Introduction

The use of hydrogen fuel as a major source of fuel in the near future has gained a tremendous recognition in the past few years [1]. It is now being used as an energy carrier in fuel cells and mobile vehicles because it is a clean fuel that does not emit carbon dioxide when burned or used. It can be stored as a liquid or gas, and has been described as a long replacement of fossil fuels [1]. While several works have been done on economical ways of producing hydrogen from hydrocarbons such as alcohols, gasoline, natural gas and many other ways [2, 3], steam reforming of methanol has been preferred because of the advantage of high energy density, availability and safe handling, and lack of inter-carbon bonds in methanol, hence relatively low reforming temperatures (433–573 K). In addition, steam reforming of methanol produces limited CO, typically less than 5000 ppm and a higher hydrogen fraction in the reformate compared to that of partial oxidation of methanol. Methanol is rich in hydrogen as the ratio of carbon to hydrogen is 1:4 compared to other candidates of hydrogen production [4]. Therefore, methanol is often converted into a hydrogen-rich gas before it is fed to the fuel cell stack, and steam reforming (SR) of methanol is suitable for this process.

Many investigators have proposed the reaction network for modelling this process, with the kinetic expressions given, based on their extensive experimental studies [4,5,6,7,8,9,10,11,12,13,14]. However, in addition to the kinetics, estimation of the effect of intra-particle diffusion resistance in catalyst particles is also significant in the design of the catalytic reformer. These will generally have to be included in any realistic model of a reactor and explicitly explained. Major works have not been done in the literature on the effect of intra-particle diffusion resistance on this complex reaction system, thus the need to study this.

The objective of this study was to numerically compute non-isothermal effectiveness factor in a spherical catalyst pellet for catalytic steam reforming of methanol on Cu/ZnO/Al2O3 catalyst. Thus, a computational procedure using the method of orthogonal collocation was presented for the numerical solution of the governing differential mass and energy balance equations with a view to estimate the concentration profiles of key components (methanol and hydrogen) and temperature profile inside the catalytic pellet using the non-linear power law and highly non-linear LHHW kinetics of the steam reforming of methanol, and thus computing the non-isothermal effectiveness factor. The effects of reaction kinetics, catalyst size, thermal conductivity, diffusion coefficient and surface temperature on effectiveness factor were considered. The kinetics for the steam reforming of methanol on Cu/ZnO/Al2O3 catalyst used in this study was obtained by Lee et al. [4] in a regime of intra-particle diffusion limitations, as external mass and heat transfer resistances were negligible during the catalytic steam reforming of methanol [1, 4, 13, 15], and thus appropriate for use in this investigation.

Development of mathematical model

The steady-state mass and energy balance equations for a spherical catalyst pellet are given by the following equations, respectively:

$$\frac{{\left( {D_{e} } \right)_{i} }}{{\rho_{p} }}\left( {\frac{{d^{2} C_{i} }}{{dr^{2} }} + \frac{2}{r}\frac{{dC_{i} }}{dr}} \right) = \left[ { - r^{\prime}_{i} \,\left( {C_{i} \,,\,T} \right)} \right]\,\,\,\,\,\,\,\,\,\,i\, = \,{\text{CH}}_{{3}} {\text{OH,}}\;{\text{H}}_{{2}} {\text{O,}}$$
(1)
$$\frac{{\lambda_{e} }}{{\rho_{p} \left( {\Delta H_{rxn,\,l} } \right)}}\left( {\frac{{d^{2} T}}{{dr^{2} }} + \frac{2}{r}\frac{dT}{{dr}}} \right) = \,\,\left[ { - r^{\prime}_{l} \,\left( {C\,,\,T} \right)} \right]\,\,\,\,,$$
(2)

where \(\left[ { - r^{\prime}_{i} \left( {C_{i} \,,\,T} \right)} \right]\) is the rate of disappearance of species \(i\)[ =] kmol/(kg cat s) and \(\left( {D_{e} } \right)_{i}\) the effective diffusivity of component \(i\), which is often approximately constant inside a catalyst particle and \(l\) is the number of reaction.

In Eqs. (1) and (2), it was assumed that external mass and heat transfer resistances were negligible while the resistances to mass and heat transfer inside the catalyst (i.e., intra-particle resistance) were significant.

These equations are subject to the following boundary conditions:

  1. (i)

    At the pore centre,

    $$r = 0,\quad \frac{{\partial C_{i} }}{\partial r} = 0\,,\,\,\,\,\frac{\partial T}{{\partial r}}\, = \,0.$$
    (3)
  2. (ii)

    At the pore mouth,

    $$r = \frac{{d_{p} }}{2}\, = \,R\,,\,\,\,\,C_{i} \, = \,\left( {C_{i} } \right)_{s} \,,\,\,\,\,\,T\, = T_{s} .$$
    (4)

