Simulating and Optimizing Hydrogen Production by Low-pressure Autothermal Reforming of Natural Gas using Non-dominated Sorting Genetic Algorithm-II

Environmental considerations will probably change automobile fuels from gasoline and gas-oil to hydrogen (as fuel cell) in the future. Problems of fossil fuels include producing gaseous pollutants, such as NOx, CO, and even SO2 (from incomplete-hydrotreated fuels), which need catalytic converters and greenhouse gas emission (such as CO2, CH4, N2O) from the exhaust with a drastic effect on global warming1. Hydrogen usage in fuel cells generates only harmless water vapor2. Another future promising application of hydrogen fuel cells may be in small power plants without pollutants or greenhouse gas emission. Nowadays, power plants of industrial countries are coal-based with high SO2, NOx, CO2, mercury, and fly-ash emissions. Flue gas desulfurization of such power plants with CaO (dry method) or Ca(OH)2 (wet method) is a costly process3. In addition, SO2 reaction with lime shows pore mouth closure and incomplete conversion problems due to high molar volume ratio of CaSO4 to CaO4. The environmental preference of natural gas in terms of producing hydrogen over coal is a very interesting issue. Natural gas sweetening is very simple, whereas coal desulfurization is almost impossible. Moreover, the amount of greenhouse gas (CO2) emissions from natural gas-based combustion is about 45 % of coal-based power plants5,6. Industrial hydrogen production plants are usually based on catalytic steam reforming (SR) of natural gas to synthesis gas (CO+3H2), water-gas shift reaction for converting CO into CO2, CO2 absorption, and methanation of trace-remaining carbon oxides7. Conventional steam reforming is a highly endothermic reaction and needs a big top-fired or side-fired furnace for external heating of Ni-packed tubes8. Steam reforming by methane as the feed stock has been extensively studied in the literature9–12. Another method for producing hydrogen or synthesizing gas from natural gas is partial oxidation method. This method includes a highly exothermic reaction with hot spot, sintering of active catalyst sites, and even run-away problems13–15. An interesting alternative for steam reforming is the autothermal reforming method with internal heat generation by oxygen or air injection to the steam reforming system and elimination of its costly furnace16,17. ATR (autothermal reforming) combines thermal effects of partial oxidation and steam reforming reactions by feeding natural gas, water vapor, and air (or oxygen) into the reactor. Steam reforming and oxidation of methane occur simultaSimulating and Optimizing Hydrogen Production by Low-pressure Autothermal Reforming of Natural Gas using Non-dominated Sorting Genetic Algorithm-II


