DOI: 10.3303/CET24114182 Paper Received: 22 July 2024; Revised: 28 August 2024; Accepted: 01 December 2024 Please cite this article as: Bellal M.N., Saouli O., Masbah A., Drede A., 2024, Modeling of a Non-Isothermal Fixed Bed Reactor: 1D Model for Hydrogen Production by Steam Methane Reforming, Chemical Engineering Transactions, 114, 1087-1092 DOI:10.3303/CET24114182 CHEMICAL ENGINEERING TRANSACTIONS VOL. 114, 2024 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Petar S. Varbanov, Min Zeng, Yee Van Fan, Xuechao Wang Copyright © 2024, AIDIC Servizi S.r.l. ISBN 979-12-81206-12-0; ISSN 2283-9216 Modeling of a Non-Isothermal Fixed Bed Reactor: 1D Model for Hydrogen Production by Steam Methane Reforming Bellal Mohamed Nazim*, Saouli Ouacil, Masbah Aflah, Drede Abdlehamid Laboratoire de Génie des Procédés pour le Développement Durable et les Produits de Santé (GPDDPS), Département Génie des Procédés, Ecole Nationale Polytechnique de Constantine, Algeria. nazimnazim2@gmail.com A numerical study was conducted to assess the overall heat transfer coefficient for exterior/reactor convection energy transfer in a steady-state, one-dimensional, pseudo-homogeneous, non-isothermal model of hydrogen production by steam methane reforming (SMR). The study primarily focuses on comparing a catalytic packed- bed reactor (PBR) with a Pd-based catalytic membrane reactor (MR). A selectively permeable Pd-based membrane is utilized to extract hydrogen from the reaction zone, promoting a shift in the thermodynamic equilibrium towards hydrogen production and facilitating higher methane conversion. The hydrogen is then transported away by a sweep gas, typically H2O. The mathematical models were simulated using MATLAB. Additionally, the influence of key operating parameters was investigated using comprehensive reactor models. 1. Introduction Hydrogen is gaining increasing significance due to its use in automobile and household fuel cells. The most common method of hydrogen production involves utilizing fossil fuels such as natural gas and coal, due to their high efficiency, low heating value, and cost-effectiveness. However, there is growing interest in using renewable feedstocks like biomass, biogas, and bioethanol. Literature contains studies on isothermal membrane reactors (Kuncharam and Dixon, 2020).However, considering the highly endothermic nature of the methane steam reforming process, investigations have been conducted on non-isothermal models (Daymo et al., 2024). Continuous heat supply is required to maintain an appropriate process temperature, typically achieved by hot gas flowing along the outer wall of the reactor. Previous studies have adopted different assumptions for the overall heat transfer coefficient (UOR). (Madia et al., 1999) assumed a constant UOR of 817.2 kJ/(m²·h·K) across the reactor, irrespective of operating conditions. Similarly, (Wu et al., 2023) performed a numerical investigation of steam methane reforming in a packed bed