The following dimensionless variables are defined:

$$\overline{C}_{i} \,\, = \,\,\frac{{C_{i} }}{{\left( {C_{i} } \right)_{s} }}\,,\,\,\,\,\,\overline{T}\,\, = \,\,\frac{T}{{T_{s} }}\,,\,\,\,\,\,\sigma \,\, = \,\,\frac{r}{R}\,,\,\,\,\,\overline{R}_{i} \left( {\overline{C}_{i} \,,\,\overline{T}} \right)\,\, = \frac{{\left[ { - r^{\prime}_{i} \,\left( {C_{i} \,,\,T} \right)} \right]}}{{\left( { - r^{\prime}_{i} \,} \right)_{s} \left[ {\left( {C_{i} } \right)_{s} \,,\,T_{s} } \right]}}.$$
(5)

Combining Eqs. (1) to (5), we have the following equation:

$$\overline{T}\,\, = \,\,1\,\, + \,\,\beta_{l} \left( {1\,\, - \,\,\overline{C}} \right),$$
(6)

where \(\beta_{l} \,\,\left[ { = \,\,\frac{{\left( { - \Delta H_{rxn,\,l} } \right)\left( {D_{e} } \right)_{i} \left( {C_{i} } \right)_{s} }}{{\lambda_{e} T_{s} }}} \right]\) is a heat of reaction parameter with respect to reaction \(l\), which characterizes the potential for temperature gradients inside the particle.

Equation (1) in dimensionless form yields:

$$\frac{{d^{2} \overline{C}_{i} }}{{d\sigma^{2} }} + \frac{2}{\sigma }\frac{{d\overline{C}_{i} }}{d\sigma } - \,\,\phi^{2} \overline{R}_{i} \left( {\overline{C}_{i} \,,\,\overline{T}} \right)\,\, = \,\,0\,\,\,\,\,\,\,\,\,\,i\, = \,{\text{CH}}_{{3}} {\text{OH,}}\;{\text{H}}_{{2}} {\text{O,}}$$
(7)

where \(\varphi \,\,\left\{ { = \,\,R\sqrt {\frac{{\rho_{p} \left[ {\left( { - r^{\prime}_{i} } \right)_{s} \left( {\left( {C_{i} } \right)_{s} \,,\,T_{s} } \right)} \right]}}{{\left( {D_{e} } \right)_{i} \left( {C_{i} } \right)_{s} }}} } \right\}\) is called the Thiele modulus, and \(\overline{R}_{i} \left( {\overline{C}_{i} \,,\,\overline{T}} \right)\) is the dimensionless rate of reaction of species \(i\), normalized with respect to the reaction rate at the external conditions.

The boundary conditions of Eq. (7) are \(\overline{C}_{i} \, = \,1\) at \(\sigma \, = \,1\) and \(\frac{{d\overline{C}_{i} }}{d\sigma }\, = \,0\) at \(\sigma \, = \,0\).

The developed model Eqs. (6) and (7) can now be applied to any reaction system or network.

Application of the model to the reaction kinetics

The steam reforming of methanol on a commercial Cu/ZnO/Al2O3 catalyst to produce hydrogen was considered. This reforming reaction was studied at atmospheric pressure and in a temperature range between 433 and 533 K by [4]. The main reaction is

$${\text{CH}}_{{\text{3}}} {\text{OH}} + {\text{H}}_{{\text{2}}} {\text{O}} \to {\text{CO}}_{{\text{2}}} + {\text{3H}}_{{\text{2}}} \quad \Delta {\text{H}} = 57\,{\text{kJ/mol}}\,{\text{at}}\,200^\circ {\text{C}}$$
(8)

The chief products of steam reforming of methanol are hydrogen and carbon dioxide with a small amount of CO, usually less than 1%, as a by-product [16,17,18]. However, Lee et al. [4] obtained a power rate law and Langmuir–Hinshelwood–Hougen–Watson (LHHW) rate expressions for the reforming reaction by fitting the rate expressions to their experimental data. An excellent fitting of the data by the LHHW expression indicated that the dehydrogenation of the adsorbed methoxy to the adsorbed oxymethylene is the rate-determining step, and that adsorption of all reacting species other than methoxy and hydrogen on the active sites was negligible.

The power-law expression obtained for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst was of the form given by Lee et al. [4]:

$$\left( { - r^{\prime}_{M} } \right) = k_{0} \exp \left( { - \frac{{E_{1} }}{{R_{u} T}}} \right)p_{M}^{{\alpha_{1} }} \left( {A + p_{{H_{2} }} } \right)^{{\alpha_{2} }} \left[ = \right]{\text{ mol}}/\left( {\text{kg cat s}} \right),$$
(9)

where \(k_{0} \,\, = \,\,2.19\, \times \,10^{9}\) mol Pa0.083/(kg cat s), \(E_{1} \,\, = \,\,1.03\, \times \,10^{5}\) J/mol, \(\alpha_{1} \, = \,0.564\), \(\alpha_{2} \, = \, - 0.647\) and \(A\, = \,1.16\, \times \,10^{4}\) Pa.

