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Stress and plastic deformation of MEA in running fuel cell
international journal of hydrogen energy 33 (2008) 5703–5717
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Stress and plastic deformation of MEA in running fuel cell Daniil Bogracheva, Mikael Gueguenb, Jean-Claude Grandidierb, Serguei Martemianovc,* a
Frumkin Institute of Physical Chemistry and Electrochemistry RAN, Moscow 117071, Leninski prospekt 31, Russia Laboratoire de Physique et Me´canique des Mate´riaux, LMPM UMR CNRS 6617, ENSMA, Te´le´port 2, 1 av. Cle´ment Ader, BP 40109 86962 Futuroscope Ce´dex, France c Laboratoire d’Etudes Thermiques, LET UMR CNRS 6608, ESIP – University of Poitiers and ENSMA 40 av. du Recteur Pineau, 86022 Poitiers, France b
article info
abstract
Article history:
Numerical modelling of mechanical stresses in running fuel cell is provided. The evolution
Received 4 April 2008
of stresses and plastic deformations in the membrane has been obtained during the turn-
Received in revised form
on phase. The operating conditions have been taken into account by imposing the heating
17 June 2008
sources and the humidity field. The results have been presented on two scales: the global
Accepted 20 June 2008
scale reflects the stress evolution in the entire fuel cell and the local one corresponds to the
Available online 14 September 2008
tooth/channel structure. It has been shown that the stresses are strongly heterogeneous on the both scales and time dependent. From the mechanical point of view, the most sensible
Keywords:
zone is under the GDL/seal joint interface. In the running fuel cell the heterogeneity
Fuel cell
decreases while it rises in the fuel cell just after the assembly. The stresses reach the
Proton exchange membrane
maximum values during the humidification step, the magnitude of these stresses is
Mechanical behaviour
sufficient for initiation of the plastic deformations in the Nafion membrane.
ABAQUS
ª 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
It is well stated in the literature that Nafion (Nafion is registered trademark of sulfonated tetrafluoroethylene copolymer by DuPont) membrane failures are one of the main reasons determining proton exchange membrane fuel cell (PEMFC) durability. Increasing of the durability is a significant challenge for the development of fuel cell technology. The lifetime limitations of Membrane Electrode Assembly (MEA) are caused by chemical and thermo-mechanical factors. The observed MEA damages can be pinholes in the membrane or delamination between polymer membrane and the electrode [1,2]. Mechanical stresses which limit MEA durability have two origins. Firstly, this is the stresses arising during fuel cell assembly (bolt assembling) [3]. The bolts provide the tightness and the electrical conductivity between the contact elements.
The influence of the contact pressure on the fuel cell performance has been studied in Refs. [4,5]. Secondly, additional mechanical stresses occur during fuel cell running because PEMFC components have different thermal expansion and swelling coefficients. Thermal and humidity gradients in the fuel cell produce dilatations obstructed by tightening of the screw-bolts. Compressive stress increasing with the hygro-thermal loading can exceed the yield strength which causes the plastic deformation [6–8]. The mechanical behaviour of Nafion depends strongly on hydration and temperature [9]. Damage mechanisms in the running fuel cell are accelerated by hydration–dehydration and thermal cycles [7]. A number of works have been devoted to numerical simulations of mechanical stresses in MEA during fuel cell running [6–8]. These simulations deal with the local effects on the single channel scale. In these studies the tightening
* Corresponding author. Tel.: þ33 5 49 45 39 04. E-mail address: [email protected] (S. Martemianov). 0360-3199/$ – see front matter ª 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2008.06.066
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Nomenclature a dij
the linear thermal expansion coefficient, the Kroneker symbols,
3EL ij
the elastic strain tensor,
3PL ij
the plastic strain tensor,
3Tij
the temperature expansion tensor,
3RH ij
the humidity expansion tensor, the Poisson’s ratio, the stress tensor, the plastic multiplier, the Young’s modulus, the deviator stress tensor, the current temperature, the reference temperature.
n sij dl E Sij T T0
effects have been taken into account introducing either fixed displacements or fixed forces. This approach is not sufficiently realistic because in this case only 1D external mechanical loading is used. The stress state in the running fuel cell is related to tightening which has been applied initially during the assembly procedure. The stiffness of the screw-bolts restricts the thermal and the humidity expansions and provokes heterogeneities in the stresses distribution. The local nature of the previous models is a serious limitation because of their incapacity to apprehend the global effect in fuel cell, in particular, the edge effect near the seal joint. In the recent work [3], a 2D model of the fuel cell has been proposed. All elements of the cell have been schematized, and tightening system has been represented by beams with the mechanical properties which correspond to the ones in real assembly procedure. This model has described the tightening effects. 2D numerical simulations have shown a significant increase of stresses near the seal joints during the assembly procedure. The stress in the cold fuel cell is smaller than the yield stress, but it approaches the critical value under the seal joints. This result demonstrates that the seal joint properties and the real clamping conditions are non-negligible and should be considered. The simulations have shown the evolution of the stresses at the local (single channel structure) and the global (entire cell) levels. The complete mechanical description of systems needs 3D model, nevertheless the developed 2D model [3] is a good starting point for realistic estimation of the mechanical stresses arising in running fuel cell. In the present paper we propose to extend the 2D model for the description of the mechanical effects in fuel cell during the transient stage (turnon). The modelling of coupled thermal, mechanical and humidification effects will be provided at the local (single channel structure) and the global (entire cell) levels. In the present article mechanical stresses arising in running fuel cell are calculated taking into account the initial stresses generated during cell assembly. An isotropic hardening plastic law for the membrane and the mechanical properties of Nafion are employed [9]. The turn-on stage of the fuel cell running is studied in details taking into consideration the presence of the temperature gradients. The expanded model from Ref. [3] and the ABAQUS code are used for
understanding the thermo-mechanical effects on the local and the global scales. The numerical simulations are provided up to a steady state when the heat and the humidity parameters reach permanent working regime. The ABAQUS package does not allow solving simultaneously the transient thermal and humidification problems. This limitation has been overcome partially by imposing a uniform and time-dependent humidity field. This procedure has also been applied in Refs. [6,7]. The evolution of the humidity field in the membrane corresponds to the real conditions during the fuel cell turn-on. The mechanical behaviour of the fuel cell components and the hygro-thermal and the mechanical boundary conditions are detailed in Section 2. The results of the numerical simulations are discussed on the local and on the global scales in Section 3. At first, the time evolution of temperature, stress and deformation fields are established on the entire fuel cell scale. Then these fields are studied on the scale of a single channel. In Section 4, some parameters’ effects have been studied.
2.
Model
2.1.
Geometry and mesh definition
A two-dimensional approach is used for the model with an entire cross-section representation of a single cell (see Fig. 1). This model is similar to the one developed in paper [3] and presents the entire fuel cell including two steel plates, two graphite plates, two gas diffusion layers (GDL), a proton exchange membrane and two seal joints (see Fig. 1a and b). The catalytic layers are thinner than the membrane by one order of magnitude and it is assumed that these layers are integrated into the GDL. There are two fastening elements (bolts) allowing a realistic simulation of applied load. The schema of a half-cell is shown in Fig. 1b. Some sizes are defined in this figure, other parameters are the following: the thickness of the membrane – 0.05 mm, the thickness of the steel plates – 10 mm, the thickness of the GDE – 0.28 mm, the length of the bolt – 80 mm, the diameter of the bolt – 5 mm, the thickness of the seal joint – 0.3 mm, the width of the tooth – 0.7 mm, the width of the channel – 0.7 mm, the depth of the channel – 1 mm. A symmetry condition is used in order to reduce the size problem, i.e. there is the central axe of symmetry for all physical quantities. The dimension which is perpendicular to a plane of cross-section (2D model) equals 100 mm. Thermo-mechanical finite elements (CPEG8T and CPEG6MT) based on the generalized plane strain hypotheses are applied. The mesh consists of 13 267 elements and 41 143 nodes. Two beams (type B21 in ABAQUS) are connected with one steel plate; these connectors are CONN2D2 type in ABAQUS. The opposite steel plate is used as the support. The displacement of the beams has been calculated in order to generate the mean pressure of 1 MPa between the steel and the graphite plates in the assembly procedure [3].
2.2.
