Thermodynamic Insight in Design of Methanation Reactor with Water Removal Considering Nexus between CO₂ Conversion and Irreversibilities

Abstract

The inevitable nexus between energy use and CO₂ emission necessitates the development of sustainable energy systems. The conversion of CO₂ to CH₄ using green H₂ in power-to-gas applications in such energy systems has attracted much interest. In this context, the present study provides a thermodynamic insight into the effect of water removal on CO₂ conversion and irreversibility within a CO₂ methanation reactor. A fixed-bed reactor with one intermediate water removal point, representing two reactors in series, was modeled by a one-dimensional pseudo-homogeneous model. Pure CO₂ or a mixture of CO₂ and methane, representing a typical biogas mixture, were used as feed. For short reactors, both the maximum conversion and the largest irreversibilities were observed when the water removal point was located in the middle of the reactor. However, as the length of the reactor increased, the water removal point with the highest conversion was shifted towards the end of the reactor, accompanied by a smaller thermodynamic penalty. The largest irreversibilities in long reactors were obtained when water removal took place closer to the inlet of the reactor. The study discusses the potential benefit of partial water removal and reactant feeding for energy-efficient reactor design.

1. Introduction

Economic growth, energy use, and CO₂ emissions are associated with one another. He et al. [1] stated that rapid growth in the economy and energy use has caused an increase in CO₂ emissions. Pao and Tsai [2] investigated the economy-energy-sustainability nexus. They illustrated that the reduction of CO₂ emission without negative effect on the economic growth could obtain by increasing energy efficiency. Balsalobre et al. [3] pointed out that the implementation of energy strategies that emphasizes shifting towards renewable energy sources could effectively reduce the dependency on fossil fuels, and thereby also reduce CO₂ emissions.

In modern energy systems, an increased share of renewable energy sources like solar and wind has been seen as a solution for mitigating CO₂ emissions [4]. However, the intermittent nature of renewable energy sources entails a need to consider energy storage when renewable power generation does not match the demand [5]. Among possible energy storage technologies [5,6], Power-to-Gas (PtG) concepts provide the possibility of converting surplus renewable electricity to chemical energy through the production of energy carrier components such as hydrogen (H₂), methane (CH₄), and methanol (CH₃OH) [7].

Essentially, the production of H₂ from water electrolysis run by surplus renewable electricity is considered the first step in these PtG applications. The production of H₂ from renewable energy has multiple advantages such as low harmful emissions fewer steps of energy conversion to produce alternative fuel from renewable electricity, and direct utilization in fuel cell vehicles. Nonetheless, lacking storage capacity and distribution infrastructure limit the use of H₂ [8]. An alternative is the further chemical conversion of the H₂ into other energy carriers, e.g., CH₄ or CH₃OH. Although the overall process efficiency is reduced by additional chemical conversion steps, higher energy density and mature infrastructure motivate the use of alternatives [9]. Moreover, CH₄ is used for power generation, transportation, and as a precursor for other chemicals [9,10]. In PtG applications, the methane is produced through catalytic or biological methanation using CO₂ from renewable carbon sources such as biogas, or captured CO₂ from industrial processes, and H₂ from water electrolysis.

In catalytic methanation, CO₂ reacts with H₂, in the presence of a catalyst, through a reversible and highly exothermic reaction, known as the Sabatier reaction, producing CH₄ and water (H₂O) [11].

$${\mathrm{CO}}_2+4{\mathrm{H}}_2\rightleftharpoons {\mathrm{CH}}_4+2{\mathrm{H}}_2\mathrm{O} (\mathsf{\Delta }{H}_{298\mathrm{K}}^{°}=-165\mathrm{kJ}/\mathrm{mol})$$

(1)

Although methanation has been in use for many years in different industries like ammonia production plants and synthetic natural gas (SNG) production, the development of the methanation reactor design is still an attractive topic in research to improve the performance of the methanation process in terms of CO₂ conversion, cost efficiency, and stability of the reactor with respect to the load fluctuation from renewable sources [12]. The traditional methanation reactors are fixed-bed reactors (FBR) [13,14,15,16,17,18] and fluidized bed reactors [19,20,21]. Recently, three-phase slurry reactors [22,23,24] and microchannel reactors [25,26] have also been developed for the methanation process. Inclusive reviews of the mentioned reactor concepts can be found in the literature [7,12]. The majority of commercial CO₂ methanation processes are based on catalytic FBR, operating under adiabatic conditions in a series of reactors with intermediate cooling or operating under isothermal conditions [12]. Hashemi et al. [27] indicated that the operation of the methanation process in a series of adiabatic reactors would reduce the irreversibility rate within the reactors, in comparison with isothermal reactors, improving the process integration potentials.

One important aspect of the methanation reactor design is the highly exothermic nature of the CO₂ methanation reaction, and many reactor concepts have been developed to overcome challenges regarding heat management and temperature control along the reactor [12]. Optimal heat management within the reactor can lead to higher CO₂ conversion. Sun et al. [28] performed a simulation-based study considering a kinetic model for the CO₂ methanation in an FBR to observe the effect of heat removal on the methane yield. They proposed a new design configuration with a molten salt-cooled heat exchanger to improve the cooling rate within the reactor, leading to increased methane yield. In another work by Sun et al. [29], it was observed that the molten salt flow rate, which indicated the cooling rate, was a crucial parameter for the reactor performance. They observed that the methane yield improved with reduced temperature (increased cooling rate), but also that overcooling would hinder the reaction. Moreover, they demonstrated that although the methane yield improved by increasing the space velocity, there was a threshold above which further increase in the space velocity led to reduced conversion. Kiewidt and Thöming [30] proposed a method to optimize the temperature profile within an FBR by balancing the heat production rate and the cooling rate. They illustrated that the methane yield improved by optimizing the temperature profile. Their results demonstrated that the optimal temperature profile was located between the temperature profiles obtained from isothermal and adiabatic operations, when balancing the kinetics and thermodynamic limitations along the reactor.

