Rigorous Equations Research Articles

An Equation-of-State Compositional In-Situ Combustion Model: A Study of Phase Behavior Sensitivity

In order to facilitate the study of reactive-compositional porous media processes we develop a virtual kinetic cell (single-cell model) as well as a virtual combustion tube (one-dimensional model). Both models are fully compositional based on an equation of state. We employ the models to study phase behavior sensitivity for in situ combustion, a thermal oil recovery process. For the one-dimensional model we first study the sensitivity to numerical discretization errors and provide grid density guidelines for proper resolution of in situ combustion behavior. A critical condition for success of in situ combustion processes is the formation and sustained propagation of a high-temperature combustion front. Using the models developed, we study the impact of phase behavior on ignition/extinction dynamics as a function of the operating conditions. We show that when operating close to ignition/extinction branches, a change of phase behavior model will shift the system from a state of ignition to a state of extinction or vice versa. For both the rigorous equation of state based and a simplified, but commonly used, K-value-based phase behavior description we identify areas of operating conditions which lead to ignition. For a particular oil we show that the simplified approach overestimates the required air injection rate for sustained front propagation by 17% compared to the equation of state-based approach.

Stochastic dynamics of bionanosystems: Multiscale analysis and specialized ensembles.

An approach for simulating bionanosystems such as viruses and ribosomes is presented. This calibration-free approach is based on an all-atom description for bionanosystems, a universal interatomic force field, and a multiscale perspective. The supramillion-atom nature of these bionanosystems prohibits the use of a direct molecular dynamics approach for phenomena such as viral structural transitions or self-assembly that develop over milliseconds or longer. A key element of these multiscale systems is the cross-talk between, and consequent strong coupling of processes over many scales in space and time. Thus, overall nanoscale features of these systems control the relative probability of atomistic fluctuations, while the latter mediate the average forces and diffusion coefficients that induce the dynamics of these nanoscale features. This feedback loop is overlooked in typical coarse-grained methods. We elucidate the role of interscale cross-talk and overcome bionanosystem simulation difficulties with (1) automated construction of order parameters (OPs) describing suprananometer scale structural features, (2) construction of OP-dependent ensembles describing the statistical properties of atomistic variables that ultimately contribute to the entropies driving the dynamics of the OPs, and (3) the derivation of a rigorous equation for the stochastic dynamics of the OPs. As the OPs capture hydrodynamic modes in the host medium, "long-time tails" in the correlation functions yielding the generalized diffusion coefficients do not emerge. Since the atomic-scale features of the system are treated statistically, several ensembles are constructed that reflect various experimental conditions. Attention is paid to the proper use of the Gibbs hypothesized equivalence of long-time and ensemble averages to accommodate the varying experimental conditions. The theory provides a basis for a practical, quantitative bionanosystem modeling approach that preserves the cross-talk between the atomic and nanoscale features. A method for integrating information from nanotechnical experimental data in the derivation of equations of stochastic OP dynamics is also introduced.

Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers

Multiple four-wave-mixing (FWM) processes with multifrequency pumps in optical fibers are investigated theoretically and experimentally. The propagation equations describing the FWM nonlinear interactions between pump waves and ``first-order'' sideband waves are derived. Based on our theoretical model, the rigorous nonlinear equations governing arbitrary pumps can be developed when the pump waves have equal frequency spacing. The proposed theory can generalize the traditional theory of FWM processes. The rigorous nonlinear equations governing three pumps and three cases of simplified equations governing three, four, and five pumps are achieved. Numerical simulations for three, four, and five pumps are experimentally tested by using a piece of highly nonlinear photonic-crystal fiber, and the experimental observations confirmed our theoretical predictions. The stabilization of waves by multiple FWM processes is proved theoretically and confirmed experimentally.

Rigorous and Phenomenological Equations of State

AbstractFor solar and stellar modeling, a high-quality equation of state is crucial. But the inverse is also true: the astrophysical data (helioseismic today, asteroseismic tomorrow) put constraints on the physical formalisms, making the Sun and the stars laboratories for plasma physics. One of the main astrophysical benefits from a good equation of state is an improved abundance determination. Recent theoretical progress in the equation of state has involved both rigorous and phenomenological approaches, giving the user a considerable choice.

