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Experimental investigation on the kinetics of catalytic recombination of hydrogen with oxygen in air
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Experimental investigation on the kinetics of catalytic recombination of hydrogen with oxygen in air K.C. Sandeep*, Rupsha Bhattacharyya, Chandrashekhar Warghat, Kalyan Bhanja, Sadhana Mohan Heavy Water Division, BARC, Mumbai, 400 085, India
article info
abstract
Article history:
Catalytic recombination of hydrogen with oxygen is one of the most attractive options to
Received 11 April 2014
control the hydrogen concentration in air. The basic pre-requisite for the process design of
Received in revised form
any catalytic reactor is the knowledge of kinetic data. In the present study, the kinetic data
27 August 2014
for the catalytic recombination of hydrogen in presence of 0.5% Pd on alumina catalyst
Accepted 31 August 2014
were generated using a packed bed reactor with complete recycle. The experiments were
Available online 23 September 2014
conducted using low concentration of hydrogen in air at different temperatures and the apparent rate constants were estimated assuming a first order reaction with respect to
Keywords:
hydrogen. The resistances due to internal and external mass transfer were decoupled from
Catalytic recombination
the apparent kinetics and estimated separately. The activation energy and frequency factor
Hydrogen mitigation
were found out using the slope and intercept of the Arrhenius plot. The effect of different
Catalytic combustion
process parameters such as temperature, superficial velocity and the catalyst particle size
Intrinsic kinetics
on the overall reaction rate was also studied. The knowledge of the intrinsic kinetics along
Palladium catalyst
with the mass transfer can be easily extended for the design of catalytic recombination
Recycle
reactors during scale up. Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydrogen and its isotopes have wide applications in the nuclear industry apart from chemical and manufacturing industries [1,2]. Deuterium and tritium will be used as the fuel for International Thermonuclear Experimental Reactor (ITER). Heavy water (D2O) is used as moderator in Pressurized Heavy Water Reactors (PHWR). In most of the nuclear reactors
currently in operation, the coolant used to remove heat from the core is light water or heavy water. Hydrogen is generated in a nuclear reactor core due to reaction of the clad metal (e.g. zirconium) with the coolant water or by the gamma radiolysis of water [3]. The zircaloy clad of the fuel tube in thermal reactors, which contains the fuel pins, is in continuous contact with the coolant water and there is always a chemical interaction of the coolant and clad material. These interactions lead to the production of hydrogen gas and a surface layer of
* Corresponding author. Tel.: þ91 22 25592955. E-mail addresses: [email protected], [email protected] (K.C. Sandeep). http://dx.doi.org/10.1016/j.ijhydene.2014.08.148 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
the metal oxide which adhere to the tube or clad. Such metal water reactions are the primary source for the production of hydrogen inside a nuclear reactor vessel. While the corrosion of the zircaloy with production of hydrogen is a regular phenomenon in the reactor, the problem is of much greater severity in case of a severe accident like coolant failure. The overheating causes fuel and clad temperatures to increase drastically and this in turn leads to enhanced clad-coolant reactions, leading to the production of large amount of hydrogen. The released hydrogen can cause overpressurization of the containment and potentially an explosion. It can also lead to the formation of an explosive mixture inside the containment and a major fire hazard subsequently [4]. The other source of hydrogen inside the reactor is the radiolysis of water due to the intense radiation field. The interaction of gamma radiation with the coolant and moderator causes its decomposition into molecular hydrogen along with free radical and ionic species [5]. Hydrogen is a very light, mobile gas and its flammability limits in air are 4e76% (v/v) under standard temperature and pressure. In addition, it has very low ignition energy (0.018 mJ) when present with air in optimal proportions [6]. All these factors raise the possibility of a hydrogen induced fire hazard in any facility that handles hydrogen. Severe nuclear accidents have been reported due to hydrogen release, the most recent one being in the Fukushima Daiichi power plant in Japan [7]. In the Three Mile Island accident in USA, partial melt down of the core lead to chemical reactions that caused buildup of a large quantity of hydrogen inside the containment, but there was no explosion because of the absence of oxygen inside the core [8]. There are several methods for hydrogen control in case of any accidental release. The concentration of hydrogen in the atmosphere can be reduced through dilution below the flammability limits by using inert gases such as nitrogen or by increasing the containment volume. Hydrogen can be removed from the atmosphere by purposeful flame ignition at low concentrations or by catalytic recombination with oxygen in air. Hydrogen removal by catalytic recombination is considered in the present study as an effective method, as hydrogen can be oxidized at a temperature much lower than that required for thermal oxidation [3]. Moreover, such reactions can be carried out in passive mode as well especially inside the primary containment of nuclear reactors. In catalytic combustion, emission of nitrogen oxides is completely eliminated as against thermal oxidation and there is no risk of preliminary fire [2]. The recombination reaction is represented by Ref. [3] 1 kJ H2 ðgÞ þ O2 ðgÞ / H2 OðgÞ; DHR ¼ 244:5 2 mol
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Experimental study and analysis Procedure of the experiments A packed bed reactor with complete recycle was used to study the performance of the catalyst. The internal diameter and height of the reactor is 15.76 mm and 300 mm respectively. The catalyst used in the present study was 0.5% Pd (w/w) loaded on spherical alumina particles of 3e5 mm diameter. The fixed bed reactor was filled with 30 g of the catalyst. The process flow diagram of the catalyst test facility is shown in Fig. 1. The other main components in the experimental system were a mixing vessel and a recirculation pump. The reactor was provided with an external heater to carry out the reaction at different temperatures. The temperature of the catalytic bed was monitored at three different locations viz. top, middle and bottom. The average temperature for each run was used for different calculations. The recycle flow rate was adjusted using the bypass valve provided across the recirculation pump and the flow rate was monitored using a rotameter. The operating pressure was measured by bourdon type pressure gauge at the inlet of the reactor. The concentration of hydrogen in the mixing vessel was continuously monitored by a thermal conductivity based gas analyzer. The catalytic reactor was provided with two isolation valves and one bypass valve. During the start-up of a typical experiment, the reactor was isolated initially by closing the inlet and outlet valves. The bypass valve of the reactor was kept open to allow the air to circulate in the loop and the flow rate was adjusted to the required value using the bypass valve of the recirculation pump. Hydrogen was added into the mixing vessel from the cylinder with precise flow control and the concentration of hydrogen was continuously monitored using the hydrogen gas analyzer. The maximum hydrogen concentration in air was limited to 1.5% v/v in the present system. This was mainly to keep the concentration well below the lower explosive limit of hydrogen (4% in air). This also served to limit the heat evolved during the recombination reaction and to maintain nearly isothermal condition of the bed. The packed bed reactor was maintained at a particular temperature by using the heater with a temperature controller. Once the hydrogen concentration was stabilized while the reactor was in isolation, the reactor was taken into the flow path by opening the inlet and outlet valves. The bypass valve to the reactor was
(1)
In order to design a suitable reactor for the catalytic recombination of hydrogen and oxygen to form water vapour, the thorough understanding of the recombination kinetics is essential. Therefore, a bench scale experimental system consists of a packed bed reactor with complete recycle has been designed and installed. A large set of experiments have been carried out at different operating conditions. The generation of experimental data and the analysis of the results are presented in detail in the subsequent sections.
