Isentropic Bulk Modulus Research Articles

Oscillating motions of slug flow in capillary tubes

A mathematical model describing the oscillation characteristics of slug flow in a capillary tube is presented. The model considers the vapor bubble as the gas spring for the oscillating motions in a capillary tube including effects of capillary force, gas spring constant, dimensions, gravitational force, and initial pressure distribution of the working fluid. Numerical results indicate that the isentropic bulk modulus generates stronger oscillations than the isothermal bulk modulus. While it demonstrates that the capillary tube diameter, bubble size, and unit cell numbers determine the oscillation, the capillary force, gravitational force, and initial pressure distribution of the working fluid significantly affect the frequency and amplitude of oscillating motion in the capillary tube.

Speed of sound and isentropic bulk modulus of alkyl monoesters at elevated temperatures and pressures

AbstractBiodiesel is a biodegradable, sulfur‐free, oxygenated, and renewable alternative diesel fuel consisting of the alkyl monoesters of FA from vegetable oils and animal fats. Biodiesel can be used in existing diesel engines without significant modifications. However, differences in physical properties between biodiesel and petroleum‐based diesel fuel may change the engine's fuel injection timing and combustion characteristics. These altered physical and chemical properties also may cause the exhaust emissions and performance to differ from the optimized settings chosen by the engine manufacturer. In particular, the density, speed of sound, and isentropic bulk modulus have a significant effect on the fuel injection system and combustion. The objective of this study was to measure these three properties for biodiesel (and the pure esters that are the constituents of biodiesel) at temperatures from 20 to 100°C and at pressures from atmospheric to 32.5 MPa. Ten different biodiesel fuels, 16 different pure FA esters, three hydrocarbons, and one diesel fuel were tested. The measured values of density, speed of sound, and isentropic bulk modulus are presented. Correlations between pressure and temperature are demonstrated. Speed of sound and isentropic bulk modulus tend to increase as the degree of unsaturation increases and as the chain length increases. However, density increased with shorter chain length and decreased with saturation.

Effect of temperature and pressure on the speed of sound and isentropic bulk modulus of mixtures of biodiesel and diesel fuel

AbstractThe density and speed of sound of blends of biodiesel with No. 2 and No. 1 diesel fuels were measured from atmospheric pressure to 32.46 MPa at temperatures of 20 and 40°C. The isentropic bulk modulus was calculated from these quantities. The results show that the density and isentropic bulk modulus can be accurately modeled as having a linear variation with blend percentage. Speed of sound is better correlated by a second‐order equation. Correlation equations are given and a blending rule is developed that allows the density, speed of sound, and isentropic bulk modulus of blends to be calculated from the properties of the biodiesel and diesel fuel.

Single-crystal elastic constants of fluorite (CaF 2 ) to 9.3 GPa

The second-order elastic constants of CaF2 (fluorite) have been determined by Brillouin scattering to 9.3 GPa at 300 K. Acoustic velocities have been measured in the (111) plane and inverted to simultaneously obtain the elastic constants and the orientation of the crystal. A notable feature of the present inversion is that only the density at ambient condition was used in the inversion. We obtain high-pressure densities directly from Brillouin data by conversion to isothermal conditions and iterative integration of the compression curve. The pressure derivative of the isentropic bulk modulus and of the shear modulus determined in this study are 4.78 ± 0.13 and 1.08 ± 0.07, which differ from previous low-pressure ultrasonic elasticity measurements. The pressure derivative of the isothermal bulk modulus is 4.83 ± 0.13, 8% lower than the value from static compression, and its uncertainty is lower by a factor of 3. The elastic constants of fluorite increase almost linearly with pressure over the whole investigated pressure range. However, at P ≥ 9 GPa, C 11 and C 12 show a subtle structure in their pressure dependence while C 44 does not. The behavior of the elastic constants of fluorite in the 9–9.3 GPa pressure range is probably affected by the onset of a high-pressure structural transition to a lower symmetry phase (α-PbCl2 type). A single-crystal Raman scattering experiment performed in parallel to the Brillouin measurements shows the appearance of new features at 8.7 GPa. The new features are continuously observed to 49.2 GPa, confirming that the orthorhombic high-pressure phase is stable along the whole investigated pressure range, in agreement with a previous X-ray diffraction study of CaF2 to 45 GPa. The high-pressure elasticity data in combination with room-pressure values from previous studies allowed us to determine an independent room-temperature compression curve of fluorite. The new compression curve yields a maximum discrepancy of 0.05 GPa at 9.5 GPa with respect to that derived from static compression by Angel (1993). This comparison suggests that the accuracy of the fluorite pressure scale is better than 1% over the 0–9 GPa pressure range.

