Tait Equation Research Articles

Solution of cavitating compressible flows using Discontinuous Galerkin discretisation

A methodology for modelling cavitating flows using a high-order Adaptive Mesh Refinement (AMR) approach based on the Discontinuous Galerkin method (DG) is presented. The AMR implementation used features on-the-fly adaptive mesh refinement for unstructured hybrid meshes. The specific implementation has been developed for the resolution of complex multi-scale phenomena where high accuracy p-adaptive discretisations are combined with an h-adaptive data structure. This approach accommodates the fine spatial resolution for the interface discontinuities and the shock waves observed in compressible cavitating flows. The Tait equation of state is used for the modelling of the liquid phase while an isentropic path is assumed for the liquid/vapour mixture. Second order spatial and a third order non-oscillatory temporal discretisation are used for the integration of the mass and momentum conservation equations, in order to resolve the flow structures responsible for the formation of cavitation bubbles and the resulting compression waves. Assessment of the developed methodology is performed for the one-dimensional advancement of a compressible liquid-vapour interface and the symmetric collapse of a spherical vapour bubble.Following, results obtained with the developed multi-scale modelling AMR approach has revealed a complex bubble collapse mechanism near a rigid wall, providing evidence of processes that have been unknown before due to reduced resolution and dissipative nature of past simulations. The impinging jet accompanying the collapse of a bubble near a wall, was found to induce vortical structures, which result to the formation of a secondary cavitation of a wall-attached bubble at the vicinity of the impingement jet shear layer. At the final stages of the initial bubble collapse, the impinging jet was found to penetrate the centre-line of the wall bubble inducing its partial collapse. This secondary collapse results to a rich spatial structure of shock waves, interacting with the secondary bubbles. Moreover, the calculated pressure level are found to be much higher than those reported from previous methodologies.

Effect of curved rigid surface on the collapsing cavitating bubble in cryogenic environment

The high impact erosive pressure resulting from collapsing cavitating bubble has advantages in stone fragmentation and shock wave lithotripsy. They carry damage potential also and can erode the curved hydrofoil of any turbo-machinery, and, still somewhat obscure in cryogenic liquids. Once the material loses its surface smoothness, the flow-field surrounding the collapsing bubbles is affected by the newly formed irregular surface geometry. In this way, the bubble collapse is significantly influenced by the curvature of the rigid boundary. In this work, a collapsing cavitating bubble near a curved rigid surface dipped in cryogenic fluid has been investigated numerically to illustrate the effect of different surface configurations (i.e. Convex i.e. ξ < 1 and Flat ξ = 1 surface) using volume-of-fluid (VOF) method in a compressible framework for different standoff distance (γs ). TAIT equation for the surrounding liquid is used to model shock perturbations during the collapse, whereas ideal gas equation is used for the modeling of compressibility of vapor (or gas) bubble. Here, different dynamical features such as jet formation, bubble-shape evolution (necking and splitting) and vortex ring motion (free and wall vortex) is shown for collapsing cavitating bubble in cryogenic environment and compared with the room-temperature fluid combination i.e. water-air for flat and convex wall configurations for different stand-off distance.

Influence of head structure on hydrodynamic characteristics of transonic motion projectiles

The hydrodynamic characteristic of transonic motion projectiles with different head diameters are investigated by numerical simulation. Compressibility effect in liquid-phase water are modeled using the Tait state equation. The result shows that with increasing of velocity the compression waves transfer to shock waves，which cause the significant increasing of pressure and decreasing the dimensions of supercavities. While the increasing of head diameter, the thickness, the vapor volume fraction and the drag coefficient of supercavities are all enhanced, which is conducive to the stability of transonic-speed projectiles. The cavity dynamics of the different head projectiles are compared, and the results shows when Mach number is in high region, the truncated cone head projectile is enveloped by a cavity which results in less drag and better stability.