The LHHW rate expression obtained for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst had the form given by Lee et al. [4]:

$$\left( { - r^{\prime}_{M} } \right)\,\, = \,\,\frac{{k_{01} \exp \left( { - \frac{{E_{2} }}{{R_{u} T}}} \right)K_{1} \frac{{p_{M} }}{{\sqrt {p_{{H_{2} }} } }}}}{{\left( {1\,\, + \,\,K_{1} \frac{{p_{M} }}{{\sqrt {p_{{H_{2} }} } }}} \right)\left( {1\,\, + \,\,\sqrt {K_{2} p_{{H_{2} }} } } \right)}}\left[ = \right]{\text{ mol}}/\left( {\text{kg cat s}} \right),$$
(10)

where

$$k\,\, = \,\,k_{01} \exp \left( { - \frac{{E_{2} }}{{R_{u} T}}} \right) = 3.13\,\, \times 10^{10} \exp \left( { - \frac{{1.11\, \times \,10^{5} }}{{R_{u} T}}} \right)\left[ = \right]{\text{ mol}}/\left( {\text{kg s}} \right),$$
$$K_{1} \,\, = \,\,K_{01} \exp \left( { - \frac{{\Delta H_{1} }}{{R_{u} T}}} \right) = 1.186\, \times \,10^{ - 4} \exp \left( {\frac{{2.0\, \times \,10^{4} }}{{R_{u} T}}} \right)\left[ = \right]{\text{ Pa}}^{{ - 0.{5}}} ,$$
$$K_{2} \,\, = \,\,K_{02} \exp \left( { - \frac{{\Delta H_{2} }}{{R_{u} T}}} \right) = 6.34\, \times \,10^{ - 10} \exp \left( {\frac{{5.0\, \times \,10^{4} }}{{R_{u} T}}} \right)\left[ = \right]{\text{ Pa}}^{{ - {1}}} .$$

To estimate the non-isothermal effectiveness factor of the catalyst particle, the intra-particle concentration profiles of the reacting species that affect the reaction rate should be determined. The concentration profiles depend on the effective diffusivities of the reacting species and on the reaction rate. However, the components identified in the steam reforming of methanol are methanol (M), steam (S), carbon dioxide (C), hydrogen (H2) and nitrogen (N2). Carbon monoxide was neglected in the reforming mixture since its concentration was insignificant. Moreover, the reaction rate for the steam reforming of methanol was found to depend on the partial pressures of methanol and hydrogen, as evident in Eqs. (11) and (12); therefore, the concentration profiles of the two components (methanol and hydrogen) are needed to evaluate the non-isothermal effectiveness factor of the catalyst particle. Using the modified Maxwell–Stefan equation to formulate the multicomponent diffusion in the catalyst [19, 20], we have

$$\overline{C}_{{H_{2} }} \, = \,1\,\, + \,\,\frac{{3\left( {D_{e} } \right)_{M} \left( {C_{M} } \right)_{s} }}{{\left( {D_{e} } \right)_{{H_{2} }} \left( {C_{{H_{2} }} } \right)_{s} }}\left( {1\,\, - \,\,\overline{C}_{M} } \right),$$
(11)
$$\overline{C}_{{CO_{2} }} \, = \,1\,\, + \,\,\frac{{\left( {D_{e} } \right)_{M} \left( {C_{M} } \right)_{s} }}{{\left( {D_{e} } \right)_{{CO_{2} }} \left( {C_{{CO_{2} }} } \right)_{s} }}\left( {1\,\, - \,\,\overline{C}_{M} } \right),$$
(12)
$$\overline{C}_{S} \, = \,1\,\, - \,\,\frac{{\left( {D_{e} } \right)_{M} \left( {C_{M} } \right)_{s} }}{{\left( {D_{e} } \right)_{S} \left( {C_{S} } \right)_{s} }}\left( {1\,\, - \,\,\overline{C}_{M} } \right),$$
(13)

where \(\left( {D_{e} } \right)_{M}\) and \(\left( {D_{e} } \right)_{{H_{2} }}\) are the respective effective diffusivities of methanol and hydrogen in the catalyst, and \(\left( {C_{M} } \right)_{s}\) and \(\left( {C_{{H_{2} }} } \right)_{s}\) the dimensionless concentrations of methanol and hydrogen at the external surface of the catalyst. Equation (11) shows that there is a mutual relation between concentration profile of methanol and hydrogen inside a catalyst particle, which is in agreement with Schneider [19, 20] relation.

The non-isothermal effectiveness factor, \(\eta\), of the catalyst is given by

$$\eta \,\, = \,\,\frac{3}{{\phi^{2} }}\left. {\frac{{d\overline{C}_{M} }}{d\sigma }} \right|_{\sigma \, = \,1} .$$
(14)

Hence, to estimate the effectiveness factor by Eq. (14), it is necessary to solve governing Eq. (7) to obtain the dimensionless concentration profile of methanol inside the catalyst pellet, \(\overline{C}_{M} (\sigma )\), and then obtain the dimensionless temperature profile inside the catalyst pellet via Eq. (6).

