2.2 Single-Phase Thermodynamics
2.2.1 Cubic Multi-Component Equation of State
Cubic EOS are widely used due to their simplicity and overall good accuracy for pure substances and mixtures. Especially in computational intensive calculations such as real-gas CFD simulations, a trade-off between accuracy and efficiency must be found. Here, cubic EOS allow for a rapid calculation of thermodynamic prop-erties while accounting for the full non-linear pressure-volume-temperature (PVT) behavior of the fluid/mixture. Furthermore, liquid and vapor phase properties can be modeled with a single EOS and so there is no need for dedicated phase-specific EOS. In this thesis, single- and two-phase models are therefore formulated for cubic EOS, which can be expressed in a generalized pressure explicit form as
p(v, T,z) = RT
v−b − aα(T)
v2+u b v+w b2, (2.12) where the pressure p is a function of the molar volume v, temperature T and if Nc>1 the molar composition z = {z1. . . zNc}. Here and in the following, all in-tensive thermodynamic properties are expressed as molar quantities, denoted by?.
2.2 Single-Phase Thermodynamics
R is the universal gas constant. Introducing the non-dimensional compressibility factor
Z = pv
RT (2.13)
together with dimensionless forms of the EOS parameters A = aαp/(RT)2 and B =bp/(RT), Eq. (2.12) can also be expressed as
Z = 1
1−B/Z − A B
B/Z
1 +u B/Z+w(B/Z)2 (2.14) which can further be rearranged in a cubic polynomial in Z, see Sec. 2.2.2.
In all subsequent simulations, we use the Peng-Robinson (PR) (Peng and Robin-son, 1976) EOS for which u= 2 and w=−1.1 The function
α=h
1 +c0(1−p
T /Tc)i2
(2.15) accounts for the polarity of a fluid and is a correlation of temperature T, critical temperature Tc and acentric factor ω via
c0 = 0.37464 + 1.54226ω−0.2699ω2. (2.16) The parameter
a= 0.45724 R2Tc2/pc
(2.17) represents attractive forces between molecules and the effective molecular volume is represented by
b= 0.0778 (RTc/pc). (2.18) The critical properties and the acentric factor of nitrogen, hydrogen, n-dodecane and n-hexane are given in Tab. 2.1.
We use conventional mixing rules to extend the PR EOS to a mixture composed
1The Soave-Redlich-Kwong (SRK) (Soave, 1972) EOS is obtained with u= 1 andw = 0and the corresponding definitions foraα(T)andb, see, e.g., Poling et al. (2000).
of Nc components. The parameters required in the EOS are calculated from aα =
Nc
X
i Nc
X
j
zizjaijαij and b=
Nc
X
i
zibi, (2.19) with zi being the mole fraction of component i. The coefficients aij and αij are calculated with combination rules. There are two common used combination rules for calculating the coefficientaijαij:
Combination rule 1: A direct adjustment of the cross-parameter
aijαij =√aiαiajαj(1−kij) (2.20) wherekij is the binary interaction parameter.
Combination rule 2 (or the pseudo-critical method): Adjustment of pseudo-critical properties. Off-diagonal elements are calculated using the same expression as for the diagonals, i.e., Eq. (2.15)-(2.17) for the PR EOS, and
aijαij =f(Tc,ij, pc,ij, ωij) (2.21) with the pseudo-critical parameters
Tc,ij =p
Tc,iTc,j(1−kij0 ), pc,ij =Zc,ij(RTc,ij/vc,ij), vc,ij = 1 8
h
vc,i1/3+v1/3c,j i3
, ωij = 0.5 (ωi+ωj), Zc,ij = 0.5 (Zi+Zj). (2.22) Here,kij0 is the binary interaction parameter. Note thatkij0 andkij are numerically not the same. The binary interaction parameter affects the PVT properties of the mixture and the accuracy of vapor-liquid equilibria (VLE). As pointed out by Reid et al. (1987), it is important to realize that mixing- and combining rules are essentially empirical2. Only a comparison against experimental data can give confidence that the employed mixture model is appropriate for the calculation of volumetric mixture properties. Typically, the binary interaction parameters in Eq. (2.20) and (2.22) are regressed using available experimental VLE or PVT data.
