Simple Gas Mixtures Research Articles

Chapman–Enskog solutions to arbitrary order in Sonine polynomials V: Summational expressions for the viscosity-related bracket integrals

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper on simple gases (I), we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error. In two subsequent papers (II–III), we extended our original simple gas work to encompass binary gas mixture computations of the viscosity, thermal conductivity, diffusion, and thermal diffusion coefficients to high-order. In a fourth paper (IV) we derived general summational representations for the diffusion- and thermal conductivity-related bracket integrals and provided compact, explicit expressions for all of these bracket integrals needed to compute the diffusion- and thermal conductivity-related transport coefficients up to order 5 in the Sonine polynomial expansions used. In all of this previous work we retained the full dependence of our solutions on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals up to the final point of solution via matrix inversion. The elements of the matrices to be inverted are, in each case, determined by appropriate combinations of bracket integrals which contain, in general form, all of the various dependencies. Since accurate expressions for the needed bracket integrals have not previously been available in the literature beyond orders 2 or 3, and since such expressions are necessary for any extensive program of computations of the transport coefficients involving Sonine polynomial expansions to higher orders, we have investigated alternative methods of constructing appropriately general bracket integral expressions that do not rely on the term-by-term, expansion and pattern matching techniques that we developed for our previous work. It is our purpose in this paper to report the results of our efforts to obtain useful, alternative, general expressions for the bracket integrals associated with the viscosity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order viscosity coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.

Chapman–Enskog solutions to arbitrary order in Sonine polynomials IV: Summational expressions for the diffusion- and thermal conductivity-related bracket integrals

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper on simple gases, we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error. In two subsequent papers, we have extended our original simple gas work to encompass binary gas mixture computations of the viscosity, thermal conductivity, diffusion, and thermal diffusion coefficients to high-order. In all of this previous work we retained the full dependence of our solutions on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals up to the final point of solution via matrix inversion. The elements of the matrices to be inverted are, in each case, determined by appropriate combinations of bracket integrals which contain, in general form, all of the various dependencies. Since accurate, explicit, general expressions for bracket integrals are not available in the literature beyond order 3, and since such general expressions are necessary for any extensive program of computations of the transport coefficients involving Sonine polynomial expansions to higher orders, we have investigated alternative methods of constructing appropriately general bracket integral expressions that do not rely on the term-by-term, expansion and pattern matching techniques that we developed for our previous work. It is our purpose in this paper to report the results of our efforts to obtain useful, alternative, general expressions for the bracket integrals associated with the diffusion- and thermal conductivity-related Chapman–Enskog solutions for gas mixtures. Specifically, we have obtained such expressions in summational form that are conducive to use in high-order transport coefficient computations for arbitrary gas mixtures and have computed and reported explicit expressions for all of the orders up to 5.

Chapman–Enskog solutions to arbitrary order in Sonine polynomials III: Diffusion, thermal diffusion, and thermal conductivity in a binary, rigid-sphere, gas mixture

The Chapman–Enskog solutions of the Boltzmann equation provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper (I), for simple, rigid-sphere gases (i.e. single-component, monatomic gases) we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are free of numerical error and, in a second paper (II), we have extended our initial simple gas work to modeling the viscosity in a binary, rigid-sphere, gas mixture. In this latter paper we reported an extensive set of order 60 results which are believed to constitute the best currently available benchmark viscosity values for binary, rigid-sphere, gas mixtures. It is our purpose in this paper to similarly report the results of our investigation of relatively high-order (order 70), standard, Sonine polynomial expansions for the diffusion- and thermal conductivity-related Chapman–Enskog solutions for binary gas mixtures of rigid-sphere molecules. We note that in this work, as in our previous work, we have retained the full dependence of the solution on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals. For rigid-sphere gases, all of the relevant omega integrals needed for these solutions are analytically evaluated and, thus, results to any desired precision can be obtained. The values of the transport coefficients obtained using Sonine polynomial expansions for the Chapman–Enskog solutions converge and, therefore, the exact diffusion and thermal conductivity solutions to a given degree of convergence can be determined with certainty by expanding to sufficiently high an order. We have used Mathematica® for its versatility in permitting both symbolic and high-precision computations. Our results also establish confidence in the results reported recently by other authors who used direct numerical techniques to solve the relevant Chapman–Enskog equations. While in all of the direct numerical methods more-or-less full calculations need to be carried out with each variation in molecular parameters, our work has utilized explicit, general expressions for the necessary matrix elements that retain the complete parametric dependence of the problem and, thus, only a matrix inversion at the final step is needed as a parameter is varied. This work also indicates how similar results may be obtained for more realistic intermolecular potential models and how other gas-mixture problems may also be addressed with some additional effort.

