On the Modeling of Steam Methane Reforming

Esmail M. A. Mokheimer1 Professor Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 279, Dhahran 31261, Saudi Arabia e-mail: [email protected]

Muhammad Ibrar Hussain Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 279, Dhahran 31261, Saudi Arabia e-mail: [email protected]

Shakeel Ahmed Research Scientist Center for Refining and Petrochemicals, Research Institute, King Fahd University of Petroleum and Minerals, P.O. Box 279, Dhahran 31261, Saudi Arabia e-mail: [email protected]

Mohamed A. Habib Professor Mechanical Engineering Department, KACST TIC on CCS, King Fahd University of Petroleum and Minerals, P.O. Box 279, Dhahran 31261, Saudi Arabia e-mail: [email protected]

On the Modeling of Steam Methane Reforming Modeling and simulations of steam methane reforming (SMR) process to produce hydrogen and/or syngas are presented in this article. The reduced computational time with high model validity is the main concern in this study. A volume based reaction model is used, instead of surface based model, with careful estimation of mixture’s physical properties. The developed model is validated against the reported experimental data and model accuracy as high as 99.75% is achieved. The model is further used to study the effect of different operating parameters on the steam and methane conversion. General behaviors of the reaction are obtained and discussed. The results showed that increasing the conversion thermodynamic limits with the decrease of the pressure results in a need for long reformers so as to achieve the associated fuel reforming thermodynamics limit. It is also shown that not only increasing the steam to methane molar ratio is favorable for higher methane conversion but the way the ratio is changed also matters to a considerable extent. [DOI: 10.1115/1.4027962] Keywords: computational fluid dynamics modeling, hydrogen, steam methane reforming, syngas, volume based reaction model

Amro A. Al-Qutub Professor Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, P.O. Box 279, Dhahran 31261, Saudi Arabia e-mail: [email protected]

Introduction Greenhouse gases emitted by the combustion of fossil fuels have severe environmental and health issues [1]. There is a need to look for availability of clean energy resources [2]. A lot of research is being carried out for wind, wave, and hydro energy resources [3–5]. Another available choice is the hydrogen that has lot of advantages over fossil fuels such as clean combustion and higher heating value [6]. Hydrogen is also used in refineries for desulfurization, hydrotreating, and for production of chemicals [7]. Hydrogen is produced by a number of ways such as electrolysis, SMR, auto CO2 methane reforming, partial oxidation reforming, and some extensions of these processes. The production of hydrogen form electrolysis is much expensive because of the high production cost of electricity [8,9]. Other processes use hydrocarbons as the main reactant for hydrogen production [10]. Among all of these processes, SMR process is the cheapest and is used 1 Corresponding author. Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received June 4, 2014; final manuscript received June 23, 2014; published online July 29, 2014. Editor: Hameed Metghalchi.

Journal of Energy Resources Technology

commercially to meet the hydrogen demands worldwide [11]. Operating temperature for SMR is low as compared to partial oxidation reforming and CO2 reforming due to absence of oxygen. It also has higher H2/CO ratio for higher H2 production [12]. Usual hydrocarbon used for SMR reaction is methane due to its high hydrogen contents and low capital cost compared to other hydrocarbon such as coal for which the capital cost is three times that of methane [13]. SMR reaction is represented by following chemical equations: CH4 þ H2 O Ð CO þ 3H2 CO þ H2 O Ð CO2 þ H2 CH4 þ 2H2 O Ð CO2 þ 4H2

DH298 ¼ 206 kJ=mol

DH298 ¼ 41:1 kJ=mol

DH298 ¼ 164:9 kJ=mol

(1) (2) (3)

First and third reactions are endothermic while the second reaction is exothermic as it is evident from their standard enthalpies of . However, the overall reaction is endothermic in reactions, DH298 nature. It is worth mentioning here that standard enthalpies of , are enthalpy changes that occur when quantities reactions, DH298 of species given by chemical equations react under standard conditions. SMR faces certain problems such as low methane conversion due to reversibility, coke formation, catalyst deactivation,

