On equations of state in a lattice Boltzmann method

We explore the properties of a simple analytic equation of state, proposed by Heyes and Woodcock [D. M. Heyes and L. V. Woodcock, Mol. Phys. 59, 1369 (1986)] which is a development of the van der Waals equation of state, improved with an analytically simple yet accurate hard-core component for the compressibility factor. We call this the hard-sphere van der Waals (HSvdW) equation of state. We show, for the first time, that the HSvdW equation of state gives analytic values for the critical parameters and that it gives significantly more realistic values for the critical density than the van der Waals and other cubic equations of state. For single-component square wells of arbitrary well width, the HSvdW equation of state also produces a critical temperature and pressure in agreement with those from a recent Gibbs Monte Carlo simulation study. The chief advantage over a similar equation of state with an alternative form for the hard-core compressibility factor, such as Carnahan–Starling, is that it leads to more tractable analytic expressions for thermodynamic properties without loss of accuracy. Extensive Monte Carlo computer simulations of square-well (SW) and square-shoulder (SS) fluids with variable well width have been performed. We have calculated the compressibility factor, internal energy, pair distribution functions, and coordination number (average number of particles within the soft-shell interaction range). We have also determined the gas–liquid coexistence curve using the particle insertion chemical potential method for the square-well case, λ=1.5σ, where λ is the diameter of the soft interaction shell. This agrees well with the Gibbs ensemble simulation coexistence curve. For the purpose of improving the attractive component of the HSvdW equation of state, we have made use of the SW and SS simulation data and that of other workers to develop another new simple equation of state based on the quasichemical local coordination number approximation (QCA) which is a lattice model of a fluid. The QCA is improved to include the local coordination of the hard-sphere fluid. Present adaptations extend the range of earlier QCA models to arbitrary well width and sign of the well (either attractive well or repulsive shoulder). The simulation coordination numbers are reproduced well by the present model, although there is little improvement in the overall equation of state over the simpler HSvdW.

The semiempirical method was suggested for construction of a simple equation of state for liquid and gas. With this purpose, some new conditions and restrictions, which should be satisfied by the “right” equation of state, were formulated. In particular, the preferable structure of the equation of state was determined. The equations for description of thermodynamic properties of real gases and liquid phase were derived with involvement of additional restrictions. The second virial coefficient calculated on the basis of the modified step potential of intermolecular interactions was used as the temperature functions in the derived equations, and the density functions were determined from analysis of experimental data. A new equation of state for liquid and gas with six individual adjustable parameters was suggested. This new equation describes the thermodynamic properties of substances much better than the known Benedict−Webb−Rubin equation of state. Some caloric properties of substances were calculated.

The study of cavitation inception is crucial in several hydraulic machines, e.g. pumps, nozzles and sprays. In the present study, a multiphase lattice Boltzmann method (LBM) is implemented for simulation of the cavitation bubbles dynamics and characteristics of cavitating flows. The effect of employing various equations of state (EoS) is investigated on the computing of interaction forces and the phase separation between the liquid and its vapor in the cavitating flows. Herein, the cubic EoSs of Shan-Chen (SC) and Carnahan-Starling (CS) and the non-cubic EoS of Peng-Robinson (PR) are applied. The exact difference method is imposed to improve the numerical stability for simulation of two-phase flow systems. The accuracy and efficiency of the present method are examined by comparison of the results obtained for the homogeneous and heterogeneous cavitation with those of reported in the literature. Then, the implemented multiphase LBM is used for studying the inception and growth of the cavitation bubbles in the throat of a venturi. The effect of hydrophobicity and hydrophobicity of the nozzle wall on the cavitation dynamics is investigated and a detail discussion is made for the results from the numerical and physical point of view. Evaluation of the present results shows that the multiphase LBM with incorporating an appropriate equation of state has excellent capability for prediction of the bubble dynamics and cavitating flow characteristics in applied geometries.