Introduction
Environmental considerations will probably change automobile fuels from gasoline and gas-oil to hydrogen (as fuel cell) in the future.Problems of fossil fuels include producing gaseous pollutants, such as NOx, CO, and even SO 2 (from incomplete-hydrotreated fuels), which need catalytic converters and greenhouse gas emission (such as CO 2 , CH 4 , N 2 O) from the exhaust with a drastic effect on global warming 1 .Hydrogen usage in fuel cells generates only harmless water vapor 2 .Another future promising application of hydrogen fuel cells may be in small power plants without pollutants or greenhouse gas emission.Nowadays, power plants of industrial countries are coal-based with high SO 2 , NOx, CO 2 , mercury, and fly-ash emissions.Flue gas desulfurization of such power plants with CaO (dry method) or Ca(OH) 2 (wet method) is a costly process 3 .In addition, SO 2 reaction with lime shows pore mouth closure and incomplete conversion problems due to high molar volume ratio of CaSO 4 to CaO 4 .
The environmental preference of natural gas in terms of producing hydrogen over coal is a very interesting issue.Natural gas sweetening is very simple, whereas coal desulfurization is almost impossi-ble.Moreover, the amount of greenhouse gas (CO 2 ) emissions from natural gas-based combustion is about 45 % of coal-based power plants 5,6 .
Industrial hydrogen production plants are usually based on catalytic steam reforming (SR) of natural gas to synthesis gas (CO+3H 2 ), water-gas shift reaction for converting CO into CO 2 , CO 2 absorption, and methanation of trace-remaining carbon oxides 7 .Conventional steam reforming is a highly endothermic reaction and needs a big top-fired or side-fired furnace for external heating of Ni-packed tubes 8 .Steam reforming by methane as the feed stock has been extensively studied in the literature [9][10][11][12] .
Another method for producing hydrogen or synthesizing gas from natural gas is partial oxidation method.This method includes a highly exothermic reaction with hot spot, sintering of active catalyst sites, and even run-away problems [13][14][15] .
An interesting alternative for steam reforming is the autothermal reforming method with internal heat generation by oxygen or air injection to the steam reforming system and elimination of its costly furnace 16,17 .ATR (autothermal reforming) combines thermal effects of partial oxidation and steam reforming reactions by feeding natural gas, water vapor, and air (or oxygen) into the reactor.Steam reforming and oxidation of methane occur simulta-Simulating and Optimizing Hydrogen Production by Low-pressure Autothermal Reforming of Natural Gas using Non-dominated Sorting Genetic Algorithm-II neously in the presence of nickel catalyst.The heat generated from partial oxidation is absorbed by steam reforming; hence, the overall temperature which is decreased with respect to partial oxidation is acceptable.In recent years, considerable attention has been paid to autothermal reforming which has an interesting potential in industrial applications [18][19][20][21][22] ; but, there have been only few works on reactor simulation and optimization.Behroozsarand et al. 23 performed multi-objective optimization of an industrial autothermal reformer, which consisted of a noncatalytic partial oxidation chamber and a two-section catalytic steam reformer in order to produce syngas with H 2 /CO molar ratio of close to 1 for application in oxo-processes.Akbari et al. 24 presented a numerical investigation of catalytic autothermal reforming of methane in a surface microreactor for mobile applications of hydrogen fuel cells.In Mohanty's 25 study, real parameter non-dominated sorting genetic algorithm was used to obtain a Pareto optimal set of process parameters for producing synthesis gas from combined carbon dioxide reforming and partial oxidation of natural gas over a Pt/Al 2 O 3 catalyst.
In this work, low-pressure autothermal fixed bed reactor was simulated for producing hydrogen for fuel cells based on a heterogeneous and one-dimensional mathematical model.In most previous studies, the Xu and Froment (XF) kinetic model for steam reforming reaction was used but the Numaguchi and kikuchi (NK) kinetic model was used in this study.In addition, the effects of operating parameters on reactor behavior and outlet hydrogen mole fraction were studied.The non-dominated sorting genetic algorithm-II (NSGA-II) was then used for optimizing this system.Finally, the best conditions were determined for obtaining the maximum molar flow rate of hydrogen with reasonable unreacted methane.The hydrogen molar flow rate was calculated from the outlet gaseous mole fractions from simulation program by the equation below.Multiplication of mass flow rate of feed and outlet hydrogen weight fraction was used in this equation.The optimization parameters were H 2 O/CH 4 and O 2 /CH 4 in the feed, as well as feed stream flux and temperature.

Literature review
There are very few references to the use of Evolutionary Algorithm (EA) techniques in chemis-try and catalysis, which includes Genetic Algorithm (GA), Evolutionary Strategy (ES), Genetic Programming (GP), etc.The Evolutionary Strategy (ES) has been used for the selection and optimization of heterogeneous catalytic materials [26][27][28] .Genetic Programming has been employed very rarely.Baumes et al. 29,30 showed two examples of this very powerful technique.Genetic Algorithms (GA) have been done by various groups, such as Pereira et al. 31 They reported a study of the effect of Genetic Algorithm (GA) configurations on the performance of heterogeneous catalyst optimization.Also, Gobin et al. 32,33 used multi-objective experimental design of experiments based on a genetic algorithm to optimize the combinations and concentrations of solid catalyst systems.Moreover, the genetic algorithm has been merged with a knowledge-based system 34 , and boosted on a GPU hardware to solve a zeolite structure 35,36 .In addition, GA has been used for crystallography and XRD measurements 37,38 , and as an Active Learning method for effective sampling 39 .
Although a number of papers on the simulation of autothermal reforming are available, very few have tried to optimize the process conditions in order to obtain maximum benefit.Behroozsarand et al. performed the multi-objective optimization of an industrial autothermal reformer consisting of a non-catalytic partial oxidation (POX) chamber and a two-section catalytic steam reformer in order to produce syngas with H 2 /CO ratio of near 1 for application in Oxo-processes 23 .The autothermal reformer that was appropriate for methanol production and Fischer-Tropsch reactions was optimized using NSGA-II method in Azarhoosh et al. study 40 .The optimization parameters were determined for the hydrogen production of 2 and maximum methane conversion.

