Polyatomic Gases Research Articles

Thermal conductivity prediction of Trans-1-Chloro-3,3,3-Trifluoropropene (R1233zd (E))

R1233zd(E), trans-1-chloro-3,3,3,-trifluoropropene, is a fluorinated propene isomer which may be considered as an alternative working fluids in the field of heat pump and organic Rankine cycle. R1233zd(E) has a much lower GWP than hydrochlorofluorocarbons (HCFC) and their mixtures. In this paper an extension of a previously developed predictive methods for thermal conductivity to a new family of organic compounds, namely R1233zd(E) is considered. A study of the correlation of thermal conductivities of polyatomic gases in the limit of zero density of R1233zd(E) is presented. A theorically correlation scheme based on the formalism Mason-Monchik-Parker theory has been examined and found to be useful for prediction of thermal conductivity data of dilute gas. An attempt is made in this work on the theoretical approach by Predvoditelev, Vargaftik and Filippov proposed for prediction of thermal conductivity for liquid state. The scheme has been tested against the limited amount of experimental data available and shown to be capable of reproducing the thermal conductivities to within few percent. This predicting approach appears promising as an assist in the judgments in the area of this new generation of working fluids data studies.

Multiscale simulation of molecular gas flows by the general synthetic iterative scheme

The in-depth knowledge of rarefied gas dynamics is crucial to address challenges in a wide range of engineering problems, where gas flows are usually multiscale, i.e., covering a wide range of Knudsen numbers. As the traditional Navier–Stokes equations fail, gas kinetic equations are required to model the flows. So far, very few numerical methods are designed to efficiently solve the multiscale gas dynamics and reveal the role of internal degrees of freedom of gas molecules. In this work, a general synthetic iterative scheme (GSIS) is proposed to find steady-state solutions of the gas kinetic equations for molecular gas flows accurately and efficiently, where the gas kinetic equations are solved together with the macroscopic synthetic equations that expedite solutions towards the steady state. In the macroscopic synthetic equations, while the momentum equation is the same as that used in the GSIS for monatomic gas, two energy equations are introduced here for polyatomic gases: one is for the translational energy and the other for the internal energy; these equations are derived exactly from the gas kinetic equations hence no approximation is made in final solutions. The Fourier stability analysis is performed to show that the GSIS permits fast convergence to steady-state solutions in the entire flow regime; meanwhile the asymptotic analysis shows that the GSIS recovers the Navier–Stokes equations when the Knudsen number is small, even on the spatial grid with cell size much larger than the molecular mean free path. With all these unique features, several challenging numerical examples are given to show that the proposed GSIS is a promising tool to simulate multiscale molecular gas flows and investigate the effects of internal degrees of freedom.

Effect of the internal degrees of freedom of the gas molecules on the heat and mass transfer in long circular capillaries

The study of the mass and heat transfer phenomena in micro-nano devices and systems (MEMS/NEMS) very often involves polyatomic gas flows in long capillaries. In the case of polyatomic gases, the internal degrees of freedom of gas molecules may have a significant effect on the macroscopic quantities of practical interest. The scope of the present study is two-fold. First, an accurate methodology for calculating the mass and heat flow rate coefficients is proposed based on a simple kinetic model proposed by Holway. The proposed methodology is based on the available experimental data in the literature and allows overcoming the well-known limitation of the Holway model to describe all the transport coefficients simultaneously. Second, the proposed methodology is applied to investigate the rarefied polyatomic gas flows through long micro- and nanotubes under temperature and pressure differences taking into account the translational, rotational and vibrational degrees of freedom of the gas molecules on the basis of the Holway kinetic model. Results are presented for N2, CO2, CH4 and SF6 representing linear and non-linear, but always non-polar molecules in a wide range of the gas temperature. The mass flow rates of the polyatomic gases in the temperature-driven flow differ significantly from the corresponding monatomic ones, with the difference reaching a maximum of about 25% for the examined gases and temperatures varied between 300 and 1000 K. The maximum relative deviation between monatomic and polyatomic heat flow rates can reach 39%, 64%, 73% and 87% for N2, CO2, CH4 and SF6, respectively. The results are compared against the corresponding published ones based on the Rykov model highlighting the very good agreement between the two models and the accuracy of the proposed methodology for the calculation of the mass and heat flow coefficients is confirmed.