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heat transfer issues, and diffusion limitations [7,14–29]. The physical structure of the catalyst affects the SMR reaction largely. Complex dependence of SMR reaction on the flow variable makes the simulative studies inevitable. Different models dealing with kinetics as well as heat transfer phenomena are available [30–50]. Improvement of heat transfer in reformers to get less radial gradients has always been the main concerns in these models [51]. Most of the models are either too complex or take too much execution time that make the simulative purpose meaningless. Flow modeling of SMR is performed by using two approaches. The first one is surface based approach and the second is a volume based approach. Modeling the SMR using surface based approach takes lot of computational time. On the other hand, volume based approach requires less computational time. The choice of the modeling approach depends on the details one interested in. The surface based modeling approach provides minor details of variations of flow variables, whereas the volume based approach gives averaged distribution of flow variables. Volume based approach is used in the present study due to its simplicity and less computational time. Models developed using volume based approach can also be used easily for the further studies of SMR reaction such as flow optimization or shape optimization of steam methane reformer. In these studies, general dependence of SMR on flow variable is required that is obtained using volume based approach. Use of surface based approach makes the study complex due to its minor details. Results may also have considerable noise. Thus in the present study, a volumetric reaction model is developed. In earlier publications, simple kinetic rate expression was developed where rate of reaction is proportional to methane partial pressure and similar expression is developed by many authors even in recent years [52–54]. Agnelli et al. [55] argued that experiments done previously were in the range of diffusion limitations that was not taken care of at that time. Moreover the partial pressure of hydrogen was low for those experiments. Experiments were done for lower value of PH2 O =PH2 ratio to check the dependence on other species’ concentrations. Model developed was similar to first order kinetic rate expression. Some power based kinetics are also developed such as over nickel aluminate (Ni/Al2O3) catalyst, reaction rate was found close to one for methane and negative one for steam [56]. Kinetic models, depending on the scheme of reaction, are also developed by recognizing the rate determining step for proposed reaction’s scheme. In these models, formation of CO, CO2, dissociation of methane or reaction of some carbon intermediates are taken to be rate determining steps. Allen et al. [57] developed a model by taking the CO and CO2 formation as rate determining step, over pressure range of 1–18 atm and for constant temperature of 1180 F. The average error, between simulated and experimental conversion, was 7% with maximum error as high as 23%. Adsorption of methane was found to be the rate determining step with the competitive adsorption of steam over coprecipitated Ni/Al2O3 catalyst that highlighted the inhibiting effect caused by steam [58]. Taking the reaction of carbon intermediate with absorbed oxygen as rate determining step, Xu and Froment [59] developed a model over pressure range of 1–10 bar and temperature of 778–848 K. This model has non monatomic reaction rate order with respect to partial pressure of steam [60]. A complex model developed by Avetisov et al. [61] converges to the model developed by Xu and Froment [59] under usual operating conditions of SMR. Moreover, the same reaction path is successfully used to develop the kinetic rate expressions for supported nickel catalysts by many authors [62–64]. Model developed by Xu and Froment [59] is also used extensively for modeling of SMR process [65–74] and is used in this work also. Several models have been developed for SMR process to predict the extent of reaction as well as to serve for optimization purposes. These models vary from 1D homogeneous models to 3D heterogeneous models. In heterogeneous models, effectiveness factor is determined locally everywhere in the reactor by

