Under adequate conditions on $g$, we show the density in $L^2((0,\pi),(0,2\pi))$ of the set of functions $p$ for which \begin{equation*} u_{tt}(x,t)-u_{xx}(x,t)= g(u(x,t)) + p(x,t) \end{equation*} has a weak solution subject to \begin{equation*} \begin{aligned} u(x,t)&=u(x,t+2\pi)\\ u(0,t)&=u(\pi,t)=0. \end{aligned} \end{equation*} To achieve this, we prove a Saddle Point Principle by means of a refined variant of the deformation lemma of Rabinowitz. Generally, inf-sup techniques allow the characterization of critical values by taking the minimum of the maximae on some particular class of sets. In this version of the Saddle Point Principle, we introduce sufficient conditions for the existence of a saddle-structure which is not restricted to finite-dimensional subspaces.

The knowledge of the phase behaviour of heavy oils and bitumen is important in order to understand the phenomenon of coke formation. Computation of their phase behaviour, using an equation of state, faces problems due to their complex composition. Hence n-alkane binaries of polyaromatic hydrocarbons are used to approximate the phase behaviour of heavy oils and bitumen. Appropriate values of binary interaction parameters are required for an equation of state to predict the correct phase behaviour of these model binary fluids. This thesis deals with fitting of the binary interaction parameter for the Peng-Robinson equation of state using landmarks in the binary phase space such as K- and L-points. A K- or an L-point is a point in the phase space where two phases become critical in the presence of another phase in equilibrium. An algorithm to calculate K- and L-points using an equation of state was developed. The variation of calculated K- and L-points with respect to the binary interaction parameter was studied and the results were compared with the experimental data in the literature. The interaction parameter was then fitted using the best match of experimental results with the computed ones. The binary interaction parameter fitted using a K- or an L-point was then used to predict the P-T projection of the binary system in phase space. Also, the qualitative effect of the binary interaction parameter on the P-T projection was studied. A numerical and thermodynamic study of the algorithm was done. Numerical issues like the initial guesses, convergence criterion and numerical techniques were studied and the thermodynamic constraints in the generalization of the algorithm are discussed. It was observed that the binary interaction parameter not only affects the location of K- and L-points in the phase space but also affects the calculation procedure of K- and L-points. Along with the propane binaries of polyaromatic hydrocarbons, K- and L-points were also calculated for systems like methane binaries of higher n-alkanes and the ethane + ethanol binary. In the case of the ethane + ethanol system, K- and L-points, matching the experimental results were calculated with different values of the binary interaction parameter. But the Peng-Robinson equation of state was unable to predict the correct type of phase behaviour using any value of the binary interaction parameter. The Peng-Robinson equation of state was able to predict the correct type of phase behaviour with the binary interaction parameter, fitted using K- and/or L-points for methane + n-alkane systems. The systems studied were the methane binaries of n-pentane, n-hexane and n-heptane. For the propane binaries of polyaromatic hydrocarbons, no value of the binary interaction parameter was able to predict the K-point with a good accuracy. The binary interaction parameter which gave the best possible results for a K-point failed to predict the correct type of phase behaviour. The binary interaction parameter fitted using the P-T projection enabled the Peng-Robinson equation of state to give a qualitative match for the high pressure complex phase behaviour of these systems. Solid phase equilibria were not taken into consideration.

