**2. Model equations**

out. These experiments showed encouraging results. The solid oxide electrolysis cell (SOEC) is the reverse operation of the same SOFC cell structure. Both are composed of two porous composite ceramic electrodes surrounding a gas-tight electrolyte. SOECs can rely on the interest that SOFCs have received for the past decades and thus utilize similar technology and materials. Water is reduced at the cathode Eq. (1), releasing hydrogen and oxygen ions. After crossing the electrolyte, the ions are then being oxidized to form oxygen within the anode Eq. (2). Both half reac-

� ! *<sup>H</sup>*<sup>2</sup> <sup>þ</sup> *<sup>O</sup>*2� (1)

� (2)

<sup>2</sup>*O*<sup>2</sup> (3)

*H*2*O* þ 2*e*

*<sup>O</sup>*2� ! <sup>1</sup>*=*<sup>2</sup>*O*<sup>2</sup> þ 2*e*

*H*2*O* ! *H*<sup>2</sup> þ <sup>1</sup>

offer the advantage to allow the co-electrolysis of water and carbon dioxide to produce syngas [5, 6]. According to AlZahrani and Dincer [7] a SOEC system can achieve energy and exergy efficiencies of 53 and 60%, respectively. However, the high operating temperature (>1000 K) is still considered the major limiting factor of these device. Jiang [8] has shown that the hydrogen production by SOECs coupled with renewable energy sources is a promising route for the sustainability of energy in the future. The solution of a real commercially competitive SOC technologies is the materials: reliability and stability of the electrode and electrolyte materials under the reversible electrolysis and fuel cell operation modes. The reversible fuel cells have the ability to switch between electrolysis cell and fuel cell modes, and it is one of the foremost features that facilitate storing/generating energy in a costeffective manner. The optimized parameters on the designs of fell cells or of electrolyzer, solely, would not necessarily result in high performance of regenerative devices because these devices differ in electrode kinetics, gas environment, heat generation, and chemical stability. It is well known that high-temperature operation of SOECs offers inherent advantages, in terms of thermodynamics and kinetics compared with low-temperature electrolysis. In this context reversible solid oxide cells (RSOCs) are still at an early stage of development [9, 10]. Unfortunately, there is a general consensus that the performance and stability of SOECs are inferior to those of SOFCs [11], which is mainly due to the high-temperature operation. Computing simulation appears to be one of the most efficient approaches to analyze the coupled mechanisms of SOEC operation. It can predict the SOEC behavior under various operating conditions. Mathematical modeling is an essential tool in the design of SOEC cells, as it is important to understand the limiting process of steam electrolysis. Recent literature shows a significant research and development effort focusing on the modeling of SOEC. The models developed by Udagawa et al. [12, 13] strive to describe all significant processes affecting the performance of a unit cell. These authors proposed one-dimensional or pseudo-2D simulations based on a planar geometry and taking into account mass transport. Ni et al. [14, 15] have described the mass transport within the electrode along with the electrochemical kinetics. The principal results of these investigations [16, 17] lead to a parametric control of the SOEC operation. In addition, Jin and Xue [18, 19] have developed a 2D model for a planar SOEC. The simulation results lead to a better understanding of the internal mechanisms for regenerative SOFCs. This model has been used to

Due to the high operating temperature, SOECs do not need expensive catalysts, but these must meet strict thermal, chemical, and structural requirements imposed by temperature, and hydrogen and oxygen partial pressures. However, performance still remains limited in electrolysis mode compared to fuel cell mode [4]. Comparing to PEM electrolyzers for which carbon monoxide is a poison, SOECs

*=*

tions are balanced by general Eq. (3):

*Electrodialysis*

**94**

In the present model, mass and charge transport phenomena coupled with chemical and electrochemical reactions have been investigated within the SOEC cell. This mathematical approach is based on classical SOFC assumptions, and thus the model should depend on operating conditions, intrinsic conductivities of materials, and geometric parameters such as porosity or grain size [21–22]. Additionally, according to the 2D model of Kenney and Karan [23], the interconnects play a critical role. Thus, a 2D approach was performed in this work. In this study, a finite element method has been used to solve mass and charge balances including transport through porous media and electrochemical reactions within the porous electrodes. The set of resulting conservation equations has been solved using the commercial software Comsol Multiphysics®. In this computational approach, steady-state conditions have been imposed. This model of SOEC is based on the following assumptions:


Equations are detailed in this section one balance at a time.