solving species transport equations over the catalyst particle [66,67,70,71,73,75–77]. Pseudo homogeneous models assume a profile of effectiveness factor along the length of catalyst bed. In these models, a correlation for effectiveness factor is used that usually depends on the catalyst particle’s shape. Thus for one catalyst bed profile of effectiveness factor remains same [69,78–80]. Homogenous models assume the constant value of effectiveness factors [65,68,72]. In most of the models, diffusion limitations are neglected, i.e., taking the effectiveness factor equal to one [81]. When the rate of diffusion of species in the reaction mixture is low as compared to the rate of reaction, then, effectiveness factor becomes important. The larger the catalyst’s particle size, the smaller will be the effectiveness factor [52,73]. When catalyst particle size is fairly small the assumption of constant effectiveness factor is valid. In most of the models, a general continuity equation representing the whole flow along with one momentum and one energy equation is solved. Estimation of pressure drop along the length of reactor is usually done by using Fanning friction factor [66,67,73,75,76,78,79] but separate pressure drop correlation is also used solely or in combination with complete momentum conservation equation [65,68]. In most of the 1D energy equations solved for various models, only energy losses and energy sink due to endothermic nature of reaction are taken into account [67,69,73,75,76,78]. Where as in recently developed 3D models, energy dissipation by shear stresses and by expansion or contraction of fluid, energy diffusion due to mass diffusion caused by concentration gradient as well as temperature gradient along with convection are considered [65,70]. Separate species transport equations are solved for the species in reaction mixture. Specie transport equations vary from simple 1D equation without diffusion [67,76] to complex 3D ones with mass and thermal diffusion [68,70,75]. Estimation of main fluid properties such as thermal conductivity, density, viscosity, and specific heat varies largely from model to model. In simple 1D models, porosity of catalyst bed only affects the diffusion coefficients [75,76]. In 3D models, apart from affecting the diffusion taking place, porosity affects the mass conservation by changing the flow velocity. It also reduces the energy diffusion due to diffusion of species’ flux and the volumetric rate of appearance or disappearance of species [68,70]. Mass diffusion coefficient is usually determined by taking the Knudsen and molecular diffusion into account [66–70,75–78]. The main objective of this article is to present the development of a comprehensive computational fluid dynamics (CFD) model for SMR process that has reduced computational time with high model validity. The developed model is validated against the reported experimental data. The validated model is further used to study the effect of different operating parameters on the steam and methane conversion.

Detailed Model The geometry of SMR reformer is shown in Fig. 1. It is a tubular reformer. Instead of modeling the catalyst pellet, a continuum of catalyst of reduced density is modeled inside the reformer [65,68]. Chemical reaction is assumed to take place everywhere in

Fig. 1 Steam methane reformer with operating conditions

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Transactions of the ASME

the reformer domain. This approach is justified as long as catalyst particle size is small. As well, this approach is justified when interest is focused on changes in concentrations of species at length scale larger than the size of catalyst particle because, in this case, a stretched distribution of concentrations of species is attained. Modeling the catalyst bed as continuum requires special attention while solving temperature, pressure and velocity field throughout the domain, because there are no catalyst pellets to offer the inertial and viscous resistance to the flow. The effect of accelerating the coming flow due to difference in flow area and decreased convection of momentum and energy as well as diffusion of energy due to difference in the thermal conductivities of main fluid flow and catalyst particle cannot be obtained in the absence of catalyst pellets. In actual reformers, porosity of catalyst bed is a decreasing factor for cross sectional area of the reformer to accelerate the fluid. In the present model, porosity is used as a decreasing factor for velocity to achieve the same mass flow rate through the reformer as the flow area is not being decreased due to absence of catalyst pellets. This decreased velocity is also called the superficial velocity. It reduces the convection and shear stresses of fluid in such a way that, per unit change in length, convection, and shear stresses at a scale larger than the scale of catalyst particle size remain the same as expected in original reformer filled with catalyst pellets. Conductivity of main fluid is determined by mass weighted law. Volume averaged conductivity of main fluid and catalyst pellets is used for diffusion of energy. Density of main fluid flow is determined by volume averaged value of temperature and pressure dependent densities of individual species. For species transport equations, superficial velocity stretches the convection and diffusion of species due to concentration and temperature gradients by multiplying the diffusive constants by porosity. The rate of chemical reaction expressed in kg/(m3s) is corrected by multiplying it by one minus porosity that is actually the reducing factor for density of catalyst bed. In some experiments, some diluents such are quartz or alumina are used. In this case, chemical reaction rate should be corrected such that reaction rate is always expressed per unit reformer volume. Instead of solving the whole Stephen–Maxwell equation for molecular diffusion, an equivalent binary diffusion equation was solved where the binary diffusion coefficients were calculated by Wilke Chang equation for dilute fluids [82]. As the particle size was small as compared to the particle size of the commercially used catalyst, binary diffusion coefficients were of order of 1014. Finally, the effective diffusion coefficient was calculated by using parallel pore model.

