Enhancement of the extended corresponding states techniques for thermodynamic modeling. II. Mixtures

Moving from a former work for pure fluids a new modeling technique has been developed for obtaining a fundamental mixture equation of state in the Helmholtz energy form. This model can be considered an evolution of the extended corresponding states method, which is modified from the conventional analytical mode to a heuristic one through the integration of a general function approximator for the representation of the scale factor functions of a target mixture. The assumed approximator is a multilayer feedforward neural network (MLFN) with two outputs, one for each scale factor. A reference pure fluid, conformal with the components of the studied mixture, is chosen and the independent variables of its dedicated equation of state (DEoS) are distorted by the scale factors, which are individual functions of temperature, density, and composition. The MLFN scale factor functions can be obtained from regression on any kind of thermodynamic data of the target mixture. The model capability to accurately represent the thermodynamic surfaces of five binary and two ternary haloalkane mixtures is studied assuming data generated from the corresponding DEoSs. The obtained prediction accuracies for the mixture thermodynamic properties are competitive with those of the available conventional DEoSs. The proposed modeling technique is then robust and straightforward for the effective development of a mixture DEoS from thermodynamic quantities distributed in the range of interest.     

An Extended Equation of State Modeling Method I. Pure Fluids

A new technique is proposed here to represent the thermodynamic surface of a pure fluid in the fundamental Helmholtz energy form. The peculiarity of the present method is the extension of a generic equation of state for the target fluid, which is assumed as the basic equation, through the distortion of its independent variables by individual shape functions, which are represented by a neural network used as function approximator. The basic equation of state for the target fluid can have the simple functional form of a cubic equation, as, for instance, the Soave–Redlich–Kwong equation assumed in the present study. A set of nine fluids including hydrocarbons, haloalkane refrigerants, and strongly polar substances has been considered. For each of them the model has been regressed and then validated against volumetric and caloric properties generated in the vapor, liquid, and supercritical regions from highly accurate dedicated equations of state. In comparison with the underlying cubic equation of state, the prediction accuracy is improved by a factor between 10 and 100, depending on the property and on the region. It has been verified that about 100 density experimental points, together with from 10 to 20 coexistence data, are sufficient to guarantee high prediction accuracy for different thermodynamic properties. The method is a promising modeling technique for the heuristic development of multiparameter dedicated equations of state from experimental data.     

Perturbation equation of state of pure fluids

The statistical thermodynamic perturbation equation of state, originally developed for fluids with Lennard-Jones intermolecular potential energy is improved, and then it is modified for the prediction of real fluid properties. The improved equation of state, when applied to Lennard-Jones fluid, predicts the properties of this fluid consistently better than the original perturbation equation of state.In order to modify the perturbation equation of state for the prediction of real fluid propertes both pair-, and triplet-interaction potential functions are considered in the formulation of this equation of state. For the pair-interaction potential function the Kihara spherical-core function is used, and for the triplet-interaction potential function the Axilord-Teller function is used. Comparisions are made between the calculated and the experimental thermodynamic properties of argon, methane and neopentane in liquid and in vapor states. These comparisions indicate that the perturbation equation of state is very promising in predicting the properties of real fluids in condensed states.     

An Extended Equation of State Modeling Method II. Mixtures

This work is the extension of previous work dedicated to pure fluids. The same method is extended to the representation of thermodynamic properties of a mixture through a fundamental equation of state in terms of the Helmholtz energy. The proposed technique exploits the extended corresponding-states concept of distorting the independent variables of a dedicated equation of state for a reference fluid using suitable scale factor functions to adapt the equation to experimental data of a target system. An existing equation of state for the target mixture is used instead of an equation for the reference fluid, completely avoiding the need for a reference fluid. In particular, a Soave–Redlich–Kwong cubic equation with van der Waals mixing rules is chosen. The scale factors, which are functions of temperature, density, and mole fraction of the target mixture, are expressed in the form of a multilayer feedforward neural network, whose coefficients are regressed by minimizing a suitable objective function involving different kinds of mixture thermodynamic data. As a preliminary test, the model is applied to five binary and two ternary haloalkane mixtures, using data generated from existing dedicated equations of state for the selected mixtures. The results show that the method is robust and straightforward for the effective development of a mixture- specific equation of state directly from experimental data.     

