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**Politecnico di Milano sede Bovisa 15 - 16 Ottobre, 2007**

MESOSCOPIC NUMERICAL MODELING OF REACTIVE MIXTURE FLOWS IN SOLID OXIDE FUEL CELLS BY LATTICE BOLTZMANN METHOD AND HIGH PERFORMANCE PARALLEL COMPUTING Pietro Asinari, Romano Borchiellini, Michele Calì Politecnico di Milano sede Bovisa Ottobre, 2007

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Outline Mesoscopic modeling of SOFC electrodes by Lattice Boltzmann Method (LBM) Mixture modeling: MRT Gross & Krook model Numerical scheme: semi-implicit linearized backward Euler formulation (SILBE) LABORA Code Cluster facilities and scaling performances Reconstruction techniques Gas permeation and diffusion: direct numerical simulation of tortuosity

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**Lattice Boltzmann Method – LBM**

Microscopic Theory Kinetic Theory Macroscopic Theory (Continuum) and Thermodynamics (Equilibrium) Deterministic Newton’s Law Statistical Mechanics Euler Equations Navier – Stokes Equations Molecular Dynamics Liouville Equation Hilbert and Chapman – Enskog Analysis (Singular Perturbation Analysis) Finite Moments Multiple Relaxation Times Lattice Boltzmann Equation LBM Lattice Gas Automata Boltzmann Equation Discretized Distribution Functions (DDFs)

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**Why Mesoscopic Modeling and LBM**

No linear system of algebraic equations must be solved no need of iterative procedures. Explicit time numerical process transient simulations can be naturally performed. No need for staggered grids unphysical solutions are automatically avoided. Additional local information are available the computed variables of a single cell are enough to estimate higher order derivatives. Complex topologies can be efficiently included the models are stable for quite rough meshes.

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**Application to SOFC Electrodes**

Mesoscopic Modeling is a very powerful tool for SOFC technology because it allows one to go deeply in the reaction core for investigating fuel cell portions, which are actually not accessible by direct measurement (spatial distribution of the concentration polarization, local fluid flow,…). However the reliability of numerical results strongly depends on the reliability of the microscopic structure used in the simulations, the reliability of the input parameters, particularly the transport coefficients effecting the reaction rate.

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Mixture Modeling Self collisions involve particles of the same type while cross collisions involve particles of different type

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**Parallel Algorithm Calculations and communications at the same time !**

Collision Step (Internal Layer) Collision Step (Core) Streaming Step (Core) Moment Calculation Step (Core) Streaming Step (Internal Layer) Moment Calculation Step (Internal Layer) Non – Blocking Send (Internal Layer) Calculations and communications at the same time ! Non – Blocking Receive (External Layer)

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LABORA POLITO The LABORA code (Lattice Boltzmann for Raster Applications) was developed from scratch at “Politecnico di Torino” (Italy), for solving mainly the fluid flow of reactive mixtures in porous media. The project started in 2003 (now 10,000 lines in C++). Main code features are: fully three dimensional formulation (D3Q19 lattice); optimized memory storage; parallelization based on automatic and arbitrary domain decomposition (open source MPI package); different tuning strategies.

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**HPC Facility: System X @ Virginia Tech**

System X is a supercomputer assembled by Virginia Tech faculty members, staff, and students in the summer of 2003, comprising 1,100 Apple PowerMac G5 computers. System X is currently running at Teraflops, (20.24 peak), and was last ranked #47 (November, 2006) in the TOP500 list of the world's most powerful supercomputers. At that time, it was still the most powerful system categorized by TOP500 as "self made" at any university.

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**HPC (?) Facility: ClusterLinux @ POLITO**

“Politecnico di Torino” (Italy): ClusterLinux, scalable grid computing facility, currently 64 Pentium single processor nodes (64 CPUs, 2.8 GHz, 512 MB RAM, 40 GB HD), LAN 100 Megabit Ethernet, up to 102 CPUs. This facility is based on PC from student laboratories which are under-used during night and/or vacations. The main goal is to fruitfully use computational resources which are already available in order to maximize the investment outcome.

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**Scaling Performances of LABORA**

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**Comparison on Different Facilities (1)**

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**Comparison on Different Facilities (2)**

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**The (near) future: EnerGRID project**

EnerGRID: design and development of a grid infrastructure for high performance computing in modeling energy networks based on widespread sources of heat and power generation On-going collaborations with research groups of Computer Science Department at Stuttgart (GE) in the framework of the program HPC – Europa.

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**Reconstruction Techniques**

Non-destructive X-ray computed micro-tomography is not enough for SOFC application, this resolution is not sufficient reconstructions from reliable 2D techniques, such as standard and back scanning electron microscopy (SEM/BSEM), is the only viable alternative. (1) granulometry law grain shapes are assumed; (2) multiple–point statistics neighboring information are processed for more reliable reconstruction.

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**Multiple-point Statistics**

Multiple-point statistics were used, based on two-dimensional (2D) thin sections as training images, to generate 3D pore space representations (Okabe & Blunt, Journal of Petroleum Science & Engineering, 2005). A 3D image can be generated that preserves typical patterns of the void space seen in the thin sections. The use of multiple-point statistics predicts long-range connectivity of the structures better than granulometry law. Essentially the algorithm is based on three steps: Borrowing multiple-point statistics from training images, Pattern reproduction, Image processing-noise reduction and smoothing.

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Reconstructed Domain

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**Fluid Flow at the Bottom**

Hexahedral mesh 2563=16.7 MCell MDof for binary mixture (H2O/H2) in 3D porous medium. 100,000 collisions. Wall clock time 57 hours with a 64 CPU cluster. Parallelization efficiency 85 % with non-optimized domain decomposition.

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**Surface Averaged Quantities**

Surface averaged quantities must be introduced for comparing the mesoscopic fluid flow with the macroscopic measurements and user-level expectations. <Concentration> <Mass Flux>

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**Optimal Refinement: Fluid Flow**

In order to recover the desired accuracy (<3%), the finest computational mesh, i.e (refinement X8) must be considered. Unfortunately, this means to simulate a portion too small of the anode, which is not representative of the whole electrode.

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**Optimal Refinement: Tortuosity**

Fortunately the tortuosity has a small dependence on the mesh resolution (<5%). It depends on the path of the considered species flowing in the porous medium and even very coarse meshes allow one to at least estimate the path of the species with acceptable accuracy. This means that larger physical domains can be simulated.

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**Spatial Dependence of Tortuosity**

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Conclusions Direct numerical simulation of tortuosity for SOFC application is promising for comparing the performances of different materials and highlighting the possible ways to improve them The required mesh resolution for solving the fluid flow with regards to the tortuosity calculation is not too demanding The simulation of the local electro-chemical reaction must be improved the ion and electron flows in the solid phases must be accurately solved too Different sintering technologies can be compared

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**Further Documentation**

P. Asinari, M.R. von Spakovsky, M. Calì, B.V. Kasula, “Direct numerical calculation of the kinematic tortuosity of reactive mixture flow in the anode layer of solid oxide fuel cells by the Lattice Boltzmann Method”, Journal of Power Sources, 170, pp , 2007. P. Asinari, “Semi-implicit-linearized Multiple-relaxation-time formulation of Lattice Boltzmann Schemes for Mixture Modeling”, Physical Review E, 73, , 2006. P. Asinari, “Viscous coupling based Lattice Boltzmann model for binary mixtures”, Physics of Fluids, 17, , 2005. P. Asinari, “Asymptotic analysis of multiple-relaxation-time lattice Boltzmann schemes for mixture modeling”, Computers and Mathematics with Applications, 2007 (in press).

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