On the Fisk–Tait equation for spin-3/2 fermions interacting with an external magnetic field in noncommutative space-time

In this paper, we investigated the Fisk-Tait equation in interaction with an external magnetic field in noncommutative space-time. Knowing that the space-time noncommutativity is introduced through the Moyal-Weyl product known method. Consequently, we studied the continuity equation in both commutative and noncommutative space-time; there we examined the influence of the space-time noncommutativity on the current density quadri-vector. Moreover, we find that the total charge obtained from the probability density still indefinite even when space does not commute. Furthermore, we found the spin current density in the two different spin directions. We also investigated the linking between the fermions and the bosons in the Fock space using the Holstein-Primakoff transformation.

Pressure–Volume–Temperature Relationships of Trimethoxymethylsilane and Tetramethyl Orthosilicate at Temperatures Ranging from 303.2 to 363.2 K and Pressures up to 160 MPa

The pressure–volume–temperature (PVT) relationships for trimethoxymethylsilane (MTMS, CH3(CH3O)3Si) and tetramethyl orthosilicate (TMOS, (CH3O)4Si) were measured at temperatures ranging from 303.2 to 363.2 K and pressures up to 160 MPa. Density measurements were carried out with an experimental apparatus based on the variable volume method. The standard uncertainties u(T) = 0.17 K and u(P) = 0.14 MPa, and the relative combined expanded uncertainty Ur (0.95 level of confidence) were Ur(ρ) = (0.0016 and 0.0022) for MTMS and TMOS, respectively. The obtained PVT relationships were correlated using the Tait equation and the Sanchez–Lacombe equation of state to determine the characteristic parameters.

A Two-Dimensional Axisymmetric Finite Element Analysis of Coupled Inertial-Viscous-Frictional-Elastic Transients in Magnetorheological Dampers Using the Compressible Herschel-Bulkley Fluid Model

It has been challenging to accurately predict the unique characteristics of magnetorheological (MR) dampers, due to their inherent nonlinear nature. Multidimensional flow simulation has received increasing attentions because it serves as a general methodology for modelling arbitrary MR devices. However, the compressibility of MR fluid which greatly affects the hysteretic behavior of an MR damper is neglected in previous multidimensional flow studies. This paper presents a two-dimensional (2D) axisymmetric flow of the compressible Herschel-Bulkley fluid in MR dampers. We simulated the fully coupled inertial-viscous-frictional-elastic transients in MR dampers under low-, medium- and high frequency excitations. An arbitrary Lagrangian-Eulerian kinematical description is adopted, with the piston movements represented by the moving boundaries. The viscoplasticity and compressibility of MR fluid are respectively modeled by the modified Herschel-Bulkley model and the Tait equation. The streamline-upwind Petrov–Galerkin finite element method is used to solve the model equations including the conservation laws and mesh motion equation. We tested the performances of an MR damper under different electric currents and different frequency displacement excitations, and the model predictions agree well with the experimental data. Results showed that the coupled transients of an MR damper are frequency dependent. The weak compressibility of MR fluid, which mainly happens in the chamber rather than in the working gap, is crucial for accurate predictions. A damper’s transition from the pre-yield to the post-yield is essentially a step-response of a second order mass-spring-viscous system, and we give such step-response a detailed explanation in terms of mass flow rate.

Viscosity Measurements of Dialkyl Adipates in the Temperature Range of (283 to 363) K and up to 40 MPa

Viscosity is one of the most important properties for many scientific and technological applications, and therefore it has been the subject of an enormous effort of measurement, correlation, and interpretation. Reliable reference data can be of great help to improve viscosity accuracy, while for viscosity, the only accepted standard is the viscosity of pure water at 20 °C (the reference value recommended by ISO is 1.0016 mPa·s). With the technology and industry advancing, there is a new requirement for reference materials of viscosity covering wide temperature, pressure, and viscosity ranges in order to face the challenge that the order of magnitude of viscosity turns to 1014. Considering the dissatisfaction of the only standard, we put forward these candidates to meet the need of appropriate viscosity standard materials: dimethyl adipate (DMA), diethyl adipate (DEA), dipropyl adipate (DPA), and dibutyl adipate (DBA). These materials have the characteristics of easy to obtain, high purity, high thermal chemical stability, and especially larger viscosity value. In this work, the viscosity was experimentally measured by the vibrating wire viscometer over the temperature range of (283 to 363) K and at pressures up to 40 MPa. The combined expanded uncertainty of the reported viscosity is about 2 % with a confidence level of 0.95 (k = 2). The empirical Andrade–Tait equation was used to correlate the experimental results.