Fig. 1 e Schematic diagram of the catalyst test facility.
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
also closed simultaneously. Hydrogen was allowed to react with oxygen in air in presence of catalyst in the reactor, which was maintained at a specified temperature. The outlet stream from the reactor was completely mixed with the gas in the mixing vessel and recycled back to the reactor through the recirculation pump. The reduction in hydrogen concentration in the mixing vessel was continuously monitored and recorded with time. The data thus generated were used to estimate the reaction rate constant. The experiments were carried out at different temperature varying from 35 C to 85 C and a typical plot of dimensionless hydrogen concentration in the mixing vessel versus time, at different temperatures is given in Fig. 2.
Determination of the apparent reaction kinetics from experimental data The existing literature indicates that the order of the recombination reaction could be between 0 and 1 with respect to O2 and between 1 and þ2 with respect to H2 [9]. In the present study, a first order rate expression with respect to hydrogen was assumed due to the presence of excess oxygen during the experiment. The entire reaction loop is modelled as a combination of plug flow packed bed reactor (PBR) and a complete mixing vessel. In the mixing vessel, the gas is assumed to be completely mixed at ambient temperature without any reaction. The volume of interconnecting pipe lines was negligibly small in comparison to the volume of mixing vessel. The simplified schematic of the loop, for the purpose of data analysis is shown in Fig. 3. The hydrogen concentration at the inlet of the packed bed reactor is CA,1 and the concentration at the outlet is CA,2 which vary with time. In every pass through the reactor, it is assumed that a constant fraction of the feed hydrogen is converted to water vapour which is further diluted in the rest of the gas in the mixing vessel [10]. The packed bed reactor is assumed to be with plug flow pattern and no axial mixing. The pressure drop in the bed is negligibly small and the bed is at nearly isothermal condition during each run. Under pseudo-steady state conditions in the PBR,
Fig. 3 e Simplified schematic diagram of experimental loop for kinetic data analysis.
the mole balance for hydrogen (i.e. species A) with apparent order of one with respect to hydrogen is given as [11]. VPBR CA;2 ¼ CA;1 exp kapp QPBR
(2)
The volumetric flow rate (QPBR) through the reactor is assumed to be constant as the change in number of moles due to reaction has negligible effect on the total gas flow rate. The concentration inside the mixing vessel is equal to the concentration at the outlet assuming that the mixing is complete and instantaneous. The dynamic mole balance of hydrogen in the mixing vessel is given by. dðVM CA;1 Þ ¼ QM CA;2 QM CA;1 dt
(3)
The initial condition is given as. at t ¼ 0; CA;1 ¼ CA;initial
(4)
Substituting the expression for CA,2 from (2) in (3) gives.
VM
dCA;1 VPBR ¼ QM CA;1 exp kapp CA;1 dt QPBR
(5)
After integration with initial conditions, the following expression is obtained. ln
CA;initial
¼
QM VPBR exp kapp 1 t VM QPBR
(6)
ln
CA;1
CA;1 CA;initial
versus t at different temperatures is plotted and the
slope is estimated. The estimated slope is equal to QVMM exp kapp VQPBR 1 and the apparent rate constant kapp PBR
Fig. 2 e Dimensionless hydrogen concentration in the mixing vessel versus time.
is determined from the slope of the best fit line. The same procedure is repeated at different temperatures and apparent rate constant is estimated at the respective temperatures. A plot of ln(kapp) versus 1/T is made and shown in Fig. 4. Based on the nature of the plot, it is observed that it follows the Arrhenius expression
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
€ ssling The mass transfer coefficient is estimated using Fro correlation as [12] Sh ¼ 2 þ 0:6Sc1=3 Re1=2 p
(13)
The value of external surface area per unit volume of the reactor is calculated as. a¼
6ð1 εÞ dp
(14)
where dp is the particle size and ε is the void fraction of the bed. The correlation 2 dp dp þ 0:412 ε ¼ 0:4 þ 0:05 D D
for
dp < 0:5 D
(15)
is used to estimate the void fraction ε in the bed [13]. The internal effectiveness factor (h) is given by, Fig. 4 e Arrhenius plot for the determination of apparent kinetics: ln(kapp) versus 1/T.
h¼
3 ½f cothðfÞ 1 f f
(16)
for first order reactions on spherical porous catalyst, where f2 is Thiele modulus given by [12] Eapp kapp ¼ kapp;o exp RT
(7)
The slope and intercept is estimated using the linear least square method. The apparent activation energy and the apparent frequency factor are found to be 73.77 kJ mol1 and 1.96 1011 s1 respectively.