Governing equations
The mechanical properties of steel, graphite and carbon paper have been taken from Refs. [17–19]. It is assumed that these
international journal of hydrogen energy 33 (2008) 5703–5717
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Fig. 1 – (a) Fuel cell components and assembled fuel cell. (b) Schematization of fuel cell: local, global paths and characteristic points illustrating the results of calculation. 1 – Seal joints, 2 – MEA, 3 – Graphite plates, 4 – Steel plates, 5 – Bolts. a – Characteristic point in GDL (centre zone of fuel cell), b – Characteristic point in the membrane (centre zone of fuel cell), c – Characteristic point in the GDL/seal joint interface (edge zone of fuel cell), d – Characteristic point in the membrane under GDL/seal joint interface, e – Characteristic point in the seal joint.(c) Mesh used for the model.
materials have linear-elastic behaviour (see Table 1) which does not depend on the temperature and the relative humidity. The seal joint (Taconic) is assumed to be composed of three identical layers of woven fiberglass coated with PTFE (polytetrafluoroethylene). Linear elastic, perfectly plastic model following von Mises yield criterion with isotropic hardening is accepted for Nafion membrane. The hardening curve depends on the temperature and the humidity (see Table 2). These data have been included in
the model in order to consider the temperature and the humidity effects. In the classical elasto-plastic model, assuming that all deformations are small, it is possible to present the tensor of deformations as the sum: PL RH T 3ij ¼ 3EL ij þ 3ij þ 3ij þ 3ij ;
(1)
PL here 3EL ij is the elastic strain tensor and 3ij is the plastic strain tensor. Variations of the temperature and the humidity
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Table 1 – General properties Property
Table 2 – Set of the hardening curve of Nafion [9] Value
Source
Nafion membrane Density, r [kg/m3] Young’s modulus: E [MPa] Poisson’s ratio, n Yield strength, s0 [MPa] Expansion coefficient: a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
2000 See Table 3 0.25 See Table 2 123 106 1050 0.259
[7] [7] [7] [7] [7] [7] [7]
Graphite plate Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
1800 10 0.25 5 106 750 95
[7] [7] [7] [7] [7] [7]
GDE (carbon paper) Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
400 10 0.25 0.8 106 500 0.3
[17] [17] [17] [17] [17] [17]
Seal joint (SJ) (40% Woven fiberglass E – 60% PTFE) PTFE 2200; Density, r [kg/m3] Fiberglass E 2540; SJ: 2336 Young’s modulus, E [GPa] PTFE 0.8; Fiberglass E 72; SJ: 30 Poisson’s ratio, n PTFE 0.46; Fiberglass E 0.22; SJ: 0.364 Expansion coefficient, a [K1] PTFE 100 106; Fiberglass E 5 106; SJ: 62 106 1 1 Specific heat [Jkg K ] PTFE 1000; Fiberglass E 830; SJ: 932 PTFE 0.195; Conductivity [W m1 K1] Fiberglass E 1; SJ: 0.517 Steel Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
7800 209 0.25 12 106 460 0.3
[18]
[18]
[18]
[18]
[18]
[18]
[19] [19] [19] [19] [19] [19]
generate expansions which are considered to be isotropic. These expansions are represented by two tensors 3Tij and 3RH ij , correspondingly. The membrane behaviour is supposed to be elastic and isotropic; in elastic region the Hook’s law is accepted. The constitutive law can be written as follows: ! X EL E EL (2) 1 2n 3kk dij ; n3ij þ sij ¼ ð1 þ nÞð1 2nÞ k where sij is the stress tensor, n is Poisson’s ratio, E is Young’s modulus, dij is the Kroneker symbols. Young’s module
Equivalent plastic stress (Mpa) 6.60 6.60 15.40 6.51 6.51 22.20 5.65 5.65 11.60 4.20 8.54 6.14 6.14 14.90 5.21 5.21 20.80 5.00 5.00 10.80 3.32 7.56 5.59 5.59 14.40 4.58 4.58 20.10 4.16 4.16 9.96 2.97 7.31 4.14 4.14 12.90 3.44 3.44 11.20 3.07 3.07 8.87 2.29 6.63
Equivalent plastic strain
Temperature (K)
Humidity (RH%)
0 0.167 2.09 0 0.194 2.06 0 0.139 2.06 0 1.93 0 0.167 2.09 0 0.194 1.99 0 0.139 2.04 0 1.91 0 0.167 2.06 0 0.194 2 0 0.139 2.03 0 1.88 0 0.167 2.06 0 0.194 2.03 0 0.139 2 0 1.85
298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358
30 30 30 30 30 30 30 30 30 30 30 50 50 50 50 50 50 50 50 50 50 50 70 70 70 70 70 70 70 70 70 70 70 90 90 90 90 90 90 90 90 90 90 90
depends on the temperature and the humidity; the corresponding data are presented in Table 3. Thermal strain is determined by the classical linear expansion law: 3Tij ¼ aðT T0 Þdij ;
(3)
where a is the linear thermal expansion coefficient, T is the current temperature, T0 is the reference temperature. The thermal expansion coefficients are given in Table 1. Following the work [7], the swelling-expansion strain is expressed as a polynomial function of humidity and temperature in order to take into account the expansion of the membrane from the initial state (35%RH and 20 C) to the work
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Table 3 – Set of the Young’s modulus curve of Nafion [7] Young’s modulus (Mpa) 197 161 148 121 192 137 117 85 132 103 92 59 121 70 63 46
Temperature (K)
Humidity (RH%)
297 317 337 357 297 317 337 357 297 317 337 357 297 317 337 357
30 30 30 30 50 50 50 50 70 70 70 70 90 90 90 90
3RH ij ¼ dij
Ckl T4k RH4l ;
(4)
k;l¼1
where RH is the relative humidity, Ckl is the polynomial constants, see Ref. [7]. The subroutine UEXPAN has been developed for introducing the swelling strains in the membrane. This subroutine takes into account Eq (4). The similar moralization of swelling strain has been used in works [7,8]. The plasticity behaviour of the membrane is described by Prandtl–Reuss theory. The von Mises yield function is defined as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 (5) Sij Sij s0 ; f sij ¼ 2 and the von Mises yield criterion is presented as: f sij ¼ 0;
(6)
where sij is the stress tensor. The term s0 ¼ s0(T, RH) is the yield strength depending on the temperature and the relative humidity (see Table 4); Sij ¼ sij (1/3)skkdij is the deviator stress tensor. When the inequality f(sij) < 0 is verified, the material has elastic behaviour. The plasticity evolutions are described by isotropic hardening curves of Nafion which are taken from the work [9]. Due to the lag of data concerning the dependence of the hardening curve on the humidity, the interpolation has been accepted taking into account the data for a reference temperature (see Table 2). According to the Prandtl–Reuss theory, the plastic increment tensor is proportional to the derivative of the yield function (5): vf d3ij ¼ dl; vsij
Yield strength (Mpa) 6.60 6.51 5.65 4.20 6.14 5.21 5.00 3.32 5.59 4.58 4.16 2.97 4.14 3.44 3.07 2.29
state (100%RH and 85 C). The swelling strain tensor can be written down as follows: 4 X
Table 4 – Yield strength at various temperatures and humidities of Nafion [7]
(7)
where dl is the plastic multiplier calculated by the consistent relation (df ¼ 0) during the hardening. The plastic multiplier corresponds to the increment of the equivalent plastic strain.
2.3.
Temperature (K)
Humidity (RH%)
297 317 337 357 297 317 337 357 297 317 337 357 297 317 337 357
30 30 30 30 50 50 50 50 70 70 70 70 90 90 90 90
Hygro-thermal and mechanical load
In this part the main operating parameters, namely the temperature and the humidity, are introduced considering the fuel cell working regime. The mechanical loading is defined in the same manner as in the paper [3].
2.3.1.
Mechanical load
This value of loading 1 MPa has been accepted in Refs. [6–8] and is used as the reference in the present work. The mechanical loading in the model is simulated by the displacement of the bolt heads. It is assumed that this displacement remains constant during the fuel cell running. This displacement reflects the loading caused by application of a moment of forces to the bolts during the cell assembly. It has been calculated [3] that the displacement dIMP ¼ 0.038 mm corresponds to the normal stress 1 MPa on the interface between graphite and steel plates. The type of bolts M6 80 has been assumed in our calculation. The required displacements corresponding to the normal stress 1 MPa for others types of bolt are presented in Table 5.
2.3.2.
Thermal load
Heat generation in fuel cells have been studied in numerous works, as for example Refs. [10–12]. At working regime, the main source of heat generation concerns irreversible reactions and entropic losses in the cathode catalytic layer (85–90%) and Joule heating of the membrane (8–12%) [10]. Thus, almost all heat is produced near or inside the membrane. The intensity
Table 5 – The bolt displacements corresponding to the normal stress 1 MPa for different types of bolts Type of bolts DIN558 M5 80 M6 80 M8 100 M10 100
Displacement [mm] 0.0380 0.0264 0.0186 0.0119
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of the heat source Q ¼ kP is directly related to the power generation P and the efficiency of the fuel cell k. In our model the continuity of all physical parameters along the fuel cell is accepted. This assumption does not permit to introduce surface heat sources in ABAQUS package. That is why we have interpreted the heat sources as the volume ones. Using the thickness of the Nafion 112 membrane (D ¼ 50 mm) and accepting that k ¼ 0.4 and P ¼ 0.75 W/cm2, it is possible to find the volume heat density h ¼ 1.4 108 W/m3. In our model the time-dependent variation of the heat generation is accepted h ¼ h(t), see Fig. 2. The initial temperature is equal to T ¼ 300 K. The boundary condition at the external surface of the fuel cell is determined by natural convection. The convective heat exchange coefficient between air and external fuel cell surface is set to 10 Wm2 K1 [13]. An external heating is applied for obtained and maintaining of the operating fuel cell temperature (T ¼ 80 C). In real experiments, this function is attained by using resistors integrated in the steel plates. They generate heat by Joule effect. It is assumed that this heat source is distributed by the steel plate volume and its intensity is equal to 8.9 106 W/m3. The external heating is imposed before the fuel cell turn-on. After reaching the operational temperature the intensity of the external source decreases to the value which allows compensation of the heat losses due to the heat exchange with environment (see Fig. 2). The temperature of moving gases in graphite plates varies during turn-on regime; this evolution is specified in Fig. 2. In the beginning the temperature of the gases is equal to 300 K. According to experimental data [5], the temperature reaches the value of 320 K in the steady-state conditions. The convective heat transfer coefficient between the gases and the channel wall (hf) is assumed to be equal to hf ¼ 110 Wm2 K1 [13].
2.3.3.
Humidity load
Water transport in the proton exchange membrane is a complicated process. It is governed by nonlinear diffusion and by electro-osmotic drag, see for example Refs. [14–16]. In the present model, the moisture is set gradually from an initial value of 35% up to 100%. The humidity is imposed after all heat sources reach steady state, see Fig. 2. The characteristic
time of the membrane humidification is set to 40 s, taking into account the real experimental data [17]. It is suggested that the imposed moisture is uniformly distributed in the membrane. This questionable assumption leads to overestimation of the maximal stresses in the membrane during turn-on stage.