In PtG applications, it is essential to convert as much H₂ as possible since it is a major driving factor for the cost [29]. Improving the hydrogen conversion is equivalent to improving the CO₂ conversion for a given amount of hydrogen. Besides the reactor design aspects concerning heat management within the reactor, many studies regarding reactor design focus on CO₂ conversion improvement by manipulating thermodynamic equilibrium within the reactor.

Water removal in the methanation process can shift the thermodynamic equilibrium of the Sabatier reaction towards the product side, enhancing the CO₂ conversion. Recently, this has triggered reactor concepts such as sorption enhanced methanation (SEM) and water-selective membrane reactors (MR) [9,31]. Within SEM and MR concepts, the produced H₂O in the methanation process is locally removed from the gas phase stream within the reactor by means of sorption materials such as zeolite and silica. A complete overview regarding SEM and MR with respect to sorption materials and reactor configuration design can be found in the work by van Kampen et al. [32] and Diban et al. [33], respectively.

Walspurger et al. [34] investigated the effect of water removal in an SEM reactor experimentally. They indicated that a CO₂ conversion near to 100% was possible in an SEM reactor with a commercial nickel-based catalyst and zeolite 4A as H₂O adsorbent when the operating temperature was between 250 and 350 °C. Based on Gibbs’ free energy minimization, Faria et al. [35] studied in-situ water removal in an equilibrium model methanation reactor. In addition to the species present in the Sabatier reaction, they also included carbon monoxide and coke in their simulations. They illustrated that the CO₂ conversion increased with increasing water removal fraction, independently of the operating temperature and pressure. However, the methane yield was maximized at an optimal water removal fraction, depending on the temperature and pressure, above which coke formation was observed [35]. Najari et al. [36] investigated the effect of in-situ water removal from a methanation reactor using kinetic models. They illustrated that removing water locally improved the reactor performance in terms of conversion, but this also increased the risk of hot spots within the reactor, which could have a negative impact on the functionality of the catalysts [36]. Their study suggested that further examination of the effects of kinetics and temperature on the reactor performance was required [36].

Although in-situ water removal provides apparent advantages for the methanation process from a thermodynamic point of view, practical aspects, such as heat management and membrane characterization, must also be considered [32]. For reactors operated in series, as previously mentioned, water may be removed in between the stages, as an alternative to in-situ water removal. This suggests investigating the effect of water removal in stages can be considered an alternative to continuous water removal through SEM or MR. Hillestad [37] proposed a systematic staging method for the design of chemical reactors. He demonstrated that staging provided additional degrees of freedom to obtain better performance in the reactor.

Even though water removal between the methanation reactor stages has been considered as a method to increase the CO₂ conversion, investigation regarding how the design of staging with intermediate water removal should consider the reactor length (i.e., representing reactor volume) is missing in the literature. In this regard, the optimal water removal location point is expected to be dependent on the length of the reactor. Further, the energy efficiency of methanation reactors with intermediate water removal has received limited attention in the literature; most research studies in the field of reactor design emphasize performance improvement in terms of CO₂ conversion and heat management. The present study aims to fill these research gaps by conducting a fundamental thermodynamic study to examine the concept of water removal for different reactor lengths. The optimal water removal point within a reactor is determined for different operating conditions. The performance of the reactor is assessed in terms of improvement in the CO₂ conversion. Moreover, alternative objectives related to the total irreversibility within the reactor and the minimum required work for water removal are also discussed.

In Section 2, the reactor model development, reaction kinetics, and numerical solution strategy for the model are given. The methods employed to investigate the effect of water removal at different locations along the reactor are explained in Section 3. Further, a description regarding the calculation of irreversibility rate and minimum work requirements for water removal is presented. Results for the optimal location for water removal in the reactor with respect to conversion and irreversibility are presented in Section 4. Remarks and suggestions for further studies are provided in Section 5. Finally, conclusions are presented in Section 6.

2. Model Development

According to the conclusions drawn by Fischer et al. [38], a simplified one-dimensional pseudo-homogenous model, known as a plug flow reactor (PFR) model, provides sufficient accuracy to predict the CO₂ conversion within an FBR. In the present study, the pseudo-homogeneous model was combined with an effectiveness factor to accommodate the intra-particle mass and heat transport limitations between solid (catalyst pellets) and fluid (gas mixtures) phases [30]. Here, it was assumed that a concept similar to the isothermal reactor concept developed by Linde can provide isothermal operating conditions for the methanation [39]. However, it should be noted that operating under isothermal conditions is a challenging task in practice, and might not be economically feasible, as the highly active catalyst causes large heat production and potential hot spots in the reactor. The irreversibility associated with heat transfer may contribute significantly to overall process irreversibility. Test simulations within this study showed that the pressure drop along the reactor length (obtained from the Ergun equation) was on a scale of 0.1 kPa for the studied dimension and flow rates. Therefore, the pressure drop in the reactor was neglected in the PFR model, to focus on the variation of driving forces caused by reaction along the reactor. Further, the ideal gas law was applied as an equation of state. The mathematical model for the FBR is presented in the following subsections.

2.1. Reactor Modeling

The FBR in the present study was modeled by considering a plug flow reactor [40] assuming steady-state conditions. In the plug flow assumption, gradients of temperature and concentration are only considered in the axial direction, not the radial direction or the angular direction. Material balances for all the chemical substances involved along the length of the reactor (CO₂, H₂, CH₄, and H₂O) can be expressed as

$$\frac{\mathrm{d}{F}_i}{\mathrm{d}x}={\rho }_{\mathrm{c}}·\left(1-\epsilon \right)·A_{\mathrm{c}}·\eta ·{\nu }_i·r.$$

(2)

Here, F_i is the molar flow rate of component i in the direction x along the reactor, while ρ_c, ε, and A_c denote the catalyst density, the void fraction, and the cross-sectional area of the reactor, respectively. Further, ν_i is the stoichiometric coefficient of component i in reaction (1). Explicit expressions for the reaction rate (r) and the effectiveness factor (η) are given in the following subsections.

