Mixture Equations Research Articles

An experimental study on applying spatial TDR to determine bentonite suspension penetration

In slurry shield tunnelling, the penetration of the supporting bentonite suspension must be reduced to a critical value to ensure safety and cost-efficiency during construction. Aiming to measure bentonite suspension penetration, this study adopted the spatial time domain reflectometry (spatial TDR) technique. Although traditional TDR can detect point-wise changes in bentonite suspension concentration of pore fluid, this technique has rarely been extended to spatial profile detection. Spatioscale tests with a flat ribbon cable TDR sensor demonstrated the potential of TDR waveform analysis for determining penetration depth. Relationships between penetration depth and waveform characteristics were established. The travel time specified by the dual tangents method decreased with increasing slurry penetration, and the determined travel time agrees well with that calculated by a newly proposed mixture equation. This novel approach enables the determination of penetration depth without visual observation, providing a powerful measuring solution for laboratory studies and slurry shield tunnelling.

The outcome of collisions between gaseous clumps formed by disk instability

The disk instability model is a promising pathway for giant planet formation in various conditions. At the moment, population synthesis models are used to investigate the outcomes of this theory, where a key ingredient of the disk population evolution are collisions of self-gravitating clumps formed by the disk instabilities. In this study, we explored the wide range of dynamics between the colliding clumps by performing state-of-the-art smoothed particle hydrodynamics simulations with a hydrogen-helium mixture equation of state and investigated the parameter space of collisions between clumps of different ages, masses (1–10 Jupiter mass), various impact conditions (head-on to oblique collisions) and a range of relative velocities. We find that the perfect merger assumption used in population synthesis models is rarely satisfied and that the outcomes of most of the collisions lead to erosion, disruption or a hit-and-run. We also show that in some cases collisions can initiate the dynamical collapse of the clump. We conclude that population synthesis models should abandon the simplifying assumption of perfect merging. Relaxing this assumption will significantly affect the inferred population of planets resulting from the disk instability model.

Modelling of wall-bounded cavitating flow and spray mixing in multi-component environments using the PC-SAFT equation of state

This work introduces a numerical multiphase model for multi-component mixtures, utilizing tabulated data for physical and transport properties across a spectrum of conditions from near-vacuum pressures to supercritical states. The property data are derived using Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT), vapor-liquid equilibrium (VLE) calculations, entropy scaling methodologies, and Group Contribution (GC) methods. These techniques accurately reflect the thermodynamic behaviors of real fluids, avoiding the empirical estimation of Equation of State (EoS) input parameters. Implemented in OpenFOAM, the fluid dynamics solver is designed to address the three-dimensional Navier-Stokes equations for multi-component mixtures. The methodology integrates operator splitting to manage hyperbolic and parabolic steps distinctively. Hyperbolic terms are solved using the HLLC (Harten-Lax-van Leer-Contact) solver with temporal integration performed via a third-order Strong-Stability-Preserving Runge–Kutta (SSP-RK3) method. Viscous stress tensor contributions in the momentum equation are handled through an implicit velocity correction equation, while parabolic terms in the energy equation are explicitly solved. The simulation efficiency is further enhanced by adaptive Local Time Stepping and the Immersed Boundary (IB) method, which addresses interactions between the fluid and solid boundaries. Turbulence is resolved using the Wall Adaptive Large Eddy (WALE) model. Applied to high-pressure diesel fuel spray injections into non-reacting (nitrogen) gas environments, the model has been validated against Engine Combustion Network (ECN) data for the Spray-C configuration, featuring a fully cavitating multi-hole orifice. Results demonstrate that the model achieves accurate predictions across a broad range of tested conditions without the need for tuning or calibration parameters.

On thermodynamic processes, state equations and critical phenomena for homogeneous mixtures

In this paper, we study the thermodynamics of homogeneous mixtures in equilibrium. From the perspective of thermodynamics, substances are understood as Legendre submanifolds, which are equipped with a Riemannian structure in addition. We refer to these as Legendre-Riemannian manifolds. This Legendre structure reflects the law of conservation of energy, while the Riemannian structure corresponds to the second central moment of measurement of extensive quantities, indicating that we only consider stable states. Thermodynamic processes, such as chemical reactions, correspond to contact vector fields that preserve the law of energy conservation, or are contact. The presence of a Riemannian structure distinguishes between three classes of processes: positive, which increase the metric; neutral, which preserve the metric; and negative, which decrease the metric. We provide a detailed description of the processes and suggest a method for finding state equations for a homogeneous mixture in mechanical or chemical equilibrium.

