Gas Equation Research Articles

Finite Difference and Reinitialization Methods with Level Set to Interfacial Area Transport Equations for Gas–Liquid Two-Phase Flows

For the direct simulation of liquid atomization processes affected by surface tension, the level-set method was suitable, owing to the ease of estimating normal vectors and interface curvatures. However, the mass is not conserved in the original method because of errors in the interfacial advection evaluation. Therefore, previous studies have developed the interfacial advection equation and the interfacial evaluation method for reinitialization. The reinitialization method can be organized in terms of control parameters. Therefore, we modified and optimized the control parameters of the high-order constrained reinitialization scheme developed previously. Furthermore, the difference method in the convection term was adapted to the interfacial advection equation to evaluate the best method for simulating the droplet atomization process. Consequently, the fifth-order accuracy weighted essentially non-oscillatory scheme was observed to provide the best mass conservation for both the reinitialization and the advection equations.

Dependence of [formula omitted] on spin-amplified phonons in pressurized hydrides

The realization of room temperature superconductivity in an under-high-pressure carbonaceous sulphur hydride has sparked off interest in pressurized superconductivity. The next and perhaps major hurdle is to achieve room temperature superconductivity (RTS) at ambient pressure. However, the quantum environment surrounding the pressurized RTS may give a glimpse of what is required to attain the phenomenon under normal pressure. This study is devoted to projecting the source of attraction of charges that keeps superconductivity afloat in hydrides under ultra-high pressure. Starting with the quantum equation of a normal-state ideal gas, we determine the dependence of Tc on pressure, spin and phonons. We show that plasmon mediation charges fails at ultra-high pressures. It is also found that bare phonons can sustain the mediation of charge pairing under pressure in the order of 1-100GPa but much less than the experimentally observe pressure, pcexp. However, the cooperation between spin and phonons sustains phonon mediation of charge pairing under much higher pressure pcexp. This is an indicator that the pairing of charges in pressurized superconductivity is mediated by enhanced phonons – spin-amplified phonons.

Numerical simulations of Richtmyer–Meshkov instability of [formula omitted] square bubble in diatomic and polyatomic gases

The Richtmyer–Meshkov instability of a shock-driven SF6 square bubble in monatomic, diatomic, and polyatomic gases is investigated numerically. The focus was placed on presenting more intuitive details of the flow-fields visualizations, vorticity production, degree of thermal non-equilibrium, enstrophy and dissipation rate evolutions, and interface structures. A mixed-type modal discontinuous Galerkin method is employed for solving the two-dimensional system of physical conservation laws derived from the Boltzmann–Curtiss kinetic equation of diatomic and polyatomic gases. For validation, the numerical results were compared with the existing experimental results. The results revealed that diatomic and polyatomic gases provoke considerable changes in the flow-fields, resulting in complex wave patterns, bubble deformation, and outward SF6 jets formation in contrast to monatomic gas. A detailed investigation on the effects of diatomic and polyatomic gases is carried out through the vorticity production, degree of nonequilibrium, and evolution of enstrophy as well as dissipation rate. Moreover, the length and height of the interface structures are investigated quantitatively. Finally, the effects of thermal non-equilibrium parameters, such as inverse power-law index and bulk viscosity ratio are examined. The present work attempts to enhance the understanding of the RM instability studies in monatomic, diatomic, and polyatomic gases.

A Three-dimensional Multi-species Flow Solver for the Euler Equations Combined with a Stiffened Gas Equation of State

Although numerical simulation in fluid mechanics is undergoing a significant development due to the dazzling evolution of computing means, complex physical phenomena, such as multidimensional viscous effects in turbomachinery and cavitation, remain mysterious and attract the curiosity of several researchers. Highresolution shock captures are often obtained by the WENO family of schemes, except that in problems that depend on discontinuities and shocks, an appearance of numerical oscillations weakens its ability to provide adequate captures. The use of the characteristic construction methods prevents this type of oscillation. The present paper contributes to the numerical resolution of multi-species flows of viscous, compressible, or incompressible fluids with shocks and discontinuities. The proposed numerical model can handle various configurations with a unique method based on a conservative and consistent threedimensional finite volume scheme with an aligned mesh. The system of equations is a set of Euler equations coupled with a two-parameters generalized state equation of state in three-dimensional Cartesian coordinates. This system is solved using a Roe type approximate Riemann solver, and second-order precision is obtained using limiters. The obtained numerical results maintain a nonoscillatory flow near the discontinuities, which makes the method satisfactory and shows its accuracy and robustness in different cases.

