Helmholtz Energy Equation Research Articles

On the transition between two-phase and single-phase interface dynamics in ammonia sprays

This study explores the application of the density gradient theory to assess the validity of the two-phase spray atomization theory in multi-component ammonia sprays under conditions relevant to practical applications. The research focuses on analyzing the liquid–vapor interface structure at thermodynamic equilibrium. The most recent Helmholtz energy equation of state was employed to compute density profiles for ammonia and the surrounding nitrogen gas. The study investigates species distribution at equilibrium by examining minimal Helmholtz free energy states and evaluates critical parameters, such as interface thickness. The Knudsen number was also calculated as a temperature function to characterize the spray regime. The findings indicate that when the reduced temperature exceeds 0.95, a diffuse interface forms in ammonia-nitrogen systems, supporting using phase-field models for simulating ammonia injection in practical scenarios.

Enhancement of the Lee–Kesler–Plöcker Equation of State for Calculating Thermodynamic Properties of Long-Chain Alkanes

Two optimization approaches to correct the physical limitations of the Lee–Kesler–Plöcker equation of state for the application to long-chain hydrocarbons are presented. The quality of the approaches is evaluated by comparisons with experimental data and reference equations of state. The calculation of thermodynamic properties for alkanes up to squalane is possible with both approaches, for some of which no highly accurate fundamental equation of state is available in the literature. For the first approach, the original parameter set was refitted with constraints guaranteeing correct behavior of the equation in the liquid state. The densities of long-chain hydrocarbons are reproduced with small deviations, while the results for some short-chain alkanes are worsened. For the second approach, existing Helmholtz energy equations of state are utilized, while keeping the linear interpolation scheme via the acentric factor. Significant improvements are achieved for all fluids considered.

A Helmholtz Energy Equation of State for cis-1-Chloro-2,3,3,3-tetrafluoro-1-propene [R-1224yd(Z)

A Helmholtz Energy Equation of State for cis-1-Chloro-2,3,3,3-tetrafluoro-1-propene [R-1224yd(Z)

Direct measurements of the enthalpy of vaporization for R290 + R1216 and R1234ze(E) binary mixtures by differential scanning calorimetry

In this work, the enthalpy of vaporization (ΔvapH) of refrigerants and mixtures has been investigated experimentally and theoretically, and an accurate and simple method for directly measuring ΔvapH has been developed. First of all, the ΔvapH of R290, R1216, R1234ze(E) and their mixtures were measured at temperatures from 260 to 290 K using a modified high-pressure differential scanning calorimeter combined with a pressure balance unit and a sample injection system. Seven temperature points were measured for each component. After that, compared the experimental results with commonly used estimation methods, such as Clausius-Clapeyron equation and Helmholtz-energy equation of state. The experimental results were in good agreement with Helmholtz equation, with the maximum deviation within 1.89%. In the end, the empirical formula was used to fit the experimental results. The average absolute deviations between the experimental values and the fitting values were between 0.07% and 0.40%, and the maximum absolute deviations were between 0.11% and 1.05%.

A Helmholtz Energy Equation of State for trans-1,1,1,4,4,4-Hexafluoro-2-butene [R-1336mzz(E)] and an Auxiliary Extended Corresponding States Model for the Transport Properties

A Helmholtz Energy Equation of State for trans-1,1,1,4,4,4-Hexafluoro-2-butene [R-1336mzz(E)] and an Auxiliary Extended Corresponding States Model for the Transport Properties

An International Standard Formulation for trans-1-Chloro-3,3,3-trifluoroprop-1-ene [R1233zd(E)] Covering Temperatures from the Triple-Point Temperature to 450 K and Pressures up to 100 MPa

A new Helmholtz energy equation of state is presented for trans-1-chloro-3,3,3-trifluoroprop-1-ene [R1233zd(E)], which is expressed with temperature and density as independent variables. Experimental data in the range of temperatures from 215 to 444 K and pressures up to 35 MPa form the basis of the new equation. In this range, expected uncertainties (k = 2) of the new equation of state are 0.07% for vapor pressures at temperatures above the normal boiling point temperature (≈291K), 0.2% for vapor pressures at lower temperatures, 0.05% for liquid densities, 0.15% for vapor densities, 0.1% for saturated liquid densities, 0.05% for liquid-phase sound speeds, and 0.08% for vapor-phase sound speeds. The new equation is valid at temperatures from the triple-point temperature (165.75 K) to 450 K and pressures up to 100 MPa with reasonable uncertainties outside the available range of data because it fully extrapolates with correct physical behavior to higher temperatures and pressures as well as to lower temperatures. The equation of state presented here has been recommended as an international standard by the working group presently revising ISO 17584 (Refrigerant Properties).

