Thermochemical Performance Analysis of the Steam Reforming of Methane in a Fixed Bed Membrane Reformer: A Modelling and Simulation Study

Pd-based membrane reformers have been substantially studied in the past as a promising reformer to produce high-purity H2 from thermochemical conversion of methane (CH4). A variety of research approaches have been taken in the experimental and theoretical fields. The main objective of this work is a theoretical modelling to describe the process variables of the Steam Reforming of Methane (SRM) method on the Pd-based membrane reformer. These process variables describe the specific aims of each equation of the mathematical model characterizing the performance from reformer. The simulated results of the mole fractions of components (MFCs) at the outlet of the Fixed Bed Reformer (FBR) and Packed-Bed Membrane Reformer (PBMR) have been validated. When the H2O/CH4 ratio decreases in PBMR, the Endothermic Reaction Temperature (ERT) is notably increased (998.32 K) at the outlet of the PBMR’s reaction zone. On the other hand, when the H2O/CH4 ratio increases in PBMR, the ERT is remarkably decreased (827.83 K) at the outlet of the PBMR’s reaction zone. An increase of the spatial velocity (Ssp) indicates a reduction in the residence time of reactant molecules inside PBMR and, thus, a decrease of the ERT and conversion of CH4. In contrast, a reduction of the Ssp shows an increase of the residence time of reactant molecules within PBMR and, therefore, a rise of the ERT and conversion of CH4. An increase of the H2O/CH4 ratio raises the conversion rate (CR) of CH4 due to the reduction of the coke content on the catalyst particles. Conversely, a reduction of the H2O/CH4 ratio decreases the CR of CH4 owing to the increase of the coke content on the catalyst particles. Contrary to the CR of CH4, the consumption-based yield (CBY) of H2 sharply decreases with the increase of the H2O/CH4 ratio. An increase of the ERT raises the thermochemical energy storage efficiency (ηtese) from 68.96% (ERT = 1023 K), 63.21% (ERT = 973 K), and 48.12% (ERT = 723 K). The chemical energy, sensible heat, and heat loss reached values of 384.96 W, 151.68 W, and 249.73 W at 973 K. The selectivity of H2 presents higher amounts in the gaseous mixture that varies from 60.98 to 73.18 while CH4 showed lower values ranging from 1.41 to 2.06. Our work is limited to the SRM method. In terms of future uses of this method, new works can be undertaken using novel materials (open-cell foams) and the physical-mathematical model (two-dimensional and three-dimensional) to evaluate the concentration polarization inside membrane reactors.


Introduction
The production of hydrogen (H 2 ) can be carried out through different methods such as thermochemical method (heat and chemical reactions to produce H 2 ), reforming of hydrocarbons, biomass gasification, coal gasification, electrolytic method, and biological method. Usually, the thermochemical reforming methods are used to study the Thermochemical Energy Storage (TES) technology of H 2 . The TES of H 2 can be produced from reforming topic is a very relevant in the literature. In comparison with traditional methodologies such as finite element, finite volume, etc., that have already been used before in the literature, our methodology can provide results faster than traditional methods and, therefore, the novelty of the present work lies in the determination of the solution method. A comparative analysis had been driven to investigate ERT, CR of CH 4 , and feed-based yield (FBY) of H 2 inside FBR and PBMR. The effects of the H 2 O/CH 4 ratio and S sp on the ERT were numerically investigated in PBMR. After checking the effects of the H 2 O/CH 4 ratio and S sp on the ERT, the effects of these parameters were also studied on the CR of CH 4 and CBY of H 2 . In addition, the ERT's effect was verified on the η tese , chemical energy, sensible heat, and heat loss. In addition, the selectivity of components (H 2 , CO, CO 2 , and CH 4 ) was computed in PBMR.

Physical-Mathematical Model
A schematic setup is used to study the SRM method's thermochemical conversion in PBMR according to Figure 1. The simplified setup from Figure 1 involves a heating module (electric furnace), input reagents (CH 4 , H 2 O), Sweep gas (N 2 ), reaction zone, permeation zone, and outlet products (CH 4 , H 2 O, H 2 , CO, and CO 2 ). The physical setup of the PBMR is built by two concentric tubes according to Figure 1. The inner tube consists of a thin palladium (Pd) dense membrane which contains a permeation zone receiving H 2 from the reaction zone through the Pd-based dense membrane. The catalyst loading is placed between the tubes in the annular zone, named the fixed-bed.
Membranes 2020, 10, x FOR PEER REVIEW 3 of 28 With the purpose of reducing the research cost and project time, mathematical modelling and computer simulation are extensively used to obtain a better understanding of design parameters in reformers. The approach and solution of physical-mathematical models are still a novelty of membrane reformers to obtain sustainable clean H2 and, thus, the topic is a very relevant in the literature. In comparison with traditional methodologies such as finite element, finite volume, etc., that have already been used before in the literature, our methodology can provide results faster than traditional methods and, therefore, the novelty of the present work lies in the determination of the solution method. A comparative analysis had been driven to investigate ERT, CR of CH4, and feedbased yield (FBY) of H2 inside FBR and PBMR. The effects of the H2O/CH4 ratio and Ssp on the ERT were numerically investigated in PBMR. After checking the effects of the H2O/CH4 ratio and Ssp on the ERT, the effects of these parameters were also studied on the CR of CH4 and CBY of H2. In addition, the ERT's effect was verified on the ηtese, chemical energy, sensible heat, and heat loss. In addition, the selectivity of components (H2, CO, CO2, and CH4) was computed in PBMR.

