Equation of State Models
AFT xStream uses the Ideal Gas model with a corrected compressibility factor, Z, to model AFT Standard fluids. The Z-factor can be calculated by numerous correlations. xStream offers 6 options: Constant Z, Ideal Gas, Peng-Robinson, Redlich-Kwong, Soave Redlich-Kwong, and Three-Parameter.
The Ideal Gas option leaves the Z-factor defined at 1, meaning the fluid is modeled as an Ideal Gas. The Constant Z option allows the user to specify a value for Z which xStream then uses for all property calculations. The other Equation of State options are discussed in turn below.
Equations of State Nomenclature
There are a few characterizing fluid properties that are common to all of the Equations of state below.
The reduced temperature (Tr) and reduced pressure (Pr) are the ratios of the thermodynamic properties to the critical properties. For example,
is the definition of the reduced pressure. The reduced temperature could be written similarly.
ω is the acentric factor, for which values can be found in chemical engineering data books. It is a correction factor that can be set to zero if you cannot find data for the fluid. However, accuracy will be reduced.
Redlich-Kwong
The Redlich-Kwong Equation of State is given by the following cubic equation.
where
And Pc and Tc are the critical pressure and temperature, respectively. V is the molar volume.
An alternative form of the Redlich-Kwong equation can be obtained by multiplying the Redlich-Kwong equation by V/RT.
where
Elimination of the "a" and "b" terms gives
where Tr and Pr are the reduced temperature and reduced pressure.
Soave-Redlich-Kwong (SRK)
Soave proposed a modified version of the Redlich-Kwong method as follows
where
where Tr is the reduced temperature, and ω is the acentric factor.
The SRK equation can also be written as in terms of Z as
where
The largest root of Z for the cubic equation is the compressibility factor for the vapor form of the fluid.
Peng-Robinson
The Peng-Robinson Equation of State uses a similar form to the Redlich-Kwong equation of state cubic equation. The Peng-Robinson cubic equation is given as follows
where
And Pc and Tc are the critical pressure and temperature, respectively, V is the molar volume, Tr is the reduced temperature and ω is the acentric factor.
The above equation for pressure can be rewritten as
where
The largest root of Z for the above equation is the compressibility factor of the vapor form of the fluid.
Three-parameter
xStream implements a Three-Parameter Equation of State derived by Pitzer for the second coefficient in the Virial Equation of State. The Three-Parameter model as used by xStream is taken from page 102 of Smith, van Ness, and AbbottSmith, J.M., van Ness, H.C., and Abbott, M.M., Introduction to Chemical Engineering Thermodynamics, 6th edition, McGraw-Hill, New York, NY, 2001. and shown below.
where Z is the compressibility factor, and Pr and Tr are the reduced pressure and temperature, respectively.
The constants B0 and B1 are given by the following
whereω is the acentric factor.
