Mixture Of Ideal Gases Research Articles

Fisher Information, Entropy, and the Second and Third Laws of Thermodynamics

We propose Fisher information as a new calculable thermodynamic property that can be shown to follow the second and third laws of thermodynamics. However, Fisher information is qualitatively different from entropy and potentially possesses much more structure. Hence, a mathematical expression is derived for computing the Fisher information of a system of many molecules from the canonical partition function. This development is further illustrated through the derivation of Fisher information expressions for a pure ideal gas and an ideal gas mixture. Some of the unique properties of Fisher information are then explored through the classic experiment of the isochoric mixing of two ideal gases. Note that, although the entropy of isochorically mixing two ideal gases is always positive and is dependent only on the respective mole fractions of the two gases, the Fisher information of mixing has far more structure, involving the mole numbers, molecular masses, temperature, and volume. Although the application of Fisher information to molecular systems is clearly in its infancy, it is hoped that the present work will catalyze further investigation into a new and truly unique line of thought on thermodynamics. The second law of thermodynamics is based on entropy. The second law states that the total entropy of an isolated system increases over time, approaching a maximum value as time goes to infinity. Entropy in the truest classical sense applies to systems in equilibrium as defined by Clausius, Caratheodory, and others for the case of heat engines with the expression dS ) δQ T (1) where dS is the change in entropy caused by the flow of heat δQ into the system at temperature T. From the microscopic statistical thermodynamics point of view, entropy was defined by Boltzmann as S ) κ ln Ω (2) which can be shown to be equal to Shannon information, which is given by

Multicomponent fluid flow analysis using a new set of conservation equations

In this work hydrodynamics of multicomponent ideal gas mixtures have been studied. Starting from the kinetic equations, the Eulerian approach is used to derive a new set of conservation equations for the multicomponent system where each component may have different velocity and kinetic temperature. The equations are based on the Grad's method of moment derived from the kinetic model in a relaxation time approximation (RTA). Based on this model which contains separate equation sets for each component of the system, a computer code has been developed for numerical computation of compressible flows of binary gas mixture in generalized curvilinear boundary conforming coordinates. Since these equations are similar to the Navier–Stokes equations for the single fluid systems, the same numerical methods are applied to these new equations. The Roe's numerical scheme is used to discretize the convective terms of governing fluid flow equations. The prepared algorithm and the computer code are capable of computing and presenting flow fields of each component of the system separately as well as the average flow field of the multicomponent gas system as a whole. Comparison of the present code results with those of a more common algorithm based on the mixture theory in a supersonic converging–diverging nozzle provides the validation of the present formulation. Afterwards, a more involved nozzle cooling problem with a binary ideal gas ( helium – xenon ) is chosen to compare the present results with those of the ordinary mixture theory. The present model provides the details of the flow fields of each component separately which is not available otherwise. It is also shown that the separate fluids treatment, such as the present study, is crucial when considering time scales on the order of (or shorter than) the intercollisions relaxation times.

Temperature in ideal gas mixtures in Couette flow: A maximum-entropy approach

A binary mixture of ideal gases in Couette flow is studied in the framework of information theory. It is shown that if the total shear viscous pressure is imposed as a constraint, there is equipartition of energy between the different chemical species but not amongst the different spatial directions. In contrast, if a same shear rate is imposed on both species, there is no equipartition between species neither amongst spatial directions. In both situations, the thermodynamic temperature, defined from the Lagrange multiplier conjugated to the internal energy, is the same for both species, independently of the equality or not of their local-equilibrium temperature.

Uniform Magnetic Fields and Equilibrium Flame Temperatures

The impact of a constant uniform magnetic field on equilibrium flame temperature under constant volume conditions was examined. The equilibrium combustion compositions were found using an expression for Helmholtz free energy that included a magnetic-field contribution. Changes in the Helmholtz free energy for a mixture of paramagnetic and diamagnetic ideal gases were minimized using the method of Lagrange multipliers. The magnetic susceptibility for the paramagnetic gases was calculated using the Curie law. Equilibrium flame temperatures were obtained from the first law of thermodynamics for a closed adiabatic system with no work interactions. A model reaction of methane in air was used to quantitatively examine changes in the equilibrium flame temperature and product mole fractions as a function of volume and magnetic induction. Equilibrium flame temperature and product mole fractions were plotted as a function of magnetic-field strength and volume for all product species. The results indicate that with the increase in magnetic-field strength, the equilibrium flame temperature increases for all volumes considered.

