Thermodynamics and Energy Conversion
Springer, Heidelberg 2014
597 pages
ISBN: 978-3-662-43714-8
doi: 10.1007/978-3-662-43715-5

My lectures on Thermodynamics and Energy Conversion are now a book (597 pages, good for 2 or 3 courses). If you access through a subscribing library, you can download the e-book for free through SpringerLink, where you can also buy a printed e-book for USD 25.

 





  

Macroscopic Transport Equations for Rarefied Gas Flows
Approximation Methods in Kinetic Theory
Springer Series Interaction of Mechanics and Mathematics
Springer, Heidelberg 2005
258 pages, ISBN: 3-540-24542-1
doi:10.1007/3-540-32386-4

H. Struchtrup, A. Frezzotti: Twenty-six moment equations for the Enskog–Vlasov equation
J. Fluid Mechanics 940, A40 (2022) [pdf] [doi:10.1017/jfm.2022.98]

The Enskog–Vlasov equation is a phenomenological kinetic equation that extends the Enskog equation for the dense (non-ideal) hard-sphere fluid by adding an attractive soft potential tail to the purely repulsive hard-sphere contribution. Simplifying assumptions about pair correlations lead to a Vlasov-like self-consistent force field that adds to the Enskog non-local hard-sphere collision integral. Within the limitations imposed by the underlying assumptions, the extension gives the Enskog–Vlasov equation the ability to give a unified description of ideal and non-ideal fluid flows as well as of those fluid states in which liquid and vapour regions coexist, being separated by a resolved interface. Furthermore, the Enskog–Vlasov fluid can be arbitrarily far from equilibrium. Thus the Enskog–Vlasov model equation provides an excellent, although approximate, tool for modelling processes with liquid–vapour interfaces and adjacent Knudsen layers, and allows us to look at slip, jump and evaporation coefficients from a different perspective. Here, a set of 26 moment equations is derived from the Enskog–Vlasov equation by means of the Grad moment method. The equations provide a meaningful approximation to the underlying kinetic equation, and include the description of Knudsen layers. This work focuses on the – rather involved – derivation of the moment equations, with only a few applications shown.

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R. Long-Innes, H. Struchtrup: Thermodynamic loss analysis of a liquid-sorbent direct air carbon capture plant
Cell Reports Physical Science 3(3), 100791 (2022)   [pdf][doi:10.1016/j.xcrp.2022.100791]

Direct air capture of CO2 is often presented as a promising technology to help mitigate climate change, although proposed processes are highly energy intensive. We analyze Carbon Engineering’s 1 Mt-CO2/year natural-gas-powered direct air capture (DAC) process, which requires 273.2 MW per plant, where we find that 252 MW are irreversibly lost, corresponding to a second-law efficiency of 7.8%. Our component-level analysis details the mechanisms by which these losses of thermodynamic work potential occur in the most energy-intensive plant segments. Here, we emphasize the effects of chemical exergy dissipation in the air contactor, where stored chemical exergy is released as low-grade heat into the environment. Other major losses occur in the calciner and its preheat cyclones due to the high temperature demanded by its internal chemical reaction, as well as in the water knockout system, CO2 compression system, and power island. Finally, we illustrate the issues arising from the use of natural gas as a feedstock for heat and power, and suggest directions to pursue for further analysis and process improvements, which we consider imperative to make this DAC process a viable option for large-scale CO2 removal toward IPCC targets.

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H. Struchtrup, H.C. Öttinger: Thermodynamically admissible 13-moment equations
Phys. Fluids 34, 017105 (2022)  [pdf] [doi:10.1063/5.0078780]

Grad's 13-moment equations describe transport in mildly rarefied gases well, but are not properly embedded into nonequilibrium thermodynamics since they are not accompanied by a formulation of the second law. In this work, the Grad-13 equations are embedded into the framework of GENERIC (general equation for the nonequilibrium reversible–irreversible coupling), which demands additional contributions in the equations to guarantee thermodynamic structure. As GENERIC building blocks, we use a Poisson matrix for the basic convection behavior and antisymmetric friction matrices to correct for additional convective transport terms. The ensuing GENERIC-13 equations completely match the Grad-13 equations up to second-order terms in the Knudsen number and fulfill all thermodynamic requirements.

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H. Struchtrup: Entropy and the Second Law of Thermodynamics: The Nonequilibrium Perspective
entropy 22, 793 (2020)  [pdf] [doi:10.3390/e22070793]

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.

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H. Struchtrup, B. Nadler: Are waves with negative spatial damping unstable?
Wave Motion 97, 102612 (2020)  [pdf] [doi:10.1016/j.wavemoti.2020.102612]

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.

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H.C. Oettinger, H. Struchtrup, M. Torrilhon: Formulation of moment equations for rarefied gases within two frameworks of nonequilibrium thermodynamics: RET and GENERIC
Philosophical Transactions A 378, 20190174 (2020)  [pdf] [doi:10.1098/rsta.2019.0174]

In this work, we make a further step in bringing together different approaches to nonequilibrium 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 coupling (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 nonentropy-producing irreversible processes associated with Casimir symmetry. Detailed results are given for the special case of ten moments.

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H. Jiang, L. Xu, H. Struchtrup, J. Li, Q. Gan, X. Xu, Z. Hu, M. Ouyang: Modeling of Fuel Cell Cold Start and Dimension Reduction Simplification Methods
J. Electrochem. Soc. 167, 044501 (2020) [pdf] [doi:10.1149/1945-7111/ab6ee7]

Sub-zero startup ability remains a key barrier for commercial application of polymer electrolyte fuel cells (PEMFC), especially for automotive applications. In order to improve the startup ability and durability of fuel cells, understanding of the characteristics and mechanisms of cold start is essential, and here modeling of fuel cell cold start plays an important role. In this study, a one-dimensional model is developed to simulate the fuel cell cold start. The model includes mass transport and phase change, heat transfer and electrochemical reaction. Key features such as membrane water and local current distributions are analyzed. Based on the one-dimensional model and simulation results, a spatial reduced simplified model is developed that distinguished only n states across the cell. The simplified model inherits the key features of the one-dimensional model, while the computational cost is significantly reduced to 10% (from 216 s to 20.88 s). The one-dimensional model and simplified model are both validated by the cold start experiment and the voltage error and temperature error are within 15% and 1.2 K respectively. Thus, the proposed simplified model could be used in dynamic simulation and in further multi-scale modeling study to build a stack model..

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H. Struchtrup and A. Frezzotti: Grad's 13 moments approximation for Enskog-Vlasov equations
31st International Symposium on Rarefied Gas Dynamics
AIP Conf. Proc. 2132, 120007 (2019)  [pdf] [doi: 10.1063/1.5119620]

Hydrodynamic models of liquid-vapor flows have to face the difficulty of describing non-equilibrium regions next to interfaces. Depending on the flow regimes and the underlying theoretical models, different answers have been given. In particular, diffuse interface models (DIMs) provide, in principle, a unified description of the whole flow field by a set of PDE’s, not much more complex than Navier-Stokes-Fourier classical equations. Unfortunately, DIMs fail to provide a proper description of kinetic layers next to interfaces. In order to develop a model incorporating kinetic effects while keeping the relative simplicity of DIMs, macroscopic transport equations—moment equations—are derived from the Enskog-Vlasov equation. The Enskog-Vlasov equation extends the Enskog equation by accounting for the attractive forces between the gas molecules. Hence, it gives a van-der-Waals-like kinetic description of a non-ideal gas, including liquid-vapor phase change. Specifically, the equation describes the liquid phase, the vapor phase, and a diffusive transition region connecting both phases. While not the most accurate model, solutions of the Enskog-Vlasov equation exhibits all relevant phenomena occurring in the evaporation and condensation of rarefied or dense vapors. In this work, Grad’s moment method is used to derive a closed set of 13 moment equations. In the appropriate limits, these reduce to the Navier-Stokes-Fourier system for liquid and vapor. Our main interest is to study non-hydrodynamic effects, in particular transport in and across the transition region, and the interplay between the transition region and Knudsen layers. We present first results of this program, including the closed transport equations for 13 moments, discussion of the limits, and solutions in simple geometries.

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M. Timokhin, H. Struchtrup, A. Kokhanchik, and Ye. Bondar: R13 moment equations applied to supersonic flow with solid wall interaction
31st International Symposium on Rarefied Gas Dynamics
AIP Conf. Proc. 2132, 120001 (2019)  [pdf] [doi: 10.1063/1.5119614]

This paper studies the applicability of various versions of the regularized 13-moment system (R13) as applied to the problem of the shock wave structure in a monatomic Maxwell gas in a wide range of Mach numbers (1.0<M<8.0). Over time, several versions of the R13 equations were presented, which differ in non-linear contributions for high-order moments. The challenge of this study is to determine the range of applicability of each variant of the moment equations as applied to non-equilibrium supersonic flows, depending on the Mach number and local Knudsen number. Numerical results obtained for the R13 system are compared to DSMC data computed by the SMILE++ software system.

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H. Struchtrup: Efficiencies and Work Losses for Cycles Interacting with Reservoirs of Apparent Negative Temperatures
entropy 21, 749 (2019)  [pdf]  [doi:10.3390/e21080749]

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.

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R. Soltani and H. Struchtrup: Modeling and Simulation of the Dual Stage Pressure Retarded Osmosis Systems
Desalination 460, 28-40 (2019) [pdf] [doi:10.1016/j.desal.2019.02.010]

Utilization of renewable energy sources, as an approach to reduce Greenhouse Gas (GHG) Emissions, have been globally popular in the last few decades. Among renewable energy sources, pressure retarded Osmosis (PRO) has been scrutinized by scientists since the mid 70's. However, even today, the existing PRO systems can only marginally meet the generally approved criterion of 5 W/m2 power density, a threshold for an economically feasible river-sea PRO system. As an approach to increase the performance of PRO systems, multi-staging of PRO modules are investigated in this paper.
For this purpose, a number of models for scaled up dual stage PRO power plants with different configurations and target functions are presented and compared to a single stage system with the same membrane area. These models consider the pressure and flow drop as well as the salinity change along the membrane. The results indicate that overall performance of the system could
be improved by up to 8 % with a dual stage PRO in the case of specific energy. Finally, a thermodynamic analysis addresses the sources for irreversible losses, and the contribution from each source.

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F. Di Michele, E. Felaco, I. Gasser, A. Serbinovskiy, and H. Struchtrup: Modeling, simulation and optimization of a pressure retarded osmosis power station
Appl. Math. Comp. 353, 189-207 (2019) [pdf] [doi:10.1016/j.amc.2019.01.046]

Pressure retarded osmosis (PRO) power plants generate power from mixing of saltwater and freshwater by means of membrane systems. In this paper we present a model which describes the complete power station, suitable to optimize the power station both with respect to system parameters and in operating conditions. Special attention is dedicated to the flow model of the “core” membrane unit. It considers the relevant water and salt flows in the system. It also accounts for irreversible losses in the flow across the membrane as well as through the membrane unit, and in the surrounding pump-turbine system. The model represents a compromise between needed complexity (including the most relevant chemophysics) and simplicity to allow rapid simulations which is an important prerequisite for optimisation. Finally, we optimise numerically, i.e., the net power output (per membrane area) with respect to geometric parameters, membrane parameters as well as operational parameters such as the applied pressure settings during operation.

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A.S. Rana, V.K. Gupta, and H. Struchtrup: Coupled constitutive relations: A second law based higher order closure for hydrodynamics
Proc. Roy. Soc. A (2018)  [pdf]  [doi:10.1098/rspa.2018.0323]

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 description 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 can not 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.

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A.F. Beckmann, A.S. Rana, M. Torrilhon, and H. Struchtrup: Evaporation Boundary Conditions for the Linear R13 Equations Based on Onsager Theory
entropy 20, 680 (2018)  [pdf]  [doi: 10.3390/e20090680]

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.

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H. Struchtrup: Work Storage in States of Apparent Negative Thermodynamic Temperature
Phys. Rev. Lett. 120, 250602 (2018) [pdf] [doi:10.1103/PhysRevLett.120.250602] [supplemental material]

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)]. 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..

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H. Struchtrup, A. Beckmann, A.S. Rana, and A. Frezzotti: Evaporation Boundary Conditions for the R13 Equations of Rarefied Gas Dynamics
Phys. Fluids 29, 092004 (2017)  [pdf]  [doi: 10.1063/1.4989570]

The regularized 13 moment (R13) equations are a macroscopic model for the description of rarefied gas flows in the transition regime. The equations have been shown to give meaningful results for Knudsen numbers up to about 0.5. Here, their range of applicability is extended by deriving and testing boundary conditions for evaporating and condensing interfaces. The macroscopic interface conditions are derived from the microscopic interface conditions of kinetic theory. Tests include evaporation into a half-space and evaporation/condensation of a vapor between two liquid surfaces of different temperatures. Comparison indicates that overall the R13 equations agree better with microscopic solutions than classical hydrodynamics.

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Lei Wu and H. Struchtrup: Assessment and development of the gas kinetic boundary condition for the Boltzmann equation
J. Fluid Mechanics 823, 511-537 (2017)  [pdf] [doi: 10.1017/jfm.2017.326]

Gas-surface interactions play important roles in internal rarefied gas flows, especially in micro-electro-mechanical systems with large surface area to volume ratios. Although great progresses have been made to solve the Boltzmann equation, the gas kinetic boundary condition (BC) has not been well studied. Here we assess the accuracy of the Maxwell, Epstein, and Cercignani-Lampis BCs, by comparing numerical results of the Boltzmann equation for the Lennard-Jones potential to experimental data on Poiseuille and thermal transpiration flows. The four experiments considered are: Ewart et al. [J. Fluid Mech. 584, 337-356 (2007)], Rojas-Cardenas et al. [Phys. Fluids, 25, 072002 (2013)], and Yamaguchi et al. [J. Fluid Mech. 744, 169-182 (2014); 795, 690-707 (2016)], where the mass flow rates in Poiseuille and thermal transpiration flows are measured. This requires the BC has the ability to tune the effective viscous and thermal slip coefficients to match the experimental data. Among the three BCs, the Epstein BC has more flexibility to adjust the two slip coefficients, and hence in most of the time it gives good agreement with the experimental measurement. However, like the Maxwell BC, the viscous slip coefficient in the Epstein BC cannot be smaller than unity but the Cercignani-Lampis BC can. Therefore, we propose to combine the Epstein and Cercignani-Lampis BCs to describe gas-surface interaction. Although the new BC contains six free parameters, our approximate analytical expressions for the slip coefficients provide a useful guidance to choose these parameters.