Influence of contact friction on static characteristics of multileaf bump foil bearing

In order to reveal the influence mechanism of foil contact friction on the multi-leaf bump foil bearing performance, a deformation simulation model is established under the contact constraint of multi top foils and bump foils, and the analysis of the fluid-structure interaction is realized by combining with the Reynolds equation of compressible gas. The influence of the nonlinear variation of foil stiffness caused by contact friction on the static characteristics of the bearing under different bearing numbers is numerically analyzed. The results show: the foil stiffness increases nonlinearly with the foil deformation and presents the characteristic of circumferential non-uniformity. With the increase of the deformation of foil, the influence of friction coefficient on the foil stiffness is more significant. Within the parameters studied in this paper, the friction coefficient increases from 0 to 0.5, the bearing capacity increases first and then decreases when the bearing numbers is small; the bearing capacity increases monotonically when the bearing numbers is large, the higher the bearing numbers, the more significant the growth rate. The attitude angle increases with the increase of friction coefficient, and the bearing becomes more and more unstable. When the bearing numbers is large, we should pay more attention to the influence of friction on bearing performance. The computed model emphasizing nonlinear stiffness in this paper is close to the realities, and the results can provide guidance in the design of multi-leaf bump foil bearing.

Short nozzles design for real gas supersonic flow using the method of characteristics

Finding a supersonic nozzle minimum length is a classical application of the method of characteristics (MOC) for aerospace propulsion technology among many other applications, such as air and vapor handling processes. That methodology allows design contoured nozzle free of oblique shock waves created at sharp area changes in short nozzles. While the designing technique is well established in textbooks for ideal gas flow, the use of the MOC technique for real gas is problematic due to the highly complex fluid behavior, captured by modern real gas equations-of-state (EOS). This work presents the MOC devised for real gases using multi-parameter equations-of-state (MPEOS) for different substances and compositions. This paper also compares the MPEOS solution obtained from other classical EOS, such as Peng–Robinson, and the ideal gas equation to establish the optimal application range for each EOS and their effect on the nozzle wall construction. The study was carried out for a commercial fluid refrigerant, pure carbon dioxide, and a CO2−CH4 mixture. The methodology can be used for designing industrial pieces of equipment, such as turbo-machineries and supersonic gas separators or supersonic ejectors, to evaluate the isentropic flow expansion within those devices.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.

From Boltzmann Equation for Granular Gases to a Modified Navier–Stokes–Fourier System

In this paper, we give an overview of the results established in Alonso (http://arxiv.org/org/abs/2008.05173, 2020) which provides the first rigorous derivation of hydrodynamic equations from the Boltzmann equation for inelastic hard spheres in 3D. In particular, we obtain a new system of hydrodynamic equations describing granular flows and prove existence of classical solutions to the aforementioned system. One of the main issue is to identify the correct relation between the restitution coefficient (which quantifies the rate of energy loss at the microscopic level) and the Knudsen number which allows us to obtain non trivial hydrodynamic behavior. In such a regime, we construct strong solutions to the inelastic Boltzmann equation, near thermal equilibrium whose role is played by the so-called homogeneous cooling state. We prove then the uniform exponential stability with respect to the Knudsen number of such solutions, using a spectral analysis of the linearized problem combined with technical a priori nonlinear estimates. Finally, we prove that such solutions converge, in a specific weak sense, towards some hydrodynamic limit that depends on time and space variables only through macroscopic quantities that satisfy a suitable modification of the incompressible Navier–Stokes–Fourier system.

Energy relaxation approximation for compressible multicomponent flows in thermal nonequilibrium

This work concerns the numerical approximation with an explicit first-order finite volume method of inviscid, nonequilibrium, high-temperature flows in multiple space dimensions. It is devoted to the analysis of the numerical scheme for the approximation of the hyperbolic system in homogeneous form. We derive a general framework for the design of numerical schemes for this model from numerical schemes for the monocomponent compressible Euler equations for a polytropic gas. Under a very simple condition on the adiabatic exponent of the polytropic gas, the scheme for the multicomponent system enjoys the same properties as the one for the monocomponent system: discrete entropy inequality, positivity of the partial densities and internal energies, discrete maximum principle on the mass fractions, and discrete minimum principle on the entropy. Our approach extends the relaxation of energy (Coquel and Perthame in SIAM J. Numer. Anal. 35:2223–2249, 1998) to the multicomponent Euler system. In the limit of instantaneous relaxation we show that the solution formally converges to a unique and stable equilibrium solution to the multicomponent Euler equations. We then use this framework to design numerical schemes from three schemes for the polytropic Euler system: the Godunov exact Riemann solver, and the HLL and pressure relaxation based approximate Riemann solvers. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and convergence of the schemes.