Soret Diffusion and Non‐Ideal Dufour Conduction in Macroporous Catalysts with Exothermic Chemical Reaction at Large Intrapellet Damköhler Numbers

AbstractThe adiabatic temperature rise in catalytic pellets is predicted from a modified version of the Prater equation. Onsager reciprocal relations for coupled heat and mass transfer are violated in an analysis of thermal diffusion in macroporous catalysts with exothermic chemical reaction when Dufour conduction (i.e., the diffusion‐thermo effect) is neglected. In this contribution, Dufour conduction is analyzed for both ideal and non‐ideal pseudo‐binary gas mixtures that simulate the production of methanol from carbon monoxide and hydrogen. In the diffusion‐controlled regime at large intrapellet Damköhler numbers where intermolecular collisions provide the dominant resistance to mass transfer within the catalytic pores, temperatures in the catalytic core could be much greater than predictions based on the original Prater equation when the Prater number exceeds 0.30. The molecular flux of thermal energy includes Fourier's law, the interdiffusional flux, and Dufour conduction. Diffusional mass flux includes Fick's law and the Soret effect. All physicochemical properties of the reactive gas mixture exhibit temperature dependence. There is essentially no difference between maximum intrapellet temperature predictions that include or neglect ideal Dufour conduction when external catalytic surface temperatures range from 300‐400 K and thermal diffusion enhances the flux of “smaller” reactants toward the centre of the catalyst. For “large‐molecule reactants” that participate in exothermic reactions, thermal diffusion opposes Fick's law and Dufour conduction opposes Fourier's law. Under these conditions, it is demonstrated that core temperatures are overestimated by neglecting both off‐diagonal coupling mechanisms (i.e., Soret diffusion and Dufour conduction). Prater numbers greater than unity and unrealistically high gas pressures are required to distinguish between maximum intrapellet temperatures for ideal and real gas simulations, where the latter consider two‐body interactions for Lennard‐Jones molecules in the virial equation of state.

Soret‐modified hydrocarbon mass transport across compressed nonisothermal gases

AbstractWhile engineering methods employed to predict mass transport rates across carrier fluids are often limited to either ideal gas mixtures or constant density liquids, we deal here with species mass transport across nonisothermal compressed gas “films”, with special reference to hydrocarbon fuel vapor transport across nonisothermal N2 or H2O boundary layers at pressures up to ca. 300 atm, and at temperatures often above 1000 K. We show that because of the pressure sensitivity of the Soret factor (which quantifies the relative importance of mass transfer due to temperature gradients compared to that due to concentration gradients) mass transfer coefficients become far more sensitive to pressure level than would have been anticipated from the pressure sensitivity of ρD. Motivated, in part, by combustion applications, we first examine the expected pressure dependence of the binary Soret factor, αT,12 for each of the n‐alkanes (CH4 to C20H42) dilute in compressed N2 or H2O, exploiting a rational formulation for “correcting” Chapman‐Enskog‐derived Soret factors (α) to higher pressures based on the Thermodynamics of Irreversible Processes (TIP) combined with a Virial Equation of State (VES). Our TIP‐VES‐predicted Soret factors are used to demonstrate the pressure sensitivity of expected “Soret‐modified” mass transfer coefficients (Sherwood numbers) for the illustrative case of C12H26(g) transport across N2(g) at temperature ratios between 0.3 (“cold”‐wall) and 2.0 (“hot”‐wall) at pressures up to 300 atm. Because of the growing importance of compact, high‐pressure systems in fields beside combustion, including supercritical fluid extraction and even distillation, our present results suggest that reliable mass transfer rate predictions in nonisothermal dense vapor systems of engineering importance will generally require systematic inclusion of non‐Fickian molecular mass transport. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Adiabatic Phase Diagram of an Ultracold Atomic Fermi Gas with a Feshbach Resonance

