Numerical Simulation of Methane and Propane Reforming Over a Porous Rh/Al₂O₃ Catalyst in Stagnation-Flows: Impact of Internal and External Mass Transfer Limitations on Species Profiles

Abstract

Hydrogen production by catalytic partial oxidation and steam reforming of methane and propane towards synthesis gas are numerically investigated in stagnation-flow over a disc coated with a porous Rh/Al₂O₃ layer. A one-dimensional flow field is coupled with three models for internal diffusion and with a 62-step surface reaction mechanism. Numerical simulations are conducted with the recently developed computer code DETCHEM^STAG. Dusty-Gas model, a reaction-diffusion model and a simple effectiveness factor model, are alternatively used in simulations to study the internal mass transfer inside the 100 µm thick washcoat layer. Numerically predicted species profiles in the external boundary layer agree well with the recently published experimental data. All three models for internal diffusion exhibit strong species concentration gradients in the catalyst layer. In partial oxidation conditions, a thin total oxidation zone occurs close to the gas-washcoat interface, followed by a zone of steam and dry reforming of methane. Increasing the reactor pressure and decreasing the inlet flow velocity increases/decreases the external/internal mass transfer limitations. The comparison of reaction-diffusion and Dusty-Gas model results reveal the insignificance of convective flow on species transport inside the washcoat. Simulations, which additionally solve a heat transport equation, do not show any temperature gradients inside the washcoat.

1. Introduction

Steam reforming (SR) of methane (CH₄) (Equation (1)) is the major process for synthesis gas (CO, H₂) production today. Steam reforming of propane (C₃H₈) (Equation (2)) is also realized, because it can be easily stored and distributed [1]. Tubular reactors packed with supported Ni catalysts are largely used for synthesis gas production. Operating temperature and pressure of these tubular reactors are around 800–900 °C and 20–30 bar, respectively. This is an energy-intensive process with long residence time (1 s or more), and industrial production requires large scale operation [2,3,4]. The process is limited by the low catalyst effectiveness factors, weak heat transport capabilities, and significant initial capital expenditures [5,6]. Microchannel reactors have been investigated as an alternative to tubular reactors for SR of CH₄ [7,8,9,10,11,12] and SR of C₃H₈ [1,13,14,15,16]. In microchannel reactors, the active catalyst material is adhered, possibly in a porous layer, to the inner walls of channels. Nobel metal catalysts are the most active catalysts for hydrocarbon reforming in microchannels for a high syngas yield [9,17]. In this respect, rhodium (Rh) is a very active metal and guarantees high conversion at short contact times. It is slightly prone to carbon-deposition, and stable at extreme, cyclic conditions without loss of activity [18]. Microchannel reactors with a Rh catalyst offer enhanced heat and mass transfer, safe control in explosive regime, high surface area, low pressure drop, and short residence time (10 ms or less) [5,19].

$${\mathrm{CH}}_4+{\mathrm{H}}_2\mathrm{O}\to \mathrm{CO}+3{\mathrm{H}}_2\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=+205.9\mathrm{kJ}/\mathrm{mol}$$

(1)

$${\mathrm{C}}_3{\mathrm{H}}_8+3{\mathrm{H}}_2\mathrm{O}\to 3\mathrm{CO}+7{\mathrm{H}}_2\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=+498\mathrm{kJ}/\mathrm{mol}$$

(2)

Catalytic partial oxidation (CPOX) of CH₄ (Equation (3)) and CPOX of C₃H₈ (Equation (4)) in dry air is an attractive alternative synthesis gas production to larger SR reactors. The system is well-suited for small scale systems, such as foam catalysts, monolithic reactors and micro fuel-cells. For instance, CPOX of CH₄ produce H₂/CO ratio of two which is suitable as feedstock for methanol synthesis and the Fischer–Tropsch reaction [20]. Rh is an active and stable noble metal for CPOX applications, as well.

$$\mathrm{C}{\mathrm{H}}_4+1/2{\mathrm{O}}_2\to \mathrm{C}\mathrm{O}+2{\mathrm{H}}_2\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=-35.9 \mathrm{kJ}/\mathrm{mol}$$

(3)

$${\mathrm{C}}_3{\mathrm{H}}_8+3/2{\mathrm{O}}_2\to 3\mathrm{CO}+4{\mathrm{H}}_2\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=-227.6\mathrm{kJ}/\mathrm{mol}$$

(4)

Aforementioned catalytic reactors for the SR and CPOX of CH₄ and C₃H₈ exhibit a complex interaction between the catalytically active surface and the surrounding flow field. Understanding the physical and chemical steps of a heterogeneous catalytic processes at a fundamental level aids optimizing the process and the catalyst. Microkinetic models are incorporated into computational fluid dynamics (CFD) codes to model the reforming reactors, and validate them in an operating range relevant to industrial applications. In this respect, detailed reaction mechanisms for methane reforming have been studied since early 1990s. Direct and indirect reaction pathways were proposed for the partial oxidation of CH₄ on Rh catalysts. Direct reaction pathways assume that methane is oxidized to form CO and H₂, along with the total oxidation [21,22,23]. Indirect mechanisms on the other hand suggest that CH₄ reacts initially with O₂ to form CO₂ and H₂O (total oxidation) followed by steam and dry reforming of CH₄ [24,25,26,27,28]. Reaction kinetics for CPOX of C₃H₈ on Rh catalysts were also proposed [16,29]. Similar to the CPOX, microkinetic studies were employed for SR on Rh catalysts [3,16,28,30,31].

One important feature that is used in the catalytic reactors is the inclusion of a porous layer, called washcoat, that is coated on the surface [8,15,19,32,33]. The catalyst is distributed inside this washcoat to increase the internal surface area. In this case, reactants in the bulk flow diffuse from the gas-washcoat interface through pores and react at active sites of the catalyst. After reaction, products diffuse from the washcoat back to the bulk flow [34]. Finite diffusion rates of reactants and products towards and away from active sites may lead to a reduced overall reaction rate. Because of the very exothermic (CPOX) and endothermic (SR) nature of methane reforming processes, heat and mass-transport and chemical kinetics are tightly coupled. At low temperatures, catalytic reactions are slow, and therefore they are the rate-limiting step of the process. At higher temperatures, when the rate of diffusion is slow compared to the intrinsic rate of reaction, mass transport affects the rate of reaction, and the process becomes diffusion-limited [35]. Therefore, it is important to account for internal mass transfer limitations in catalytic reactor modeling in case of a thick catalyst layer [34,36,37,38,39,40].

The stagnation-flow reactor (SFR) offers a simple geometry to investigate the aforementioned chemical and physical processes and their interactions in a heterogeneous catalytic process that are relevant in practical reactor applications [41,42]. Its fluid-mechanical properties enable measuring and modeling the gas-phase boundary layer adjacent to the zero-dimensional catalytic surface [41,43]. One-dimensional (1D) formulation of the SFR facilitates computational modeling and simulation of processes dealing with catalytic combustion/oxidation.

In the present study, CPOX and SR of CH₄ and C₃H₈ are investigated numerically in stagnation-flow over a Rh/Al₂O₃ catalyst. The experimental conditions are taken from the recent studies of Karakaya et al. [16,28] as a basis for numerical predictions. Reaction-diffusion equation and effectiveness factor approach are used in the simulations to account for internal mass transfer resistances inside the washcoat. The Dusty-Gas model is used exemplarily for the CPOX and SR of CH₄ to investigate the effect of convective flow on species transport inside the washcoat layer. Energy balance inside the washcoat is included to analyze heat transport effects. The results give an insight into chemical and physical processes inside the washcoat and the interaction of active catalytic surface with the surrounding flow field. The possible reaction routes, internal and external mass transfer limitations, the effect of convective flow and heat transport inside the catalyst layer are investigated in detail. It is discussed if the surface models are adequate enough to predict the experiments. The results of this study are expected to help in understanding the complex processes in practical catalytic reactors for CPOX and SR of CH₄ and C₃H₈.

2. Results and Discussion

2.1. Cases Studied

In this study, we use the experimental stagnation-flow reactor data of Karakaya et al. [16,28] to investigate CPOX and SR of CH₄ and C₃H₈ over Rh/Al₂O₃. In this respect,

Table 1. Reaction conditions for catalytic partial oxidation (CPOX) of CH₄.