A new approach of CO2 separation by applying rapid expansion of supercritical CO2 rich natural gas

The previous methods for CO2 separation from CO2 rich natural gas led to expensive production costs. This work was implemented to overcome the problems utilizing a new approach economically. The cooling and rapid expansion processes were integrated for the CO2 separation from CO2 rich natural gas on the supercritical condition. The experimental apparatus was newly constructed to perform the experiments, and the results were simulated using a various equation of state. The result reveals that the inlet temperature of supercritical expansion diminished the outlet temperature and the gas condensed easily. The simulation indicated that the 70% CO2 in natural gas was condensed easier than 45% CO2. We found that the outlet temperature of −42 °C and the vapor fraction of 0.69 was attained at the CO2 composition of 70%. Besides, the pressure drop change influences the vapor fraction at various CO2 compositions. The vapor fraction under supercritical diminished significantly compared with the non-supercritical condition. The expansion coefficient determined utilizing the equation of state escalates by the enhancement of expansion inlet temperature based on CO2 composition in natural gas. The acid gas equation of state was the perfect equation to estimate the expansion coefficient with the absolute average error of 4.83%. This work suggests that the CO2 separation from CO2 rich natural gas with the cooling and rapid expansion method promotes the new approach to overcome the disadvantages of previous methods.

State variable, entropy, should be negative

<p indent="0mm">The positive-and-negative property of a physical quantity is usually related to its physical meaning. The general convention is to use positive values to represent positive things and negative values to represent negative things. However, the state variable, entropy, is a strange physical quantity. From a macroscopic perspective, entropy represents the unavailable energy of the heat output of a thermodynamic system as it changes from an arbitrary state to a state in equilibrium with the environment, in short, the unavailable heat of the system. From a microscopic point of view, entropy represents the degree of disorder (chaos) in a system. Both the macroscopic and microscopic physical meanings of entropy indicate that it represents a negative thing. Therefore, entropy should be negative by its nature. However, historically, since Planck’s statement of Nernst’s heat theorem, entropy has always been taken for granted as a positive value. This is contrary to people’s cognitive habits, which adds to the mystery of entropy. Analyses indicate that Planck’s statement of Nernst’s heat theorem is not general and is valid only for single-component substances, such as perfect crystals. It is not an absolute fact that “entropy cannot be negative”. The Planck’s choice of a zero-temperature perfect crystal as the zero point of entropy is a non-physical operation and involves only a matter of convenience. Recently, based on the symmetry of reversible thermodynamics, a new equation of state for ideal solids has been established. Since entropy is explicitly included in the equation of state for ideal solids, the zero point of entropy cannot be chosen arbitrarily. The analysis from the thermodynamic stability and the verification of the property data show that entropy in the ideal solid equation of state must have a negative value. This physically specifies that the zero point of entropy should be the point of maximum entropy, i.e., the zero-pressure ideal gas limiting state. In fact, the ideal gas equation of state has a zero point of volume. That is the ideal solid limiting state (zero-volume state). At this zero point, the volume always takes a positive value. In brief, the zero point of volume of the ideal gas equation of state is the zero-temperature ideal solid, while the zero point of the entropy of the ideal solid equation of state is the zero-pressure ideal gas. The ideal solid equation of state is a creation of thermodynamic symmetry. The zero point of entropy, i.e., zero-pressure ideal gas is likewise a requirement of thermodynamic symmetry. The smaller the absolute value of negative entropy, the greater the unavailable energy of the system (the smaller the available energy) and the higher the degree of disorder of the system (the lower the degree of order). This is consistent with people’s cognitive habits. In addition, the zero-pressure ideal gas can be used as the zero point of entropy for both the ideal gas and the ideal solid equations of state, thus allowing a uniform orientation and division of the applicability of various practical equations of state between these two extreme equations of state. Unlike the choice of zero-temperature perfect crystal as the zero point of entropy, when zero-pressure ideal gas is used as the zero point of entropy, entropy takes a negative value, which is in accordance with the physical meaning of entropy and is also the requirement of thermodynamic symmetry. Taking a negative value for entropy is the physical, essential choice.