The speed of sound and isentropic bulk modulus of biodiesel at 21°C from atmospheric pressure to 35 MPa

AbstractBiodiesel, an alternative diesel fuel consisting of the alkyl monoesters of fatty acids from vegetable oils and animal fats, can be used in existing diesel engines without modification. However, property changes associated with the differences in chemical structure between biodiesel and petroleumbased diesel fuel may change the engine's injection timing. These injection timing changes can change the exhaust emissions and performance from the optimized settings chosen by the engine manufacturer. This study presents the results of measurements of the speed of sound and the isentropic bulk modulus for methyl and ethyl esters of fatty acids from soybean oil and compares them with No. 1 and No. 2 diesel fuel. Data are presented at 21±1°C and for pressures from atmospheric to 34.74 MPa. The results indicate that the speed of sound and bulk modulus of the monoesters of soybean oil are higher than those for diesel fuel and these can cause changes in the fuel injection timing of diesel engines. Linear equations were used to fit the data as a function of pressure, and the correlation constants are given.

Composition and temperature of Earth's inner core

We compare a theoretical prediction of the equation of state of iron at high pressures and temperatures to the properties of the Earth's inner core. The theoretical result is based on a first principles treatment of the static pressure and the pressure due to thermal excitation of electrons and an approximate ab initio (cell model) treatment of the Vibrational pressure. The density of iron is found to be greater than that of the inner core even for unrealistically high temperatures of 8000 K. The isentropic bulk modulus of iron is found to be consistent with that of the inner core over a wide range of temperatures (4000–8000 K). We conclude on the basis of these comparisons that the inner core contains a substantial fraction of elements lighter than iron. Assuming ideal solutions, we find the temperature and light component mass fraction required to simultaneously match the density and bulk modulus of the inner core. For a temperature of 7000 K, 1 wt % O as FeO, satisfies the inner core observations. The temperature and mass fraction of S required depend on whether S is included as pyrite (2 wt % S, 5500 K) or as Fe0.9S (>8 wt % S, <3500 K). On the basis of this result and empirical mixing rules for Fe‐O solutions, we argue that Fe‐light element solid solutions at inner core conditions may be significantly nonideal. We derive expressions for the properties of nonideal multicomponent solutions that are valid in the limit of small amounts of impurities. These lead to general results for the properties of the alloy fraction that are required by comparisons of our equation of state of iron with seismological models.

An equation of state for liquid iron and implications for the Earth's core

An equation of state is presented for liquid iron based on published ultrasonic, thermal expansion, and enthalpy data at 1 bar and on pulse‐heating and shock wave compression and sound speed data up to 10 Mbar. The equation of state parameters, centered at 1 bar and 1811 K (the normal melting point of iron), are density, ρ0 = 7019 kg/m3, isentropic bulk modulus, KS0 = 109.7 GPa, and the first‐and second‐pressure derivatives of KS, K′S0 = 4.66 and K″S0 = −0.043 GPa−1. A parameterization of the Grüneisen parameter γ as a function of density ρ and specific internal energy E is γ = γ0 + γ′(ρ/ρ0)n(E ‐ E0) where γ0 = 1.735, γ′ = −0.130 kg/MJ, n = −1.87, and E0 is the internal energy of the liquid at 1 bar and 1811 K. The model gives the temperature dependence of γ at constant volume as (∂γ/∂T)v|1bar,1811K = −8.4 × 10−5 K−1. The constant volume specific heat of liquid Fe at core conditions is 4.0–4.5 R. The model gives excellent agreement with measured temperatures of Fe under shock compression. Comparison with a preliminary reference Earth model indicates that the light component of the core does not significantly affect the magnitude of the isentropic bulk modulus of liquid Fe but does decrease its pressure derivative by ∼10%. Pure liquid Fe is 3–6% more dense than the inner core, supporting the presence of several percent of light elements in the inner core.

