abstract: Production from a high pressure gas well at a high production-rate encounters the risk of operating near the choking condition for a compressible flow in porous media. The unbounded gas pressure gradient near the point of choking, which is located near the wellbore, generates an effective tensile stress on the porous rock frame. This tensile stress almost always exceeds the tensile strength of the rock and it causes a tensile failure of the rock, leading to wellbore instability. In a porous rock, not all pores are choked at the same flow rate, and when just one pore is choked, the flow through the entire porous medium should be considered choked as the gas pressure gradient at the point of choking becomes singular. This thesis investigates the choking condition for compressible gas flow in a single microscopic pore. Quasi-one-dimensional analysis and axisymmetric numerical simulations of compressible gas flow in a pore scale varicose tube with a number of bumps are carried out, and the local Mach number and pressure along the tube are computed for the flow near choking condition. The effects of tube length, inlet-to-outlet pressure ratio, the number of bumps and the amplitude of the bumps on the choking condition are obtained. These critical values provide guidance for avoiding the choking condition in practice. / Dissertation/Thesis / M.S. Mechanical Engineering 2013

New unsaturated coupled models have been developed for fluid transport in deformable rock by using modified mixture theory rather than a fully mechanics-based approach. These models include the following: an unsaturated hydro-mechanical coupled model for both non-swelling and swelling materials, in which a new coupled formulation for hydration swelling rock has been included; and an unsaturated hydro-mechanical-chemo coupled model, incorporating a new coupled formulation including osmosis flow and an unsaturated version of Darcy's law which has been extended by including osmosis effects.Modified mixture theory is mainly based on non-equilibrium thermodynamics. Helmholtz free energy is used to give the energy relationship between the fluids and solid and, by using the Gibbs-Duhem equation, the interactions between different fluids such as gas, water and chemical can be obtained. In this research, general coupled formulations for both large small and deformations have been obtained. For swelling rocks, the water between the clay platelets can be modeled by including the difference between the free energy of whole domain and that of the pore water plus the solid skeleton. By assuming small deformations, the final equations can be compared with those derived using the mechanics approach.The new coupled models have been tested by carrying out simple benchmark numerical simulations using finite elements. Problems analyzed include: (1) the consolidation of saturated swelling rocks in which the hydration swelling effects on consolidation have been analysed in detail; (2) the desaturation and resaturation of seasonally affected rocks around tunnels; (3) the desaturation stage for swelling rocks used in the containment of nuclear waste disposal; (4) chemical transport in very low permeability rock used for nuclear waste disposal, in which particular attention has been focused on osmosis flow and chemical consolidation. In summary, this thesis extends modified mixture theory and develops new coupled formulations which can be applied to deep nuclear waste disposal, including tunnelling, drilling and chemical transport in low permeability host rock.

In this thesis, we study the nonlinear diffusion equation especially Porous Medium Equation (PME). u_t= \Delta(u^m) + f(u), Parameter m>1 in the case of slow diffusion, m=1 means linear model and $0

