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.
This document discusses vapor/liquid equilibrium (VLE) and provides models for predicting VLE using simple models like Raoult's law and Henry's law. It defines key terms like mass fraction, mole fraction, molar concentration. Duhem's theorem is introduced which states that the equilibrium state is determined by fixing any two independent variables for a closed system. Simple calculations are shown for using Raoult's law to determine the bubble point and dew point temperatures and pressures of a binary system from its phase compositions or known temperature. P-x-y and T-x-y diagrams are used to illustrate the VLE behavior between the phases.
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.
Liquid liquid equilibrium for the ternary system of isopropyl acetate 2 propa...Josemar Pereira da Silva
The document presents experimental data on liquid-liquid equilibrium for the ternary system of isopropyl acetate, 2-propanol, and glycerol at temperatures of 298.15 K, 308.15 K, and 318.15 K under atmospheric pressure. Triangular phase diagrams were obtained at each temperature showing the two-phase region. Distribution coefficients and selectivity parameters were calculated to evaluate glycerol's capacity as an extractive solvent. The NRTL and UNIQUAC models were applied to correlate the experimental data with low deviations.
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.
This document compares different methods for predicting the density of liquid refrigerants, including correlations by ISH, Rackett, Yamada & Gunn, Spencer & Danner, Hankinson & Thomson, Reidel, and NM. It finds that the NM correlation best predicts densities for R22, R32, R134a, R152a, R600, and R12, while the Hankinson & Thomson method works best for R290, R600a, and R1270. The Reidel correlation accurately models R143a and R125, and the Yamada & Gunn and Spencer & Danner modifications are suitable for R123 and R718, respectively. Overall, the document evaluates methods for various refriger
1. The document examines a three-parameter representation of the equation of state that provides reasonably accurate results for modeling the thermodynamic properties of various substances.
2. It presents a "main term" equation of state based solely on critical constants like temperature and compressibility factor, which approximates behaviors near the critical point. Additional terms are needed to reduce discrepancies.
3. The author defines several auxiliary functions and divides the temperature-pressure space into regions, applying a different additional term to each region to capture deviations piecewise and improve accuracy overall. Mean deviations confirm a satisfactory algebraic representation using three individual parameters for most substances.
heat capacity of sitric acid0c96051e8eb63eea58000000Tika Ningsih
This document summarizes thermodynamic properties of the citric acid-water binary system determined through various experimental methods. Key findings include:
- The phase diagram exhibits eutectic and peritectic behavior with solid-liquid and vapor-liquid equilibria measured.
- Enthalpies of formation for citric acid monohydrate, solution, transition, and vaporization were determined via calorimetry and vapor pressure measurements.
- Specific heat capacity and solubility data were also collected to characterize the phase behavior and thermodynamic functions of citric acid and its monohydrate over a range of temperatures.
Liquid-Vapor Equilibria in Binary SystemsKarnav Rana
1) The document discusses liquid-vapor equilibria in binary systems, specifically measuring the compositions of chloroform and acetone mixtures using refractometry.
2) It introduces concepts like Raoult's law and Henry's law to describe ideal and non-ideal behavior in binary solutions, and how vapor pressure varies with composition.
3) Temperature-composition diagrams are used to visualize ideal and non-ideal behavior, including positive and negative deviations from ideality and the possibility of azeotropes.
This document discusses vapor/liquid equilibrium (VLE) and provides models for predicting VLE using simple models like Raoult's law and Henry's law. It defines key terms like mass fraction, mole fraction, molar concentration. Duhem's theorem is introduced which states that the equilibrium state is determined by fixing any two independent variables for a closed system. Simple calculations are shown for using Raoult's law to determine the bubble point and dew point temperatures and pressures of a binary system from its phase compositions or known temperature. P-x-y and T-x-y diagrams are used to illustrate the VLE behavior between the phases.
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.
Liquid liquid equilibrium for the ternary system of isopropyl acetate 2 propa...Josemar Pereira da Silva
The document presents experimental data on liquid-liquid equilibrium for the ternary system of isopropyl acetate, 2-propanol, and glycerol at temperatures of 298.15 K, 308.15 K, and 318.15 K under atmospheric pressure. Triangular phase diagrams were obtained at each temperature showing the two-phase region. Distribution coefficients and selectivity parameters were calculated to evaluate glycerol's capacity as an extractive solvent. The NRTL and UNIQUAC models were applied to correlate the experimental data with low deviations.
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.
This document compares different methods for predicting the density of liquid refrigerants, including correlations by ISH, Rackett, Yamada & Gunn, Spencer & Danner, Hankinson & Thomson, Reidel, and NM. It finds that the NM correlation best predicts densities for R22, R32, R134a, R152a, R600, and R12, while the Hankinson & Thomson method works best for R290, R600a, and R1270. The Reidel correlation accurately models R143a and R125, and the Yamada & Gunn and Spencer & Danner modifications are suitable for R123 and R718, respectively. Overall, the document evaluates methods for various refriger
1. The document examines a three-parameter representation of the equation of state that provides reasonably accurate results for modeling the thermodynamic properties of various substances.
2. It presents a "main term" equation of state based solely on critical constants like temperature and compressibility factor, which approximates behaviors near the critical point. Additional terms are needed to reduce discrepancies.
3. The author defines several auxiliary functions and divides the temperature-pressure space into regions, applying a different additional term to each region to capture deviations piecewise and improve accuracy overall. Mean deviations confirm a satisfactory algebraic representation using three individual parameters for most substances.
heat capacity of sitric acid0c96051e8eb63eea58000000Tika Ningsih
This document summarizes thermodynamic properties of the citric acid-water binary system determined through various experimental methods. Key findings include:
- The phase diagram exhibits eutectic and peritectic behavior with solid-liquid and vapor-liquid equilibria measured.
- Enthalpies of formation for citric acid monohydrate, solution, transition, and vaporization were determined via calorimetry and vapor pressure measurements.
- Specific heat capacity and solubility data were also collected to characterize the phase behavior and thermodynamic functions of citric acid and its monohydrate over a range of temperatures.
Liquid-Vapor Equilibria in Binary SystemsKarnav Rana
1) The document discusses liquid-vapor equilibria in binary systems, specifically measuring the compositions of chloroform and acetone mixtures using refractometry.
2) It introduces concepts like Raoult's law and Henry's law to describe ideal and non-ideal behavior in binary solutions, and how vapor pressure varies with composition.
3) Temperature-composition diagrams are used to visualize ideal and non-ideal behavior, including positive and negative deviations from ideality and the possibility of azeotropes.