Vapor-liquid equilibria and mixture densities for 2,2,4,4,6,8,8-heptamethylnonane + N2 and n-hexadecane + N2 binary mixtures up to 535 K and 135 MPa

In this work, we report high-pressure, high-temperature (HPHT) mixture density and T-p isopleth (bubble (BP) and dew (DP) point) data for hexadecane (HXD) + N2 and heptamethylnonane (HMN) + N2 mixtures from ~323 to 523 K and pressures to ~100 MPa. Isothermal, mixture density data for both mixtures are measured in the single–phase region from the BP pressure to ~135 MPa and with ~14–90 mol% N2. A HPHT variable-volume, windowed view cell is used for both density and phase behavior measurements using the synthetic method. Mixture densities are correlated with the modified Tait equation and isothermal BP/DP data are correlated with an Antoine-type equation to allow for reliable interpolation of the data sets. Mixture densities and BP/DP pressures are modeled with the PC-SAFT equation coupled with pure component parameters calculated with two different group contribution methods. Although fairly reasonable predictions of liquid mixture densities are obtained when the binary interaction parameter, kij, is set to zero for both HXD + N2 and HMN + N2 mixtures, a value of kij equal to at least 0.119 is needed for both systems to obtain reasonable predictions of isothermal p-x behavior.

Effect of Composition, Temperature, and Pressure on the Viscosities and Densities of Three Diesel Fuels

In this work, a rolling-ball viscometer/densimeter is used to measure high-pressure, high-temperature (HPHT) density and viscosity data from 298.2 to 532.6 K and pressures up to 300.0 MPa for three different diesel fuels. The densities and viscosities have combined expanded uncertainties of 0.6 and 2.5%, respectively, with a coverage factor, k = 2. Two of the diesels contain either a larger paraffinic or a larger aromatic content relative to the others and are standard engine test fuels. The third is an ultralow sulfur diesel that resembles an unfinished commercial diesel. Detailed compositional information is also reported for each diesel that provides a basis for interpreting the impact of composition on density and viscosity at high pressures. Both density and viscosity data are correlated to Tait-type equations with uncertainties of 0.6 and 4.0%, respectively. The Tait equations provide a facile means to compare observed differences in the density–pressure and viscosity–pressure profiles of the three different diesels. Density data are modeled with the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) with pure component parameters calculated representing diesel as a single, pseudo-component only requiring average molecular weight (Mave) and hydrogen-to-carbon ratio (RH/C) as inputs. Viscosity data are modeled reasonably well using entropy scaling coupled with the PC-SAFT EoS and information on the diesel Mave and RH/C. The HPHT viscosity data are also modeled reasonably well with free volume theory with model parameters correlated to Mave and RH/C.

Experimental High-Temperature, High-Pressure Density Measurement and Perturbed-Chain Statistical Associating Fluid Theory Modeling of Dimethyl Sulfoxide, Isoamyl Acetate, and Benzyl Alcohol

Experimental high-temperature, high-pressure (HTHP) density of three solvents including dimethyl sulfoxide, benzyl alcohol, and isoamyl acetate was measured at temperatures of 293.15–473.15 K and pressures up to 35–37.5 MPa. The density was measured with an Anton Paar density meter (DMA HP) at 20 K and 2.5 MPa intervals. The calibration fluids were water and benzene. The combined expanded uncertainty of density taking into account the device specification and the impurities of the materials was 1 kg·m–3 (0.95 level of confidence). The experimental density data were correlated with the Tait and new modified Tait equations. The thermal expansion coefficient (αp) and isothermal compressibility (κT) were calculated with the new modified Tait equations. Moreover, the perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state was used to model the density data at HTHP. In most cases, the PC-SAFT parameters are obtained from saturated vapor pressure and liquid phase density; however, these ...

