[Paper Review] When theories and experiments meet: rarefied gases as a benchmark of non-equilibrium thermodynamic models

Róbert Kovács, Patrizia Rogolino+1 more|arXiv (Cornell University)|Dec 4, 2019

Phase Equilibria and Thermodynamics58 references4 citations

TL;DR

This paper benchmarks Extended Irreversible Thermodynamics (EIT) and Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) by analyzing ultrasound propagation in rarefied gases, using experimental data from Meyer and Sessler. NET-IV provides a more accurate and general fit across the full frequency-pressure range than EIT, particularly by capturing the density-dependent speed of sound through phenomenological coefficients, while EIT relies on kinetic theory-based assumptions with fewer adjustable parameters but reduced precision in complex regimes.

ABSTRACT

The role of thermodynamics in deriving constitutive equations is unique, and various approaches have been developed in the last decades. In the present paper, the frameworks of Extended Irreversible Thermodynamics (EIT) and Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) are discussed and compared to each other on the basis of a particular problem of rarefied gases. In this comparison, both theoretical and experimental aspects are taken into account. Eventually, an experiment by Meyer and Sessler covering a wide range of pressures and frequencies is investigated. Here, concentrating on the scaling properties and the density dependence of parameters, the change of speed of sound in terms of frequency and pressure is recovered using NET-IV, and this fitting is compared to the results of Lebon and Cloot using EIT.

Motivation & Objective

• To compare the predictive accuracy of Extended Irreversible Thermodynamics (EIT) and Non-Equilibrium Thermodynamics with Internal Variables (NET-IV) in modeling ultrasound propagation in rarefied gases.
• To evaluate the performance of both frameworks against the experimental data of Meyer and Sessler, which shows significant frequency- and pressure-dependent changes in the speed of sound.
• To assess the role of phenomenological coefficients in NET-IV versus kinetic-theory-based assumptions in EIT, particularly regarding the fitting of density-dependent transport behavior.
• To determine whether NET-IV’s additional degrees of freedom improve modeling fidelity over EIT’s constrained parameter set, especially in high-frequency and low-pressure regimes.
• To examine the structural differences in the resulting generalized Navier-Stokes-Fourier equations, including hyperbolic versus parabolic character, and their implications for physical consistency.

Proposed method

• Theoretical derivation of generalized Navier-Stokes-Fourier equations using both EIT and NET-IV frameworks, with entropy production inequality as the central constraint.
• In EIT, heat flux and viscous pressure are treated as independent variables in the entropy density and flux, leading to hyperbolic or parabolic evolution equations depending on the entropy flux form.
• In NET-IV, internal variables such as dissipative fluxes are introduced phenomenologically, allowing for greater flexibility in modeling coupling and transport coefficients.
• The constitutive equations derived from both frameworks are applied to the experimental data of Meyer and Sessler on ultrasound propagation in rarefied gases across varying pressures and frequencies.
• Parameter fitting is performed using experimental scaling properties and density dependence of the speed of sound, with EIT relying on two fitted parameters (τd and λ3) and NET-IV using a broader set of phenomenological coefficients.
• The models are compared based on their ability to reproduce the observed frequency- and pressure-dependent speed of sound, with emphasis on the spherical part of the viscous pressure tensor in NET-IV.

Experimental results

Research questions

• RQ1 How do EIT and NET-IV compare in their ability to reproduce the experimentally observed frequency- and pressure-dependent speed of sound in rarefied gases?
• RQ2 What is the impact of using kinetic-theory-based coefficients (in EIT) versus phenomenological coefficients (in NET-IV) on the accuracy and generality of the model predictions?
• RQ3 Does the inclusion of the spherical part of the viscous pressure tensor in NET-IV improve the model’s performance across the entire range of frequency and pressure ratios compared to EIT?
• RQ4 How do the structural differences—hyperbolic vs. parabolic character—of the resulting evolution equations affect the physical consistency and predictive power of the models?
• RQ5 Can NET-IV provide a better fit than EIT without requiring a priori assumptions from kinetic theory, particularly in regimes where kinetic theory predictions may break down?

Key findings

• NET-IV provides a more accurate and general fit to the Meyer and Sessler experimental data across the entire range of frequency and pressure ratios compared to EIT.
• The improved fitting in NET-IV is attributed to its ability to account for the spherical part of the viscous pressure tensor, which is not fully captured in the standard EIT formulation.
• EIT’s reliance on two fitted parameters (τd and λ3) and kinetic-theory-based assumptions limits its precision in high-frequency and low-pressure regimes, where the experimental deviations are most pronounced.
• In contrast, NET-IV’s phenomenological coefficients allow for a richer structure and better adaptation to experimental scaling laws, particularly the density dependence of the speed of sound.
• The comparison shows that while EIT can reproduce experimental trends in certain f/p regions, NET-IV covers the full range without apparent restrictions, demonstrating its broader validity domain.
• The study confirms that both EIT and NET-IV can yield hyperbolic or parabolic systems depending on the entropy flux form, but NET-IV’s flexibility allows for a more accurate representation of complex transport phenomena without requiring strict kinetic-theory compatibility.