(8)
The overall rate of reaction based on unit volume of reactor is given as. (9)
where h is the internal effectiveness factor accounting for the internal diffusion limitations and kRS is the intrinsic first order reaction rate constant. At steady state, the rate of hydrogen diffusing from the bulk fluid to the catalyst surface will be equal to the rate of hydrogen reacted in the catalyst. Therefore; kc aðCA CAs Þ ¼ hkRS CAS
(10)
Using the expression for CAS from (10), the rate of reaction in terms of bulk concentration can be expressed as. rA ¼ kapp CA
(11)
where kapp is the apparent rate constant, given by 1 1 1 ¼ þ kapp hkRS kc a
sffiffiffiffiffiffiffiffiffi kRS De
(17)
The effective diffusivity De is evaluated using the equation 1 1 1 ¼ þ De DAB DK
DK ¼ 97rp
Hydrogen is assumed to be diffusing through a hypothetical film near the spherical catalyst from the bulk fluid to the external surface of the catalyst. Hydrogen is then diffusing through the internal pores where diffusion and reaction take place on the catalyst surface. The molar flux can be expressed in terms of external film mass transfer coefficient kc as
rA ¼ hkRS CAS
dp 2
(18)
and
Evaluation of mass transfer coefficient, internal effectiveness factor and determination of intrinsic kinetics
NA $a ¼ kc aðCA CAs Þ
f¼
(12)
rffiffiffiffiffi T M
(19)
where DK is the Knudsen diffusivity inside the catalyst [14]. Using the experimental data, the values of kapp was estimated at the respective temperatures. The value of kc was calculated using the correlation given by (13) at the specific temperature and superficial velocity. Thus equation (12) was used to calculate the values of hkRS. Using the value of hkRS and equation (16), the effectiveness factor h and intrinsic rate constant kRS were estimated iteratively at the respective temperatures. The Arrhenius plot of ln(kRS) versus 1/T was plotted and is shown in Fig. 5. The activation energy (Ea) and frequency factor (kRS,o) are estimated to be 76.42 kJ mol1and 5.37 1011 s1respectively. The literature on recombination kinetics varies from purely empirical relations to kinetics with multi-step mechanisms. The empirical relations, which give the reaction rates, are typically valid for a given system and for a particular geometry [15]. Kwang et al. has reported the apparent kinetics using 0.5% Pd on alumina catalyst in the temperature range of 50 Ce150 C [16]. The apparent activation energy and frequency factor is reported as 13.25 kJ mol1 and 36.67 s1 respectively, which are less than the intrinsic parameters obtained in the present case. The present experimental study is in the operating range of 35e85 C and at higher temperature above 110 C, very strong transport limitations are possible which are illustrated in Figs. 6e8 in the subsequent sections. This can lead to a reduced slope and intercept of Arrhenius plot and finally a lower apparent activation energy
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i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
Fig. 5 e Arrhenius plot for the determination of intrinsic kinetics: ln(kRS) versus 1/T.
and frequency factor. Deutschmann et al. has reported a detailed multi-step mechanism, where the activation energy of different individual steps are varying from 11.5 kJ mol1 to 230 kJ mol1 and the activation energy in the present study lies within the above range [17].
Results and discussions
Fig. 7 e Effect of superficial velocity on apparent kinetics: ln(kapp) versus 1/T at different superficial velocities.
effectiveness factor h varies from about 0.99 to 0.85. The value of h decreases with rise in temperature for a given particle size. This implies that pore diffusion resistances were not significantly affecting the overall kinetics of the reaction under the conditions used in the present study. It is evident from the calculated values of h that internal diffusion becomes slightly more significant at higher temperatures as the controlling regime for the reaction tends to shift from kinetic control to mass transfer control.
The intrinsic rate equation is given by. 9192 CA ðrA Þ ¼ 5:37 1011 exp T
(20)
It was observed from the calculations that the values of hkRS and kRS are quite close for the process conditions employed in the present study. The value of internal
Fig. 6 e Effect of temperature on apparent kinetics, ln(kapp) versus 1/T in the extended temperature range.
Sensitivity analysis on the apparent reaction kinetics Using Arrhenius expression for intrinsic kinetic rate constants € ssling correlation for mass transfer coefficients, the and Fro apparent rate constants were estimated at different
Fig. 8 e Effect of particle size on apparent kinetics: ln(kapp) versus 1/T at different particle sizes.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
superficial velocities and particle sizes for temperature varying from 35 C to 300 C using the equations from 12 to 20. It has been presumed that the mechanism of reaction does not change in the above temperature range and the same intrinsic rate expression holds good in the extended temperature range also.
Effect of temperature The apparent rate constants were estimated at the superficial velocity of 0.40 m s1corresponding to the average superficial velocity used in the experiment, in the temperature range of 35 Ce300 C. The plot of ln(kapp) versus 1/T is shown in Fig. 6. It is observed that upto a temperature of about 110 C, the overall rate constant rises exponentially according to Arrhenius equation. Nearly above 110 C, the rate of increase of the apparent rate constant with temperature decreases. In this region, the internal diffusion of hydrogen inside catalyst particle may be limiting the overall rate of reaction. Above temperature of 280 C, the plot is tending towards a plateau, indicating the predominance of external mass transfer limitations on the overall kinetics at higher temperatures. The external mass transfer coefficient increases nearly linearly with temperature according to the correlations used. In this region, the intrinsic surface reaction is so fast that the rate determining step becomes the external mass transfer of hydrogen from bulk fluid to the catalyst surface.
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Effect of particle size The particle size is varied from 3 to 5 mm and the effect of particle size on apparent rate constant at a constant superficial velocity of 0.40 m s1 is studied. The variation in the bulk density of the catalyst with particle size is neglected. The results are shown in Fig. 8. It is observed that the apparent rate constant for different particle sizes is nearly independent of particle size up to a temperature of 85 C, as in this region the overall reaction rate is controlled by the surface reaction. Above 85 C, the value of apparent kinetic rate constant is higher for a lower particle size. The external surface area per unit volume and the external mass transfer coefficient is increased when the particle size is decreased. The value of Thiele modulus is less for a lower particle size. In this case, the internal effectiveness factor will be more close to one and the concentration inside the pores will be approaching the concentration at the external surface. Thus, any reduction in the particle size can reduce the limitations due to both the internal and external mass transfer and can improve the overall reaction kinetics. When the reactor is operating in high temperature regions, this analysis can be used to select the optimum particle size combined with pressure drop considerations. While operating in the lower temperature regime, the pressure drop may be taken as the governing factor for selection of particle size as the apparent kinetics are not affected by particle size under these conditions.
