# Mechanical stability.pdf

Mechanical stability A. Atkinson and A. J. Marquis Imperial College, London, UK 1 INTRODUCTION The aim of this article is to describe the mechani- cal stability issues that arise in solid oxide fuel cells (SOFCs), how they are modeled and how the likeli- hood of mechanical damage to the SOFC’s structure is assessed. It focuses on the ceramic components because they fail in a brittle manner and the result can be com- plete malfunction of the cell or stack. The areas covered are sources of stress in the components that are gen- erated by thermal or chemical gradients, or differences in coefficients of thermal expansion (CTE); the theoret- ical background to modeling stress distributions; failure processes if stresses are too high; and data for critical mechanical properties. Finally, a case study is described as an example to illustrate how these different aspects are integrated in assessing the reliability of a typical SOFC structure. 2 SOURCES OF STRESS IN SOFCs Here we consider stresses that are driven by differential strains within, or between, materials. In addition, stresses arise from externally applied loads, such as when compres- sion seals are employed. These are specific to particular stack designs and are not considered here. However, they should be added to the strain-driven stresses to obtain the total stress distribution in the structure. Thermal stresses are the result of differences in thermally induced stress-free strain between different components or in a single component in a temperature gradient. The stress-free thermal strain is given by ε t = α(T ? T 0 )(1) where α is the CTE and T 0 is a reference temperature. The constraints applied to any particular component by the rest of the structure then convert the stress-free strains into stresses. 2.1 Residual stresses from manufacturing Residual stresses are due entirely to differences in thermal expansion coefficients of the different component mate- rials as they cool from high manufacturing temperatures where they are almost stress free. The residual stresses are maximum at room temperature. It is often the case that one component in the structure is far stiffer mechanically than the rest; for example, the substrate in supported fuel cell structures. In such cases, the other components will be forced (as long as they are adherent) to be compati- ble with the stress-free strains and displacements of the stiffest component that remains almost stress free because of its greater stiffness. In planar structures (that are also constrained to remain planar), the stress state remote from edges and discontinuities is equibiaxial in the plane. The resulting stresses in the other components give an elastic strain contribution ε e = σ(1 ? ν) E (2) Handbook of Fuel Cells – Fundamentals, Technology and Applications. Edited by Wolf Vielstich, Hubert A. Gasteiger, Arnold Lamm and Harumi Yokokawa. ? 2010 John Wiley G c , fracture energy; σ f , failure stress). Material CTE ppm K ?1 (25–1000 ? C) E (GPa) G c (plane strain) J m ?2 σ f (MPa) 8YSZ [1] 10.5 215 12 416 3YSZ [2] 10.8 211 140 770 Porous LSM [3] 12.4 35 13 46 40% Porous Ni/YSZ [4, 5] 13.1 50 100 51 where E is Young’s modulus and ν is Poisson’s ratio. The total strain is ε tot = ε t + ε e (3) and must be equal for all the components in the planar structure. In applying this simplified approach, it is necessary to identify the stiffest component and the nature of the con- straint. This clearly depends on the fuel cell configuration. Illustrative values for thermomechanical properties of some typical SOFC materials are given in Table 1. When the approximation of a dominant stiff component is not valid, then the conditions for mechanical equilibrium must be explicitly satisfied. These are that the total force and total moment acting on an element of the structure must be zero. If a planar structure is constrained to remain flat by application of external moments, then the problem simplifies to maintaining zero total force. In a planar structure, this becomes integraldisplay h 0 σ 11 dx 3 = 0 (4) where x 3 is the position coordinate perpendicular to the plane of the cell and h is the total thickness. If the cell is allowed to bend, then the requirement for zero total moment brings the additional condition: integraldisplay h 0 σ 11 x 3 dx 3 = 0 (5) These equations can be solved [6] to give the stress distributions and curvature of unconstrained cells. Residual stresses can be measured using various tech- niques, of which the most reliable is probably X-ray diffrac- tion. For example, Fischer et al. [7] report compressive stress of 520 MPa in the electrolyte of free-standing anode- supported cells at room temperature, measured by X-ray diffraction, which is broadly consistent with the thermome- chanical properties they measured for their materials and assuming a stress-free temperature of 1200 ? C. Seals in planar SOFCs can be subjected to particularly high residual stresses. These arise not only from the thermal strains where the seal is in contact with different materials (e.g., electrolyte and bipolar plate) but also from the differential displacements between the materials in contact with the seal. These are larger in larger cells and therefore sealing stresses are also larger for larger cells. 