Problem statement and objectives of the present study
In this study, the fixed-bed autothermal reactor was simulated.It only had a catalyst bed, in which combustion and steam reforming reactions occurred simultaneously.First, the appropriate steady-state one-dimensional model was selected and presented for the autothermal reforming.Then, the differential equations of the related model were solved by suitable numerical methods for the simulation of the fixed-bed reactor and low-pressure autothermal reformer was simulated.The simulation framework was then used for predicting the effects of major operating variables, such as feed temperature, O 2 /CH 4 molar ratio, and steam-to-methane (H 2 O/CH 4 ) molar ratio on the behavior of the autothermal reactor.Finally, optimization of the autothermal reforming reactor was considered using non-dominated sorting genetic algorithm II (NSGA-II).The main objec-tives of autothermal reforming were maximum methane conversion and hydrogen production in the exit synthesis gas.
Specific objectives of this study were: -Simulating a low-pressure auto-thermal reformer -Predicting effects of main operating variables such as temperature, etc.
-Optimizing the ATR reformer to produce maximum hydrogen.

Model description and solution procedure
In this study, the fixed bed autothermal reactor, illustrated in Fig. 1, was simulated.It only had a catalyst bed on which combustion and steam reforming reactions occurred simultaneously.
The mathematical modeling for autothermal reforming with internal heating by oxidation reaction was done by a steady-state one-dimensional heterogeneous model 2,41 .The following assumptions were made for modeling the autothermal reforming reactor: -The feed gas has an ideal gas behavior.
-The catalyst pellets are isothermal: The main transport resistance inside the catalyst pellet is due to mass transfer, even in the case of highly exothermic reactions 2,41 .
-The reactor is at plug flow regime: Axial dispersion of heat and mass can be neglected.
-The packed bed reactor is assumed to be adiabatic.
-Effective binary diffusivities are used in the catalyst pellet.
-The catalyst pellet is semi-spherical and of uniform size.
-Total pressure of the reformer is constant: It is negligible in the investigated conditions 41 .
-The bed porosity is constant.

Reaction kinetics model
In a reforming process of natural gas, many reactions are likely to occur.If methane is considered to be the major dominating species in natural gas, the following set of reactions shown in Table 1 will be involved 2 .
To reduce the complexity in the development and solution of the mathematical model, only the reactions with significant rates will be considered.The first three reactions prove to have significant rates 41,42 .Therefore, other reactions were ignored in this modeling study.
As the reactor simulations concern the autothermal reforming of methane to synthesis gas on supported Ni catalysts, the rate equations for total combustion, steam reforming, and water-gas shift reactions have to be combined in the calculations.Kinetics of methane combustion (reaction 3 in Table 1) was taken from Trimm and Lam 43 .Since this kinetic model was derived for supported Pt catalysts, the corresponding adsorption parameters were adjusted for Ni (see Equation 4in Table 2).For reactions 1 and 2 in Table 1, two of the most important models were the reforming model proposed by Xu and Froment, and the kinetic model taken from Numaguchi and Kikuchi 9,42 .These models will be referred to hereinafter as XF and NK models, respectively.The XF reforming model was obtained using relatively low temperatures ( 773 848 < < T s K ).During the indirect partial oxidation of methane to synthesis gas, however, high catalyst temperatures Ta b l e 1 -Autothermal reactions 2 Reaction DH j 0 (kJ mol -1 )

