Dissertations / Theses on the topic 'Phase rule and equilibrium'

We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle and matrix, or on the nature of misfit. The particle shape is parameterized using a set of design variables which are the magnitudes of vectors from a reference point inside the particle to points on the particle/matrix interface. We use a sequential quadratic programming approach to carry out the optimization. Although this approach can be used to find equilibrium shapes of particles in three dimensional systems, we have presented the details of our formulation for two dimensional systems under plane strain conditions. Using our formulation, we have studied the equilibrium shapes in two dimensional systems with cubic anisotropy; the precipitate and matrix phases may have different elastic moduli, and the misfit may be dilatational or non-dilatational. The equilibrium shapes and their size dependence are analysed within the framework of symmetry-breaking shape transitions. These transitions are further characterized in terms their dependence on the cubic elastic anisotropy parameter, defined by A = 2C44/(C11 – C12), and on the modulus mismatch, defined by δ=μp/μm, where /μp and μm are the effective shear moduli of the precipitate and matrix phases, respectively. Depending on the type of misfit, the systems may be classified into the following four cases: Case A: For dilatational misfit, the equilibrium shapes in isotropic systems are circular (with an isotropic or I symmetry) at small sizes and undergo a transition at a critical size to become ellipse-like (with an orthorhombic or O symmetry). This I --O transition is continuous and is obtained only when the precipitate phase is softer than the matrix. These results are in good agreement with the analytical results of Johnson and Cahn. In cubic systems with dilatational misfit, the particles exhibit a transition from square-like shapes (with a tetragonal or T symmetry) at small sizes to rectangle-like shapes (with an O symmetry) at large sizes. This T -- O transition is continuous. It occurs even in systems with stiffer precipitates; however, it is forbidden for systems with δ >δC, where δ C represents a critical modulus mismatch. The critical size decreases with increasing cubic anisotropy (i.e., with increasing values of (A-1)/(A+1). The sides of the square-like and rectangle-like shapes are along the elastically soft directions. Case B: In these systems, the principal misfits e*xx and e*yy differ in magnitude but have the same sign. The precipitates at small sizes become elongated along the direction of lower misfit; this shape has an O symmetry. In systems with A > 1, they continue to become more elongated along the same direction, exhibiting no symmetry-breaking transition. However, in systems with A < 1, particles at large sizes are elongated along an intermediate direction between the direction of lower misfit and one of the elastically soft <11> directions; this shape has only a monoclinic or M symmetry. This O - M transition, in which the mirror symmetries normal to the x and y axes are lost, may be discontinuous or continuous. The critical size increases with δ (in the range 0.8 < δ <1.25), indicating that this transition would also be forbidden for systems with δ > δC. In systems with A < 1, the critical size decreases with increasing values of A-1/ A+1 Case C: In these systems, the principal misfits differ in both magnitude and sign, and the misfit strain tensor allows an invariant line along which the normal strain is zero. The precipitates at small sizes are elongated along the direction of lower absolute misfit, and possess an 0 symmetry. At large sizes, the mirror symmetries normal to the x and y axes are broken to yield shapes which are elongated along a direction between that of lower misfit and the invariant line. This 0 -> M transition is continuous and occurs in all the systems irrespective of the value of A The critical size increases with A and decreases with δ. Case D; The misfit in this case is a special form of that in Case C; the principal misfits have the same magnitude but opposite signs. The precipitates at small sizes have a square-like shape with its sides normal to the < 11 > axes, irrespective of the type of cubic anisotropy. At large sizes, they become rectangle-like with the long axis oriented along one of the <11> directions. Similar to Case C, this T - 0 transition is continuous and occurs in all the systems irrespective of the values of A. The critical size increases with A and decreases with δ. Thus, we have identified all the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two dimensional systems. We have identified their origin and nature, and characterized them in terms of their dependence on the anisotropy parameter and modulus mismatch.

