Carnahan-Starling Equation Research Articles

Innovative molecular descriptors in QSPR modeling: Integrating Carnahan-Starling EoS for predicting diffusion coefficients in hydrocarbons and mixtures

The rapid and accurate prediction of self-diffusion coefficients for pure fluids and mutual diffusion coefficients for binary mixtures remains a focal point of current research. In this work, molecular descriptors are innovatively defined by introducing molecular repulsive and attractive pressures based on the Carnahan-Starling equation of state. This approach aims to improve the quantitative structure–property relationship (QSPR) models for predicting thermophysical properties that vary with thermodynamic conditions. The shapely additive explanation method is employed for a comprehensive analysis of the model mechanism. Among the six molecular descriptors selected, our defined attractive pressure (patt), MATS2e and SpMin3_Bh(v) contribute significantly to the self-diffusion coefficient QSPR model. Based on this approach, QSPR models are constructed for 15 hydrocarbons and five of their binary mixtures. The results demonstrate that the GA-BPNN-based QSPR model for pure fluids achieves high prediction accuracy, with an external Rext2 of 0.978 and an external RMSEext of 5.349 × 10−9·m2·s−1. The optimal mixture descriptor type is x1d1+x2d2, and the GS-SVM-based mixture QSPR model predicts 98.24 % of values within a 5 % error margin, with Rext2 of 0.991 and RMSEext of 0.042 × 10−9·m2·s−1.

Study on the evolution of heterogeneous double-cavity induced by near-wall and the fluctuation characteristics of load field

It is well known that the collapse of heterogeneous multi-cavity near the wall will induce the fluctuation of the load field. To address this problem, the Lattice Boltzmann Method (LBM) is applied to model the three-phase coupling between gas-liquid-solid. The objective is to investigate the evolution of heterogeneous double bubbles and the spatial-temporal distribution characteristics of wall loads induced near the wall. In this study, the pseudopotential Multi-Relaxation-Time Lattice Boltzmann Model (MRT-LBM) and the Carnahan-Starling Equation of State (C-S-EOS) with an extended format for the external force term are used. The effects of the distance of the bubble to the wall, the pressure differences between the inside and outside of the bubble, and the relative size of the bubble on the dynamic evolution and the load distribution characteristics of heterogeneous multi-bubbles near the wall are investigated in order to determine the influence of these factors. Under a two-dimensional pressure field, the collapse process of double cavitation bubbles is visualized. Through the flow field, the morphological changes of the cavitation bubble collapse near the wall are also described. Various parameters are found to have an influence on the evolution of double cavitation bubbles near the wall and the resulting load field. The study employs the Lattice Boltzmann Method and the Potential Model for the analysis of the heterogeneous bubble collapses in the near wall region.

Confinement Effects in Droplet Formation on a Solid Particle.

Formation of a droplet around a spherical solid particle in supersaturated vapor is considered. The number and stability of equilibrium solutions in a closed small system are studied in the canonical ensemble in comparison to an open system in the grand canonical ensemble. Depending on the system's parameters, two modes exist in the canonical ensemble: the first one with only one solution and the second one with three solutions; the presence of the third solution is due to confinement. The analysis is conducted first on a macroscopic thermodynamic level of description, and then the results are supported by studies within two versions of classical density functional theory: the square-gradient approximation with the Carnahan-Starling equation of state for hard spheres on a completely wettable particle and the random-phase approximation with the fundamental measure theory on a poorly wettable particle. In the latter case, a solution breaking the spherical symmetry is observed at a small total number of molecules.

