Mathematical model of the point kinetic equations of a fast reactor in LabVIEW

The reactor point kinetics equations have been subjected to intense research in an effort to find simple yet accurate numerical solution methods. Many solution techniques have been presented on the point kinetics equations with varying degrees of complexity including Power Series Solutions, CORE, PCA, Ganapol and Taylor series methods. In this paper, fast and simple algorithms were developed based on the first and second order Taylor series expansion and simulated in LabVIEW to solve the Reactor Point Kinetics. Simulations for a fast reactor are presented. At 1× 10-8 s the neutron population was 1.000220 neutrons / cm3, at 1× 10-2 s it was 2.007681 neutrons / cm3 and at 1× 10-1 s it was 2.075317 neutrons / cm3. Keywords: reactor point kinetics equations, taylor series approximations, neutron population density, fast reacto

Thermodynamics and kinetics of nucleation of phase-separated water bubbles in synthetic quartz (a review)

Nucleation is a physical process characterized by localized budding of a different thermodynamic phase by precipitation of vapour from bulk to form a nucleus containing liquid. In nucleation by diffusive transformati-on, the system evolves through clustering of kinetic (growth) units, embryos and nucleus formation; and enlargem-ent or expansion to a microbubble, which grows into stable bubble. Interfacial free energy is the driving force for thermodynamics of nucleation, while cluster dynamics describe the kinetics of nucleation controlled by diffusion and hydrodynamic constraints. When grown-in water in synthetic quartz becomes supersaturated upon heating, the excess dissolved water phase separates upon annealing at critical level of supersaturation, and by interconversi-on reaction, hydroxyl (OH) vapour molecules aggregate and spherical water bubbles nucleate and grow by diffusi-on of OH vapour from bulk quartz in response to Laplace pressure release.Nucleation in quartz is initiated by aggregation and clustering of OH molecules, controlled by rate of OH vapour diffusion and condensation or precipitation, and rate at which vacancies are supplied at the nucleus surface in order to maintain the nucleus water pressure in equilibrium with the surface tension restraint during the period of annealing. In heterogeneous nucleation, formation of nucleus takes place on defects, surfaces or minute particl-es, where the contact angle becomes a controlling factor. Since hydroxyl groups are almost insoluble in quartz, the OH vapour precipitation and cluster incubation at grown-in defects are assumed to be stable. Homogeneous nucleation requires higher supersaturation than heterogeneous nucleation, as the nucleation involves initiation of new phase directly from supersaturated state by thermal fluctuations.The main factors which influence nucleation are supersaturation, phase transition, thermal transformation and presence of defects; however, thermodynamics and kinetics models have not been examined in details for nu-cleation of phase-separated water bubbles in annealed synthetic quartz. In the review, the clustering of OH kinetic units leading to nucleation are analyzed from the perspectives of thermodynamics of Gibbs free energy and kineti-cs of OH dynamics. The two different formulations adopted for examining the nucleation energetics are formation energy and Gibbs free energy.Keywords: Synthetic Quartz, Supersaturation, Phase Separation, Hydroxyl (OH) Kinetic (Growth) Units, Cluster-ing, Gibbs Free Energy, Formation Energy, Homogeneous and Heterogeneous Nucleatio

Nucleation and growth of spherical water bubbles in hydrothermal grown quartz (A Review)

Formation of water bubbles by precipitation of supersaturated water vapour and interactions of the bubbl-es with dislocations are fundamental for understanding the phenomenon of hydrolytic weakening of quartz during high temperature deformation. Annealing deformed synthetic quartz crystals at temperature in excess of 500 °C leads to phase-separation of grown-in water into hydroxyl (OH) species and molecular water (H2Om). By homoge-neous inter-conversion hydroxyl reaction of H2Om and O (anhydrous oxygen), the excess OH groups agglomerate into clusters to form embryos. Embryonic-nuclei of radii less than Gibbs critical size dissolve, while embryos of larger sizes grow to form stable nuclei in equilibrium with the supersaturated vapour. Ostwald ripening dissoluti-on-diffusion growth of nuclei at near critical state is limited by OH diffusion and surface vacancy kinetics. Chemi-cal potential difference of OH at the nucleus-bulk interface leads to changes in interfacial energy and higher OH concentration for growth of larger nucleus, as described by Gibbs-Thompson equation. For size evolution of nuclei, concentration gradients provide additional driving force for growth of larger nuclei at expense of smaller nuclei. La Mer model accounts for separation of nucleation and growth. Irreversible aggregation of stable nuclei form macrobubbles at the supercritical stage, which coalesce into spherical water bubbles of equilibrium sizes.Hydroxyl groups of low solubility in quartz are considered as interstitial defects, which are primarily tra-pped by vacancies. The spherical bubbles consisting of OH-vacancy complexes grow by absorbing OH monomers and coalescing with other bubbles under annealing conditions. The rate-controlling steps for bubble growth are migration by volume, surface diffusion, and interaction with lattice defects. By coalescence events of random co-llisions and OH capture, the bubbles gain energy for growth due to reduction of surface energy, limited by vacancy relaxation of strain fields produced by excess pressure in the bubble.The review examines mechanisms of nucleation and growth of water bubbles in quartz and presents com-prehensive models describing the processes occurring during bubble formation from evolution of clusters, embryonic-nuclei dissolution and growth, stable nuclei aggregation into nanobubbles, and coalescence to form macro-bubbles, which grow into equilibrium bubbles. The mathematical formulations governing clustering, dissolution-diffusion growth, coalescence events, interfacial hydroxyl mass diffusion, and surface adsorption kinetics for the physical processes of nucleation and growth of water-filled bubbles in quartz crystals are also presented.Keywords: Embryonic-Nucleus, La Mer Model, Ostwald Ripening, Interfacial Energy, Chemical Potential, Coale-scence Event, Rate Theor