Efficient three-equation two-phase model for free surface and water impact flows on a general curvilinear body-fitted grid

In this work, a simple and efficient three-equation two-phase model is transformed to a multi-dimensional general curvilinear coordinate system for simulation of free surface and water impact flows on curvilinear body-fitted grids. The two-phase model is a reduced version of the Baer–Nunziato model for compressible multiphase flows, in which the governing equations for only the liquid phase are active, whereas those for the air phase are neglected. The CPU cost is significantly reduced by solving the equation system only in the liquid regions, instead of solving the entire computational domain as in classical Euler approaches. The physical system is formulated in a hyperbolic vector form in a curvilinear coordinate system and solved using a monotonic upstream-centered scheme for conservation laws (MUSCL)/Godunov-type finite volume scheme. Compressive MUSCL limiters combined with the high-order finite volume approach result in a sharp interface solution. Validations with numerical grid refinement and convergence studies on dam-break and water impact problems are addressed, and the capability and robustness of the present model for accurate simulation of free surface and water impact flows are demonstrated. The implementation of the present system on a curvilinear body-fitted grid and the use of the Tait equation of state for water for the closure of the compressible system allow the method to solve water impact flow problems with both low-speed and high-speed complex geometries in industry and engineering.

Viscosity measurements of 1-ethyl-3-methylimidazolium trifluoromethanesulfonate (EMIM OTf) at high pressures using the vibrating wire technique

The goal of the present work is to contribute to the characterization of ionic liquids by measuring their viscosity at high pressures. As 1-ethyl-3-methylimidazolium trifluoromethanesulfonate (EMIM OTf) has been used as a solvent in CO2 capture processes, the temperature and pressure ranges of the measurements cover the intervals used in those processes.Measurements of the viscosity of EMIM OTf along five isotherms in the range (298–358) K and at pressures up to 50 MPa, have been performed using the vibrating wire technique in the forced mode of operation. As far as the authors are aware, these are the first measurements of this ionic liquid at pressures higher than 0.1 MPa, to be published. The viscosity results were correlated with the molar volume, using a modified hard-spheres model. The root mean square (σ) deviation of the data from the correlation is less than 0.5% The expanded uncertainty of the present viscosity data is estimated as ±2.0% at a 95% confidence level. As a complement, the pressure-viscosity coefficient has been calculated within the temperature range of the present results.Previous studies of the influence of the electric conductivity of ionic liquids, including EMIM OTf, in the vibrating wire method, have been taken into account for the present work.Complementary measurements of the density have been performed along seven isotherms in the temperature range from (298–363) K and pressures from (0.1–70) MPa. The density measurements were carried out with an Anton Paar vibrating U-tube densimeter and the raw data were corrected for viscosity effects. The density results were correlated with the temperature and pressure using a modified Tait equation. The expanded uncertainty of the present density data is estimated as ±0.2% at a 95% confidence level.

Study of shape, size and temperature-dependent elastic properties of nanomaterials

The impact of shape, size and temperature on elastic properties of nanomaterials is studied in this work. We have extended the melting temperature expression for nanostructures formulated by Guisbiers et al. and obtained the expression of elastic moduli and thermal expansivity for nanomaterials. An isobaric Tait equation of state is combined with Guisbiers model and the model so obtained is applied to analyze the shape, size and temperature effect on Young’s modulus and thermal expansivity in nanomaterials. The present computed results are compared with the simulated results and available experimental data. The Young’s modulus is observed to decrease as particle size is reduced while thermal expansivity increases with decrease in the size of nanomaterial. The Young’s modulus shows decrease with increase in temperature and decrement is observed maximum in spherical nanomaterials and minimum in nanofilms (NFs). Rate at which modulus is decreasing is found to increase as particle size is reduced. Good consistency of present predicted results with the available theoretical and experimental data is observed. The present calculated results are thus found consistent with the general trend of variation.

