Mixture Equations Research Articles

Small dust grain dynamics on adaptive mesh refinement grids

Context. Small dust grains are essential ingredients of star, disk and planet formation. Aims. We present an Eulerian numerical approach to study small dust grain dynamics in the context of star and protoplanetary disk formation. It is designed for finite volume codes. We use it to investigate dust dynamics during the protostellar collapse. Methods. We present a method to solve the monofluid equations of gas and dust mixtures with several dust species in the diffusion approximation implemented in the adaptive-mesh-refinement code RAMSES. It uses a finite volume second-order Godunov method with a predictor-corrector MUSCL scheme to estimate the fluxes between the grid cells. Results. We benchmark our method against six distinct tests, DUSTYADVECT, DUSTYDIFFUSE, DUSTYSHOCK, DUSTYWAVE, SETTLING, and DUSTYCOLLAPSE. We show that the scheme is second-order accurate in space on uniform grids and intermediate between second- and first-order on non-uniform grids. We apply our method on various DUSTYCOLLAPSE simulations of 1 M⊙ cores composed of gas and dust. Conclusions. We developed an efficient approach to treat gas and dust dynamics in the diffusion regime on grid-based codes. The canonical tests were successfully passed. In the context of protostellar collapse, we show that dust is less coupled to the gas in the outer regions of the collapse where grains larger than ≃100 μm fall significantly faster than the gas.

Diffusion models for mixtures using a stiff dissipative hyperbolic formalism

We consider a system of fluid equations for mixtures with a stiff relaxation term of Maxwell–Stefan diffusion type. We use the formalism developed by Chen et al. and derive a limiting system of Fick type, in which the species velocities tend to align with a bulk velocity when the relaxation parameter remains small.

Modeling of radio frequency heating of egg white powder continuously moving on a conveyor belt

The radio frequency (RF) heating of low moisture foods continuously moving on a conveyor belt is a promising approach for preheating to reduce come-up time and/or pasteurization to ensure food safety. Three-dimensional multiphysics-based models were developed to simulate the RF heating process using two approaches (moving mesh and discrete moving steps) to incorporate the movement of the food product. The models were validated by RF heating of a package of egg white powder (EWP) continuously moving on a conveyor belt for 47 min through the system. The dielectric properties of the porous product were determined by using four mixture equations and their effects on model prediction accuracy were evaluated. The model using dielectric properties calculated by the Complex Refractive Index mixture equation showed best prediction accuracy. The simulated point temperatures from the model using moving mesh approach matched best with the experiment at eight locations inside the food product, with a smallest average Root-Mean-Square-Error value of 3.9 °C. The heating uniformity of the moving EWP was not improved or deteriorated when compared to the stationary product which was heated at the center of the system after 47 min heating.

Thermophysical properties of 1-ethyl-3-methylimidazolium bromide ionic liquid in water + ethylene carbonate mixtures at T = (298.2, 308.2 and 318.2) K

In this research, we studied the effect of ethylene carbonate (EC) on thermodynamic, transport and optical properties such as electrical conductance, refractive indices and activity coefficients of aqueous 1-ethyl-3-methylimidazolium bromide, [EMIm]Br solutions at T = (298.2, 308.2 and 318.2) K and 0.1 MPa. Electrical conductance was measured for [EMIm]Br ionic liquid from 0.0029 to 0.2500 mol kg−1 in different mass fractions of ethylene carbonate (EC) in water + EC mixtures (wEC/wmixture% = 10, 20 and 30). Limiting molar conductivity (Λ°) and ion association constant (KA) were calculated through Fuoss-Onsager equation and used to obtain the thermodynamic functions. Refractive index measurements were carried out for the binary and ternary water + [EmIm]Br + EC mixtures. The measured data were used to derive the refractive index changes of mixing (ΔnD) and the values were correlated by Redlich-Kister and the Cibulka equations for binary and ternary mixtures, respectively. Moreover, the results relating to the mean activity coefficient measurements for [EMIm]Br in the EC + water solvent mixtures were presented by potentiometric method. The potentiometric measurements were performed on the electrochemical cell of the type Br-ISE | [EMIm]Br (m), EC (wt.%), H2O (1-wt)%| [EMIm]-ISE, in various mass fractions of EC in water (0, 10, 20 and 30%) over total ionic strength from 0.0085 to 2.500 mol kg−1. The experimental results were correlated by the Pitzer equation and the values of Pitzer ion-interaction parameters β(0), β(1) and Cϕ were evaluated. Finally the parameters were utilized to calculate the values of the osmotic coefficients, excess Gibbs free energies and the solvent activity for the whole series of investigated system.

