Thermodynamic Equation Of State Research Articles

Analytic thermodynamic properties of the Lieb-Liniger gas

We present a comprehensive review on the state-of-the-art of the approximate analytic approaches describing the finite-temperature thermodynamic quantities of the Lieb-Liniger model of the one-dimensional (1D) Bose gas with contact repulsive interactions. This paradigmatic model of quantum many-body-theory plays an important role in many areas of physics-thanks to its integrability and possible experimental realization using, e.g., ensembles of ultracold bosonic atoms confined to quasi-1D geometries. The thermodynamics of the uniform Lieb-Liniger gas can be obtained numerically using the exact thermal Bethe ansatz (TBA) method, first derived in 1969 by Yang and Yang. However, the TBA numerical calculations do not allow for the in-depth understanding of the underlying physical mechanisms that govern the thermodynamic behavior of the Lieb-Liniger gas at finite temperature. Our work is then motivated by the insights that emerge naturally from the transparency of closed-form analytic results, which are derived here in six different regimes of the gas and which exhibit an excellent agreement with the TBA numerics. Our findings can be further adopted for characterising the equilibrium properties of inhomogeneous (e.g., harmonically trapped) 1D Bose gases within the local density approximation and for the development of improved hydrodynamic theories, allowing for the calculation of breathing mode frequencies which depend on the underlying thermodynamic equation of state. Our analytic approaches can be applied to other systems including impurities in a quantum bath, liquid helium-4, and ultracold Bose gas mixtures.

Microscopic state equation for oscillator chains

Systems allowing anomalous transport of mass, momentum energy, etc., such as low-dimensional particles systems or highly confining media, are hard to characterize thermodynamically. Indeed, local thermodynamic equilibrium may not be established and their behaviour often strongly depends on many microscopic parameters, including the symmetry of the interaction potentials. Thermodynamic state equations, on the other hand, involve a small set of observables, which are obtained averaging in time and over the large number of particles that populate mesoscopic cells in which local equilibrium can be realized. In this work we show that a linear relation discovered earlier, that connects the average distance between pairs of consecutive particles with their kinetic energy, applies to quite a large set of 1-dimensional particle systems known to produce anomalous transport. This relation is microscopic in nature, since the quantities involved are neither averaged over many particles, neither over very large times. Nevertheless, its robustness is under variations of the external parameters, and the limited set of quantities it involves qualify it as a state equation, analogously to thermodynamic relations. We provide conditions for which the relation can be violated within a limited range of parameters values, and we find that it can be extended to two-dimensional networks of coupled oscillators. The validity of this relation further shows that the states of aggregation of matter in low-dimensional systems are often different from standard macroscopic ones.

Revisiting the dry friction-like magnetic vector hysteresis model

Physically correct simulations of magnetic devices require precise and energy-consistent hysteresis models. Electrical steel sheets of, e.g., power transformers or rotating machines represent one crucial element in such a simulation since the hysteresis is revealed uni- and multi-axial, and the steel sheets may have anisotropy. A dedicated vector hysteresis model based on dry friction-like pinning is revisited and derived in terms of thermodynamic state equations in a theoretical manner. The model is then linked to an incremental energy minimization procedure used in elasto-plasticity theory. A numerically efficient two-dimensional solution scheme of the hysteresis model is extended to the three-dimensional case. Rotational losses are also in the scope of this paper since the model cannot describe this loss type by nature. Therefore, numerical adaptations are discussed to account for vanishing rotational losses in saturation.

Use the Thermodynamic State Equations to Analyze the Non-ideality of Gas Mixtures

The assessment of gas behavior in chemical engineering systems necessitates a profound understanding of thermodynamic principles that govern the interactions among the components within a given system. To this end, the deviation from ideality in a single gas or gas mixture is associated with the disparity between the actual behavior of the gas or gas mixture and the behavior anticipated by the ideal gas model. This study is aimed at scrutinizing the deviation from ideal behavior in a gas mixture composed of CH4 and CO2. The analysis employs the cubic equations of state: Van Der Waals, Soave-Redlich-Kwong, and generalized Virial equations, truncated to the third term. These equations are widely recognized for their utility in characterizing substance behavior under specific thermodynamic conditions. The investigation involves an evaluation of the mixture's behavior by assessing variations in the compressibility factor concerning pressure, volume, and pressure, using a thermodynamic calculator at 296.15 K and 15 bar. The findings of this study reveal the prevalence of attractive intermolecular forces at higher pressures and repulsive interactions at lower pressures. An analogous examination of the effect of altering the composition of CH4 was undertaken using the Soave-Redlich-Kwong equation, which incorporates parameters allowing for an evaluation of the impact of molecule size and intermolecular interactions within the mixture. Furthermore, experimental data were employed to validate the results obtained in this study. Consequently, it can be inferred that these equations provide insight into the influence of pressure on molecular interaction forces, encompassing repulsive and attractive forces, which in turn can define the new volume of a real system. Thus, based on the corroboration established herein, these equations demonstrate a high degree of consistency and applicability, thereby expanding the realm of thermodynamic inquiry.

