Carnahan-Starling Equation Of State Research Articles

Pore-scale modelling of water sorption in nanopore systems of shale

Water vapor sorption in nanoporous media with complex pore structures, like shale, is not well understood. To address this, a pseudopotential lattice Boltzmann method (LBM) is developed to investigate water sorption behavior. The LBM model incorporates long-range molecular forces using a modified Shan-Chen model based on the Carnahan-Starling equation of state. The simulation results show that the presence of water films in nanopores can create a liquid pressure difference of up to 100 MPa between confined and free states. The adsorption theories based on simple pore shapes are not applicable to nanoporous systems with complex pore geometries. Additionally, when the relative humidity exceeds 1, water vapor condenses inside hydrophobic nanostructures while being attracted by neighboring liquid, leading to cluster formation. In water-wet nanoporous media, capillary condensation occurs progressively from small throats to large pores, and the sorption curves vary smoothly with pressure variation. In mixed-wet nanoporous media, the sorption curves exhibit no hysteresis in the early desorption stage, where the adsorption and desorption processes occur in the region around the hydrophobic particles and are reversible. Overall, this study provides valuable insights into water sorption behavior in nanopore systems of shale and sets the foundation for modelling liquid-vapor distribution in nanoporous media at the pore scale.

Equationsof state in multiphase lattice Boltzmann method revisited.

The single-component multiphase fluids can be described by a single equationof state (EOS), and various EOSs have been employed in the multiphase lattice Boltzmann (LB) method. In this work, we revisit five commonly used EOSs, including the van der Waals EOS, the Redlich-Kwong EOS, the Redlich-Kwong-Soave EOS, the Peng-Robinson EOS, and the Carnahan-Starling EOS. The recent multiphase LB model with self-tuning EOS is employed because of its thermodynamic consistency in a strict sense and clear physical picture at the microscopic level. First, the way to incorporate these multiphase EOSs is proposed. Two scaling factors are introduced to independently adjust the surface tension and interface thickness, and the lattice sound speed is EOS-dependent to ensure the numerical stability. Then, numerical tests are conducted to validate the incorporations of these EOSs and compare their numerical performances. The surface tension and interface thickness are set to the same values for different EOSs in the comparisons. The liquid and gas densities, surface tension, and interface thickness by the LB simulation agree well with the thermodynamic results. The maximum density ratios achieved with different EOSs are at the same level and could be very close to each other when the interface thickness is relatively small. The effects of multiphase EOS, density ratio, and dimensionless relaxation time on the spurious current are discussed in detail. It is interesting to find the van der Waals EOS shows the best numerical performance in reducing the spurious current.

The dynamic response of He bubble in bicrystal copper under uniaxial compression and tension

He bubbles can exhibit different behaviors due to their different positions in the nanocrystalline materials. Using molecular dynamics simulations, we study the dynamic response of He bubble in bicrystal Cu under uniaxial compression and tension. The differences between He bubbles in grain interiors and grain boundaries (GBs) are explored, for different loading orientations. Our results reveal that the He bubble-induced dislocations dominate the initial yielding behaviors of the samples, and will induce the earlier dislocations emission from the GB due to the GB-dislocation interaction. Different reductions in yielding stress are observed for different loading directions and He bubble positions, leading to the changes in the compression/tension asymmetry. Particularly, the effect of He bubble positions on the yield stress is more significant under the loading direction that perpendicular to the GB plane than parallel to the GB plane. Three stages of He bubble evolution are interpreted during both compression and tension loading. The GB-bubble distance is found to have little effect on the yielding behavior of the material, but plays an important role in the He bubble evolution at the later stage of tension. Furthermore, the Carnahan-Starling equation of state is used to describe the evolution of He bubble embedded in bicrystal Cu during compression and tension, by fitting the parameters in the Brearley and MacInnes hard-sphere model.