reactor Installed with Metal Foam with a fixed UOR for each foam model, (Dong et al., 2019) performed an optimization analysis within the range of 241.56–946.8 kJ/(m²·h·K), while. (Park et al., 2019) used a fixed value of 360 kJ/(m²·h·K). However, these approaches do not account for the dynamic variations in heat transfer influenced by fluid dynamics and varying operating conditions along the reactor length. The objective of this study is to improve the accuracy of the steam methane reforming simulation by integrating a variable UOR into a comparative numerical simulation of two reactor types: a packed-bed reactor (PBR) and a membrane reactor (MR) with co-current flow. By focusing on methane (CH₄) conversion, and the effects of multiple parameters such as inlet flow rate, steam to methane ratio, and the inlet temperature, this approach aims to enhance operational conditions using a non-isothermal model to account for temperature variations. 2. Mathematical Model Reaction Kinetics In general, steam methane reforming is modeled through three key global reactions: methane reforming reaction (Endothermic): CH4 + H2O ↔ CO + 3H2 ∆H0298= 206kJ/mol. (1) 1087 water-gas shift reaction (Exothermic): CO + H2O ↔ CO2 + H2 ∆H0298= −41kJ/mol. (2) Overall Reaction: CH4 + 2H2O ↔ CO2 + 4H2 ∆H0298= 165kJ/mol. (3) Since the reforming process is endothermic, thermal energy input is necessary to sustain the conversion of methane into hydrogen. These reactions are modeled using the Langmuir-Hinshelwood mechanism (Xu and Froment, 1989), with rate expressions as follows: 𝑟𝑟1 = 𝑘𝑘1 𝑃𝑃𝐻𝐻2 2.5 ∙ �𝑃𝑃𝐶𝐶𝐻𝐻4 ∙ 𝑃𝑃𝐻𝐻2𝑂𝑂 − 𝑃𝑃𝐻𝐻2 3 ∙ 𝑃𝑃𝑐𝑐𝑐𝑐 𝐾𝐾1 � 𝐷𝐷2� (4) 𝑟𝑟2 = 𝑘𝑘2 𝑃𝑃𝐻𝐻2 ∙ �𝑃𝑃𝐶𝐶𝑂𝑂 ∙ 𝑃𝑃𝐻𝐻2𝑂𝑂 − 𝑃𝑃𝐻𝐻2 ∙ 𝑃𝑃𝐶𝐶𝑂𝑂2 𝐾𝐾2 � 𝐷𝐷2� (5) 𝑟𝑟3 = 𝑘𝑘3 𝑃𝑃𝐻𝐻2 3.5 ∙ �𝑃𝑃𝐶𝐶𝐻𝐻4 ∙ 𝑃𝑃𝐻𝐻2𝑂𝑂2 − 𝑃𝑃𝐻𝐻2 4 ∙ 𝑃𝑃𝐶𝐶𝑂𝑂2 𝐾𝐾3 � 𝐷𝐷2� (6) Where: 𝐷𝐷 = 1 +𝐾𝐾𝐶𝐶𝑂𝑂𝑃𝑃𝐶𝐶𝑂𝑂 +𝐾𝐾𝐻𝐻2𝑃𝑃𝐻𝐻2 +𝐾𝐾𝐶𝐶𝐻𝐻4𝑃𝑃𝐶𝐶𝐻𝐻4 + 𝐾𝐾𝐻𝐻2𝑂𝑂 𝑃𝑃𝐻𝐻2𝑂𝑂 𝑃𝑃𝐻𝐻2 (7) 𝑘𝑘𝑖𝑖 = 𝐾𝐾𝑖𝑖 = 𝐾𝐾𝑗𝑗 = 𝑎𝑎 ∙ exp (−𝑏𝑏 𝑇𝑇 + 𝑐𝑐) (8) The kinetic parameters are presented in Table 1, knowing that K3= K1·K2 Table 1: Kinetic variables (Xu and Froment, 1989; Saw et al., 2016) variable k1 k2 k3 K1 K2 KCH4 KH2O KCO KH2 a 9.49·1016 4.39·104 2.29·1016 10,266.76 1 6.65·10-6 1.77·103 6.12·10-11 8.23·10-7 b 28,879 8,074.3 29,336 26,830 4,400 4,604.28 10,666.35 9,971.13 8,497.71 c 0 0 0 30.11 -4.063 0 0 0 0 To represent the reactor design, a one-dimensional (1D) configuration was employed similar to the catalytic membrane reactor (MR) setups in previous studies (Iulianelli et al., 2016; Wu et al., 2020). As illustrated in Figure 1, the MR consists of a tubular permeation zone (fed with sweep gas in a co-current mode) and a reaction zone filled with catalyst particles, separated by a membrane. Figure 1: Schematic of catalytic membrane reactor (MR) in co-current mode. The PBR is a tube packed with catalyst particles and heated through the wall. The model for MR consists of mass and energy balances and is based on the following simplifying assumptions: -Steady state condition, -Plug flow