Chapman–Enskog solutions to arbitrary order in Sonine polynomials II: Viscosity in a binary, rigid-sphere, gas mixture

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosity, the thermal conductivity, and the diffusion coefficient. In a preceding paper (I), for simple, rigid-sphere gases (i.e. single-component, monatomic gases) we have shown that the use of higher-order Sonine polynomial expansions enables one to obtain results of arbitrary precision that are error free. It is our purpose in this paper to report the results of our investigation of relatively high-order, standard, Sonine polynomial expansions for the viscosity-related Chapman–Enskog solutions for binary gas mixtures of rigid-sphere molecules. We note that in this work we have retained the full dependence of the solution on the molecular masses, the molecular sizes, the mole fractions, and the intermolecular potential model via the omega integrals. For rigid-sphere gases, all of the relevant omega integrals needed for these solutions are analytically evaluated and, thus, results to any desired precision can be obtained. The values of viscosity obtained using Sonine polynomial expansions for the Chapman–Enskog solutions converge monotonically from below and, therefore, the exact viscosity solution to a given degree of convergence can be determined with certainty by expanding to sufficiently high an order. We have used Mathematica® for its versatility in permitting both symbolic and high precision computations. Our results also establish confidence in the results reported recently by other authors who used direct numerical techniques to solve the relevant Chapman–Enskog equations. While in all of the direct numerical methods more-or-less full calculations need to be carried out with each variation in molecular parameters, our work utilizes explicit, general expressions for the necessary matrix elements that retain the complete parametric dependence of the problem and, thus, only a matrix inversion at the final step is needed as a parameter is varied. This work also indicates how similar results may be obtained for more realistic intermolecular potential models and how other gas-mixture problems may also be addressed with some additional effort.

The velocity slip problem: Accurate solutions of the BGK model integral equation

The velocity slip problem (also known as the Kramers problem) in the kinetic theory has been solved by a variety of techniques, and some accurate numerical as well as general variational results are available for both simple gases and gas mixtures. For the model equations, reasonably accurate solutions (within about 1% of the exact results) have been obtained by solving the related integral equations, but such techniques have not previously been pushed far enough to explore the possibilities of obtaining benchmark solutions. Some of the reasons for not further pursuing these techniques were limitations on the efficiencies and accuracies of the computer routines that could be constructed for the needed Abramowitz functions and limitations in terms of computational resources with respect to available precision and storage capabilities. These limitations have now effectively been removed with modern advances in computer technology and the computational tools that have become available, and it is the purpose of the present paper to report on our explorations in this regard within the specific context of the velocity slip problem as based on the BGK model which we have approached using the singularity subtraction technique in conjunction with numerical quadratures.