Finite ion size and ion permittivity effects on gel electrophoresis of a soft particle

Electrophoresis of a soft particle embedded in a hydrogel medium is studied by considering the hydrated ions as charged dielectric spheres of finite radius. The Nernst-Planck equation for the transport of ions is modified to take into account the ion steric repulsion and the ion-solvent dielectric interactions (MNP-model). The effective polarizability of the hydrated ions lowers the dielectric permittivity of the ionic solution and consequently, the permittivity of the medium varies with local ionic concentration. This spatial variation of the permittivity creates a Born energy difference in ions leading to a Born force experienced by the ions. In addition, ions experience a dielectrophoretic force due to the polarization under the imposed external electric field. The ion steric repulsion are modeled by the Carnahan-Starling equation of state. The convection of ions are governed by the Brinkman extended Navier-Stokes equations. The modified Nernst-Planck equation coupled with the Brinkman Navier-Stokes equations and the equation for electric field are solved numerically through a control volume approach. The ion steric interactions and ion-solvent interactions create a saturation of mobile ions in the polyelectrolyte layer (PEL) of the soft particle, which leads to an attenuation of the counterion condensation of the PEL. This short range steric repulsion and ion-solvent interactions have significance for higher bulk ionic concentration as well as higher range of charge density of the soft particle. These interactions are elucidated by comparing with the mobility obtained by the standard model (PNP-model). The Dukhin number and the effective charge density of the PEL are determined to illustrate the mobile ion saturation. The discrepancy between the present modified model from the standard model magnifies for the case of salts of multivalent counterions.

Monte Carlo simulations and mean-field modeling of electric double layers at weakly and moderately charged spherical macroions.

Monte Carlo simulations are employed to determine the differential capacitance of an electric double layer formed by small size-symmetric anions and cations in the vicinity of weakly to moderately charged macroions. The influence of interfacial curvature is deduced by investigating spherical macroions, ranging from flat to moderately curved. We also calculate the differential capacitance using a previously developed mean-field model where, in addition to electrostatic interactions, the excluded volumes of the ions are taken into account using either the lattice-gas or the Carnahan-Starling equation of state. For both equations of state, we compare the mean-field model for arbitrary curvature with a recently developed second-order curvature expansion. Our Monte Carlo simulations predict an increase in the differential capacitance with growing macroion curvature if the surface charge density is small, whereas for moderately charged macroions the differential capacitance passes through a local minimum. Both mean-field models tend to somewhat overestimate the differential capacitance as compared with Monte Carlo simulations. At the same time, they do reproduce the curvature dependence of the differential capacitance, especially for small surface charge density. Our study suggests that the quality of mean-field modeling does not worsen when weakly or moderately charged macroions exhibit spherical curvature.

Pore-scale investigation on multiphase reactive transport for the conversion of levulinic acid to γ-valerolactone with Ru/C catalyst

Hydrogenation of levulinic acid (LA) into γ-valerolactone (GVL) with Ru/C catalytic system involves homogeneous/heterogeneous reactions, multiphase mass transfer and hydrogen dissolution. The understanding of these physiochemical phenomena is fundamental to propose the performance improvement strategies in such reaction system. In this study, a pore-scale multiphase reactive transport model based on the lattice Boltzmann method (LBM) is developed to simulate the conversion of LA to GVL over Ru/C catalyst. A Carnahan-Starling equation of state (CS EOS) modified Shan-Chen pseudopotential model is applied for gas–liquid flow. The model is validated by comparing the simulation predictions with existing experimental results. Simulation results show that the intra-particle diffusion distance of hydrogen is shorter than that of LA. The LA consumption rate is found to be determined by temperature and dissolved hydrogen when the average concentration of LA is higher than a critical value (about 0.5 mol/L). Notable negative effect of LA intra-particle mass transfer on the reaction has been observed for the LA average concentration lower than 0.5 mol/L. The reaction-generated water would result in a little inhibition on LA hydrogenation. To enhance the utilization of catalyst and reduce the local transport resistant, hollow spheres or coating catalyst are more effective for this system. We demonstrate the usefulness of the present pore-scale LB model in providing guidance for efficient hydrogenation LA into GVL.

