Reduced cell and stack modeling of planar solid oxide fuel cells
Date of Issue2014
School of Mechanical and Aerospace Engineering
A solid oxide fuel cell (SOFC) is an electrochemical energy conversion device operating at high temperatures from 500 to 1000 ℃ fueled by hydrogen or hydrocarbon, and is developed for environmentally friendly and sustainable energy production. The cell performance typically involves complicated multiphysical behaviours, such as transport phenomena and electrochemical performance. To understand the characteristics and mechanism of the SOFC, mathematical modeling is an efficient approach, compared with repetitive and costly experimental techniques. In general, however, three-dimensional (3D) modeling of SOFCs is computationally expensive, especially for stacks, due to the highly coupled and nonlinear nature of the mathematical formulation as well as a large number of functional domains in the cell. Although there are simplified P-SOFC models from 3D to 2D, the assumptions on dimensionality tend to lower the fidelity of model prediction. In this thesis, novel reduced models are developed for the planar SOFC (P-SOFC) and stack equipped with parallel plain flow channels to significantly reduce computational cost with desired numerical accuracy, while capturing both the average properties and the variability of the dependent variables in the 3D model. The model reduction is performed based on a full 3D cell model that is validated with experimental measurements from the literature, where the 3D model consists of a set of governing equations for the conservation of mass, momentum, species, charge and energy, and provides a detailed description of the transport phenomena and electrochemical performance of a P-SOFC. The first achievement of the present work is the development of novel correlation factors to spatially smooth the flow fields of parallel plain channels and solid ribs based on the 3D single-cell model under isothermal conditions. The factors are derived based on the full set of governing equations over a cell cross section, such that they can handle not only variations in diffusion pathways due to ribs but also the coupling effects between governing equations. They are specified for the cathode and anode separately. The complexity of the mathematical model is reduced via spatial smoothing and asymptotic analysis without sacrificing the leading-order physics. The numerical computation is sped up further via a space-marching algorithm. As a result of spatial smoothing with the developed correlation factors, the isothermal 3D single-cell model is reduced to a two-dimensional (2D) counterpart with effective physical parameters, which is solved with respect to directions along the cell length (streamwise direction, ex) and through the cell thickness (normal direction, ez). When performing spatial smoothing for model reduction, there is no loss in physical information except that the local pointwise information is replaced with spanwise averaged counterparts. The 2D spatially-smoothed counterpart is then asymptotically reduced to a set of parabolic partial differential equations (PDEs) and ordinary ones (ODEs) with a space marching algorithm. This reduced model, called the 2D asymptotic spatially-smoothed isothermal (ASSI) P-SOFC model, is finally verified with the 3D counterpart in terms of global and local cell properties. Good agreement is achieved for the verification in view of the relative errors between the solutions from the 3D and reduced models. The fidelity of model prediction is guaranteed by the formulated correlation factors and effective parameters. Moreover, the computing time and memory requirement are reduced by three and two orders of magnitude, respectively, in comparison with the original 3D model. The second achievement is the development of novel variation factors for spatial smoothing to capture the variability of dependent variables in the spanwise direction (ey) along the cell width, although the spanwise direction is reduced for the 3D model in this study. The resulting 2D spatially-smoothed model is verified in terms of mass fractions, allowing for the study of depletion of reactants underneath the ribs. The spanwise variability of mass fractions is quantified in terms of their standard deviations along the cell width. The variation factors are obtained by normalizing the standard deviations with respect to the concentration drop across the electrode, such that they only vary with geometrical parameters and are applicable at different current densities. The resulting variation factors are expressed in the form of polynomial expressions in terms of dimensionless geometrical parameters for the cathode and anode, respectively. Good agreement between the 3D and 2D models verifies that the developed variation factors can capture the variability of mass fractions underneath ribs in the original 3D model for various cell designs with parallel flow channels. Furthermore, the present method for the development of the variations factors of mass fractions can be extended to the other dependent variables in the 3D domain. In terms of sensitivity analysis of isothermal single-cell performance, apart from the development of the above correlation and variation factors for spatial smoothing, Monte Carlo simulation (MCS) is conducted for an advanced 2D ASSI P-SOFC model, in which the previously achieved 2D ASSI model for the leading-order average predictions is coupled with the variation factors for the spanwise variability of mass fractions, in order to statistically assess the effects of stochastic material properties, operating conditions, and geometrical parameters on cell behaviours. The advanced 2D ASST model is computationally cost-efficient and takes about 1 second for each run and thus allows MCS with a sample size on the order of magnitude of 103. Sensitivity analysis of cell performance to the stochastic modeling parameters is conducted in two scenarios: only one parameter varying at a time, and all parameters varying together. The former investigates the influence of a single modeling parameter on the variation of cell performance, while the latter studies the impact of a combination of varying parameters. The analysis is elaborated in terms of the average current density, the average value of oxygen mass fractions and their spanwise variation underneath the ribs. According to the contribution of each parameter to the variation of the cell performance, the modeling parameters are ranked for each scenario and then two distinct rankings are obtained, which is not captured by conventional parametric studies. The last achievement is the development of 2D spatially-smoothed non-isothermal single-cell and stack models as well as their asymptotic counterparts that are called the asymptotic spatially-smoothed thermal (ASST) cell and stack models, in which the spatially smoothed energy equations are in particular developed subject to both the local thermal equilibrium (LTE) and non-equilibrium (LTNE) conditions in the volume-averaged flow field. The variation in the pathways of conductive heat transfer between the flow field and adjacent layers is captured by functional correlation factors that are derived based on a Laplace equation. The selection of the reduced models under either the LTE or LTNE condition for use depends on the temperature difference between the working fluid in channels and the interconnect of the original 3D model. Verifying the reduced models with the 3D counterparts, it is found that the accuracy of the LTE model decreases when the temperature difference in the 3D flow field increases. When the temperature difference is more than 10 K, the LTNE model is preferable for high accuracy. Moreover, good agreements are obtained for single-cell modeling, and for stack modeling with negligible perturbations of dependent variables between cells. If the perturbations are not negligible, the relative error between the 2D ASST and 3D stack models increases significantly.
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