Since all the reacting species (methanol, M, steam, S, carbon dioxide and hydrogen) involved in the steam reforming of methanol are in the gaseous phase, ideal gas law, \(p\, = \,CR_{u} T\), is applied to the reaction rate of disappearance of methanol, Eq. (9), to obtain

$$\left( { - r^{\prime}_{M} } \right) = k_{0} \exp \left( { - \frac{{E_{1} }}{{R_{u} T}}} \right)\left( {C_{M} R_{u} T} \right)^{{\alpha_{1} }} \left( {\frac{A}{{R_{u} T}} + C_{{H_{2} }} R_{u} T} \right)^{{\alpha_{2} }} .$$
(15)

Applying ideal gas law to LHHW rate expression (10), we have

$$\left( { - r^{\prime}_{M} } \right) = \frac{{k_{01} \exp \left( { - \frac{{E_{2} }}{{R_{u} T}}} \right)K_{1} \sqrt {R_{u} T} \left( {\frac{{C_{M} }}{{\sqrt {C_{{H_{2} }} } }}} \right)}}{{\left[ {1 + K_{1} \sqrt {R_{u} T} \left( {\frac{{C_{M} }}{{\sqrt {C_{{H_{2} }} } }}} \right)} \right]\left( {1 + \sqrt {K_{2} C_{{H_{2} }} R_{u} T} } \right)}}.$$
(16)

Rendering the power rate law model and LHHW rate expression for the steam reforming of methanol in dimensionless forms, and using the respective expressions obtained in Eq. (9), we have the following equations for the power rate law model and LHHW rate expression, respectively:

$$\frac{{d^{2} \overline{C}_{M} }}{{d\sigma^{2} }} + \frac{2}{\sigma }\frac{{d\overline{C}_{M} }}{d\sigma } - \,\,\varphi_{PL}^{2} \exp \left[ { - \frac{{\gamma_{PL} }}{{1\,\, + \,\beta \left( {1\,\, - \,\overline{C}_{M} } \right)\,}}} \right]\left[ {\frac{A}{{R_{u} T_{s} \overline{T}}}\,\, + \,R_{u} \,\left( {C_{{H_{2} }} } \right)_{s} T_{s} \overline{C}_{{H_{2} }} \overline{T}} \right]^{{\alpha_{2} }} \left( {\overline{C}_{M} \overline{T}} \right)^{{\alpha_{1} }} \,\, = \,\,0,$$
(17)
$$\frac{{d^{2} \overline{C}_{M} }}{{d\sigma^{2} }} + \frac{2}{\sigma }\frac{{d\overline{C}_{M} }}{d\sigma } - \,\,\frac{{\varphi_{L}^{2} \exp \left[ { - \frac{{\left( {\gamma_{L} \,\, + \,\,\gamma_{1} } \right)}}{{1\,\, + \,\beta \left( {1\,\, - \,\,\overline{C}_{M} } \right)\,}}} \right]\overline{C}_{M} \sqrt {\frac{{\overline{T}}}{{\overline{C}_{{H_{2} }} }}} }}{{\left( {DEN1} \right)\,\left( {DEN2} \right)}}\,\, = \,\,0,$$
(18)

where

$$\gamma_{PL} \,\, = \,\,\frac{{E_{1} }}{{R_{u} T_{s} }},$$
(19)
$$\beta \,\, = \,\,\frac{{\left( { - \Delta H_{rxn} } \right)\left( {D_{e} } \right)_{M} \left( {C_{M} } \right)_{s} }}{{\lambda_{e} T_{s} }},$$
(20)
$$\phi_{PL} \,\, = \,\,R\sqrt {\frac{{\rho_{p} k_{0} \left( {R_{u} T_{s} } \right)^{{\alpha_{1} }} }}{{\left( {D_{e} } \right)_{M} \left( {C_{M} } \right)_{s}^{{1\, - \,\alpha_{1} }} }}} ,$$
(21)
$$\phi_{L} \,\, = \,\,R\sqrt {\frac{{\rho_{P} k_{01} K_{01} }}{{\left( {D_{e} } \right)_{M} }}\left[ {\frac{{R_{u} T_{s} }}{{\left( {C_{{H_{2} }} } \right)_{s} }}} \right]^{0.5} } ,$$
(22)
$$DEN1\,\, = \,\,1\,\, + \,\,K_{01} \sqrt {\frac{{R_{u} T_{s} }}{{\left( {C_{{H_{2} }} } \right)_{s} }}} \,\left( {C_{M} } \right)_{s} \exp \left[ { - \frac{{\gamma_{1} }}{{1\,\, + \,\beta \left( {1\,\, - \,\,\overline{C}_{M} } \right)\,}}} \right]\overline{C}_{M} \left( {\frac{{\overline{T}}}{{\overline{C}_{{H_{2} }} }}} \right),$$
(23)
$$DEN2\,\, = \,\,1\,\, + \,\,\sqrt {K_{02} R_{u} \left( {C_{{H_{2} }} } \right)_{s} T_{s} \exp \left[ { - \frac{{\gamma_{2} }}{{1\,\, + \,\beta \left( {1\,\, - \,\,\overline{C}_{M} } \right)\,}}} \right]\overline{C}_{{H_{2} }} \overline{T}} ,$$
(24)
$$\gamma_{L} \,\, = \,\,\frac{{E_{2} }}{{R_{u} T_{s} }},$$
(25)
$$\gamma_{i} \,\, = \,\,\frac{{\Delta H_{i} }}{{R_{u} T_{s} }}\,,\,\,\,\,\,\,\,\,i\, = \,1\,,\,2.$$
(26)

The boundary conditions of Eqs. (17) and (18) are \(\overline{C}_{M} \, = \,1\) at \(\sigma \, = \,1\) and \(\frac{{d\overline{C}_{M} }}{d\sigma }\, = \,0\) at \(\sigma \, = \,0\).