Shock wave equation of state of muscovite

Shock wave data to provide an equation of state of muscovite (initial density: 2.835 g/cm3) were determined up to a pressure of 141 GPa. The shock velocity (Us) versus particle velocity (Up) data are fit with a single linear relationship: Us=4.62(±0.12) +1.27(±0.04)Up (km/s). Third‐order Birch‐Murnaghan equation of state parameters (isentropic bulk modulus and isentropic pressure derivative of bulk modulus) are Kos=52±4GPa and K'os=3.2±0.3. The pressure‐temperature relation along the Hugoniot suggests that muscovite may dehydrate to KAlSi3O8 (hollandite), corundum, and water, with a small volume change, above 80 GPa. Thermodynamic calculations of the equilibrium pressure for the dehydration yields a significantly lower value. Observed unloading paths from shock pressures up to about 80 GPa are steeper in a density‐pressure plane than the Hugoniot and become shallower with increasing shock pressure above that pressure. The changing slope may indicate that devolatilization occurs during unloading above 80 GPa. The present equation of state data for muscovite are compared with results of previously reported recovery experiments.

In-Situ Measurement of the Wavespeed and Bulk Modulus in Hydraulic Lines

The dynamic performance of a hydraulic system is dependent on the fluid bulk modulus, which can be affected by several factors including air release and pipe compliance. A technique is described for the in situ measurement of the speed of sound from which the bulk modulus may be evaluated. It entails analysis of pressure ripple recorded at three locations in the hydraulic line. Measured values of the bulk modulus in high-pressure pipes were found to be close to the isentropic tangent bulk modulus predicted from the fluid manufacturer's data.

The equation of state of a molten komatiite: 1 Shock wave compression to 36 GPa

The equation of state (EOS) of an initially molten (1550°C) komatiite (27 wt % MgO) was determined in the 5–36 GPa pressure range via shock wave compression. Shock wave velocityUsand particle velocityUp(kilometers/second) follow the linear relationshipUs= 3.13(±0.03) + 1.47(±0.03)Up. Based on a calculated density at 1550°C, 0 bar of 2.745±0.005 g/cm3, thisUs‐Uprelationship gives the isentropic bulk modulusKS= 27.0 ± 0.6 GPa, and its first and second isentropic pressure derivatives,K′S= 4.9 ±0.1 andK″S= −0.109 ± 0.003 GPa−1. The calculated liquidus compression curve agrees within error with the static compression results of Agee and Walker (1988) to 6 GPa but is less dense than their extrapolated values at higher pressures. We determine that olivine (Fo94) will be neutrally buoyant in komatiitic melt of the composition that we studied near 8.2 GPa. Clinopyroxene would also be neutrally buoyant near this pressure. Liquidus garnet‐majorite may be less dense than this komatiitic liquid in the 20–24 GPa interval; however, pyropic‐garnet and perovskite phases are denser than this komatiitic liquid in their respective liquidus pressure intervals to 36 GPa. Liquidus perovskite may be neutrally buoyant near 70 GPa. At 40 GPa, the density of shock‐compressed molten komatiite would be approximately equal to the calculated density of an equivalent mixture of dense solid oxide components. This observation supports the model of Rigden et al. (1989) for compressibilities of liquid oxide components. Using their theoretical EOS for liquid forsterite and fayalite, we calculate the densities of a spectrum of melts from basaltic through peridotitic that are related to the experimentally studied komatiitic liquid by addition or subtraction of olivine. At low pressure, olivine fractionation lowers the density of basic magmas, but above 13–14 GPa this trend is reversed. All of these basic to ultrabasic liquids are predicted to have similar densities at 13–14 GPa, and this density is approximately equal to the density of the bulk (preliminary reference Earth model) mantle in this pressure range. This suggests that melts derived from a peridotitic mantle may be inhibited from ascending from depths greater than 400 km.

A Hugoniot theory for solid and powder mixtures

A model is presented from which one can calculate the Hugoniot of solid and porous two-component mixtures up to moderate pressures using only static thermodynamic properties of the components. The model does not presuppose either the relative magnitude of the thermal and elastic energies or temperature equilibrium between the two components. It is shown that for a mixture, the conservation equations define a Hugoniot surface and that the ratio of the thermal energy of the components determines where the shocked state of the mixture lies on this surface. This ratio, which may strongly affect shock-initiated chemical reactions and the properties of consolidated powder mixtures, is found to have only a minor effect on the Hugoniot of a mixture. It is also found that the Hugoniot of solids and solid mixtures is sensitive to the pressure derivative of the isentropic bulk modulus at constant entropy.