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M.Yu.Timokhin, H. Struchtrup, A.A. Kokhanchik, and Ye.A. Bondar: Different Variants of R13 Moment Equations Applied to the Shock-Wave Structure
Phys. Fluids 29, 037105 (2017)  [pdf] [doi: 10.1063/1.4977978]

Various versions of the regularized 13-moment system (R13) are applied to the problem of the shock wave structure in a monatomic Maxwell gas for Mach numbers up to M = 10. Numerical solutions are compared to direct simulation Monte Carlo results computed by the SMILE++ software system, in order to identify applicability and limitations of the variants. Over time, several versions of the R13 equations were presented, which differ in non-linear contributions for high-order moments but agree in asymptotic expansion to the third order in the Knudsen number. All variants describe typical subsonic microflows well, for which the non-linear contributions only play a minor role. The challenge of the present study is to determine the real boundaries of applicability of each variant of the moment equations as applied to non-equilibrium supersonic flows, depending on the Mach number and local Knudsen number.

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D. Bharadwaj and H. Struchtrup: Large scale energy storage using multistage osmotic processes: Approaching high efficiency and energy density
Sustainable Energy & Fuels 1, 599-614 (2017) [pdf] [doi: 10.1039/C6SE00013D]

With the increase in ocean levels due to global warming, there is a desperate need for clean and renewable energy at this time, more than ever before. Although the economic front of technologies, such as wind and solar power, has shown improvement, the fact remains that these energy sources are intermittently available in nature. This calls for a reliable energy storage technology that can bridge the gap between the supply and demand of electricity, leading us to a world driven by clean and renewable energy. Here, we propose a process for storing electrical energy using engineered osmosis. To store electrical energy, a salty solution is separated into brine and fresh water streams using modified reverse osmosis. When there is a demand for electricity, the chemical potential is converted back into electrical work by mixing the solutions using a modified version of pressure retarded osmosis. With modelling and simulations, we demonstrate that the proposed process can achieve round trip efficiencies of 50–60% and energy densities equivalent to that of a 500 m high pumped-hydro plant. The results demonstrate a promising process to store electrical energy, which, unlike pumped-hydro, is unconstrained by geography..

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A. Mohammadzadeh and H. Struchtrup: A moment model for phonon transport at room temperatures
30th International Symposium on Rarefied Gas Dynamics, A. Ketsdever, H. Struchtrup (Eds.)
AIP Conf. Proc. 1786, 140009 (2016)  [pdf] [doi: 10.1063/1.4967640]

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, its diameter depends on the temperature of the system. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of 3 directional and 7 frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained.

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B. Rahimi and H. Struchtrup: Heat transfer in polyatomic gases
30th International Symposium on Rarefied Gas Dynamics, A. Ketsdever, H. Struchtrup (Eds.)
AIP Conf. Proc. 1786, 070006 (2016)  [pdf] [doi: 10.1063/1.4967582]

A high-order macroscopic model for the accurate description of rarefied polyatomic gas flows is used to explore heat transfer in rarefied polyatomic gases. The unsteady heat conduction of a gas at rest is studied numerically and analytically. The full boundary conditions are obtained for the macroscopic models of the refined Navier-Stokes-Fourier (RNSF) equations and the R19 equations. The results for different gases are given and effects of Knudsen numbers, degrees of freedom and temperature dependent properties are investigated. For some cases, the higher order effects are very dominant and the widely used first order set of the Navier-Stokes-Fourier equations fails to accurately capture the gas behavior and should be replaced by the proposed higher order set of equations.

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M.Yu.Timokhin, H. Struchtrup, A.A. Kokhanchik, and Ye.A. Bondar: The Analysis of Different Variants of R13 Equations Applied to the Shock-Wave Structure
30th International Symposium on Rarefied Gas Dynamics, A. Ketsdever, H. Struchtrup (Eds.)
AIP Conf. Proc. 1786, 140006 (2016)  [pdf] [doi: 10.1063/1.4967637]

This paper studies the applicability of various versions of the regularized 13-moment system (R13) as applied to the problem of the shock wave structure in a monatomic Maxwell gas in a wide range of Mach numbers (1.0<M<8.0). Over time, several versions of the R13 equations were presented, which differ in non-linear contributions for high-order moments. The challenge of this study is to determine the range of applicability of each variant of the moment equations as applied to non-equilibrium supersonic flows, depending on the Mach number and local Knudsen number. Numerical results obtained for the R13 system are compared to DSMC data computed by the SMILE++ software system.

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H. Struchtrup and A. Frezzotti: Evaporation/Condensation Boundary Conditions for the Regularized 13 Moment Equations
30th International Symposium on Rarefied Gas Dynamics, A. Ketsdever, H. Struchtrup (Eds.)
AIP Conf. Proc. 1786, 140002 (2016)  [pdf] [doi: 10.1063/1.4967633]

The regularized 13 moment equations (R13) are a macroscopic model for the description of rarefied gas flows in the transition regime. The equations have been shown to give meaningful results for Knudsen numbers up to about 0.5. Here, their range of applicability is extended by boundary conditions for evaporating and condensing interfaces, derived from the microscopic interface conditions of kinetic theory. Simple 1-D problems are used to test the R13 equations with evaporation and condensation.

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B. Rahimi and H. Struchtrup: Macroscopic and kinetic modelling of rarefied polyatomic gases
J. Fluid Mechanics 806, 437-505 (2016)  [pdf] [doi: 10.1017/jfm.2016.604]

A kinetic model and corresponding high-order macroscopic model for the accurate description of rarefied polyatomic gas flows are introduced. The different energy exchange processes are accounted for with a two term collision model. The proposed kinetic model, which is an extension of the S-model, predicts correct relaxation of higher moments and delivers the accurate Prandtl (Pr) number. Also, the model has a proven linear H-theorem. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad’s 36 moment equations for polyatomic gases. At the first order, a modification of the Navier–Stokes–Fourier (NSF) equations is obtained. At third order of accuracy, a set of 19 regularized partial differential equations (R19) is obtained. Furthermore, the terms associated with the internal degrees of freedom yield various intermediate orders of accuracy, a total of 13 different orders. Thereafter, boundary conditions for the proposed macroscopic model are introduced. The unsteady heat conduction of a gas at rest is studied numerically and analytically as an example of a boundary value problem. The results for different gases are given and effects of Knudsen numbers, degrees of freedom, accommodation coefficients and temperature-dependent properties are investigated. For some cases, the higher-order effects are very dominant and the widely used first-order set of the NSF equations fails to accurately capture the gas behaviour and should be replaced by the proposed higher-order set of equations.

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A. Mohammadzadeh and H. Struchtrup: A moment model for phonon transport at room temperature
Cont. Mech. Thermodyn. 29, 117- 144 (2017) [pdf] [doi: 10.1007/s00161-016-0525-y]

Heat transfer in solids is modeled by deriving the macroscopic equations for phonon transport from the phonon-Boltzmann equation. In these equations, the Callaway model with frequency-dependent relaxation time is considered to describe the Resistive and Normal processes in the phonon interactions. Also, the Brillouin zone is considered to be a sphere, and its diameter depends on the temperature of the system. A simple model to describe phonon interaction with crystal boundary is employed to obtain macroscopic boundary conditions, where the reflection kernel is the superposition of diffusive reflection, specular reflection
and isotropic scattering. Macroscopic moments are defined using a polynomial of the frequency and wave vector of phonons. As an example, a system of moment equations, consisting of three directional and seven frequency moments, i.e., 63 moments in total, is used to study one-dimensional heat transfer, as well as Poiseuille flow of phonons. Our results show the importance of frequency dependency in relaxation times and macroscopic moments to predict rarefaction effects. Good agreement with data reported in the literature is obtained.

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D. Bharadwaj, T.M. Fyles, and H. Struchtrup:, Multistage Pressure Retarded Osmosis
J. Non-Equilibrium Thermodynamics 41, 327 - 347 (2016)  [pdf] [doi:10.1515/jnet-2016-0017]

One promising sustainable energy source is the chemical potential difference between salt and freshwater. The membrane process of pressure-retarded osmosis (PRO) has been the most widely investigated means to harvest salinity gradient energy. In this report, we analyse the thermodynamic efficiency of multistage PRO systems to optimize energy recovery from a salinity gradient. We establish a unified description of the efficiencies of the component pumps (P), turbines (T), pressure exchangers (PX), and membrane modules (M) and exploit this model to determine the maximum available work with respect to the volume of the brine produced, the volume of the sea water consumed, or the volume of the freshwater that permeates the membrane. In an idealized series configuration of 1–20 modules (P–M–T), the three optimization conditions have significantly different intermediate operating pressures in the modules, but demonstrate that multistage systems can recover a significantly larger fraction of the available work compared to single-stage PRO. The biggest proportional advantage occurs for one to three modules in series. The available work depends upon the component efficiencies, but the proportional advantage of multistage PRO is retained. We also optimize one- and two-stage PX–M–T and P–M–T configurations with respect to the three volume parameters, and again significantly different optimal operating conditions are found. PX–M–T systems are more efficient than P–M–T systems, and two-stage systems have efficiency advantages that transcend assumed component efficiencies. The results indicate that overall system design with a clear focus on critical optimization parameters has the potential to significantly improve the near-term practical feasibility of PRO.

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V.K. Gupta, H. Struchtrup, and M. Torrilhon: Regularized moment equations for binary gas mixtures: Derivation and linear analysis
Phys. Fluids 28,042003 (2016) [pdf] [doi: 10.1063/1.4945655]

The applicability of the order of magnitude method [H. Struchtrup, “Stable transport equations for rarefied gases at high orders in the Knudsen number,” Phys. Fluids 16, 3921–3934 (2004)] is extended to binary gas mixtures in order to derive various sets of equations—having minimum number of moments at a given order of accuracy in the Knudsen number—for binary mixtures of monatomic-inert-ideal gases interacting with the Maxwell interaction potential. For simplicity, the equations are derived in the linear regime up to third order accuracy in the Knudsen number. At zeroth order, the method produces the Euler equations; at first order, it results into the Fick, Navier–Stokes, and Fourier equations; at second order, it yields a set of 17 moment equations; and at third order, it leads to the regularized 17-moment equations. The transport coefficients in the Fick, Navier–Stokes, and Fourier equations obtained through order of magnitude method are compared with those obtained through the classical Chapman–Enskog expansion method. It is established that the different temperatures of different constituents do not play a role up to second order accurate theories in the Knudsen number, whereas they do contribute to third order accurate theory in the Knudsen number. Furthermore, it is found empirically that the zeroth, first, and second order accurate equations are linearly stable for all binary gas mixtures; however, although the third order accurate regularized 17-moment equations are linearly stable for most of the mixtures, they are linearly unstable for mixtures having extreme difference in molecular masses.

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A.S. Rana and H. Struchtrup: Thermodynamically admissible boundary conditions for the regularized 13 moment equations
Phys. Fluids 28, 027105 (2016) [pdf] [doi:10.1063/1.4941293]

A phenomenological approach to the boundary conditions for linearized R13 equations is derived using the second law of thermodynamics. The phenomenological coefficients appearing in the boundary conditions are calculated by comparing the slip, jump, and thermal creep coefficients with linearized Boltzmann solutions for Maxwell’s accommodation model for different values of the accommodation coefficient. For this, the linearized R13 equations are solved for viscous slip, thermal creep, and temperature jump problems and the results are compared to the solutions of the linearized Boltzmann equation. The influence of different collision models (hard-sphere, Bhatnagar–Gross–Krook, and Maxwell molecules) and accommodation coefficients on the phenomenological coefficients is studied.

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A. Mohammadzadeh, A.S. Rana, and H. Struchtrup: DSMC and R13 Modeling of the Adiabatic Surface
Int. J. Thermal Sciences 101, 9-23 (2016) [pdf] [doi:10.1016/j.ijthermalsci.2015.10.007]

Adiabatic wall boundary conditions for rarefied gas flows are described with the isotropic scattering model. An appropriate sampling technique for the direct simulation Monte Carlo (DSMC) method is presented, and the corresponding macroscopic boundary equations for the regularized 13-moment system (R13) are obtained. DSMC simulation of a lid driven cavity shows slip at the wall, which, as a viscous effect, creates heat that enters the gas while there is no heat flux in the wall. Analysis with the macroscopic equations and their boundary conditions reveals that this heat flux is due to viscous slip heating, and is the product of slip velocity and shear stress at the adiabatic surface. DSMC simulations of the driven cavity with adiabatic walls are compared to R13 simulations, which both show this non-linear effect in good agreement for Kn < 0.3.