We present a general approach for determining the equilibrium shape of isolated, coherent, misfitting particles by minimizing the sum of elastic and interfacial energies using a synthesis of finite element and optimization techniques. The generality derives from the fact that there is no restriction on the initial or final shape, or on the elastic moduli of the particle and matrix, or on the nature of misfit. The particle shape is parameterized using a set of design variables which are the magnitudes of vectors from a reference point inside the particle to points on the particle/matrix interface. We use a sequential quadratic programming approach to carry out the optimization. Although this approach can be used to find equilibrium shapes of particles in three dimensional systems, we have presented the details of our formulation for two dimensional systems under plane strain conditions. Using our formulation, we have studied the equilibrium shapes in two dimensional systems with cubic anisotropy; the precipitate and matrix phases may have different elastic moduli, and the misfit may be dilatational or non-dilatational. The equilibrium shapes and their size dependence are analysed within the framework of symmetry-breaking shape transitions. These transitions are further characterized in terms their dependence on the cubic elastic anisotropy parameter, defined by A = 2C44/(C11 – C12), and on the modulus mismatch, defined by δ=μp/μm, where /μp and μm are the effective shear moduli of the precipitate and matrix phases, respectively. Depending on the type of misfit, the systems may be classified into the following four cases: Case A: For dilatational misfit, the equilibrium shapes in isotropic systems are circular (with an isotropic or I symmetry) at small sizes and undergo a transition at a critical size to become ellipse-like (with an orthorhombic or O symmetry). This I --O transition is continuous and is obtained only when the precipitate phase is softer than the matrix. These results are in good agreement with the analytical results of Johnson and Cahn. In cubic systems with dilatational misfit, the particles exhibit a transition from square-like shapes (with a tetragonal or T symmetry) at small sizes to rectangle-like shapes (with an O symmetry) at large sizes. This T -- O transition is continuous. It occurs even in systems with stiffer precipitates; however, it is forbidden for systems with δ >δC, where δ C represents a critical modulus mismatch. The critical size decreases with increasing cubic anisotropy (i.e., with increasing values of (A-1)/(A+1). The sides of the square-like and rectangle-like shapes are along the elastically soft directions. Case B: In these systems, the principal misfits e*xx and e*yy differ in magnitude but have the same sign. The precipitates at small sizes become elongated along the direction of lower misfit; this shape has an O symmetry. In systems with A > 1, they continue to become more elongated along the same direction, exhibiting no symmetry-breaking transition. However, in systems with A < 1, particles at large sizes are elongated along an intermediate direction between the direction of lower misfit and one of the elastically soft <11> directions; this shape has only a monoclinic or M symmetry. This O - M transition, in which the mirror symmetries normal to the x and y axes are lost, may be discontinuous or continuous. The critical size increases with δ (in the range 0.8 < δ <1.25), indicating that this transition would also be forbidden for systems with δ > δC. In systems with A < 1, the critical size decreases with increasing values of A-1/ A+1 Case C: In these systems, the principal misfits differ in both magnitude and sign, and the misfit strain tensor allows an invariant line along which the normal strain is zero. The precipitates at small sizes are elongated along the direction of lower absolute misfit, and possess an 0 symmetry. At large sizes, the mirror symmetries normal to the x and y axes are broken to yield shapes which are elongated along a direction between that of lower misfit and the invariant line. This 0 -> M transition is continuous and occurs in all the systems irrespective of the value of A The critical size increases with A and decreases with δ. Case D; The misfit in this case is a special form of that in Case C; the principal misfits have the same magnitude but opposite signs. The precipitates at small sizes have a square-like shape with its sides normal to the < 11 > axes, irrespective of the type of cubic anisotropy. At large sizes, they become rectangle-like with the long axis oriented along one of the <11> directions. Similar to Case C, this T - 0 transition is continuous and occurs in all the systems irrespective of the values of A. The critical size increases with A and decreases with δ. Thus, we have identified all the symmetry-breaking transitions in equilibrium shapes of coherent precipitates in two dimensional systems. We have identified their origin and nature, and characterized them in terms of their dependence on the anisotropy parameter and modulus mismatch.

Thesis (MScEng)--University of Stellenbosch, 2000.<br>ENGLISH ABSTRACT: Supercritical extraction is being investigated as a possible alternative to the processes currently used in the fractionation of paraffinic waxes. By removing the lighter carbon fractions from the wax, the wax hardness will be improved and its melting temperature range reduced, hence improving the performance of the wax product in certain applications. In order to evaluate and operate such an extraction process optimally, it is necessary to have a thermodynamic model that accurately represents the process system. There are, however, currently no predictive models available for these systems. In order to fit present models to the systems, accurate phase equilibrium data of the supercritical solvent - n-alkane systems are needed. Unfortunately, the amount of reliable published data on these systems in the required operating range is very limited. A view cell was designed and developed with which these high pressure equilibria could be studied. The binary phase equilibria of supercritical CO2 with n-CI2, n-CI6, n-C20, n-C24, n-C28 and n-C36 and of supercritical ethane with n-CI6, n-C24 and n-C28 were measured in the temperature range 313 - 367 K. It was found that the systems with these two solvents have very different types of phase behaviour. The n-alkane solubility is much higher in ethane, but supercritical CO2 will provide a much better degree of control over the selectivity achieved in an extraction process. Of the various equations of state investigated, it was found that the Patel Teja equation of state provided the best fit of the CO2 - n-alkane systems and that the Soave-Redlich- Kwong equation fitted the ethane - n-alkane systems the best. The interaction parameters of both these equations of state display a functional relationship with temperature and nalkane acentric factor, making it possible to determine parameter values for application at other operating temperatures and with other n-alkane systems. It was found that the current equations of state were not able to represent the phase equilibria accurately over the entire range of operating conditions. The poor performance of the equations of state can be attributed to inherent flaws in the existing equations of state.<br>AFRIKAANSE OPSOMMING: Superkritiese ekstraksie word tans ondersoek as 'n moontlike altematief vir die prosesse wat huidiglik gebruik word om paraffiese wasse te fraksioneer. Die Iigter koolstofwasse word verwyder om die washardheid te verhoog en die temperatuurgebied waaroor die was smelt te verklein. Dit verbeter dan die was se kwaliteit en werkverrigting. Modelle wat die superkritiese ekstraksie proses akkuraat kan voorstel word egter benodig om die ekstraksie proses te kan evalueer en optimaal te bedryf. Daar is tans geen modelle beskikbaar wat die proses direk kan voorstel nie. Akkurate fase-ewewigsdata word benodig om bestaande modelle aan te pas vir gebruik in hierdie sisteme. Daar is egter baie min betroubare faseewewigsdata vir die superkritiese oplosmiddel - n-alkaan sisteme beskikbaar in die literatuur. 'n Sig-sel, waarrnee hierdie hoe druk data gemeet kan word, is ontwerp en ontwikkel. Die volgende binere fase ewewigte is in die temperatuur gebied 313 - 367 K gemeet: superkritiese CO2 met n-CI2, n-CI6, n-C20, n-C24, n-C28 en n-C36, en superkritiese Etaan met n-CI6, n-C24 en n-C28. Daar is gevind dat hierdie twee superkritiese oplosmiddelsisteme verskillende tipes fase-ewewigsgedragte openbaar. Die n-alkane het 'n baie boer oplosbaarheid in Etaan, maar deur superkritiese C02 in 'n ekstraksie kolom te gebruik, sal tot beheer oor die selektiwiteit van die ekstraksieproses lei. Uit die verskillende toestandsvergelykings wat ondersoek is, is daar gevind dat die Patel- Teja vergelyking die CO2 sisteme die beste kon beskryf en dat die Soave-Redlich-Kwong vergelyking die beste vergelyking was om die Etaan sisteme mee te modelleer. Beide die toestandsvergelykings se interaksie parameters het 'n funksionele verband met temperatuur en die n-alkaan asentrise faktor getoon. Dit is dus moontlik om waardes vir die parameters vir sisteme by ander temperature en met ander n-alkaan tipes te bepaal. Daar was gevind dat die bestaande toestandsvergelykings nie die die fase-ewewigte oor die hele eksperimenele gebied akkuraat kon voorstel nie. Dit kan toegeskryf word aan foute wat inherent is aan die vergelykings.