On derivation and verification of a kinetic model for quantum vibrational energy of polyatomic gases in the gas-kinetic unified algorithm

In this paper, a kinetic Boltzmann model equation involving internal degrees of freedom is constructed for thermodynamic non-equilibrium polyatomic gases, in which the continuous distribution mode of rotational energy and discrete quantum effect of the vibrational energy are both considered, respectively. By theoretically analyzing the “recurrence feature” of this model, it is found that the root of the quantum effect comes from the translational-rotational-vibrational relaxation term (G3t,r,v), because of the additional term relative to the continuous distribution mode of vibrational energy. Besides, some important properties of this model are also analyzed and proved, including the conservative property and entropy inequality. Then, the balance equations on each vibrational energy level and balance equations about macroscopic variables are derived by the moment method. The corresponding thermodynamic relationship in the hydrodynamic limit is also analyzed by the simplified version of Chapman-Enskog method, and the correct Prandtl number is obtained. Numerical methods and wall surface boundary condition to solve this model equation are presented here. Finally, planar flows past a cylinder with different Mach numbers are numerically simulated to test and verify the reliability of this model, and to analyze the influence rules of the excited quantum vibrational energy for polyatomic gas flows.

Two-temperature Navier-Stokes equations for a polyatomic gas derived from kinetic theory.

A polyatomic gas with slow relaxation of the internal modes is considered, and the Navier-Stokes equations with two temperatures, the translational and internal temperatures, are derived for such a gas on the basis of the ellipsoidal-statistical (ES) model of the Boltzmann equation for a polyatomic gas, proposed by Andries etal. [Eur. J. Mech. B, Fluids 19, 813 (2000)10.1016/S0997-7546(00)01103-1], by the Chapman-Enskog procedure. Then, the derived equations are applied to numerically investigate the structure of a plane shock wave in CO_{2} gas, which is known to have slowly relaxing internal modes. The results show good agreement with those obtained by the direct numerical analysis of the ES model for moderately strong shock waves. In particular, the results perfectly reproduce the double-layer structure of the shock profiles consisting of a thin front layer with rapid change and a thick rear layer with slow relaxation of the internal modes.

Thermal transpiration in molecular gas

The thermal transpiration of molecular gas is investigated based on the model of Wu et al. [“A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases,” J. Fluid Mech. 763, 24–50 (2015)], which is solved by a synthetic iterative scheme efficiently and accurately. A detailed investigation of the thermal slip coefficient, Knudsen layer function, and mass flow rate for molecular gas interacting with the inverse power-law potential is performed. It is found that (i) the thermal slip coefficient and Knudsen layer function increase with the viscosity index determined by the intermolecular potential. Therefore, at small Knudsen number, gas with a larger viscosity index has a larger mass flow rate; however, at late transition and free molecular flow regimes, this is reversed. (ii) The thermal slip coefficient is a linear function of the accommodation coefficient in Maxwell’s diffuse–specular boundary condition, while its variation with the tangential momentum accommodation coefficient is complicated in Cercignani–Lampis’s boundary condition. (iii) The ratio of the thermal slip coefficients between monatomic and molecular gases is roughly the ratio of their translational Eucken factors, and thus, molecular gas always has a lower normalized mass flow rate than monatomic gas. (iv) In the transition flow regime, the translational Eucken factor continues to affect the mass flow rate of thermal transpiration, but in the free molecular flow regime, the mass flow rate converges to that of monatomic gas. Based on these results, accommodation coefficients were extracted from thermal transpiration experiments of air and carbon dioxide, which are found to be 0.9 and 0.85, respectively, rather than unity used in the literature. The methodology and data presented in this paper are useful, e.g., in the pressure correction of capacitance diaphragm gauge when measuring low gas pressures.

Testing a Mathematical Model of the Radiative Heat Transfer of Polyatomic Gases in Vacuum Hydrocarbon Dehydrogenation Plants

Testing a Mathematical Model of the Radiative Heat Transfer of Polyatomic Gases in Vacuum Hydrocarbon Dehydrogenation Plants

Efficient supersonic flow simulations using lattice Boltzmann methods based on numerical equilibria.