Droplet coalescence by molecular dynamics and phase-field modeling

Coalescence of argon droplets with a radius of 25, 50, and 100 nm is studied with computational methods. Molecular dynamics (MD) simulations are carried out to generate reference data. Moreover, a phase-field model resting on a Helmholtz energy equation of state is devised and evaluated by computational fluid dynamics (CFD) simulations. Exactly the same scenarios in terms of geometry, fluid, and state are considered with these approaches. The MD and CFD simulation results show an excellent agreement over the entire coalescence process, including the decay of the inertia-induced oscillation of the merged droplet. Theoretical knowledge about the asymptotic behavior of coalescence process regimes is confirmed. All considered scenarios cross from the inertially limited viscous regime over to the inertial regime because of the low shear viscosity of argon. The particularly rapid dynamics during the initial stages of the coalescence process in the thermal regime is also captured by the phase-field model, where a closer look at the liquid density reveals that metastable states associated with negative pressure are attained in the emerging liquid bridge between the coalescing droplets. This demonstrates that this model is even capable of adequately handling the onset of coalescence. To speed up CFD simulations, the phase-field model is transferred to coarser grids through an interface widening approach that retains the thermodynamic properties including the surface tension.

Apparatus for the measurement of the thermodynamic speed of sound of diethylene glycol and triethylene glycol

The speed of sound is a thermodynamic equilibrium property which relates pressure to density variations at constant entropy. Because of its thermal and caloric nature, thermodynamic speed of sound data are crucial for the parametrization of highly accurate Helmholtz energy equations of state, which are in great demand. In the present work, the thermodynamic speed of sound of diethylene glycol and triethylene glycol is measured with an apparatus based on the pulse-echo technique. The measurements are performed along five isotherms, covering a temperature range between 300 K and 500 K, while the pressure was varied from 0.1 MPa to 45 MPa. The maximum relative expanded uncertainty of the reported data is 0.11% for both glycols. A double polynomial equation is fitted to the present data with a maximum deviation of 0.2% for diethylene glycol and 0.04% for triethylene glycol. A comprehensive comparison with literature data is carried out, confirming the present data at ambient conditions. However, for high temperature isotherms, deviations are up to 2% for diethylene glycol and up to 3% for triethylene glycol.

Combination of Gibbs and Helmholtz Energy Equations of State in a Multiparameter Mixture Model Using the IAPWS Seawater Model as an Example

For carbon capture and storage (CCS) applications different sets of equations of state are used to describe the whole CCS-chain. While for the transport and pipeline sections highly accurate equations of state (EOS) explicit in the Helmholtz energy are used, properties under typical geological storage conditions are described by more simple, mostly cubic EOS, and brines are described by Gibbs energy models. Combining the transport and storage sections leads to inconsistent calculations. Since the used models are formulated in different independent variables (temperature and density versus temperature and pressure), mass and energy balances are challenging and equilibria in the injection region are difficult to model. To overcome these limitations, a predictive combination of the Gibbs energy-based IAPWS seawater model (IAPWS R13-08, 2008) with Helmholtz energy-based multi-parameter EOS is presented within this work. The Helmholtz energy model used in this work is based on the EOS-CG-2016 of Gernert and Span (J Chem Thermodyn 93:274–293, https://doi.org/10.1016/j.jct.2015.05.015, 2016). The results prove that a consistent combination of the two different models is possible. Furthermore, it is shown, that a more complex brine model needs to be combined with Helmholtz energy EOS for calculations at storage conditions.

