Comparative Analysis of Loss Mechanism Localization in a Semi-2D SOEC Single Cell Modell: Non-Equilibrium Thermodynamics versus Monocausal-Based Approach

This study aims to theoretically analyze the local entropy production rate in a SOEC single cell at T = 1123.15 K and p = 1 bar. Local entropy rates signify loss mechanisms, crucial for cell design and optimization. A semi-2D SOEC model based on non-equilibrium thermodynamics is developed, supplemented by monocausal correlations for direct comparison. The model is validated using KeraCell III data and grid independence analysis. Simulations of electric current density, temperature, heat flow, and local entropy production for various SOEC operating modes are presented. Coupled transport mechanisms significance is discussed, highlighting the pronounced impact of the Peltier effect on heat flux and temperature. The importance of the Peltier effect in SOECs compared to SOFCs is emphasised. The effects of the Seebeck effect on the potential distribution are superimposed by the dominant ohmic losses in the electrolyte. The localization of entropy production rates shows for exothermic operation that 66.6% of the total losses are due to the predominantly dominant irreversible ion transport in the electrolyte, while the entanglements in the reaction layers contribute 33% and GDLs less than 1%.

The demand for electrical energy has been steadily increasing in recent years, and along with it the importance of environmentally friendly energy converters.Electrochemical systems, such as fuel cells, are such alternative technologies for directly converting the chemical internal energy of a fuel into electrical energy.The solid oxide fuel cell (SOFC) is a promising technology due to its superior performance indicators.These include the temporary integration into the current energy supply system through the use of carboncontaining fuels, like natural gas, the variability of the purity of the hydrogen required and high efficiency due to the high operating temperatures.According to the current state of development, the electrical efficiency of SOFCs is up to 60%-65%. 1 Hydrogen as an energy carrier becomes inexhaustible due to its simple production and enables its use without direct CO 2 emissions environmentally friendly, also with regard to the Paris Climate Agreement.The production of "clean" hydrogen can be achieved, among other things, by solid oxide electrolysis (SOEC).With an electrical efficiency of up to 100% and a hydrogen production efficiency of about 90%, the reverse reaction of the SOFC offers an efficient way to produce hydrogen. 2he structure of a SOEC is similar to that of a SOFC, but the thermal management of the two systems must be differentiated.The reaction layers of the SOFC behave like heat sources or heat sinks due to the electrochemical reactions taking place there.The transport of electrons and ions also leads to many local heat sources through dissipation.This results in an inhomogeneous temperature distribution and thus in temperature gradients, which lead to thermal stresses within the cell.Aguiar et al. state in their study that the temperature gradient in a planar SOFC should not exceed 10 K/cm to avoid structural failure. 3Accurate knowledge about the temperature distribution is of great importance for successful thermal management. 47][8] The influence of changes in the channel design 9 and the flow mechanisms, 10 the integration of heat pipes 11 and internal endothermic chemical reactions 12 on the temperature field are investigated in a large number of studies.
In a SOEC, the temperature can be used to determine the course of the electrochemical reaction (exothermic, endothermic, thermoneutral) within in the cell via the temperature profile.Depending on the operating mode, individual thermal management is necessary.The relevance of the temperature field is shown in the work of 13 and. 14In electrolysis operation, the supply of thermal energy is useful as the electrical energy to be supplied is then reduced according to the molar free reaction enthalpy Δ R G m which decreases with increasing temperature.This thermal energy can be coupled into the process e.g. by renewable thermal energy.For such thermal transients, the temperature field in the oxide ceramic cell must be well known. 15Knowledge of the temperature field is also of great interest for the integration of chemical-technical processes for providing chemical energy carriers such as hydrogen, ammonia or methane through electrical energy (power-to-gas) in order to control these reactions. 16Thus the focus of this paper is the calculation of the temperature field in a SOEC.
The theory of non-equilibrium thermodynamics (NET) is a suitable approach to model these irreversible processes with dominated the temperature field.From the thermodynamic fundamental equation, the generalized fluxes J i can be described as the temporal change of extensive state variables and their conjugate forces X j as spatial gradients of intensive state variables. 17The product of flux and its conjugate force gives a contribution to the local entropy production rate σ: By applying Onsagers reciprocity relation, it is shown that the different transport mechanisms of the same tensorial order in a system are coupled with each other.Accordingly, a heat transport within an electrolysis cell cannot be exclusively caused by the corresponding force, i.e. by a local temperature gradient, but also by further forces in the system, such as a gradient in the electrical or in the material potential.The following linearised phenomenological equations between the generalised flows J i and the generalised forces X j result in: 18 Here L ij are phenomenological coefficients.The thermodynamic consideration of non-equilibrium ultimately leads to the quantification of the local volume-specific entropy production rate, see Eq. 1.
z E-mail: gedik@ift.uni-hannover.deECS Advances, 2024 3 014501 For the local processes within a SOEC near thermodynamic equilibrium, this function can thus be minimised for different operating conditions.The corresponding variables of this optimization function are defined by the boundary conditions of the system and the phenomenological coefficients.
0][21][22][23][24] As mentioned before, the local transport processes for a SOFC have already been determined by a 1D model based on the NET. 5 It is shown that under the prevailing temperature gradients, the Soret effect (coupling the mass flow and a temperature gradient) can be neglected.However, the temperature and heat flux curves clearly illustrate the importance of the Peltier effect, especially in the electrolyte.At T= 1123.15K and j = 8000A m −2 , 64.4% of the total losses occur in the electrolyte.Furthermore, it is shown that at the gas difusion layer (GDL) of the cathode the heat flux is directed from a lower to a higher temperature.In view of these results, the coupling effects cannot be neglected for a reliable description of the temperature field and transport mechanisms in electrochemical systems and it is of great importance to adopt an integrated approach such as the theory of NET.
As there are still no studies on the coupled transport processes in a SOEC in liteature and, above all, it is possible to operate the SOEC in three different modes, knowledge of the temperature and heat distribution is of great importance for successful thermal management.For this reason, a semi-2D SOEC model based on the NET is presented below, in which the effects of the coupled approach are compared with the standard monocausal approach known from literature, 12,25,26 which postulates the dependence of a flux exclusively on a single deterministic force.Especially in the area of exothermic, endothermic and thermoneutral operation, the differences are of great importance.The individual loss mechanisms are then analysed through the entropy production rates according to Eq. 1.

