Proposing a novel theoretical optimized model for the combined dry and steam reforming of methane in the packed-bed reactors

Abstract

A packed-bed reactor was modeled for combined dry and steam reforming (CDSR) and simulated using a two-dimensional heterogeneous model at steady-state condition. The model outputs showed a good agreement with experimental data. The effects of important operational parameters such as feed temperature, pressure, molar flow, and CO₂/CH₄ and H₂O/CH₄ ratios on methane conversion and H₂/CO ratio in synthesis gas were also evaluated. Afterward, the modified artificial neural network (ANN) model was used for approximating the results of a simulation with high accuracy. The outputs of ANN model show that the predicted values of ANN model are in good agreement with those of the heterogeneous model, suggesting that the model was successfully developed to capture the correlation between operation conditions, methane conversion, and H₂/CO ratio in the synthesis gas. Finally, a multi-objective optimization based on the hybrid of ANN and non-dominated sorting genetic algorithm-II (NSGA-II) was carried out to find the best-operating conditions for the methanol production and Fischer–Tropsch synthesis reaction with the desired H₂/CO molar ratio of about two in synthesis gas. So, the main objectives for CDSR are providing a high methane conversion and also H₂/CO ratio of two in the output synthesis gas.

Appendices

Appendix: Governing gas property and transport coefficients

Average molecular weight

The average molecular weight of gas phase was gained using the following formula (Murmura et al. 2017):

$$M_{\text{av}} = \mathop \sum \limits_{i = 1}^{n} y_{i} M_{i} \left( {n:{\text{number of components}}} \right).$$

(A1)

Fluid effective radial mass diffusivity

The fluid effective radial mass diffusivity ($D_{\text{er}}$) was obtained from the following equation (Froment et al. 2010):

$$D_{\text{er}} = \left( {u_{s} d_{p} } \right)/{\text{Pe}}_{\text{rf}} .$$

(A2)

The radial Peclet number in radial coordinate of the reactor (${\text{Pe}}_{\text{rf}} )$ was achieved from a graph that is illustrated against Reynolds number in Froment et al. (2010).

The Reynolds number was calculated as follows (Froment et al. 2010):

$$\text{Re} = \frac{{d_{\text{p}} \rho_{\text{g}} u_{\text{s}} }}{\mu }.$$

(A3)

The viscosity of the gas-phase mixture (μ) was obtained by Ermley and Wilke method as follows (Green and Perry 2007):

$$\mu = \mathop \sum \limits_{i = 1}^{n} \frac{{\mu_{i} }}{{1 + \mathop \sum \nolimits_{j = 1}^{n} \left( {Q_{ij} \frac{{y_{j} }}{{y_{i} }}} \right)}},$$

(A4)

where

$$Q_{ij} = \frac{{1 + \left[ {\left( {\frac{{\mu_{i} }}{{\mu_{j} }}} \right)^{0.5} \left( {\frac{{M_{j} }}{{M_{i} }}} \right)^{0.25} } \right]^{2} }}{{\sqrt 8 \left[ {1 + \frac{{M_{i} }}{{M_{j} }}} \right]^{0.5} }}.$$

(A5)

Different components Viscosity was reported as a function of temperature as follows (Green and Perry 2007):

$$\mu = \frac{{1000{\text{AT}}^{B} }}{{1 + \frac{C}{T} + \frac{D}{{T^{2} }}}}.$$

(A6)

Constants A, B, C and D are accessible in the references.

By modifying the ideal gas law, the following equation was obtained to determine the gas-phase density (Green and Perry 2007):

$$\rho_{g} = \frac{{P_{t} M_{\text{av}} }}{\text{RT}}.$$

(A7)

Radial effective thermal conductivity

The radial effective thermal conductivity was obtained from the following equation (Dixon and Cresswell 1979):

$$\lambda_{\text{er}} = k_{\text{rf}} + k_{\text{rs}} \left[ {\frac{{1 + 8k_{\text{rf}} /\left( {h_{\text{wf}} d_{p} } \right)}}{{1 + 16/3k_{\text{rs}} \left( {\frac{1}{{h_{\text{sf}} d_{p} }} + \frac{0.1}{{k_{p} }}} \right)}}} \right]\left( {1 - \varepsilon_{s} } \right)\left( {\frac{{d_{t,i} }}{{d_{p} }}} \right)^{2} .$$

(A8)

The wall–fluid thermal convection coefficient ($h_{\text{wf}}$) was determined as follows (Dixon and Cresswell 1979):

$$h_{\text{wf}} = \frac{{{\text{Nu}}_{\text{wf}} k_{g} }}{{d_{p} }},$$

(A9)

where the Nusselt number of the fluid in touch with the wall (${\text{Nu}}_{\text{wf}}$) is determined as follows (Dixon and Cresswell 1979):

$${\text{Nu}}_{\text{wf}} = 0.22{\text{Re}}^{0.75} .$$

(A10)

Catalyst–fluid thermal convective coefficient was calculated by Heggs and Handley method as follows (Handley and Heggs 1968):

$$h_{\text{sf}} = \frac{0.255}{{\varepsilon_{s} }} \times \frac{{k_{\text{af}} }}{{d_{p} }}\Pr^{1/3} \text{Re}^{2/3} .$$

(A11)

The gas-phase radial thermal conductivity coefficient can be achieved from the following equation (Othmer 1995):

$$k_{\text{rf}} = \frac{{k_{f} \text{Re} \Pr }}{{{\text{Pe}}_{\text{rf}} }}.$$

(A12)

The thermal conductivity of catalyst (k_rs) can be achieved by Zehner and Sculunder method (Dixon and Cresswell 1979).

Wall heat-transfer coefficient

The wall heat-transfer coefficient was reported as the following equation (Dixon and Cresswell 1979):

$$\alpha_{w} = \frac{{{\text{Nu}} \times k_{f} }}{{d_{p} }},$$

(A13)

where

$${\text{Nu}} = {\text{Nu}}_{\text{wf}} \left( {1 + \frac{{10\left( {{\text{k}}_{\text{rs}} /{\text{k}}_{\text{f}} } \right)}}{{{\text{Re}}.{ \Pr }}}} \right),$$

(A14)

where Nu is the modified Nusselt number of the fluid.

Catalyst effective radial mass diffusivity

The catalyst effective radial mass diffusivity was determined as follows (De Smet et al. 2001):

$$\frac{1}{{D_{e,i} }} = \left( {\frac{\tau }{{\varepsilon_{s} }}\left( {\frac{1}{{D_{i,m} }} + \frac{1}{{D_{k,i} }}} \right)} \right).$$

(A15)

For calculation of $D_{i,m}$, effective binary diffusion coefficient were determined from Wike. Then, by the Stefan–Maxwell equation the effective diffusion coefficient of component i in the multicomponent mixture was calculated equation (Froment et al. 2010).

The Knudsen diffusivity can be obtained from the kinetic theory of gases (Bird et al. 2002):

$$D_{k,i} = \frac{{d_{p} }}{3}(8\frac{\text{RT}}{{\pi M_{i} }})^{0.5} .$$

(A16)