Single Equation Of State Research Articles

Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law

In this paper, we are interested in modelling the flow of the coolant (water) in a nuclear reactor core. To this end, we use a monodimensional low Mach number model supplemented with the stiffened gas law. We take into account potential phase transitions by a single equation of state which describes both pure and mixture phases. In some particular cases, we give analytical steady and/or unsteady solutions which provide qualitative information about the flow. In the second part of the paper, we introduce two variants of a numerical scheme based on the method of characteristics to simulate this model. We study and verify numerically the properties of these schemes. We finally present numerical simulations of a loss of flow accident (LOFA) induced by a coolant pump trip event.

Structure and density of molten fayalite at high pressure

The structure of molten fayalite was studied up to 7.5GPa by means of in situ energy-dispersive X-ray diffraction. The pressure-range studied covers the fayalite–spinel–liquid triple point at 6.2GPa. For pure molten fayalite, Fe–O coordination increases gradually from 4.8(2) at ambient pressure (P) to 7.2(3) at 7.5GPa. Compressibility of the melt is derived from the extrapolation of the structure factor to q=0Å−1, enabling the determination of density as a function of P with an unprecedented P-resolution. This is a promising method to extract the equation of state of non-crystalline materials at moderate P. The link between observed structural changes and density increase and the fact that structural changes occur over a broad but limited P-range in silicate melts implies that the equation of state should not be extrapolated at P-values higher than obtained in measurements, and that a single equation of state cannot accurately describe the density evolution over the whole terrestrial mantle P–temperature (T) range. Fe-rich melts are expected to have a higher densification rate than their Mg counterparts in the 0–10GPa range due to the increase of Fe–O coordination number. As a consequence, Fe-rich melts are more likely to be trapped at depth.

Equation of state of silicate melts with densified intermediate-range order at the pressure condition of the Earth’s deep upper mantle

The knowledge from the compression behavior of densified SiO2 glass suggests that SiO2 melt may behave as a single phase having a densified network structure (intermediate-range order) at the pressure condition of the Earth’s deep upper mantle, including the transition zone. A simple and easy-to-use equation of state of silicate melts which is applicable to a wide range of chemical composition at the pressure condition of the deep upper mantle is proposed based on the assumption that SiO2 component is in its densified state (or phase). The equation of state proposed in this study is consistent with all the available density data of silicate melts with an SiO2 content of about 35–55 mol% measured with large-volume presses at pressures between 8 and 22 GPa. The equations of state in previous studies differ considerably from each other. The main reason for the discrepancies seems to be that the compression behavior of multiple states (or phases) of silicate melts has been described in most cases with a single equation of state. It is necessary to consider that silicate melts are in their densified states (or phases) in the deep upper mantle.

Evolution of the cosmological horizons in a concordance universe

The particle and event horizons are widely known and studied concepts, but the study of their properties, in particular their evolution, have only been done so far considering a single state equation in a decelerating universe. This paper is the first of two where we study this problem from a general point of view. Specifically, this paper is devoted to the study of the evolution of these cosmological horizons in an accelerated universe with two state equations, cosmological constant and dust. We have obtained simple expressions in terms of their respective recession velocities that generalize the previous results for one state equation only. With the equations of state considered, it is proved that both velocities remain always positive.

THE EFFECT OF CLIMATE CHANGE ON WETLANDS AND WATERFOWL IN WESTERN CANADA: INCORPORATING CROPPING DECISIONS INTO A BIOECONOMIC MODEL

Abstract We extend an earlier bioeconomic model of optimal duck harvest and wetland retention in the Prairie Pothole Region of Western Canada to include cropping decisions. Instead of a single state equation, the model has two state equations representing the population dynamics of ducks and the amount of wetlands. We use the model to estimate the impact of climate change on wetlands and waterfowl, including direct climate effects as well as land use change due to biofuel policies aimed at mitigating climate change. The model predicts that climate change will reduce wetlands by 37–56% from historic levels. Land use change due to biofuel policies is expected to reduce wetlands by between 35% and 45% from historic levels, whereas direct climate effects will range from a reduction of 2–11%, depending on the future climate scenario. This result indicates that models that neglect the effect of land use changes underestimate the effect of climate change on wetlands. Further, wetlands loss is geographically heterogeneous, with losses being the largest in Saskatchewan.

