Centrifugal compressor performance prediction relies heavily on accurate modelling of thermodynamic properties using Equations of State (EOS); In particular, the gas compressibility factor (Z) and ratio of specific heat (k). There have been efforts to develop more generalised EOS such as GERG, but the challenge remains on identifying the best EOS fit for specific duties.
More recent EOS including AGA8 and REFPROP’s NIST EOS haven been explored in this paper, along with some earlier ones. The boundary limits of the various EOS are herein described with comparison of the results of all of these equations on various gas mixtures encountered in real applications.
The purpose of this work is to explore the more thermodynamically challenging heavy gas and mixtures. Operating points are selected to cover typical duties that are commonly encountered in LNG and offshore compression. Z and k derived from the EOS are then compared with REFPROP’s EOS as a reference and the deviations are tabulated.
More specifically, Mixed Refrigerant gases are typically used for LNG liquefaction applications while CO2 gas are common in sour gas fields, hence relevant for the intended investigation.
Discharge temperature is not calculated and compared between EOS in this paper; a reliable model for calculating polytropic exponents is open for further research.
This document summarizes modifications that have been made to the Redlich-Kwong equation of state since its development in 1949. It discusses early modifications that focused on improving predictions of vapor phase properties and volumetric behavior. It then outlines modifications made by researchers like Wilson, Soave, and others that introduced temperature-dependent parameters to better predict vapor-liquid equilibrium behavior of pure components and mixtures. The document reviews modifications made to both the attractive and repulsive terms of the equation of state, including the introduction of third parameters. It compares the performance of different modified equations of state in predicting properties like critical compressibility factors, second virial coefficients, and liquid densities.
The document presents a new two-constant equation of state that modifies the attractive pressure term of the van der Waals equation. Examples show the new equation provides improved predictions of liquid densities and phase behavior for single- and multi-component systems compared to the Soave-Redlich-Kwong equation. The development outlines determining parameters from critical properties and vapor pressure data, and using mixing and interaction rules to extend the equation to mixtures. Comparisons demonstrate the new equation performs as well or better than the Soave-Redlich-Kwong equation in all cases tested, with its greatest advantage in predicting liquid phase densities.
FIRST-PRINCIPLES KINETIC MONTE CARLO STUDY OF NO OXIDATION ON Pd SurfaceNi Zhenjuan
The document provides background information on first-principles kinetic Monte Carlo simulations of NO oxidation on Pd(100) and PdO(101) surfaces. It discusses that Pd has shown better performance than Pt for NO oxidation in NOx storage reduction catalysts, but Pd may form an oxide under reaction conditions. Previous studies have suggested a thin film surface oxide, (√5×√5)R27°PdO(101), as an appropriate model. The document outlines the theory and computational setup for the first-principles kinetic Monte Carlo simulations, including the rate constants calculated using DFT for adsorption, desorption, diffusion, and reaction processes. Lattice models of Pd(100) and the P
Simultaneousnonlinear two dimensional modeling of tubular reactor of hydrogen...Arash Nasiri
This paper develops two mathematical models of a packed tubular reactor for methane steam reforming to produce hydrogen. The models generate 2D radial and axial plots of component concentrations and temperature over time. Both steady state and transient flow regimes are considered. The models consist of two coupled partial differential equations, one for material balance and one for energy balance, with initial and boundary conditions. The equations include terms for convection, diffusion, reaction, and heat transfer. Simplifying assumptions are made to reduce complexity, such as ignoring pressure drop and the water-gas shift reaction.
The document discusses further developing the solvent model parameterization in the ONETEP software for recent exchange-correlation functionals. It aims to re-examine the parameters established previously using outdated approximations and to compare results using functionals like VV10 and B97M-V to PBE and LDA. The author finds the exchange functional makes little difference to solvation energy accuracy. Parameters may need minor adjustments and more research is needed, particularly for anions. Validation of the model for 60 molecules using different functionals finds B3LYP and B97M-V have the lowest error compared to experiment.
This document summarizes an experiment that uses optical pumping to measure the nuclear spins and energy levels of rubidium-85 and rubidium-87 isotopes. Helmholtz coils are used to apply a magnetic field to samples, inducing Zeeman splitting of the energy levels. Resonance frequencies are measured and analyzed using the Breit-Rabi equation to determine the nuclear spins match predicted values of 5/2 for Rb-85 and 3/2 for Rb-87. The earth's magnetic field is also measured and found to be consistent with known values. Sources of error are discussed and results show strong agreement with theoretical predictions.
The document discusses properties of pure substances, including their phases of vapor, liquid, and solid and how they relate on a pressure-temperature diagram. It explains concepts like saturation temperature and pressure, triple points, and that the state of a pure substance is determined by two independent properties. Tables of thermodynamic properties are commonly available for many pure substances.
This document summarizes modifications that have been made to the Redlich-Kwong equation of state since its development in 1949. It discusses early modifications that focused on improving predictions of vapor phase properties and volumetric behavior. It then outlines modifications made by researchers like Wilson, Soave, and others that introduced temperature-dependent parameters to better predict vapor-liquid equilibrium behavior of pure components and mixtures. The document reviews modifications made to both the attractive and repulsive terms of the equation of state, including the introduction of third parameters. It compares the performance of different modified equations of state in predicting properties like critical compressibility factors, second virial coefficients, and liquid densities.
The document presents a new two-constant equation of state that modifies the attractive pressure term of the van der Waals equation. Examples show the new equation provides improved predictions of liquid densities and phase behavior for single- and multi-component systems compared to the Soave-Redlich-Kwong equation. The development outlines determining parameters from critical properties and vapor pressure data, and using mixing and interaction rules to extend the equation to mixtures. Comparisons demonstrate the new equation performs as well or better than the Soave-Redlich-Kwong equation in all cases tested, with its greatest advantage in predicting liquid phase densities.
FIRST-PRINCIPLES KINETIC MONTE CARLO STUDY OF NO OXIDATION ON Pd SurfaceNi Zhenjuan
The document provides background information on first-principles kinetic Monte Carlo simulations of NO oxidation on Pd(100) and PdO(101) surfaces. It discusses that Pd has shown better performance than Pt for NO oxidation in NOx storage reduction catalysts, but Pd may form an oxide under reaction conditions. Previous studies have suggested a thin film surface oxide, (√5×√5)R27°PdO(101), as an appropriate model. The document outlines the theory and computational setup for the first-principles kinetic Monte Carlo simulations, including the rate constants calculated using DFT for adsorption, desorption, diffusion, and reaction processes. Lattice models of Pd(100) and the P
Simultaneousnonlinear two dimensional modeling of tubular reactor of hydrogen...Arash Nasiri
This paper develops two mathematical models of a packed tubular reactor for methane steam reforming to produce hydrogen. The models generate 2D radial and axial plots of component concentrations and temperature over time. Both steady state and transient flow regimes are considered. The models consist of two coupled partial differential equations, one for material balance and one for energy balance, with initial and boundary conditions. The equations include terms for convection, diffusion, reaction, and heat transfer. Simplifying assumptions are made to reduce complexity, such as ignoring pressure drop and the water-gas shift reaction.
The document discusses further developing the solvent model parameterization in the ONETEP software for recent exchange-correlation functionals. It aims to re-examine the parameters established previously using outdated approximations and to compare results using functionals like VV10 and B97M-V to PBE and LDA. The author finds the exchange functional makes little difference to solvation energy accuracy. Parameters may need minor adjustments and more research is needed, particularly for anions. Validation of the model for 60 molecules using different functionals finds B3LYP and B97M-V have the lowest error compared to experiment.
This document summarizes an experiment that uses optical pumping to measure the nuclear spins and energy levels of rubidium-85 and rubidium-87 isotopes. Helmholtz coils are used to apply a magnetic field to samples, inducing Zeeman splitting of the energy levels. Resonance frequencies are measured and analyzed using the Breit-Rabi equation to determine the nuclear spins match predicted values of 5/2 for Rb-85 and 3/2 for Rb-87. The earth's magnetic field is also measured and found to be consistent with known values. Sources of error are discussed and results show strong agreement with theoretical predictions.
The document discusses properties of pure substances, including their phases of vapor, liquid, and solid and how they relate on a pressure-temperature diagram. It explains concepts like saturation temperature and pressure, triple points, and that the state of a pure substance is determined by two independent properties. Tables of thermodynamic properties are commonly available for many pure substances.
1. The phase rule describes the relationship between the number of phases, components, and degrees of freedom in a system at equilibrium. It states that the degrees of freedom F equals the number of components C minus the number of phases P plus two.
2. A phase diagram graphically represents the phase equilibria of a system. The phase diagram for ice, water, and water vapor shows three single-phase regions and three phase boundary lines where two phases coexist in equilibrium.