A two-phase SPH model for massive sediment motion in free surface flows

Massive sediment motion in water with a free surface is an important kind of geophysical flows such as hyper-concentrated sediment laden river flows discharging into estuarine delta and turbidity currents generated by subaqueous landslides. One of the key and common characteristics of such flows is that interactions between water and sediment as well as those among sediment particles are equally important in affecting the sediment motion and the fluid flow. This paper presents a numerical model that builds on and extends an earlier two-phase SPH model based on a continuum formulation of solid-liquid mixtures (Shi et al., 2017) to provide a unified description of massive sediment motion in free surface flows. In the model, a constitutive law based on the rheology of dense granular flow is introduced to express the intergranular stresses while the interphase drag force is determined by combining the Ergun equation for dense solid-fluid mixtures and the power law for dilute suspensions. The proposed model is firstly applied to the study of collapse of loosely or densely packed granular columns submerged in water. The computed surface profiles of the granular column are found to be in good agreement with the experimental data. It shows that the loosely packed and the densely packed columns behave rather differently due to the differences in water-sediment interaction processes. The model is then used to simulate a dam-break flow over a mobile sediment bed. The computed configurations of the flow and the movable bed also agree well with the measured data. The predicted position on the leading edge of the flow has a mean error of 0.8% while the mean error for the maximum bed height is 12.9%. To further identify the dynamic processes involved, effects of water-sediment interactions on the motion of bed materials are investigated by examining the spatial and temporal variations of pressure and flow velocity. As shown in the applications, the proposed two-phase SPH model can successfully represent both the gravity-driven underwater granular flows and the shear flow driven intense sediment transport, implying its potential use in practical scenarios in which the two kinds of flows exist simultaneously, such as landslides triggered by storm in shallow sea and flows resulted in barrier or dam breaks.

Two-Dimensional General Rate Model of Liquid Chromatography Incorporating Finite Rates of Adsorption–Desorption Kinetics and Core–Shell Particles

A two-dimensional general rate model of liquid chromatography incorporating slow rates of adsorption–desorption kinetics, axial and radial dispersions, and core–shell particles is formulated. Radial concentration gradients are generated inside the column by considering different regions of injection at the inlet. Analytical solutions are obtained for a single-component linear model by simultaneously utilizing the Laplace and Hankel transformations for the considered two sets of boundary conditions. These linear solutions are useful for simulating liquid-chromatographic columns with diluted or small-volume samples and those in which radial concentration gradients are significant. To gain further insight into the process, analytical moments are also deduced from the Laplace–Hankel-domain solutions. For situations of concentrated and large-volume samples, which are not solvable analytically, formulation of nonlinear models is necessary. In this study, a semidiscrete, high-resolution, finite-volume scheme is extended to approximate the resulting nonlinear-model equations for multicomponent mixtures. The performance of the column is analyzed by implementing a specified criterion of performance. A few numerical case studies are conducted to inspect the effects of the model parameters on the elution profiles.

Heat transfer of ice slurry flows in a horizontal pipe: A numerical study

The thermal behavior of ethylene glycol-water ice slurry flowing through a horizontal heated tube is investigated numerically. The ice slurry is assumed to behave as a Newtonian fluid with effective properties depending on temperature and ice fraction. The 3D steady-state model includes the mass, momentum and energy conservation equations for the liquid-solid mixture as well as a transport equation for the ice particles which characterizes the balance between convective and diffusive fluxes. The diffusive flux includes those induced by the shear rate gradients, the spatial variation in viscosity and the particle settling. The heat and mass exchanges between the ice particles and the carrier liquid are modeled through source terms. The model is validated against experimental data from the open literature and applied to laminar and turbulent flows. Some important heat transfer characteristics of melting are obtained including the axial distributions of the heat transfer coefficient, the bulk and wall temperatures, the ice volume fraction and the melting rate. They show significant differences from corresponding distributions for non-melting flows. The effects of the wall heat flux and other operating parameters on these quantities are presented and analyzed.