PREDICTION OF PHYSICAL & THERMODYNAMICAL PROPERTIES FOR BINARY SYSTEMS USING EQUATION OF STATE

A single equation of state (EOS) such as Soave – Redlich Kwong EOS can accurately describe both the liquid and vapour phase, therefore it is used for binary systems to predict some physical and thermodynamical properties. Two methods, which are Soave- Redlich Kwong and Generalized Compressibility Factor Correlation are compared and adopted for the cubic equation of state to calculate molar volume, density, viscosity, thermal conductivity, specific heat and compressibility factor.
 In this paper a computer program is developed requiring critical properties to perform these calculations. The results are compared with some available literature data, and we find that the computer programs are shown to be adequately reliable for this purpose, with deviation in some properties equal to (3.6%) as other predictive programs and procedures. Also from this comparison we notice that the Generalized Compressibility Factor Correlation method is better and more general than the Soave- Redlich Kwong

Analysis of melting of pyrope compound at high pressures

The pressure–volume–temperature relationships have been studied for the pyrope mineral. Values of volumes at simultaneously elevated temperatures and pressures have been evaluated using the Rydberg-Vinet equation of state for isothermal compression and the Singh–Suzuki equation for isobaric thermal expansion. It has been found that the results obtained from these two conventional equations can be reproduced very well by a single equation of state which covers both temperature and pressure changes in solids namely the Hartmann equation of state. Values of volumes at melting temperatures corresponding to elevated pressures have been obtained and found to compare well with the experimental values. It is also found that there exists a simple linear relationship between relative change in melting volume and melting temperature in case of pyrope at different pressures. This is supported by the earlier findings of Anderson in case of forsterite mineral.

New accurate elastic parameters for the forsterite-fayalite solid solution

Three natural olivines with Fo80Fa20, Fo71Fa29, and Fo62Fa38 compositions were investigated in situ at high pressure by single-crystal X‑ray diffraction using a diamond-anvil cell up to -8 GPa at room temperature. The bulk modulus, KT0, and its first pressure derivative, K′, do not show any significant variation among the compositions investigated and, using the data on a further sample with Fo92Fa8 composition recently investigated in the same laboratory and using the same experimental technique, we obtain, for the first time, a single equation of state for the entire Fo92Fa8-Fo62Fa38 compositional range. The equation has the following coefficients: KT0 = 124.7(9) GPa and K′ = 5.3(3) and can be used for thermodynamic calculations involving the most common mantle olivine compositions.

Critical behavior in single-crystallineLa0.67Sr0.33CoO3

The critical behavior of ${\text{La}}_{0.67}{\text{Sr}}_{0.33}{\text{CoO}}_{3}$ single crystal has been investigated from the bulk magnetization measurements around the Curie temperature $({T}_{C})$. The detailed analysis of the magnetization indicates the occurrence of a continuous ferromagnetic to paramagnetic phase transition at 223.0 K. The critical exponents $\ensuremath{\beta}=0.361\ifmmode\pm\else\textpm\fi{}0.007$, $\ensuremath{\gamma}=1.31\ifmmode\pm\else\textpm\fi{}0.001$, and $\ensuremath{\delta}=4.64\ifmmode\pm\else\textpm\fi{}0.01$ characterizing this second order phase transition, have been estimated using different techniques such as the Kouvel-Fisher plot, the Arrott-Noaks plot, and critical isotherm analysis. With these values of ${T}_{C}$, $\ensuremath{\beta}$, and $\ensuremath{\gamma}$, one can scale the magnetization below and above ${T}_{C}$ following a single equation of state. The consistency in the values of the critical exponents obtained from different methods and the well-obeyed scaling behavior confirm that the calculated exponents are unambiguous and purely intrinsic to the system. These values of the exponents match well with those theoretically predicted for the three-dimensional Heisenberg model with nearest-neighbor interaction.

