Cubic equations of state are a convenient means for predicting real fluid behavior. They are highly useful from an engineering standpoint because they generally only require that the user know the fluid critical point properties, and for some versions, the acentric factor. These properties are well known for many pure substances or can be estimated if not available. They are called cubic equations of state because, when rearranged as a function volume they are cubic in volume. This means that cubic state equations can be used to predict both liquid and vapor volumes at a given pressure and temperature. Generally the lowest root is the liquid volume and the higher root is the vapor volume.
Three versions of cubic state equations are available. Standard and Aungier Redlich Kwong as well as Peng Robinson.
The Redlich-Kwong equation of state was first published in 1949 [85 (p. 284)] and is considered one of the most accurate two-parameter corresponding states equations of state. More recently, Aungier (1995) [96 (p. 285)] has modified the Redlich-Kwong equation of state so that it provides much better accuracy near the critical point. The Aungier form of this equation of state is the default cubic equation used by ANSYS CFX. The Peng Robinson equation of state was first published in 1976 [157 (p. 292)]] and was developed to overcome the short comings of the Redlich Kwong equations to accurately predict liquid properties and vapor-liquid equilibrium.
Redlich Kwong Models
The Redlich Kwong variants of the cubic equations of state are written as:
(Eq. 1.91)
= − + − p νR Tb c a T+
ν ν b ( )
( )
where v is the specific volume (v =1/ )ρ .
The Standard Redlich Kwong model sets the parameter c to zero, and the function a to:
(Eq. 1.92)
= ⎛⎝⎜
⎞
⎠⎟
−
a a TT
n
0 c
where n is 0.5 and
(Eq. 1.93)
=
a0 0.42747pR T2 c2
c
Equations of State
(Eq. 1.94)
=
b 0.08664p R Tc
c
The Aungier form differs from the original by a non-zero parameter c which is added to improve the behavior of isotherms near the critical point, as well as setting the exponent n differently. The parameter c in Equation 1.91 (p. 20) is given by:
(Eq. 1.95)
= + −
+ +
c R T b ν
p a c
ν ν( b)
c
c 0
c c
and the standard Redlich Kwong exponent of n=0.5 is replaced by a general exponent n. Optimum values of n depend on the pure substance. Aungier (1995) [96 (p. 285)] presented values for twelve experimental data sets to which he provided a best fit polynomial for the temperature exponent n in terms of the acentric factor, ω:
(Eq. 1.96)
= + +
n 0.4986 1.1735ω 0.4754ω2
The acentric factor must be supplied when running the Redlich Kwong model and is tabulated for many common fluids in Poling et al. [84 (p. 284)]. If you do not know the acentric factor, or it is not printed in a common reference, it can be estimated using knowledge of the critical point and the vapor pressure curve with this formula:
(Eq. 1.97)
= − ⎛
⎝⎜ ⎞
⎠⎟ − ω log10 ppv 1
c
where the vapor pressure, pv, is calculated at T=0.7Tc. In addition to the critical point pressure, this formula requires knowledge of the vapor pressure as a function of temperature.
Peng Robinson Model
The Peng Robinson model also gives pressure as a function of temperature and volume:
(Eq. 1.98)
= − −
− +
p v bRT a T
v bv b ( )
2 2 2
where
= + −
a T( ) a0
(
1 n(
1 T T/ c) )
2
=
a0 0.45724pR Tc
c 2 2
=
b 0.0778p RTc
c
= + −
n 0.37464 1.54226ω 0.26993ω2 Real Gas Constitutive Relations
In order to provide a full description of the gas properties, the flow solver must also calculate enthalpy and entropy.
These are evaluated using slight variations on the general relationships for enthalpy and entropy that were presented in the previous section on variable definitions. The variations depend on the zero pressure, ideal gas, specific heat capacity and derivatives of the equation of state. The zero pressure specific heat capacity must be supplied to ANSYS CFX while the derivatives are analytically evaluated from Equation 1.91 (p. 20)and Equation 1.98 (p. 21).
Internal energy is calculated as a function of temperature and volume (T,v) by integrating from the reference state (Tref,vref) along path 'amnc' (see diagram below) to the required state (T,v) using the following differential relationship:
(Eq. 1.99)
= + ⎛
⎝⎜ − ⎞
du c dTv T
( )
dTdp p dv⎠⎟v
Equations of State
First the energy change is calculated at constant temperature from the reference volume to infinite volume (ideal gas state), then the energy change is evaluated at constant volume using the ideal gas cv. The final integration, also at constant temperature, subtracts the energy change from infinite volume to the required volume. In integral form, the energy change along this path is:
(Eq. 1.100)
∫ ∫ ∫
− =
⎛
⎝⎜ − ⎞
⎠⎟ + − ⎛
⎝⎜ − ⎞
⎠⎟
∞ ∞
u T v u T v
T
( )
p dv c dT T( )
p dv( )
( , ) ,
v
dp
dT T
T T
v dp
dT T
ref ref
v v0
v
ref ref
ref
Once the internal energy is known, then enthalpy is evaluated from internal energy:
= + h u p ν
The entropy change is similarly evaluated:
(Eq. 1.101)
∫ ∫ ∫
− =
+ − ⎛
⎝⎜ ⎞
⎠⎟ −
∞ ∞
s T v s T v
dv dT R dv
( ) ( )
( )
( , ) ,
v ln
dp
dT T
T T c
T
p
p v
dp
dT T
ref ref
v v
p
ref ref
ref 0
ref
where cp o is the zero pressure ideal gas specific heat capacity. By default, ANSYS CFX uses a 4th order polynomial for this and requires that coefficients of that polynomial are available. These coefficients are tabulated in various references including Poling et al. [84 (p. 284)].
In addition, a suitable reference state must be selected to carry out the integrations. The selection of this state is arbitrary, and can be set by the user, but by default, ANSYS CFX uses the normal boiling temperature (which is provided) as the reference temperature and the reference pressure is set to the value of the vapor pressure evaluated using Equation 1.107 (p. 23) at the normal boiling point. The reference enthalpy and entropy are set to zero at this point by default, but can also be overridden if desired.
Other properties, such as the specific heat capacity at constant volume, can be evaluated from the internal energy.
For example, the Redlich Kwong model uses:
(Eq. 1.102)
= ∂∂ = ∂∂ − + +
cν
( )
Tu ν log 1( )
u T
n n a
b T
b v ( 1)
0
Equations of State
where u0 is the ideal gas portion of the internal energy:
(Eq. 1.103)
∫
= − = −
u0 u uref TT
(
cpo( )T R dT)
ref
specific heat capacity at constant pressure, cp, is calculated from cv using:
(Eq. 1.104)
= + cp cv ν T βκ2
where β and κ are the volume expansivity and isothermal compressibility, respectively. These two values are functions of derivatives of the equation of state and are given by:
(Eq. 1.105)
= −
∂
∂∂
∂
β ν
( ) ( )
p T ν
p ν T
(Eq. 1.106)
= − ∂
∂
κ ν