Fatigue equation for asphalt mixture under low temperature and low loading frequency conditions

Abstract In order to reveal the thermal fatigue behavior of asphalt mixtures under low temperature conditions, the four-point bending beam tests of strength and fatigue were carried out. The asphalt mixtures are typical viscoelastic materials whose fatigue properties are significantly influenced by low loading frequency. Thus, the fatigue tests under low loading frequency were used to simulate the thermal fatigue of asphalt mixtures. Based on the nominal stress ratio (not considering the effect of loading rate on the strength value) and the real stress ratio (the strength value corresponding to the loading rate of fatigue test), the fatigue equations of asphalt mixtures under low temperature and low frequency conditions were established. The flexural strengths of the four-point bending beam tests at different loading rates at 0 °C, −10 °C and −20 °C were studied. The results show that the flexural strength of asphalt mixtures expresses a power function on the pattern of variation with the loading rates. The intersection of fatigue curve characterized by the nominal stress ratio and the abscissa axis is much larger than 1. The fatigue curve based on the nominal stress ratio cannot be extended to fatigue failure point (1,1). But the fatigue curve characterized by the real stress ratio can be extended to the strength failure point, whose fatigue life is 1. So, the behavior of strength failure and fatigue failure was integrated. The uniform characterization model of fatigue equations for asphalt mixture under low temperature and low loading frequency conditions was realized, which means that the fatigue equations and curves under different loading frequencies could be characterized by one equation and one curve. The results can provide a theoretical basis for predicting the thermal fatigue performance of asphalt mixtures under low temperature conditions more accurately.

Large-Eddy Simulation of Sandia Flame D with Efficient Explicit Filtering

A uniform Gaussian filter has been applied explicitly to the LES conservation equations for mass, momentum and mixture to simulate a piloted non-premixed methane-air flame (Sandia Flame D). Using a basic property of exponential functions, the three dimensional Gaussian filter is decomposed into the product of three one dimensional filters, greatly reducing the cost of filtering. Seven simulations on three different grids have been performed to investigate the influence of grid refinement with a purely implicit filter, the effects of explicit filtering with increasing filter width and the effect of grid refinement at constant filter-size. Overall, consistent results have been achieved at a cost that is moderate with implicit or explicit filtering, so that explicit filtering can be applied in cases where the numerical error should be independent of the modelling error.

Characterization of the dielectric properties of biological tissues using mixture equations and correlations to different states of hydration

The aim of this study was to characterize the hydration fractions of biological tissues and to model these accurately from mixture equations. Hydration fractions, better known as volume fractions, are based on quantification of tissue hydration and accurate knowledge about the physiological composition of tissue fluids. Data on weight loss percentages for excised muscle and adipose tissue from a previous study were utilized for this purpose. The Bruggeman and Maxwell Garnett equations were then used to characterize the dielectric properties of the tissues in terms of mixtures of dry biological constituents and physiological saline solutions. It is shown that these models are accurate in modelling in vivo and ex vivo tissue in different states of hydration. This is based on precise knowledge of the physiological composition of biological fluids and their corresponding percentage contents.

A Kriging Surrogate Model for Computing Gas Mixture Equations of State

Accurate simulation of the complex flow following the detonation of an explosive material is a challenging problem. In these flows, the detonation products of the explosive must be treated as a real gas while the surrounding air is treated as an ideal gas. As the detonation process unfolds and the blast wave moves into the surrounding ambient air, the products of detonation expand outward and interact with the air creating a mixture region. In this region, both of the state equations for air and the products must be satisfied. One of the most accurate, yet computationally expensive, methods to handle this problem is an algorithm that iterates between the equations of state until both pressure and temperature reach an equilibrium inside of a computational cell. Since this mixture region moves and grows over time, this algorithm must be performed millions, or even billions, of times in a typical detonation simulation. As such, these calculations can account for a large percentage of the overall solution time. This work aims to use a kriging surrogate model to replace this process. The iterative method solves a nonlinear system of equations created from the gas mixture density, internal energy, and composition using a Broyden iterative solver to obtain an output pressure and temperature. Kriging is used to produce curve fits which interpolate selected pressures and temperatures from this solver from appropriate ranges of the mixture input quantities. Using a finite volume hydrocode, the performance of the model with respect to the iterative solver is demonstrated in the simulation of a pentaerythritol tetranitrate (PETN) charge detonation. The model's computational speed and accuracy are quantified as a function of the choice of sampling points in order to try optimize the combination as well as to show the benefits of this novel approach.