Compressed liquid densities of binary mixtures of difluoromethane (R32) and 2,3,3,3-tetrafluoroprop-1-Ene (R1234yf) at temperatures from (283 to 363) K and pressures up to 100 MPa

The compressed liquid densities of binary mixtures difluoromethane (R32) and 2,3,3,3-tetrafluoroprop-1-ene (R1234yf) were measured using a high pressure vibrating-tube densimeter over the temperatures from (283 to 363) K and pressures up to 100 MPa at five different compositions, xR32 = (0.189, 0.268, 0.500, 0.723, 0.904) in the present work. The combined expanded uncertainties of the temperature, pressure, mole fraction and the density with a level of confidence of 0.95 (k = 2) is estimated to be 16 mK, 0.062 MPa (p ≤ 60 MPa), 0.192 MPa (60 MPa < p < 100 MPa), 0.01 and up to 0.6 kg·m−3 depending on the temperature and pressure ranges, respectively. The experimental data were correlated using the modified Tait equation. The isothermal compressibility, κT, and the isobaric thermal expansivity, αp, were determined using liquid densities of mixtures and modified-Tait equation. Furthermore, the relative deviations of the experimental data with the equation of state (EOS) from Akasaka were lower than 1% from (283 to 363) K with the pressure up to 40 MPa.

High-Pressure Soybean Oil Biodiesel Density: Experimental Measurements, Correlation by Tait Equation, and Perturbed Chain SAFT (PC-SAFT) Modeling

Biodiesel can easily become the crucial solution for environmental problems. The high production rate of soybean oil has been the subject of several research works to transform it into biodiesel. K...

Physico-chemical properties of ionic liquids: Density, viscosity, density at high pressure, surface tension, octan-1-ol/water partition coefficients and thermodynamic models

Ionic liquids (ILs) are an important new media and one of the most attractive solvents used in extraction and separation processes. This work presents some important physico-chemical properties and topics related to ILs, which were found by us as good extractive solvents for the desulfurization of fuels and for the separation of water/biobutan-1-ol. It consists of the most important series of experimental measurements of density and viscosity of pure ILs and in binary systems with thiophene and butan-1-ol: (N-hydroxyethylpyridinium tricyanomethanide, [C2OHPy][TCM] + thiophene), or (N-hydroxypropylpyridinium tricyanomethanide, [C3OHPy][TCM] + thiophene) or (N-benzyl-N-dodecyl-N,N-dimethylammonium bis(trifluoromethyl-sulfonyl)imide, [N1,1,12,Bz][NTf2] + butan-1-ol). These systems exhibit mainly negative molar excess volumes, VE in systems with thiophene and slightly positive and negative in the system with butan-1-ol. The Redlich – Kister equation was used for correlation of the liquid density, excess molar volumes and dynamic viscosity deviations as a function of temperature and composition, whereas Vogel-Fulcher-Tammann, VFT was used for describing the temperature dependence of dynamic viscosities. Very often, the high pressure is proposed for new technologies of separation and that is why we present the influence of pressure on the density of [C2OHPy][TCM], [C3OHPy][TCM] and [N1,1,12,Bz][NTf2] in a wide range of temperature from T = (298.15–348.15) K and pressure from p = (0.1–40) MPa. The Tait equation was used for the correlation of density as a function of mole fraction, temperature, and pressure. The influence of pressure for ILs, is as usual, not significant. The thermal expansion coefficient α and isothermal compressibility coefficient κ as a function of temperature T, and pressure p are presented. Herein, surface tension of new ILs was measured for the collection of physico-chemical properties of the ILs used in this work. Also, the octan-1-ol/water, Kow partition coefficients in ternary liquid-liquid phase equilibrium were determined for the systems {[C2OHPy][TCM], or [C3OHPy][TCM] (1) + octan-1-ol (2) + water (3)} at temperature T = 298.15 K and the discussion of the biocompatibility of the used ILs is presented.

Modeling of pvT behavior of semi-crystalline polymer based on the two-domain Tait equation of state for injection molding

The absence of the influence of cooling rate in the pressure–specific volume–temperature (pvT) models makes them inadequate in the process simulation of injection molding. This work derived new equations of state (EoS) from the traditional two-domain Tait model that describes the pvT behavior of semi-crystalline polymers in both molten and solid state. Especially, the cooling rate was considered. The second aim of this work is to solve the discontinuity problem of the traditional two-domain Tait model. A continuous two-domain Tait EoS was developed. A good agreement between the calculations and experimental results was achieved. The new pvT models were successfully applied in the prediction of specific volume as a function of temperature, pressure and cooling rate during injection molding. A higher value of the specific volume was predicted when the cooling rate was considered. Combining the continuous two-domain Tait EoS, more accurate results can be obtained especially at the transition phase from melt to solid of the semi-crystalline polymer.