Investigation of dimensional accuracy of metal droplet deposition under repulsion using a lattice Boltzmann approach

Purpose This paper aims to investigate the influence of droplet infiltration and sliding on the deposition size and make a uniform deposition by controlling the interaction between droplets, using the three-dimensional lattice Boltzmann method (LBM) based on the actual working condition. Design/methodology/approach D3Q19 Shan-Chen LB approach is developed and optimized based on the metal droplet deposition. The Carnahan-Starling equation of state and transition layers are introduced to maintain the greater stability and low pseudo velocities. In addition, an additional collision term is adopted to implement immersed moving boundary scheme to deal with no-slip boundaries on the front of the phase change. Findings The numerical results show that the new¬ incoming droplet wet and slide off the solidified surface and the rejection between droplets are the reasons for the deviation of the actual deposition length. The total length of the longitudinal section negatively correlates with the deposition distance. To improve the dimensional accuracy, the deposition distance and repulsion rate need to be guaranteed. The optimal deposition distance is found to have a negative linear correlation with wettability. Originality/value The numerical model developed in this paper will help predict the continuous metal droplet deposition and provide guidance for the selection of deposition distance.

Thermodynamic Properties of the Parabolic-Well Fluid

The thermodynamic properties of the parabolic-well fluid are considered. The intermolecular interaction potential of this model, which belongs to the class of the so-called van Hove potentials, shares with the square-well and the triangular well potentials the inclusion of a hard-core and an attractive well of relatively short range. The analytic second virial coefficient for this fluid is computed explicitly and an equation of state is derived with the aid of the second-order thermodynamic perturbation theory in the macroscopic compressibility approximation and taking the hard-sphere fluid as the reference system. For this latter, the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation of state as derived within the rational function approximation method, is employed. The results for the reduced pressure of the parabolic-well fluid as a function of the packing fraction and two values of the range of the parabolic-well potential at different temperatures are compared with Monte Carlo and Event‐driven molecular dynamics simulation data. Estimates of the values of the critical temperature are also provided.

Equation of state of hard-sphere fluid in nanoporous media

The present work is devoted to the study of fluid equations of state in nanoporous media. The modelling is carried out in the framework of the molecular dynamics method using the potential of hard spheres. The dependences of the compressibility factor on packing fraction are studied for various characteristic pore sizes, porosities, and types of grains packing of a porous medium. It is shown that for all cases considered, deviations of the obtained equations of state from the Carnahan-Starling equation of state are insignificant. These deviations may be due to the presence of restricted zones in a porous medium inaccessible to fluid molecules that is also discussed in the work.

Lattice Boltzmann simulations of thermal flows beyond the Boussinesq and ideal-gas approximations.

In this work, the recent lattice Boltzmann model with self-tuning equation of state (EOS) [R. Huang et al., J. Comput. Phys. 392, 227 (2019)]JCTPAH0021-999110.1016/j.jcp.2019.04.044 is improved in three aspects to simulate the thermal flows beyond the Boussinesq and ideal-gas approximations. First, an improved scheme is proposed to eliminate the additional cubic terms of velocity, which can significantly improve the numerical accuracy. Second, a local scheme is proposed to calculate the density gradient instead of the conventional finite-difference scheme. Third, a scaling factor is introduced into the lattice sound speed, which can be adjusted to effectively enhance numerical stability. The thermal Couette flow of a nonattracting rigid-sphere fluid, which is described by the Carnahan-Starling EOS, is first simulated, and the better performance of the present improvements on the numerical accuracy and stability is demonstrated. As a further application, the turbulent Rayleigh-Bénard convection in a supercritical fluid slightly above its critical point, which is described by the van der Waals EOS, is successfully simulated by the present lattice Boltzmann model. The piston effect of the supercritical fluid is successfully captured, which induces a fast and homogeneous increase of the temperature in the bulk region, and the time evolution from the initiation of heating to the final turbulent state is analyzed in detail and divided into five stages.