Le besoin d’accès aux énantiomères purs a largement augmenté depuis les années 90. Les procédés ayant recours à la cristallisation sont souvent moins cher mais tout aussi efficace que les autres procédés. Deux études ont été réalisées dans cette thèse : -La première concerne l’amélioration d’un procédé déjà établi. En effet, l’efficacité de la Cristallisation Préférentielle (PC) de conglomérat est connue. Néanmoins, son application à des solutions solides images l’une de l’autre dans un miroir est plus complexe. Est présenté le premier exemple de PC réalisée sur un système à solution solide complète démixant à basse température : le sel d’hydrogénomaléate de baclofène (un API). De plus, il s’agit de la première preuve expérimentale d’un tel équilibre de phases et de PC de solutions solides dont les résultats sont comparables à la cristallisation préférentielle traditionnelle. -Le second sujet ne traite pas directement de la séparation d’énantiomère mais de prometteuses discriminations chirales ont été obtenues. Le diagramme d’équilibre de phases entre les compositions racémiques de deux couples d’atropoisomères a été étudié. Deux résultats en lien avec la discrimination chirale ont été établis : le premier est la possibilité de déracémiser un composé A initialement non-discriminé (composé racémique) à l’intérieur de la matrice d’un composé B discriminé (conglomérat). Le second est l’observation d’un nouveau type de chiralité : Chiralité Induite par Dissymmétrie de Distribution (CIDD)<br>The need for access to pure enantiomers has largely increased since the nineties. Processes using crystallization are often cheaper and as efficient as other processes. Thus, the improvement of existing techniques and the need for new processes are attractive for scientist and industry. Two studies have been performed in this thesis:The first one concerns the improvement of an existing technique. Indeed, conglomerate Preferential Crystallization efficiency is well known. Nonetheless, its application to mirror image partial solid solutions is more difficult. Here is presented the first example of PC performed on a complete solid solution at high temperature which shows a solid solutions demixion at low temperature: the baclofen hydrogenomaleate salt (an API). Moreover, it is the first experimental observation of such phase equilibria and PC of mirror image solid solutions whose results are comparable to classical PC. -The second topic did not involve directly enantiomer separation but promising chiral discriminations have been obtained. The phase diagram between the racemic composition of two couples of atropisomers has been defined. Beside multiples equilibria observed, two results concerning the chiral discrimination can be highlighted: the first one is the possibility to deracemize a compound A initially not discriminated (racemic compound) inside the matrix of a compound B (which forms a stable conglomerate). The second is the observation of a new kind of chirality: Chirality Induced by Dissymmetric Distribution (CIDD)

The thermophysical properties of supercritical CO₂ and H₂O-CO₂ mixtures are reviewed and their computation and prediction improved through theory and experiment. A resolution is attempted among inconsistencies between and within data sets, including P-V-T measurements, phase equilibrium experiments and equations of state. Pure carbon dioxide: Equations of state for CO₂ (Kerrick & Jacobs, 1981; Bottinga & Richet, 1981; Holloway, 1977) are based solely on P-V-T data up to 8 kbar and lead to deviations from phase equilibrium data at pressures greater than 10-20 kbar. Mathematical programming analysis has been applied to the fitting of parameters for an equation of state using simultaneously constraints from phase equilibrium and P-V-T data. Phase equilibrium data up to 42 kbar are used to define a feasible region for the adjustable parameters in free energy space. Each half-bracket places an inequality constraint on the fugacity of CO₂ provided the thermophysical properties of the solid phases are known. Except for magnesite thermophysical data from the mineral data base of Berman (1988) were used. A least squares objective function served to optimize parameters to P-V-T data. The enthalpy of formation of magnesite was revised on the basis of recent low pressure phase equilibrium experiments by Philipp (1988) to —1112.505 kj/mole. Piston-cylinder experiments were performed to constrain the equilibrium magnesite ⇌ periclase + CO₂ at high pressure. The equilibrium boundary is located at 12.1(±1) kbar, 1173-1183 °C (±10), and at 21.5(±1) kbar, 1375-1435 °C (±10). A van der Waals type equation of state with five adjustable parameters has been developed for CO₂. The function is smooth and continous above the critical region, behaves well in the high and low pressure limits, and the calculation of ʃ VdP for free energy does not require numerical integration. Computed free energies are consistent with all phase equilibrium data at high pressure, and computed volumes agree reasonably with P-V-T measurements. The proposed equation is: [ Equation omitted ] with B₁ = 28.0647, B₂ = 1.7287.10⁻⁴, B3 = 83653, A₁ = 1.0948.10⁹, A₂ = 3.3 7 47.10⁹, and R = 83.147, in units of Kelvin, bar and cm³/mole. The equation is recommended up to 50 kbar and above 400 K with reasonable extrapolation capabilities. A FORTRAN source code to evaluate the volume and fugacity is provided. Thermophysical properties for the calcium carbonate polymorphs calcite-I, IV, V, and aragonite were derived that are consistent with phase equilibrium experiments. Data required for further improvement include high pressure phase equilibria involving CO₂, constraints on the thermal expansion of magnesite, and P-V-T data to resolve inconsistencies among existing measurements. Water-carbon dioxide mixtures: The two widely used equations of state for H₂O-CO₂ mixtures are those proposed by Kerrick & Jacobs (1981) and by Holloway (1977)-Flowers (1979). Evaluation of existing equations and data is difficult due to inconsistencies among experimental studies. P-V-T-X data by Franck ＆ Todheide (1959) are inconsistent with data by Greenwood (1973) and Gehrig (1980), and cannot be reconciled with measured phase equilibria in H₂O-CO₂ fluid mixtures. Data by Greenwood and Gehrig are in loose agreement but extend only to 600 bar and do not constrain activities at higher pressures. A procedure is developed for using experimental phase equilibrium constraints to put limits on the fugacities of components of the fluid mixture. Inconsistencies among phase equilibrium studies are discussed. It is concluded that the data base available is not yet adequate to derive a reliable equation of state for H₂O-CO₂ mixtures. Future work must include P-V-T-X measurements to 8 kbar and phase equilibrium studies to resolve inconsistencies. These can constrain deviations from ideal mixing in the fluid phase, and constrain specific volumes at high pressures where P-V-T-X data connot be obtained.<br>Science, Faculty of<br>Earth, Ocean and Atmospheric Sciences, Department of<br>Graduate