Temperature- and pressure-dependent lattice behaviour ofRbFe(MoO4)2

Trigonal RbFe(MoO4)2 is a quasi-two-dimensional antiferromagnet on a triangular lattice belowTN = 3.8 K, The crystal exhibits also a structural phase transition atTc = 190 K related to symmetry change from to . We present the temperature- and pressure-dependent characteristics of this material inthe context of ambiguous opinions on the symmetry and crystal properties belowTc. A single-crystal x-ray diffraction shows that the temperature-dependent evolution of the unitcell in the range 100–300 K is strongly anisotropic with markedly discontinuous changes atTc. The transition is connected with a spontaneous strain developing in effect of the volumedecrease. The structure releases the strain by rotation of corner-sharing rigidMoO4 andFeO6 polyhedrain the (a,b) basal plane. The temperature dependence of the IR vibrational wavenumbers exhibits weak changesnear Tc, which are consistent with the symmetry transformation from to . High-pressure x-ray powder diffraction indicates that the material is extremely softbut with some stiffening at high pressure. The zero-pressure bulk modulus isB0 = 7.9(6) GPa and the pressurederivative is B0′ = 10(1). The compression curve can be described by a single equation of state, corresponding tothe trigonal cell, up to 5 GPa. An amorphization appearing above 5 GPa and increasinggradually on further pressure increase suggests the thermodynamic instability of thehigh-pressure structure.

Pressure Induced Critical Behavior of Ferromagnetic Phase Transition in Sm-Nd-Sr Manganites

We report on the hydrostatic pressure dependence of the order of ferromagnetic (FM) to paramagnetic (PM) phase transition in a (Sm(0.7)Nd(0.3))(0.52)Sr(0.48)MnO(3) single crystal. At ambient pressure, the system undergoes a first-order FM-PM phase transition at 146 K. The application of pressure increases the T(C), suppresses the hysteresis width, and thus makes the transition second order. We have analyzed the critical behavior associated with the second-order FM-PM transition in the presence of an external pressure (12.1 kbar) and obtained the critical exponents beta=0.358, gamma=1.297, and delta=4.536, which are close to those predicted for the three-dimensional Heisenberg system. Using these values of beta, gamma, and T(C) ( approximately 176 K), one can scale the magnetization data below and above T(C) following a single equation of state.

A vapor–liquid phase equilibrium model for binary CO 2–H 2O and CH 4–H 2O systems above 523 K for application to fluid inclusions

Accurate prediction of both volumetric and vapor–liquid phase equilibria of binary CO 2–H 2O and CH 4–H 2O mixtures with a single equation of state proves to be difficult. In this study we use an activity–fugacity model to predict the vapor–liquid phase equilibria above 523 K and adapt a Helmoholtz model to calculate volumetric properties of these two binary systems. The average deviations of water content in the vapor phase from experimental data are 3.25% and 3.19% for the CO 2–H 2O and CH 4–H 2O mixtures, respectively, and the average deviations of gas solubility in liquid phase from experimental data are 4.29% and 3.50%, respectively. The model can find wide applications, and an example is given for the analysis of fluid inclusions in geochemistry.

Critical parameters near the ferromagnetic–paramagnetic phase transition in La 0.7A0.3(Mn 1−xb x)O 3 ( A=Sr; B=Ti and Al; x=0.0 and 0.05) compounds

The critical parameters provide important information concerning the interaction mechanisms near the paramagnetic-to-ferromagnetic transition. In this paper, we present a thorough study for the critical behavior of La 0.7 A 0.3(Mn 1− xB x )O 3 ( A=Sr; B=Ti and Al; x=0.0 and 0.05) polycrystalline samples near ferromagnetic–paramagnetic phase transition temperature by analyzing isothermal magnetization data. We have analyzed our dc-magnetization data near the transition temperature with the help of the modified Arrot plot, Kouvel–Fisher method. We have determined the critical temperature T C and the critical parameters β, γ and δ. With the values of T C, β and γ, we plot M×(1− T/ T C) − β vs. H×(1− T/ T C) − γ . All the data collapse on one of the two curves. This suggests that the data below and above T C obey scaling, following a single equation of state. Critical parameters for x=0 and x Ti=0.05 samples are between those predicted for a 3D-Heisenberg model and mean-field theory and for x Al=0.05 samples the values obtained for the critical parameters are close to those predicted by the mean-field theory.