-Scheme of a typical autothermal reformer
could occur due to the exothermal total combustion reaction 9,41 .Hence, in this study, the kinetic model proposed by Numaguchi and Kikuchi, derived at higher catalyst temperatures (up to 1000 K), was used to simulate adiabatic fixed-bed autothermal reactors 41,42 .Langmuir-Hinshelwood rate equations for steam reforming towards CO and for the watergas shift reaction were proposed by the authors based on a rate-determining surface reaction step (see Equations 2 and 3 in Table 2) 42 .Table 2 shows the reaction kinetic equations for nickel catalyst.
Rate coefficients k 1 , k 2 , k 3a , and k 3b depended on temperature according to an Arrhenius type equation: Values of pre-exponential factors and activation energies are presented in Table 3.
The first term of the oxidation rate equation (see Equation 4 in Table 2) accounts for the reaction between molecularly adsorbed methane and oxygen.The second term describes Eley-Rideal reaction between molecularly adsorbed methane and gaseous oxygen.The adsorption constant is written as: Standard CH 4 and O 2 adsorption enthalpies in the case of Pt and Ni were taken from Shustorovich (see Table 4) 44 .

Adsorption coefficient
A i (atm -1 ) DH i (kJ mol -1 ) In Equations 2 and 3 (in Table 2), K 1 and K 2 are equilibrium constants of reactions 1 and 2. The equilibrium constants calculated at each temperature by Von't Hoff method and their parameters are presented in Table 5 45 : Rates of consumption or formation of species i (r i ) is determined by summing up the reaction rates of that species in all the reactions (r j ).Effectiveness factors η j are used to account for the intra-particle transport limitation 2,13,46 .Therefore, the reaction rate of each species becomes: R k P P P Effectiveness factors are computed along the reactor length using the following equation by integrating internal reaction rates 47 : It is a measure of the relative importance of diffusion to reaction limitations and defined as the actual overall reaction rate to rate if the entire surface is exposed to C i s s , .

Mass and energy balances
The gas-phase and solid-phase mass and energy balance equations are presented in Table 6 along with the corresponding boundary conditions.

Gas phase
Mass balance: Energy balance: Solid phase Gas phase boundary condition z = 0 ,

Mass transfer coefficient
Mass transfer coefficient (k g ) is calculated using Chilton-Colburn j-factors (J D ) [48][49][50] : where: Re   Density of the gas mixtures is calculated by modifying the perfect gas law as follows 49 : Viscosity of the components is accurately corrected as a function of temperature 49 : in which A, B, C, and D constants are available in 49,51 .In order to predict the viscosity of gaseous mixtures, the Bromley and Wilke method was used 49 .

Heat transfer coefficient
Heat transfer coefficient is calculated using Chilton-Colburn j-factors [48][49][50] : where: Heat capacity of the components is accurately corrected as a function of temperature 45 : in which A, B, C, and D constants are available in reference 45 .

Effective diffusion coefficients
The corresponding effective diffusion coefficients were calculated from the molecular and Knudsen diffusion coefficients corrected for the pellet porosity and tortuosity (0.53 and 4.0, respectively) 41 D i,m is molecular diffusion coefficient of species i in multi-component mixtures.For calculating D i,m , the molecular binary diffusion coefficients were calculated according to Wilke equation 50,52 .
From the kinetic theory of gases, the so-called Knudsen diffusivity can be formulated in Eq. 31 50 : , 4 2 3 where M i is molecular weight of the diffusing species, and r is the average pore radius.

Numerical solution procedure
Step 1: Axial coordinate of reactor length is divided into the uniform grid with 1000 intervals.
Step 2: In each grid of the reactor length, the solid-phase (catalyst) is divided into the uniform grid with 40 intervals in radial coordinate.The solid-phase continuity equation (see Equation 18in Table 6) is then solved in the first grid of the axial direction by means of the central finite deference method in MATLAB software.Thus, in first radial grid, concentration and partial pressure of each component in all the grids of the catalyst pellet are calculated by solving Equation 18.
Step 3: Using the calculated partial pressures, the effectiveness factor of each reaction is specified by numerical integration (see Equation 15).
Step 4: Ordinary equations for gas phase (Equations 16 and 17 in Table 6) are solved using Runge-Kutta method.Also, mole fraction and temperature of components are specified in the bulk gas for the second grid of the axial direction.
Step 5: The catalyst temperature for the second grid of the axial direction is calculated using Equation 19 (in Table 6).
This cycle is performed until the end of the reactor length.Therefore, using this method in all grids of the reactor, the dependent variables are calculated at all nodal points.The concentration profiles versus reactor length as well as pellet radius coordinates can be determined by this procedure.