#### **2.1 Charge balance**

The electrode material is a mixed electronic and ionic conductor. For modeling purposes, this electrode is considered as a porous gas diffusion electrode wherein the electrochemical reaction occurs at the triple phase boundary, i.e., at the interface between the electronic conductor, ionic conductor, and gas phase. The current in a porous electrode can be split into two parts: one part flowing through the ionic phase and the other through the electronic phase of the porous matrix. During

electrochemical reactions, electrons are then transferred from the ionic phase to the electronic phase or vice versa. The transport of each kind of charges (e�, O2�) can be described using Ohm's law. To account for charge transfer between electronic and ionic materials, a current source term *Qi*,*a=<sup>c</sup>* (A m�<sup>3</sup> ) is employed in the charge balance in Eq. (4):

$$-\nabla \cdot \left(\sigma\_{\text{S/M},a/c}^{\text{eff}} \nabla \phi\_{\text{S/M}}\right) = \mathbf{Q}\_{i,a/c} \tag{4}$$

*ibv*,*<sup>a</sup>* ¼ *i*0\_*<sup>a</sup>* exp

*DOI: http://dx.doi.org/10.5772/intechopen.90352*

have a thermodynamic potential equal to 0 V:

current densities can be expressed by Eq. (10):

performance, and are gathered in **Table 2**.

Cathodic exchange current density, *i*

Anodic exchange current density, *i*

*Parameters used to apply the Butler-Volmer equation.*

the inlet feed.

**Table 2.**

**97**

*i*0,*a=<sup>c</sup>* ¼ *i*

*ref*

2*αaFη<sup>a</sup> RT* � � � *CO*<sup>2</sup>

*Solid Oxide Steam Electrolyzer: Gas Diffusion Steers the Design of Electrodes*

*C*0 *O*<sup>2</sup>

where *ηa=<sup>c</sup>* are the overpotentials defined as the difference between the electronic potential on one hand and the ionic and equilibrium potentials on the other hand, displayed in Eq. (9). In the present model, hydrogen electrode is assumed to

*<sup>η</sup>a=<sup>c</sup>* <sup>¼</sup> *<sup>ϕ</sup>electronic*,*a=<sup>c</sup>* � *<sup>ϕ</sup>ionic*,*a=<sup>c</sup>* � *<sup>E</sup>*<sup>0</sup>

0,*a=<sup>c</sup>* <sup>1</sup> � *<sup>ε</sup><sup>a</sup>=<sup>c</sup>* � <sup>0</sup>*:*<sup>26</sup>

Laurencin et al. [16] suggested that Butler-Volmer expression is suitable to describe electrochemical reaction involving the exchange of 2 electrons. The values for electrochemical geometric symmetric coefficients *α<sup>c</sup>* and *α<sup>a</sup>* are usually considered to be close to 0.5 in the literature [26, 28]. The exchange current densities *i*

are critical parameters to describe the current generated by the cell. Both parameters take into account the virtual specific surface area linked to the vicinity of the triple point boundaries (TPB) where the charge transfer occurs. These parameters have been deduced from previous work [16, 24] in order to observe usual SOEC

Finally, the electrolyte material is YSZ, a suitable ionic conductor at high tem-

*Vcell* ¼ *Eeq* þ *ηcell* ¼ *Eeq* þ *Vpol* � *ϕelectronic*,*<sup>c</sup>*

**Parameter Value References**

Symmetrical factors, *α<sup>a</sup>=<sup>c</sup>* [�] 0.5 [24] Reference grain diameter, *dgref* [μm] 3 This work