Physical-Mathematical Model
A schematic setup is used to study the SRM method's thermochemical conversion in PBMR according to Figure 1. The simplified setup from Figure 1 involves a heating module (electric furnace), input reagents (CH4, H2O), Sweep gas (N2), reaction zone, permeation zone, and outlet products (CH4, H2O, H2, CO, and CO2). The physical setup of the PBMR is built by two concentric tubes according to Figure 1. The inner tube consists of a thin palladium (Pd) dense membrane which contains a permeation zone receiving H2 from the reaction zone through the Pd-based dense membrane. The catalyst loading is placed between the tubes in the annular zone, named the fixed-bed.

Electric Power of the Electric Furnace
In Figure 1, a resistive loading inside of an electric furnace has been used to heat the FBMR's reaction zone and therefore the thermal energy storage is used to drive the reforming reactions. The electric power provided by the electric furnace is given for Equation (1) as follows.
( ) 2 elet. elet. ghe she er g P Ri The thermochemical energy storage is obtained by subtracting the product enthalpy (reaction heat) from the reagent enthalpy at room temperature. Thus, the PBMR's chemical energy is obtained using Equation (2) as follows.

Electric Power of the Electric Furnace
In Figure 1, a resistive loading inside of an electric furnace has been used to heat the FBMR's reaction zone and therefore the thermal energy storage is used to drive the reforming reactions. The electric power provided by the electric furnace is given for Equation (1) as follows. P elet. = R i 2 elet. = U ghe S she T er − T g (1) Membranes 2021, 11, 6 4 of 26 The thermochemical energy storage is obtained by subtracting the product enthalpy (reaction heat) from the reagent enthalpy at room temperature. Thus, the PBMR's chemical energy is obtained using Equation (2) as follows.
After cooling the products to the room temperature, the sensible heat of the products (ranging from outer temperature to the room temperature) can be used and therefore the sensible heat can be computed from Equation (3) as follows.
From Equations (1)- (3), it is possible to estimate the heat loss using Equation (4) as follows.

Thermochemical Kinetic Model
The reforming reactions of CH 4 are used to produce syngas (H 2 and CO) and they are highly endothermic [3]. The SRM method has a limited equilibrium and it comprises three major reactions as follows.
The global rate equations of the three reactions, Equations (5)- (7), are based on the Langmuir-Hinshelwood kinetic model [3]. The kinetic rates from Equations (5)-(7) are considered more general for nickel (Ni) catalyst and, therefore, the equations of the SRM method are presented as: where β is given by Equation (11)  The partial pressures of chemical components i, i = CH 4 , H 2 O, CO, CO 2 and H 2 , from Equations (8)- (11) are computed from Equations (12)-(16) below. where, From Equation (18), the net rates of each chemical component i are obtained by Equations (19)-(23) as follows.

PBMR's Mathematical Modelling
The mathematical modelling inside PBMR's reaction zone is described through the NIPHD model. The development of the NIPHD model takes into account the following assumptions: (1) the NIPHD model is described under non-isothermal conditions inside the reaction zone, (2) the NIPHD model in the reaction zone is plug-flow with axial dispersion under transient condition, (3) the radial dispersion is negligible inside the reaction zone, (4) the gaseous mixture has constant density inside the reaction zone from PBMR, (5) the membrane is considered to be 100% H 2 -permselectivity, i.e., the selectivity of H 2 is typically very high in dense metallic membranes, (6) the heat exchange between the reaction zone and permeation zone is negligible, (7) the molar flow rates in the reaction zone and permeation zone are constant, (8) the deposition effect of carbon at the surface of catalytic particles has been neglected, (9) the gas behavior in the reaction zone from PBMR was considered as an ideal gas mixture, (10) the bed porosity in the axial direction is considered constant, and (11) chemical reactions are assumed to take place at the surface of catalyst particles.
These premises are used to build the governing equations of the NIPHD model in PBMR's reaction zone and permeation zone as follows.

