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 \equiv u - T s

Gleichung 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) = f^{0} (ρ, T) + f^{r} (ρ, T)

Gleichung 2

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

Φ (δ, τ) = Φ^{0} (δ, τ) + Φ^{r} (δ, τ)

Gleichung 3

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

f^{0} = h^{0} - R T - T s^{0} = \int_{T_{0}}^{T} {c_{P}}^{0} dT + {h_{0}}^{0} - R T - T [\int_{T_{0}}^{T} \frac{{c_{P}}^{0}}{T} dT - R \ln \frac{ρ}{ρ_{0}} - R \ln \frac{T}{T_{0}} + {s_{0}}^{0} - R \sum_{i = 1}^{n} x_{i} \ln x_{i}]

Gleichung 4

Φ^{0} = - τ \int_{τ_{0}}^{τ} \frac{{c_{P}}^{0}}{R τ^{2}} dτ + \frac{{h_{0}}^{0}}{R} τ - 1 + \int_{τ_{0}}^{τ} \frac{{c_{P}}^{0}}{R τ} dτ + \ln \frac{δ}{δ_{0}} + \ln \frac{τ_{0}}{τ} - \frac{{s_{0}}^{0}}{R} + \sum_{i = 1}^{n} x_{i} \ln x_{i}

Gleichung 5

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

Symbols

c_p⁰

molar ideal gas specific isobaric heat capacity [kJ/(kmol·K)]

c_p

molar isobaric heat capacity [kJ/(kmol·K)]

c_v

molar isochoric heat capacity [kJ/(kmol·K)]

f⁰

ideal gas contribution to the molar Helmholtz energy [kJ/kmol]

molar Helmholtz free energy [kJ/kmol]

f^r

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]

molar mass [kg/kmol]

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)]

molar entropy [kJ/(kmol·K)]

absolute temperature [K]

molar internal energy [kJ/kmol]

speed of sound [m/s]

x_i

mole fraction of component i [-]

δ⁰

molar density at the reference state of 298.15 K and 0.101325 MPa [-]

reduced density δ=K³·ρ [-], where K is a mixture size parameter using the constants from annex D of ISO 20765-1

isentropic exponent [-]

Joule-Thomson coefficient [K/MPa]

Φ⁰

ideal gas contribution to the reduced Helmholtz energy [-]

reduced Helmholtz energy f/(RT) [-]

Φ^r

residual part contribution to the reduced Helmholtz energy [-]

molar density [kmol/m³]

inverse reduced temperature (T_r/T) [-], where T_r = 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

z = {δ Φ}_{δ}

Gleichung 6

Internal Energy

\frac{u}{R T} = τ Φ_{τ}

Gleichung 7

Enthalpy

\frac{h}{R T} = {τ Φ}_{τ} + {δ Φ}_{δ} = \frac{u}{R T} + z

Gleichung 8

Entropy

\frac{s}{R} = {τ Φ}_{τ} - Φ = \frac{u}{R T} - Φ

Gleichung 9

Isochoric Heat Capacity

\frac{c_{v}}{R} = - τ^{2} Φ_{τ τ}

Gleichung 10

Isobaric Heat Capacity

\frac{c_{p}}{R} = - τ^{2} Φ_{τ τ} + \frac{{Φ_{2}}^{2}}{Φ_{1}}

Gleichung 11

Joule Thomson Coefficient

μ R ρ = \frac{Φ_{2} - Φ_{1}}{{Φ_{2}}^{2} - τ^{2} Φ_{τ τ} Φ_{1}} = \frac{R}{c_{p}} (\frac{Φ_{2}}{Φ_{1}} - 1)