The MgSiO3system at high pressure: Thermodynamic properties of perovskite, postperovskite, and melt from global inversion of shock and static compression data

We present new equation‐of‐state (EoS) data acquired by shock loading to pressures up to 245 GPa on both low‐density samples (MgSiO3glass) and high‐density, polycrystalline aggregates (MgSiO3perovskite + majorite). The latter samples were synthesized using a large‐volume press. Modeling indicates that these materials transform to perovskite, postperovskite, and/or melt with increasing pressure on their Hugoniots. We fit our results together with existingP‐V‐Tdata from dynamic and static compression experiments to constrain the thermal EoS for the three phases, all of which are of fundamental importance to the dynamics of the lower mantle. The EoS for perovskite and postperovskite are well described with third‐order Birch‐Murnaghan isentropes, offset with a Mie‐Grüneisen‐Debye formulation for thermal pressure. The addition of shock data helps to distinguish among discrepant static studies of perovskite, and for postperovskite, constrain a value ofK′ significantly larger than 4. For the melt, we define for the first time a single EoS that fits experimental data from ambient pressure to 230 GPa; the best fit requires a fourth‐order isentrope. We also provide a new EoS for Mg2SiO4liquid, calculated in a similar manner. The Grüneisen parameters of the solid phases decrease with pressure, whereas those of the melts increase, consistent with previous shock wave experiments as well as molecular dynamics simulations. We discuss implications of our modeling for thermal expansion in the lower mantle, stabilization of ultra‐low‐velocity zones associated with melting at the core‐mantle boundary, and crystallization of a terrestrial magma ocean.

A General Internal Potential Including Molecular Volume. Internal Energy and Equation of State of Fluids in Gas and Liquid Phases and under Critical Conditions: Study over a Wide Range of Temperature and Pressure

The energy of molecular interaction for all fluids is represented by means of a single relation which takes into account both the excluded volume V τ and the free volume (V − V τ). From this expression, we obtain a single equation of state which is in very close agreement with the representation of experimental results for all fluids both in the liquid and gaseous phase and over wide ranges of temperature and pressure. The generalisation of the study on different fluids allows following the variation of the excluded volume V τ as function of molar volume and temperature. This variation, which we write as (∂V τ/∂V) T , becomes pronounced in the neighbourhood of critical conditions. It must be taken into consideration in the explanation of observed anomalies of the pressure and C v near critical conditions.

Applications of the Soave-Redlich-Kwong Equation of State Using Mathematica

A single equation of state (EOS) such as the Soave–Redlich–Kwong EOS can accurately describe both the liquid and vapor phase. We present several applications of this equation of state including estimation of pure component properties, computation of vapor–liquid equilibrium (VLE) diagram and phase envelope for binary mixtures. Finally, we solve a flash distillation problem for a mixture of hydrocarbons and show how one can obtain the true vapor pressure (TVP). All these computations are available in the form of notebooks upon request from authors or from the Wolfram Research Information Center (Binous, 2006a). These problems can serve as teaching material for a chemical engineering thermodynamics undergraduate level course.

Equation of State of a Complex Fluid Column and Prediction of Contacts in Orocual Field, Venezuela

SummaryThe Orocual field is located in the northern Monagas state of Venezuela and is owned and operated by Petroleos de Venezuela S.A. (PDVSA), the national oil company of Venezuela.Reservoir compartmentalization adds complexity to the field, and structurally equivalent, noncommunicating fluid regions exist. An equation of state (EOS) is needed for reservoir modeling, requiring a review of available data and area fluid distributions.A seven-pseudocomponent EOS with a single characterization defining the compositional gradient of the hydrocarbon column from gas to black oil is defined. The method demonstrates that composition relative to depth can be predicted in those parts of the reservoir in which samples do not exist but in which production and test data must be matched, and thus where a gas-to-oil transition occurs.This paper demonstrates a technique to identify representative samples for use in developing an EOS and for initializing fluids in place. A method is presented to adjust the component composition vs. depth, providing consistent vertical composition distribution and compositional-model stability. This method meets the objectives of matching field production observations. A method is presented to quickly initialize a full-field model using a 1D compositional simulator to give full-field-model stability using the local high-temperature gradient.Results of compositional simulation show a single EOS, and vertical compositional and thermal variation reproduce the complex character of the field hydrocarbon column, matching field-measured observations of saturation pressure (ps), gas/oil ratio (GOR), and fluid densities.