Energy Balance of the Gas Phase in Reaction Zone
The developed equation provides clear information to drive the temperature distribution of the gas phase in porous medium from PBMR's reaction zone. The energy transport in the gas phase inside the reaction zone is characterized by a balance equation in PBMR's axial direction. Thus, a one-dimensional dynamic equation is modelled for the temperature of the gas phase as follows.
-Energy balance in the gas phase ρ g,mix. C p,g,mix.
∂T g ∂t The gas phase's effective thermal conductivity is defined as a function of the gaseous mixture thermal conductivity as follows.
The suitable initial and boundary conditions from Equations are given as follows.

Energy Balance of the Solid Phase in Reaction Zone
The spherical particle's tortuous structure in the reaction zone could give rise to turbulences with an increase in heat transfer between the solid and gas phases. The thermal energy storage takes place on the solid particles to ensure sufficient energy for processing the endothermic reactions from the SRM method. A promising point is reported by thermal interactions at the surface of catalyst particles where SRM reactions are thermochemically converted. However, the energy balance for the temperature of reforming reactions at the surface of catalytic particles is given as follows.
-Energy balance at the surface of catalytic particles The solid phase's effective thermal conductivity is defined as a function of the thermal conductivity of the gaseous mixture according to Equation below. (31) The suitable initial and boundary conditions from Equations are given as follows.
-Initial condition at t = 0 The suitable initial and boundary conditions from Equations are presented as follows.
-Initial condition, i.e., t = 0 -At the inlet face surface (upper) of the reaction zone from PBMR, i.e., z = 0 + -At the outlet face surface (bottom) of the reaction zone from PBMR, i.e., z = L The permeation rate of H 2 through the membrane from the high-pressure zone into the permeation zone is assumed to obey the half power pressure law. However, the permeation rate of H 2 from the reaction zone into the permeation zone is given as follows.
A differential model allows us to quantify the amount of H 2 in the permeation side, but the model has to be consistent with the permeation rate which passes through the Pdbased dense membrane. Thus, a transport equation is developed to estimate the production of H 2 in the permeation zone from PBMR as follows. The CIEA method can be considered as a powerful technique because of its low computer time relative to traditional methods (finite difference, finite volume, finite element, etc.). The CIEA methodology has been used to transform the NPDE system (Equations (24), (30), (35) and (39)) into an NODE system using the boundary conditions (Equations (28), (29), (33), (34), (37), (38), (41) and (42)) of each NPDE. The coefficients of Equations (47)-(50) can be found in Appendix A of this work. Thus, NODEs (Equations (47)-(50)) are reported as follows.
-Transformed NODE for chemical components i in reaction zone -Transformed NODE for H 2 in reaction zone

Approximation of the Full Solution
Several numerical methods have been proposed to solve NPDE systems [15]. The numeric methodology's selection is limited to the desired accuracy on the consistency and robustness of numerical data of the NPDE system. Regarding NODEs, Equations (47)-(50) have been solved by the Runge-Kutta Gill method as well as the NODE in the permeation zone (Equation (44)). On the other hand, the full solution is obtained from Equations (51)-(54) as follows.
-Gas phase's full solution -Full solution for chemical components H 2 .

Model Parameters for Simulations
A physical-mathematical model has been used to investigate the SRM method's thermochemical conversion in PBMR using an external energy loading. A mathematical model is developed to simulate the energy transfer of the gaseous and solid phases and transport of chemical components coupled to the SRM method's thermochemical kinetic model in PBMR. A computational algorithm using the FORTRAN 95 language has been elaborated by the authors to compute the results as in the model equations of this work. In Tables 1 and 2, the geometrical characteristics from PBMR, catalytic bed's properties, and operating conditions at the inlet from PBMR are shown.  Table 2. Operating conditions at the inlet from PBMR.

PBMR Sources
Operating conditions Ref. [3] In order to ensure good results of the SRM method on PBMR, the convergence criterion for all results of this work has been secured using the ratio between the new variable value and the old variable value according to the new variable value as follows.
After specifying the main geometrical characteristics from PBMR, the catalytic bed's properties and operating conditions at the inlet from PBMR are shown in the above tables. Tables 3 and 4 show the numerical values of kinetic constants, equilibrium adsorption constants, equilibrium constants, thermophysical parameters, and dispersion coefficients of chemical components i in the reaction zone from PBMR. The parameter values of Tables 1-4 are used to feed the developed computational algorithm for this work. Table 3. Numeric values of kinetic constants, adsorption constants, and equilibrium constants for simulating of the SRM method on PBMR. Table 4. Numerical values of thermophysical parameters and dispersion coefficients of chemical components for simulating the SRM method in PBMR.