2.2. Reaction Kinetics

The kinetic model of Koschany et al. [41], where the CO₂ methanation reaction over a Ni-based catalyst is considered, was used in this work. The reaction kinetics, and thereby the CO₂ conversion, is influenced by the operating temperature, pressure, and inlet gas mixture composition. The reaction rate model of Koschany et al. [41] can be applied in temperature and pressure ranges of 180–340 °C and 1–15 bar, respectively. Under these conditions, the CO methanation through reverse-water-gas-shift (RWGS) is limited; hence considering only the CO₂ methanation reaction is reasonable [42]. The reaction rate is based on a Langmuir-Hinshelwood-Hougen-Watson approach as follows:

$$r=\frac{k·p_{{\mathrm{H}}_2}^{0.5}·p_{{\mathrm{CO}}_2}^{0.5}·\left(1-\frac{p_{{\mathrm{CH}}_4}·p_{{\mathrm{H}}_2\mathrm{O}}^2}{p_{{\mathrm{CO}}_2}·p_{{\mathrm{H}}_2}^4·K_{\mathrm{eq}}}\right)}{{\left(1+K_{\mathrm{OH}}·\frac{p_{{\mathrm{H}}_2\mathrm{O}}}{p_{{\mathrm{H}}_2}^{0.5}}+K_{{\mathrm{H}}_2}·p_{{\mathrm{H}}_2}^{0.5}+K_{\mathrm{mix}}·p_{{\mathrm{CO}}_2}^{0.5}\right)}^2}.$$

(3)

Here, p_i is the partial pressure of component i. The rate constant (k) and adsorption constants (K_j) are calculated in accordance with Arrhenius and van ’t Hoff-type equations, respectively, as follows:

$$k=k_{\mathrm{ref}}·\exp \left(\frac{E_{\mathrm{a}}}{R}·\left(\frac{1}{T_{\mathrm{ref}}}-\frac{1}{T}\right)\right),$$

(4)

$$K_{\mathrm{j}}=K_{\mathrm{j},\mathrm{ref}}·\exp \left(\frac{\mathsf{\Delta }{H}_j}{R}·\left(\frac{1}{T_{\mathrm{ref}}}-\frac{1}{T}\right)\right).$$

(5)

Here, E_a and ΔH are activation energy and enthalpy of adsorption, respectively. T and R are the temperature and the universal gas constant, respectively. The equilibrium constant (K_eq) is approximated as [43]

$$K_{\mathrm{eq}}=137·T^{-3.998}·\exp \left(\frac{158.7\frac{\mathrm{kJ}}{\mathrm{mol}}}{RT}\right).$$

(6)

The values of the kinetic model parameters are given in

Table 1. Values of kinetics model parameters [41].

Variable	Tref	kref	Ea	KOH,ref	ΔHOH	KH2,ref	ΔHH2	Kmix,ref	ΔHmix
Unit	K	mol/bar·s·kgcat	kJ/mol	bar−0.5	kJ/mol	bar−0.5	kJ/mol	bar−0.5	kJ/mol
Value	555	3.46·10−1	77.5	0.5	22.4	0.44	−6.2	0.88	−10

.

2.3. Effectiveness Factor

In comparison to CO₂, H₂ diffuses much faster into the catalyst pellets. Thereby, an adopted Thiele modulus can be used to calculate the effectiveness factor, assuming CO₂ to be the limiting species for intra-particle mass transport [30]. The effectiveness factor for spherical catalyst pellets is given as

$$\eta =\frac{3}{\varphi }\left(\frac{1}{\tanh \varphi }-\frac{1}{\varphi }\right),$$

(7)

where the Thiele modulus ($\varphi$) can be calculated considering CO₂ as the key species in the determination of the mass transfer limitations:

$$\varphi =\frac{D_{\mathrm{p}}}{2}\sqrt{\frac{r·{\rho }_{\mathrm{c}}·\left(1-\epsilon \right)·R·T}{D_{{\mathrm{CO}}_2}^{eff}·y_{{\mathrm{CO}}_2}·p·{10}^5}}.$$

(8)

Here, D_p and y_CO2 denote the catalyst pellets diameter and the mole fraction of CO₂ in the gas mixture, respectively. The effective CO₂ diffusivity ($D_{{\mathrm{CO}}_2}^{eff}$) is calculated according to the Bosanquet equation taking into account molecular diffusion ($D_{{\mathrm{CO}}_2}^m$) for gas-gas collisions and Knudsen diffusion ($D_{{\mathrm{CO}}_2}^{kn}$) for gas-wall collisions [30]:

$$\frac{1}{D_{{\mathrm{CO}}_2}^{eff}}=\frac{{\tau }_{\mathrm{p}}}{{\epsilon }_{\mathrm{p}}}\left(\frac{1}{D_{{\mathrm{CO}}_2}^m}+\frac{1}{D_{{\mathrm{CO}}_2}^{kn}}\right).$$

(9)

The effective CO₂ diffusivity ($D_{{\mathrm{CO}}_2}^{eff}$) takes into account the catalyst pellet configuration through particle porosity (τ_p), tortuosity (ε_p), average pore diameter (D_pore), and molecular interaction between different species [17]. The molecular diffusion is based on a simplified form of the mixture-averaged diffusion coefficient by Maxwell-Stefan [44]:

$$\frac{1}{D_{{\mathrm{CO}}_2}^m}={\displaystyle \sum }_i\frac{y_i}{D_{ij}}+\frac{y_j}{1-w_j}{\displaystyle \sum }_i\frac{w_i}{D_{ij}}.$$

(10)