	Tdisc (K)	Tinlet (K)	CH4 (% vol.)	O2 (% vol.)	C/O -	Ar (% vol.)	Inlet Velocity (cm/s)	Reactor Pressure (mbar)
Case 1	873	313	5.30	2.57	1.03	92.13	51	500
Case 2	973	313	5.32	2.78	0.99	91.90	51	500
Case 3	973	313	5.20	4.90	0.53	89.90	51	500
Case 4	973	313	4.38	7.80	0.28	87.82	51	500
Case 5	1023	313	5.20	2.83	0.93	91.97	51	500

and

Table 2. Reaction conditions for CPOX of C₃H₈.

	Tdisc (K)	Tinlet (K)	C3H8 (% vol.)	O2 (% vol.)	C/O -	Ar (% vol.)	Inlet Velocity (cm/s)	Reactor Pressure (mbar)
Case 6	883	313	6.60	7.93	1.25	85.47	53	500
Case 7	933	313	5.70	8.85	0.97	85.45	53	500

give the investigated cases for the CPOX of CH₄ and C₃H₈, respectively.

Table 3. Reaction conditions for steam reforming (SR) of CH₄.

	Tdisc (K)	Tinlet (K)	CH4 (% vol.)	H2O (% vol.)	S/C -	Ar (% vol.)	Inlet Velocity (cm/s)	Reactor Pressure (mbar)
Case 8	973	423	5.06	5.38	1.06	89.56	71	500
Case 9	1008	423	5.16	5.38	1.04	89.46	71	500

and

Table 4. Reaction conditions for SR of C₃H₈.

	Tdisc (K)	Tinlet (K)	C3H8 (% vol.)	H2O (% vol.)	S/C -	Ar (% vol.)	Inlet Velocity (cm/s)	Reactor Pressure (mbar)
Case 10	883	423	2.45	7.38	1.00	90.17	77	500
Case 11	923	423	2.44	7.42	0.99	90.14	77	500

give the investigated cases for the SR of CH₄ and C₃H₈, respectively.

2.2. Input Data for Numerical Simulations

The inlet conditions of the numerical simulations are based on the experimental conditions that are given in Table 1, Table 2, Table 3 and Table 4. The inlet velocity is taken as 51 cm/s and 53 cm/s for CPOX of CH₄ and C₃H₈, respectively. The inlet velocity is taken as 71 cm/s and 77 cm/s for SR of CH₄ and C₃H₈, respectively. The finite gap between the inlet and catalytic surface is 3.9 cm. The reactor inlet temperature is taken as 313 K and 423 K for CPOX and SR cases, respectively.

In this study, simulations are performed with two different surface models, i.e., with the η-approach and RD-approach. CH₄ and C₃H₈ are chosen as the rate-limiting species for the η-approach simulations. DGM simulations are performed exemplarily for the CPOX and SR of CH₄ for evaluating the effect of convective flow on species transport inside the washcoat layer. The thickness, mean pore diameter, tortuosity, and porosity of the washcoat are the parameters that are used in the η-approach, RD-approach, and DGM simulations. In DGM, particle diameter is also needed. The values of these parameters are given in

Table 5. The parameters used in the surface models.

Thickness of the Washcoat (µm)	Mean Pore Diameter (nm)	Porosity (%)	Tortuosity (-)	Particle Diameter (nm) (DGM Only)
100	10	40	8	100

. The washcoat temperature is kept constant in simulations.

2.3. CPOX of CH₄ and C₃H₈

2.3.1. CPOX of CH₄

The results of this section is based on the studies of Karadeniz [44]. Experimental and simulation results are given in Figure 1. In the experiments, each data point is given as an average of four repeated measurements. Differences between these four measurements is less than 10%. According to the results, η-approach, RD-approach, and DGM simulations show relatively good agreement with the experiments for species profiles in the gas-phase boundary layer for Cases 1–5 (Figure 1a,c,e,g,i). Gas-phase boundary layers are around 6–7 mm for Cases 1–5, relative to the external catalyst surface.

Surface reactions are fast and internal mass transfer limitations are observed. Φ and η values, which are obtained from η-approach, confirm the strong diffusion limitations (

Table 6. Φ and η values in CPOX of CH₄ cases.

	Φ	η
Case 1	17.39	0.058
Case 2	27.40	0.037
Case 3	30.01	0.033
Case 4	32.20	0.030
Case 5	32.53	0.030

). Therefore, the rate-limiting process is the internal diffusion. RD-approach and DGM simulations give an insight to realize the physical and chemical processes (reaction routes) inside the washcoat (Figure 1b,d,f,h,j). According to DGM simulations, the pressure gradient inside the washcoat is low (

Table 7. The pressure difference in the washcoat in CPOX of CH₄ cases.

	Case 1	Case 2	Case 3	Case 4	Case 5
Pressure difference (Pa)	494	440	104	45	403

). Therefore, DGM yields identical species profiles with the RD-approach.

Further, the reaction routes inside the washcoat is explained below for each individual case.

Case 1: Case 1 is stoichiometric for partial oxidation and total oxidation products (CO₂ and H₂O) are the main products at the catalyst surface (Figure 1a). Zones one to three in Figure 1b show that there are complex reaction routes inside the washcoat. For a better explanation, only the first and second reaction zones in the washcoat are considered in Figure 2. In zone one, there is a thin total oxidation layer near the external catalyst surface. After this thin total oxidation zone, mainly SR of CH₄ occurs in zone two. Dry reforming (DR) of CH₄ (Equation (5)) is observed simultaneously in this zone, as well, but to a much lesser extent. In zone three (Figure 1b), only a slight DR of CH₄ is observed. Beyond zone three, there is not any reaction in the rest of the washcoat and the system reaches equilibrium. The whole reaction layer is around 50 µm and it is given here as relative to the external catalyst surface (gas-washcoat interface). The species composition at the 50 µm of the washcoat is used in DETCHEM^EQUIL code [45] to realize if the composition has reached the thermodynamic equilibrium. Further, DETCHEM^EQUIL code calculations show that the species composition (mole fractions (mole frac.)) reaches equilibrium at 50 µm of the washcoat (

Table 8. Equilibrium composition (mole frac.) in the washcoat, relative to external catalyst surface.

Case/Species	CH4	O2	H2O	CO2	H2	CO	AR
Case 1	1.1 E-02	6.7 E-21	6.2 E-05	2.7 E-04	3.1 E-02	5.0 E-02	9.1 E-01
Case 2	6.6 E-03	1.6 E-20	8.4 E-06	2.0 E-05	3.7 E-02	5.5 E-02	9.0 E-01
Case 3	2.1 E-06	1.1 E-20	1.2 E-02	2.0 E-05	3.2 E-02	4.10 E-02	8.9 E-01
Case 4	2.2 E-09	4.6 E-20	5.3 E-02	5.2 E-02	1.0 E-02	6.0 E-03	8.8 E-01
Case 5	6.0 E-03	1.7 E-20	2.5 E-06	4.6 E-06	3.8 E-02	5.4 E-02	9.0 E-01

).

$${\mathrm{CH}}_4+{\mathrm{CO}}_2\to 2\mathrm{CO}+2{\mathrm{H}}_2\hspace{1em}\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=+247.0\mathrm{kJ}/\mathrm{mol}$$

(5)

Case 2: Case 2 is stoichiometric for partial oxidation and experiments show that O₂ is almost completely consumed on the surface. The main products are synthesis gas and total oxidation products (Figure 1c). In Figure 1d, only 30 µm of the washcoat is shown for Case 2, because the reactions occur only in this section. According to RD-approach and DGM simulations, total oxidation is a weak process due to too little amount of O₂ inside the catalyst. There exist SR and DR of CH₄ in zone one inside the washcoat. However, DR occurs in a much lesser extent than SR. There is just a slight DR process within zone two. DETCHEM^EQUIL code shows that the chemical composition reaches thermodynamic equilibrium at the 30 µm of the washcoat as given in Table 8.