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A. Mohammadzadeh, A.S. Rana, and H. Struchtrup: Thermal stress vs. thermal transpiration: A competition in thermally driven cavity flows
Phys. Fluids 27, 112001 (2015) [pdf] [doi:10.1063/1.4934624]

The velocity dependent Maxwell (VDM) model for the boundary condition of a rarefied gas, recently presented by Struchtrup [“Maxwell boundary condition and velocity dependent accommodation coefficient,” Phys. Fluids 25, 112001 (2013)], provides the opportunity to control the strength of the thermal transpiration force at a wall with temperature gradient. Molecular simulations of a heated cavity with varying parameters show intricate flow patterns for weak, or inverted transpiration force. Microscopic and macroscopic transport equations for rarefied gases are solved to study the flow patterns and identify the main driving forces for the flow. It turns out that the patterns arise from a competition between thermal transpiration force at the boundary and thermal stresses in the bulk.

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A. Mohammadzadeh and H. Struchtrup: Velocity dependent Maxwell boundary conditions in DSMC
Int. J. Heat and Mass Transfer 87, 151-160 (2015) [pdf] [doi:10.1016/j.ijheatmasstransfer.2015.03.045]

Recently Struchtrup (2013) proposed an extension to the original Maxwell boundary conditions for the Boltzmann equation which introduces velocity dependent accommodation coefficients. These boundary conditions are implemented into the direct simulation Monte Carlo (DSMC) method. The effect of the velocity dependent Maxwell (VDM) boundary conditions on thermal transpiration phenomena is studied for two-dimensional micro-cavities. Variation of the three microscopic parameters provided by the VDM boundary condition yields changes in slip velocity, temperature jump and the thermal transpiration effect. The results indicate that the strength of thermal transpiration can change and, depending on the values of the coefficients, the rarefied flow can be driven from warmer toward colder regions.

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B. Rahimi and H. Struchtrup: Kinetic model and moment method for polyatomic gases
29th International Symposium on Rarefied Gas Dynamics 2014
AIP Conf. Proc. 1628, 618-625 (2014) [pdf] [doi:10.1063/1.4902650]

A model kinetic equation for accurate description of rarefied polyatomic gases is introduced. The collisions of polyatomic gas particles are modeled by a two term collision operator, one modeling only exchange of translational energy and the other modeling exchange of both translational and internal energies. The proposed kinetic model, which is an extension of the Rykov and Shakov models, predicts correct relaxation of higher moments and delivers the accurate Prandtl (Pr) number. Also, the model has a non-linear H-theorem.The model is used to construct a system of 36 moment equations, which is closed by the generalized Grad method for polyatomic gases.

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A.S. Rana, A. Mohammadzadeh, and H. Struchtrup:A numerical study of the heat transfer through a rarefied gas confined in a micro cavity
Cont. Mech. Thermodyn. 27, 433-446 (2015) [pdf] [doi:10.1007/s00161-014-0371-8]

Flow and heat transfer in a bottom-heated square cavity in a moderately rarefied gas is investigated using the R13 equations and the Navier–Stokes–Fourier equations. The results obtained are compared with those from the direct simulation Monte Carlo (DSMC) method with emphasis on understanding thermal flow characteristics from the slip flow to the early transition regime. The R13 theory gives satisfying results—including flow patterns in fair agreement with DSMC—in the transition regime, which the conventional Navier–Stokes–Fourier equations are not able to capture.

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B. Rahimi and H. Struchtrup: Refined Navier-Stokes-Fourier Equations for Rarefied Polyatomic Gases
Proceedings of the ASME 2014 12th International Conference on Nanochannels, Microchannels, and Minichannels (ICNMM 2014), August 3-7, 2014, Chicago, Illinois, USA [pdf] 

A macroscopic model for the description of rarefied poly- atomic gas flows is introduced. Grad’s moment method is used to construct a closed set of equations for 36 primary moments. The order of magnitude method is then applied to acquire optimized moment definitions. The appropriate sets of equations corresponding to the desired order of accuracy in the Knudsen number are derived by reducing the equations. A refined version of the Navier-Stokes-Fourier (NSF) equations is obtained where the dynamic temperature is an additional independent variables.

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B. Rahimi and H. Struchtrup: Capturing non-equilibrium phenomena in rarefied polyatomic gases: A high-order macroscopic model
Phys. Fluids 26, 052001 (2014) [pdf] [doi:10.1063/1.4873577]

A high-order macroscopic model for the accurate description of rarefied polyatomic gas flows is introduced based on a kinetic equation of Bhatnagar-Gross-Krook (BGK)-type, where the different energy exchange processes are accounted for by two collision terms. The order of magnitude method is applied to the primary moment equations to acquire the optimized moment definitions and the final scaled set of Grad's 36 moment equations for polyatomic gases. The two Knudsen numbers of the system are used for model reduction in terms of their powers, which yields a wide range of different reduced systems, a total of 13 different orders. These include, at lower order, a modification of the Navier-Stokes-Fourier (NSF) equations which shows considerable extended range of validity in comparison to the classical NSF equations. The highest order of accuracy considered gives a set of 18 regularized partial differential equations (PDEs) (R18). Attenuation and speed of linear waves are studied as the first application of the many sets of equations. For frequencies where the internal degrees of freedom are effectively frozen, the equations reproduce the behavior of monatomic gases.

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H. Struchtrup: Maxwell boundary condition and velocity dependent accommodation coefficient
Phys. Fluids 25, 112001 (2013) [pdf] [doi:10.1063/1.4829907]

A modification of Maxwell's boundary condition for the Boltzmann equation is developed that allows to incorporate velocity dependent accommodation coefficients into the microscopic description. As a first example, it is suggested to consider the wall-particle interaction as a thermally activated process with three parameters. A simplified averaging procedure leads to jump and slip boundary conditions for hydrodynamics. Coefficients for velocity slip, temperature jump, and thermal transpiration flow are identified and compared with those resulting from the original Maxwell model and the Cercignani-Lampis model. An extension of the model leads to temperature dependent slip and jump coefficients.
PACS: 51.10.+y, 47.45.Gx, 47.61.Fg

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M.J. Fryer and H. Struchtrup: Moment Model and Boundary Conditions for Energy Transport in the Phonon Gas
Cont. Mech. Thermodyn. 26, 593-618 (2014) [pdf] [doi:10.1007/s00161-013-0320-y]

Heat transfer in solids is modelled in the framework of kinetic theory of the phonon gas. The microscopic description of the phonon gas relies on the phonon-Boltzmann equation and the Callaway model for phonon-phonon interaction. A simple model for phonon interaction with crystal boundaries, similar to the Maxwell boundary conditions in classical kinetic theory, is proposed. Macroscopic transport equation for an arbitrary set of moments are developed and closed by means of Grad's moment method. Boundary conditions for the macroscopic equations are derived from the microscopic model and the Grad closure. As example, sets with 4, 9, 16, and 25 moments are considered, and solved analytically for one-dimensional heat transfer, and Poiseuille flow of phonons. The results show the influence of Knudsen number on phonon drag at solid boundaries. The appearance of Knudsen layers reduces the net heat conductivity of solids in rarefied phonon regimes..

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H. Struchtrup and M. Torrilhon: Regularized 13 Moment Equations for Hard Sphere molecules: Linear Bulk Equations
Phys. Fluids 25, 052001 (2013) [pdf] [doi:10.1063/1.4802041]

The regularized 13 moment equations of rarefied gas dynamics are derived for a monatomic hard sphere gas in the linear regime. The equations are based on an extended Grad-type moment system, which is systematically reduced by means of the Order of Magnitude Method [Struchtrup, Phys. Fluids 16(11), 3921-3934 (2004)]. Chapman-Enskog expansion of the final equations yields the linear Burnett and super-Burnett equations. While the Burnett coefficients agree with literature values, this seems to be the first time that super-Burnett coefficients are computed for a hard sphere gas. As a first test of the equations the dispersion and damping of sound waves is considered.

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H. Struchtrup and M. Torrilhon: Regularized 13 Moment Equations for Hard Spheres
28th International Symposium on Rarefied Gas Dynamics 2012
AIP Conf. Proc. 1501, 199-206 (2012) [pdf] [doi:10.1063/1.4769503]

The regularized 13 moment equations (R13) of rarefied gas dynamics for a monatomic hard sphere gas in the linear regime are presented. The equations are based on an extended Grad-type moment system, which was systematically reduced by means of the Order of Magnitude Method [Struchtrup, Phys. Fluids 16(11), 3921-3934 (2004)]. The linear Burnett and super-Burnett equations are derived from Chapman-Enskog expansion of the R13 equations. While the Burnett coefficients agree with literature values, this seems to be the first time that super-Burnett coefficients are computed for a hard sphere gas. The equations are considered for stability, and dispersion and damping of sound waves. Boundary conditions are given, and solutions of simple boundary value problems are briefly discussed..

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A.S. Rana, M. Torrilhon, and H. Struchtrup: Numerical Solution of the Moment Equations Using Kinetic Flux-Splitting Schemes
28th International Symposium on Rarefied Gas Dynamics 2012
AIP Conf. Proc. 1501, 287-293 (2012) [pdf] [doi:10.1063/1.4769525]

Processes in rarefied gases are accurately described by the Boltzmann equation. The solution of the Boltzmann equation using direct numerical methods and direct simulation Monte Carlo methods (DSMC) is very time consuming. An alternative approach can be obtained by using moment equations, which allow the calculation of processes in the transition regime at reduced computational cost. In the current work, a finite volume method is developed for the solution of these moment equations. The numerical scheme is based on kinetic schemes, similar to those developed for the Euler and Navier-Stokes equations by Deshpande (1986), Perthame (1990), Xu et al. (2005), Le Tallec and Perlat (1998), and others

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A.S. Rana, M. Torrilhon, and H. Struchtrup: A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity
J. Comp. Phys 236, 169-186 (2013) [pdf] [doi:10.1016/j.jcp.2012.11.023]

In this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment (R13) equations of rarefied gas dynamics. The scheme allows fast solutions for 2D and 3D boundary value problems (BVPs) with velocity slip and temperature jump boundary conditions. The scheme is applied to the lid driven cavity problem for Knudsen numbers up to 0.7. The results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Integro Moment Method (IMM) and the Direct Simulation Monte Carlo (DSMC) method. The R13 equations yield better results than the classical Navier–Stokes–Fourier equations for this boundary value problem, since they give an approximate description of Knudsen boundary layers at moderate Knudsen numbers. The R13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate Knudsen numbers.

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H. Struchtrup, S. Kjelstrup, and D. Bedeaux: Analysis of temperature difference driven heat and mass transfer in the Phillips-Onsager Cell
Int. J. Heat and Mass Transfer 58, 521-531 (2013) [pdf] [doi:10.1016/j.ijheatmasstransfer.2012.11.066]

In a series of experimental investigations Phillips and co-workers have determined the "Onsager heat of transport" in a cell with layers of a liquid and its vapor. Their results also include experimental observation of "cold to warm distillation", that is temperature difference driven mass transfer through the vapor from a cold to a warm liquid [e.g., Mills and Phillips, Chem. Phys. Lett. 372, 615-619 (2003)]. Using standard irreversible thermodynamics for evaporation, condensation and     transport, we present a theoretical analysis of the experimental set-up and discuss the reported measurements in terms of layer and interface properties. Necessary and sufficient criteria for a possible temperature difference driven cold to warm mass transfer are given. The criteria indicate that the occurrence of this phenomenon is highly unlikely.

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O.  Kastner, W.H. Müller, S. Seelecke, H. Struchtrup, M. Torrilhon, W. Weiss: Editorial: Trends in thermodynamics and materials theory
Cont. Mech. Thermodyn. 24, 267–269 (2012) [pdf] [doi:10.1007/s00161-012-0272-7]

 

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A. Rana, M. Torrilhon, and H. Struchtrup:   Heat transfer in micro devices packaged in partial vacuum
J. Physics: Conference Series 362, 012034 (2012) [pdf] [doi:10.1088/1742-6596/362/1/012034]

The influence of rarefaction effects on technical processes is studied numerically for a heat transfer problem in a rarefied gas, a box with bottom heated plate. Solutions obtained from several macroscopic models, in particular the classical Navier-Stokes-Fourier equations with jump and slip boundary conditions, and the regularized 13 moment (R13) equations [Struchtrup & Torrilhon, Phys. Fluids 15, 2003] are compared. The R13 results show significant flow patterns which are not present in the classical hydrodynamic description.

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H. Struchtrup, S. Kjelstrup, and D. Bedeaux: Temperature-difference-driven mass transfer through the vapor from a cold to a warm liquid
Phys. Rev. E 85, 061201 (2012) [pdf] [doi:10.1103/PhysRevE.85.061201]

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, 306-312 (1971)] and mass transfer from a cold to a warm liquid driven by a temperature difference across the vapor phase [Mills & Phillips, Chem. Phys. Lett. 372, 615--619 (2002)]. Careful evaluation of the thermodynamic laws has shown [Bedeaux, Hermans &Ytrehus, Physica A 169, 263-280, 1990] that the inverted temperature profile will be observed for processes with a large heat of vaporization. We show in this paper 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.

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H. Struchtrup: Unique moment set from the order of magnitude method
Kinetic and Related Models 5, 417-440 (2012) [pdf] [doi:10.3934/krm.2012.5.417]

The order of magnitude method [Struchtrup, Phys. Fluids 16, 3921-3934 (2004)] is used to construct a unique moment set for 1-D transport with scattering. Simply speaking, the method uses a series of leading order Chapman-Enskog expansions in the Knudsen number to construct the moments such that the number of moments at a given Chapman-Enskog order is minimal. For isotropic scattering, when one begins with monomials for the moments, the method constructs step by step moments of the Legendre polynomials. For anisotropic scattering, however, it constructs moments of new polynomials relevant for the particular scattering mechanism. All terms in the final moment equations are scaled by powers of the Knudsen number, which gives an easy handle to model reduction.