Here, i = H₂, CH₄ and H₂O and j = CO₂. Further, y and w are the mole fraction and the mass fraction, respectively. The binary diffusion coefficients (D_ij) are calculated by the equations from Fuller et al. [45]:

$$D_{ij}=\frac{0.00143·T^{1.75}·{\left(\frac{1}{M_i}+\frac{1}{M_j}\right)}^{\frac{1}{2}}}{p·{\left({\left({\nu }_i\right)}^{\frac{1}{3}}+{\left({\nu }_j\right)}^{\frac{1}{3}}\right)}^2}.$$

(11)

Here, M_i is the molar mass of component i, and ν_i is the specific diffusion volume of component i (26.9, 7.07, 24.42, and 12.7 for CO₂, H₂, CH_4, and H₂O, respectively) [45]. The Knudsen diffusion is computed as follows, considering only CO₂:

$$D_{{\mathrm{CO}}_2}^{kn}=\frac{D_{\mathrm{pore}}}{3}\sqrt{\frac{8·R·T}{\pi ·M_{{\mathrm{CO}}_2}}}.$$

(12)

2.4. Numerical Solution Strategy

The balance equations and the correlations for reaction rates and effectiveness factors generate a set of ordinary differential equations (ODEs). The ODEs were solved with MATLAB^® R2019a using the ode15s function. To solve the ODEs, the initial molar flow of components at the inlet of the reactor is required. A good trade-off between model precision and computational time is achieved by assuming an equidistant step size (representing the cell size along the reactor) of 0.001 m and a relative error tolerance of 10⁻⁸. Here, the relative error tolerance refers to the considered significance of digits for the computation of the ODEs. The water removal is implemented numerically by adjusting the molar flow of water to zero at the step where water removal is taking place. The reactor specifications and input values for the model are given in

Table 2. Reactor specifications in the PFR model.

Parameter	Unit	Value	Ref.
Temperature range	K	500–600	[41]
Pressure range	bar	1–15	[41]
Catalyst density	kg/m3	2355.2	[41]
Catalyst void fraction	-	0.4	[17]
H2/CO2 ratio	-	4	-
CH4/CO2 ratio range	-	0–1.5	-
Inlet CO2 molar flow rate	mol/s	0.002	-
Tube diameter	m	0.0254	-
Catalyst diameter	m	0.002	[17]
Catalyst pore diameter	nm	10	[17]
Catalyst porosity	-	0.6	[17]
Catalyst tortuosity	-	2	[17]
Ambient temperature	K	298.15	-
Ambient pressure	bar	1.01325	-

.

3. Methodology

The hypothesis behind removing water at different positions along an FBR is that the CO₂ conversion (and thereby the CH₄ yield) would change by moving the water removal position. In the present study, water is only removed at one point along the reactor, and the influence of moving the water removal point on the CO₂ conversion is investigated. For the sake of fair comparison, the water produced after the water removal point is removed at the end of the reactor. Before studying the effect of water removal, the reactor length required to reach equilibrium without water removal is identified. At equilibrium, the reaction rate approaches zero, which would require an infinitely long reactor. Here, the equilibrium length (L_EQ.1) is defined as the point at which 99.9% of the equilibrium conversion is obtained. It is worth mentioning that the equilibrium length of the rector is correlated to the tube diameter. The smaller the tube diameter, the longer is the reactor required to reach the equilibrium composition.

Figure 1a illustrates a tubular FBR, with diameter D and the equilibrium length (L_EQ.1), in which the entire water removal takes place at the end of the reactor. The case with intermediate water removal along the reactor length is demonstrated in Figure 1b. Conventional directions of material and energy streams are specified in Figure 1. When studying the effect of water removal, reactors lengths (L) shorter than the equilibrium length are examined, varying the water removal point along the reactor. Water is removed at a point z along the reactor. In practice, this assumption can be interpreted as dividing the original reactor with length L into two reactors with lengths of z and L-z, with water removal between the two reactors. In this work, it is assumed that 100% of produced water present at this location is removed. A case with no internal water removal (WR) (i.e., all water removed at the end of the reactor) is given by z = 0 or z = L. Even though 100% continuous water removal along the reactor is practically impossible, a continuous WR case is considered to account for the theoretical cases of SME and MR. In this case, the water produced in each cell of the reactor model is assumed to be removed in the same cell.

Here, the ratio between H₂ and CO₂ is equal to the stoichiometric ratio of the Sabatier reaction (i.e., 4). Moreover, the ratio between CH₄ and CO₂ (here defined as A) is zero or 1.5, representing pure CO₂ methanation and methanation of a biogas mixture with 60 mol% CH₄ and 40 mol% CO₂, respectively. In the present study, the initial CO₂ molar flow rate is kept constant; accordingly, the initial molar flow rate of H₂ and CH₄ is calculated using the stoichiometric ratio between H₂ and CO₂ and the studied ratio between CH₄ and CO₂, respectively.

The performance of the reactor is evaluated with respect to the CO₂ conversion and the irreversibilities within the reactor. The CO₂ conversion is defined as the change between inlet and outlet CO₂ molar flow in the reactor:

$$X_{{\mathrm{CO}}_2}=\frac{F_{{\mathrm{CO}}_2}^{in}-F_{{\mathrm{CO}}_2}^{out}}{F_{{\mathrm{CO}}_2}^{in}}$$

(13)

Since only the Sabatier reaction is considered, the CO₂ conversion will also represent the methane yield from the reactor. In order to compare the effect of water removal for different reactor lengths, the relative conversion improvement (RX_CO2) is defined as the CO₂ conversion improvement for the optimum water removal location compared to the case with no water removal:

$$R{X}_{{\mathrm{CO}}_2}=\frac{X_{{\mathrm{CO}}_2,@\mathrm{opt}}-X_{{\mathrm{CO}}_2,\mathrm{no}\mathrm{WR}}}{X_{{\mathrm{CO}}_2,\mathrm{no}\mathrm{WR}}}.$$

(14)