A double-distribution-function based lattice Boltzmann method (DDF-LBM) is proposed for the simulation of polyatomic gases in the supersonic regime. The model relies on a numerical equilibrium that has been extensively used by discrete velocity methods since the late 1990s. Here, it is extended to reproduce an arbitrary number of moments of the Maxwell–Boltzmann distribution. These extensions to the standard 5-constraint (mass, momentum and energy) approach lead to the correct simulation of thermal, compressible flows with only 39 discrete velocities in 3D. The stability of this BGK-LBM is reinforced by relying on Knudsen-number-dependent relaxation times that are computed analytically. Hence, high Reynolds-number, supersonic flows can be simulated in an efficient and elegant manner. While the 1D Riemann problem shows the ability of the proposed approach to handle discontinuities in the zero-viscosity limit, the simulation of the supersonic flow past a NACA0012 aerofoil confirms the excellent behaviour of this model in a low-viscosity and supersonic regime. The flow past a sphere is further simulated to investigate the 3D behaviour of our model in the low-viscosity supersonic regime. The proposed model is shown to be substantially more efficient than the previous 5-moment D3Q343 DDF-LBM for both CPU and GPU architectures. It then opens up a whole new world of compressible flow applications that can be realistically tackled with a purely LB approach.This article is part of the theme issue ‘Fluid dynamics, soft matter and complex systems: recent results and new methods’.

Master equation approach for modeling diatomic gas flows with a kinetic Fokker-Planck algorithm

In recent years the kinetic Fokker-Planck approach for modeling gas flows has become increasingly popular. In the Fokker-Planck ansatz the collision integral of the Boltzmann equation is approximated by a Fokker-Planck operator in velocity space. Instead of solving the resulting Fokker-Planck equation directly, the underlying random process is modeled, which leads to an efficient stochastic solution algorithm. Despite the attention to the Fokker-Planck ansatz, the modeling of polyatomic gases has been addressed only in a few works. In this paper a scheme is presented to extend arbitrary monatomic Fokker-Planck models to model polyatomic species. A master equation approach is used to model internal energy relaxation, but instead of solving the master equation directly, the underlying random process is simulated. Three different models are suggested to describe internal particle energies as continuous scalars or as a set of discrete energy levels. The proposed models are applied on different test cases to demonstrate their accuracy. Within the bounds of expectations, a very good agreement with reference DSMC simulations is achieved.

Hyperbolicity of first and second order extended thermodynamics theory of polyatomic rarefied gases

The balance laws of Rational Extended Thermodynamics describe well the evolution of rarefied gases in non-equilibrium. Usually, it is necessary to approximate the theory in a neighborhood of an equilibrium state and consequently, its hyperbolicity property remains valid only in a neighborhood of the equilibrium state of the field variables, called hyperbolicity region. The goal of this paper is first to determine the differential system with 14 fields for a rarefied polyatomic polytropic gas, approximated at the second-order in the non-equilibrium variables. Then, we investigate and compare the hyperbolicity property of the first-order and second-order systems. In particular, we analyze the role played by the dynamic pressure and the molecular degrees of freedom. Finally, we also show that in the monatomic singular limit the quadratic theory for a polyatomic gas converges to the corresponding quadratic theory for a monatomic gas.

Shock structure in extended thermodynamics with second-order maximum entropy principle closure

An investigation on the features of the shock structure solution of the 13-moment system of extended thermodynamics with a second-order closure based on the maximum entropy principle is presented. The results are compared to those obtained by means of the traditional first-order closure and to those obtained in the framework of kinetic theory by solving the Boltzmann equation with a BGK model for the collision term. It is seen that when adopting a second-order closure, the strength of the subshock that appears in the shock structure profile for large enough Mach numbers is remarkably reduced with respect to what is found with the first-order closure, and the overall profile of the shock structure solution is in better agreement with the results obtained with the kinetic theory approach. The analysis is extended to the case of the 14-moment system of a polyatomic gas, and some preliminary results are presented also for this case.