Case 3: Case 3 is stoichiometric for total oxidation and CH₄ consumption rate increases, compared to Case 2, due to increased amount of oxygen (Figure 1e). Therefore, more total oxidation products are obtained. The amount of synthesis gas products decreases. The reaction layer is divided into three zones in Figure 1f. Zone one, which is adjacent to the external catalyst surface, shows a thin reaction layer where total oxidation occurs. In zone two, there is SR of CH₄, where CH₄ and H₂O are consumed, CO and H₂ are produced. CO₂ is still formed in the second zone due to water-gas shift (WGS) reaction (Equation (6)). In zone three, CO₂ is not formed anymore. The little amount of remaining CH₄ reacts with H₂O (SR) to yield synthesis gas. DETCHEM^EQUIL code shows that the chemical composition reaches thermodynamic equilibrium at the 20 µm of the washcoat as given in Table 8.

$$\mathrm{CO}+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{CO}}_2+{\mathrm{H}}_2\hspace{1em}\hspace{1em}∆{\mathrm{H}}_{\mathrm{r}}=-41.1\mathrm{kJ}/\mathrm{mol}$$

(6)

Case 4: Case 4 is fuel-rich for total oxidation and total oxidation products are the main products on the surface. No synthesis gas formation is observed at this fuel-rich case (Figure 1g). RD-approach and DGM simulations reveal that there is a total oxidation zone in the washcoat (zone one in Figure 1h), near the external catalyst surface. After this total oxidation zone, there is the SR of CH₄. Since there is not any oxygen left and CO₂ is still formed, WGS occurs as well. SR and WGS occur simultaneously in the entire zone two. The total reaction layer (zone one and zone two together) is around 15 µm. The species composition reaches thermodynamic equilibrium at the 15 µm of the washcoat as given in Table 8.

Case 5: Case 5 is slightly lean for partial oxidation and CH₄ is converted more in Case 5 compared to Case 2 due to the increased surface temperature (Figure 1i). There is a slight increase of the synthesis gas products compared to Case 2. According to the RD-approach and DGM simulations, total oxidation inside the washcoat is weak (Figure 1j). Total reaction layer thickness inside the washcoat is around 30 µm. SR and DR processes take place simultaneously within zone one of the washcoat. However, SR is the dominant process and DR occurs although its much weaker compared to SR. The species composition reaches thermodynamic equilibrium at the 30 µm of the washcoat as given in Table 8.

2.3.2. CPOX of C₃H₈

The experimental and simulation results of CPOX of C₃H₈ are given in Figure 3. According to the results, η-approach and RD-approach simulations show relatively good agreement with the experiments for Case 6 and Case 7 (Figure 3a,c). Gas-phase boundary layers are around 6 mm, relative to the external catalyst surface. Surface reactions are fast and internal mass transfer limitations are observed. Φ and η values, which are obtained from η-approach, confirm the strong diffusion limitations (

Table 9. Φ and η values in CPOX of C₃H₈ cases.

	Φ	η
Case 6	25.48	0.039
Case 7	37.70	0.026

). Since DGM simulations have already revealed that the pressure gradient inside the washcoat is insignificant for CPOX cases, and DGM yields identical species profiles with RD-approach, it is not calculated here for the CPOX of C₃H₈.

Further, the reaction routes inside the washcoat is explained below for each individual case.

Case 6: Case 6 is rich for partial oxidation. Zones one to three in Figure 3b show that there are complex reaction routes inside the washcoat. For a better explanation, only the first and second reaction zones in the washcoat are considered in Figure 4. In zone one, there is a thin total oxidation zone near the external catalyst surface similar to the CPOX of CH₄ for Case 1 and Case3. After this thin total oxidation zone, mainly SR of CH₄ occurs in zone two. DR of CH₄ is observed simultaneously in this zone, as well, but in a lesser extent. In zone three, only a slight DR of CH₄ is observed (Figure 3b). Beyond zone three, there is not any reaction in the rest of the washcoat and the system reaches equilibrium. DETCHEM^EQUIL code shows that the chemical composition reaches thermodynamic equilibrium at the 50 µm of the washcoat as given in

Table 10. Equilibrium composition (mole frac.) in the washcoat, relative to external catalyst surface.

Case/Species	CH4	O2	H2O	CO2	H2	CO	C3H8	AR
Case 6	2.9 E-04	1.6 E-21	2.1 E-03	1.3 E-02	5.3 E-02	1.3 E-01	1.1 E-13	8.0 E-01
Case 7	4.0 E-05	5.2 E-22	1.1 E-02	4.3 E-02	4.3 E-02	1.3 E-01	1.2 E-18	8.2 E-01

.

Case 7: Case 7 is stoichiometric for partial oxidation and experiments show that O₂ is almost completely consumed on the surface (Figure 3c). The main products are synthesis gas and total oxidation products. In Figure 3d, only 30 µm of the washcoat is shown for Case 7, because reactions occur only in this section. In zone one, there is a thin total oxidation zone again near the external catalyst surface similar to Case 6. There exist mainly SR of CH₄ inside the second reaction zone of the washcoat. There is DR of CH₄, as well, but in a much lesser extent than SR. The species composition reaches thermodynamic equilibrium at the 30 µm of the washcoat as given in Table 10.

2.4. SR of CH₄ and C₃H₈

2.4.1. SR of CH₄

The results of this section is based on the studies of Karadeniz [44]. Experimental and simulation results are given in Figure 5. According to the results, η-approach, RD-approach, and DGM simulations show relatively good agreement with the experiments for species profiles in the gas-phase boundary layer for Case 8 and Case 9 (Figure 5a,c). Gas-phase boundary layers are around 9 mm. It is observable that the CO/H₂ ratio on the surface obtained from SR of CH₄ at 973 K differs from the CO/H₂ ratio obtained from CPOX of CH₄ at 973 K. Surface reactions are fast and internal mass transfer limitations are observed for SR of CH₄, as well. Φ and η values confirm the strong diffusion limitations (

Table 11. Φ and η values in SR of CH₄ cases.

	Φ	η
Case 8	26.46	0.038
Case 9	30.10	0.030

). Therefore, the rate-limiting process is the internal diffusion. According to the DGM simulations, the pressure difference between the gas-washcoat interface and the washcoat support side is low for SR of CH₄ (

Table 12. The pressure difference in the washcoat in SR of CH₄ cases.

	Case 8	Case 9
Pressure difference (Pa)	472	464

). Therefore, DGM yields identical species profiles with the RD-approach.

Further, the reaction routes inside the washcoat is explained below for each individual case.

Case 8: The reaction layer inside the washcoat is divided into two zones (Figure 5b). There is a very slight WGS kinetics within zone one. However, the driving process here is SR of CH₄, where most of the CH₄ and H₂O are converted to synthesis gas. In zone two, there is no more WGS kinetics, but a slight SR of CH₄. The whole reaction layer is only 20 µm due to strong internal mass transfer limitations. The chemical composition reaches thermodynamic equilibrium at the 20 µm of the washcoat according to the DETCHEM^EQUIL code calculations (

Table 13. Equilibrium composition (mole frac.) in the washcoat, relative to external catalyst surface.

Case/Species	CH4	O2	H2O	CO2	H2	CO	AR
Case 8	1.8 E-04	4.0 E-21	1.6 E-03	2.2 E-03	6.4 E-02	5.6 E-02	8.8 E-01
Case 9	1.1 E-04	5.2 E-21	1.2 E-03	1.5 E-03	6.6 E-02	6.0 E-02	8.8 E-01

).

Case 9: Case 9 considers SR of CH₄ at 1008 K. An increased reaction rate is observed for CH₄ and O₂, compared to Case 8, due to increased surface temperature. Therefore, a higher synthesis gas yield is obtained (Figure 5c). Figure 5d shows that the reaction layer is around 15 µm. There is only SR of methane within the whole reaction layer. The chemical composition reaches thermodynamic equilibrium at the 15 µm of the washcoat according to the DETCHEM^EQUIL code calculations (Table 13).

2.4.2. SR of C₃H₈

Experimental and simulation results are given in Figure 6. Simulations predict the experiments reasonably well (except CO₂ formation). Gas-phase boundary layers are around 7–8 mm. Surface reactions are fast and internal mass transfer limitations are observed for SR of C₃H₈, as well. Φ and η values confirm the strong diffusion limitations (

Table 14. Φ and η values in SR of C₃H₈ cases.

	Φ	η
Case 10	14.90	0.067
Case 11	19.00	0.053

). Since DGM simulations have already revealed that the pressure gradient inside the washcoat is insignificant for SR cases, and DGM yields identical species profiles with RD-approach, it is not calculated here for the CPOX of C₃H₈.

Further, the reaction routes inside the washcoat is explained below for each individual case with RD-approach.