3. The triple point is the only condition where ice, water, and water vapor can coexist in equilibrium, with zero degrees of freedom. It occurs at a temperature of 0.01 degrees C and a pressure of 4.
Science Express Paper by: Kevin B. Stevenson et al.GOASA
- Spectroscopic phase curve observations of the exoplanet WASP-43b using the Hubble Space Telescope revealed a distinct increase in flux as the dayside rotated into view, peaking prior to secondary eclipse.
- Analysis of the spectrally resolved phase curves showed wavelength-dependent amplitudes, phase shifts, and eclipse depths, allowing inference of the temperature structure and molecular abundances at 15 orbital phases.
- Atmospheric modeling found water to be the dominant absorber influencing the phase-resolved emission spectra. The data showed large day-night temperature variations at all measured altitudes and a monotonically decreasing temperature with pressure.
1) Gibbs phase rule determines the number of intensive properties (F) that can be independently varied for a system with N chemical species and P phases. For a single phase of a pure substance, F=2. For two coexisting phases, F=1. For three coexisting phases, F=0.
2) The lever rule determines the mole fraction of each phase in a binary equilibrium phase diagram. It relates the composition of an alloy to the compositions of its constituent phases.
3) The lever rule equation for the weight percentage of an α phase is: Xα = (c - b) / (a - b), where a, b, c are the weight percentages of an element in
The weight of a 1 kg mass where the acceleration due to gravity is 9.75 m/s^2 is 9.75 N.
Weight = mass x acceleration due to gravity
= 1 kg x 9.75 m/s^2
= 9.75 N
So the weight of a 1 kg mass in the given conditions is 9.75 N.
Phase equilibria: phase, components and degrees of freedom. The phase rule and its
thermodynamic derivation. The phase diagrams of water and sulphur systems, partially
miscible liquid pairs: the phenol and water and nicotine-water systems. Completely
miscible liquid pairs and their separation by fractional distillation. Freeze drying
(lyophilization).
This document describes the solid-gas equilibria of the CuSO4-H2O system. It discusses how copper sulfate forms different hydrates - monohydrate, trihydrate, and pentahydrate - at increasing vapor pressures when water vapor is added at a constant temperature of 55°C. The phase diagram plots vapor pressure against water vapor composition, showing invariant equilibrium lines where two solid phases coexist with vapor, and monovariant lines where a single solid phase coexists with vapor. It also describes the dehydration of copper sulfate pentahydrate upon decreasing vapor pressure, passing through trihydrate and monohydrate before forming anhydrous copper sulfate.
This document discusses work and heat in thermodynamics. It defines work and heat, and explains that they are both forms of energy transfer across system boundaries. Work is defined as energy transfer due to a force acting through a distance, while heat is energy transfer due to a temperature difference. The document also discusses different modes of heat transfer and provides comparisons between work and heat. Figures and examples are provided to illustrate key concepts.
This document summarizes computational chemistry research on 1,2-dioxetane chemiluminescent molecules. Gaussian '09 was used to optimize molecule structures in gas phase and aqueous solution, and calculate atomic charges. Results showed oxygen charges became more negative from O1 to O3, and group "b" molecules had the most negative oxygen charges. Carbon charges varied without pattern, but C1 was consistently partially negative and C2 partially positive for group "b" molecules. The goal is to develop chemiluminescent probes for detecting cancer cells.
This document evaluates the performance of an air heater that was installed at a power plant as part of an efficiency improvement project. It describes the standard ASME method for evaluating air heater performance and discusses issues with applying that method in practice. The document then presents a new method for applying the vendor's performance curves to evaluate the air heater performance according to the ASME standard. It derives equations to calculate temperature correction factors needed for the evaluation and discusses estimating the uncertainty of the results.
This document discusses properties of pure substances and phase changes. It defines a pure substance as having a fixed chemical composition and describes the three common phases as solid, liquid, and gas. Phase changes between these three states are explained, including saturated and supersaturated states. Key concepts introduced are the saturation temperature and pressure, which define conditions for phase changes. Diagrams are presented to illustrate phase equilibria, including temperature-volume, pressure-volume, and pressure-temperature diagrams. Common thermodynamic properties like latent heat and reference states are also defined. Ideal gas behavior is discussed along with limitations of the ideal gas model.
Master Thesis Total Oxidation Over Cu Based Catalystsalbotamor
The evolution in the oxidation state of Cu and Ce in a benchmark catalyst is studied
under different conditions: temperature programmed reduction with propane and hydrogen,
and isothermal reduction with propane and hydrogen.
Analytical methods used involve operando X-ray absorption spectroscopy (XAS) in
transmission mode at the Cu K edge and Ce LIII edge, as well as online mass spectrometry
(MS) at the outlet of the reactor.
In situ XAFS studies of carbon supported Pt and PtNi(1:1) catalysts for the o...qjia
it\'s a presentation for APS Meeting In Iowa. It mainly introduces our work of rationalizing the superior reactivity of certain commercial alloy nanocatalysts by probing their physical and chemical properties through x-ray experiments and theoretical model simulation.
High pressure structural properties of rare earth antimonideAlexander Decker
1) The document investigates the high-pressure structural phase transition of rare-earth antimonide DySb using a three-body potential model with electronic polarizability.
2) This model predicts a first-order phase transition from the NaCl (B1) structure to the CsCl (B2) structure in DySb at 22.6 GPa, with a 3.8% volume collapse.
3) The predicted transition pressure and volume change are in good agreement with available experimental data.
This document discusses solid solutions and phase diagrams in engineering metallurgy. It defines key terms like alloy, solute, solvent and phase. It explains that solid solutions occur when elements are atomically dispersed in a single crystal structure, following Hume-Rothery rules regarding size, structure, valence and electronegativity. Phase diagrams graphically represent phases present at various temperatures, pressures and compositions, and the Gibbs phase rule determines the maximum number of phases based on components and system. Examples are given of phase diagrams for pure substances and alloys with partially solid solutions.
This document provides an overview of phase diagrams and their components. It discusses that a phase diagram shows the phases that are present at equilibrium under different temperature and composition conditions. It outlines the key components of phase diagrams, including phase boundaries, triple points, solidus and liquidus lines. It also describes the different types of phase diagrams - unary, binary, and ternary - as well as common binary phase diagrams like eutectic, peritectic, and solid solution types. The document emphasizes that phase diagrams are important for understanding phase transformations and determining properties of materials at different temperatures and compositions.
This document discusses the properties of pure substances and their phase changes. Some key points:
1) A pure substance has a fixed chemical composition throughout and may exist in different phases including solid, liquid, and gas.
2) The molecular arrangements and bonding strengths differ between phases, with solids having the strongest bonds and gases having the weakest.
3) A pure substance undergoes phase changes as it is heated or cooled, passing through solid, liquid, and gas states depending on the temperature and pressure conditions. Phase change diagrams depict these relationships.
4) Properties of saturated liquid-vapor mixtures are averages based on the fraction of each phase present. Superheated vapors and compressed liquids
This document discusses energy analysis of closed systems, including specific heats at constant volume and pressure, internal energy, enthalpy, and specific heats of ideal gases and incompressible substances like solids and liquids. It provides equations relating these concepts and explains how to calculate changes in internal energy and enthalpy using data tables, functional forms of specific heats, or average specific heat values. Examples are also included.
The document discusses phase diagrams and the phase rule. It begins by defining key terms like phase, components, and degrees of freedom. It then explains Gibbs phase rule which relates the number of phases (P), components (C), and degrees of freedom (F) as F=C-P+2. Several examples of 1-component and 2-component phase diagrams are discussed. The document also covers concepts like true equilibrium, metastable equilibrium, and phase transitions. Phase diagrams show the conditions under which different phases can exist in equilibrium and include lines separating single phase and two phase regions.
This document summarizes a study that used the SAFT-VR Mie equation of state to model transport properties like viscosity and interfacial tension of CO2-rich systems relevant to carbon capture and storage. The SAFT-VR Mie EoS was used to calculate densities, from which a viscosity model and density gradient theory were used to predict viscosity and interfacial tension, respectively. Results for five binary mixtures and two multicomponent mixtures were compared to experimental data and showed good agreement, supporting the capabilities of the models.
232372441 correlation-and-prediction-of-vle-and-lle-by-empirical-eosJohn Barry
This document provides a summary of recent advances from 1980 to 1985 in the use of empirical equations of state for correlating and predicting fluid phase equilibria. It discusses developments in nonanalytical equations of state, virial-type equations of state, and van der Waals-type equations of state, with a focus on cubic equations of state. Key advances discussed include combining classical and nonclassical treatments in nonanalytical equations of state, generalizing parameters in virial equations to reduce input data needs, and improving temperature dependence and volume translations in cubic equations of state to better model vapor pressure and liquid volumes.