A numerical study of the early-stage dynamics of a bubble cluster

The dynamics of multiple cavitating bubbles is numerically simulated, with the ambient pressure lower than the saturated vapor pressure, using a pseudopotential lattice Boltzmann method (LBM) coupled with the Carnahan-Starling equation of state. Dual-bubble and multi-bubble systems are tested, and the method for the bubble cluster is validated. It is found that the bubble can either grow or collapse in the early stage, depending on the configuration of the bubble cluster, characterized by the bubble number, the inter-bubble distance and the initial radii. In the induced flow, the bubbles are mutually interacted. Scaling relations of the interaction are proposed according to the numerical results. With consideration of the interactions, the simplified Rayleigh-Plesset equations (RPEs) for multiple bubbles can describe the evolution of the bubbles approximately. The results may serve as the basis for improved cavitation models.

Study of Cavitation Bubble Collapse near a Wall by the Modified Lattice Boltzmann Method

In this paper, an improved lattice Boltzmann Shan‒Chen model coupled with Carnahan-Starling equation of state (C-S EOS) and the exact differential method (EDM) force scheme is used to simulate the cavitation bubble collapse in the near-wall region. First, the collapse of a single cavitation bubble in the near-wall region was simulated; the results were in good agreement with the physical experiment and the stability of the model was verified. Then the simulated model was used to simulate the collapse of two cavitation bubbles in the near-wall region. The main connection between the two cavitation bubble centre lines and the wall surface had a 45° angle and parallel and the evolution law of cavitation bubbles in the near-wall region is obtained. Finally, the effects of a single cavitation bubble and double cavitation bubble on the wall surface in the near-wall region are compared, which can be used to study the method to reduce the influence of cavitation on solid materials in practical engineering. The cavitation bubble collapse process under a two-dimensional pressure field is visualized, and the flow field is used to describe the morphological changes of cavitation bubble collapse in the near-wall region. The improved lattice Boltzmann Method (LBM) Shan‒Chen model has many advantages in simulating cavitation problems, and will provide a reference for further simulations.

Modeling the camel-to-bell shape transition of the differential capacitance using mean-field theory and Monte Carlo simulations

Mean-field electrostatics is used to calculate the differential capacitance of an electric double layer formed at a planar electrode in a symmetric 1:1 electrolyte. Assuming the electrolyte is also ion-size symmetric, we derive analytic expressions for the differential capacitance valid up to fourth order in the surface charge density or surface potential. Our mean-field model accounts exclusively for electrostatic interactions but includes an arbitrary non-ideality in the mixing entropy of the mobile ions. The ensuing criterion for the camel-to-bell shape transition of the differential capacitance is analyzed using commonly used mixing models (one based on a lattice gas and the other based on the Carnahan-Starling equation of state) and compared with Monte Carlo simulations. We observe a reasonable agreement between all our mean-field models and the simulation data for the camel-to-bell shape transition. The absolute value of the differential capacitance for an uncharged (or weakly charged) electrode is, however, not reproduced by our mean-field approaches, not even upon introducing a Stern layer with a thickness equal of the ion radius. We show that, if a Stern layer is introduced, its thickness dependence on the ion size is non-monotonic or, depending on the salt concentration, even inversely proportional.