Isobaric vapor-liquid equilibrium for ethanol/water and binary linear siloxane mixtures at 100 kPa

Isobaric vapor-liquid equilibrium (VLE) measurements are performed for ethanol/water, hexamethyldisiloxane (MM)/octamethyltrisiloxane (MDM), hexamethyldisiloxane (MM)/decamethyltetrasiloxane (MD2M) and octamethyltrisiloxane (MDM)/decamethyltetrasiloxane (MD2M) mixtures at 100 kPa using a recirculation still and Raman-spectroscopy for sample analysis. The experimental results for ethanol/water show a good agreement with the calculated values of NRTL, UNIQUAC, UNIFAC and Refprop v.10.0. For the first time, measurement data for MDM/MD2M at 100 kPa are recorded and the results of MM/MDM, MDM/MD2M and MM/MD2M reveal a good accordance with the NRTL, UNIFAC model and Refprop v.10.0. Furthermore, binary interaction parameters for NRTL and UNIQUAC are determined for the first time for the mixtures MM/MDM, MDM/MD2M and MM/MD2M. Thereby, a better description of the vapor-liquid phase equilibrium could be achieved for all three mixtures than with the conventional models using estimated binary interaction parameters from the UNIFAC model. With new binary interaction parameters from Keulen et al. for the multi-fluid Helmholtz energy equation of state for mixtures for Refprop, also a better consistency could be achieved for MM/MDM and MM/MD2M compared to the other calculation models.

A new model combining Helmholtz energy equations of state with excess Gibbs energy models to describe reactive mixtures

A new model is presented, which consistently combines equations of state in terms of the Helmholtz energy with excess Gibbs-energy models. It enables the description of reactive mixtures by solving the phase and chemical equilibrium based on Helmholtz-energy equations of state, that are commonly used to describe non-reactive systems. The consistent calculation of all thermodynamic properties in the complete fluid region, including gas, liquid, and supercritical states as well as phase equilibria, is possible. The fundamental property is the Helmholtz energy with the independent variables temperature, density, and composition. The new model reproduces well experimental vapor–liquid equilibrium, homogeneous density, and speciation data of the binary system of monoethanolamine + carbon dioxide as well as of the ternary system of water + monoethanolamine + carbon dioxide.

The isobaric heat capacity of 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) at temperatures from (230 to 285) K and pressures up to 8 MPa using a new flow calorimeter

In this paper, a new flow calorimeter was developed to measure the isobaric specific heat capacity (cp) of liquids. The uncertainties of temperature, pressure, and cp were estimated to be less than 11 mK, 0.02 MPa and 1.26%, respectively. The pure 1,1,1,2-tetrafluoroethane (R134a) was measured at temperatures from (230 to 285) K to quantitatively calculated the heat leakage at different temperatures. In order to verify the reliability of apparatus, the cp for R134a and difluoromethane (R32) were obtained at temperatures from (230 to 285) K and pressures up to 8 MPa. Measurements of R134a and R32 show good agreement with the reported data and the high-precision Helmholtz energy equations of state (EoSs). Finally, a total of 33 cp data for 2,3,3,3-Tetrafluoroprop-1-ene (R1234yf) were obtained at temperatures from (230 to 285) K and pressures up to 8 MPa. The presented cp data show good agreement with the published data and two Helmholtz energy EoSs developed by Akasaka et al. and Richter et al. with the average absolute relative deviations (AARDs) of 0.92% and 1.80%, respectively. The Helmholtz energy EoS developed by Akasaka et al. gives a better prediction performance for the cp of R1234yf at low temperature than the other EoS.

New pressure and density based methods for isothermal-isobaric flash calculations

Isobaric-isothermal flash calculations are at the heart of any process calculation. In this paper, we propose two new methods for flash calculations at a specified temperature and pressure. The governing equations derived can not only be solved rapidly but also converge correctly even very close to the critical point with the methods presented in this paper. The derived equations show that simplified equations have been used in the past by other workers. The use of full equations results in faster convergence. Two different approaches, the pressure and density-based, are presented in this work.The density-based approach is particularly useful in reducing the computational time substantially in the case of multiparameter equations of state such as the PC-SAFT and the Helmholtz energy equations of state. Many examples are provided to demonstrate the efficacy of the methods. The methods presented have immediate use in process simulation software and thermodynamic programs.