1. The phase rule describes the relationship between the number of phases, components, and degrees of freedom in a system at equilibrium. It states that the degrees of freedom F equals the number of components C minus the number of phases P plus two.
2. A phase diagram graphically represents the phase equilibria of a system. The phase diagram for ice, water, and water vapor shows three single-phase regions and three phase boundary lines where two phases coexist in equilibrium.
3. The triple point is the only condition where ice, water, and water vapor can coexist in equilibrium, with zero degrees of freedom. It occurs at a temperature of 0.01 degrees C and a pressure of 4.
Science Express Paper by: Kevin B. Stevenson et al.GOASA
- Spectroscopic phase curve observations of the exoplanet WASP-43b using the Hubble Space Telescope revealed a distinct increase in flux as the dayside rotated into view, peaking prior to secondary eclipse.
- Analysis of the spectrally resolved phase curves showed wavelength-dependent amplitudes, phase shifts, and eclipse depths, allowing inference of the temperature structure and molecular abundances at 15 orbital phases.
- Atmospheric modeling found water to be the dominant absorber influencing the phase-resolved emission spectra. The data showed large day-night temperature variations at all measured altitudes and a monotonically decreasing temperature with pressure.
1) Gibbs phase rule determines the number of intensive properties (F) that can be independently varied for a system with N chemical species and P phases. For a single phase of a pure substance, F=2. For two coexisting phases, F=1. For three coexisting phases, F=0.
2) The lever rule determines the mole fraction of each phase in a binary equilibrium phase diagram. It relates the composition of an alloy to the compositions of its constituent phases.
3) The lever rule equation for the weight percentage of an α phase is: Xα = (c - b) / (a - b), where a, b, c are the weight percentages of an element in
The weight of a 1 kg mass where the acceleration due to gravity is 9.75 m/s^2 is 9.75 N.
Weight = mass x acceleration due to gravity
= 1 kg x 9.75 m/s^2
= 9.75 N
So the weight of a 1 kg mass in the given conditions is 9.75 N.
Phase equilibria: phase, components and degrees of freedom. The phase rule and its
thermodynamic derivation. The phase diagrams of water and sulphur systems, partially
miscible liquid pairs: the phenol and water and nicotine-water systems. Completely
miscible liquid pairs and their separation by fractional distillation. Freeze drying
(lyophilization).
This document describes the solid-gas equilibria of the CuSO4-H2O system. It discusses how copper sulfate forms different hydrates - monohydrate, trihydrate, and pentahydrate - at increasing vapor pressures when water vapor is added at a constant temperature of 55°C. The phase diagram plots vapor pressure against water vapor composition, showing invariant equilibrium lines where two solid phases coexist with vapor, and monovariant lines where a single solid phase coexists with vapor. It also describes the dehydration of copper sulfate pentahydrate upon decreasing vapor pressure, passing through trihydrate and monohydrate before forming anhydrous copper sulfate.
This document discusses work and heat in thermodynamics. It defines work and heat, and explains that they are both forms of energy transfer across system boundaries. Work is defined as energy transfer due to a force acting through a distance, while heat is energy transfer due to a temperature difference. The document also discusses different modes of heat transfer and provides comparisons between work and heat. Figures and examples are provided to illustrate key concepts.
This document summarizes computational chemistry research on 1,2-dioxetane chemiluminescent molecules. Gaussian '09 was used to optimize molecule structures in gas phase and aqueous solution, and calculate atomic charges. Results showed oxygen charges became more negative from O1 to O3, and group "b" molecules had the most negative oxygen charges. Carbon charges varied without pattern, but C1 was consistently partially negative and C2 partially positive for group "b" molecules. The goal is to develop chemiluminescent probes for detecting cancer cells.
This document evaluates the performance of an air heater that was installed at a power plant as part of an efficiency improvement project. It describes the standard ASME method for evaluating air heater performance and discusses issues with applying that method in practice. The document then presents a new method for applying the vendor's performance curves to evaluate the air heater performance according to the ASME standard. It derives equations to calculate temperature correction factors needed for the evaluation and discusses estimating the uncertainty of the results.
This document discusses properties of pure substances and phase changes. It defines a pure substance as having a fixed chemical composition and describes the three common phases as solid, liquid, and gas. Phase changes between these three states are explained, including saturated and supersaturated states. Key concepts introduced are the saturation temperature and pressure, which define conditions for phase changes. Diagrams are presented to illustrate phase equilibria, including temperature-volume, pressure-volume, and pressure-temperature diagrams. Common thermodynamic properties like latent heat and reference states are also defined. Ideal gas behavior is discussed along with limitations of the ideal gas model.
Master Thesis Total Oxidation Over Cu Based Catalystsalbotamor
The evolution in the oxidation state of Cu and Ce in a benchmark catalyst is studied
under different conditions: temperature programmed reduction with propane and hydrogen,
and isothermal reduction with propane and hydrogen.
Analytical methods used involve operando X-ray absorption spectroscopy (XAS) in
transmission mode at the Cu K edge and Ce LIII edge, as well as online mass spectrometry
(MS) at the outlet of the reactor.
In situ XAFS studies of carbon supported Pt and PtNi(1:1) catalysts for the o...qjia
it\'s a presentation for APS Meeting In Iowa. It mainly introduces our work of rationalizing the superior reactivity of certain commercial alloy nanocatalysts by probing their physical and chemical properties through x-ray experiments and theoretical model simulation.
High pressure structural properties of rare earth antimonideAlexander Decker
1) The document investigates the high-pressure structural phase transition of rare-earth antimonide DySb using a three-body potential model with electronic polarizability.
2) This model predicts a first-order phase transition from the NaCl (B1) structure to the CsCl (B2) structure in DySb at 22.6 GPa, with a 3.8% volume collapse.
3) The predicted transition pressure and volume change are in good agreement with available experimental data.
This document discusses solid solutions and phase diagrams in engineering metallurgy. It defines key terms like alloy, solute, solvent and phase. It explains that solid solutions occur when elements are atomically dispersed in a single crystal structure, following Hume-Rothery rules regarding size, structure, valence and electronegativity. Phase diagrams graphically represent phases present at various temperatures, pressures and compositions, and the Gibbs phase rule determines the maximum number of phases based on components and system. Examples are given of phase diagrams for pure substances and alloys with partially solid solutions.
This document provides an overview of phase diagrams and their components. It discusses that a phase diagram shows the phases that are present at equilibrium under different temperature and composition conditions. It outlines the key components of phase diagrams, including phase boundaries, triple points, solidus and liquidus lines. It also describes the different types of phase diagrams - unary, binary, and ternary - as well as common binary phase diagrams like eutectic, peritectic, and solid solution types. The document emphasizes that phase diagrams are important for understanding phase transformations and determining properties of materials at different temperatures and compositions.
This document discusses the properties of pure substances and their phase changes. Some key points:
1) A pure substance has a fixed chemical composition throughout and may exist in different phases including solid, liquid, and gas.
2) The molecular arrangements and bonding strengths differ between phases, with solids having the strongest bonds and gases having the weakest.
3) A pure substance undergoes phase changes as it is heated or cooled, passing through solid, liquid, and gas states depending on the temperature and pressure conditions. Phase change diagrams depict these relationships.
4) Properties of saturated liquid-vapor mixtures are averages based on the fraction of each phase present. Superheated vapors and compressed liquids
This document discusses energy analysis of closed systems, including specific heats at constant volume and pressure, internal energy, enthalpy, and specific heats of ideal gases and incompressible substances like solids and liquids. It provides equations relating these concepts and explains how to calculate changes in internal energy and enthalpy using data tables, functional forms of specific heats, or average specific heat values. Examples are also included.
The document discusses phase diagrams and the phase rule. It begins by defining key terms like phase, components, and degrees of freedom. It then explains Gibbs phase rule which relates the number of phases (P), components (C), and degrees of freedom (F) as F=C-P+2. Several examples of 1-component and 2-component phase diagrams are discussed. The document also covers concepts like true equilibrium, metastable equilibrium, and phase transitions. Phase diagrams show the conditions under which different phases can exist in equilibrium and include lines separating single phase and two phase regions.
This document summarizes a study that used the SAFT-VR Mie equation of state to model transport properties like viscosity and interfacial tension of CO2-rich systems relevant to carbon capture and storage. The SAFT-VR Mie EoS was used to calculate densities, from which a viscosity model and density gradient theory were used to predict viscosity and interfacial tension, respectively. Results for five binary mixtures and two multicomponent mixtures were compared to experimental data and showed good agreement, supporting the capabilities of the models.