Materials and Methods
The thermodynamic system-solid oxide electrolysis cell (SOEC).-The1D SOFC model from 5 is used as the initial model.Here, the SOEC is divided into nine different layers in the ydirection, see Fig. 1.The membrane electrode assembly comprises a homogeneous bulk phase, which is separated to the adjacent bulk phases by two surfaces.These 2-dimensional surfaces represent the reaction layer.Boundary to these two surfaces are the anode and the cathode GDL.Via the flow field plates the reactants are homogeneously distributed to the anode and cathode GDL.The surfaces also form interfaces between the three homogeneous bulk phases.The five homogeneous bulk phases, i.e. flow channel (fc) at the cathode, cathode GDL (c).electrolyte membrane (e), anode GDL (a) and flow channel at the anode each have a thickness of Δy i .The system is bounded by the interconnectors (ic) on both sides.To reach a higher degree of realization of the existing model, the gas flows in a flow channel are considered in x-direction.y is defined as the spatial coordinate perpendicular to the active surface, while x is defined as the spatial coordinate in the flow channel direction of the SOEC.
The hydrogen evolution reaction (HER) takes place on the surface of the cathode catalyst layer and at the surface of the anode catalyst layer, the negative oxygen ions react in the oxygen evolution reaction (OER) to give oxygen: Atomic vacancies in the electrolyte allow for the diffusion of the negative oxygen ions.In this study, 8 mol% yttria-stabilized zirconia (8YSZ) is investigated as the electrolyte material.The relevant molar fluxes J i are defined according to their flow direction and the heat fluxes J q are defined positively in the direction of the y-axis.
The direction of the electric current density j is assumed to be the technical direction of the current against the electron flow.
Throughout the simulations, an equimolar mixture of hydrogen and water is supplied on the anode side and air on the cathode side.Ideal gas mixtures are assumed.The thermodynamic reference state point is set at T = 1123.15K and p = 1 bar.
Theory.-The generally valid equation of the 1D SOFC model from 5 are used for modelling the SOEC.The equations are defined in such a way that both modellings differ by the sign of the electric current density j.This approach is derived from simulations of reversible solid oxide cells (rSOC). 27For this reason, only the new equations relevant for the 2D modelling are listed below.The equation for an SOFC are defined for the completion.
Equation for the flow channels.-Tointegrate the flow channels into the simulation, different heat transfer mechanisms between the cell, the flowing gases and the interconnectors were implemented using the model of Laurencin et al. 25 Convective heat transfer occurs between the gases and the GDLs, as well as within the interconnectors.Furthermore, there is radiative heat exchange between the interconnectors and GDLs.The results of Sánchez et al. 28 and Damm and Fedorov 29 show that radiation between gases and solids can be neglected.
The resulting heat flows are emitted at the edges of the GDL of the cathode and anode via convection and radiation.To elucidate this association, the energy conservation equation for the GDLs at their initial and terminal discretization points are augmented with an additional term accounting for convection as well as a term accounting for radiation: with the first, second, penultimate and last y-position in sandwich direction y 0 , y 1 , y −2 and y −1 and the heat flux densities of convection J q,conv i and the heat flux densities of radiation J q,rad i .The heat flux density of the radiation J q,rad i is approximated by the expression of the radiation exchange of two infinitely parallel surfaces: with the surface temperatures of the GDLs T T y with the heat transfer coefficient h g and the gas temperatures T g for g = fuel, air.The convection term is calculated on the anode and cathode side for convection from the GDL to the gas J q,conv i,GDL and for convection from the interconnectors to the gas J q,conv i,ic .T s i describes the surface temperature of the GDL or the interconnector.The heat transfer coefficients h g are calculated via the Nusselt number Nu, the For the hydraulic diameter applies: 30 with the flowed-through area of the gas channel A d and the gaswetted circumference of the gas channel P d .Since for the flow of the gases through the channels Re < 2300 applies and the thermal gradients can be assumed to be very low, the following Nu = 3.68 is assumed. 30,31The thermal conductivities of the gas mixtures λ g are calculated using the approach of Todd and Young: 32 The Mason and Saxena modification A kj is defined as: 32 with the dynamic viscosity η.The dynamic viscosities and thermal conductivities of the components are calculated employing power series methodologies, following the exposition provided by. 32quations 5-7 are iteratively solved at each discrete point along the x-axis.Furthermore, the heat transfer coefficients h g , thermal conductivities λ g , and dynamic viscosities η k are evaluated for each specific x-coordinate, given their temperature-dependent nature and partial dependence on substance quantities.The changes in the molar flows of the components dn k along the flow channels are determined by applying the mass balance equation for each individual component: with the contact area dS between gas and GDL.This is defined as dS = dx • b, with the width of the gas channel b.In addition to the energy balances of the GDLs given above, the energy balance is established for the flowing gas mixtures along the flow channel: Modelling of the gas diffusion layers (GDL).-Thefocus of the previous study in Ref. 5 is the investigation of mass transport in GDLs.Following these results, the NET approach for modelling the GDLs is chosen here.Assuming ideal gases, the following applies: with the Ficks diffusion coefficient D k eff of component k.This can be calculated using the Bosanquet formula: Model of the solid oxide electrolysis cell (SOEC) including the electric current density j, all heat fluxes J q and molar fluxes J i .
ECS Advances, 2024 3 014501 with the molar mass M k and the dimensionless diffusion volume v k of component k as well as the porosity ϵ i and the tortuosity τ i .The effective Knudsen diffusion coefficients are defined as: 33 with the pore diameter d p i .The thermal diffusion coefficient D kj T is expressed by the thermal diffusion factor kj T α : The thermal diffusion factor is determined with the aid of the kinetic theory of gases, which establishes a relationship between the macroscopic transport coefficients and the intermolecular interactions of ideal gases.The calculation approaches for the determination of the thermal diffusion factor are comprehensively documented in Ref. 5.
Modelling of the reaction layers.-Inthis model, reactions are assumed to take place on the surfaces (s) of the electrodes only.A homogeneous temperature profile is additionally assumed at the electrode surfaces.However, jumps in the y-direction occur in the course of the electrical potential and the heat flux density.Two different approaches are considered.