Effect of superficial velocity
Conclusion The apparent rate constant is calculated at three different superficial velocities such as 0.40, 0.25 and 0.10 m s1, for a temperature range of 35e300 C and the results are shown in Fig. 7. It is observed that the temperature up to which the apparent rate constant rises monotonically is about 110 C for all the different velocities considered in the present study. The value of apparent rate constant is independent of the flow rate from 35 to 110 C. This indicates that in the low temperature regime, the apparent kinetics is dominated by the intrinsic surface reaction kinetics and external mass transfer has only negligibly small influence on it. Above 110 C, the apparent rate constant rises at a lower rate and the value of the apparent rate constant becomes weakly dependent on temperature, for all the velocities considered. The apparent rate constant is lower for a lower velocity in the high temperature region, thus clearly demonstrating the fact that the overall rate is controlled mainly by the external mass transfer limitations. Thus, if the recombiner is to be operated at low superficial velocities, as in the case of passive recombiners, there is no significant additional benefit with respect to improved kinetics to be obtained by operation at temperatures above 110 C. When the operating temperature is less than 110 C, the apparent rate constant is not significantly increased by increasing the superficial velocity. The operation of the recombiner at a higher superficial velocity may not be beneficial in this region, considering the high pumping cost due to the high pressure drop. The above results can be used to determine the optimal temperature of operation for different velocities and a given particle size.
The kinetics of catalytic recombination of hydrogen in presence of 0.5% Pd on Alumina catalyst was studied in a fixed bed reactor operated with complete recycle. In order to extract the overall kinetics of the reaction, the experimental loop was modelled as a combination of plug flow packed bed reactor operating in pseudo-steady state and a complete mixing vessel under transient conditions. The apparent rate constant was estimated based on the least square method using the experimental data. The intrinsic rate constant was calculated by decoupling the effect of external and internal mass transfer. The intrinsic rate constants at different temperature were found out based on the experimental data generated at the respective temperatures. Using Arrhenius plot, the true activation energy and frequency factor were also estimated. The effects of different process variables such as temperature, superficial velocity and particle size on the apparent rate constant were studied. In the lower temperature region, the overall reaction was controlled by surface reaction and in the higher temperature regions, the overall kinetics was controlled by external mass transfer. A lower particle size and a higher superficial velocity were found to be beneficial while operating at higher temperature, but effect of pressure drop should also be considered. Variations in superficial velocity and particle size do not significantly affect the overall reaction kinetics in the low temperature region. Once the intrinsic kinetics data is available, the apparent kinetics can be estimated for different operating conditions (e.g. temperature, superficial gas velocity, catalyst particle size) during the reactor design for scale up.
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Acknowledgement The authors wish to acknowledge the support received from Shri. S A Prabhu and Shri. D Y Gaikwad, HWD, BARC throughout the experiments.
Nomenclature a CA CA,initial CAs CA,1 CA,2 D DAB De DK dp Ea Eapp kc kapp kapp,o kRS kRS,o M QM QPBR rA rp R Rep Sc Sh t T
external surface area of the catalyst pellet per unit reactor volume, m2 m3 concentration of hydrogen gas (or species A) in the bulk gas stream, mol m3 initial concentration of hydrogen gas in the mixing vessel, mol m3 concentration of hydrogen gas on the catalyst pellet surface, mol m3 inlet concentration of hydrogen to PBR, mol m3 outlet concentration of hydrogen from PBR, mol m3 reactor inner diameter, m diffusivity of species A (hydrogen) through B (air), m2 s1 effective diffusivity of hydrogen through the porous catalyst, m2 s1 Knudsen diffusivity of hydrogen inside catalyst, m2 s1 diameter of catalyst, m true activation energy of the recombination reaction, J mol1 apparent activation energy of the recombination reaction, J mol1 mass transfer coefficient in the fixed bed reactor, m s1 apparent reaction rate constant, s1 frequency factor for apparent reaction rate constant, s1 intrinsic reaction rate constant, s1 frequency factor for intrinsic reaction rate constant, s1 molecular weight of gas diffusing inside the porous catalyst, g mol1 volumetric flow rate of gas through the mixing vessel, m3 s1 volumetric flow rate of gas through the reactor, m3 s1 rate of reaction of species A, mol m3 s1 pore radius in catalyst, m universal gas constant, 8.314 J mol1 K1 particle Reynolds number, dimensionless Schmidt number, dimensionless Sherwood number, dimensionless time, s absolute temperature, K
u VM VPBR ε h f2
superficial velocity of gas through the reactor, m s1 volume of mixing vessel, m3 volume of the packed bed reactor, m3 void fraction in the packed bed reactor, dimensionless internal effectiveness factor, dimensionless Thiele modulus, dimensionless
references
[1] Ramachandran R. An overview of industrial uses of hydrogen. Int J Hydrogen Energy 1998;23:593e8. [2] Kaske G, Plenard F. High-purity hydrogen distribution network for industrial use in Western Europe. Int J Hydrogen Energy 1985;10:479e82. [3] Prabhudharwadkar DM, Aghalayam PA, Iyer KN. Simulation of hydrogen mitigation in catalytic recombiner: part-I: surface chemistry modelling. Nucl Eng Des 2011;241:1746e57. [4] Karwat H. Ignitors to mitigate the risk of hydrogen explosions e a critical review. Nucl Eng Des 1990;118:267e71. [5] Kabakchi SA, Arkhipov OP, Lukashenko ML. Specific features of the radiolysis of water and aqueous solutions of H2 and O2 by mixed n,g-radiation with a high portion of the neutron component. High Energy Chem 2013;47:147e51. [6] Crowl DA, Jo Y-D. The hazards and risks of hydrogen. J Loss Prev Process Ind 2007;20:158e64. [7] Michael Weightman, Jaymet Philippe LJE. IAEA International fact finding expert mission of the Fukushima Dai-Ichi NPP accident following great east Japan earthquake and tsunami. 2011. Vienna. [8] Zhong W. Mitigation of hydrogen hazards in water cooled power reactors, IAEA-TECDOC-1196. 2001. Vienna. [9] Homsy RV, Glatron CA. Review of literature on catalytic recombination of hydrogeneoxygen. 1968. Oak Ridge. [10] Sherwood AE, Monahan BG, McWilliams RA, Uribe FS, Griffith CM. Catalytic oxidation of tritium in air at ambient temperature. 1979. California. [11] Levenspiel Octave. Chemical reaction engineering. 3rd ed. Wiley India Pvt. Ltd; 2006. [12] Scott Fogler H. Elements of chemical reaction engineering. 4th ed. Prentice Hall of India Pvt Ltd.; 2006. [13] Yang Wen-Ching. Handbook of fluidization and fluid-particle systems. 1st ed. Marcel Dekker Inc; 2003. [14] Satterfield Charles N. Heterogeneous catalysis in industrial practice. 2nd ed. McGraw-Hill Book Co; 1993. [15] Bachellerie E, Arnould F, Auglaire M, Boeck B De. Generic approach for designing and implementing a passive autocatalytic recombiner PAR-system in nuclear power plant containments. Nucl Eng Des 2003;221:151e65. [16] Kwang-Rag K, Seung-Woo P, Heui-Joo C, Hongsuk C. Catalytic recombination of hydrogen and oxygen in air stream. J Ind Eng Chem 2001;7:116e20. [17] Deutschmann O, Schmidt R, Behrendt F, Warnatz J. Numerical modeling of catalytic ignition. In: Twenty Sixth Symp Combust Combust Inst; 1996. p. 1747e54.