3. Stress and strain distributions in running fuel cell In this article calculations and data treatment of stresses are provided on the single channel scale (local effects) and on the entire fuel cell level (global effects). The local analysis has been performed in the cell centre region (near the axis of symmetry) and in the cell edge region (see Fig. 1b, the small dots in the zoomed pictures). Both regions include two channels to emphasize a periodic character of the local fields. The stresses distribution on the scale of the entire cell is presented with a particular global path. This path goes through the membrane centre points which are situated in front of the lowest corners of the channels (see Fig. 1b, the big dots in the zoomed pictures). The path goes also under the seal joint in order to visualize the stresses in the edge zone. At last, the special points are selected for observing the temporal evolution of stresses in MEA and the seal joints (see ‘a–e’ Fig. 1b in the zoomed pictures). The normal direction 1 is perpendicular to the MEA and the transverse directions 2 and 3 are situated in the plane of the MEA; the direction 3 is perpendicular to the plane of the model (see Fig. 1b). The calculations are provided up to the establishing of the steady state. In our model, the steady-state regime is achieved after about 1400 s (see Fig. 3a). In the steady-state regime the heat balance between the fuel cell and the environment is attained and the temperature changes become negligible. At the first moments (between 4 and 30 s) the temperature evolution is governed by the external heating of the steel plates. After the working temperature is reached (40–50 s), the intensity of the external heating decreases, and the supply of hydrogen starts. Then, the temperature evolution is governed mainly by chemical heat generation and by the heat exchange
Fig. 2 – Fuel cell loading curves.
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Fig. 3 – (a) Time evolution of temperature at characteristic points. (b) Time evolution of Mises stress at characteristic points.
between the gas channels and the MEA. Before supplying of hydrogen, the temperature gradients exist in the active part of the cell (GDL, membrane). The temperature difference between the seal joint (the hottest point) and the centre of MEA is about 5 K. While hydrogen supply is turned on, the inversion between the hot and the cold zones takes place and the membrane warms up, while the temperature of the seal joints remains constant (Fig. 3a, around 50 s). After this stage, a homogeneous evolution of the temperature in the joints and in the membrane occurs. The stress distributions have been calculated for the entire cell scale during all the transition stage. The variations of the Mises stresses are shown in Figs. 3b and 4a for different locations (see points ‘a–e’ in Fig. 1b). The Mises stresses change in the graphite in accordance with the temperature evolution. At the same time, the peak can be observed in the membrane for a short time with the following weak decrease
which asymptotically tends to be about 2.7 MPa. The stresses in the joint correspond to the global changes in the temperature but in less remarkable way than the ones in the GDL. The stresses change strongly during the first 100 s of the cell running; afterwards they stabilize with the achievement of the temperature steady-state regime. The step of cold assembling procedure which has been studied in the paper [3] corresponds to the period between 0 and 1 s. The strongest changes correspond to the phase of pre-heating of the steel plate (between 4 s and 30 s). During this phase, the membrane stresses under the GDL rise in accordance with the temperature. On the contrary, in the very beginning of the heating, the stresses under the seal joint decline, but increase later. This is a result of the competition between the expansion of the different elements and the membrane Young modulus decrease caused by temperature increase. During pre-heating, the stresses in the membrane
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a
b
0.25
center point b
under point e
edge point d
equivalent plastic strain
0.2
0.15
0.1
0.05
0 0
50
100
150
200
time (s) Fig. 4 – (a) Time evolution of Mises stress in the membrane for the beginning of turn-on. (b) Evolution of equivalent plastic strain in the membrane at the edge (d) and at the centre (b).
under the seal joint increase more significantly than in the assembly step. After pre-heating, the heat exchange caused by the fuel cell turn-on (about 50 s) leads to a very small increase of the stresses in the MEA elements. This is a second order effect with respect to the humidification. In our model the humidification load starts at 60 s and generates strong variations of the stresses in the membrane which are the same order of the magnitude as the ones during the pre-heating stage. The swelling strain generates stresses which are modified also due to dependence of the mechanical properties on the humidity. The new effect appears: the stresses are governed by plastic deformation as well. The first plastic deformations in the membrane emerge during the stage of the humidification, see Fig. 4b. In the beginning of this stage the stresses are not sufficient and the yield stress is not achieved. The plasticity becomes visible at about 70 s at the joint/GDL interface (point d). The plastic deformation in the centre of the membrane arises later;
however, it reaches a higher level. The plasticity appears also in the membrane under the seal joint; but this effect takes place later and its magnitude is less important (see Fig. 4b, the curve e). The plastic deformation remains constant at the end of the humidification stage; it is important to note that its level is very high. Distribution of the Mises stress in the middle plane of the membrane is shown on Fig. 5a and b for the scale of the entire fuel cell. The first curve corresponds to the end of the assembly step (t ¼ 4 s on Fig. 2) where heterogeneous stresses on the joint/GDL interface are already presented. During the period between 4 s and 30 s the fuel cell pre-heating works. At the first moments, the Mises stress decreases along all the membrane except the borders under the seal joints. After 14 s, the Mises stress increases along all the membrane and the heterogeneity becomes more and more important, particularly under the interface joint/GDL. During the humidity loading the Mises stress in the centre of the membrane
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a
b
t=59 heating in membrane
t=64
t=74
t=84
t=94
t=104 transient state
humidity loading
4 seal joint
diffusion layer
3.5
Mises stress (MPa)
3
2.5
2
1.5
1
0.5
0
edge 0
center 0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m) Fig. 5 – (a) Space evolution of Mises stress along the membrane (global path) before humidification loading. (b) Space evolution of Mises stress along the membrane (global path) during humidification loading.
increases (see Fig. 5b); at the same time, the Mises stress declines in the membrane edge under the seal joint. Around 74–84 s the discontinuity at the GDL/seal joint interface is very prominent. After that, the stress field becomes homogeneous very rapidly. The stress distribution along the membrane has a complex character during the transient phase. This distribution stabilizes at the end of the humidification step and becomes quasi-
homogeneous when the temperature field reaches steady state (see t ¼ 104 s, Fig. 5b). It can be noted that the maximal stresses (higher than 3.5 MPa) occur during the humidification step. These peak stresses are localized under the joint/GDL interface; in this part of MEA the failures are especially often observed. After the end of the humidification (100–1500 s) the stresses decrease, and the discontinuity on the joint/GDL interface becomes visible again (Fig. 6a).
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a
t=161
t=261
t=661
t=1161
t=1661
transient to permanent state
4 seal joint
diffusion layer
3.5
Mises stress (MPa)
3 2.5 2 1.5 1 0.5 0
edge 0
center 0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
Fig. 6 – (a) Space evolution of Mises stress along the membrane (global path) after humidification loading. (b) Space evolution of the normal stress s11 along the membrane (global path) during all turn-on stages.
So, the fuel cell turn-on produces complex distribution of stresses along the membrane. This distribution is affected by expansions of different MEA components, increase of the temperature and, for the most part, by the membrane moisture which leads to the membrane plasticity (see Fig. 5b, curves corresponding to t ¼ 64 s and t ¼ 94 s). The evolutions of the stress tensor components in the middle of the membrane are presented in Figs. 6b and 7a, b. The stresses correspond to a biaxial state. The normal and the tangential components have the same order of magnitude. Note that the tangential component is slightly higher. This result should be considered with some caution because the model ignores any slipping between MEA elements.
The normal stress (Fig. 6b) has large discontinuity under the joint/GDL interface in the beginning of the cell running (59 s) which decreases later. The significant value of the compression stress near the free surface of the cell at the steady state should be noted. This is an interesting result with respect to the airtight problem. Firstly, in the centre of the cell the normal stress in the membrane increases gradually and afterwards it decreases slowly during the reaching the steady state. The tangential stress rising from the centre to the edge of the membrane (Fig. 7a) increases in time during all the steps. This stress is directly related to the expansion of the MEA components. The discontinuity increases also with time (Fig. 7a and b).
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a
t=4
t=59
t=84
t=104
seal joint
5713
t=1661
diffusion layer
0
Tangential stress (MPa)
-1 -2 -3 -4 -5 -6 -7 -8 edge
center
-9 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
t=4 0.3
t=59
t=84
seal joint
t=104
t=1661
diffusion layer
0.2
Shear stress (MPa)
0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 edge
center
-0.6 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m) Fig. 7 – (a) Space evolution of the tangential stress s22 along the membrane (global path) during all turn-on stages. (b) Space evolution of the shear stress s12 along the membrane (global path) during all turn-on stages.
The shear stresses (Fig. 7b) are smaller than the normal and the tangential ones (Fig. 7a); their level is lower than 1 MPa. The changes of stresses sign can be noted between the phase of the assembling and the step of the pre-heating; this is the consequence of the thermal gradients in the perpendicular to the membrane direction. The stresses decline during the last period (100–1400 s), however, the discontinuity is present constantly under joint/GDL interface. Plastic deformations occur under joint/GDL interface around 70 s during the phase of the membrane humidification (Fig. 8a). These deformations develop considerably (at first under the GDL and after under the joint) up to the end of the humidification when they reach the maximal magnitude. Afterward the plastic deformations remain constant.