\(\overline{T}\) and \(\overline{C}_{{H_{2} }}\) are as given in Eqs. (6) and (11), respectively.

The Ergun equation was used to calculate the pressure drop through the catalyst beds. It is represented as follows:

$$\Delta P = \,\left( {\frac{150}{{\text{Re}}} + 1.75} \right)\frac{{\rho_{f} u_{o}^{2} }}{{d_{p} }}\left( {\frac{{1 - \in_{p} }}{{ \in_{p}^{3} }}} \right)\,\,L,$$

where \(\Delta P\) is the pressure drop, Re is the Reynolds number, \(\rho_{f}\) is the density of the fluid, \(u_{o}\) is the superficial velocity, \(d_{p}\) is the catalyst pellet diameter, \(\in_{p}\) is the bed voidage and \(L\) is the length of the packed bed.

Numerical method of solution

Orthogonal collocation

The numerical method adopted for the solution of the developed models for the catalytic steam reforming of methanol with a view to predicting the key component concentration profile inside the pellet and thus computing non-isothermal effectiveness factor values is the orthogonal collocation method [21, 22].

Parameters and operating conditions used for the computation

In the computational procedure of this study, the effective diffusivity values of methanol and hydrogen in the catalyst were estimated using the modified Maxwell–Stefan equation [19] and the cross-linked pore model [23], recommended by Haugaard and Livbjerg [24] based on experimental comparison of the pore-diffusion models. The respective effective diffusivities of methanol, \(\left( {D_{e} } \right)_{M}\), hydrogen, \(\left( {D_{e} } \right)_{{H_{2} }}\), carbon dioxide, \(\left( {D_{e} } \right)_{{CO_{2} }}\), steam, \(\left( {D_{e} } \right)_{S}\), are given by

$$\left( {D_{e} } \right)_{M} \,\, = \,\,\frac{{ \in_{p} }}{\tau }\left( {\frac{{y_{S} \, - \,y_{M} }}{{D_{M - S} }}\,\, + \,\,\frac{{y_{{CO_{2} }} \, + \,y_{M} }}{{D_{{M - CO_{2} }} }}\,\, + \,\,\frac{{y_{{H_{2} }} \, + \,3y_{M} }}{{D_{{M - H_{2} }} }}\,\, + \,\frac{{y_{{N_{2} }} }}{{D_{{M - N_{2} }} }}\,\, + \,\frac{1}{{\left( {D_{K} } \right)_{M} }}\,} \right)^{ - 1} ,$$
(27)
$$\left( {D_{e} } \right)_{{H_{2} }} \,\, = \,\,\frac{{ \in_{p} }}{\tau }\left( {\frac{{y_{M} \, + \,y_{{H_{2} }} }}{{3D_{{H_{2} - M}} }}\,\, + \,\,\frac{{y_{S} \, + \,y_{{H_{2} }} }}{{3D_{{H_{2} - S}} }}\,\, + \,\,\frac{{y_{{CO_{2} }} \, - \,y_{{H_{2} }} }}{{3D_{{H_{2} - CO_{2} }} }}\,\, + \,\frac{{y_{{N_{2} }} }}{{D_{{H_{2} - N_{2} }} }}\,\, + \,\frac{1}{{\left( {D_{K} } \right)_{{H_{2} }} }}\,} \right)^{ - 1} ,$$
(28)
$$\left( {D_{e} } \right)_{{CO_{2} }} \,\, = \,\,\frac{{ \in_{p} }}{\tau }\left( {\frac{{y_{{H_{2} }} \, - \,3y_{{CO_{2} }} }}{{D_{{CO_{2} - H_{2} }} }}\,\, + \,\,\frac{{y_{M} \, + \,y_{{CO_{2} }} }}{{D_{{M - CO_{2} }} }}\,\, + \,\,\frac{{y_{S} \, - \,y_{{CO_{2} }} }}{{D_{{CO_{2} - S}} }}\,\, + \,\frac{{y_{{N_{2} }} }}{{D_{{CO_{2} - N_{2} }} }}\,\, + \,\frac{1}{{\left( {D_{K} } \right)_{{CO_{2} }} }}\,} \right),$$
(29)
$$\left( {D_{e} } \right)_{S} \,\, = \,\,\frac{{ \in_{p} }}{\tau }\left( {\frac{{y_{M} \, - \,y_{S} }}{{D_{M - S} }}\,\, + \,\,\frac{{y_{{CO_{2} }} \, + \,y_{S} }}{{D_{{S - CO_{2} }} }}\,\, + \,\,\frac{{y_{{H_{2} }} \, + \,3y_{S} }}{{D_{{S - H_{2} }} }}\,\, + \,\frac{{y_{{N_{2} }} }}{{D_{{S - N_{2} }} }}\,\, + \,\frac{1}{{\left( {D_{K} } \right)_{S} }}\,} \right),$$
(30)