As the amount of water removed from the reactor and the CO₂ conversion change with the water removal location, the work requirement for water removal and the irreversibility rate within the rector also change. These changes are investigated using exergy analysis. The total irreversibility rate ($\dot{I}$) within the reactor at steady-state operation, illustrated in Figure 1 is calculated based on changes in exergy of material streams ($\mathsf{\Delta }{\dot{\mathrm{E}}}_{\mathrm{x},\mathrm{streams}}$), the exergy of heat rejected from the reactor (${\dot{\mathrm{E}}}_{\mathrm{x}}\left(\dot{Q}\right)$) and the work required for water removal within the reactor ($\dot{W}$) according to the methodology described by Kotas [46]:

$$\dot{I}=\mathsf{\Delta }{\dot{\mathrm{E}}}_{\mathrm{x},\mathrm{streams}}+{\dot{\mathrm{E}}}_{\mathrm{x}}\left(\dot{Q}\right)-\dot{W},$$

(15)

$$\mathsf{\Delta }{\dot{\mathrm{E}}}_{\mathrm{x},\mathrm{streams}}={\displaystyle \sum }{\dot{n}}_i·{\overline{\epsilon }}_{x,i}-{\displaystyle \sum }{\dot{n}}_e·{\overline{\epsilon }}_{x,e},$$

(16)

$${\dot{\mathrm{E}}}_{\mathrm{x}}\left(\dot{Q}\right)={\displaystyle \int }\left(1-\frac{T_0}{T}\right)·\delta \dot{Q}.$$

(17)

Here, ${\dot{n}}_i$ is the molar flow rate and ${\overline{\epsilon }}_x$ is the molar exergy, calculated for inlet streams i and outlet streams e. Further, $\dot{Q}$ is the heat flow transferred to the reactor at reactor temperature T and $\dot{W}$ is the power delivered from the reactor. The subscript “0” denotes environment state (here T₀ = 298.15 K, p₀ = 1 atm = 1.01325 bar). In this study, it is assumed that water removal takes place in a reversible process; hence, $\dot{W}$ represents the minimum work required for water removal.

The molar exergy of material streams can be decomposed into physical exergy (${\overline{\epsilon }}_x^{phy}$) and chemical exergy (${\overline{\epsilon }}_x^{chem}$):

$${\overline{\epsilon }}_x={\overline{\epsilon }}_x^{phy}+{\overline{\epsilon }}_x^{chem}.$$

(18)

By neglecting kinetic and potential energy effects, ${\overline{\epsilon }}_x^{phy}$ can be expressed as

$${\overline{\epsilon }}_x^{phy}=\left(\overline{h}-{\overline{h}}_0\right)-T_0·\left(\overline{s}-{\overline{s}}_0\right).$$

(19)

Based on ideal gas and ideal mixture assumptions, the molar enthalpy and entropy of the mixtures can be calculated as

$$\left(\overline{h}-{\overline{h}}_0\right)={\displaystyle \sum }{y}_i·\left({\overline{h}}_i\left(T\right)-{\overline{h}}_i\left(T_0\right)\right),$$

(20)

$$\left(\overline{s}-{\overline{s}}_0\right)={\displaystyle \sum }{y}_i·\left({\overline{s}}_i\left(T,p_i\right)-{\overline{s}}_i\left(T_0,p_{0,i}\right)\right).$$

(21)

Here, y_i is the molar fraction of component i in the mixture, while $\overline{h}$ and $\overline{s}$ refer to the molar enthalpy and entropy of the material stream, respectively. The chemical exergy of an ideal mixture can be calculated as

$${\overline{\epsilon }}_x^{chem}={\displaystyle \sum }{y}_i·{\overline{\epsilon }}_{x,i}^{std}+T_0·\overline{R}·{\displaystyle \sum }{y}_i·\ln y_i.$$

(22)

The standard chemical exergy of component i in the mixture (${\overline{\epsilon }}_x^{chem}$) is obtained from the reference tables provided by Kotas [46]. Further, $\overline{R}$ is the universal gas constant.

4. Results

4.1. Conversion

According to thermodynamic principles, operating the reactor at a lower temperature and higher pressure is favorable for improving the CO₂ conversion due to the highly exothermic nature of the Sabatier reaction and the volume reduction after conversion. Figure 2 illustrates the CO₂ conversion within the FBR operating under different conditions when reaction (1) reaches equilibrium. As is expected, higher CO₂ conversion is obtained by increasing the operating pressure and lowering the reactor temperature. Moreover, additional CH₄ in the inlet gas mixture (i.e., cases with A = 1.5) results in a reduction of the CO₂ conversion. Since CH₄ is one of the products in the reaction, the presence of CH₄ in the feed shifts the equilibrium composition towards the reactant side, leading to reduced CO₂ conversion.

Figure 3 demonstrates the influence of water removal on the CO₂ conversion and reaction rate along a reactor that is operating at 600 K and 1 bar. In Figure 3, the solid red line (EQ.1) illustrates the point at which the reaction is assumed to have reached equilibrium (99.9% of the CO₂ conversion at equilibrium). The dotted red line (EQ.2) illustrates the point at which the reaction reaches a new equilibrium if all produced water is removed at EQ.1. The solid black line (“No WR”) illustrates a case with no water removal along the reactor, while the black dotted lines represent cases with water removal at different points in the reactor. Further, the solid blue line demonstrates a case where water is removed continuously along the reactor. In all cases, the reactor is long enough to reach equilibrium after water removal. It is worth mentioning that the cases with no water removal and continuous water removal define the limiting cases with minimum and maximum CO₂ conversion, respectively.