Heat‐flux‐specified boundary treatment for gas flow and heat transfer in microchannel using direct simulation Monte Carlo method

AbstractDirect simulation Monte Carlo (DSMC) method has been widely used to study gaseous flow and heat transfer in micro‐fluidic devices. For flows associated with microelectromechanical systems (MEMS), the heat‐flux‐specified (HFS) boundary condition broadly exists. However, problems with HFS boundary have not been realized in the simulation of microchannel flows with DSMC method. To overcome this problem, a new technique named as inverse temperature sampling (ITS) is developed. This technique provides an approach to calculate the molecular reflective characteristic temperature from the specified heat flux at the wall boundary. Coupling with DSMC method, the ITS technique can treat the HFS boundary condition in DSMC method for both simple gas and gas mixtures. For validation, heat flux obtained from two‐dimensional Poiseuille flows with wall‐temperature‐specified (WTS) boundary condition is employed as the initial thermal boundary condition of our new method. Sampled wall temperature by the ITS method agrees well with the expected value. Pressure, velocity and temperature distributions under these two thermal boundary conditions (WTS and HFS) are compared. Effects of molecule collision model and gas–surface interaction model are also investigated. Results show that the proposed ITS method could accurately simulate gaseous flow and heat transfer in MEMS. Copyright © 2007 John Wiley & Sons, Ltd.

Chapman–Enskog solutions to arbitrary order in Sonine polynomials I: Simple, rigid-sphere gas

The Chapman–Enskog solutions of the Boltzmann equations provide a basis for the computation of important transport coefficients for both simple gases and gas mixtures. These coefficients include the viscosities, the thermal conductivities, and diffusion coefficients. The Chapman–Enskog solutions are also useful for computation of the associated slip and jump coefficients near surfaces. Generally, these solutions are expressed in terms of Sonine polynomial expansions. While it has been found that relatively, low-order expansions (of order 4) can provide reasonable precision in the computation of the transport coefficients (to about 1 part in 1000), the adequacy of the low-order expansions for computation of the slip and jump coefficients still needs to be explored. Also of importance is the fact that such low-order expansions do not provide good convergence (in velocity space) for the actual Chapman–Enskog solutions even though the transport coefficients derived from these solutions appear to be reasonable. Thus, it is of some interest to explore Sonine polynomial expansions to higher orders. It is our purpose in this paper to report the results of our investigation of high-order, standard, Sonine polynomial expansions for the viscosity and the thermal conductivity related Chapman–Enskog solutions for a simple, rigid-sphere gas where we have carried out our calculations using expansions to order 150 and where our reported values for the transport coefficients have been demonstrated to converge to at least 25 significant digits. We note that, for a rigid-sphere gas, all of the relevant integrals needed for these solutions are evaluated analytically as pure fractions and, thus, results to any desired precision may be obtained. This work also indicates how results may be obtained in a similar fashion for realistic intermolecular potential models, and how gas-mixture problems may also be addressed with some additional effort.

Modelling gas mixture adsorption in active carbons

We review recent progress made concerning the modelling of equilibrium gas mixture adsorption in activated carbons. Much of the discussion focuses on modern statistical mechanical methods, such as classical density functional theory and Monte-Carlo simulation, as well as the surface models employed, i.e. the surface characterisation, and we confine our attention to work that has been compared quantitatively with experiment. We will see that for less demanding scenarios, i.e. relatively simple gas mixtures adsorbed at supercritical temperatures, current methods are satisfactory. But further developments in our models and theories are probably needed to describe the adsorption of more complex adsorbates such as those involving water at room temperature.

Gas Adsorption in Active Carbons and the Slit-Pore Model 2: Mixture Adsorption Prediction with DFT and IAST

We use a fast density functional theory (a "slab-DFT") and the polydisperse independent ideal slit-pore model to predict gas mixture adsorption in active carbons. The DFT is parametrized by fitting to pure gas isotherms generated by Monte Carlo simulation of adsorption in model graphitic slit-pores. Accurate gas molecular models are used in our Monte Carlo simulations with gas-surface interactions calibrated to a high surface area carbon, rather than a low surface area carbon as in all previous work of this type, as described in part 1 of this work. We predict the adsorption of binary mixtures of carbon dioxide, methane, and nitrogen on two active carbons up to about 30 bar at near-ambient temperatures. We compare two sets of results; one set obtained using only the pure carbon dioxide adsorption isotherm as input to our pore characterization process, and the other obtained using both pure gas isotherms as input. We also compare these results with ideal adsorbed solution theory (IAST). We find that our methods are at least as accurate as IAST for these relatively simple gas mixtures and have the advantage of much greater versatility. We expect similar results for other active carbons and further performance gains for less ideal mixtures.