232372441 correlation-and-prediction-of-vle-and-lle-by-empirical-eosJohn Barry
This document provides a summary of recent advances from 1980 to 1985 in the use of empirical equations of state for correlating and predicting fluid phase equilibria. It discusses developments in nonanalytical equations of state, virial-type equations of state, and van der Waals-type equations of state, with a focus on cubic equations of state. Key advances discussed include combining classical and nonclassical treatments in nonanalytical equations of state, generalizing parameters in virial equations to reduce input data needs, and improving temperature dependence and volume translations in cubic equations of state to better model vapor pressure and liquid volumes.
The document describes the development of a dynamic model of an industrial packed bed multi-tubular reactor used to manufacture ethylene oxide through the catalytic oxidation of ethylene with oxygen. A system of non-linear partial differential equations is used to model the highly exothermic gas phase reactions. The model is benchmarked against plant data and reasonably predicts the reactor behavior. The heterogeneous two phase model is reduced to a single phase homogeneous model for simplicity. A comparison shows the homogeneous model provides accurate predictions of the reactor performance.
The document describes the development of a dynamic model of an industrial packed bed multi-tubular reactor used for producing ethylene oxide. Ethylene oxide is produced through the catalytic oxidation of ethylene with oxygen over a silver-based catalyst. The model is developed using a system of non-linear partial differential equations and is benchmarked against plant data from an industrial ethylene oxide reactor. Both a heterogeneous two-phase model and a reduced homogeneous single-phase model are considered and compared against plant data.
The document describes a numerical study of two-phase flows in a model ladle containing molten metal agitated by nitrogen gas injection. An Eulerian-Eulerian model is used to simulate the gas and liquid phases. Different drag and lift force coefficients are examined and compared to experimental data on liquid velocity, bubble velocity, and gas volume fraction. Results show reasonable agreement with experiments in the liquid region but larger deviations in the gas-liquid plume region. An improved drag coefficient correlation is proposed.
The document reviews equations of state and their applicability, focusing on phase equilibrium modeling. It discusses several common equations of state (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson), outlining their development, strengths/weaknesses, and applicability. The key intention is to use the Peng-Robinson equation of state to compute thermodynamic interactions of volatile organic compounds in biodiesel polymers to aid in separation processes by simulating phase equilibria without costly actual measurements.
Stellar Measurements with the New Intensity FormulaIOSR Journals
In this paper a linear relationship in stellar optical spectra has been found by using a
spectroscopical method used on optical light sources where it is possible to organize atomic and ionic data.
This method is based on a new intensity formula in optical emission spectroscopy (OES). Like the HR-diagram ,
it seems to be possible to organize the luminosity of stars from different spectral classes. From that organization
it is possible to determine the temperature , density and mass of stars by using the new intensity formula. These
temperature, density and mass values agree well with literature values. It is also possible to determine the mean
electron temperature of the optical layers (photospheres) of the stars as it is for atoms in the for laboratory
plasmas. The mean value of the ionization energies of the different elements of the stars has shown to be very
significant for each star. This paper also shows that the hydrogen Balmer absorption lines in the stars follow
the new intensity formula.
Efficient estimation of natural gas compressibility factor usingAbelardo Contreras
This document presents a new method for estimating natural gas compressibility factor (Z-factor) using least square support vector machine (LSSVM) modeling. The LSSVM model is developed and tested using a database of over 2,200 samples of sour and sweet gas compositions. The model predicts Z-factor as a function of gas composition, molecular weight, pressure, and temperature. Statistical analysis shows the LSSVM model outperforms existing empirical correlations with an average absolute relative error of 0.19% and correlation coefficient of 0.999. The accurate prediction of Z-factor is important for natural gas engineering calculations.
A density correction for the peng robinson equationLuis Follegatti
This document presents a density correction for the Peng-Robinson equation of state. The correction involves adding a simple empirical term that requires one parameter per component. It improves the prediction of liquid densities by 2-4% and vapor densities slightly. The correction retains the internal consistency between vapor and liquid properties predicted by equations of state. It provides a reliable way to enhance density predictions without significantly affecting other properties.
This document proposes a modification to the Redlich-Kwong equation of state by making the temperature-dependent parameter a(T) instead of a constant. This improves the equation's ability to model vapor pressures of pure substances and phase equilibria of mixtures. The modified equation represents vapor pressure data for hydrocarbons more accurately than the original equation. When combined with the original Redlich-Kwong mixing rules, the modified equation can also predict vapor-liquid equilibrium for mixtures of nonpolar fluids like hydrocarbons, with some limitations for hydrogen-containing mixtures.
Heat Capacity of BN and GaN binary semiconductor under high Pressure-Temperat...IOSR Journals
In this paper, we have calculated the molar heat capacity for cubic zinc blende (cZB) BN and GaN binary semiconductors at high pressure-temperature (PT). For the calculation of heat capacity, we firstly obtained the Debye temperature (ϴD) variation with temperature and at higher temperature it becomes constant with temperature in quasi-harmonic approximation limits. We have also calculated the static Debye temperature (ϴD) from elastic constant for the both BN and GaN binary semiconductors. The elastic constants are calculated from the energy-strain relation using plane wave method in DFT approach. All the calculated results are well consistence with experimental and reported data
This document presents a new intensity formula for optical emission spectroscopy that has been applied to stellar spectra. The formula relates spectral line intensity to wavelength, frequency, electron temperature, and ionization energies. The author analyzed spectra from the literature for 17 elements and 11 ions and found linear relationships between the logarithm of intensity and the inverse of frequency times ionization energies, supporting the new formula. Stellar spectra from classes O-M were also analyzed and found to follow similar linear relationships, allowing the determination of electron temperatures, mean ionization energies, and effective temperatures for different stellar classes. Intensity ratios of Balmer lines from various stars correlated well between theoretical predictions using the formula and experimental measurements.
FlowVision CFD - Verification Calculations as per CFD FlowVision Code for Sod...capvidia
Application of FlowVision CFD Software for Analytical Validation of Sodium-Cooled Fast Reactor Structure Components
Verification of LMS (Liquid Metals Sodium) Turbulent Heat Transfer Model
Streaming and Mixing of Coolant Flows within "OKBM Afrikantov, BN-600 Reactor with Integral Layout of Equipment
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This document summarizes a first-principles constrained thermodynamics study of H2 oxidation at Pd(100). It examines various surface structures and compositions using DFT calculations linked to ab initio thermodynamics to build a thermodynamic map of surface termination under different O2 and H2 partial pressures. Key findings include:
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1. Equations For Heavy Gases in Centrifugals
Lee Chong Jin (Team leader)
Mohd Zakiyuddin Mohd Zahari
Cheah Cang To
James Bryan
- Rotating Equipment Department, Technip Geoproduction Malaysia
August 12, 2014
2. Abstract
Centrifugal compressor performance prediction relies heavily on accurate modelling of thermodynamic properties using Equations of State (EOS); In particular, the gas compressibility factor (Z) and ratio of specific heat (k). There have been efforts to develop more generalised EOS such as GERG, but the challenge remains on identifying the best EOS fit for specific duties.
More recent EOS including AGA8 and REFPROP’s NIST EOS haven been explored in this paper, along with some earlier ones. The boundary limits of the various EOS are herein described with comparison of the results of all of these equations on various gas mixtures encountered in real applications.
The purpose of this work is to explore the more thermodynamically challenging heavy gas and mixtures. Operating points are selected to cover typical duties that are commonly encountered in LNG and offshore compression. Z and k derived from the EOS are then compared with REFPROP’s EOS as a reference and the deviations are tabulated.
More specifically, Mixed Refrigerant gases are typically used for LNG liquefaction applications while CO2 gas are common in sour gas fields, hence relevant for the intended investigation.
Discharge temperature is not calculated and compared between EOS in this paper; a reliable model for calculating polytropic exponents is open for further research.
3. Nomenclature
Symbols
( ) ( )
Abbreviations
The following are abbreviations of the different EOS names used throughout the report:
( )
4. Eq. 1
Eq. 2
Fundamentals
The Compressibility Factor, Z, is the fundamental thermodynamic property for modifying the ideal gas law to account for the real gas behavior. Z is introduced into the Ideal Gas equation [1]:
Due to the various factors involved such as having infinite possible combinations of ratios between each component in a gas mixture, it is infeasible to develop an EOS that will accurately calculate Z across a wide combination of operating conditions (in terms of gas compositions, temperatures and pressures).
With the specific heat ratio, k, polytropic exponents can be obtained and in turn gas compression can then be expressed in terms of pressure and temperature variation [2]: ( )
The power required to compress a gas is directly proportional to the gas compressibility factor, Z. For an ideal gas, Z=1 regardless of the gas’ state. Since in practice Z changes depending on the gas conditions P and T, power calculation will deviate between a real gas and ideal gas calculation by as much as the Z deviates. Similarly, k affects the accuracy of the head and power equations.