Available online at www.sciencedirect.com
ScienceDirect journal homepage: www.elsevier.com/locate/he
Experimental investigation on the kinetics of catalytic recombination of hydrogen with oxygen in air K.C. Sandeep*, Rupsha Bhattacharyya, Chandrashekhar Warghat, Kalyan Bhanja, Sadhana Mohan Heavy Water Division, BARC, Mumbai, 400 085, India
article info
abstract
Article history:
Catalytic recombination of hydrogen with oxygen is one of the most attractive options to
Received 11 April 2014
control the hydrogen concentration in air. The basic pre-requisite for the process design of
Received in revised form
any catalytic reactor is the knowledge of kinetic data. In the present study, the kinetic data
27 August 2014
for the catalytic recombination of hydrogen in presence of 0.5% Pd on alumina catalyst
Accepted 31 August 2014
were generated using a packed bed reactor with complete recycle. The experiments were
Available online 23 September 2014
conducted using low concentration of hydrogen in air at different temperatures and the apparent rate constants were estimated assuming a first order reaction with respect to
Keywords:
hydrogen. The resistances due to internal and external mass transfer were decoupled from
Catalytic recombination
the apparent kinetics and estimated separately. The activation energy and frequency factor
Hydrogen mitigation
were found out using the slope and intercept of the Arrhenius plot. The effect of different
Catalytic combustion
process parameters such as temperature, superficial velocity and the catalyst particle size
Intrinsic kinetics
on the overall reaction rate was also studied. The knowledge of the intrinsic kinetics along
Palladium catalyst
with the mass transfer can be easily extended for the design of catalytic recombination
Recycle
reactors during scale up. Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
Introduction Hydrogen and its isotopes have wide applications in the nuclear industry apart from chemical and manufacturing industries [1,2]. Deuterium and tritium will be used as the fuel for International Thermonuclear Experimental Reactor (ITER). Heavy water (D2O) is used as moderator in Pressurized Heavy Water Reactors (PHWR). In most of the nuclear reactors
currently in operation, the coolant used to remove heat from the core is light water or heavy water. Hydrogen is generated in a nuclear reactor core due to reaction of the clad metal (e.g. zirconium) with the coolant water or by the gamma radiolysis of water [3]. The zircaloy clad of the fuel tube in thermal reactors, which contains the fuel pins, is in continuous contact with the coolant water and there is always a chemical interaction of the coolant and clad material. These interactions lead to the production of hydrogen gas and a surface layer of
* Corresponding author. Tel.: þ91 22 25592955. E-mail addresses: [email protected], [email protected] (K.C. Sandeep). http://dx.doi.org/10.1016/j.ijhydene.2014.08.148 0360-3199/Copyright © 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
the metal oxide which adhere to the tube or clad. Such metal water reactions are the primary source for the production of hydrogen inside a nuclear reactor vessel. While the corrosion of the zircaloy with production of hydrogen is a regular phenomenon in the reactor, the problem is of much greater severity in case of a severe accident like coolant failure. The overheating causes fuel and clad temperatures to increase drastically and this in turn leads to enhanced clad-coolant reactions, leading to the production of large amount of hydrogen. The released hydrogen can cause overpressurization of the containment and potentially an explosion. It can also lead to the formation of an explosive mixture inside the containment and a major fire hazard subsequently [4]. The other source of hydrogen inside the reactor is the radiolysis of water due to the intense radiation field. The interaction of gamma radiation with the coolant and moderator causes its decomposition into molecular hydrogen along with free radical and ionic species [5]. Hydrogen is a very light, mobile gas and its flammability limits in air are 4e76% (v/v) under standard temperature and pressure. In addition, it has very low ignition energy (0.018 mJ) when present with air in optimal proportions [6]. All these factors raise the possibility of a hydrogen induced fire hazard in any facility that handles hydrogen. Severe nuclear accidents have been reported due to hydrogen release, the most recent one being in the Fukushima Daiichi power plant in Japan [7]. In the Three Mile Island accident in USA, partial melt down of the core lead to chemical reactions that caused buildup of a large quantity of hydrogen inside the containment, but there was no explosion because of the absence of oxygen inside the core [8]. There are several methods for hydrogen control in case of any accidental release. The concentration of hydrogen in the atmosphere can be reduced through dilution below the flammability limits by using inert gases such as nitrogen or by increasing the containment volume. Hydrogen can be removed from the atmosphere by purposeful flame ignition at low concentrations or by catalytic recombination with oxygen in air. Hydrogen removal by catalytic recombination is considered in the present study as an effective method, as hydrogen can be oxidized at a temperature much lower than that required for thermal oxidation [3]. Moreover, such reactions can be carried out in passive mode as well especially inside the primary containment of nuclear reactors. In catalytic combustion, emission of nitrogen oxides is completely eliminated as against thermal oxidation and there is no risk of preliminary fire [2]. The recombination reaction is represented by Ref. [3] 1 kJ H2 ðgÞ þ O2 ðgÞ / H2 OðgÞ; DHR ¼ 244:5 2 mol