This document derives thermodynamic relations using the Redlich-Kwong-Soave equation of state for nitrogen. It starts by evaluating the constants a(T) and b in the equation of state. It then expresses the equation of state in reduced form and calculates the critical compressibility factor. Using the equation of state, it derives expressions for departure enthalpy, entropy, and internal energy. It also expresses other properties like speed of sound, isothermal expansion exponent, heat capacities, and viscosity in reduced form. The purpose is to estimate various thermodynamic relations for nitrogen using the Redlich-Kwong-Soave equation of state.
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.
Gases are highly compressible and expand to fill their containers, with pressure inversely proportional to volume according to Boyle's Law. The properties and behavior of gases can be explained by the kinetic molecular theory, which models gases as large numbers of molecules in random motion. Real gases deviate from ideal gas behavior at high pressures and low temperatures due to intermolecular forces and molecular volumes.
Vapor-liquid equilibrium (VLE) describes the distribution of a chemical species between the gas and liquid phases at equilibrium. The concentration of a vapor in contact with its liquid depends on temperature, with vapor pressure strongly dependent on temperature. At equilibrium, the concentrations or partial pressures of vapor components and liquid component concentrations are related. VLE is described thermodynamically, with temperature, pressure, and chemical potentials equal between phases for single-component and multicomponent systems. VLE diagrams graphically represent vapor and liquid compositions. VLE is important for distillation column design in separation processes.
This document discusses an equation of state for gaseous solutions and the calculation of fugacities from pressure-volume-temperature data.
The key points are:
1) An equation of state containing two coefficients (a and b) is proposed that provides satisfactory results for calculating fugacities of gaseous solutions above their critical temperature over a wide pressure range.
2) The coefficients a and b are shown to depend on the composition of the gas mixture in a nonlinear way based on the mole fractions and properties of the pure components.
3) Calculating fugacities requires differentiating the equation of state with respect to mole fraction, so an algebraic representation using the proposed equation of state is preferable
This document provides information about an advanced chemical engineering thermodynamics course, including:
1) The course covers basic definitions, concepts, relationships for pure components and mixtures including pvT relationships and thermodynamic property relationships.
2) Relevant textbooks are listed for reference.
3) Methods for determining pvT properties of pure components and mixtures are discussed, including experimental determination, databases, equations of state, and process simulators.
4) The Lydersen and Pitzer methods for corresponding states are summarized, which use critical compressibility factor and acentric factor respectively as third parameters to determine compressibility factor from reduced temperature and pressure.
This document defines key terms used to explain phase equilibria and the phase rule. It discusses:
- A system, phase, components, number of phases (P), and number of components (C)
- The phase rule relationship that the degree of freedom (F) of a system equals the number of components (C) minus the number of phases (P) plus two.
- Examples of applying the phase rule to systems with different numbers of phases and components, such as a one-component system with ice, liquid water, and water vapor having three phases and zero degrees of freedom.
The document discusses various thermodynamic concepts including:
1) Reversible and irreversible processes, with reversible processes proceeding in both directions and irreversible only proceeding in one direction.
2) Extensive and intensive properties, with extensive depending on amount and intensive independent of amount.
3) Types of processes like isothermal, isobaric, isochoric, cyclic, and adiabatic classified based on constant temperature, pressure, volume, state functions, and no heat transfer.
4) First law of thermodynamics stating energy is conserved and can be converted between forms.
5) Free energy and how it relates to available work for a system.
This document discusses applying nonparametric regression to existing ideal gas specific heat correlations. It aims to identify inconsistencies in the coefficients of existing correlations and generalize the correlations. It analyzes data on four groups of gases (n-Alkane, Alkene, Naphthene, Alcohol) and finds some coefficients within groups lay off the trend, indicating inconsistencies. A nonparametric regression technique is used to correlate the coefficients to physical parameters of the gases, smoothing inconsistencies. When applied to n-Alkanes, the new correlation reproduces existing coefficients with less than 3% average error and specific heat values with about 1% error, validating the approach.
This document discusses thermodynamic properties of fluids, including:
1) Derivations of equations relating the primary thermodynamic properties of pressure, volume, temperature, internal energy, and entropy for homogeneous phases and fluids.
2) Calculations of changes in enthalpy, entropy, and internal energy based on changes in pressure and temperature.
3) The thermodynamic properties of Gibbs energy and residual properties.
4) An example problem calculating the enthalpy and entropy of saturated isobutane vapor at a specified temperature and pressure using compressibility factor and ideal gas heat capacity data.
The characteristics of SecondaryCharged Particlesproduced in 4.5 A GeV/c 28Si...IOSR Journals
To study the characteristics of secondary charged particles produced in 4.5GeV/c 28Si-nucleus interactions a lot of rigorous attempts have been made. The results reveal that the multiplicity correlations are not linear. The findings do not agree with those reported by several earlier workers. However, these correlations may be reproduced quite well by second order polynomial. It is also observed that the dependence of mean normalized, RA and reduced multiplicity, RS on the multiplicity of different charged secondaries is linear up to a certain value and then acquire almost a constant value. Results also reveal that the Kth root of central moment increases with the increase of <ns> and the values of normalized moments do not depend on the nature and the energy of the projectiles. Finally, it is observed that the integral multiplicity distribution of heavily ionizing tracks provide a method for selecting the disintegrations caused by the projectile due to different target nuclei of nuclear emulsion.
This document discusses the third law of thermodynamics. It states that the entropy of a perfectly crystalline substance is zero at absolute zero temperature. The mathematical expressions for determining absolute entropy are provided. The document also discusses Nernst's heat theorem, which states that the change in Gibbs free energy of a reaction approaches the change in enthalpy as temperature approaches absolute zero. Exceptions to the third law for certain gases with non-ordered crystal structures are also noted.
Partial gibbs free energy and gibbs duhem equationSunny Chauhan
Partial gibbs free energy and gibbs duhem equation,relation between binary solution,relation between partiaL properties,PARTIAL PROPERTIES,PARTIAL PROPERTIES IN BINARY SOLUTION,RELATIONS AMONG PARTIAL PROPERTIES,Maxwell relation,Examples
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity at any temperature within the studied ranges.
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity based on fundamental physical property data.
The document provides notes on equilibrium chemistry concepts and calculations. It defines equilibrium constants Kc and Kp, and explains how to calculate them using concentration or pressure data for reversible reactions at equilibrium. It also discusses how the reaction quotient Q relates to the direction a reaction will shift to reach equilibrium. Sample equilibrium problems are worked through step-by-step to demonstrate setting up reaction tables and solving for unknown concentrations and constants.
