction system can be minimized and easily
x y
independent variable (time) vector of a dependent variable (temperature or concentration) wall capacitance (J/°C)
ws
Greek letters q density of liquid mixture (kg/m3) (DT)max maximum reactor temperature (°C) (DT)ad adiabatic reactor temperature (°C) (DH) heat of reaction (J/mol) u represents the input parameter of the system Subscripts A acetic anhydride B acetic acid–water–sulphuric acid mixture cin coolant inlet temperature exp experimental P specific heat s stainless steel wall of the batch reactor sim simulated f feed c coolant temperature in the jacket i denoting coolant temperature in jacket or stainless steel wall
realizable in the laboratory by theoretically identifying its existence. Through selecting a set of process variables combinations from input parameters, the trial runs required for experimentally confirming the existence of parametric sensitivity of a reaction system can be reduced considerably. The unsteady state mass balance for first order catalyzed hydrolysis of acetic anhydride can be written as:
dNA ¼ kC A dt dT ðV qC p þ ws Þ ¼ ðDHÞVkC A UaðT T c Þ Ua1ðT T A Þ1:25 dt
V
ð3Þ ð4Þ
The unsteady state energy balance for reactor temperature T in Eq. (4) takes into account the heat generation due to chemical reaction and the energy losses by cooling water and through natural convection. The cooling coil dynamics for cooling water exit temperature Tc can be written as:
ðV c qc C pc Þ
dT c ¼ UaðT T c Þ þ qc qc C pc ðT cin T c Þ dt
ð5Þ
The differential equation describing the dynamic behavior in a chemical reactor for parametric sensitivity analysis can be written as:
dy ¼ Fðx; y; /Þ dx
ð6Þ
where y is the vector of a dependent variable (reactor temperature or concentration or cooling water temperature), x is an independent variable (time), u represents the input parameters of the system such as cooling water flow rate, coolant feed temperature and wall capacitance. The elements of the first order sensitivity coefficient matrix can be defined as:
sj/i ¼
dyj d/i
ð7Þ
By directly differentiating Eq. (5), one can obtain the following:
ds/i @Fðx; y; /Þ ¼ Js/i þ @/i dx
ð8Þ
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where J is the Jacobian matrix of the system. The initial condition for Eq. (8) is s/i ¼ 0 at x ¼ 0. The dynamic equation of batch reactor model for parameter sensitivity is:
T T d @F 1 @F 1 @F 1 dC A dT dT c @F 1 @F 2 @F 3 ½s/i ¼ þ dt @C A @T @T c d/i d/i d/i @/i @/i @/i
ð9Þ
The dynamic differential Eqs. (3)–(5) for the batch reactor along with sensitivity Eq. (9) can be rewritten for the input parameters such as cooling water flow rate qc cooling water feed temperature Tcin and wall capacitances ws.
Using Eqs. (3)–(5) for the batch reactor along with sensitivity Eq. (9) for cooling water flow rate qc, i.e. ui = qc can be rewritten as:
@F 1 @T @F 2 @T @F 3 @T
@F 1 dC A @T c dqc @F 2 dC A @T c dqc @F 3 dC A @T c dqc
dT dqc dT dqc dT dqc
T dT c @F 1 þ dqc @qc T dT c @F 2 þ dqc @qc T dT c @F 3 þ dqc @qc
dT c dqc
ðDHVC A EÞ ðV qC P þ ws ÞRT
2
d dT @F 2 @F 2 @F 2 ¼ dt dT cin @C A @T @T c
dC A dT dT c dT cin dT cin dT cin
T
þ
@F 2 @T cin
ð22Þ
ð13Þ ð14Þ ð15Þ
ð16Þ
dC A dT dT c dT cin dT cin dT cin
T
ð23Þ T d dC A dC A dT dT c ¼ k 0 þ ½0 ð24Þ dt dT cin dT cin dT cin dT cin RT 2 T d dT DHVk Ua dC A dT dT c ¼ q1 dt dT cin ðV qC P þ ws Þ ðV qC P þ ws Þ dT cin dT cin dT cin Ua þ ð25Þ ðV qC P þ ws Þ T d dT c Ua ðUa qc Þ dC A dT dT c ¼ 0 dt dT cin ðV c qc C pc Þ ðV c qc C pc Þ dT cin dT cin dT cin qc qc C pc ð26Þ þ ðV c qc C pc Þ
ð17Þ
þ
@F 3 @T cin
kC A E
The differential equations, Eqs. (24)–(26) along with Eqs. (10)–(12), are solved together using the values of parameters show in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 3.3. Parametric sensitivity analysis for wall capacitance
Ua 1:25Ua1ðT T A Þ0:25 ðV qC P þ ws Þ ðV qC P þ ws Þ
ðT T c Þ ðT T A Þ1:25 ðV qC P þ ws Þ d dT c Ua ðUa qc Þ dC A ¼ 0 dt dqc Vc Vc dqc ðT cin TÞ þ Vc
ð12Þ
where q1 and q2 can be expressed as:
q1 ¼
d dT c @F 3 @F 3 @F 3 ¼ dt dT cin @C A @T @T c
T þ ½0
ð21Þ
ð11Þ
Eqs. (13)–(15) can be rewritten in matrix notation as:
d dC A E dC A dT ¼ k kC A 0 dt dqc dqc dqc RT 2 d dT DHVk Ua ¼ q1 dt dqc ðV qC P þ ws Þ ðV qC P þ ws Þ T dC A dT dT c þ ½q2 dqc dqc dqc
1050 3533 1000 4180 220 3.14–13.05 0.0185 1019 11,239 58,520 298
ð10Þ
The sensitivity Eq. (9) can be rewritten in vector form as:
d dC A @F 1 ¼ dt dqc @C A d dT @F 2 ¼ dt dqc @C A d dT c @F 3 ¼ dt dqc @C A
Density of liquid mixture q (kg m3) Specific heat of liquid mixture, Cp (J kg1 K1) Density of cooling water qc (kg m3) Specific heat of cooling water Cpc (J kg1 K1) Concentration of sulphuric acid Cs (mol m3) Overall heat transfer coefficient range Ua (W/°C) Frequency factor ko (s1) Activation energy E/R (K) Heat of reaction (DH) (J mol1) Reference temperature Tref (K)
T d dC A @F 1 @F 1 @F 1 dC A dT dT c @F 1 ¼ þ dt dT cin @C A @T @T c dT cin dT cin dT cin @T cin
3.1. Parametric sensitivity analysis for cooling water flow rate
dC A ¼ kC A ¼ F 1 dt dT ðDHÞVkC A UaðT T c Þ Ua1ðT T c Þ1:25 ¼ ¼ F2 dt ðV qC p þ ws Þ ðV qC p þ ws Þ ðV qC p þ ws Þ dT c UaðT T c Þ qc ¼ þ ðT cin T c Þ ¼ F 3 ðV c qc C pc Þ V c dt
Table 1 System parameters used in batch reactor simulation.