Thesis (Ph.D.)--Aerospace Engineering, Georgia Institute of Technology, 2008.<br>Committee Chair: Seitzman, Jerry; Committee Member: Jagoda, Jechiel; Committee Member: Lieuwen, Tim; Committee Member: Menon, Suresh; Committee Member: Tan, David.

The high temperature BCC phase (b) of titanium undergoes a martensitic transformation to HCP phase (a) upon cooling, but can be stabilized at room temperature by alloying with BCC transition metals such as Mo. There exists a metastable composition range within which the alloyed b phase separates into a + b upon equilibrium cooling but not when rapidly quenched. Compositional partitioning of the stabilizing element in as-quenched b microstructure creates nanoscale precipitates of a new simple hexagonal w phase, which considerably reduces ductility. These phase transformation reactions have been extensively studied experimentally, yet several significant questions remain: (i) The mechanism by which the alloying element stabilizes the b phase, thwarts its transformation to w, and how these processes vary as a function of the concentration of the stabilizing element is unclear. (ii) What is the atomistic mechanism responsible for the non-Arrhenius, anomalous diffusion widely observed in experiments, and how does it extend to low temperatures? How does the concentration of the stabilizing elements alter this behavior? There are many other w forming alloys that such exhibit anomalous diffusion behavior. (iii) A lack of clarity remains on whether w can transform to a -phase in the crystal bulk or if it occurs only at high-energy regions such as grain boundaries. Furthermore, what is the nature of the a phase embryo? (iv) Although previous computational results discovered a new wa transformation mechanism in pure Ti with activation energy lower than the classical Silcock pathway, it is at odds with the a / b / w orientation relationship seen in experiments. First principles calculations based on density functional theory provide an accurate approach to study such nanoscale behavior with full atomistic resolution, allowing investigation of the complex structural and chemical effects inherent in the alloyed state. In the present work, a model Ti-Mo system is investigated to resolve these fundamental questions. Particular attention is paid to how Mo- (i) influences the bonding in Ti, (ii) distorts the local structure in the Ti lattice, (iii) impacts the point and interfacial defect formation and migration energies, and (iv) affects the mechanism and energetics of b w and wa transformations. Our results are correlated with appropriate experimental results of our collaborators and those in open literature. The modification of Ti bonding by Mo solutes and the attendant distortion of the lattice hold the key to answering the diverse questions listed above. The solutes enhance electron charge density in the <111> directions and, consequently, stiffen the lattice against the displacements necessary for b w transformation. However, Ti atoms uncoordinated by Mo remain relatively mobile, and locally displace towards w lattice positions. This effect was further studied in a metastable Ti-8.3 at.% Mo system with an alternate cell geometry which allows for either b w or $\betaa transformation, and it was found that after minimization Ti atoms possessed either a or w coordination environments. The creation of this microstructure is attributed to both the disruption of uniform b w transformation by the Mo atoms and the overlap of Ti-Mo bond contractions facilitating atomic displacements to the relatively stable a or w structures in Mo-free regions. The vacancy migration behavior in such a microstructure was then explored. Additionally, several minimized configurations were created with planar interfaces between Mo-stabilized b region and its adjacent a- or w- phases, and it was found that the positioning of Mo at the interface strongly dictates the structure of the adjacent Mo depleted region.