Approach via the activation and concentration overvoltages.-Thecalculation equations for the reaction layers can be found in Ref. 5. The exchange current density describes the current density at the same reaction rates of the forward and reverse reaction.Therefore, the same factors can be used for SOFC and SOEC modelling. 34It is important to acknowledge that this represents a simplification, as the exchange current density can be subject to modifications due to irreversible processes, potentially resulting in distinct exchange current densities as indicated by Fukumoto et al. 35 However, if different electrode materials are used for SOFC and SOEC operation, other activation energies and pre-exponential factors must be taken into account to calculate the exchange current density.Fukumoto et al. 35 have determined the exchange current densities of different cathode and anode materials for solid oxide cells in SOEC and SOFC operation.
Approach via NET.-The modelling of the reaction layers is carried out from the approach via the definition of the chemical affinity A: with ξ as the reaction evolution number and v as the reaction velocity, which characterises the flow and can also be assumed as in the steady state for SOFC operation.The chemical potentials of the oxygen ion O 2 μ − and the electron e μ − are combined to the chemical potential of an oxygen atom μ O according to the approach of Kjelstrup et al. 36 The effective potential difference dφ eff /dy is an electric potential which is present over an infinitesimally small interval dy normal to the surface.It follows by the application of the coupled forces exemplary for the anode for SOFC operation: With the definition for the Peltier coefficient for a SOFC 17 the conditions dT = 0, dμ i,T = 0 and assuming that no electroosmosis takes place, the following equation for the Peltier coefficient applies: with S m,k as the transported molar entropy of all n − 1 components.
The n-th component is chosen as the reference of frame.In this case, this is the positive ion lattice of the electrolyte as an unmoved reference wall, resulting in the following equation for an SOFC: The molar entropy S m,k of the component k at the temperature T can be taken from. 5The conductivities k s λ and resistances r s for this infinite thin volume are given by means of a scaling factor k and the thickness of the reaction layer Δδ: The heat flows J q a,e and J q e,c are calculated via the entropy balance so that the temperatures at the reaction layer can be determined from the equations shown.The entropy production rates result from the activation overvoltages at the respective electrode.Butler-Volmer kinetics is used to calculate the activation overvoltages.For more details see. 5 The following applies for the dissipation function:  Simulation.-ThePython GEKKO software is used to perform these necessary calculations.This is based on the open-source library GEKKO, which accesses APMonitor that compiles the model into a Fortran application.The model equations derived in the previous section are used to calculate 1D profiles of partial pressures, temperature, heat flow, electric potential and local entropy production rates.The equations for the flow channels add a semi-2D direction to the existing model.As experimental investigations of NET influences on SOFC/SOEC modelling and on the appropriate phenomenological coefficients are running at the Institute of Thermodynamics in Hannover, the simulation of cell operation in a hot temperature test apparatus is considered.As already shown in Ref. 5 the NET influences are best visible when the temperature gradient across the cell is low.Therefore it is assumed in the simulation that the interconnectors are at a constant operating temperature (T T ic i = ) and also the temperatures of the inflowing gases are heated accordingly to the operating temperature (T These temperatures are given to the simulation as boundary conditions.The heat conduction in the direction of the flow channel can be assumed to be negligibly small.This is typically not true in reality, but it is a simplifying assumption for discussion.Given the presumption of the electric current density being insignificantly small in comparison to the direction perpendicular to the flow channel, the Peltier effect is hypothesized to possess no discernible impact in the direction of the flow channel.Because the Seebeck and Soret effects are influenced by temperature, they are disregarded in the direction of the flow channel as well.In the context of comprehensive 2D analysis, employing modern CFD programs with the finite volume method is suggested due to their access to greater computing capabilities and utilization of superior algorithms compared to the solver employed in this study.The operating conditions of the cell are fixed, including the operating temperature T, the operating pressure p, the composition of the input gases x k in and the cell voltage Δφ.The cell voltage results in a local electrical current density distribution across the cell in the x-direction, as the cell voltage must be constant across the individual computational layers.Consequently, the cell voltage serves as a determinant for the type of model being considered, whether a SOFC or a SOEC, contingent upon whether it surpasses (SOEC) or falls short (SOFC) of the OCV.At y = 0, the cells potential is set to 0V, while at y = Δy c + Δy e + Δy a , the potential must align with the designated voltage value.As an alternative, it is possible to define the electric current density for the initial computational layer in the x-direction.
Given the relevance of discerning distinctions from the monocausal approach when examining cells within the context of NET, the computer program was enhanced to facilitate the setting of all phenomenological coefficients in the transport equations to zero.This enables an exclusive depiction of the cell using monocausal relationships.These variations are of interest due to the limited consideration of coupled transport processes in the NET literature.The absence of literature substantiating the insignificance of these coupled processes was noted.
General parameters.-Theparameters for a specific example mode are summarised in Table I.The volume flows of the inflowing gases are V 0.5 k in = NL/min, unless otherwise stated.Given that the simulation is confined to an individual gas duct from among 17 gas ducts of equivalent dimensions per interconnector, the volume flow is partitioned by a factor of 17.This rationale rests upon the assumption of homogenous gas distribution across the various channels.All other parameters that are not given in the previous chapters have already been used in the 1D model and are documented extensively in Ref. 5.
Operating modes.-Forevaluation, the results of the modes specified in Table II are considered in SOEC operation mode.Mode (a), (c) and (d) simulate the cell under the assumption that both the interconnectors and the gases correspond exactly to the test environmental oven temperature.Considering the dependence of the Peltier effect on electric current density and the dependence of Fouriers heat conduction on temperature gradients, it is hypothesized that the most significant influence of the Peltier effect will be observed on both the fuel and air sides when they are held at the same temperature.The KeraCell III is being investigated experimentally at the Institute of Thermodynamics in Hannover.Cell operation in a furnace at 1123.15 K is considered.Due to heat losses occurring on the air side of the furnace, temperature measurements were conducted on the gas mixtures and interconnectors on both the fuel and air sides as part of this study.These measurements revealed an approximate temperature difference of 2 K at the interconnectors and between the gases on the fuel and air sides.This difference is considered in mode b).These simulations are interesting in that the Soret effect and the Seebeck effect are dependent on the temperature gradients.