*PH*2,*cP* 1*=*2 *O*2,*a PH*2*O*,*<sup>c</sup>*

1

] <sup>4</sup> � <sup>10</sup><sup>8</sup> [16, 24]

] <sup>4</sup> � <sup>10</sup><sup>7</sup> [16, 24]

0 @

*RT* 2*F* ln

*Eeq* <sup>¼</sup> *<sup>E</sup>*<sup>0</sup> <sup>þ</sup>

*ref* 0,*<sup>c</sup>* [A m�<sup>3</sup>

*ref* 0,*<sup>a</sup>* [A m�<sup>3</sup>

peratures. The electrolyte potential is thus expressed by a classical Ohm's law Eq. (4) without any current source term (*QS*,*a=<sup>c</sup>* ¼ 0) within the electrolyte. The Nernst law Eq. (12) is used to assess the open circuit voltage (OCV) at operating temperature *T* of 1173.15 with an E<sup>0</sup> of 1.1 V which, in turn, is used to obtain the cell voltage through Eq. (11) with *Vpol* the parametric input of the polarization computation. Vpol has been defined in order to stabilize the numerical convergence near the OCV operating point. The partial pressures are according to

According to [24], the influence of the electrode microstructure on the exchange

<sup>1</sup> � <sup>0</sup>*:*<sup>26</sup> � � *dgref*

exp �2 1ð Þ � *<sup>α</sup><sup>a</sup> <sup>F</sup>η<sup>a</sup> RT* ! � � (8)

*dga=<sup>c</sup>*

� � (11)

A (12)

!<sup>3</sup>

*<sup>a</sup>=<sup>c</sup>* (9)

(10)

*ref* 0,*a=c*

The effective conductivity (*σ eff S=M*,*a=c* ) depends on the material and the microstructure of each electrode. Their values can be computed using Eq. (5) [20], where S and M subscript are, respectively, ionic material and electronic material.

$$
\sigma\_{\text{S/M,a/c}}^{\text{eff}} = \mathbf{Y}\_{\text{S/M,a/c}} \frac{\left(\mathbf{1} - \varepsilon\_{a/c}\right)}{\mathbf{1}.\mathsf{G}} \sigma\_{\text{S/M,a/c}} \tag{5}
$$

The bulk conductivities and molar fractions that have been used throughout this work are gathered in **Table 1**. Nickel has been used as the electronic material at the cathode, whereas the anode was made of LSM-type perovskite [24, 25]. YSZ allows the transport of ions in both electrodes and the electrolyte. Within the electrolyte ceramic membrane, there are no current sources. Therefore, pure and dense YSZ electrolyte is considered (*σYSZ*) in Eq. (4). Similarly, current collectors are assumed to be ideal electronic conductors. The charge balance ensures that the current produced at the cathode is consumed at the anode. Additionally, in each electrode, the electronic current is the opposite of the ionic one. According to Costamagna et al. [26], the current source terms *Qi*,*a=<sup>c</sup>* can be described by the classical Butler-Volmer expression Eqs. (7) and (8), and the current sources are expressed as follows:

$$\mathbf{Q}\_{i,a/c} = \pm \mathbf{i}\_{bv,a/c} \tag{6}$$

The selected parameters of Butler-Volmer equation remain valid at high fuel utilization and low value of hydrogen concentration corresponding to the SOEC mode. The current sources and therefore the expression of the electrochemical reactions are expressed by the following Butler-Volmer Eqs. (7) and (8), *F* being the Faraday's constant (*F* = 95,485 C mol�<sup>1</sup> ) and R the ideal gas constant (*R* = 8.314 J mol�<sup>1</sup> K�<sup>1</sup> ):

$$\dot{\mathbf{u}}\_{bv,\varepsilon} = \dot{\mathbf{u}}\_{0\cdot\varepsilon} \left( \frac{\mathbf{C}\_{H\_2}}{C\_{H\_2}^0} \exp\left(\frac{2a\_c F \eta\_c}{RT}\right) - \frac{\mathbf{C}\_{H\_2O}}{C\_{H\_2O}^0} \exp\left(\frac{-\mathbf{2}(\mathbf{1} - a\_c)F \eta\_c}{RT}\right) \right) \tag{7}$$