Three-dimensional numerical study of heat and mass transfer in fluidized beds with spray nozzle

The numerical computations of temperature and concentration distributions inside a fluidized bed with spray injection in three-dimensions are presented. A continuum model, based on rigorous mass and energy balance equations developed from Nagaiah et al., is used for the three-dimensional simulations. The three-dimensional model equation for nozzle spray is reformulated in comparison to Heinrich. For solving the non-linear partial differential equations with boundary conditions a finite element method is used for space discretization and an implicit Euler method is used for time discretization. The time-dependent behavior of the air humidity, air temperature, degree of wetting, liquid film temperature and particle temperature is presented using two different sets of experimental data. The presented numerical results are validated with the experimental results. Finally, the parallel numerical results are presented using the domain decomposition methods.

Direct observation of excitons in polymer/carbon nanotube composites at room temperature: The influence of nanotube concentration

In this work, we have employed spectroscopic ellipsometry technique to study the optical properties of polymer/carbon nanotube (CN) composites as a function of nanotube concentration. Using a two-layer structural model based on Airy rigorous equations, the optical constants in various CN concentrations have been extracted. In the optical absorption spectra, we have observed a tuning of the excitonic transitions with the addition of single wall CN concentration. Based on theoretical calculations for the interaction of CNs with the surrounding effective media, the spectral evolution of the binding energy of optically active excitonic transitions with the addition of CNs suggests that the effective dielectric function of surrounding media decreases. Furthermore, using an oscillator Lorentz model, the absorption peaks, oscillator strengths and energy broadenings of the excitonic transitions have been extracted.

Calculation of thermodynamic properties of substances in metastable and labile regions of real gas states

Questions related to the concepts of the equilibrium and stability of a thermodynamic system are considered. Metastable states are defined as fully equilibrium but relatively unstable states. A preferable structure of the equation of state (ES) of a single-component substance has been determined. Equations for the second virial coefficient and ES of a real gas describing the thermodynamic characteristics in a wide range of state parameters within the limits of the error in experimental data have been obtained. This was done with the help of a combined spherically symmetric potential of interaction. With the use of rigorous equations of thermodynamics, it has been found that the isochoric heat capacity in the metastable and labile regions of states remains finite and positive (with the exception of the critical point). It has been shown that, at the transition from the stable region to the region of metastable and labile states, some thermodynamic characteristics such as the isochoric heat capacity, internal energy, enthalpy, and entropy do not have any singularities. Some examples of calculation of these characteristics in the stable, metastable, and labile regions with the help of equations presented by the authors are given. They confirm the validity of the analysis performed in this paper.

Diffraction of plane waves incident on a grated dielectric slab: An entire domain integral equation analysis

Diffraction of electromagnetic waves by periodic grating waveguides is investigated by using a rigorous integral equation method, which combines semianalytical techniques and the Method of Moments with entire domain basis functions. The electric field integral equation is employed with unknown function the electric field on the grooves. This equation is subsequently solved by applying an entire domain Galerkin's technique. The proposed analysis provides high numerical stability and controllable accuracy. All the involved computations are analytically carried out, leading to an analytic solution with the sole approximation of the final truncation of the expansion functions sets. The computed results exhibit superior accuracy and numerical efficiency compared with those already derived by applying different methods. The effect of the incident field's and grating's characteristics on the diffraction process as well as the grating structure's efficient operation as a narrow band reflection filter are thoroughly investigated. The numerical results obtained provide design guidelines, which may be exploited appropriately in the development of millimeter and optical waveguide structures.

Relativistic Radiation Hydrodynamical Accretion-Disk Winds

Abstract Accretion-disk winds blowing off perpendicular to a luminous disk are examined within the framework of fully special relativistic radiation hydrodynamics. The wind is assumed to be steady, vertical, and isothermal. Using a velocity-dependent variable Eddington factor, we can solve the rigorous equations of relativistic radiative hydrodynamics, and can obtain radiatively driven winds accelerated up to relativistic speeds. For less-luminous cases, disk winds are transonic types passing through saddle-type critical points, and the final speeds of the winds increase as the disk flux and/or the isothermal sound speed increase. For luminous cases, on the other hand, disk winds are always supersonic, since the critical points disappear due to the characteristic nature of the disk gravitational fields. The boundary between the transonic and supersonic types is located at around $\hat{F}_{\rm c} \sim 0.1 (\varepsilon+p)/(\rho c^2)/\gamma_{\rm c}$, where $\hat{F}_{\rm c}$ is the radiative flux at the critical point normalized by the local Eddington luminosity, $(\varepsilon+p)/(\rho c^2)$ is the enthalpy of the gas divided by the rest-mass energy, and $\gamma_{\rm c}$ is the Lorentz factor of the wind velocity at the critical point. In transonic winds, the final speed becomes 0.4–0.8$c$ for typical parameters, while it can reach $\sim c$ in supersonic winds.