q2 ¼
ð18Þ ð19Þ
dT dqc
dT c dqc
T
ð20Þ
The differential equations, Eqs. (16)–(20) along with Eqs. (10)–(12) are solved together using the values of the parameters shown in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 3.2. Parametric sensitivity analysis for cooling water feed temperature Similarly, dynamic equations for cooling water feed temperature as input parameter i.e. ui = Tcin using Eqs. (3)–(5), along with sensitivity Eq. (9) can be rewritten as:
The procedure adopted for cooling water flow rate and feed temperature is implemented for wall capacitance as input parameters, i.e. /i ¼ ws can be rewritten as:
d dC A @F 1 ¼ dt dws @C A d dT @F 2 ¼ dt dws @C A d dT c @F 3 ¼ dt dws @C A
@F 1 @T @F 2 @T @F 3 @T
dC A dws @F 2 dC A @T c dws @F 3 dC A @T c dws @F 1 @T c
dT dws
dT c dws
dT dws
dT c dws
dT dws
dT c dws
T
þ
T
þ
T
þ
@F 1 @ws @F 2 @ws @F 3 @ws
ð27Þ ð28Þ ð29Þ
The differential Eqs. (27)–(29) is rewritten by using Eqs. (10)–(12) as:
d dC A dC A E dT ¼ k kC A 2 ð30Þ dt dws dws RT dws " # dF 2 ðDHÞVkC A UaðT T c Þ Ua1ðT T A Þ1:25 ð31Þ ¼ dws ðV qC p þ ws Þ2 ðV qC p þ ws Þ2 ðV qC p þ ws Þ2
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Fig. 1. Experimental set up for batch reactor study.
Table 2 Batch reactor geometry used in the present study. Inside diameter (mm) Height (mm) Volume of reactor (m3) Number of baffles Type of agitator Speed of agitator (rpm) Inside diameter of cooling coil (mm) Number of turns of cooling coil Coil diameter (mm) Pitch of the coil (mm)
90 120 400 106 4 4 bladed turbine 900 3 9 65 7.5
Table 3 Experimental conditions used in batch reactor study. Volume of the reactor (m3) Volume of acetic anhydride (m3) Volume of acetic acid (m3) Volume of water (m3) Volume of sulphuric acid (m3) Initial temperature range of the reactor (°C) Air temperature range (°C) Cooling water feed temperature range Tcin (°C) Cooling water flow rate range (m3/s)
400 106 189 106 76 106 134.5 106 0.5 106 21–24.5 21–25 21–25 0.4 106–0.567 106
d dT dF 2 ðDHÞVk dC A ¼ þ dt dws ðV qC p þ ws Þ dws dws " # qqC p þ ððDHÞVkC A RTE2 Ua 1:25Ua1ðT T A Þ0:25 dT dws ðV qC p þ ws Þ Ua dT C þ ð32Þ ðV qC p þ ws Þ dws d dT c dC A Ua dT c ¼0 þ dt dws ðV c qc C pc Þ dws dws ðUa þ qC Þ dT C dF 3 þ0 ð33Þ ðV c qc C pc Þ dws dws
The differential Eqs. (30)–(33) along with Eqs. (10)–(12) are solved together using the values of the parameters shown in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 4. Experimental set-up and technique A closed batch steel reactor vessel of inside diameter 90 mm, height 120 mm, and thickness 1 mm was used for carrying out the acid catalyzed hydrolysis of acetic anhydride in the presence of solvent acetic acid. A schematic diagram of the experimental
N.S. Jayakumar et al. / International Journal of Heat and Mass Transfer 54 (2011) 439–446
setup is as shown in Fig. 1. The reactor is made of stainless steel and has a capacity of 500 ml. The reactor was provided with facilities for pouring reactants, measuring temperature and stirring using four bladed turbine running at 900 rpm to ensure homogeneous gradient less condition in temperature and concentration. It was initially filled with 400 ml of reaction mixture at an initial concentration of acetic anhydride equal to 5000 mol/m3 at 21.5 °C. The transient temperature of the reaction mixture was recorded using a thermometer with an accuracy of ±0.1 °C. The geometry of the reactor and the experimental conditions used in the present study are given in Tables 2 and 3 respectively. The experiments and theoretical simulations were conducted for the hydrolysis of acetic anhydride by varying input parameters such as cooling water flow rate, cooling water feed temperature and wall capacitance for studying parametric sensitivity.
443
The theoretical simulation was developed by solving coupled unsteady state Eqs. (16)–(20) numerically using ODE15s solver of Matlab software (version 7.02). The simulated temperature profile was compared to the experimental data, as illustrated in Fig. 2. Experimentally, the reactor reached a maximum temperature of 69 °C in 28 min as against the simulated steady state temperature of 59 °C in 42 min. The reactor did not exhibit any sensitivity when the coolant flow rate changed to 0.550 106 m3/s. Similarly, the effect of changing the coolant flow rate at 0.550 106 m3/s, caused the maximum temperature to reach 40.1 °C in 31 min whereas the simulated results gave a maximum of 31.7 °C in 53 min. Thus, the experiments showed remarkable sensitivities in time for the attainment of maximum temperature.
5.2. Effect of cooling water feed temperature on parametric sensitivity 5. Results and discussion 5.1. Effect of cooling water on parametric sensitivity The cooling water was set at a flow rate of at 0.483 106 m3/s and the stirrer speed was kept at 900 rpm. The reactor was filled with 400 106 m3 of the reaction mixture with the initial concentration of acetic anhydride of 5000 mol/m3 and a temperature of 21.5 °C. As shown in Fig. 2, the reactor then reached a final temperature of 69 °C. Keeping all other experimental conditions the same, a small change of 0.550 106 m3/s in the cooling water flow rate was introduced and it was observed that the reactor reached a final temperature of 40.1 °C. The reactor did not exhibit any sensitivity when the coolant flow rate changed to 0.550 106 m3/s.