Gleichung 12

Isentropic Exponent

κ = \frac{Φ_{1} - \frac{{Φ_{2}}^{2}}{τ^{2} Φ_{τ τ}}}{{δ Φ}_{δ}}

Gleichung 13

Speed of Sound

\frac{w^{2} M}{R T} = Φ_{1} - \frac{{Φ_{2}}^{2}}{τ^{2} Φ_{τ τ}} = z κ

Gleichung 14

with

Φ_{1} = {(δ^{2} Φ_{δ})}_{δ} = 2 {δ Φ}_{δ} + δ^{2} Φ_{δ δ}

Gleichung 15

and

Φ_{2} = - τ^{2} {(\frac{{δ Φ}_{δ}}{τ})}_{τ} = {δ Φ}_{δ} - {τ δ Φ}_{τ δ}

Gleichung 16

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Functional Forms

The common functional forms of the fundamental equations are

Φ = Φ^{0} + \frac{B}{K^{3}} δ - δ \sum_{n = 13}^{18} C_{n}^{*} τ^{u_{n}} + \sum_{n = 13}^{58} C_{n}^{*} τ^{u_{n}} δ^{b_{n}} \exp (- c_{n} δ^{k_{n}})

Gleichung 17

τ Φ_{τ} = τ {(\frac{\partial Φ}{\partial τ})}_{δ x} = τ {Φ_{τ}}^{0} + \frac{δ}{K^{3}} \sum_{n = 1}^{18} u_{n} τ^{u_{n}} B_{n}^{*} - δ \sum_{n = 13}^{18} u_{n} τ^{u_{n}} C_{n}^{*} + \sum_{n = 13}^{58} u_{n} τ^{u_{n}} C_{n}^{*} δ^{b_{n}} \exp (- c_{n} δ^{k_{n}})

Gleichung 18

τ^{2} Φ_{τ τ} = τ^{2} {(\frac{\partial^{2} Φ}{\partial^{2} τ})}_{δ x} = τ^{2} Φ_{τ τ}^{0} + \frac{δ}{K^{3}} \sum_{n = 1}^{18} ({u_{n}}^{2} - u_{n}) τ^{u_{n}} B_{n}^{*} - δ \sum_{n = 13}^{18} ({u_{n}}^{2} - u_{n}) τ^{u_{n}} C_{n}^{*} + \sum_{n = 13}^{58} ({u_{n}}^{2} - u_{n}) τ^{u_{n}} C_{n}^{*} δ^{b_{n}} \exp (- c_{n} δ^{k_{n}})

Gleichung 19

δ Φ_{δ} = δ {(\frac{\partial Φ}{\partial δ})}_{τ x} = 1 + \frac{B}{K^{3}} δ - δ \sum_{n = 13}^{18} C_{n}^{*} τ^{u_{n}} + \sum_{n = 13}^{58} C_{n}^{*} τ^{u_{n}} δ^{b_{n}} (b_{n} - c_{n} k_{n} δ^{k_{n}}) \exp (- c_{n} δ^{k_{n}})

Gleichung 20

Φ_{1} = {(δ^{2} Φ_{δ})}_{δ} = {(\frac{\partial (δ^{2} Φ_{δ})}{\partial δ})}_{τ x} = 1 + 2 \frac{B}{K^{3}} δ - 2 δ \sum_{n = 13}^{18} C_{n}^{*} τ^{u_{n}} + \sum_{n = 13}^{58} C_{n}^{*} τ^{u_{n}} δ^{b_{n}} (b_{n} - (1 + k_{n}) c_{n} k_{n} δ^{k_{n}} + {(b_{n} - c_{n} k_{n} δ^{k_{n}})}^{2}) \exp (- c_{n} δ^{k_{n}})

Gleichung 21

Φ_{2} = - τ^{2} {(\frac{{δ Φ}_{δ}}{τ})}_{τ} = - τ^{2} {(\frac{\partial (\frac{{δ Φ}_{δ}}{τ})}{\partial τ})}_{δ x} = 1 + \frac{δ}{K^{3}} \sum_{n = 1}^{18} B_{n}^{*} (1 - u_{n}) τ^{u_{n}} - δ \sum_{n = 13}^{18} C_{n}^{*} (1 - u_{n}) τ^{u_{n}} + \sum_{n = 13}^{58} C_{n}^{*} τ^{u_{n}} δ^{b_{n}} (1 - u_{n}) (b_{n} - c_{n} k_{n} δ^{k_{n}}) \exp (- c_{n} δ^{k_{n}})

Gleichung 22

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

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Fluid Technology

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Designer

Axel Ebert

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