Equation of state measurements of chlorite, pyrophyllite, and talc

The compressibilities of five sheet silicates-three chlorite compositions, pyrophyllite, and talc-have been measured in situ using energy-dispersive powder diffraction in a multi-anvil apparatus at the synchrotron radiation source, Daresbury Laboratory, U.K. Some high-temperature data were also collected. The three chlorite samples were a natural Mg-rich chlorite [Mg 4 . 4 9 Fe 0 . 2 8 Cr 0 . 1 2 Al 2 . 1 0 Si 2 . 9 5 O 1 0 (OH) 8 ], a natural Fe-rich chlorite [Mg 0 . 2 8 Fe 4 . 1 1 Al 2 . 7 1 Si 2 . 7 7 O 1 0 (OH) 8 ], and a synthetic clinochlore [Mg 5 Al 2 Si 3 O 1 0 (OH) 8 ]. The compressibility data for all three samples, measured up to between 3 and 6 GPa, can be fit to a single equation of state with a bulk modulus of 86.9(16) GPa (using K' = 4). The thermal expansivity measured for the natural Mg-chlorite [α 0 = 2.5(6) x 10 - 5 /K] is similar to previous measurements on various chlorite compositions, suggesting that the thermal expansivity of chlorite also can be described by a single equation, independent of composition. The bulk modulus of the natural pyrophyllite [A[Si 4 O 1 0 (OH) 2 ], measured up to 6.3 GPa, is 37(3) GPa, with K' = 10(1). The few high-temperature points agree with previous measurements. The natural talc [Mg 3 Si 4 O 1 0 (OH) 2 ], measured up to 6.2 GPa, has a bulk modulus of 41(4) GPa, with K' = 6(2). Our bulk modulus of chlorite is much higher than previously determined using X-ray diffraction. This finding explains why calculations of chlorite reactions using the previous data considerably overestimate its stability when compared to experimentally determined reaction positions. The pyrophyllite compressibility data should be useful in calculations of metamorphic reactions involving this mineral. The talc data are essentially identical to previous measurements.

Double Lattice Inversion Technique ? Application to the EAM Potential Construction

The embedded atom model (EAM) has been proved to be a very effective and powerful approach to energy calculation in metals. In the current work, we present a new approach to the construction of the EAM potential. The EAM functions are determined in a way to reproduce exactly two state equations defined for unit cells of two different metal structures. Our method is in some sense an extension of the lattice inversion algorithm proposed by Carlsson et al. (Phil Mag. A 41, 241 (1980)), where the resulting pair potential reconstructs only one given state equation. The inversion technique has been applied before to the construction of a pair potential and embedded atom potential from a single state equation. Using two input state equations we can determine only two functions, thus, we construct the pair potential and embedding function. The atomic charge density is taken from first-principle calculation. As an example, we apply our method to the determination of the EAM potential for solid lead. In order to estimate the quality of this potential, some bulk and defect properties are also calculated.

Critical behavior inLa0.5Sr0.5CoO3

We have studied the critical behavior in ${\mathrm{La}}_{0.5}{\mathrm{Sr}}_{0.5}{\mathrm{CoO}}_{3}$ near the paramagnetic to ferromagnetic transition temperature. We have analyzed our dc magnetization data near the transition temperature with the help of the modified Arrot plot, Kouvel-Fisher method. We have determined the critical temperature ${T}_{c}$ and the critical exponents $\ensuremath{\beta},$ $\ensuremath{\gamma},$ and $\ensuremath{\delta}.$ With the values of ${T}_{c},$ $\ensuremath{\beta},$ and $\ensuremath{\gamma},$ we plot $M/(1\ensuremath{-}{T/T}_{c}{)}^{\ensuremath{\beta}}$ vs $H/(1\ensuremath{-}{T/T}_{c}{)}^{\ensuremath{\gamma}}.$ All the data collapse on one of the two curves. This suggests that the data below and above ${T}_{c}$ obeys scaling, following a single equation of state. The exponents are close to Heisenberg values.