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H. Struchtrup: Resonance in Rarefied Gases
Cont. Mech. Thermodyn. 24, 361-376 (2012) [pdf] [doi:10.1007/s00161-011-0202-0]

Dispersion and damping of ultrasound waves is a standard test for mathematical models of rarefied gas flows. Normally, one considers waves in semi-inifinite systems in relatively large distance of the source. For a more complete picture, ultrasound propagation in finite closed systems of length L is studied by means of several models for rarefied gas flows: the Navier-Stokes-Fourier equations, Grad's 13 moment equations, the regularized 13 moment equations, and the Burnett equations. All systems of equations are considered in simple 1-D geometry with their appropriate jump and slip boundary conditions. Damping and resonance are studied in dependence of frequency and length. For small L all wave modes contribute to the solution.

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P. Taheri and H. Struchtrup: Poiseuille Flow of Moderately Rarefied Gases in Annular Channels
Int. J. Heat and Mass Transfer 55,1291-1303 (2012) [pdf] [doi:10.1016/j.ijheatmasstransfer.2011.09.012]

In this study, rarefaction effects in pressure-driven gas flows in annular microchannels are investigated. The influence of gas rarefaction, aspect ratio of the annulus, and surface accommodation coefficient on wall friction, mass flow rate, and thermal energy flow rate is studied. For this, the linearized Navier-Stokes-Fourier (NSF) and regularized 13-moment (R13) equations are solved analytically. The results are compared to available solutions of the Boltzmann equation to highlight the advantages of the R13 over the NSF equations in describing rarefaction effects in the process. Moreover, a second-order slip boundary condition is proposed to improve the accuracy of the classical NSF equations.

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J.P. Caputa and H. Struchtrup: Interface model for non-equilibrium evaporation
Physica A 390, 31-42 (2011) [pdf] [doi:10.1016/j.physa.2010.09.019]

A microscopic interface condition for condensing/evaporating interfaces is developed by combining a velocity dependent condensation probability [Tsuruta et al., Int. J. Heat Mass Transfer 42, 4107 (1999)] and Maxwell type interface conditions with accommodation. Using methods from kinetic theory, macroscopic interface conditions for mass and energy transport across the phase boundary are derived. The results are compared to classical non-equilibrium thermodynamics. The interface conditions are considered for the limit of small deviation from equilibrium, and the corresponding Onsager coefficients are computed. These results are useful as boundary conditions for non-equilibrium evaporation and condensation problems, as done previously by our group [Bond et al., Phys. Rev. E 70, 061605 (2004)].

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P. Taheri and H. Struchtrup: An Extended Macroscopic Transport Model for Rarefied Gas Flows in Long Capillaries with Circular Cross Section
Phys. Fluids 22, 112004 (2010)  [pdf] [doi:10.1063/1.3500681]

Pressure-driven and thermally-driven rarefied gas flows in long capillaries with circular cross sections are investigated. For both, Poiseuille and thermal transpiration flows, a unified theoretical approach is presented based on the linear form of regularized 13-moment (R13) equations. The captured nonequilibrium effects in the processes are compared to available kinetic solutions, and the shortcomings of classical hydrodynamics, i.e., the Navier-Stokes-Fourier (NSF) equations, are highlighted. Breakdown of Onsager's symmetry is proposed as a criterion to determine the range of applicability of extended macroscopic models. Based on Onsager's reciprocity relation it is shown that linearized R13 equations provide agreement with kinetic data for moderate Knudsen numbers, Kn<0.25. Two-way flow pattern and thermomolecular pressure difference in simultaneous pressure-driven and temperature-driven flows are analyzed. Moreover, second-order boundary conditions for velocity slip and temperature jump are derived for the Navier-Stokes-Fourier system. The proposed boundary conditions effectively improve classical hydrodynamics in the transition flow regime.

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H. Struchtrup and M. Torrilhon: Comment on ''Thermodynamically admissible 13 moment equations from the Boltzmann equation''
Phys. Rev. Lett.  105, 128901 (2010) [pdf] [doi:10.1103/PhysRevLett.105.128901]

(no abstract)

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H. Struchtrup and P. Taheri: Macroscopic Transport Models for Rarefied Gas Flows: A Brief Review
IMA J. Apppl. Math. 76(5), 672-697 (2011) [pdf] [doi: 10.1093/imamat/hxr004]

Efficient modelling of gas microflows requires accurate, yet fast to solve, models. For finite but moderate Knudsen numbers, extended macroscopic transport equations offer an alternative to the Boltzmann equation, from which they are derived. Classical and modern approaches for the derivation of these models are reviewed, and the resulting equations are compared for their ability to describe the multitude of known rarefaction phenomena. Among the equations discussed are the Burnett and super-Burnett equations, Grad's 13 moment equations, and the regularized 13 and 26 moment equations.

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H. Struchtrup, P. Taheri and A. Rana: Analytical and Numerical Solutions of Boundary Value Problems for the Regularized 13 Moment Equations
Proc. 27th International Symposium on Rarified Gas Dynamics, AIP Conf. Proc. 1333, 627-634 (2011) [pdf] [doi:10.1063/1.3562717]

Classical hydrodynamics---the laws of Navier-Stokes and Fourier---fails in the description of processes in rarefied gases. For not too large Knudsen numbers, extended macroscopic models offer an alternative to the solution of the Boltzmann equation. Analytical and numerical solutions show that the regularized 13 moment equations can capture all important linear and non-linear rarefaction effects with good accuracy.

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P. Taheri and H. Struchtrup: Rarefaction Effects in Thermally-driven Microflows
Physica A 389, 3069-3080 (2010) [pdf] [doi:10.1016/j.physa.2010.03.050]

Rarefied gas flow in a parallel-plate micro-channel is considered, where a streamwise constant temperature gradient is applied in the channel walls. An analytical approach to the problem is conducted based on linearized and semi-linearized forms of the regularized 13-moment equations (R13 equations), which are a set of macroscopic transport equations for rarefied gases at super-Burnett order. Typical nonequilibrium effects at the boundary, i.e., velocity slip, temperature jump, and formation of Knudsen boundary layers are investigated. Nonlinear contributions lead to temperature, density, and normal stress profiles across the channel which are not reported elsewhere in the literature.

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T. Burdyny and H. Struchtrup: Hybrid membrane/cryogenic separation of oxygene from air for use in the oxyfuel process
Energy 35, 1884-1897 (2010) [pdf]  [doi:10.1016/j.energy.2009.12.033]

The process of oxy-fuel combustion requires the separation of oxygen from air on a large scale for use in the combustion chamber. This separation is currently done through energy intensive cryogenic distillation. To reduce the overall energy requirements for air separation it is examined whether a hybrid membrane and cryogenic process be utilized instead. The examined process uses an O2/N2 permeable membrane to create oxygen enriched air. This enriched air is then turned into high purity oxygen using cryogenic distillation. Several arrangements of such a system are investigated and compared on a practical and thermodynamic level to the current cryogenic process in use. It is found that using a vacuum pump arrangement to draw air through the membrane has potential to reduce energy requirements from the current standard. It is also found that the hybrid system is more productive in small to medium scale applications than in large scale applications because of the increased irreversibilities in the cryogenic process at smaller scales.

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P. Taheri and H. Struchtrup: Effects of rarefaction in microflows between coaxial cylinders
Phys. Rev. E 80, 066317 (2009) [pdf] [doi:10.1103/PhysRevE.80.066317]

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 new 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 (DSMC) data.

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H.C. Öttinger, H. Struchtrup and M. Liu: Inconsistency of a dissipative contribution to the mass flux in hydrodynamics
Phys. Rev. E 80, 056303 (2009) [pdf] [doi:10.1103/PhysRevE.80.056303]

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_bar. However, this field 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.

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P. Taheri, A.S. Rana, M. Torrilhon and H. Struchtrup: Macroscopic description of steady and unsteady rarefaction effects in boundary value problems of gas dynamics
Cont. Mech. Thermodyn. 21, 423-443 (2009) [pdf] [doi:10.1007/s00161-009-0115-3]

Four basic flow configurations are employed to investigate steady and unsteady rarefaction effects in monatomic ideal gas flows. Internal and external flows in planar geometry, namely, viscous slip (Kramer's problem), thermal creep, oscillatory Couette, and pulsating Poiseuille flows are considered. A characteristic feature of the selected problems is the formation of the Knudsen boundary layers, where non-Newtonian stress and non-Fourier heat conduction exist. The linearized Navier-Stokes-Fourier and regularized 13-moment equations are utilized to analytically represent the rarefaction effects in these boundary-value problems. It is shown that the regularized 13-moment system correctly estimates the structure of Knudsen layers, compared to the linearized Boltzmann equation data.

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D. Lockerby, J. Reese and H. Struchtrup: Switching Criteria for Hybrid Rarefied Gas Flow Solvers
Proc. Roy. Soc. A, Vol. 465 no. 2105, 1581-1598 (2009) [pdf] [doi:10.1098/rspa.2008.0497]

Switching criteria for hybrid hydrodynamic/molecular gas flow solvers are developed, and are demonstrated to be more appropriate than conventional ones for the purposes of identifying thermodynamic non-equilibrium. For switching from a molecular/kinetic solver to a hydrodynamic (continuum-fluid) solver, the criterion is based on the difference between the hydrodynamic non-equilibrium fluxes (i.e. the Navier-Stokes stress and Fourier heat flux) and the actual values of stress and heat flux as computed from the molecular solver. For switching from hydrodynamics to molecular/kinetic, a similar criterion is used but the values of stress and heat flux are approximated through higher-order constitutive relations; in this case, we use the R13 equations [Struchtrup & Torrilhon, Phys. Fluids 15(9), 2668-2680 (2003)]. The efficacy of our proposed switching criteria is tested within an illustrative hybrid BGK/Navier-Stokes solver. For the test cases investigated, the results from the hybrid procedure compare very well with the full kinetic solution, and are obtained at a fraction of the computational cost.

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D.A. Lockerby, H. Struchtrup, and J.M. Reese: Switching Criteria for Hybrid Rarefied Gas Flow Solvers
Proc. 26th International Symposium on Rarified Gas Dynamics, AIP Conf. Proc. 1084, 434-440 (2008) [pdf] [doi:10.1063/1.3076516]

A set of local Knudsen numbers are defined, which are demonstrated to be more appropriate than conventional ones for the purposes of identifying gas flow non-equilibrium. The problematic area of choosing an appropriate switching criteria is addressed by adopting a local Knudsen number definition based on higher-order constitutive relations; the R13 equations are chosen. A procedure is then described that allows the estimation of the R13 local Knudsen number within a Navier-Stokes solver, and the efficacy of this as a switching criterion is tested within an illustrative hybrid BGK/Navier-Stokes procedure. For the test case investigated, the results from the hybrid procedure compare very well with the full BGK solution, and are obtained at a fraction (depending on the global Kn) of the computational cost.

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H. Struchtrup: Higher Order Bulk and Boundary Effects in Channel Flows
Proc. 26th International Symposium on Rarefied Gas Dynamics, AIP Conf. Proc. 1084, 75-80 (2008) [pdf] [doi:10.1063/1.3076577]

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 to describe the characteristic dip in the temperature profile observed in force driven Poiseuille flow.

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H. Struchtrup and M. Torrilhon: Higher-order effects in rarefied channel flows
Phys. Rev. E 78, 046301 (2008) [pdf] [doi:10.1103/PhysRevE.78.046301]
Erratum: Phys.Rev. E 78, 069903 (2008) [pdf] [doi:10.1103/PhysRevE.78.069903]

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 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.

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P. Taheri, M. Torrilhon and H. Struchtrup: Couette and Poiseuille Microflows: Analytical Solutions for Regularized 13-moment Equations
Phys. Fluids 21, 017102 (2009) [pdf] [doi:10.1063/1.3064123]

The regularized thirteen moment equations (R13 equations) for rarefied gas flows are considered for planar micro-channel flows. The governing equations and corresponding kinetic boundary conditions are partly linearized, such that analytical solutions become feasible. The non-linear terms include contributions of the shear stress and shear rate, which describe the coupling between velocity and temperature fields. Solutions for Couette and force-driven Poiseuille flows show good agreement with direct simulation Monte Carlo data. Typical rarefaction effects, e.g. heat-flux parallel to the wall and the characteristic dip in the temperature profile in Poiseuille flow, are reproduced accurately. Furthermore, boundary effects such as velocity slip, temperature jump, and Knudsen boundary layers are predicted correctly.

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M. Torrilhon and H. Struchtrup: Modelling Micro Mass and Heat Transfer for Gases Using Extended Continuum Equations
ASME J. Heat Transfer 131, 033103 (2009)  [pdf] [doi:10.1115/1.3056598]

This paper presents recent contributions to the development of macroscopic continuum transport equations for micro gas flows and heat transfers. Within kinetic theory of gases a combination of the Chapman-Enskog expansion and Grad's moment method yields the regularized 13 moment equations (R13 equations) which are of high approximation order. In addition, a complete set of boundary conditions can be derived from the boundary conditions of the Boltzmann equation. The R13 equations are linearly stable and their results for moderate Knudsen numbers stand in excellent agreement to DSMC simulations. We give analytical expressions for heat and mass transfer in micro-channels. These expressions help to understand the complex interaction of fluid variables in micro-scale systems. Additionally, we compare interesting analogies like a mass flux and energy Knudsen paradox. In particular, the R13 model is capable to predict and explain detailed features of Poiseuille micro-flows.

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G. Elfring and H. Struchtrup: Thermodynamics of Pore Wetting and Swelling in Nafion
J. Membrane Sci. 315, 125-132 (2008) [pdf] [doi:10.1016/j.memsci.2008.02.016]

A model for the wetting and swelling of pores with water within a Nafion membrane, based on minimizing all contributions to the total free energy, is developed. We find that equilibrium state depends on entropic mixing forces and energetic surface forces. The wetting of the pore relies on the entropic forces exceeding the energetic forces. Specifically this indicates a critical pore size in which liquid is the favorable state. If the pore fills with liquid it will swell until balanced by the energy of the deforming membrane. Several factors including pressure relative to saturation and the phase which bounds the membrane are shown to dramatically affect the final equilibrium state of the system.

