# A leakage model to design seals for solid oxide fuel and electrolyser cell stacks

### Abstract

Although planar solid oxide fuel cell and electrolyser technology is a key perspective for the next energy systems, it still suffers from a lack of efﬁcient tightness solutions due to the need for the use of a mix of brittle ceramics and stiff metallic materials at high tempera-tures. In order to design new well adapted metallic seals, an original computational model is proposed. It links the evolution of the local mechanical ﬁelds to the leakage rate. It re-mains purely macroscopic and does not require a ﬁne description of roughness. As the model is designed to deal with high temperature systems, it takes into account the strain rate dependence of the seal materials.

High temperature leakage tests are realized under load control conditions using a seal consisting of a 0.3 mm thick Fecralloy sheet lying between two elastic bearings made of Udimet 720 nickel alloy. One of the bearings presents a boss for which several geometries are used. Finite element calculations are performed to describe the mechanical state of the seal as a function of time. These results are post-processed using the proposed model to derive an estimation of the leakage rate. The model is tuned against the experimental results. Finally the validity of the model is checked by comparing its predictions to addi-tional experimental results in which seal geometry, loading history, gas pressure or gas composition are varied.

Copyright ª 2014, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights reserved.

### Introduction

Hydrogen has been identiﬁed as an energy carrier to store renewable and intermittent energies and as an alternative fuel [1]. But presently, it is mainly produced by natural gas reforming which involves CO_{2} emission. Hydrogen can also be produced in a cleaner manner by water electrolysis. When coupled to low cost heat sources, High Temperature Steam Electrolysis (HTSE) is one of the most promising ways for hydrogen mass production as it presents a high efﬁciency due to less electrical consumption compared to conventional low temperature water electrolysis [2].

Although HTSE presents more favourable thermodynamic conditions, high temperature components are expensive and may age too rapidly. Therefore, performance and durability, in association with cost-effective stack and system components are the key points to develop this technology [3—12].

Gas tightness largely drives the performances of Solid Oxide Electrolysis Cells (SOEC) and Solid Oxide Fuel Cells(SOFC) stacks which present leakage related problems [13,14]. Any leakage significantly decreases the efficiency and increases overheating risks. This gas tightness is difficult to achieve and maintain at high temperature (about 800° C) between metallic components (interconnects) and brittle ceramic materials (cells) because of the thermal expansion mismatch. The electrochemical cell and more particularly the thin electrolyte has to be tight enough to separate the two hydrogen and oxygen chambers. If this brittle membrane fails due to unsuited thermo-mechanical loading, the entire stack will fail as hydrogen will recombine with oxygen causing very high temperature spots in the stack. Several seals must be used in SOEC or SOFC stacks. The first one is required to separate both chambers; the second one has to tight the inner part of the stack from the outside; and the last one allows the correct gas circulation from the collectors to the cells. This first seal is often seen as being the more critical and is usually located between the cell electrolyte and the interconnect. For this seal, suitable loading must be used in order to preserve the electrochemical cells on the one hand, and to obtain the targeted tightness on the other hand.

Considerable efforts have been paid to develop suitable seals for SOEC/SOFC stacks in recent years [15—19]. The typical sealing solution relies on glass-based seals, despite a large number of drawbacks such as their low creep and thermal cycling resistance or the high temperature required to correctly form the glass. Metallic solutions have also been studied [18,20]. However without speciﬁc developments, they may be too stiff and too hard to be used on ceramic cells. Nevertheless, it has been shown elsewhere [19] that the electrolyte can sustain a relatively low seating load.