It is important to emphasize the model limitations. Reducing of the membrane behaviour to the elasto-plastic law with isotropic hardening leads to the inherent stabilisation of the stress distributions. In the future it is important to introduce the models with viscosity in order to take into account the creep-relaxation effects. In particular, it is possible to introduce a rate dependent viscoplastic model using the curves presented in the paper [20]. As pointed by these authors, a viscoelastic – viscoplastic model is more adapted for description on Nafion behaviour. The existence of the periodic changes on the scale of a single channel (see Fig. 8b) has been already discussed in the paper [3]. The growing temperature causes (see Fig. 8b, t ¼ 59 s) the inversion of the stress distribution: the maximum values
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a
t=72
t=80
t=84
sealjoint
0.25
t=94
t=104
diffusion layer
equivalent plastic strain
0.2
0.15
0.1
0.05
edge
center
0 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
t=4
t=59
t=84
t=104
t=1661
3.5
Mises stress (MPa)
3
2.5
2
1.5
1
0.5
0 0
0.0005
0.001
0.0015
0.002
0.0025
local path (m) Fig. 8 – (a) Space evolution (global path) of the plastic equivalent strain in the membrane. (b) Space evolution (local path) of Mises stress in the centre of the membrane for different times.
become the minimum ones and vice versa. The stress distribution becomes more homogeneous along the channels but a strong discontinuity remains under the joint/GDL interface. The local variations of stresses become negligible compared to the mean value. The magnitude of the Mises stresses in the centre and at the edge of the cell increases gradually up to the end of the membrane humidification (around t ¼ 84 s). Later on, when the fuel cell tends to the steady state, the Mises stress in the middle plane of the membrane decreases and reaches the level of about 2.75 MPa at the centre and at the edge. The local distribution of the stress under the joint/GDL interface during the transitional phase is presented in Fig. 9a. The maximal value of 3.5 MPa is achieved during the humidification step.
The stress maximums are localized in this zone at the steady state as well. It should be noted that significant changes of the stresses exist under the joint/GDL interface before the humidification of the membrane. Apparently, this is a source of the fatigue limitations generated by the sequence of turnons and shutdowns of the fuel cell. After reaching the steady state, all stress components change on the local scale, see Fig. 9b. Nevertheless, the stress changes in the centre are smaller than at the edge of the cell. The shear stresses are small both in the centre and at the edge. The tangential stress is more evident at the boarder of the cell. The normal stress demonstrates the same evolution with the moderate magnitude as the mean value near the edge of the cell (under the seal joint); this magnitude becomes
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a
b
s11 center s11 edge
s12 center s12 edge
s22 center s22 edge
0 -1
stress (MPa)
-2 -3 -4 -5 -6 -7 -0.002
-0.001
0
0.001
0.002
0.003
local path (m) Fig. 9 – (a) Space evolution (local path) of Mises stress at the edge of fuel cell during transient period. (b) Space evolution (local path) of stresses at the centre and at the edge of fuel cell in the steady state.
relatively small in the centre. The presented results illustrate once more the heterogeneities of the stress distributions which arise in the membrane during fuel cell running.
4. Influence of model parameters on mechanical stresses The goal of this paragraph is the study of the influence of some model parameters on mechanical stresses arising in fuel cell.
4.1.
thus the membrane stiffness is higher and swelling generates higher stresses. This peak decreases more slowly and stabilizes on the higher level in the case when the pre-heating is absent. The evolution of the stresses in the centre of the membrane and under the GDL/seal joint interface is identical, contrary to the case of pre-heated fuel cell. It should be noted that in our simulations the steady-state thermal regime is not reached for the cell without pre-heating, that is why a slow variations of stresses can be observed at the end of the simulation. To summarize, the mode of the cell heating seriously influences the maximal stresses distribution and stimulates the mechanical fatigue of the system.
Influence of pre-heating 4.2.
Pre-heating can influence significantly the lifetime of fuel cells. Numerical simulations have been provided for estimation of the influence of this action on the evolution of the mechanical stresses; the results are presented on Fig. 10. It can be noted that excluding of pre-heating provokes more important peak of stresses during the first moments of the turn-on stage. Indeed, in this case the temperature is lower,
Influence of the seal joints stiffness
The provided simulations point towards the increasing of stresses on the GDL/seal joint interface. These significant variations are caused by the particularities of the materials properties, in particular, by the stiffness differences. Thus, it is interesting to investigate the influence of seal joints stiffness on stress distribution. We have investigated different
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Fig. 11 – (a) Time evolution of Mises stress in Nafion during steady-state stage. Influence of seal joint stiffness. (b) Time evolution of Mises stress in Nafion during transient stage. Influence of seal joint stiffness. Fig. 10 – (a) Time evolution of Mises stress in Nafion during steady-state stage. Influence of pre-heating. (b) Time evolution of Mises stress in Nafion during transient stage. Influence of pre-heating.
tend to the same level at steady state. During the humidification stage, the stresses are not influenced by joint stiffness.
5. types of seal joints: for the soft ones the material properties are similar to PTFE and for the rigid joints the material properties correspond to fiberglass. A uniform thickness of the joints is assumed in our simulations. Time evolution of Mises stresses at critical points is presented on Fig. 11 for three types of joints. In the steady state the effect is not important; a small diminishing of Mises stress can be noted with the decreasing of the joint stiffness. On the other hand, in the transient regime the influence of the joint stiffness is more pronounced. During the pre-heating stage, the stresses increase with the rising of the joint stiffness and reach the maximal level of 4 MPa near the border of the cell. This level surpasses by the factor 4 the initial loading and is 30% higher than steady-state value. In the centre of the fuel cell the stresses are independent on the Young’s modulus. For the soft joint (PTFE) the results are different. The maximal value of stresses at the fuel cell border is reached during the humidification stage due to membrane swelling. In this case, the maximal stresses arise in the centre of the membrane; their level exceeds the steady-state value less than 20%. It is interesting to note that for all joints, stresses
Conclusions
Two-dimensional numerical modelling of mechanical stresses in MEA of a running fuel cell is developed. The evolution of stresses and plastic deformations in the Nafion membrane has been studied during the start-up phase. The model considers the initial mechanical loading arising during the cold assembly procedure and takes into account the mechanical action of the flexible bolts. The mechanical stresses in MEA have been simulated by imposing the heating sources and humidity field with respect to real operating conditions in running fuel cell. The results have been obtained for both the global and the local scales. The global scale reflects the stress evolution in the entire fuel cell and the local one corresponds to the tooth/channel structure. The calculations show that the stresses are strongly heterogeneous on the both scales and evolve in time during the transient stage. The stresses arising in the membrane under the joint/GDL interface due to the assembly decrease during the transition (turn-on) stage. The temperature field reaches the steady-state conditions around 1200–1400 s. The maximal stresses in the membrane take place during the
international journal of hydrogen energy 33 (2008) 5703–5717
humidification step before the temperature comes to its steady-state value. The plastic deformations in the membrane develop during the entire humidification step. At the steady state the stresses have the highest value in the centre of the membrane; the Mises stress is equal to 2.5 MPa. The mechanical charging of the membrane corresponds mainly to biaxial compression which is accompanied by the small transversal shear stress. After the beginning of the cell running the stresses on the local scale appear to be more homogeneous than the ones after the assembly. The peak stresses have been revealed at the transient stage. This is an interesting result, which allows understanding of the mechanical aging phenomena of MEA structures while multiple turn-ons and shutdowns of the fuel cell. The effects of joint stiffness and pre-heating on Mises stress in Nafion have been established. For the quantitative estimation of the mechanical durability of the MEA during the cycle loading the model should be improved, namely by more precise simulation of the humidification loading. Moreover, the viscous effects should be introduced in the modelling of the polymer membrane behaviour in order to predict the MEA aging during work cycles.
Acknowledgment The authors thank the Federation of laboratories P’PRIMME of Poitiers and the Region Poitou – Charentes for the financial support of this work.
references
[1] Hinds G. Performance and durability of PEM fuel cells: a review. NPL; 2004. [2] Kundu S, Fowler MW, Simon LC, Grot S. Morphological features (defects) in fuel cell membrane electrode assemblies. J Power Sources 2006;157(2):650–6. [3] Bograchev D, Gueguen M, Grandidier J-C, Martemianov S. Stress and plastic deformation of MEA in fuel cells. Stresses generated during cell assembly. J Power Sources 2008;180(1): 393–401. [4] Kadjo J-JA, Garnier J-P, Maye J-P, Relot F, Martemianov S. Performance and instabilities of proton exchange membrane fuel cells. Russian J Electrochem 2006;42:467–75.