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H. Struchtrup:  Boundary conditions and Knudsen layers for moment equations of rarefied gas dynamics
Oberwolfach Reports 4(4), 3405-3406 (2007)

(extended abstract).

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H. Struchtrup: Linear Kinetic Heat Transfer: Moment Equations, Boundary Conditions, and Knudsen layers
Physica A 387, 1750-1766 (2008) [pdf] [doi:10.1016/j.physa.2007.11.044]

A linear kinetic equation for heat transfer is solved by means of the method of moments. The moment equations are solved with Maxwell-type boundary conditions for steady state energy transport. The results exhibit marked Knudsen boundary layers. The accuracy of the description is examined, and it is shown that already a relatively small number of moments can give satisfactory resolution of Knudsen layers for Knudsen numbers ɛ≤1. The implications for moment equations for more complicated kinetic equations (such as the Boltzmann equation) are discussed.

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M. Torrilhon and H. Struchtrup : Gas micro-flow modeling based on regularized 13 moment equations
Proceedings of MNHT 2008, ASME Micro/Nanoscale Heat Transfer International Conference, January 6-9, 2008, Tainan, Taiwan [pdf]

We summarize our recent contributions to the development of macroscopic transport equations for gas micro-flows. A combination of the Chapman-Enskog expansion and Grad's moment method in kinetic theory of gases yields the Regularized 13-Moment-Equations (R13 equations). These equations overcome deficiencies of Grad's equations or Burnett models. They are asymptotically of super-Burnett order, i.e., of third order in the Knudsen number and linearly stable for all wave frequencies. In addition, a complete set of boundary conditions can derived from the accommodation boundary conditions of the Boltzmann equations. Mathematically, more boundary conditions are required and they can be derived from the R13 system itself through coherence relations. We present micro-channel and shock wave simulations to prove that R13 is a reliable and efficient continuum model for micro-flows of gases with moderate Knudsen numbers.

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H. Struchtrup and M. Torrilhon: Regularization and Boundary Conditions for the 13 Moment Equations
Proc.14th Conference on Waves and Stability in Continuous Media, Scicli 2007, 548-563, World Scientific, Singapore (2008) [pdf]

We summarize our recent contributions to the development of macroscopic transport equations for rarefied gas flows. A combination of the Chapman-Enskog expansion and Grad's moment method, termed as the order of magnitude method, yields the regularized 13 moment equations (R13 equations) which are of super-Burnett order. A complete set of boundary conditions is derived from the boundary conditions of the Boltzmann equations. The R13 equations are linearly stable and their results for Knudsen numbers below 0.5 stand in excellent agreement to DSMC simulations.

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H. Struchtrup: What does an ideal wall look like?
Cont. Mech. Thermodyn. 19, 493-498 (2008) [pdf] [doi:10.1007/s00161-007-0066-5]

This paper deals with the interface between a solid and an ideal gas. The surface of the solid is considered to be an ideal wall, if the flux of entropy is continuous, i.e. if the interaction between wall and gas is non-dissipative. The concept of an ideal wall is discussed within the framework of kinetic theory. In particular it is shown that a non-dissipative wall must be adiabatic and does not exerts shear stresses to the gas, if the interaction of a gas atom with the wall is not influenced by the presence of other gas atoms. It follows that temperature jumps and slip will be observed at virtually all walls, although they will be negligibly small in the hydrodynamic regime (i.e. for small Knudsen numbers).

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M. Torrilhon and H. Struchtrup: Boundary Conditions for Regularized 13-Moment-Equations for Micro-Channel-Flows
J. Comp. Phys. 227, 1982-2011 (2008) [pdf] [doi:10.1016/j.jcp.2007.10.006]

Boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro-flows. In this paper we present a theory how to combine the regularized 13-moment-equations derived from Boltzmann's equation with boundary conditions obtained from Maxwell's accommodation model. Our hypothesis is that the equations have to be adapted to the boundary conditions in a way that the number of boundary conditions required does not depend on the process. To achieve this continuity condition, the equations need to be properly transformed while keeping their asymptotic accuracy with respect to Boltzmann's equation.
After finding a suitable set of boundary conditions and equations, a numerical method for generic shear flow problems is formulated. Several test simulations demonstrate the stable and oscillation-free performance of the new approach.

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H. Struchtrup and G. Elfring: External losses in high-bypass turbo fan air engines
Int. J. Exergy, 5(4), 400-412 (2008) [pdf]

The external irreversible losses of air engines, due to equilibration of the hot and fast exhaust with the environment, are discussed based on the second law of thermodynamics. The effect of the bypass ratio on thermomechanical exergy destruction in the exhaust stream is demonstrated. The analysis gives a strong motivation for the use of high bypass turbo fan engines in modern aircraft.
Key words: bypass air engine, second law analysis, engineering education.

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H. Struchtrup and M. Torrilhon: H-theorem, regularization, and boundary conditions for linearized 13 moment equations
Phys. Rev. Lett. 99, 014502 (2007) [pdf] [doi:10.1103/PhysRevLett.99.014502]

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 to DSMC simulations. The Knudsen minimum for the relative mass flow rate is reproduced.

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T. Thatcher, Y. Zheng, and H. Struchtrup: Boundary conditions for Grad's 13 moment equations
Progress in Computational Fluid Dynamics 8(1-4), 69-83 (2008) [pdf] [doi:10.1504/PCFD.2008.018080]

A complete set of boundary conditions for Grad's 13 moment equations is derived from Maxwell's boundary conditions for the Boltzmann equation. The equations are solved for plane Couette flow. The results exhibit temperature jump and slip, and agree well with DSMC calculations for Knudsen numbers Kn≤0.1. Non-linear effects lead to unphysical results at larger Knudsen numbers, and for very fast flows. A simplified version of the Grad 13 equations, the so-called bulk equations, gives meaningful results in conditions where the full set of equations fails.

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H. Struchtrup and T. Thatcher: Bulk equations and Knudsen layers for the regularized 13 moment equations
Cont. Mech. Thermodyn. 19, 177-189 (2007) [pdf] [doi:10.1007/s00161-007-0050-0]

The order of magnitude method offers an alternative to the methods of Chapman-Enskog and Grad to derive macroscopic transport equations for rarefied gas flows. This method yields the regularized 13 moment equations (R13) and a generalization of Grad's 13 moment equations for non-Maxwellian molecules. Both sets of equations are presented and discussed. Solutions of these systems of equations are considered for steady state Couette flow. The order of magnitude method is used to further reduce the generalized Grad equations to the non-linear bulk equations, which are of second order in the Knudsen number. Knudsen layers result from the linearized R13 equations, which are of third order. Superpositions of bulk solutions and Knudsen layers show excellent agreement with DSMC calculations for Knudsen numbers up to 0.5.

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G. Elfring and H. Struchtrup: Thermodynamic Considerations on the Stability of Water in Nafion
J. Membrane Sci. 297, 190-198 (2007) [pdf] [doi:10.1016/j.memsci.2007.03.044]

This work entails modeling the thermodynamic forces contributing to the total free energy of a Nafion membrane to find how liquid water equilibrates and agglomerates inside the membrane. Since the sulfonate acid sites attract water to dissociate, there is a mixture of water and ions within the Nafion membrane. This mixture contributes an entropic term to the free energy of the system which decreases with increasing water content. The hydrophobic Nafion backbone yields a contact angle greater than 90 degrees when in contact with water. This curvature in the surface of the water induces a capillary pressure which can be very high if the size of the agglomeration is small. As the membrane takes on water, polymer strands of the Nafion are stretched and straightened reducing their configurational entropy. Minimizing the free energy under the influence of all these forces at varying pressures and temperatures gives insight into the nature of liquid or vapor filled pores throughout a Nafion membrane. Specifically it indicates a critical pore size in which liquid or vapor is the favorable state.
Keywords: Nafion, Water, Sorption, Equilibrium, Stability

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H. Struchtrup, T. Thatcher, and M. Torrilhon: Couette flow solution for regularized 13 moment equations
in 25th International Symposium on Rarefied Gas Dynamics, M.S. Ivanov, A.K. Rebrov (Eds.), Publishing House of the Siberian Branch of the Russian Academy of Sciences, 91-96 (2007) [pdf]

The order of magnitude method offers an alternative to the methods of Chapman-Enskog and Grad to derive macroscopic transport equations for rarefied gas flows. This method yields the regularized 13 moment equations (R13) which are presented and discussed. Approximate solutions of the R13 equations are considered for steady state Couette flow. The order of magnitude method is used to derive the non-linear bulk equations, which are of second order in the Knudsen number. Knudsen layers result from the linearized R13 equations, which are of third order. Superpositions of bulk solutions and Knudsen layers show excellent agreement with DSMC calculations.

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H.C. Öttinger and H. Struchtrup: The Mathematical Procedure of Coarse Graining: From Grad's Ten-Moment Equations to Hydrodynamics
Multiscale Model. Simul. 6, 53-69 (2007) [pdf] [doi:10.1137/060654700]

We employ systematic coarse graining techniques to derive hydrodynamic equations from Grad's ten-moment equations. The coarse graining procedure is designed such that it manifestly preserves the thermodynamic structure of the equations. The relevant thermodynamic structure and the coarse graining recipes suggested by statistical mechanics are described in detail and are illustrated by the example of hydrodynamics. A number of mathematical challenges associated with structure-preserving coarse graining of evolution equations for thermodynamic systems as a generalization of Hamiltonian dynamic systems are presented. Coarse graining is a key step that should always be considered before attempting to solve an equation.l.

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H. Struchtrup: Scaling and expansion of moment equations in kinetic theory
J. Stat. Phys. 125(3), 565-587 (2006) [pdf] [doi:10.1007/s10955-006-9199-3]

The set of generalized 13 moment equations for molecules interacting with power law potentials [Struchtrup, Multiscale Model. Simul. 3, 211 (2004)] forms the base for an investigation of expansion methods in the Knudsen number and other scaling parameters. The scaling parameters appear in the equations by introducing dimensionless quantities for all variables and their gradients. Only some of the scaling coefficients can be chosen independently, while others depend on these chosen scales--their size can be deduced from a Chapman-Enskog expansion, or from the principle that a single term in an equation cannot have a larger order of magnitude than all other terms.
    It is shown that for the least restrictive scaling the new order of magnitude expansion method [Struchtrup, Phys. Fluids 16(11), 3921 (2004)] reproduces the original equations after only two expansion steps, while the classical Chapman-Enskog expansion would require an infinite number of steps. Both methods yield the Euler and Navier-Stokes-Fourier equations to zeroth and first order. More restrictive scaling choices, which assume slower time scales, small velocities, or small gradients of temperature, are considered as well.

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H. Struchtrup: Model Reduction in Kinetic Theory
in Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, A. Gorban et al. (Eds.), Springer 2006 [pdf] [doi:10.1007/3-540-35888-9_14]

Methods to derive macroscopic transport equations for rarefied gases from the Boltzmann equation are presented. Featured methods include the Chapman-Enskog expansion, Grad's moment method, and the author's order of magnitude method. The resulting macroscopic equations are compared and discussed by means of simple problems, including linear stability, shock wave structures, and Couette flow.

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J. Fimrite, B. Carnes, H. Struchtrup and N. Djilali: Coupled Proton and Water Transport Modelling in Polymer Electrolyte Membranes
in Device and Materials Modeling in PEM Fuel Cells, Topics in Applied Physics Vol. 113, S. Paddison, K. Promislow (Eds.), Springer 2008 [pdf]

This Chapter presents a critical examination and analysis of classical and recently proposed models for transport phenomena in polymer electrolyte membranes, and proposes a new macroscopic model based on the generalized Stefan-Maxwell relations.
    First, key experimental observations related to membrane conductivity, membrane hydration and sorption isotherms are reviewed, and proton transport mechanisms in bulk water, and the influence of the membrane phase on these mechanisms are examined.
    Then, various formulations and underlying assumptions to account for macroscopic transport are reviewed, and an analysis of the Binary Friction Model (BFM) and Dusty Fluid Model (DFM) is performed to show that the BFM provides a physically consistent modelling framework and implicitly accounts for viscous transport (i.e. Schloegl equation), whereas the Dusty Fluid Model erroneously accounts twice for viscous transport.
    Next, the BFM framework is applied to develop a general transport model for perfluorosulfonic acid membranes. As a tool for investigating the unknown parameters in the general membrane transport model, a simplified conductivity model is derived to represent conditions found in AC impedance conductivity measurements. This Binary Friction Conductivity Model (BFCM) is applied to 1100 EW Nafion, and compared to other established membrane models, it is shown to provide a more consistent fit to the data over the entire range of water contents and at different temperatures. The subset of transport coefficients in the BFCM are the same as in the general Binary Friction Membrane Model (BFM2), and thus with additional data on water transport, the BFM2 model and all its required parameters can be fully specified.
    The Chapter closes with illustrative predictions obtained from numerical simulations coupling the BFM2 with a fuel cell model. The simulations highlight the predictive abilities of the model, particularly under low hydration conditions characteristic of ambient air-breathing fuel cells.