Influence of hydrogen bond networks in Glycerol / N-Methyl-2-Pyrrolidone mixtures studied by dielectric relaxation spectroscopy

In this paper, we report the dielectric permittivity of the Glycerol (Gly) with N-Methyl-2-Pyrrolidone (NMP) binary mixtures in the microwave frequency region at different temperatures. The dipole moments of Gly, NMP and their equimolar binary mixtures are calculated by using Higasi's method in the temperature range 298.15K-323.15K. The dielectric relaxation spectra of the binary mixtures are calculated using Cole-Cole and Cole-Davidson equation and shows an unsymmetrical relaxation behaviour. The excess parameters of volume, permittivity, refractive index, polarization and relaxation times are fitted with Redlich-Kister polynomial equation. The molecular association and their hydrogen bond interactions between the components in the mixture are discussed in terms of Kirkwood correlation geff factor and excess Helmholtz energy (ΔFE) equation. The mean molecular polarizability (αM) of the individual and their binary mixture are calculated using Lippincott δ- function potential model and compared with the LeFevre method of polarizability values. The enthalpy of activation ΔH*, entropy of activation ΔS* and Gibbs free energy of activation ΔG* are also evaluated and the results are discussed in terms of the orientation of the dipoles. The presence of hydrogen bonding between Gly and NMP is confirmed from the FT-IR spectra.

Experimental and Numerical Study of Supersonic Non-ideal Flows for Organic Rankine Cycle Applications

Abstract The organic Rankine cycle (ORC) is low-grade heat recovery technology, for sources as diverse as geothermal, industrial, and vehicle waste heat. The working fluids used within these systems often display significant real-gas effects, especially in proximity of the thermodynamic critical point. Three-dimensional (3D) computational fluid dynamics (CFD) is commonly used for performance prediction and flow field analysis within expanders, but experimental validation with real gases is scarce within the literature. This paper therefore presents a dense-gas blowdown facility constructed at Imperial College London, for experimentally validating numerical simulations of these fluids. The system-level design process for the blowdown rig is described, including the sizing and specification of major components. Tests with refrigerant R1233zd(E) are run for multiple inlet pressures, against a nitrogen baseline case. CFD simulations are performed, with the refrigerant modeled by ideal gas, Peng–Robinson, and Helmholtz energy equations of state. It is shown that increases in fluid model fidelity lead to reduced deviation between simulation and experiment. Maximum and mean discrepancies of 9.59% and 8.12% in nozzle pressure ratio with the Helmholtz energy EoS are reported. This work demonstrates an over-prediction of pressure ratio and power output within commercial CFD packages, for turbomachines operating in non-ideal fluid environments. This suggests a need for further development and experimental validation of CFD simulations for highly non-ideal flows. The data contained within this paper are therefore of vital importance for the future validation and development of CFD methods for dense-gas turbomachinery.

Thermodynamic properties of trifluoroethene (R1123): (p, ρ, T) behavior and fundamental equation of state

Abstract The (p, ρ, T) behavior of trifluoroethene (R1123) was investigated with the isochoric method. A total of 87 (p, ρ, T) data were measured along nine isochores (42, 98, 151, 197, 296, 493, 691, 790, and 888 kg · m − 3 ) at temperatures from 300 K to 430 K and pressures up to 6.9 MPa. The uncertainties in temperature are estimated to be within 10 mK below 380 K and 20 mK at higher temperatures, and the uncertainties in pressure are estimated to be within 1 kPa below 380 K and 2 kPa at higher temperatures. The estimated relative uncertainties in the density measurements are 0.15 % or less. A new Helmholtz energy equation of state was formulated based on the (p, ρ, T) data and recently published experimental data for the vapor pressure, liquid and vapor densities including those at saturation, vapor-phase sound speed, and ideal-gas isobaric heat capacity. The equation is applicable at temperatures from the triple-point temperature (195.15 K) to 480 K and pressures up to 20 MPa. Typical uncertainties in calculated properties are 0.1 % for vapor pressures, 0.1 % for liquid densities, and 0.2 % for vapor densities, except in the critical region where larger deviations up to about 1.5 % are sometimes observed in densities. The uncertainties in calculated vapor-phase sound speeds and ideal-gas isobaric heat capacities are 0.03 % and 1 %, respectively. The equation shows reasonable extrapolation behavior at extremely low and high temperatures, and at high pressures.