Pyrolysis of complex organics following high-energy proton irradiationof a simple inorganic gas mixture

Complex organics formed by 3MeV proton irradiation of a simple inorganic gas mixture with a composition representative of the primitive earth atmosphere (carbon monoxide, nitrogen, and water) were characterized by application of Curie-point pyrolysis. Pyrolysis products consisted of a wide variety of organic compounds including amide compounds, heterocyclic, and polycyclic aromatic hydrocarbons. The present data showed that primary and primitive organic matter serving as “precursors” to fundamental building blocks associated with life might have been formed in a gaseous mixture of a similar composition to that of the primitive earth atmosphere. Large numbers of endogenous organic compounds and protocatalysis components produced by cosmic rays may have contributed significantly to the early stages of chemical evolution on the primitive earth.

Electrochemical reduction of the anaesthetic agent sevoflurane (fluoromethyl 2,2,2-trifluoro-1-[trifluoromethyl] ethyl ether) in the presence of oxygen and nitrous oxide

The electrochemical reduction of the inhalation anaesthetic agent sevoflurane was investigated at a variety of microelectrode substrates (Au, Ag and Cu) in dimethyl sulphoxide (DMSO) solvent, individually, and also as a component of a simple tertiary gas mixture with oxygen and nitrous oxide. In a gas mixture with oxygen, sevoflurane is shown to react with the superoxide anion radical, formed from the electro-reduction of oxygen complicating their simultaneous detection. However, the apparent timescale of reaction between the electrochemically produced superoxide and sevoflurane is shown to be dependent of the electrode size and the concentration of the anaesthetic agent, sevoflurane. Using microelectrodes with diameters ≤5 μm and low sevoflurane concentration ([SEVO]<0.2%, v/v), the reduction is shown to be a one-electron process, the superoxide/sevoflurane reaction kinetics are essentially “out run”. Using larger microelectrodes (diameters≥10 μm), or with higher concentrations of sevoflurane (0.5<[SEVO]<2.0%, v/v) the reduction current is augmented by the catalytic regeneration of oxygen, formed by the reaction between superoxide and sevoflurane. By contrast, in a tertiary gas mixture with nitrous oxide, sevoflurane reduction is shown to be unaffected by nitrous oxide or its reduction products. Unlike other anaesthetic agents, the reduction wave for sevoflurane no longer lies concomitant with the reduction wave of nitrous oxide. This work is a continuation of the development towards a rapid, accurate and inexpensive electrochemical gas sensor for the delivery of anaesthetic agents during surgery.

Electrochemical studies of the anaesthetic agent isoflurane (1-chloro-2,2,2-trifluoroethyl difluoromethyl ether) in binary gas mixtures with oxygen and nitrous oxide

The electrochemical reduction of the inhalation anaesthetic agent isoflurane was investigated at a variety of microelectrode substrates (Au, Ag and Cu) in DMSO solvent, individually, and also as a component of a simple binary gas mixture with oxygen or nitrous oxide. In a binary gas mixture with oxygen, isoflurane is shown to react with the superoxide anion radical, formed from the electro-reduction of oxygen complicating their simultaneous detection. However, the electro-reduction of oxygen is shown to be dependent of the electrode size and the concentration of the anaesthetic agent, isoflurane. Using microelectrodes with diameters ≤5 μm and low isoflurane concentration ([ISO]<0.2% (v/v)), the reduction is shown to be a one-electron process, since the homogeneous reaction kinetics are essentially “out-run”. Using larger microelectrodes (diameters≥10 μm), or with higher concentrations of isoflurane, (0.5>[ISO]<2.0% (v/v)) the reduction current is augmented by the catalytic regeneration of oxygen, formed by the reaction between superoxide and isoflurane. By contrast, in a binary gas mixture with nitrous oxide, isoflurane reduction is shown to be unaffected by nitrous oxide or it is reduction products. The reduction waves for isoflurane and nitrous oxide are concomitant which complicates their simultaneous detection somewhat. In essence, this work provides scope for the development of a rapid, accurate and inexpensive electrochemical gas sensor for measuring the concentration of anaesthetic agents during surgery.

