Solid-fluid thermal interactions during unsteady flow in porous media play an important role in the regenerators of pulse tube cryocoolers. Pore-level thermal processes in porous media under unsteady flow conditions are poorly understood. The objective of this investigation is to study the pore-level thermal phenomena during pulsating flow through a generic, two-dimensional porous medium by numerical analysis. Furthermore, an examination of the effects of flow pulsations on the thermal dispersion and heat transfer coefficient that are encountered in the standard, volume-average energy equations for porous media are carried out. The investigated porous media are periodic arrays of square cylinders. Detailed numerical data for the porosity range of 0.64 to 0.84, with flow Reynold's numbers from 0-1000 are obtained. Based on these numerical data, the instantaneous as well as cycle-average thermal dispersion and heat transfer coefficients, to be used in the standard unsteady volume-average energy conservation equations for flow in porous media, are derived. Also, the adequacy of current applied cycle-average correlations for heat transfer coefficients and the inclusion of the thermal dispersion in the definition of an effective fluid thermal conductivity are investigated.

This study addressed the numerical modelling of flow and diffusion in packed beds of mono-sized spheres. Comprehensive research was conducted in order to implement various numerical approaches in explicit1 and implicit2 simulations of flow through packed beds of uniform spheres. It was noted from literature that the characterization of a packed bed using porosity as the only geometrical parameter is inadequate (Van Antwerpen, 2009) and is still under much deliberation due to the lack of understanding of different flow phenomena through packed beds. Explicit simulations are not only able to give insight into this lack of understanding in fluid mechanics, but can also be used to develop different flow correlations that can be implemented in implicit type simulations. The investigation into the modelling approach using STAR-CCM+®, presented a sound modelling methodology, capable of producing accurate numerical results. A new contact treatment was developed in this study that is able to model all the aspects of the contact geometry without compromising the computational resources. This study also showed, for the first time, that the LES (large eddy simulation) turbulence model was the only model capable of accurately predicting the pressure drop for low Reynolds numbers in the transition regime. The adopted modelling approach was partly validated in an extensive mesh independency test that showed an excellent agreement between the simulation and the KTA (1981) and Eisfeld and Schnitzlein (2001) correlations' predicted pressure drop values, deviating by between 0.54% and 3.45% respectively. This study also showed that explicit simulations are able to accurately model enhanced diffusion due to turbulent mixing, through packed beds. In the tortuosity study it was found that the tortuosity calculations were independent of the Reynolds number, and that the newly developed tortuosity tests were in good agreement with techniques used by Kim en Chen (2006), deviating by between 2.65% and 0.64%. The results from the TMD (thermal mixing degree) tests showed that there appears to be no explicit link between the porosity and mixing abilities of the packed beds tested, but this could be attributed to relatively small bed sizes used and the positioning and size of the warm inlet. A multi-velocity test showed that the TMD criterion is also independent of the Reynolds number. It was concluded that the results from the TMD tests indicated that more elaborate packed beds were needed to derive applicable conclusions from these type of mixing tests. The explicit BETS (braiding effect test section) simulation results confirmed the seemingly irregular temperature trends that were observed in the experimental data, deviating by between 5.44% and 2.29%. From the detail computational fluid dynamics (CFD) results it was possible to attribute these irregularities to the positioning of the thermocouples in high temperature gradient areas. The validation results obtained in the effective thermal conductivity study were in good agreement with the results of Kgame (2011) when the same fitting techniques were used, deviating by 5.1%. The results also showed that this fitting technique is highly sensitive for values of the square of the Pearson product moment correlation coefficient (RSQ) parameter and that the exclusion of the symmetry planes improved the RSQ results. It was concluded that the introduction of the new combined coefficient (CC) parameter is more suited for this type of fitting technique than using only the RSQ parameter. / Thesis (M.Ing. (Mechanical Engineering))--North-West University, Potchefstroom Campus, 2012