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[5] Kadjo AJ-J, Brault P, Caillard A, Coutanceau C, Garnier J-P, Martemianov S. Improvement of proton exchange membrane fuel cell electrical performance by optimization of operating parameters and electrodes preparation. J Power Sources 2007;172(2):613–22. [6] Tang Y, Santare MH, Karlsson AM, Cleghorn S, Johnson WB. Stresses in proton exchange membranes due to hygro-thermal loading. J Fuel Cell Sci Technol 2006;3(2): 119–24. [7] Kusoglu A, Karlsson AM, Santare MH, Cleghorn S, Johnson WB. Mechanical response of fuel cell membranes subjected to a hygro-thermal cycle. J Power Sources 2006; 161(2):987–96. [8] Kusoglu A, Karlsson AM, Santare MH, Cleghorn S, Johnson WB. Mechanical behavior of fuel cell membranes under humidity cycles and effect of swelling anisotropy on the fatigue stresses. J Power Sources 2007;170(2):345–58. [9] Tang Y, Karlsson AM, Santare MH, Gilbert M, Cleghorn S, Johnson WB. An experimental investigation of humidity and temperature effects on the mechanical properties of perfluorosulfonic acid membrane. Mat Sci Eng A 2006; 425(1–2):297–304. [10] Ju H, Meng H, Wang C-Y. A single-phase, non-isothermal model for PEM fuel cells. Int J Heat Mass Transfer 2005;48(7): 1303–15. [11] Wang Y, Wang C-Y. A nonisothermal, two-phase model for polymer electrolyte fuel cells. J Electrochem Soc 2006;153(6): A1193–200. [12] Bao C, Ouyang M, Yi B. Analysis of the water and thermal management in proton exchange membrane fuel cell systems. Int J Hydrogen Energy 2006;31(8):1040–57. [13] Necati Ozisik M. Heat transfer a basic approach. McGraw-Hill Company; 1985. [14] Hyun D, Kim J. Study of external humidification method in proton exchange membrane fuel cell. J Power Sources 2004; 126(1–2):98–103. [15] Inoue G, Yoshimoto T, Matsukuma Y, Minemoto M, Itoh H, Tsurumaki S. Effect of flow pattern of gas and cooling water on relative humidity distribution in polymer electrolyte fuel cell. J Power Sources 2006;162(1):94–104. [16] Carnes B, Djilali N. Analysis of coupled proton and water transport in a PEM fuel cell using the binary friction membrane model. Electrochim Acta 2006;52(3):1038–52. [17] Product information of Toray Industries, Inc.; 2003. [18] Product information of Tonoga Inc.; 2005. [19] ABAQUS analysis user’s manual, V6.6, HKS; 2006. [20] Solasi Roham, Zou Yue, Huang Xinyu, Reifsnider Kenneth, Condit David. On mechanical behavior and in-plane modeling of constrained PEM fuel cell membranes subjected to hydration and temperature cycles. J Power Sources 2007; 167(2):366–77.
Available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/he
Stress and plastic deformation of MEA in running fuel cell Daniil Bogracheva, Mikael Gueguenb, Jean-Claude Grandidierb, Serguei Martemianovc,* a
Frumkin Institute of Physical Chemistry and Electrochemistry RAN, Moscow 117071, Leninski prospekt 31, Russia Laboratoire de Physique et Me´canique des Mate´riaux, LMPM UMR CNRS 6617, ENSMA, Te´le´port 2, 1 av. Cle´ment Ader, BP 40109 86962 Futuroscope Ce´dex, France c Laboratoire d’Etudes Thermiques, LET UMR CNRS 6608, ESIP – University of Poitiers and ENSMA 40 av. du Recteur Pineau, 86022 Poitiers, France b
article info
abstract
Article history:
Numerical modelling of mechanical stresses in running fuel cell is provided. The evolution
Received 4 April 2008
of stresses and plastic deformations in the membrane has been obtained during the turn-
Received in revised form
on phase. The operating conditions have been taken into account by imposing the heating
17 June 2008
sources and the humidity field. The results have been presented on two scales: the global
Accepted 20 June 2008
scale reflects the stress evolution in the entire fuel cell and the local one corresponds to the
Available online 14 September 2008
tooth/channel structure. It has been shown that the stresses are strongly heterogeneous on the both scales and time dependent. From the mechanical point of view, the most sensible
Keywords:
zone is under the GDL/seal joint interface. In the running fuel cell the heterogeneity
Fuel cell
decreases while it rises in the fuel cell just after the assembly. The stresses reach the
Proton exchange membrane
maximum values during the humidification step, the magnitude of these stresses is
Mechanical behaviour
sufficient for initiation of the plastic deformations in the Nafion membrane.
ABAQUS
ª 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.
1.
Introduction
It is well stated in the literature that Nafion (Nafion is registered trademark of sulfonated tetrafluoroethylene copolymer by DuPont) membrane failures are one of the main reasons determining proton exchange membrane fuel cell (PEMFC) durability. Increasing of the durability is a significant challenge for the development of fuel cell technology. The lifetime limitations of Membrane Electrode Assembly (MEA) are caused by chemical and thermo-mechanical factors. The observed MEA damages can be pinholes in the membrane or delamination between polymer membrane and the electrode [1,2]. Mechanical stresses which limit MEA durability have two origins. Firstly, this is the stresses arising during fuel cell assembly (bolt assembling) [3]. The bolts provide the tightness and the electrical conductivity between the contact elements.
The influence of the contact pressure on the fuel cell performance has been studied in Refs. [4,5]. Secondly, additional mechanical stresses occur during fuel cell running because PEMFC components have different thermal expansion and swelling coefficients. Thermal and humidity gradients in the fuel cell produce dilatations obstructed by tightening of the screw-bolts. Compressive stress increasing with the hygro-thermal loading can exceed the yield strength which causes the plastic deformation [6–8]. The mechanical behaviour of Nafion depends strongly on hydration and temperature [9]. Damage mechanisms in the running fuel cell are accelerated by hydration–dehydration and thermal cycles [7]. A number of works have been devoted to numerical simulations of mechanical stresses in MEA during fuel cell running [6–8]. These simulations deal with the local effects on the single channel scale. In these studies the tightening
* Corresponding author. Tel.: þ33 5 49 45 39 04. E-mail address: [email protected] (S. Martemianov). 0360-3199/$ – see front matter ª 2008 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijhydene.2008.06.066
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Nomenclature a dij
the linear thermal expansion coefficient, the Kroneker symbols,
3EL ij
the elastic strain tensor,
3PL ij
the plastic strain tensor,
3Tij
the temperature expansion tensor,
3RH ij
the humidity expansion tensor, the Poisson’s ratio, the stress tensor, the plastic multiplier, the Young’s modulus, the deviator stress tensor, the current temperature, the reference temperature.
n sij dl E Sij T T0
effects have been taken into account introducing either fixed displacements or fixed forces. This approach is not sufficiently realistic because in this case only 1D external mechanical loading is used. The stress state in the running fuel cell is related to tightening which has been applied initially during the assembly procedure. The stiffness of the screw-bolts restricts the thermal and the humidity expansions and provokes heterogeneities in the stresses distribution. The local nature of the previous models is a serious limitation because of their incapacity to apprehend the global effect in fuel cell, in particular, the edge effect near the seal joint. In the recent work [3], a 2D model of the fuel cell has been proposed. All elements of the cell have been schematized, and tightening system has been represented by beams with the mechanical properties which correspond to the ones in real assembly procedure. This model has described the tightening effects. 2D numerical simulations have shown a significant increase of stresses near the seal joints during the assembly procedure. The stress in the cold fuel cell is smaller than the yield stress, but it approaches the critical value under the seal joints. This result demonstrates that the seal joint properties and the real clamping conditions are non-negligible and should be considered. The simulations have shown the evolution of the stresses at the local (single channel structure) and the global (entire cell) levels. The complete mechanical description of systems needs 3D model, nevertheless the developed 2D model [3] is a good starting point for realistic estimation of the mechanical stresses arising in running fuel cell. In the present paper we propose to extend the 2D model for the description of the mechanical effects in fuel cell during the transient stage (turnon). The modelling of coupled thermal, mechanical and humidification effects will be provided at the local (single channel structure) and the global (entire cell) levels. In the present article mechanical stresses arising in running fuel cell are calculated taking into account the initial stresses generated during cell assembly. An isotropic hardening plastic law for the membrane and the mechanical properties of Nafion are employed [9]. The turn-on stage of the fuel cell running is studied in details taking into consideration the presence of the temperature gradients. The expanded model from Ref. [3] and the ABAQUS code are used for
understanding the thermo-mechanical effects on the local and the global scales. The numerical simulations are provided up to a steady state when the heat and the humidity parameters reach permanent working regime. The ABAQUS package does not allow solving simultaneously the transient thermal and humidification problems. This limitation has been overcome partially by imposing a uniform and time-dependent humidity field. This procedure has also been applied in Refs. [6,7]. The evolution of the humidity field in the membrane corresponds to the real conditions during the fuel cell turn-on. The mechanical behaviour of the fuel cell components and the hygro-thermal and the mechanical boundary conditions are detailed in Section 2. The results of the numerical simulations are discussed on the local and on the global scales in Section 3. At first, the time evolution of temperature, stress and deformation fields are established on the entire fuel cell scale. Then these fields are studied on the scale of a single channel. In Section 4, some parameters’ effects have been studied.
2.
Model
2.1.
Geometry and mesh definition
A two-dimensional approach is used for the model with an entire cross-section representation of a single cell (see Fig. 1). This model is similar to the one developed in paper [3] and presents the entire fuel cell including two steel plates, two graphite plates, two gas diffusion layers (GDL), a proton exchange membrane and two seal joints (see Fig. 1a and b). The catalytic layers are thinner than the membrane by one order of magnitude and it is assumed that these layers are integrated into the GDL. There are two fastening elements (bolts) allowing a realistic simulation of applied load. The schema of a half-cell is shown in Fig. 1b. Some sizes are defined in this figure, other parameters are the following: the thickness of the membrane – 0.05 mm, the thickness of the steel plates – 10 mm, the thickness of the GDE – 0.28 mm, the length of the bolt – 80 mm, the diameter of the bolt – 5 mm, the thickness of the seal joint – 0.3 mm, the width of the tooth – 0.7 mm, the width of the channel – 0.7 mm, the depth of the channel – 1 mm. A symmetry condition is used in order to reduce the size problem, i.e. there is the central axe of symmetry for all physical quantities. The dimension which is perpendicular to a plane of cross-section (2D model) equals 100 mm. Thermo-mechanical finite elements (CPEG8T and CPEG6MT) based on the generalized plane strain hypotheses are applied. The mesh consists of 13 267 elements and 41 143 nodes. Two beams (type B21 in ABAQUS) are connected with one steel plate; these connectors are CONN2D2 type in ABAQUS. The opposite steel plate is used as the support. The displacement of the beams has been calculated in order to generate the mean pressure of 1 MPa between the steel and the graphite plates in the assembly procedure [3].