In order not to exceed this critical load, a speciﬁc seal shape and a speciﬁc loading pattern must be deﬁned. The aim of this study is to achieve this goal by developing a computational strategy in order to design speciﬁc metallic seals for high temperature SOEC/SOFC stacks. Note that existing models developed at predict tightness are limited to purely elastic or elasto-plastic materials (see e.g. Refs. [21—23]) and are not suited to deal with creeping materials at high temperature. A possible design for a stack element incorporating FeCrAl seals is schematically represented on Fig. 1. Two designs for the seals can be envisaged. In the ﬁrst case, the metallic seal is machined so as to obtained a sharp boss which is deformed by the other stack elements thus establishing tightness. In the second case, the metallic seal is ﬂat and is indented by bosses machined in the stack elements. For that purpose, tests are conducted at 800 °C using seals made of a FeCrAl sheet lying between two Udimet 720 bearings (Section 2). One of the bearings presents a machined boss which deforms the soft sheet thus establishing tightness. These tests are simulated using the ﬁnite element method (Section 3). Results are post-processed using the new model developed in Section 4. The model is tuned against experimental results. Finally the model is applied to describe a series of new tests to evaluate the ef-fect of loading history, seal geometry, gas pressure and gas composition and therefore to test the model robustness (Section 5).

### Experiments

**Materials and seal geometry**

The investigated seal consists of a thin metal sheet, com-pressed between two surfaces of superior hardness, one ﬂat and one with a rounded boss respectively referred to as lower and upper bearings in the following. Sealing is achieved due to the high temperature viscoplastic deformation of the thin sheet. A schematic view of the experimental setup is shown in Fig. 2.

The sheet consists of a FeCrAl alloy (OC404^{®}) composed of 72.8 wt% Fe, 22 wt% Cr, 5 wt% Al, 0.1 wt% yttria (Y_{2}O_{3}). Its thickness is equal to 0.3 mm. This material is supplied in the bright annealed state, characterized by a very ﬁne grain size (8 μm). An ageing treatment (900°C in air for 30 h and then cooling in the oven) is applied to the as received material in order to increase its grain size (25 μm) and to increase its creep resistance. The treatment also allows forming a protective alumina layer. A comprehensive description of the FeCrAl sheet material can be found in Ref. [24]. In particular the mechanical behaviour of the material was tested along the longitudinal and transverse directions showing no signiﬁca-tive anisotropy. Work hardening was not observed at high temperature. The roughness (R_{α}, measured by an Inﬁnite Focus Microscope) of the sheet after annealing is between 0.4 and 0.8 m.

The bearings are made of Udimet 720^{®}. This material is a creep resistant nickel base superalloy which is selected to make sure that the bearings remain elastic during testing up to 800 °C and that they can be reused in order to be able to perform several tests with a single setup. Both upper and lower bearings are machined by turning and present an average roughness equal to R_{a} = 0.8 μm. The lower bearing is ﬂat and the upper bearing contains a boss and an air inlet at its centre which is used to introduce the gas overpressure (see Fig. 2). Five different upper bearings were machined with various boss geometries which are plotted in Fig. 3. Bosses referred to as “large”, “medium 1”, “medium 2” and “sharp” have a diameter equal to 50 mm whereas boss referred to as “R10” has a radius equal to 10 mm. The sharp boss is designed to generate large local plastic strains and high stresses but a limited contact length. On the other hand, the large boss will generate a larger contact zone while keeping lower strains and stresses. Medium 1 and 2 bosses are intermediate geometries. The proﬁle of boss “R10” is close to that of boss medium 1.

**Testing device and experimental procedure**

The sealing tests are performed on a speciﬁc test machine, which allows studying the relationship between applied load, seal deformation and leak at high temperature [19].It includesa 50 kN electromechanical testing device associated to a furnace to perform experiments up to 1000 °C in air. A MTS load cell of 5 kN with a ball-and-socket joint is used to measure the load. The leak measurement device has been developed to allow a relative pressure control between 1 and 1000 mbar. Two different sensors are used. The measurement precision is equal to 0.1 mbar in the range 1e50 mbar and 1.5 mbar between 50 and 1000 mbar. To monitor the leak ﬂow, two mass ﬂow controllers from Brooks are used. Their range is 0.06e3 N ml/min and 2e100 Nml/min with an error equal to 0.7% of the measurement plus 0.2% of the full scale. Three K-type thermocouples ﬁxed on the bearing surfaces allow recording the temperature.