The knowledge of the phase behaviour of heavy oils and bitumen is important in order to understand the phenomenon of coke formation. Computation of their phase behaviour, using an equation of state, faces problems due to their complex composition. Hence n-alkane binaries of polyaromatic hydrocarbons are used to approximate the phase behaviour of heavy oils and bitumen. Appropriate values of binary interaction parameters are required for an equation of state to predict the correct phase behaviour of these model binary fluids. This thesis deals with fitting of the binary interaction parameter for the Peng-Robinson equation of state using landmarks in the binary phase space such as K- and L-points. A K- or an L-point is a point in the phase space where two phases become critical in the presence of another phase in equilibrium. An algorithm to calculate K- and L-points using an equation of state was developed. The variation of calculated K- and L-points with respect to the binary interaction parameter was studied and the results were compared with the experimental data in the literature. The interaction parameter was then fitted using the best match of experimental results with the computed ones. The binary interaction parameter fitted using a K- or an L-point was then used to predict the P-T projection of the binary system in phase space. Also, the qualitative effect of the binary interaction parameter on the P-T projection was studied. A numerical and thermodynamic study of the algorithm was done. Numerical issues like the initial guesses, convergence criterion and numerical techniques were studied and the thermodynamic constraints in the generalization of the algorithm are discussed. It was observed that the binary interaction parameter not only affects the location of K- and L-points in the phase space but also affects the calculation procedure of K- and L-points. Along with the propane binaries of polyaromatic hydrocarbons, K- and L-points were also calculated for systems like methane binaries of higher n-alkanes and the ethane + ethanol binary. In the case of the ethane + ethanol system, K- and L-points, matching the experimental results were calculated with different values of the binary interaction parameter. But the Peng-Robinson equation of state was unable to predict the correct type of phase behaviour using any value of the binary interaction parameter. The Peng-Robinson equation of state was able to predict the correct type of phase behaviour with the binary interaction parameter, fitted using K- and/or L-points for methane + n-alkane systems. The systems studied were the methane binaries of n-pentane, n-hexane and n-heptane. For the propane binaries of polyaromatic hydrocarbons, no value of the binary interaction parameter was able to predict the K-point with a good accuracy. The binary interaction parameter which gave the best possible results for a K-point failed to predict the correct type of phase behaviour. The binary interaction parameter fitted using the P-T projection enabled the Peng-Robinson equation of state to give a qualitative match for the high pressure complex phase behaviour of these systems. Solid phase equilibria were not taken into consideration.

A computational tool that uses an automated and reliable procedure for systematic generation of global phase equilibrium diagram (GPED) is developed for binary system using equation of state and its extension to the ternary system is discussed. The proposed algorithm can handle solid phase and also can predict all major six types of phase diagrams. The procedure enables automatic generation of GPED which incorporates calculations of all important landmarks such as critical endpoints, quadruple point (if any), critical line, liquid-liquid-vapor line (if any), solid-liquid-liquid line (if any) and solid-liquid-vapor line. The method is also capable of locating all azeotropic phenomena such as azeotropic endpoint, critical azeotrope, pure azeotropic point and azeotropic lines. Although, we demonstrated the methodology for cubic equation of state, the proposed strategy is completely general that doesn't require any knowledge about the type of phase diagram and can be applied to any pressure explicit equation of state model. Newton homotopy based global method has been applied for phase stability test and critical point calculations to ensure reliability. Having computed the binary phase diagrams, the methodology to generate global phase diagrams for ternary system is discussed that can locate all important thermodynamic landmarks such as tricritical point, quadruple critical endpoint, quadruple azeotropic endpoint, quintuple point and critical azeotropic endpoint. The procedure to trace ternary phenomena having two degree of freedom such as critical surface, solid-liquid-vapor surface and liquid-liquid-vapor surface has been discussed. Finally, applications of reliable global methods to solve the fluid-fluid phase equilibrium problem using SAFT equation for binary system and the solid-fluid phase equilibrium problem for binary and ternary systems have been demonstrated through representative computations.