2.2.
Governing equations
The mechanical properties of steel, graphite and carbon paper have been taken from Refs. [17–19]. It is assumed that these
international journal of hydrogen energy 33 (2008) 5703–5717
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Fig. 1 – (a) Fuel cell components and assembled fuel cell. (b) Schematization of fuel cell: local, global paths and characteristic points illustrating the results of calculation. 1 – Seal joints, 2 – MEA, 3 – Graphite plates, 4 – Steel plates, 5 – Bolts. a – Characteristic point in GDL (centre zone of fuel cell), b – Characteristic point in the membrane (centre zone of fuel cell), c – Characteristic point in the GDL/seal joint interface (edge zone of fuel cell), d – Characteristic point in the membrane under GDL/seal joint interface, e – Characteristic point in the seal joint.(c) Mesh used for the model.
materials have linear-elastic behaviour (see Table 1) which does not depend on the temperature and the relative humidity. The seal joint (Taconic) is assumed to be composed of three identical layers of woven fiberglass coated with PTFE (polytetrafluoroethylene). Linear elastic, perfectly plastic model following von Mises yield criterion with isotropic hardening is accepted for Nafion membrane. The hardening curve depends on the temperature and the humidity (see Table 2). These data have been included in
the model in order to consider the temperature and the humidity effects. In the classical elasto-plastic model, assuming that all deformations are small, it is possible to present the tensor of deformations as the sum: PL RH T 3ij ¼ 3EL ij þ 3ij þ 3ij þ 3ij ;
(1)
PL here 3EL ij is the elastic strain tensor and 3ij is the plastic strain tensor. Variations of the temperature and the humidity
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Table 1 – General properties Property
Table 2 – Set of the hardening curve of Nafion [9] Value
Source
Nafion membrane Density, r [kg/m3] Young’s modulus: E [MPa] Poisson’s ratio, n Yield strength, s0 [MPa] Expansion coefficient: a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
2000 See Table 3 0.25 See Table 2 123 106 1050 0.259
[7] [7] [7] [7] [7] [7] [7]
Graphite plate Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
1800 10 0.25 5 106 750 95
[7] [7] [7] [7] [7] [7]
GDE (carbon paper) Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
400 10 0.25 0.8 106 500 0.3
[17] [17] [17] [17] [17] [17]
Seal joint (SJ) (40% Woven fiberglass E – 60% PTFE) PTFE 2200; Density, r [kg/m3] Fiberglass E 2540; SJ: 2336 Young’s modulus, E [GPa] PTFE 0.8; Fiberglass E 72; SJ: 30 Poisson’s ratio, n PTFE 0.46; Fiberglass E 0.22; SJ: 0.364 Expansion coefficient, a [K1] PTFE 100 106; Fiberglass E 5 106; SJ: 62 106 1 1 Specific heat [Jkg K ] PTFE 1000; Fiberglass E 830; SJ: 932 PTFE 0.195; Conductivity [W m1 K1] Fiberglass E 1; SJ: 0.517 Steel Density, r [kg/m3] Young’s modulus, E [GPa] Poisson’s ratio, n Expansion coefficient, a [K1] Specific heat [Jkg1 K1] Conductivity [W m1 K1]
7800 209 0.25 12 106 460 0.3
[18]
[18]
[18]
[18]
[18]
[18]
[19] [19] [19] [19] [19] [19]
generate expansions which are considered to be isotropic. These expansions are represented by two tensors 3Tij and 3RH ij , correspondingly. The membrane behaviour is supposed to be elastic and isotropic; in elastic region the Hook’s law is accepted. The constitutive law can be written as follows: ! X EL E EL (2) 1 2n 3kk dij ; n3ij þ sij ¼ ð1 þ nÞð1 2nÞ k where sij is the stress tensor, n is Poisson’s ratio, E is Young’s modulus, dij is the Kroneker symbols. Young’s module
Equivalent plastic stress (Mpa) 6.60 6.60 15.40 6.51 6.51 22.20 5.65 5.65 11.60 4.20 8.54 6.14 6.14 14.90 5.21 5.21 20.80 5.00 5.00 10.80 3.32 7.56 5.59 5.59 14.40 4.58 4.58 20.10 4.16 4.16 9.96 2.97 7.31 4.14 4.14 12.90 3.44 3.44 11.20 3.07 3.07 8.87 2.29 6.63
Equivalent plastic strain
Temperature (K)
Humidity (RH%)
0 0.167 2.09 0 0.194 2.06 0 0.139 2.06 0 1.93 0 0.167 2.09 0 0.194 1.99 0 0.139 2.04 0 1.91 0 0.167 2.06 0 0.194 2 0 0.139 2.03 0 1.88 0 0.167 2.06 0 0.194 2.03 0 0.139 2 0 1.85
298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358 298 298 298 318 318 318 338 338 338 358 358
30 30 30 30 30 30 30 30 30 30 30 50 50 50 50 50 50 50 50 50 50 50 70 70 70 70 70 70 70 70 70 70 70 90 90 90 90 90 90 90 90 90 90 90
depends on the temperature and the humidity; the corresponding data are presented in Table 3. Thermal strain is determined by the classical linear expansion law: 3Tij ¼ aðT T0 Þdij ;
(3)
where a is the linear thermal expansion coefficient, T is the current temperature, T0 is the reference temperature. The thermal expansion coefficients are given in Table 1. Following the work [7], the swelling-expansion strain is expressed as a polynomial function of humidity and temperature in order to take into account the expansion of the membrane from the initial state (35%RH and 20 C) to the work
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Table 3 – Set of the Young’s modulus curve of Nafion [7] Young’s modulus (Mpa) 197 161 148 121 192 137 117 85 132 103 92 59 121 70 63 46
Temperature (K)
Humidity (RH%)
297 317 337 357 297 317 337 357 297 317 337 357 297 317 337 357
30 30 30 30 50 50 50 50 70 70 70 70 90 90 90 90
3RH ij ¼ dij
Ckl T4k RH4l ;
(4)
k;l¼1
where RH is the relative humidity, Ckl is the polynomial constants, see Ref. [7]. The subroutine UEXPAN has been developed for introducing the swelling strains in the membrane. This subroutine takes into account Eq (4). The similar moralization of swelling strain has been used in works [7,8]. The plasticity behaviour of the membrane is described by Prandtl–Reuss theory. The von Mises yield function is defined as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffi 3 (5) Sij Sij s0 ; f sij ¼ 2 and the von Mises yield criterion is presented as: f sij ¼ 0;
(6)
where sij is the stress tensor. The term s0 ¼ s0(T, RH) is the yield strength depending on the temperature and the relative humidity (see Table 4); Sij ¼ sij (1/3)skkdij is the deviator stress tensor. When the inequality f(sij) < 0 is verified, the material has elastic behaviour. The plasticity evolutions are described by isotropic hardening curves of Nafion which are taken from the work [9]. Due to the lag of data concerning the dependence of the hardening curve on the humidity, the interpolation has been accepted taking into account the data for a reference temperature (see Table 2). According to the Prandtl–Reuss theory, the plastic increment tensor is proportional to the derivative of the yield function (5): vf d3ij ¼ dl; vsij
Yield strength (Mpa) 6.60 6.51 5.65 4.20 6.14 5.21 5.00 3.32 5.59 4.58 4.16 2.97 4.14 3.44 3.07 2.29
state (100%RH and 85 C). The swelling strain tensor can be written down as follows: 4 X
Table 4 – Yield strength at various temperatures and humidities of Nafion [7]
(7)
where dl is the plastic multiplier calculated by the consistent relation (df ¼ 0) during the hardening. The plastic multiplier corresponds to the increment of the equivalent plastic strain.
2.3.
Temperature (K)
Humidity (RH%)
297 317 337 357 297 317 337 357 297 317 337 357 297 317 337 357
30 30 30 30 50 50 50 50 70 70 70 70 90 90 90 90
Hygro-thermal and mechanical load
In this part the main operating parameters, namely the temperature and the humidity, are introduced considering the fuel cell working regime. The mechanical loading is defined in the same manner as in the paper [3].
2.3.1.
Mechanical load
This value of loading 1 MPa has been accepted in Refs. [6–8] and is used as the reference in the present work. The mechanical loading in the model is simulated by the displacement of the bolt heads. It is assumed that this displacement remains constant during the fuel cell running. This displacement reflects the loading caused by application of a moment of forces to the bolts during the cell assembly. It has been calculated [3] that the displacement dIMP ¼ 0.038 mm corresponds to the normal stress 1 MPa on the interface between graphite and steel plates. The type of bolts M6 80 has been assumed in our calculation. The required displacements corresponding to the normal stress 1 MPa for others types of bolt are presented in Table 5.
2.3.2.