New Thermodynamics: Rethinking the Science of Climate Change

Statistical analysis shows that climate change is due to human activities. The accepted reason being the greenhouse effect, which is based on the erroneous assumption that homonuclear gases are opaque to thermal energy. The reality is that all polyatomic gases absorb and then radially radiate thermal energy, as proven by their heat capacities. The greenhouse effect, then becomes secondary. Thermal energy generated by human activities is part of Earth’s anthroposphere which is where climate change is measured. Such human-generated thermal energy is absorbed by all our atmosphere’s polyatomic gases, which is then radially radiated by those very gases. Therefore, our whole atmosphere forms Earth’s thermal blanket, and human’s activities becomes the root cause of climate change. Our Sun’s influx starts outside of Earth’s thermal blanket, with much of its visible light reaching Earth’s surface. However, the majority of the Sun’s insolation is at thermal wavelengths, which are readily absorbed by Earth’s atmosphere, and whose energy are then radially radiated by its polyatomic gases.

New Thermodynamics: Rethinking the Science of Climate Change

Statistical analysis shows that climate change is due to human activities. The accepted reason being the greenhouse effect, which is based on the erroneous assumption that homonuclear gases are opaque to thermal energy. The reality is that all polyatomic gases absorb and then radially radiate thermal energy, as proven by their heat capacities. The greenhouse effect, then becomes secondary. Thermal energy generated by human activities is part of Earth’s anthroposphere which is where climate change is measured. Such human-generated thermal energy is absorbed by all our atmosphere’s polyatomic gases, which is then radially radiated by those very gases. Therefore, our whole atmosphere forms Earth’s thermal blanket, and human’s activities becomes the root cause of climate change. Our Sun’s influx starts outside of Earth’s thermal blanket, with much of its visible light reaching Earth’s surface. However, the majority of the Sun’s insolation is at thermal wavelengths, which are readily absorbed by Earth’s atmosphere, and whose energy are then radially radiated by its polyatomic gases.

Viscous multi-species lattice Boltzmann solver for simulating shock-wave structure

The Onsager Bhatnagar Gross Krook kinetic model proposed by Mahendra et al. [32, 33] has been extended to arrive at the viscous compressible flow equations for a multi-species diffusion problem. The multi-species model ensures (i) correct exchange coefficients, (ii) positivity, (iii) that it follows indifferentiability principle and (iv) that correct amount of entropy get added as per the Onsager’s maximum entropy production principle (MEPP). A Lattice Boltzmann method (LBM) has been developed using the derived methodology within the framework of finite volume method (FVM). This solver incorporates the physics of polyatomic gases with internal degrees of freedom with flexible Prandl number. This solver can simulate the problem with species having different specific heat ratio and molecular weights. Although the equations are derived for multi-species, only binary test cases have been considered in this paper. The solver has been validated against binary shock-wave structure problems using experimental and theoretical data. Two dimensional Richtmyer Meshkov instability test case has been compared with and without viscous effects. The diffusive (viscous and mass diffusion) effects are manifested in the reduction in amplitude and increment in width of the mushroom structure. In the framework of FVM, the proposed scheme of LBM is efficient with good resolution of solutions and is easily adaptable.

Effect of the dynamic pressure on the similarity solution of cylindrical shock waves in a rarefied polyatomic gas

The similarity solution for a strong cylindrical shock wave in a rarefied polyatomic gas is analyzed on the basis of Rational Extended Thermodynamics with six independent fields; the mass density, the velocity, the pressure and the dynamic pressure. A new ODE system for the similarity solution is derived in a systematic way by using the method based on the Lie group theory proposed in the context of the spherical shock wave in a rarefied monoatomic gas in Donato and Ruggeri (J Math Anal Appl 251:395, 2000). The boundary conditions are also specified from the Rankine–Hugoniot conditions for the sub-shock. The derived similarity solution is characterized by only one dimensionless parameter $$\alpha $$ related to the relaxation time for the dynamic pressure. The numerical analysis of the similarity solution is also performed. The solution agrees with the well-known Sedov–von Neumann–Taylor (SNT) solution when $$\alpha $$ is small. When $$\alpha $$ is larger, due to the presence of the dynamic pressure, the deviation from the SNT solution is evident; the strength of a peak near the shock front becomes smaller and the profile becomes broader.

