Mixture Equations Research Articles

Regularized moment equations for binary gas mixtures: Derivation and linear analysis

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

A multiphysics and multiscale model for low frequency electromagnetic direct-chill casting

Simulation and control of macrosegregation, deformation and grain size in low frequency electromagnetic (EM) direct-chill casting (LFEMC) is important for downstream processing. Respectively, a multiphysics and multiscale model is developed for solution of Lorentz force, temperature, velocity, concentration, deformation and grain structure of LFEMC processed aluminum alloys, with focus on axisymmetric billets. The mixture equations with lever rule, linearized phase diagram, and stationary thermoelastic solid phase are assumed, together with EM induction equation for the field imposed by the coil. Explicit diffuse approximate meshless solution procedure [1] is used for solving the EM field, and the explicit local radial basis function collocation method [2] is used for solving the coupled transport phenomena and thermomechanics fields. Pressure-velocity coupling is performed by the fractional step method [3]. The point automata method with modified KGT model is used to estimate the grain structure [4] in a post-processing mode. Thermal, mechanical, EM and grain structure outcomes of the model are demonstrated. A systematic study of the complicated influences of the process parameters can be investigated by the model, including intensity and frequency of the electromagnetic field. The meshless solution framework, with the implemented simplest physical models, will be further extended by including more sophisticated microsegregation and grain structure models, as well as a more realistic solid and solid-liquid phase rheology.

Asymptotic-preserving Boltzmann model equations for binary gas mixture.

An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations.

An examination of compositional dependencies for Y2O3-SrO-Ga2O3-SiO2 glasses: Evaluation of CT imageability and chemical durability for radioembolization

Event Abstract Back to Event An examination of compositional dependencies for Y2O3-SrO-Ga2O3-SiO2 glasses: Evaluation of CT imageability and chemical durability for radioembolization Xiaofang Zhang1, Elena Tonkopi2, Robert J. Abraham1, 2, 3, Daniel Boyd1, 2, 3, 4 and Sharon Kehoe1, 4 1 ABK Biomedical Inc., Research & Development, Canada 2 QE II Health Sciences Centre, Department of Diagnostic Imaging and Interventional Radiology, Canada 3 Dalhousie University, School of Biomedical Engineering, Canada 4 Dalhousie University, Applied Oral Sciences, Canada Introduction: The inability to directly monitor embolic microspheres during transarterial embolization (TAE) “preclude[s] ideal spatial delivery” of the embolic particles by limiting the accurate assessment of their terminal locations within target tissues[1]. A potential advantage associated with ‘imageable’ microspheres in selective internal radiation therapy (SIRT) expands to the provision of accurate dosimetry determinations; a feature that may enhance and optimize clinical outcomes. A primary design requirement for glass microspheres in this indication is that it be durable enough to limit the release of 90Y in vivo. Intersecting with this design requirement is the necessity that the material be imageable. This work seeks to show how such a design requirement may be achieved through a design of mixtures (DOM) experimental approach, and presents data for potential glass microspheres for use in SIRT. Materials and Methods: Sixteen glass-based compositions (mol. fraction) (0.100-0.170)Y2O3-(0.025-0.050)SrO-(0.100-0.300)Ga2O3-(0.50-0.75)SiO2 were synthesized using conventional melt-quenching techniques (fired at 1530°C, 3 h). Further processing (drying, grinding and sieving) retrieved particulates with a particle size distribution (PSD) of 20-32µm for subsequent evaluation. Quantitative radiopacity measurements of the experimental materials and control (n=5) were determined by taking axial CT scans (1mm slice thickness, pitch=0.5, 70 and 120kVp) through glass vials using a Somatom Definition AS+ scanner (Siemens Healthcare, Erlangen, Germany). Extracts were derived from the experimental materials over time periods of 1, 7 and 14days at an extraction ratio of 0.2g/mL (per ISO10993-12[2]) in polar medium (DI water). Ionic concentrations were analyzed via inductively coupled plasma atomic emission spectroscopy (ICP-AES, Perkin Elmer Optima 7300, MA, USA). Using a DOM experimental design approach (Design–Expert Ver. 8.0, Stat-Ease Inc.) the ordinary linear Scheffé polynomial[3] mixture equations were fitted to Yttrium (Y) release and CT imageability for each composition to determine the variant to impart the highest level of statistical significance on each response. Results and Discussion: Y release levels from all compositions tested were observed to not exceed 0.255 ±0.076 (SE) ppm over the 14day period. The regression outputs (mathematical models) developed for Y release and CT radiopacity yielded R2 values ranging from 0.7833-0.9485 and 0.9865-0.9950, respectively. In summary, the top 3 significant (positive and negative) main or interaction compositional factors (ranked highest to lowest based on L_Pseudo components) were noted as follows: Y release at 1day: +Y2O3*SrO > -SrO > +Ga2O3 Y release at 7 and 14days: -SrO > +Y2O3 > +Ga2O3 CT Imageability at 70kVp: -SrO > -Y2O3 > +Ga2O3 CT Imageability at 120kVp: +SrO > +Y2O3 > -Ga2O3 Overall, reduced concentrations of SrO relative to both Y2O3 and Ga2O3 were observed to impart the most signigifacnt effects (main and interaction) on all properties evaluated with the exception of CT imageability at 120kVp. Fig. 1 illustrates this contrasting effect for each element on subsequent CT imageability at 70 and 120kVp, a feature attributable to the K-absorption edge of these elements. Conclusions: Imageable glass microspheres synthesized and evaluated herein using DOM methodologies could potentially enhance dosimetry determinations to target tissues for SIRT. Atlantic Canada Opportunity Agency (ACOA) through the Atlantic Innovation Fund (AIF); John Fraser for his technical assistance with the synthesis of glass

