Nonequilibrium liquid-vapor interfaces: Linear and nonlinear descriptions.

While it is often assumed that liquid-vapor interfaces in nonequilibrium processes are in states of local thermodynamic equilibrium, this might not be the case for strong deviations from equilibrium. Clausius-Clapeyron equations for bulk properties yield a consistently defined temperature of the interface that is close to the liquid bulk temperature. The alternative interface temperature defined through the surface tension will be different for stronger nonequilibrium processes. Structural variables are introduced to extend the thermodynamic description of interfaces to a wider range of processes. Interfacial resistivities will depend on interface temperature as well as mass and heat flux through the interface.

Entropy and the Second Law of Thermodynamics-The Nonequilibrium Perspective.

An alternative to the Carnot-Clausius approach for introducing entropy and the second law of thermodynamics is outlined that establishes entropy as a nonequilibrium property from the onset. Five simple observations lead to entropy for nonequilibrium and equilibrium states, and its balance. Thermodynamic temperature is identified, its positivity follows from the stability of the rest state. It is shown that the equations of engineering thermodynamics are valid for the case of local thermodynamic equilibrium, with inhomogeneous states. The main findings are accompanied by examples and additional discussion to firmly imbed classical and engineering thermodynamics into nonequilibrium thermodynamics.

Formulation of moment equations for rarefied gases within two frameworks of non-equilibrium thermodynamics: RET and GENERIC.

In this work, we make a further step in bringing together different approaches to non-equilibrium thermodynamics. The structure of the moment hierarchy derived from the Boltzmann equation is at the heart of rational extended thermodynamics (RET, developed by Ingo Müller and Tommaso Ruggeri). Whereas the full moment hierarchy has the structure expressed in the general equation for the nonequilibrium reversible-irreversible coup- ling (GENERIC), the Poisson bracket structure of reversible dynamics postulated in that approach is a major obstacle for truncating moment hierarchies, which seems to work only in exceptional cases (most importantly, for the five moments associated with conservation laws). The practical importance of truncated moment hierarchies in rarefied gas dynamics and microfluidics motivates us to develop a new strategy for establishing the full GENERIC structure of truncated moment equations, based on non-entropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of 10 moments. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Efficiencies and Work Losses for Cycles Interacting with Reservoirs of Apparent Negative Temperatures.

Inverted quantum states of apparent negative temperature store the work required for their creation [Struchtrup. Phys. Rev. Lett. 2018, 120, 250602]. Thermodynamic cycles operating between a classical reservoir and an inverted state reservoir seem to have thermal efficiencies at or even above unity. These high efficiencies result from inappropriate definition adopted from classical heat engines. A properly defined efficiency compares the work produced in the cycle to the work expended in creating the reservoir. Due to work loss to irreversible processes, this work storage based efficiency always has values below unity.

Work Storage in States of Apparent Negative Thermodynamic Temperature.

Inverted quantum states provide a challenge to classical thermodynamics, since they appear to contradict the classical formulation of the second law of thermodynamics. Ramsey interpreted these states as stable equilibrium states of negative thermodynamic temperature, and added a provision to allow these states to the Kelvin-Planck statement of the second law [N. F. Ramsey, Phys. Rev. 103, 20 (1956)PHRVAO0031-899X10.1103/PhysRev.103.20]. Since then, Ramsey's interpretation has prevailed in the literature. Here, we present an alternative option to accommodate inverted states within thermodynamics, which strictly enforces the original Kelvin-Planck statement of the second law, and reconciles inverted states and the second law by interpreting the former as unstable states, for which no temperature-positive or negative-can be defined. Specifically, we recognize inverted quantum states as temperature-unstable states, for which all processes are in agreement with the original Kelvin-Planck statement of the second law, and positive thermodynamic temperatures in stable equilibrium states. These temperature-unstable states can only be created by work done to the system, which is stored as energy in the unstable states, and can be released as work again, just as in a battery or a spring.

Evaporation Boundary Conditions for the Linear R13 Equations Based on the Onsager Theory.

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier-Stokes-Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed.

Coupled constitutive relations: a second law based higher-order closure for hydrodynamics.

In the classical framework, the Navier-Stokes-Fourier equations are obtained through the linear uncoupled thermodynamic force-flux relations which guarantee the non-negativity of the entropy production. However, the conventional thermodynamic descrip- tion is only valid when the Knudsen number is sufficiently small. Here, it is shown that the range of validity of the Navier-Stokes-Fourier equations can be extended by incorporating the nonlinear coupling among the thermodynamic forces and fluxes. The resulting system of conservation laws closed with the coupled constitutive relations is able to describe many interesting rarefaction effects, such as Knudsen paradox, transpiration flows, thermal stress, heat flux without temperature gradients, etc., which cannot be predicted by the classical Navier-Stokes-Fourier equations. For this system of equations, a set of phenomenological boundary conditions, which respect the second law of thermodynamics, is also derived. Some of the benchmark problems in fluid mechanics are studied to show the applicability of the derived equations and boundary conditions.

Temperature-difference-driven mass transfer through the vapor from a cold to a warm liquid.