Fluid mixture’s thermal conductivity 15Rli 5Cpi 1 ki ¼ þ 3 4Mi R X kf ¼ Yi ki Effective conductivity ke ¼ ekf þ ð1 eÞks

r ðqi VYi Þ ¼ r Ji þ gi Ri

Ji ¼ qi Di;e rYi þ DT;i

rT T

Energy equation r ðqf VE þ VPÞ ¼ r þ

ke rT

X

X

(11)

1 1 1 ¼ þ De;i Dn;i Dm;i

(12)

X Yi Mi i

(13)

Equation of state P ¼ qf RT Effective density 1 qf ¼ X i

Yi qi

(14)

Effective specific heat Cpi ¼ A1 þ A2 T þ A3 T 2 þ A4 T 3 þ A5 T 4 X Yi Cpi Cp ¼

(15) (16)

i

Effective viscosity n T To X Yi li l¼

(17) (18)

i

The kinetic reaction rates for reactions 1, 2, and 3 are as follows: 3 k1 pCH4 pH2 O pCO pH2 =Ke;1 (19) r1 ¼ 2:5 pH2 ðDENÞ2

(4)

150 ð1 eÞ2 lf V dp2 e3

(10)

where

k2 pCO pH2 O pCO2 pH2 =Ke;2 r2 ¼ pH2 ðDENÞ2

Momentum equation rðqf VVÞ ¼ rP þ rðsÞ

(9)

Species transport equation

Mathematical and CFD Model

rðqf VÞ ¼ 0

(8)

i

li ¼ li;o The mathematical model adopted in the present work to simulate the SMR process comprise the conservation equations of mass, momentum, and energy along with (n 1) species transport equations and one species conservation equation are solved where n is the total number of species in reaction mixture. The governing equations are solved under the assumptions of steady flow conditions and the absence of gravitational effects. Continuity equation

(7)

2 4 p p p p =K CH CO e;3 4 H2 O 2 H2 k3 r3 ¼ 3:5 p H2 ðDENÞ2

(5) !

(20)

(21)

where

hi Ji þ ðs VÞ

i

hi gi Ri

i

(6)

ðDENÞ ¼ 1 þ KCH4 pCH4 þ KCO pCO þ KH2 pH2 þ PH2 O KH2 O =pH2 (22)



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Fig. 2 Axisymmetric solution domain

where k1, k2, and k3 are forward rate constants for reactions 1, 2, and 3, respectively. On the other hand, reverse reaction rate constants are obtained using ki/Ki, while r1, r2, and r3 are the net rates (forward-backward) of reactions 1, 2, and 3, respectively. It is worth mentioning here that reverse reaction rate constants should be estimated carefully to avoid inconsistency and some levels of errors such as the possible negative productions of entropy during kinetic transients as discussed by Janbozorgi et al. [83]. The accurate estimations of the forward and reverse reaction rates are beyond the scope of the present article. Thus, the forward and reverse reaction rate constants are calculated from equations that are given in the literature as indicated in Parameters for Kinetic Model of SMR. Parameters for Kinetic Model of SMR Ei Rate constants for Arrhenius equation ki ¼ Ai exp RT Table 1 Rate constant for Arrhenius equation [59]

Activation energy, Ei (kJ mol 1) Pre-exponential factor, Ai (dimension of ki)

E1 240.1

E2 67.13

E3 243.9

A1 (kmol bar5 kgcat1h1) 4.225 1015



Constants for Van’t Hoff equation Ki ¼ Bi exp

Hi RT

Table 2 Constant for Van’t Hoff equation [59]

Absorption enthalpy change, Hi (kJ mol1) Pre-exponential factor, Bi (dimension of Ki)

HH2 O 88.68

HCH4 38.28

HCO 70.61

HH2 82.90

BH2 O

BCO, BH2 , BCH4 , (bar1) (bar1) (bar1) 1.77 105 6.65 10 4 8.23 105 6.12 109

Solution Procedure The solution domain is shown in Fig. 2. Constant value of porosity allows modeling of reformer as 2D axisymmetric domain. Mass flow rate boundary condition is used at the inlet. Wall of the reformer is isothermally heated with zero wall thickness and the exit is maintained at constant pressure. The governing equations are solved numerically using ANSYS FLUENT 13.0. A pressure-correction based solver is used, where the momentum equations are solved separately by taking each velocity component as a scalar. After that, continuity and pressure correction equations are solved to correct the velocity and pressure fields. The energy equation is, then, solved and certain data such as pressure, temperature, and mass fraction of species are extracted and used to edit the kinetic reaction rate using a user defined function (UDF). UDFs are used to customize ANSYS FLUENT code. These UDFs are built in macros in ANSYS FLUENT. ANSYS FLUENT provides a variety of UDF. Most of UDFs are

