Query: Ideal gas non-constant temperature state equation

title: Exact solution of shock wave structure in a non ideal gas under constant and variable coefficient of viscosity and heat conductivity
abstract: This paper investigates the structure of a normal shock wave using the continuum model for steady one dimensional flow of a viscous non ideal gas under heat conduction. The coefficients of viscosity and heat conductivity are assumed to be directly proportional to a power of the temperature. The simplified van der Waals equation of state for the non ideal gas has been assumed in this work. The smooth and rough sphere models of the gas molecules in the kinetic theory of gases are used for the viscosity of a non ideal gas. Assuming the Prandtl number to be 3 4, the complete integral of the energy equation, exact velocity, density, pressure, Mach number, change in entropy, viscous stress, and heat flux across the shock transition zone have been obtained in a perfect and a non ideal gas under both constant and variable properties of the medium. The validity of the continuum hypothesis with respect to Mach number is examined for the study of shock wave structure in both the smooth and rough sphere models of ideal and non ideal gas molecules. It has been shown that the continuum theory gives reasonably valid results for flows with higher Mach numbers in the case of a non ideal gas in comparison with an ideal gas. The inverse thickness of the shock wave is calculated and compared for constant and variable properties of the gases. The shock wave thickness is also discussed as a function of mean free path of the gas molecules computed at different points between the boundary states. It is found that the inverse shock thickness decreases with the increase in non idealness of the gas. In the rough sphere model of gas molecules, the increase in the non idealness of the gas and the temperature exponent in the coefficients of viscosity and heat conductivity significantly increases the validity limit of the continuum model.

title: Similarity solution for the flow behind a shock wave in a non ideal gas with heat conduction and radiation heat flux in magnetogasdynamics
abstract: The propagation of a spherical (or cylindrical) shock wave in a non ideal gas with heat conduction and radiation heat flux, in the presence of a spacially decreasing azimuthal magnetic field, driven out by a moving piston is investigated. The heat conduction is expressed in terms of Fourier's law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity K and the absorption coefficient a R are assumed to vary with temperature and density. The gas is assumed to have infinite electrical conductivity and to obey a simplified van der Waals equation of state. The shock wave moves with variable velocity and the total energy of the wave is non constant. Similarity solutions are obtained for the flow field behind the shock and the effects of variation of the heat transfer parameters, the parameter of the non idealness of the gas, both, decreases the compressibility of the gas and hence there is a decrease in the shock strength. Further, it is investigated that with an increase in the parameters of radiative and conductive heat transfer the tendency of formation of maxima in the distributions of heat flux, density and isothermal speed of sound decreases. The pressure and density vanish at the inner surface (piston) and hence a vacuum is form at the center of symmetry. The shock waves in conducting non ideal gas with conductive and radiative heat fluxes can be important for description of shocks in supernova explosions, in the study of central part of star burst galaxies, nuclear explosion, chemical detonation, rupture of a pressurized vessels, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or reentry vehicles, etc. The findings of the present works provided a clear picture of whether and how the non idealness parameter, conductive and radiative heat transfer parameters and the magnetic field affect the flow behind the shock front.

title: Astrophysical fluid simulations of thermally ideal gases with non constant adiabatic index: numerical implementation
abstract: An Equation of State \textit{EoS} closes the set of fluid equations. Although an ideal EoS with a constant \textit{adiabatic index} \Gamma$ is the preferred choice due to its simplistic implementation, many astrophysical fluid simulations may benefit from a more sophisticated treatment that can account for diverse chemical processes. Here, we first review the basic thermodynamic principles of a gas mixture in terms of its thermal and caloric EoS by including effects like ionization, dissociation as well as temperature dependent degrees of freedom such as molecular vibrations and rotations. The formulation is revisited in the context of plasmas that are either in equilibrium conditions (local thermodynamic or collisional excitation equilibria) or described by non equilibrium chemistry coupled to optically thin radiative cooling. We then present a numerical implementation of thermally ideal gases obeying a more general caloric EoS with non constant adiabatic index in Godunov type numerical schemes.We discuss the necessary modifications to the Riemann solver and to the conversion between total energy and pressure (or vice versa) routinely invoked in Godunov type schemes. We then present two different approaches for computing the EoS.The first one employs root finder methods and it is best suited for EoS in analytical form. The second one leans on lookup table and interpolation and results in a more computationally efficient approach although care must be taken to ensure thermodynamic consistency. A number of selected benchmarks demonstrate that the employment of a non ideal EoS can lead to important differences in the solution when the temperature range is $500 10^4$ K where dissociation and ionization occur. The implementation of selected EoS introduces additional computational costs although using lookup table methods can significantly reduce the overhead by a factor $3\sim 4$

title: On the structure of MHD shock waves in a viscous non ideal gas
abstract: The structure of one dimensional magnetohydrodynamics (MHD) shock waves is studied using the Navier Stokes equations for the non ideal gas phase. The exact solutions are obtained for the flow variables (i.e. particle velocity, temperature, pressure and change in entropy) within the shock transition region. The equation of state for a non ideal gas is considered as given by Landau and Lifshitz. The effects of the non idealness parameter and coefficient of viscosity of the gas are analysed on the flow variables assuming the magnetic field having only constant axial component. The findings confirm that the thickness of MHD shock front increases with decreasing values of the non idealness parameter.