Oblique impact of droplets on microstructured superhydrophobic surfaces

This paper studies the oblique impact of droplets on microstructured superhydrophobic surfaces with the lattice Boltzmann method. Three-dimensional simulations are performed based on the pseudopotential multiphase model, with the Carnahan-Starling equation of state and the multiple-relaxation-time collision operator adopted to overcome the numerical instability problems at high liquid/gas density ratio and low fluid viscosity. The droplet impact dynamics is simulated in a wide range of normal Weber numbers (0–30), Ohnesorge numbers (0.002–3), and impact angles (15°–90°), with particular interest in understanding different types of bouncing behavior and their effects on the contact time. The results show that, among the four possible types of bouncing identified in the simulation, including conventional retracting bouncing, incomplete-retracting bouncing, impaled retracting bouncing, and tumbling bouncing, incomplete-retracting bouncing and tumbling bouncing have less contact time and could be facilitated by increasing the impact obliqueness. The increase in the normal Weber number leads to a transition from conventional retracting bouncing, to incomplete-retracting bouncing, and then to impaled retracting bouncing. The contact time shows a non-monotonic trend of “decrease”-“increase”-“decrease” with enlarged Ohnesorge number, and the sheer drop in the contact time at high Ohnesorge numbers is due to the occurrence of tumbling bouncing.

Study on multicomponent pseudo-potential model with large density ratio and heat transfer

Large density ratio multiphase flow is a persistent challenge within the field of computational fluid dynamics. This paper investigates improvements to the lattice Boltzmann based large density ratio multicomponent multiphase pseudo-potential model. The improvements include: the multiple-relaxation-time (MRT) collision operator; the exact difference method scheme; the Carnahan-Starling equation of state; and an addition of correction factor k to the equation of state. The improved model can be used for simulating large density ratio (O(1000)) multiphase flow with small spurious current and better numerical stability. The smaller spurious current can be obtained by decreasing k value, and yet interface thickness increases. The density ratio is 1284 for k=0.1 and the spurious current is reduced to 0.0069, which is much smaller than that of 0.033 in literature. The interfacial tension can be adjusted independently from density ratio by changing k value. A thermal multiphase flow model is developed based on the large density ratio pseudo-potential model. The model is validated by using static heat conduction and dynamic flow simulations. The result of the static heat conduction of the flat interface has smaller error with the theoretical solution than that of droplet. The result of thermocapillary migration is comparable with the theoretical prediction. Finally, the heat conduction melting is simulated by coupling the enthalpy-based method.

Three-Dimensional Cavitation Bubble Simulations based on Lattice Boltzmann Model Coupled with Carnahan-Starling Equation of State

AbstractThe Shan-Chen multiphase lattice Boltzmann model (LBM) coupled with Carnahan-Starling real-gas equation of state (C-S EOS)was proposed to simulate three-dimensional (3D) cavitation bubbles. Firstly, phase separation processes were predicted and the inter-phase large density ratio over 2×104was captured successfully. The liquid-vapor density ratio at lower temperature is larger. Secondly, bubble surface tensions were computed and decreased with temperature increasing. Thirdly, the evolution of creation and condensation of cavitation bubbles were obtained. The effectiveness and reliability of present method were verified by energy barrier theory. The influences of temperature, pressure difference and critical bubble radius on cavitation bubbles were analyzed systematically. Only when the bubble radius is larger than the critical value will the cavitation occur, otherwise, cavitation bubbles will dissipate due to condensation. According to the analyses of radius change against time and the variation ratio of bubble radius, critical radius is larger under lower temperature and smaller pressure difference condition, thus bigger seed bubbles are needed to invoke cavitation. Under higher temperature and larger pressure difference, smaller seed bubbles can invoke cavitation and the expanding velocity of cavitation bubbles are faster. The cavitation bubble evolution including formation, developing and collapse was captured successfully under various pressure conditions.

A variational approach to the liquid-vapor phase transition for hardcore ions in the bulk and in nanopores.

We employ a field-theoretical variational approach to study the behavior of ionic solutions in the grand canonical ensemble. To describe properly the hardcore interactions between ions, we use a cutoff in Fourier space for the electrostatic contribution of the grand potential and the Carnahan-Starling equation of state with a modified chemical potential for the pressure one. We first calibrate our method by comparing its predictions at room temperature with Monte Carlo results for excess chemical potential and energy. We then validate our approach in the bulk phase by describing the classical "ionic liquid-vapor" phase transition induced by ionic correlations at low temperature, before applying it to electrolytes at room temperature confined to nanopores embedded in a low dielectric medium and coupled to an external reservoir of ions. The ionic concentration in the nanopore is then correctly described from very low bulk concentrations, where dielectric exclusion shifts the transition up to room temperature for sufficiently tight nanopores, to high concentrations where hardcore interactions dominate which, as expected, modify only slightly this ionic "capillary evaporation."