conditions apply to both retentate and permeate streams, -Pseudo-homogeneous catalyst bed, -All gas species follow the ideal gas law, -The inlet temperature is the same as the wall temperature: Treact,0=Tperm,0=Tw. The differential equations governing mass and heat balances for both retentate zone and permeate zone are listed below, with Table 2 summarizing their form: 1088 Table 2: differential equation used for the simulation (Wu et al., 2020). Balance zone Equation Mass balance Reaction zone 𝑑𝑑𝑋𝑋𝐶𝐶𝐶𝐶4 𝑑𝑑𝑑𝑑 = 𝜌𝜌𝑐𝑐∙𝐴𝐴𝑐𝑐∙𝐿𝐿 𝐹𝐹0𝐶𝐶𝐶𝐶4 ∙ (𝜂𝜂1 r1 + 𝜂𝜂3 r3) (9) 𝑑𝑑𝑋𝑋𝐶𝐶𝐶𝐶2 𝑑𝑑𝑑𝑑 = 𝜌𝜌𝑐𝑐∙𝐴𝐴𝑐𝑐∙𝐿𝐿 𝐹𝐹0𝐶𝐶𝐶𝐶4 ∙ (𝜂𝜂2 r2 + 𝜂𝜂3 r3) (10) 𝐹𝐹𝐶𝐶𝐻𝐻4=𝐹𝐹 0 𝐶𝐶𝐻𝐻4, 𝑋𝑋𝐶𝐶𝐻𝐻4 = 0 at 𝜙𝜙=0. Permeate zone 𝑄𝑄𝑝𝑝𝑑𝑑 = 6.3227 10−3 ∙ 𝑒𝑒(−15630𝑅𝑅.𝑇𝑇 ) (11) 𝑑𝑑𝑌𝑌𝐶𝐶2 𝑑𝑑𝑑𝑑 = 𝑄𝑄𝑝𝑝𝑝𝑝∙2𝜋𝜋∙(𝑟𝑟0+δ)∙L 𝐹𝐹0𝐶𝐶𝐶𝐶4∙δ∙ 22.4 ∙(𝑃𝑃0.5 𝐻𝐻2 𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑃𝑃0.5 𝐻𝐻2 𝑝𝑝𝑟𝑟𝑟𝑟) (12) 𝐹𝐹𝐻𝐻2 = 0 at 𝜙𝜙=0. Heat balance Reaction zone 𝑑𝑑𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 𝑑𝑑𝑑𝑑 = �∑ �−∆𝐻𝐻𝑟𝑟,𝑖𝑖(𝑇𝑇)� 𝑅𝑅𝑖𝑖 𝑖𝑖 𝜌𝜌𝑐𝑐� ∙𝐴𝐴∙ 𝐿𝐿−∑ 𝐶𝐶𝑝𝑝𝑗𝑗 𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 𝑝𝑝𝑑𝑑𝑗𝑗 𝑝𝑝𝑑𝑑 ∙�𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟−𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟�𝑗𝑗 ∑ 𝐶𝐶𝑝𝑝𝑗𝑗 𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 𝐹𝐹𝑗𝑗𝑗𝑗 − ℎ𝑚𝑚𝑟𝑟𝑚𝑚𝑚𝑚∙ 2𝜋𝜋 (𝑟𝑟0+𝛿𝛿)∙ 𝐿𝐿∙ �𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟−𝑇𝑇𝑝𝑝𝑟𝑟𝑟𝑟𝑚𝑚� ∑ 𝐶𝐶𝑝𝑝𝑗𝑗 𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 𝐹𝐹𝑗𝑗𝑗𝑗 + 𝑈𝑈𝐶𝐶𝑅𝑅∙ 2∙𝜋𝜋∙𝐿𝐿∙(𝑟𝑟𝑠𝑠+𝑙𝑙)∙(𝑇𝑇𝑊𝑊−𝑇𝑇𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟) ∑ 𝐶𝐶𝑝𝑝𝑗𝑗 𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 𝐹𝐹𝑗𝑗𝑗𝑗 (13) T= Treact –TRef 𝑇𝑇𝑝𝑝𝑒𝑒𝑟𝑟𝑝𝑝 = 𝑇𝑇0 at 𝜙𝜙 = 0. Permeate zone 𝑑𝑑𝑇𝑇𝑝𝑝𝑟𝑟𝑟𝑟𝑚𝑚 𝑑𝑑𝑑𝑑 = 𝐶𝐶𝑝𝑝𝐶𝐶2 𝑟𝑟𝑟𝑟𝑟𝑟𝑐𝑐𝑟𝑟 ∙𝐹𝐹𝐶𝐶𝐶𝐶4 0 ∙ 𝑝𝑝𝑑𝑑𝐶𝐶2 𝑝𝑝𝑑𝑑 ∙(𝛥𝛥𝑇𝑇)+ℎ𝑚𝑚𝑟𝑟𝑚𝑚𝑚𝑚∙ 2𝜋𝜋 ∙(𝑟𝑟0+𝛿𝛿)∙ 𝐿𝐿 ∙(𝛥𝛥𝑇𝑇) 𝐶𝐶𝑝𝑝𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟𝑝𝑝 𝑝𝑝𝑟𝑟𝑟𝑟𝑚𝑚 ∙𝐹𝐹𝑠𝑠𝑠𝑠𝑟𝑟𝑟𝑟𝑝𝑝+𝐶𝐶𝑝𝑝𝐶𝐶2 𝑝𝑝𝑟𝑟𝑟𝑟𝑚𝑚 ∙𝐹𝐹𝐶𝐶2 (14) 3. Results and discussion The system of ordinary differential equations (ODEs), consisting of equations (9, 10, 12, 13, and 14), was solved numerically in MATLAB using the ode15s function. The steam methane reforming conditions are as follows: the system operates with initial reaction, permeation, and wall temperature of 500 °C. Methane is fed into the reactor at a rate of 1 kmol /h. The reactor is 1 m long, and the operating pressure is maintained at 1 bar. The reactor has an outer shell radius of 8.89 cm and an inner tube radius of 6.35 cm, with a catalyst density of 2100 kg/m³ and a bed porosity of 0.53. The permeate zone pressure is also set at 1 bar. The feed composition includes 24.99 % methane, 74.99 % water, 0.01% hydrogen, and no carbon monoxide or carbon dioxide. The membrane used has a radius of 2.54 cm and a thickness of 5∙10-6 m, and operates with a sweep ratio of 20. Figure 2 Variation of overall heat transfer coefficient in PBR and MR PT = 1 bar. The system's performance is evaluated based on key operating conditions, namely feed flow rate, temperature, steam/methane ratio. The most important parameters to assess are CH4 conversion and temperature. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 8 8.5 9 9.5 10 10.5 U M R a nd U P B R (J /(m 2 .h. K)) 10 5 U M R U P B R 1089 As shown in Figure 2, when the operating pressure is fixed at 1 bar, the overall heat transfer coefficient in the membrane reactor (UMR) decreases at a faster rate compared to the overall heat transfer coefficient in the packed-bed reactor (UPBR). Specifically, UMR declined from 1,010 kJ/(m²·h·K) to 813 kJ/(m²·h·K), while UPBR decreased from 984 kJ/(m²·h·K) to 871 kJ/(m²·h·K). The overall heat transfer coefficient is influenced by the overall flow rate, which is related to the Reynolds number. In the PBR, the low permeation rate of hydrogen through the membrane, due to the low-pressure difference between the reaction zone and the permeation zone, prevents significant reversals in the reaction. As a result, it has a smaller impact on the Reynolds number and, consequently, on the overall heat transfer coefficient. Additionally, thermal conductivity decreases in both reactors due to the steam methane reforming (SMR) and water-gas shift (WGS) reactions, with CO2 having the lowest thermal conductivity. As a consequence of this behavior, the temperature in the MR is lower than in the PBR as shown in Figure 3. Figure 3. PBR and MR temperature profile (Treact0 = 500 °C). Figure 3 shows the temperature profiles of the PBR and MR along the reactor length. The initial temperature decrease is due to the major consumption of reactants at the beginning of the reactor (up to ϕ = 0.02). After this point, the heat generated by convection compensates for the temperature drop, causing an increase in temperature until thermal equilibrium is reached (Treact = Twall). Figure 4 shows the methane conversion for the PBR and MR along the reactor length. Figure 4: Methane conversion in PBR and MR. It is clear that under the same conditions, the MR exhibits better performance than the PBR. In the membrane reactor, increasing the reaction pressure leads to a greater difference between the partial pressure of hydrogen on the reaction zone and the permeate zone. This increase in hydrogen permeation further enhances methane conversion. MR showed better results compared to PBR so to investigate the mole fraction variation in the membrane reactor, the mole fraction percentages of the reactants and products are displayed in Figure 5 (co-current flow, steam/methane ratio m = 3). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 400 410 420 430 440 450 460 470 480 490 500 T(° C) Treact-MR- Treact-PBR- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 X CH 4 MR PBR 1090 Figure 5: Molar fraction of each component (FCH4=1 kmol/h) H2 is the hydrogen flow that goes through the membrane. The inlet gas temperature was 500 °C and the steam to methane ratio was 3 for this simulation. The methane mole fraction declined from 24.9 % to 13.8 %, while the hydrogen mole fraction increased from 0.01 % to 32.2 %. The CO and CO2 mole fractions increased slightly. The increase in hydrogen mole fraction was consistently higher than the other products (CO and CO2). The water mole fraction decreased from 74.99 % to 53.8 % due to the SMR reaction. To improve methane conversion rates and increase hydrogen production, several parameters in the membrane reactor (MR) can be adjusted. Figure 6 highlights the effect of the steam-to-methane molar ratio m on methane conversion. This ratio, representing the balance between reactants, plays a significant role in driving the reforming reaction forward. Figure 6: Steam-to-methane ratio effect on methane conversion (FCH4=1 kmol/h). As shown in Figure 6, an increase in m enhances conversion, indicating that an excess of steam improves methane conversion. Beyond a ratio of m = 3, the rate of increase in conversion is reduced, suggesting that further increases in steam provide diminishing returns. This supports the initial selection of the chosen ratio. The temperature plays a major role in the steam methane reforming reaction. Figure 7 shows that an increase in inlet temperature leads to a higher methane conversion. Figure 7: Inlet temperature effect on methane conversion (FCH4=1 kmol/h). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X CH 4 H 2 O CO CO 2 H 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 X C H 4 m=1 m=3 m=5 m=10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 X C H 4 T=300 °C T=400 °C T=500 °C T=600 °C 1091 The Figure shows that higher inlet temperatures correlate strongly with increased conversion: at an inlet temperature of Tin=600 °C, the conversion reaches its highest level, whereas at Tin=300 °C, the conversion is minimal. 4. Conclusions This study has demonstrated that the assimilation of a variable overall heat transfer coefficient (UOR) in a non- isothermal model simulation of steam methane reforming (SMR) improves the predictive accuracy of the simulation for both packed-bed reactors (PBR) and membrane reactors (MR) Results show that membrane reactors outperform PBRs by reaching a higher methane conversion under similar operating conditions. Key parameters, including inlet temperature, methane flow rate, and the steam-to-methane ratio, have significant effects on conversion efficiency. The minimum reactor temperature must be at least 500 °C or higher to achieve a decent conversion rate with an H₂O/CH₄ ratio of 3 and provides diminishing returns beyond this ratio. The H₂ production rate can be improved by increasing the temperature. For methane conversion, the inlet temperature has a positive effect, while the methane flow rate has a negative effect; their interaction, however, has a positive effect. Nomenclature PBR – Packed Bed Reactor MR – Membrane reactor 𝑟𝑟i – kinetic rate of ith reaction, 𝑘𝑘𝑝𝑝𝑘𝑘𝑘𝑘 𝑘𝑘𝑘𝑘𝑐𝑐𝑐𝑐𝑐𝑐.