Production of Light Naphtha by Flash Distillation of Crude Oil

The escalating global demand for sustainable, renewable energy sources has catalyzed research into advanced refining processes, particularly focusing on the generation of essential raw materials like light naphtha. This paper delves into the production of light naphtha via the flash distillation of crude oil, underlining its critical role in fulfilling the petrochemical industry's requirements. The research delineates the fundamentals of flash distillation, incorporating both physical and mathematical models, with a specific emphasis on Raoult's Law, to demystify the hydrocarbon separation process in crude oil. The mathematical framework encompasses material and energy balances, coupled with the application of liquid-vapor equilibrium equations, thereby shedding light on the thermodynamic principles steering this procedure. The progression of this study involved analyzing the distillation curve of the referenced crude oil and identifying the pseudo-components representative of the involved hydrocarbons. Utilizing the DWSIM commercial simulator, the flash distillation process of Nemba crude oil was simulated, taking into account diverse operational conditions and thermodynamic models. The outcomes were juxtaposed with industrial datasets, demonstrating a significant alignment and affirming the simulation's predictive efficacy. A parametric sensitivity analysis was conducted to refine light naphtha yield, elucidating the effect of variables like the temperature of crude oil feed and naphtha separator feed on the process efficiency. The response surface methodology underscored the feasibility of augmenting naphtha production under certain conditions. Furthermore, this study assessed the impacts of employing two distinct thermodynamic models – the Soave-Redlich-Kwong (SRK) and Peng-Robinson (PR) equations of state. This analysis revealed minimal discrepancies in physical parameters, confirming the simulation's robustness and the reliability of both models. The findings of this paper validate the efficacy of flash distillation in light naphtha production, with numerical simulation emerging as a potent tool for enhancing process optimization and predictive analysis. These insights not only contribute to a deeper understanding of environmental sustainability but also offer strategies for maximizing light naphtha production in the oil industry. In conclusion, the results from this investigation significantly advance our comprehension of phase equilibrium profiles and their relationship with the applied thermodynamic state equations, thereby enriching the quality of the results.

A fast method for the calculation of refrigerant thermodynamic properties in a refrigeration cycle

Modeling and simulating dynamic and complex energy systems that are composed of a large number of components where every component model is described by mathematical equations can take high computational time. The calculation of the refrigerant thermodynamic properties has a big impact on the computational speed of the model. In an attempt to speed up simulations, two implementation methods for the calculation of refrigerants’ thermodynamic properties were investigated, the Coolprop thermodynamic equations of state and the Implicit fitting method. A complex dynamic cascade refrigeration cycle was developed and simulated using the two fluids implementation methods. The advantages and disadvantages of each method were discussed. The results show that using the implicit fitting method instead of Coolprop for the simulations of complex dynamic systems in Dymola will accelerate the simulation by about 30 times. In addition, the implicit method is accurate and offers more flexibility for modeling complex realistic energy systems.

Physics-Constrained Deep Learning Postprocessing of Temperature and Humidity

Abstract Weather forecasting centers currently rely on statistical postprocessing methods to minimize forecast error. This improves skill but can lead to predictions that violate physical principles or disregard dependencies between variables, which can be problematic for downstream applications and for the trustworthiness of postprocessing models, especially when they are based on new machine learning approaches. Building on recent advances in physics-informed machine learning, we propose to achieve physical consistency in deep learning–based postprocessing models by integrating meteorological expertise in the form of analytic equations. Applied to the postprocessing of surface weather in Switzerland, we find that constraining a neural network to enforce thermodynamic state equations yields physically consistent predictions of temperature and humidity without compromising performance. Our approach is especially advantageous when data are scarce, and our findings suggest that incorporating domain expertise into postprocessing models allows the optimization of weather forecast information while satisfying application-specific requirements. Significance Statement Postprocessing is a widely used approach to reduce forecast error using statistics, but it may lead to physical inconsistencies. This outcome can be problematic for trustworthiness and downstream applications. We present the first machine learning–based postprocessing method intentionally designed to strictly enforce physical laws. Our framework improves physical consistency without sacrificing performance and suggests that human expertise can be incorporated into postprocessing models via analytic equations.