Investigation of cavitation bubble collapse near rigid boundary by lattice Boltzmann method

The dynamics of the bubble collapse near a rigid boundary is a fundamental issue for the bubble collapse application and prevention. In this paper, the bubble collapse is modeled by adopting the lattice Boltzmann method (LBM) and is verified, and then the dynamic characteristics of the collapsing bubble with the second collapse is investigated. The widely used Shan-Chen model in the LBM multiphase community is modified by coupling with the Carnahan-Starling equation of state (C-S EOS) and the exact difference method (EDM) for the forcing term treatment. The simulation results of the bubble profile evolution by the LBM are in excellent agreements with the theoretical and experimental results. From the two-dimensional pressure field evolution, the dynamic characteristics of the different parts during the bubble collapse stage are studied. The role of the second collapse in the rigid boundary damage is discussed, and the impeding effect between two collapses is demonstrated.

Simple cubic equation of state applied to hard-sphere, Lennard-Jones fluids, simple fluids and solids

Based on the expansion and extension of the virial equation of state (EOS) of hard-sphere fluids solved by the Percus–Yevick integration equation, a universal cubic (UC) EOS is developed. The UC EOS is applied to model hard-sphere and Lennard-Jones (LJ) fluids, simple Ar and N2 liquids at low temperatures, and supercritical Ar and N2 fluids at high temperatures, as well as ten solids, respectively. The three parameters are determined for the hard-sphere fluid by fitting molecular dynamics (MD) simulation data of the third to eighth virial coefficients in the literature; for other fluids by fitting isothermal compression data; and for solids by using the Einstein model. The results show that the UC EOS gives better results than the Carnahan–Starling EOS for compressibility of hard-sphere fluids. The Helmholtz free energy and internal energy for LJ fluids are predicted and compared with MD simulation data. The calculated pressures for simple Ar and N2 liquids are compared with experimental data. The agreement is fairly good. Eight three-parameter EOSs are applied to describe isothermals of ten typical solids. It is shown that the UC EOS gives the best precision with correct behavior at high-pressure limitation. The UC EOS considering thermal effects is used to analytically evaluate the isobaric thermal expansivity and isothermal compressibility coefficients. The results are in good agreement with experimental data.

Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model

Owing to its conceptual simplicity and computational efficiency, the pseudopotential multiphase lattice Boltzmann (LB) model has attracted significant attention since its emergence. In this work, we aim to extend the pseudopotential LB model to simulate multiphase flows at large density ratio and relatively high Reynolds number. First, based on our recent work [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)], an improved forcing scheme is proposed for the multiple-relaxation-time pseudopotential LB model in order to achieve thermodynamic consistency and large density ratio in the model. Next, through investigating the effects of the parameter a in the Carnahan-Starling equation of state, we find that the interface thickness is approximately proportional to 1/√a. Using a smaller a will lead to a wider interface thickness, which can reduce the spurious currents and enhance the numerical stability of the pseudopotential model at large density ratio. Furthermore, it is found that a lower liquid viscosity can be gained in the pseudopotential model by increasing the kinematic viscosity ratio between the vapor and liquid phases. The improved pseudopotential LB model is numerically validated via the simulations of stationary droplet and droplet oscillation. Using the improved model as well as the above treatments, numerical simulations of droplet splashing on a thin liquid film are conducted at a density ratio in excess of 500 with Reynolds numbers ranging from 40 to 1000. The dynamics of droplet splashing is correctly reproduced and the predicted spread radius is found to obey the power law reported in the literature.