Results and Discussion
Grid-independence study and validation.-Anetwork independence study ensures that the results of the simulation only change slightly when the computational network is refined.In addition, it is checked whether there is a network convergence.Based on the The model is validated using a U, j-characteristic curve of the KeraCell III SOFC published by the manufacturer and a selfrecorded U, j-characteristic curve.A SOFC/SOEC test stand (Evaluator C1000-HT) from HORIBA FuelCon is available for this purpose, which is suitable for analysing individual cells.Additional details pertaining to the experimental configuration and the measurement methodology can be found in Ref. 38.To ensure a direct comparison of the U, j-characteristic curves, the same conditions were selected in the experiment as for Kerafol, cf. 37igure 2 shows the own experimentally determined (exp) and the simulated (sim) U, j-characteristic curves for T = 1123 ± 1 K as well as values from Kerafol GmbH (Kerafol) for SOEC and SOFC operation.For visualisation reasons, all data have been fitted.
The open circuit voltage (OCV) determined from the simulation with U OCV sim = 0.925 V is 0.0973% lower than the OCV of Kerafol with U OCV Kerafol = 0.926 V.The two characteristic curves diverge with increasing current densities in both process directions and lead to a cell voltage deviation of 1.65% at a current density of j = 5000 A/m 2 and 0.923% at a current density of j = − 5000 A/m 2 .The areaspecific resistance (ASR) is 0.3808 Ωcm 2 for the simulated curve and 0.3584 Ωcm 2 for the Kerafol curve.The ASR determined experimentally by Kerafol deviates by ∼5.882 % from the simulated ASR.It can therefore be deduced that the irreversible ion transport is assumed to be too high in the model.In addition, the activation losses in the electrodes can also contribute to the ASR.From this it can be concluded that the calculated overvoltages are also overestimated.The parameters for calculating the exchange current density for the air electrode consisting of LSCF are currently still taken from the literature.To achieve a higher degree of realism of the model, the values can be determined empirically by means of impedance spectroscopy and the creation of an Arrhenius plot for the exchange current densities, which then enables direct adaptation to the cell. 39he OCV of the simulation is 0.703% greater than the experimentally measured value of U OCV exp = 0.919 V.A divergent behavior with increasing current density can also be observed between these two characteristic curves.The deviation increases to a value of 2.49% at j = 2749 A/m 2 in SOFC mode.In SOEC mode, the relative deviation increases to a maximum of 0.359% at j = −2000 A/m 2 .For the experimentally determined U, j-characteristic curve, this results in an ASR of 0.4315 Ωcm 2 .The relative deviation of 20.41% to the ASR value of the simulation can be identified for several possible reasons.On the one hand, slight gas leaks in the experimental setup could lead to changes in the gas composition.On the other hand, the deviation could be due to activation and concentration overvoltages, as these parameters originate from the literature and were integrated into the model.The impact of temperature deviation during the recording of the characteristic curve at 1123 K is considered low.This is due to the furnace heating being regulated to a thermocouple in close proximity to the cell for each measuring point, and the cell voltage value being measured after a 15 min hold time.However, there are heat losses on the air side in the furnace, which leads to a cooling of the cell by about 2 K.This leads to a slight increase in irreversibilities due to ion transport in the electrolyte.
As the deviations according to 40 are within an acceptable range, it is concluded that the simulation model can successfully reproduce the U, j characteristic curve.
Analysis of the reaction layers in SOFC mode.-In the following, the calculation method is analysed first using the NET approach in SOFC operation and compared with the electrode losses consisting of the activation overvoltages and the concentration losses from. 5Figure 3 shows the potential curves for j = 4000 A/m 2 for three different modes.Modes a) and b) show the potential curves with calculation according to the NET approach using the scaling factors k = 10 −5 and k = 10 −3 (see Eqs. 29-30) and mode c) the potential curve with previously implemented calculation method.The primary selection of the scaling factors can be attributed to Kjelstrup et al. 36 as well as the approach.
It is evident that the selection of scaling factor significantly impacts the potential curve.When utilizing k = 10 −5 , the effective electrode loss on the anode side is Δφ s,a,eff = 0.138 V, while the actual overpotential is η a = 0.033 V.This suggests that the losses are relatively high.Conversely, a scaling factor of k = 10 −3 results in a   ECS Advances, 2024 3 014501 significantly lower anode-side electrode loss with Δφ s,a,eff = 0.001 V. From this analysis, it can be deduced that there is a certain scaling factor at which analogue losses result for both approaches.In order to determine the optimum value, the scaling factor was first introduced into the model as a variable.This determination was made using the first law of thermodynamics as an additional equation.The resulting anode-side overvoltages (k -1.LoT) are shown in Fig. 4. In addition, the anode-side overvoltages are calculated using the scaling factors k = 10 −3 , 10 −4 and 10 −5 and using the previously implemented calculation method. 5From this, two facts can be established regarding the NET approach.
If the energy balances are observed, the model can no longer be solved for j < 1500 A/m 2 with the scaling factor as a variable.As already shown in Fig. 3, the choice of scaling factor as a function of the current density has a major influence on the effective electrode loss on the anode side.Furthermore, no scaling factor can be used for the model independently of the current density without underestimating or overestimating the electrode losses.In compliance with the energy balances, different scaling ratios result for the effective electrode losses on the anode side (k -1.LoT) depending on the current density, e.g.k(j = 1500 A/m 2 ) = 3.5 • 10 −4 and k(j = 4000 A/m 2 ) = 6.87 • 10 −5 .For these reasons, the following simulation uses the previously implemented approach. 5ectrical current density j in x-direction for all modes.-Figure 5 illustrates the computed electric current density j distributions of both calculation approaches for a voltage of U = 1.1 V (a) and (b), 1.2882V (c) and 1.4 V (d).In both calculation approaches, there is a reduction in the magnitude of the electric current density along the flow channel direction.This phenomenon is attributed to the increasing hydrogen content within the fuel gas.Consequently, the OCV rises across the flow channel direction, resulting in a diminished electric current density magnitude.This effect is due to the constancy of the cell voltage across all computational layers in the flow channel direction. 41The difference between the two calculation approaches considered is small.The maximum difference of −0.09% for mode (d) is very negligible.It can be stated that the coupled transport processes in the sense of NET have a small influence on the electric current density and thus on the performance of the cell.