Predicting the Adsorption of Gas Mixtures: Adsorbed Solution Theory versus Classical Density Functional Theory

Accurate prediction of the adsorption properties of fluid mixtures in equilibrium with surfaces and/or nanoporous structures is of considerable scientific and practical importance. Often, while the pure fluid adsorption isotherms are known for each component, those for the mixture are not. Using data from Monte Carlo simulations of model mixtures (including hydrogen and carbon dioxide) adsorbed in graphitic slit pores, for a range of pressures to 1000 bar, we compare theories for mixed adsorption which require pure fluid isotherm data as input. In particular, we develop and evaluate methods based on adsorbed solution theory (AST) and classical density functional theory (DFT). We find that a novel approximate DFT-based model is generally more accurate than AST methods in predicting the adsorption isotherms of mixtures of simple gases.

Characterization of Complex Organic Compounds Formed by Irradiation of Simple Gas Mixtures

Characterization of Complex Organic Compounds Formed by Irradiation of Simple Gas Mixtures

Analysis of the Thermal Conductivity Coefficients for Gas Mixtures: Application of the Modified Enskog Theory

The method for the calculation of transport coefficients for mixtures of high-density simple gases, which is a generalization of the modified Enskog theory for pure (individual) gases, is used in a simplified form suggested by the author. The simplification is derived using the perturbation method. Computer codes are compiled for performing an analysis of the coefficients of viscosity and thermal conductivity for multicomponent mixtures of high-density gases. The results of calculation of the thermal conductivity coefficient of a nitrogen–carbon dioxide mixture are compared with relevant experimental data: the deviation does not exceed 7% in the experimentally treated region of states.

Periodic Control for Improved Low-Temperature Catalytic Activity

The influence of transient changes in the gas composition on the low-temperature activity of a commercial three-way catalyst and a Pt/Al2O3 model catalyst has been studied. By introducing well-controlled periodic O2 pulses to simple gas mixtures of CO or C3H6 (in N2), a substantial improvement of the low temperature oxidation activity was observed for both catalysts. The reason for low activity at low temperatures is normally attributed to self-poisoning by CO or hydrocarbons. The improved catalytic performance observed here is suggested to origin from the transients causing a surface reactant composition that is favourable for the reaction rate.

Time evolution of ion energy distributions and optical emission in pulsed inductively coupled radio frequency plasmas

This article reports the results of time-resolved measurements of ion energy distributions (IEDs), relative ion densities, as well as optical emissions and electrical characteristics in pulsed inductively coupled plasmas for the simple gas mixture of oxygen and argon (50%Ar:50%O2). The peak radio frequency power, frequency, repetition rate, and duty cycle were 200 W, 13.56 MHz, 500 Hz, and 85%, respectively. Examination of the time evolution of the data over the pulse cycle indicates that when the plasma is energized, it begins in the dim (E) mode undergoing at first slow changes in certain plasma parameters. After about 1.2 ms, a sudden transition to the bright (H) mode occurs. The characteristics of the IEDs are consistent with an average plasma potential profile that has (a) a relatively large collisionless sheath and narrow presheath during the dim mode and (b) a very narrow sheath and greatly extended presheath during the bright mode; the ion mean free path influences the peak position of the IED during the bright mode. For most of the pulse cycle, the relative abundances of the ions Ar+, O2+, and O+ maintain the relation %O2+⩾%Ar+&amp;gt;%O+.