used for a specific purpose but there are some general purpose UDFs as well. Each UDF has certain inputs and outputs. Outputs of these UDF depend upon the purpose they are used for. To incorporate the kinetic model by Xu and Froment [59] into ANSYS FLUENT code, a UDF named “DEFINE_VR_RATE” is used. By using this UDF, mass fractions, molar masses, and reaction rates are accessed for each cell of grid. Reaction rates are edited using mass fraction, molar mass, pressure, and temperature. Pressure and temperature are obtained using flow variable macros. Diffusion coefficient for each species is also edited using UDF named “DEFINE_DIFFUSIVITY”. This UDF provides access to species diffusion coefficients for each cell of grid. These reaction rates and diffusion coefficients data are, then, used in the specie transport equation. Another UDF named “DEFINE_PROPERTY” is also used for properties of individual species. These property UDFs are used before the solution of momentum equations to edit the material properties. Convection criteria for momentum and energy equation are upwind first order.

Grid Independence Test To check the independence of the solution on the grid, a grid independence study is conducted and a convergence criterion of flow field variables of 107 is applied. In this regard, five meshes are developed. Details of meshes containing the number of nodes and elements are shown in Table 3. The fractional methane conversion is selected as a judging variable to decide that either the solution is independent of the grid or it needs further refinement. It can be seen from Fig. 3 that initially a very high value of fractional methane conversion is achieved for very course mesh but as the mesh is refined, the relative difference in the fractional methane conversion decreases and finally vanishes for the mesh containing up to 3000 elements. Thus, the grid was not refined further to save the computational time. Therefore, mesh no. 5 with 3000 elements and 3146 nodes have been used to generate all the results presented in this article.

Model Validation To check the validity of the model, simulations are carried out at the operating conditions reported by Xu and Froment [59]. In this regard, Fig. 4 depicts the comparison between the simulated methane conversion and experimental methane conversion reported by Xu and Froment [59]. Simulations are carried out at temperatures of 848 K, 823 K, 798 K, and 773 K at the exit pressure of 10 bar with FH2 O =FCH4 value of 3 and FH2 =FCH4 value of 1.25. To quantize the quality of fit, a correlation factor is calculated. This correlation factor is the measure of fit of a straight line between simulated and experimental methane conversion. Correlation factor is found to be 0.9929, 0.9919, 0.9954, and 0.9975 for operating temperatures of 848 K, 823 K, 798 K, and 773 K, respectively. A value very close to one indicates a high quality validation of developed model against the experimental data as shown in Fig. 4. Thus, the developed model can be effectively used for further studies. X

ðx xÞðy yÞ Correlation factor ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X ðy yÞ2 ðx xÞ2

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(23)


Table 3 Details of meshes and respective fractional methane conversion Mesh no. No. of elements No. of nodes Fractional methane conversion 1 2 3 4 5

200 600 1200 2000 3000

246 671 1296 2121 3146

0.153 0.144 0.145 0.145 0.145

Fig. 5 Usual temperature distribution inside SMR reformer [84]

explained by the variation of rates of reactions 1, 2, and 3 in the later part of this section.

Fig. 3 size

Fractional methane conversion variation with the mesh

Results and Discussion Temperature Distribution. Temperature in catalyst bed of SMR reformers has its unique distribution. The temperature, first, decreases at the start of reformer then increases in the later part. As the reactants enter the reformer, only the reactions 1 and 3 can take place because there is no CO and H2 for the reaction 2 to happen. Both reactions 1 and 3 are endothermic in nature, therefore, decreases the temperature. As the some of the reactants for the reaction 2 forms, it takes place quickly and raises the temperature sharply as shown in Fig. 5. This behavior of the temperature distribution is also used as a measure of the catalyst quality. The greater the decrease in the temperature the better is the catalyst. As the catalyst ages, less decrease in the temperature takes place and reaction 2 takes large time to recover the temperature [84]. The present SMR model can capture this temperature distribution effectively. Profile of temperature distribution in the reformer is shown in Fig. 6. To capture this phenomenon, the inlet feed is supplied at the same temperature at which wall of reformer is maintained. This effect is also