Case 10: Case 10 considers the SR of C₃H₈ at 883 K. WGS reaction occurs within the catalyst, where CO₂ is formed (Figure 6b). However, the driving process here is SR of C₃H₈ at this temperature, where most of the C₃H₈ and H₂O are converted to synthesis gas. The chemical composition reaches thermodynamic equilibrium at the 35 µm of the washcoat according to the DETCHEM^EQUIL code calculations (

Table 15. Equilibrium composition (mole frac.) in the washcoat, relative to external catalyst surface.

Case/Species	CH4	O2	H2O	CO2	H2	CO	C3H8	AR
Case 10	1.0 E-04	5.2 E-21	1.2 E-02	2.4 E-02	4.7 E-02	3.6 E-02	6.4 E-18	8.8 E-01
Case 11	2.5 E-05	9.6 E-21	1.4 E-02	2.2 E-02	4.7 E-02	3.7 E-02	1.7 E-19	8.8 E-01

).

Case 11: Case 11 considers SR of C₃H₈ at 923 K. Reaction rate increases for C₃H₈ and O₂ slightly, compared to Case 10, due to increased surface temperature. Therefore, slightly higher synthesis gas yield is obtained (Figure 6c). WGS reaction occurs within the catalyst (Figure 6d). However, the driving process here is again SR of C₃H₈, where most of the C₃H₈ and H₂O are converted to synthesis gas. Figure 6d shows that the reaction layer is around 25 µm relative to the external catalyst surface. The chemical composition reaches thermodynamic equilibrium at the 25 µm of the washcoat according to the DETCHEM^EQUIL code calculations (Table 15).

2.5. The Effect of Pressure and Flow Rates on External and Internal Mass Transfer Limitations and Syngas Production via CPOX and SR of CH₄ and C₃H₈

In this section, the effect of the pressure and flow rates on syngas production is investigated for CPOX and SR of CH₄ and C₃H₈. Surface temperature, inlet temperature, carbon to oxygen (C/O) ratio, and steam to carbon (S/C) ratio are given for CPOX and SR cases in

Table 16. Reaction conditions for CPOX of CH₄ and C₃H₈.

	Tdisc (K)	Tinlet (K)	C/O -
CPOX of CH4	973	313	0.99
CPOX of C3H8	973	313	0.97

and

Table 17. Reaction conditions for SR of CH₄ and C₃H₈.

	Tdisc (K)	Tinlet (K)	S/C -
SR of CH4	973	419	1.04
SR of C3H8	973	419	0.99

, respectively. Simulations are performed initially with varying pressures from 0.5 to 4 bar, and varying inlet velocities from 0.1 to 1.0 m/s. Internal mass transfer limitations are discussed based on the effectiveness factor (η). It should be mentioned here that internal mass transfer limitation increases with decreasing η. External mass transfer limitations are discussed based on Damkohler (Da) number. External mass transfer limitation is considered to be important, if Da exceeds three [46].

2.5.1. CPOX of CH₄ and C₃H₈

Figure 7 shows that external mass transfer limitations become important with the increasing reactor pressure and decreasing inlet flow velocity both for CPOX of CH₄ and C₃H₈. External mass transfer limitations are higher for CPOX of CH₄ than C₃H₈ at lower inlet flow velocities and higher reactor pressures. They remain similar at higher flow velocities.

Figure 8 shows that internal mass transfer limitations decrease with the increasing reactor pressure and decreasing inlet flow velocity both for CPOX of CH₄ and C₃H₈. Internal mass transfer limitations are slightly higher for CPOX of CH₄ than C₃H₈ at higher reactor pressures and lower inlet flow velocities. On the other hand, CPOX of CH₄ shows less internal mass transfer limitations than CPOX of C₃H₈ at lower reactor pressures and higher inlet flow velocities.

The mole fraction of H₂ at the surface increases with the increasing reactor pressure and decreasing inlet flow velocity both for CPOX of CH₄ and C₃H₈ (Figure 9). This indicates here that internal mass transfer limitations outweigh external mass transfer limitations. As explained above, internal and external mass transfer limitations exhibit opposite behavior to reactor pressure and inlet flow velocities. However, H₂ production increases in regimes, where internal mass transfer limitations decrease, even though external mass transfer limitations become important.

2.5.2. SR of CH₄ and C₃H₈

Figure 10 shows that external mass transfer limitations become important with increasing reactor pressure and decreasing inlet flow velocity in SR of CH₄ and C₃H₈. Similar to CPOX, external mass transfer limitations are slightly higher for SR of CH₄ than C₃H₈ at higher reactor pressures and lower inlet flow velocities.

Figure 11 shows that internal mass transfer limitations decrease with the increasing reactor pressure and decreasing inlet flow velocity. Internal mass transfer limitations are higher for SR of CH₄ than C₃H₈ at higher reactor pressures and lower inlet flow velocities.

The mole fraction of H₂ at the surface increases with the increasing reactor pressure and decreasing inlet flow velocity both for SR of CH₄ and C₃H₈ (Figure 12). This indicates here again that internal mass transfer limitations outweigh external mass transfer limitations.

2.6. The Effect of Heat Transport Limitations in the Washcoat

In order to study the effect of the heat transport limitations in the washcoat, the experimental configuration of SFR which was used by Karakaya [47] is examined. In the experiments, the resistive heater (FeCrAl alloy) was used for supplying the required heat to the washcoat. There is the ceramic support between the resistive heater and the washcoat. The details of the corresponding heat transfer modeling were explained in Section 3.3 and Section 3.4.

Further, simulations with the energy balance equations are performed exemplarily for the CPOX and SR of CH₄. The results indicate that the temperature gradient inside the washcoat is negligible for CPOX and SR cases. Temperature gradient inside the washcoat is obtained less than 0.5 K for all CPOX of CH₄ cases and less than 0.3 K for all SR of CH₄ cases, respectively. It implies that the thermal conductivity of the alumina is adequate enough to obtain a uniform temperature distribution inside the thin Rh/Al₂O₃ catalyst layer.

3. Materials and Methods

3.1. Modeling Approach

The modeling approach of this study is based on the consideration of the SFR configuration in one dimension (1D). In the SFR configuration, reactants are directed from the inlet manifold to the active catalytic surface through a finite gap, with a uniform flow velocity, as illustrated in Figure 13. The corresponding mathematical model considers flow with thermal energy and potential gas-phase reactions, heterogeneous surface reactions on the catalytic plate, and boundary conditions for flow equations. The governing equations defining this system are briefly given in the following three sections (Section 3.2, Section 3.3 and Section 3.4). A more detailed explanation for the governing equations were given elsewhere [34,43,44].

3.2. Gas-Phase Equations

In the stagnation-flow field, scalar quantities (temperature and species mass fractions) depend only on the distance from the surface, not on the radial position [48,49]. This allows us to draw the attention to the center of the catalytic plate, and the system can be modeled by a one-dimensional representation of Navier–Stokes equations. The flow equations (mass and momentum) are coupled with the thermal energy and species conservation equations in their 1D form.

Thermal energy

$$\frac{\partial \mathrm{T}}{\partial \mathrm{t}}=-\left[\frac{{\mathsf{\rho }\mathrm{v}}_{\mathrm{x}}}{\mathsf{\rho }}+\frac{1}{{\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}}{\displaystyle \sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{g}}}{\mathrm{c}}_{\mathrm{p},\mathrm{i}}{\mathrm{j}}_{\mathrm{i}}\right]\frac{\partial \mathrm{T}}{\partial \mathrm{x}}-\frac{1}{{\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}}{\displaystyle \sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{g}}}{\dot{\mathsf{\omega }}}_{\mathrm{i}}{\mathrm{M}}_{\mathrm{i}}{\mathrm{h}}_{\mathrm{i}}+\frac{1}{{\mathsf{\rho }\mathrm{c}}_{\mathrm{p}}}\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\lambda }\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)$$

(7)

Species continuity

$$\frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{t}}=-\frac{{\mathsf{\rho }\mathrm{v}}_{\mathrm{x}}}{\mathsf{\rho }}\frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{x}}+\frac{1}{\mathsf{\rho }}{\dot{\mathsf{\omega }}}_{\mathrm{i}}{\mathrm{M}}_{\mathrm{i}}-\frac{1}{\mathsf{\rho }}\frac{\partial {\mathrm{j}}_{\mathrm{i}}}{\partial \mathrm{x}}$$

(8)