Speciation and phase separation of water in quartz (A review)

Speciation, supersaturation and phase separation of water in hydrothermal grown (synthetic) quartz cryst-als have been examined as fundamental issues determining clustering, nucleation and bubble formation, which have profound influence on hydrolytic weakening, dislocation-bubble interactions and high temperature deformat-ion of quartz. Details of water speciation are necessary for explaining cluster dynamics and dislocation motion in quartz, and are also relevant in understanding magma evolution and eruption dynamics, since quartz and other sil-icates are important minerals of the earth crust. Infrared absorption studies strongly suggest that water bubbles in quartz contain hydrous species of hydroxyl ion (OH-) and molecular water (H2Om), originating from phase separat-ion of supersaturated water vapour after prolonged heat treatment of quartz at temperatures in excess of 500 °C. leading to decomposition of the hydrolyzed bonds. The hydrous species interconnect through a homogeneous spe-ciation reaction, condense, cluster and nucleate to form water bubbles, either homogeneously by thermal fluctuati-ons or heterogeneously on defects and other grown-in occlusions. The nucleation is a first order phase transition of creating liquid nucleus within the vapour phase, and chemical potential difference (related to the level of super-saturation) is the thermodynamic driving force.The unique properties of water that affect the mechanical deformation of quartz is due not only to the di-pole character, but even more to the geometrical structure of the molecules which form extended four coordinated networks. Incorporation of water tetrahedra in silicate framework [SiO4] produce electron-unsaturated chemical bonds. By sp3-hybridization of electron bonds, the water molecule exhibits a tetrahedral charged structure, leading to various water point defects in the form of substitution of SiO2 by two H2O substitutional defects (4H)Si, interstit-ial water molecules in channels parallel to the c-axis, hydrolyzed Si-O-Si bonds, and H-like alkali ions associated with substitutional impurities (such as Al3+ or Fe3+) to maintain charge balance. Water-related defects in quartz have complex structures and compositions, and the aqua-complexes are either bonded with different cations or are structurally bonded in the quartz matrix. The formation of combined defect [SiO4]-H2O-M+ [M3+O4] upon water speciation (M+ is metallic ion), is an indication of the presence of hydrous species.Keywords: Speciation, Supersaturation, Phase separation, Phase transition, Infrared absorption, Nucleatio

Molecular dynamics simulation of mechanical deformation of austenitic stainless steels (Fe-Ni-Cr alloys) at supercritical water conditions

Austenitic stainless steel (Fe-Ni-Cr alloy) has been identified as potential candidate material for fabrication of str-uctural components of supercritical water-cooled reactor (SCWR), because the alloy has proven nuclear, physical, and mechanical properties for the construction of prototype fast breeder reactor. The variation of mechanical pro-perties of stainless steel grades SS 304, 308, 309 and 316 at supercritical water (SCW) conditions of 300 - 500 ºC and 25 MPa, were simulated by molecular dynamics to examine the thermo-mechanical behavior of the alloys tai-lored for in-core structural components and pressure vessel design of SCWR. Large-scale Atomic/Molecular Mas-sively Parallel Simulator (LAMMPS) deformation of the alloys in uniaxial tensile tests were performed at strain rate of 5.0 x 1010 s-1, using Velocity Verlet Algorithm, Periodic Boundary Conditions and Isobaric-Isothermal En-sembles. The stress-stain data were imported into MATLAB for graphical representation, from which values of Ultimate Tensile Strength (UTS), Young’s Modulus (YM), Yield Strength (YS) and Breaking Strength (BS) were extracted. Compared with ambient values, the mechanical properties of the alloys decreased with increase in tem-perature by about 30 - 40 %. Stainless steel grades SS 304 and 308 showed comparatively higher UTS and BS at SCW conditions. The values of thermo-mechanical properties of Fe-Ni-Cr alloys would augment the data base for material design of SCWR.Keywords: Supercritical Water (SCW) Condition, Molecular Dynamics (MD) Simulation, LAMMPS, Velocity Verlet Algorithm, Embedded Atom Model (EAM), Periodic Boundary Condition (PBC), Isobaric-Isothermal Ens-embles, Austenitic Stainless Stee