The ability of coated particles of enriched uranium dioxide (UO2) fuel to withstand high temperatures and contain the fission products in the case of a loss of cooling event is a vital passive safety measure over traditional nuclear fuel requiring active safety systems to provide cooling. As a possible solution towards enhancing the safety of light-water reactors (LWRs), it is envisaged that the fuel in the form of loose-coated particles in a helium atmosphere can be introduced inside Silicon-Carbide nuclear reactor fuel cladding tubes of the fuel elements. The coated particles in this investigation were treated as a bed from where heat was transferred to the cladding tube by means of helium gas and the gas movement was by natural convection. Hence, it is proposed that light-water reactors (LWR) could be made safer by redesigning the fuel in the fuel assembly (see Fig. 1.3b). As a first step towards the implementation of this proposal, a proper understanding of the mechanisms of heat transfer, fluid flow and pressure drop through a packed bed of spheres during natural convection was of utmost importance. Such an understanding was achieved through a review of existing literature on porous media. However, most heat transfer correlations and models in heated packed beds are for forced convectional conditions and as such characterise porous media as a function of Reynolds number only rather than expressing media heat transfer performance as a function of thermal properties of the bed in combination with the various components of the overall heat transfer. The media heat transfer performance considered as a function of thermal properties of the bed in the proposed design is found to be a more appropriate approach than the media as a function of Reynolds number. The quest to examine the particle-to-fluid heat transfer characteristics expected in the proposed new fuel design led to implementing this research work in three phases, namely experimental, theoretical and numerical simulation. An experimental investigation of fluid-to-particle natural convection heat transfer characteristics in packed beds heated from below was carried out. Captured data readings from the experiment were analysed and heat transfer characteristics in the medium evaluated by applying the first principle heat transfer concept. A basic unit cell (BUC) model was developed for the theoretical analysis and applied to determine the heat transfer coefficient, h, of the medium. The model adopted a concept in which a single unit of the packed bed was analysed and taken as representative of the entire bed; it related the convective heat transfer effect of the flowing fluid with the conduction and radiative effect at the finite contact spot between adjacent unit cell particles. As a result, the model could account for the thermophysical properties of sphere particles and the heated gas, the interstitial gas effect, gas temperature, contact interface between particles, particle size and particle temperature distribution in the investigated medium. Although the heat transfer phenomenon experienced in the experimental set-up was a reverse case of the proposed fuel design, the study with the achievement in the validation with the Gunn correlation aided in developing the appropriate theoretical relations required for evaluating the heat transfer characteristics in the proposed nuclear fuel design. A slender geometrical model mimicking the proposed nuclear fuel in the cladding was numerically simulated to investigate the heat transfer characteristics and flow distribution under the natural convective conditions anticipated in beds of randomly packed spheres (coated fuel particles) using a commercial code. Random packing of the particles was achieved by discrete element method (DEM) simulation with the aid of Star CCM+ while particle-to-particle and particle-to-wall contacts were achieved through the combined use of the commercial code and a SolidWorks CAD package. Surface-to-surface radiative heat transfer was modelled in the simulation reflecting real-life application. The numerical results obtained allowed for the determination of parameters such as particle-to-fluid heat transfer coefficient, Nusselt number, Grashof number and Rayleigh number. These parameters were of prime importance when analysing the heat transfer performance of a fixed bed reactor. A comparison of three approaches indicated that the application of the CFD combined with the BUC model gave a better expression of the heat transfer phenomenon in the medium mimicking the heat transfer in the new fuel design / Thesis (PhD)--University of Pretoria, 2016. / Mechanical and Aeronautical Engineering / PhD / Unrestricted

L'objectif de cette thèse est de proposer une modélisation du rayonnement thermique dans un réacteur nucléaire au cours d'un accident grave conduisant à la dégradation des crayons combustibles. Un réacteur étant refroidi par de l'eau, le rayonnement se fait en présence de vapeur et de gouttes d'eau. Le modèle de rayonnement est construit à partir d'expériences de dégradation de crayons fossiles, réalisées sur le réacteur expérimental PHEBUS.Les configurations géométriques accidentelles de grappes de 21 crayons dégradés ont pu être caractérisées en trois dimensions à partir d'images issues de tomographies. Les propriétés radiatives homogénéisées de ces configurations ont été complètement caractérisées à partir de la fonction de distribution cumulée d'extinction Gext et de la fonction de phase de diffusion p. Ces fonctions ont été précisément calculées par une méthode de Monte Carlo. Gext, qui n'est pas de type exponentiel, ne suit pas la loi de Beer. p dépend fortement des angles d'incidence et de diffusion. A partir de l'équation de transfert radiatif généralisée à des milieux non Beeriens, introduite par Taine et al., un tenseur des conductivités radiatives a été déterminé par une méthode de perturbations, en supposant dans une première étape la phase fluide transparente. Les conductivités radiatives axiales et radiales ont été exprimées avec précision en fonction de la porosité, de la surface spécifique et de l'absorptivité locale du milieu poreux. Dans une deuxième étape, une équation de transfert radiatif à trois températures a été établie. Dans ce modèle, les effets de la phase fluide sur le rayonnement ont été couplés aux effets des parois. Les propriétés radiatives de la vapeur et des gouttes d'eau sont calculées en utilisant respectivement le modèle CK et la théorie de Mie, dans les conditions thermohydrauliques typiques des accidents de réacteur. Les flux radiatifs s'expriment en fonction de flux conductifs couplés caractérisés par des conductivités radiatives associées aux champs de températures de chaque phase. Les puissances volumiques échangées par rayonnement entre les phases sont aussi calculées à partir de ce modèle. / This work aims at modelling thermal radiation in a nuclear reactor, in the course of a severe accident leading to its degradation. Because the reactor coolant is water, radiative heat transfer occurs in presence of steam and water droplets. The 3D geometry of a fuel bundle with 21 damaged rods has been characterized from tomography images. The degradation of the rods has been simulated in the experimental small-scale facility PHEBUS.The homogenized radiative properties of the considered configurations with a transparent fluid phase have been completely characterized by both the extinction cumulated distribution function Gext and the scattering phase functions p. Gext strongly differs from the exponential function associated with the Beer law and p strongly depends on both the incidence and the scattering directions. By using the radiative transfer equation generalized for non Beerian porous media by Taine et al. the radiative conductivity tensor has been first determined with a transparent fluid phase, by a numerical perturbation method. Only the diagonal radial and axial components of this tensor are not equal to zero. They have been fitted by a simple law only depending on the porosity, the specific area and the wall absorptivity. In a second step, a radiative transfer equation based on three temperatures is established. This model takes into account a semi transparent fluid phase by coupling the radiative properties of fluid and solid phases. The radiative properties of water steam and droplets are calculated respectively with the CK approach and Mie theory, in typical thermal hydraulics conditions of reactor accidents. The radiative fluxes verify the Fourier law and are characterized by radiative coupled conductivity tensors associated with the temperatures of each phase. The radiative powers exchanged between phases per unit volume are also calculated from this model.