Chemical treatments using solvents and surfactants can be used to increase the productivity of gas-condensate wells with condensate banks. CMG’s compositional simulator GEM was used to simulate such treatments to gain a better understanding of design questions such as how much treatment solution to inject and to predict the benefits of such treatments. GEM was used to simulate treatments in vertical wells with and without hydraulic fractures and also horizontal wells. However, like other commercial compositional simulators, the flash calculations used to predict the phase behavior is limited to two phases whereas a three-phase flash is needed to accurately model the complex phase behavior that occurs during and after the injection of treatment solutions. UTCOMP is a compositional simulator with three-phase flash routine and attempts were made to use it to simulate such well treatments. However, this is a very difficult problem to simulate and all previous attempts failed because of numerical problems caused by inconsistent phase labeling (so called phase flipping) and the discontinuities this causes in the relative permeability values. In this research, a new relative permeability model based on molar Gibbs free energy was developed, implemented in a compositional simulator and applied to several difficult three-phase flash problems. A new way of modeling the residual saturations was needed to ensure a continuous variation of the residual saturations from the three-phase region to the two-phase region or back and was included in the new model. The new relative permeability model was implemented in the compositional reservoir simulator UTCOMP. This new relative permeability model makes it is unnecessary to identify and track the phases. This method automatically avoids the previous phase flipping problems and thus is physically accurate as well as computationally faster due to the improved numerical performance. The new code was tested by running several difficult simulation problems including a CO2 flood with three-hydrocarbon phases and a water phase. A new framework for doing flash calculations was also developed and implemented in UTCOMP to account for the multiple roots of the cubic equation-of-state to ensure a global minimum in the Gibbs free energy by doing an exhaustive search for the minimum value for one, two and three phases. The purpose was to determine if the standard method using a Gibbs stability test followed by a flash calculation was in fact resulting in the true minimum in the Gibbs free energy. Test problems were run and the results of the standard algorithm and the exhaustive search algorithm compared. The updated UTCOMP simulator was used to understand the flow back of solvents injected in gas condensate wells as part of chemical treatments. The flow back of the solvents, a short-term process, affects how well the treatment works and has been an important design and performance question for years that could not be simulated correctly until now due to the limitations of both commercial simulators and UTCOMP. Different solvents and chase gases were simulated to gain insight into how to improve the design of the chemical treatments under different conditions. / text

System of HACCP (Hazard Analysis and Critical Control Points) is the main and unbroken system in the field of food safety. There is a duty for food producers in the Czech Republic to implement a system of critical control points into their business. It means to identify points (critical) in the process of production where can be the highest hazard for food safety, to monitor these points, to control and register data regularly. Production of safety food, protection of consumers, and also protection of a producer is the principle and result of proper and functional system of HACCP. Documents, an indivisible part of system of HACCP, were drafted for a selected company. Analysis of hazards was also done with the result of setting control points (CP) and two of critical control points (CCP) in a production step of cooling and packing.

In the first part of this thesis, we study critical points of random polynomials. We choose two deterministic sequences of complex numbers, whose empirical measures converge to the same probability measure in complex plane. We make a sequence of polynomials whose zeros are chosen from either of sequences at random. We show that the limiting empirical measure of zeros and critical points agree for these polynomials. As a consequence we show that when we randomly perturb the zeros of a deterministic sequence of polynomials, the limiting empirical measures of zeros and critical points agree. This result can be interpreted as an extension of earlier results where randomness is reduced. Pemantle and Rivin initiated the study of critical points of random polynomials. Kabluchko proved the result considering the zeros to be i.i.d. random variables. In the second part we deal with the spectrum of products of Ginibre matrices. Exact eigenvalue density is known for a very few matrix ensembles. For the known ones they often lead to determinantal point process. Let X1, X2,..., Xk be i.i.d Ginibre matrices of size n ×n whose entries are standard complex Gaussian random variables. We derive eigenvalue density for matrices of the form X1 ε1 X2 ε2 ... Xk εk , where εi = ±1 for i =1,2,..., k. We show that the eigenvalues form a determinantal point process. The case where k =2, ε1 +ε2 =0 was derived earlier by Krishnapur. In the case where εi =1 for i =1,2,...,n was derived by Akemann and Burda. These two known cases can be obtained as special cases of our result.