Effect of Temperature on SMR. SMR reaction is endothermic and proceeds with increase in entropy. Thus, increase in temperature makes the reaction more favorable, therefore, having more conversion of reactants to products. Figure 7 shows the variation of fractional methane conversion with Wcat =FCH4 , where the fractional methane conversion is obtained from carbon balance and is defined as follows: Fractional methane conversion ¼

nCO þ nCO2 nCO þ nCO2 þ nCH4

(24)

where ni is the mole fraction of specie i. Wcat is the mass of catalyst that is present inside the reformer. FCH4 is the initial molar flow rate of methane. It can be seen from Fig. 7 that methane conversion increases as the temperature increases. It can be also seen that, at maximum achieved value of Wcat =FCH4 , the gradient of methane conversion with respect to Wcat =FCH4 is almost zero at 1000–1200 K. This indicates that the mixture has reached equilibrium. While at low temperature of 800–1000 K, a nonzero value of gradient indicates that mixture is far from equilibrium. Figure 8 presents the methane conversion at Wcat =FCH4 value of 0.4. It is clear that, at low temperature, conversion is lower than the equilibrium conversion value. However, the reaction approaches the equilibrium conversion values at high temperature. Thus, it can be concluded that the reaction is kinetically limited at low temperature but thermodynamically limited at high temperature.

Fig. 4 Variation of fractional methane conversion versus Wcat =FCH4 at FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and P 5 10 bar



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Fig. 6 Temperature distribution inside SMR reformer

Fig. 9 Pressure distribution inside the reformer

Fig. 7 Fractional methane conversion variation with Wcat =FCH4 at different temperature for FH2 O =FCH4 5 3, FH2/FCH4 5 1.25, and P 5 10 bar Fig. 10 Fractional methane conversion versus exit pressure for FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and T 5 800 K

Fig. 8 Fractional methane conversion versus operating temperature at FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and P 5 10 bar

Fig. 11 Fractional methane conversion versus Wcat =FCH4 at different operating pressure for FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and T 5 800 K

Pressure Distribution. Pressure drop due to porous nature of catalyst is estimated using Blake kozeny correlation DP=L ¼ 150ð1 eÞ2 lf V=dp2 e3 [85]. It is a momentum sink and offers viscous resistance to flow. Pressure field does not highlight the minor pressure variations at the scale smaller than the size of catalyst particle. According to the used correlation, the pressure field depends on the inlet feed velocity, size of catalyst particle and

porosity of the catalyst bed. A catalyst bed with high porosity and large catalyst particles give low pressure gradient across the reformer. Similarly, the low value of inlet feed velocity gives high pressure gradient across the reformer. For the present study, the variation of pressure drop is shown in Fig. 9. In the present study, a reformer of small length (0.114 m) is used; therefore, over all pressure drop is small.

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Fig. 12 Fractional distance from equilibrium versus Wcat =FCH4 at 2 bar and 10 bar for FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and T 5 800 K

Effect of Pressure on SMR. As stated earlier, steam methane reaction is favored by expansion, thus, decreasing the pressure results in an increase in the methane conversion as shown by Fig. 10. If one compares the gradient of fractional methane conversion with Wcat =FCH4 at different pressure, it is clear that the gradient deceases as the pressure increases, indicating that reaction mixture is closer to equilibrium at high pressure but far from equilibrium at low pressure as shown in Fig. 11. To elaborate more on this effect, a comparison of distance from equilibrium at the pressure of 2 bar and 10 bar is conducted and the results are shown in Fig. 12. More contact time or longer reformer is needed to reach equilibrium conversion at low pressure. Moreover a threshold minimum value of distance from equilibrium can be used as an optimization limit so that process can always be thermodynamically limited. To further elaborate on the effect of pressure on the mole fraction of individual species, Fig. 13 shows the increase in mole fraction of CO, CO2 and hydrogen and decrease

Fig. 13 Comparison of species mole fraction along the reformer length at 2 bar and 10 bar for FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, and T 5 800 K