2.2 Operating stresses due to thermal gradients Temperature gradients arise in operating SOFCs (cells and stacks) due to the electrochemical reactions taking place and the cooling effects of gas flows. The temperature gradients cause different amounts of thermal expansion, and hence differential strains and resulting stresses, even in a single material. 2.3 Thermal cycling Stresses during thermal cycling arise from both thermal expansion mismatches and also thermal gradients. [8] The latter depend on the size and geometry of the stack, the rate of cooling and the heat-transfer processes occurring within the stack. The thermal gradient stresses are likely to be highest at the start of the cooling or heating parts of the cycle. The thermal expansion mismatch stresses, on the other hand, are present even at very slow cooling rates and are highest at the lowest temperature in the thermal cycle. To a first approximation, they are the same as the residual stresses, provided no stress relief (plastic deformation) has occurred at the operating temperature. 2.4 Chemical stresses (e.g., redox and chemical expansion) The bulk dimensions of many SOFC materials depend on their chemical environment, and in SOFCs this is primar- ily the thermodynamic activity of oxygen. For example, many oxides lose oxygen when the temperature increases or when the oxygen activity is reduced and the result- ing oxygen vacancies cause an expansion of the lattice, the so-called chemical expansion. Cobaltite cathodes are particularly susceptible in this regard in oxygen, and ceria- based electrolytes and oxide anodes in fuel conditions. The changes with temperature can be described by an abnor- mally high thermal expansion coefficient, but changes with oxygen activity must be accounted for by adding an extra strain term to equation (3). Ni-based composite anodes expand on oxidation when the Ni is oxidized to NiO. This can also be described by a chemical strain, but this is now a property of the composite and depends on the structure Mechanical stability 3 of the composite and the number of cycles of reduction and oxidation (redox). [9] 2.5 Stress-relieving processes (e.g., creep) The damage caused by stresses is discussed in detail later in this article. Processes that can relieve stress without fracture are plasticity and creep (although they do result in irreversible distortions). Plasticity is an instantaneous strain that occurs (by slip of dislocations in crystalline materials) when the stress exceeds the yield stress. In ceramics, this is a difficult process, because the yield stress is high and crystallographic slip systems are limited, and is usually neglected. Creep is a time-dependent process and is assisted by grain boundaries that provide relatively fast pathways for transport of atoms and for sliding between grains. It is therefore favored by high temperatures and small grain sizes. Viscoplastic creep typically passes through three stages. Initially, the strain rate decreases with increasing strain; this is known as primary creep. The strain rate eventually reaches a minimum and becomes near-constant; this is known as secondary or steady-state creep and this regime is best understood and accounts for the largest proportion of total creep deformation. In the third and final stage, tertiary creep, the strain rate increases exponentially with stress until failure occurs. 3 THEORY OF STRESS MODELING 3.1 Timescales of key physical processes (e.g., thermomechanical and creep) The temperature distribution in the SOFC depends on the balance between and location of the net heat production (electrochemical, and ohmic heating) and heat losses (radi- ation, convection, and conduction) throughout the fuel cell. The rate of change of the temperature distribution depends on the imbalance between the heat production and losses and the thermal inertia of the cell. These processes occur over timescales that vary from nearly instantaneous for the electrochemical and radiation to 1–100 ms for the conduc- tion and 0.1–10 s for the convection processes; in contrast the timescales for creep are in the range 300–3000 h at working temperatures, depending on the temperature and stress level. 3.2 Elastic behavior The constitutive relationship for an elastic material at equilibrium at temperature T 0 subject to a temperature rise (T ? T 0 )isgivenby σ ij = αE(T ? T 0 )δ ij + E ijkl ε kl (6) where E ijkl is the elasticity tensor defined for isotropic materials. This represents the combined effects of strain due to thermal expansion and elastic strain (Hooke’s Law). 