Model Parameters Sources
Thermophysical parameters  The CIEA method has been used to simulate the results of the SRM method in PBMR and FBR. The CIEA method can be considered as a potential candidate for solving an NPDE system at lower CPU time. Table 5 shows the results of the SRM method in PBMR and FBR and, thus, these results are compared with the Finite Volume (FV) method against the CIEA method.

Temperature Profiles of Endothermic Reactions
The Temperature Profiles of Endothermic Reactions (TPERs) have been computed inside FBR and within the reaction zone from PBMR and can be seen in Figure 3. It was shown that TPERs tend to assume inflection points of minimum values at which the SRM method's minimum temperatures  The SRs for these two cases of the MFCs i are in agreement good with the literature data available in Ref. [23]. Slight differences can be found due to the deviation between the literature results and simulating results. An average relative error (ARE), Equation (57), was used to compute the consistency criterion between the results obtained and this ARE is given as follows.

ARE = MFCs
Ref. [ The good accordance between the SRs and literature data show that the developed model is acceptable. Considering the studied cases in Figure 2a,b, we obtained consistent satisfactory results from MFCs i against experimental results from the literature [23], resulting in AREs of 7.35% ≤ ARE CH 4 The CIEA method has been used to simulate the results of the SRM method in PBMR and FBR. The CIEA method can be considered as a potential candidate for solving an NPDE system at lower CPU time. Table 5 shows the results of the SRM method in PBMR and FBR and, thus, these results are compared with the Finite Volume (FV) method against the CIEA method. The Temperature Profiles of Endothermic Reactions (TPERs) have been computed inside FBR and within the reaction zone from PBMR and can be seen in Figure 3. It was shown that TPERs tend to assume inflection points of minimum values at which the SRM method's minimum temperatures are found due to the effect of endothermic reactions. The location of the minimum values of these TPERs could be due to the interaction of many factors as the catalytic bed's composition of the reaction zone from PBMR, initial temperature, operating pressure, and the thermodynamic equilibrium of endothermic reactions. It is clearly shown that the ERT of the FBR is much higher than the ERT in the reaction zone from PBMR. As it was reported in Figure 3, the TPER in PBMR's reaction zone reached the stable state (at about z/L = ±0.5) faster than FBR. After achieving the stable state, ERT is kept constant up to z/L = 1.0 for FMBR. On the other hand, the TPER in FBR achieved the stable state at about z/L = ±0.85. As an advantage from PBMR compared to FBR, the thermodynamic equilibrium of reforming reactions in PBMR is obtained for a lesser reaction temperature due to the removal of H 2 through Pd-based dense membrane.
Membranes 2020, 10, x FOR PEER REVIEW 14 of 28 are found due to the effect of endothermic reactions. The location of the minimum values of these TPERs could be due to the interaction of many factors as the catalytic bed's composition of the reaction zone from PBMR, initial temperature, operating pressure, and the thermodynamic equilibrium of endothermic reactions. It is clearly shown that the ERT of the FBR is much higher than the ERT in the reaction zone from PBMR. As it was reported in Figure 3, the TPER in PBMR's reaction zone reached the stable state (at about z/L = ± 0.5) faster than FBR. After achieving the stable state, ERT is kept constant up to z/L = 1.0 for FMBR. On the other hand, the TPER in FBR achieved the stable state at about z/L = ± 0.85. As an advantage from PBMR compared to FBR, the thermodynamic equilibrium of reforming reactions in PBMR is obtained for a lesser reaction temperature due to the removal of H2 through Pd-based dense membrane.  Figure 4 describes the PBMR's inlet H2O/CH4 ratio as an important parameter on the ERT inside reaction zone from PBMR. For PBMR, the decrease of the H2O/CH4 ratio has a negative effect on the ERT within the reaction zone. Incidentally, the main objective of PBMR is to carry out the thermochemical conversion of CH4 at moderate temperature because of the shift of the thermodynamic equilibrium on account of the removal of H2 through the membrane. Different values of the H2O/CH4 ratio were chosen for the comparison. When the H2O/CH4 ratio at the inlet from PBMR is low (H2O/CH4 = 0.95), the heat absorption in reaction zone section is notably increased and therefore, the ERT is favored. the heat absorption in reaction zone section is notably increased and therefore, the ERT is favored.
3.3.2. Effect of H2O/CH4 Ratio on the ERT Figure 4 describes the PBMR's inlet H2O/CH4 ratio as an important parameter on the ERT inside reaction zone from PBMR. For PBMR, the decrease of the H2O/CH4 ratio has a negative effect on the ERT within the reaction zone. Incidentally, the main objective of PBMR is to carry out the thermochemical conversion of CH4 at moderate temperature because of the shift of the thermodynamic equilibrium on account of the removal of H2 through the membrane. Different values of the H2O/CH4 ratio were chosen for the comparison. When the H2O/CH4 ratio at the inlet from PBMR is low (H2O/CH4 = 0.95), the heat absorption in reaction zone section is notably increased and therefore, the ERT is favored.   Figure 4 shows a symmetry region in relation to inflection points of TPERs. Before achieving the inflection points, there is a decrease of the heat rate (q heat < 0) due to the heat consumption for reforming reactions. After passing the inflection points, there is an increase of the heat rate (q heat > 0) to keep the thermodynamic equilibrium of reforming reactions.

