Tait Equation Research Articles

Development and Application of a Cooling Rate Dependent PVT Model for Injection Molding Simulation of Semi Crystalline Thermoplastics.

This technical paper delves into the creation and application of an enhanced mathematical model for semi crystalline thermoplastics based on the Pressure-Volume-Temperature (PVT) Two Domain Tait Equation. The model is designed to incorporate the impact of the cooling rate on the specific volume of the material. This is achieved by utilizing Flash differential scanning calorimetry (fDSC) measurements, thereby ensuring a direct correlation to the actual behavior of the material in reality. The practical application of the model in the context of injection molding simulation was also considered. This was done by integrating the mathematical model into the Moldflow software via the Solver API. The paper underscores the discontinuity issue inherent in the traditional Tait equation with cooling rates and proposes a solution that guarantees a correct transition from the liquid to the solid phase, even at high cooling rates and pressures. The results demonstrated a realistic PVT curve across a wide range of cooling rates and high pressures. The model was put to the test using a 3D tetrahedron meshed calculation model in the injection molding simulation. This study marks a significant step forward in the simulation of injection molding processes, as it successfully bridges the gap between real material properties and simplified simulation, paving the way for more accurate and efficient simulations in the future.

Numerical Simulation of Shock Wave in Gas–Water Interaction Based on Nonlinear Shock Wave Velocity Curve

In a gas–water interaction problem, the nonlinear relationship between shock wave velocity is introduced into a Hugoniot curve, and a Mie–Grüneisen Equation of state (EOS) is established by setting the Hugoiot curve as the reference state. Unlike other simple EOS based on the thermodynamics laws of gas (such as the Tait EOS), the Mie–Grüneisen EOS uses reference states to cover an adiabatic impact relationship and considers the thermodynamics law separately. However, the expression of the EOS becomes complex, and it is not adaptive to many methods. A multicomponent Mie–Grüneisen mixture model is employed in this study to conquer the difficulty of the complex form of an EOS. In this model, some coefficients in the Mie–Grüneisen EOS are regarded as variables and solved using newly constructed equations. The performance of the Mie–Grüneisen mixture model in the gas–water problem is tested by low-compression cases and high-compression cases. According to these two tests, it is found that the numerical solutions of the shock wave under the Mie–Grüneisen EOS agrees with empirical data. When compared to other simple-form EOSs, it is seen that the Mie–Grüneisen EOS has slight advantages in the low-compression case, but it plays an important role in the high-compression case. The comparison results show that the solution of the simple-form EOS clearly disagrees with the empirical data. A further study shows that the gap between the Mie–Grüneisen EOS and other simple-form EOSs becomes larger as the initial pressure and particle velocity increase. The impact effects on the pressure, density and particle velocity are studied. Moreover, the gas–water interaction in a spherical coordinate plane and a two-dimensional coordinate is a significant part of our work.

Determination of Compressed Liquid Densities for CO2 + n-Decane Using a Vibrating Tube Densimeter

Understanding the density of CO2 + n-decane is crucial for designing and operating CO2 capture, transport, and storage. The safety and effectiveness of CO2 burial is directly affected by the density of CO2 + n-decane mixtures. The liquid densities of CO2(1) + n-decane(2) mixtures with mole fractions of CO2 x1 = 0, 0.2032, 0.4434, 0.7589, and 0.8947 were measured using a vibrating tube densimeter. The combined expanded uncertainties of density with a level of confidence of 0.95 are estimated to be 0.6 kg·m−3. A total of 221 compressed liquid densities of CO2(1) + n-decane(2) mixtures along the five isotherms between T = (283 and 363) K with pressures up to 100 MPa were presented. The densities of mixtures were correlated by the modified Tait equation, resulting in absolute average deviations between the experimental and calculated values of 0.028%, 0.013%, 0.017%, 0.044%, and 0.042%. In addition, the isothermal compressibility, isobaric thermal expansivity, and excess molar volume were derived from the modified Tait equation.