The document discusses chemical equilibrium, including:
1) All physical and chemical changes tend toward a state of equilibrium according to Le Chatelier's principle.
2) At dynamic equilibrium, the rates of the forward and reverse reactions are equal and the reaction quotient equals the equilibrium constant.
3) Equilibrium constants can be calculated using concentration values and reaction quotients can indicate the direction a reaction will proceed to reach equilibrium.
This document presents a new correlation for predicting the thermal conductivity of liquids based on their molar polarization. The correlation relates two parameters (A and b) from an existing thermal conductivity equation to molar polarization. Molar polarization takes into account molecular structure, polarity, and temperature effects. Experimental thermal conductivity data for various substances was used to develop a correlation between the parameters and molar polarization. The new correlation was found to predict thermal conductivity with average errors less than 5% for most substances tested, outperforming some existing methods.
This document derives thermodynamic relations using the Redlich-Kwong-Soave equation of state for nitrogen. It starts by evaluating the constants a(T) and b in the equation of state. It then expresses the equation of state in reduced form and calculates the critical compressibility factor. Using the equation of state, it derives expressions for departure enthalpy, entropy, and internal energy. It also expresses other properties like speed of sound, isothermal expansion exponent, heat capacities, and viscosity in reduced form. The purpose is to estimate various thermodynamic relations for nitrogen using the Redlich-Kwong-Soave equation of state.
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.
Gases are highly compressible and expand to fill their containers, with pressure inversely proportional to volume according to Boyle's Law. The properties and behavior of gases can be explained by the kinetic molecular theory, which models gases as large numbers of molecules in random motion. Real gases deviate from ideal gas behavior at high pressures and low temperatures due to intermolecular forces and molecular volumes.
Vapor-liquid equilibrium (VLE) describes the distribution of a chemical species between the gas and liquid phases at equilibrium. The concentration of a vapor in contact with its liquid depends on temperature, with vapor pressure strongly dependent on temperature. At equilibrium, the concentrations or partial pressures of vapor components and liquid component concentrations are related. VLE is described thermodynamically, with temperature, pressure, and chemical potentials equal between phases for single-component and multicomponent systems. VLE diagrams graphically represent vapor and liquid compositions. VLE is important for distillation column design in separation processes.
This document discusses an equation of state for gaseous solutions and the calculation of fugacities from pressure-volume-temperature data.
The key points are:
1) An equation of state containing two coefficients (a and b) is proposed that provides satisfactory results for calculating fugacities of gaseous solutions above their critical temperature over a wide pressure range.
2) The coefficients a and b are shown to depend on the composition of the gas mixture in a nonlinear way based on the mole fractions and properties of the pure components.
3) Calculating fugacities requires differentiating the equation of state with respect to mole fraction, so an algebraic representation using the proposed equation of state is preferable
This document provides information about an advanced chemical engineering thermodynamics course, including:
1) The course covers basic definitions, concepts, relationships for pure components and mixtures including pvT relationships and thermodynamic property relationships.
2) Relevant textbooks are listed for reference.
3) Methods for determining pvT properties of pure components and mixtures are discussed, including experimental determination, databases, equations of state, and process simulators.
4) The Lydersen and Pitzer methods for corresponding states are summarized, which use critical compressibility factor and acentric factor respectively as third parameters to determine compressibility factor from reduced temperature and pressure.
This document defines key terms used to explain phase equilibria and the phase rule. It discusses:
- A system, phase, components, number of phases (P), and number of components (C)
- The phase rule relationship that the degree of freedom (F) of a system equals the number of components (C) minus the number of phases (P) plus two.
- Examples of applying the phase rule to systems with different numbers of phases and components, such as a one-component system with ice, liquid water, and water vapor having three phases and zero degrees of freedom.
The document discusses various thermodynamic concepts including:
1) Reversible and irreversible processes, with reversible processes proceeding in both directions and irreversible only proceeding in one direction.
2) Extensive and intensive properties, with extensive depending on amount and intensive independent of amount.
3) Types of processes like isothermal, isobaric, isochoric, cyclic, and adiabatic classified based on constant temperature, pressure, volume, state functions, and no heat transfer.
4) First law of thermodynamics stating energy is conserved and can be converted between forms.
5) Free energy and how it relates to available work for a system.
This document discusses applying nonparametric regression to existing ideal gas specific heat correlations. It aims to identify inconsistencies in the coefficients of existing correlations and generalize the correlations. It analyzes data on four groups of gases (n-Alkane, Alkene, Naphthene, Alcohol) and finds some coefficients within groups lay off the trend, indicating inconsistencies. A nonparametric regression technique is used to correlate the coefficients to physical parameters of the gases, smoothing inconsistencies. When applied to n-Alkanes, the new correlation reproduces existing coefficients with less than 3% average error and specific heat values with about 1% error, validating the approach.
This document discusses thermodynamic properties of fluids, including:
1) Derivations of equations relating the primary thermodynamic properties of pressure, volume, temperature, internal energy, and entropy for homogeneous phases and fluids.
2) Calculations of changes in enthalpy, entropy, and internal energy based on changes in pressure and temperature.
3) The thermodynamic properties of Gibbs energy and residual properties.
4) An example problem calculating the enthalpy and entropy of saturated isobutane vapor at a specified temperature and pressure using compressibility factor and ideal gas heat capacity data.
The characteristics of SecondaryCharged Particlesproduced in 4.5 A GeV/c 28Si...IOSR Journals
To study the characteristics of secondary charged particles produced in 4.5GeV/c 28Si-nucleus interactions a lot of rigorous attempts have been made. The results reveal that the multiplicity correlations are not linear. The findings do not agree with those reported by several earlier workers. However, these correlations may be reproduced quite well by second order polynomial. It is also observed that the dependence of mean normalized, RA and reduced multiplicity, RS on the multiplicity of different charged secondaries is linear up to a certain value and then acquire almost a constant value. Results also reveal that the Kth root of central moment increases with the increase of <ns> and the values of normalized moments do not depend on the nature and the energy of the projectiles. Finally, it is observed that the integral multiplicity distribution of heavily ionizing tracks provide a method for selecting the disintegrations caused by the projectile due to different target nuclei of nuclear emulsion.
This document discusses the third law of thermodynamics. It states that the entropy of a perfectly crystalline substance is zero at absolute zero temperature. The mathematical expressions for determining absolute entropy are provided. The document also discusses Nernst's heat theorem, which states that the change in Gibbs free energy of a reaction approaches the change in enthalpy as temperature approaches absolute zero. Exceptions to the third law for certain gases with non-ordered crystal structures are also noted.