Fig. 2. The transient batch reactor temperature profile for cooling water flow rate qc as parameter.
The reactor was filled with 400 106 m3 of the reaction mixture with the initial concentration of acetic anhydride of 5000 mol/m3. The effect of cooling water temperature on the existence of parametric sensitivity was tested by changing the feed temperature from 24.5 °C to 23.5 °C for both the experiments and theoretical simulations. The theoretical simulation developed by solving coupled unsteady state Eqs. (10)–(13) and Eqs. (22)– (24) numerically using ODE15s solver of Matlab software (version 7.02). In Fig. 3, the simulated temperature profiles were compared to the experimental data. The coolant feed temperature was 24.5 °C and the cooling water flow rate set at 0.567 106 m3/s. Fig. 3 illustrated the transient temperature of the reaction mixture and the reactor exhibited maximum temperature of 82 °C experimentally at 20 min and 71.6 °C theoretically at 29 min. Keeping all other experimental
Fig. 3. The transient batch reactor temperature profile for cooling water feed temperature Tcin as parameter.
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conditions the same, a small change of 1 °C in the cooling water feed temperature was introduced and it was observed that the reactor exhibited a maximum temperature of 49.4 °C in 29 min experimentally and 49 °C in 43 min theoretically. Thus, the reactor exhibited parametric sensitivity behavior with the respect to the coolant flow feed temperature.
5.4. Analysis of sensitivity variables Investigations on the effect of input parameters, namely coolant flow rate, coolant feed temperature and wall capacitance on
5.3. Effect of wall capacitance on parametric sensitivity The effect of wall capacitance on parametric sensitivity were studied experimentally and simulated by varying wall capacitance values as 121 and 1793 J/°C at constant cooling water flow rate of 0.4 106 m3/s. The theoretical simulation developed by solving coupled unsteady state Eqs. (10)–(13) and Eqs. (28)–(31) numerically using ODE15s solver of Matlab software (version 7.02). As illustrated in Fig. 4. parametric sensitivity occurs for wall capacitance value at 121 and 1793 J/°C. The wall capacitance has an effect on the transients, thus making the system sluggish. It was observed that the reactor stabilized when the wall capacitance value was 1793 J/°C. Fig. 4 further illustrates that the reactor exhibits a maximum temperature of 88 °C in 21 min experimentally and 79 °C in 31 min theoretically for wall capacitance value of 121 J/°C. Keeping all other experimental conditions the same, the wall capacitance was increased to 1793 J/°C by having a jacket of water with capacity 4 104 m3. From Fig. 4, it can be observed that the reactor exhibits a maximum temperature of 34.1 °C in 40 min experimentally and 37 °C in 67 min theoretically. Thus, the reactor showed the existence of parametric sensitivity behavior with respect to the wall capacitance as input parameter.
Fig. 4. The transient batch reactor temperature profile with wall capacitance ws as parameter.
Fig. 5. Time profile of parameter sensitivity variable Sj/i for qc as parameter developed in the batch reactor at (a) qc = 0.483 106 m3/s; (b) qc = 0.55 106 m3/s.
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445
Fig. 6. Time profile of parameter sensitivity variable Sj/i for Tcin as parameter developed in the batch reactor at (a) Tcin = 24.5 °C; (b) Tcin = 23.5 °C.
Fig. 7. Time profile of parameter sensitivity variable Sj/i for ws as parameter developed in the batch reactor at (a) ws = 121 J/°C; (b) ws = 1793 J/°C.
the sensitivity variable Sj/i were carried out. As illustrated in the respective Figs. 5–7, each of this input parameter was analysed according to the transient variations of Sj/i corresponding to the reactor concentration, reactor temperature, and the cooling water temperature. The figures clearly illustrate that Sj/i have
peaks and the reactor temperature, reactor concentration, and cooling water outlet temperature attain a maximum at the points of zero slope. This information is of significance from the point of view of implementing a control scheme on the reactor.
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Table 4 Summary of the results obtained in the present study. Sl. no.
ws (J/°C)
qc 106 (m3/s)
Tcin (°C)
Initial temperature (°C)
(DT)max (°C)
(DT)ad (°C)
Time to observe maximum temperature (min)
Remarks
1 2 3 4 5 6 7 8 9 10 11 12
121 121 121 121 121 121 121 121 121 121 1793 1793
0.483 (exp) 0.483 (sim) 0.55 (exp) 0.55 (sim) 0.567 (exp) 0.567 (sim) 0.567 (exp) 0.567 (sim) 0.40 (exp) 0.40 (sim) 0.40 (exp) 0.40 (sim)
21.5 21.5 21.5 21.5 24.5 24.5 23.5 23.5 22.0 22.0 22.0 22.0
22.0 22.0 21.5 21.5 24.5 24.5 24.5 24.5 22.0 22.0 22.0 22.0
46.7 38.5 40.1 31.7 57.5 47.1 24.8 24.53 65 57.1 12.1 15.44
70 70 – – 70 70 – – 70 70 – –
28 42 31 53 20 29 26 43 20.7 31 40 75
Sensitive (Figs. 2 and 5) Sensitive (Figs. 2 and 5) Non-Sensitive (Figs. 2 and 5) Non-sensitive (Figs. 2 and 5) Sensitive (Figs. 3 and 6) Sensitive (Figs. 3 and 6) Non-Sensitive (Figs. 3 and 6) Non-sensitive (Figs. 3 and 6) Sensitive (Figs. 4 and 7) Sensitive (Figs. 4 and 7) Non-Sensitive (Figs. 4 and 7) Non-sensitive (Figs. 4 and 7)
6. Conclusions The sulphuric acid catalyzed hydrolysis of acetic anhydride has been used to prove the existence of parametric sensitivity in a batch mode. Simulations were successfully carried out using published kinetic parameters. The effect of extraneous wall capacitance showed that the reactor exhibited non-parametric sensitivity behavior dynamics experimentally, and it has been illustrated that the presence of extraneous wall capacitance tends to stabilize the reactor by exhibiting insensitivity. Table 4 provides a summary of the results obtained. The transients are in good agreement with the experimental temperature data when the reactor is operating at high coolant water flow rate. The predicted transient temperatures are in satisfactory agreement with the experimental temperature data when the reactor is operating at low coolant water flow rate. References [1] P.K. Shukla, S. Pushpavanam, Parametric sensitivity, runaway, and safety in batch reactors: experiments and models, Ind. Eng. Chem. Res. 33 (1994) 3202– 3208. [2] C.H. Barkelew, Stability of chemical reactors, Chem. Eng. Prog. Symp. Ser. 55 (1959) 37–46. [3] J. Adler, J.W. Enig, The critical thermal explosion theory with reactant consumption, Combust. Flame 8 (1964) 97–103.