Non-dominated Sorting Genetic Algorithm-II (NSGA-II)
The principles, on which the NSGA-II relies, are the same as those of the single-objective optimization.The strongest individuals (or chromosomes) are combined to create the offspring by crossover and mutation, and this scheme is repeated over many generations.However, the multi-objective optimization algorithm must consider the fact that there are many "best solutions", which modify the selection process.NSGA-II sorts individuals based on the non-domination rank and the crowding distance to ensure a high level of performance as well as good dispersion of results 41,[56][57][58][59] .Fig. 2 shows the flowchart of non-dominated sorting genetic algorithm-II 41,53,56,60 .

Reactor dimensions and operating conditions
The autothermal reactor and catalyst specifications were the same as reported by de Smet et al. 41 .The catalyst specifications are presented in Table 7.The reactor dimensions as well as operating conditions for hydrogen-for-fuel-cell production are demonstrated in Table 8 41 .In the case of fuel-cell application, the fixed-bed catalytic autothermal reactor operated at atmospheric pressure.Beside methane and air, water vapor was added as a reactant in the reactor inlet to increase hydrogen formation and suppress coke deposition.In the applied mass-flux and catalyst pellet diameter, the pressure drop was assumed negligible.

Simulation results
A fixed-bed reactor suitable for the operation of a fuel-cell was investigated.The calculated catalyst and bulk gas temperature profiles along the reactor co-ordinate are shown in Fig. 3 for NK reforming kinetics.In this figure, catalyst temperature increased first due to the large amount of heat produced in the methane oxidation reaction.Then, the catalyst temperature reached 1001 K and decreased consecutively as a result of the prevailing endothermic reforming reaction.The final temperature increase was due to the heat produced from the water-gas shift reaction.Relatively low value of the maximum catalyst temperature indicated that catalyst deactivation would not be very serious in this case.The catalyst temperature in the reactor inlet was approximately 40 K higher than the inlet gas phase temperature.
The calculated axial mole fraction profiles, obtained in case of the NK reforming kinetics, are shown in Fig. 4.This figure indicates that both methane and oxygen mole fractions decreased rapidly.As a result, significant amounts of H 2 and CO were observed in the first part of the reactor.The simulated results of this work were in good agreement with those of De Smet et al. 41 final temperature and mole fraction values, as demonstrated in Figs. 2 and 3.
Typical intra-particle mole-fraction profiles are shown in Fig. 5 in the case of simulations with NK reforming kinetics for CH 4 and H 2 , respectively.The calculated mole fractions were plotted as a function of axial reactor coordinate as well as dimensionless pellet axis.
In these simulations, the calculated effectiveness factors varied considerably along the axial reactor coordinate, which indicated that it was necessary to consider intra-particle concentration gradients, and these gradients varied considerably along the reactor indeed.These effects were accounted for by solving the solid-phase continuity equations in each position in the fixed-bed reactor.