Vlasov-Maxwell equilibria are characterised by the self-consistent descriptions of the steady-states of collisionless plasmas in particle phase-space, and balanced macroscopic forces. We study the theory of Vlasov-Maxwell equilibria in one spatial dimension, as well as its application to current sheet and flux tube models. The ‘inverse problem' is that of determining a Vlasov-Maxwell equilibrium distribution function self-consistent with a given magnetic field. We develop the theory of inversion using expansions in Hermite polynomial functions of the canonical momenta. Sufficient conditions for the convergence of a Hermite expansion are found, given a pressure tensor. For large classes of DFs, we prove that non-negativity of the distribution function is contingent on the magnetisation of the plasma, and make conjectures for all classes. The inverse problem is considered for nonlinear ‘force-free Harris sheets'. By applying the Hermite method, we construct new models that can describe sub-unity values of the plasma beta (βpl) for the first time. Whilst analytical convergence is proven for all βpl, numerical convergence is attained for βpl = 0.85, and then βpl = 0.05 after a ‘re-gauging' process. We consider the properties that a pressure tensor must satisfy to be consistent with ‘asymmetric Harris sheets', and construct new examples. It is possible to analytically solve the inverse problem in some cases, but others must be tackled numerically. We present new exact Vlasov-Maxwell equilibria for asymmetric current sheets, which can be written as a sum of shifted Maxwellian distributions. This is ideal for implementations in particle-in-cell simulations. We study the correspondence between the microscopic and macroscopic descriptions of equilibrium in cylindrical geometry, and then attempt to find Vlasov-Maxwell equilibria for the nonlinear force-free ‘Gold-Hoyle' model. However, it is necessary to include a background field, which can be arbitrarily weak if desired. The equilibrium can be electrically non-neutral, depending on the bulk flows.

This PhD thesis is focused in structural aspects of alkali nitrates. In the first chapter, we recollected the most important advancement in this field in alkali nitrates compounds published since 1970. Along this lines, crystal structure, polymorphism and phase transition is discussed first; second, crystal growth, and third, the morphology. And, finally a section concerning solid state miscibility and binary phase diagrams between these compounds is presented. The following chapters present a deeper study of growth morphology and how the impurities affect to the morphology and growth rate. These two topics are closely related because the growth morphology is a consequence of the growth rate ratio of different faces. Evidently, we chose only one compound of all the alkali nitrate family. It turned to be sodium nitrate, nitratine or NaNO3. First, in chapter 2, we calculate the morphology of nitratine by applying the two most common approaches to determine the theoretic morphology of crystals: the Bravais-Friedel-Donnay-Harker (BFDH) methodology and the periodic bond chains (PBCs) procedure proposed by Hartman and Perdok (HP). Then we compare the obtained morphology with the experimental growth shape. This compound is interesting from the morphological and structural point of view because it is isostructural with calcite, a calcium carbonate (CaCO3) polymorph. This fact will permit us to discuss how the charges affect the final crystal morphology; an introduction to this question is in chapter 2. In chapter 3 we sought to determination of the normal growth rates (R104) of {104} faces of sodium nitrate single crystals under isothermal conditions in the temperature interval 288 K  297.5 K and hence find the most probable growth mechanisms. Chapter 4 is devoted to the morphology change in NaNO3 crystals by the effect of impurities. It is followed by an atomic force microscopy (AFM) investigation of heterogeneous nucleation of nitratine on calcite in chapter 5. Finally, the NaNO3 – KNO3 phase diagram has been studied from both experimental and theoretical aspects in chapter 6.