Pyrite: Shock compression, isentropic release, and composition of the Earth's core

New shock wave data (to 180 GPa) for pyrite (FeS2) shocked along (001) demonstrate that this mineral, in contrast to other sulfides and oxides, does not undergo a major pressure‐induced phase change over the entire pressure range (320 GPa) now explored. (This is probably so because of the initial, low‐spin 3‐d, orbital configuration of Fe+2). The primary evidence which indicates that a large phase change does not occur is the approximate agreement of the shock velocity when extrapolated to zero particle velocity, 5.4 km/s, with the expected zero‐pressure bulk sound speed of pyrite (5.36 to 5.43 km/s on the basis of previous ultrasonic data). Pyrite displays a prominent elastic shock (or Hugoniot elastic limit) of 8 ± 1 GPa. The velocity of the elastic shock approaches 8.72 km/s with decreasing shock pressure, the longitudinal elastic wave velocity. As shock pressure increases, the elastic shock velocity approaches 9.05 km/s and the elastic shock becomes overdriven for shock pressures greater than about 120 GPa. Analysis of release isentrope data obtained via the pressure‐particle velocity buffer method indicates that buffer particle velocities in all experiments are from 1.7% to 20% greater than expected for a Grüneisen ratio given by 1.56 (V/Vo)1.0. This discrepancy appears to result from volume increases upon pressure release of 0.04% to 4.5% which may result from shock‐induced partial melting. The normalized pressure, finite‐strain formalism for reducing Hugoniot data is extended to take into account initial porosity and shock‐induced phase transitions. A least squares fit to the present and previous shock data for pyrite yields an isentropic bulk modulus, Ks, of 162 ± 9 GPa and a value of dKs/dP = 4.7 ± 0.3. This is close to the 145 ± 3 GPa bulk modulus observed ultrasonically. If the slight discrepancy in zero‐pressure modulus is taken into account in the normalized pressure finite‐strain formalism, a zero‐pressure density of the shock‐induced high‐pressure phase having a density some 2% to 3% less than pyrite is inferred to occur in the high‐pressure shocked state. We suggest from this result, the release isentrope results, and limited phase diagram data that the Hugoniot states probably correspond to material which is partially to completely melted. Using the above derived equation of state and previous shock wave data for iron, both the seismologically determined density and bulk modulus distribution in the outer core are fit to models with various temperature distributions and varying weight percent sulfur. Good agreement between the shock wave derived equation of state and the density/bulk modulus relations of the liquid outer core are obtained for temperatures of ∼3000 K at the core/mantle boundary extending to 4400 K at the outer core‐inner core boundary. For this thermal model a calculated sulfur content of 11 ± 2% is obtained.

Shock wave compression of single‐crystal forsterite

Hugoniot equation of state measurements have been performed on pure synthetic single‐crystal forsterite (Mg2SiO4) in the pressure range 70–160 GPa (0.7–1.6 Mbar). These and earlier data for polycrystalline forsterite are compared with theoretical Hugoniots for the assemblages 2MgO (rocksalt) + SiO2 (stishovite) and MgO (rocksalt) + MgSiO3 (perovskite). The densities attained by single‐crystal forsterite at pressures in excess of 120 GPa are greater than those expected in the event of shock‐induced transformation to the isochemical oxide mixture. A similar test of the hypothesis of shock‐induced transformation to the perovskite‐bearing assemblage is sensitive to the choice of MgSiO3 (perovskite) bulk modulus. Recent static compression measurements of Yagi et al. (1978) yield a K0T of 286 GPa (for K0T′ = 5), which, along with other elastic and thermodynamic parameters, suggests that shocked forsterite may be more dense than the perovskite‐bearing assemblage. Crystalline phases of up to 5% greater zeropressure density or equally dense short‐range‐order‐only phases may well be involved. Alternatively, the use of an isentropic bulk modulus of 250 GPa (estimated by Liebermann et al., 1977) for MgSiO3 (perovskite) allows consistency between the data and the calculated MgO + MgSiO3 (perovskite) Hugoniot for a reasonable choice (∼3.8) of K0s′ for the latter phase. The new forsterite data along with high‐pressure Hugoniot data for other olivines and olivinitic rocks define a smooth isobaric variation of Hugoniot density with composition. It is shown that an estimated pyrolite (Ringwood, 1975) Hugoniot density of 5.31 g/cm3 at 120 GPa is ∼2% less dense than inferred from typical lower mantle density profiles.