Through the years different models have been proposed to calculate the total effective thermal conductivity in packed beds. The purpose amongst others of these models is to calculate the temperature distribution and heat flux in high temperature pebble bed reactors. Recently a new model has been developed at the North–West University in South Africa and is called the Multi–Sphere Unit Cell (MSUC) model. The unique contribution of this model is that it manages to also predict the effective thermal conductivity in the near wall region by taking into account the local variation in the porosity. Within the MSUC model the thermal radiation has been separated into two components. The first component is the thermal radiation exchange between spheres in contact with one another, which for the purpose of this study is called the short range radiation. The second, which is defined as the longrange radiation, is the thermal radiation between spheres further than one sphere diameter apart and therefore not in contact with each other. Currently a few shortcomings exist in the modelling of the long–range radiation heat transfer in the MSUC model. It was the purpose of this study to address these shortcomings. Recently, work has been done by Pitso (2011) where Computational Fluid Dynamics (CFD) was used to characterise the long–range radiation in a packed bed. From this work the Spherical Unit Nodalisation (SUN) model has been developed. This study introduces a method where the SUN model has been modified in order to model the long–range radiation heat transfer in an annular reactor packed with uniform spheres. The proposed solution has been named the Cylindrical Spherical Unit Nodalisation (CSUN, pronounced see–sun) model. For validation of the CSUN model, a computer program was written to simulate the bulk region of the High Temperature Test Unit (HTTU). The simulated results were compared with the measured temperatures and the associated heat flux of the HTTU experiments. The simulated results from the CSUN model correlated well with these experimental values. Other thermal radiation models were also used for comparison. When compared with the other radiation models, the CSUN model was shown to predict results with comparable accuracy. Further research is however required by comparing the new model to experimental values at high temperatures. Once the model has been validated at high temperatures, it can be expanded to near wall regions where the packing is different from that in the bulk region. / Thesis (M.Ing. (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2012.

Through the years different models have been proposed to calculate the total effective thermal conductivity in packed beds. The purpose amongst others of these models is to calculate the temperature distribution and heat flux in high temperature pebble bed reactors. Recently a new model has been developed at the North–West University in South Africa and is called the Multi–Sphere Unit Cell (MSUC) model. The unique contribution of this model is that it manages to also predict the effective thermal conductivity in the near wall region by taking into account the local variation in the porosity. Within the MSUC model the thermal radiation has been separated into two components. The first component is the thermal radiation exchange between spheres in contact with one another, which for the purpose of this study is called the short range radiation. The second, which is defined as the longrange radiation, is the thermal radiation between spheres further than one sphere diameter apart and therefore not in contact with each other. Currently a few shortcomings exist in the modelling of the long–range radiation heat transfer in the MSUC model. It was the purpose of this study to address these shortcomings. Recently, work has been done by Pitso (2011) where Computational Fluid Dynamics (CFD) was used to characterise the long–range radiation in a packed bed. From this work the Spherical Unit Nodalisation (SUN) model has been developed. This study introduces a method where the SUN model has been modified in order to model the long–range radiation heat transfer in an annular reactor packed with uniform spheres. The proposed solution has been named the Cylindrical Spherical Unit Nodalisation (CSUN, pronounced see–sun) model. For validation of the CSUN model, a computer program was written to simulate the bulk region of the High Temperature Test Unit (HTTU). The simulated results were compared with the measured temperatures and the associated heat flux of the HTTU experiments. The simulated results from the CSUN model correlated well with these experimental values. Other thermal radiation models were also used for comparison. When compared with the other radiation models, the CSUN model was shown to predict results with comparable accuracy. Further research is however required by comparing the new model to experimental values at high temperatures. Once the model has been validated at high temperatures, it can be expanded to near wall regions where the packing is different from that in the bulk region. / Thesis (M.Ing. (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2012.

L'objectif de cette thèse est d'étudier l'impact de la structure interne des matériaux isotropes granulaires sur leurs propriétés thermiques macroscopiques. Nous avons développé un code de calcul pour résoudre les transferts thermiques au sein d'un matériau hétérogène tridimensionnel. Ce code est couplé avec un algorithme de génération de structures aléatoires. Après validation expérimentale, nous avons généré des géométries granulaires dont nous avons caractérisé la structure interne, puis nous avons étudié l'impact de cette structure sur la conductivité thermique. Nous avons également développé une nouvelle méthode de mesure expérimentale de la capacité thermique surfacique d'un élément de paroi à structure interne complexe. L'originalité de cette méthode est le couplage d'un modèle analytique de la capacité thermique, indépendant des propriétés thermiques des constituants, et d'une étude expérimentale. / The objective of this thesis is to study the impact of the internal structure of isotropic granular materials on the macroscopic thermal properties. We have developed a model to solve the heat transfer problem within a heterogeneous three-dimensional material. This code is coupled with an algorithm generating random structure. After an experimental validation, we first generated granular materials and we characterized their internal structure; then we studied the impact of this structure on the thermal conductivity. We also developed a new experimental method for measuring the heat capacity area of a wall element with complex internal structure. The originality of this method is the coupling of an analytical model of heat capacity area, which is independent of the thermal properties of the constituents, and an experimental study.