Исследование баротропной квазигидродинамической модели двухфазной смеси с учетом поверхностных эффектов

A study of the barotropic quasi-hydrodynamic system of equations for the two-phase twocomponent mixture involving the surface tension and stationary potential force is accomplished. Under fairly general assumptions on the Helmholtz free energy of the mixture, the energy balance equation with non-positive energy production together with its corollary, the law of nonincreasing total energy, are derived; in particular, both the isothermal and isentropic cases are covered. Under additional assumptions, the necessary and sufficient conditions for the linearized stability of constant solutions are derived (it does not take place always). A finite-difference approximation of the problem is constructed in the 2D periodic case for a non-uniform rectangular mesh. Approximation of the capillary stress tensor is improved in the momentum balance equation thus increasing the quality of numerical solutions. The results of numerical experiments are presented. They demonstrate the ability of the model to ensure the qualitatively correct description for the dynamics of surface effects as well as applicability of the linearized stability criterion in the original nonlinear statement.

Analysis of accumulated lubricant for air–helium gas mixture in HDDs

For higher areal density, full helium as well as air---helium gas mixtures are currently being investigated as candidates for use in advanced drives. Moreover, the head mechanical spacing and the amount of accumulated lubricant on the bottom surface of the slider should be reduced for higher areal density. To analyze the characteristics of accumulated lubricant under these conditions, gas property equations for binary gas mixtures as functions of helium fraction ratio and temperature were derived. Accumulated lubricant equations were derived by considering the attractive forces, such as the dispersive and polar effects. An increase in the helium fraction ratio reduces the small shear stress on the bottom surface of the slider, which leads to a decrease in lubricant transfer capability. As a result, the amount of total lubricant transfer and accumulation are increased markedly. Also, with a decrease in lubricant thickness, the amount of accumulated lubricant increases greatly with a high helium fraction ratio. In conclusion, the HDI problems induced by lubricant transfer should be considered in full helium-filled or air---helium gas mixture drives.