The experimental procedure is as follows: the heating up to 800 °C is realised at 5 °C/min without any loading. Then, the load is applied in 50 s to reach the targeted value deﬁned as the applied force divided by the seal length (from 3 N/mm to 10 N/mm for a seal length of 157 mm). Finally, the gas over-pressure is introduced while the mechanical test is realised under load control conditions. The leak ﬂow is recorded dur-ing the whole test. Otherwise stated, tests are performed using air with an overpressure equal to 200 mbar at 800 °C. As the FeCrAl material was heat treated at 900 °C (see above) it can be regarded as stable for the test duration (i.e. no oxide or grain growth). This study is limited to isothermal conditions (800 °C). It is however important to keep in mind that that the interconnects (Inconel) and the seal (FeCrAl) materials have different coefﬁcients of thermal expansion (CTE): 14. 10^{-6} K^{-1} for FeCrAl and (16. 10^{-6} K^{-1} for Inconel. In cases where tem-perature changes, the CTE mismatch generates shear stresses in the contact zone which can lead to loss of tightness. For each boss geometry, a mean of three tests is realized to ensure the reproducibility of the results. Economically, the produced hydrogen leakage must be as low as possible. However, a loss of 1% of the produced hydrogen in the stacks seems today reasonable. It corresponds to a leak rate of about 10^{-3} N ml/min/mm. This value determines the targeted order of magnitude to be achieved during tests as well as the goal to be reached in new sealing solutions to be designed.

**Results**

The leakage is expressed as the volume ﬂow in a reference state (0 U°C under 1 atm) per seal unit length. Fig. 4 presents the leak evolution as a function of time for different load levels (3, 5 and 10 N/mm) and the four boss geometries for which the seal diameter is 50 mm. Mean values and experimentally observed standard deviations are presented. The tests condi-tions allow leak measurements in the right targeted range from 10^{-4} to 10^{-2} N ml/min/mm. At 5 and 10 N/mm, the sharper the boss is, the lower the leak. This result shows that high strains and stresses play a more important role than contact length on the reduction of leakage. High deformations are expected to reduce the gap between both surfaces and therefore to reduce the leak ﬂow. For the lowest load (3 N/mm) the large boss presents a signiﬁcantly more important leak whereas the measurements for the other three geometries are close. This low load level is not sufﬁcient to discriminate the geometry effect. In all cases, a global tendency is observed: the leakage rate decreases as a function of time. A strong leak decrease is observed at the beginning of the test whereas it reaches a relatively stable stage after 10 h. These observations are very consistent with the fact that the contact surface and the local contact conditions evolve with time as a conse-quence of the rate dependent material behaviour (i.e. visco-plasticity) at high temperature.

### Simulation of the seal mechanical behaviour

**Simulation procedure**

Finite Element (FE) simulations of the various seals are per-formed using the Cast3M FE software (see http://www-cast3m.cea.fr). Axisymmetric ﬁnite strain eight-node ele-ments with reduced integration (4 Gauss points) are used. Contact between the FeCrAl sheet and the lower and upper bearings are taken into account. A uniform pressure is applied on the upper bearing so as to produce the applied load per unit length. This load is corrected to account for pressure applied in the space between the upper bearing and the FeCrAl sheet (end load effect). The mesh size was chosen small enough to ensure convergence in terms of local stress and strain ﬁelds and in terms of predicted leakage rate. The convergence cri-terion in FE simulations is based on the euclidean norm of the difference between external and internal forces. The critical value of the norm below which the solution is accepted was chosen small enough to assure convergence in terms of leakage rate. The example of mesh (medium 1 boss) is given in Fig. 5 together with the applied boundary conditions.

The bearing material is described as an elastic material (Young’s modulus: 165 MPa, Poisson’s ratio: 0.3) as stresses are small enough not to cause noticeable creep. This assumption was validated by checking that the boss proﬁles were not macroscopically deformed after several tests.