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Experimental study and analysis Procedure of the experiments A packed bed reactor with complete recycle was used to study the performance of the catalyst. The internal diameter and height of the reactor is 15.76 mm and 300 mm respectively. The catalyst used in the present study was 0.5% Pd (w/w) loaded on spherical alumina particles of 3e5 mm diameter. The fixed bed reactor was filled with 30 g of the catalyst. The process flow diagram of the catalyst test facility is shown in Fig. 1. The other main components in the experimental system were a mixing vessel and a recirculation pump. The reactor was provided with an external heater to carry out the reaction at different temperatures. The temperature of the catalytic bed was monitored at three different locations viz. top, middle and bottom. The average temperature for each run was used for different calculations. The recycle flow rate was adjusted using the bypass valve provided across the recirculation pump and the flow rate was monitored using a rotameter. The operating pressure was measured by bourdon type pressure gauge at the inlet of the reactor. The concentration of hydrogen in the mixing vessel was continuously monitored by a thermal conductivity based gas analyzer. The catalytic reactor was provided with two isolation valves and one bypass valve. During the start-up of a typical experiment, the reactor was isolated initially by closing the inlet and outlet valves. The bypass valve of the reactor was kept open to allow the air to circulate in the loop and the flow rate was adjusted to the required value using the bypass valve of the recirculation pump. Hydrogen was added into the mixing vessel from the cylinder with precise flow control and the concentration of hydrogen was continuously monitored using the hydrogen gas analyzer. The maximum hydrogen concentration in air was limited to 1.5% v/v in the present system. This was mainly to keep the concentration well below the lower explosive limit of hydrogen (4% in air). This also served to limit the heat evolved during the recombination reaction and to maintain nearly isothermal condition of the bed. The packed bed reactor was maintained at a particular temperature by using the heater with a temperature controller. Once the hydrogen concentration was stabilized while the reactor was in isolation, the reactor was taken into the flow path by opening the inlet and outlet valves. The bypass valve to the reactor was
(1)
In order to design a suitable reactor for the catalytic recombination of hydrogen and oxygen to form water vapour, the thorough understanding of the recombination kinetics is essential. Therefore, a bench scale experimental system consists of a packed bed reactor with complete recycle has been designed and installed. A large set of experiments have been carried out at different operating conditions. The generation of experimental data and the analysis of the results are presented in detail in the subsequent sections.
Fig. 1 e Schematic diagram of the catalyst test facility.
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also closed simultaneously. Hydrogen was allowed to react with oxygen in air in presence of catalyst in the reactor, which was maintained at a specified temperature. The outlet stream from the reactor was completely mixed with the gas in the mixing vessel and recycled back to the reactor through the recirculation pump. The reduction in hydrogen concentration in the mixing vessel was continuously monitored and recorded with time. The data thus generated were used to estimate the reaction rate constant. The experiments were carried out at different temperature varying from 35 C to 85 C and a typical plot of dimensionless hydrogen concentration in the mixing vessel versus time, at different temperatures is given in Fig. 2.
Determination of the apparent reaction kinetics from experimental data The existing literature indicates that the order of the recombination reaction could be between 0 and 1 with respect to O2 and between 1 and þ2 with respect to H2 [9]. In the present study, a first order rate expression with respect to hydrogen was assumed due to the presence of excess oxygen during the experiment. The entire reaction loop is modelled as a combination of plug flow packed bed reactor (PBR) and a complete mixing vessel. In the mixing vessel, the gas is assumed to be completely mixed at ambient temperature without any reaction. The volume of interconnecting pipe lines was negligibly small in comparison to the volume of mixing vessel. The simplified schematic of the loop, for the purpose of data analysis is shown in Fig. 3. The hydrogen concentration at the inlet of the packed bed reactor is CA,1 and the concentration at the outlet is CA,2 which vary with time. In every pass through the reactor, it is assumed that a constant fraction of the feed hydrogen is converted to water vapour which is further diluted in the rest of the gas in the mixing vessel [10]. The packed bed reactor is assumed to be with plug flow pattern and no axial mixing. The pressure drop in the bed is negligibly small and the bed is at nearly isothermal condition during each run. Under pseudo-steady state conditions in the PBR,
Fig. 3 e Simplified schematic diagram of experimental loop for kinetic data analysis.
the mole balance for hydrogen (i.e. species A) with apparent order of one with respect to hydrogen is given as [11]. VPBR CA;2 ¼ CA;1 exp kapp QPBR
(2)
The volumetric flow rate (QPBR) through the reactor is assumed to be constant as the change in number of moles due to reaction has negligible effect on the total gas flow rate. The concentration inside the mixing vessel is equal to the concentration at the outlet assuming that the mixing is complete and instantaneous. The dynamic mole balance of hydrogen in the mixing vessel is given by. dðVM CA;1 Þ ¼ QM CA;2 QM CA;1 dt
(3)
The initial condition is given as. at t ¼ 0; CA;1 ¼ CA;initial
(4)
Substituting the expression for CA,2 from (2) in (3) gives.
VM
dCA;1 VPBR ¼ QM CA;1 exp kapp CA;1 dt QPBR
(5)
After integration with initial conditions, the following expression is obtained. ln
CA;initial
¼
QM VPBR exp kapp 1 t VM QPBR
(6)
ln
CA;1
CA;1 CA;initial
versus t at different temperatures is plotted and the
slope is estimated. The estimated slope is equal to QVMM exp kapp VQPBR 1 and the apparent rate constant kapp PBR
Fig. 2 e Dimensionless hydrogen concentration in the mixing vessel versus time.
is determined from the slope of the best fit line. The same procedure is repeated at different temperatures and apparent rate constant is estimated at the respective temperatures. A plot of ln(kapp) versus 1/T is made and shown in Fig. 4. Based on the nature of the plot, it is observed that it follows the Arrhenius expression
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€ ssling The mass transfer coefficient is estimated using Fro correlation as [12] Sh ¼ 2 þ 0:6Sc1=3 Re1=2 p
(13)
The value of external surface area per unit volume of the reactor is calculated as. a¼
6ð1 εÞ dp
(14)
where dp is the particle size and ε is the void fraction of the bed. The correlation 2 dp dp þ 0:412 ε ¼ 0:4 þ 0:05 D D
for
dp < 0:5 D
(15)
is used to estimate the void fraction ε in the bed [13]. The internal effectiveness factor (h) is given by, Fig. 4 e Arrhenius plot for the determination of apparent kinetics: ln(kapp) versus 1/T.
h¼
3 ½f cothðfÞ 1 f f
(16)
for first order reactions on spherical porous catalyst, where f2 is Thiele modulus given by [12] Eapp kapp ¼ kapp;o exp RT
(7)
The slope and intercept is estimated using the linear least square method. The apparent activation energy and the apparent frequency factor are found to be 73.77 kJ mol1 and 1.96 1011 s1 respectively.