Study of the positional and orientational contributions to the Helmholtz free energy of a finite hard-disk system. A molecular dynamics analysis of its hexatic transition

Based on an equilibrium thermodynamic scheme, this work studies the separation of the Helmholtz free energy of a hard-disk system into its positional and orientational contributions. For the positional part, an accurate two dimensional version of the Carnahan-Starling equation of state is proposed, while a simple ansatz of the one particle partition function for the orientational part is proposed. The latter expression, which contains information about the probability of the norm of the six-fold orientational order parameter, is obtained through molecular dynamics simulations. Perhaps due to finite size effects, the ordering of the fluid is not a sudden phenomenon: the fluid-hexatic phase transition is preceded by a linear increase in the orientational contribution to the pressure of the fluid. A similar result is obtained for the hexatic-solid phase transition. Thus, the hexatic first-order phase transition is located between two linear ordering processes, consistent with the KTHNY theory. The proposed separation also predicts the existence of first order fluid-hexatic phase transition by using two procedures: the first, based on a purely bimodal probability density function, produces a pressure with a van del Waals-like first-order phase transition, while the second, considering density fluctuations, results in a flattened pressure phase transition.

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.

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.

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.

On Using the BMCSL Equation of State to Renormalize the Onsager Theory Approach to Modeling Hard Prolate Spheroidal Liquid Crystal Mixtures

Modifications to the traditional Onsager theory for modeling isotropic–nematic phase transitions in hard prolate spheroidal systems are presented. Pure component systems are used to identify the need to update the Lee–Parsons resummation term. The Lee–Parsons resummation term uses the Carnahan–Starling equation of state to approximate higher-order virial coefficients beyond the second virial coefficient employed in Onsager’s original theoretical approach. As more exact ways of calculating the excluded volume of two hard prolate spheroids of a given orientation are used, the division of the excluded volume by eight, which is an empirical correction used in the original Lee–Parsons resummation term, must be replaced by six to yield a better match between the theoretical and simulation results. These modifications are also extended to binary mixtures of hard prolate spheroids using the Boublík–Mansoori–Carnahan–Starling–Leland (BMCSL) equation of state.

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.

BoltzmaNN: Predicting effective pair potentials and equations of state using neural networks.

Neural networks (NNs) are employed to predict equations of state from a given isotropic pair potential using the virial expansion of the pressure. The NNs are trained with data from molecular dynamics simulations of monoatomic gases and liquids, sampled in the NVT ensemble at various densities. We find that the NNs provide much more accurate results compared to the analytic low-density limit estimate of the second virial coefficient and the Carnahan-Starling equation of state for hard sphere liquids. Furthermore, we design and train NNs for computing (effective) pair potentials from radial pair distribution functions, g(r), a task that is often performed for inverse design and coarse-graining. Providing the NNs with additional information on the forces greatly improves the accuracy of the predictions since more correlations are taken into account; the predicted potentials become smoother, are significantly closer to the target potentials, and are more transferable as a result.

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.

Carnahan-Starling type equations of state for stable hard disk and hard sphere fluids

The well-known Carnahan-Starling (CS) equation of state (EoS) [N.F. Carnahan and K.E. Starling. J. Chem. Phys. 51 (2), 635–636 (1969). doi:10.1063/1.1672048] for hard sphere (HS) fluid was derived from a quadratic relationship between the integer portions of the virial coefficients, , and their orders, . In this paper, the method is extended to cover the full virial coefficients, , for the general D-dimensional case. We propose a (D-1)th order polynomial for the virial coefficients starting from the 4th order and EoS’s are derived from it. For the hard rod (D = 1) case, the exact solution is obtained. For the stable hard disk fluid (D = 2), the most recent virial coefficients, up to the 10th order, [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0] and accurate compressibility data [J. Kolafa and M. Rottner. Mol. Phys. 104 (22–24), 3435–3441 (2006). doi:10.1080/00268970600967963; J.J. Erpenbeck and M. Luban. Phys. Rev. A. 32 (5), 2920–2922 (1985). doi:10.1103/PhysRevA.32.2920] are employed to construct and to test the EoS. For the stable hard sphere (D = 3) fluid, a new CS-type EoS is obtained by using the most recent virial coefficients [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0; A.J. Schultz and D.A. Kofke. Physical Review E. 90 (2) (2014). doi:10.1103/PhysRevE.90.023301], up to the 11th order, along with highly-accurate compressibility data [S. Pieprzyk et al. Phys. Chem. Chem. Phys. 21 (13), 6886–6899 (2019). doi:10.1039/C9CP00903E; M.N. Bannerman et al. J. Chem. Phys. 132 (8), 084507 (2010). doi:10.1063/1.3328823; J. Kolafa, et al. Phys. Chem. Chem. Phys. 6 (9), 2335–2340 (2004). doi:10.1039/B402792B]. The simple new EoS’s prove to be as accurate as the Padé approximations based on all available virial coefficients, which significantly improve the accuracy of the CS-type EoS in the hard sphere case. This research also reveals that as long as the virial coefficients obey a polynomial function, any EoS derived from it will diverge at the non-physical packing fraction, .