Equation of State for the Thermodynamic Properties of 1,1,2,2,3-Pentafluoropropane (R-245ca)

An equation of state for the calculation of the thermodynamic properties of 1,1,2,2,3-pentafluoropropane (R-245ca), which is a hydrofluorocarbon refrigerant, is presented. The equation of state (EOS) is expressed in terms of the Helmholtz energy as a function of temperature and density, and can calculate all thermodynamic properties through the use of derivatives of the Helmholtz energy. The equation is valid for all liquid, vapor, and supercritical states of the fluid, and is valid from the triple point to 450 K, with pressures up to 10 MPa. Comparisons to experimental data are given to verify the stated uncertainties in the EOS. The estimated uncertainty for density is 0.1 % in the liquid phase between 243 K and 373 K with pressures up to 6.5 MPa; the uncertainties increase outside this range, and are unknown. The uncertainty in vapor-phase speed of sound is 0.1 %. The uncertainty in vapor pressure is 0.2 % between 270 K and 393 K. The uncertainties in other regions and properties are unknown due to a lack of experimental data.     

Dynamics of oscillating viscous droplets immersed in viscous media

Summary Damped oscillations of a viscous droplet immersed in a viscous medium are considered in detail. The characteristic equation is solved numerically for arbitrary, finite fluid properties. The cylinder functions in the characteristic equation are solved using an accurate continued fraction algorithm, and the complex decay factor is searched using a minimization scheme. Oscillation frequency and damping rate results are presented for the fundamental mode, for various cases of practical interest (liquid-gas, and liquid-liquid systems), and the effect of the external medium properties are discussed. Results are compared to exact solutions for limiting cases, and to existing experimental data for both the fundamental and higher order modes. It is shown that the theoretical frequency prediction matches well with the experimental observation. Damping rate predictions, however, underestimate experimental observation in some cases, and this is thought to be due to surface impurities. The application of these results to the measurement of surface tension and viscosity of liquid droplets from single-droplet levitation experiments is also discussed.     

Extension of nonlinear Onsager theory of irreversibility

Nonlinear Onsager theory in combination with Wang’s representation theorem is utilized to obtain nonlinear constitutive equations for fluid. The constitutive equations can be derived using the derivative of a dissipative function according to nonlinear Onsager’s theory. The requirement of the dissipative function is convexity in thermodynamic force and continuity in thermodynamic flux. An isothermal fluid is provided as an example. Objectivity is required for the constitutive equations of the fluid. Such an axiom permits that all response functions are isotropic functions and can be expressed by Wang’s representation theorem. Therefore, the dissipative part of Cauchy stress is obtained using (i) Wang’s representation theorem only and (ii) both nonlinear Onsager theory and Wang’s representation theorem. In method (i), the coefficients for the constitutive equations are only constrained by the Clausius–Duhem inequality, while in method (ii), these are not only constrained by Clausius–Duhem inequality but also by the positive semidefiniteness of the Hessian matrix of the dissipative function (convexity of the dissipative function).     

A new method for measuring fluid densities using the Taylor dispersion experiment

The density of a pure fluid or a fluid mixture is of fundamental importance in the design of equipment for fluid processing and in the development of theory describing thermodynamic and transport properties of the liquid state. A new experimental technique for measuring fluid densities is presented, which is based on the well-known Taylor dispersion experiment for measuring mutual diffusion coefficients. The equipment and working equation are both simple, yet experimental results show that the method is accurate to at least 0.1%. An analysis of errors indicates that the accuracy could be improved to 0.01%. This new technique is of particular virtue since it can be used to obtain simultaneous measurements of the fluid density and diffusion coefficient of a solute in the fluid, using data from a single experiment.     

A Thermodynamic Property Model for Fluid-Phase Propane

A fundamental equation of state for propane (R-290), formulated in terms of the non-dimensional Helmholtz free energy, is presented. It was developed based on selected reliable measurements for pressure-volume-temperature (PVT), isochoric and isobaric heat capacities, speed of sound, and the saturation properties which were all converted to ITS-90. Supplementary input data calculated from a virial equation for the vapor-phase PVT properties at lower temperatures and other correlations for the saturated vapor pressures and saturated vapor- and liquid-densities have also been used. The present equation of state includes 19 terms in the residual part and represents most of the reliable experimental data accurately in the range of validity from 85.48 K (the triple point temperature) to 623 K, at pressures to 103 MPa, and at densities to 741 kg·m^–3. The smooth behavior of the derived thermodynamic properties in the entire fluid phase is demonstrated. In addition, graphical and statistical comparisons between experimental data and the available thermodynamic models, including the present one, showed that the present model can provide a physically sound representation of all the thermodynamic properties of engineering importance.     