Mixture continuity

$$0=\frac{\mathrm{p}}{\mathrm{R}}\frac{{\overline{\mathrm{M}}}^2}{{\mathrm{T}}^2}\left[\mathrm{T}{\displaystyle \sum }_{\mathrm{i}}\frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{t}}\frac{1}{{\mathrm{M}}_{\mathrm{i}}}+\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\frac{1}{\overline{\mathrm{M}}}\right]-2\mathsf{\rho }\mathrm{V}-\frac{\partial \left({\mathsf{\rho }\mathrm{v}}_{\mathrm{x}}\right)}{\partial \mathrm{x}}$$

(9)

Radial momentum

$$0=-\frac{{\mathsf{\rho }\mathrm{v}}_{\mathrm{x}}}{\mathsf{\rho }}\frac{\partial \mathrm{V}}{\partial \mathrm{x}}-{\mathrm{V}}^2-\frac{\Lambda }{\mathsf{\rho }}+\frac{1}{\mathsf{\rho }}\frac{\partial }{\partial \mathrm{x}}\left(\mathsf{\mu }\frac{\partial \mathrm{V}}{\partial \mathrm{x}}\right)$$

(10)

Eigenvalue of the radial momentum

$$0=\frac{\partial \Lambda }{\partial \mathrm{x}}$$

(11)

Ideal gas law

$$\mathsf{\rho }=\frac{\mathrm{p}\overline{\mathrm{M}}}{\mathrm{RT}}$$

(12)

Dependent variables in the governing equations are the temperature T, the species mass fraction ${\mathrm{Y}}_{\mathrm{i}}$, the axial mass flux ${\mathsf{\rho }\mathrm{v}}_{\mathrm{x}}$, the scaled radial velocity $\mathrm{V}$, and the eigenvalue of the momentum equation $\Lambda$. Independent variables are the axial distance from the surface x and the time t.

The diffusion flux of each gas-phase species considers diffusive mass flux caused by concentration and temperature gradients in a mixture-average formulation

$${\mathrm{j}}_{\mathrm{i}}=-\left({\mathsf{\rho }\mathrm{D}}_{\mathrm{i},\mathrm{M}}\frac{{\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{X}}_{\mathrm{i}}}\frac{\partial {\mathrm{X}}_{\mathrm{i}}}{\partial \mathrm{x}}+\frac{{\mathrm{D}}_{\mathrm{i}}^{\mathrm{T}}}{\mathrm{T}}\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)$$

(13)

in which (${\mathrm{D}}_{\mathrm{i},\mathrm{M}}$) is the averaged diffusion coefficient and (${\mathrm{D}}_{\mathrm{i}}^{\mathrm{T}}$) is the thermal diffusion coefficient of the species i. We account for thermal diffusion effect in the diffusion velocity calculation, because it may play a vital role for light species (namely for H₂).

3.3. Chemical and Physical Processes at the Catalytic Surface

Heterogeneous chemical processes at the surface are taken into account by considering finite diffusion inside the washcoat. Therefore, three different surface models are used, i.e., simple effectiveness factor approach, one-dimensional reaction-diffusion equations, and Dusty-Gas model.

Effectiveness factor approach (η-approach) is a simple method to account for diffusion limitations in the washcoat. It is based on the assumption that one target species determines overall reactivity [50]. An effectiveness factor is calculated for the chosen species based on the Thiele modulus [51], and all reaction rates are multiplied by this factor in the species governing equation at the gas-surface interface. In this approach, the effectiveness factor is defined as the ratio of the effective surface reaction rate inside the washcoat to the surface reaction rate without considering the diffusion limitation [34,52].

One-dimensional reaction-diffusion equations (RD-approach) offer a more adequate model to account for internal mass transfer limitations than the effectiveness factor approach. The model calculates spatial variations of concentrations and surface reaction rates inside the washcoat and each gas-phase species leads to one reaction-diffusion equation [34,52].

The most sophisticated surface model in our study is the Dusty-Gas model (DGM). In the DGM, species transport inside the washcoat accounts for ordinary and Knudsen diffusion as well as the pressure-driven convective flow (Darcy flow) [53,54]. The species mass conservation inside the washcoat is given in a conservative form as

$$\mathsf{\epsilon }\frac{\partial \left({\mathsf{\rho }}_{\mathrm{g}}{\mathrm{Y}}_{\mathrm{k}}\right)}{\partial \mathrm{t}}=-\nabla {\mathrm{j}}_{\mathrm{i}}^{\mathrm{DGM}}+{\mathsf{\gamma }\dot{\mathrm{s}}}_{\mathrm{k}}{\mathrm{W}}_{\mathrm{k}}$$

(14)

where $\mathsf{\gamma }$ stands for the catalyst density (active catalytic surface area per washcoat volume) as

$$\mathsf{\gamma }=\frac{{\mathrm{F}}_{\mathrm{cat}/\mathrm{geo}}}{\mathrm{L}}$$

(15)

Total mass density inside the washcoat is given as

$$\mathsf{\epsilon }\frac{\partial \left({\mathsf{\rho }}_{\mathrm{g}}\right)}{\partial \mathrm{t}}=-{\displaystyle \sum }_{\mathrm{k}=1}^{{\mathrm{K}}_{\mathrm{g}}}\nabla {\mathrm{j}}_{\mathrm{i}}^{\mathrm{DGM}}+{\displaystyle \sum }_{\mathrm{k}=1}^{{\mathrm{K}}_{\mathrm{g}}}{\mathsf{\gamma }\dot{\mathrm{s}}}_{\mathrm{k}}{\mathrm{W}}_{\mathrm{k}}$$

(16)

In the DGM, the fluxes of each species are coupled with one another [55]. The species molar fluxes are evaluated here using the DGM as it is given in [56]

$${\mathrm{j}}_{\mathrm{i}}^{\mathrm{DGM}}=-\left[{\displaystyle \sum }_{\mathrm{n}=1}^{{\mathrm{K}}_{\mathrm{g}}}{\mathrm{D}}_{\mathrm{in}}^{\mathrm{DGM}}\nabla {\mathrm{C}}_{\mathrm{n}}+\left({\displaystyle \sum }_{\mathrm{n}=1}^{{\mathrm{K}}_{\mathrm{g}}}\frac{{\mathrm{D}}_{\mathrm{in}}^{\mathrm{DGM}}{\mathrm{C}}_{\mathrm{n}}}{{\mathrm{D}}_{\mathrm{n},\mathrm{Knud}}}\right)\frac{{\mathrm{B}}_{\mathrm{g}}}{{\mathsf{\mu }}_{\mathrm{w}}}\nabla {\mathrm{p}}_{\mathrm{w}}\right]$$

(17)

where ${\mathrm{C}}_{\mathrm{n}}$ is the concentration of gas-phase species n, and $\mathsf{\mu }$ is the viscosity of the mixture. The pressure ($\mathrm{p}$) inside the washcoat is calculated from the ideal-gas law. ${\mathrm{D}}_{\mathrm{in}}^{\mathrm{DGM}}$ in Equation (17) are the matrix of diffusion coefficients. Diffusion coefficients (${\mathrm{D}}_{\mathrm{in}}^{\mathrm{DGM}}$) can be calculated from the inverse matrix [54]

$${\mathrm{D}}_{\mathrm{in}}^{\mathrm{DGM}}={\mathrm{H}}^{-1}$$

(18)

where the elements of the H matrix are determined as [54]

$${\mathrm{h}}_{\mathrm{in}}=\left[\frac{1}{{\mathrm{D}}_{\mathrm{i},\mathrm{Knud}}}+{\displaystyle \sum }_{\mathrm{M}\ne \mathrm{i}}\frac{{\mathrm{X}}_{\mathrm{M}}}{{\mathrm{D}}_{\mathrm{i},\mathrm{M}}}\right]{\mathsf{\delta }}_{\mathrm{in}}+\left({\mathsf{\delta }}_{\mathrm{in}}-1\right)\frac{{\mathrm{X}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{i},\mathrm{n}}}$$

(19)

where ${\mathrm{D}}_{\mathrm{i},\mathrm{knud}}$ is the Knudsen diffusion coefficient of ith species and determined as

$${\mathrm{D}}_{\mathrm{i},\mathrm{Knud}}=\frac{{\mathrm{d}}_{\mathrm{p}}}{3}\sqrt{\frac{8\mathrm{RT}}{{\mathsf{\pi }\mathrm{M}}_{\mathrm{i}}}}$$

(20)

in which ${\mathrm{d}}_{\mathrm{p}}$ is the mean pore diameter. Averaged diffusion coefficients (${\mathrm{D}}_{\mathrm{i},\mathrm{M}}$) in Equation (19) are calculated in DETCHEM library [45]. The permeability in Equation (17) is calculated from the Kozeny–Carman relationship [56] as

$${\mathrm{B}}_{\mathrm{g}}=\frac{{\mathsf{\epsilon }}^3{\mathrm{d}}_{\mathrm{pt}}^2}{72\mathsf{\tau }{\left(1-\mathsf{\epsilon }\right)}^2}$$

(21)

where ${\mathrm{d}}_{\mathrm{pt}}$ is the particle diameter. Surface coverages (i.e., fraction of the surface sites covered by surface species i) in the washcoat are calculated via

$$\frac{\partial {\Theta }_{\mathrm{i}}}{\partial \mathrm{t}}=\frac{{\dot{\mathrm{s}}}_{\mathrm{i},\mathrm{w}}{\mathsf{\sigma }}_{\mathrm{i}}}{\Gamma }$$

(22)

in which ${\mathsf{\sigma }}_{\mathrm{i}}$ is the coordination number and $\Gamma$ is the surface site density.