Dielectric properties of Pouteria sapota compounds in terahertz frequency range

The analysis and a mathematical approximation of the dielectric properties of Pouteria sapota (Mamey) pulp compounds in the Terahertz (THz) range are presented. Since THz electromagnetic waves show a high sensitivity to detect water content when interacting with organic materials, this technique was used to determine the effective dielectric function (EDF) of a wet and a dry sample from the THz spectrum data. By fitting the EDF to the Landau-Lifshitz-Looyenga (LLL) dielectric mixture equation, the theoretical dielectric functions of each pulp compounds were approximated. The approach proposed here not only considered dry matter and water content in the Pouteria sapota pulp, as has been reported in the literature for organic samples, but also included the contribution of the volatile components (VC) that allowed a better adjustment to the experimental results. The water and VC content increase the value of the real and imaginary part of the calculated EDF of the wet sample. The use of the non-destructive THz spectroscopy technique allows determining the dielectric properties of fruits that have been dehydrated. Therefore, the methodology proposed here can be used as a method of in situ and ex situ quality control in the agri-food industry.

A new fatigue equation for asphalt mixtures considering loading sequence effects

The fatigue life of asphalt mixture has been found to significantly depend on loading sequences. However, the widely used traditional linear fatigue equations are difficult to determine the fatigue life of asphalt mixture subjected to different loading sequences. This study aims to improve these fatigue equations by establishing a fatigue equation considering loading sequence effects. First, the semi-circular bending (SCB) constant amplitude fatigue tests were conducted under the high loading amplitude (σhigh) and low (σlow) loading amplitude, respectively. Then, two of the loading sequences, namely the low-high stress (σlow-σhigh) and high-low stress (σhigh-σlow), were selected to carry out the SCB variable amplitude fatigue tests. Furthermore, the loading sequence effects on the life fractions and fatigue life of asphalt mixtures were analyzed. Finally, the damage equivalent criterion was successfully applied to establish the nonlinear fatigue damage accumulation model and fatigue equation considering loading sequence effects. It was found that the cumulative life fractions of asphalt mixtures under constant amplitude loading are equal to 1, indicating the linear fatigue damage accumulation. The cumulative life fractions corresponding to the σlow-σhigh and σhigh-σlow are greater than 1 and less than 1, respectively, indicating the nonlinear fatigue damage accumulation. The established nonlinear fatigue damage accumulation model can accurately characterize the dependence of fatigue damage accumulation on loading sequences. The established fatigue equations can improve the traditional linear fatigue equations to some extent and accurately predict the fatigue life of asphalt mixtures corresponding to the loading sequences of the σlow-σhigh and σhigh-σlow.

On the Influence of Humidity on a Thermal Conductivity Sensor for the Detection of Hydrogen.

Thermal conductivity sensors face an omnipresent cross-influence through varying humidity levels in real-life applications. We present the results of investigations on the influence of humidity on a hydrogen thermal conductivity sensor and approaches for predicting the behavior of thermal conductivity towards humidity. A literature search and comparison of different mixing equations for binary gas mixtures were carried out. The theoretical results were compared with experimental results from three different thermal conductivity sensors with mixtures of water vapor in nitrogen. The mixing equations show a large discrepancy between each other. Some of the models predict a continuously decreasing thermal conductivity and some predict an increasing thermal conductivity for increasing levels of humidity. Our measurements indicate an increase in thermal conductivity followed by a decrease after reaching a peak value. It is shown that the measured behavior is reproducible with different sensors. Depending on the sensor, this corresponds to an error up to 2 vol.% in the measured hydrogen value. The measured behavior is consistent with only one of the three models. Compared to this model, our own sensor shows a maximum deviation of 1.4%. Mixing equations for gas mixtures must be chosen carefully, taking into consideration whether mixing partners include polar or non-polar molecules. Some simplified mixing equations cannot be used to calculate the thermal conductivity of water vapor in air or nitrogen.