Densities and excess molar volumes of the binary system N-methyldiethanolamine + (2-aminoethyl)ethanolamine and its ternary aqueous mixtures from 283.15 to 363.15 K

ABSTRACTExperimental density data of the binary mixtures of N-methyldiethanolamine + (2-aminoethyl)ethanolamine and the ternary mixtures of N-methyldiethanolamine + (2-aminoethyl)ethanolamine + water were reported at atmospheric pressure over the entire composition range at temperatures from 283.15 to 363.15 K. Density measurements were performed using an Anton Paar digital vibrating U-tube densimeter. Excess molar volumes were calculated from the experimental data and correlated as the Redlich-Kister equation for the binary mixtures, and as the Nagata-Tamura equation for the ternary mixtures. Several empirical models were applied to predict the excess molar volumes of ternary mixtures from the corresponding binary mixture values. It indicates that the best agreement with the experimental data was achieved by the Redlich-Kister, Kohler, and Jacob-Fitzner models.

Shock structure for macroscopic multi‐temperature model of binary mixtures: comparison with kinetic models

AbstractThe present study deals with the shock wave profiles in the macroscopic multi‐temperature (MT) model of binary gaseous mixtures. For that purpose we have adopted the hyperbolic model developed within the framework of extended thermodynamics [1] in which we include viscosity and thermal conductivity to eliminate restriction on parameters. This allows us to compare our results with more sophisticated kinetic solutions, which were computed for hypothetic mixtures of gases [2]. Numerical implementation of the MT model is considerably simpler than the one for Boltzmann equations for mixtures or the direct simulation Monte Carlo method (DSMC). (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the well-posedness of an inhomogeneous boundary value problem for the equations of mixtures of viscous compressible fluids

Under consideration is the inhomogeneous boundary value problem for the equations of a mixture of viscous compressible fluids modelling a steady flow around an obstacle. The statement of the problem presumes the possibility of varying the shape of obstacle. The well-posedness of such a problemin the class of strong solutions is proved. The results of the paper can be used for analyzing the optimal shape of an obstacle under a flow around it by a mixture of compressible viscous fluids.

Sub-shock formation in Grad 10-moment equations for a binary gas mixture

The shock structure problem for Grad 10-moment equations for an inert binary mixture is investigated: necessary conditions for the formation of sub-shocks in fields of only one gas or of both components are rigorously obtained, and a detailed comparison with the shock-wave structure of its principal sub-system (deduced assuming vanishing viscous stress tensors) and of the equilibrium Euler sub-system is performed. Some numerical simulations for a mixture of argon and helium are presented.

About the Divergent form of Equations for a One-Velocity Multicomponent Mixture

The divergent form of the equations of a generalized equilibrium model of a multicomponent mixture, in which the interfraction-interaction forces are taken into account, is discussed. The modified Godunov method developed for integration of model equations is described. A number of one-dimensional and two-dimensional problems were calculated with the use of this method.

Velocity profile of turbulent sediment-laden flows in open-channels

In this paper, a study was carried out on the velocity profile of sediment-laden flows in open channels using a two-phase mixture model for two-phase flows. The governing equations for water-sediment mixtures were derived based on the two-fluid equations for solid–liquid two-phase flows. The drift velocity, a key variable involved in the two-phase mixture equations, was derived from the equation of momentum conservation for the solid phase. The drift velocity shows that the inertia of flow, particle turbulence, and collisions effect contribute to the dispersion of the sediment particles in turbulent flows. Using the two-phase mixture equation, the vertical velocity profile of open channel flows was obtained. Further analysis indicated that the distribution of the velocity over depth of water-sediment mixtures, composed of two different phases, is significantly affected by the turbulence of water-sediment mixtures and the density stratification. However, the velocity distribution is also affected by other factors including collisions between particles and particle turbulence as a basic feature of two-phase flows where interphase interactions inevitably mark their influence on the velocity distribution. Comparisons of this approach with observations for a wide range of experimental conditions are presented in this paper, which show that this approach agrees well with the experiments.