Analysis of effective parameters
The effects of variation on operating conditions versus base case on the system performance were studied.Performance of the reactor can be evaluated based on outlet hydrogen molar flow rate and final conversion of methane.Fig. 6 shows the effect of feed mass flux on final methane conversion (X CH 4 ) and hydrogen molar flow.By increasing inlet gas mass flux with respect to the base case, final methane conversion was decreased due to less gas flow residence time and also reduced gas temperature along the reactor.In addition, as shown in this figure, from the base case (0.15 kg m -2 s -1 feed mass flux) to about 2.6 kg m -2 s -1 , the outlet hydrogen production increased and then decreased by further increase in the feed mass flux.Fig. 7 demonstrates the effect of inlet gas temperature on the final methane conversion and outlet hydrogen molar flow rate, respectively.Both of them were enhanced with increasing inlet gas temperature, and maximum conversion occurred in the maximum inlet operating gas temperature.
Variations of the outlet H 2 molar flow and conversion of methane with the feed H 2 O/CH 4 ratio are presented in Fig. 8 along with other parameters at their baseline values (given in Table 8).At first, increasing the inlet water content promoted the steam reforming reaction; thus, both methane conversion and hydrogen production increased by increasing water content of the feed mixture 24 .With the further increase in water content of the feed, methane conversion and produced hydrogen molar flow were decreased, which was due to decreasing air content of the feed, and exothermic oxidation rate.The optimization problem was considered for a fuel-cell unit.It can logically search for operating scenarios that will maximize methane conversion and produced hydrogen molar flow simultaneously.Performing an NSGA-II optimization method with both of them as objectives can identify such scenarios.The optimization problem can be expressed mathematically as follows: H O 1 ( Other parameters were the same as the base case.Higher limit of the feed temperature constraint of the autothermal reformer (973 K) was selected based on the lifetime of Ni catalyst in the fixed-bed reactor.Very high inlet gas temperatures were not recommended because of economic aspects, producing high catalyst temperature, and thus catalyst deactivation due to coke formation and even catalyst destruction by sintering 23 .Also, at low temperatures (below 773 K), the first objective function (CH 4 conversion) was decreased; therefore, the lower limit (773 K) was appropriate.
At low O 2 /CH 4 (below 0.4), methane conversion decreased and catalysts were prone to coking.Also, at high O 2 /CH 4 , hot spot formation in the catalyst increased due to the fast oxidation reaction; therefore, the higher limit (0.6) was appropriate.According to the literature and industrial autothermal reformer 41 , the lower limit of H 2 O/CH 4 was set at 1 to avoid carbon formation on the catalyst, which occurred at low value of H 2 O/CH 4 .Also, very high H 2 O/CH 4 ratio adversely affected the process economics because of the energy requirement for heating up the feed.
Different operations were performed for 200 generations to obtain non-dominated Pareto-optimal solutions.A Pareto-optimal set is a series of solutions that are non-dominated with respect to each other.While moving from one Pareto solution to another, there is always a certain amount of sacri-fice in one objective(s) to achieve a certain amount of gain in other(s) 23 .The Pareto-optimal solution sets after 200 generations are shown in Table 10 and Fig. 10.
By comparing the solutions (chromosomes) in Table 10, it seems that chromosome 2 was the best solution, because it had the maximum molar rate and reasonable methane conversion.It is obvious that there was a huge increase in the produced hydrogen molar flow to the base case (from 0.0223 to 0.1382 mol s -1 ), which showed the importance of optimizing the autothermal reformers for hydrogen production.