ℎ⁄ 𝐹𝐹j𝑖𝑖𝑖𝑖 – inlet jth component molar flow rate, 𝑘𝑘𝑝𝑝𝑘𝑘𝑘𝑘 ℎ⁄ 𝐹𝐹𝑗𝑗 – outlet jth component molar flow rate, 𝑘𝑘𝑝𝑝𝑘𝑘𝑘𝑘 ℎ⁄ 𝑃𝑃𝑗𝑗 – partial pressure of jth component, bar 𝜂𝜂𝑖𝑖 – effectiveness factor of ith reaction, - 𝐴𝐴𝑐𝑐 – cross section of the reactor 𝑝𝑝2 𝛿𝛿 – membrane thickness, 𝜇𝜇𝑝𝑝 𝜙𝜙 – dimensionless length 𝜙𝜙 = 𝑧𝑧 𝐿𝐿 , - k1, k2, k3 – rate coefficients, kmol·kPa0.5/(kg·h), kmol·kPa- 1/(kg·h), kmol·kPa0.5/(kg·h) 𝐿𝐿 – length of the reactor, m 𝐶𝐶𝑝𝑝𝑗𝑗 – molar specific heat for jth component, 𝐽𝐽 𝑝𝑝𝑘𝑘𝑘𝑘. 𝑘𝑘⁄ 𝑇𝑇𝑟𝑟𝑟𝑟𝑐𝑐𝑐𝑐𝑐𝑐 – temperature of reaction zone, K 𝑇𝑇𝑝𝑝𝑟𝑟𝑟𝑟𝑝𝑝 – temperature of permeation zone, K ℎ𝑝𝑝𝑟𝑟𝑝𝑝𝑏𝑏 – membrane heat transfer coefficient 𝐽𝐽.𝑝𝑝−1𝐾𝐾−1 Kj – absorption constant of component j - K1, K2, K3 – equilibrium constants, References Daymo, E., Tonkovich, A., Hettel, M., Shirsath, A., 2024. 1D and 2D porous media fixed bed reactor simulations with DUO: Steam Methane Reforming (SMR) validation test. IOP Conf Ser Mater Sci Eng 1312, 012004. https://doi.org/10.1088/1757-899x/1312/1/012004 Dong, X., Peng, L., Xiaoyou, Y., 2019. Process modeling, optimization, and heat integration of ethanol reforming process for syngas production with high H2/CO ratio. Processes 7. https://doi.org/10.3390/PR7120960 Iulianelli, A., Liguori, S., Wilcox, J., Basile, A., 2016. Advances on methane steam reforming to produce hydrogen through membrane reactors technology: A review. Catal Rev Sci Eng 58, 1–35. https://doi.org/10.1080/01614940.2015.1099882 Kuncharam, B.V.R., Dixon, A.G., 2020. Multi-scale two-dimensional packed bed reactor model for industrial steam methane reforming. Fuel Processing Technology 200. https://doi.org/10.1016/j.fuproc.2019.106314 Madia, G.S., Barbieri, G., Drioli, E., 1999. Theoretical and experimental analysis of methane steam reforming in a membrane reactor. Canadian Journal of Chemical Engineering 77, 698–706. https://doi.org/10.1002/cjce.5450770411 Park, H.G., Han, S.Y., Jun, K.W., Woo, Y., Park, M.J., Kim, S.K., 2019. Bench-scale steam reforming of methane for hydrogen production. Catalysts 9, 1–14. https://doi.org/10.3390/catal9070615 Saw, S.Z., Nandong, J., Ghosh, U.K., 2016. Comparative Study of Homogeneous and Heterogeneous Modelling of Water-Gas Shift Reaction with Macro-or Micro-kinetics, in: Procedia Engineering. Elsevier Ltd, pp. 949– 956. https://doi.org/10.1016/j.proeng.2016.06.467 Wu, H.C., Rui, Z., Lin, J.Y.S., 2020. Hydrogen production with carbon dioxide capture by dual-phase ceramic- carbonate membrane reactor via steam reforming of methane. J Memb Sci 598, 117780. https://doi.org/10.1016/j.memsci.2019.117780 Wu, Z., Yang, J., Wang, Q., 2023. Numerical Investigation of Methane Steam Reforming in the Packed Bed Installed with Metal Foam. Chem Eng Trans 103, 67–72. https://doi.org/10.3303/CET23103012 Xu, J., Froment, G.F., 1989. Methane steam reforming: II. Diffusional limitations and reactor simulation. AIChE Journal 35, 97–103. https://doi.org/10.1002/aic.690350110 1092 387Nazim-PAGATO.pdf Modeling of a Non-Isothermal Fixed Bed Reactor: 1D Model for Hydrogen Production by Steam Methane Reforming