L’augmentation de la puissance électrique des composants électroniques pose le problème de la dissipation de la chaleur générée. Les boîtiers électriques doivent permettre la dissipation de cette chaleur en conservant une isolation électrique. La solution retenue pour évacuer la chaleur par transfert thermique consiste en matériaux composites dont les renforts par leur structuration vont améliorer la conductivité thermique. Des composites à matrice polymère ont été choisis pour leur aptitude de mise en forme. La conductivité thermique et l’isolation électrique sont assurées par des charges céramiques. Les méthodes d’homogénéisation donnent des pistes d’amélioration du comportement de composites en fonction des propriétés de leurs constituants, de leur géométrie et de leur distribution. Elles fournissent ainsi une formulation optimisée de matériaux répondant à certaines caractéristiques issues de cahiers des charges émanant du partenaire industriel (Institut de Soudure). La conductivité thermique attendue des composites impose une forte fraction volumique de charges pour compenser le caractère isolant de la matrice polymère. Des méthodes d’homogénéisation ont été développées pour prédire la conductivité thermique effective pour de forts taux de charges (supérieur à 20%) et des contrastes élevés de conductivité thermique. La présence d’une interphase provenant d’incompatibilités fortes entre les composants doit également être modélisée / The increase of electronic components in the integrated circuits and the required electrical power set the question of the dissipation of the heat generated. The electrical box must favor the heat dissipation while maintaining electrical insulation. The solution chosen to transfer the heat is to develop composite materials whose reinforcements by their structure will improve the thermal conductivity. Polymer-based composite materials were chosen for their building ability. Thermal conductivity and electrical insulation are insured by ceramic reinforcements. The homogenization methods allow to improve the composites’ design according to the properties of their constituents, their geometry and their distribution. They thus provide an optimized formulation of materials satisfying the characteristics emanating from the industrial partner (‘Institut de Soudure’). The expected thermal conductivity of the composites imposes a high volume fraction of reinforcements to counterbalance the insulating polymer matrix. Homogenization methods have been developed to provide predictions of effective thermal conductivity for high (greater than 20%) reinforcement rates and high thermal conductivity contrasts. The presence of an interphase resulting from strong physico-chemical incompatibilities between the components must also be modeled