Blending biomass-based liquid biofuels for a circular economy: Measuring and predicting density for biodiesel and hydrocarbon mixtures at high pressures and temperatures by machine learning approach

Circular economy is an efficient approach to deal with the rapidly increasing demand for energy and its environmental impact. It involves switching to renewable and sustainable energy sources. By providing an alternative to traditional energy resources, biodiesel promotes a circular economy paradigm that embraces renewable energies rather than fossil fuels. As an alternative to fossil fuels, liquid biodiesel helps to improve energy efficiency and reduce pollution. Furthermore, blending biodiesel has proven to be economically and environmentally viable. Knowledge of the thermodynamic properties of biodiesel, such as densities and coefficients of expansivity and compressibility, plays an important role in understanding and determining the behavior of the fuel under different operating conditions. The blend densities of Soybean Oil Biodiesel (SOB) with hexane and Waste Cooking Oil Biodiesel (WCOB) with heptane in four volume ratios were experimentally studied. An Anton Paar vibrating tube densimeter HPM DMA has been used for accurate measurements of densities of the biofuels and their blends. Density measurements were performed at temperatures ranging from 298.15 K to 393.15 K, and pressures between 0.1 MPa and 140 MPa. This work also aims to develop accurate prediction models of the properties of biodiesel blends under various conditions, such as changes in temperature and pressure, which are essential for optimizing energy production processes and engine performance. In this context, the dataset was first investigated with Tait Equation of State (Eos) to determine thermodynamic parameters which including isothermal compressibility and isobaric expansivity. It showed generally low standard deviations. Furthermore, these new experimental densities were also modeled with Artificial Neural Network (ANN) as a machine learning method. This paper models proved that the machine learning method is a suitable tool to model the thermo-physical characteristics of liquid fuels. The findings of this study have the potential to provide valuable guidance to engine engineering applications developers regarding operating conditions and the selection of suitable biodiesel for their system.

PρT measurements of 3-aminopropan-1-ol and N-(2-hydroxyethyl)morpholine from (293.15 to 473.15) K and up to 40 MPa and modeling with modified Tait and PC-SAFT equations

New density data has been reported for 3-aminopropan-1-ol (AP) and N-(2-hydroxyethyl)morpholine (NHEM) at temperatures ranging from (293.15–473.15) K at 19 pressures ranging from (0.1–40) MPa. The experimental measurements were carried out using an Anton Paar high-pressure vibrating tube densimeter with a combined expanded uncertainty of 1 kg m−3 at a 95 % confidence level. Contributions considered in the uncertainty analysis included the impurities in the materials used and apparatus specification. In addition, the density and speed of sound at ambient pressure (81.5 kPa) and temperatures (293.15–343.15) K were measured. The experimental density data were correlated with the modified Tait equation. Values of thermal expansion coefficient (αP ) and isothermal compressibility (κT) were calculated. The work is completed with the modeling of the experimental data using the perturbed-chain statistical associating fluid theory equation of state (PC-SAFT EOS). The parameters of the PC-SAFT equation of state, for the pure compounds, were determined by fitting the equation to the liquid PρT experimental data. Thermodynamic properties such as thermal expansion coefficient (αP), isothermal compressibility (κT), isobaric heat capacity (CP), and speed of sound (u) were calculated with the obtain parameters. Good agreement between experimental data and derived properties represented the modeling accuracy with the obtained parameters.

A multiscale model to predict the equation of state of a saturated cement paste

An advanced multiscale analytical approach for predicting the Equation of State (EOS) of fully saturated cement pastes is presented. This approach models the matrix as an elastic–plastic material with linear hardening that includes capillary pores filled with water. The model takes into account both the stochastic variation of pore sizes and their spatial distribution. The microscale level is represented by a spherical medium having the mechanical properties of the cement paste matrix. A single spherical pore that is filled with compressible water is located at the center of the micro-domain. The behavior of the compressible water is modeled by the nonlinear Tait equation of state. The EOS at the macro level is obtained by averaging the micro level domain strains over the pore sizes and pore wall thicknesses. It is shown that the solution depends only on the ratio between the minimum pore wall thickness and the lower limit of capillary pore radius and is independent of each of these parameters separately. The EOS obtained by the proposed model shows good agreement with experimental data for cement paste with water/cement ratio w/c = 0.50. The performed parametric study shows that the bulk modulus and yield stress of undisturbed cement gel matrix significantly affect the EOS, while the variation of Poison ratio, plastic hardening parameter and minimum limit pore size and wall thickness (within physically accepted range) slightly affect the cement paste bulk behavior.