Towards Understanding of the Plastic Zone Thermal Residual Stresses in Functionally Graded Materials

Abstract: Functionally graded materials (FGM’s) are advanced composites consisting of two or more materials whose composition and structure changes gradually over the volume, leading to gradual change in its property. Typically one component is a ceramic, while the other is metal or metal alloy. In common embodiments, the composition can change gradually from all ceramic on one side to all metal on the other side, the ceramic contributing high resistance to temperature with the metal contributing high ductility. It’s a way to incorporate favorable properties of two materials into different locations of a single structure. In ideal cases the composition changes gradually but for manufacturing reasons, the changes occurs in homogeneous layers. The boundaries between the layers cause thermal residual stresses to arise, especially during material processing and in cases of cyclic loading. These thermal residual stresses are a result of differences in the thermal expansion coefficient of the varying compositions of materials. Thermal residual stresses may lead to cracking and ultimate failure of FGM’s. This study investigates the formation of thermal residual stresses for a Nickel-Alumina Ni-Al₂O₃ FGM. The Ni-Al₂O₃ system is chosen because it is one of the most common systems used in practice. This work explores the formation, impact and minimization of thermal residual stresses for a number of practical conditions that may arise during processing. All relevant properties are calculated using the rule of mixture equations.

Minimizing the Formation of Thermal Residual Stresses of the Elastic Zone in Functionally Graded Materials

Abstract: Functionally graded materials (FGM’s) are advanced composites consisting of two or more materials whose composition and structure changes gradually over the volume, leading to gradual change in its property. Typically one component is a ceramic, while the other is metal or metal alloy. In common embodiments, the composition can change gradually from all ceramic on one side to all metal on the other side, the ceramic contributing high resistance to temperature with the metal contributing high ductility. It’s a way to incorporate favorable properties of two materials into different locations of a single structure. In ideal cases the composition changes gradually but for manufacturing reasons, the changes occurs in homogeneous layers. The boundaries between the layers cause thermal residual stresses to arise, especially during material processing and in cases of cyclic loading. These thermal residual stresses are a result of differences in the thermal expansion coefficient of the varying compositions of materials. Thermal residual stresses may lead to cracking and ultimate failure of FGM’s. This study investigates the formation of thermal residual stresses for a Nickel-Alumina Ni-Al₂O₃ FGM. The Ni-Al₂O₃ system is chosen because it is one of the most common systems used in practice. This work explores the formation, impact and minimization of thermal residual stresses for a number of practical conditions that may arise during processing. All relevant properties are calculated using the rule of mixture equations. Abaqus, a non-linear FEA solver, is used for all the simulation work. In addition, the study highlights the need for examining elasto-plastic analysis to illuminate the process of crack development which is our future research goal.

Finite Element Modeling of the Plastic Zone Thermal Residual Stresses in Functionally Graded Materials

Abstract: Functionally graded materials (FGM’s) are advanced composites consisting of two or more materials whose composition and structure changes gradually over the volume, leading to gradual change in its property. Typically one component is a ceramic, while the other is metal or metal alloy. In common embodiments, the composition can change gradually from all ceramic on one side to all metal on the other side, the ceramic contributing high resistance to temperature with the metal contributing high ductility. It’s a way to incorporate favorable properties of two materials into different locations of a single structure. In ideal cases the composition changes gradually but for manufacturing reasons, the changes occurs in homogeneous layers. The boundaries between the layers cause thermal residual stresses to arise, especially during material processing and in cases of cyclic loading. These thermal residual stresses are a result of differences in the thermal expansion coefficient of the varying compositions of materials. Thermal residual stresses may lead to cracking and ultimate failure of FGM’s. This study investigates the formation of thermal residual stresses for a Nickel-Alumina Ni-Al₂O₃ FGM. The Ni-Al₂O₃ system is chosen because it is one of the most common systems used in practice. This work explores the formation, impact and minimization of thermal residual stresses for a number of practical conditions that may arise during processing. All relevant properties are calculated using the rule of mixture equations. Abaqus, a non-linear FEA solver, is used for all the simulation work. In addition, the study includes an elasto-plastic analysis to illuminate the process of crack development.

Hilbert expansion of Boltzmann equations for gas mixture

The aim of this paper is to study the hydrodynamic limit of Boltzmann equations for a gas mixture containing two kinds of particles, where one is assumed to be near a vacuum and the other is near a Maxwellian equilibrium state. By the method of Hilbert expansion, we could derive the compressible Euler systems together with the terms of Boltzmann equations according to the different orders of the Knudsen number. Next, we truncate the expansion and establish the uniform estimates of the remainder term by using L 2 − L ∞ estimate.

Coupled-cluster theory for trapped bosonic mixtures.