Conclusions
A steady-state one-dimensional heterogeneous reactor model was used to simulate adiabatic fixed bed catalytic autothermal reactor for hydrogen production.Intra-particle concentration gradients were explicitly taken into account by solving the corresponding continuity equations in each position along the reactor coordinate.Effects of operating parameters such as inlet gas temperature, inlet gas mass flux, inlet O 2 /CH 4 , and H 2 O/CH 4 ratios on the system behavior were also studied.Finally, the autothermal reformer for hydrogen production was optimized using NSGA-II method.The optimization parameters were determined for the maximum hydrogen production rate and methane conversion.These parameters were determined for feed temperature, feed mass flux, inlet O 2 /CH 4 , and H 2 O/ CH 4 ratios as 957.2 K, 0.960 kg m -2 s -1 , 0.540, and 1.410, respectively.In this optimum condition, the produced outlet hydrogen molar flow and final methane conversion were 0.138 mol s -1 and 0.961, respectively.It can be seen that there was a huge increase in the produced hydrogen molar flow to the base case and optimization was necessary for this system.Ta b l e 1 0 -Non-dominated Pareto-optimal solutions after 200 generations  A i -Pre-exponential factor of species i, atm A i ' -Pre-exponential factor of reaction i, reaction dependent a v -External pellet surface area per unit reactor volume, m -1 C i -Molar concentration of species i at bulk gas, mol m -3 C i o -Inlet molar concentration of species i at bulk gas, mol m -3 C i,s -Intra-particle molar concentration of species i, mol m -3 C i,s s -Molar concentration of species i at the external pellet surface, mol m -3 C p -Specific heat at constant pressure, J kg -1 K -1 C p,i -Specific heat of component i, J kg -1 K -1 D e,i -Effective diffusion coefficient of species i in catalyst, m 2 s -1 D k,i -Knudsen diffusion coefficients, m 2 s -1 D i,m -Molecular diffusion coefficients, m 2 s -1 d r -Reactor diameter, m E act -Activation energy, J mol -1 G -Superficial feed mass flux, kg m -2 s -1 h f -Gas-to-solid heat transfer coefficient, W m -2 K -1 J D -Mass transfer Chilton-Colburn j-factor J H -Heat transfer Chilton-Colburn j-factor K -Conduction heat transfer coefficient, W m s -1 K i ox -Adsorption constant for component i in combustion reaction, atm -1 K j -Equilibrium constant of reaction j, reaction dependent K j o -Equilibrium constant of reaction j at 25 °C, reaction dependent k i -Reaction rate constant of reaction i, reaction dependent K g -Gas-to-solid mass transfer coefficient, m s -1 l r -Reactor length, m M ave -Molecular weight of gaseous mixture, g mol -1 M i -Molecular weight of species i, g mol -1  n H 2 -Outlet hydrogen molar flow P i -Partial pressure of component i, atm P tot -Total pressure, atm R -Universal gas constant, atm m 3 mol -1 K -1 r -Radius reactor co-ordinate, m r i -Rate of component i production, mol kg -1 s -1 r j -Rate of reaction j (j = 1,2,3), mol kg -1 s -1 r i s -Rate of reaction j (j = 1,2,3) at the external pellet surface, mol kg -1 s - o -Gibbs free energy of component i at 25 °C, J mol -1 DG j -Gibbs free energy of reaction j, J mol -1 DG j o -Gibbs free energy of reaction j at 25 °C, J mol -1 -D f H i -Heat of formation of species i, J mol -1 R e f e r e n c e s

Ta b l e 2 -
Reaction kinetic equations for nickel catalyst

Ta b l e 7 -
Catalyst specifications Numaguchi and Kikuchi Ni/Al 2 O 3 Catalyst 1970 Density (kg m -3 ) 3.6 Metal surface area (m 2 g -1 )F i g . 2 -Flowchart of non-dominated sorting genetic algorithm-II programTa b l e 8 -Reactor and catalyst dimensions, operating conditions, and feed composition41

F i g . 3 -
Bulk gas and catalyst temperatures versus length of the reactorF i g .4 -Components mole fraction profiles versus length of the reactor F i g .5 -Intra-particle mole-fraction profiles of (a) methane and (b) hydrogen versus axial reactor coordinate and the dimensionless pellet coordinate

F i g . 6 -
Effect of feed mass flux on (a) final methane conversion and (b) outlet hydrogen molar flow rateF i g .7 -Effect of feed temperature on (a) hydrogen molar flow rate and (b) final methane conversion F i g .8 -Effect of feed H 2 O/CH 4 on (a) final methane conversion and (b) hydrogen molar flow rate

Fig. 9
Fig.9shows the effect of O 2 /CH 4 ratio on methane conversion and outlet hydrogen molar flow rate, respectively.By increasing oxygen in the feed, combustion of methane was increased.Therefore, methane conversion and hydrogen production would increase.

4. 4 .
Optimization results using NSGA-II Non-dominated sorting genetic algorithm-II (NSGA-II) was used to optimize and obtain Pareto-optimal solutions.Population size of 100 was chosen with crossover of 0.7 and mutation probability of 0.05.Input parameters of NSGA-II are given in Table9.Ta b l e 9 -Input parameters of NSGA-II feed temperature  973 K (34) 0.1 kg m -2 s -1  superficial feed mass flux  1 kg m -2 s -1

F
i g . 1 0 -Non-dominated Pareto optimal solutions after 200 generations N o m e n c l a t u r e