Stability of freely cooling granular mixtures at moderate densities

The formation of velocity vortices and density clusters is an intriguing phenomenon of freely cooling granular flows. In this work, the critical length scale Lc for the onset of instability is determined via stability analysis of the linearized Navier–Stokes hydrodynamic equations of d-dimensional granular binary mixtures at moderate densities. In contrast to previous attempts, the analysis is not restricted to nearly elastic systems since it takes into account the nonlinear dependence of the transport coefficients and the cooling rate on the collisional dissipation. As expected from previous results obtained in the very dilute regime, linear stability shows d−1transversal (shear) modes and a longitudinal (“heat”) mode to be unstable with respect to long enough wavelength excitations. The theoretical predictions also show that the origin of the instability is driven by the transversal component of the velocity field that becomes unstable when the system length L > Lc. An explicit expression of Lc is obtained in terms of the masses and diameters of the mixture, the composition, the volume fraction and the coefficients of restitution. Previous results derived in the limit of both mechanically equivalent particles and low-density mixtures are consistently recovered. Finally, a comparison with previous theoretical works which neglect the influence of dissipation on the transport coefficients shows quantitative discrepancies for strong dissipation.

PLUME-MoM 1.0: A new integral model of volcanic plumes based on the method of moments

Abstract. In this paper a new integral mathematical model for volcanic plumes, named PLUME-MoM, is presented. The model describes the steady-state dynamics of a plume in a 3-D coordinate system, accounting for continuous variability in particle size distribution of the pyroclastic mixture ejected at the vent. Volcanic plumes are composed of pyroclastic particles of many different sizes ranging from a few microns up to several centimeters and more. A proper description of such a multi-particle nature is crucial when quantifying changes in grain-size distribution along the plume and, therefore, for better characterization of source conditions of ash dispersal models. The new model is based on the method of moments, which allows for a description of the pyroclastic mixture dynamics not only in the spatial domain but also in the space of parameters of the continuous size distribution of the particles. This is achieved by formulation of fundamental transport equations for the multi-particle mixture with respect to the different moments of the grain-size distribution. Different formulations, in terms of the distribution of the particle number, as well as of the mass distribution expressed in terms of the Krumbein log scale, are also derived. Comparison between the new moments-based formulation and the classical approach, based on the discretization of the mixture in N discrete phases, shows that the new model allows for the same results to be obtained with a significantly lower computational cost (particularly when a large number of discrete phases is adopted). Application of the new model, coupled with uncertainty quantification and global sensitivity analyses, enables the investigation of the response of four key output variables (mean and standard deviation of the grain-size distribution at the top of the plume, plume height and amount of mass lost by the plume during the ascent) to changes in the main input parameters (mean and standard deviation) characterizing the pyroclastic mixture at the base of the plume. Results show that, for the range of parameters investigated and without considering interparticle processes such as aggregation or comminution, the grain-size distribution at the top of the plume is remarkably similar to that at the base and that the plume height is only weakly affected by the parameters of the grain distribution. The adopted approach can be potentially extended to the consideration of key particle–particle effects occurring in the plume including particle aggregation and fragmentation.

A conservative multi-group approach to the Boltzmann equations for reactive gas mixtures

Starting from a simple kinetic model for a quaternary mixture of gases undergoing a bimolecular chemical reaction, multi-group integro-differential equations are derived for the particle distribution functions of all species. The procedure takes advantage of a suitable probabilistic formulation, based on the underlying collision frequencies and transition probabilities, of the relevant reactive kinetic equations of Boltzmann type. Owing to an appropriate choice of a sufficiently large number of weight functions, it is shown that the proposed multi-group equations are able to fulfil exactly, at any order of approximation, the correct conservation laws that must be inherited from the original kinetic equations, where speed was a continuous variable. Future developments are also discussed.

Normalization method for asphalt mixture fatigue equation under different loading frequencies

In order to analyze the effect of different loading frequencies on the fatigue performance for asphalt mixture, the changing law of asphalt mixture strengths with loading speed was revealed by strength tests under different loading speeds. Fatigue equations of asphalt mixtures based on the nominal stress ratio and real stress ratio were established using fatigue tests under different loading frequencies. It was revealed that the strength of the asphalt mixture is affected by the loading speed greatly. It was also discovered that the fatigue equation based on the nominal stress ratio will change with the change of the fatigue loading speed. There is no uniqueness. But the fatigue equation based on the real stress ratio doesn’t change with the loading frequency. It has the uniqueness. The results indicate the fatigue equation based on the real stress ratio can realize the normalization of the asphalt mixture fatigue equation under different loading frequencies. It can greatly benefit the analysis of the fatigue characteristics under different vehicle speeds for asphalt pavement.