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This thesis focuses on a certain type of nonlinear parabolic partial differential equations, i.e. PME and FDE. Chapter 1 consists of a survey on results related to PME and FDE, and a short review on some works about deriving gradient estimates in probabilistic ways. In Chapter 2 we estimate gradient on space variables of solutions to the heat equation on Euclidean space. The main idea is to construct two semimartingales by letting the solution and its gradient running backward on the path space of a diffusion process. Estimates derived from decompositions of those two semimartingales are then combined to give rise to an upper bound on gradient that only involves the maximum of the initial data and time variable. In particular, it is independent of the dimension. In Chapter 3 we carry the idea in Chapter 2 onto the study of positive solutions to PME or FDE, and obtained a similar type of bound on |∇u| for local solutions to PME or FDE on Euclidean space. In existing literature there have always been constraints on m. By considering a more general form of transformation on u and introducing a family of equivalent measures on path space, we add more flexibility to our method. Thus our result is valid for a larger range of m. For global solutions, when m violates our constraint, we need two-sided bound on u to control |∇u|. In Chapter 4 we utilize maximum principle to derive Li-Yau type gradient estimate for PME on a compact Riemannian manifold with Ricci curvature bounded from below. Our result is able to yield a Harnack inequality possessing the right order in time variable when the lower bound of Ricci curvature is negative.

In this thesis, we consider four different problems in membrane filtration, using a different mathematical approach in each instance. We account for the fluid-driven deformation of a filtercake using nonlinear poroelasticity in Chapter 2. By considering feeds with very high and very low particle concentrations, we introduce a quasi-static caking model that provides a suitable approximation to the full model for the physically realistic concentration regimes. We illustrate the agreements and differences between our model and the existing conventional cake-filtration law. In Chapter 3, we introduce a stochastic model for membrane filtration based on the quantised nature of the particles and show how it can be applied for feeds with different particle types and membranes with an interconnected pore structure. This allows us to understand the relation between the effects of clogging on the level of an individual pore and on the macroscopic level of the entire membrane. We conclude by explaining the transition between the discrete and continuous model based on the Fokker--Planck equation. In Chapter 4, we consider the inverse problem of determining the underlying filtration law from the spreading speed of a particle-laden gravity current. We first couple the theory of gravity currents with the stochastic model developed in Chapter~3 to determine a filtration law from a given set of experiments. We then generalise this idea for the porous medium equation, where we show that the position of the front follows a power law for the conventional filtration laws, which allows us to infer the clogging law in certain instances. We conclude the thesis by showing in Chapter 5 how we can combine experimental measurements for the clogging of a depth filter and simple fluid dynamics to accurately predict the pressure distribution in a multi-capsule depth filter during a filtration run.

Cette étude porte sur les systèmes de stockage thermique régénératif dont le principe consiste à stocker de l’énergie sous forme de chaleur sensible dans un lit fixe. Le système est chargé et déchargé à l’aide d’un fluide caloporteur circulant à travers le lit fixe. Ce type de système est prometteur pour réduire le coût des infrastructures de stockage, par exemple dans les centrales solaires thermodynamiques. Cependant, le pilotage de ces systèmes est relativement complexe car leur fonctionnement est régi par divers phénomènes et met en jeux plusieurs modes de transfert de chaleur. Leur identification a permis de développer un modèle numérique monodimensionnel constitué d’une équation de continuité et de trois équations d’énergie : une pour le fluide caloporteur, une pour le solide du lit fixe et une pour les parois du réservoir. Les études expérimentales réalisées sur trois systèmes différents (un système huile/roches+sable, un système gaz/roches et un système gaz/céramique structuré en canaux), ainsi que des résultats issus de la littérature ont permis de valider le modèle proposé dans une large gamme de configurations. Le modèle s’avère notamment capable de traiter les fluides caloporteurs liquides ou gazeux et les lits fixes structurés en canaux ou constitués de milieux granulaires à granulométrie simple ou double. / This study deals with regenerative heat storage systems which aim to store sensible heat in a packed bed. The system is charged and discharged thanks to a heat transfer fluid which circulates through the packed bed. This kind of system is promising to reduce the cost of heat storage facilities, for example in concentrated solar power plants. However, the operation of these systems is relatively complex because their thermal behavior is governed by several phenomena and heat transfer modes. Thanks to the identification of these mechanisms, a one-dimensional numerical model consisting of one continuity equation and three energy conservation equations has been developed. There is one energy equation for the heat transfer fluid, one for the packed bed and one for the walls of the tank. The experimental studies carried out on three different systems (an oil/rock+sand system, a gas/rock system and a structured gas/ceramic system) and experimental results from the literature have enabled to validate the model in various configurations. The model is able to deal with liquid or gaseous heat transfer fluids and with structured packed beds with channels or granular packed beds with uniformly sized particles or particles of two different sizes.