**Table 1.**

*Conductivities and molar composition of the SOEC.*

*Solid Oxide Steam Electrolyzer: Gas Diffusion Steers the Design of Electrodes DOI: http://dx.doi.org/10.5772/intechopen.90352*

$$i\_{bv,a} = i\_{0\\_d} \left( \exp\left(\frac{2\alpha\_d F \eta\_a}{RT}\right) - \frac{C\_{O\_2}}{C\_{O\_2}^0} \exp\left(\frac{-2(1-\alpha\_d)F\eta\_a}{RT}\right) \right) \tag{8}$$

where *ηa=<sup>c</sup>* are the overpotentials defined as the difference between the electronic potential on one hand and the ionic and equilibrium potentials on the other hand, displayed in Eq. (9). In the present model, hydrogen electrode is assumed to have a thermodynamic potential equal to 0 V:

$$
\eta\_{a/c} = \phi\_{\text{electronic},a/c} - \phi\_{\text{ionic},a/c} - E\_{a/c}^0 \tag{9}
$$

According to [24], the influence of the electrode microstructure on the exchange current densities can be expressed by Eq. (10):

$$i\_{0,a/c} = i\_{0,a/c}^{ref} \left( 1 - \frac{\varepsilon\_{a/c} - 0.26}{1 - 0.26} \right) \left( \frac{d g^{ref}}{d g\_{a/c}} \right)^3 \tag{10}$$

Laurencin et al. [16] suggested that Butler-Volmer expression is suitable to describe electrochemical reaction involving the exchange of 2 electrons. The values for electrochemical geometric symmetric coefficients *α<sup>c</sup>* and *α<sup>a</sup>* are usually considered to be close to 0.5 in the literature [26, 28]. The exchange current densities *i ref* 0,*a=c* are critical parameters to describe the current generated by the cell. Both parameters take into account the virtual specific surface area linked to the vicinity of the triple point boundaries (TPB) where the charge transfer occurs. These parameters have been deduced from previous work [16, 24] in order to observe usual SOEC performance, and are gathered in **Table 2**.

Finally, the electrolyte material is YSZ, a suitable ionic conductor at high temperatures. The electrolyte potential is thus expressed by a classical Ohm's law Eq. (4) without any current source term (*QS*,*a=<sup>c</sup>* ¼ 0) within the electrolyte.

The Nernst law Eq. (12) is used to assess the open circuit voltage (OCV) at operating temperature *T* of 1173.15 with an E<sup>0</sup> of 1.1 V which, in turn, is used to obtain the cell voltage through Eq. (11) with *Vpol* the parametric input of the polarization computation. Vpol has been defined in order to stabilize the numerical convergence near the OCV operating point. The partial pressures are according to the inlet feed.

$$\mathcal{V}\_{cell} = E\_{eq} + \eta\_{cell} = E\_{eq} + \left(\mathcal{V}\_{pol} - \phi\_{electronic,c}\right) \tag{11}$$

$$E\_{eq} = E^0 + \frac{RT}{2F} \ln\left(\frac{P\_{H\_2,c} P\_{O\_2,a}^{1\_2}}{P\_{H\_2O,\varepsilon}}\right) \tag{12}$$


#### **Table 2.**

*Parameters used to apply the Butler-Volmer equation.*

electrochemical reactions, electrons are then transferred from the ionic phase to the electronic phase or vice versa. The transport of each kind of charges (e�, O2�) can be described using Ohm's law. To account for charge transfer between electronic

> ∇*ϕS=<sup>M</sup>* � �

structure of each electrode. Their values can be computed using Eq. (5) [20], where