High-pressure phase transformations and compressions of ilmenite and rutile, I. Experimental results

Abstract Natural ilmenite (Fe,Mg)TiO3 has been found to transform to the perovskite structure and then to disproportionate into its component oxides, (Fe,Mg)O plus a cubic phase of TiO2, at loading pressures of 140 and 250 kbar respectively, and at temperatures of 1,400 to 1,800°C. Samples were compressed in a diamond-anvil press and heated by irradiation with a YAG laser. The lattice parameters of the perovskite phase of (Fe,Mg)TiO3 at room temperature and 1 bar are a 0 = 4.471 ± 0.004, b 0 = 5.753 ± 0.005, and c 0 = 7.429 ± 0.006 A with 4 molecules per cell. The zero-pressure volume change is 8.0% for the ilmenite-perovskite transition, 13.3% for the perovskite-mixed-oxides transition, and 20.2% for the ilmenite-mixed-oxides transition. The cubic phase of TiO2 can be indexed on the basis of space group Fm3m with Z = 4 and a 0 = 4.455 ± 0.008 A at room temperature and 1 bar, which corresponds to a decrease in zero-pressure volume of 29.2% for the rutile-cubic-phase transition. An isentropic bulk modulus at zero pressure of 5.75 ± 0.30 Mbar and a pressure derivative greater than 8 were calculated for the high-pressure cubic phase. The calculated bulk modulus for the mixture of (Fe,Mg)O and cubic TiO2 is 2.48 ± 0.25 Mbar. All the phase transformations, the calculated lattice parameters, and the bulk moduli observed in this study are in good agreement with published shock-Hugoniot data for ilmenite and rutile.

On the Compression of Stishovite

Summary Currently available shock-wave compression data for single-crystal α-quartz and pore-free quartzite have been used to determine values for the isentropic bulk modulus of stishovite and its first pressure derivative. The shock-wave data reduction scheme makes use of the observed linearity in the shock-velocity-particle-velocity field over the pressure range 0.4–2.0 Mbar and the first-order Murnaghan equation of state. In addition, an accurate form of the temperature independent Gruneisen parameter (γ), consistent with the linear Us–up relation, has been used in the reduction of the fundamental shock-wave data to the metastable Hugoniot. The temperature derivative of the isentropic bulk modulus is provided by a least-square fit of the γ relation to available Hugoniot data directly measured on porous and fused-quartz samples in the high-pressure regime. The pertinent results are: KS= 3.35±0.19 Mbar; (∂KS/∂P)T= 5.5±0.6; and (∂KS/∂T)p= (−0.35±0.08)× 10−3 Mbar/°K. These results are consistent with recent static-compression and ultrasonic measurements on stishovite and are related systematically to corresponding data for isostructural GeO2 and TiO2. Comparison of the present results with recent shock-wave analyses by other investigators suggests that the first-order Murnaghan form of the equation of state is more appropriate than the first-order Birch equation in reproducing the compression of stishovite in the zero to 2 Mbar pressure range. An evaluation of the composition of the lower mantle in terms of the fundamental oxides, FeO, MgO, and SiO2, using the present results for the elastic properties of stishovite, support marginally the conclusions of previous investigators regarding enrichment of FeO and SiO2 relative to the upper mantle; however, such an interpretation is predicated on the mixed oxide assumption and should be considered in relation to alternative models.

On the accuracy of the Wachtman-Anderson Relation

The Wachtman-Anderson relation has consistently shown itself to be an accurate interpolation equation for the experimental isobaric relationship between the isentropic bulk (Bs) or Young's (Ys) modulus and the temperature T. Here the equation is analyzed in some detail, and it is shown that, although the equation is an approximation, a high degree of consistency is maintained between the relation itself and the Gruneisen equations from which it has been derived. In particular it is shown that the relation may be used to obtain reasonable estimates of the derivative (∂2Bs/∂T2) along the zero-pressure isobar.

Experimental Verification at High Pressure of the Relationship between Compression, Density and Sonic Velocity

THE equation relating density, compressibility and sonic velocity can be expressed as where C is the velocity of sonic wave propagation, V is the volume of unit mass, Ks is the isentropic bulk modulus = − V (∂P/∂V)s: and s is entropy.