Le bicarbonate de soude raffiné, produit industriellement par la société Solvay, est fabriqué dans des colonnes à bulles de grande taille, appelées les colonnes BIR.<p>Dans ces colonnes, une phase gazeuse contenant un mélange d’air et dioxyde de carbone (CO2) est dispersée sous forme de bulles dans une solution aqueuse de carbonate et de bicarbonate de sodium (respectivement Na2CO3 et NaHCO3). Cette dispersion donne lieu à un transfert de CO2 des bulles vers la phase liquide. Au sein des colonnes, la phase gazeuse se répartit dans deux populations de bulles :des petites bulles (diamètre de quelques mm) et des grandes bulles (diamètre de quelques cm). Le transfert bulle-liquide de CO2 est couplé à des réactions chimiques prenant place en phase liquide, qui conduisent à la conversion du Na2CO3 en NaHCO3. Une fois la concentration de saturation dépassée le NaHCO3 précipite sous forme de cristaux et un mélange liquide-solide est recueilli à la sortie de ces colonnes.<p>Ce travail, réalisé en collaboration avec la société Solvay, porte sur l’étude et la modélisation mathématique des phénomènes de transfert de matière entre phases, couplés à des réactions chimiques, prenant place au sein d’une colonne BIR. L’association d’études sur des colonnes à bulles à l’échelle industrielle ou réduite (pilote) et d’études plus fondamentales sur des dispositifs de laboratoire permet de développer une meilleure compréhension du fonctionnement des colonnes BIR et d’en construire un modèle mathématique détaillé.<p>L’objectif appliqué de ce travail est la mise au point d’un modèle mathématique complet et opérationnel d’une colonne BIR. Cet objectif est supporté par trois blocs de travail, dans lesquels différents outils sont développés et exploités.<p><p>Le premier bloc est consacré à la modélisation mathématique du transfert bulle-liquide de CO2 dans une solution aqueuse de NaHCO3 et de Na2CO3. Ce transfert est couplé à des réactions chimiques en phase liquide qui influencent sa vitesse. Dans un premier temps, des modèles sont développés selon des approches unidimensionnelles classiquement rencontrées dans la littérature. Ces approches passent par une idéalisation de l’écoulement du liquide autour des bulles. Une expression simplifiée de la vitesse du transfert bulle-liquide de CO2, est également développée et validée pour le modèle de colonne BIR.<p>Dans un second temps, une modélisation complète des phénomènes de transport (convection et diffusion), couplés à des réactions chimiques, est réalisée en suivant une approche bidimensionnelle axisymétrique. L’influence de la vitesse de réactions sur la vitesse de transfert est étudiée et les résultats des deux approches sont également comparés.<p><p>Le deuxième bloc est consacré à l’étude expérimentale du transfert gaz-liquide de CO2 dans des solutions aqueuses de NaHCO3 et de Na2CO3. A cette fin, un dispositif expérimental est développé et présenté. Du CO2 est mis en contact avec des solutions aqueuses de NaHCO3 et de Na2CO3 dans une cellule transparente. Les phénomènes provoqués en phase liquide par le transfert de CO2 sont observés à l’aide d’un interféromètre de Mach-Zehnder.<p>Les résultats expérimentaux sont comparés à des résultats de simulation obtenus avec un des modèles unidimensionnels développés dans le premier bloc. De cette comparaison, il apparaît qu’une mauvaise estimation de la valeur de certains paramètres physico-chimiques apparaissant dans les équations de ce modèle conduit à des écarts significatifs entre les grandeurs observées expérimentalement et les grandeurs estimées par simulation des équations du modèle.<p>C’est pourquoi une méthode d’estimation paramétrique est également développée afin d’identifier les valeurs numériques de ces paramètres physico-chimiques sur base des résultats expérimentaux. Ces dernières sont également discutées.<p><p>Dans le troisième bloc, nous apportons une contribution à l’étude des cinétiques de précipitation du NaHCO3 dans un cristallisoir à cuve agitée. Cette partie du travail est réalisée en collaboration avec Vanessa Gutierrez (du service Matières et Matériaux de l’ULB).<p>Nous contribuons à cette étude par le développement de trois outils :une table de calcul Excel permettant de synthétiser les résultats expérimentaux, un ensemble de simulations de l’écoulement au sein du cristallisoir par mécanique des fluides numérique et une nouvelle méthode d’extraction des cinétiques de précipitation du NaHCO3 à partir des résultats expérimentaux. Ces trois outils sont également utilisés de façon combinée pour estimer les influences de la fraction massique de solide et de l’agitation sur la cinétique de germination secondaire du NaHCO3.<p><p>Enfin, la synthèse de l’ensemble des résultats de ces études est réalisée. Le résultat final est le développement d’un modèle mathématique complet et opérationnel des colonnes BIR. Ce modèle est développé en suivant l’approche de modélisation en compartiments, développée au cours du travail de Benoît Haut. Ce modèle synthétise les trois blocs d’études réalisées dans ce travail, ainsi que les travaux d’Aurélie Larcy (du service Transferts, Interfaces et Procédés de l’ULB) et de Vanessa Gutierrez. Les équations modélisant les différents phénomènes sont présentées, ainsi que la méthode utilisée pour résoudre ces équations. Des simulations des équations du modèle sont réalisées et discutées. Les résultats de simulation sont également comparés à des mesures effectuées sur une colonne BIR. Un accord raisonnable est observé.<p>A l’issue de ce travail, nous disposons donc d’un modèle opérationnel de colonne BIR. Bien que ce modèle doive encore être optimisé et validé, il peut déjà être utilisé pour étudier l’effet des caractéristiques géométriques des colonnes BIR et des conditions appliquées à ces colonnes sur le comportement des simulations des équations du modèle et pour identifier des tendances.<p>//<p>The refined sodium bicarbonate is produced by the Solvay company using large size bubble columns, called the BIR columns.<p>In these columns, a gaseous phase containing an air-carbon dioxyde mixture (CO2) is dispersed under the form of bubbles in an aqueous solution of sodium carbonate and sodium bicarbonate (Na2CO3 and NaHCO3, respectively). This dispersion leads to a CO2 transfer from the bubbles to the liquid phase. Inside these columns, the gaseous phase is distributed in two bubbles populations :small bubbles (a few mm of diameter) and large bubbles (a few cm of diameter).<p>The bubble-liquid CO2 transfer is coupled with chemical reactions taking places in the liquid phase that leads to the conversion of Na2CO3 to NaHCO3. When the solution is supersaturated in NaHCO3, the NaHCO3 precipitates under the form of crystals and a liquid-solid mixture is extracted at the outlet of the BIR columns.<p>This work, realized in collaboration with Solvay, aims to study and to model mathematically the mass transport phenomena between the phases, coupled with chemical reactions, taking places inside a BIR column. Study of bubble columns at the industrial and the pilot scale is combined to a more fundamental study at laboratory scale to improve the understanding of the BIR columns functioning and to develop a detailed mathematical modeling.<p>The applied objective of this work is to develop a complete and operational mathematical modeling of a BIR column. This objective is supported by three blocks of work. In each block, several tools are developed and used.<p><p>The first block is devoted to the mathematical modeling of the bubble-liquid CO2 transfer in an NaHCO3 and Na2CO3 aqueous solution. This transfer is coupled with chemical reactions in liquid phase, which affect the transfer rate.<p>In a first time, mathematical models are developed following the classical one-dimensional approaches of the literature. These approaches idealize the liquid flow around the bubbles. A simplified expression of the bubble-liquid CO2 transfer rate is equally developed and validated for the BIR column model.<p>In a second time, a complete modeling of the transport phenomena (convection and diffusion) coupled with chemical reactions is developed, following an axisymmetrical twodimensional approach. The chemical reaction rate influence on the bubble-liquid transfer rate is studied and the results of the two approaches are then compared.<p><p>The second block is devoted to the experimental study of the gas-liquid CO2 transfer to NaHCO3 and Na2CO3 aqueous solutions. An experimental set-up is developed and presented. CO2 is put in contact with NaHCO3 and Na2CO3 aqueous solutions in a transparent cell. The phenomena induced in liquid phase by the CO2 transfer are observed using a Mach-Zehnder interferometer.<p>The experimental results are compared to simulation results that are obtained using one of the one-dimensional model developed in the first block. From this comparison, it appears that a wrong estimation of some physico-chemical parameter values leads to significative differences between the experimentally observed quantities and those estimated by simulation of the model equations. Therefore, a parametric estimation method is developed in order to estimate those parameters numerical values from the experimental results. The found values are then discussed.<p><p>In the third block is presented a contribution to the NaHCO3 precipitation kinetic study in a stirred-tank crystallizer. This part of the work is realized in collaboration with Vanessa Gutierrez (Chemicals and Materials Department of ULB).<p>Three tools are developed :tables in Excel sheet to synthetize the experimental results, a set of simulations of the flow inside the crystallizer by Computational Fluid Dynamic (CFD) and a new method to extract the NaHCO3 precipitation kinetics from the experimental measurements. These three tools are combined to estimate the influences of the solid mass fraction and the flow on the NaHCO3 secondary nucleation rate.<p><p>Finally, the synthesis of all these results is realized. The final result is the development of a complete and operational mathematical model of BIR columns. This model is developed following the compartmental modeling approach, developed in the PhD thesis of Benoît Haut. This model synthetizes the three block of study realized in this work and the studies of Aurélie Larcy (Transfers, Interfaces and Processes Department of ULB) and those of Vanessa Gutierrez. The equations modeling the phenomena taking place in a BIR column are presented as the used method to solve these equations. The equations of the model are simulated and the results are discussed. The results are equally compared to experimental measurement realized on a BIR column. A reasonable agreement is observed.<p>At the end of this work, an operational model of a BIR column is thus developed. Although this model have to be optimized and validated, it can already be used to study the influences of the geometrical characteristics of the BIR columns and of the conditions applied to these columns on the behaviour of the model equation simulations and to identity tendencies.<br>Doctorat en Sciences de l'ingénieur<br>info:eu-repo/semantics/nonPublished