In the present work, the pseudo-critical method is used withk120 = 0. As it will be discussed in more detail in Sec. 3.4.2 and Sec. 4.2, a reasonably good agreement is obtained to experimental VLE data at relevant pressures and temperatures. Also note that binary interaction parameters are assumed independent of temperature,
2With one exception being the truncated virial equation of state, for which an exact relation is known for mixture coefficients, see Reid et al. (1987) Chapter 4.
2.2 Single-Phase Thermodynamics
pressure and composition.
Departure Function Formalism
In addition to the thermal EOS, expressions for caloric properties that account for their pressure dependence are needed. The departure function formalism pro-vides such expressions and only requires relationships provided by the EOS. The departure function, e.g., for the internal energy, can be written as
e(T, v,z) = eig(T,z) +
Using the generalized cubic EOS (Eq. 2.12), the solution of the integral reads e−eig The ideal reference state denoted as ig is evaluated using the 9-coefficient NASA polynomials (Goos et al., 2009). There are a number of thermodynamic derivatives needed and the most important ones will be given in the following. Helpful details, derivations and analytical solutions to caloric and derived properties not listed below are available in literature, see, e.g., Firoozabadi (1999), Poling et al. (2000), and Elliott and Lira (2012).
The enthalpy h is obtained from h−hig
= e−eig
+pv− RT. (2.26)
The entropy departure function s is defined as
s(T, v,z) = sig(T,z) +
and the solution to the integral reads
A general expression for the Gibbs energy g can be written by combining the enthalpy and entropy departures, i.e.,
g−gig
For pure substances, the Gibbs energy is needed to determine the most stable root when three volume (or equivalently compressibility factor) roots are found as solution to Eq. (2.12) for given temperature and pressure, see Sec. 2.2.2. By differentiating Eq. (2.24) with respect to the temperature, we obtain an expression for the heat capacity at constant volume
cv−cigv
=−T∂2aα
∂T2 K. (2.31)
The heat capacity at constant pressure
cp =cv−T ∂p
The coefficient of thermal expansionαp(also known as expansivity) and isothermal
2.2 Single-Phase Thermodynamics
compressibility βT are defined as αp = 1
respectively. The isentropic or thermodynamic speed of sound c is defined as
c= which can be recast into
c=
with M being the molar mass of the pure substance or mixture.
Partial Molar Properties
In order to calculate the interdiffusional enthalpy flux in Eq. (2.11), the partial enthalpy on a mass basis hi is required. In the following, a brief introduction in the concept of partial properties is given. Let F be any extensive property (e.g.
volumeV, enthalpyH or Gibbs free energy G) of a homogeneous phase. F can be expressed as a function of the two independent intensive properties temperature and pressure and the size of the system n = {n1, . . . , nNc} where ni denotes the number of moles of each component:
F =F(T, p,n). (2.40)
The molar specific property f is defined by f(T, p,z) = F
n (2.41)
and because f is an intensive property it is a function of only the intensive proper-ties temperatureT, pressurepand mole fractionsz={z1, . . . , zNc}. Per definition,
the derivative at constant temperature, pressure and mole numbernj6=i is called the partial molar property ofF. A partial property tells us how an extensive property of the mixture changes with an infinitesimal change in the number of moles of species i at constant temperature, pressure and mole number nj6=i (Elliott and Lira, 2012). Partial properties are intensive properties. They are a function of temperature, pressure and composition, and the relation between a partial mass propertyfi and a partial molar propertyf
i is the molecular weight Mi of the component i
fi =Mifi. (2.43)
It can further be shown that F(T, p,n) =
Using the generalized cubic EOS the partial molar volume vi and enthalpy hi yield
2.2 Single-Phase Thermodynamics
The partial molar Gibbs energy, commonly referred to as chemical potential µ
i, is
As it will be discussed in Sec. 2.3.1, the chemical potential is needed in phase equilibria calculations.
Note that the EOS parameters u and w and the binary interaction parameter kij0 are assumed to be no function of the composition. Very helpful details on the calculation of partial molar properties can be found in Masquelet (2013) and Elliott and Lira (2012). For a more detailed theoretical background it is referred to Michelsen and Mollerup (2007) or Shavit and Gutfinger (2008).