Partial gibbs free energy and gibbs duhem equationSunny Chauhan
Partial gibbs free energy and gibbs duhem equation,relation between binary solution,relation between partiaL properties,PARTIAL PROPERTIES,PARTIAL PROPERTIES IN BINARY SOLUTION,RELATIONS AMONG PARTIAL PROPERTIES,Maxwell relation,Examples
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity at any temperature within the studied ranges.
This document presents a new two-parameter model for predicting the thermal conductivity of liquids. The model is derived based on theories of molecular dynamics, liquid free volume, and the authors' previous model for estimating heat of vaporization. Thermal conductivity data for 68 liquids over a wide range of temperatures are fitted using the model and compared to other existing models, showing good agreement. Key parameters A and B are determined for different liquids to allow calculation of thermal conductivity based on fundamental physical property data.
The document provides notes on equilibrium chemistry concepts and calculations. It defines equilibrium constants Kc and Kp, and explains how to calculate them using concentration or pressure data for reversible reactions at equilibrium. It also discusses how the reaction quotient Q relates to the direction a reaction will shift to reach equilibrium. Sample equilibrium problems are worked through step-by-step to demonstrate setting up reaction tables and solving for unknown concentrations and constants.
The document discusses chemical equilibrium, including:
1) All physical and chemical changes tend toward a state of equilibrium according to Le Chatelier's principle.
2) At dynamic equilibrium, the rates of the forward and reverse reactions are equal and the reaction quotient equals the equilibrium constant.
3) Equilibrium constants can be calculated using concentration values and reaction quotients can indicate the direction a reaction will proceed to reach equilibrium.
This document presents a new correlation for predicting the thermal conductivity of liquids based on their molar polarization. The correlation relates two parameters (A and b) from an existing thermal conductivity equation to molar polarization. Molar polarization takes into account molecular structure, polarity, and temperature effects. Experimental thermal conductivity data for various substances was used to develop a correlation between the parameters and molar polarization. The new correlation was found to predict thermal conductivity with average errors less than 5% for most substances tested, outperforming some existing methods.
This document provides an overview of wellbore hydraulics and the theoretical basis for calculating static bottomhole pressures in gas wells. It discusses the energy relationships involved in fluid flow, including terms for internal energy, kinetic energy, potential energy, pressure energy, heat, and work. It also covers irreversibility losses due to friction. The document presents equations that relate these factors and can be used to directly calculate static bottomhole pressures based on known surface conditions, using tables of pseudocritical properties to account for gas compressibility with varying pressure.
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
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.
Equations For Heavy Gases In Centrifugalsdieselpub
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 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.
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.
This document discusses thermodynamic properties of fluids and covers several key topics:
1) It outlines the objectives of studying thermodynamic properties, including developing property relations from the first and second laws of thermodynamics.
2) Fundamental thermodynamic properties like pressure, volume, temperature, and entropy are defined.
3) The first and second laws of thermodynamics are summarized, including equations for closed systems and ideal gases.
4) Methods for determining thermodynamic properties from tables or correlations are introduced.
SU(3) Symmetry in hafnium isotopes with even neutron N=100-108IJAAS Team
In this paper, we have reviewed the calculation of ground states energy level up to spin 14+, electric quadrupole moments up to spin 12+, and reduced transition probabilities of Hafnium isotopes with even neutron N = 100-108 by Interacting Boson Model (IBM-1). The calculated results are compared with previous available experimental data and found good agreement for all nuclei. Moreover, we have studied potential energy surface of those nuclei. The systematic studies of quadrupole moments, reduced transition strength, yrast level and potential energy surface of those nuclei show an important property that they are deformed and have dynamical symmetry SU(3) characters.
This document presents a study of new Bianchi type-IX cosmological models with a binary mixture of anisotropic dark energy and perfect fluid. The models are analyzed using the statefinder diagnostic pair (sr,). Field equations are derived for the metric and energy-momentum tensors. Assuming the energy conservation equations vanish separately and a special law for the Hubble parameter, the field equations yield power law and exponential type solutions. The statefinder parameters are used to characterize different phases of the universe, and properties of the models such as anisotropy and expansion rates are discussed.
This document provides an overview of a chemical engineering course on thermodynamics 2. The course covers topics such as thermodynamic properties of fluid mixtures, phase equilibria, solution thermodynamics, and chemical reaction equilibria. It lists recommended reference books and the grading breakdown. Key concepts that will be discussed include thermodynamic properties of fluids mixtures, Maxwell equations, residual properties, Gibbs energy as a generating function, phase equilibrium criteria for vapor-liquid systems, and the Clapeyron equation. Assignments and exams make up 10% and 90% of the final grade respectively.
GDQ SIMULATION FOR FLOW AND HEAT TRANSFER OF A NANOFLUID OVER A NONLINEARLY S...AEIJjournal2
This paper presents the generalized differential quadrature (GDQ) simulation for analysis of a nanofluid
over a nonlinearly stretching sheet. The obtained governing equations of flow and heat transfer are
discretized by GDQ method and then are solved by Newton-Raphson method. The effects of stretching
parameter, Brownian motion number (Nb), Thermophoresis number (Nt) and Lewis number (Le), on the
concentration distribution and temperature distribution are evaluated. The obtained results exhibit that
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This document summarizes research on the structure and phase behavior of Cu-Ni nanoalloys. It uses thermodynamic modeling to predict how the phase diagrams and mixing/demixing behavior depend on nanoparticle size, shape, and temperature. The modeling indicates Cu-Ni nanoalloys can form either mixed particles or Janus particles, depending on the synthesis temperature, with nickel preferentially segregating to surfaces. Phase maps are provided to guide experimentalists on controlling particle structure.
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Advanced control scheme of doubly fed induction generator for wind turbine us...IJECEIAES
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6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
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Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
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Low power architecture of logic gates using adiabatic techniquesnooriasukmaningtyas
The growing significance of portable systems to limit power consumption in ultra-large-scale-integration chips of very high density, has recently led to rapid and inventive progresses in low-power design. The most effective technique is adiabatic logic circuit design in energy-efficient hardware. This paper presents two adiabatic approaches for the design of low power circuits, modified positive feedback adiabatic logic (modified PFAL) and the other is direct current diode based positive feedback adiabatic logic (DC-DB PFAL). Logic gates are the preliminary components in any digital circuit design. By improving the performance of basic gates, one can improvise the whole system performance. In this paper proposed circuit design of the low power architecture of OR/NOR, AND/NAND, and XOR/XNOR gates are presented using the said approaches and their results are analyzed for powerdissipation, delay, power-delay-product and rise time and compared with the other adiabatic techniques along with the conventional complementary metal oxide semiconductor (CMOS) designs reported in the literature. It has been found that the designs with DC-DB PFAL technique outperform with the percentage improvement of 65% for NOR gate and 7% for NAND gate and 34% for XNOR gate over the modified PFAL techniques at 10 MHz respectively.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
1. Palmer, H. J., Berg, J. C., J. FIuidMech., 51, 385 (1972).
Palmer, H. J.. Berg, J. C., A./.Ch.€. J.,19, 1082 (1973).