The gas equation of state (EOS) serves as a critical tool for analyzing the thermal effects within the hydrogen storage tank during refueling processes. It quantifies the dynamic relationships among pressure, temperature and volume, playing a vital role in numerical simulations of hydrogen refueling, the development of refueling protocols, and ensuring refueling safety. This study first establishes a lumped-parameter thermodynamic model for the hydrogen refueling process, which combines a zero-dimensional gas model with a one-dimensional tank wall model (0D1D). The model’s accuracy was validated against experimental data and will be used in combination with different EOSs to simulate hydrogen temperature and pressure. Subsequently, parameter values are derived for the van der Waals EOS and its modified forms—Redlich–Kwong, Soave, and Peng–Robinson. The accuracy of the modified forms is evaluated using the Joule–Thomson inversion curve. A polynomial EOS is formulated, and its parameters are numerically determined. Finally, the hydrogen temperatures and pressures calculated using the van der Waals EOS, Redlich–Kwong EOS, polynomial EOS, and the National Institute of Standards and Technology (NIST) database are compared. Within the initial and boundary conditions set in this study, the results indicate that among the modified forms for van der Waals EOS, the Redlich–Kwong EOS exhibits higher accuracy than the Soave and Peng–Robinson EOSs. Using the NIST-calculated hydrogen pressure as a benchmark, the relative error is 0.30% for the polynomial EOS, 1.83% for the Redlich–Kwong EOS, and 17.90% for the van der Waals EOS. Thus, the polynomial EOS exhibits higher accuracy, followed by the Redlich–Kwong EOS, while the van der Waals EOS demonstrates lower accuracy. This research provides a theoretical basis for selecting an appropriate EOS in numerical simulations of hydrogen refueling processes.

Thermodynamic inconsistencies, particularly the prominent spurious currents near three-phase contact lines, present a major obstacle in pseudopotential lattice Boltzmann (LB) simulations of contact angles. Previous solutions have predominantly focused on equations of state (EOSs) and forcing schemes (FSs). In this work, a hierarchical force computation architecture (HFCA) is established through a comprehensive assessment of EOSs, interparticle interaction force terms (IFTs), and FSs, aiming to suppress spurious currents at three-phase contact lines in lattice Boltzmann simulations. The results demonstrate that the Peng-Robinson EOS shows excellent numerical stability at high density ratios, yet it leads to elevated spurious currents when used alone. More significantly, Gong’s term, when incorporated into the HFCA, ensures universal stability, full compatibility with all EOSs and FSs, and systematically reduces the maximum spurious currents through hierarchical coordinated optimization. Additionally, the relaxation time τ is identified as a sensitive system-level parameter whose optimal value depends on the configuration established by the higher layers of the HFCA. By leveraging the synergistic effects within the HFCA, specifically by optimizing τ in a configured framework, spurious currents can be significantly reduced. Evidently, this approach is more effective than optimizations based solely on schemes and EOSs. Based on these findings, specific combinations of τ, forcing terms, schemes, and EOSs are recommended to minimize spurious currents in LB simulations.

In 1873, van der Waals proposed a simple equation of state that qualitatively described the phase behavior of fluids. Since then, the principles behind the van der Waals equation have been used and refined countless times in the quest for an accurate equation of state. Despite the enormous amount of work reported, the goal of a simple and accurate equation of state for even relatively simple systems, such as monatomic or polyatomic gases, has proved elusive. Therefore, the analysis of phase equilibria with equations of state is more often than not an exercise in data fitting rather than genuine prediction. In this work, we revisit the early work of Dieterici. When coupled with modern developments in equations of state, this little used approach has the potential of generating useful equations of state. To illustrate the usefulness of the Dieterici formula, we use it to derive a simple equation of state based on the Carnahan–Starling hard-sphere term and van der Waals interactions. This simple approach yields more accurate predictions of the phase co-existence of fluids than the traditional additive approach used to develop equations of state.

Homogeneous equilibrium model (HEM) has been widely used in cavitating flow simulations. The major feature of this model is that a single equation of state (EOS) is proposed to describe the thermal behavior of bubbly liquid, where both kinematic and thermal equilibrium is assumed between two phases. In this paper, the HEM was coupled with multi-relaxation-time lattice Boltzmann model (MRT-LBM) and the acoustic behavior was simulated. Two approaches were applied alternatively: adjusting speed of sound (Buick, J. Phys. A, 2006, 39:13807-13815) and setting real gas EOS. Both approaches result in high accuracy in acoustic speed predictions for different void (gas) volume of fractions. It is demonstrated that LBM could be successfully applied as a Navier-Stokes equation solver for industrial applications. However, further dissipation and dispersion analysis shows that Shan-Chen type approaches of LBM are deficient, especially in large wave-number region.