(8)
The overall rate of reaction based on unit volume of reactor is given as. (9)
where h is the internal effectiveness factor accounting for the internal diffusion limitations and kRS is the intrinsic first order reaction rate constant. At steady state, the rate of hydrogen diffusing from the bulk fluid to the catalyst surface will be equal to the rate of hydrogen reacted in the catalyst. Therefore; kc aðCA CAs Þ ¼ hkRS CAS
(10)
Using the expression for CAS from (10), the rate of reaction in terms of bulk concentration can be expressed as. rA ¼ kapp CA
(11)
where kapp is the apparent rate constant, given by 1 1 1 ¼ þ kapp hkRS kc a
sffiffiffiffiffiffiffiffiffi kRS De
(17)
The effective diffusivity De is evaluated using the equation 1 1 1 ¼ þ De DAB DK
DK ¼ 97rp
Hydrogen is assumed to be diffusing through a hypothetical film near the spherical catalyst from the bulk fluid to the external surface of the catalyst. Hydrogen is then diffusing through the internal pores where diffusion and reaction take place on the catalyst surface. The molar flux can be expressed in terms of external film mass transfer coefficient kc as
rA ¼ hkRS CAS
dp 2
(18)
and
Evaluation of mass transfer coefficient, internal effectiveness factor and determination of intrinsic kinetics
NA $a ¼ kc aðCA CAs Þ
f¼
(12)
rffiffiffiffiffi T M
(19)
where DK is the Knudsen diffusivity inside the catalyst [14]. Using the experimental data, the values of kapp was estimated at the respective temperatures. The value of kc was calculated using the correlation given by (13) at the specific temperature and superficial velocity. Thus equation (12) was used to calculate the values of hkRS. Using the value of hkRS and equation (16), the effectiveness factor h and intrinsic rate constant kRS were estimated iteratively at the respective temperatures. The Arrhenius plot of ln(kRS) versus 1/T was plotted and is shown in Fig. 5. The activation energy (Ea) and frequency factor (kRS,o) are estimated to be 76.42 kJ mol1and 5.37 1011 s1respectively. The literature on recombination kinetics varies from purely empirical relations to kinetics with multi-step mechanisms. The empirical relations, which give the reaction rates, are typically valid for a given system and for a particular geometry [15]. Kwang et al. has reported the apparent kinetics using 0.5% Pd on alumina catalyst in the temperature range of 50 Ce150 C [16]. The apparent activation energy and frequency factor is reported as 13.25 kJ mol1 and 36.67 s1 respectively, which are less than the intrinsic parameters obtained in the present case. The present experimental study is in the operating range of 35e85 C and at higher temperature above 110 C, very strong transport limitations are possible which are illustrated in Figs. 6e8 in the subsequent sections. This can lead to a reduced slope and intercept of Arrhenius plot and finally a lower apparent activation energy
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Fig. 5 e Arrhenius plot for the determination of intrinsic kinetics: ln(kRS) versus 1/T.
and frequency factor. Deutschmann et al. has reported a detailed multi-step mechanism, where the activation energy of different individual steps are varying from 11.5 kJ mol1 to 230 kJ mol1 and the activation energy in the present study lies within the above range [17].
Results and discussions
Fig. 7 e Effect of superficial velocity on apparent kinetics: ln(kapp) versus 1/T at different superficial velocities.
effectiveness factor h varies from about 0.99 to 0.85. The value of h decreases with rise in temperature for a given particle size. This implies that pore diffusion resistances were not significantly affecting the overall kinetics of the reaction under the conditions used in the present study. It is evident from the calculated values of h that internal diffusion becomes slightly more significant at higher temperatures as the controlling regime for the reaction tends to shift from kinetic control to mass transfer control.
The intrinsic rate equation is given by. 9192 CA ðrA Þ ¼ 5:37 1011 exp T
(20)
It was observed from the calculations that the values of hkRS and kRS are quite close for the process conditions employed in the present study. The value of internal
Fig. 6 e Effect of temperature on apparent kinetics, ln(kapp) versus 1/T in the extended temperature range.
Sensitivity analysis on the apparent reaction kinetics Using Arrhenius expression for intrinsic kinetic rate constants € ssling correlation for mass transfer coefficients, the and Fro apparent rate constants were estimated at different
Fig. 8 e Effect of particle size on apparent kinetics: ln(kapp) versus 1/T at different particle sizes.
i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 9 ( 2 0 1 4 ) 1 7 9 0 6 e1 7 9 1 2
superficial velocities and particle sizes for temperature varying from 35 C to 300 C using the equations from 12 to 20. It has been presumed that the mechanism of reaction does not change in the above temperature range and the same intrinsic rate expression holds good in the extended temperature range also.
Effect of temperature The apparent rate constants were estimated at the superficial velocity of 0.40 m s1corresponding to the average superficial velocity used in the experiment, in the temperature range of 35 Ce300 C. The plot of ln(kapp) versus 1/T is shown in Fig. 6. It is observed that upto a temperature of about 110 C, the overall rate constant rises exponentially according to Arrhenius equation. Nearly above 110 C, the rate of increase of the apparent rate constant with temperature decreases. In this region, the internal diffusion of hydrogen inside catalyst particle may be limiting the overall rate of reaction. Above temperature of 280 C, the plot is tending towards a plateau, indicating the predominance of external mass transfer limitations on the overall kinetics at higher temperatures. The external mass transfer coefficient increases nearly linearly with temperature according to the correlations used. In this region, the intrinsic surface reaction is so fast that the rate determining step becomes the external mass transfer of hydrogen from bulk fluid to the catalyst surface.