We determine the adiabatic phase diagram of a resonantly-coupled system of Fermi atoms and Bose molecules confined in the harmonic trap by using the local density approximation. The adiabatic phase diagram shows the fermionic condensate fraction composed of condensed molecules and Cooper pair atoms. The key idea of our work is conservation of entropy through the adiabatic process, extending the study of Williams et al. [Williams et al., New J. Phys. 6, 123 (2004)] for an ideal gas mixture to include the resonant interaction in a mean-field theory. We also calculate the molecular conversion efficiency as a function of initial temperature. Our work helps to understand recent experiments on the BCS-BEC crossover, in terms of the initial temperature measured before a sweep of the magnetic field.

Coherent Rayleigh–Brillouin scattering in binary gas mixtures

In this paper we present the first theoretical study of the coherent Rayleigh–Brillouinscattering in binary mixtures of monatomic ideal gases. The line shape of the coherentspectrum is determined by solving a kinetic model of the Boltzmann equation that replacesthe collision operators by simple relaxation-time terms which are compatible with thetwo-fluid hydrodynamic theory. Theoretical predictions for the coherent light scatteringspectrum in dilute argon–krypton gas mixtures are presented and its dependence onconcentration and pressure is analysed.

On the equipartition of kinetic energy in an ideal gas mixture

A refinement of an argument due to Maxwell for the equipartition of translational kinetic energy in a mixture of ideal gases with different masses is proposed. The argument is elementary, yet it may work as an illustration of the role of symmetry and independence postulates in kinetic theory.

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

A numerical method is described for inviscid, compressible, multi-material flow in two space dimensions. The flow is governed by the multi-material Euler equations with a general mixture equation of state. Composite overlapping grids are used to handle complex flow geometry and block-structured adaptive mesh refinement (AMR) is used to locally increase grid resolution near shocks and material interfaces. The discretization of the governing equations is based on a high-resolution Godunov method, but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard, conservative shock-capturing schemes. The energy correction is constructed based on a uniform-pressure–velocity flow and is significant only near the captured interface. A variety of two-material flows are presented to verify the accuracy of the numerical approach and to illustrate its use. These flows assume an equation of state for the mixture based on the Jones–Wilkins–Lee (JWL) forms for the components. This equation of state includes a mixture of ideal gases as a special case. Flow problems considered include unsteady one-dimensional shock–interface collision, steady interaction of a planar interface and an oblique shock, planar shock interaction with a collection of gas-filled cylindrical inhomogeneities, and the impulsive motion of the two-component mixture in a rigid cylindrical vessel.

Configuration-Specific Kinetic Theory Applied to an Ideal Binary Gas Mixture

This paper is the second in a two-part series dealing with the configuration-specific analyses for molecular collision events of hard, spherical molecules at thermal equilibrium. The first paper analyzed a single-component system, and the reader is referred to it for the fundamental concepts. In this paper, the expressions for the configuration-specific collision frequencies and the average line-of-centers collision angles and speeds are derived for an ideal binary gas mixture. The analyses show that the average line-of-centers quantities are all dependent upon the ratio of the masses of the two components, but not upon molecular size. Of course, the configuration-specific collision frequencies do depend on molecular size. The expression for the overall binary collision frequency is a simple sum of the configuration-specific collision frequencies and is identical to the conventional expression.