Thermal load
Heat generation in fuel cells have been studied in numerous works, as for example Refs. [10–12]. At working regime, the main source of heat generation concerns irreversible reactions and entropic losses in the cathode catalytic layer (85–90%) and Joule heating of the membrane (8–12%) [10]. Thus, almost all heat is produced near or inside the membrane. The intensity
Table 5 – The bolt displacements corresponding to the normal stress 1 MPa for different types of bolts Type of bolts DIN558 M5 80 M6 80 M8 100 M10 100
Displacement [mm] 0.0380 0.0264 0.0186 0.0119
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of the heat source Q ¼ kP is directly related to the power generation P and the efficiency of the fuel cell k. In our model the continuity of all physical parameters along the fuel cell is accepted. This assumption does not permit to introduce surface heat sources in ABAQUS package. That is why we have interpreted the heat sources as the volume ones. Using the thickness of the Nafion 112 membrane (D ¼ 50 mm) and accepting that k ¼ 0.4 and P ¼ 0.75 W/cm2, it is possible to find the volume heat density h ¼ 1.4 108 W/m3. In our model the time-dependent variation of the heat generation is accepted h ¼ h(t), see Fig. 2. The initial temperature is equal to T ¼ 300 K. The boundary condition at the external surface of the fuel cell is determined by natural convection. The convective heat exchange coefficient between air and external fuel cell surface is set to 10 Wm2 K1 [13]. An external heating is applied for obtained and maintaining of the operating fuel cell temperature (T ¼ 80 C). In real experiments, this function is attained by using resistors integrated in the steel plates. They generate heat by Joule effect. It is assumed that this heat source is distributed by the steel plate volume and its intensity is equal to 8.9 106 W/m3. The external heating is imposed before the fuel cell turn-on. After reaching the operational temperature the intensity of the external source decreases to the value which allows compensation of the heat losses due to the heat exchange with environment (see Fig. 2). The temperature of moving gases in graphite plates varies during turn-on regime; this evolution is specified in Fig. 2. In the beginning the temperature of the gases is equal to 300 K. According to experimental data [5], the temperature reaches the value of 320 K in the steady-state conditions. The convective heat transfer coefficient between the gases and the channel wall (hf) is assumed to be equal to hf ¼ 110 Wm2 K1 [13].
2.3.3.
Humidity load
Water transport in the proton exchange membrane is a complicated process. It is governed by nonlinear diffusion and by electro-osmotic drag, see for example Refs. [14–16]. In the present model, the moisture is set gradually from an initial value of 35% up to 100%. The humidity is imposed after all heat sources reach steady state, see Fig. 2. The characteristic
time of the membrane humidification is set to 40 s, taking into account the real experimental data [17]. It is suggested that the imposed moisture is uniformly distributed in the membrane. This questionable assumption leads to overestimation of the maximal stresses in the membrane during turn-on stage.
3. Stress and strain distributions in running fuel cell In this article calculations and data treatment of stresses are provided on the single channel scale (local effects) and on the entire fuel cell level (global effects). The local analysis has been performed in the cell centre region (near the axis of symmetry) and in the cell edge region (see Fig. 1b, the small dots in the zoomed pictures). Both regions include two channels to emphasize a periodic character of the local fields. The stresses distribution on the scale of the entire cell is presented with a particular global path. This path goes through the membrane centre points which are situated in front of the lowest corners of the channels (see Fig. 1b, the big dots in the zoomed pictures). The path goes also under the seal joint in order to visualize the stresses in the edge zone. At last, the special points are selected for observing the temporal evolution of stresses in MEA and the seal joints (see ‘a–e’ Fig. 1b in the zoomed pictures). The normal direction 1 is perpendicular to the MEA and the transverse directions 2 and 3 are situated in the plane of the MEA; the direction 3 is perpendicular to the plane of the model (see Fig. 1b). The calculations are provided up to the establishing of the steady state. In our model, the steady-state regime is achieved after about 1400 s (see Fig. 3a). In the steady-state regime the heat balance between the fuel cell and the environment is attained and the temperature changes become negligible. At the first moments (between 4 and 30 s) the temperature evolution is governed by the external heating of the steel plates. After the working temperature is reached (40–50 s), the intensity of the external heating decreases, and the supply of hydrogen starts. Then, the temperature evolution is governed mainly by chemical heat generation and by the heat exchange
Fig. 2 – Fuel cell loading curves.
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Fig. 3 – (a) Time evolution of temperature at characteristic points. (b) Time evolution of Mises stress at characteristic points.
between the gas channels and the MEA. Before supplying of hydrogen, the temperature gradients exist in the active part of the cell (GDL, membrane). The temperature difference between the seal joint (the hottest point) and the centre of MEA is about 5 K. While hydrogen supply is turned on, the inversion between the hot and the cold zones takes place and the membrane warms up, while the temperature of the seal joints remains constant (Fig. 3a, around 50 s). After this stage, a homogeneous evolution of the temperature in the joints and in the membrane occurs. The stress distributions have been calculated for the entire cell scale during all the transition stage. The variations of the Mises stresses are shown in Figs. 3b and 4a for different locations (see points ‘a–e’ in Fig. 1b). The Mises stresses change in the graphite in accordance with the temperature evolution. At the same time, the peak can be observed in the membrane for a short time with the following weak decrease
which asymptotically tends to be about 2.7 MPa. The stresses in the joint correspond to the global changes in the temperature but in less remarkable way than the ones in the GDL. The stresses change strongly during the first 100 s of the cell running; afterwards they stabilize with the achievement of the temperature steady-state regime. The step of cold assembling procedure which has been studied in the paper [3] corresponds to the period between 0 and 1 s. The strongest changes correspond to the phase of pre-heating of the steel plate (between 4 s and 30 s). During this phase, the membrane stresses under the GDL rise in accordance with the temperature. On the contrary, in the very beginning of the heating, the stresses under the seal joint decline, but increase later. This is a result of the competition between the expansion of the different elements and the membrane Young modulus decrease caused by temperature increase. During pre-heating, the stresses in the membrane
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a
b
0.25
center point b
under point e
edge point d
equivalent plastic strain
0.2
0.15
0.1
0.05
0 0
50
100
150
200
time (s) Fig. 4 – (a) Time evolution of Mises stress in the membrane for the beginning of turn-on. (b) Evolution of equivalent plastic strain in the membrane at the edge (d) and at the centre (b).
under the seal joint increase more significantly than in the assembly step. After pre-heating, the heat exchange caused by the fuel cell turn-on (about 50 s) leads to a very small increase of the stresses in the MEA elements. This is a second order effect with respect to the humidification. In our model the humidification load starts at 60 s and generates strong variations of the stresses in the membrane which are the same order of the magnitude as the ones during the pre-heating stage. The swelling strain generates stresses which are modified also due to dependence of the mechanical properties on the humidity. The new effect appears: the stresses are governed by plastic deformation as well. The first plastic deformations in the membrane emerge during the stage of the humidification, see Fig. 4b. In the beginning of this stage the stresses are not sufficient and the yield stress is not achieved. The plasticity becomes visible at about 70 s at the joint/GDL interface (point d). The plastic deformation in the centre of the membrane arises later;
however, it reaches a higher level. The plasticity appears also in the membrane under the seal joint; but this effect takes place later and its magnitude is less important (see Fig. 4b, the curve e). The plastic deformation remains constant at the end of the humidification stage; it is important to note that its level is very high. Distribution of the Mises stress in the middle plane of the membrane is shown on Fig. 5a and b for the scale of the entire fuel cell. The first curve corresponds to the end of the assembly step (t ¼ 4 s on Fig. 2) where heterogeneous stresses on the joint/GDL interface are already presented. During the period between 4 s and 30 s the fuel cell pre-heating works. At the first moments, the Mises stress decreases along all the membrane except the borders under the seal joints. After 14 s, the Mises stress increases along all the membrane and the heterogeneity becomes more and more important, particularly under the interface joint/GDL. During the humidity loading the Mises stress in the centre of the membrane
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a
b
t=59 heating in membrane
t=64
t=74
t=84
t=94
t=104 transient state
humidity loading
4 seal joint
diffusion layer
3.5
Mises stress (MPa)
3
2.5
2
1.5
1
0.5
0
edge 0
center 0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m) Fig. 5 – (a) Space evolution of Mises stress along the membrane (global path) before humidification loading. (b) Space evolution of Mises stress along the membrane (global path) during humidification loading.
increases (see Fig. 5b); at the same time, the Mises stress declines in the membrane edge under the seal joint. Around 74–84 s the discontinuity at the GDL/seal joint interface is very prominent. After that, the stress field becomes homogeneous very rapidly. The stress distribution along the membrane has a complex character during the transient phase. This distribution stabilizes at the end of the humidification step and becomes quasi-
homogeneous when the temperature field reaches steady state (see t ¼ 104 s, Fig. 5b). It can be noted that the maximal stresses (higher than 3.5 MPa) occur during the humidification step. These peak stresses are localized under the joint/GDL interface; in this part of MEA the failures are especially often observed. After the end of the humidification (100–1500 s) the stresses decrease, and the discontinuity on the joint/GDL interface becomes visible again (Fig. 6a).
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a
t=161
t=261
t=661
t=1161
t=1661
transient to permanent state
4 seal joint
diffusion layer
3.5
Mises stress (MPa)
3 2.5 2 1.5 1 0.5 0
edge 0
center 0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
Fig. 6 – (a) Space evolution of Mises stress along the membrane (global path) after humidification loading. (b) Space evolution of the normal stress s11 along the membrane (global path) during all turn-on stages.
So, the fuel cell turn-on produces complex distribution of stresses along the membrane. This distribution is affected by expansions of different MEA components, increase of the temperature and, for the most part, by the membrane moisture which leads to the membrane plasticity (see Fig. 5b, curves corresponding to t ¼ 64 s and t ¼ 94 s). The evolutions of the stress tensor components in the middle of the membrane are presented in Figs. 6b and 7a, b. The stresses correspond to a biaxial state. The normal and the tangential components have the same order of magnitude. Note that the tangential component is slightly higher. This result should be considered with some caution because the model ignores any slipping between MEA elements.