Effect of the S sp on the ERT
After investigating the H 2 O/CH 4 ratio's effect on the ERT, authors also studied the effect of the S sp on the ERT inside reaction zone from PBMR. The S sp is inversely proportional to the residence time. Thus, an increase of the S sp indicates a decrease in the residence time of reactants within reaction zone from PBMR and, therefore, a reduction in the thermochemical conversion of CH 4 . In contrast, a decrease of the S sp indicates an increase of the residence time (higher contact time between catalyst and reactants) of reactants in reaction zone from PBMR and, thus, a rise in the thermochemical conversion of CH 4 . As a result, Figure 5 shows that the reduction of the S sp increases the ERT. An increase of the ERT has a positive effect on the thermochemical conversion of CH 4 and yield of H 2 , i.e., the conversion of CH 4 and yield of H 2 increase with the rise of the ERT [3]. As it was analyzed in in Figure 4, three different values of the S sp were used to check the sensibility of the ERT to the S sp . Figure 5 reports that the S sp is inversely proportional to the ERT, i.e., when the S sp is lower (S sp,ref. = 15,000 h −1 ), the ERT is notably increased. In contrast, when S sp is higher (S sp = 1. i.e., the conversion of CH4 and yield of H2 increase with the rise of the ERT [3]. As it was analyzed in in Figure 4, three different values of the Ssp were used to check the sensibility of the ERT to the Ssp. Figure 5 reports that the Ssp is inversely proportional to the ERT, i.e., when the Ssp is lower (Ssp,ref. = 15,000 h −1 ), the ERT is notably increased. In contrast, when Ssp is higher (Ssp = 1.6 Ssp,ref = 24,000 h −1 ), the ERT is remarkably reduced. After achieving the stable state (z/L = ± 0.70), ERTs are kept constant in reaction zone up to z/L = 1.0 from PBMR with values of 729.35 K (Ssp = 24,000 h −1 ), 816.47 K (Ssp = 19,500 h −1 ), and 885.98 K (Ssp,ref. = 15,000 h −1 ), respectively.  Figure 6 investigates the profiles of mole fractions for each component i of the SRM method in reaction zone and permeation zone from PBMR at the following operating conditions: H2O/CH4 = 3.00, 950 kPa, 973 K, and 4.473 × 10 −6 (m 3 /h). In this figure, the profiles of mole fractions from consumed reactants (H2O and CH4) and produced products (CO, CO2 and H2) have been reported in the reaction zone as well as the profile of mole fraction of H2 in the permeation zone. H2O was not fully consumed for the SRM method within the reaction zone from PBMR, i.e., after reaching a stable consumption (z/L = ± 0.17), only 18.67% was spent. On the other hand, CH4 was completely consumed for the SRM method inside the reaction zone from PBMR, after achieving a stable consumption (z/L = ± 0.37), an amount of 97.26% was consumed. After reaching the stable mole fractions (z/L = ± 0.52)  Figure 6 investigates the profiles of mole fractions for each component i of the SRM method in reaction zone and permeation zone from PBMR at the following operating conditions: H 2 O/CH 4 = 3.00, 950 kPa, 973 K, and 4.473 × 10 −6 (m 3 /h). In this figure, the profiles of mole fractions from consumed reactants (H 2 O and CH 4 ) and produced products (CO, CO 2 and H 2 ) have been reported in the reaction zone as well as the profile of mole fraction of H 2 in the permeation zone. H 2 O was not fully consumed for the SRM method within the reaction zone from PBMR, i.e., after reaching a stable consumption (z/L = ±0.17), only 18.67% was spent. On the other hand, CH 4 was completely consumed for the SRM method inside the reaction zone from PBMR, after achieving a stable consumption (z/L = ±0.37), an amount of 97.26% was consumed. After reaching the stable mole fractions (z/L = ±0.52) of H 2 in reaction zone and permeation zone, we obtained quantities of 29.49% (reaction zone) and 29.48% (permeation zone), respectively. Similarly, after attaining the stable mole fractions of CO 2 (z/L = ±0.57) and CO (z/L = ±0.4) in the reaction zone, we computed the amounts of 14.23% and 2.76%, respectively.