Simulation mode (a): endotherm, U = 1.1 V, T fuel = T air .-Figure6 depicts the temperature distributions as functions of the x and y coordinates.It should be noted that the boundary temperatures are maintained constant in the x-direction.The air reaction layer is the coldest due to the endothermic half-cell reaction on the air side of the SOEC, while the exothermic reaction on the fuel side makes it the hottest layer.In both representations, no gradients are visible across the flow channel, as the temperatures of the interconnectors are kept constant and the electrical current density along the flow channel changes only slightly.The temperature distributions of the NET and the monocausal approach differ greatly.In the monocausal approach, the temperature of the GDL surface facing the air side is inferior to that of the GDL surface facing the fuel side.Conversely, under the NET approach, it is evident that the surface of the air-side GDL exhibits higher temperature.This phenomenon can be attributed to the impact of the Peltier effect.Heat is conveyed to the air side through the charge carriers, resulting in cooling on the fuel side and warming on the air side.This phenomenon significantly influences the temperature gradients present at the reaction layers.In this regard, the Peltier effect becomes significant, giving rise to  ECS Advances, 2024 3 014501 amplified temperature gradients between the reaction layer facing the air side and the GDL, predominantly due to the elevated Peltier coefficient observed within the GDL facing the air side.
The influence of the Peltier effect becomes particularly clear when considering the heat flux density distribution, cf Fig. 7.In accordance with the heat flux density magnitudes depicted in both figures, it is apparent that energy is generated during the half-cell reaction on the fuel side, whereas energy is absorbed during the corresponding half-cell reaction on the air side.The overall reaction in this considered mode is endothermic, since the summed heat flux density at both GDL surfaces is positive.The transport of oxygen ions from the fuel reaction layer to the air reaction layer releases heat through the irreversible ion transport.The divergence between the two approaches is discernible through the distinct heat flux densities.As a result of the heat transfer driven by the Peltier effect toward the air side, the heat flux density here within the NET approach exceeds that of the monocausal approach by approximately 1400 W/m 2 .This is also due to the high Peltier coefficient with π a = -0.74J/C.As a result, the Peltier effect has a stronger influence than the Fourier heat conduction.This phenomenon is evident in both the fuel-side and air-side GDLs.Within both GDLs, heat is directed toward the air side, even though the temperature rises with increasing y-direction.When examining the monocausal processes, the presence of Fouriers heat conduction is apparent.In the monocausal perspective, the necessary energy from the overall reaction is taken in at the air-side GDLs surface.In contrast, the NET perspective reveals a substantial heat flux entering the cell at the fuel-side GDLs surface.This is altered at both reaction layers by the energy released or absorbed during the half-cell reactions, resulting in a positive heat flux being dissipated at the surface of the air-side GDL.Accordingly, when considering the NET approach, the necessary energy from the airside half-cell reaction enters the cell through the fuel-side GDL.This association bears significant implications for the cells thermal management, given the necessity of providing heat to the cell on the fuel side, despite its consumption in the air-side half-cell reaction.
In this operating mode the temperature of the electrolyte here remains lower than the temperature of the interconnector and the gas, since the heat generation of the cell is not yet in equilibrium with the energy demand of the overall endothermic reaction.
Simulation mode (b): endotherm, U = 1.1 V, T fuel − T air = 2 K .-Despite the imposed temperature gradient, in this mode the differences between the two calculation approaches are minimal, as can be seen in Fig. 8.In both distributions, the temperatures decrease from the fuel side to the air side due to the forced temperature gradient.Since the fuel half-cell reaction is exothermic, the temperature jump is larger than that at the air-side reaction layer.The discrepancies between the NET approach and the monocausal approach are marginal due to the increased heat flux density resulting from Fourier heat conduction through the boundary conditions, which mitigates the effects of the Peltier effect.Nevertheless, the transport of charge carriers creates a heat flux density toward the air side.As a result, the temperature of the airside GDL surface increases by approximately 0.14 K, while the temperature of the fuel-side GDL surface decreases by approximately 0.04 K.
Figure 9 presents the distribution of the Seebeck effect and ohmic losses across the cell.The influence of the Seebeck effect is consistently negative throughout the cell.This shows a favorable  influence of the Seebeck effect on cell performance, as it results in a reduction of voltage across all layers.Furthermore, it can be observed that the Seebeck effect displays similarity between the fuel-side GDL and the electrolyte, even though the Peltier coefficients follow an inverse trend.This suggests that the dominance of temperature gradients outweighs the influence of Peltier coefficients.A comparison between the Seebeck effect and ohmic losses reveals the heightened influence of the former in the fuel-side GDL, which exceed the impact of ohmic losses and leading to opposing effects.Consequently, the Seebeck effect counteracts the ohmic losses slightly due to the good electrical conductivity of the GDL, resulting in diminished ohmic losses.On a comprehensive scale, the pronounced prominence of irreversible ion transport in the electrolyte is evident, attributed to its inherently low ionic conductivity.This insight implies that the Seebeck effect has a marginal role in governing the operation of a SOEC.Correspondingly, the minimal influence of thermal diffusion or the Soret effect is corroborated by findings in Ref. 5.
Simulation mode (c): thermoneutral, U = 1.2882V,T fuel = T air .-Theendothermic and exothermic nature of the half-cell reactions is also apparent in the temperature profiles for this operational mode, as shown in Fig. 10.Overall, an increase in cell temperature is deducible due to amplified losses accompanying a higher electric current density.In comparison to mode (a), the temperature disparity between the two approaches has intensified, attributable to the Peltier effect rendering heat transport proportional to the electric current density.