where \(\in_{p}\) is the porosity of the catalyst, \(\tau\) the tortuosity of pores in the catalyst, \(y_{i}\) the mole fraction of component \(i\), \(D_{i - j}\) the diffusivity for a binary mixture of \(i\) and \(j\), and \(\left( {D_{K} } \right)_{M}\), \(\left( {D_{K} } \right)_{{H_{2} }}\), \(\left( {D_{K} } \right)_{{CO_{2} }}\) and \(\left( {D_{K} } \right)_{{S_{{}} }}\) the Knudsen diffusivities of methanol, hydrogen, carbon dioxide and steam, respectively. The tortuosity factor, \(\tau\), is a parameter, which corrects deviations of the real pore structure from the ideal cylindrical straight pores. \(\tau\) = 3 was assumed, a theoretical value for a random pore system [23]. The binary diffusivity was estimated by the method of Reid et al. [26]. The Knudsen diffusivity of component \(i\) is given by Smith [25]:

$$\left( {D_{K} } \right)_{i} \,\, = \,\,97\overline{r}\sqrt {\frac{{T_{rxn} }}{{M_{i} }}} \left[ = \right]{\text{ m}}^{{2}} /{\text{s,}}$$
(31)

where \(\overline{r}\) is the mean pore radius [ =] m, \(T_{rxn}\) the reaction temperature [ =] K and \(M_{i}\) the molecular weight of species \(i\).

The effective diffusivities, \(\left( {D_{e} } \right)_{M}\), \(\left( {D_{e} } \right)_{{H_{2} }}\), \(\left( {D_{e} } \right)_{{CO_{2} }}\) and \(\left( {D_{e} } \right)_{S}\), are seen to depend on the mole fractions of all components of the reaction mixture at a point in the catalyst. Schneider [19, 20] has shown that the change of the effective diffusivity along the pores in the catalyst is not very marked even under strong diffusion limitations, and that the effective diffusivity evaluated at the outer surface of a catalyst particle can be used throughout the particle without introducing any serious error into the concentration profile.

Effectiveness radial thermal conductivity computation [27]:

$$\frac{{\lambda_{e} }}{{\lambda_{f} }} = \alpha + \beta \Pr {\text{Re}} .$$
(32)

For metal spheres:

$$\alpha = 13,\quad \beta = 0.11,$$

0.021 < \(\frac{{d_{p} }}{{d_{t} }}\) < 0.086,

$$\Pr = \frac{{C_{pf} \mu }}{{\lambda_{f} }} = \frac{48.9926 \times 2.71E - 4}{{3.08E - 2}} = 0.4311,$$
(33)
$${\text{Re}} = \frac{{d_{p} G}}{\mu } = \frac{3.62E - 4 \times 3.42}{{2.71E - 4}} = 4.5744,$$
(34)
$$\frac{{\lambda_{e} }}{{\lambda_{f} }} = \left[ {13 + 0.118 \times 0.4311 \times 4.5747} \right],$$
(35)

\(\lambda_{e} = 0.4071\) W/mK.

Results and discussion

The results of the study are presented and discussed in this section. The case study depicts the results of the effect of intra-particle diffusion resistance on methanol steam reforming on a commercial Cu/ZnO/Al2O3 catalyst using power rate law and LHHW kinetics, respectively (Tables 1 and 2).

Table 1 The geometric, kinetic and parameters used for the steam reforming of methanol on a commercial Cu/ZnO/Al2O3 catalyst
Full size table
Table 2 Calculated values of Knudsen and effective diffusion coefficients of species for the reaction system
Full size table

Effect of temperature on effectiveness factor

Figure 1 shows the effect of change in surface temperature on effectiveness factor of steam reforming of methanol on a commercial Cu/ZnO/Al2O3 catalyst. At temperature below 473 K, the effect on intra-particle diffusion limitation is reduced to the minimum indicated by the effectiveness factor being almost equal to one. At this point, almost all the catalyst surfaces are available for reaction. As the temperature increases above 473 K, there is considerable increase in the diffusion limitation effect. This implies that at temperature above 473 K, effect of intra-particle diffusion resistance becomes predominant providing only small area of the catalyst for reaction with most of the reactions taking place very close to the outer surface of the catalyst. The trend is in agreement with Lee et al. [4] with an absolute average deviation (AAD) of 0.111 and standard deviation (SD) of 0.131.