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the mole fraction of CH4 and H2O as the pressure is decreased from 10 bar to 2 bar. Effect of Mass Flow Rate. Increasing the mass flow rate of gas, reduces the contact time over a fixed rector length. Eventually the conversion of the reactants decreases but the way the mass flow rate is decreased also matters. Changing the mass flow rate by changing the flow velocity of the reacting mixture can produce different curves than those resulting from changing the pressure of the system. If the mass flow rate of the reacting mixture is changed by changing the flow pressure of the system, a combined effect of change in pressure and contact time will be obtained on the conversion of reaction. Changing the mass flow rate by changing the flow velocity at the constant pressure gives the sole variation of conversion with respect to contact time. Thus instead of mass flow rate, the variation of conversion with respect to the flow velocity should be studied. Figure 14 shows the variation of methane conversion with respect to the average inlet flow velocity of reacting gas. The general trends of the SMR reaction rates are important. The rate of reactions 1 and 3 are very high at the start of the reformer. This is due to the presences of excess of reactants for both of these reactions. Rates of reactions 1 and 3 decrease gently along the reformer as shown in Fig. 15. The rate of reaction 2 is zero at the start of the reformer due to absence of reactants of this reaction. As reactions 1 and 3 proceed considerably producing reactants for reaction 2, its rate reaches its peak and, then,

decreases gently along the reformer as shown in Fig. 15. Thus, the low concentrations of CO2 and H2 with excess of steam cause the rate of reaction 2 to increase but as the considerable amount of CO2 and H2 is produced by the parallel reactions, rate of reaction 2 starts to decrease. Effect of Relative Molar Ratios of Species. To study the effect of reacting species at the inlet, three ratios FH2 O =FCH4 ; FH2 =FCH4 , and 1=nCH4 have been defined as below FCH4 þ FH2 O þ FH2 ¼ FMIX FCH4 FH2 O FH2 FMIX þ þ ¼ FCH4 FCH4 FCH4 FCH4 1þ

FH2 O F H2 FMIX þ ¼ FCH4 FCH4 FCH4

FH2 O =FCH4 and FH2 =FCH4 ratios are varied from 1 to 4 and 1=nCH4 ratio is varied from 6 to 12. Figures 16 and 17 show the variation of methane conversion with respect to FH2 =FCH4 ratio. It is clear that methane conversion increases with increase in FH2 O =FCH4 ratio but the value of increment depends on either FH2 O =FCH4 ratio is increased on the expense of FH2 =FCH4 ratio or 1=nCH4 ratio. If FH2 O =FCH4 ratio is increased by keeping the FH2 =FCH4 ratio constant as in Fig. 16, the mole fractions of CH4 and hydrogen decrease in the inlet feed and mole fraction of H2O increases. Thus, the methane conversion is always increasing but increase is less because along with hydrogen, methane is also decreasing. On the other hand, if 1=nCH4 ratio and FH2 O =FCH4 ratio are kept constant by decreasing the FH2 =FCH4 ratio as in Fig. 17, mole fraction

Fig. 14 Fractional methane conversion versus average inlet feed velocity for FH2 O =FCH4 5 3, FH2 =FCH4 5 1.25, T 5 800 K, and P 5 10 bar Fig. 16 Fractional methane conversion versus FH2 =FCH4 ratio for constant FH2 =FCH4 ratio, T 5 800 K, and P 5 10 bar

Fig. 15 Kinetic reaction rate versus Wcat =FCH4 for FH2 O = FCH4 5 3, FH2 =FCH4 5 1.25, T 5 800 K, and P 5 10 bar

Fig. 17 Fractional methane conversion versus FH2 O =FCH4 ratio for constant 1=nCH4 ratio, T 5 800 K and P 5 10 bar

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by KFUPM-MIT Clean Water and Clean Energy Research Collaboration center.

Nomenclature

Fig. 18 Fractional methane conversion versus FH2 =FCH4 ratio for constant FH2 O =FCH4 ratio, T 5 800 K and P 5 10 bar

of the methane remains constant. Increasing the FH2 O =FCH4 ratio decreases the hydrogen mole fraction that in turn is going to move all the reaction in forward direction. Thus, methane conversion increases sharply as compared to the precious case. Moreover the variation of methane conversion for constant value of FH2 O =FCH4 ratio is also studied that is varying the FH2 =FCH4 ratio with 1=nCH4 ratio. It is clear from Fig. 18 that for the constant value of FH2 O =FCH4 ratio, increase in FH2 =FCH4 ratio requires a decrease in mole fraction of CH4 and H2O. Thus, the mole fraction of H2 increases. This will make the reaction to proceed in reverse direction. Similarly, if some inert gases are used along with feed, the effect of change in mole fraction of these gases on methane conversion depends on the way their mole fraction is decreased.