Lattice Boltzmann simulation of low-Reynolds-number cavitating contracting-nozzle flow interacting with a moving valve

Investigating internal-injector cavitating flow dynamics is difficult but important. The interaction of nozzle cavitation with the moving needle valve dictates the fuel mass flow rate and therefore spray combustion performance and emissions. In the present study, a two-dimensional low-Reynolds-number cavitating contracting-nozzle flow interacting with a moving valve is simulated using the lattice Boltzmann (LB) method. The Bhatnagar–Gross–Krook algorithm coupled with the immersed boundary method and an improved pseudo-potential multiphase flow model are employed and further developed based on the open-source LB code PALABOS. The performance of the immersed boundary method is first verified in a case where an oscillating cylinder moves according to a sine function in water. In order to improve the pseudo-potential model on its limitation of the density ratio, so to be used in engineering multiphase flow, the Carnahan–Starling equation of state is incorporated together with the exact difference method force scheme and an upgraded interaction force term. The upgraded pseudo-potential model proves via validations to be effective in improving numerical stability at large density ratios. With a seamless cooperation of the improved Shan–Chen model and the immersed boundary method achieved in PALABOS, cavitation in a contracting nozzle is simulated for a whole cycle of the valve motion. Cavitation dynamics under different fuel mass flow rates is investigated. It is found that cavitation dynamics, including interface conditions, cavitation bubble distributions, and inside-bubble vapor-phase flow fields, is distinctly different when the flow path is widely open and completely shut by the valve.

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.

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.

Impact of ion-steric and ion-partitioning effects on electrophoresis of soft particles.

A theoretical study on the electrophoresis of a soft particle is made by taking into account the ion steric interactions and ion partitioning effects under a thin Debye layer consideration with negligible surface conduction. Objective of this study is to provide a simple expression for the mobility of a soft particle which accounts for the finite-ion-size effect and the ion partitioning arise due to the Born energy difference between two media. The Donnan potential in the soft layer is determined by considering the ion steric interactions and the ion partitioning effect. The volume exclusion due to the finite ion size is considered by the Carnahan-Starling equation and the ion partitioning is accounted through the difference in Born energy. The modified Poisson-Boltzmann equation coupled with Stokes-Darcy-Brinkman equations are considered to determine the mobility. A closed-form expression for the electrophoretic mobility is obtained, which reduces to several existing expressions for mobility under various limiting cases.

Kenneth E. Starling

Kenneth E. Starling, 84, died on Nov. 26, 2019, in Norman, Oklahoma. “Ken joined the University of Oklahoma in 1966 and taught there for nearly 3 decades. He was well known in the natural gas industry, and his papers have been cited more than 8,000 times. He is perhaps best known for two equations of state: the Benedict-Webb-Rubin-Starling equation and the Carnahan-Starling equation. He was inducted into the Oklahoma Higher Education Hall of Fame in 2012.”—Brian Grady, friend and colleague Most recent title: Professor of chemical engineering, University of Oklahoma Education: BS, chemistry and chemical engineering, Texas A&I University (now Texas A&M University–Kingsville), 1957; PhD, chemical engineering, Illinois Institute of Technology, 1962 Survivors: Wife, Barbara; daughters, Stephani and Suzanne; son, Scott; nine grandchildren; six great-grandchildren To recognize your late loved one or colleague, submit obituary information at cenm.ag/obits.