Compressibility effects on cavity dynamics and shock waves in high-speed water entry

The importance of high-speed water entry is acknowledged within the defense industry. This study numerically investigates the water entry of a high-speed rectangle projectile, focusing on cavity dynamics and shock wave generation. A computational model is employed to accurately simulate the intricate fluid dynamics of compressible multiphase flows. This model integrates a dual-phase flow algorithm with a thermally sensitive Tait equation of state for the liquid phase. The primary focus lies in understanding the effects of fluid compressibility on cavity evolution and shock wave propagation across different Froude numbers. The findings reveal that compressibility induces changes in cavity formation size, leading to significant variations in phase composition within the cavity. Furthermore, compressibility enhances the air cushion effect upon surface impact, resulting in delayed water entry and concurrent reduction in projectile drag. Moreover, a prognostic model is proposed, correlating shock pressure with propagation distance, thereby validating theoretical hypotheses advanced by Lee et al. [J. Fluid Struct., 11, 819–844 (1997)].

Liquid density of binary mixtures of n-pentane and n-dodecane at temperatures from 283K to 363K and pressures up to 100MPa

The compressed liquid densities of n-pentane(1) + n-dodecane(2) binary mixtures were measured using a high pressure vibrating-tube densimeter at temperatures from 283 K to 363 K, pressures up to 100 MPa, and five different compositions (x1 = 0.3778, 0.6230, 0.7968, 0.9051, and 1). The densities of binary mixtures were correlated using the modified Tait equation, and the absolute average deviations of the experimental and calculated values were 0.010 %, 0.019 %, 0.016 %, 0.025 %, and 0.017 %, respectively. In addition, the isothermal compressibility and the isobaric thermal expansivity were calculated from the obtained densities.

Effects of the crystallinity of fluoropolymer binders components in polymer-bonded explosives on shock Hugoniots: A computational study

Fluoropolymers play a crucial role as binders in polymer-bonded explosive (PBX) formulations. However, there is a lack of clear understanding of the effects of increased fluoropolymer crystallinity on the shock response of PBXs in the service environment. This study investigated the shock Hugoniots of two widely applied fluoropolymer binders: (1) F2314 from China—a copolymer with a molar ratio of vinylidene fluoride (VDF) to chlorotrifluoroethylene (CTFE) of 1:4 and (2) F2313 from the United States, also known as Kel F-800, with a VDF to CTFE molar ratio of 1:3. The Hugoniot curves of both fluoropolymers were calculated based on equilibrium molecular dynamics (MD) and a mixing rule. Furthermore, the corresponding P–V curves were obtained through fitting using the Tait equation of state (EOS). Their calculated parameters, including zero-pressure bulk modulus (κo) and sound velocity (co), agreed well with experimental data. The results reveal that the Hugoniots of amorphous F2314 and F2313 exhibited negligible differences. However, increasing crystallinity significantly impacted the Hugoniot curves of both fluoropolymers, especially for F2314 with high crystallinity. The obtained macroscopic characteristic parameters, namely κo and co, exhibited an exponential dependence on crystallinity. Physically, this phenomenon can be attributed to a reduction in the compressible free volume of the fluoropolymers due to a more orderly chain arrangement. Additionally, under the same compression ratio, the shock temperature of the fluoropolymers increased with the crystallinity, posing potential safety risks to explosives. These findings establish a correlation between the crystallinity of fluoropolymers and the shock properties of PBXs, providing a theoretical reference for the formulation design of fluoropolymer-based PBXs.