The sheet material creeps at 800°C. Its viscoplastic behaviour is described by a SellarseTegart law [24,25]. The equivalent plastic strain rate, p_ is then expressed as:

where ε_{0} = 0:51 s^{-1} is a reference strain rate, σ_{0} = 221 MPa a reference stress, R_{0} = 1 MPa the ﬂow stress and m = 4.58 the stress exponent [24]. σ_{eq} is the applied von Mises stress. The material is therefore assumed to be isotropic based on results published in Ref. [24]. The Young’s modulus is 82 GPa and the Poisson’s ratio is 0.3.

Finally the friction coefﬁcient between the FeCrAl sheet and both bearings needs to be determined. This is done by comparing the depth of the indentation created in the sheet by the hard boss to the results of the various simulations. An example of such a comparison is shown in Fig. 6 for the me-dium 2 boss (5 N/mm, 10 h). Considering the whole database, the best agreement is obtained for a friction coefﬁcient equal to 0.2 (thick line in Fig. 6).

**Simulation results and correlation with experiments**

can possibly be used to determine the leakage rate. The con-tact area is evaluated by ﬁrst determining the maximum value of the normal contact stress σ_{n} over the surface of the metal sheet. The contact area is then deﬁned as the zone where the normal contact stress is higher than 1/10 of the maximum value. Quantities such as the mean plastic strain or the mean contact stress over the contact length can then be evaluated.

Correlations between the experimental leakage rate, the contact area, the mean plastic strain and the mean contact stress are plotted in Fig. 7 for boss “medium 2” and load levels equal to 3, 5 and 10 N/mm. A very good correlation is obtained considering the contact area (Fig. 7(a)) and the mean plastic strain (Fig. 7(b)). On the other hand, there is no correlation between the mean contact stress and the leakage rate as shown in Fig. 7(c). Similar trends are obtained for the other boss geometries.

At this point, the contact length or the mean plastic strain appear as potential candidates to determine the leakage rate of the high temperature seals. Fig. 8(a) shows the experi-mental leakage rate as a function of the calculated contact length for a ﬁxed load (5 N/mm) and various boss geometries. In that case a correlation cannot be established between both quantities. A better correlation is obtained using the mean plastic strain (Fig. 8(b)). Similar trends are obtained for the other load levels.

### Model for the estimation of the leakage rate

In the previous section, it is shown that using quantities such a the contact length or the mean plastic strain does not allow forecasting the leakage rate for all tested boss geometries and load levels. In the following, a model is developed which al-lows post-processing macroscopic simulations to predict the leakage rate as a function of geometry, load and time. The model is a priori developed at the macroscopic level so that it can be easily implemented contrarily to more complex ap-proaches which integrate a microscopic description of the contact surfaces [21—23]. Note that these approaches are either dealing with elastic [22] or elasto-plastic materials [23] but not with rate dependent materials undergoing creep deformation.

**Local gap closure**

The model developed in the following assumes that both plastic deformation and high stresses are required to ensure gas tightness. Stresses normal to the contact cause the visco-plastic deformation of the soft metal sheet thus leading to the closure of asperities existing between both contacting sur-faces. The degree of closure at a given point along the contact area is measured by a variable referred to as D in the following. The variation rate of D is assumed to depend on the normal stress σ_{n} and the cumulated plastic strain of the soft metal, p, according to the following equation:

σ_{R} is a reference quantity which can be arbitrarily chosen to normalize the dependence of the F function on stress. λ, a and β are model parameters which need to be adjusted on the experimental results. The normal stress, σ_{n} is considered to be positive in the case of contact. It is otherwise equal to zero (no contact). In practice values of p and σ_{n} used for the calculation result from an extrapolation of the values at Gauss points to the nodes at the surface of the sheet. The process may result in slightly negative values for σ_{n} which are considered to be equal to zero. The equation selected to describe the degree of closure (eq. (2)) is directly inspired from models describing damage growth by cavitation at grain boundaries during creep [26,27]. In the context of this study, it is used to describe the process of void or asperity closure. It is important to outline that the model assumes that tightness is obtained due to the viscoplastic deformation of the sheet metal only and that elastic deformation of the contact area, which plays an important role at low temperature, can be neglected in the case of high temperature metallic seals. Note also that active plastic deformation (i.e. ṗ > 0) is assumed to be required to achieve tightness as the rate of variation of D (Ḋ) is propor-tional to the rate of p (ṗ).