The Energy-Momentum Tensor in Relativistic Kinetic Theory: The Role of the Center of Mass Velocity in the Transport Equations for Multicomponent Mixtures

Abstract Relativistic kinetic theory is applied to the study of the balance equations for relativistic multicomponent mixtures, comparing the approaches corresponding to Eckart’s and Landau–Lifshitz’s frames. It is shown that the concept of particle velocity relative to the center of mass of the fluid is essential to establish the structure of the energy-momentum tensor in both cases. Different operational definitions of the center of mass velocity lead either to the inclusion of heat in the energy-momentum tensor (particle/Eckart frame) or to strictly relativistic contributions to the diffusion fluxes (energy/Landau–Lifshitz frame). The results here obtained are discussed emphasizing the physical features regarding each approach.

Numerical simulation of air ionization processes in plasma-actuator electric field

Mathematical model based on drift-diffusion approximation of hydrodynamic method of plasma description with two-temperature approximation was carried out for air ionization case in order to model process of interaction between dielectric barrier discharge plasma actuator and gas flow. Electron-induced ionization-recombination mechanism was used. Air was considered as mixture of molecular oxygen and nitrogen in proportion at normal conditions. Process kinetics was described by electron-induced reactions with molecular nitrogen and oxygen, ions and electron components. Equations system consisted of transport equations for concentrations of all mixture components, electron temperature and equation for electric potential was solved. Extended Scharfetter-Gummel scheme for large quantity of transport equations was developed. It was shown, that main influence on charge distribution has electron temperature. This model could be used for volume and surface charge density estimations necessary for volume force calculation in the vicinity of electrodes.

Numerical simulations of complex nonequilibrium flows in finite regions on the basis of the Boltzmann kinetic equation

A new formulation of the inner boundary problem for description complex nonequilibrium flows between membranes is presented. Solutions by means of the Boltzmann equation and the model kinetic equation for mixtures with chemical reactions demonstrate interesting physical properties in this open nonequilibrium system. Possibility of the anomalous thermal transport where the heat flux and the temperature gradient are of the same sign, is studied. Influence of boundary conditions on the parameters of the structure is considered as well.

Numerical investigation of a mixture two-phase flow model in two-dimensional space

A two-dimensional two-phase flow model for gas-liquid mixture is presented. The model takes into account the relative velocity between the gas and liquid phases and is based on conservation equations for gas-liquid mixtures. The mixture model involves balance equations for the relative velocity and is able to handle it without any physical or artificial stabilization in the source terms. The novel aspect of the mixture model is that it is written in a conservative form and ensures the hyperbolicity of the two-phase flow equations. With this regard, the governing equations are solved with finite volume methods. We extend and apply the framework of Godunov methods of centred-type, namely, the FirstOrder Centered (FORCE) and the Slope Limiter Centered (SLIC) methods to the two-dimensional governing equations without any loss of generality in the numerical solutions. An efficient assessment of both the mixture model and the numerical methods is carried out by simulating physical problems available in the literature. Simulations agree well with those in the literature and include new insights that could be used to explain the relative velocity observations. The favourable results suggest that the two-dimensional mixture model simulations can be employed for practical engineering problems of the non-equilibrium type.

