SUMMARYBoundary conditions come from Nature. Therefore these conditions exist at natural boundaries. Often, owing to limitations in computing power and means, large domains are truncated and confined between artificial synthetic boundaries. Then the required boundary conditions there cannot be provided naturally and there is a need to fabricate them by intuition, experience, asymptotic behaviour and numerical experimentation. In this work several kinds of outflow boundary conditions, including essential, natural and free boundar conditions, are evaluated for two flow and heat transfer model problems. A new outflow boundary condition, called hereafter the f?ee boundary condition, is introduced and tested. This free boundary condition is equivalent to extending the validity of the weak form of the governing equations to the synthetic outflow instead of replacing them there with unknown essential or natural boundary conditions. In the limit of zero Reynolds number the free boundary condition minimizes the energy functional among all possible choices of outflow boundary conditions. A review of results from applications of the same boundary conditions to several other flow situations is also presented and discussed.KEY WORDS Open boundary conditions Backward-facing step Unbounded flow Free boundary condition
and cracks between the Li metal and the SE (such as Li 7 La 3 Zr 2 O 12 , LLZO, and Li phosphorus oxynitride, LiPON), and eventually penetrate the SE. [5][6][7] Another critical problem is the interfacial instability arising from the contact loss at the Li/SE interface during stripping, which lowers the battery's cyclability and ultimately causes cell failure. [8][9][10][11][12] Thus, the dynamic behavior of the mechanical contact at the Li/SE interface needs to be understood to design a better battery cell.A challenge to maintain mechanical contact at the Li/SE interface is void formation. [13][14][15][16] Void formation leads to interfacial porosity, surface roughness, and consequently contact loss. [17][18][19] Recently, experimental characterization has shown that the stack pressure is an important factor in preventing void formation during stripping in SSBs. [18,20,21] It has been proposed that the pressuredriven creep deformation of Li metal replenishes the void at the Li/SE interface. [18,22] However, void formation at the solid-solid interface involves stress, contact, reaction, and Li/Li + transport, which are challenging to observe and measure experimentally. Therefore, the understanding of contact issues during stripping is still in its infancy. Specifically, the fundamental questions as to how the external pressure and current as well as intrinsic material properties impact the internal void formation at the Li/SE interface are unanswered.Taking a deeper insight into the mechanism of interfacial void formation, when applying current density and stack pressure, the stripping current removes electrons from Li metal and releases Li + into the SE to migrate away from the interface (i.e., the flux of Li + migration away from the interface, J migration ). This generates a large number of vacancies in Li metal near the interface. The flux of the vacancies contributed by the Li metal creep, J creep , and diffusion, J diffusion , can transport the vacancies away from the interface and towards the bulk Li metal, as illustrated in Figure 1a. Recent kinetic Monte Carlo (KMC) simulations [23] show that for an ideal flat Li/SE interface, J diffusion is high enough to transport the vacancies away from the interface and maintain a smooth Li/Li 2 O surface even without the stack pressure (i.e., J diffusion > J migration where J # represents the magnitude of the flux), as illustrated in Figure 1b. However, such an ideal flat interface is unlikely due to the limitation of the experimental conditions and techniques, and pre-existing interfacial defects such as Interfacial instability from void formation at the solid-solid interface is one of the crucial challenges in solid-state batteries. However, the fundamental mechanism as to how stress is generated in lithium and thus impacts void formation has not been established. A general creep/contact electro-chemomechanical model is herein developed to reveal the mechanisms of void formation at the Li/solid electrolyte (SE) interface during stripping. Li stress calculation is ...
Lithium dendrite penetration has been widely evidenced in ceramic solid electrolytes (SEs), which are expected to suppress Li dendrite formation due to their ultrahigh elastic modulus. This work aims to reveal the mechanism of Li penetration in polycrystalline SEs through electro‐chemo‐mechanical phase‐field model, using Li7La3Zr2O12 (LLZO) as the model material. The results show the Li penetration patterns are influenced by both mechanical and electronic properties of the microstructures, i.e., grain boundaries (GBs). Li nucleates at the GB junctions on the Li/SE interface and propagates along the GB, at which the interfacial compressive stress is small due to the GB softening. Moreover, the excess trapped electrons at the GB may trigger isolated Li nucleation sites, abruptly increasing the Li penetration depth. High‐throughput simulations yield a phase map of Li penetration patterns under different trapped electrons concentrations and GB/grain elastic modulus mismatch. The map can quantitatively inform whether the mechanical or electronic properties dominate Li penetration morphologies, providing a strategy for the design of improved SE materials.