Due to the higher electric current density, the heat released from the irreversible charge transport within the electrolyte becomes significant.Amplified by the intensified Peltier effect, the heat flux density calculated via the NET approach surpasses that of the monocausal relationships by approximately 4000 W/m 2 , cf.Fig. 11.This operating state considerably simplifies the cells thermal management, as the net heat consumption of the cell approximates 0 W. In the simulation, the disparity between the summed heat flux densities at the two GDL surfaces is 6.1 W/m 2 for the NET calculation and 6.3 W/m 2 for the monocausal calculation.Consequently, it can be inferred that the influence of coupled transport processes under NET terms on thermoneutral stress remains minimal for this operational mode.
Simulation mode (d): exotherm, U = 1.4 V, T fuel = T air .-Asapparent from the contrasts between mode (a) and (c), a similar trend emerges here: with a rising operating voltage, the cells temperature elevates owing to augmented losses accompanied by a higher electric current density, as shown in Fig. 12.In addition, this also leads to an increase in the maximum and minimum temperatures at the reaction layers.The disparity between the two approaches has also grown, reflecting the proportional relationship between the heat transported by the Peltier effect and the electric current density.The exothermic operation results in cell temperatures surpassing the operating temperature.
Similar to the previous instances, the influence of the Peltier effect is also apparent in this scenario, as shown in Fig. 13.In this operational mode, the cells heat generation, primarily attributed to the irreversible ion transport within the electrolyte, surpasses the energy demand of the overall endothermic reaction.Consequently, effective thermal management is imperative for proficient heat dissipation in exothermic operation, aiming to prevent the emergence of hotspots and potential cell damage.Flow channels in SOEC mode for the simulation mode (a).-Figure 14 depicts the temperature profiles of the GDL surfaces and gas mixtures along the length of the flow channel for simulation mode (a).The temperature gradients are minor due to the presumed constant interconnector temperatures.Given the overall endothermic nature of the SOEC reaction and the substantial heat flux transported to the air side as a consequence of the Peltier effect, the fuel-side GDL exhibits lower temperatures compared to the interconnectors.This results in a warming of the air-side GDL.With the cells energy requirement for the overall reaction diminishing alongside decreasing electric current density, the temperature of the fuel-side GDL rises.The emergence of temperature gradients is attributed to the localized electric current density.
If the boundary conditions are altered so that only the gas temperatures at the inlet are set to T g,in = 1123.15K, the resulting configuration is depicted in Fig. 15.To achieve this, the heat flux densities in the energy balance equations are expanded to account for the flow channel direction.In this particular mode, the average electric current density measures −1869.82A/m 2 at an operating voltage of 1 V. Since this voltage is lower than the thermoneutral voltage, the cell experiences cooling due to the fully endothermic reaction occurring in the flow channel.The highest temperature gradient across the cell reaches 5 K/cm.This gradient decreases as the local electric current density decreases within the flow channel.Regarding the y-direction, the temperature variations are relatively minor compared to those in the flow channel direction.Overall, the cells average cooling effect amounts to around 12 K.This result aligns with the results made by Laurencin et al. at an operating voltage of 1 V. 25 However, a direct comparison is hindered by various factors, including the influence of the Peltier effect and the use of different cells.
Entropy production rate for mode (a).-With the use of the following equation resulting from NET, the individual entropy production rates can be localised: The results are shown in Fig. 16 for mode a).For better comparability of the different entropy production rates and the profiles across the layers and along the flow channel direction, the values are related to the value at the left edge of the layer.The reference values are given in Table III.The reference values indicate that the irreversible ion transport within the electrolyte contributes significantly to a notable local entropy production rate of 3123.92W m −3 K. Within the air-side GDL, this entropy production rate related to charge transport σ̇ϕ exceeds that of the fuel-side GDL.This discrepancy arises from two factors: the air-side GDLs specific electrical resistance is twice as high, and it is influenced by the Seebeck effects impact on potential distribution.In terms of the heat flux density, local entropy production rates are negative within the GDLs and positive within the electrolyte.The Peltier effect is responsible for the negative production rates observed in the GDLs.Within the GDLs, mass transfer contributes most significantly to the overall local entropy production rate.Similar magnitudes are noted between the anode and cathode.The overall production rate is, of course, positive as required by the second law of thermodynamics.
Once more, Fig. 16 demonstrates the decreasing trend of all local entropy production rates along the flow channel direction.This phenomenon arises from the declining magnitude of the electric current density in the x-direction.Since mass current densities are directly proportional to the electric current density, the local entropy production rates also decrease due to mass transport k σ̇.The heat flux densities are intimately connected to electric current densities through the energy balance equation, resulting in similar behavior.Along the y-direction, differences are mainly observed in the entropy production rate attributed to heat flux q σ̇.This discrepancy is a consequence of heat generation due to irreversible ion transport.In the direction of ion flow, the heat flux density increases due to irreversibilities, which concurrently elevates the temperature    gradient.This mechanism amplifies both the thermodynamic flux and the corresponding force.
When examining the integrated losses across both the y-direction and the x-direction, it is apparent that roughly 66.6% of the overall losses occur within the electrolyte.The cumulative losses from the reaction layers constitute approximately 33.02%, with a marginal contribution of about 0.3% from the GDLs.Such a distribution aligns with the characteristic behavior of electrolyte-reinforced cells.The GDLs play a minor role owing to their exceptionally low specific electrical resistances and effective facilitation of mass transport through gas utilization.