Irreversible thermodynamics provides interface conditions that yield temperature and chemical potential jumps at phase boundaries. The interfacial jumps allow unexpected transport phenomena, such as the inverted temperature profile [Pao, Phys. Fluids 14, 306 (1971)] and mass transfer from a cold to a warm liquid driven by a temperature difference across the vapor phase [Mills and Phillips, Chem. Phys. Lett. 372, 615 (2002)]. Careful evaluation of the thermodynamic laws has shown [Bedeaux et al., Physica A 169, 263 (1990)] that the inverted temperature profile is observed for processes with a high heat of vaporization. In this paper, we show that cold to warm mass transfer through the vapor from a cold to a warm liquid is only possible when the heat of evaporation is sufficiently small. A necessary criterium for the size of the mass transfer coefficient is given.

Inconsistency of a dissipative contribution to the mass flux in hydrodynamics.

The possibility of dissipative contributions to the mass flux is considered in detail. A general thermodynamically consistent framework is developed to obtain such terms, the compatibility of which with general principles is then checked-including Galilean invariance, the possibility of steady rigid rotation and uniform center-of-mass motion, the existence of a locally conserved angular momentum, and material objectivity. All previously discussed scenarios of dissipative mass fluxes are found to be ruled out by some combinations of these principles but not a new one that includes a smoothed velocity field v[over ] . However, this field v[over ] is nonlocal and leads to unacceptable consequences in specific situations. Hence, we can state with confidence that a dissipative contribution to the mass flux is not possible.

Effects of rarefaction in microflows between coaxial cylinders.

Microscale gas flows between two rotating coaxial circular cylinders of infinite length with different temperatures are investigated. Navier-Stokes-Fourier (NSF) and regularized 13-moment (R13) equations in their linear form are used to independently analyze velocity and temperature fields in shear-driven rotary flows, i.e., cylindrical Couette flows. Knudsen boundary layers, which present non-Newtonian stress and non-Fourier heat flow, are predicted as the dominant rarefaction effects in the linear theory. We show that the R13 system yields more accurate results for this boundary value problem by predicting the Knudsen boundary layers, which are not accessible for NSF equations. Furthermore, a set of second-order boundary conditions for velocity slip and temperature jump are derived for the NSF system. It is shown that the proposed boundary conditions effectively improve the classical hydrodynamics. The accuracy of NSF and R13 equations is discussed based on their comparison with available direct simulation Monte Carlo data.

Higher-order effects in rarefied channel flows.

The regularized 13 moment (R13) equations and their boundary conditions are considered for plane channel flows. Chapman-Enskog scaling based on the Knudsen number is used to reduce the equations. The reduced equations yield second-order slip conditions, and allow us to describe the characteristic dip in the temperature profile observed in force driven flow. Due to the scaling, the R13 equations' ability to describe Knudsen layers is lost. Solutions with Knudsen layers are discussed as well, and it is shown that these give a better match to direct solutions of the Boltzmann equations than the reduced equations without Knudsen layers. For a radiatively heated gas the R13 equations predict a dependence of the average gas temperature on the Knudsen number with a distinct minimum around Kn = 0.2 , similar to the well-known Knudsen minimum for Poiseuille flow.

H theorem, regularization, and boundary conditions for linearized 13 moment equations.

An H theorem for the linearized Grad 13 moment equations leads to regularizing constitutive equations for higher fluxes and to a complete set of boundary conditions. Solutions for Couette and Poiseuille flows show good agreement with direct simulation Monte Carlo calculations. The Knudsen minimum for the relative mass flow rate is reproduced.

Mean evaporation and condensation coefficients based on energy dependent condensation probability.

A generalization of the classical Hertz-Knudsen and Schrage laws for the evaporation mass and energy fluxes at a liquid-vapor interface is derived from kinetic theory and a simple model for a velocity dependent condensation coefficient. These expressions, as well as the classical laws and simple phenomenological expressions, are then considered for the simulation of recent experiments [Phys. Rev. E 59, 419 (1999)]]. It is shown that mean condensation and evaporation coefficients in the mass flow influence the results only if they are small compared to unity and that the expression for evaporation mass flow determines the temperature of the liquid. Moreover, it is shown that the expression for evaporation energy flow plays the leading role in determining the interface temperature jump, which can be obtained in good agreement with the experiment from the generalized kinetic theory model and phenomenological approaches, but not from the classical kinetic-theory-based Hertz-Knudsen and Schrage laws. Analytical estimates show that the interface temperature jump depends strongly on the temperature gradient of the vapor just in front of the interface, which explains why much larger temperature jumps are observed in spherical geometry and the experiments as compared to planar settings.

Heat transfer in the transition regime: solution of boundary value problems for Grad's moment equations via kinetic schemes.

This paper presents a systematic approach to the calculation of heat transfer in rarefied gases (Knudsen numbers between 0.01 and 1) by means of Grad's moment method with high moment numbers, based on the Boltzmann equation with linearized collision term. The problem of describing boundary conditions for the moments is solved by the use of the so-called kinetic schemes that allow the implementation of the boundary condition for the Boltzmann equation. The results, obtained with up to 48 one-dimensional moment equations, exhibit temperature jumps at the walls with adjacent Knudsen boundary layers. For given wall temperatures and Knudsen number, the results change with the number of moments, and converge if the number of moments is increased.