New Thermodynamic Equation of State for Refrigerant HFO-1243zf

R-1243zf is a new refrigerant that could replace R-134a. Its thermodynamic properties represented in the equation of state (EOS) play an essential role in analyzing and designing thermal systems. The EOS exists without including caloric property data due to unavailable data during the development time. New EOS was developed explicitly in Helmholtz free energy and optimized to represent the experimental data accurately and maintain thermodynamic consistency. The optimization process undergoes using a genetic algorithm and weighted-least squares regression. The experimental data used in the optimization have a range of 233–430 K and 0.106–34.6 MPa and were validated from the extrapolation and consistency to confirm the reliability. The average absolute deviation from the data is 0.48% for the ideal gas isobaric specific heat, 1.7% for the isochoric specific heat, 0.33% for the speed of sound, 0.22% for the liquid density in single-phase, 0.49% for the vapor density in single-phase, 0.96% for the vapor pressure, 2.2% for the saturated liquid density, and 3.2% for the saturated vapor density. The EOS has a reasonable extrapolation behavior from the triple point up to 700 K and 100 MPa.

Nonlinear Interaction of Modes in a Planar Flow of a Gas with Viscous and Thermal Attenuation

The nonlinear interaction of wave and non-wave modes in a gas planar flow are considered. Attention is mainly paid to the case when one sound mode is dominant and excites the counter-propagating sound mode and the entropy mode. The modes are determined by links between perturbations of pressure, density, and fluid velocity. This definition follows from the linear conservation equations in the differential form and thermodynamic equations of state. The leading order system of coupling equations for interacting modes is derived. It consists of diffusion inhomogeneous equations. The main aim of this study is to identify the principle features of the interaction and to establish individual contributions of attenuation (mechanical and thermal attenuation) in the solution to the system.

Derivation of the Penman–Monteith Equation with the Thermodynamic Approach. I: A Review and Theoretical Development

A review of the derivation of the Penman–Monteith equation, with the thermodynamic approach of Monteith, is presented. Evapotranspiration (henceforth referred to simply as evaporation) is described as a process consisting of a pair of formal thermodynamic subprocesses (adiabatic cooling and diabatic heating) that leads to changes in the energy states of the ambient air in ways that are readily quantifiable. The Penman–Monteith equation is derived in two steps. As an initial approximation, first a form of the equation that models evaporation into a quiescent ambient air, from a wet source/sink surface, is developed based on thermodynamic equations of state applied to a suitably defined system. In a subsequent step, resistance parameters are introduced into the basic equations, accounting for the dynamic effects of wind-surface interactions and those of (bulk) canopy system effects on evaporation, leading to the Penman–Monteith equation. Although less compact than the conventional approach, the thermodynamic approach to the derivation of the Penman–Monteith equation exhibits a benefit of revealing key assumptions and concepts that are not explicit in the conventional approach. The initial step of the thermodynamic formulation accentuates the notion that the Penman–Monteith equation is fundamentally a description of the process of vapor and heat transfer between a wet source/sink surface and a stationary ambient air. The subsequent step, on the other hand, emphasizes the fact that wind and crop effects on evaporation are taken into account in an approximate sense. The thermodynamic approach also shows that evaporation is essentially a process driven by energy supply and as such, each term of the Penman–Monteith equation represents a separate heat source for evaporation. Furthermore, the approach reveals some useful mathematical/physical attributes of the key parameters of the Penman–Monteith equation. The derivation here emphasizes basic assumptions and mathematical/physical interpretations of results and fills conceptual gaps left from the original discussion. The resultant equations for latent heat flux, sensible heat flux, and the final air temperature represent a coupled set. Thus, numerical solutions to this system of equations are presented and evaluated in a companion manuscript.

Improved Singum Model Based on Finite Deformation of Crystals with the Thermodynamic Equation of State

The recently published simplified singum model has been improved by using the thermodynamics-based equation of state (EOS) of solids to derive a new interatomic potential based on elastic constants. The finite deformation formulation under hydrostatic load has been used to evaluate the pressure-volume (p-v) relationship for the EOS of a solid. Using the bulk modulus and its derivatives at the free-stress state, one can construct the EOS, from which a new form of interatomic potential is derived for the singum, which exhibits much higher accuracy than the previous one obtained from the Fermi energy and provides a general approach to construct the interatomic potential. The long-range atomic interactions are approximated to be proportional to the pressure. This improved singum model is demonstrated for the face-centered cubic (FCC) lattice of single-crystalline aluminum. The elastic properties at different pressures are subsequently predicted through the bond length change and compared with the available experimental data. The model can be straightforwardly extended to higher-order terms of EOS with better accuracy and other types of lattices.