Optimizing Thermodynamic Parameters to Match Methane and Ethane Structural Transition in Natural Gas Hydrate Equilibria

Abstract: A simple natural gas mixture (methane and ethane) when mixed with water to form natural gas hydrates can exhibit complex phase equilibria. In this case, two structure I hydrate formers when combined form a structure II hydrate under certain conditions. Optimized Kihara parameters were derived from experimental data using the statistical thermodynamics model based on the van der Waals and Platteeuw theory and using the Kihara spherical core potential for the guest‐water interactions. The parameters were based upon both pure component data and the structural transition point, with the intent of better predictions over the remainder of the phase diagram.

The Nonmethane Hydrocarbon Intercomparison Experiment (NOMHICE): Task 3

The Nonmethane Hydrocarbon Intercomparison Experiment (NOMHICE) was designed to evaluate current analytical methods used to determine mixing ratios of atmospheric nonmethane hydrocarbons (NMHCs). A series of planned experiments, or tasks, were implemented to test the analytical methods in a graduated fashion. Tasks 1 and 2 involved relatively simple standard gas mixtures prepared by the National Institute for Standards and Technology (NIST). Results are presented here for task 3 in which a complex mixture containing 60 commonly observed NMHCs at concentrations of 1–30 parts per billion by volume (ppbv) in nitrogen diluent gas was prepared and distributed for analysis to 29 participating laboratories throughout the world. Reference mixing ratios were determined jointly by scientists from the National Center for Atmospheric Research (NCAR) and from the U.S. Environmental Protection Agency (EPA). Participants were asked to identify and quantify the hydrocarbons present in the mixture and submit their results to NCAR‐NOMHICE scientists. The results were encouraging overall. Some laboratories performed extremely well during this exercise whereas other laboratories experienced problems in either identification or quantification or both. It is evident from the comparison of the NCAR‐NOMHICE results with both the EPA analysis and the top 11 analyses in the study that very good agreement is achievable between laboratories for mixtures in this concentration range. Some of the largest analytical discrepancies were from laboratories that used in‐house standards for their calibration and/or syringe sample injection techniques. A major conclusion from this study is that the use of high‐quality gas phase standards, introduced into the measurement instrument in a similar manner to air samples, is an important prerequisite for an accurate analysis.

Experimental and theoretical study of the response of ZrO2 oxygen sensors to simple one-reducing-gas mixtures

The emf of commercial automotive zirconia oxygen sensors exposed to O 2 /CO/CO 2 /N 2 or O 2 /H 2 /H 2 O/N 2 gas mixtures was measured as a function of R ′=(2P O 2 /P CO ) or (2P O 2 /P H 2 ) at several temperatures. A step-like transition from high to low emf occurs at the switch-point R s0 ′ that is almost independent of temperature. The high emf (at say R ′=0.5) first increases and then decreases with temperature for O 2 /CO/CO 2 /N 2 gas mixture, whereas it only decreases for O 2 /H 2 /H 2 O/N 2 . These results may be analyzed using the steady state model of gas sensors [A.D. Brailsford, M. Yussouff, E.M. Logothetis, M. Shane, Steady-state model of a zirconia oxygen sensor in a simple gas mixture, Sensors and Actuators, B 24/25 (1995) 362–365]. Two lumped parameters, R s ′ and ξ , are needed in the model to compute the emf. R s ′ represents the theoretical switch-point and ξ determines the theoretical high emf. To obtain good fit with experiment, we choose R s ′ close to the observed R s0 ′ and then vary ξ . The quantity ξ depends upon the fundamental processes taking place on the sensor electrode; the temperature dependence of its fitted values (and hence the high emf) may be understood in terms of rate constants for adsorption, desorption and reaction.