188
L. F. Vega et al.
a
polar
=
a 2
1 − a 3 /a 2
(5)
where a 2 and a 3 are the second- and third-order terms in the perturbation expansion.
Two additional parameters are added in this case, the quadrupole or dipole moment
and x p , the fraction of polar segments in the chain. These parameters can be treated
as adjustable or can be taken from the experimental values. Detailed descriptions
of the different terms of the model can be found in previous contributions, and the
reader is referred to them for additional information [46–50].
A holistic evaluation of the performance of any EoS should involve the description
of second order thermodynamic derivative properties, of special relevance for refrigerants and other systems. Properties such as the thermal expansion coefficient, the
Joule–Thomson coefficient and the isobaric heat capacity can be directly obtained
through deriving a thermodynamic potential function [51]. However, the computations for these properties are more sensitive to errors than the original function. In the
case of SAFT equations, they are calculated by direct derivation of the Helmholtz
free energy. Predictions of second order derivative properties can be obtained by
using the parameters fitted to vapour-liquid equilibrium data or they can be included
in the fitting procedure to obtain more robust parameters [52].
Moreover, extending the equation to compute new properties is possible through
its integration with other theories. For example, the density gradient theory (DGT) is
a successful approach to treat inhomogeneous systems. This theory is used to provide
information on the surface tension [53], which is a macroscopical consequence of the
density profile. Assuming a planar interface along with neglecting the dependence of
the influence parameter on the density, the expression relating the interfacial tension
to the square of the density gradient is [54, 55]:
γ =
i
j
∞
−∞
c i j
dρ i
dz
dρ j
dx
dz
= 2
∞
−∞
a 0 (ρ) +
i
ρ i μ 0i − p 0
dz
(6)
where a 0 is the Helmholtz free energy of the homogeneous system at density ρ, μ 0i
and p 0 represent the equilibrium chemical potential and pressure, respectively, z is the
direction perpendicular to the interface, and c ij is the so-called influence parameter,
describing the effect of the density gradients on the local Helmholtz energy. Details
on the theory and its coupling with soft-SAFT [56, 57] and PC-SAFT [58] can be
found in the original works.
In addition, the free-volume theory (FVT) developed by Allal et al. [59, 60] can
be coupled with SAFT to calculate transport properties. FVT is developed on the
basis of the link between the idea of empty space between molecules, viscosity and
the diffusion models, and is devised to appropriately capture the transition from the
L. F. Vega et al.
a
polar
=
a 2
1 − a 3 /a 2
(5)
where a 2 and a 3 are the second- and third-order terms in the perturbation expansion.
Two additional parameters are added in this case, the quadrupole or dipole moment
and x p , the fraction of polar segments in the chain. These parameters can be treated
as adjustable or can be taken from the experimental values. Detailed descriptions
of the different terms of the model can be found in previous contributions, and the
reader is referred to them for additional information [46–50].
A holistic evaluation of the performance of any EoS should involve the description
of second order thermodynamic derivative properties, of special relevance for refrigerants and other systems. Properties such as the thermal expansion coefficient, the
Joule–Thomson coefficient and the isobaric heat capacity can be directly obtained
through deriving a thermodynamic potential function [51]. However, the computations for these properties are more sensitive to errors than the original function. In the
case of SAFT equations, they are calculated by direct derivation of the Helmholtz
free energy. Predictions of second order derivative properties can be obtained by
using the parameters fitted to vapour-liquid equilibrium data or they can be included
in the fitting procedure to obtain more robust parameters [52].
Moreover, extending the equation to compute new properties is possible through
its integration with other theories. For example, the density gradient theory (DGT) is
a successful approach to treat inhomogeneous systems. This theory is used to provide
information on the surface tension [53], which is a macroscopical consequence of the
density profile. Assuming a planar interface along with neglecting the dependence of
the influence parameter on the density, the expression relating the interfacial tension
to the square of the density gradient is [54, 55]:
γ =
i
j
∞
−∞
c i j
dρ i
dz
dρ j
dx
dz
= 2
∞
−∞
a 0 (ρ) +
i
ρ i μ 0i − p 0
dz
(6)
where a 0 is the Helmholtz free energy of the homogeneous system at density ρ, μ 0i
and p 0 represent the equilibrium chemical potential and pressure, respectively, z is the
direction perpendicular to the interface, and c ij is the so-called influence parameter,
describing the effect of the density gradients on the local Helmholtz energy. Details
on the theory and its coupling with soft-SAFT [56, 57] and PC-SAFT [58] can be
found in the original works.
In addition, the free-volume theory (FVT) developed by Allal et al. [59, 60] can
be coupled with SAFT to calculate transport properties. FVT is developed on the
basis of the link between the idea of empty space between molecules, viscosity and
the diffusion models, and is devised to appropriately capture the transition from the