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Effect of particle size The particle size is varied from 3 to 5 mm and the effect of particle size on apparent rate constant at a constant superficial velocity of 0.40 m s1 is studied. The variation in the bulk density of the catalyst with particle size is neglected. The results are shown in Fig. 8. It is observed that the apparent rate constant for different particle sizes is nearly independent of particle size up to a temperature of 85 C, as in this region the overall reaction rate is controlled by the surface reaction. Above 85 C, the value of apparent kinetic rate constant is higher for a lower particle size. The external surface area per unit volume and the external mass transfer coefficient is increased when the particle size is decreased. The value of Thiele modulus is less for a lower particle size. In this case, the internal effectiveness factor will be more close to one and the concentration inside the pores will be approaching the concentration at the external surface. Thus, any reduction in the particle size can reduce the limitations due to both the internal and external mass transfer and can improve the overall reaction kinetics. When the reactor is operating in high temperature regions, this analysis can be used to select the optimum particle size combined with pressure drop considerations. While operating in the lower temperature regime, the pressure drop may be taken as the governing factor for selection of particle size as the apparent kinetics are not affected by particle size under these conditions.
Effect of superficial velocity
Conclusion The apparent rate constant is calculated at three different superficial velocities such as 0.40, 0.25 and 0.10 m s1, for a temperature range of 35e300 C and the results are shown in Fig. 7. It is observed that the temperature up to which the apparent rate constant rises monotonically is about 110 C for all the different velocities considered in the present study. The value of apparent rate constant is independent of the flow rate from 35 to 110 C. This indicates that in the low temperature regime, the apparent kinetics is dominated by the intrinsic surface reaction kinetics and external mass transfer has only negligibly small influence on it. Above 110 C, the apparent rate constant rises at a lower rate and the value of the apparent rate constant becomes weakly dependent on temperature, for all the velocities considered. The apparent rate constant is lower for a lower velocity in the high temperature region, thus clearly demonstrating the fact that the overall rate is controlled mainly by the external mass transfer limitations. Thus, if the recombiner is to be operated at low superficial velocities, as in the case of passive recombiners, there is no significant additional benefit with respect to improved kinetics to be obtained by operation at temperatures above 110 C. When the operating temperature is less than 110 C, the apparent rate constant is not significantly increased by increasing the superficial velocity. The operation of the recombiner at a higher superficial velocity may not be beneficial in this region, considering the high pumping cost due to the high pressure drop. The above results can be used to determine the optimal temperature of operation for different velocities and a given particle size.
The kinetics of catalytic recombination of hydrogen in presence of 0.5% Pd on Alumina catalyst was studied in a fixed bed reactor operated with complete recycle. In order to extract the overall kinetics of the reaction, the experimental loop was modelled as a combination of plug flow packed bed reactor operating in pseudo-steady state and a complete mixing vessel under transient conditions. The apparent rate constant was estimated based on the least square method using the experimental data. The intrinsic rate constant was calculated by decoupling the effect of external and internal mass transfer. The intrinsic rate constants at different temperature were found out based on the experimental data generated at the respective temperatures. Using Arrhenius plot, the true activation energy and frequency factor were also estimated. The effects of different process variables such as temperature, superficial velocity and particle size on the apparent rate constant were studied. In the lower temperature region, the overall reaction was controlled by surface reaction and in the higher temperature regions, the overall kinetics was controlled by external mass transfer. A lower particle size and a higher superficial velocity were found to be beneficial while operating at higher temperature, but effect of pressure drop should also be considered. Variations in superficial velocity and particle size do not significantly affect the overall reaction kinetics in the low temperature region. Once the intrinsic kinetics data is available, the apparent kinetics can be estimated for different operating conditions (e.g. temperature, superficial gas velocity, catalyst particle size) during the reactor design for scale up.
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Acknowledgement The authors wish to acknowledge the support received from Shri. S A Prabhu and Shri. D Y Gaikwad, HWD, BARC throughout the experiments.
Nomenclature a CA CA,initial CAs CA,1 CA,2 D DAB De DK dp Ea Eapp kc kapp kapp,o kRS kRS,o M QM QPBR rA rp R Rep Sc Sh t T
external surface area of the catalyst pellet per unit reactor volume, m2 m3 concentration of hydrogen gas (or species A) in the bulk gas stream, mol m3 initial concentration of hydrogen gas in the mixing vessel, mol m3 concentration of hydrogen gas on the catalyst pellet surface, mol m3 inlet concentration of hydrogen to PBR, mol m3 outlet concentration of hydrogen from PBR, mol m3 reactor inner diameter, m diffusivity of species A (hydrogen) through B (air), m2 s1 effective diffusivity of hydrogen through the porous catalyst, m2 s1 Knudsen diffusivity of hydrogen inside catalyst, m2 s1 diameter of catalyst, m true activation energy of the recombination reaction, J mol1 apparent activation energy of the recombination reaction, J mol1 mass transfer coefficient in the fixed bed reactor, m s1 apparent reaction rate constant, s1 frequency factor for apparent reaction rate constant, s1 intrinsic reaction rate constant, s1 frequency factor for intrinsic reaction rate constant, s1 molecular weight of gas diffusing inside the porous catalyst, g mol1 volumetric flow rate of gas through the mixing vessel, m3 s1 volumetric flow rate of gas through the reactor, m3 s1 rate of reaction of species A, mol m3 s1 pore radius in catalyst, m universal gas constant, 8.314 J mol1 K1 particle Reynolds number, dimensionless Schmidt number, dimensionless Sherwood number, dimensionless time, s absolute temperature, K
u VM VPBR ε h f2
superficial velocity of gas through the reactor, m s1 volume of mixing vessel, m3 volume of the packed bed reactor, m3 void fraction in the packed bed reactor, dimensionless internal effectiveness factor, dimensionless Thiele modulus, dimensionless
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