[4] M. Morbidelli, A. Varma, Parametric sensitivity and runaway in tubular reactors, AIChE J. 28 (1982) 705–713. [5] M. Morbidelli, A. Varma, On parametric sensitivity and runaway criteria of pseudo homogeneous tubular reactors, Chem. Eng. Sci. 40 (1985) 2165– 2169. [6] M. Morbidelli, A. Varma, A generalised criterion for parametric sensitivity: application to thermal explosion theory, Chem. Eng. Sci. 43 (1988) 91–102. [7] N.N. Semonov, Zur theorie des verbrennungs prozessess, Z. Phys. 48 (1928) 571–582. [8] Y.B. Zeldovich, G.I. Brenblatt, V.B. Librovich, G.M. Makhiviladze, The Mathematical Theory of Combustion and Explosions, Plenum Press, New York, 1985. [9] E. von Westerholt, J.N. Beard, S.S. Melsheimer, Time-optimal startup control algorithm for batch processes, Ind. Eng. Chem. Res. 30 (6) (1991) 1205– 1212. [10] D.R. Lewin, R. Lavie, Designing and implementing trajectories in an exothermic batch chemical reactor, Ind. Eng. Chem. Res. 29 (1) (1990) 89–96. [11] A. Varma, M. Morbidelli, H. Wu, Parametric Sensitivity in Chemical Systems, Cambridge University Press, Cambridge, 1999. [12] J. Adler, J.W. Enig, The critical conditions in thermal explosion theory with reactant consumption, Combust. Flame 8 (1964) 97–103. [13] E.D. Gilles, H. Schuler, Early detection of hazardous states in chemical reactors, German Chem. Eng. 5 (1982) 69–78. [14] P. Hugo, J. Steinbach, F. Stossel, Calculation of the maximum temperature in stirred tank reactors in case of a breakdown of cooling, Chem. Eng. Sci. 43 (8) (1989) 2147–2152. [15] R.M. Chemburkar, M. Morbrdelli, A. Varma, Parametric sensitivity of a CSTR, Chem. Eng. Sci. 41 (6) (1986) 1647–1654. [16] R. Haldar, D.P. Rao, Experimental studies on limit cycle behavior of the sulphuric acid catalysed hydrolysis of acetic anhydride in a CSTR, Chem. Eng. Sci. 46 (1991) 1197–1200.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Modeling the effect of wall capacitance on the dynamics of an exothermic reaction system in a batch reactor N.S. Jayakumar a,⇑, A. Agrawal b, M.A. Hashim a, J.N. Sahu a,c a
Department of Chemical Engineering, Faculty of Engineering, University of Malaya, Kuala Lumpur 50603, Malaysia Department of Chemical Engineering, Dharmsinh Desai Institute of Technology, Nadiad 387001, Gujarat, India c Department of Chemical Engineering, Indian Institute of Technology (IIT), Kharagpur, P.O. Kharagpur Technology, West Bengal 721302, India b
a r t i c l e
i n f o
Article history: Received 28 April 2010 Received in revised form 18 August 2010 Accepted 18 August 2010 Available online 8 October 2010 Keywords: Runaway reactions Parametric sensitivity Wall capacitance Input parameters
a b s t r a c t The chemical reactors in certain range of operating conditions may exhibit parametric sensitivity where small changes in one or more of the input parameters lead to large changes in the output variable. This is a form of critical behavior that leads to runaway conditions, resulting in hazardous reactor operation. In the present work, hydrolysis of acetic anhydride reaction was used to investigate the existence and effect of parametric sensitivity with respect to the input parameters, namely cooling water flow rate, cooling water feed temperature and wall capacitance. Parametric sensitivity was observed for a small change in coolant water flow rate and feed temperature. It is found experimentally that with the introduction of extraneous wall capacitance, the batch reactor showed non-sensitivity under parametric sensitivity conditions. A mathematical model for the reactor was developed by incorporating both mass and energy balance with ordinary coupled differential equations. This dynamic model was solved numerically using ODE15s (gear method) of Matlab software, and the numerically simulated results are in satisfactory agreement with the experimental data. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Batch reactors have etched a permanent and important place for themselves in the chemical industry. Some of the industries where batch reactors are predominately used include drugs, pharmaceuticals, polymers and speciality chemicals [1]. Batch reactors are popular because of their flexibility in operation, and hence widely used in the manufacturing of products of low volume but of high added value. They are operated in closed batch system. The reactor may produce side reactions if it is not properly controlled. In an out of control system, the reactor may exhibit runaway conditions, thus causing high temperature and pressure beyond its endurance. It is known that batch reactors, in certain regions of operating conditions may exhibit a parametric sensitive behavior where small changes in one or more of the reactor input parameters lead to much larger changes in output variables [2–8]. 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 the optimal temperature profiles in batch systems [9]. Lewin and Lavie [10] have estimated the safe start-up time in terms of design ⇑ Corresponding author. Fax: +60 3 79675319. E-mail address: [email protected] (N.S. Jayakumar). 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.09.025
parameters for 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. An excessive temperature excursion may adversely affect conversion of exothermic equilibrium reaction and the selectivity of the reaction process and, in the case of catalytic reaction, catalyst activity and durability and even reactor safety [11]. Knowledge of the circumstances leading to thermal runway and knowledge of the process chemistry and of the stored reagents are essential factors for the safe operation of chemical reactors and the storage of explosive chemicals. A parametric sensitivity and runaway are in fact related terms since one is the consequence of the other. A parametrically sensitive chemical reaction involves high risk of unleashing a runaway reaction by accidental causes [8,12,13]. Semonov [7] proposed a criterion to evaluate the thermal instability of a reagent by assuming the reaction is zero order and that the fluid is well mixed. The priori criterion for runaway condition was proposed by Barkelew [2] based on the examination of a number of reaction kinetics in which heat and mass balance equations were taken into account. The technique of early detection by hazardous states in chemical reactors has been developed by Gilles and Schuler [13] who predicted a safe operation detection system by applying one order and second order differentiation of reaction temperature with time. Hugo et al. [14] concluded that as far as safe operation problem in reaction is concerned, reactors should