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Y. Zheng and H. Struchtrup: A linearization of Mieussens's discrete velocity model for kinetic equations
Eur. J. Mechanics B/Fluids 26(2), 182-192 (2007) [pdf] [doi:10.1016/j.euromechflu.2006.08.003]

An approximate numerical method is developed for Mieussens' discrete velocity model [see, e.g., L. Mieussens, Journal of Computational Physics 162 (2000) 429-466]. The basic idea is to use a linearized expression of the reference distribution function in the kinetic equation, instead of its exact expression, in the numerical scheme. This modified scheme is applied to various kinetic models, which include the BGK model, the ES-BGK model, the BGK model with velocity-dependent collision frequency, and the recently proposed ES-BGK model with velocity-dependent collision frequency. One-dimensional stationary shock waves and stationary planar Couette flow, which are two benchmark problems for rarefied gas flows, are chosen as test examples. Molecules are modeled as Maxwell molecules and hard sphere molecules. It is found that results from the modified scheme are very similar to results from the original Mieussens' numerical scheme for almost all tests, while 20-40 percent of computational time can be saved.
Keywords: rarefied gas dynamics / kinetic equation / discrete velocity model / shock waves / Couette flow

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Y. Zheng, J. Reese, and H. Struchtrup: Comparison of macroscopic continuum models in rarefied gas dynamics: A test method
J. Comp. Phys. 218(2), 748-769 (2006) [pdf] [doi:10.1016/j.jcp.2006.03.005]

In this work, a test method is presented to examine which macroscopic continuum model among the many existing models gives a good description of rarefied gas flows, e.g., in relation to the values of the Knudsen number. The merits of the proposed method are that no boundary conditions for continuum models are needed, and no coupled governing equations are solved, while the Knudsen layer in complex flows is considered nevertheless. This distinguishes the proposed test method from other existing techniques for the comparison of macroscopic continuum models in rarefied gas dynamics, such as stability analysis in time and space, computations of sound speed and dispersion, and the computation of shock wave structures.
     The method relies on accurate noise-free solutions of the basic microscopic kinetic equation, e.g. the Boltzmann equation or a kinetic model equation, and in this paper the BGK model and the ES-BGK model equations are considered.
     The method is applied to test the feasibility for the description of one-dimensional stationary Couette flow of the following macroscopic transport models: Navier-Stokes-Fourier equations, Burnett equations, Grad's 13 moment equations, and Regularized 13 moment equations. Gas molecules are modeled as Maxwell molecules. For not too large Knudsen numbers (Kn<=0.1) in the transition regime, it is found that the Regularized 13 moment equations give better results than Grad's original 13 moment equations, which, however, give better results than the Burnett equations, while the Navier-Stokes-Fourier equations give the worst results, which is in agreement with expectation based on the order of the Knudsen number of the models tested. For larger Knudsen numbers, i.e. Kn>0.1, all macroscopic continuum equations tested fail in the accurate description of flows. It is also realized that conclusions from the tests are general, independent of the kinetic model used.

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Y. Zheng and H. Struchtrup: Ellipsoidal statistical Bhatnagar-Gross-Krook model with velocity-dependent collision frequency
Phys. Fluids 17, 127103 (2005) [pdf] [doi:10.1063/1.2140710]

In this paper, an ellipsoidal statistical (ES) Bhatnagar-Gross-Krook (BGK)-type kinetic model with velocity-dependent collision frequency is proposed and further numerically tested for one-dimensional shock waves and planar Couette flow at steady state for hard sphere molecules. In this new kinetic model, a physically meaningful expression for the velocity-dependent collision frequency derived from the Boltzmann equation is used, while the important properties for a kinetic model are retained at the same time. This kinetic model can be simplified to the classical ES-BGK model and the BGK model with velocity-dependent collision frequency for suitable choices of parameters. The H theorem for this new kinetic model has so far been proven only for small Knudsen numbers. The numerical method used here for kinetic models is based on Mieussens's discrete velocity model [L. Mieussens, J. Comput. Phys. 162, 429 (2000)]. Computational results from the kinetic models (including the BGK model, the ES-BGK model, the BGK model with velocity-dependent collision frequency, and this new kinetic model) are compared to results obtained from the direct simulation Monte Carlo (DSMC) method. It is found that results obtained from this new kinetic model lie in between results from the ES-BGK model and results from the BGK model with velocity-dependent collision frequency. For one-dimensional shock waves, results from this new kinetic model fit best with results from the DSMC, while for planar Couette flow, the classical ES-BGK model is suggested.

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M.W. Bond and H. Struchtrup: Mean evaporation and condensation coefficients based on energy dependent condensation probability
Phys. Rev. E 70, 061605 (2004) [pdf] [doi:10.1103/PhysRevE.70.061605]

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 [Fang&Ward, Phys. Rev. E 59, 419]. 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 to the experiment from the generalized kinetic theory model, and the 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.

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J. Fimrite, B. Carnes, H. Struchtrup, and N. Djilali: Transport phenomena in polymer electrolyte membranes. II. Binary friction membrane model
J. Electrochem. Soc. 152(9), A1804-A1814 (2005) [pdf] [doi:10.1149/1.1952647]

The insight gained from the analysis conducted in Part I is used in the development of a general transport model for water and protons in perfluorosulfonic acid membranes based on the Binary Friction Model.  As a tool for investigating the unknown parameters in the general membrane transport model, a simplified conductivity model is derived to represent conditions found in AC impedance conductivity measurements. This Binary Friction Conductivity Model (BFCM) is applied to 1100 EW Nafion, and compared to other established membrane models, it is shown to provide a more consistent fit to the data over the entire range of water contents and at different temperatures.
The subset of transport coefficients in the BFCM are the same as in the general Binary Friction Membrane Model (BFM2), and thus with additional data on water transport, the BFM2 model and all its required parameters can be fully specified. The paper discusses possible experimental investigations and fundamental simulations to determine the model parameters required to apply the general BFM2 to predict coupled proton and water transport in PEM fuel cells.

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J. Fimrite, H. Struchtrup and N. Djilali: Transport phenomena in polymer electrolyte membranes. I. Modeling framework
J. Electrochem. Soc. 152(9), A1815-A1823 (2005) [pdf] [doi:10.1149/1.1952627]

This paper presents a critical examination and analysis of classical and recently proposed models for transport phenomena in polymer electrolyte membranes. Key experimental observations related to membrane conductivity, membrane hydration and sorption isotherms are first reviewed. Proton transport mechanisms in bulk water, and the influence of the membrane phase on these mechanisms are examined. Finally various formulations and underlying assumptions to account for macroscopic transport are reviewed, and an analysis of the Binary Friction Model (BFM) and Dusty Fluid Model (DFM) is performed to resolve an outstanding formulation issue. It is shown that the BFM provides a physically consistent modeling framework and implicitly accounts for viscous transport (i.e. Schloegl equation), whereas the Dusty Fluid Model erroneously accounts twice for viscous transport. In Part II we apply the BFM framework to develop a general transport model for perfluorosulfonic acid membranes. 

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H. Struchtrup and M. Torrilhon:  Regularized 13 Moment Equations for Rarefied Gas Flows   
in Trends and Applications of Mathematics to Mechanics, S. Rionero, G. Romano (Eds), pp. 247- 267
Springer, Milano (2005), ISBN: 88-470-0269-9 [pdf]  [doi:10.1007/88-470-0354-7_19]

A new closure for Grad's 13 moment equations is presented that adds terms of Super-Burnett order to the balances of pressure deviator and heat flux vector. The resulting system of equations contains the Burnett and Super-Burnett equations when expanded in a series in the Knudsen number. However, other than the Burnett and Super-Burnett equations, the new set of equations is linearly stable for all wavelengths and frequencies. Dispersion relation and damping for the new equations agree better with experimental data than those for the Navier-Stokes-Fourier equations, or the original 13 moments system. The new equations allow the description of Knudsen boundary layers., and yield smooth shock structures for all Mach numbers in good agreement with experiments and DSMC simulations.

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H. Struchtrup:  Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials
Multiscale Model. Simul. 3(1), 221-243 (2005) [pdf] [doi:10.1137/040603115]

A recent approach to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number (Struchtrup, 2003) is used to derive a set of 13 moment equations for arbitrary molecular interaction potentials. It is shown that the new set of equations is accurate to second order, while Grad's original 13 moment equations are of second order accuracy only for Maxwell molecules and BGK models.

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H. Struchtrup:  Macroscopic models for rarefied gas flows
Oberwolfach Reports 1(4), 3014-3017 (2004) [pdf]

(extended abstract).

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H. Struchtrup:  Stable transport equations for rarefied gases at high orders in the Knudsen number
Phys. Fluids 16(11), 3921-3934 (2004) [pdf] [doi:10.1063/1.1782751]

A new approach is presented to derive transport equations for rarefied gases from the Boltzmann equation within higher orders of the Knudsen number. The method focuses on the order of magnitude of the moments of the phase density, and the order of accuracy of the transport equations, both measured in powers of the Knudsen number. The method is developed up to the third order, and it is shown that it yields the Euler equations at zeroth order, the Navier-Stokes-Fourier equations at second order, Grad's 13 moment equations (with omission of a non-linear term) at second order, and a regularization of these at third order. The method is discussed in detail, and compared with the classical methods of kinetic theory, i.e. Chapman-Enskog expansion and Grad moment method. The advantages of the new method above the classical approaches are discussed conclusively. An important feature of the method presented is that the equations of any order are stable, other than in the Chapman-Enskog method, where the second and third approximation - Burnett and super-Burnett equations - are unstable.

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H. Struchtrup:  Failures of the Burnett and Super-Burnett equations in steady state processes
Cont. Mech. Thermodyn. 17(1), 43-50 (2005) [pdf] [doi:10.1007/s00161-004-0186-0]

Linearized Burnett and Super-Burnett equations are considered for steady state Couette flow. It is shown that the linear Super-Burnett equations lead to periodic velocity and temperature curves, i.e. unphysical solutions. The problem is discussed as well for the so-called augmented Burnett equations by Zhong et al. (AIAA Journal 31, 1036-1043 (1993)), and for the recently introduced regularized 13 moment equations (R13) of Struchtrup and Torrilhon (Phys. Fluids 15(9), 2668-2680 (2003) ). It is shown that both theories exhibit proper Knudsen boundary layers for velocity and temperature. However, the heat flux parallel to the wall has different signs for the Burnett and the R13 equations, and a comparison with DSMC results shows that only the R13 equations predict the proper sign.

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Y. Zheng and H. Struchtrup:  Burnett Equations for the Ellipsoidal Statistical BGK Model
 Cont. Mech. Thermodyn. 16(1-2), 97-108 (2004) [pdf] [doi:10.1007/s00161-003-0143-3]

In order to discuss the agreement of the ellipsoidal statistical BGK (ES-BGK) model with the Boltzmann equation, Burnett equations are computed by means of the second order Chapman-Enskog expansion of the ES-BGK model. It is found that the Burnett equations for the ES-BGK model with proper Prandtl number are identical to the Burnett equations for the Boltzmann equation for Maxwell molecules (fifth order power potentials). However, for other types of particle interaction, the Boltzmann Burnett equations can not be reproduced from the ES-BGK model.
Furthermore, the linear stability of the ES-BGK-Burnett equations is discussed. It is shown that the ES-BGK Burnett equation are linearly stable for Prandtl numbers in 1<=Pr <= 5/4 and for Pr = infinity , while they are linearly unstable for 2/3 < Pr < 1 and 5/4 < Pr < infinity

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L. Mieussens and H. Struchtrup:  Numerical comparison of BGK-models with proper Prandtl number
Phys. Fluids 16(8), 2797-2813 (2004) [pdf] [doi:10.1063/1.1758217]

While the standard BGK model leads to the wrong Prandtl number, the BGK-model with velocity dependent collision frequency as well as the Ellipsoidal Statistical BGK model can be adjusted to give its proper value of 2/3. In this paper, the BGK model with velocity dependent collision frequency is considered in some detail. The corresponding thermal conductivity and viscosity are computed from the Chapman-Enskog method, and several velocity-dependent collision frequencies are introduced which all give the proper Prandtl number. The models are tested for Couette flow and the shock structure problem, and the results are compared to solutions obtained with the ES-BGK model, and the Direct Simulation Monte Carlo method. The simulations rely on a numerical scheme that ensures positivity of solutions, conservation of moments, and dissipation of entropy. The advantages and disadvantages of the various BGK models are discussed.

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M. Torrilhon and H. Struchtrup:  Regularized 13-Moment-Equations: Shock Structure Calculations and Comparison to Burnett Models
J. Fluid Mech. 513, 171-198 (2004) [pdf] [doi:10.1017/S0022112004009917]

Only recently a new system of field equations for the accurate description of flows in rarefied gases, called regularized 13-moment-equations, was obtained by means of a hybrid gas kinetic approach. The first part of this paper discusses the relationship of the new system to classical high order theories like Burnett and super-Burnett equations as well as to modified models like the augmented and regularized Burnett equations. In the second part, shock structure calculations with the new theory are presented and compared to DSMC solutions and to solutions of the Burnett models. Due to additional higher order dissipation in the system, the profiles are smooth for any Mach number in contrast to the results of Grad's 13-moment-case. The results show reliable quantitative agreement with DSMC simulations for Mach numbers up to M₀≈3.0. The agreement is better for Maxwell molecules than for hard spheres. The results of the augmented Burnett equations are comparable, but these equations are shown to be spatially unstable. Additionally, a validiation procedure for the new equations is presented by investigating the positivity of Grad's distribution function.

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H. Struchtrup and M. Torrilhon: Regularization of Grad's 13 Moment Equations: Derivation and Linear Analysis
Phys. Fluids 15(9), 2668-2680 (2003) [pdf] [doi:10.1063/1.1597472]

A new closure for Grad's 13 moment equations is presented that adds terms of Super-Burnett order to the balances of pressure deviator and heat flux vector. The additional terms are derived from equations for higher moments by means of the distribution function for 13 moments. The resulting system of equations contains the Burnett and Super-Burnett equations when expanded in a series in the Knudsen number. However, other than the Burnett and Super-Burnett equations, the new set of equations is linearly stable for all wavelengths and frequencies. Dispersion relation and damping for the new equations agree better with experimental data than those for the Navier-Stokes-Fourier equations, or the original 13 moments system. The new equations also allow the description of Knudsen boundary layers.