HEAT TRANSFER AT NATURAL CONVECTION OF VAN DER WAALS GAS

This paper focuses on a study of natural convection in a Van der Waals gas over a vertical heated plate. An approximate analytical solution of the problem was obtained using an integral method for momentum and energy equations. A simplified form of the van der Waals equation for real gases enabled estimating the effects of the dimensionless van der Waals parameters on the normalized heat transfer coefficient. The effects of the dimensionless Waa and Wab numbers on the normalized Nusselt number in the real gas compared to the ideal gas were estimated. The analysis of the calculation results showed that, with an increase in the Waa number (which characterizes the additional pressure in the real gas), the normalized Nusselt number increases. The effect of additional volume, which shows in an increase in the Wab number, causes a deterioration in the conditions interaction between gas molecules and the wall. This is accompanied by a decrease in the Archimedes force and flow rate in the boundary layer, which leads to a weakening of heat transfer in comparison with an ideal gas.

On the Comparative Analysis of Different Phase Coexistences in Mesoporous Materials.

Alterations of fluid phase transitions in porous materials are conventionally employed for the characterization of mesoporous solids. In the first approximation, this may be based on the application of the Kelvin equation for gas–liquid and the Gibbs–Thomson equation for solid–liquid phase equilibria for obtaining pore size distributions. Herein, we provide a comparative analysis of different phase coexistences measured in mesoporous silica solids with different pore sizes and morphology. Instead of comparing the resulting pore size distributions, we rather compare the transitions directly by using a common coordinate for varying the experiment’s thermodynamic parameters based on the two equations mentioned. Both phase transitions in these coordinates produce comparable results for mesoporous solids of relatively large pore sizes. In contrast, marked differences are found for materials with smaller pore sizes. This illuminates the fact that, with reducing confinement sizes, thermodynamic fluctuations become increasingly important and different for different equilibria considered. In addition, we show that in the coordinate used for analysis, mercury intrusion matches perfectly with desorption and freezing transitions.

Simulations of water-vapor two-phase flows with non-condensable gas using a Noble-Able-Chemkin stiffened gas equation of state

In order to simulate some accidental scenarii which might affect a pressurized water reactor, a homogeneous two-phase water-vapor flow model taking into account non-condensable gas is considered. The liquid phase and the gaseous phase are assumed to be immiscible, but the gaseous phase is composed of two miscible components. Due to these hybrid hypotheses of miscibility, the configurations with at least one missing field are carefully examined. A semi-analytical equation of state is chosen for the liquid water, which is an extension of the Noble-Able stiffened gas equation of state. Its accuracy is assessed with respect to the reference equation of state IAPWS. The homogeneous model is first verified thanks to Riemann problems. Then, some simulations aiming at reproducing SUPERCANON experiment are presented. The amount of air dissolved in liquid water is shown to have a strong influence on numerical results, in particular on the sound speed. Some out of thermodynamical equilibrium simulations are also presented, using two time scales describing the return towards equilibrium.

Study on the two-component gas apparent permeability model in shale nanopores considering Knudsen number correction

At present, the technology of carbon dioxide flooding shale gas is a hot topic for scholars. The numerical simulation of shale gas driven by carbon dioxide is inseparable from the accurate understanding of gas percolation mechanism and the construction of two-component gas motion equation. During the flooding process, the gas apparent permeability and the Knudsen number both change with time and space. However, this has not been fully discussed in previous studies. In this paper, the percolation mechanisms of two-component gas are discussed. On this basis, the calculation formula of Knudsen number in porous media is modified, and the transport equation of two-component gas is established. According to the derivation of apparent permeability, the two-component gas apparent permeability model is deduced. Then, the correction effect of Knudsen number and the related properties of two-component apparent permeability are analyzed. The results show that: (1) the effect of Knudsen number correction is significant at low pressure. Under high pressure, that is, when the pressure is greater than 10 MPa, the effect can be ignored; (2) the effect of Knudsen number correction is mainly significant in the range of medium pores, which is generally 10 nm–60 nm. And the peak value range decreases with the increase of pressure; (3) for the relative component apparent permeability curve, it has the properties of both relative permeability and apparent permeability. Based on the data in this paper, for the pressure change, it can be concluded that the maximum change rate of methane is 47.79% and that of carbon dioxide is 51.36%. For the change of pore size, it can be concluded that the maximum change rate of methane is 223.45% and that of carbon dioxide is 106.19%. It can be shown that the relative permeability of carbon dioxide is sensitive to pressure and the relative permeability of methane is sensitive to pore size.

Generalized Energy-Conserving Dissipative Particle Dynamics with Reactions.