Measurement of the gradient of viscosity with composition of mixtures of non-ideal gases

A perturbation viscometer is a differential capillary viscometer that measures the logarithmic viscosity gradient of the viscosity–composition curve for gas mixtures. Measurements are made at different gas mixture compositions. Integration of the logarithmic viscosity gradients measured over the full composition range gives the mixture viscosity relative to the viscosity of one of the pure components of the gas mixture. This method is attractive because, for measurements of equal precision, integration of the gradients is potentially an order of magnitude more precise than measurement of the viscosities directly. It can also work at high and low temperatures and perhaps high pressures. The perturbation viscometer has been used to make measurements on ideal gas mixtures at ambient and elevated temperatures. The situation is more complicated when the gas mixtures are non-ideal. Extra effects due to density differences, molar volume change on mixing and differential thermal expansion may be measured in addition to the desired viscosity change producing systematic errors in the results. Thus, a more sophisticated apparatus is required. The standard perturbation viscometer has been modified to separate out the extra effects to permit measurement of the true change in viscosity. In addition, the theoretical operation of the modified apparatus has been revised to account for the design changes to permit calculation of the viscosity–composition profiles from the results. The apparatus has been tested using helium–HFC-125 mixtures and two new viscosity–composition profiles are presented for these mixtures at 23 and 0 ∘ C . Internal consistency tests have been used to confirm that the data produced are of high quality with an estimated uncertainty in the viscosity ratio data at 23 ∘ C of 0.9% and at 0 ∘ C of 1.5%.

A simple mixture of gases is a mixture of ideal gases

If the thermodynamics of fluid mixture has a classical form, then the simple fluid mixture (Muller [1, 2]) has the special form of the state equation (see Eq. (3.5)). Moreover, if such mixture is gaseous, then it is equivalent to the mixture of ideal gases.

Effect of Uniform Magnetic Field on Equilibrium Combustion Compositions: Constant Volume

The impact of a uniform magnetic field on equilibrium combustion characteristics under constant volume conditions has been explored. An expression for Helmholtz free energy in terms of Gibbs free energy, which includes a magnetic field contribution, has been developed. With use of the method of Lagrange multipliers, changes in the Helmholtz free energy for a mixture of paramagnetic and diamagnetic ideal gases are minimized. The magnetic susceptibility of paramagnetic ideal gases has been calculated using the Curie‐Weiss law. A model reaction of methane in air is used to examine quantitatively the changes in mole fractions as a function of temperature and magnetic induction for all product species. Equilibrium mole fractions have been plotted as a function of temperature and magnetic field strength for all product species. The variation of equilibrium pressure as a function of temperature and magnetic field strength has also been examined. As expected, the results indicate that the equilibrium pressure increases with increasing temperature. At a specified temperature, the equilibrium pressure decreases with increasing magnetic field strength. The results indicate that within the temperature range considered, the magnetic field decreases the mole fraction of most of the product species except for CO2 ,N 2 ,H 2O, and CH4. Fo ra nincrease in magnetic induction of 0.00‐0.04 T, the maximum equilibrium mole fraction of NO was observed to decrease considerably. Nomenclature A = Helmholtz free energy a ij = atoms of element j in product i B = magnetic induction b j = atoms of element j in reaction C = damping constant E = iteration error G = Gibbs free energy gL = Lande’s g factor

Simulations of pulsating one-dimensional detonations with true fifth order accuracy

A novel, highly accurate numerical scheme based on shock-fitting coupled with fifth order spatial and temporal discretizations is applied to a classical unsteady detonation problem to generate solutions with unprecedented accuracy. The one-dimensional reactive Euler equations for a calorically perfect mixture of ideal gases whose reaction is described by single-step irreversible Arrhenius kinetics are solved in a series of calculations in which the activation energy is varied. In contrast with nearly all known simulations of this problem, which converge at a rate no greater than first order as the spatial and temporal grid is refined, the present method is shown to converge at a rate consistent with the fifth order accuracy of the spatial and temporal discretization schemes. This high accuracy enables more precise verification of known results and prediction of heretofore unknown phenomena. To five significant figures, the scheme faithfully recovers the stability boundary, growth rates, and wave-numbers predicted by an independent linear stability theory in the stable and weakly unstable regime. As the activation energy is increased, a series of period-doubling events are predicted, and the system undergoes a transition to chaos. Consistent with general theories of non-linear dynamics, the bifurcation points are seen to converge at a rate for which the Feigenbaum constant is 4.66 ± 0.09, in close agreement with the true value of 4.669201…. As activation energy is increased further, domains are identified in which the system undergoes a transition from a chaotic state back to one whose limit cycles are characterized by a small number of non-linear oscillatory modes. This result is consistent with behavior of other non-linear dynamical systems, but not typically considered in detonation dynamics. The period and average detonation velocity are calculated for a variety of asymptotically stable limit cycles. The average velocity for such pulsating detonations is found to be slightly greater than the Chapman–Jouguet velocity.