**Darcy’s law with varying local permeability**

The gas ﬂow through the contact area is described using Darcy’s law for a purely axisymmetric radial ﬂow. The gas mass ﬂow, q_{m} is then given by:

where_{ p} is the gas density, A the cross-sectional area to ﬂow, * _{k}* the permeability of the medium, μ the gas viscosity and

*P*the pressure. The radial coordinate is referred to as

*r*. The gas is compressible and is assumed to obey the ideal gas law so that the gas density can be rewritten as:

where *M* is the molar mass, *R* the gas constant and* T* the ab-solute temperature. The cross-sectional area is given by:

where *h* is the local height between both surfaces. In the case of rough surfaces,* h* should be considered as an effective height. Using both previous equations (4) and (5), eq. (3) can be rewritten as:

In this equation M/μRT can be considered as constant under isothermal conditions whereas k(r) = h(r)k(r) will depend on the position (r) and on the local loading history. Equation (6) can then easily be integrated as:

where P_{i} and P_{e} are respectively the internal and external pressures. r_{i} and r_{e} are the internal and external radii of the contact zone. It is important to note that both r_{i }and r_{e} change with time as the contact area increases. Similarly the local effective height h will depend on the history: it tends to decrease with increasing plastic deformation of the sheet. To represent this effect the evolution of 1/*k* is assumed to follow eq. (2) so that 1/*k*(r) can be replaced by D(r). Finally the leakage rate is given by:

Values of M and μ for gases and gas mixtures used in this study are listed in Table 1. The previous equation expresses the mass ﬂux through the entire seal and the mass leakage rate per unit length is simply given by:

where r_{0} is the nominal seal radius (see Fig. 2). Finally, the volume ﬂux per unit length in the reference conditions (i.e. 0 °C under 1 atm) is then given by:

where V_{0} is the molar volume at 0 °C under 1 atm. In the following v0 is assumed to correspond to the molar volume of an ideal gas: v_{0} = 22.4 l/mol. Assuming that the gas leaks through a gap of constant height over the entire seal perim-eter, the Reynolds number is equal to R_{e} = Q_{m}/μ. In the case of air for a leakage rate equal to 0.001 ml/min/mm, ones gets: R_{e} ¼ 0.5$10^{-3} << 1. This indicates that using Darcy’s law is a valid assumption.

**Model implementation and model identiﬁcation**

The model is used to post-process results of the FE simula-tions (Section 3). Each loading step is post-processed so that the evolution of the leakage rate with time can be evaluated. The position of the contact area must ﬁrst be determined by ﬁnding r_{i} and r_{e} (see 3.2). At each node along the contact area D(r) is then computed by integrating eq. (2). The initial value of D is assumed to be equal to 0. This corresponds to a very large gap between both surfaces. Nodal values of σ_{n} and p are ob-tained by extrapolating values at Gauss points within the metal sheet. Finally the overall leakage rate is obtained after integrating U (eq. (8)) P_{i} and P_{e} are assumed to follow the prescribed experimental values.

The model to describe the leakage rate has three unknown parameters: α, β and λ. These parameters were tuned by minimizing (Simplex method) the quadratic difference be-tween the measured leakage rate and the one predicted by the model for all test cases presented in Section 2.3 (see Fig. 4). Optimized parameters are listed in Table 2.

