La nucléation, la croissance et la dissociation des hydrates de gaz au voisinage d’un ménisque eau – gaz dans des capillaires de verre sont étudiées par vidéo-microscopie et spectroscopie Raman à température, pression, mouillabilité et géométrie du substrat contrôlées. Dans ce travail, deux hydrates simples de structure I (hydrate de CO2 et hydrate de CH4), deux hydrates simples de structure II (N2 et Cyclopentane) et un hydrate double (cyclopentane + CO2) sont examinés. En baissant la température bien au-dessous de 0 °C, i.e., sous un fort sous-refroidissement, tous ces hydrates, excepté l’hydrate de cyclopentane, nucléent sans que la glace soit formée. L’hydrate forme d’abord une croûte polycristalline sur le ménisque eau-molécule invitée (guest). Ensuite, il se propage rapidement à partir de ce ménisque dans l’eau sous forme de fibres ou dendrites et le long de la paroi capillaire sous forme d’une croûte fine et polycristalline appelée ”halo”. Sur un substrat hydrophile, ce halo avance du côté de la phase invitée, alimenté par une couche d’eau entre le halo et la paroi interne du capillaire. Symétriquement, sur un verre hydrophobe (traité au silane), le halo et une couche de la phase invitée se propagent du côté eau. Aucun halo n’est observé sur un substrat de mouillabilité intermédiaire. La croissance et la morphologie du halo d’hydrate et l’épaisseur de sa couche sous-jacente d’eau (ou de phase invitée) dépendent fortement du sous-refroidissement. Grâce au faible volume du capillaire utilisé et à la vitesse rapide de refroidissement, la limite de métastabilité de l’hydrate de CO2 est approchée pour différentes pressions et mouillabilité. Le régime des faibles sous-refroidissements est également étudié : une nouvelle morphologie d’hydrate de CO2 est découverte pour des sous-refroidissements inférieurs à 0,5 °C, constituée d’un cristal creux, générée au niveau du ménisque eau – guest et avançant du côté guest le long du verre, alimenté par une épaisse couche d’eau prise en sandwich entre le verre et ce cristal. Une nouvelle procédure est proposée pour détermination des conditions d’équilibre des hydrates de gaz dans une large plage de température et de pression, y compris l’extension métastable de la ligne triphasique (eau liquide – hydrate – guest) jusqu’à des températures bien inférieures à 0 °C. Enfin, les mécanismes par lesquels le CO2 et le cyclopentane agissent en synergie pour former l’hydrate de structure II sont discutés. / The nucleation, growth and dissociation of gas hydrate across a water – gas meniscus in glass capillaries are investigated by means of video-microscopy and confocal Raman spec- troscopy under controlled temperature, pressure, cooling rate and substrate wettability and geometry. Structure I and II hydrates are examined, with the following guest molecules: CO2, CH4, N2, cyclopentane, and cyclopentane + CO2. By lowering the temperature well below 0 °C, i.e., under strong subcooling, all these hydrates but the cyclopentane hydrate nucleate without forming ice on the liquid water – guest meniscus, which is rapidly covered with a polycrystalline crust. The hydrate then propagates from this meniscus as fast-growing fibers or dendrites in bulk water and as a thin polycrystalline crust, or halo, along the capillary wall. On water-wet substrates, this halo advances on the guest side of the meniscus, fed by a water layer sandwiched between the halo and glass. Symmetrically, on guest-wet (silane-treated) glass, the halo and an underlying guest layer grow on the water side of the interface. No halo is observed on intermediate-wet glass. The hydrate halo growth and morphology and the thickness of its underlying water (or guest) layer strongly depend on subcooling. Thanks to the small capillary volume and the rapid temperature descent, the metastability limit of CO2 hydrate is approached for various pressures and substrate wettabilities. The low subcooling regime is investigated as well: a novel CO2 hydrate morphology is discovered for subcoolings below 0.5 °C, which consist of a hollow hydrate crystal originating from the water – guest meniscus and advancing on the guest side along glass, fed by a thick water layer sandwiched between glass and this crystal. A new procedure is proposed to determine gas hydrate dissociation conditions in a large temperature and pressure range, including the metastable extension of the three-phase (liquid water – hydrate - guest) down to temperatures well below 0 °C. Finally, the mechanisms by which CO2 and cyclopentane synergistically act to form the structure II hydrate are discussed.