As can be seen in Figure 3a, removing water from the reactor increases the CO₂ conversion compared to the case with no water removal. As expected, maximum CO₂ conversion is obtained when the water is removed after reaching the first equilibrium. It should be noticed that most of the conversion takes place before reaching the first equilibrium and that the length extension required to reach the second equilibrium point is smaller than the length required to reach the first equilibrium. This can be also seen in Figure 3b, where the reaction rate along the reactor is illustrated. It can be observed that water removal causes an abrupt increase in the reaction rate, increasing the average reaction rate compared to the case with no water removal. When the reaction reaches the first equilibrium, no further conversion takes place due to zero reaction rate. No matter where water is removed after reaching the first equilibrium length, the reaction rate reaches an identical maximum value and then approaches zero where the second equilibrium is obtained (see the identical lines for water removal at the first equilibrium and 1.7 m in Figure 3b). If the water removal point is located before reaching equilibrium, the total CO₂ conversion is reduced. However, the reaction rate is higher, and the conversion can be improved if the reactor is not long enough to reach equilibrium.

Figure 4 illustrates the CO₂ conversion improvement and relative length extension required to obtain maximum CO₂ conversion under different operating conditions. The length required to reach the first equilibrium point depends on the kinetics of the reaction. Operating the CO₂ methanation reactor at a higher pressure and temperature increases the reaction rate, thereby reducing the length required to reach equilibrium. Moreover, the presence of CH₄ in the inlet gas mixture reduces the reaction rate, hence increasing the length required to reach equilibrium. These trends also apply to the relative length extension required to reach the second equilibrium after removing water.

As can be observed in Figure 4, the relative conversion improvement is higher under conditions where CO₂ conversion is limited due to the thermodynamic equilibrium (i.e., low pressure, high temperature, and CH₄ in the feed). This suggests that the water removal is of more significance when the final CO₂ conversion at the first equilibrium length is lower. For instance, the highest CO₂ conversion improvement of approximately 8% is achieved for the case with the lowest conversion at the first equilibrium point (corresponding to the situation in Figure 2, i.e., at p = 1 bar, T = 600 K and A = 1.5).

As mentioned earlier in this section, both the CO₂ conversion and the conversion rate changes when moving the point of the water removal. This becomes important when the reactor is not long enough to reach equilibrium. The effect of the water removal location on the CO₂ conversion for a reactor with a length equal to 50% of the first equilibrium length (L_EQ.1) is illustrated in Figure 5. It should be noted that the results for water removal at either the inlet or the outlet of the reactor are the same since no water removal takes place along the reactor in both cases. As can be seen in Figure 5, there is an optimal location for the water removal at which the CO₂ conversion is maximized. Corresponding to Figure 3b, the optimal point for water removal should be where the average reaction rate is highest. The reaction rate is reduced as the number of products increases. After removing water, the driving force for the reaction, and thereby the reaction rate, increases. The later the water removal, the higher the conversion will be at equilibrium. However, if the water is removed too late, there will not be enough length left to take advantage of the increased reaction rate.

Figure 6 illustrates the effect of operating conditions on the optimum water removal location for different lengths of the reactor. The optimal location is here defined as the water removal point that gives the maximum CO₂ conversion for a given reactor length. However, the optimal point is only refined to a resolution of 0.1 in z/L. As can be observed in Figure 6, for a given operating pressure, the optimum water removal location is independent of the reactor temperature. The shorter the length of the reactor, the earlier the water removal must take place to maximize the conversion. At higher pressure, the results indicate that the optimal point of water removal is at a higher relative length of z/L.

As can be observed in Figure 3b, the reaction rate drops faster after removing the water. To overcome this limitation, the optimal value for the relative point for water removal will not be smaller than 0.5 even in short reactors. For the cases where CH₄ is present in the inlet gas mixture, the optimal value for the relative length of z/L is generally smaller. With A = 1.5 and low pressure, the optimal point is the same for all relative reactor lengths below 0.75. Corresponding to Figure 4, the presence of CH₄ in the feed gas reduces the reaction rate, thereby increasing the length required to reach equilibrium.

The relative conversion improvement for different reactor lengths under different operating conditions is demonstrated in Figure 7. The water removal leads to larger CO₂ conversion improvement when the length of the reactor is shorter than the first equilibrium length. At low operating pressure, the strongest effect on the relative conversion improvement due to water removal is seen at the relative length (L/L_EQ.1) of 0.5, whereas the strongest effect is observed at the relative length (L/L_EQ.1) of 0.25 at higher operating pressure. For short reactors, the reaction will not reach equilibrium. Nonetheless, an effort to increase the average reaction rate is favorable as this gives a higher conversion. The largest relative conversion improvement due to water removal is achieved in the case without CH₄ in the inlet gas mixture, operating at high pressure and high temperature. Referring to Figure 3b, it was observed that water removal boosted the average reaction rate, and the new equilibrium condition would be obtained earlier. Moreover, increasing the operating temperature and pressure improves the reaction rate and this will be amplified by removing water in a short reactor.

4.2. Irreversibility

The present section focuses on assessing the performance of the methanation reactor in terms of energy efficiency. This is realized by analyzing the irreversibility rate of a reactor with or without intermediate water removal. In cases without intermediate water removal, the analysis is presented for the final composition at the end of a reactor sufficiently long to reach equilibrium.

Figure 8 illustrates the specific irreversibility (i.e., the ratio between the total irreversibility rate and the amount of CH₄ produced) within a reactor operating under different conditions for cases without intermediate water removal. As can be observed in Figure 8, operating at lower pressure and higher temperature decreases the specific irreversibility within the reactor. Also, the existence of CH₄ in the inlet gas mixture reduces the specific irreversibility within the reactor. Contrarily, results in Figure 2 illustrated that the CO₂ conversion decreased if the reactor operated at low pressure, high temperature, and in the presence of CH₄ in the inlet gas mixture. These two perspectives suggest that the reactor design should reflect a compromise between CO₂ conversion and irreversibility. From a reversibility point of view, to reduce the thermodynamic losses within the reactor the reaction should follow a path where the magnitude of the reaction driving forces along the reactor approaches zero (i.e., theoretically proceeding the chemical reaction infinitesimally close to equilibrium). Here, in cases with no intermediate water removal, the final CO₂ conversion and the specific irreversibility are only influenced by the fixed operating conditions. Therefore, the degrees of freedom to manipulate the driving forces, and thereby the extent of reaction, are limited.