**Results**

Normal stress (σ_{n}), plastic strain (p) and D proﬁles for various holding times are plotted in Fig. 9 for the medium 1 boss (load: 10 N/mm). These graphs show the evolution of the contact and deformed zones through time. It is shown that stress and plastic strain are maximum close to both ends of the contact area whereas they are minimum in the region where contact ﬁrst occurs (position ≈25 mm). The D proﬁle follows these trends as a result of the integration of eq. (2). This proﬁle shows that tightness is achieved due to the presence of two peaks close to both ends of the contact area. Similar results are consistently obtained for all load levels and boss geometries.

Experimental and simulated leakage rates are compared in Fig. 10 for all boss geometries and load levels. A good agree-ment is reached in all cases with the best results being ob-tained for high load levels (10 N/mm).

### Application of the model

In this section the model developed to forecast leakage rates is applied to various test cases which were not included in the experimental database used to ﬁt the model parameters. This allows to validate the predictive capabilities of the model.

**Effect of the seal loading history**

Two experiments are carried out to evaluate the effect of the seal loading history on the leakage rate. In the ﬁrst one, the load is decreased from 5 N/mm to 3 N/mm and ﬁnally to 1 N/mm. Three boss geometries are tested: large, medium 2 and sharp. In the second one, the load is increased from 3 N/mm to 5 N/mm and ﬁnally to 10 N/mm. Two boss geometries are tested: large and sharp. In both cases, each load is kept con-stant during 10 h. Results are shown in Fig. 11 together with the model predictions. In Fig. 10, loading steps corresponding to a constant load are separated by vertical dashed lines. The duration needed to vary the load is 10 s.

In the case of a decreasing load (Fig. 11(a)), tightness is almost stabilized after 10 h under the maximum load. When the load is decreased from 5 to 3 N/mm the experimental leakage rate is not modiﬁed. Based on the proposed model, this evolution can be explained as follows. Decreasing the load leads to a sudden decrease of the local stresses in the FeCrAl sheet so that the plastic deformation rate, ṗ , is strongly reduced as the creep law (eq. (1)) is strongly non linear with respect to stress. The decrease of both the applied stress and the plastic deformation rate leads to a strongly reduced vari-ation rate for D (eq. (2)) so that D can be considered as con-stant. This corresponds to a constant leakage rate as experimentally observed. The experiment also proves that elastic deformation does not inﬂuence leakage at very high temperature as assumed while developing the model. When the load is decreased down to 1 N/mm, the same phenomenon (i.e. no variation of the leakage rate) is observed for the me-dium 2 boss whereas the leakage rate suddenly increases for the sharp and large bosses. This is interpreted as a partial failure of the seal due to the fact that the load is not perfectly distributed along the contact area so that the normal stress may become zero in some regions causing local failure. The proposed model does not allow describing this phenomenon.

In the case of an increasing load (Fig. 11(b)), the leakage rate decreases sharply when the load is increased. Each load in-crease causes an increase of the seal viscoplastic deformation thus improving the tightness due to the extension of the contact zone and the increase of the local gap closure (i.e. increase of D). The proposed model is able to describe these trends. The under-estimation of the leakage rate by the model is caused by boss degradation (scratches) as the same boss was used to perform all tests. One can observe that the measured leakage rates (for times smaller than 10 h) are higher than those measured in experiments with a constant load (see Fig. 4(a)).

**Effect of gas pressure**

In this set of experiments the overpressure is gradually increased from 50 mbar to 800 mbar while keeping the applied seal load equal to 10 N/mm (medium 2 boss). This implies that the force must be increased together with the over-pressure to exactly counterbalance the end load. Each pressure step has a duration of 2 h. Results are shown in Fig. 12 together with the model predictions. As expected from eq. (7) increasing the gas pressure immediately leads to an increase of the leakage rate. A the beginning of the test (duration less than 4 h), the leakage rate decreases due to the formation of the contact area. After 4 h, the leakage rate is nearly constant during each pressure step as the contact zone hardly evolves. All these trends are well reproduced by the model predictions.