The bulk conductivities and molar fractions that have been used throughout this work are gathered in **Table 1**. Nickel has been used as the electronic material at the cathode, whereas the anode was made of LSM-type perovskite [24, 25]. YSZ allows the transport of ions in both electrodes and the electrolyte. Within the electrolyte ceramic membrane, there are no current sources. Therefore, pure and dense YSZ electrolyte is considered (*σYSZ*) in Eq. (4). Similarly, current collectors are assumed to be ideal electronic conductors. The charge balance ensures that the current produced at the cathode is consumed at the anode. Additionally, in each electrode, the electronic current is the opposite of the ionic one. According to Costamagna et al. [26], the current source terms *Qi*,*a=<sup>c</sup>* can be described by the classical Butler-Volmer expression Eqs. (7) and (8), and the current sources are expressed as

The selected parameters of Butler-Volmer equation remain valid at high fuel utilization and low value of hydrogen concentration corresponding to the SOEC mode. The current sources and therefore the expression of the electrochemical reactions are expressed by the following Butler-Volmer Eqs. (7) and (8), *F* being the

> � *CH*2*<sup>O</sup> C*0 *H*2*O*

! � �

2*αcFη<sup>c</sup> RT* � �

**Parameter Value**

Cathodic volume fraction of nickel, *YNi*,*<sup>c</sup>* [24] 0.4 Anodic volume fraction of LSM, *YLSM*,*<sup>a</sup>* [24] 0.5

1 � *εa=<sup>c</sup>* � � ) is employed in the charge

¼ *Q <sup>i</sup>*, *<sup>a</sup>=<sup>c</sup>* (4)

<sup>1</sup>*:*<sup>6</sup> *<sup>σ</sup><sup>S</sup>=M*,*a=<sup>c</sup>* (5)

) depends on the material and the micro-

*Qi*,*a=<sup>c</sup>* ¼ �*ibv*,*a=<sup>c</sup>* (6)

exp �2 1ð Þ � *<sup>α</sup><sup>c</sup> <sup>F</sup>η<sup>c</sup> RT*

(7)

*=T*

0.6 0.5

) and R the ideal gas constant

] [24] 4.5 � <sup>10</sup><sup>5</sup>

] [24] 1.6 � <sup>10</sup><sup>5</sup>

] [27] *<sup>σ</sup>YSZ* <sup>¼</sup> <sup>0</sup>*:*<sup>334</sup> � <sup>10</sup><sup>5</sup> exp ð Þ �<sup>10300</sup>

and ionic materials, a current source term *Qi*,*a=<sup>c</sup>* (A m�<sup>3</sup>

*σ eff* �∇*: <sup>σ</sup>eff*

*eff S=M*,*a=c*

*<sup>S</sup>=M*,*a=<sup>c</sup>* ¼ *YS=M*,*a=<sup>c</sup>*

*S=M*,*a=c*

S and M subscript are, respectively, ionic material and electronic material.

balance in Eq. (4):

*Electrodialysis*

follows:

**Table 1.**

**96**

The effective conductivity (*σ*

Faraday's constant (*F* = 95,485 C mol�<sup>1</sup>

Nickel electronic conductivity, *σNi* [S m�<sup>1</sup>

LSM electronic conductivity, *σLSM* [*S* m�<sup>1</sup>

Cathodic volume fraction of YSZ, *YYSZ*,*<sup>c</sup>* [24] Anodic volume fraction of YSZ, *YYSZ*,*<sup>a</sup>* [24]

*Conductivities and molar composition of the SOEC.*

YSZ ionic conductivity, *σYSZ* [S m�<sup>1</sup>

*ibv*,*<sup>c</sup>* ¼ *i*0\_*<sup>c</sup>*

):

*CH*<sup>2</sup> *C*0 *H*<sup>2</sup> exp

(*R* = 8.314 J mol�<sup>1</sup> K�<sup>1</sup>


**Table 3.**

*Microstructural parameters used in this study.*

#### **2.2 Mass balance**

As stated in the assumptions of the model, since electrodes are porous media with low permeability, mass transport is only due to diffusion process. Binary gas interaction and pore wall effect should be taken into account using the following general expression of the mass balance:

$$\nabla \cdot \left( -D\_k^{\sharp \mathcal{F}} \nabla \mathbf{C}\_k \right) = \Gamma\_k \tag{13}$$