We have fully classified the momentum polytopes of the SU(3) action on CP(2)xCP(2) and CP(2)xCP(2) xCP(2), both actions with weighted symplectic forms, and their corresponding transition momentum polytopes. For CP(2)xCP(2) the momentum polytopes are distinct line segments. The action on CP(2)xCP(2) xCP(2), has 9 different momentum polytopes. The vertices of the momentum polytopes of the SU(3) action on CP(2)xCP(2) xCP(2), fall into two categories: definite and indefinite vertices. The reduced space corresponding to momentum map image values at definite vertices is isomorphic to the 2-sphere. We show that these results can be applied to assess the dynamics by introducing and computing the space of allowed velocity vectors for the different configurations of two-vortex systems.

Agência Nacional do Petróleo / O comportamento de fases para sistemas binários com um hidrocarboneto leve e um pesado é muito importante tanto para o projeto real de um processo quanto para o desenvolvimento de modelos teóricos. Para atender a crescente demanda por informação experimental de equilíbrio de fases a altas pressões, o objetivo deste estudo é obter uma metodologia que substitua parcialmente ou maximize a pouca informação experimental disponível. Para isto propõe-se a modelagem do equilíbrio de fases em misturas de hidrocarboneto leve com um pesado, sem o conhecimento da estrutura molecular do pesado, inferindo-se os parâmetros do modelo a partir da modelagem de dados de ponto de bolha obtidos na literatura. Esta metodologia implica não só na descrição do equilíbrio de fases de um sistema como na estimação das propriedades críticas do pesado, de difícil obtenção devido ao craqueamento destes a altas temperaturas. Neste contexto, este estudo apresenta uma estratégia que estima indiretamente as propriedades críticas dos compostos pesados. Para isto, foram correlacionados dados experimentais de ponto de bolha de misturas binárias contendo um hidrocarboneto leve e um pesado, usando-se dois modelos: o de Peng-Robinson e o TPT1M (Teoria da Polimerização Termodinâmica de primeira ordem de Wertheim modificada). Os parâmetros ajustados com o modelo de Peng-Robinson correspondem diretamente às propriedades críticas do composto pesado, enquanto os ajustados com o modelo TPT1M foram usados para obtê-las. Esta estratégia fornece parâmetros dependentes do modelo, porém permite o cálculo de outras propriedades termodinâmicas, como a extrapolação da temperatura dos dados estudados. Além disso, acredita-se que a correlação dos parâmetros obtidos com as propriedades críticas disponíveis ajudará na caracterização de frações pesadas de composição desconhecida / The phase behavior of binary systems with a light hydrocarbon and a heavy one is very important both for the design of a process and the development of theoretical models. To meet the growing demand for information from experimental phase equilibria at high pressures, this study aims to obtain a methodology to replace partially or maximize the little experimental data available. Therefore it is proposed to model the phase equilibria in hydrocarbon mixtures with a heavy compound, without the knowledge of its molecular structure, inferring the model parameters from the data modeling of bubble point from the literature. This methodology involves not only the description of phase equilibrium of a system as the estimation of critical properties of heavy hydrocarbons, difficult to obtain due to its cracking at high temperatures. In this context, this work presents a strategy that indirectly estimates critical properties of heavy compounds. Here, bubble point experimental data of binary mixtures containing a light HC and a heavy one are correlated by using a TPT1 (Wertheims first order thermodynamic polymerization theory) model of freely joined tangent spheres employing the square-well potential with variable well width as segment interaction potential. The fitted parameters are used in order to obtain the heavy compound critical properties. The Peng-Robinson model was also used for comparison. This strategy provides model dependent parameters, but allows the calculation of other thermodynamic properties, i.e. temperature extrapolation of the studied data. Besides, the correlation of the obtained parameters with available critical properties will help in the characterization of the heavy fractions

