Governing Equations

According to the descriptions and assumptions above, the basic transport equations for the two-dimensional and three-dimensional plate methanol steam micro-reformer are as follows:

Continuity equation:

u v w

2 2 2 terms for the reactant gas flow in the porous material of the catalyst layer of the micro-reformer. The Su, Sv and Sw are zero in the flow channel region. While in the catalyst layer, Su, Sv and Sw are different in each computation domain due to the difference in pressure when fluids pass through a porous medium. So, Su, Sv and Sw in the catalyst layer are [76]:

where kp is the permeability and β is the inertial loss coefficient in each component direction[76].

and where Dp is the catalyst particle diameter.

The viscosity of the gas mixture can be calculated from Wilke’s mixture rule [77] as follows:

28

In the species equation, mi denotes the mass fraction of the i^th species; the calculations have included CH3OH, H2O, H2, CO2 and CO. In this work, the porosity ε is expressed as 0.38 and 1.00, in the catalyst layer and the flow channel, respectively. In Eq(2-12), Deff is the effective diffusion coefficient based on the Stefan-Maxwell equations [51]. Eq(2-13) is employed to describe the influence of the porosity on the diffusion coefficient

eff k

D =D ε ^τ (2-13)

The diffusion coefficient Dk for the methanol steam micro-reformer was derived from the Stefan-Maxwell equations which were used to calculate the mean effective binary diffusivity [51].

Sc represents the source terms due to the chemical reaction in the catalyst layer.

Therefore, Sc is zero in the flow channel. Furthermore, Sc differs according to the reactant respectively, in the reaction.

According to the chemical kinetics of Hotz et al. [78], the steam reforming reaction is much faster than the decomposition and water-gas shift reaction. Purnama et al. [79] and

Agrell et al. [80] proposed that using a Cu/ZnO/AlO3 catalyst for methanol steam reforming gives rise to two main chemical reactions, the steam reforming and the reverse water gas shift reactions. They also indicated that CO was generated by the reverse water gas shift reaction.

Therefore, only the steam reforming reaction, Eq. (2-15), and the reverse water-gas shift reaction, Eq. (2-16), are considered in this study.

k1

In this study, the model for methanol steam reforming is that used by Hsueh and collaborators [56, 57], and the Arrhenius equation is used to calculate the concentration of reactant gases generated by the chemical reaction.

3 2

where the steam reforming reaction is a non-reversible reaction and the reverse water-gas shift reaction is reversible. The constants k1 and k2 are the forward rate constants for the steam reforming reaction and the reverse water-gas shift reaction, respectively. The constant k-2 is the backward rate constant for the water-gas shift reaction.

To calculate the local temperature, the energy equations must be solved.

Energy equation:

30

eff f s

k = ε + − εk (1 )k (2-20) where kf is the fluid phase thermal conductivity, ks the solid medium thermal conductivity and ε the porosity of the medium.

In the energy equation, St is the source term due to the chemical reactions, which in the channel is zero. The catalyst layer experiences exothermic and endothermic chemical reactions, so St can be described as:

t SR SR rWGS rWGS

S = − Δ( H R + ΔH R ) (2-21)

where ΔH_SR is the enthalpy of reaction of the steam reforming reaction, and ΔH_rWGS is the enthalpy of reaction of the reverse water-gas shift reaction.

In the solid regions, the energy transport equation can be written as

2 2 2

The boundary conditions of the present computation include those at the inlet, outlet, wall, and the interfaces between the flow channel and the catalyst layer.

(1) The boundary conditions for inlets at the flow channel and the catalyst layer: The inlet flow velocity is constant, the inlet gas composition is constant, and the inlet temperature is constant.

(2) The boundary conditions for outlets at the flow channel and the catalyst layer: The gauge pressure is zero.

(3) The boundary conditions for the interface between the solid wall and the insulated walls: The temperature gradients are zero.

(4) The boundary conditions for the interface between the flow channel and the insulated

walls: The velocities, temperature, temperature gradient, species concentration and species flux are zero.

(5) The boundary conditions for the interface between the flow channel and solid wall.

No slip and zero fluxes hold the velocities and the concentration gradients at zero.

(6) The boundary conditions for the interface between the flow channel and the catalyst layer: The velocities, temperature, species concentration and species flux are continuous.

(7) The boundary conditions for the interface between the heated wall and the catalyst layers: The velocities and the concentration gradient are assumed to be zero, and the temperature is assumed to be equal to the constant wall temperature.

2.2 The Model of a Plate Methanol Steam Micro-Reformer with Methanol Catalytic Combustor

2.2.1 Model Description

In order to simplify the multifarious changes due to the wall temperature variation, the model above did not consider the wall thermal boundary condition to be a non-uniform temperature. To keep everything isothermal there must be a continual input of heat because the reaction is endothermic. However, in actual experimental operations, a key design consideration for the reformer is how to supply the heat for the reaction. The supply of heat will result in a non-uniform temperature along the length of the flow channel. The heat consumed by the reaction will cause the temperature to decrease near the inlet of the channel

32

which is probably the most important results we could obtain from the reactor analysis.

Consequently, a methanol steam micro-reformer with a methanol catalytic combustor would be considered in order to simulate the characteristics of the non-uniform temperature along the channel. Now, a reactor model with consideration of the methanol steam micro-reformer and methanol catalytic combustor has been developing.