*Deff* \_*<sup>F</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.90352*

10<sup>5</sup> Pa:

*<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> <sup>1</sup> 1 *D<sup>K</sup> H*2*O*

*Solid Oxide Steam Electrolyzer: Gas Diffusion Steers the Design of Electrodes*

*β<sup>c</sup>* ¼ 1 �

coefficient, and the Fick diffusion form coefficient, respectively:

*<sup>O</sup>*<sup>2</sup> <sup>¼</sup> *dpAε<sup>A</sup>* 3*τ<sup>A</sup>*

*<sup>O</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 1 *DK O*2

*β<sup>a</sup>* ¼ 1 �

*D<sup>K</sup>*

*Deff* \_*<sup>F</sup>*

use this value as boundary conditions of mass balance.

**3. Simulation conditions**

implemented.

**3.1 Geometry**

**99**

<sup>þ</sup> <sup>1</sup> � *<sup>β</sup>cCH*2*<sup>O</sup> Ctot* � � <sup>1</sup>

q

In the set of mass balance equations, the ideal gas law is applicable Eq. (21). This work has been done at a working temperature *T* of 1173.15 K and a pressure *P* of

*Ctot* <sup>¼</sup> *<sup>P</sup>*

Similar modeling has been performed at the anode side to describe mass flow through the porous anode. O2 and N2 are substituted to H2O and H2, respectively. The binary diffusion coefficient *DO*2\_*N*<sup>2</sup> is presented in Annex under Eq. (27). Eqs. (22)–(24) display the Knudsen diffusion coefficient, the effective diffusion

s

<sup>þ</sup> <sup>1</sup> � *<sup>β</sup>aCO*<sup>2</sup> *Ctot* � � <sup>1</sup>

q

The commercial software Comsol Multiphysics® has been used to investigate the behavior of a simplified serial repeating unit (SRU) of SOEC. The modeled geometry will thus include both electrodes, the electrolyte, and the current collectors. The resulting set of conservation equations is solved using the commercial software. This section will focus on the description of how the model was

This study objective has been to investigate the influence of the electrode microstructure considering a realistic geometry of the SRU. **Figure 1** displays the geometry that forms the basis of the model. For simplicity's sake, a two-dimensional model of the SRU was used according to its cross section. The cell dimensions are gathered in **Table 4**. A mapped mesh has been used to obtain a reasonable number of degrees of freedom allowing calculation convergence within an acceptable computation time. Its parameters are also listed in **Table 4**. The studied geometry is a cross section of SRU which is perpendicular to the main direction of the gas flow. Considering that channels allow an ideal distribution of reactants on each electrode, it is possible to consider them as boundary conditions. This means that the produced hydrogen and oxygen are perfectly collected and exhausted from the SRU by the gas manifolds. Therefore, constant gas concentrations within the gas channels have been considered, in order to

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *MO*2*=MN*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *MH*2*<sup>O</sup>=MH*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8*RT <sup>π</sup>MO*<sup>2</sup> � <sup>10</sup><sup>3</sup>

> *Deff O*2\_*N*2

*Deff H*2*O*\_*H*2

*RT* (21)

(19)

(20)

(22)

(23)

(24)

where Γ*<sup>k</sup>* is the mass source term, directly linked to the current sources *Qa=<sup>c</sup>* by Faraday's law as expressed below:

$$
\Gamma\_{O\_2} = \frac{i\_{bv,a}}{4F} \tag{14}
$$

$$
\Gamma\_{H\_2} = -\Gamma\_{H\_2O} = \frac{i\_{bv,c}}{2F} \tag{15}
$$

The binary diffusion coefficient of water in hydrogen was computed through Eq. (26), presented in appendix. Thus, at the cathode, only hydrogen and steam are taken into account. The effective diffusion coefficient takes into consideration the microstructure of the electrode according to Eq. (16) [24]:

$$D\_{H\_2O\_-H\_2}^{\sharp f} = D\_{H\_2O\_-H\_2} . \mathcal{E}\_C^{\pi c} \tag{16}$$