Esta dissertação aborda a teoria dos jogos diferenciais em sua estreita relação com a teoria das equações de Hamilton-Jacobi (HJ). Inicialmente, uma revisão da noção de solução em teoria dos jogos é empreendida. Discutem-se nesta ocasião as idéias de equilíbrio de Nash e alguns de seus refinamentos. Em seguida, tem lugar uma introdução à teoria dos jogos diferenciais, onde noções de solução como a função de valor de Isaacs e de Friedman são discutidas. É nesta altura do trabalho que fica evidente a conexão entre este conceito de solução e a teoria das equações de Hamilton-Jacobi. Por ocasião desta conexão, é explorada a noção de solução clássica e é exposta uma demonstração do fato de que se um jogo diferencial possuir uma função de valor pelo menos continuamente diferenciável, esta será uma solução da equação de Hamilton-Jacobi associada ao jogo. Este resultado faz uso do princípio da programação dinâmica, devido a Bellman, e cuja demonstração está presente no texto. No entanto, quando a função de valor do jogo é apenas contínua, então embora esta não seja uma solução clássica da equação HJ associada a jogo, vemos que ela será uma solução viscosa, ou solução no sentido da viscosidade - e a esta altura são discutidos os elementos e propriedades desta classe de soluções, um teorema de existência e unicidade e alguns exemplos. Por fim, retomamos o estudo dos jogos diferenciais à luz das soluções viscosas da equação de Hamilton-Jacobi e, assim, expomos uma demonstração de existência da função de valor e do princípio da programação dinâmica a partir das noções da viscosidade / This dissertation aims to address the topic of Differential Game Theory in its connection with the Hamilton-Jacobi (HJ) equations framework. Firstly we introduce the idea of solution for a game, through the discussion of Nash equilibria and its refinements. Secondly, the solution concept is then translated to the context of Differential Games and the idea of value function is introduced in its Isaacs\'s as well as Friedman\'s version. As the value function is discussed, its relationship with the Hamilton-Jacobi equations theory becomes self-evident. Due to such relation, we investigate the HJ equation from two distinct points of view. First of all, we discuss a statement according to which if a differential game has a continuously differentiable value function, then such function is a classical solution of the HJ equation associated to the game. This result strongly relies on Bellman\'s Dynamic Programming Principle - and this is the reason why we devote an entire chapter to this theme. Furthermore, HJ is still at our sight from the PDE point of view. Our motivation is simple: under some lack of regularity - a value function which is continuous, but not continuously differentiable - a game may still have a value function represented as a solution of the associated HJ equation. In this case such a solution will be called a solution in the viscosity sense. We then discuss the properties of viscosity solutions as well as provide an existence and uniqueness theorem. Finally we turn our attention back to the theory of games and - through the notion of viscosity - establish the existence and uniqueness of value functions for a differential game within viscosity solution theory.

In this thesis, parameterizations of non-linear interactions in quasigeostrophic (QG) flows for severely truncated models (STM) and Large Eddy Simulations (LES) are studied. Firstly, using Direct Numerical Simulations (DNS), atmospheric barotropic flows over topography are examined, and it is established that such flows exhibit multiple equilibrium states for a wide range of parameters. A STM is then constructed, consisting of the large scale zonal flow and a topographic mode. It is shown that, qualitatively, this system behaves similarly to the DNS as far as the interaction between the zonal flow and topography is concerned, and, in particular, exhibits multiple equilibrium states. By fitting the analytical form of the topographic stationary wave amplitude, obtained from the STM, to the results obtained from DNS, renormalized dissipation and rotation parameters are obtained. The usage of renormalized parameters in the STM results in better quantitative agreement with the DNS.¶ In the second type of problem, subgrid-scale parameterizations in LES are investigated with both atmospheric and oceanic parameters. This is in the context of two-level QG flows on the sphere, mostly, but not exclusively, employing a spherical harmonic triangular truncation at wavenumber 63 (T63) or higher. The methodology that is used is spectral, and is motivated by the stochastic representation of statistical closure theory, with the ‘damping’ and forcing covariance, representing backscatter, determined from the statistics of DNS. The damping and forcing covariance are formulated as 2 × 2 matrices for each wavenumber. As well as the transient subgrid tendency, the mean subgrid tendency is needed in the LES when the energy injection region is unresolved; this is also calculated from the statistics of the DNS. For comparison, a deterministic parameterization scheme consisting of 2×2 ‘damping’ parameters, which are calculated from the statistics of DNS, has been constructed. The main difference between atmospheric and oceanic flows, in this thesis, is that the atmospheric LES completely resolves the deformation scale, the energy and enstrophy injection region, and the truncation scale is spectrally distant from it, being well in the enstrophy cascade inertial range. In oceanic flows, however, the truncation scale is in the vicinity of the injection scale, at least for the parameters chosen, and is therefore not in an inertial range. A lower resolution oceanic LES at T15 is also examined, in which case the injection region is not resolved at all.¶ For atmospheric flows, it is found that, at T63, the matrix parameters are practically diagonal so that stratified atmospheric flows at these resolutions may be treated as uncoupled layers as far as subgrid-scale parameterizations are concerned. It is also found that the damping parameters are relatively independent of the (vertical) level, but the backscatter parameters are proportional to the subgrid flux in a given level. The stochastic and deterministic parameterization schemes give comparably good results relative to the DNS. For oceanic flows, it is found that the full matrix structure of the parameters must be used. Furthermore, it is found that there is a strong injection of barotropic energy from the subgrid scales, due to the unresolved, or partially resolved, baroclinic instability injection scales. It is found that the deterministic parameterization is too numerically unstable to be of use in the LES, and instead the stochastic parameterization must be used to obtain good agreement with the DNS. The subgrid tendency of the ensemble mean flow is also needed in some problems, and is found to reduce the available potential energy of the flow.

