Transport properties of solid foams having circular strut cross section using pore scale numerical simulations Yann Jobic, Prashant Kumar, Frederic Topin, René Occelli To cite this version: Yann Jobic, Prashant Kumar, Frederic Topin, René Occelli. Transport properties of solid foams having circular strut cross section using pore scale numerical simulations. Heat and Mass Transfer, Springer Verlag, 2017, 10.1007/s00231-017-2193-2. hal-01792823 HAL Id: hal-01792823 https://hal.archives-ouvertes.fr/hal-01792823 Submitted on 3 Apr 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Heat Mass Transfer https://doi.org/10.1007/s00231-017-2193-2 ORIGINAL Transport properties of solid foams having circular strut cross section using pore scale numerical simulations 1 1 1 1 Yann Jobic & Prashant Kumar & Frédéric Topin & René Occelli Received: 28 April 2017 /Accepted: 8 October 2017 # Springer-Verlag GmbH Germany 2017 Abstract Light cellular materials are increasingly used in many characteristics and friction factor vs. Reynolds number relation- engineering applications as they present several attractive prop- ship. Similarly, heat transfer results were used to derive heat erties including heat transfer enhancement, low pressure drop exchange coefficient between solid and fluid phases of foam compared to packed bed of spheres. Transport properties are material. This length scalehasprovedtodetermine‘universal’ dependent on foam morphology and thus, their precise knowl- thresholds to identify thermo-hydraulic regimes and is found to edge is required for efficient designing and optimization of in- be independent of foam morphology. Correlations to predict hy- dustrial devices. Discrepancies and ambiguities in definitions, draulic characteristics and heat transfer coefficients/Nusselt num- interpretation of various parameters, limited experimental ber for circular strut open cell foams were derived. The correla- methods and non-consistencies inmeasurementsaresomecriti- tions proposed in this work appear to be very generic taking into cal factors that lead to highly scattered morphological and trans- account variability in foams for variable porosities. The predicted port properties in the literature. These properties are, however, results were validated against numerical and experimental data linked with the strut cross-sectional shape and thus, no relation- and an excellent agreement has been obtained. ship exists in the literature that bears a reasonable applicability in multidisciplinary domains. In this context, virtual foams based on Nomenclature based on tetrakaidecahedron unit-cell have been constructed and Latin symbols −1 geometrically characterized. These periodic idealized foam struc- ac Specific surface area (m ). ∗ −1 tures are constituted of circularstrutswhosediameterscouldbe ac Specific surface area, Eq. 4 (m ). varied arbitrary. The thermo-hydraulic properties of open-cell cF Form drag coefficient (−). −1 foams in relation with morphology are systematically studied CFor Forchheimer inertia coefficient (m ). using 3-D direct pore-scale numerical simulations of single- CL Characteristic length, Eq. 9 (m). −1 −1 phase flow in the virtual samples. A comprehensive database Cp Specific heat capacity (J. kg .K ). (more than 100 samples/cases) of flow and heat transfer charac- dc Circular strut diameter (m). ∗ teristics has been generated. Mathematical formulation has been dc Circular strut diameter, Eq. 3a, b (m). developed to predict accurately the morphological characteristics dcell Cell size (m). and discussed with the findings intheliteraturedata.Anoriginal dp Pore diameter (m). ∗ definition of pore diameter is proposed and its uniqueness to use dp Dimensionless pore diameter (−). as a characteristic length scale has been obtained. Flow regime dph Equivalent hexagonal face diameter, Eq. 6a, b (m). transitions were identified by analyzing the pressure drop dps Equivalent square face diameter, Eq. 6a, b (m). f Friction factor (−). hs − f Interstitial (strut-fluid) heat transfer coefficient * Yann Jobic (W. m−2.K−1). [email protected] −2 −1 hconv Wall heat transfer coefficient (W. m .K ). hvol Volumetric heat transfer coefficient, Eq. 16 (W. −3 −1 1 Aix-Marseille Université, IUSTI, CNRS UMR 7343, 5, Rue Enrico m .K ). Fermi, 13453 Marseille Cedex 13, France k Constant of a geometrical parameter, Eq. 1a, b (−). Heat Mass Transfer −1 −1 keff Effective thermal conductivity (W. m K ). 