La problématique suivie dans ce travail est la détermination des propriétés effectives, élastiques et conductivité, de matériaux poreux hétérogènes tels que des roches, les roches oolithiques en particulier, en relation avec leur microstructure. Le cadre théorique adopté est celui de l’homogénéisation des milieux hétérogènes aléatoires et on suit les approches par tenseurs d’Eshelby. Ces approches sont basées sur la résolution des problèmes d’Eshelby : le problème de l’inclusion (premier problème) et le problème de l’inhomogénéité (second problème) isolées dans un milieu infini. La solution de ces problèmes de référence est analytique, en élasticité linéaire isotrope et en diffusion linéaire stationnaire, dans le cas d’inhomogénéités 2D ou 3D de type ellipsoïde. Elle conduit à la définition de tenseurs caractérisant les interactions entre l’inclusion/inhomogénéité et le milieu environnant. On utilise dans ce travail les tenseurs de contribution relatifs à une inhomogénéité isolée, définis par Kachanov et Sevostianov 2013, contributions à la souplesse (élasticité) et à la résistivité (conductivité). Ces tenseurs au cœur des méthodes d’homogénéisation de type EMA (Effective Medium Approximation), et en particulier les schémas NIA (Non Interaction Approximation), Mori Tanaka et Maxwell. Ce travail est centré sur la caractérisation des paramètres géométriques microstructuraux dont l’influence sur les propriétés effectives est majeure. On étudie en particulier les effets de forme des inhomogénéités, la nouveauté est l’aspect 3D. Les observations microstructurales de roches oolithiques, dont le calcaire de référence de Lavoux, mettent en évidence des hétérogénéités de forme 3D et concave. En particulier les matériaux de remplissage inter-oolithes, pores ou calcite syntaxiale. Ces formes peuvent être observées sur d’autres matériaux hétérogènes et ont été peu étudiées dans le cadre micromécanique. Cela nécessite de considérer des formes non ellipsoïdales et de résoudre numériquement les problèmes d’Eshelby. Le cœur de ce travail est consacré à la détermination des tenseurs de contribution d’inhomogénéités 3D convexes ou concaves de type supersphère (à symétrie cubique) ou supersphéroïde (à symétrie de révolution). Le premier problème d’Eshelby a été résolu, dans le cas de la supersphère, par intégration numérique de la fonction de Green exacte (solution de Kelvin dans le cas isotrope) sur la surface de l’inclusion. Des modélisations 3D aux éléments finis ont permis de résoudre le second problème d’Eshelby et d’obtenir les tenseurs de contribution à la souplesse et à la résistivité pour les superphère et supersphéroïde. Sur la base des résultats numériques, des relations analytiques simplifiées ont été proposées pour les tenseurs de contribution sous forme de fonctions des paramètres élastiques des constituants et du paramètre adimensionnel p caractérisant la concavité. Un résultat important, dans le cas de la superphère et dans le domaine concave, est l’identification d’un même paramètre géométrique pour les tenseurs de contribution à la souplesse et à la résistivité. Les résultats numériques et théoriques obtenus sont appliqués à deux problèmes : l’estimation de la conductivité thermique effective de roches calcaires oolithiques d’une part et l’étude de l’extension des relations dites de substitution définies par Kachanov et Sevostianov 2007 au cas non ellipsoïdal d’autre part. Pour le premier problème, un modèle micromécanique de type Maxwell, à deux échelles a permis de retrouver les résultats expérimentaux disponibles dans la littérature, en en particulier l’influence de la porosité sur la conductivité thermique effective dans les cas sec et humide. Dans le cas du second problème, les résultats obtenus ont permis de montrer que la validité de relations de substitution est restreinte, dans le cas non ellipsoïdal et en considérant une forme d’inhomogénéité de type supersphère, au domaine convexe seulement / Focusing on the effect of shape factor on the overall effective properties of heterogeneous materials, the 1st and the 2nd Eshelby problem related to 3-D non-ellipsoidal inhomogeneities with a specific application to oolitic rocks have been discussed in the current work. Particular attention is focused on concaves shapes such as supersphere and superspheroid. For rocks, they may represent pores or solid mineral materials embbeded in the surrounding rock matrix. In the 1st Eshelby problem, Eshelby tensor interrelates the resulting strain about inclusion and eigenstrain that would have been experienced inside the inclusion without any external contraire. Calculations of this tensor for superspherical pores– both concave and convex shapes – are performed numerically. Results are given by an integration of derivation of Green’s tensor over volume of the inclusion. Comparisons with the results of Onaka (2001) for convex superspheres show that the performed calculations have an accuracy better than 1%. The current calculations have been done to complete his results. In the 2nd Eshelby problem, property contribution tensors that characterizes the contribution of an individual inhomogeneity on the overall physical properties have been numerically calculated by using Finite Element Method (FEM). Property contribution tensors of 3D non ellipsoidal inhomogeneities, such as supersphere and superspheroid, have been obtained. Simplified analytical relations have been derived for both compliance contribution tensor and resistivity contribution tensor. Property contribution tensors have been used to estimate effective elastic properties and effective conductivity of random heterogeneous materials, in the framework of Non-Interaction Approximation, Mori-Tanaka scheme and Maxwell scheme. Two applications in the field of geomechanics and geophysics have been done. The first application concerns the evaluation of the effective thermal conductivity of oolitic rocks is performed to complete the work of Sevostianov and Giraud (2013) for effective elastic properties. A two step homogenization model has been developed by considering two distinct classes of pores: microporosity (intra oolitic porosity) and meso porosity (inter oolitic porosity). Maxwell homogenization scheme formulated in terms of resistivity contribution tensor has been used for the transition from meso to macroscale. Concave inter oolitic pores of superspherical shape have been taken into account by using resistivity contribution tensor obtained thanks to FEM modelling. Two limiting cases have been considered: ‘dry case’ (air saturated pores) and ‘wet case’ (water liquid saturated pores). Comparisons with experimental data show that variations of effective thermal conductivity with porosity in the most sensitive case of air saturated porosity are correctly reproduced. Applicability of the replacement relations, initially derived by Sevostianov and Kachanov (2007) for ellipsoidal inhomogeneities, to non-ellipsoidal ones has been investigated. It it the second application of newly obtained results on property contribution tensors. We have considered 3D inhomogeneities of superspherical shape. From the results, it has been seen that these relations are valid only in the convex domain, with an accuracy better than 10%. Replacement relations can not be used in the concave domain for such particular 3D shape