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Nomenclature a CA CA0 Cp Cpc Cs E F1 F2 F3 J k qc rA Sj/i t T TA Tcin Ua Ua1 V Vc
heat transfer area (m2) concentration of acetic anhydride (mol/m3) initial concentration of acetic anhydride (mol/m3) specific heat of reaction mixture (J/(kg °C)) specific heat of coolant water (J/(kg °C)) initial concentration of sulphuric acid (mol/m3) activation energy (J/mol) function defined in Eq. (10) function defined in Eq. (11) function defined in Eq. (12) Jacobian matrix of the system rate constant (s1) coolant water flow rate (m3/s) reaction rate (s1) first order sensitivity coefficient matrix time (min) temperature of the batch reactor (°C) air temperature (°C) coolant water feed temperature (°C) over all heat transfer coefficient on the cooling side (W/°C) over all heat transfer coefficient due to natural convection heat loss from the reactor (W/°C) volume of the batch reactor (m3) volume of coolant (m3)
not reach high temperatures. Chemburkar et al. [15] proposed the generalized parametric criterion to determine the safe operation in a continuous stirred tank reactor while Morbidelli and Varma [6] applied the same method to determine the safe operating region for batch reactor operation. Unlike the theoretical development, experimental studies addressing parametric sensitivity of exothermic reaction in bath reactor are scarce, hence the present study on the phenomena of parametric sensitivity using exothermic reaction system is under taken. In this work, the dynamics of a batch reactor using the acid catalyzed hydrolysis of acetic anhydride was carried out. 2. Reaction pathway The overall hydrolysis of acetic anhydride reaction can be represented as: H2 SO4 Catalyst ðCH3 COÞ2 O þH2 O ! acetic acid solvent Acetic anhydride
2CH3 COOH
ð1Þ
Acetic acid
The mechanisms of the hydrolysis of acetic anhydride are well documented. It is well known that the reaction proceeds via a substitution, in which an attacking nucleophile replaces a substituent group on the central carbon. The reaction is of first order with respect to acetic anhydride. The kinetic equation for as given by Haldar and Rao [16] is:
r A ¼ ð1:85 1010 C s expð11243:9=TÞÞC A ; ðDHÞ ¼ 58520 J=mol
ð2Þ
The kinetic parameters of the above reaction were used to simulate the batch reactor experimental runs and the simulations were compared with the experimental results. 3. Modeling The quantum of investigations required for confirming parametric sensitivity in a reaction system can be minimized and easily
x y
independent variable (time) vector of a dependent variable (temperature or concentration) wall capacitance (J/°C)
ws
Greek letters q density of liquid mixture (kg/m3) (DT)max maximum reactor temperature (°C) (DT)ad adiabatic reactor temperature (°C) (DH) heat of reaction (J/mol) u represents the input parameter of the system Subscripts A acetic anhydride B acetic acid–water–sulphuric acid mixture cin coolant inlet temperature exp experimental P specific heat s stainless steel wall of the batch reactor sim simulated f feed c coolant temperature in the jacket i denoting coolant temperature in jacket or stainless steel wall
realizable in the laboratory by theoretically identifying its existence. Through selecting a set of process variables combinations from input parameters, the trial runs required for experimentally confirming the existence of parametric sensitivity of a reaction system can be reduced considerably. The unsteady state mass balance for first order catalyzed hydrolysis of acetic anhydride can be written as:
dNA ¼ kC A dt dT ðV qC p þ ws Þ ¼ ðDHÞVkC A UaðT T c Þ Ua1ðT T A Þ1:25 dt
V
ð3Þ ð4Þ
The unsteady state energy balance for reactor temperature T in Eq. (4) takes into account the heat generation due to chemical reaction and the energy losses by cooling water and through natural convection. The cooling coil dynamics for cooling water exit temperature Tc can be written as:
ðV c qc C pc Þ
dT c ¼ UaðT T c Þ þ qc qc C pc ðT cin T c Þ dt
ð5Þ
The differential equation describing the dynamic behavior in a chemical reactor for parametric sensitivity analysis can be written as:
dy ¼ Fðx; y; /Þ dx
ð6Þ
where y is the vector of a dependent variable (reactor temperature or concentration or cooling water temperature), x is an independent variable (time), u represents the input parameters of the system such as cooling water flow rate, coolant feed temperature and wall capacitance. The elements of the first order sensitivity coefficient matrix can be defined as:
sj/i ¼
dyj d/i
ð7Þ
By directly differentiating Eq. (5), one can obtain the following:
ds/i @Fðx; y; /Þ ¼ Js/i þ @/i dx
ð8Þ
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where J is the Jacobian matrix of the system. The initial condition for Eq. (8) is s/i ¼ 0 at x ¼ 0. The dynamic equation of batch reactor model for parameter sensitivity is:
T T d @F 1 @F 1 @F 1 dC A dT dT c @F 1 @F 2 @F 3 ½s/i ¼ þ dt @C A @T @T c d/i d/i d/i @/i @/i @/i
ð9Þ
The dynamic differential Eqs. (3)–(5) for the batch reactor along with sensitivity Eq. (9) can be rewritten for the input parameters such as cooling water flow rate qc cooling water feed temperature Tcin and wall capacitances ws.