A thermodynamic property formulation for nitrogen from the freezing line to 2000 K at pressures to 1000 MPa

A new fundamental equation explicit in Helmholtz energy for thermodynamic properties of nitrogen from the freezing line to 2000 K at pressures to 1000 MPa is presented. A new vapor pressure equation and equations for the saturated liquid and vapor densities as functions of temperature are also included. The techniques used for development of the fundamental equation are those reported in a companion paper for ethylene. The fundamental equation and the derivative functions for calculating internal energy, enthalpy, entropy, isochoric heat capacity (C _v), isobaric heat capacity (C _p), and velocity of sound are also included in that paper. The property formulation using the fundamental equation reported here may generally be used to calculate pressures and densities with an uncertainty of ±0.1%, heat capacities within ± 2%, and velocity of sound values within ±2%. The fundamental equation is not intended for use near the critical point.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Enhancement of the extended corresponding states techniques for thermodynamic modeling. I. Pure fluids

This work, limited to pure fluids' modeling, has two main goals. The first one is to discuss in a rigorous way the modeling background of the conventional extended corresponding states (ECS) methods proposed in the literature. A critical review is in the meantime developed allowing to point out the limits of the methods. The second goal is to propose a practicable and plain solution for the use of ECS as the basic framework to develop a fundamental equation of state (EoS) for a target fluid in a totally correlative mode. For this purpose the conventional analytical procedure was left out and an optimization procedure, based on a general function approximator, was applied. The model capability to accurately represent several thermodynamic surfaces of a number of haloalkanes is verified assuming data generated from the corresponding EoSs. The achieved results show that the method is robust and straightforward, while the obtained prediction accuracies for the thermodynamic functions are competitive with those of the available conventional EoSs.     

A thermodynamic property formulation for ethylene from the freezing line to 450 K at pressures to 260 MPa

A new thermodynamic property formulation based upon a fundamental equation explicit in Helmholtz energy of the form A=A(, T) for ethylene from the freezing line to 450 K at pressures to 260 MPa is presented. A vapor pressure equation, equations for the saturated liquid and vapor densities as functions of temperature, and an equation for the ideal-gas heat capacity are also included. The fundamental equation was selected from a comprehensive function of 100 terms on the basis of a statistical analysis of the quality of the fit. The coefficients of the fundamental equation were determined by a weighted least-squares fit to selected P--T data, saturated liquid and saturated vapor density data to define the phase equilibrium criteria for coexistence, C _v data, velocity of sound data, and second virial coefficients. The fundamental equation and the derivative functions for calculating internal energy, enthalpy, entropy, isochoric heat capacity (C _v), isobaric heat capacity (C _p), and velocity of sound are included. The fundamental equation reported here may be used to calculate pressures and densities with an uncertainty of ±0.1%, heat capacities within ±3 %, and velocity of sound values within ±1 %, except in the region near the critical point. The fundamental equation is not intended for use near the critical point. This formulation is proposed as part of a new international standard for thermodynamic properties of ethylene.Paper presented at the Ninth Symposium on Thermophysical Properties, June 24–27, 1985, Boulder, Colorado, U.S.A.     

Asymptotic properties of ideal curves on a thermodynamic surface

The configurations and properties of the characteristic curves on the thermodynamic surface of a real gas, that is, of locus points where the configuration component of a thermodynamic property becomes zero, have been analyzed. The relation of asymptotic properties of the curves with a virial equation of state at ρ → 0 and with a hypothetical fluid system overcooled to 0 K at T → 0 has been studied. The correlation of asymptotic properties of the curves manifesting themselves, in particular, in the direct relationship of coordinates of characteristic points of ideal curves on a thermodynamic surface with an intermolecular interaction potential has been demonstrated.     

A Fundamental Equation for Calculation of the Thermodynamic Properties of Ethanol

A formulation for the thermodynamic properties of ethanol (C₂H₅OH) in the liquid, vapor, and saturation states is presented. The formulation is valid for single-phase and saturation states from 250 to 650K at pressures up to 280MPa. The formulation includes a fundamental equation and ancillary functions for the estimation of saturation properties. The experimental data used to determine the fundamental equation include pressure-density-temperature, ideal gas heat capacity, speed of sound, and vapor pressure. Saturation values computed from the ancillary functions were used to ensure thermodynamic consistency at the vapor-liquid phase boundary. Comparisons between experimental data and values computed using the fundamental equation are given to verify the uncertainties in the calculated properties. The formulation presented may be used to compute densities to within ±0.2%, heat capacities to within ±3%, and speed of sound to within ±1%. Saturation values of the vapor pressure and saturation densities are represented to within ±0.5%, except near the critical point.     