A Pressure-based Compressible-Liquid Flow Model for Computation of Instantaneous Valve Closure in Pipes

A generalized, pressure-based compressible-liquid flow model is proposed for the isothermal low-speed flow of liquids. The flow model incorporates dedicated equations of state for liquids into the pressure-based solvers to simulate the compressible effects in liquids. The model’s capability to handle compressible liquid flow problems is evaluated against the well-established density-based water-hammer model. The isothermal flow problem of an instantaneous valve closure in an irrigation pipe and the associated flow transients are numerically solved using the proposed model by employing the Tait equation of state in conjunction with a segregated pressure-based solver algorithm. The transient flow problem is solved for a range of operating pressures, and the surge in pressure and variation of other flow properties and their interrelations are studied in detail. The proposed model could successfully capture the entire physics of the problem, including the compressible modeling of the liquid involved and could produce high accuracy numerical results. The results suggest that the pressure-based compressible-liquid flow model is a reliable and computationally inexpensive numerical tool for isothermal low-speed compressible liquid flow computations.

High-pressure densities of n-decane+o-xylene mixtures: Measurement and modelling

In this work, the liquid densities of n-decane/o-xylene mixtures were measured at four different component proportions, the mole fractions of n-decane are 0.2001, 0.4065, 0.5835, and 0.8041, respectively. The measurements were performed based on the high-pressure vibrating-tube densimeter, and the measured temperatures were from 283.15 K to 363.15 K and the pressures were from 0.1 MPa to 60 MPa. The Tammann-Tait equation was used to correlate the experimental densities, and the simplified Perturbed Associating Fluid Theory (sPC-SAFT) and Cubic Plus Associating (CPA) equation were used to predict the high-pressure densities of the mixtures with the binary interaction parameters. Derived properties, including isothermal compressibilities and isobaric thermal expansivities, were calculated from Tait equation, sPC-SAFT equation, and CPA equation, respectively. In addition, excess molar volumes of the considered mixtures were obtained and the effects of pressure and temperature on the excess molar volume were discussed.

Characterization and determination of thermodynamic properties of waste cooking oil biodiesel: Experimental, correlation and modeling density over a wide temperature range up to 393.15 and pressure up to 140 MPa

Biodiesel can be described as mono-alkyl esters of long chain fatty acids, derived from a vegetable oil in reactions with alcohols. The physical properties of biodiesel are largely determined by the length of the hydrocarbon chain, the degree of unsaturation, and the effect of molecular packing, while the calculation of the thermodynamic properties of biodiesel is difficult in reason they are high molar weight components and complex structures. The density data (136 points) for waste cooking oil biodiesel at several temperatures between 298.15 K and 393.15 K and pressures up to 140 MPa are reported. The chemical constituents were analyzed by GC/MS, 1HNMR and 13CNMR in order to determine the fatty acid composition of waste cooking oil biodiesel. An Anton Paar vibrating tube densimeter, calibrated with an uncertainty of ±0.7 kg m−3 was used to perform these measurements between 298.15 K and 393.15 K. In this work, the density experimental data of biodiesel were correlated with the Tait equation of state (EoS), an empirical equation widely used for high pressure density correlation. The PC-SAFT equation of state using group contribution methods was employed for the calculation of parameters and prediction of biodiesel densities.The statistical parameters of empirical Tait-correlation are much lower than the experimental uncertainty of the density measurements with the AAD (0.01%), MD (0.04%), Bias (7.1 × 10−4%), and standard deviation, σ (1.5 × 10−4g.cm−3), and from PC-SAFT, density of sample was reasonably well predicted with an average relative deviation 0.083%In addition, the isobaric thermal expansivity and the isothermal compressibility have been derived from the Tait-like equation.