Conclusions
The aim of this study is to determine theoretically the local entropy production rate of a SOEC single cell at an operating temperature of T = 1123.15K and an operating pressure of p = 1 bar.Local entropy  production rates can be identified as loss mechanisms, so that this knowledge is of great importance for cell development and the optimisation of operating strategies.Therefore, a semi-2D SOEC model is developed.The modelling is based on the NET approach.The phenomenological coefficients in the transport equations are described as a function of the empirical coefficients, as the true values are not yet known.In addition, the cell description is supplemented by the monocausal correlations to determine the differences to the NET approach.For validation, data for an U, j-characteristic curve of the single cell KeraCell III from the manufacturer Kerafol GmbH and additionally own-measured data for the same single cell are used.The calculated curves of temperature, heat flux, electric potential and local entropy production rates are discussed.
The assumption of constant interconnector temperatures leads to minimal temperature gradients across the flow channel.The impact of coupled transport processes on the current-voltage behavior and, consequently, the cells performance is minor.In contrast, the discernible influence of the Peltier effect on the heat flux density distribution and temperature profile of the cell is significant.In both GDLs, heat migrates toward the air side due to charge carrier transport, despite the air side being the coldest layer of the cell due to the endothermic halfcell reaction.This discrepancy results in a higher heat flux density in the NET approach compared to the monocausal approach.Notably, the Peltier effect is more pronounced in SOEC operation than in SOFC operation, as reported in Ref. 5. The NET model is therefore suitable for the detailed design of cells if the assessment of local temperature gradients is of interest, e.g. when testing the compatibility of different electrode materials/electrolyte materials.The influence of the Seebeck effect on the potential distribution of the cell is overshadowed by the dominant ohmic losses occurring in the electrolyte.Although the direct influence of the Seebeck effect can overtake the ohmic losses due to the GDLs low electrical resistance, the losses stemming from irreversible ion transport in the electrolyte remain dominant.
Below the thermoneutral voltage, the overall reaction exhibits an endothermic nature, resulting in the cells temperature being lower than that of the gases and interconnectors.Conversely, above the thermoneutral voltage, the irreversible ion transport in the electrolyte generates more heat than the overall reaction consumes, causing the cells temperature to exceed that of the gases and interconnectors.
Once again, the examination of local entropy production rates is carried out.The predominant sources of irreversibility are localized within the electrolyte, accounting for approximately 66.6%.The combined losses within the reaction layers contribute around 33%, with the GDLs contributing less than 1%.This distribution pattern is in line with the typical behavior observed in electrolyte-reinforced solid oxide cells.