Density, viscosity, and CO2 solubility in ether-functionalized phosphonium-based bis(trifluoromethanesulfonyl)amide ionic liquids

We measured the densities and viscosities of two ionic liquids, triethyl(methoxymethyl)phosphonium bis(trifluoromethanesulfonyl)amide ([P222(1O1)][TFSA]) and triethyl(2-methoxyethyl)phosphonium bis(trifluoromethanesulfonyl)amide ([P222(2O1)][TFSA]), at atmospheric pressure and 273.15–363.15 K. The high-pressure density at the pressures up to 50 MPa was also measured at 298.15–353.15 K. Meanwhile, CO2 solubility in these phosphonium-based ionic liquids was determined using a magnetic suspension balance at 303.15–333.15 K and pressures up to 6 MPa. The experimental density and viscosity at atmospheric pressure were fitted using quadratic and Vogel-Fulcher-Tammann equations, respectively. The high-pressure density was fitted using the Tait equation and Sanchez-Lacombe equation of state. The properties of the two ether-functionalized ionic liquids were compared to those of triethylpentylphosphonium bis(trifluoromethanesulfonyl)amide [P2225][TFSA]. The density at atmospheric pressure increased in the order [P2225][TFSA] < [P222(2O1)][TFSA] < [P222(1O1)][TFSA]. The introduction of the ether group effectively reduced viscosity in the following order: [P222(1O1)][TFSA] < [P222(2O1)][TFSA] < [P2225][TFSA]. The CO2 solubilities in [P222(1O1)][TFSA] and [P222(2O1)][TFSA] were slightly lower than those in [P2225][TFSA]. In contrast, the molality of CO2 (m1) increased in the order [P2225][TFSA] < [P222(2O1)][TFSA] < [P222(1O1)][TFSA]. Ether functionalized phosphonium-based cations are effective in enhancing the physical absorption of CO2, particularly the molality of CO2.

A reinterpretation of the results for melting curves of metals based on the force – Heat equivalence energy density principle

A phenomenological model for describing the melting of metals at high pressures has recently been formulated by Ma et al. using the force – heat equivalence energy density principle. This theory represents an alternative to the Lindemann law of melting. Ma et al. have derived a formulation for the pressure dependence of melting temperature using the Murnaghan approximation for the linear dependence of bulk modulus on pressure. Since this approximation has proven to be not accurate for solids at high pressures, we have revised the expressions obtained by Ma et al. using the third order Birch- Murnaghan equation of state and the usual Tait equation of state. The revised expressions are used to describe the melting curves for twenty metals viz. Al, Ti, V, Cr, γ-Fe, Co, Ni, Cu, Zn, Mo, Pd, Ag, Ta, W, Re, Ir, Pt, Au, Pb and U. We have thus presented a reinterpretation of previous results for melting using the method proposed by Ma et al.

A level-set method for ultrasound-driven bubble motion and tissue deformation

Ultrasound-driven bubble collapse and resulting tissue deformation are numerically investigated by combining a level-set method for determining the liquid–gas and liquid–solid interfaces and a full Eulerian method to solve the unified governing equations for both solid and fluid phases. The fluid compressibility effect is considered by employing the van der waals and Tait equations of state for gas and liquid, respectively, and a semi-implicit pressure correction method is used to effectively capture ultrasound and shock waves. Two tissue models, treating tissue as a viscoelastic solid or as a fluid with large surface tension, are compared in terms of the bubble expansion and compression processes and tissue deformation patterns. Computations are also carried out to investigate the influences of ultrasonic pulse amplitude, ultrasound frequency, and bubble-tissue distance on the bubble-tissue interaction.

Thermodynamic Properties and Equation of State for Tungsten

A high-temperature equation of state for tungsten was constructed in this study using experimental data on its thermodynamic properties, thermal expansion, compressibility, and bulk compression modulus. The totality of experimental data were optimized by the temperature-dependent Tait equation over a pressure range from 0 up to 1000 kbar and over a temperature range from 20 K to the melting point. An extended Einstein model was used to describe the temperature dependence of thermodynamic and thermophysical parameters. A linear temperature dependence was embraced for the derivative of the isothermal bulk modulus. The resultant equation of state provides a good fit to the whole set of experimental data within measurement uncertainties associated with individual quantities.