Therefore, it is worth investigating the different EOS that can be used to obtain Z and k for a specified mixture and operating condition. The EOS that are investigated in this report are tabulated in Table 1:
Equation of State
General Form of Equation
Redlich-Kwong (RK) [1]
( )√
Soave-Redlich-Kwong (SRK) [1]
( )
Peng-Robinson (PR) [1]
( ) ( )
Benedict-Webb-Rubin- Starling & Han modified by Nishiumi & Saito (BWRS- NS) [3][4]
( ) ( ) ( ) ( ) ( ) ( )
Lee-Kesler-Plöcker (LKP) [1]
( ) ( )
AGA8 [5]
Σ Σ ( ) ( )
GERG (2004) [6]
(Σ ( ) ΣΣ ( ) )
NIST [7]
Based on GERG2008 which is an updated GERG2004 EOS
5. Eq. 3
Eq. 4
Table 1: Equations of States analysed and the general form of the equation.
REFPROP, a commercially available program developed by the National Institute of Standards and Technology (NIST), performs estimation of real gas thermodynamic properties based on three models for the thermodynamic properties of pure fluids: EOS explicit in Helmholtz energy, the modified Benedict-Webb-Rubin equation of state, and an extended corresponding states (ECS) model [7]. Equation of state modules available from the REFPROP package are:-
1. AGA 8 (for pipelines)
2. GERG 2008
3. Peng-Robinson (PR)
4. NIST
The NIST EOS is primarily based on the GERG 2008 EOS (which is used in [8;9;10;11]), in turn expanded from GERG 2004 to include additional fluids (e.g. ethylene, propylene, methanol, etc.). NIST's database is widely recognised as a reliable source of reference in terms of real gas behaviour, as can be traced in both the academic and turbo-machinery industry [12;13;14;15]. Thus, with the established database in REFPROP software, the default NIST EOS will be the benchmark EOS which other EOS will be referred to for the purpose of this paper.
The standalone EOS (not included in REFPROP) compiled by the authors for the purpose of this discussion are as follows:-
1. Redlich-Kwong (RK)
2. Lee-Kesler-Plӧcker (LKP)
3. Modified Benedict, Webb, Rubin, Starling and Han by Nishiumi and Saito (BWRS-NS)
4. Soave-Redlich-Kwong (SRK)
RK and SRK EOS are relatively straightforward to model as they are cubic EOS. Virial EOS such as BWRS-NS and LKP are developed as an improvement to the former; These are EOS which represents a power series of density with temperature coefficients [1]:
The roots in these virial EOS are evaluated using the Newton-Raphson method where the initial guess for compressibility factor is set to be 0.8 for the vapor phase [1]. AGA8 which is an extended virial equation is even lengthier where it contains summations of 58 polynomial terms.
BWRS-NS is selected over the standard BWRS model for its wider range of operations; specifically in the cryogenic range [3]. Nevertheless, BWRS would still suffice for noncryogenic CO2 duties.
In the absence of REFPROP, the Multiparameter EOS such as GERG could also be modelled. GERG is represented in the Helmholtz Free Energy form in terms of reduced density and inverse reduced temperature [6]: ( ) ( ) ( )
6. Comparison of Z & k between different EOS
For a given pressure, temperature and gas mixture, different EOS will yield different values of Z and k. The goal would be to tabulate and understand the differences between each EOS for various gas compositions. One way to demonstrate the differences is by plotting graphs of Z and k versus pressure (at a specific temperature) for different EOS. This plot shows how the different EOS varies with each other over the range of pressures. Note that we can also choose to plot Z versus temperature for specific pressures instead, however for convention sake plots of Z versus pressure will be used (i.e. in the standard form of Nelson-Obert compressibility charts). [16]
By investigating the different EOS, if results are very similar for specific EOS for certain common gasses then the results can be interchangeable for future comparisons. In general, at lower pressure and higher temperature it is expected that the different EOS should corroborate better among each other as the conditions are approaching that of an Ideal Gas, thus yielding a more accurate and predicable model.
To ensure accurate and consistent results, the operating ranges for a given gas mixture are selected to ensure the fluid is in a pure gas state, and not as a multi-phase, liquid-phase fluid or near the critical point. This is verified by plotting a Phase Map using REFPROP NIST (i.e. vapor-liquid equilibrium curves) and ensuring the operating points chosen are in the gas phase region. Operating points are selected based on typical compressor applications of each gas, such as refrigerant compressors and high-pressure CO2 reinjection duties.
As stated previously, NIST’s EOS from the REFPROP software is widely recognized as a reliable EOS and thus will be used as the reference for a comparison datum for other EOS. REFPROP software is used to calculate Z and k for NIST, GERG, AGA8 and PR EOS. RK, SRK, BWRS-NS and LKP are not available in REFPROP and thus are compiled individually by the authors. Deviations will be quoted in terms of % deviation from NIST. Since NIST is an updated form of the GERG (2004) EOS, it is expected that the NIST and GERG2004 will have negligible deviation.
The different EOS are utilized for the calculation of Z and k for Mixed Refrigerant Gas, Pure CO2 gas and a CO2 gas mixture. For reference, Pure Methane and a Natural Gas Mixture is also investigated as the properties of methane are well-established.
Note that for mixture comparisons, RK is not used due to lack of interaction parameters on-hand. Instead, AGA8 is used. AGA8 is not used for pure fluids because in REFPROP, AGA8 is only used for mixtures and will revert back to NIST EOS when calculating pure fluids.
7. Compressibility and Cp/Cv vs Pressure Graphs
Figure 1: Compressibility Z vs Pressure for Methane gas at T=210K and T=300K
Figure 1 illustrates a typical compressibility graph of Z vs P for pure Methane using the various EOS. On the left side of the graph close to P=0 bara, it can be seen that all the EOS converges to 1. This signifies the point where the gas behaves closest to an ideal gas; the lower the P, the less collisions and force interactions between the gas molecules. As P increases, gas intermolecular forces become prevalent. This causes the gas molecules to occupy a denser space (Z reduces) than predicted by the ideal gas model. Each EOS notably branches off from each other; the different EOS models have their own set of parameters to estimate Z with varying degrees of accuracy. As P is increased even further, Z slowly increases due to the physical size of the molecules (Ideal gas model neglects gas molecule size).
At lower temperatures, the gas molecules’ kinetic energy is low enough that the interaction forces between molecules are prevalent. Thus, there is a huge variation in Z as P increases. As T is increased however, the kinetic energy of the gas molecules renders the interaction forces to be less significant which approaches the ideal gas model. Thus the curve starts to approach a flatter, straight line closer to Z=1 throughout the range of P. This trend is visible in Figure 1 by comparing the two curves at T=210K and T=300K.
8. The Critical Point of Methane is at P=46bara and T=190.6k. Thus, data to the right of the critical P line in Figure 1 in this case are within the Supercritical Region.
Figure 2: Specific Heat Ratio (k=Cp/Cv) vs Pressure for Methane gas at T=210K and T=300K
Similarly, the Specific Heat Ratios, k can be plotted versus P, as seen in Figure 2. As the ratio k=Cp/Cv and Cp>Cv in all cases, the graph for k is always above k=1. A notable feature is that near the critical point, the EOS spikes to infinity yielding erroneous results. This issue is not apparent on the Z graph, thus it is possible to obtain operating points near the critical point where values of Z appear sensible while k becomes overly sensitive. The curve slowly flattens out as Temperature is increased beyond Tc. Deviations of k also increase when P increases as the gas deviates from the Ideal Gas model.
It is not practical to present the entire range of data on this paper as this requires a 3D graph to effectively plot Z for various P and T; even then, it will be difficult to compare multiple 3d surfaces representing each EOS. Therefore, selected operating points applicable for the gas examined will be used to compare Z and k to evaluate how the EOS differ from each other.
9. Pure Methane
Methane is the simplest alkane molecule and the main constituent in natural gas, serving as a reference to establish the comparisons between EOS. Selected points are chosen rather than presenting all the data on a graph here as it is impractical to overlay all the data of the various EOS.
Operating Point 1 represents boil-off gas conditions. Operating points 2-7 represent typical values of a natural gas compressor.
Operating Point
1
2
3
4
5
6
7
Pressure, bar a
1.01
13.82
24.41
24.6
105.66
56.42
128.94
Temp, K
115
320
320
350
350
380
380
Table 2: Selected Operating Points for Pure Methane
Note that the critical point of methane is at 46bara, 191K. None of the operating points are selected near this value.