Le développement du modèle prédictif E-PPR78, basé sur une méthode de contribution de groupe, a été entrepris depuis plus de dix ans pour prédire le comportement de systèmes multiconstituants. Ce modèle repose sur l'équation d'état de Peng-Robinson dans sa version de 1978 et les règles de mélanges de Van der Waals. Il utilise un seul paramètre d'interaction binaire, kij, qui dépend de la température. Afin de permettre au modèle E-PPR78 de prédire le comportement du gaz naturel, trois nouveaux groupes sont ajoutés : le monoxyde de carbone, l'hélium et l'argon. Pour cela, il a été nécessaire de former une base de données expérimentales la plus large possible contenant les mesures d'équilibres de phase et d'enthalpies de mélange pour les systèmes binaires constitués par ces trois groupes ainsi que ceux définis dans les études précédentes et présents dans le gaz naturel. Après une description de la classification des diagrammes de phase de Van Konynenburg et Scott, le modèle E-PPR78 est présenté. La troisième partie est consacrée à l'ajout des trois nouveaux groupes au sein du modèle. Les résultats sont obtenus avec une précision satisfaisante. Il apparaît clairement que le modèle E PPR78 est capable de prédire le comportement du gaz naturel dans des conditions de températures et de pressions particulièrement larges<br>The development of the predictive E-PPR78 model, based on a contribution group method, has been undertaken since ten years to predict accurately the behaviour of multi-component systems. This model lies on the Peng-Robinson equation of state with classical Van der Waals mixing rules. It uses a unique binary interaction parameter, kij, which is temperature dependant. To enable the E-PPR78 model to predict the behavior of natural gases, three new groups are added: carbon monoxide, helium and argon. It was necessary to build an experimental database, as exhaustive as possible, containing phase equilibrium and enthalpies of mixing data for binary systems formed by these groups and those defined in previous studies and present in natural gases. After a description of the classification scheme of Van Konynenburg and Scott, the E-PPR78 model is described. The third part is about the addition of the three new groups to the model. It clearly appears that the E-PPR78 model is able to predict the fluid-phase behavior of natural gases over wide ranges of temperatures and pressures

The equilibrium properties of polymers end-grafted to an impenetrable interface, the "polymer brush", are investigated. Relevant concepts and techniques of statistical polymer physics are discussed; in particular, a simulation technique that is very efficient for studying polymer brushes is introduced. This technique is demonstrated through simulations of a well characterized polymer brush system. The results of original investigations of phase separation in polymer brushes are also presented. An instability in the lateral monomer density of a polymer brush is observed under sufficiently poor solvent conditions. The onset of this instability is found to agree with a previous prediction. A compositional instability is found in the lateral densities of a two-component polymer brush under conditions of sufficient immiscibility between the two components. The effects of varying solvent conditions are considered. Finally, the onset of the compositional instability is determined using the technique of the self consistent mean field, and the results compared to simulation.

The partition behavior of aqueous and organic ternary solutions into two immiscible phases at ambient temperatures was studied. A classification system of the types of materials which phase partition water and organic solvents was developed and a comprehensive list of systems for which phase diagram information is available in the literature has been presented.<br>Systems in three different classes of materials were studied. Equilibrium phase diagrams, tie-lines and plait points were generated for the ternary systems: poly(ethylene glycol)-1,000/sodium sulfate/water at 28$ sp circ$C; poly(ethylene glycol)-3,350/sodium sulfate/water at 28$ sp circ$C and 35$ sp circ$C; poly(ethylene glycol)-8,000/sodium sulfate/water at 28$ sp circ$C; poly(vinyl alcohol)/methylcellulose/water at 25$ sp circ$C; and poly(propylene glycol)/polystyrene/cyclohexane at 25$ sp circ$C. All measurements were done at 1 atm. The influence of temperature and polymer molecular weight on the shape and location of the equilibrium phase cure and the location of the critical point was analyzed for the poly(ethylene glycol)/sodium sulfate/water system.

