Anatomy of a Thermodynamic Property Formulation Properties of Air

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• Anatomy of a Thermodynamic Property Formulation
  --Properties of Air
• Richard T JacobsenVivek Utgikar
• October 25, 2007

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BRIEF REVIEW

• PRINCIPLES OF PROPERTY FORMULATION PROCESS

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The Process

• Locate and select experimental data
• Select a functional form
• Select a fitting method
• Develop a model that fits the data
• Compare calculated/predicted values to data
• Develop a computer formulation for engineering
  applications

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Properties Calculated from an EOS

• Temperature
• Pressure
• Density
• Heat capacity
• Speed of sound
• Energy
• Entropy
• Enthalpy

• Fugacity
• Second virial coef.
• Joule-Thomson coef.
• Volume expansivity
• Compressibility
• Vapor-liquid equilibrium

Cannot calculate viscosity and thermal
conductivity directly
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Fundamental Equation

• All thermodynamic properties can be calculated
  as derivatives from the fundamental equation
• Helmholtz energy as a function of temperature and
  density
• Both temperature and density are measurable.
• Continuous across two-phase region.

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Fixed Points Needed in the Development of an
Equation of State

• Temperature, density, and pressure at the
  critical point (maxcondentherm for air)
• Triple point temperature
• Molecular weight
• Molar gas constant
• Enthalpy and entropy reference values

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Experimental Data Used to Develop an Equation of
State

• Ideal gas heat capacity data
• Pressure-density-temperature data
• Vapor pressure data (dew and bubble point
  pressure and density data for air)
• Isochoric heat capacity data
• Speed of sound data
• Isobaric heat capacity data
• Second virial coefficients
• Shock tube data

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Techniques Used in Fitting

• Linear fitting
• Fast
• Selects an optimum set of terms from a large bank
  of terms
• Can fit multiple properties simultaneously, but
  isobaric heat capacity, sound speed, and phase
  boundary data must be linearized first with a
  preliminary equation
• Results in equations with 25-50 terms

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Techniques Used in Fitting (continued)

• Nonlinear fitting
• Time consuming
• Allows the exponents of the terms to float,
  decreasing the number of terms needed in the
  equation
• Can fit multiple properties simultaneously
  without the need to linearize
• Results in equations with 15-25 terms
• Both require many iterations using different
  selected data sets before the final equation is
  determined.

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Approximate Accuracies

• State of the Art Accuracy to be
• Calculated Property Experimental Expected from an
• Accuracy Equation of State
• Density 0.02 0.1
• Pressure 0.02
• Temperature 1 mK
• Isochoric rgtrc 0.5 0.5 Heat
  Capacity rltrc 1 1
• Isobaric rgtrc 0.5 1 Heat Capacity rltrc
  2 1
• Speed of Sound rgtrc 0.1 0.5 rltrc
  0.001 0.01
• Vapor Pressure plt0.1 MPa 0.05 0.5 pgt0.1
  MPa 0.01 0.1

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The Fundamental Equation
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Contributions to EOS

• Ideal gas Helmholtz energy
• Real fluid Helmholtz energy

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Ideal Gas Heat Capacity (for air)
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Thermodynamic Properties
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Accuracy and Thermodynamic Consistency

• All thermodynamic properties can be calculated
  within the limits of experimental uncertainty
• The equation of state reduces to the ideal gas
  equation of state in the limit as ?? 0
• The equation of state obeys the Maxwell criterion
  (equal pressures and Gibbs Functions for liquid
  and vapor states in equilibrium)
• The critical region behavior is reasonably
  consistent with experimental measurements and
  theoretical considerations except at and very
  near the critical point

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Accuracy and Thermodynamic Consistency (continued)

• The behavior of calculated constant property
  lines on the surface of state is consistent with
  available experimental data and with theoretical
  predictions (e.g., isotherms plotted using
  calculated values from the model should not
  intersect at high pressures)

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Equations of State for Mixtures

• Virial expansion
• Gas phase only
• Extended corresponding states
• Slow and sometimes nonconvergent
• Has the ability for high accuracy
• Coefficient mixing of multiparameter equations
• Requires fixed functional form for pure fluids
• Excess Helmholtz energy using ?,T
• High accuracy with high convergence

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• Recall that a virial equation is a power series
  in pressure or density
• Corresponding states is based upon the fact that
  the surfaces of state for different fluids appear
  to have similar geometric shapes and that
  conformal mapping could be used to map one fluid
  surface onto another with the right parameters
  and transformations.

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Excess Helmholtz Energy Mixture Model

• Excess property model explicit in Helmholtz
  energy
• Independent parameters are density and
  temperature
• Generalized/Predictive
• High accuracy
• Quicker than ECS models
• Requires accurate pure fluid equations of state

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Excess Helmholtz Energy Mixture Model (continued)

• Allows mixing of Helmholtz and BWR equations, and
  ECS models for the pure fluids.
• Calculates all thermodynamic properties,
  including heat capacities, speed of sound,
  vapor-liquid equilibria, liquid-liquid
  equilibria, and critical lines.

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Excess Helmholtz Energy Mixture Model (continued)

• Mixture model applied to systems containing
  normal hydrocarbons, cryogens, refrigerants, and
  carbon dioxide.
• With some modification of the reducing parameters
  for binary mixtures, the shapes of the "excess"
  properties are nearly identical.
• One set of coefficients is used to describe all
  binary mixtures. The coefficients model the
  "excess" properties of a mixture.

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Excess Helmholtz Energy Mixture Model (continued)

• Multicomponent mixtures do not require additional
  parameters.
• There are 10 terms in the functional form.
• Up to four additional parameters can be used to
  model a binary mixture
• Magnitude of excess properties
• Shape of the reducing line for temperature
  (especially useful for modeling azeotropes)
• Symmetry of reducing line for temperature
• Shape of reducing line for density

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Air Assumptions

• Air is dry
• Air is a ternary mixture composed of 78.12 N2,
  20.96 O2, and 0.92 Ar
• Can neglect trace elements
• Dissociation effects are negligible
• Can predict high pressure, high temperature
  values from nitrogen data

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Fixed Points for Air
Molecular weight 28.9585 g/mol Values used
for the reducing parameters in the equation of
state.
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Critical Points
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Ancillary Equations
Bubble and Dew Point Pressures
Bubble and Dew Point Densities
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Freezing Liquid Line
F 2.773234
t T/Ttrp , Ts 59.75 K, Ps 0.005265 MPa
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P-r-T Data for Air
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P-r-T Data for Air
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Second Virial Coefficients
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Ideal Curves

• Ideal curve
• Boyle curve
• Joule-Thomson inversion curve
• Joule inversion curve

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Hugoniot Curve
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Percent Deviation in Density for Air
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Percent Deviation in Density for Air
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Percent Deviation in Cv Data
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Percent Deviation in Sound Speed Data
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BREAKAfter the break, we will demonstrate the
use of REFPROP .