Heat transport between the resistive heater and the gas/washcoat interface involves different contributions. In the experiments of Karakaya et al. [28], the resistive heater (FeCrAl alloy) is used for supplying the required heat to the washcoat. There exists the ceramic support between the resistive heater and the washcoat (Figure 13). The following energy conservation equations are coupled in this study only with the RD-approach.

Heat flux from the heater to the ceramic support can be calculated from the resistive heating. Energy equation for the ceramic substrate is given as

$${\mathsf{\rho }}_{\mathrm{cr}}{\mathrm{C}}_{\mathrm{p},\mathrm{cr}}\frac{\partial {\mathrm{T}}_{\mathrm{cr}}}{\partial \mathrm{t}}={\mathsf{\lambda }}_{\mathrm{cr}}\frac{{\partial }^2{\mathrm{T}}_{\mathrm{cr}}}{\partial {\mathrm{x}}^2}$$

(23)

where the left-hand side represents the energy storage in the ceramic substrate. The right-hand side represents the conduction of the energy along the substrate. Energy equation inside the washcoat is given as

$$\overline{{\mathsf{\rho }\mathrm{C}}_{\mathrm{p}}}\frac{\partial {\mathrm{T}}_{\mathrm{wc}}}{\partial \mathrm{t}}={\mathsf{\lambda }}_{\mathrm{eff}}\frac{{\partial }^2{\mathrm{T}}_{\mathrm{wc}}}{\partial {\mathrm{x}}^2}-\mathsf{\gamma }{\displaystyle \sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{g}}+{\mathrm{N}}_{\mathrm{s}}}{\mathrm{h}}_{\mathrm{i}}{\dot{\mathrm{s}}}_{\mathrm{i}}{\mathrm{W}}_{\mathrm{i}}-{\displaystyle \sum }_{\mathrm{k}=1}^{{\mathrm{N}}_{\mathrm{g}}}{\mathrm{h}}_{\mathrm{i}}{\mathrm{j}}_{\mathrm{i}}^{\mathrm{w}}$$

(24)

where the left hand side represents the energy storage in the washcoat. The term $\overline{{\mathsf{\rho }\mathrm{C}}_{\mathrm{p}}}$ is here the effective specific heat capacity of the combined washcoat and gas mixture in each cell of the washcoat [57]. The first term on the right-hand side accounts for the conduction of the energy along the washcoat. Heat release due to surface reactions is modeled via the second term. The last term on the right-hand side considers the heat transport due to species diffusion. The effective conductivity in the washcoat is calculated as

$${\mathsf{\lambda }}_{\mathrm{eff}}=\frac{1}{\left(\left(1-\mathsf{\epsilon }\right)/3{\mathsf{\lambda }}_{\mathrm{wc}}\right)+\left(\mathsf{\epsilon }/\left(2{\mathsf{\lambda }}_{\mathrm{wc}}+{\mathsf{\lambda }}_{\mathrm{mix},\mathrm{wc}}\right)\right)}-2{\mathsf{\lambda }}_{\mathrm{wc}}$$

(25)

in which ${\mathsf{\lambda }}_{\mathrm{wc}}$ is the thermal conductivity of the washcoat and ${\mathsf{\lambda }}_{\mathrm{mix},\mathrm{wc}}$ is the thermal conductivity of the gas mixture in each cell of the washcoat [32]. In RD-approach, it is assumed that the diffusive mass flux in the washcoat is due to the concentration gradient [34]. Here, we extend the approach by assuming that the diffusive mass flux in the washcoat is due to both concentration and temperature gradients. Therefore, diffusive mass flux is given as

$${\mathrm{j}}_{\mathrm{i}}^{\mathrm{w}}=-\left({\mathrm{D}}_{\mathrm{i},\mathrm{eff}}\frac{\partial {\mathrm{c}}_{\mathrm{i},\mathrm{w}}}{\partial \mathrm{x}}+\left(\frac{\mathsf{\epsilon }}{\mathsf{\tau }}\frac{{\mathrm{D}}_{\mathrm{i},\mathrm{T}}}{{\mathrm{M}}_{\mathrm{i}}}\frac{1}{\mathrm{T}}\right)\frac{\partial \mathrm{T}}{\partial \mathrm{x}}\right)$$

(26)

where (${\mathrm{D}}_{\mathrm{i},\mathrm{eff}}$) is the effective diffusion coefficient of the ith species [34].

3.4. Boundary Conditions

Boundary conditions must be included to close the equation system. They are introduced for the washcoat support side, gas-surface/washcoat interface, and the gas-inlet. Detailed information for the boundary conditions was given in the studies of Karadeniz et al. [34,44]. Here, we additionally give the boundary conditions of the energy conservation equations for the ceramic support and washcoat.

The boundary condition between the resistive heater and the ceramic support (at x = ${\mathsf{\delta }}_{\mathrm{wc}}$ + ${\mathsf{\delta }}_{\sup }$) is given as

$${\mathsf{\rho }}_{\mathrm{cr}}{\mathrm{C}}_{\mathrm{p},\mathrm{cr}}\frac{\partial {\mathrm{T}}_{\mathrm{cr},1}}{\partial \mathrm{t}}∆{\mathrm{x}}_1^{+}={\mathrm{q}}^{″}+{\mathsf{\lambda }}_{\mathrm{cr}}\frac{\partial {\mathrm{T}}_{\mathrm{cr},1}}{\partial {\mathrm{x}}_1}$$

(27)

where ${\mathrm{q}}^{″}$ is the heat flux supplied by the heater. $∆{\mathrm{x}}_1^{+}$ is the halfway of the distance between the heater-ceramic support interface and adjacent grid point in the ceramic support. The boundary condition at the ceramic support-washcoat interface (at x = ${\mathsf{\delta }}_{\mathrm{wc}}$) is given as

$$\left({\mathsf{\rho }}_{\mathrm{cr}}{\mathrm{C}}_{\mathrm{p},\mathrm{cr}}∆{\mathrm{x}}_2^{+}+{\mathsf{\rho }}_{\mathrm{wc}}{\mathrm{C}}_{\mathrm{p},\mathrm{wc}}∆{\mathrm{x}}_3^{+}\right)\frac{\partial {\mathrm{T}}_{\mathrm{k}}}{\partial \mathrm{t}}={\mathsf{\lambda }}_{\mathrm{cr}}\frac{{\mathrm{T}}_{\mathrm{k}-1}-{\mathrm{T}}_{\mathrm{k}}}{∆{\mathrm{x}}_2}-{\mathsf{\lambda }}_{\mathrm{eff}}\frac{{\mathrm{T}}_{\mathrm{k}}-{\mathrm{T}}_{\mathrm{k}+1}}{∆{\mathrm{x}}_3}$$

