Henning Struchtrup:
Macroscopic Transport Equations for Rarefied Gas Flows
Approximation Methods in Kinetic Theory

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

(extended abstract).

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.

M. Torrilhon and

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.

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.

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).

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.

H. Struchtrup and G. Elfring:

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.

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.

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.

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.

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 in‡uence 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

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.

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.

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.

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.

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.

An approximate numerical method is developed for Mieussens’s 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’s 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

Y. Zheng, J. Reese, and H. Struchtrup:

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.

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.

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.

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.

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. 

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.

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.

(extended abstract).

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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

 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.

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.

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 .

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 ( http://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.

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

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.

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.

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.

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.

  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

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

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.

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.

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^{A₁A₂...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.

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)

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.

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.

H. Struchtrup: Zwei homogene irreversible Prozesse
Facetten der Thermodynamik, Prof. I. Müller zum 60. Geburtstag
H. Struchtrup (Ed.), Institut für 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.

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)

Back
homepage

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.

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

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.

Last updated:  November 2008
URL http://www.me.uvic.ca/~struchtr/