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Natural Gas Calculation - Helmholtz Energy

Symbols
Thermodynamic Properties
Functional Forms

The Helmholtz free energy is a thermodynamic potential that measures the »useful« work obtainable from a closed thermodynamic system at a constant temperature and volume. The Helmholtz energy is defined as

f ≡ u - T·s     (1)

All single-phase thermodynamic properties can be calculated as derivatives of the Helmholtz energy, as a function of temperature and density.

f(ρ,T) = f0(ρ,T) + fr(ρ,T)     (2)

The dimensionless Helmholtz energy Φ uses independent variables of dimensionless density and temperature.

Φ(δ,τ) = Φ0(δ,τ) + Φr(δ,τ)     (3)

For a certain composition of a mixture the ideal Helmholtz energy is

f0 = h0 - RT - Ts0 = TT0cp0dT + h00 - RT - T[TT0cp0/TdT - Rlnρ/ρ0 - RlnT/T0 + s00 - Rni=1xi·lnxi]     (4)

Φ0 = -τττ0cp0/R·τ2dτ + h00/Rτ - 1 + ττ0cp0/R·τdτ + lnδ/δ0 + lnτ0/τ - s00/R + ni=1xi·lnxi     (5)

The ideal gas part as well as the real fluid behavior is often described using empirical models.

Symbols

cp0
molar ideal gas specific isobaric heat capacity [kJ/(kmol·K)]
cp
molar isobaric heat capacity [kJ/(kmol·K)]
cv
molar isochoric heat capacity [kJ/(kmol·K)]
f0
ideal gas contribution to the molar Helmholtz energy [kJ/kmol]
f
molar Helmholtz free energy [kJ/kmol]
fr
residual part contribution to the molar Helmholtz energy [kJ/kmol]
h0
molar ideal gas entalpy [kJ/kmol]
h00
ideal gas integration constant for zero enthalpy at the reference state of 298.15 K and 0.101325 MPa [kJ/kmol]
M
molar mass [kg/kmol]
R
molar gas constant 8.31451 kJ/(kmol·K)
s0
molar ideal gas entropy [kJ/(kmol·K)]
s00
ideal gas integration constant for zero entropy at the reference state of 298.15 K and 0.101325 MPa [kJ/(kmol·K)]
s
molar entropy [kJ/(kmol·K)]
T
absolute temperature [K]
u
molar internal energy [kJ/kmol]
w
speed of sound [m/s]
xi
mole fraction of component i [-]
δ0
molar density at the reference state of 298.15 K and 0.101325 MPa [-]
δ
reduced density δ=K3·ρ [-], where K is a mixture size parameter using the constants from annex D of ISO 20765-1
κ
isentropic exponent [-]
μ
Joule-Thomson coefficient [K/MPa]
Φ0
ideal gas contribution to the reduced Helmholtz energy [-]
Φ
reduced Helmholtz energy f/(RT) [-]
Φr
residual part contribution to the reduced Helmholtz energy [-]
ρ
molar density [kmol/m3]
τ
inverse reduced temperature (Tr/T) [-], where Tr = 1K

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Natural Gas Calculation - Thermodynamic Properties

The functions for calculating compressibility factor, internal energy, enthalpy, entropy, heat capacity, speed of sound and other caloric properties are all related to the Helmholtz free energy and its derivates.

Compression Factor

equation 9.18 (9.18)


Internal Energy

equation 9.19 (9.19)


Enthalpy

equation 9.20 (9.20)


Entropy

equation 9.21 (9.21)


Isochoric Heat Capacity

equation 9.22 (9.22)


Isobaric Heat Capacity

equation 9.23 (9.23)


Joule Thomson Coefficient

equation 9.24 (9.24)


Isentropic Exponent

equation 9.25 (9.25)


Speed of Sound

equation 9.26 (9.26)


with

equation 9.27 (9.27)


and

equation 9.28 (9.28)

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Functional Forms

The common functional forms of the fundamental equations are

equation 9.29 (9.29)

equation 9.30 (9.30)

equation 9.31 (9.31)

equation 9.32 (9.32)

equation 9.33 (9.33)

equation 9.34 (9.34)

The various constants are given in the detailed documentation for the equation of state in annex D of ISO 20765-1.

Symbols

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