title: Bayesian calibration of the Peng Robinson fluid model for siloxane MDM vapor flows in the non ideal regime
abstract: (NICFD) investigates the gas dynamics of a special class of fluids, typically (but not limited to) vapors of molecular complex compounds, characterized by a thermodynamic state space containing a non ideal region. In the non ideal regime, the fluid no longer behaves as an ideal gas with state equation Pv RT (whereas P is the gas pressure, v the specific volume, R the gas constant and T the temperature) and more complex fluid models are required. Currently, the community is racing to develop reliable and predictive tools to investigate the non ideal dynamics and to ultimately improve the design of devices involving NICFD flows. A global perspective including experiments, computations, and theory is needed in order to develop sophisticated physical models as well as a systematic and comprehensive treatment of calibration and validation procedures. This work focuses on the calibration of the polytropic Peng Robinson (PR) fluid model [Peng and Robinson, 1976] for siloxane MDM (Octamethyltrisiloxane, C8H24O2Si3) vapor flows in the non ideal regime. Specifically, the goal is to calibrate the material dependent parameters appearing in the equations of state by combining experimental NICFD flows measurements with numerical simulations. The calibration process relies on a standard Bayesian inference framework and it takes advantage of the first ever experiments on non ideal expanding flows of siloxane MDM vapor [Spinelli et al. 2018] The Bayesian framework to infer the polytropic PR model parameters accounts for uncertainties in the test rig operating conditions (treated as nuisance parameters) Specifically, the inference considers the total pressure Pt and total temperature Tt at the inlet of the test section (uncertain operating conditions) and the fluid critical pressure Pcr, *giulio.gori@inria.fr, DeFI Team, CMAP Lab (Ecole Polytechnique, Inria Saclay Ile de France) 1 rue Estienne d'Orves, 91120 Palaiseau, France +LIMSI, CNRS, Universite Paris Saclay, Bat. 508, rue John von Neumann, Campus Universitaire, F 91405 Orsay, France +DeFI Team, CMAP Lab (Ecole Polytechnique, Inria Saclay Ile de France) 1 rue Estienne d'Orves, 91120 Palaiseau, France

title: Lattice Boltzmann simulation of forced condensation flow on a horizontal cold surface in the presence of a non condensable gas
abstract: A multi component/multi phase (MCMP) lattice Boltzmann method (LBM) with vapor/liquid phase change is proposed in this paper. Two equations of state (EOS) including Peng Robinson (PR) EOS for water and ideal gas EOS for the non condensable gas (NCG) are applied in the thermal equation. Based on this newly developed MCMP phase change LB model, the problem of forced condensing flow on a horizontal cold plate at constant wall temperature in the presence of NCG is simulated. Effects of the NCG inlet fraction and plate subcooled temperature on forced film condensation under the same inlet velocity are simulated. Condensate film thickness, distributions of velocity/temperature/NCG fraction in the entire flow field, as well as condensation heat transfer on the cold plate are obtained numerically. The effect of an interfacial parameter (containing inlet NCG fraction and wall temperature) on film thickness is shown in good agreement with an existing analytical model, which validates the correctness and accuracy of this newly developed MCMP phase change LB model. Since this novel model does not involve any approximations/assumptions nor the use of empirical correlations for interfacial mass transfer, the results can be considered as the first attempt in the direct numerical simulation of laminar forced condensation heat transfer on a horizontal cold plate in the presence of a NCG.

title: Equation of state in relativistic magnetohydrodynamics: variable versus constant adiabatic index
abstract: The role of the equation of state (EoS) for a perfectly conducting, relativistic magnetized fluid is the main subject of this work. The ideal constant G law EoS, commonly adopted in a wide range of astrophysical applications, is compared with a more realistic EoS that better approximates the single specie relativistic gas. The paper focuses on three different topics. First, the influence of a more realistic EoS on the propagation of fast magnetosonic shocks is investigated. This calls into question the validity of the constant G law EoS in problems where the temperature of the gas substantially changes across hydromagnetic waves. Secondly, we present a new inversion scheme to recover primitive variables (such as rest mass density and pressure) from conservative ones that allows for a general EoS and avoids catastrophic numerical cancellations in the non relativistic and ultrarelativistic limits. Finally, selected numerical tests of astrophysical relevance (including magnetized accretion flows around Kerr black holes) are compared using different equations of state. Our main conclusion is that the choice of a realistic EoS can considerably bear upon the solution when transitions from cold to hot gas (or vice versa) are present. Under these circumstances, a polytropic EoS can significantly endanger the solution.