-0.5
0
0.5
1
1.5
2
1
2
3
4
5
6
7
%Deviation Z
Operating Points
Deviation of Z for Pure Methane
Z GERG
Z LKP
Z BWRSNS
Z SRK
Z PR
Z RK
10. Figure 3 & 4: Deviation of Z and k for Pure Methane
Z: With reference to NIST, it can be seen that for Z most of the EOS agree with each other within these ranges; exhibiting deviations <0.5% except for SRK at higher P and T. LKP appears to be the most consistent throughout the range with the lowest deviations. SRK on the other hand appears to deviate more as P and T increases by up to nearly 2%. What appears promising is that even at low T of 115K (i.e. near boil-off gas conditions), the EOS all fall within 0.4%< deviation.
k: Specific heat ratios (k) on the other hand do not follow the same trend even with the same operating conditions. It is apparent that SRK has generally the least deviations this time, however the trend indicates that k for SRK gradually shifts to negative deviation as P increase. For BWRS-NS the opposite is observed; k gradually shifts to positive deviation as P increase. LKP and PR demonstrates stable deviations throughout the range. Again, at 115K, the EOS are within 0.5%< deviation.
Hence for Z, all the EOS throughout the range (even for low T) fall within +- 0.5% deviation except for SRK at higher P (>50bara). For k, all the EOS throughout the range (even for low T) fall within +- 0.5% deviation except for RK at higher P (>50bara).
-2
-1.5
-1
-0.5
0
0.5
1
1
2
3
4
5
6
7
%Deviation k
Operating Points
Deviation of k for Pure Methane
k GERG
k LKP
k BWRSNS
k SRK
k PR
k RK
11. Natural Gas Mixture
The difficulty of modeling mixtures arises from the interactions between the different components. Thus, more reliable EOS consider the interactions by using binary interaction parameters. By considering the effects between each pair of compounds in a mixture and taking an average, a more accurate result for the calculation of EOS parameters is obtained. Hence, it is expected that the EOS will result in higher deviations for mixtures than pure components.
With the analysis of EOS for pure Methane, it can be predicted that a natural gas composition will exhibit similar trends for Z and k. A sample typical natural gas mixture consisting of 82% methane is analysed using similar operating points to pure methane gas. The exact composition can be found in Appendix A.
Operating Point
1
2
3
4
5
6
Pressure, bar a
10.91
21.42
24.3423
102.8026
55.93
126.52
Temp, K
320
320
350
350
380
380
Table 3: Selected Operating Points for Natural Gas Mixture
-1
-0.5
0
0.5
1
1.5
2
2.5
1
2
3
4
5
6
%Deviation Z
Operating Points
Deviation of Z for Natural Gas
Z GERG
Z LKP
Z BWRSNS
Z SRK
Z PR
Z AGA8
12. Figure 5 & 6: Deviation of Z and k for Natural Gas Mixture
As predicted, the results of the graphs (Figures 1 & 3, 2 & 4) for natural gas are similar to the pure Methane gas graph except with slightly more deviations.
Z: For Natural Gas data, it is observed that LKP gives stable results; with less than 0.5% deviation for each data. However, the results show that AGA8 show the least deviation throughout the range; demonstrating great correlation with NIST for natural gas mixtures. As predicted, SRK again exhibits large deviations up to 2% at high P and T. PR has low deviations in this range but may overshoot at higher P and T; same goes for BWRS-NS.
k: AGA8 represents the closest EOS to NIST but deviates more significantly at the higher P>100bara (around 0.5%). SRK still models k well comparatively (around 0.25% at most) despite poor comparison with Z, however k continues to deviate more negatively as P and T increase. PR basically demonstrates to be a worse SRK in this mixture. LKP remains fairly consistent with deviations throughout the range around 0.25%.
Overall, it can be concluded that for the ranges above, AGA8 resembles closest to NIST for this methane-predominant mixture. Otherwise, LKP is also a strong contender for Z, and LKP/SRK for k. However within the P and T ranges analysed above, most EOS do agree well with each other as deviations are at most 1%. Therefore for a natural gas mixture with similar composition to the above, any of the above EOS can be used and a deviation of not more than 1% for Z and k can be expected (except for RK and SRK at higher P).
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
2
3
4
5
6
%Deviation k
Operating Points
Deviation of k for Natural Gas
k GERG
k LKP
k BWRSNS
k SRK
k PR
k AGA8
13. Mixed Refrigerant Gas
Mixed refrigerant gas consists of a wider range of heavier hydrocarbons compared to a typical natural gas mixture and therefore has a significantly heavier molecular weight. Since the longer chain hydrocarbon molecules come into the picture, their size and interaction are important to consider. A sample gas consisting primarily of Methane, Ethylene and Butane is investigated for the following operating conditions:
Operating Point
1
2
3
4
5
6
Pressure, bar a
3.35
16.73
16.73
43.48
43.48
56.86
Temp, K
300
310
390
360
400
400
Table 4: Selected Operating Points for Mixed Refrigerant Gas
Note that the critical point of this mixture is at 103bara, 334K. The operating P typically do not exceed the critical P. GERG2004 does not contain parameters for ethylene and propylene, therefore is excluded from the analysis of this mixed refrigerant composition.
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1
2
3
4
5
6
%Deviation Z
Operating Points
Deviation of Z for Mixed Refrigerant Gas
Z LKP
Z BWRSNS
Z SRK
Z PR
Z AGA8
14. Figure 7 & 8: Deviation of Z and k for Mixed Refrigerant Gas
Z: Compared to the primarily Methane mixture, some larger deviations can be observed. PR and BWRS-NS appears to be the more stable EOS; with a deviations below 1%. The other EOS demonstrates inconsistent trends even at lower P; AGA8 and LKP exceed 1% deviation at higher P to T ratios.
K: Again, trends appear inconsistent especially for AGA8. The most stable EOS appears to be BWRS-NS, however there is excellent correlation between EOS at Operating Points 1 and 3 (i.e. at relatively lower P values with sufficient T).
Overall, BWRS-NS seems like a safer option for Z and k to compare with NIST for this heavier hydrocarbon mixture especially at higher pressure and temperature conditions. However, PR performs relatively well too for computing Z. The other EOS are expected to deviate at least 1% at higher P.
-1.5
-1
-0.5
0
0.5
1
1.5
1
2
3
4
5
6
%Deviation k
Operating Points
Deviation of k for Mixed Refrigerant Gas
k LKP
k BWRSNS
k SRK
k PR
k AGA8
15. Pure Carbon Dioxide
Carbon Dioxide (CO2) is more difficult to model accurately, as its properties are far from an ideal gas. As it is a linear molecule, it has a high acentric factor (i.e. highly non-spherical). This affects the interaction between the molecules in the gas, yielding inaccurate values of Z and k if not taken into account. CO2 is of interest in oil & gas for CO2 reinjection, thus high pressure ranges are investigated for comparison.
Operating Point
1
2
3
4
5
6
7*
8
9
10
Pressure, bar a
32
48.1
54.1
62.9
79.6
95.6
100.6
416.8
481
520
Temp, K
310
350
320
370
400
420
320
400
400
430
Table 5: Selected Operating Points for Pure Carbon Dioxide
*The critical point of CO2 is at 73.77bara, 304.1K. Operating point 7 is reasonably close and therefore may result in anomalous results.
Figure 9 & 10: Deviation of Z and k for Pure CO2 Gas
-4
-2
0
2
4
6
8
10
12
14
1
2
3
4
5
6
7
8
9
10
%Deviation Z
Operating Points
Deviation of Z for Pure CO2
Z GERG
Z LKP
Z BWRSNS
Z SRK
Z PR
Z RK
-20
-15
-10
-5
0
5
10
1
2
3
4
5
6
7
8
9
10
%Deviation k
Operating Points
Deviation of k for Pure CO2
k GERG
k LKP
k BWRSNS
k SRK
k PR
k RK
16. Z: Below the critical point, LKP and BWRS-NS have the lowest deviations to NIST (<1%). At P=100.57bara and T=320K which is close to the critical point of CO2, most EOS have massive deviations with NIST. At this point, only LKP demonstrates astonishingly high correlation with NIST, while the other EOS deviates by at least 7%. RK performs surprisingly well throughout these conditions, deviating at most 2% throughout the range. RK does not take acentric factor into account when calculating Z and k, thus for CO2 it was expected that RK will have large deviations from NIST. Beyond the critical point - at supercritical conditions (Operating Points 8, 9, 10), some EOS exhibits larger inconsistencies to NIST; with SRK having up to 8% deviations. In general however, LKP and PR shows the most consistent deviation with NIST at around 1% even in supercritical regions.
k: Massive deviations can be seen throughout the range, with average deviations at least 2% among the EOS. This is because k is more sensitive than Z - especially near the critical point; k theoretically shoots to infinity while Z is not affected. Below the critical point, BWRS-NS and surprisingly RK demonstrate excellent correlation with NIST. In the supercritical region, each EOS deviates by large amounts with each other and thus, it is unsafe to draw a general conclusion as to the validity of the EOS models.