In the porous media of each electrode, and because of the presence of small pore size, the model integrates the Knudsen diffusion process, which coefficient is expressed by Eq. (17) [20]:

$$D\_{H\_2O}^K = \frac{dp\_C \varepsilon\_C}{\mathfrak{Z} \tau\_C} \sqrt{\frac{8RT}{\pi \mathbf{M}\_{H\_2O} \times \mathbf{10}^3}}\tag{17}$$

**Table 3** gathers grain diameters (*dga=<sup>c</sup>*) and tortuosity (*τ<sup>a</sup>=<sup>c</sup>*) values. Pore sizes are a function of both parameters, as displayed by Eq. (18) [20]:

$$dp\_{a/c} = \frac{2\varepsilon\_{a/c}d\mathbf{g}\_{a/c}}{\Im\left(1 - \varepsilon\_{a/c}\right)}\tag{18}$$

Usually, the dusty gas model considers Maxwell and Knudsen diffusion to describe the gas flow through a porous media. It was possible to suggest an approach equivalent to Fick's law (Eq. (13)) by considering an effective equivalent diffusion coefficient [24], illustrated by Eqs. (19) and (20), with *β<sup>c</sup>* being the Fick diffusion form coefficient:

*Solid Oxide Steam Electrolyzer: Gas Diffusion Steers the Design of Electrodes DOI: http://dx.doi.org/10.5772/intechopen.90352*

$$D\_{H\_2O}^{eff\\_F} = \frac{1}{\frac{1}{D\_{H\_2O}^K} + \left(1 - \frac{\beta\_c C\_{H\_2O}}{C^{\rm sat}}\right) \frac{1}{D\_{H\_2O\\_H}^{\rm eff}}}\tag{19}$$

$$\boldsymbol{\beta}\_{\varepsilon} = \mathbf{1} - \sqrt{\mathbf{M}\_{\mathrm{H}\_{2}\mathrm{O}}\prime\_{\mathrm{M}\_{2}}} \tag{20}$$

In the set of mass balance equations, the ideal gas law is applicable Eq. (21). This work has been done at a working temperature *T* of 1173.15 K and a pressure *P* of 10<sup>5</sup> Pa:

$$\mathbf{C}^{\text{tot}} = \frac{P}{RT} \tag{21}$$

Similar modeling has been performed at the anode side to describe mass flow through the porous anode. O2 and N2 are substituted to H2O and H2, respectively. The binary diffusion coefficient *DO*2\_*N*<sup>2</sup> is presented in Annex under Eq. (27). Eqs. (22)–(24) display the Knudsen diffusion coefficient, the effective diffusion coefficient, and the Fick diffusion form coefficient, respectively:

$$D\_{O\_2}^K = \frac{dp\_A \varepsilon\_A}{3\tau\_A} \sqrt{\frac{8RT}{\pi M\_{O\_2} \times 10^3}}\tag{22}$$

$$D\_{O\_2}^{\mathcal{G}|\!^-F} = \frac{1}{\frac{1}{D\_{O\_2}^K} + \left(1 - \frac{\beta\_a C\_{O\_2}}{C^{\text{ot}}}\right) \frac{1}{D\_{O\_2, N\_2}^{\mathcal{G}}}} \tag{23}$$

$$
\beta\_a = \mathbf{1} - \sqrt{{}^{M\_{\text{O}\_2}} \! \! \! / \! M\_{\text{N}\_2}} \tag{24}
$$

### **3. Simulation conditions**

The commercial software Comsol Multiphysics® has been used to investigate the behavior of a simplified serial repeating unit (SRU) of SOEC. The modeled geometry will thus include both electrodes, the electrolyte, and the current collectors. The resulting set of conservation equations is solved using the commercial software. This section will focus on the description of how the model was implemented.