3.3 Creep behavior In the steady-state creep region, the rate of change of strain over time may be described by ˙ε = Aσ n c L ?p P(O 2 ) m exp bracketleftbigg ? Q c RT bracketrightbigg (7) where σ is the applied stress, n c is the stress exponent, L is the grain size, p is the grain size exponent, P(O 2 )isthe oxygen partial pressure and m is the oxygen partial pressure exponent, Q c is the creep activation energy, R is the universal gas constant, and T is the absolute temperature. Experiments have shown that m = 0 (i.e., the creep rate is not dependent on oxygen partial pressure), but is dependent on the grain size in doped lanthanum gallate (LSGM) and YSZ electrolytes. Some typical parameter values are given in Table 2. There are several mechanisms by which creep can occur in polycrystalline materials such as nickel and zirconia, all based on the movement of lattice defects such as vacancies or dislocations, through the grains or around the boundaries. They can be broadly categorized as thermally activated glide, climb-assisted glide, climb, and diffusion. The diffusive processes can be further subdivided into grain boundary diffusion and lattice diffusion, known as Coble and Nabarro-Herring creep respectively. There are two factors that control which diffusive mechanism is active, the diffusion activation energy for each process, and the diffusive path length. For a given material, the lattice diffusion activation energy Q l associated with Nabarro- Herring creep is typically larger than the grain boundary diffusion activation energy Q gb associated with Coble creep, while the diffusive path length is generally shorter through the lattice than around the boundary. Coble creep Table 2. Material creep parameters used in the model (equa- tion 7). Material n c Q c (kJ mol ?1 ) A Electrolyte (YSZ) [10] 1 390–460 40 Cathode (LSM) [11] 1.3 405–521 5700 Anode (Ni/YSZ) [12] 1 390–460 40 4 Materials for high temperature fuel cells is active at lower temperatures than Nabarro-Herring creep and at SOFC operating temperatures it is considered to be the dominant creep deformation mechanism, in which case n c = 1andp = 3inequation(7). 3.4 Numerical modeling The detailed modeling of the stresses in the SOFC, except in simplified cases, requires the construction of a geo- metrically similar model consisting of elements that are interconnected and linked with the outside world via bound- ary conditions. The equations of equilibrium are writ- ten for each of these elements along with compatibility conditions and material relationships linking the stresses and strains/displacements (constitutive relationships). This results in a large system of simultaneous equations whose solution describes the displacements of each of the ele- ments that make up the SOFC. The material relationships in the case of an SOFC need to describe the thermal stresses arising from temperature gradients and differences in coef- ficients of thermal expansion and the effects of creep, all of which require knowledge of the temperature distribution in the structure. The determination of the temperature distribution throughout the SOFC requires the convolution of all the heat generation, flow, and loss processes, which in turn depend on the chemical and electrochemical reactions and the flow of heat and material. The equations describing the physical processes associated with the transport of chemical species, heat, and current flow are a set of coupled, nonlinear partial differential equations that can be modeled computationally using the finite volume method. The equations are integrated over a small (finite) volume, more often referred to as a ‘cell’, associated with each point in space. The cells are arranged such that their faces touch each other to form a continuous computational domain; this process is known as discretization. The calculated variables are then stored at the cell centroid. An interpolation procedure is used to express the variables at cell centroids as cell vertex values and any divergence of a variable in the volume integral is converted to surface fluxes on the cell faces using Gauss’ divergence theorem. For the particular conditions that exist in the SOFC electrodes, the diffusive transport of species under a con- centration gradient is complicated further because the mean free path of the gas molecules is of the same order as the mean pore diameter and therefore two transport mecha- nisms are active, Knudsen flow and continuum diffusion. Knudsen flow is active when the molecular mean free path is much greater than the pore diameter and therefore molecule–wall interactions dominate, whereas continuum diffusion is active when the mean free path is much smaller than the pore diameter and molecule–molecule interactions dominate. It would be computationally difficult to model all of these processes in full dynamic detail, i.e., resolve all the timescales of the physical processes taking place, so the approach often taken is to resolve the mechanical timescales and to consider the electrochemical and thermal processes, which prov