ECS Advances, 2024 3
014501thermal conductivities of the gas mixtures λ g and the hydraulic diameter D h :25

Figure 2 .
Figure 2. U, j-characteristics of a SOEC and SOFC of the type KeraCell III with an electrolyte of 8YSZ at T = 1123 ± 1 K.

Figure 3 .
Figure 3.Comparison of the approaches of calculating the potential jumps at the reaction layers.

Figure 4 .
Figure 4. Effective electrode losses Δφ s,a,eff and overvoltage η a at the anode as a function of the current density j for different scaling factors k.

Figure 5 .
Figure 5. Difference of the electric current density j distribution in flow channel direction x between NET and monocausality for all modes.

Figure 6 .
Figure 6.Temperature T distribution for mode (a) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 7 .
Figure 7. Heat flux density J q distribution for mode (a) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 8 .
Figure 8. Temperature distribution for mode (b) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 9 .
Figure 9. Seebeck influence on the potential distribution for mode b).

Figure 10 .
Figure 10.Temperature T distribution for mode (c) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 11 .
Figure 11.Heat flux density J q distribution for mode (c) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 14 .
Figure 14.Gas and GDL surfaces temperature T along the flow channel for mode a).

Figure 12 .
Figure 12.Temperature T distribution for mode d) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 13 .
Figure 13.Heat flux density J q distribution for mode d) according to NET approach (top) and monocausal calculation approach (bottom).

Figure 16 .
Figure16.Distribution of the local entropy production rates σ̇in the fuel-side GDL (left), in the electrolyte (centre) and in the air-side GDL (right) for mode a).

Figure 15 .
Figure 15.Gas and GDL surfaces temperature T along the flow channel with variable interconnector temperatures.

Table II .
Simulated operating modes in SOEC operation.

Table III .
Reference values of local entropy production rates.