Thus, for Z it is generally safe to use LKP as the EOS with the lowest deviation to NIST. PR may be used for Z in the supercritical region (about 1% Deviation). However for k, it is advised there will be deviations between EOS of at least 2% in the supercritical region. Otherwise, below the critical point BWRS-NS and RK does comparatively well (<1.5% Deviation).
17. CO2 Gas mixture
For CO2 reinjection purposes, a 95% CO2 gas mixture is analysed. Again, similar operating points are selected inline with the pure CO2 gas for comparison purpose; It is expected that the trends of both graphs should be similar.
Operating Point
1
2
3
4
5
6
7**
8
9
10
Pressure, bar a
30.99
46.29
52.84
60.36
75.9
90.75
100.85
415.61
475.1
536.18
Temp, K
310
350
320
370
400
420
320
400
400
430
Table 6: Selected Operating Points for CO2 Gas Mixture
**The critical point of CO2 is at 73.77bara, 304.1K. Since this gas mixture consists of 95% CO2, the mixture’s critical point is expected to be very similar to pure CO2. Operating point 7 is reasonably close and therefore may result in anomalous results.
Figure 11 & 12: Deviation of Z and k for CO2 Gas Mixture
-2
0
2
4
6
8
1
2
3
4
5
6
7
8
9
10
%Deviation Z
Operating Points
Deviation of Z for CO2 Gas Mixture
Z GERG
Z LKP
Z BWRSNS
Z SRK
Z PR
Z AGA8
-10
-8
-6
-4
-2
0
2
4
6
8
10
1
2
3
4
5
6
7
8
9
10
%Deviation k
Operating Points
Deviation of k for CO2 Gas Mixture
k GERG
k LKP
k BWRSNS
k SRK
k PR
k AGA8
18. Z: Similar to the pure CO2 graph, LKP demonstrates consistently low deviations throughout the range. However for mixtures, the correlation between AGA8 and NIST is unrivalled; only deviating notably near the critical point. At P higher than Tc, LKP and PR compares reasonably well with NIST at 1% deviation but AGA8 correlates much better.
k: Similar to pure CO2, BWRS-NS correlates well with NIST below the critical point (<1% Deviation). Near the critical point, all of the EOS examined tend to deviate significantly; PR, SRK and BWRS-NS deviates around 8-10%. In the supercritical region, again it is not possible to establish with confidence the validity of the results due to the large deviations between EOS.
A similar conclusion for CO2 mixture can be drawn; LKP is fairly reliable for Z across the range, and BWRS-NS correlates well with NIST below the critical point for k. However, AGA8 demonstrates the best comparison with NIST for both Z and k for CO2.
Conclusion
The selection of a reliable EOS ensures more accurate calculation of a compressor’s power and discharge temperature. By establishing NIST as a datum, results of Z and k of different EOS for Mixed Refrigerant and CO2 duties were compared.
For predominantly methane based mixtures, most EOS agree well with each other as the properties of methane are well established (0.5% average deviation for Z and k). However, heavier hydrocarbon mixtures such as Mixed Refrigerants and CO2 gas demonstrate larger deviations among EOS. The results are summarised in Table 7. The following EOS are therefore recommended (with some caution):
Mixture
Recommended EOS
Remarks (% Deviation with respect to NIST EOS)
Z
k
Mixed Refrigerants
BWRS-NS, PR
BWRS-NS
<0.5% for Z and k (BWRS-NS),
<1% for Z (PR)
Pure CO2 (gas)
LKP/BWRS- NS
BWRS-NS
<1% for Z and <0.5% for k
Pure CO2 (supercritical)
LKP/PR
-
1% for Z, k inconclusive
CO2 Gas Mixtures (gas)
AGA8
AGA8
<0.2% for Z and <1% for k
LKP/BWRS- NS
BWRS-NS
<1% for Z and k
CO2 Gas Mixtures (supercritical)
LKP/PR
-
1% for Z, k inconclusive
Table 7: Summary of EOS comparisons with NIST EOS
The findings are generally in agreement with those of authorities such as Sandberg [17] and Lüdtke [18], where BWRS and LKP are already shown to be reliable EOS models.
This also emphasizes the need for vendors to justify their EOS selection when providing quotes to consultant engineers - especially for mixed refrigerants and CO2. Typically, datasheets of compressors provided by vendors do not clarify the EOS used in calculation of the Z and k values. Therefore, it is not possible to verify the values as different EOS will have deviations between each other. Clarification will ensure consistency between both parties, and consequently better confidence to the operator/end-user of the compressor.
19. References
[1] Marc J. Assael, J. P. M. Trusler, Thomas F. Tsolakis (1996) Thermophysical Properties of Fluids, Imperial College Press
[2] Heinz P. Bloch (2006) A Practical Guide to Compressor Technology 2nd Edition, John Wiley & Sons, Inc.
[3] Nishiumi H., Saito S. (1975) An Improved Generalized BWR Equation of State Applicable to Low Reduced Temperature, Journal of Chemical Engineering of Japan
[4] Nishiumi H., Saito S. (1977) Correlation of the Binary Interaction Parameter of The Modified Generalized BWR Equation of State, Journal of Chemical Engineering of Japan
[5] ISO 12213-2 (2006) Natural Gas-Calculation of compression factor (Part 2: Calculation using molar composition analysis)
[6] Kunz O., Klimeck R., Wagner W., Jaeschke M. (2007) The GERG-2004 Wide-Range Equation of State for Natural Gases and Other Mixtures, Lehrstuhl für Thermodynamik Ruhr-Universität Bochum Germany
[7] Lemmon, E.W., Huber, M.L., McLinden, M.O. (2013) NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1, National Institute of Standards and Technology, Standard Reference Data Program, Gaithersburg
[8] Nimtza M., Klatta M., Joachim H. Krautza (2011) Evaluation of the GERG-2008 Equation of State for the Simulation of Oxyfuel Systems, 2nd Oxyfuel Combustion Conference
[9] Raimondi L. (2010) Rigorous calculation of LNG flow reliefs using the GERG-2004 equation of state, 4th International Conference on Safety & Environment in Process Industry
[10] Yildiz T. (1996) Analytical Gas Pipeline Design Method Using The GERG Equation of State, European Petroleum Conference 22-24 October, Milan, Italy
[11] Mark R. Sandberg, Gary M. Colby (2014) Limitations of ASME PTC 10 in Accurately Evaluating Centrifugal Compressor Thermodynamic Performance, 42nd Turbomachinery Symposium
[12] Aicher W. (1993) Test of Process Turbocompressors Without CFC Gases, 22nd Turbomachinery Symposium
[13] Moore J., Lerche A., Delgado H., Allison T., Pacheco J. (2011) Development of Advanced Centrifugal Compressors and Pumps for Carbon Capture and Sequestration Applications, 40th Turbomachinery Symposium
[14] F-Chart Software (2014) Engineering Equation Solver REFPROP Interface, <http://www.fchart.com/ees/ees-refprop.php>
[15] AspenTech (2014) Aspen Properties®, <http://www.aspentech.com/products/aspen- properties.aspx>
[16] Yunus A. Cengel, Michael A. Boles (2005) Thermodynamics: An Engineering Approach 5th Edition, McGraw-Hill College, Boston, MA
[17] Mark R. Sandberg (2005) Equation of State Influences on Compressor Performance Determination, 34th Turbomachinery Symposium
[18] H. K. Lüdtke (2004) Process Centrifugal Compressor: Basics, Function,
Operation, Design, Application, 1st Edition, Springer
20. Appendix A – Gas Mixture Compositions
Gas composition
Natural Gas Mixture
Mixed Refrigerant
CO2 Gas Mixture
Mole fraction
Methane
0.829700
0.257800
0.041018
Nitrogen
0.060810
0.079780
0.001157
Carbon dioxide
0.029040
0.000000
0.947150
Ethane
0.042890
0.002530
0.005496
Propane
0.017600
0.016730
0.002540
n-Butane
0.005328
0.047480
0.000393
i-Butane
0.003198
0.215113
0.000558
n-Pentane
0.001838
0.000078
0.000131
i-Pentane
0.001929
0.001319
0.000210
n-Hexane
0.001500
0.000000
0.000245
n-Heptane
0.004157
0.000000
0.001103
Water
0.002010
0.000000
0.000000
Ethylene
0.000000
0.378900
0.000000
Propylene
0.000000
0.000270
0.000000
Appendix B – Tabulated values of Z and k of the gas compositions for various EOS
NIST, GERG, AGA8 and PR EOS calculations are done via REFPROP software. RK, SRK, LKP and BWRS-NS calculations are compiled by the authors independantly.