This study extends the established 3-dimensional channel model of methanol steam micro-reformer to a reactor channel model with consideration of the methanol steam micro-reformer and methanol catalytic combustor in 3-dimensional model. The reactor consists of a methanol steam micro-reformer and a methanol catalytic combustion chamber. A schematic diagram of the physical system under consideration is shown in Fig. 2-4. The system consists of the solid wall, two catalyst layers and two flow channels each at the catalytic combustion/steam reforming side. It is seen that the methanol catalytic combustion chamber and the methanol steam reforming chamber are separated by a solid wall. Both sides of each channel are coated with a combustion catalyst layer and a steam reforming catalyst layer. The heat from the combustion reaction is used to drive the steam reforming reaction.

Next, the three-dimensional computational models with various flow fields have been established for methanol steam micro-reformer with methanol catalytic combustor. The flow fields in the methanol steam micro-reformer and methanol catalytic combustor include the parallel flow field and the serpentine flow field. The parallel flow field has five parallel channels and the serpentine flow field has one channel with four turns. A schematic illustration of these flow fields and associated coordinate system are shown in Fig. 2-5. The parallel flow field has five flow channels and each channel is 40mm in length. The serpentine flow field has one flow channel, the total flow channel length is five times the length of a channel in the parallel flow field, and there are four turning points. In this study, constant flow rate approach is utilized to investigate the effect of flow field on the performance of

micro-reformer. The u, v, and w are the velocity components in the x-, y -, and z-directions, respectively. The reactor consists of the solid wall, a steam reforming flow channel, a steam reforming catalyst layer, a catalytic combustion catalyst layer and a catalytic combustion flow channel.

2.3.2 Assumption

To simplify the analysis for the present study, the flowing assumptions are made:

(1) The flow is steady state;

(2) The inlet fuel is an ideal gas;

(3) The flow is laminar and incompressible;

(4) The catalyst layer is isotropic;

(5) The chemical reaction occurs only in the catalyst layer;

(6)Thermal radiation and conduction in the gas phase are negligible compared to convection.

2.3.3 Governing Equations

With the above assumptions, the gas transport equations for the three-dimensional reactor can be described as follows.

Continuity equation:

u v w

∂ +∂ +∂ =0 (2-23)

34 corrected terms of the reactant gas flow in a porous material in the catalyst layer. The source terms, Su, Sv and Sw in the momentum equations are listed in Eqs. (2-24)-(2-26), respectively.

Among them, the source terms, Su, Sv and Sw account for the Ergun equations [76] in the x-, y- and z-directions, respectively. The parameter kp stands for the permeability and β is the inertial loss coefficient in each component direction.

2 2 2

In the species equation, mi denotes the mass fraction of the i^th species, where the various species are CH3OH, H2O, H2, CO2, CO and O2. In these expressions, the concentrations of CH3OH, H2O, H2, CO2, CO are calculated on the steam reforming side and CH3OH, H2O, CO2, O2 are calculated on the combustion side. The effective diffusion coefficient, Deff

determined by the Stefan-Maxwell equations [77]. Eq. (2-33) is employed to describe the influence of the porosity on the diffusion coefficient

eff k

D =D ε ^τ (2-33)

The diffusion coefficient Dk for the methanol steam micro-reformer was derived from the Stefan-Maxwell equations which were used to calculate the mean effective binary diffusivity [51]. Sc is the source term of chemical reaction in the species equation, and differs according to the reactant gases in the catalyst layer. In the present study, there is no chemical reaction in the flow channel. Therefore, Sc is zero in the flow channel. In the catalyst layer, the source term of the species equation, Sc, can be described by the following modified concentration term:

where Mw,i is the molecular weight of species i, and Ri,r is the Arrhenius molar rate of creation and destruction of species i in the reaction. λ and ^"_i λ are the stoichometric ^'_i coefficient for reaction i and product i, respectively, in the reaction.

According to the chemical kinetics of Pepply et al. [40], the methanol steam reforming

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(2-36), and the decomposition reaction, Eq. (2-37), are considered in this study.

1

In this work, to simplify the analysis, the model of Mastalir et al. [41] for methanol steam reforming is used, and the Arrhenius equation is employed to calculate the reactant gases generated by the chemical reaction.

3 2 2 2

where the steam reforming reaction and reverse water-gas shift reaction are reversible reactions and the decomposition reaction is a non-reversible reaction. The constants k1, k2 and k3 are forward rate constants, and the constant k-1 and k-2 are the backward rate constants.

The reaction of the combustion catalyst layer can be represented by the following reaction Eq.(2-41). The reaction rate of methanol over the Pt/Al2O3 catalyst was calculated with Eq.(2-42), as proposed by Pasel et al.[81]

k4

In order to evaluate the distributions of the local temperature, the energy equations must be solved.

Energy equation:

The effective thermal conductivity is modified to account for the porous medium effect:

eff f s

k = ε + − εk (1 )k (2-44) where kf is the fluid phase thermal conductivity, ks is the solid medium thermal conductivity and ε is the porosity of the medium.

The source term St in the energy equation due to the chemical reactions is determined by

SR SR rWGS rWGS MD MD

As for the energy equation of the solid wall, one has

2 2 2

The boundary conditions of the present computation include those at the inlet, the outlet, the wall, and the interface between the flow channel and the catalyst layer.

(1) The boundary conditions for inlets at the flow channel and the catalyst layer: the inlet flow velocity is constant, the inlet gas composition is constant, and the inlet