Plevan,R. E., Quinn, J. A,. A./.Ch.€. J., 12, 894 (1966).
Rhodes, F. H.. Bridges, C. H., Ind. fng. Chem., 30, 1401 (1938).
Robb. I. D., Alexander, A. E., J. Co/Ioidlnterface Sci., 28, 1 (1968).
Rosano, H., LaMer. V. K.. J. Phys. Chem., 60, 348 (1956).
Springer, T. G., Pigford, R. L., Ind. fng. Chem.. Fundam., 9, 458 (1970).
Van Stralen, S. J. D., Nether., J. Agric. Sci., 4, 107 (1956).
Van Stralen, S.J. 0.. lnt. J. Heat Mass Transfer, I O , 1469 (1967).
Van Straien, S. J. D., 4th Intl. Heat Transfer Conf., Paris-Versailles, "Heat
Transfer 1970", Vol. VI, paper B.7.b, 1970.
Received for reuiew December 11,1974
Accepted October 14,1975
This work was supported by the Office of Saline Water under Con-
tract No. 14-30-2964 and No. 14-30-2572.
A New Two-Constant Equation of State
Ding-Yu Peng and Donald B. Robinson'
Department of Chemical Engineering, Universityof Alberta, Edmonton, Alberta, Canada
The development of a new two-constant equation of state in which the attractive pressure term of the semiem-
pirical van der Waals equation has been modified is outlined. Examples of the use of the equation for predicting
the vapor pressure and volumetric behavior of singie-component systems, and the phase behavior and volu-
metric behavior of binary, ternary, and multicomponent systems are given. The proposed equation combines
simplicity and accuracy. It performs as well as or better than the Soave-Redlich-Kwong equation in all cases
tested and shows its greatest advantages in the prediction of liquid phase densities.
Introduction
Ever since the appearance of the van der Waals equation
in 1873 (van der Waals, 1873),many authors have proposed
variations in the semiempirical relationship. One of the
most successful modifications was that made by Redlich
and Kwong (1949). Since that time, numerous modified Re-
dlich-Kwong (RK) equations have been proposed (Redlich
and Dunlop, 1963; Chueh and Prausnitz, 1967; Wilson,
1969; Zudkvitch and Joffe, 1970; and others). Some have
introduced deviation functions to fit pure substance PVT
data while others have improved the equation's capability
€or vapor-liquid equilibrium (VLE) predictions. A review
of some of the modified RK equations has been presented
(Tsonopoulos and Prausnitz, 1969).One of the more recent
modifications of the RK equation is that proposed by
Soave (1972). The Soave-Redlich-Kwong (SRK) equation
has rapidly gained acceptance by the hydrocarbon process-
ing industry because of the relative simplicity of the equa-
tion itself as compared with the more complicated BWRS
equation (Starling and Powers, 1970; Lin et al., 1972) and
because of its capability for generating reasonably accurate
equilibrium ratios in VLE calculations.
However, there still are some shortcomings which the
SHK equation and the original RK equation have in com-
mon. l h e most evident is the failure to generate satisfacto-
ry density values for the liquid even though the calculated
vapor densities are generally acceptable. This fact is illus-
trated in Figure 1 which shows the comparison of the spe-
cific volumes of n-butane in its saturated states. The litera-
ture values used for the comparison were taken from Star-
ling (1973). It can be seen that the SRK equation always
predicts specific volumes for the liquid which are greater
than the literature values and the deviation increases from
about 7% at reduced temperatures below 0.65 to about 27%
when the critical point is approached. Similar results have
been obtained for other hydrocarbons larger than methane.
For small molecules like nitrogen and methane the devia-
tions are smaller.
Although one cannot expect a two-constant equation of
state to give reliable predictions for all of the thermody-
namic properties, the demand for more accurate predic-
tions of the volumetric behavior of the coexisting phases in
VLE calculations has prompted the present investigation
into the possibility that a new simple equation might exist
which would give better results than the SRK equation. In
this paper, an equation is presented which gives improved
liquid density values as well as accurate vapor pressures
and equilibrium ratios.
Formulation of the Equation
Semiempirical equations of state generally express pres-
sure as the sum of two terms, a repulsion pressure PRand
an attraction pressure PAas follows
P = P R + P A (1)
The equations of van der Waals (1873),Redlich and Kwong
(1949),and Soave (1972) are examples and all have the re-
pulsion pressure expressed by the van der Waals hard
sphere equation, that is
RT
P R = -
V - b
The attraction pressure can be expressed as
(3)
where g(u) is a function of the molar volume u and the con-
stant b which is related to the size of the hard spheres. The
parameter a can be regarded as a measure of the intermo-
lecular attraction force. Applying eq 1 at the critical point
where the first and second derivatives of pressure with re-
spect to volume vanish one can obtain expressions for a
and b at the critical point in terms of the critical proper-
ties. While b is usually treated as temperature indepen-
dent, a is constant only in van der Waals equation. For the
RK equation and the SRK equation, dimensionless scaling
Ind. Eng. Chem., Fundam., Vol. 15, No. 1, 1976 59
2. 1----
30 I----At temperatures other than the critical, we let
23
Z
Q
4
E 10
5
:
c
>
LU
iya
0
I 1 1
0 8 0 9 1 0
REDUCED TEMPERATURE
Figure 1. Comparison of predicted molar volumes for saturated
n-butane.
factors are used to describe the temperature dependence of
the energy parameter.
A study of the semiempirical equations having the form
of eq 1 indicates that by choosing a suitable function for
g ( u ) , the predicted critical compressibility factor can be
made to approach a more realistic value. The applicability
of the equation at very high pressures is affected by the
magnitude of blu, where u, is the predicted critical volume.
Furthermore, by comparing the original RK equation and
the SRK equation, it is evident that treating the dimen-
sionless scaling factor for the energy parameter as a func-
tion of acentric factor in addition to reduced temperature
has significantly improved the prediction of vapor pres-
sures for pure substances and consequently the equilibrium
ratios for mixtures.