Conclusion It is clear from the study that modeling the continuum of catalyst inside the reformer instead of catalyst pellets is a valid assumption and can provide highly accurate results. Moreover, with a careful use of porosity as a decreasing factor for the source term in species transport equations along with the other term in momentum and energy equations, actual trends of fractional methane conversion with different operating parameters can be obtained. Validity up to 99.75% of developed model with experimental data is obtained which showed that the model can be used for further studies of SMR reaction. Parametric studies have shown that longer reformers are required at low pressure to attain the thermodynamically limited conversion. This is basically attributed to the fact that the equilibrium conversion values are high at low pressures. Thus, reactors working at low pressure can achieve high conversion values but this would need enough time (or reactor length) so as to achieve these high equilibrium conversion values. It is also recognized that fractional distance from equilibrium with some minimum threshold value can be used as an optimization limit to get thermodynamically limited conversion. The influence of ratios of reacting species at the inlet is also highlighted and it is shown that not only increasing the FH2 O =FCH4 ratio enhances the methane conversion but the way this ratio is increased also affect the conversion to a considerable extent. The methane conversion varies sharply if FH2 O =FCH4 ratio is varied on expense of FH2 =FCH4 ratio. On the other hand, the change in methane conversion is not sharp enough if FH2 O =FCH4 is varied on the expense FMIX =FCH4 ratio.

Acknowledgment The authors of this article highly appreciate and acknowledge the support of King Fahd University of Petroleum and Minerals for this work through the research Grant No. R12-CE-10 offered

A ¼ area (m2) Ai ¼ pre-exponential factor for Arrhenius equation, dimensions of ko,i Bi ¼ pre-exponential factor for Van’t Hoff equation, dimension of Ki Cp ¼ specific heat capacity of reaction mixture (kJ/kg K) Cpi ¼ specific heat capacity of reaction mixture (kJ/kg K) dp ¼ equivalent particle diameter (m) Di,e ¼ effective diffusive coefficient of specie i (m2/s) Di,T ¼ thermal diffusive coefficient of specie i, (kg/ms) Dm,i ¼ molecular diffusive coefficient of specie i (m2/s) Dn,i ¼ Knudsen diffusive coefficient of specie i (m2/s) DEN ¼ denominator Ei ¼ activation energy of reaction i (kJ/mol) Fi ¼ molar flow rate of specie i (mol/h) Fmix ¼ molar flow rate of reaction mixture (mol/h) hi ¼ enthalpy of specie i (W/mK) Hi ¼ absorption enthalpy of specie i (kJ/mol) ji ¼ diffusive flux of specie i (kg/m2s) ke ¼ effective heat conduction coefficient (W/mK) kf ¼ heat conduction coefficient of reaction mixture (W/mK) ki ¼ heat conduction coefficient of specie i (W/mK) ki ¼ rate constant of reaction i, Table 1 ks ¼ heat conduction coefficient of catalyst (W/mK) Ki ¼ Van’t Hoff constant for specie i, Table 2 Ke,i ¼ equilibrium constant for reaction i, bar2 for i ¼ 1, 3 and 0 for i ¼ 2 Mi ¼ molecular weight of specie i (kg/mol) ni ¼ molar ratio of specie i, Fi/FMIX P ¼ pressure of reaction mixture (N/m2) pi ¼ partial pressure of specie i (bar) r1,r2,r3 ¼ rate of reactions 1, 2, and 3 (kmol/kgcath) R ¼ general gas constant (J/mol K) Ri ¼ rate of reaction of specie i (mol/m3s) S/C ¼ steam to carbon molar ratio T ¼ temperature of reaction mixture (K) To ¼ reference temperature for viscosity calculation (K) V ¼ velocity of reaction mixture (m/s) Wcat ¼ mass of catalyst (kg) Yi ¼ mass fraction of specie, i

Greek Symbols e¼ gi ¼ lf ¼ li ¼ li;o ¼ qf ¼ qi ¼ s¼

porosity of catalyst bed effectiveness factor of reaction, i viscosity of reaction mixture (kg/ms) viscosity of specie i (kg/ms) viscosity of specie at reference temperature (kg/ms) density of reaction mixture (kg/m3) density of specie i (kg/m3) effective shear stress (N/m2)

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