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H. Struchtrup: Grad's Moment Equations for Microscale Flows
Symposium on Rarefied Gasdynamics 23, AIP Conference Proceedings 663, 792-799 (2003) [pdf] [doi:10.1063/1.1581623]

Grad's moment equations are discussed for application to microscale flows in the transition regime. While Grad's 13 moment equations as well as Hilbert and Chapman-Enskog expansions of the Boltzmann equation cannot resolve the Knudsen boundary layer at the wall, this is different for moment theories with extended sets of moments. Since for channel flow with Knudsen numbers above 0.1 the Knudsen layer extends over the whole channel width, theories with more than 13 moments can be expected to give a more accurate description. Various sets of moment equations (up to 48 moments in the one-dimensional case) are considered for one-dimensional heat transfer in order to show the usefulness as well as the limitations of Grad's moment equations for microscale flows. The results give evidence that approaches with increasing moment number allow for the resolution of finer details of the Knudsen layer.

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L. Mieussens and H. Struchtrup:  Numerical Solutions For The BGK-Model With Velocity-Dependent Collision Frequency
Symposium on Rarefied Gasdynamics 23, AIP Conference Proceedings 663, 320-327 (2003) [pdf] [doi:10.1063/1.1581566]

The BGK-model with velocity dependent collision frequency is discussed and applied to Couette flow and the shock structure problem. Thermal conductivity and viscosity are computed from the Chapman-Enskog method and several velocity-dependent collision frequencies are introduced which all give the proper Prandtl number. The models are tested and compared to results from the Direct Simulation Monte Carlo method. The simulations rely on a numerical scheme that ensures positivity of solutions, conservation of moments, and dissipation of entropy. The advantages and disadvantages of the various BGK models are discussed.

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M.Torrilhon, J. D. Au, and H. Struchtrup:  Explicit Fluxes and Productions for Large Systems of the Moment Method based on Extended Thermodynamics
Continuum Mech Thermodyn 15 (1) , 97-111 (2003) [pdf] [doi:10.1007/s00161-002-0107-z]

The moment method of kinetic theory solves Boltzmann's equation approximately via an infinite hierarchy of transfer equations for the moments of the distribution function. Extended themodynamics furnishes the moment method with a rational constitutive theory. Since more and more moment equations are needed to describe extreme non-equilibrium processes, there is need for an algorithmical derivation of large explicit moment equations.
This paper presents detailed techniques and formulas which are needed to implement a numerical equation generator. This includes tensorial conversion formulas as well as the core equations of the constitutive theory. In the last part of the paper the special case of a one dimensional process is discussed. In such a case only one generic polynomial evaluation needs to be implemented, whereas the coefficients may be easily calculated a priori.

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I. Müller and H. Struchtrup: Inflating a Rubber Balloon
Mathematics and Mechanics of Solids 7 (5), 569-577 (2002) [pdf] [doi:10.1177/108128650200700506]

A spherical balloon has a non-monotonic pressure-radius characteristic. This fact leads to interesting stability properties when two balloons of different radii are put in contact, see [1], [2], [3]. Here, however, we investigate what happens when a single balloon is inflated by mouth (say). We simulate that process and show how the maximum of the pressure-radius characteristic is overcome by the pressure in the lungs and how the downward sloping part of the characteristic is "bridged" while the lung pressure relaxes.
Keywords:  Rubber balloons, Mooney-Rivlin material, Non-convexity, Stability.

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H. Struchtrup and M.A. Rosen: How much work is lost in an irreversible turbine?
eXergy - An International Journal 2(3), 152-158 (2002) [pdf] [doi:10.1016/S1164-0235(02)00068-7]

The question of how much work is lost in an adiabatic turbine due to its irrerversibilities finds different answers when discussed on basis of the isentropic efficiency, or with the exergy method. In this contribution, we seek to clarify why the two viewpoints lead to quite distinct results for the lost work. In particular, we discuss how the "reversible work" of the exergy method could be realized and how to recover the "recoverable work of friction." The difference between both approaches is explained.

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Y. Efendiev, H. Struchtrup, M. Luskin, and M.R. Zachariah: A Hybrid Sectional-Moment Model for Coagulation and Phase Segregation in Binary Liquid Nanodroplets
J. Nanoparticle Research 4, 61-72 (2002) [pdf] [doi:10.1023/A:1020122403428]

We describe a new formulation of the aerosol general dynamic equation (GDE) that incorporates the phase segregation in a binary aerosol. The model assumes that complete phase segregation is the thermodynamically favored state, that no thermodynamic activation energy exists, and that the segregation process is kinetically controlled. We develop a GDE formulation that involves the solution of a distribution function N_n, s(V), where N_n,s(V) is the number density of aerosols with volume V and n phase domains (which we might think of as enclosures) with an enclosure size distribution characterized by s. The model improves our earlier efforts which did not account for the enclosure size distribution. The description of the enclosures is based on a moment approach relying on a log-normal distribution. Numerical solutions are presented.

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H. Struchtrup: Heat Transfer in the Transition Regime: Solution of Boundary Value Problems for Grad's Moment Equations via Kinetic Schemes
Phys. Rev. E 65, 041204 (2002) [pdf] [doi:10.1103/PhysRevE.65.041204]

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. The problem of describing boundary conditions for the moments is solved by the use of the so-called kinetic schemes which allow the implementation of the boundary condition for the Boltzmann equation. The results, obtained with up to 430 moments, exhibit temperature jumps at the walls with adjacent 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.
Keywords: Kinetic theory of gases, Boltzmann equation, moment method, boundary conditions, boundary layers

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H. Struchtrup: Some remarks on the equations of Burnett and Grad
Transport in Transition Regimes,  IMA Volume Series 135, Springer, New York 2003 [pdf]

Both, Grad and Burnett, derived sets of equations from the Boltzmann equation, which improve the classical laws of Navier-Stokes and Fourier for the description of rarefied gases, i.e. gases with Knudsen numbers above 0.01. Using results of other authors, it is shown that both sets of equations are closer related then is commonly thought - indeed, the Burnett equations can be derived from Grad's equations by the so-called Maxwellian iteration. This derivation allows to identify the proper form of the Burnett equations in non-inertial frames. Moreover, Grad's equations with more than 13 moments can describe linear boundary layers while these are not among the phenomena which can be described by Burnett's equations.

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H. Struchtrup, M. Luskin, and M. Zachariah:  A Model for Kinetically Controlled Internal Phase Segregation During Aerosol Coagulation
 J. Aerosol Science 32, 1479-1504 (2001) [pdf] [doi:10.1016/S0021-8502(01)00068-4]

In previous studies of particle growth, we have synthesized binary metal oxide aerosols and have observed the evolution of internal phase segregation during growth of molten nanodroplets. We describe a new formulation of the aerosol general dynamic equation (GDE) that incorporates the phase segregation in a binary aerosol. The model assumes that complete phase segregation is the thermodynamically favored state, that no thermodynamic activation energy exists, and that the segregation process is kinetically controlled. We develop a GDE formulation that involves the solution of a distribution function N_n(V), where N_n(V) is the number density of aerosols with volume V and n phase domains (which we might think of as enclosures). The GDE is solved using a 2-D sectional model and under the assumption that the phase coalescence of the minority phase is controlled by Brownian coagulation. For the purposes of these initial studies, the rate laws governing the enclosures (minority phase) assume a monodisperse particle size distribution. The dynamical behavior of such a system is presented.

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S.F. Liotta and H.Struchtrup: Comparison of spherical harmonics and moment equations for electrons in semiconductors
Proc.10th Conference on Waves and Stability in Continuous Media, Vulcano 1999
Wolrd Scientific, Singapore (2001) [pdf]

The semiclassical Boltzmann equation for electrons in semiconductors is considered together with the parabolic band approximation and interaction terms for elastic scattering with acoustic phonons and inelastic scattering with optical phonons.
Taking only scalar and vectorial moments into account, two sets of equations are derived from the Boltzmann equation: spherical harmonics equations and equations for full moments.
The equations are solved for two simple processes in an infinite semiconductor in a homogeneous electric field. The results show that both moment systems agree, if the number of full moments exceeds the usual choice of hydrodynamical models. .
Keywords: Electron transport, Boltzmann equation, Spherical harmonics, Moments

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 H. Struchtrup: Positivity of entropy production and phase density in the Chapman-Enskog expansion
J. of Thermophysics and Heat transfer 15(3), 372-373 (2001) [pdf]

In a recent paper, Comeaux et al. [Comeaux, K.A., Chapman, D.R. and MacCormack, R.W., ``An Analysis of the Burnett Equations Based on the Second Law of Thermodynamics,'' AIAA Paper 95-0415, 1995]  showed that the entropy production according to the Burnett equations may become negative. This result stands in contradiction to Boltzmann's H-theorem which states that the entropy production is positve for any distribution function f. In this short note we show that the negative production of entropy results from the use of approximative solutions of the Boltzmann equation outside their proper range, and improper mathematics, i.e. a series expansion which does not converge outside that range.

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H. Struchtrup and J.W. Dold: Surface tension in a reactive binary mixture of incompressible fluids
IMA preprint 1708 (2000) [pdf]

Based on the Cahn-Hilliard free energy, a thermodynamic model for a reactive binary mixture of incompressible and miscible fluids is derived with a distributed form of surface tension. The model describes chemistry, diffusion, viscosity and heat transfer as well as stresses produced by (and at right angles to) concentration gradients. It allows for a rich spectrum of processes and some of these are briefly discussed.
Scaling arguments based on experimental data of Abid et al. (submitted to Phys. Rev. Lett.), lead to considerable simplifications. Saffmann-Taylor stability analysis (here extended to interfaces of finite thickness) agrees with the experimental findings of Abid et al. who argued that the buoyant instability of a reaction diffusion front is controlled by a form of surface tension at the front.

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H. Struchtrup and W. Weiss: Temperature jump and velocity slip in the moment method
Cont. Mech. Thermodyn. 12, 1-18 (2000) [pdf] [doi: 10.1007/s001610050119]

The moment method of kinetic theory requires boundary conditions for the moments. It is not possible to derive these in an easy manner from the boundary conditions for the phase density. The conservation laws of mass, momentum and energy give only five relations between the moments and the properties of the wall. Additional boundary conditions may be determined from the minimax principle for the entropy production which was recently proposed by Struchtrup & Weiss [Phys. Rev. Lett. 80, 5048-5051 (1998)]. These ideas are outlined for the case of 13 and 14 moments and Maxwell's boundary conditions for the phase density which lead to temperature jumps and velocity slip at walls. In particular, one-dimensional stationary heat transfer between two walls at rest is considered. The temperature jumps at the walls are shown to depend on the values of all moments in front of the wall. The results obtained by the minimax principle are compared with results obtained for the same problem by a minimum principle for the global entropy production and by the so-called kinetic schemes .

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H. Struchtrup: Kinetic schemes and boundary conditions for moment equations
ZAMP 51, 346-365 (2000) [pdf] [doi: 10.1007/s000330050002]
also as ESI preprint No. 646 ( https://www.esi.ac.at/ESI-Preprints.html )

A numerical scheme for moment equations of kinetic theory, due to LeTallec & Perlat, is considered for the calculation of stationary heat transfer in the Grad 13 moment system and linearized extended thermodynamics of 14 moments. It is shown that the required distance of grid points must be considerably smaller than the mean free path. Thus, the kinetic scheme is useful only in the case of large Knudsen numbers. Results of the numerical calculation for 13 and 14 moments are compared with an analytical solution for heat transfer with 13 moments. The results indicate that the boundary conditions do not guarantee conservation of energy at the walls. In order to overcome this deficiency a modification of the boundary conditions is presented and discussed.

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S.F. Liotta and H. Struchtrup: Moment equations for electrons in semiconductors: comparison of spherical harmonics and full moments
Solid-State Electronics 44, 95-103 (2000) [pdf] [doi:10.1016/S0038-1101(99)00215-4]

The semiclassical Boltzmann equation for electrons in semiconductors is considered with the parabolic band approximation and interaction terms for elastic scattering with acoustic phonons and inelastic scattering with optical phonons.
Taking only scalar and vectorial moments into account, two sets of equations are derived from the Boltzmann equation: spherical harmonics equations and equations for full moments.
The equations are solved for two simple processes in an infinite semiconductor in a homogeneous electric field. The results show that both moment systems agree, if the number of full moments is sufficiently high.
Keywords: Electron transport, Boltzmann equation, Spherical harmonics, Moments

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H. Struchtrup: Extended moment method for electrons in semiconductors
Physica A 275, 229-255 (2000) [pdf] [doi:10.1016/S0378-4371(99)00418-5]

The semiclassical Boltzmann equation for electrons in semiconductors with the parabolic band approximation is considered. The interaction term models elastic scattering with acoustic phonons as well as inelastic scattering with optical phonons. From the Boltzmann equation systems of moment equations with an arbitrary number of moments are derived.
First, the paper deals with spherical harmonics in the formalism of symmetric trace-free tensors. The spherical harmonics equations are derived and the collision frequencies are carefully studied for the physical properties of silicon.
Then, the hierarchy of equations for full moments of the phase density and the corresponding closure problem is discussed. In particular, a set of 2R scalar and vectorial moments is considered; this choice of moments is appropriate in the case of the so-called SHE limit, where the phase density is almost isotropic. For R=1, the set of moments resembles the hydrodynamic model.
To answer the question which number R one has to chose in order to retain the physical contents of the Boltzmann equation or the SHE-model, the moment equations are examined in the drift-diffusion limit and in an infinite crystal in a homogeneous electric field for increasing number of moments R. The number R must be considered to be sufficient, if its further increase does not change the result considerably and the appropriate numbers for the processes are given.