Efficient Analysis and Design of Low-Loss Whispering-Gallery-Mode Coupled Resonator Optical Waveguide Bends

Waveguides that are composed of electromagnetically coupled optical microcavities [coupled resonator optical waveguides (CROWs)] can be used for light guiding, slowing, and storage. In this paper, we present a 2-D analysis of finite-size straight and curved CROW sections based on a rigorous Muller boundary integral equation method. We study the mechanisms of the coupling of whispering-gallery (WG) mode and guiding light around bends in CROWs composed of both identical and size- mismatched microdisk resonators. Our accurate analysis reveals the differences in WG modes coupling in the vicinity of bends in CROWs composed of optically large and wavelength-scale micro- cavities. We propose and discuss possible ways to design low-loss CROW bends and to reduce bend losses. These include selecting specific bend angles depending on the azimuthal order of the WG mode and tuning the radius of the microdisk positioned at the CROW bend.

GROWING HIERARCHICAL SCALE-FREE NETWORKS BY MEANS OF NONHIERARCHICAL PROCESSES

We introduce a fully nonhierarchical network growing mechanism, that furthermore does not impose explicit preferential attachment rules. The growing procedure produces a graph featuring power-law degree and clustering distributions, and manifesting slightly disassortative degree-degree correlations. The rigorous rate equations for the evolution of the degree distribution and for the conditional degree-degree probability are derived.

An extended coupled phase theory for the sound propagation in polydisperse concentrated suspensions of rigid particles

An extension of the classical coupled phase theory is proposed to account for hydrodynamic interactions between neighboring rigid particles, which are essential to describe properly the sound propagation in concentrated suspensions. Rigorous ensemble-averaged equations are derived for each phase and simplified in the case of acoustical wave propagation. Then, closure is achieved by introducing a self-consistent scheme originally developed by Buyevich and Shchelchkova [Prog. Aerosp. Sci. 18, 121-151 (1978)] for incompressible flows, to model the transfer terms between the two phases. This provides an alternative to the effective medium self-consistent theory developed by Spelt et al. [J. Fluid Mech. 430, 51-86 (2001)] in which the suspension is considered as a whole. Here, a significantly simpler formulation is obtained in the long wavelength regime. Predictions of this self-consistent theory are compared with the classical coupled phase theory and with experimental data measuring the attenuation in concentrated suspensions of silica in water. Our calculation is shown to give a good description of the attenuation variation with volume fraction. This theory is also extended to the case of polydisperse suspensions. Finally, the link between the self-consistent theory and the different orders of the multiple scattering theory is clarified.

Penetration of aerosol undergoing combined electrostatic dispersion and diffusion in a cylindrical tube

The deposition of unipolarly charged aerosol particles by simultaneous Brownian diffusion and electrostatic dispersion (space-charge) in laminar flow tubes has been modelled in two different ways. In the first approach, the rigorous transport equation has been solved numerically under the restriction of negligible axial components of the particle Brownian motion and of the space-charge electric field. The thus calculated theoretical penetrations agree satisfactorily with the experimental results obtained for air ions. Second, the system has also been modelled from a phenomenological point of view, based on the fact that deposition by diffusion and space-charge follows first- and second-order kinetics, respectively. The latter approach yields a simple and practical equation correlating particle penetration with tube geometry, aerosol flow rate, and particle mobility and concentration.

Water phases under high electric field and pressure applied simultaneously

The rigorous equation of state of an open system containing water in an electric field above 10 8 V m − 1 under pressure applied in the range 10 − 4 ≤ P o ≤ 0.8 GPa leads to phase diagrams with two possible kinds of phase transitions at 293 K. The first one is the discontinuous phase transition under pressure applied in the range 10 − 4 ≤ P o ≤ 0.05 GPa in the Π, σ coordinates ( Π is the electrostriction pressure and σ denotes the surface charge density at an adjacent charged surface). The second one represents the continuous phase transition under pressure applied in the range 10 − 4 ≤ P o ≤ 0.8 GPa; it occurs at higher values of σ than the former one.