(28)

where $∆{\mathrm{x}}_2$ is the distance between the ceramic support-washcoat interface and adjacent grid point in the ceramic substrate. $∆{\mathrm{x}}_3$ is the distance between the ceramic support-washcoat interface and adjacent grid point in the washcoat. $∆{\mathrm{x}}_2{}^{+}$ and $∆{\mathrm{x}}_3{}^{+}$ are given as $∆{\mathrm{x}}_2{}^{+}=∆{\mathrm{x}}_2/2$ and $∆{\mathrm{x}}_3{}^{+}=∆{\mathrm{x}}_3/2$, respectively. Finally, energy balance at the gas-washcoat interface (at x = 0.0) is given as

$$\begin{array}{cc}\hfill ({\mathsf{\rho }}_{\mathrm{mix}}{\mathrm{c}}_{\mathrm{p},\mathrm{mix}}∆{\mathrm{x}}_4^{+}& +{\mathsf{\rho }}_{\mathrm{wc}}{\mathrm{C}}_{\mathrm{p},\mathrm{wc}}∆{\mathrm{x}}_5^{+})\frac{\partial {\mathrm{T}}_{\mathrm{int}}}{\partial \mathrm{t}}\hfill \\ \hfill & =\mathsf{\lambda }\frac{\partial {\mathrm{T}}_{\mathrm{int}}}{\partial {\mathrm{x}}_4}-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{{\mathrm{N}}_{\mathrm{g}}}}{\mathrm{h}}_{\mathrm{i}}\left({\mathrm{j}}_{\mathrm{i}}+{\mathsf{\rho }\mathrm{Y}}_{\mathrm{i}}\mathrm{u}\right)-\mathsf{\sigma }\mathsf{ϵ}\left({\mathrm{T}}_{\mathrm{int}}{}^4-{\mathrm{T}}_{\mathrm{rad}}{}^4\right)+{\mathsf{\lambda }}_{\mathrm{eff}}\frac{\partial {\mathrm{T}}_{\mathrm{int}}}{\partial {\mathrm{x}}_5}\hfill \end{array}$$

(29)

where $∆{\mathrm{x}}_4$ is the distance between the gas-washcoat interface and adjacent grid point in the gas-phase. $∆{\mathrm{x}}_5$ is the distance between the gas-washcoat interface and adjacent grid point in the washcoat. $∆{\mathrm{x}}_4^{+}$ and $∆{\mathrm{x}}_5^{+}$ are given as $∆{\mathrm{x}}_4{}^{+}=∆{\mathrm{x}}_4/2$ and $∆{\mathrm{x}}_5{}^{+}=∆{\mathrm{x}}_5/2$, respectively. The first term on the right hand side of Equation (29) accounts for heat conduction from the interface to the gas according to the Fourier heat conductivity law. The second term describes convective and diffusive energy transport from the gas-phase to the surface, where ${\mathrm{h}}_{\mathrm{i}}$ is the enthalpy of species i. The third term is heat radiation from the surface due to the Stefan–Boltzmann law, where $\mathsf{\sigma }$ is the Stefan–Boltzmann constant and $\mathsf{\epsilon }$ is the emissivity of the outer washcoat surface. Here, ${\mathrm{T}}_{\mathrm{rad}}$ is the reference temperature to which the surface radiates. The fourth term encompasses heat conduction from washcoat to interface according to the Fourier heat conductivity law.

3.5. Numerical Solution of the Model Equations

The modeled system consists of partial-differential equations (PDE). For numerical solution, it is necessary to transform it to a system of ordinary differential and algebraic equations (DAE). This is accomplished through spatial discretization of the PDE system by using finite difference approximations on a non-equidistant grid. This results in a set of ordinary differential and algebraic equations. The equation set is solved with the LIMEX [58] solver that uses a semi-implicit extrapolation method. Implementation of the mentioned numerical problem is accomplished by the software tool, DETCHEM^STAG [45], which was also used in [34,59].

3.6. Surface Reaction Mechanism for CH₄ and C₃H₈ Partial Oxidation and Steam Reforming over Rh/Al₂O₃

The reaction mechanism presented here uses a 62 step detailed reaction steps among 7 gas-phase and 17 surface species to represent H₂/CO/H₂O/CO₂/O₂/CH₄/C₃H₈ system on a Rh/Al₂O₃ surface, which is given in the Appendix A (

Table A1. Proposed reaction mechanism for the system H₂/CO/H₂O/CO₂/O₂/CH₄/C₃H₈.

	Reaction	A (cm, mol, s)	b	Ea (kJ/mol)
R1	H2 + Rh(s) + Rh(s) → H(s) + H(s)	3.000 × 10−2	stick. coeff.
R2	O2 + Rh(s) + Rh(s) → O(s) + O(s)	1.000 × 10−2	stick. coeff.
R3	H2O + Rh(s) → H2O(s)	1.000 × 10−1	stick. coeff.
R4	CO2 + Rh(s) → CO2(s)	4.800 × 10−2	stick. coeff.
R5	CO + Rh(s) → CO(s)	4.971 × 10−1	stick. coeff.
R6	CH4 + Rh(s) → CH4(s)	1.300 × 10−2	stick.coeff.
R7	C3H8+ Rh(s) → C3H8(s)	2.000 × 10−3	stick.coeff.
R8	H(s) + H(s) → Rh(s) + Rh(s) + H2	5.574 × 1019	0.239	59.69
R9	O(s) + O(s) → Rh(s) + Rh(s) + O2	5.329 × 1022	−0.137	387.00
R10	H2O(s) → H2O + Rh(s)	6.858 × 1014	−0.280	44.99
R11	CO(s) → CO + Rh(s)	1.300 × 1013	0.295	134.07–47θCO
R12	CO2(s) → CO2 + Rh(s)	3.920 × 1011	0.315	20.51
R13	CH4 (s) → CH4 + Rh(s)	1.523 × 1013	−0.110	26.02
R14	C3H8(s) → C3H8+ Rh(s)	1.000 × 1013	−0.500	30.10
R15	H(s) + O(s) → OH(s)+ Rh(s)	8.826 × 1021	−0.048	73.37
R16	OH(s)+ Rh(s) → H(s) + O(s)	1.000 × 1021	0.045	48.04
R17	H(s) + OH(s) → H2O(s)+ Rh(s)	1.743 × 1022	−0.127	41.73
R18	H2O(s) + Rh(s) → H(s) + OH(s)	5.408 × 1022	0.129	98.22
R19	OH(s) + OH(s) → H2O(s) + O(s)	5.736 × 1020	−0.081	121.59
R20	H2O(s) +O(s) → OH(s) + OH(s)	1.570 × 1022	0.081	203.41
R21	CO2(s) + Rh(s) → CO(s) + O(s)	5.752 × 1022	−0.175	106.492
R22	CO(s) + O(s) → CO2(s) + Rh(s)	6.183 × 1021	0.034	129.98–47θCO
R23	CO(s) + Rh(s) → C(s) + O(s)	6.390 × 1021	0.000	174.76–47θCO
R24	C(s) + O(s) → CO(s) + Rh(s)	1.173 × 1022	0.000	92.14
R25	CO(s) + OH(s) → COOH(s) + Rh(s)	2.922 × 1020	0.000	55.33–47θCO
R26	COOH(s) + Rh(s) → CO(s) + OH(s)	2.738 × 1021	0.160	48.38
R27	COOH(s) + Rh(s) → CO2(s) + H(s)	1.165 × 1019	0.000	5.61
R28	CO2(s) + H(s) → COOH(s) + Rh(s)	1.160 × 1020	−0.160	14.48
R29	COOH(s) + H(s) → CO(s) + H2O(s)	5.999 × 1019	−0.188	33.55
R30	CO(s) + H2O(s) → COOH(s) + H(s)	2.258 × 1019	0.051	97.08–47θCO
R31	CO(s) + OH(s) → CO2(s) + H(s)	3.070 × 1019	0.000	82.94–47θCO
R32	CO2(s) + H(s) → CO(s) + OH(s)	2.504 × 1021	−0.301	84.77
R33	C(s) + OH(s) → CO(s) + H(s)	4.221 × 1020	0.078	30.04–120θC
R34	CO(s) + H(s) → C(s) + OH(s)	3.244 × 1021	−0.078	138.26–47θCO
R35	CH4(s) +Rh(s) → CH3(s) +H(s)	4.622 × 1021	0.136	72.26
R36	CH3(s) +H(s) → CH4(s) +Rh(s)	2.137 × 1021	−0.058	46.77
R37	CH3(s) +Rh(s) → CH2(s) +H(s)	1.275 × 1024	0.078	107.56
R38	CH2(s) +H(s) → CH3(s) +Rh(s)	1.073 × 1022	−0.078	39.54
R39	CH2(s) +Rh(s) → CH(s) +H(s)	1.275 × 1024	0.078	115.39
R40	CH(s) +H(s) → CH2(s) +Rh(s)	1.073 × 1022	−0.078	52.61
R41	CH(s) +Rh(s) → C(s) +H(s)	1.458 × 1020	0.078	23.09
R42	C(s) +H(s) → CH(s) +Rh(s)	1.122 × 1023	−0.078	170.71–120 θC
R43	CH4(s) +O(s) → CH3(s) +OH(s)	3.465 × 1023	0.051	77.71
R44	CH3(s) +OH(s) → CH4(s) +O(s)	1.815 × 1022	−0.051	26.89
R45	CH3(s) +O(s) → CH2(s) +OH(s)	4.790 × 1024	0.000	114.52
R46	CH2(s) +OH(s) → CH3(s) +O(s)	2.858 × 1021	0.000	20.88
R47	CH2(s) +O(s) → CH(s) +OH(s)	4.790 × 1024	0.000	141.79
R48	CH(s) +OH(s) → CH2(s) +O(s)	2.858 × 1021	−0.000	53.41
R49	CH(s) +O(s) → C(s) +OH(s)	5.008 × 1020	0.000	26.79
R50	C(s) +OH(s) → CH(s) +O(s)	2.733 × 1022	0.000	148.81–120θC
R51	C3H8(s)+ Rh(s) → C3H7(s)+ H(s)	1.300 × 1021	0.000	52.00
R52	C3H7(s)+ H(s) → C3H8(s)+ Rh(s)	1.349 × 1021	0.156	46.73
R53	C3H7(s) + Rh(s) → C3H6(s)+ H(s)	5.028 × 1021	−0.118	84.05
R54	C3H6(s)+ H(s) → C3H7(s)+ Rh(s)	2.247 × 1022	0.115	65.25
R55	C3H8(s)+ O(s) → C3H7(s)+ OH(s)	7.895 × 1024	−0.124	69.65
R56	C3H7(s)+ OH(s) → C3H8(s)+ O(s)	1.087 × 1024	0.124	33.24
R57	C3H7(s)+ O(s) → C3H6(s)+ OH(s)	1.276 × 1022	−0.162	88.97
R58	C3H6(s)+ OH(s) → C3H7(s)+ O(s)	1.875 × 1020	0.162	45.03
R59	C3H6(s)+ Rh(s) → C2H3(s)+ CH3(s)	1.370 × 1024	−0.280	94.63
R60	C2H3(s)+ CH3(s) → C3H6(s)+ Rh(s)	9.113 × 1024	0.279	44.88
R61	C2H3(s)+ Rh(s) → CH3(s)+ C(s)	1.370 × 1022	−0.280	46.53
R62	CH3(s)+ C(s) → C2H3(s)+ Rh(s)	1.563 × 1023	0.280	107.78–120θC