We propose an equation of the form
p=--RT a ( T )
u - b U ( U + b ) + b(u - b)
(4)
Equation 4 can be rewritten as
Z" - (1- B)Z2+ (A - 3B2- 2B)Z - (AB - B2- B3)= 0
(5)
where
PU
Z = -
RT
Equation 5 yields one or three roots depending upon the
number of phases in the system. In the two-phase region,
the largest root is for the compressibility factor of the
vapor while the smallest positive root corresponds to that
of the liquid.
Applying eq 4 at the critical point we have
R2TC2
a (T,) = 0.45724 -
pc
RT
p ,
b(T,) = 0.07780
2, = 0.307
where cy( T,, w ) is a dimensionless function of reduced tem-
perature and acentric factor and equals unity at the critical
temperature. Equation 12 was also used by Soave (1972)
for his modified RK equation.
Applying the thermodynamic relationship
to eq 4, the following expression for the fugacity of a pure
component can be derived
In-=f Z - 1- l n ( Z - B) -- A In ('+ 2'414B) (15)
2 4 B Z-0.414BP
The functional form of a(T,,w ) was determined by using
the literature vapor pressure values (Reamer et al., 1942;
Rossini et al., 1953;Reamer and Sage, 1957; Starling, 1973)
and Newton's method to search for the values of cy to be
used in eq 5 and 15such that the equilibrium condition
is satisfied along the vapor pressure curve. With a conver-
gence criterion of If', -fvl I kPa about two to four it-
erations were required to obtain a value for cy at each tem-
perature.
For all substances examined the relationship between a
and T , can be linearized by the following equation
G'"' = 1+ K ( l - Trl'*) (17)
where K is a constant characteristic of each substance. As
shown in Figure 2, these constants have been correlated
against the acentric factors. The resulting equation is
K = 0.37464 + 1.54226~- 0 . 2 6 9 9 2 ~ ~ (18)
It is interesting to note that eq 17 is similar to that ob-
tained by Soave (1972) for the SRK equation although eq
17 is arrived at for each substance using vapor pressure
data from the normal boiling point to the critical point
whereas Soave used only the critical point and the calculat-
ed vapor pressure at T , = 0.7 based on the value of acentric
factor.
The fugacity coefficient of component k in a mixture can
be calculated from the following equation
In- f k = -bk (2 - 1)-In (Z - B ) -- * x
X k P b 2fiB
The mixture parameters used in eq 5 and 19 are defined by
the mixing rules
a = CCxLxJaLJ (20)
1 J
where
a,, = (1- 6,J)aL~'~aJ1~2 (22)
In eq 22 6,,is an empirically determined binary interac-
tion coefficient characterizing the binary formed by com-
ponent i and component j . Equation 22 has been used pre-
viously by Zudkevitch and Joffe (1970) for their modified
RK equation in calculating vapor-liquid equilibrium ratios.
60 Ind. Eng. Chern., Fundarn., Vol. 15, No. 1, 1976
3. Table I. Comparison of Vapor Pressure Predictions
Absolute error, psia Relative error, %
___
BIAS RMS AAD BIAS RMSN O . Of AAD
Sub- data _____
stance points SRK E q 4 SRK E q 4 SRK E q 4 SRK E q 4 SRK E q 4 SRK E q 4
c, 28 3.08
c, 27 1.12
c3 31 2.68
i-C, 27 1.83
n-C, 28 1.45
i-C, 15 0.64
n-C, 30 1.65
n-C, 29 2.86
n-C- 18 2.29
n-C, 16 2.61
N2 1 7 0.74
CO, 30 2.77
H,S 30 1.68
1.82
0.58
1.09
0.54
0.50
0.95
0.69
1.69
1.34
1.55
0.38
1.95
1.18
2.82 1.72 4.31
0.87 -0.58 1.38
2.66 1.06 3.37
1.78 0.50 2.33
1.38 0.03 2.05
0.22 -0.95 0.86
1.56 0.28 2.26
2.81 1.53 3.97
2.29 1.30 3.24
2.61 1.54 3.30
0.60 -0.10 1.07
2.73 -0.82 3.87
1.57 -0.53 2.52
n. OCTANE -
n -HEPTANE
,-PENTANE- CARBON DIOXIDE
n - BUTANE '
- PROPANE
HYDROGEN SULFIDE
NITROGEN
- - - METHANE
0 3 1 1 , I
01 0 2 0 3 0 A 0 5
ACENTIC FACTOR
Figure 2. Relationship between characterization constantsand ac-
entric factors.
In this study all's were determined using experimental bi-
nary VLE data. The value of 6ij obtained for each binary
was the one that gave a minimum deviation in the predict-
ed bubble point pressures. The importance of the interac-
tion coefficient is illustrated in Figure 3 for the binary sys-
tem isobutane-carbon dioxide (Besserer and Robinson,
1973). It can be seen that the use of an interaction coeffi-
cient has greatly improved the predictions.
The enthalpy departure of a fluid which follows eq 4 is
given by
da
T--a
H - H* = RT(Z - 1)+- dT In ( -t 2'44B ) (23)
2 ~ 5 b Z -0.414B
This is obtained by substituting eq 4 into the thermody-
namic equation
H - H* = RT(Z - 1)+ I-''[T(%)u - P ] du (24)
Comparisons
Since two-constant equations of state have their own
purposes we do not compare the equation obtained in this
study with the more complicated BWR (Benedict et al.,
1940) or BWRS equations although in some circumstances
these may give more accurate predictions at the expense of
more computer time and computer storage space. The fol-
lowing comparisons are intended to show that in regions
where engineering calculations are most frequently encoun-
tered better results can usually be obtained with the equa-
tion presented in this study than with the SRK equation.
The symbols AAD, BIAS, and RMS are used to denote re-
spectively the average absolute deviation, the bias, and the
2.83
0.65
1.47
0.71
0.62
1.48
0.95
2.65
2.02
2.08
0.48
2.44
1.42
1.44
0.70
0.98
1.06
0.75
0.46
0.92
1.55
1.51
1.99
0.56
0.53
0.66
0.66
0.34
0.36
0.32
0.37
0.54
0.58
0.90
0.79
1.04
0.31
0.62
0.96
0.47 0.38
-0.10 -0.34
0.87 0.31
0.82 0.16
0.47 -0.22
0.17 -0.53
0.50 -0.29
1.31 0.37
1.48 0.63
1.97 1.02
0.00 -0.02
0.50 -0.49
0.34 0.42
1.57
0.95
1.10
1.18
0.86
0.49
1.02
1.75
1.88
2.24
0.75
0.63
1.00
EXPERIMENTAL
JBESSERER AND ROBINSON 19731
TEMP 'F LIQUID VAPOR
I00 e o
220 m n
PREDICTED - EQUATION 141
6,,= o--
-6,,:0 I30
0._
g 800
Lu(z
v)
YI
Lu
(z0.