#### **3.1 Geometry**

**2.2 Mass balance**

**Table 3.**

*Electrodialysis*

general expression of the mass balance:

*Microstructural parameters used in this study.*

Faraday's law as expressed below:

expressed by Eq. (17) [20]:

diffusion form coefficient:

**98**

As stated in the assumptions of the model, since electrodes are porous media with low permeability, mass transport is only due to diffusion process. Binary gas interaction and pore wall effect should be taken into account using the following

**Parameter Value** Electrodes porosity, *ε<sup>a</sup>=<sup>c</sup>* [�] 0.37 Electrodes grain diameter, *dga=<sup>c</sup>* [μm] 3 Electrodes tortuosity, *τ<sup>a</sup>=<sup>c</sup>* [�] 4.8

> *<sup>k</sup>* ∇*Ck* � �

where Γ*<sup>k</sup>* is the mass source term, directly linked to the current sources *Qa=<sup>c</sup>* by

<sup>Γ</sup>*<sup>O</sup>*<sup>2</sup> <sup>¼</sup> *ibv*,*<sup>a</sup>*

<sup>Γ</sup>*<sup>H</sup>*<sup>2</sup> ¼ �Γ*<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *ibv*,*<sup>c</sup>*

The binary diffusion coefficient of water in hydrogen was computed through Eq. (26), presented in appendix. Thus, at the cathode, only hydrogen and steam are taken into account. The effective diffusion coefficient takes into consideration the

*<sup>H</sup>*2*O*\_*H*<sup>2</sup> ¼ *DH*2*O*\_*H*<sup>2</sup> *:ε<sup>C</sup>*

size, the model integrates the Knudsen diffusion process, which coefficient is

*<sup>H</sup>*2*<sup>O</sup>* <sup>¼</sup> *dpCε<sup>C</sup>* 3*τ<sup>C</sup>*

are a function of both parameters, as displayed by Eq. (18) [20]:

In the porous media of each electrode, and because of the presence of small pore

s

**Table 3** gathers grain diameters (*dga=<sup>c</sup>*) and tortuosity (*τ<sup>a</sup>=<sup>c</sup>*) values. Pore sizes

*dpa=<sup>c</sup>* <sup>¼</sup> <sup>2</sup>*ε<sup>a</sup>=cdga=<sup>c</sup>*

Usually, the dusty gas model considers Maxwell and Knudsen diffusion to describe the gas flow through a porous media. It was possible to suggest an

approach equivalent to Fick's law (Eq. (13)) by considering an effective equivalent diffusion coefficient [24], illustrated by Eqs. (19) and (20), with *β<sup>c</sup>* being the Fick

3 1 � *ε<sup>a</sup>=<sup>c</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8*RT <sup>π</sup>MH*2*<sup>O</sup>* � <sup>10</sup><sup>3</sup>

¼ Γ*<sup>k</sup>* (13)

<sup>4</sup>*<sup>F</sup>* (14)

<sup>2</sup>*<sup>F</sup>* (15)

*<sup>τ</sup><sup>C</sup>* (16)

� � (18)

(17)

<sup>∇</sup>*:* �*Deff*

microstructure of the electrode according to Eq. (16) [24]:

*D<sup>K</sup>*

*Deff*

This study objective has been to investigate the influence of the electrode microstructure considering a realistic geometry of the SRU. **Figure 1** displays the geometry that forms the basis of the model. For simplicity's sake, a two-dimensional model of the SRU was used according to its cross section. The cell dimensions are gathered in **Table 4**. A mapped mesh has been used to obtain a reasonable number of degrees of freedom allowing calculation convergence within an acceptable computation time. Its parameters are also listed in **Table 4**. The studied geometry is a cross section of SRU which is perpendicular to the main direction of the gas flow. Considering that channels allow an ideal distribution of reactants on each electrode, it is possible to consider them as boundary conditions. This means that the produced hydrogen and oxygen are perfectly collected and exhausted from the SRU by the gas manifolds. Therefore, constant gas concentrations within the gas channels have been considered, in order to use this value as boundary conditions of mass balance.