The availability of high quality thermodynamic data on solid solutions and compounds present in multicomponent systems assists in optimizing processing parameters for synthesis, and in evaluating stability domains and materials compatibility under different conditions. Several oxide systems of technological interest, for which thermodynamic data was either not available or is inconsistent were selected for study. Thermodynamic properties of phases present in the binary systems Nb-O and Ta-O were measured in the temperature range from 1000 to 1300 K using solid state electrochemical cells based on (Y2O3) ThO2 as the electrolyte. Based on these measurements and more recent data on heat capacity and phase transitions reported in the literature, Gibbs energy of formation for NbO, NbO2, NbO2.422, Nb2O5-x and Ta2O5 were reassessed. Significant improvements in the data for NbO2, Nb2O5 and Ta2O5 are suggested. The pseudo binary system MoO2-TiO2 was investigated because of the inconsistency between the phase diagram and thermodynamic properties of the solid solution reported in the literature. Based on new electrochemical measurements, a new improved phase diagram for the system MoO2-TiO2, incorporating recently discovered monoclinic to tetragonal phase transition in MoO2 at 1533 K, is presented. Isothermal section of the phase diagram for the ternary systems Cr-Rh-O and Ta-Rh-O and thermodynamic properties of ternary oxides CrRhO3 and TaRhO4 were measured for the first time in the temperature range from 900 to 1300 K. Phase relations for these systems have been computed as a function of oxygen potential at fixed temperature and as a function of temperature at selected oxygen partial pressures. Metal-spinel-corundum three-phase equilibrium in the Ni-Al-Cr-O system at 1373 K has been explored because of its relevance to high temperature corrosion of super alloys. The Gibbs energy of mixing of spinel solid solution was derived from the tie-line data and is compared with the values calculated from cation distribution models. An oxygen potential diagram is developed for the decomposition of spinel solid solution to nickel and corundum solid solution at 1373 K under reducing conditions. The high temperature thermodynamic properties of the phases present in quaternary systems Ca-Co-Al-O and Ca-Cu-Ti-O have been measured by solid state electrochemical cells based on stabilized zirconia. Gibbs energies of formation of the quaternary oxides Ca3CoAl4O10 in the temperature range from 1150 to1500 K and CaCu3Ti4O12 in the range from 900 to 1350 K are presented. Chemical potential diagrams have been computed for the system Al2O3-CaO-CoO at 1500 K. The oxygen potential corresponding to the decomposition of the complex perovskite CaCu3Ti4O12 (CCTO) has been calculated as a function of temperature from the emf of the cell. The effect of the oxygen partial pressure on the phase relations in the pseudo-ternary system CaO-(CuO/Cu2O)-TiO2 at 1273 K has been evaluated. The phase diagrams are useful for the control of the secondary phases that form during synthesis of CCTO, a material exhibiting colossal dielectric response.

Studying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F). We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.

Die ternären Systeme Ti/P/Te, Ce/P/Te und Si/P/Te wurden in der vorliegenden Dissertation hinsichtlich der Existenz neuer Verbindungen untersucht. Diese Verbindungen wurden insbesondere in Bezug auf ihre thermochemischen Eigenschaften charakterisiert. Durch Kombination von experimentellen Untersuchungen und thermodynamischen Modellierungen der ablaufenden Festkörper-Gasphasen-Gleichgewichte konnten die Synthesen der neuen Verbindungen optimiert werden. Abschließend erfolgte zudem die physikalische und kristallographische Charakterisierung der gefundenen Phosphidtelluride.