A nuclear magnetic resonance study of the critical region has been made in ethane which was chosen as the working substance for its convenient critical temperature and pressure. Standard radio frequency pulse techniques were used to measure the spin-lattice relaxation time T₁ and the self diffusion constant D by the method of spin echoes. A spectrometer having good stability and very flexible timing circuits was designed and constructed. An automatic temperature control system capable of holding the sample temperature constant to better than 0.01° C for long periods of time was also designed and constructed. The spin-lattice relaxation time in ethane has been measured along the vapor pressure curve over the entire liquid temperature range as well as in the equilibrium vapor from 0° C to the critical temperature (Tc =32.32° C) and in the dense gas from Tc to 60°C. In the liquid T₁ rises rapidly with increasing temperature and goes through a maximum at about 0°C after which it begins to fall. In the vapor T₁ is always less than in the liquid and increases with increasing temperature. In the dense gas above Tc the relaxation time decreases slowly with increasing temperature. These results are compared with the conventional theory for relaxation in liquids and dense gases. The theory gives the relaxation rate 1/T₁ in terms of three relaxation mechanisms: the dipole-dipole intermolecular interaction the dipole-dipole intramolecular interaction and the spin-rotational interaction. In view of the gross approximations made in the theory a very reasonable fit to the experimental data is obtained. For the low temperature liquid the dipole-dipole interactions are sufficient to account for the relaxation. At high temperatures the spin-rotational interaction seems to contribute significantly to the relaxation and near the critical point it is the dominant relaxation mechanism. No anomalous behaviour was observed in the relaxation near the critical point and to within the error of measurement it is adequately described in terms of changes in density and self diffusion constant. T₁ was also measured in dilute ethane gas over a temperature range of 180°K to 300°K. It was observed that T₁ is proportional to density ρ and the temperature dependence of T/ρ is about T⁻¹˙³⁷. Measurements of the diffusion constant reveal that for low temperatures the product Dρ for liquid ethane varies approximately as T³. As the temperature approaches the critical temperature there appears to be anomalous behaviour in D. For both the liquid and vapor the product Dρ begins to decrease and goes through a minimum and then increases rapidly as the critical point is reached. Oxygen has been added to these samples to decrease their relaxation time and this may well be an impurity effect. Particular attention was devoted to the question of the equilibrium state in the critical region and measurements were made on the time taken to achieve equilibrium. The approach of the ratio of liquid to vapor density to its equilibrium value was found to vary in a roughly exponential manner with a time constant of the order of several hours. Sufficient time was allowed after changing the sample temperature for equilibrium to be established and all measurements of diffusion constant and spin-lattice relaxation time reported here are thought to be equilibrium values. / Science, Faculty of / Physics and Astronomy, Department of / Graduate

京都大学 / 0048 / 新制・論文博士 / 博士(理学) / 乙第12996号 / 論理博第1552号 / 新制||理||1604(附属図書館) / 32924 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 松田 祐司, 教授 前野 悦輝, 教授 石田 憲二 / 学位規則第4条第2項該当 / Doctor of Science / Kyoto University / DGAM

Aqueous electrolyte solutions at temperature and pressure conditions near the critical point of water are difficult to describe using traditional equations of state based upon the excess Gibbs energy. Models based upon the residual Helmholtz energy have proven more effective. Anderko and Pitzer1 developed a residual Helmholtz energy model (AP) for aqueous electrolyte solutions in which the electrolyte is assumed to be fully associated. The model has been effectively used in describing densities and vapor-liquid equilibria for simple electrolyte systems. The model is less effective for describing enthalpic properties such as heats of dilution. Oscarson and coworkers2, 3 modified the AP model for NaCl solutions by adding a term accounting for the change in Helmholtz energy as a result of aqueous dissociation reactions. This new model, called the RI model, is more accurate than the AP model at conditions where the NaCl dissociates more fully into ions. Liu et. al4, 5 modified the RI model by adding a term to describe interactions between ions in solution and by regressing new model parameter values. This new model, called the RIII model, is more accurate than both the AP model and the RI model and may be used to predict species concentrations in solution as a result of aqueous phase reactions. The RIII model has substantial thermodynamic inconsistencies, however, and is poorly suited for describing mixed solute solutions. This dissertation presents the RIV model which is an electrolyte solution model for solutions in the ranges of 350 °C to 400 °C and 18 MPa to 40 MPa. The RIV model has been applied to aqueous NaCl solutions and aqueous LiCl solutions. The RIV model is a modification of the AP model and includes aqueous phase reactions implicitly through fundamental species interactions. The RIV model is thermodynamically consistent. It is capable of describing densities and heats of dilution. Density predictions from the RIV model are less accurate than the AP model predictions (6.66 % error vs. 3.51 % error) but are reasonable. The heats of dilution predictions from the RIV model are much more accurate than those from the AP model (25.16 % error vs. 78.78 % error). Predictions of the ionic species concentration from the RIV model are likely to be poor as indicated by the poor agreement between experimental values and calculated values of equilibrium constants valid at infinite dilution. In order to provide the necessary data from which to regress the parameters of the RIV model, experimental heat of dilution values were determined using flow calorimetry techniques. These values are also reported in this dissertation.