Simulation of the Flow of a Gaseous Hydrocarbon Compound and its Thermodestruction in a Heat-Loaded Channel of Large Length

A two-dimensional mathematical model is proposed for calculating the parameters of a nonequilibrium flow of a gaseous hydrocarbon compound in a heat-loaded cylindrical channel of large length on the basis of the system of Navier–Stokes equations for a multicomponent mixture of heat-conducting viscous gases with a reduced mechanism of chemical kinetics. On the basis of processing of the results of calculations by this model, an integral one-dimensional mathematical model of a viscous flow of a gaseous hydrocarbon compound decomposed in an endothermic gross reaction, allowing one to perform a large number of computer calculations for a minimum time, has been developed. The results of calculations of the nonequilibrium flow of a gaseous hydrocarbon compound in a channel of large length by these models were compared with experimental data. Results of a parametric investigation of the cooling of channels having a large length by gaseous hydrocarbons in different regimes of their flow in them and at different heat loads on the channel walls are presented. The mathematical models developed and the data obtained with them can be used for estimating the efficiency of cooling of the channels of process installations and power plants by gaseous hydrocarbon compounds.

A Formal Passage From a System of Boltzmann Equations for Mixtures Towards a Vlasov-Euler System of Compressible Fluids

A formal asymptotics leading from a system of Boltzmann equations for mixtures towards either Vlasov-Navier-Stokes or Vlasov-Stokes equations of incompressible fluids was established by the same authors and Etienne Bernard in: A Derivation of the Vlasov-Navier-Stokes Model for Aerosol Flows from Kinetic Theory Commun. Math. Sci., 15: 1703–1741 (2017) and A Derivation of the Vlasov-Stokes System for Aerosol Flows from the Kinetic Theory of Binary Gas Mixtures. KRM, 11: 43–69 (2018). With the same starting point but with a different scaling, we establish here a formal asymptotics leading to the Vlasov-Euler system of compressible fluids. Explicit formulas for the coupling terms are obtained in two typical situations: for elastic hard spheres on one hand, and for collisions corresponding to the inelastic interaction with a macroscopic dust speck on the other hand.

On the Compatibility and Theoretical Equations for Mixtures of Tropical Soils and Bentonite for Barrier Purposes

On the Compatibility and Theoretical Equations for Mixtures of Tropical Soils and Bentonite for Barrier Purposes

Investigation of the fatigue properties of asphalt mixture designed using vertical vibration method

In this work, the flexural tensile fatigue behaviours of AC-20 asphalt mixtures were studied using beams, which were designed by the vertical vibration method (VVTM) and Marshall method. Then an analysis of variance (ANOVA) was used to examine the differences in fatigue lives, and the Weibull distribution was used to analyze the fatigue lives and establish the fatigue equations of the mixtures. Subsequently, the fatigue lives were compared and analyzed under the same stress level and repeated load force. The results showed that the test accuracy of specimens produced using the vertical vibration compaction method (VVCM) exceeds 92%. Compared to the Marshall design, the asphalt mixture fabricated with the VVTM design improved crack resistance and rutting resistance by at least 40% and 30%, respectively. The ratio of the slopes (the values of the regression coefficient b) of fatigue equations of asphalt mixtures designed by the VVTM to that of Marshall method ranges from 0.883 to 0.965, and the ratio of the intercepts (the values of the regression coefficient a) ranges from 1.278 to 1.284. The results also showed that compared to the Marshall design, asphalt mixtures fabricated with VVTM design exhibited improved fatigue lives by at least 61% (at a stress level of 0.45) and 356% (with a repeated load force of 0.378 kN).

Impedance characterization of elevated temperature carbon dioxide adsorption process on potassium-modified hydrotalcite

The adsorption process is strongly dependent on the dielectric properties of the adsorbent. In this study, electrochemical impedance spectroscopy (EIS) was applied for measuring the dielectric characteristics of CO2 adsorption on hydrotalcite at 400 °C and atmospheric pressure. Compared with the gravimetric method measuring the kinetic properties at different CO2 pressures, EIS proved to be more effective for characterizing the CO2 adsorption process in situ with high sensitivity, distinguishing between physisorption and chemisorption. A model for the dielectric properties of hydrotalcite in the CO2 adsorption process was developed using the kinetic model of CO2 adsorption on hydrotalcite and dielectric mixture equation. The simulation results agree well with the experimental data, demonstrating the potential application of EIS in the study of CO2 adsorption on hydrotalcite.