title: Analysis of p, T, r data for all fluid states via comparison with a transcribed Van der Waals equation
abstract: The old concepts of the Van der Waals equation (vdW EOS) and of the virial equation, truncated to the second virial coefficient, can be rewritten with a function defined as cc p R T M r 2 This function arises from introducing the ideal gas law into the thermodynamic equation of state, which describes the volume dependence of the internal energy. Thus both concepts can be superimposed and then complement one another. This allows an almost complete and correct description of the p/T/r behavior of real fluids in the gaseous, liquid, critical and supercritical state up to the Joule inversion curve. The function cc, used as structure parameter with constant values, reveals a hidden feature of the vdW EOS, which describes a linear function between density and temperature for iso cc lines. This is empirically correct for 16 examined regular pure fluids. Non regular fluids like He, H2, water and acetonitrile, which show more or less slightly curved iso cc lines, can be linearized using a simple factor, applied to the density, and then be described in a similar manner like regular fluids.

title: Particle Entropies and Entropy Quanta. I. The Ideal Gas
abstract: We consider an ideal gas of monatomic independent particles, which is enclosed in a cubic box. At temperature T the particles are in thermal equilibrium. All relevant properties of the gas can be deduced from the particle statistics on the assumption that each particle of the ensemble has the particle entropy s e/T ka. e is the translational energy of the particle. The non dimensional number a measures the particle entropy s in multiples of the Boltzmann constant k, which acts as an atomic entropy unit. a obeies an eigenvalue equation and satisfies boundary conditions. Eigenvalues and eigenfunctions are determined by the translational quantum numbers. Using particle entropies it is easy to calculate the Bose Einstein and the Boltzmann distribution; and in combination with the density function we immediately get the internal energy E and the Helmholtz free energy F, the total entropy S, the chemical potential m, the equation of state of the ideal gas at ordinary temperatures and at low temperatures near absolute zero, inclusively Bose Einstein condensation. Entropy quanta are used to introduce the temperature into the equations of statistical thermodynamics and to calculate the thermal and the actual de Broglie wavelength at temperature T.

title: Particle Entropies and Entropy Quanta: IV. The Ideal Gas, the Second Law of Thermodynamics, and the P t Uncertainty Relation
abstract: The quantum thermodynamics of the ideal gas is treated by means of the particle entropies s=ka. The non dimensional entropy numbers a are calculated by the extended, temperature dependent Schrodinger equation applying periodic boundary conditions. The Boltzmann constant k represents the atomic entropy unit. The quantized particle entropies a are used to calculate the Boltzmann distribution f=exp(a a) the density of entropy levels D(a) the dimensionless chemical potential a=m/kT, etc. The thermodynamics of the ideal gas is treated on the basis of entropy quanta by means of the density matrix formalism. In this way we obtain the total particle entropy A, the thermodynamic entropy S (equation of Sackur Tetrode) the internal and the Helmholtz free energy E and F, respectively, etc. Moreover, it can also be shown on the basis of entropy quanta that the entropy of a closed system in thermal equilibrium has a maximum value, S=Smax. Against this, the entropy S' of an arbitrary non equilibrium state is always smaller, S' Smax (second law of thermodynamics) Application of ordinary density matrix theory to a closed system of N particles leads to the result that the entropy S does not change in time, dS/dt=0, regardless whether we consider an equilibrium state, S=Smax, or a non equilibrium state, S=S' Consequently, S cannot irreversibly change from S' to Smax. However, any irreversible process is accompanied by a positive entropy production P=dS/dt>0. In order to overcome this obvious contradiction, we discuss the entropy evolution in time by means of the commutator equation G=i[P,t] which is deduced from very general assumptions. Here P and t are the operators of the entropy production P and the time t. Accordingly, P and t do not commute in general, and hence P and t are not sharply defined simultaneously. Instead we have uncertainties DP and Dt, which are expressed by the uncertainty relation of the N particle system, DPDt>(1/2)<G> (g/2)k. This P t uncertainty relation easily allows a discussion of the evolution of the entropy S in time t from S' to Smax. Now the irreversible steps are correctly described by the entropy production P=dS/dt 0, and the thermal equilibrium by P=0, P=0, and thus the lifetime t=