We develop a coupled-cluster theory for bosonic mixtures of binary species in external traps, providing a promising theoretical approach to demonstrate highly accurately the many-body physics of mixtures of Bose-Einstein condensates. The coupled-cluster wavefunction for the binary species is obtained when an exponential cluster operator eT, where T = T(1) + T(2) + T(12) and T(1) accounts for excitations in species-1, T(2) for excitations in species-2, and T(12) for combined excitations in both species, acts on the ground state configuration prepared by accumulating all bosons in a single orbital for each species. We have explicitly derived the working equations for bosonic mixtures by truncating the cluster operator up to the single and double excitations and using arbitrary sets of orthonormal orbitals for each of the species. Furthermore, the comparatively simplified version of the working equations are formulated using the Fock-like operators. Finally, using an exactly solvable many-body model for bosonic mixtures that exists in the literature allows us to implement and test the performance and accuracy of the coupled-cluster theory for situations with balanced as well as imbalanced boson numbers and for weak to moderately strong intra- and interspecies interaction strengths. The comparison between our computed results using coupled-cluster theory with the respective analytical exact results displays remarkable agreement exhibiting excellent success of the coupled-cluster theory for bosonic mixtures. All in all, the correlation exhaustive coupled-cluster theory shows encouraging results and could be a promising approach in paving the way for high-accuracy modeling of various bosonic mixture systems.

A large deformation multiphase continuum mechanics model for shock loading of soft porous materials

AbstractA large deformation, coupled finite‐element (FE) model is developed to simulate the multiphase response of soft porous materials subjected to high strain‐rate loading. The approach is based on the theory of porous media (TPM) at large deformations. Simplifications to the one‐dimensional regime studied in the numerical simulations follow. An overview of several different time integration schemes is presented for the purpose of solving the nonlinear dynamic coupled balance of momenta (mixture and fluid) and balance of mass of the mixture equations. Numerical examples are presented for (i) verification against closed‐form analytical solutions assuming small loads, (ii) demonstrating large deformation effects at high strain‐rate, and (iii) showing differences in deformations between a single‐phase elastodynamics model with occluded compressible pore fluid and a multiphase poroelastodynamics model at high strain‐rate. The multiphase model shows that the relative motion of the pore fluid significantly dampens the deformation response of the solid skeleton as compared to the single‐phase model, and makes it possible to extract quantitative values for the stresses of the different constituents, thereby allowing one to form preliminary conclusions about the onset of damage in the solid skeleton. The novelty of the current work is developing a multiphase, large deformation, mixture theory numerical model for high strain‐rate loading of soft porous materials. It was discovered that explicit, adaptive time‐stepping Runge–Kutta schemes offer high accuracy at relatively low cost when compared to traditional implicit or explicit central difference time‐stepping schemes for shock‐like loadings. Shock viscosity is added to the mixture momentum balance equation to regularize the shock front, and a stabilization term is added to the mixture mass balance equation to stabilize equal order interpolation finite elements for the coupled finite element solution of multiphase materials.

A reaction-cross-diffusion model derived from kinetic equations for gas mixtures

We consider a binary reacting mixture composed by a monatomic and a polyatomic gas, diffusing in a host medium, and undergoing elastic, inelastic and chemical interactions. We deduce a proper set of reaction–diffusion equations for the number densities of the two components from Boltzmann kinetic equations, in presence of a multiple time scale process. The dominant phenomenon is constituted by the elastic collision with the host medium, while other elastic collisions are less frequent; inelastic and chemical interactions are part of the slow process as well. We show that, in the diffusive limit, the total density of polyatomic constituent exhibits a cross-diffusion term. Then, we perform a preliminary Turing instability analysis in terms of macroscopic and microscopic parameters, and we investigate numerically the formation of stable patterns.

Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling

We present the arising of the Fick cross-diffusion system of equations for fluid mixtures from the multi-species Boltzmann equation in a rigorous manner in Sobolev spaces. To this end, we formally show that, in a diffusive scaling, the hydrodynamical limit of the kinetic system is the Fick model supplemented with a closure relation and we give explicit formulae for the macroscopic diffusion coefficients from the Boltzmann collision operator. Then, we provide a perturbative Cauchy theory in Sobolev spaces for the constructed Fick system, which turns out to be a dilated parabolic equation. We finally prove the stability of the system in the Boltzmann equation, ensuring a rigorous derivation between the two models.

Hydrodynamic limit of Boltzmann equations for gas mixture

In this paper, we study the hydrodynamic limit of Boltzmann equations for gas mixture by Hilbert expansion method. We formally derive all the terms in the Hilbert expansion of Boltzmann equations according to different order about the Knudsen number. Then we truncate the expansion and justify the hydrodynamic limit of Boltzmann equations by establishing the uniform estimates of the remainder term. Our approach is based on L2−L∞ framework, which is motivated by the study of the single Boltzmann equation in [22].