On the unsteady cavitation characteristics of compressible liquid nitrogen flows through a venturi tube

In this paper, the unsteady cavitating flows of liquid nitrogen (LN2) through a three-dimensional (3D) venturi tube in different thermal cavitation modes are modeled based on the mixture multiphase model with the LES method and Sauer-Schnerr cavitation model. A numerical framework considering compressibility and the thermal effects of cryogenic fluids is developed, and the modified temperature-dependent Tait equations of state provide the mathematical closure. The numerical results are compared with those from previous experiments that we have conducted, and it is indicated that the simulations and experiments agree well. Condensation shock wave propagation is observed to be initiated by the impingement of the collapse-induced pressure waves from the previously shedding cloud, and details of the process are presented. The impacts of the thermal effect on the cavitation dynamics, especially the evolution of cavitation patterns and dynamic interactions between cavitation and vortex, are analyzed. The numerical results show that the formulation of condensation shock generated at the rear of the cavity varies with temperature. In addition to the pressure waves caused by shed vapor collapse, the violent mass transfer process at the closure region of the cavity releases a substantial quantity of latent heat, initiating further condensation towards the cavity as liquid temperature decreases. The entropy production theory is derived and implemented, and different entropy production terms are established in unsteady LN2 cavitating flow with different thermal effects to understand the interactions between cavitation and entropy production. The present study provides insight into the interactions of cavitation-condensation shock in the LN2 cavitating flow and contributes to a better understanding of the influences of thermal effects on cryogenic cavitation dynamics.

Novel algorithm for compressible Newtonian axisymmetric thermal flow

In this paper, we mainly focus on the development of the Taylor-Galerkin/Pressure-Correction method (TG/PC) method to solve the flow of a compressible Newtonian fluid under nonisothermal conditions. The process of development needs applying the two-step Lax–Wendroff method on the energy equation, after which two new steps for energy equations will be obtained within the TG/PC algorithm. In addition, for the density component, the Tait equation of state is adopted to relate density to pressure, which also yields a new step within the novel method to give the correction formula of compressible pressure. To perform the analysis of the new algorithm, Poiseuille flow through axisymmetric rectangular channel for the Newtonian thermal flow is utilized as a simple problem test. The effect of nondimensional factors such as Reynolds number (Re) and Prandtl number (Pr) is discussed in this study. The influence of thermal conductivity [Formula: see text] and viscosity [Formula: see text] on the solution components is presented as well. Also, comparison for both compressible and incompressible flows is provided in addition to a comparison between both cases’ convergence, as it turns out that the convergence of the incompressible case is better than the compressible case. All the results of the effects presented in this study agree well with the approved physical trend.

Liquid densities of n-octane, ethylcyclohexane and their binary mixtures {(x) n-octane) + (1 − x) ethylcyclohexane} at temperatures from (283 to 363) K and pressures up to 60 MPa

In this work, liquid densities of n-octane, ethylcyclohexane and their binary mixtures {(x) n-octane) + (1 − x) ethylcyclohexane} with mole fraction x= (0.200, 0.400, 0.600, 0.800) are reported. The density measurements were performed with a high-pressure vibrating-tube densimeter at temperatures from (283 to 363) K and pressures up to 60 MPa. The combined expanded uncertainties (0.95 level of confidence, k = 2) are estimated to be 16 mK for temperature, 0.062 MPa for pressure, and up to 0.6 kg/m3 for density. The experimental densities were correlated using the modified Tait equation. The average absolute percentage deviations of measurement results from the modified Tait equation correlations of n-octane, ethylcyclohexane and their binary mixtures {(x) n-octane) + (1 − x) ethylcyclohexane} are all lower than 0.1%. The isothermal compressibility and isobaric expansivity have also been derived from the experimental densities.