Using Eqs. (3)–(5) for the batch reactor along with sensitivity Eq. (9) for cooling water flow rate qc, i.e. ui = qc can be rewritten as:
@F 1 @T @F 2 @T @F 3 @T
@F 1 dC A @T c dqc @F 2 dC A @T c dqc @F 3 dC A @T c dqc
dT dqc dT dqc dT dqc
T dT c @F 1 þ dqc @qc T dT c @F 2 þ dqc @qc T dT c @F 3 þ dqc @qc
dT c dqc
ðDHVC A EÞ ðV qC P þ ws ÞRT
2
d dT @F 2 @F 2 @F 2 ¼ dt dT cin @C A @T @T c
dC A dT dT c dT cin dT cin dT cin
T
þ
@F 2 @T cin
ð22Þ
ð13Þ ð14Þ ð15Þ
ð16Þ
dC A dT dT c dT cin dT cin dT cin
T
ð23Þ T d dC A dC A dT dT c ¼ k 0 þ ½0 ð24Þ dt dT cin dT cin dT cin dT cin RT 2 T d dT DHVk Ua dC A dT dT c ¼ q1 dt dT cin ðV qC P þ ws Þ ðV qC P þ ws Þ dT cin dT cin dT cin Ua þ ð25Þ ðV qC P þ ws Þ T d dT c Ua ðUa qc Þ dC A dT dT c ¼ 0 dt dT cin ðV c qc C pc Þ ðV c qc C pc Þ dT cin dT cin dT cin qc qc C pc ð26Þ þ ðV c qc C pc Þ
ð17Þ
þ
@F 3 @T cin
kC A E
The differential equations, Eqs. (24)–(26) along with Eqs. (10)–(12), are solved together using the values of parameters show in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 3.3. Parametric sensitivity analysis for wall capacitance
Ua 1:25Ua1ðT T A Þ0:25 ðV qC P þ ws Þ ðV qC P þ ws Þ
ðT T c Þ ðT T A Þ1:25 ðV qC P þ ws Þ d dT c Ua ðUa qc Þ dC A ¼ 0 dt dqc Vc Vc dqc ðT cin TÞ þ Vc
ð12Þ
where q1 and q2 can be expressed as:
q1 ¼
d dT c @F 3 @F 3 @F 3 ¼ dt dT cin @C A @T @T c
T þ ½0
ð21Þ
ð11Þ
Eqs. (13)–(15) can be rewritten in matrix notation as:
d dC A E dC A dT ¼ k kC A 0 dt dqc dqc dqc RT 2 d dT DHVk Ua ¼ q1 dt dqc ðV qC P þ ws Þ ðV qC P þ ws Þ T dC A dT dT c þ ½q2 dqc dqc dqc
1050 3533 1000 4180 220 3.14–13.05 0.0185 1019 11,239 58,520 298
ð10Þ
The sensitivity Eq. (9) can be rewritten in vector form as:
d dC A @F 1 ¼ dt dqc @C A d dT @F 2 ¼ dt dqc @C A d dT c @F 3 ¼ dt dqc @C A
Density of liquid mixture q (kg m3) Specific heat of liquid mixture, Cp (J kg1 K1) Density of cooling water qc (kg m3) Specific heat of cooling water Cpc (J kg1 K1) Concentration of sulphuric acid Cs (mol m3) Overall heat transfer coefficient range Ua (W/°C) Frequency factor ko (s1) Activation energy E/R (K) Heat of reaction (DH) (J mol1) Reference temperature Tref (K)
T d dC A @F 1 @F 1 @F 1 dC A dT dT c @F 1 ¼ þ dt dT cin @C A @T @T c dT cin dT cin dT cin @T cin
3.1. Parametric sensitivity analysis for cooling water flow rate
dC A ¼ kC A ¼ F 1 dt dT ðDHÞVkC A UaðT T c Þ Ua1ðT T c Þ1:25 ¼ ¼ F2 dt ðV qC p þ ws Þ ðV qC p þ ws Þ ðV qC p þ ws Þ dT c UaðT T c Þ qc ¼ þ ðT cin T c Þ ¼ F 3 ðV c qc C pc Þ V c dt
Table 1 System parameters used in batch reactor simulation.
q2 ¼
ð18Þ ð19Þ
dT dqc
dT c dqc
T
ð20Þ
The differential equations, Eqs. (16)–(20) along with Eqs. (10)–(12) are solved together using the values of the parameters shown in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 3.2. Parametric sensitivity analysis for cooling water feed temperature Similarly, dynamic equations for cooling water feed temperature as input parameter i.e. ui = Tcin using Eqs. (3)–(5), along with sensitivity Eq. (9) can be rewritten as:
The procedure adopted for cooling water flow rate and feed temperature is implemented for wall capacitance as input parameters, i.e. /i ¼ ws can be rewritten as:
d dC A @F 1 ¼ dt dws @C A d dT @F 2 ¼ dt dws @C A d dT c @F 3 ¼ dt dws @C A
@F 1 @T @F 2 @T @F 3 @T
dC A dws @F 2 dC A @T c dws @F 3 dC A @T c dws @F 1 @T c
dT dws
dT c dws
dT dws
dT c dws
dT dws
dT c dws
T
þ
T
þ
T
þ
@F 1 @ws @F 2 @ws @F 3 @ws
ð27Þ ð28Þ ð29Þ
The differential Eqs. (27)–(29) is rewritten by using Eqs. (10)–(12) as:
d dC A dC A E dT ¼ k kC A 2 ð30Þ dt dws dws RT dws " # dF 2 ðDHÞVkC A UaðT T c Þ Ua1ðT T A Þ1:25 ð31Þ ¼ dws ðV qC p þ ws Þ2 ðV qC p þ ws Þ2 ðV qC p þ ws Þ2
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Fig. 1. Experimental set up for batch reactor study.