The knowledge of phase equilibrium is essential for the planning and realisation of separation processes in chemical engineering. In this work an equipment for measurement of precise isothermal vapour–liquid equilibria (VLE) using the dynamic method was developed. The pool of experimental data for cyclohexylamine was extended significantly. Isothermal VLE were measured in 3 binary and 4 ternary systems, liquid-liquid equilibria (LLE) were measured in 4 ternary systems and in one quarternary system, in each case for two temperatures, whereas in 2 ternary systems and in the quarternary system even liquid-liquid-liquid equilibria (LLLE) occur. Furthermore, activity coefficients at infinite dilution in 4 binary systems and excess molar volumes in 7 binary systems have been estimated. Binary VLE and LLE data of the components water, octane, cyclohexylamine and aniline of this work and data from literature were fitted with the activity coefficient models NRTL and UNIQUAC, as well as with the equations of state Elliott-Suresh-Donohue (ESD) and Perturbed-Chain-Statistical Associating Fluid Theory (PC-SAFT) which contain both a term to consider explicit hydrogen bonds. In addition, the predictive capabilities of the equations of state (EoS) were investigated. With parameters obtained by simultaneous fitting of VLE and, if available, LLE data similar results with the models NRTL and UNIQUAC could be obtained. Each time the deviations for the vapour pressure were lower than 3 % and lower than 2 % in vapour phase composition. The deviations, in three out of the six systems for vapour pressure and vapour phase composition, were larger with the ESD-EoS than with the activity coefficient models. NRTL, UNIQUAC and ESD delivered similar results with the simultaneously fitted parameters for the LLE, whereas the deviations were lower than 5 %. Comparable results were delivered by the ESD-EoS and PC-SAFT for the fitting and the prediction in the investigated binary systems. Also a fitting for NRTL, UNIQUAC and ESD was carried out, but only to one data set. The intention was to show the effect of parameterization on prediction in ternary systems. Predictions were made for VLE and LLE in ternary systems of the above mentioned components, solely with interaction parameters fitted to binary data. For the models NRTL, UNIQUAC und ESD predictions of simultaneously and separately fitted parameters are presented. It is shown that with parameters simultaneously fitted to several data sets significantly better results could be obtained compared to the parameters separately fitted to a single data set. Additionally, for the equations of state ESD and PC-SAFT predictions for the LL(L)E in ternary systems are compared, but here only with separately fitted parameters. For three out of the four investigated ternary systems a too large miscibility gap is calculated with the models NRTL, UNIQUAC and ESD. In the system water+octane+aniline good results could be obtained for the prediction of the LLLE. In summary the equations of state deliver similar results. In the systems water+octane+CHA and octane+CHA+aniline also too large two phase regions were delivered. Better predictions could be obtained in the systems water+octane+aniline and water+CHA+aniline. The forecasts of the VLE in the ternary systems are good with the simultaneously fitted parameters. The deviations for the vapour phase compositions are as for the vapour pressures under 6 %. Larger deviations occur for the system water+octane+aniline only. As evaluation result for the thermodynamic models can be mentioned that the activity coefficient models NRTL and UNIQUAC deliver somewhat better results for the fitting of the binary data than the equations of state ESD and PC-SAFT however, with a larger number of adjustable parameters. The prediction of the VLE is satisfactorily in the ternary systems and with similar quality of all considered models. Larger deviations occur for the prediction of the LL(L)E in the ternary systems. The results of the ESD-EoS were, with one exception, each time better than those of the activity coefficient models. There is no significant difference between the prediction of the ternary systems for the ESD-EoS and the PC-SAFT.

Polyhalite is an abundant distributed mineral in rock salt formations, and considered to respond as a natural heavy metal sink because the Mg2+ ion can be substituted by other bivalent metal ions like Mn2+, Co2+, Ni2+, Cu2+ and Zn2+. One of the quantities needed to predict mineral solubilities in multi-electrolyte solutions is the solubility constant Ksol. Since polyhalite forms slowly over months or years at 298 K, the solid-liquid phase equilibria experiments were accomplished at 313 K. Enthalpies of dissolution were measured and used to extrapolate lnKsol from 313 K to 298 K. Pitzer’s equations have been applied to describe activities of solute and water, with Pitzer parameters estimated from experimental data. The solubility constants for the polyhalites were applied to calculate the solubility equilibria in the quaternary systems K+, M2+, Ca2+ / SO42- // H2O (M = Mg, Mn, Co, Cu, Zn) at 298 K and 313 K, and in case of Mg-polyhalite also in the hexary system Na+, K+, Mg2+, Ca2+ / Cl-, SO42- // H2O.