Relativistic Radiative Flow in a Luminous Disk II

Radiatively driven transfer flow perpendicular to a luminous disk is examined in the relativistic regime of $(v/c)^2$, while taking into account the gravity of the central object. The flow is assumed to be vertical, and the gas pressure as well as the magnetic field are ignored. Using a velocity-dependent variable Eddington factor, we can solve the rigorous equations of the relativistic radiative flow accelerated up to relativistic speeds. For sufficiently luminous cases, the flow resembles the case without gravity. For less-luminous or small initial radius cases, however, the flow velocity decreases due to gravity. Application to a supercritical accretion disk with mass loss is briefly discussed.

Analysis of scattering by a linear chain of spherical inclusions in an optical fiber

The scattering by a linear chain of spherical dielectric inclusions, embedded along the axis of an optical fiber, is analyzed using a rigorous integral equation formulation, based on the dyadic Green's function theory. The coupled electric field integral equations are solved by applying the Galerkin technique with Mie-type expansion of the field inside the spheres in terms of spherical waves. The analysis extends the previously studied case of a single spherical inhomogeneity inside a fiber to the multisphere-scattering case, by utilizing the classic translational addition theorems for spherical waves in order to analytically extract the direct-intersphere-coupling coefficients. Results for the transmitted and reflected power, on incidence of the fundamental HE(11) mode, are presented for several cases.

Rigorous integral equation analysis of nonsymmetric coupled grating slab waveguides

A rigorous integral equation formulation in conjunction with Green's function theory is used to analyze the waveguiding and coupling phenomena in nonsymmetric (composed of dissimilar slabs) optical couplers with gratings etched on both slabs. The resulting integral equation is solved by applying an entire-domain Galerkin technique based on a Fourier series expansion of the unknown electric field on the grating regions. The proposed analysis actually constitutes a special type of the method of moments and provides high numerical stability and controllable accuracy. The singular points of the system's matrix accurately determine the complex propagation constants of the guided waves. The results obtained improve on those derived by coupled-mode methods in the cases of large grating perturbations and highly dissimilar slabs. Numerical results referring to the evolution of the propagation constants as a function of the grating's characteristics are presented. Optimal grating parameters with respect to minimum coupling length and maximum coupling efficiency are reported. The coupler's efficient operation as an optical bandpass filter is thoroughly investigated.

Complex permittivity measurements of ferroelectrics employing composite dielectric resonator technique

Composite cylindrical TE(0n1) mode dielectric resonator has been used for the complex permittivity measurements of ferroelectrics at frequency about 8.8 GHz. Rigorous equations have been derived that allowed us to find a relationship between measured resonance frequency and Q-factor and the complex permittivity. It has been shown that the choice of appropriate diameter of a sample together with rigorous complex angular frequency analysis allows precise measurements of various ferroelectric. Proposed technique can be used for materials having both real and imaginary part of permittivity as large as a few thousand. Variable temperature measurements were performed on a PbMg(1/3)Nb(2/3)O3 (PMN) ceramic sample, and the measured complex permittivity have shown good agreement with the results of measurements obtained on the same sample at lower frequencies (0.1-1.8 GHz).

Method of describing the hydrodynamics of two-phase streams in heat-and mass-exchange units

0009-3092/06/4205–0369 © 2006 Springer Science+Business Media, Inc. The hydrodynamic characteristics of contact devices in heat exchangers are functions of a large number of parameters whose effect, with rare exceptions, cannot be evaluated with rigorous theoretical equations. The difficulties of analytically describing the characteristics of two-phase streams make it necessary to conduct laborious and expensive experimental studies whose large volume is due to the wide range of variation of the flow and physical characteristics of the stream and the design parameters of the devices in industrial units. For this reason, in conducting an experiment, two very important problems arise: determining the minimally necessary and sufficient volume of the experiment and selecting the method of generalizing the experimental data to obtain reliable calculation dependences which can be extrapolated to the real operating conditions of the equipment. Methods of design of experiments, similarity theory, and dimensionality analysis are widely used for solving these problems.

Topological vortices in a two-gap superconductor

Based on the ϕ-mapping theory, we derive a new rigorous equation describing the distribution of the magnetic field for vortices in a two-gap superconductor, of which the so-called modified London equation is just a special case in a one-flavor limit. We explicitly investigate the London penetration depth, the Meissner and mixed states and Josephson effect. A magnetic flux quantization condition for vortices in a two-gap superconductor is also derived, from which it follows that in a two-gap superconductor there exist vortices which carry an arbitrary fraction of magnetic flux quantum. The branch processes during the evolution of the vortices in a two-gap superconductor are discussed.