3
A00
0.77
0.38
0.42
0.34
0.42
0.60
0.66
1.06
1.04
1.26
0.37
0.71
1.48
0 2 0 4 06 0 8 I 0
MOLE FRACTION CARBON DIOXIDE
Figure 3. Pressure-equilibrium phase composition diagram for
isobutane-carbon dioxidesystem.
root-mean-squaredeviation
v
2 IdiI
i=l
AAD = -
N
f271 - . ,
N
where the d, are the errors (either absolute or relative) and
N is the number of data points.
Pure Substances
Vapor Pressures. Both the SRK equation and eq 4 are
designed with a view to reproduce accurately the vapor
pressures of pure nonpolar substances. Nevertheless eq 4
gives better agreement between calculated vapor pressures
and published experimental values. A comparison of the
predictions is presented in Table I for ten paraffins and
Ind. Eng. Chern., Fundarn., Vol. 15, No. 1, 1976 61
4. Table 11. Comparison of Enthalpy Departure Predictions
- -___
Error, BTU/lb
No. of AAD
data Temperature Pressure
Substance points Reference range, "F range, psia SRK Eq 4
Nitrogen 48 (Mage, 1963) -250-50 200-2000 0.57 1.13
Methane 35 (Jones, 1963) -250-50 250-2000 2.58 1.97
-
n-Pentane 160 (Lenoir, 1970) 75-700 200-1400 1.43 1.18
n-Octane 70 (Lenoir, 1970) 75-600 200-1400 2.47 2.43
Cyclohexane 113 (Lenoir, 1971) 300-680 200-1400 2.83 2.48
BIAS
SRK E q 4
0.24 -0.58
-1.13 -0.78
0.78 0.25
2.18 2.36
2.16 1.75
RMS
SRK E q 4
0.80 1.25
3.58 2.52
1.82 1.61
3.36 3.16
3.60 3.26
_ _ _ _ ~
Table 111. Compressibility Factor of the n-Butane-Carbon Dioxide System (Data by Olds et al., 1949)
Mole fraction n-Butane
Temp, Pressure,
"F psia
0.9
SRK Eq 4 Expt
0.5
SRK Eq 4 Expt SRK
100 600
1000
2000
3000
4000
280 600
1000
2000
3000
4000
460 600
1000
2000
3000
4000
0.170 0.151 0.158
0.279 0.248 0.260
0.542 0.482 0.510
0.793 0.707 0.753
1.037 0.926 0.989
0.320
0.536
0.740
0.934
0.830
0.730
0.690
0.808
0.950
0.289
0.482
0.665
0.840
0.804
0.696
0.643
0.744
0.869
0.274
0.489
0.694
0.888
0.818
0.694
0.627
0.746
0.895
0.242
0.452
0.649
0.838
0.804
0.665
0.584
0.702
0.838
0.928
0.889
0.843
0.871
0.942
three commonly encountered nonhydrocarbons. It can be
seen that the absolute errors are greatly reduced using eq 4
for all substances except isopentane. The slightly larger
overall relative errors shown for carbon dioxide and hydro-
gen sulfide resulted from the higher percentage errors at
the lower pressure regions. The consistently larger devia-
tion shown by eq 4 for isopentane could be due to errors in
the experimental results in the high-temperature region
where differences between the predicted and experimental
values as large as 4 psi (equivalent to 1%)occurred.
Densities. Generally, saturated liquid density values cal-
culated from the SRK equation are lower than literature
values. This is true except for small molecules like nitrogen
and methane at very low temperatures where the predicted
values are slightly higher. Equation 4 predicts saturated
liquid densities which are higher at low temperatures and
lower at high temperatures than the experimental values.
As an example, the prediction for n-butane is presented in
Figure 1.The fact that eq 4 gives a universal critical com-
pressibility factor of 0.307 as compared with SRK's 0.333
has improved the predictions in the critical region.
The specific volumes of saturated vapors have also been
compared. The results for n-butane are included in Figure
1. It can be seen that both equations yield acceptable
values except that in the critical region better results have
been obtained with eq 4.The improvement is more evident
for large molecules although both equations work well for
small molecules.
Enthalpy Departures. Experimental values of enthalpy
departures for five pure substances have been used to com-
pare with the values calculated from the SRK equation and
from eq 4. The results are presented in Table 11. It is fair to
say that both equations generate enthalpy values of about
the same reliability.
0.215
0.404
0.580
0.750
0.782
0.638
0.545
0.645
0.765
0,910
0.862
0.803
0.822
0.881
-
0.218
0.418
0.610
0.794
0.798
0.644
0.525
0.642
0.782
0.920
0.870
0.796
0.806
0.877
0.743
0.374
0.505
0.631
0.924
0.876
0.786
0.766
0.801
0.976
0.964
0.947
0.949
0.969
0.1
Eq 4 Expt
0.722 0.740
0.339 0.325
0.455 0.454
0.568 0.580
0.908 0.918
0.852 0.862
0.750 0.744
0.722 0.699
0.749 0.727
0.965 0.968
0.946 0.948
0.915 0.912
0.908 0.898
0.921 0.906
-
Mixtures
PVT Calculations. In order to illustrate the usefulness
of eq 4 for predicting the volumetric behavior of binary
mixtures in the single phase region, the n-butane-carbon
dioxide system studied by Olds and co-workers (1949) was
selected for comparison. For the SRK equation and eq 4
the interaction coefficients for this binary were 0.135 and
0.130, respectively. The compressibility factors at three
temperatures and five pressures for three compositions are
presented in Table 111. The improvement resulting from eq
4 is evident.
VLE Calculations. One of the advantages of using sim-
ple two-constant equations of state is the relative simplici-
ty with which they may be used to perform VLE calcula-
tions. Multi-constant equations of state, for example the
BWRS equation, require the use of iteration procedures to
solve for the densities of the coexisting phases while simple
equations like the SRK equation and eq 4 can be reduced
to cubic equations similar to eq 5 and the roots can be ob-
tained analytically.
Vapor-liquid equilibrium conditions for a number of
paraffin-paraffin binaries were predicted using eq 4. It was
found that the optimum binary interaction coefficients
were negligibly small for components with moderate differ-
ences in molecular size. However, systems involving compo-
nents having relatively large differences in molecular size
required the use of a nontrivial interaction coefficient in
order to get good agreement between predicted and experi-
mental bubble point pressures.