Experimental pvT property for the liquid HFO1234ze(E) using the isochoric method

Knowledge of the pvT property of the trans-1,3,3,3-tetrafluoropropene (HFO1234ze(E)) is essential for its industrial applications and the optimization of the energy-conversion systems that with HFO1234ze(E). The pvT property of HFO1234ze(E) in the liquid phase was obtained by means of a constant-volume pressure vessel. A total of 97 experimental state points was measured along four isochoric lines at temperatures from 280 K to 331 K with the pressures from 0.3 MPa to 27 MPa. The expanded uncertainties of the temperature and pressure were estimated to be less than 6.2 mK and 30 kPa at the confidence level of 0.95, respectively. The relative standard uncertainty of the experimental density was estimated to be within 0.1%. The experimental density shows good agreements with the high accuracy literature values. The relative deviations of the present experimental densities are less than 0.1% from the calculations of the high-precision Helmholtz energy equation of state in the liquid phase.

Residual entropy model for predicting the viscosities of dense fluid mixtures.

In this work, we have investigated the mono-variant relationship between the reduced viscosity and residual entropy in pure fluids and in binary mixtures of hydrocarbons and hydrocarbons with dissolved carbon dioxide. The mixtures considered were octane + dodecane, decane + carbon dioxide, and 1,3-dimethylbenzene (m-xylene) + carbon dioxide. The reduced viscosity was calculated according to the definition of Bell, while the residual entropy was calculated from accurate multi-parameter Helmholtz-energy equations of state and, for mixtures, the multi-fluid Helmholtz energy approximation. The mono-variant dependence of reduced viscosity upon residual molar entropy was observed for the pure fluids investigated, and by incorporating two scaling factors (one for reduced viscosity and the other for residual molar entropy), the data were represented by a single universal curve. To apply this method to mixtures, the scaling factors were determined from a mole-fraction weighted sum of the pure-component values. This simple model was found to work well for the systems investigated. The average absolute relative deviation (AARD) was observed to be between 1% and 2% for pure components and a mixture of similar hydrocarbons. Larger deviations, with AARDs of up to 15%, were observed for the asymmetric mixtures, but this compares favorably with other methods for predicting the viscosity of such systems. We conclude that the residual-entropy concept can be used to estimate the viscosity of mixtures of similar molecules with high reliability and that it offers a useful engineering approximation even for asymmetric mixtures.

Prediction of vapour-liquid equilibria of neon-nitrogen, neon-oxygen and neon-argon mixtures used in J-T refrigerators

Neon is one of the main components of the mixture used in J-T refrigerators operating below 70 K along with nitrogen, argon and oxygen. The properties of mixtures of these fluids need to be estimated accurately for designing optimum mixture compositions for these refrigerators. However, the binary interaction parameters of neon and other cryogens are not readily available in the literature for the widely used cubic equations of state. The binary interaction parameters of neon-nitrogen, neon-argon and neon-oxygen mixtures for the Peng-Robinson equation of state and that of neon-oxygen mixture for the Helmholtz energy equation of state have been regressed in this work. The vapour-liquid equilibria of neon-nitrogen, neon-argon, neon-oxygen mixtures estimated with the Helmholtz energy and the Peng-Robinson equations of states are compared with experimental data.

Phase equilibrium modelling for multi-component mixtures using highly accurate Helmholtz energy equation of state

Phase stability of a mixture at given conditions is studied. The modelling of equilibrium phase compositions is considered when the original phase of the mixture is unstable. The tangent plane distance criterion is used for stability analysis and the Gibbs energy minimisation is employed for phase equilibrium calculation when the successive substitution does not work well. With the Helmholtz free energy equation of state, thermodynamic properties in single phase and two-phase equilibrium were calculated for methane-ethane mixtures. Phase envelopes were also calculated.