FeOs: An Open-Source Framework for Equations of State and Classical Density Functional Theory

In this work, we present an open-source software package, referred to as FeOs─Framework for Equations of State and Classical Density Functional Theory. FeOs is a collection of interfaces and data types that can be used (1) to implement thermodynamic equations of state and Helmholtz energy functionals for classical density functional theory, and (2) to compute thermodynamic properties of pure substances and mixtures, phase equilibria, and interfacial properties such as surface tensions and adsorption isotherms. The framework is written in the Rust programming language with a complete Python interface and is designed with a focus on usability and extensibility. It is openly available on GitHub (https://github.com/feos-org/feos). Equations of state can be implemented in Rust, yielding performant code, or as a Python class, which is useful for prototyping and with less emphasis on execution speed. In both cases, the user has to implement a single function: the Helmholtz energy. FeOs then uses generalized (hyper-) dual numbers to evaluate the Helmholtz energy as well as the required exact partial (higher-order) derivatives. Using this type of automatic differentiation delivers performance without the need for implementing any analytical derivatives. The performance is further enhanced by a caching mechanism that avoids duplicate model evaluations. Together with the core interfaces and functionalities for equations of state and classical density functional theory, we provide implementations for multiple models such as the PC-SAFT equation of state (with homo- and heterosegmented group contribution methods) and Helmholtz energy functionals (including segment-based functionals). To showcase a selection of FeOs’ features, an example study of the adsorption of biogas in porous media using the PC-SAFT functional is provided.

Study of the large scale structure through modified gravity theory using statistical mechanics

We discuss the galaxy clustering based on thermodynamics and statistical mechanics in the expanding universe in a modified theory of gravity. The modified general relativity (MGR) is developed using the regular line element field to construct a symmetric tensor that represents the energy momentum of the gravitational field. This in turn provides a modified gravitational potential with terms that represent dark matter and dark energy effects without actually invoking the two. Based on the modified gravitational potential we calculate the distribution function of the galaxies. We also calculate various thermodynamic equations of state. We make a data analysis of the data obtained through the SDSS-III survey and check the feasibility of the theoretical model of probability distribution of galaxies in the universe.

First principle study of gravitational pressure and thermodynamics of FRW universe

We make a first principle study of gravitational pressure in cosmic thermo- dynamics. The pressure is directly derived from the unified first law, in fact the Einstein field equation in spherically symmetric spacetime. By using this pressure, we obtain the thermodynamics for the FRW universe, especially presenting the gravitational equation of state for the FRW spacetime itself, i.e. P = P (RA, T) for the first time. Furthermore, we study the Joule-Thomson expansion as an application of the thermodynamic equation of state to find the cooling-heating property of the FRW universe. We demonstrate that there is an inversion temperature for a FRW universe if its enthalpy ℋ is negative. These investigations shed insights on the evolution of our universe in view of thermodynamics.

Fractional Calculus Extension of the Kinetic Theory of Fluids: Molecular Models of Transport within and between Phases

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases.

Multi-criticality and related bifurcation in accretion discs around non-rotating black holes: an analytical study

Low angular momentum, general relativistic, axially symmetric accretion of hydrodynamic fluid onto Schwarzschild black holes may undergo more than one critical transition. To obtain the stationary integral solutions corresponding to such multi-critical accretion flow, one needs to employ numerical solutions of the corresponding fluid dynamics equations. In the present work, we develop a completely analytical solution scheme which may be used to find several trans-critical flow behaviours of aforementioned accretion, without explicitly solving the flow equations numerically. We study all possible geometric configurations of the flow profile, governed by all possible thermodynamic equations of state. We use Sturm’s chain algorithm to find out how many physically acceptable critical points the accretion flow can have, and discuss the transition from the mono to the multi-critical flow profile, and related bifurcation phenomena. We thus illustrate, completely analytically, the application of certain aspects of the dynamical systems theory in the field of large-scale astrophysical flow under the influence of strong gravity. Our work may possibly be generalized to calculate the maximal number of equilibrium points certain autonomous dynamical systems can have in general.