Influence of distributed hydrogen injection and combustion on supersonic boundary layer stability and transition

This theoretical study recorded the influence of the distributed injection of hydrogen through a permeable surface and its combustion on the stability and transition of a supersonic boundary layer (BL). The laminar base flow for a multicomponent flat-plate BL was computed using the computational fluid dynamics solver with finite-rate chemistry for free-stream Mach number (M) = 2. Linear stability theory (LST) equations for a reactive gas mixture were derived in the local parallel base flow approximation. Stability calculations based on the developed theory revealed the possibility of a decrease in the local spatial amplification rates of unstable perturbations. A double reversal in the maximal amplification rate magnitudes of perturbations was obtained, wherein at first, they increased, then decreased to zero, and then rose again. The influence of the distributed injection and combustion of hydrogen in the supersonic BL on the position of the laminar-turbulent transition was estimated using the LST-based eN-method. Calculated and experimentally obtained dependencies of the relative transition Reynolds number on the relative mass flow rate of injected hydrogen were compared. It was found that depending upon the flow conditions, the hydrogen diffusion flame could accelerate or decelerate the transition in the supersonic M = 2 BL, when compared to absence of hydrogen burning. Two counteracting effects compete: heat supply by combustion exerts stabilizing influence, while low-molecular-weight gas blowing from the surface exerts destabilizing influence. Depending on the interplay of these two factors, it would be possible to obtain acceleration or deceleration of transition.

PANI/short nylon fiber/natural rubber conducting composites: Dielectric properties

Polyaniline/Polyaniline coated short nylon fibre/natural rubber (PANI/PANI-N/NR) CPCs and PANI/NR CPCs were prepared by mechanical mixing on a two-roll mill. PANI-N provides both conductivity and good mechanical properties to the CPCs. Such CPCs with enhanced mechanical properties are expected to find many device applications. The frequency and temperature dependence of dielectric properties of PANI, NR, PANI/NR and PANI/PANI-N/NR CPCs in the frequency range 0.1 to 8 MHz and in the temperature range 303 to 393 K, were investigated. Different theoretical equations and mixture equations were applied to fit the observed dielectric data of the CPCs.

A novel concept of enhanced direct-contact condensation of vapour- inert gas mixture in a spray ejector condenser

An analytical model of direct steam condensation (DCC) in the novel idea of spray ejector condenser (SEC) in the presence of inert gas has been developed. It is based on continuity, momentum and energy equations for the steam-carbon dioxide mixture and direct contact condensation mechanisms due to heat transfer and concentration. Crucial in the process of DCC is atomisation of the motive fluid in the ejector. The effect of atomised droplet size is exhibiting a significant amplification influence with increasing size of the droplet. Motive fluid is driving the secondary fluid-mixture of steam and inert gas in the Venturi nozzle and is cold enough to cause direct condensation. The intensity of heat transfer process from steam to water when the phases are in direct contact is much higher than the heat transfer intensity in surface heat exchangers. The analytical model pertains to a subcritical flow of a mixture of steam and gas in SEC. The model exhibits satisfactory agreement with experimental data. The model of DCC predicts higher values of temperature drop between inlet and outlet from the mixing section for the case of smaller steam-CO2 flow rates. Increasing the flow rate of steam mixture from 1.2 g/s to 3.6 g/s results in a reduction of steam mixture temperature from 25 °C to 14 °C respectively, at CO2 flow rate of 6.8 m3/h. Condensation without presence of CO2 in the same range of steam flow rate, i.e. from 1.2 g/s to 3.6 g/s results in reduction of steam mixture temperature from 56 °C to 25 °C respectively, confirming in such way the effect of CO2 presence on the efficiency of DCC. Such model allow for discussion of parameters affecting process of condensation in SEC and ability of application such condenser in power plants.

Relative dynamics of quantum vortices and massive cores in binary BECs

We study the motion of superfluid vortices with filled massive cores. Previous point-vortex models already pointed out the impact of the core mass on the vortex dynamical properties, but relied on an assumption that is questionable in many physical systems where the immiscibility condition is barely satisfied: the fact that the massive core always lays at the very bottom of the effective confining potential constituted by the hosting vortex. Here, we relax this assumption and present a new point-vortex model where quantum vortices are harmonically coupled to their massive cores. We thoroughly explore the new dynamical regimes offered by this improved model; we then show that the functional dependence of the system normal modes on the microscopic parameters can be correctly interpreted only within this new generalized framework. Our predictions are benchmarked against the numerical simulations of coupled Gross–Pitaevskii equations for a realistic mixture of atomic Bose–Einstein condensates.