Pure Methane
Pressure, bar a
Temp, K
Z NIST
Z GERG
Z LKP
Z BWRS-NS
Z SRK
Z PR
Z RK
1.01
115
0.9671
0.9674
0.9667
0.9685
0.9704
0.9699
0.9691
%Deviation to NIST
0.00
0.03
-0.05
0.14
0.34
0.29
0.20
13.82
320
0.9820
0.9820
0.9826
0.9806
0.9830
0.9790
0.9808
%Deviation to NIST
0.00
0.00
0.06
-0.14
0.10
-0.30
-0.12
24.41
320
0.9687
0.9687
0.9697
0.9665
0.9706
0.9639
0.9667
%Deviation to NIST
0.00
0.00
0.11
-0.22
0.20
-0.49
-0.20
24.59628
350
0.9786
0.9786
0.9793
0.9763
0.9809
0.9750
0.9766
%Deviation to NIST
0.00
0.00
0.07
-0.24
0.24
-0.36
-0.20
105.6605
350
0.9280
0.9280
0.9315
0.9306
0.9429
0.9260
0.9248
%Deviation to NIST
0.00
0.00
0.38
0.28
1.61
-0.21
-0.34
56.42
380
0.9706
0.9706
0.9718
0.9675
0.9772
0.9667
0.9668
%Deviation to NIST
0.00
0.00
0.12
-0.31
0.68
-0.39
-0.39
128.94
380
0.9525
0.9525
0.9563
0.9583
0.9700
0.9532
0.9479
%Deviation to NIST
0.00
0.01
0.40
0.61
1.84
0.07
-0.48
Pure Methane
Pressure, bar a
Temp, K
k NIST
k GERG
k LKP
k BWRS-NS
k SRK
k PR
k RK
1.01
115
1.3693
1.3697
1.3760
1.3733
1.3680
1.3627
1.3709
%Deviation to NIST
0.00
0.03
0.49
0.29
-0.10
-0.48
0.12
13.82
320
1.3250
1.3250
1.3221
1.3222
1.3266
1.3293
1.3229
%Deviation to NIST
0.00
0.00
-0.21
-0.21
0.12
0.32
-0.16
24.41
320
1.3504
1.3503
1.3464
1.3469
1.3539
1.3579
1.3472
%Deviation to NIST
0.00
-0.01
-0.30
-0.26
0.26
0.56
-0.24
24.60
350
1.3232
1.3231
1.3197
1.3206
1.3252
1.3288
1.3201
%Deviation to NIST
0.00
-0.01
-0.26
-0.20
0.15
0.42
-0.23
105.66
350
1.4822
1.4818
1.4761
1.4844
1.4803
1.4905
1.4597
%Deviation to NIST
0.00
-0.02
-0.41
0.15
-0.13
0.56
-1.52
56.42
380
1.3445
1.3443
1.3401
1.3433
1.3465
1.3521
1.3380
%Deviation to NIST
0.00
-0.02
-0.33
-0.09
0.15
0.56
-0.48
128.94
380
1.4462
1.4456
1.4412
1.4502
1.4398
1.4490
1.4223
%Deviation to NIST
0.00
-0.04
-0.35
0.28
-0.45
0.19
-1.66
Natural Gas Mixture
21. Pressure, bar a
Temp, K
Z NIST
Z GERG
Z LKP
Z BWRS-NS
Z SRK
Z PR
Z AGA8
10.91
320
0.9811
0.9811
0.9819
0.9802
0.9817
0.9781
0.9811
%Deviation to NIST
0.00
0.00
0.08
-0.09
0.06
-0.31
0.00
21.42
320
0.9632
0.9632
0.9647
0.9617
0.9647
0.9578
0.9634
%Deviation to NIST
0.00
0.00
0.15
-0.16
0.16
-0.56
0.01
24.3423
350
0.9711
0.9711
0.9722
0.9693
0.9734
0.9665
0.9713
%Deviation to NIST
0.00
0.00
0.12
-0.19
0.24
-0.47
0.02
102.8026
350
0.9003
0.9004
0.9032
0.9049
0.9164
0.8972
0.9025
%Deviation to NIST
0.00
0.01
0.32
0.51
1.78
-0.35
0.24
55.93
380
0.9579
0.9579
0.9596
0.9558
0.9650
0.9528
0.9585
%Deviation to NIST
0.00
0.00
0.17
-0.22
0.74
-0.54
0.05
126.52
380
0.9286
0.9287
0.9315
0.9365
0.9480
0.9285
0.9305
%Deviation to NIST
0.00
0.01
0.32
0.85
2.09
-0.01
0.21
Natural Gas Mixture
Pressure, bar a
Temp, K
k NIST
k GERG
k LKP
k BWRS-NS
k SRK
k PR
k AGA8
10.91
320
1.2921
1.2921
1.2898
1.2899
1.2938
1.2958
1.2921
%Deviation to NIST
0.00
-0.01
-0.18
-0.18
0.13
0.28
0.00
21.42
320
1.3206
1.3205
1.3168
1.3172
1.3247
1.3279
1.3204
%Deviation to NIST
0.00
-0.01
-0.29
-0.26
0.31
0.55
-0.02
24.3423
350
1.2991
1.2990
1.2958
1.2964
1.3023
1.3053
1.2986
%Deviation to NIST
0.00
-0.01
-0.26
-0.21
0.25
0.47
-0.04
102.8026
350
1.4820
1.4817
1.4783
1.4798
1.4840
1.4923
1.4743
%Deviation to NIST
0.00
-0.02
-0.25
-0.14
0.14
0.70
-0.52
55.93
380
1.3258
1.3256
1.3215
1.3235
1.3296
1.3345
1.3241
%Deviation to NIST
0.00
-0.02
-0.32
-0.17
0.29
0.65
-0.13
126.52
380
1.4414
1.4410
1.4387
1.4411
1.4374
1.4448
1.4342
%Deviation to NIST
0.00
-0.03
-0.19
-0.02
-0.27
0.24
-0.50
Mixed Refrigerant
Pressure, bar a
Temp, K
Z NIST
Z LKP
Z BWRS-NS
Z SRK
Z PR
Z AGA8
3.35
300
0.9762
0.9781
0.9754
0.9765
0.9747
0.9722
%Deviation to NIST
0.00
0.19
-0.08
0.03
-0.16
-0.42
16.73
310
0.8870
0.8963
0.8835
0.8890
0.8808
0.8664
%Deviation to NIST
0.00
1.04
-0.40
0.22
-0.70
-2.33
16.73
390
0.9504
0.9552
0.9487
0.9526
0.9457
0.9440
%Deviation to NIST
0.00
0.50
-0.18
0.23
-0.49
-0.68
43.48
360
0.8225
0.8371
0.8180
0.8302
0.8149
0.7985
%Deviation to NIST
0.00
1.78
-0.55
0.94
-0.92
-2.92
43.48
400
0.8838
0.8945
0.8806
0.8919
0.8773
0.8705
%Deviation to NIST
0.00
1.21
-0.36
0.92
-0.73
-1.51
56.86
400
0.8498
0.8630
0.8470
0.8621
0.8450
0.8346
%Deviation to NIST
0.00
1.55
-0.34
1.45
-0.57
-1.79
Mixed Refrigerant
Pressure, bar a
Temp, K
k NIST
k LKP
k BWRS-NS
k SRK
k PR
k AGA8
3.35
300
1.1959
1.1943
1.1959
1.1955
1.1960
1.1859
%Deviation to NIST
0.00
-0.13
0.00
-0.03
0.01
-0.84
16.73
310
1.2778
1.2692
1.2798
1.2799
1.2796
1.2908
%Deviation to NIST
0.00
-0.67
0.16
0.17
0.14
1.02
16.73
390
1.1823
1.1801
1.1827
1.1859
1.1854
1.1750
%Deviation to NIST
0.00
-0.18
0.03
0.30
0.26
-0.62
43.48
360
1.3471
1.3331
1.3517
1.3606
1.3577
1.3659
%Deviation to NIST
0.00
-1.04
0.34
1.00
0.78
1.40
43.48
400
1.2513
1.2446
1.2530
1.2598
1.2589
1.2496
%Deviation to NIST
0.00
-0.54
0.14
0.68
0.61
-0.13
56.86
400
1.2989
1.2901
1.3023
1.3096
1.3074
1.2988
%Deviation to NIST
0.00
-0.67
0.26
0.82
0.65
-0.01
Pure Carbon Dioxide
Pressure, bar a
Temp, K
Z NIST
Z GERG
Z LKP
Z BWRS-NS
Z SRK
Z PR
Z RK