A thermodynamic property formulation for cyclohexane

A formulation for the thermodynamic properties of cyclohexane is presented. The equation is valid for single-phase and saturation states from the melting line to 700 K at pressures up to 80 MPa. It includes a fundamental equation explicit in reduced Helmholtz energy with independent variables of reduced density and temperature. The functional form and coefficients of the ancillary equations were determined by weighted linear regression analyses of evaluated experimental data. An adaptive regression algorithm was used to determine the final equation. To ensure correct thermodynamic behavior of the Helmholtz energy surface the coefficients of the fundamental equation were determined with multiproperty fitting, Pressure-density-temperature (P-p-T) and isobaric heat capacity (C _p -P-T) data were used to develop the fundamental equation, SaturationP-p-T values, calculated from the estimating functions, were used to ensure thermodynamic consistency at the vapor-liquid phase boundary. Separate functions were used for the vapor pressure, saturated liquid density, saturated vapor density. ideal-gas heat capacity. and pressure on the melting curve, Comparisons between experimental data and values calculated using the fundamental equation are given to verify the accuracy of the formulation. The formulation given here may be used to calculate densities within ±0.1 %, heat capacities to within ±2 %. and speed of sound to within ± 1 %, except near the critical point.Paper presented at the Twelfth Symposium on Thermophysical Properties, June 19–24, 1994, Boulder, Colorado, U.S.A.     

与权函数对应的积分方程及其联立解法

边界元法一般采用控制方程的基本解作为权函数,这往往能在控制方程为齐次时可避免域积分。但当问题复杂,基本解不能求得时,此法便产生了困难。虽有人也曾偶尔用非基本解函数作为权函数,但本文将系统地探讨函娄与边界积分方程的关系,所涉及的各类定解问题都用Laplace基本解和kelvin基本解作为权函数,并提出边界点公式和内点公式联立求解的方法。这不但避免了求基本解的困难,同时也为编制能求解多种问题的多功能电算程序提供了方便,使程序的长度缩短了,编程和调试的难度也降低了。     

On Failures of van der Waals’ Equation at the Gas–Liquid Critical Point

This comment is in response to a recent “new comment” by Umirzakov on the article “Gibbs density surface of fluid argon: revised critical parameters.” It was incorrectly asserted that van der Waals equation “proves” the existence of a scaling singularity with a divergent isochoric heat capacity (C_v). Van der Waals’ equation, however, is inconsistent with the universal scaling singularity concept; it erroneously predicts, for instance, that C_v is a constant for all fluid states. Van der Waals hypothetical singular critical point is based upon a common misconception that van der Waals equation represents physical reality of fluids. A comparison with experimental properties of argon shows that state functions of van der Waals’ equation fail to describe the thermodynamic properties of low-temperature gases, liquids and of gas–liquid coexistence. The conclusion that there is no “critical point” singularity on Gibbs density surface remains scientifically sound.     

On compressible Korteweg fluid-like materials

We provide a thermodynamic basis for compressible fluids of a Korteweg type that are characterized by the presence of the dyadic product of the density gradients ∇? ⊗ ∇? in the constitutive equation for the Cauchy stress. Our approach does not need to introduce any new or non-standard concepts such as multipolarity or interstitial working and is based on prescribing the constitutive equations for two scalars: the entropy and the entropy production. In comparison with the Navier-Stokes-Fourier fluids we suppose that the entropy is not only a function of the internal energy and the density but also of the density gradient. The entropy production takes the same form as for a Navier-Stokes-Fourier fluid. For a Navier-Stokes-Fourier fluid one can express the entropy production equivalently in terms of either thermodynamic affinities or thermodynamic fluxes. Following the ideas of K.R. Rajagopal concerning the systematic development of implicit constitutive theory and primary role of thermodynamic fluxes (such as force) that are cause of effects in thermodynamic affinities (such as deformation) in considered processes, we further proceed with a constitutive equation for entropy production expressed in terms of thermodynamic fluxes. The constitutive equation for the Cauchy stress is then obtained by maximizing the form of the rate of entropy production with respect to thermodynamic fluxes keeping as the constraint the equation expressing the fact that the entropy production is the scalar product of thermodynamic fluxes and thermodynamic affinities. We also look at how the form of the constitutive equation changes if the material in question is incompressible or if the processes take place at constant temperature. In addition, we provide several specific examples for the form of the internal energy and make the link to models proposed earlier. Starting with fully implicit constitutive equation for the entropy production, we also outline how the methodology presented here can be extended to non-Newtonian fluid models containing the Korteweg tensor in a straightforward manner.     

Thermodynamic properties of methane in the critical region

A scaled fundamental equation is presented for the thermodynamic properties of methane in the critical region. The equation supplements the international formulation for the thermodynamic properties of methane issued by IUPAC.