Modeling Low Mach Number Reacting Flow with Detailed Chemistry and Transport

An efficient projection scheme is developed for the simulation of reacting flow with detailed kinetics and transport. The scheme is based on a zero-Mach-number formulation of the compressible conservation equations for an ideal gas mixture. It relies on Strang splitting of the discrete evolution equations, where diffusion is integrated in two half steps that are symmetrically distributed around a single stiff step for the reaction source terms. The diffusive half-step is integrated using an explicit single-step, multistage, Runge---Kutta---Chebyshev (RKC) method. The resulting construction is second-order convergent, and has superior efficiency due to the extended real-stability region of the RKC scheme. Two additional efficiency-enhancements are also explored, based on an extrapolation procedure for the transport coefficients and on the use of approximate Jacobian data evaluated on a coarse mesh. We demonstrate the construction in 1D and 2D flames, and examine consequences of splitting errors. By including the above enhancements, performance tests using 2D computations with a detailed C1C2 methane-air mechanism and a mixture-averaged transport model indicate that speedup factors of about 15 are achieved over the starting split-stiff scheme

Kinetic model analysis of light scattering in binary mixtures of monatomic idealgases

The spectral distribution of light scattered by spontaneous density fluctuations in abinary mixture of monatomic ideal gases is studied by using a kinetic model of theBoltzmann equation which is compatible with the two-fluid hydrodynamic theory.Comparison of the theoretical results with available light scattering experiments inhelium–xenon gas mixtures shows that the two-fluid model equation can be applied todescribe the light scattering spectra at wavenumbers such that species temperatureseparation produces strong deviations from the classical hydrodynamic predictions.

Large Eddy Simulation of an Ammonia Jet

A compressible jet flow of ammonia was studied through a series of Large Eddy Simulations. Simulations were focused in the zone where the inertial effects are much bigger than the gravitational ones. The ammonia jet is discharged into an air medium and the gases are treated as a two non-reactive ideal gas mixture. Different simulations were performed, going from the use of constant transport properties to the use of kinetic theory of gases relations for calculating them as a function of the resolved variables. The main goal of the present study is to investigate the effect of varying density as well as variable transport properties on the turbulent flow behavior.

Computational studies of the laminar burning velocity of a producer gas and air mixture under typical engine conditions

This paper discusses computational results concerning the laminar burning velocity of a biomass-derived producer gas and air mixture at pressures and temperatures typical of the unburned mixture in a reciprocating engine. The computations are based on solving conservation equations describing laminar one-dimensional, multicomponent, chemically reacting, and ideal gas mixtures that have been formulated by earlier researchers. Based on a number of calculations at varying initial pressures and temperatures, and equivalence ratios, an expression for estimating the laminar burning velocity with the recycled gas mass fraction has been obtained. Also, the effect of varying amounts of recycled gas on the burning velocity has been determined. These data on laminar burning velocities will be useful in predicting the burnrate in a spark ignition (SI) engine fuelled with a producer gas and air mixture.

Gibbs function continuation for the stable computation of chemical equilibrium

We consider the computation of the chemical equilibrium state of an ideal gas mixture of given elemental composition, with and without general linear constraints on the species present. While computer programs to solve these problems have been available for more than 30 years, it is found that they are not always successful in determining the chemical equilibrium state in the presence of constraints. We present here the new method of Gibbs function continuation, which is guaranteed to determine the chemical equilibrium state for all well-posed constrained and unconstrained problems.