Table 2 Batch reactor geometry used in the present study. Inside diameter (mm) Height (mm) Volume of reactor (m3) Number of baffles Type of agitator Speed of agitator (rpm) Inside diameter of cooling coil (mm) Number of turns of cooling coil Coil diameter (mm) Pitch of the coil (mm)
90 120 400 106 4 4 bladed turbine 900 3 9 65 7.5
Table 3 Experimental conditions used in batch reactor study. Volume of the reactor (m3) Volume of acetic anhydride (m3) Volume of acetic acid (m3) Volume of water (m3) Volume of sulphuric acid (m3) Initial temperature range of the reactor (°C) Air temperature range (°C) Cooling water feed temperature range Tcin (°C) Cooling water flow rate range (m3/s)
400 106 189 106 76 106 134.5 106 0.5 106 21–24.5 21–25 21–25 0.4 106–0.567 106
d dT dF 2 ðDHÞVk dC A ¼ þ dt dws ðV qC p þ ws Þ dws dws " # qqC p þ ððDHÞVkC A RTE2 Ua 1:25Ua1ðT T A Þ0:25 dT dws ðV qC p þ ws Þ Ua dT C þ ð32Þ ðV qC p þ ws Þ dws d dT c dC A Ua dT c ¼0 þ dt dws ðV c qc C pc Þ dws dws ðUa þ qC Þ dT C dF 3 þ0 ð33Þ ðV c qc C pc Þ dws dws
The differential Eqs. (30)–(33) along with Eqs. (10)–(12) are solved together using the values of the parameters shown in Table 1. The equations are solved numerically using ODE15s (Gear method) of Matlab. 4. Experimental set-up and technique A closed batch steel reactor vessel of inside diameter 90 mm, height 120 mm, and thickness 1 mm was used for carrying out the acid catalyzed hydrolysis of acetic anhydride in the presence of solvent acetic acid. A schematic diagram of the experimental
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setup is as shown in Fig. 1. The reactor is made of stainless steel and has a capacity of 500 ml. The reactor was provided with facilities for pouring reactants, measuring temperature and stirring using four bladed turbine running at 900 rpm to ensure homogeneous gradient less condition in temperature and concentration. It was initially filled with 400 ml of reaction mixture at an initial concentration of acetic anhydride equal to 5000 mol/m3 at 21.5 °C. The transient temperature of the reaction mixture was recorded using a thermometer with an accuracy of ±0.1 °C. The geometry of the reactor and the experimental conditions used in the present study are given in Tables 2 and 3 respectively. The experiments and theoretical simulations were conducted for the hydrolysis of acetic anhydride by varying input parameters such as cooling water flow rate, cooling water feed temperature and wall capacitance for studying parametric sensitivity.
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The theoretical simulation was developed by solving coupled unsteady state Eqs. (16)–(20) numerically using ODE15s solver of Matlab software (version 7.02). The simulated temperature profile was compared to the experimental data, as illustrated in Fig. 2. Experimentally, the reactor reached a maximum temperature of 69 °C in 28 min as against the simulated steady state temperature of 59 °C in 42 min. The reactor did not exhibit any sensitivity when the coolant flow rate changed to 0.550 106 m3/s. Similarly, the effect of changing the coolant flow rate at 0.550 106 m3/s, caused the maximum temperature to reach 40.1 °C in 31 min whereas the simulated results gave a maximum of 31.7 °C in 53 min. Thus, the experiments showed remarkable sensitivities in time for the attainment of maximum temperature.
5.2. Effect of cooling water feed temperature on parametric sensitivity 5. Results and discussion 5.1. Effect of cooling water on parametric sensitivity The cooling water was set at a flow rate of at 0.483 106 m3/s and the stirrer speed was kept at 900 rpm. The reactor was filled with 400 106 m3 of the reaction mixture with the initial concentration of acetic anhydride of 5000 mol/m3 and a temperature of 21.5 °C. As shown in Fig. 2, the reactor then reached a final temperature of 69 °C. Keeping all other experimental conditions the same, a small change of 0.550 106 m3/s in the cooling water flow rate was introduced and it was observed that the reactor reached a final temperature of 40.1 °C. The reactor did not exhibit any sensitivity when the coolant flow rate changed to 0.550 106 m3/s.
Fig. 2. The transient batch reactor temperature profile for cooling water flow rate qc as parameter.
The reactor was filled with 400 106 m3 of the reaction mixture with the initial concentration of acetic anhydride of 5000 mol/m3. The effect of cooling water temperature on the existence of parametric sensitivity was tested by changing the feed temperature from 24.5 °C to 23.5 °C for both the experiments and theoretical simulations. The theoretical simulation developed by solving coupled unsteady state Eqs. (10)–(13) and Eqs. (22)– (24) numerically using ODE15s solver of Matlab software (version 7.02). In Fig. 3, the simulated temperature profiles were compared to the experimental data. The coolant feed temperature was 24.5 °C and the cooling water flow rate set at 0.567 106 m3/s. Fig. 3 illustrated the transient temperature of the reaction mixture and the reactor exhibited maximum temperature of 82 °C experimentally at 20 min and 71.6 °C theoretically at 29 min. Keeping all other experimental
Fig. 3. The transient batch reactor temperature profile for cooling water feed temperature Tcin as parameter.
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conditions the same, a small change of 1 °C in the cooling water feed temperature was introduced and it was observed that the reactor exhibited a maximum temperature of 49.4 °C in 29 min experimentally and 49 °C in 43 min theoretically. Thus, the reactor exhibited parametric sensitivity behavior with the respect to the coolant flow feed temperature.
5.4. Analysis of sensitivity variables Investigations on the effect of input parameters, namely coolant flow rate, coolant feed temperature and wall capacitance on
5.3. Effect of wall capacitance on parametric sensitivity The effect of wall capacitance on parametric sensitivity were studied experimentally and simulated by varying wall capacitance values as 121 and 1793 J/°C at constant cooling water flow rate of 0.4 106 m3/s. The theoretical simulation developed by solving coupled unsteady state Eqs. (10)–(13) and Eqs. (28)–(31) numerically using ODE15s solver of Matlab software (version 7.02). As illustrated in Fig. 4. parametric sensitivity occurs for wall capacitance value at 121 and 1793 J/°C. The wall capacitance has an effect on the transients, thus making the system sluggish. It was observed that the reactor stabilized when the wall capacitance value was 1793 J/°C. Fig. 4 further illustrates that the reactor exhibits a maximum temperature of 88 °C in 21 min experimentally and 79 °C in 31 min theoretically for wall capacitance value of 121 J/°C. Keeping all other experimental conditions the same, the wall capacitance was increased to 1793 J/°C by having a jacket of water with capacity 4 104 m3. From Fig. 4, it can be observed that the reactor exhibits a maximum temperature of 34.1 °C in 40 min experimentally and 37 °C in 67 min theoretically. Thus, the reactor showed the existence of parametric sensitivity behavior with respect to the wall capacitance as input parameter.