A lattice Boltzmann model for the conjugate heat transfer

Zeolitic encapsulation of small non-polar gas molecules under appropriate p–T conditions (Vr > 1) can be conveniently dealt with in terms of van der Waals' equation of state for real gases. Thus, the amount of gas trapped in the zeolitic cages can be predicted if the cage volumes are known; otherwise, void-volume values may be obtained from encapsulation experiments. These void-volume values, as obtained from hydrogen encapsulation in A-type zeolites, are ca. 20% larger than those reported in the literature, but in agreement with calculated void volumes derived from crystallographic data and from a refined structural model of zeolite A. The ‘jump’ in encapsulation capacity at ca. 2.5 univalent blocking cations per unit cell is explained in terms of the percolation theory as transition from β-cage encapsulation to encapsulation in both the α- and β-cages. Based on the van der Waals equation, calculated available void volumes and the percolation theory, a theoretical encapsulation curve is proposed which agrees fairly well with experimental results reported so far for H2 in Cs, Na-A zeolites and for Ar, Kr and CH4 in K, Na-A zeolites.

Evaluation of force implementation in pseudopotential-based multiphase lattice Boltzmann models

The pseudopotential lattice Boltzmann (LB) model is a widely used multiphase model in the LB community. In this model, an interaction force, which is usually implemented via a forcing scheme, is employed to mimic the molecular interactions that cause phase segregation. The forcing scheme is therefore expected to play an important role in the pseudoepotential LB model. In this paper, we aim to address some key issues about forcing schemes in the pseudopotential LB model. First, theoretical and numerical analyses will be made for Shan-Chen's forcing scheme [Shan and Chen, Phys. Rev. E 47, 1815 (1993)] and the exact-difference-method forcing scheme [Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)]. The nature of these two schemes and their recovered macroscopic equations will be shown. Second, through a theoretical analysis, we will reveal the physics behind the phenomenon that different forcing schemes exhibit different performances in the pseudopotential LB model. Moreover, based on the analysis, we will present an improved forcing scheme and numerically demonstrate that the improved scheme can be treated as an alternative approach to achieving thermodynamic consistency in the pseudopotential LB model.

To describe the thermodynamic properties of xenon, a new fundamental low-parametric equation of state (in the form of reduced Helmholtz energy) is obtained with the help of the methods and approaches developed by the authors. It allows us to describe the thermal properties of gas, liquid, and fluid with a sufficiently high accuracy close to the accuracy of experiment in a range from the density in the ideal gas state to the density at the triple point, excluding the critical region. The caloric properties and speed of xenon sound are calculated without involving any caloric data, with the exception of ideal gas enthalpy. The values of isobaric heat capacity, sound speed, and other thermodynamic properties obtained by calculations are in good agreement with the experimental data.

A simple equation of state for polymer liquids and solids, previously reported, is modified for simple liquids. Values of the parameters in the equation of state are given for 18 liquids. Comparison of the predictions of this equation with experimental data is made for all of the liquids.

Simulation of two-phase fluid mixture flow in rectangular two-inlet cavity using lattice Boltzmann method

The study of detonation processes in gaseous media is of interest for the coal mining, aviation, aerospace industries, and hydrogen energy. Under high pressure conditions during the passage of a detonation wave, it is necessary to take into account the equation of state of a real gas, the effects of endothermicity and exothermicity. The paper considers the flow of real gas through a plane detonation wave. A modified Rankine-Hugoniot equation has been obtained to describe the dynamics of changes in the parameters of a van der Waals gas flow when it passes through a detonation wave. The relations that determine the limiting pressure for the existence of detonation in a Van der Waals gas were obtained. The dependence of the limiting pressure on the parameters A and B of the Van der Waals equation of state is shown. An increase in parameter A slows down the growth of pressure in the detonation wave, and an increase in parameter B intensifies it. An equation for determining the velocity of combustion products in a real gas is obtained. Compared with the velocity of combustion products in an ideal gas. For an ideal gas, combustion products flow from the detonation front at a critical (sonic) velocity. For a Van der Waals gas, the velocity of combustion products can be greater than the critical one. Moreover, both parameters characterizing the additional pressure (A) and the additional volume (B) lead to the acceleration of combustion products. The additional pressure transfers its energy to the flow and thus causes acceleration. The additional volume leads to a decrease in the degree of compression in the same way as the limiting compression (28). Therefore, the flow velocity increases.