The theory of vapour-liquid equilibria is reviewed, as is the present status or prediction methods in this field. After discussion of the experimental methods available, development of a recirculating equilibrium still based on a previously successful design (the modified Raal, Code and Best still of O'Donnell and Jenkins) is described. This novel still is designed to work at pressures upto 35 bar and for the measurement of both isothermal and isobaric vapour-liquid equilibrium data. The equilibrium still was first commissioned by measuring the saturated vapour pressures of pure ethanol and cyclohexane in the temperature range 77-124oC and 80-142oC respectively. The data obtained were compared with available literature experimental values and with values derived from an extended form of the Antoine equation for which parameters were given in the literature. Commissioning continued with the study of the phase behaviour of mixtures of the two pure components as such mixtures are strongly non-ideal, showing azeotopic behaviour. Existing data did not exist above one atmosphere pressure. Isothermal measurements were made at 83.29oC and 106.54oC, whilst isobaric measurements were made at pressures of 1 bar, 3 bar and 5 bar respectively. The experimental vapour-liquid equilibrium data obtained are assessed by a standard literature method incorporating a themodynamic consistency test that minimises the errors in all the measured variables. This assessment showed that reasonable x-P-T data-sets had been measured, from which y-values could be deduced, but that the experimental y-values indicated the need for improvements in the design of the still.

A total pressure apparatus has been developed to measure vapour-liquid equilibrium data on binary mixtures at atmospheric and sub-atmospheric pressures. The method gives isothermal data which can be obtained rapidly. Only measurements of total pressure are made as a direct function of composition of synthetic liquid phase composition, the vapour phase composition being deduced through the Gibbs-Duhem relationship. The need to analyse either of the phases is eliminated. As such the errors introduced by sampling and analysis are removed. The essential requirements are that the pure components be degassed completely since any deficiency in degassing would introduce errors into the measured pressures. A similarly essential requirement was that the central apparatus would have to be absolutely leak-tight as any leakage of air either in or out of the apparatus would introduce erroneous pressure readings. The apparatus was commissioned by measuring the saturated vapour pressures of both degassed water and ethanol as a function of temperature. The pressure-temperature data on degassed water measured were directly compared with data in the literature, with good agreement. Similarly the pressure-temperature data were measured for ethanol, methanol and cyclohexane and where possible a direct comparison made with the literature data. Good agreement between the pure component data of this work and those available in the literature demonstrates firstly that a satisfactory degassing procedure has been achieved and that secondly the measurements of pressure-temperature are consistent for any one component; since this is true for a number of components, the measurements of both temperature and pressure are both self-consistent and of sufficient accuracy, with an observed compatibility between the precision/accuracy of the separate means of measuring pressure and temperature. The liquid mixtures studied were of ethanol-water, methanol-water and ethanol-cyclohexane. The total pressure was measured as the composition inside the equilibrium cell was varied at a set tmperature. This gave P-T-x data sets for each mixture at a range of temperatures. A standard fitting-package from the literature was used to reduce the raw data to yield y-values to complete the x-y-P-T data sets. A consistency test could not be applied to the P-T-x data set as no y-values were obtained during the experimental measurements. In general satisfactory agreement was found between the data of this work and those available in the literature. For some runs discrepancies were observed, and further work recommended to eliminate the problems identified.

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Chemical Engineering, 2013.<br>Cataloged from PDF version of thesis.<br>Includes bibliographical references.<br>Colloidal dispersion of nanoparticles (CNPs) has interesting properties both in terms of fundamental studies and industrials applications. Particular focus on the phase equilibrium and separation dynamics of CNPs has been necessary for understanding how exactly and fast CNPs are assembled and for controlling the assembly structure and dynamic properties. For understanding and controlling assembly structure and dynamics of CNPs, theoretical analysis in conjunction with computational approaches supported by experimental validation is necessary. In this thesis, studies on the phase-equilibrium-mediated assembly of CNPs are performed by using various computational tools accompanied by theoretical modeling to cover wide range of spatio and temporal dimensions of the desired system containing CNPs. To address the phase separation of CNPs, we studied on two main mechanisms; (1) cluster formation and (2) spinodal decomposition. In each mechanism, we developed novel, effective, and efficient computational algorithms to elucidate phase-equilibrium assembly structure and formation dynamics of CNPs: (1) a kinetic Monte Carlo (KMC) algorithm for cluster formation in microscopic dimensions and spinodal decomposition of homogeneous mixture of CNPs in mesoscopic scale, (2) a self-consistent mean-field (SCMF) model for surface-directed separation of a binary mixture of CNPs in mesoscopic-macroscopic scale, and (3) the spectral method for spinodal decomposition of a binary or ternary mixture of CNPs in macroscopic scale. All the algorithms and results from the simulations were verified by either mathematical proofs or comparisons to other computational methods. In particular, proof-of-concept experimental results of the fabricition of a functional thin film in which a binary mixture of CNPs form the controlled gradient concentrations profile across the thickness direction were presented. On the basis of the experimental demonstration, we showed the validity of the computational model and possible future applications of the fabricated thin film as an optically-functional material. The computational algorithms and numerical tools developed in this thesis supported by theoretical analysis and experimental demonstration can be applicable to various dynamic problems regarding CNPs, especially, for the complicated cases including multi-component, multi-phase systems. We expect that the work performed in this thesis can provide a substantial advantage for future research, such as controlled cluster formation of CNPs by polymer gel mesh, cluster formation of Janus CNPs, and physically controlled spinodal decomposition of CNPs in thin films, as well as progressive application to preparation of novel devices.<br>by Seok Joon Kwon.<br>Ph. D.