We present an extension of the generalized energy-conserving dissipative particle dynamics method (J. Bonet Avalos, et al., Phys Chem Chem Phys, 2019, 21, 24891-24911) to include chemical reactivity, denoted GenDPDE-RX. GenDPDE-RX provides a means of simulating chemical reactivity at the micro- and mesoscales, while exploiting the attributes of density- and temperature-dependent many-body force fields, which include improved transferability and scalability compared to two-body pairwise models. The GenDPDE-RX formulation considers intra-particle reactivity via a coarse-grain reactor construct. Extent-of-reaction variables assigned to each coarse-grain particle monitor the temporal evolution of the prescribed reaction mechanisms and kinetics assumed to occur within the particle. Descriptions of the algorithm, equations of motion, and numerical discretization are presented, followed by verification of the GenDPDE-RX method through comparison with reaction kinetics theoretical model predictions. Demonstrations of the GenDPDE-RX method are performed using constant-volume adiabatic heating simulations of three different reaction models, including both reversible and irreversible reactions, as well as multistep reaction mechanisms. The selection of the demonstrations is intended to illustrate the flexibility and generality of the method but is inspired by real material systems that span from fluids to solids. Many-body force fields using analytical forms of the ideal gas, Lennard-Jones, and exponential-6 equations of state are used for demonstration, although application to other forms of equation of states is possible. Finally, the flexibility of the GenDPDE-RX framework is addressed with a brief discussion of other possible adaptations and extensions of the method.

A Modified Solid-Liquid-Gas Phase Equation of State.

The solid–liquid–gas equation of state (SLV-EOS) is based on the initial cubic equation of state, the van der Waals equation. Since the van der Waals equation is not accurate enough to predict gas–liquid properties, SLV-EOS cannot better predict the gas–liquid properties of hydrocarbons in actual gas reservoirs. Therefore, a modified solid–liquid–gas unified equation of state was constructed inthis paper, which was developed using the material’s actual critical compressibility factor Zc. The minimum liquid-phase volume at the triple point is also introduced to limit the value of c in the equation, which effectively avoids the solution of Maxwell’s equal-area rule in the solid–liquid transformation process. The model extends the classical Peng–Robinson equation of state for fluid-only (liquid and vapor) states. The predicted p-T and p-ρ phase transition diagrams are reported in this paper for methane, ethane, propane, carbon dioxide, hydrogen sulfide, and sulfur, and they are in good agreement with the experimental data. This methodology is suitable for any substance for which the density of the solid phase is higher than that of the liquid phase. Additionally, the modified SLV equation can be used to estimate the solubility of solid sulfur in the absence of relevant experimental data.

Kinetic Equation for Soliton Gas: Integrable Reductions

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2times 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

Data driven forecast of concurrent flame spread in micro-gravity

This paper presents a methodology that combines data assimilation and physical modelling of flame spread for fire growth forecast. The concurrent flame spreading over a flat solid sample in the absence of buoyant flows is considered as a simple canonical fire spread configuration. To predict flame spread rate and flame length evolution, the analytical solution to the two-dimensional boundary layer non-premixed combustion problem is combined with a CFD model which delivers the structure of the flow away from the fuel surface. In the process, two coefficients which intervene in the gas and solid heat transfer equations are assimilated using the pyrolysis length data as a flame spread rate surrogate input to absorb the shortcomings of the modelling. The robustness of the overall method is evaluated through convergence assessment of these two assimilated variables for different initial guesses, and for different values of the input invariants. Validation over large-scale microgravity data provides confidence in this approach, and demonstrates its potential to deliver flame spread predictions from initial measurements at a reduced computational cost. However, discrepancies between flame length measurements and predictions question the degree of correlation between pyrolysis and flame lengths.

Vanishing flux perturbation, pressure, and magnetic field limit in a Chaplygin magnetogasdynamics

The Riemann problem for a Chaplygin magnetogasdynamics system of compressible fluid flow with a triple parameter perturbation containing flux, pressure, and magnetic field is first solved. Second, it is shown that as the triple parameter flux perturbation, pressure, and magnetic field vanish, any Riemann solution containing two shock waves tends to a delta-shock solution to the transport equations and any Riemann solution containing two rarefaction waves tends to a vacuum solution to the transport equations. Third, it is also proved that as the double parameter flux perturbation and magnetic field vanish, any Riemann solution containing two shock waves approaches a delta-shock solution to the Chaplygin gas equations and any Riemann solution containing two rarefaction waves tends to a solution containing two contact discontinuities of the Chaplygin gas equations. Finally, some numerical results are presented to be consistent with the theoretical analysis.