Conversion Rate of CH4 and Feed-Based Yield of H2 in FBR and FDMR
Thermodynamic limitations of FBRs are considered as a great problem to increase the thermochemical conversion of the SRM method. To solve this gap, we used PBMRs as innovating equipment that act with permselective membranes to overcome thermodynamic limitations and thus get a high conversion rate of CH4 at lower temperature [24]. The thermochemical performance of FBR and PBMR is analyzed from the Conversion Rate (CR) of CH4, and Feed-Based Yield (FBY) of H2 as follows.

Conversion Rate of CH 4 and Feed-Based Yield of H 2 in FBR and FDMR
Thermodynamic limitations of FBRs are considered as a great problem to increase the thermochemical conversion of the SRM method. To solve this gap, we used PBMRs as innovating equipment that act with permselective membranes to overcome thermodynamic limitations and thus get a high conversion rate of CH 4 at lower temperature [24]. The thermochemical performance of FBR and PBMR is analyzed from the Conversion Rate (CR) of CH 4  (59) Figure 7a compares the CR of CH 4 inside FBR and CR of CH 4 in reaction zone from PBMR. It is clearly shown that the thermochemical CR rate of CH 4 in FBR is lower while a substantial improvement in the thermochemical CR of CH 4 is achieved by PBMR. As a result, after achieving the stable state at about z/L = ±0.20 (see Figure 7a), the CR of CH 4 is kept constant until z/L = 1.0 of FBR with a value of ±0.58 at the operating conditions of H 2 O/CH 4 = 3.00, 950 kPa, 973 K, and 4.473 × 10 −6 m 3 /h. A higher CR of CH 4 in reaction zone from PBMR is favored by the shift in the thermodynamic equilibrium according to LeChatelier's principle, sweep gas flux, change of concentration, operating temperature, and reduction of the partial pressure of H 2 in the separation side. On the other hand, it is clearly shown that the CR of CH 4 in reaction zone from PBMR is remarkably enhanced. After reaching the stable state at about z/L = ±0.80 (see Figure 7a), the CR of CH 4 is maintained constant up to z/L = 1.0 in reaction zone from PBMR with a value of ±0.95 at the same operating conditions of FBR.

Effect of the Ssp on the CR of CH4
The process of reforming reactions of CH4 includes the adsorption of the reactant composition in the gaseous phase, desorption of mixture gas, and residence time of gaseous reactants.
As the Ssp is inversely proportional to the residence time, an increase on the Ssp will reduce the residence time for the gaseous reactants on the catalyst particles. Conversely, a reduction of the Ssp points to an increase of the residence time of gaseous reactants to react on the catalyst particles and, thus, a rise in the CR of CH4. In order to understand the effect of the Ssp on the CR of CH4 (see Equation

Effect of the S sp on the CR of CH 4
The process of reforming reactions of CH 4 includes the adsorption of the reactant composition in the gaseous phase, desorption of mixture gas, and residence time of gaseous reactants.
As the S sp is inversely proportional to the residence time, an increase on the S sp will reduce the residence time for the gaseous reactants on the catalyst particles. Conversely, a reduction of the S sp points to an increase of the residence time of gaseous reactants to react on the catalyst particles and, thus, a rise in the CR of CH 4 . In order to understand the effect of the S sp on the CR of CH 4 (see Equation (58) Figure 9a shows the profiles of the CR of CH4 at three different values of the H2O/CH4 ratio in reaction zone from PBMR. The CR of CH4 can be notably reduced with the increase of the coke formation on Ni catalysts. Based on the profiles from Figure 9a, it has been noted that an increase of the H2O/CH4 ratio raises the CR of CH4 due to the reduction of the coke content on the external surface of the catalyst particles. On the other hand, a decrease of the H2O/CH4 ratio reduces the CR of CH4 owing to the increase of the coke formation on the outside surface of catalyst particles. As a result, the CR of CH4 was reduced to be at z/L = 1.0 with the values of 94.75% (H2O/CH4 = 3.25), 86.57% (H2O/CH4 = 2.95), and 72.06% (H2O/CH4 = 0.95), respectively. Figure 9b reports the profiles of the CBY of H2 (Equation (60)) at three different values of the H2O/CH4 ratio in reaction zone from PBMR. Unlike of the CR of CH4, the profiles of the CBY of H2 sharply decrease with the increase of the H2O/CH4 ratio. An increase of steam content has a negative effect on the production of H2, i.e., the generating rate of H2 isn't strengthened according to