The rate coefficients are given in the form of k = AT^β exp(−E_a/RT); adsorption kinetics is given in the form of sticking coefficients; the surface site density is Г = 2.72 × 10⁻⁹ mol cm⁻¹.

). The mechanism takes the thermodynamically consistent H₂/CO/H₂O/CO₂/O₂/CH₄ system [28] and extends it by an additional 14 reactions to account for partial oxidation and steam reforming of propane [16]. The reaction mechanism was developed and validated against a zero-dimensional stagnation-flow reactor data. The fidelity of the reaction kinetics was further tested against a lab-scale annular flow reactor for propane partial oxidation for various temperature and C/O range [29]. The temperature-dependent forward rate coefficients of (${\mathrm{k}}_{\mathrm{fj}}$) are written in the Arrhenius form. The net production rate of each chemical species in the gas phase is balanced with the diffusive flux of that species in the gas phase at steady-state conditions by assuming that no deposition or ablation of chemical species occurs on/from the catalyst surface:

$${\mathsf{\rho }\mathrm{Y}}_{\mathrm{k}}{\mathrm{V}}_{\mathrm{k}}={\mathrm{F}}_{\mathrm{cat}/\mathrm{geo}}\dot{{\mathrm{s}}_{\mathrm{k}}}{\mathrm{W}}_{\mathrm{k}}{\mathsf{\eta }}_{\mathrm{k}}$$

(30)

The molar reaction rates ($\dot{{\mathrm{s}}_{\mathrm{k}}}$) of adsorbed and gas-phase species are then a function of the local coverage and gas-phase concentrations at the gas-catalyst [43,60].

$${\dot{\mathrm{s}}}_{\mathrm{k}}={\displaystyle \sum }_{\mathrm{j}=1}^{{\mathrm{K}}_{\mathrm{s}}}{\mathrm{v}}_{\mathrm{kj}}{\mathrm{k}}_{\mathrm{fj}}{\displaystyle \prod }_{\mathrm{n}=1}^{{\mathrm{N}}_{\mathrm{g}}+{\mathrm{N}}_{\mathrm{s}}+{\mathrm{N}}_{\mathrm{b}}}{\mathrm{c}}_{\mathrm{n}}^{{\mathrm{v}}_{\mathrm{nj}}}$$

(31)

Here, ${\mathrm{K}}_{\mathrm{s}}$ is the number of surface reactions and ${\mathrm{c}}_{\mathrm{n}}$ are the species concentrations, given in mol/m² for ${\mathrm{N}}_{\mathrm{s}}$ surface species and in mol/m³ for ${\mathrm{N}}_{\mathrm{g}}$ gas phase species and ${\mathrm{N}}_{\mathrm{b}}$ bulk species. The concentration of adsorbed species c_i is related to the coverage, $\mathsf{\theta }$_i, by the surface site density Γ (c_i = θ _I × Γ). The surface site density, i.e., the number of adsorption sites per surface area of the Rh particle, is estimated to be 2.72 × 10⁻⁹ mol cm⁻². For the reaction $\mathrm{j}$ with its stoichiometric coefficients of ${\mathrm{v}}_{\mathrm{kj}}$.

The term ${\mathrm{F}}_{\mathrm{cat}/\mathrm{geo}}$, which is derived from the CO chemisorption measurements, is introduced as a scaling factor for the active catalytic surface area ${\mathrm{A}}_{\mathrm{catalyst}}$ and the geometric surface area ${\mathrm{A}}_{\mathrm{geometric}}$ of the stagnation disk [32],

$${\mathrm{F}}_{\mathrm{cat}/\mathrm{geo}}=\frac{{\mathrm{A}}_{\mathrm{catalyst}}}{{\mathrm{A}}_{\mathrm{geometric}}}$$

(32)

4. Conclusions

This study investigated CPOX and SR of CH₄ and C₃H₈ numerically in stagnation-flow over a porous Rh/Al₂O₃ catalytic disk. Numerically predicted species profiles in the external boundary layer were validated with the recently published experimental data. Physical and chemical processes inside the washcoat were discussed at a fundamental level. The fundamental findings of this study are relevant for practical conditions for catalytic reactors and they are summarized below:

• The simulations with all three surface models (η-approach, RD-approach, and DGM) indicated strong diffusion limitations inside the washcoat for all studied CPOX and SR cases. Therefore, internal mass transfer limitations must be considered for accurately predicting the experiments for CPOX and SR of CH₄ and C₃H₈ over Rh/Al₂O₃ catalyst layers.
• The RD-approach and DGM gave an insight into the reaction routes inside the washcoat. According to the RD-approach and DGM simulations, there is not a direct reaction mechanism in the catalyst for the oxidation of CH₄ and C₃H₈ in particular cases. Total oxidation, steam and dry reforming of CH₄ and C₃H₈, and WGS reactions occur in the catalyst. In steam reforming, the main reaction route is SR of CH₄ and C₃H₈ in the catalyst.
• DGM simulations gave almost identical species profiles with the RD-approach for all CPOX and SR cases for CH₄, which indicates that the species transport inside the washcoat due to pressure-driven convective flow can be neglected.
• The simulations showed that increasing the reactor pressure and decreasing the inlet flow velocity increases the external mass transfer limitations and decreases the internal mass transfer limitations both for CPOX and SR of CH₄ and C₃H₈. However, internal mass transfer limitations outweigh external mass transfer limitations. Hence, increasing the reactor pressure and decreasing the inlet flow velocity increases the hydrogen production significantly.
• The results showed that the temperature gradient inside the washcoat is negligible, which implies that the thermal conductivity of the alumina is adequate enough to obtain a uniform temperature distribution inside a thin Rh/Al₂O₃ catalyst layer. Therefore, it is a proper assumption to consider the washcoat as isothermal.

It is expected that the fundamental findings of this study can help to understand the complex processes in practical catalytic reactors for CPOX and SR of CH₄ and C₃H₈.