L'utilisation de pile à combustible à bord d'un avion impose d'extraire des espèces légères (telles que l'hydrogène et les hydrocarbures légers) du combustible liquide qui est stocké et utilisé, éventuellement à des températures où se produit une pyrolyse du carburant. La porosité d’un matériau composite pourrait être utilisée pour filtrer les espèces sélectionnées. L'efficacité de séparation d’un matériau poreux dépend de deux facteurs qui sont: la perméance et la sélectivité.Ces facteurs sont souvent déterminés avec une configuration classique utilisant un échantillon en forme d’un disque d’un matériau poreux. Cependant, cette configuration est loin de la réalité qui est composée de tubes. Par conséquent, une étude est réalisée en considérant les deux configurations en utilisant différents types de disques poreux et un tube composite poreux. Ensuite, les résultats obtenus sont comparés et les différents facteurs affectant le processus de perméation sont étudiés.Après cela, un banc d'essai innovant est développé et utilisé afin de déterminer la distribution axiale des deux propriétés d'un tube poreux en acier inoxydable (c'est-à-dire la perméance et la sélectivité). Les effets des conditions opératoires (débit massique d'entrée et pression d'entrée) ont été étudiés. Une nouvelle forme radiale de l'équation de perméabilité aux gaz a été développée pour ce travail et sa relation avec la perméabilité de Darcy est établie. La variation de pression le long de l'axe central du tube est déterminée. Les effets de cette variation de pression sur les propriétés physiques des gaz tels que la densité et la viscosité sont déterminés et leur influence sur la sélectivité est étudiée en utilisant différents gaz tels que l'azote, le dioxyde de carbone, le méthane et l'hélium.Plus tard, un mélange binaire de dioxyde de carbone (CO2) et d'Azote (N2) est considéré sous trois compositions volumétriques différentes (50/50%, 60/40% et 70/30%) afin d'évaluer la propriété de séparation de gaz d’un tube poreux (effet de membrane). La perméabilité au gaz pur, la perméabilité du mélange, la sélectivité idéale et la sélectivité de séparation de ce tube sont déterminées pour un débit massique et une pression d'entrée différents. Les facteurs affectant les distributions de CO2 et de N2 à l'intérieur du tube poreux sont étudiés.Les résultats obtenus peuvent être utiles pour comprendre les facteurs affectant la séparation des gaz dans le cas d'un tube poreux pour des processus industriels continus. / Using Fuel Cell on board of aircraft imposes to extract light species (such as Hydrogen and light hydrocarbons) from the liquid fuel which is stored and used, possibly at temperatures where a fuel pyrolysis occurs. Porosity of a composite material could be used to filtrate the selected species. The separation efficiency of a porous material depends upon two factors which are: Permeance and Selectivity.These factors are often determined with a classical configuration using a porous disk sample. However, this configuration is far from the realistic one consisting of tubes. Therefore, a study is performed considering both configurations using different types of porous disks and a porous composite tube. Then, the obtained results are compared and the different factors affecting the permeation process are studied.After that, an innovative permselectivity test bench is developed and used in order to determine the axial distribution of the two properties of a stainless steel porous tube (i.e. permeance and selectivity). The effects of the operating conditions (inlet mass flowrate and inlet pressure) have been studied. A new radial form of the gas permeability equation has been developed for this work and its relationship with Darcy‘s permeability is established. The pressure variation along the centre axis of the tube is determined. The effects of this pressure variation on the physical properties of gases such as density and viscosity are determined and their influence on the selectivity is studied using different gases such as Nitrogen, Carbon dioxide, Methane, and Helium. Later, a binary mixture of Carbon Dioxide (CO2) and of Nitrogen (N2) is considered under three different volumetric compositions (50/50%, 60/40% and 70/30%) in order to evaluate the separation property of the porous stainless steel tube (membrane effect). The pure gas permeability, the mixture permeability, the ideal selectivity and the separation selectivity of this tube are determined for a different mass flowrate and inlet pressure. The factors affecting the distributions of CO2 and N2 inside the porous tube are investigated. The obtained results can be useful to understand the factors affecting gas separation in case of a porous tube for continuous industrial processes