Title: Anatomy of a Thermodynamic Property Formulation Properties of Air
1- Anatomy of a Thermodynamic Property Formulation
--Properties of Air - Richard T JacobsenVivek Utgikar
- October 25, 2007
2BRIEF REVIEW
- PRINCIPLES OF PROPERTY FORMULATION PROCESS
3The 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
4Properties 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
5Fundamental 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.
6Fixed 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
7Experimental 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
8Techniques 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
9Techniques 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.
10Approximate 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
11The Fundamental Equation
12Contributions to EOS
- Ideal gas Helmholtz energy
- Real fluid Helmholtz energy
13Ideal Gas Heat Capacity (for air)
14Thermodynamic Properties
15Accuracy 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
16Accuracy 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)
17Equations 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
18 - 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.
19Excess 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
20Excess 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.
21Excess 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.
22Excess 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
23Air 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
24Fixed Points for Air
Molecular weight 28.9585 g/mol Values used
for the reducing parameters in the equation of
state.
25Critical Points
26Ancillary Equations
Bubble and Dew Point Pressures
Bubble and Dew Point Densities
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28Freezing Liquid Line
F 2.773234
t T/Ttrp , Ts 59.75 K, Ps 0.005265 MPa
29P-r-T Data for Air
30P-r-T Data for Air
31Second Virial Coefficients
32Ideal Curves
- Ideal curve
- Boyle curve
- Joule-Thomson inversion curve
- Joule inversion curve
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34Hugoniot Curve
35Percent Deviation in Density for Air
36Percent Deviation in Density for Air
37Percent Deviation in Cv Data
38Percent Deviation in Sound Speed Data
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46BREAKAfter the break, we will demonstrate the
use of REFPROP .