Multiparameter Equations of State

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Multiparameter Equations of State

• Richard T JacobsenVivek Utgikar
• October 25, 2007

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MOTIVATION

• Where does thermodynamic data for design and
  analysis come from?
• How are tables and charts produced?
• What is the value of accuracy in properties for
  use in design?
• Where can you go to get information?

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Determination of Properties for Engineering
Design and Analysis
Modern Formulations equation of state correlation
Experimental Data
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Definitions

• The word "data" is used to refer to experimental
  measurements.
• The term "property formulation" is the set of
  equations used to calculate properties of a fluid
  at specified thermodynamic states defined by an
  appropriate number of independent variables.

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Definitions (continued)

• A typical thermodynamic property formulation is
  based on an equation of state.
• The general term "equation of state" is used to
  refer to an empirical model developed for
  calculating fluid properties in engineering
  applications.

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Definitions (continued)

• A modern equation of state allows the correlation
  and computation of all thermodynamic properties
  of the fluid, including properties such as
  entropy that cannot be measured directly.

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Equations of State--Variety

• Some are for limited or restricted ranges of
  temperature and pressure
• Some are limited in size for industrial use
• Some are wide range and cover liquid, vapor and
  two-phase states as well as supercritical fluid
  states.
• Some are based upon theoretical
  considerationsothers are empirical

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Modeling/Correlation

• Some equations of state are based upon literally
  thousands of experimental data points over the
  entire surface of state
• The model or correlation is produced by some form
  of data selection and analysis followed by
  least-squares fitting of a selected data
  seteither linear or nonlinear least squares may
  be used

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The Equation of State

• The data considered are experimental
  thermodynamic pressure-density-temperature
  (p-r-T) data, denoted as
• (p1, r1,T1), (p2, r2,T2), ..., (pn, rn,Tn).
  
• A functional relationship between the data
  variables is assumed, i.e.,
• f (?1, ..., ?j, r, T), where (?1, ..., ?j) are
  the coefficients of the terms in the equation
  used to represent the data.

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Linear Least Squares

• Numerical estimates of the coefficients may be
  calculated by minimizing the sum of the squares
  of the residuals between pi (a function of pi)
  and the value of f(?1, ..., ?j, ri, Ti). If the
  exact functional relationship between r, T, and p
  is known, the minimum-variance estimate of ?j is
  obtained by minimizing

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pi

• The value of pi in the expression for the least
  squares fit variable is a function of the
  measured pressure and perhaps other constants or
  properties
• The coefficients are the values to be determined
  by the fitting process

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Linear Least Squares (continued)

• This function is minimized using
• i 1, 2, ?, j
• where j is the number of terms (each with one
  adjustable coefficient, ?i) in the equation being
  developed. This results in j equations with j
  unknowns.

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Weighting

• Since errors of measurement in p-r-T data vary
  among experiments and are different in various
  thermodynamic regions (e.g., liquid or vapor
  phases), data are generally weighted for use in
  fitting.
• Weights are arbitrary or are calculated using
  the law of propagation of variance (the
  error-propagation formula).

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Weighting (continued)

• Where is the standard deviation of a
• particular data point from a preliminary or
• standard calculated value.

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Linear Least Squares (continued)

• The new function to be minimized is then

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Linear Least Squares (continued)

• In Matrix Form

where
are estimates of the coefficients for the
weighted least-squares fit.
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Equations of State

• Variables are frequently non-dimensionalized.
• Nondimensional parameters generally vary less in
  size than the measurements and are better behaved
  in statistical calculations.
• Constraints may be applied to least squares fits
  to force the equation to represent a state or
  property that is very well known nearly exactly.

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Constraints

• Where would you think a constraint might be
  appropriate on the fluid surface of state?

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REVIEW

• Some previous details

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Thermophysical Properties
Solid
Liquid-Solid
Triple Point
Liquid
DENSITY
Critical Point
Liquid-Vapor
Gas
Solid-Vapor
Critical Temperature
Vapor
TEMPERATURE
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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
• Present and publish resultschange as needed
• Develop a computer formulation for engineering
  applications

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Types of Equations of State

• Fundamental equations
• Pressure explicit equations
• Cubic equations
• Virial equations
• Extended corresponding states
• Statistical models

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Fixed Points Needed in the Development of an EOS

• Temperature, density, and pressure at the
  critical point
• 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
• Isochoric heat capacity data
• Speed of sound data
• Isobaric heat capacity data
• Second virial coefficients
• Shock tube data

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Back to the Modeling Process and Constraints

• Modeling is a complex process.
• There is considerable art as well as good science
  in the process.
• There are models for various applications that
  may or may not be suitable for other purposes.
• The user must be aware of both limitations and
  strengths of any model.

