Homogenization of a Conductive, Convective and Radiative Heat Transfer Problem in a Heterogeneous Domain

International audienceWe are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order than the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection and radiative transfer in the fluid part (the cylinders). A non-local boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a 3D to 2D asymptotic analysis since the radiative transfer in the limit cell problem is purely two-dimensional. Eventually, we provide some 3D numerical results in order to show the convergence and the computational advantages of our homogenization method

Thermophysical Phenomena in Metal Additive Manufacturing by Selective Laser Melting: Fundamentals, Modeling, Simulation and Experimentation

Among the many additive manufacturing (AM) processes for metallic materials, selective laser melting (SLM) is arguably the most versatile in terms of its potential to realize complex geometries along with tailored microstructure. However, the complexity of the SLM process, and the need for predictive relation of powder and process parameters to the part properties, demands further development of computational and experimental methods. This review addresses the fundamental physical phenomena of SLM, with a special emphasis on the associated thermal behavior. Simulation and experimental methods are discussed according to three primary categories. First, macroscopic approaches aim to answer questions at the component level and consider for example the determination of residual stresses or dimensional distortion effects prevalent in SLM. Second, mesoscopic approaches focus on the detection of defects such as excessive surface roughness, residual porosity or inclusions that occur at the mesoscopic length scale of individual powder particles. Third, microscopic approaches investigate the metallurgical microstructure evolution resulting from the high temperature gradients and extreme heating and cooling rates induced by the SLM process. Consideration of physical phenomena on all of these three length scales is mandatory to establish the understanding needed to realize high part quality in many applications, and to fully exploit the potential of SLM and related metal AM processes

Fluid flow and heat transfer in dual-scale porous media

Porous media are omnipresent in various natural and engineered systems. The study of transport phenomena in porous media has attracted the attention of researchers from a wide variety of disciplines. In many applications such as hydrogeology, petroleum engineering and thermochemistry, porous media are encountered, in which heterogeneity exists at a multitude of length-scales. In solar thermochemical reactors, a promising approach to accomplish the thermochemical cycle is to form the reactive solid into a porous structure to promote efficient solid-gas reactions through a high specific surface area, while simultaneously achieving desired transport characteristics. Recently, in light of the apparent trade-offs between rapid reaction kinetics and efficient radiation absorption, reticulated porous ceramics (RPCs) featuring dual-scale porosity have been engineered. These structures are capable of combining the desired properties, namely uniform radiative absorption and high specific surface area. Therefore, investigations are required to understand and analyse different transport phenomena in such structures. This dissertation is motivated by the need for understanding and analysing transport phenomena dual-scale porous media appear and used in many applications such as hydrogeology, petroleum engineering, chemical reactors, and in particular, energy technologies in high-temperature thermochemistry. The main objective of this thesis is to theoretically formulate and numerically demonstrate the fluid flow and heat transfer phenomena in dual-scale porous media. The theoretical and numerical results are used to propose models in forms of effective flow and heat transfer coefficients. The models are capable of estimating the fluid flow and heat transfer phenomena taking place in dual-scale porous media with appropriate fidelity and lower computational cost. The physical understanding of the models of transport phenomena in dual-scale porous structures allows us to tailor and optimise the morphology to accomplish optimal transport characteristics for the desired applications. To determine the flow coefficients, numerical simulations are performed for the fluid flow in a dual-scale porous medium. Two numerical procedures are considered. Firstly, we perform direct pore-level simulations by solving the traditional mass and momentum conservation equations for a fluid flowing through the dual-scale porous structure. Secondly, numerical simulations are performed at the Darcy level. For this purpose, the permeability and Forchheimer coefficient of the small-scale pores are numerically determined. Then, they are implemented in Darcy-level simulations in which the volume-averaged and traditional conservation equations are solved for the small- and large-scale pores, respectively. The results of the two approaches are separately used to determine and compare the permeability and Forchheimer coefficient of the dual-scale porous media. To analyse the energy transport phenomena in dual-scale porous media, a mathematical model is developed by applying volume-averaging method to the convective-conductive energy conservation equation to derive the large-scale equations with effective coefficients. The closure problems are introduced along with the closure variables to establish the closed form of the two-equation model for heat transfer of dual-scale porous media. The closure problems are numerically solved for specific cases of dual-scale porous medium consisting of packed beds of porous spherical particles. The effective coefficients appearing in the two-equation model of heat transfer in dual-scale porous media are determined using the solution of the closure problems. The velocity field in the dual-scale porous structure is calculated using the solution of the fluid flow simulations in dual-scale porous medium. Finally, "numerical experiment" is performed to qualitatively and quantitatively analyse the accuracy of the up-scaled model.The support by the Australian Research Council through Prof Wojciech Lipiński’s Future Fellowship, award no. FT14010121

Precise Derivations of Radiative Properties of Porous Media Using Renewal Theory

This work uses the mathematical machinery of Renewal/Ruin (surplus risk) theory to derive preliminary explicit estimations for the radiative properties of dilute and disperse porous media otherwise only computable accurately with Monte Carlo Ray Tracing (MCRT) simulations. Although random walk and Levy processes have been extensively used for modeling diffuse processes in various transport problems and porous media modeling, relevance to radiation heat transfer is scarce, as opposed to other problems such as probe diffusion and permeability modeling. Furthermore, closed form derivations that lead to tangible variance reduction in MCRT are widely missing. The particular angle of surplus risk theory provides a richer apparatus to derive directly related quantities. To the best of the authors' knowledge, the current work is the only work relating the surplus risk theory derivations to explicit computations of ray tracing results in porous media. The paper contains mathematical derivations of the radiation heat transfer estimates using the extracted machinery along with proofs and numerical validation using MCRT

HYDRA: Macroscopic modeling of hybrid ablative thermal protection system

In the framework of HYDRA, an European funded program, technological solutions of hybrid Thermal Protection System (TPS) are developed. This advanced shielding relies on the hybridization of upper lightweight porous ablative material and inner Ceramic Matrix Composite (CMC) bonded together with a novel high temperature adhesive. The aerial mass optimization of the full TPS requires a controlled reduction in the ablative material thickness to reach high operating temperature configuration of the CMC. Therefore, radiative heat transfer takes place in a thin layer of ablator and becomes a major contributor to the elevation of the interface temperature. In this paper we develop an high fidelity radiative transfer in porous carbon fibers charring and ablative material. Specific elementary characterization, plasma test campaign and numerical simulation are scheduled to feed this radiative heat transfer model

Effective heat conductivity of a composite with hexagonal lattice of perfectly conducting circular inclusions: An analytical solution

Paper is devoted to the effective stationary heat conductivity for the fibre composite materials. We are aimed on getting on analytical expression for effective thermal conductivity coefficient. Asymptotic homogenization approach, based on the multiple scale perturbation method, is used. This allows to reduce the original boundary value problem in multiply connected domain to the sequence of boundary value problems in simply connected domains. These problems include: the local problem for the periodically repeated cell and homogenized problem with effective coefficient. It is shown that for densely packed and high contrast fibre composites, the cell problem can be solved analytically. For this aim, lubrication approach (asymptotics of thin layer) has been employed. We also generalise obtained solution to the case of medium-sized inclusions in the framework of the Padé approximants