Fig. 4. The transient batch reactor temperature profile with wall capacitance ws as parameter.
Fig. 5. Time profile of parameter sensitivity variable Sj/i for qc as parameter developed in the batch reactor at (a) qc = 0.483 106 m3/s; (b) qc = 0.55 106 m3/s.
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Fig. 6. Time profile of parameter sensitivity variable Sj/i for Tcin as parameter developed in the batch reactor at (a) Tcin = 24.5 °C; (b) Tcin = 23.5 °C.
Fig. 7. Time profile of parameter sensitivity variable Sj/i for ws as parameter developed in the batch reactor at (a) ws = 121 J/°C; (b) ws = 1793 J/°C.
the sensitivity variable Sj/i were carried out. As illustrated in the respective Figs. 5–7, each of this input parameter was analysed according to the transient variations of Sj/i corresponding to the reactor concentration, reactor temperature, and the cooling water temperature. The figures clearly illustrate that Sj/i have
peaks and the reactor temperature, reactor concentration, and cooling water outlet temperature attain a maximum at the points of zero slope. This information is of significance from the point of view of implementing a control scheme on the reactor.
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Table 4 Summary of the results obtained in the present study. Sl. no.
ws (J/°C)
qc 106 (m3/s)
Tcin (°C)
Initial temperature (°C)
(DT)max (°C)
(DT)ad (°C)
Time to observe maximum temperature (min)
Remarks
1 2 3 4 5 6 7 8 9 10 11 12
121 121 121 121 121 121 121 121 121 121 1793 1793
0.483 (exp) 0.483 (sim) 0.55 (exp) 0.55 (sim) 0.567 (exp) 0.567 (sim) 0.567 (exp) 0.567 (sim) 0.40 (exp) 0.40 (sim) 0.40 (exp) 0.40 (sim)
21.5 21.5 21.5 21.5 24.5 24.5 23.5 23.5 22.0 22.0 22.0 22.0
22.0 22.0 21.5 21.5 24.5 24.5 24.5 24.5 22.0 22.0 22.0 22.0
46.7 38.5 40.1 31.7 57.5 47.1 24.8 24.53 65 57.1 12.1 15.44
70 70 – – 70 70 – – 70 70 – –
28 42 31 53 20 29 26 43 20.7 31 40 75
Sensitive (Figs. 2 and 5) Sensitive (Figs. 2 and 5) Non-Sensitive (Figs. 2 and 5) Non-sensitive (Figs. 2 and 5) Sensitive (Figs. 3 and 6) Sensitive (Figs. 3 and 6) Non-Sensitive (Figs. 3 and 6) Non-sensitive (Figs. 3 and 6) Sensitive (Figs. 4 and 7) Sensitive (Figs. 4 and 7) Non-Sensitive (Figs. 4 and 7) Non-sensitive (Figs. 4 and 7)
6. Conclusions The sulphuric acid catalyzed hydrolysis of acetic anhydride has been used to prove the existence of parametric sensitivity in a batch mode. Simulations were successfully carried out using published kinetic parameters. The effect of extraneous wall capacitance showed that the reactor exhibited non-parametric sensitivity behavior dynamics experimentally, and it has been illustrated that the presence of extraneous wall capacitance tends to stabilize the reactor by exhibiting insensitivity. Table 4 provides a summary of the results obtained. The transients are in good agreement with the experimental temperature data when the reactor is operating at high coolant water flow rate. The predicted transient temperatures are in satisfactory agreement with the experimental temperature data when the reactor is operating at low coolant water flow rate. References [1] P.K. Shukla, S. Pushpavanam, Parametric sensitivity, runaway, and safety in batch reactors: experiments and models, Ind. Eng. Chem. Res. 33 (1994) 3202– 3208. [2] C.H. Barkelew, Stability of chemical reactors, Chem. Eng. Prog. Symp. Ser. 55 (1959) 37–46. [3] J. Adler, J.W. Enig, The critical thermal explosion theory with reactant consumption, Combust. Flame 8 (1964) 97–103.
[4] M. Morbidelli, A. Varma, Parametric sensitivity and runaway in tubular reactors, AIChE J. 28 (1982) 705–713. [5] M. Morbidelli, A. Varma, On parametric sensitivity and runaway criteria of pseudo homogeneous tubular reactors, Chem. Eng. Sci. 40 (1985) 2165– 2169. [6] M. Morbidelli, A. Varma, A generalised criterion for parametric sensitivity: application to thermal explosion theory, Chem. Eng. Sci. 43 (1988) 91–102. [7] N.N. Semonov, Zur theorie des verbrennungs prozessess, Z. Phys. 48 (1928) 571–582. [8] Y.B. Zeldovich, G.I. Brenblatt, V.B. Librovich, G.M. Makhiviladze, The Mathematical Theory of Combustion and Explosions, Plenum Press, New York, 1985. [9] E. von Westerholt, J.N. Beard, S.S. Melsheimer, Time-optimal startup control algorithm for batch processes, Ind. Eng. Chem. Res. 30 (6) (1991) 1205– 1212. [10] D.R. Lewin, R. Lavie, Designing and implementing trajectories in an exothermic batch chemical reactor, Ind. Eng. Chem. Res. 29 (1) (1990) 89–96. [11] A. Varma, M. Morbidelli, H. Wu, Parametric Sensitivity in Chemical Systems, Cambridge University Press, Cambridge, 1999. [12] J. Adler, J.W. Enig, The critical conditions in thermal explosion theory with reactant consumption, Combust. Flame 8 (1964) 97–103. [13] E.D. Gilles, H. Schuler, Early detection of hazardous states in chemical reactors, German Chem. Eng. 5 (1982) 69–78. [14] P. Hugo, J. Steinbach, F. Stossel, Calculation of the maximum temperature in stirred tank reactors in case of a breakdown of cooling, Chem. Eng. Sci. 43 (8) (1989) 2147–2152. [15] R.M. Chemburkar, M. Morbrdelli, A. Varma, Parametric sensitivity of a CSTR, Chem. Eng. Sci. 41 (6) (1986) 1647–1654. [16] R. Haldar, D.P. Rao, Experimental studies on limit cycle behavior of the sulphuric acid catalysed hydrolysis of acetic anhydride in a CSTR, Chem. Eng. Sci. 46 (1991) 1197–1200.