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Constraints
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Pressure Explicit Equation of State

• with the inverse reduced temperature
  t  Tc / T
• and the reduced density
• d  r / rc

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Thermodynamic Properties from Pressure-Explicit
Equations

• Enthalpy

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Thermodynamic Properties from Pressure-Explicit
Equations
Entropy
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Thermodynamic Properties from Pressure-Explicit
Equations Internal Energy
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Thermodynamic Properties from Pressure-Explicit
EquationsHeat Capacity at Constant Volume
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Thermodynamic Properties from Pressure-Explicit
Equations Heat Capacity at Constant Pressure
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The Benedict-Webb-Rubin Equation of State
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Strobridge Equation
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The Martin-Hou Equation of State
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Jacobsen-Stewart Equation
where g 1/rc2

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Fundamental Relations
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Fundamental Equation

• Explicit in Helmholtz Energy

where the are arbitrary functional
forms used to improve the representation of
properties in the critical region and which
depend on a vector of internal parameters.

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Equation of Keenan, Keyes, Hill and MooreSteam
t 1000 K/T
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Equation of Jacobsen, Stewart, Penoncello and
Jahangiri
t Tc /T, d r/rc
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The Unified Equation of Hill
is the "far field" function.
is the function "near" the critical point.
The "switching" function,
is unity at the critical point, and
is zero everywhere outside the critical region.
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The Unified Equation of Hill(continued)
The far-field equation,
, is given by
is
W1, W2, W3, and W4 are empirical correlations
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Equation of Span et al.Nitrogen
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Equation of Setzmann WagnerCarbon Dioxide
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Thermodynamic Properties from the Fundamental
EquationPressure
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Thermodynamic Properties from the Fundamental
Equation Internal Energy
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Thermodynamic Properties from the Fundamental
Equation
Entropy
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Thermodynamic Properties from the Fundamental
Equation Enthalpy
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Thermodynamic Properties from the Fundamental
Equation Gibbs Energy
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Thermodynamic Properties from the Fundamental
Equation Heat Capacity at Constant Volume
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Thermodynamic Properties from the Fundamental
Equation Heat Capacity at Constant Pressure
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Thermodynamic Properties from the Fundamental
Equation Speed of Sound
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Standard Property Formulations

• The next three slides contain the references for
  the most accurate available thermodynamic
  property models for the pure fluids of
  engineering interest as system working fluids.
  (As of 2000)
• References numbers in these slides are those in
  Equations of State for Fluids and Fluid
  Mixtures, Part 1, Chapter 18. You have a
  reprint of this chapter.

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Standard Property Formulations
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Standard Property Formulations(continued)
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Standard Property Formulations(continued)
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NIST REFPROP Database(Incorporates CATS ALLPROPS)

• Pure fluid equations and mixture models have been
  incorporated into one program.
• Properties can be calculated using an Excel
  spreadsheet in addition to the graphical
  interface.
• Version for argon, normal hydrogen, methane,
  nitrogen, oxygen, parahydrogen, propane, R134a,
  water, and air on the class web. Download and
  use as you wish for this course.

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Comparisons of Calculated Properties to
Experimental Data

• DATA VALUES ARE THE BASIS OF COMPARISONS.
• CALCULATED PROPERTY VALUES FROM MODELS ARE
  COMPARED TO DATA VALUES, NOT VICE-VERSA.
• STATISTICAL ANALYSIS OF COMPARISONS IS BASED ON
  STANDARDIZED TESTS WHICH GIVE A MEASURE OF THE
  UNCERTAINTY OF CALCULATED OR PREDICTED PROPERTIES.

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Statistical ComparisonsPercent Difference
Calculations
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Statistical ParametersAbsolute Average Deviation
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Statistical ParametersBias Standard Deviation
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Statistical ParametersRoot Mean Squared
Deviations
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Statistical Comparisons

• High values of AAD indicate a systematic or large
  random difference between the equation of state
  and the data.
• The BIAS is the average deviation of the
  calculated values from the data set.
• The standard deviation SDV gives an indication of
  the systematic or random dispersion of the data
  set about the BIAS value.
• The root-mean-square deviation gives a similar
  indication of systematic or random dispersion of
  the data.
• The data sets are accurately represented when the
  values of the parameters are near zero.

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BREAKAfter the break

• We will discuss comparisons of calculated
  properties to experimental data to evaluate and
  analyze the data and to estimate the
  uncertainties of calculated properties using the
  model being evaluated.
• Graphical comparisons are the most expedient
  format. The example will be the standard
  equation of state for air.
• We will also look at formulations for properties
  of fluid mixtures in the context of air as a
  mixture of nitrogen, oxygen and argon.