L'étude du transfert de chaleur et de masse dans les milieux poreux saturés et insaturés fortement chauffés à leur surface possèdent de nombreuses applications, notamment en archéologie, en agriculture et en géothermie. La première partie de ce travail concerne l'amélioration de la méthode AHC (Accumulation de chaleur latente) qui permet de traiter le changement de phase, dans un milieu homogène : l'intervalle de changement de température au moment du changement de phase apparaît comme un paramètre important, et il doit être choisi proportionnel à la taille des mailles. Des résultats à la fois précis et lisses sont obtenus grâce à un raffinement du maillage localisé près de l'interface de changement de phase. La deuxième partie se rapporte à l'estimation des propriétés thermophysiques du sol par problème inverse à l'aide de données à la fois synthétiques et expérimentales. La méthode de Gauss-Newton avec relaxation et l'algorithme de Levenberg-Marquardt sont utilisés pour résoudre le problème inverse. Le choix de l'intervalle de température de la méthode AHC apparaît crucial : la convergence n'est obtenue parfois qu'au prix d'un enchaînement de plusieurs problèmes inverses. La troisième partie présente un modèle simple pour calculer la conductivité thermique effective d'un milieu granulaire contenant une faible quantité d'eau liquide. La forme exacte de ces ménisques est calculée à l'équilibre. Les résultats montrent un phénomène très net d'hystérésis quand on étudie la variation de la conductivité thermique effective en fonction de la quantité d'eau liquide ; un futur travail concernant un nouveau modèle insaturé, limité au cas du régime pendulaire et présenté à la fin de cette thèse, devrait pouvoir utiliser ces résultats. / In this work, we are interested in studying heat and mass transfer in water saturated and unsaturated porous medium with a strong heating at the surface. Applications concerned are archaeology, agriculture and geothermal engineering. The first part of this work concerns the improvement of the AHC (Apparent Heat Capacity) method used in the numerical resolution of phase change problem in a homogeneous medium: the phase change temperature interval, over which the heat capacity varies, appears as a key parameter which must be chosen proportional to the mesh size. Accurate and smooth results are obtained thanks to a local refinement of the mesh near the phase change interface. The second part is about the estimation of the thermophysical properties of the soil by inverse problem using both synthetic and experimental data. The Damped Gauss-Newton and the Levenberg-Marquardt algorithms are used to solve the problem. In relation with the AHC method, the choice of the phase change temperature interval caused convergence problems which have been fixed by chaining many inverse problems. The obtained results show good convergence to the desired solution. The third part presents a simple model to calculate the effective thermal conductivity of a granular medium which contains a small quantity of liquid water. The exact shape of the liquid menisci between the grains is calculated at equilibrium. The effective thermal conductivity experiences a hysteresis behavior with respect to the liquid volume. A future work that concerns a new unsaturated model, restricted to the pendular regime and detailed at the end of this thesis, should be able to use this result.

Zellulare metallische Strukturen zeichnen sich durch ihre hohe Porosität und ihre komplexe geometrische Struktur aus. Dadurch ist die Bestimmung ihrer Wärmeleitfähigkeit eine komplizierte und aufwändige Messaufgabe. Für die vorliegende Arbeit wurden Materialien ausgewählt, die sich im Grundmaterial, in der Herstellungsmethode und im strukturellen Aufbau unterscheiden. Dabei wurden Materialien mit unterschiedlicher Porosität und Porengröße untersucht. Die effektive Wärmeleitfähigkeit wurde mit drei unterschiedlichen Messverfahren bestimmt. Die Messungen wurden bei normalem Druck bei Raumtemperatur und für ausgewählte Materialien bis maximal 800 °C durchgeführt. Dabei wurden die Bedingungen für die optimale Durchführung der Messungen und die Probleme jedes Verfahrens herausgearbeitet. Daraus wurden Empfehlungen für die optimale Messungsdurchführung abgeleitet. Die gewonnenen Messergebnisse wurden miteinander, mit Werten aus der Literatur und mit vorhandenen mathematischen Modellen verglichen.

Zellulare metallische Strukturen zeichnen sich durch ihre hohe Porosität und ihre komplexe geometrische Struktur aus. Dadurch ist die Bestimmung ihrer Wärmeleitfähigkeit eine komplizierte und aufwändige Messaufgabe. Für die vorliegende Arbeit wurden Materialien ausgewählt, die sich im Grundmaterial, in der Herstellungsmethode und im strukturellen Aufbau unterscheiden. Dabei wurden Materialien mit unterschiedlicher Porosität und Porengröße untersucht. Die effektive Wärmeleitfähigkeit wurde mit drei unterschiedlichen Messverfahren bestimmt. Die Messungen wurden bei normalem Druck bei Raumtemperatur und für ausgewählte Materialien bis maximal 800 °C durchgeführt. Dabei wurden die Bedingungen für die optimale Durchführung der Messungen und die Probleme jedes Verfahrens herausgearbeitet. Daraus wurden Empfehlungen für die optimale Messungsdurchführung abgeleitet. Die gewonnenen Messergebnisse wurden miteinander, mit Werten aus der Literatur und mit vorhandenen mathematischen Modellen verglichen.