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H. Struchtrup: The BGK Model for an Ideal Gas with an Internal Degree of Freedom
Transp. Theory Stat. Phys. 28(4), 369-385 (1999) [pdf] [doi:10.1080/00411459908205849]

The BGK model for an ideal gas with an internal degree of freedom introduces two characteristic times into the Boltzmann equation, accounting for the times between elastic and inelastic collisions, respectively. Moment equations are derived from the Boltzmann equation and the influence of the characteristic times is studied for some processes, such as heating, sound waves and shocks.

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H. Struchtrup: Projected Moments in Relativistic Kinetic Theory
Physica A 253, 555-593 (1998) [pdf] [doi:10.1016/S0378-4371(98)00037-5]

In this paper a new set of moment equations in relativistic kinetic theory is presented. The moments under consideration are the projections of particle 4-flux and energy momentum tensor with respect to the Eckart velocity or the Landau-Lifshitz velocity, alternatively. The moment equations follow from integrations of the relativistic Boltzmann equation in which the interactions of the particles are described by the relativistic BGK model for reasons of simplicity.
The projected moment formalism is extended to an arbitrary number of moments and moment equations and it is shown that the non-relativistic limit of moments and moments equations leads to the so-called central moments of non-relativistic theory.
The moment equations may be closed by means of the entropy maximum principle. After this method has been outlined, the closure is performed for the case of 14 moments, i.e. the projections of particle 4-flux and energy momentum tensor.
Moreover local thermal equilibrium is considered where the projected moment formalism is used for the derivation of the relativistic Navier-Stokes and Fourier laws. Different choices of moment equations for this task are compared and it is shown that the proper choice of moment equations depends on the interaction term in the relativistic Boltzmann equation.

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H. Struchtrup: On the Number of Moments in Radiative Transfer Problems
Annals of Physics 266, 1-26 (1998) [pdf] [doi:10.1006/aphy.1998.5791]

In a recent paper we have set up an extended moment method for radiative transfer problems, which involves matrices of mean absorption and scattering coefficients. In the present paper, we examine the resulting moment equations for one-dimensional radiative transfer problems. In particular we are interested in the number of moments one has to choose in order to have a satisfactory agreement between solutions of the moment equations and solutions of the radiative transfer equation. We show that the moment theory will describe a one-dimensional beam properly if moments with a tensorial rank of about 30 are taken into account.

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  H. Struchtrup and W. Weiss: Struchtrup and Weiss Reply on a comment by Castillo and Hoover [Phys. Rev. Lett 81, 5700 (1998)]
Phys. Rev. Lett. 81, 5701 (1998) [doi:10.1103/PhysRevLett.81.5701]

Comment and Reply concern the application of the minimax principle "The Maximum of the local entropy production becomes minimal in stationary processes" [Struchtrup & Weiss, Phys. Rev. Lett. 80, 5048-5051 (1998)] to unstable flows described by the Navier-Stokes equations.
The Reply may be summarized in three statements:
(1) The minimax principle cannot be used for the determination of boundary conditions for the Navier-Stokes equations, since all necessary boundary conditions are controlled in experiment
(2) The minimax principle does not replace any stability criterion and should not be used for stability analysis
(3) The future description of convection flows by extended moment methods will require the use of the minimax principle and a stability analysis
PACS number(s): 47.27.Cn, 05.70.Ln, 47.15.Fe, 47.27.Eq

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H. Struchtrup and W. Weiss: The Maximum of the local entropy production becomes minimal in stationary processes
Phys. Rev. Lett. 80, 5048-5051 (1998) [pdf] [doi:10.1103/PhysRevLett.80.5048]

In this paper we propose a new principle for stationary thermodynamic processes: The maximum of the local entropy production becomes minimal in stationary processes. In order to show the usefulness of the principle, we consider one-dimensional stationary heat transfer in monatomic gases. Here we solve extended moment schemes that follow from the Boltzmann equation. Such schemes require boundary conditions for all moments under consideration and these are constructed by means of the new principle. Moreover we show that the minimum principle for the global entropy production does not lead to good results in this case.
PACS number(s): 05.70.Ln, 47.45.-n, 51.10.+y

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H. Struchtrup: Extended Thermodynamics of Phonons
Chapter 14 in I. Müller, T. Ruggeri: Rational Extended Thermodynamics, Springer New York (1998)

Phonons are much like photons which - as we have seen - are much like atoms of a gas. Infact phonons are much more like particles of a gas than photons, because they can exchange energy and momentum among themselves, which photons cannot do. Because of the similarity to gases it is possible to formulate a kinetic theory of the phonon gas, based on a phonon transfer equation. Peierls was the first to do that. And that equation may serve as the starting point for an extended thermodynamics of phononic moments. This theory was developed by Struchtrup and Dreyer & Struchtrup.
Extended Thermodynamics of phonons is used here to describe the thermal propagation that occurs in low-temperature crystals, in particular in the so-called heat-pulse experiments. The theory covers most phenomena that are observed in that experiment, namely: ballistic phonons, second sound and thermal diffusion. Either one of these phenomena has its proper range and their ranges depend on the mean free paths of phonons between interactions among themselves and the impurities of the body.

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H. Struchtrup: Extended Thermodynamics of Radiation
Chapter 13 in I. Müller, T. Ruggeri: Rational Extended Thermodynamics, Springer New York (1998)

It is useful to think of radiation as a photon gas which is governed by a transfer equation much like the Boltzmann equation for an atomic gas, albeit with a different production term. This makes it possible to develop an extended thermodynamics of photons.
In many respects this theory is similar to extended thermodynamics of moments of atoms. Thus we have moments and moment equations, we have a closure problem and this may be solved by maximizing the entropy. Given absorption and scattering mechanisms, we obtain an explicit set of field equations which can be applied to non-equilibrium situations.
The photonic moments lend themselves for an easy distinction of isotropic non-equilibrium, in which the frequency distribution of the photons deviates from the Planck distribution, and non-equilibrium through anisotropy like in a sun-beam. Both are generally coupled, of course, but there are illustrative phenomena where they are not.
As usual the essential question is where to close the hierarchy - or hierarchies - of moments and we answer that question for three suggestive problems: local equilibrium, sudden compression of a photon gas and penetration of a light beam into matter.
It is not common in thermodynamics to deal with radiation as a part of thermodynamic systems; in particular the role which the photons play in the entropy, the entropy flux and the entropy production is not always well-understood. This problem is discussed in the last section of this chapter, albeit only for gray bodies.
While the bulk of this chapter is due to the work of Struchtrup , the last section represents work by Müller.

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H. Struchtrup: An Extended Moment Method in Radiative Transfer: The Matrices of Mean Absorption and Scattering Coefficients
Annals of Physics 257, 111-135 (1997) [pdf] [doi:10.1006/aphy.1997.5684]

In extension of the ideas of Anderson & Spiegel the radiative transfer equation is replaced by moment equations for the moments

A_r^A₁^A₂^...A_N = Integral[( p⁰_RL)^r+1-N p^A₁ p^A₂ ... p^A_N f ]dP

r=0,1,...,R. Here p^A is the photon 4-momentum, p⁰_RL is the photon energy in the rest Lorentz frame and f is the photon phase density. From these follow moment equations for the projected symmetric trace free (PSTF) moments introduced by Thorne. The required closure of the equations is achieved by use of a series expansion of the phase density which is motivated by the entropy maximum principle. This procedure provides a coupling of the moment equations by means of matrices of mean absorption and scattering coefficients. It is shown that the extension from r=1 (Anderson & Spiegel, Thorne, Schweizer) to r=0,1,...,R (this paper) gives reasonable results: In the limit of local radiative equilibrium (LRE) the well-known Rosseland mean of the absorption coefficient is recovered. For a simple non-LRE experiment, the homogeneous compression and relaxation of radiation, the radiative transfer equation and the moment equations are solved. The comparison of the results in the case of pure bremsstrahlung (free-free) absorption shows an excellent agreement for R>=6.

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H. Struchtrup: Levermore Eddington Factor and Entropy Maximization
(unpublished) [pdf]

We discuss the physical significance of a special Eddington factor, introduced into radiative transfer theory by Levermore (1983) and obtained also by Anile et.al. (1991) and Kremer&Müller (1992). Since that Eddington factor follows from the maximization of entropy, it is not suitable for the description of radiation beams. Indeed, there are only few physical situations in which it may play a role.
PACS 95.30.Jx/42.86.Ay/44.40.+a (Radiative Transfer)

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H. Struchtrup: The BGK-model with velocity-dependent collision frequency (short version)
Proceedings of the IX Intenational Conference on Waves and Stability in Continuous Media, Bari 1997
Rendiconti Del Circolo Matematico di Palermo, Serie II, Suppl. 57, 471-475 (1998)

We consider the BGK-model with velocity dependent collision frequency. By use of the Chapman-Enskog method we calculate thermal conductivity and viscosity. We show that a simple power law for the collision frequency may lead to the proper Prandtl number.

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H. Struchtrup: The BGK-model with velocity-dependent collision frequency
Cont. Mech. Thermodyn. 9(1), 23-32 (1997) [pdf] [doi: 10.1007/s001610050053]

We consider the BGK-model with velocity dependent collision frequency. By use of the Chapman-Enskog method we calculate thermal conductivity and viscosity. We show that a simple power law for the collision frequency may lead to the proper Prandtl number. Moreover we use Grad's moment method to calculate thermal conductivity and viscosity. We show that the results of both methods coincide if Grad's method is based on a large number of moments.

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H. Struchtrup: Zwei homogene irreversible Prozesse
Facetten der Thermodynamik, Prof. I. Müller zum 60. Geburtstag
H. Struchtrup (Ed.), Institut fuer Verfahrenstechnik der TU Berlin, 1997 [pdf]

Von Zeit zu Zeit gibt Prof. Ingo Müller seinen Mitarbeitern kleinere Aufgaben, deren Lösung ihn interessiert. Zwei dieser Probleme werden im vorliegenden Artikel behandelt. Bei den betrachteten Prozessen handelt es sich um homogene irreversible Prozesse, die nicht zuletzt auch für den Anfänger der Thermodynamik irreversibler Prozesse einigen didaktischen Wert haben mögen. Behandelt werden die Plötzliche Kompression von Naßdampf und ein auf einem Gas schwingender Kolben mit Dämpfung durch irreversiblen Wärmeübergang.

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H. Struchtrup: Zur irreversiblen Thermodynamik der Strahlung
Dissertation, Technische Universität Berlin, 1996 [pdf]

Das wesentliche Ziel der irreversiblen Strahlungsthermodynamik ist die Berechnung des Transportes von Energie und Impuls durch Strahlung sowie des Austausches von Energie und Impuls zwischen Strahlung und Materie. Ein weiteres Ziel ist die Berechnung anderer in der Thermodynamik wichtiger Grössen, etwa der Entropie, des Entropieflusses und der Entropieproduktion der Strahlung...
Als Folge der aus dem Entropiemaximumprinzip bestimmten Verteilungsfunktion, die einer Reihenentwicklung mit den Momenten als Koeffizienten entspricht, sind die Momentengleichungen durch Matrizen von mittleren Absorptions- und Streukoeffizienten gekoppelt. Diese Kopplung ist bisher in der Literatur der Strahlungsthermodynamik nicht bekannt und als das wichtigste Ergebnis der Arbeit anzusehen. Im Verlauf der Arbeit wird gezeigt, dass erst diese Kopplung eine Übereinstimmung zwischen Lösungen der Momentengleichungen und Lösungen der Strahlungs-Transport-Gleichung liefert. (mehr)

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H. Struchtrup: A New Moment Method in Radiation Thermodynamics
Proceedings of the VIII International Conference on Waves and Stability in Continuous Media, Palermo 1996
Rendiconti del Circolo Matematico di Palermo, Serie II, Suppl. 45, 627-636 (1996)

We introduce a new set of moment equations in radation thermodynamics. It is based on the moments

u^r_i1i2...in = Integral[k^r n _i1 n_i2 ... n_iN f ]dk

Here, f is the photon distribution function, k is the photon wave number and n_i is the photon direction vector. The closure by means of the entropy maximum principle leads to a coupling of the equations by matrices of mean absorption and scattering coefficients. The effectiveness of the theory is shown for a simple example, the homogenous compression and relaxation of radiation.

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H. Struchtrup: On the Number of Field Equations in Extended Thermodynamics of Phonons
Proc. 7th Conference on Waves and Stability in Continuous Media, Bologna 1993
World Scientific, Singapore 1994

Extended Thermodynamics of phonons which is based on the kinetic theory of phonons provides a set of N field equations for the description of heat transport phenomena in dielectric crystals. These equations contain only two materially dependent quantities, the collision frequencies 1/tau _R and 1/tau_N for R- and N-processes, respectively. It is shown that the number N of field equations necessary for a proper description of the phenomena depends on the value of omega_max / (1/tau_R + 1/tau_N) where omega_max is the highest frequency of the essential harmonics of the process. If ( omega_max / (1/tau_R + 1/tau_N) )^N-2 <<1 , one will expect satisfactory agreement between a theory with N equations and experiments

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W. Dreyer and H. Struchtrup: Heat Pulse Experiments Revisited
Cont. Mech. Thermodyn. 5(1), 3-50 (1993) [pdf] [doi:10.1007/BF01135371]

This is a review of heat propagation - theory and experiment - in dielectric solids at low temperatures where the phenomenon of second sound occurs. The review does not merely present a list of the various explanations of the observed phenomena. Rather it views them as special cases of a unified theory which is formulated within the framework of extended thermodynamics of phonons. Field equations are derived by averaging over the phonon-Boltzmann equation and initial and boundary value problems are solved. Thus it became possible to achieve a full explanation of the observations of the heat-pulse experiments in which ballistic phonons, second sound and ordinary heat conduction compete.

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URL https://www.me.uvic.ca/~struchtr/