The steady and transient behavior of jets generated by circular and slit nozzles are analyzed by the Galerkin finite-element method with free-surface parametrization and Newton iteration. A novel constitutive equation is used to approximate Bingham liquids that is valid uniformly in yielded and unyielded domains and which approximates the ideal Bingham model and the Newtonian liquid in its two limiting behaviors. At steady state the influence of yield stress on the die swell is equivalent to that of surface tension; that is, suppression of jet diameter at low Reynolds numbers and necking at high Reynolds number. The predictions at high Reynolds numbers agree with the asymptotic behavior at infinite Reynolds number of the jet far downstream. In the transient analysis, surface tension destabilizes round jets and increases the size of satellite drops. Yield stress was found to retard jet breakup times in addition to producing smaller satellites. Shear thinning was found to result in shorter collapse times than those for Newtonian fluid; furthermore, the satellite drop size increased with increasing shear thinning. The nonlinear analysis predicts that, although round jet breakup may occur spontaneously by surface tension, an external factor, commonly air shear, must be applied to break a planar jet at Reynolds numbers below its transition to a turbulent jet.
Recent demonstrations of direct utilization of hydrocarbon fuels have stimulated an automotive interest in solid oxide fuel cells for reformerless auxiliary power units with high power density, high chemical-to-electrical efficiency, and low exhaust emissions. Furthermore, recent designs with small-diameter oxide tubes appear to be well-suited to accommodate repeated cycling under rapid changes in electrical load and in cell operating temperatures. To understand the limiting transient processes in these small-tube fuel cell designs, we applied an analysis approach which requires no a priori equivalent circuit model assumptions. This approach was applied to the electrochemical impedance spectroscopy ͑EIS͒ data measured from such cells in the temperature range from 585 to 888°C. In this way, the complex, overlapping arc EIS details ͑seen in Cole-Cole plots͒ were transformed in a network-model-independent way into a spectrum of relaxation times. We extended the deconvolution method to allow peak fitting and integration to calculate the resistances of individual processes within the cathode polarization, which becomes limiting in comparison to either anode or electrolyte at temperatures below about 700°C. With the new results, the process with the highest apparent activation energy can be targeted to improve cathode development.Solid oxide fuel cells ͑SOFCs͒ are established as electrochemical devices for stationary electrical power generation. 1 Because of their high power conversion efficiency and low exhaust emissions, 2,3 these energy conversion devices have also been suggested for use in automotive auxiliary power units ͑APUs͒ to meet the increasing demands for on-board electrical power. Futhermore, recent laboratory-based demonstrations have given evidence of direct oxidation of hydrocarbon fuels 4,5 at the SOFC anode, without carbon deposition, at sufficiently low temperatures. Consequently, the likelihood of developing a compact, reformerless APU that can operate on the same fuel as the internal combustion engine now seems to be more realistic.The development of practical SOFCs for automotive use can benefit from fundamental knowledge about ac impedance related interfacial processes and associated microstructures. Since a SOFC is a complicated electrochemical system consisting of heterogeneous phases of the electrodes 6,7 and the electrolyte, 8 improvements in cell performance and manufacturing technology can be better focused if there is a means to separate the individual impedance-related processes within a single cell. These processes include the transport of ions and electrons through ͑and reacting at͒ the interphase boundaries. Each of these processes may depend upon temperature.Therefore, the main objective of the present work is to estimate the influence of temperature ͑580-888°C͒ on the separate component impedances in fast-thermal-response cells of potential automotive interest. ApproachWe adopt and extend an approach that does not require any a priori assumptions of an equivalent network model to rep...
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