Figure 9 illustrates the specific work required for water removal, the specific exergy of heat rejected from the reactor, and the change in specific exergy of material streams for a reactor operating under different conditions. As can be seen in Figure 9a, the work required to remove water from the reactor increases with increasing temperature. The presence of CH₄ in the inlet gas mixture also leads to an increase in the work requirement for water removal, as part of the work would be used to separate water from the additional CH₄ in the gas mixture. However, an increase in operating pressure decreases the work required to remove water. Figure 9b illustrates that the exergy of heat extracted from the reactor is independent of the operating pressure and the inlet gas composition. Running the reactor at higher temperature results in increased available exergy of heat, which can be utilized in the overall process design when potential process integration options are considered, contributing to reduced total irreversibility rate at higher temperatures. As can be observed in Figure 9c, operating the reactor at higher temperature and pressure enables extracting more exergy from the material streams, which is also observed in the CO₂ conversion in Figure 2. For a given temperature and pressure, additional CH₄ in the inlet gas mixture leads to a reduction in the exergy extraction from the material streams. The results in Figure 9c are reflected in the specific irreversibility presented in Figure 8.

Here, the effect of the intermediate water removal location on the irreversibility rate within the reactor is studied. In cases with intermediate water removal, the water produced after the removal point is removed at the end of the reactor, to provide a fair comparison between the different cases. Figure 10 demonstrates the effect of the water removal location on the accumulated irreversibility rate and specific irreversibility along a reactor operating at T = 600 K, p = 1 bar, and A = 0. It is worth mentioning that the results in Figure 10 are considered for the same cases as in Figure 3. Here, the specific irreversibility refers to the ratio between the irreversibility rate and the amount of produced CH₄ in each cell of the reactor. As can be observed in Figure 10a, compared to the case with no intermediate water removal, removing water at the intermediate point increases the total irreversibility rate within the reactor. It can be observed that the increase in the irreversibility rate reaches a maximum value when the water is removed at a certain intermediate point. Continuous water removal yields the highest irreversibility rate within the reactor (solid blue line).

In Figure 10b, the intermediate water removal point is observed as a sudden jump in the curves, which leads to higher average specific irreversibility within the reactor. The areas below the curves in Figure 10b indicate the total irreversibility rate within the reactor. As can be observed in Figure 10b, the CO₂ conversion improvement is accompanied by a smaller increase in the irreversibility rate when the intermediate water removal point moves towards the end of the rector. A smaller increase in the average specific irreversibility can be seen when the water is removed after the first equilibrium length (i.e., where the specific irreversibility becomes zero since no reaction occurs in the reactor). In this particular case study, with no length limitation and fixed operating conditions, the results of the irreversibility analysis suggest that water should be removed when the reaction reaches equilibrium.

Results for the specific irreversibility within a reactor operating at T = 600 K, p = 1 bar, and A = 0 are tabulated in

Table 3. Summarized results for the effect of water removal location (z) along a reactor operating at T = 600 K, p = 1 bar, and A = 0, considering an FBR with a length of 3 m.

x	XCO2	$\Delta {\dot{\mathbf{E}}}_{\mathbf{x},\mathbf{streams}}/{\dot{\mathit{m}}}_{{\mathbf{CH}}_{\mathbf{4}}\mathbf{prod}.}$	${\dot{\mathbf{E}}}_{\mathbf{x}}\left(\dot{\mathit{Q}}\right)/{\dot{\mathit{m}}}_{{\mathbf{CH}}_{\mathbf{4}}\mathbf{prod}.}$	$\dot{\mathit{W}/}{\dot{\mathit{m}}}_{{\mathbf{CH}}_{\mathbf{4}}\mathbf{prod}.}$	$\dot{\mathit{I}}/{\dot{\mathit{m}}}_{{\mathbf{CH}}_{\mathbf{4}}\mathbf{prod}.}$
(m)	(%)	(kWh/kg CH4 Prod.)	(kWh/kg CH4 Prod.)	(kWh/kg CH4 Prod.)	(kWh/kg CH4 Prod.)
No WR	92.9	1.976	1.560	0.098	0.514
0.15	94.2	1.972	1.560	0.155	0.567
0.3	95.3	1.968	1.560	0.166	0.574
0.8	97.6	1.958	1.560	0.132	0.530
1.27 *	98.4	1.954	1.560	0.109	0.502
1.7	98.4	1.954	1.560	0.109	0.502
Continuous WR	100.0	1.942	1.560	0.678	1.060

* The intermediate water removal occurs at the first equilibrium length (EQ.1).

when the water is removed at different intermediate locations. An increase in the CO₂ conversion due to moving the water removal point towards the end of the reactor length is accompanied by a decrease in the change of specific exergy of material streams. Since the temperature and heat of the reaction are constant, the specific exergy of heat remains constant when moving the water removal point. When the water is removed after reaching the first equilibrium, a smaller total specific irreversibility is obtained compared to the case with no intermediate water removal. This is because the average specific irreversibility before the removal point would be higher than the obtained peak value for the specific irreversibility due to the water removal. Therefore, the amount of total specific irreversibility would reduce.

As it is assumed that the water separation is achieved through a reversible process, no additional irreversibility caused by the water removal process is expected. As can be seen in Table 3, in comparison with no water removal case, the increase in the total specific irreversibility due to water removal is accompanied by additional reversible specific work. This suggests that considering the specific work required for water removal from a reactor operating at fixed conditions can be an alternative approach to investigate how the water removal location would influence the irreversibility rate.