Numerical Prediction of Internal Flows in He/LOx Seals for Liquid Rocket Engine Cryogenic Turbopumps

Cryogenic turbopumps are used in high-performance, lightweight liquid rocket engines for space applications. The development of bearings and shaft seals for cryogenic turbopumps requires detailed characterization of the internal flow, taking into account the effects of boiling and multi-component two-phase flow. Here, a flow network solver was developed to analyse the secondary flow circuit of a cryogenic turbopump where the propellant is mixed with high-temperature helium after bearing cooling. The network solver is based on an extension of a classic 1D homogeneous model, originally developed for a pure substance, to the case of two-phase multi-component flow. The solver is capable of predicting pressures, temperatures, flow rates, and species concentrations in a complex two-phase flow in the presence of non-condensable gases. The unsteady mass, momentum, and energy conservation equations are implemented in conjunction with the thermodynamic equations of state using a general-purpose finite volume formulation, where the pressure drop and the heat transfer are calculated using correlations. The numerical tool was validated by comparing its predictions with experimental data obtained during tests on the secondary circuit of an oxygen turbopump developed at Avio S.p.A. A number of engine operating conditions were considered (inlet helium temperature in the range of 250–280 K, helium/liquid oxygen drain in the range of 165–230 K). The predicted temperature values showed good agreement with the experimental data in most conditions.

Thermal conductivity prediction of pure refrigerants and mixtures based on entropy-scaling concept

The residual entropy scaling (RES) concept builds up connection between the transport properties and thermodynamic equation of state directly. However, the low-density divergence issue of the reduced dimensionless thermal conductivity might not be fixed properly through current literature models. This study proposes a new RES model coupled with the original perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) for the thermal conductivities of 11 typical refrigerants (including hydrocarbons, hydrofluorocarbons, hydrofluoroolefins and carbon dioxide) and their multicomponent mixtures. Model parameters are adjusted to saturated vapor and liquid data obtained from REFPROP. The combination of a modified reference thermal conductivity term as an extension of the original work by Rosenfeld et al. and the new formulation of the reduced term could provide reasonable predictive results for the investigated pure and mixture systems with AAD% of 3.95% and 5.76%, respectively. As expected, quasi-linear behaviors in both dilute and dense fluid regions could be observed for the refrigerant systems under study. The extended corresponding states (ECS) and the other two RES approaches reported in the literatures are used as reliable tools for assessing the predictive capability of the proposed model. However, the critical enhancement of thermal conductivity and the correction of the reference EoS especially near the critical point will be the priority in the future.

Numerical study of real gas effects during bubble collapse using a disequilibrium multiphase model

An explicit density-based solver of the Euler equations for inviscid and immiscible gas–liquid flow media is coupled with real-fluid thermodynamic equations of state supporting mild dissociation and calibrated with shock tube data up to 5000 K and 28 GPa. The present work expands the original 6-equation disequilibrium method by generalising the numerical approach required for estimating the equilibrium pressure in computational cells where both gas and liquid phases co-exist while enforcing energy conservation for all media. An iterative numerical procedure is suggested for taking into account the properties of the gas content as derived from highly non-linear real gas equations of state and implemented in a tabulated form during the numerical solution. The developed method is subsequently used to investigate gaseous bubble collapse cases considering both spherical and 2D asymmetric arrangements as induced by the presence of a rigid wall. It is demonstrated that the predicted maximum temperatures are strongly influenced by the equations of state used; the real gas model predicts a temperature reduction in the bubble interior up to 41% space-averaged and 50% locally during the collapse phase compared to the predictions obtained with the aid of the widely used ideal gas approximation.

Liquid-Phase Speed of Sound and Vapor-Phase Density of Difluoromethane.

Difluoromethane (HFC-32, DFM), with a global warming potential (GWP) of 677, is of interest as a pure refrigerant and as a component in low-GWP refrigerant mixtures. Additionally, difluoromethane has recently been identified as a safe, liquefied-gas electrolyte material in batteries. Using state-of-the-art instruments for measurements, this paper presents new liquid-phase speed of sound and vapor-phase density data for difluoromethane. Two hundred and nine liquid-phase speed of sound values were measured using a dual-path pulse-echo instrument at temperatures from 230 to 345 K and pressures from 2.1 to 70 MPa. Accounting for all sources of uncertainty, the relative expanded combined uncertainty (k = 2) in the speed of sound ranged from 0.035 to 0.17%. One hundred and thirty-eight vapour-phase density values were measured using a two-sinker densimeter at temperatures from 240 to 340 K and pressures from 0.1 to 1.61 MPa with an uncertainty of 0.011 to 0.12%. These experimental data will be valuable in the ongoing development of a new fundamental thermodynamic equation of state for difluoromethane.