Thermochemical Energy Storage Efficiency
The reforming reaction's thermochemical conversion of CH4 is a new technology which provides the advantage of high storage densities and minor thermal losses. The ηteses in PBMRs using the SRM method play an important role in energy storage and this will be discussed in this section. The ηtese of the SRM process in PBMR is computed as the ratio between the net chemical energy produced per the input electric power as follows. Figure 10 describes the curves of the ηtese (Equation (61)) at three different values of the ERT in reaction zone from PBMR. A raise of the ERT has a positive effect on the ηtese, i.e., the ηtese is improved according to Figure 10. As a result, it is clearly noted that an increase of the ERT will lead to higher ηteses. The curves of ηteses trend to assume inflection points of maximum values (Pelet. = ±150 W) in which the effective ηtese are optimum. After reaching the maximum values, the ηteses decrease and then they are maintained until Pelet. = 1000 W with values of 68.96% (ERT = 1023 K), 63.21% (ERT = 973 K), and 48.12% (ERT = 723 K), respectively.

Thermochemical Energy Storage Efficiency
The reforming reaction's thermochemical conversion of CH 4 is a new technology which provides the advantage of high storage densities and minor thermal losses. The η teses in PBMRs using the SRM method play an important role in energy storage and this will be discussed in this section. The η tese of the SRM process in PBMR is computed as the ratio between the net chemical energy produced per the input electric power as follows. Figure 10 describes the curves of the η tese (Equation (61)) at three different values of the ERT in reaction zone from PBMR. A raise of the ERT has a positive effect on the η tese , i.e., the η tese is improved according to Figure 10. As a result, it is clearly noted that an increase of the ERT will lead to higher η teses . The curves of η teses trend to assume inflection points of maximum values (P elet. = ±150 W) in which the effective η tese are optimum. After reaching the maximum values, the η teses decrease and then they are maintained until P elet. = 1000 W with values of 68.96% (ERT = 1023 K), 63.21% (ERT = 973 K), and 48.12% (ERT = 723 K), respectively.
Membranes 2020, 10, x FOR PEER REVIEW 20 of 28 Figure 10. Effect of the ERT on the ηtese of the SRM method in PBMR on Ni/Mg/γ-Aℓ2O3 with 6% Ni loading.

Energy Storage Performance
The energy storage's importance has motivated researchers of this work to study the energetic aspects of the storing technology as from thermochemical conversion. The chemical energy storage, sensible heat, and heat loss play important roles in the energy storage process. Figure 11 shows the energy storage performances of the thermochemical reforming method of CH4 for different reaction temperatures in reaction zone from PBMR. As it can be seen in this figure, as the ERT increases the chemical energy and the heat loss increase drastically. On the other hand, the sensible heat gradually increases as the ERT rises. After reaching the ERT of 973 K, the chemical energy, sensible heat, and heat loss had values of 384.96 W, 151.68 W, and 249.73 W, respectively.

Selectivity of Components of the SRM Method
The loading process of thermal energy on PBMR is used to drive the endothermic reactions of CH4. The thermal energy is thermochemically employed to convert reactants (CH4 and H2O) into products (H2, CO, and CO2). As a result, the performance of components (CH4, H2, CO, and CO2) of the SRM method were analyzed in terms of the CR of CH4, FBY of H2, CBY of H2, and selectivity of H2, CO, CO2 and CH4. A set of corresponding expressions are used to compute the selectivity of H2, CO, CO2, and CH4 as follows.

Energy Storage Performance
The energy storage's importance has motivated researchers of this work to study the energetic aspects of the storing technology as from thermochemical conversion. The chemical energy storage, sensible heat, and heat loss play important roles in the energy storage process. Figure 11 shows the energy storage performances of the thermochemical reforming method of CH 4 for different reaction temperatures in reaction zone from PBMR. As it can be seen in this figure, as the ERT increases the chemical energy and the heat loss increase drastically. On the other hand, the sensible heat gradually increases as the ERT rises. After reaching the ERT of 973 K, the chemical energy, sensible heat, and heat loss had values of 384.96 W, 151.68 W, and 249.73 W, respectively.
Membranes 2020, 10, x FOR PEER REVIEW 20 of 28 Figure 10. Effect of the ERT on the ηtese of the SRM method in PBMR on Ni/Mg/γ-Aℓ2O3 with 6% Ni loading.

Energy Storage Performance
The energy storage's importance has motivated researchers of this work to study the energetic aspects of the storing technology as from thermochemical conversion. The chemical energy storage, sensible heat, and heat loss play important roles in the energy storage process. Figure 11 shows the energy storage performances of the thermochemical reforming method of CH4 for different reaction temperatures in reaction zone from PBMR. As it can be seen in this figure, as the ERT increases the chemical energy and the heat loss increase drastically. On the other hand, the sensible heat gradually increases as the ERT rises. After reaching the ERT of 973 K, the chemical energy, sensible heat, and heat loss had values of 384.96 W, 151.68 W, and 249.73 W, respectively.