Fig. 1
figure 1

Effect of surface temperature on effectiveness factor for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Effectiveness factor and Thiele modulus plot of both kinetics

Figure 2 shows the variation of effectiveness factor, \(\eta\), with Thiele modulus, \(\varphi\), for the power law kinetics of the reaction process. The reaction being endothermic, both concentration and temperature profiles decrease towards the centre of the pellet. For large values of the Thiele modulus, the rate of reaction is much greater than the rate of diffusion; thus, the effectiveness factor is much less than unity, and the process is said to be diffusion-limited, that is, the surface reaction is rapid and that the reactant is consumed very close to the external pellet surface and very little penetrates into the interior of the pellet. However, when the diffusion rate is much larger than the reaction rate, the effectiveness factor is near unity and the process is said to be reaction-limited, and a significant amount of the reactant diffuses well into the pellet interior without reacting. From the plot, it can been that the internal diffusion effectiveness factor is approximately equal to 1.0 when the Thiele modulus is lower than 1.1, which means majority of the reactants diffuse quickly into the catalyst centre and the effect of internal diffusion can be neglected. When the value of Thiele modulus increased from 1.3 to 3.4, the value of the internal diffusion effectiveness factor decreased from 0.8332 to 0.507, which indicates that the effect of internal diffusion resistance is pronounced. At Thiele modulus greater than 4.0, the internal diffusion effectiveness factor is very small which means severe intra-particle diffusion resistance meaning most of the catalyst is not being utilized.

Fig. 2
figure 2

Effectiveness factor versus Thiele modulus using power kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Figure 3 shows the power law and Langmuir–Hinshelwood–Hougen–Watson (LHHW) kinetics superimposed, it is seen that there is only a slight difference in the values of the effectiveness factor variation with Thiele modulus for both the power law and LHHW kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst, indicating that the two reaction kinetics can be used to predict the reaction process. Figure 4 shows the variation of the values of effectiveness factor with Thiele modulus at different heat of reaction parameter, β, using power law kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst. The plot shows decrease in the magnitude of effectiveness factor as β decreases from − 0.01 to − 0.8, all having very close values of effectiveness factor at different Thiele modulus. This is because the reaction being endothermic, the temperature decreases toward the centre of the pellet and since the rate decreases with drop in temperature, the effect of the heat-transfer resistance is diminished thus resulting in the closeness of the non-isothermal \(\eta\) profiles for various values of heat of reaction parameter for methanol, \(\beta\). Therefore, the decrease in temperature (and of course the reaction rate) in going into the pellet is indicative of the little significance of the mass transfer.

Fig. 3
figure 3

Effectiveness factor versus Thiele modulus for both power law and LHHW kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Fig. 4
figure 4

Effectiveness factor versus Thiele modulus using LHHW kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst at different heat of reaction parameter

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Effect of diffusion coefficient and thermal conductivity

Figures 5 and 6 show the effect of diffusion coefficient and thermal conductivity on effectiveness factor respectively. From Fig. 5, as the diffusion coefficient increases, the effectiveness factor becomes limited by surface reaction effect and thus increasing the effectiveness factor. At a diffusion coefficient of 4.60 \(\times \,\) 10–7 m2/s, the effectiveness factor is 0.6571; at 7.50 \(\times \,\) 10–7 m2/s, the effectiveness factor increases by almost 30% to 0.6691. Increasing the diffusion coefficient to 1.50 \(\times \,\) 10–6 m2/s shows a small increase of about 0.5% to 0.6721. Further increase in diffusion coefficient does not show appreciable increase in the value of effectiveness factor. This is because for endothermic reaction which is the reaction we are considering, temperature and concentration gradients both reduce the reaction rate below that evaluated at surface conditions. The decrease in rate going into the pellet for endothermic reaction means that the mass transfer is of little importance which thus makes the diffusion coefficient to have minimal effect on effectiveness factor. Therefore, operating with the diffusion coefficient of 1.50 \(\times \;\) 10–6 m2/s for this reaction process will reduce to minimal value the effect of intra-particle diffusion resistance. As shown in Fig. 6, as the thermal conductivity increases, the diffusion effect becomes the limiting effect and thus decreasing the effectiveness factor. At thermal conductivity of 0.204 W/m2K, the effectiveness factor is 0.6598; at 0.4071 W/m2K, the effectiveness factor decreases by almost 1.2% to 0.6521. Increasing the thermal conductivity to 1.080 W/m2K shows a small decrease of about 1.1% to 0.644. Further increase in thermal conductivity does not show appreciable change in the value of effectiveness factor. This is because the effective thermal conductivities of catalyst pellets are very low. For endothermic reaction, there is a decrease in temperature and diffusion rate into the pellets as stated earlier. Thus, the effect of heat transfer resistance is diminished which then makes the effect of effective thermal conductivity on the effectiveness factor not to be pronounced. Therefore, operating with the thermal conductivity 1.080 W/m2K will reduce to minimal value the effect of intra-particle diffusion resistance in the reaction process.