Maximum entropy principle closure for 14-moment system for a non-polytropic gas

In this paper, we consider a rarefied polyatomic gas with a non-polytropic equation of state. We use the variational procedure of maximum entropy principle (MEP) to obtain the closure of the binary hierarchy of 14 moments associated with the Boltzmann equation in which the distribution function depends also on the energy of internal modes. The closed partial differential system is symmetric hyperbolic and the Cauchy problem is well-posed. In the limiting case of polytropic gas in which the internal energy is a linear function of the temperature, we find, as a special case, the previous results of Pavic et al. (Physica A 392:1302–1317, 2013). This paper, therefore, completes the equivalence between the closure obtained in the phenomenological rational extended thermodynamics theory and the one obtained by the MEP for general non-polytropic gas.

Rational extended thermodynamics of dense polyatomic gases incorporating molecular rotation and vibration.

The paper aims to construct a rational extended thermodynamics (RET) theory of dense polyatomic gases by taking into account the experimental evidence that the relaxation time of molecular rotation and that of molecular vibration are quite different from each other. For simplicity, we focus on gases with only one dissipative process due to bulk viscosity. In fact, in some polyatomic gases, the effect of bulk viscosity is much larger than that of shear viscosity and heat conductivity. The present theory includes the previous RET theory of dense gases with six fields as a particular case, and it also includes the RET theory of rarefied polyatomic gases with seven fields in the rarefied-gas limit. The closure is carried out by using the universal principles, that is, Galilean invariance and objectivity, entropy principle, and thermodynamic stability (convexity of entropy), where the duality principle connecting rarefied gases to dense gases also plays an important role. A detailed discussion is devoted to the expression of the production terms in the system of balance equations. As typical examples, we study a gas with virial equations of state and a van der Waals gas. Lastly the dispersion relation of a linear wave is derived, and its comparison with experimental data is made. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Classical Limit of Relativistic Moments Associated with Boltzmann–Chernikov Equation: Optimal Choice of Moments in Classical Theory

The moment system associated with the Boltzmann equation is largely used in many applications and is the main ingredient of Rational Extended Thermodynamics (RET). The choice of truncation of moments is arbitrary and many possibilities are present in the literature depending in particular on many contracted indexes in each moment tensor considered. Moreover, in a polyatomic gas, we have two hierarchies of moments with different indexes of truncation. As any classical theory can be considered as a limiting case of a relativistic one, we study in this paper the moments associated with the relativistic Boltzmann–Chernikov equation until the index of truncation N. We take the classical limit for both polyatomic and monatomic rarefied gases and prove that there exists only unique possible choice of the moments in the classical case for a given N.

A BGK model for mixtures of monoatomic and polyatomic gases with discrete internal energy

We generalize the BGK model proposed in Bisi and Cáceres (2016) to a mixture constituted by both monoatomic and polyatomic gas species with each polyatomic one being characterized by its own number of discrete internal energy levels, to model its non-translational degrees of freedom. We prove that all disposable parameters appearing in the Maxwellian attractors may be determined in terms of the actual macroscopic fields in such a way that correct Maxwellian equilibria and collision invariants are preserved, as well as the validity of the H-theorem. Evolution equations for species densities, velocities and temperatures are also derived, and some numerical examples are shown in space homogeneous conditions.

Topology of the second-order constitutive model based on the Boltzmann–Curtiss kinetic equation for diatomic and polyatomic gases

The topological aspects of fluid flows have long been fascinating subjects in the study of the physics of fluids. In this study, the topology of the second-order Boltzmann–Curtiss constitutive model beyond the conventional Navier–Stokes–Fourier equations and Stokes’s hypothesis was investigated. In the case of velocity shear, the topology of the second-order constitutive model was shown to be governed by a simple algebraic form. The bulk viscosity ratio in diatomic and polyatomic gases was found to play an essential role in determining the type of topology: from an ellipse to a circle, to a parabola, and then finally to a hyperbola. The topology identified in the model has also been echoed in other branches of science, notably in the orbits of planets and comets and Dirac cones found in electronic band structures of two-dimensional materials. The ultimate origin of the existence of the topology was traced to the coupling of viscous stress and velocity gradient and its subtle interplay with the bulk viscosity ratio. In the case of compression and expansion, the topology of the second-order constitutive model was also found to be governed by a hyperbola. The trajectories of solutions of two representative flow problems—a force-driven Poiseuille gas flow and the inner structure of shock waves—were then plotted on the topology of the constitutive model, demonstrating the indispensable role of the topology of the constitutive model in fluid dynamics.