A fast spectral method for the Boltzmann equation for monatomic gas mixtures

Although the fast spectral method has been established for solving the Boltzmann equation for single-species monatomic gases, its extension to gas mixtures is not easy because of the non-unitary mass ratio between the different molecular species. The conventional spectral method can solve the Boltzmann collision operator for binary gas mixtures but with a computational cost of the order mr3N6, where mr is the mass ratio of the heavier to the lighter species, and N is the number of frequency nodes in each frequency direction. In this paper, we propose a fast spectral method for binary mixtures of monatomic gases that has a computational cost O(mrM2N4log⁡N), where M2 is the number of discrete solid angles. The algorithm is validated by comparing numerical results with analytical Bobylev–Krook–Wu solutions for the spatially-homogeneous relaxation problem, for mr up to 36. In spatially-inhomogeneous problems, such as normal shock waves and planar Fourier/Couette flows, our results compare well with those of both the numerical kernel and the direct simulation Monte Carlo methods. As an application, a two-dimensional temperature-driven flow is investigated, for which other numerical methods find it difficult to resolve the flow field at large Knudsen numbers. The fast spectral method is accurate and effective in simulating highly rarefied gas flows, i.e. it captures the discontinuities and fine structures in the velocity distribution functions.

Density and viscosity of aqueous mixtures of N-methyldiethanolamines (MDEA), piperazine (PZ) and ionic liquids

The density and viscosity measurements of aqueous mixtures of N-methyldiethanolamine (MDEA), piperazine (PZ) and the ionic liquids (ILs) 1-n-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF4]), 1-butyl-3-methylimidazolium dicyanamide ([bmim][DCA]), and 1-ethyl-3-methylimidazolium trifluoromethanesulfonate ([emim][OTf]) were conducted in this work. It was found that an increase in IL concentration resulted in an increase in the densities and viscosities of the aqueous mixtures. The density and viscosity equations for MDEA, PZ and IL aqueous mixtures as a function of temperature and concentration of ILs were derived as well. The density and viscosity equations obtained adequate accuracies with less than 0.34% and 4.013% of deviation from experimental data respectively.

Adiabatic Processes in the Vapor–Liquid Two-Phase Region. 2. Binary Mixtures

The phase equilibrium conditions and entropy balance equations for multicomponent fluid mixtures are expressed with a density-based formalism (“isochoric thermodynamics”), and isentropes in the one- and two-phase region are computed from equations of state; here the Peng–Robinson equation is used as an example. Griffiths’ theorem—one- and two-phase isentropes meet at a maxcondenbar point (pressure maximum of an isopleth) with equal slopes—could be confirmed. For chemically similar compounds at subcritical conditions, the resulting isentrope patterns are similar to those of pure fluids. If one of the components is supercritical, it is possible that, along a part of a two-phase isentrope, the liquid phase has a higher molar entropy than the vapor phase (“entropic inversion”). The phenomenon not only poses a numerical problem, but is also relevant for the question whether a two-phase isentrope can run into the llg three-phase curve.

Fluctuating hydrodynamics of multi-species reactive mixtures.

We formulate and study computationally the fluctuating compressible Navier-Stokes equations for reactive multi-species fluid mixtures. We contrast two different expressions for the covariance of the stochastic chemical production rate in the Langevin formulation of stochastic chemistry, and compare both of them to predictions of the chemical master equation for homogeneous well-mixed systems close to and far from thermodynamic equilibrium. We develop a numerical scheme for inhomogeneous reactive flows, based on our previous methods for non-reactive mixtures [Balakrishnan , Phys. Rev. E 89, 013017 (2014)]. We study the suppression of non-equilibrium long-ranged correlations of concentration fluctuations by chemical reactions, as well as the enhancement of pattern formation by spontaneous fluctuations. Good agreement with available theory demonstrates that the formulation is robust and a useful tool in the study of fluctuations in reactive multi-species fluids. At the same time, several problems with Langevin formulations of stochastic chemistry are identified, suggesting that future work should examine combining Langevin and master equation descriptions of hydrodynamic and chemical fluctuations.