The normal stress (Fig. 6b) has large discontinuity under the joint/GDL interface in the beginning of the cell running (59 s) which decreases later. The significant value of the compression stress near the free surface of the cell at the steady state should be noted. This is an interesting result with respect to the airtight problem. Firstly, in the centre of the cell the normal stress in the membrane increases gradually and afterwards it decreases slowly during the reaching the steady state. The tangential stress rising from the centre to the edge of the membrane (Fig. 7a) increases in time during all the steps. This stress is directly related to the expansion of the MEA components. The discontinuity increases also with time (Fig. 7a and b).
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a
t=4
t=59
t=84
t=104
seal joint
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t=1661
diffusion layer
0
Tangential stress (MPa)
-1 -2 -3 -4 -5 -6 -7 -8 edge
center
-9 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
t=4 0.3
t=59
t=84
seal joint
t=104
t=1661
diffusion layer
0.2
Shear stress (MPa)
0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 edge
center
-0.6 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m) Fig. 7 – (a) Space evolution of the tangential stress s22 along the membrane (global path) during all turn-on stages. (b) Space evolution of the shear stress s12 along the membrane (global path) during all turn-on stages.
The shear stresses (Fig. 7b) are smaller than the normal and the tangential ones (Fig. 7a); their level is lower than 1 MPa. The changes of stresses sign can be noted between the phase of the assembling and the step of the pre-heating; this is the consequence of the thermal gradients in the perpendicular to the membrane direction. The stresses decline during the last period (100–1400 s), however, the discontinuity is present constantly under joint/GDL interface. Plastic deformations occur under joint/GDL interface around 70 s during the phase of the membrane humidification (Fig. 8a). These deformations develop considerably (at first under the GDL and after under the joint) up to the end of the humidification when they reach the maximal magnitude. Afterward the plastic deformations remain constant.
It is important to emphasize the model limitations. Reducing of the membrane behaviour to the elasto-plastic law with isotropic hardening leads to the inherent stabilisation of the stress distributions. In the future it is important to introduce the models with viscosity in order to take into account the creep-relaxation effects. In particular, it is possible to introduce a rate dependent viscoplastic model using the curves presented in the paper [20]. As pointed by these authors, a viscoelastic – viscoplastic model is more adapted for description on Nafion behaviour. The existence of the periodic changes on the scale of a single channel (see Fig. 8b) has been already discussed in the paper [3]. The growing temperature causes (see Fig. 8b, t ¼ 59 s) the inversion of the stress distribution: the maximum values
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a
t=72
t=80
t=84
sealjoint
0.25
t=94
t=104
diffusion layer
equivalent plastic strain
0.2
0.15
0.1
0.05
edge
center
0 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
global path (m)
b
t=4
t=59
t=84
t=104
t=1661
3.5
Mises stress (MPa)
3
2.5
2
1.5
1
0.5
0 0
0.0005
0.001
0.0015
0.002
0.0025
local path (m) Fig. 8 – (a) Space evolution (global path) of the plastic equivalent strain in the membrane. (b) Space evolution (local path) of Mises stress in the centre of the membrane for different times.
become the minimum ones and vice versa. The stress distribution becomes more homogeneous along the channels but a strong discontinuity remains under the joint/GDL interface. The local variations of stresses become negligible compared to the mean value. The magnitude of the Mises stresses in the centre and at the edge of the cell increases gradually up to the end of the membrane humidification (around t ¼ 84 s). Later on, when the fuel cell tends to the steady state, the Mises stress in the middle plane of the membrane decreases and reaches the level of about 2.75 MPa at the centre and at the edge. The local distribution of the stress under the joint/GDL interface during the transitional phase is presented in Fig. 9a. The maximal value of 3.5 MPa is achieved during the humidification step.
The stress maximums are localized in this zone at the steady state as well. It should be noted that significant changes of the stresses exist under the joint/GDL interface before the humidification of the membrane. Apparently, this is a source of the fatigue limitations generated by the sequence of turnons and shutdowns of the fuel cell. After reaching the steady state, all stress components change on the local scale, see Fig. 9b. Nevertheless, the stress changes in the centre are smaller than at the edge of the cell. The shear stresses are small both in the centre and at the edge. The tangential stress is more evident at the boarder of the cell. The normal stress demonstrates the same evolution with the moderate magnitude as the mean value near the edge of the cell (under the seal joint); this magnitude becomes
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a
b
s11 center s11 edge
s12 center s12 edge
s22 center s22 edge
0 -1
stress (MPa)
-2 -3 -4 -5 -6 -7 -0.002
-0.001
0
0.001
0.002
0.003
local path (m) Fig. 9 – (a) Space evolution (local path) of Mises stress at the edge of fuel cell during transient period. (b) Space evolution (local path) of stresses at the centre and at the edge of fuel cell in the steady state.
relatively small in the centre. The presented results illustrate once more the heterogeneities of the stress distributions which arise in the membrane during fuel cell running.
4. Influence of model parameters on mechanical stresses The goal of this paragraph is the study of the influence of some model parameters on mechanical stresses arising in fuel cell.
4.1.
thus the membrane stiffness is higher and swelling generates higher stresses. This peak decreases more slowly and stabilizes on the higher level in the case when the pre-heating is absent. The evolution of the stresses in the centre of the membrane and under the GDL/seal joint interface is identical, contrary to the case of pre-heated fuel cell. It should be noted that in our simulations the steady-state thermal regime is not reached for the cell without pre-heating, that is why a slow variations of stresses can be observed at the end of the simulation. To summarize, the mode of the cell heating seriously influences the maximal stresses distribution and stimulates the mechanical fatigue of the system.
Influence of pre-heating 4.2.
Pre-heating can influence significantly the lifetime of fuel cells. Numerical simulations have been provided for estimation of the influence of this action on the evolution of the mechanical stresses; the results are presented on Fig. 10. It can be noted that excluding of pre-heating provokes more important peak of stresses during the first moments of the turn-on stage. Indeed, in this case the temperature is lower,
Influence of the seal joints stiffness
The provided simulations point towards the increasing of stresses on the GDL/seal joint interface. These significant variations are caused by the particularities of the materials properties, in particular, by the stiffness differences. Thus, it is interesting to investigate the influence of seal joints stiffness on stress distribution. We have investigated different
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Fig. 11 – (a) Time evolution of Mises stress in Nafion during steady-state stage. Influence of seal joint stiffness. (b) Time evolution of Mises stress in Nafion during transient stage. Influence of seal joint stiffness. Fig. 10 – (a) Time evolution of Mises stress in Nafion during steady-state stage. Influence of pre-heating. (b) Time evolution of Mises stress in Nafion during transient stage. Influence of pre-heating.
tend to the same level at steady state. During the humidification stage, the stresses are not influenced by joint stiffness.
5. types of seal joints: for the soft ones the material properties are similar to PTFE and for the rigid joints the material properties correspond to fiberglass. A uniform thickness of the joints is assumed in our simulations. Time evolution of Mises stresses at critical points is presented on Fig. 11 for three types of joints. In the steady state the effect is not important; a small diminishing of Mises stress can be noted with the decreasing of the joint stiffness. On the other hand, in the transient regime the influence of the joint stiffness is more pronounced. During the pre-heating stage, the stresses increase with the rising of the joint stiffness and reach the maximal level of 4 MPa near the border of the cell. This level surpasses by the factor 4 the initial loading and is 30% higher than steady-state value. In the centre of the fuel cell the stresses are independent on the Young’s modulus. For the soft joint (PTFE) the results are different. The maximal value of stresses at the fuel cell border is reached during the humidification stage due to membrane swelling. In this case, the maximal stresses arise in the centre of the membrane; their level exceeds the steady-state value less than 20%. It is interesting to note that for all joints, stresses
Conclusions
Two-dimensional numerical modelling of mechanical stresses in MEA of a running fuel cell is developed. The evolution of stresses and plastic deformations in the Nafion membrane has been studied during the start-up phase. The model considers the initial mechanical loading arising during the cold assembly procedure and takes into account the mechanical action of the flexible bolts. The mechanical stresses in MEA have been simulated by imposing the heating sources and humidity field with respect to real operating conditions in running fuel cell. The results have been obtained for both the global and the local scales. The global scale reflects the stress evolution in the entire fuel cell and the local one corresponds to the tooth/channel structure. The calculations show that the stresses are strongly heterogeneous on the both scales and evolve in time during the transient stage. The stresses arising in the membrane under the joint/GDL interface due to the assembly decrease during the transition (turn-on) stage. The temperature field reaches the steady-state conditions around 1200–1400 s. The maximal stresses in the membrane take place during the
international journal of hydrogen energy 33 (2008) 5703–5717
humidification step before the temperature comes to its steady-state value. The plastic deformations in the membrane develop during the entire humidification step. At the steady state the stresses have the highest value in the centre of the membrane; the Mises stress is equal to 2.5 MPa. The mechanical charging of the membrane corresponds mainly to biaxial compression which is accompanied by the small transversal shear stress. After the beginning of the cell running the stresses on the local scale appear to be more homogeneous than the ones after the assembly. The peak stresses have been revealed at the transient stage. This is an interesting result, which allows understanding of the mechanical aging phenomena of MEA structures while multiple turn-ons and shutdowns of the fuel cell. The effects of joint stiffness and pre-heating on Mises stress in Nafion have been established. For the quantitative estimation of the mechanical durability of the MEA during the cycle loading the model should be improved, namely by more precise simulation of the humidification loading. Moreover, the viscous effects should be introduced in the modelling of the polymer membrane behaviour in order to predict the MEA aging during work cycles.
Acknowledgment The authors thank the Federation of laboratories P’PRIMME of Poitiers and the Region Poitou – Charentes for the financial support of this work.
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