As an example of the use of eq 4 to predict the phase be-
havior of a ternary paraffin hydrocarbon system, the data
of Wichterle and Kobayashi (1972) on the methane-eth-
ane-propane system were compared to predicted values. As
62 Ind. Eng. Chem.,Fundam.,Vol. 15, No. 1, 1976
5. -PREDICTED EOUATION [ 4 !
v 200 p i D 4 600 p i
PROPANE 1 1
0 1 - r t -
1 1
0 06
0 04
0 0 3 ‘ I . ! ! ,
MOLE FRACTION OF ETHANE IN LIQUID PHASE
0 2 0 4 0 6 0 8 I O
Figure 4. Comparison of equilibrium ratios for methane-ethane-
propane system at -75°F.
indicated in Figure 4, agreement was excellent. No interac-
tion coefficients were used.
In a previous study (Peng et al., 1974), the binary inter-
action coefficients required for use with the SRK equation
were determined and used to predict the phase and volu-
metric behavior of selected systems used in a study by Yar-
borough (1972).Good agreement was obtained between the
predicted equilibrium ratios and the experimental values.
In this study these systems have been tested using eq 4 and
good results have also been obtained. The equilibrium ra-
tios for a mixture containing only paraffins are shown in
Figure 5. The volumetric behavior of this mixture is pre-
sented in Figure 6. Although both the SRK equation and
eq 4 generate reliable equilibrium ratios, the new equation
predicts much more accurate volumetric behavior. A paper
concerning the binary interaction coefficients required for
use in eq 4 for systems involving nonhydrocarbons is cur-
rently in preparation.
Conclusions
By modifying the attraction pressure term of the semi-
empirical van der Waals equation a new equation of state
has been obtained. This equation can be used to accurately
predict the vapor pressures of pure substances and equilib-
rium ratios of mixtures.
While the new equation offers the same simplicity as the
SRK equation and although both equations predict vapor
densities and enthalpy values with reasonable accuracy,
more accurate liquid density values can be obtained with
the new equation. In regions where engineering calculations
are frequently required the new equation gives better
agreement between predictions and experimental PVT
data.
Since two-constant equations have their inherent limita-
tions, and the equation obtained in this study is no excep-
tion, the justification for the new equation is the compro-
mise of its simplicity and accuracy.
0 EXPERIMENIAL
100
IYARBOROUGH 1972! -
PREDICTED. EQUATION 14) 1
40
20 -
10
6 -
-
% 4 -
9 2 -
5
5 1 0
g 0 6
s
-
- -
5 0 4 -3
0 2 -
0 1
006 -
004 -
-
-
e .
0 0 0 4
, I , , , , , , 1 , , , , , , I , , , I
10 30 50 io0 300 500 1000 3000
PRESSURE, psi0
Figure 5. Comparison of equilibrium ratios for Yarborough
ture no. 8 at ZOOOF.
300C
0._
2 2ooc
YI
Ly
3
YI
v)
u
Lya
1000
0
I I I I I I I--
,
/
/
0 EXPERIMENTAL
[YAREOROUGH,1972 1
-PREDICTED. EQUATION idl
I--PREDlClED S R K
1 l 1 , , , , ! , , , ,
2 4 6 8 io 12
VOLUME PERCENT LIQUID
mix-
Figure 6. Volumetric behavior of Yarborough mixture no. 8 at
ZOOOF.
Acknowledgment
The financial support provided by the Alberta Research
Council and the National Research Council of Canada for
this work is sincerely appreciated.
Nomenclature
A = constant defined by eq 6
a = attraction parameter
B = constant defined by eq 7
Ind. Eng. Chem., Fundam., Vol. 15,No. 1, 1976 63
6. b = van der Waals covolume
f = fugacity
H = enthalpy
P = pressure
R = gasconstant
T = absolute temperature
u = molar volume
z = mole fraction
2 = compressibility factor
Greek Letters
a = scaling factor defined by eq 12
6 = interaction coefficient
K = characteristic constant in eq 17
o = acentric factor
Superscripts
L = liquid phase
V = vapor phase
* = idealgas state
Subscripts
A = attraction
c = critical property
R = repulsion
r = reducedproperty
i,j , k = component identifications
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Received for review May 5, 1975
Accepted September 26,1975
Creation and Survival of Secondary Crystal Nuclei. The Potassium
Sulfate-Water System
Alan D.Randolph' and Subhas K. Sikdar
Department of ChemicalEngineering, Universityof Arizona, Tucson, Arizona 85727
Formation of secondary nuclei of K2S04 was observed in a continuous flow, mixed-magma crystallizer by
counting particles in the crystal effluent with a Coulter Counter. Spontaneous birth of secondary nuclei occurs
over at least the 1-5 p m size range. Only a fraction of originally formed nuclei survive to populate the larger
size ranges. The fraction of such surviving nuclei increases with the supersaturation level in the growing envi-
ronment. The number of originally formed nuclei depends on stirrer RPM, supersaturation, and the fourth mo-
ment of the parent crystal size distribution.
Introduction
In the past several years there has been an increasing
recognition of the importance of secondary nucleation as
grain source in typical crystallizers of the mixed-magma
type. The so-called MSMPR crystallizer with its simple
distribution form (Randolph and Larson, 1971) provided a
means of quantitatively measuring the effective nucleation
rate under realistic mixed-magma conditions. This led to
the correlation of such nucleation data in simple power-law
forms of the type
Bo = kx(T, RPM)M+si (Class I System) (la)
or
Bo = ky(T, RPM)M+Gi (Class I1 System) (Ib)
The dependence of these kinetics on agitation level and sol-
ids concentration together with a low-order supersatura-
tion dependence confirm a secondary mechanism which is
at variance with homogeneous nucleation theory.
Clontz and McCabe (1971) conducted a now-classical ex-
periment in which they demonstrated that nuclei could be
generated in a slightly supersaturated solution by low ener-
gy metal/crystal or crystal/crystal contacts. No visible
damage to the contacting crystals could be determined
even after continued secondary nuclei breeding in the ex-
periment. Ottens and de Jong (1973) and Bennett et al.
(1973) take the contact nucleation mechanisms detailed by
McCabe and hueristically derive the form of power-law ki-
netics to be expected in a mixed-magma crystallizer. These
formulations were supported with additional MSMPR
data.
64 Ind. Eng. Chem., Fundam.,Vol. 15,No. 1, 1976