The minimum work comprises the work required for water removal at the intermediate location and the end of the reactor. The effect of the water removal location on the amount of removed water and the specific work is demonstrated in Figure 11a,b, respectively, for a reactor operating at T = 600 K, p = 1 bar, and A = 0. By moving the water removal point from the inlet of the reactor towards the outlet of the reactor, the amount of removed water at the intermediate location increases, reaching a maximum amount when the intermediate water removal occurs at the first equilibrium length. As illustrated in Figure 11, the amount of water removed at the intermediate point would not necessarily reflect the required specific work at the intermediate point. The specific work at the intermediate point decreases as the intermediate point moves towards the end of the reactor, reaching a constant value when equilibrium is obtained. Moreover, the larger the amount of water removed at the intermediate point, the higher the specific work required at the end of the reactor to separate the additional produced water after the water removal point. The work required for water removal depends on the composition of the gas mixture, and the specific work requirement for water removal increases as the water fraction reduces.

As discussed earlier, removing water from the reactor increases the average reaction rate. Hence the same conversion can be obtained in a shorter reactor. However, the increase in the average reaction rate would cause an unavoidable increase in irreversibility when the operating conditions for the reaction are fixed. Here, the aim is to investigate how the water removal location can influence the irreversibility rate within reactors shorter than the required equilibrium length (similar to the case studies in Section 4.1.). The effect of the water removal point on the specific irreversibility for different relative reactor lengths operating at T = 600 K, p = 1 bar, and A = 0 is illustrated in Figure 12. Further, Figure 13 demonstrates the specific work required for water removal, the specific exergy of heat rejected from the reactor, and the changes in specific exergy of material streams for the cases presented in Figure 12.

As can be observed in Figure 12, for a short reactor (e.g., L/L_EQ.1 = 0.25), the highest specific irreversibility is obtained when the water removal occurs in the middle of the reactor. This coincides with where the maximum CO₂ conversion is obtained (at z/L = 0.5). As the reactor length is increased, the optimal water removal point is shifted towards the outlet of the reactor, while the largest irreversibilities are observed for water removal points closer to the inlet of the reactor. Moreover, for the optimal water removal point (in terms of conversion), the increase in specific irreversibility compared to the case with no water removal is reduced. When the reactor length is equal to the equilibrium length EQ.1, there is only a small penalty compared to the irreversibilities in the case with no water removal. This can be explained by the direct effect of the water removal location on the average specific irreversibility and average reaction rate. As discussed earlier, the benefits from reduced specific irreversibility for longer reactors at the optimal water removal point are reflected in the required specific work and the change in specific exergy of the streams (see Figure 13).

5. Remarks

The present study illustrates that considering intermediate water removal increases the average reaction rate within the reactor and, as a result, increases the CO₂ conversion and the irreversibility rate within a reactor operating at fixed conditions. As mentioned earlier, obtaining a fixed isothermal condition for a highly activated catalytic reaction is a challenging task in practice. Besides, the fixed operating conditions limit the degrees of freedom to manipulate the driving forces within the reactor to obtain an energy-efficient reactor design. Therefore, an extended study should consider adding more complexity to the reactor model, e.g., non-isothermal operating modes.

In the present study, the examination of removing water at an intermediate point demonstrated the potential possibility of manipulation of the average reaction rate through changing driving forces leading to different irreversibility rates. To avoid a significant increase in the reaction rate at the water removal point, and thereby a significant jump in the specific irreversibility, partial water removal (instead of complete water removal at the intermediate point) or water removal at several points can be considered as alternatives. It is worth mentioning that partial water removal results in longer reactors while the effect of multiple removal points will be reduced reactor length Nonetheless, the length of the reactor might not be an important factor if an energy-efficient reactor with a high CO₂ conversion is guaranteed.

The intermediate water removal concept can also be accompanied by a partial feeding strategy of reactants along the reactor. On the one hand, water removal increases the average reaction rate, while on the other hand, a limited concentration of the reactants would reduce the average reaction rate. This can aid in controlling the reaction rate within the reactor. Considering a combination of water removal and partial feeding of the reactants can lead to efficient heat management within the reactor, at which high CO₂ conversion can be obtained even though the irreversibility rate is low. It is observed that the work required for removing water at an intermediate location could be used as an alternative approach to investigate the irreversibility rate within the reactor. However, it should be noted that the actual work for water removal will be higher than what is presented here, as, in the present study, the produced water was removed in reversible processes.

6. Conclusions

A comprehensive thermodynamic study was performed in order to investigate the effect of water removal on the CO₂ conversion and the irreversibility in an isothermal fixed bed methanation reactor. A one-dimensional pseudo-homogenous model was applied for the reactor, considering the kinetics of the Sabatier reaction. An effectiveness factor was used to consider the intra-particle mass and heat transport limitations between the two phases. At one intermediate point in the reactor, all the water produced up until that point was is removed, assuming a reversible process.

With respect to the CO₂ conversion, for short reactors, the optimal water removal point was located in the middle of the reactor. As the length of the reactor approached the length required to reach equilibrium, the water removal point resulting in the highest CO₂ conversion moved closer to the end. The location of the optimal removal point was found to be independent of the operating temperature, while the presence of CH₄ in the feed gas resulted in moving the optimal water removal point upstream.

With respect to the irreversibility within the reactor, for a short reactor, it was observed that the water removal point that gave the highest CO₂ conversion in the middle of the reactor also provided the largest irreversibility rate. For a longer reactor (i.e., reactor lengths close to the required equilibrium length), the improvement in the CO₂ conversion was accompanied by a smaller penalty in the irreversibilities compared to the case with no water removal.

The results illustrate that the length of the reactor is essential to the optimal water removal point. Further, it is demonstrated that the reactor length should not necessarily be close to the equilibrium length when the water removal takes place to obtain the best performance; the additional gains in terms of both the CO₂ conversion and the irreversibilities are not significant.

Possible conceptual ideas such as partial water removal and partial reactant feeding are suggested for future studies to develop an energy-efficient reactor. These ideas are based on manipulating the reaction rate within the reactor to control the driving forces along the reactor. The proposed model should be examined using process simulators, and for cases operating under non-isothermal conditions in future work. In this context, additional concerns such as runaway temperature limitations should be considered.