Selectivity of Components of the SRM Method
The loading process of thermal energy on PBMR is used to drive the endothermic reactions of CH4. The thermal energy is thermochemically employed to convert reactants (CH4 and H2O) into products (H2, CO, and CO2). As a result, the performance of components (CH4, H2, CO, and CO2) of the SRM method were analyzed in terms of the CR of CH4, FBY of H2, CBY of H2, and selectivity of H2, CO, CO2 and CH4. A set of corresponding expressions are used to compute the selectivity of H2, CO, CO2, and CH4 as follows.

Selectivity of Components of the SRM Method
The loading process of thermal energy on PBMR is used to drive the endothermic reactions of CH 4 . The thermal energy is thermochemically employed to convert reactants (CH 4 and H 2 O) into products (H 2 , CO, and CO 2 ). As a result, the performance of components (CH 4 , H 2 , CO, and CO 2 ) of the SRM method were analyzed in terms of the CR of CH 4

Conclusions and Future Work
The present work has been focused on a numerical analysis of physical-mathematical modelling and computer simulation to describe the performance of reformers for the production of H 2 using a reference method of steam reforming CH 4 . The model equations that describe the gas temperature in the reaction zone, endothermic reaction's temperature in the reaction zone, molar flow of the components i in the reaction zone, molar flow of H 2 in the reaction zone, and molar flow of H 2 in the permeation zone have been reported and discussed. As a solution to the model equations, the main focus has been the CIEA method as a powerful technique to reduce the NPDE system of this work into a NODE system using the boundary conditions of each NPDE. The work's results highlighted the importance of the mathematical model developed to describe the performance from FBR and PBMR. In this context, the main conclusions are summarized as follows. In future research, the exploration of novel materials like open-cell foams can be explored in the context of reforming reactions. Solid open-cell foams constitute a class of porous materials with low density and improved thermal properties. In addition, open-cell foams can be considered as potential candidates for catalyst support in the gassolid reaction field due to their high external surface area, high porosity, and low drop pressure. Thus, solid open-cell foams are future trends for reforming methods such as steam reforming of CH 4 , dry reforming of CH 4 , etc.
Author Contributions: This work has not been published elsewhere. We attest to the fact that all Authors reported on the title page have contributed significantly to the work, have read the manuscript, attest to the validity and legitimacy of the data and agree to its publication in the Journal of Membranes. All authors have read and agreed to the published version of the manuscript.
Funding: The authors of this paper would like to thank CNPq (National Council of Scientific and Technological Development) for the financial support given (Process 48354/2012).

Conflicts of Interest:
Authors have no affiliation with any institution with a direct or indirect financial interest in the matter reported in the manuscript. However, the interest of authors is simply academic. Permeation rate of H 2 from the reaction zone into the permeation zone (kmol/m 2 h) k 1 Reaction constant of the R 1 (kmol kPa 0.5 /kgcat h) k 2 Reaction constant of the R 2 (kmol kPa −1 /kgcat h) k 3 Reaction constant of the R 3 (kmol kPa 0.5 /kgcat h) K eq., 1 Equilibrium constant of Equation (8), (kPa 2 ) K eq., 2 Equilibrium constant of Equation (9), (-) K eq., 3 Equilibrium constant of Equation (10), (kPa 2 ) k gs,eff.
Electric Power of the electric furnace (W) P i Partial pressures of components i, i = CH 4 , H 2 O, CO, CO 2 and H 2 , (kPa) P op.
Operating pressure inside the reaction zone (kPa) P op,per.
Operating pressure within the permeation zone (kPa) P H 2 ,rz Partial pressure of H 2 in the reaction zone (kPa 0.5 ) P H 2 ,per.
Partial pressure of H 2 in the permeation zone (kPa 0.5 ) Q che.
Chemical energy of the PBMR (W) Q 0 Pre-exponential factor of the Arrhenius law (kPa 0.5 /m h) Q eq. Chemical energy storage as enthalpy (W) q g Flow rate of the gas phase (m 3  Fixed-bed porosity (m 3 gas/m 3 reformer) ε p Particle porosity (m 3 particle/m 3 reformer) η j Effectiveness factors from the reforming reactions j (-) λ g,mix.
Thermal conductivity of the gaseous mixture (W/m K) λ g,eff Effective thermal conductivity of the gas phase (W/m K) λ s,eff Effective solid thermal conductivity of the solid phase (W/m K) ρ g, mix.
Gas mixture density of components i (kg/m 3 ) ρ s Solid phase density (kg/m 3 ) σ Dimensionless parameter given by Equation (17)