Fig. 5
figure 5

Effect of diffusion coefficient variation on effectiveness factor for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Fig. 6
figure 6

Effect of thermal conductivity variation on effectiveness factor for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Effect of catalyst size

Figure 7 presents the effect that change in catalyst size will have on the magnitude of effectiveness factor. As the catalyst size increases, the effectiveness factor decreases. At the catalyst size of 8.41 \(\times\) 10–4 m, the effectiveness factor is 0.3954, decreasing the catalyst size to 2.36 \(\times \,\) 10–4 m the value of effectiveness factor increased to 0.6522. The catalyst size of 1.623 \(\times\) 10–4 m was able to produce effectiveness factor of 1, thus removing the effect of intra-particle resistance on the catalyst with minimal pressure drop of 8% which is still within the allowed 10% pressure drop for fixed bed reactor [28]. This shows that catalyst size of 1.623 \(\times\) 10–4 m is still acceptable for the reaction process as it is still with accepted pressure drop value. This effect is very important in the design and optimization of the reactor. Figure 8 shows the variation of pressure drop with catalyst size at reactor length of 0.00635 m considered for the reaction process. The pressure drop varies linearly with catalyst size with the difference from the initial value still less than 10%.

Fig. 7
figure 7

Effect of catalyst size on effectiveness factor

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Fig. 8
figure 8

Variation of pressure drop in the reactor with catalyst size

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Figure 9 shows the concentration profiles of the reactant (methanol) at different Thiele moduli, 1.2. 2.2, 3.4, 4.6, 5.0 and 5.9 inside the catalyst pellet using the power law kinetics. At Thiele modulus of 1.2, the intra-particle diffusion resistance has no major effect on the rate of reaction per pellet. Almost all the entire volume of the catalyst is reacting at the same rate as about 80% of the methanol is able to diffuse quickly into the centre of the catalyst pellet. At Thiele modulus of 2.2, the effect of intra-particle diffusion started becoming predominant with almost 60% of the reactant being converted before diffusing to the centre of the catalyst pellet. When the Thiele modulus is 3.4, 75% of the reactant was converted at the surface of the catalyst pellet before reaching the centre. While for large Thiele moduli of 4.6, 5.0 and 5.9, by the time reactant gets to the centre of the catalyst pellet, almost 97% of it has been converted. This is in agreement with Fig. 2. For these large values of Thiele moduli, the rate of reaction is greater than the diffusion rate; thus, the effectiveness factor is much less than unity and the process is said to be diffusion limited, that is, the surface reaction is rapid and the reactant is consumed very close to external pellet surface and vert little penetrates into the interior of the pellet. Figure 10 depicts the temperature profile at different Thiele moduli inside the catalyst pellet. The temperature drops are minimal at different Thiele moduli which shows that the reaction process is isothermal. This may be due to the endothermic nature of steam reforming process and also because of slow heat transfer into the reactor from the external heating system [29, 30]. Thus, to increase heat transfer into the process, internal heating of the system can be adopted which can increase the temperature in the catalyst and leads to increase in the rate of conversion [30], or by combining endothermic steam reforming with exothermic combustion part of methanol which supplies the required heat [29].

Fig. 9
figure 9

Dimensionless concentration profiles of methanol at different Thiele modulus along the radial distance within the catalyst pellet

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Fig. 10
figure 10

Temperature of methanol at different Thiele modulus along the radial distance within the catalyst pellet

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Figure 11 shows the partial pressure of the key components inside the catalyst pellet. The two reactants methanol and steam partial pressure decreases from the outer catalyst surface to the centre of the catalyst as they reacted to form the products carbon dioxide and hydrogen. The products have their partial pressure increase from the centre of the pellet to the outer surface. Table 3 depicts the effect of molecular diffusion and Knudsen diffusion on the overall effective diffusivity inside the catalyst pellet. The molecular diffusivity is greater than Knudsen diffusivity which shows that the diffusion of the reactants into the catalyst pellet is more of molecular diffusion than Knudsen diffusion. This is due to the smaller size of the methanol particles which allow easy passage of the reactants into the catalyst pellet.

Fig. 11
figure 11

Partial pressure of reactants and products inside the catalyst pellet for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst

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Table 3 Molecular, Knudsen and effective diffusion coefficients of methanol inside the catalyst pellet
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Conclusions

A mathematical model was developed for a diffusion–reaction process in a spherical catalyst pellet contained in a non-isothermal heterogeneous packed bed reactor. The model developed was solved to predict the non-isothermal effectiveness factor and also to perform sensitivity analysis for steam reforming of methanol on Cu/ZnO/Al2O3 catalyst, a source of hydrogen fuel. Results show that reducing the catalyst size to 1.623 \(\times\) 10–4 m eliminates the effect of intra-particle diffusion effect. An increase in diffusion coefficient leads to an increase in the effectiveness factor while increase in thermal conductivity leads to a decrease in effectiveness factor. Both power law and the developed LHHW kinetics for steam reforming of methanol on Cu/ZnO/Al2O3 were able to predict the reaction process adequately. Also at temperature below 473 K, the effect on intra-particle diffusion limitation is reduced to the barest minimum.