**Effect of gas composition**

In this series of experiments, two gas compositions are used in lieu of air as in the previous tests. Helium and Argon con-taining 2 wt% di-hydrogen (H_{2}) are used. The R10 boss (see Fig. 3) is used with a load equal to 5 N/mm and an over-pressure of 200 mbar. Results are shown in Fig. 13 and are compared with model predictions. Following the model, the volume leakage rate, Q_{v} (see eq. (10)), is only affected by the variations of the inverse of the gas viscosity (1/μ) when changing the gas composition. Note that the mass leakage rate, Q_{m} depends on the M/μ ratio (see eq. (9)). Consequently the predicted volume leakage rate (Q_{v}) is straightforwardly obtained from that computed for air applying a constant correction factor equal to the ratio of μ for air to μfor the considered gas. Values of μ at 800 °C are reported in Table 1.

**Effect of seal geometry**

In all previous experiments, the boss is machined in the creep-resistant upper bearing. In this experiment, the FeCrAl sheet is machined to create a circular D-shaped boss and the upper bearing is ﬂat. The boss has a diameter equal to 50 mm. The sharp ∆ allows creating localized high strains and stresses which were proved to be effective in creating tight seals as in the case of the sharp boss presented above. The average proﬁle of the ∆-shaped boss is shown in Fig. 14(a). The upper part of the ∆ is truncated and its width is about 100 mm. Following the modelling procedure proposed above, the machined sheet and the upper and lower bearings were meshed (see Fig. 14(b)) to simulate the deformation of the seal under load. The leakage rate prediction model is then applied and compared to test results. The comparison ((see Fig. 14(c)) shows good agreement even if the model is applied on a totally different geometry from these on which it was opti-mized. This constitutes a strong validation of the model applied on a seal structure to estimate the predicted leakage at high temperature.

### Conclusions

SOEC and SOFC represent promising components in the power generation chain. However, in spite of important pro-gresses, this solid oxide technology still suffers from a lack of efﬁcient technological solutions to achieve required tightness levels. Moreover, very few simulation tools are available to design adapted seals.

In this work a computational modelling strategy was pro-posed that allows linking the time evolution of local me-chanical ﬁelds to leakage rates. The proposed tool uses ﬁnite element calculations of seals which are post-processed using a new approach to derive an estimation of the leakage rates. This model aims to be simple enough to allow numerous simulations and therefore parametric studies to improve the design of new seals. To keep the model simple, it is purely macroscopic so that a ﬁne description of roughness is not required. It is however based on microscopic considerations as it describes the local gap closure between opposite sur-faces. The model is also designed to be applied at high tem-perature and takes therefore into account the rate dependent behaviour of the seal materials. The model introduces pa-rameters which need to be tuned against experimental data. In this work several experiments were carried out at 800 °C using various seals geometries and load levels. The tuned model well represents experimental results. In particular it reproduces the fact that a sharp boss generating large strains but a small contact zone is more efﬁcient in reducing leakage than a large boss. The model also well reproduces the time evolution of the leakage rate which is directly related to the high temperature viscoplasticity of the seal materials.

The model was ﬁnally checked using speciﬁcally designed additional experiments. It reproduces well the effect of gas composition, gas pressure, seal geometry and seal loading history on leakage rate. While studying the effect of the loading history, it was shown, as predicted by the model, that a moderate load decrease can be tolerated without any vari-ation of the leakage rate. This indicates that a limited relax-ation of the clamping system could be accepted in high temperature systems. Based on its prediction capabilities, this patented approach [29] could be used to design new seal ge-ometries as well as to deﬁne seal operation conditions. The model could also be improved by implementing a seal failure condition in order to describe the loss of tightness during se-vere unloading. The model could also be extended to deal with non-isothermal conditions. Developing the model further to take into account seal failure and non-isothermal conditions is needed to study the entire life cycle of the stack including heating up, high temperature and cooling down.

### Acknowledgements

This work has been supported by French Research National Agency (ANR) through Hydrogen and Fuel Cells Programs (EMAIL project ANR-07-PANH-004-01). The authors wish to thank C. Perret, B. Oresic and L. Bruguie` re for their help in conducting the experiments.

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