Numerical simulation of subsonic and transonic water entry with compressibility effect considered

The purpose of this study is to analyze the subsonic and transonic water entry of the axisymmetric body with compressibility effect considered. The water-entry processes of axisymmetric body are numerically investigated, with Tait state equation and energy equation used to describe the compressibility of the liquid. Reasonable agreement has been obtained between the numerical results and the experimental measurements of trajectory, velocity and acceleration. The cavity evolution and dynamic characteristics of subsonic cases are investigated for vertical and oblique water-entry. The results show that the maximum drag coefficient of the oblique water entry is lower, and it is affected by a large moment which causes pitch angle and trajectory inclination angle to change significantly. In the oblique transonic conditions, the effect of compressibility of the liquid on the dynamic flow characteristics is analyzed. On this basis, the influence of entry angle on the hydrodynamic characteristics is investigated. It is concluded that the water-entry angle has a more significant influence on the trajectory characteristics.

The Suppression of Hump Instability inside a Pump Turbine in Pump Mode Using Water Injection Control

The occurrence of hump instability in pump mode within a pump turbine poses a significant challenge to the safe and stable operation of Pumped Storage Power Plants (PSPP). To achieve more precise numerical simulations, this paper establishes a weakly compressible model of water based on the Tait equation. Using this model, it is discovered that the onset of hump instability is closely linked to an increase in hydraulic losses induced by stalled rotation within the diffuser. Then, a flow control approach employing water injection into the guide vanes of a pump turbine is proposed in order to suppress flow instabilities and optimize the hump region. The findings reveal that the water injection approach can mitigate hydraulic losses, suppress unstable structures, and diminish the pulsation amplitude within the diffuser, ultimately delaying the emergence of the hump region to lower flow mass conditions. This study is helpful in widening the range of the safe and stable operation of pump turbines in pump mode.

PVT behaviour of hydrophilic and hydrophobic eutectic solvents

Among the basic principles of green chemistry is the search for less harmful alternative solvents than conventional solvents. Knowing the thermophysical properties of fluids under different pressure and temperature conditions is essential to propose them. Herein, we present data on the densities at several pressures (from 0.1 to 65 MPa) and temperatures (from 283.15 to 338.15 K) of two deep eutectic solvents with hydrophilic characteristics (choline chloride + ethylene glycol or glycerol) and two eutectic solvents with hydrophobic characteristics (camphor + thymol or menthol). We used the Tait equation of state to correlate and calculate derived properties. Moreover, we modelled the mixtures with the PC-SAFT equation of state. The results showed that the hydrophilic solvents were more compact than the hydrophobic ones. The former exhibited an abnormal thermal behaviour of the isobaric thermal expansibility. The deviations in the correlation of densities with the thermodynamic model were between 0.5 and 3%. They were lower for the mixtures with weaker interactions.

Density Measurements and PC-SAFT Modeling for Beef Tallow Fatty Acid Methyl Esters (FAME Mixture) + Alkane up to 70 MPa

Biodiesel is mainly constituted by fatty acid alkyl esters, and the effect of adding alkane compounds is of great importance in order to analyze the changes in the thermophysical properties under high pressure that is of relevance in diesel engines. This work is aimed at the study of density behavior of two systems formed by beef tallow fatty acid methyl esters (FAME mixture) + alkane (decane or dodecane) in the temperature range of 303–393 K and in the pressure range of 0.2–68.29 MPa. Experiments were carried out by means of two vibrating tube densimeters with an expanded uncertainty in density of 0.7 kg·m–3. Transesterification of waste beef tallow by using methanol at supercritical conditions was used to obtain biodiesel. The FAME mixture was constituted by methyl tetradecanoate, methyl hexadecanoate, methyl octadecenoate, and methyl (Z)-octadec-9-enoate. Density behavior was correlated using the Tammann–Tait equation and modeled with the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state. Density correlation deviations using the Tammann–Tait equation were found to be less than 0.05% and 4.15 kg·m–3 for the average absolute deviation (AAD) and standard deviation (STD), respectively. The PC-SAFT modeling yielded an AAD = 0.11% and an STD = 1.15 kg·m–3. The behavior for density and the derived thermodynamic properties were explained in terms of the fatty acid methyl ester content.