1 Introduction eff k f Fluid phase effective conductivity, Eq. 16 (W. m−1.K−1). Kelvin-like cell foams constitute a model of classic replication eff ks Solid phase effective conductivity, Eq. 16 (W. foams as well as a new class of industrial material. Transport m−1.K−1). properties are dependent on foam morphology and thus, their −1 −1 kf Thermal conductivity of fluid (W. m .K ). precise knowledge is required for efficient designing and op- −1 −1 ks Intrinsic solid phase conductivity (W. m .K ). timization of industrial devices. Discrepancies and ambigui- 2 KD Darcian permeability (m ). ties in definitions, interpretation of various parameters, limited LN Node-to-node length (m). experimental methods and non-consistencies in measurements m Geometrical parameter, Eq. 1a, b (m). are some critical factors that lead to highly scattered morpho- Nuconv Nusselt number (solid channel wall-fluid) (−). logical and transport properties in the literature. These prop- Nus − f Nusselt number (strut-fluid) (−). erties are, however, linked with the strut cross-sectional shape ΔP/Δx Pressure drop per unit length (Pa. m−1). and thus, no relationship exists in the literature that bears a ∇〈P〉 Pressure gradient (Pa. m−1). reasonable applicability in multidisciplinary domains. Pr Prandtl number (−). The thermo-hydraulic behaviour of open-cell foams de- Re Reynolds number in generic notion (−). pends on their microscopic structure and recent studies re- ReCL Reynolds number based on characteristic length vealed that experimental characterization of open-cell foams (−). can be time-consuming and sometimes very expensive (e.g. Redcell Reynolds number based on cell size (−). Application of BET, MRI, μ-CT, resolution quality etc.). Such Redp Reynolds number based on pore diameter (−). detailed approach could prove to be quite time consuming, RepKD Reynolds number based on square root of Darcian since the morphological and transport properties of a specific permeability (−). foam structure are obtained individually on a case to case basis Sligamentffiffiffiffiffi Total surface area of ligaments inside a cubic cell, which actually prevents to perform systematic studies to ob- Eq. 5a (m2). tain a general tendency of relations between different param- Snode Total surface area of nodes inside a cubic cell, Eq. eters (e.g. Kumar et al., [1]). It is thus cumbersome to study 5a (m2). systematically for a given foam texture obtained from recon- 2 sFoam Fluid-Solide interface surface, Eqs. 4 and 18 (m ). structed foam sample. V Superficial velocity (m. s−1). In this context, numerous empirical correlations with and Vc Volume of tetrakaidekahedron unit cell, Eq. 4 without fitting parameter as well as mathematical formula- (m3). tions to predict morphological parameters of the foam struc- 3 VT Volume of cubic unit cell, Eq. 5a (m ). tures have been presented in the literature. The aim was to 3 Vligament Volume of one ligament, Eq. 1a (m ). predict the complete set of morphological characteristics by 3 Vnode Volume of one node, Eq. 1b (m ). knowing at least two easily measurable structural parameters. X Constant and intrinsic property of foam material, Usually, these correlations or formulations were derived using Eq. 11 (−). the suitable 3-D foam structures that constituted mainly rep- Y Constant and intrinsic property of foam material, resentative unit cell (e.g. Du Plessis and Masliyah [2]), cubic Eq. 11 (−). unit cell (e.g. Giani et al., [3]), pentagonal dodecahedron (e.g. Greek symbols Huu et al., [4]), Weaire-Phelan (e.g. Grosse et al., [5]) and εo Open porosity (−). tetrakaidecahedron or Kelvin-like foam structure (e.g. μ Dynamic fluid viscosity (Pa. s). Kumar et al., [6]). Many authors reviewed different models ρ Fluid density (kg. m−3). of cellular foam structure and preferred the tetrakaidecahedron Ωc Dimensionless strut diameter (or Kelvin-like cell) structure since it is a space filling struc- (−). ture contrary to pentagonal dodecahedron structure (that is not Abbreviations a space filling structure one) and gave the most consistent BET Brunauer–Emmett–Teller theory agreement with observed morphological properties (e.g. CAD Computer aided design Inayat et al., [7]). CFD Computation fluid dynamics While deriving the correlations, most of the authors (e.g. ETC Effective thermal conductivity Richardson et al., [8]; Inayat et al., [7]) have ignored the im- LTE Local thermal equilibrium pact of node junction. Moreover, the relationship between LTNE Local thermal non-equilibrium specific surface area and pore diameter (according the author’s MRI Magnetic resonance imaging definition) appears