NTNU
Norges teknisk-naturvitenskapelige
universitet
Fakultet for naturvitenskap og teknologi
Institutt for kjemisk prosessteknologi
Calculation of Vapor-Liquid Equilibria
for Methanol-Water Mixture using
Cubic-Plus-Association
Equation of State
Project work in the subject KP8108
”Advanced Thermodynamics”
Ardi Hartono
Inna Kim
2004
Table of content
INTRODUCTION
1. THERMODYNAMIC FRAMEWORK
1.1. Thermodynamics of vapor-liquid equilibria
1.2. Fugacity criteria for phase equilibria
1.3. Fugacity coefficient
1.4. Calculation of fugacity
1.5. Fugacity of pure fluid
2. VAPOR-LIQUID EQUILBRIA WITH EQUATIONS OF STATE
2.1. Virial equations of state
2.2. Analytical equations of state
2.2.1. Estimation of a, b and c parameters for the Soave-Redlich-Kwong EoS
2.2.2. Mixing rules
2.3. Nonanalytic equations of state
2.3.1. Associating and polar fluids (Chemical theory EoS)
2.4. Summary on equations of state
3. THERMODYNAMIC PROPERTIES FROM VOLUMETRIC DATA
3.1. The Helmholz energy and its derivatives
3.2. Calculation of the derivatives of the Helmholz energy
4. THE CPA EQUATION OF STATE
4.1. Association energy and volume parameters
4.2. Association term in CPA EoS
4.2.1. Fraction of nonbonded associating molecules, XA
4.2.2. Association schemes
4.3. Chemical potential and fugacity coefficient from CPA EoS
4.3.1. Volume from CPA EoS
4.3.2. Newton-Raphson technique in calculating volume in CPA EoS
4.4. Flash calculation
4.4.1. Bubble-point pressure calculation
4.5. Results and discussion
LIST OF SYMBOLS
REFERENCES
APPENDICES
2
INTRODUCTION
Vapor-liquid equilibria are the fundamental properties whose knowledge is required, for
example, in the design of separation columns in chemical industries. Many experiments are
necessary to obtain such equilibrium data, at least for binary systems, where non-idealities in
both phases must be determined. Therefore further improvements to theoretical models for
describing and predicting these non-idealities are indispensable.
At low pressure, deviations from ideal behaviour are due mainly to the liquid phase. The
association of one or more components in a liquid mixture and the chemical forces due to
electrical charge exchange between an associating and an active compound influence strongly
the excess properties of associated solutions and the fluid phase equilibria. These effects are
in many cases stronger than those due to physical forces. It is therefore advantageous to treat
chemical and physical interactions separately in theoretical models for the excess properties
of associated solutions (Nath A, 1981).
Equations of state have traditionally been applied to modelling systems with non-polar and
slightly polar compounds. For associating compounds, however, a new concept has evolved
in recent years with the development of equations of state combining the physical effects from
the classical models and a chemical contribution. An example of this new concept is an
equation of state abbreviated CPA – Cubic Plus Association presented by Kontogeorgis et al.
(1996). CPA has been applied extensively to the modelling of vapor-liquid equilibria (VLE)
for alcohol-hydrocarbon systems, in correlating liquid-liquid equilibria (LLE) for alcoholhydrocarbon mixtures, as well as for binary aqueous systems containing hydrocarbons It has
also been applied to the multicomponent systems, namely prediction of VLE and LLE for
ternary mixtures consisting of water-alcohol-hydrocarbons, including the prediction of the
partitioning of methanol between water and hydrocarbons. (Derawi et al., 2003).
In this work we will correlate the experimental VLE data for the methanol-water system using
simplified CPA EoS.
3
1. THERMODYNAMIC FRAMEWORK
1.1. Thermodynamics of vapor-liquid equilibria
We are concerned with a liquid mixture that, at temperature T and pressure P, is in
equilibrium with a vapor mixture at the same temperature and pressure. The quantities of
interest are the temperature, the pressure, and the composition of both phases. Given some of
these quantities, our task is to calculate the others.
Phase equilibria govern the distribution of molecular species between the vapor and liquid
phases. The equilibrium conditions for phase equilibria can be derived in the simplest way
using the Gibbs energy G. The Gibbs free energy of a mixture is a function of temperature,
pressure and composition, and its total derivative can be written in terms of partial derivatives
in the independent variables as (Stromberg, 2003):
⎛ ∂G ⎞⎟
⎛ ∂G ⎞⎟
⎛ ∂G ⎞
⎛ ∂G ⎞
⎟⎟
⎟⎟
dG = ⎜⎜ ⎟⎟⎟ dT + ⎜⎜ ⎟⎟⎟ dP + ⎜⎜⎜
dn1 + ⎜⎜⎜
dn2 ...
⎜⎝ ∂T ⎠
⎝⎜ ∂P ⎠T , ni
⎝ ∂n1 ⎠⎟P ,T ,n
⎝ ∂n2 ⎠⎟P ,T , n
P , ni
j
(1.1)
j
where ni is the number of moles of all components, nj is the number of moles of all
components except one which is under consideration (i). At the constant temperature and
pressure Eq (1.1) is reduced to
dGP ,T = μ1dn1 + μ2 dn2 .... or dGP ,T = (∑ μi dni )
P ,T
(1.2)
⎛ ∂G ⎞
here μi is the chemical potential of component i , μi = ⎜⎜ ⎟⎟⎟
.
⎜⎝ ∂ni ⎠⎟P ,T ,n
j≠ì
For the equilibrium at constant P and T, the Gibbs energy is minimized and mathematically
the minimum means dG=0 (Elliott, 1999). Therefore, Eq (1.1) is equal to 0 at a minimum and
for a closed system all dni are zero. Thus,
dGT , P = 0
(1.3)
In the two-phase system is at equilibrium, then application of the Eq (1.2) yields
dGT , P = ∑ μiV dniV + ∑ μiL dniL = 0
i
(1.4)
i
here superscripts V and L denote the vapor and liquid phases respectively.
4
For the closed system without chemical reaction dniV = −dniL , so it follows that
μiV = μiL
(1.5)
Thus, the general equilibrium criteria for a closed, heterogeneous system consisting of π
phases and n components is that at equilibrium:
T (1) = T (2) = ... = T ( π )
(1.6)
P (1) = P (2) = ... = P ( π )
(1.7)
μ1(1) = μ1(2) = ... = μ1( π )
…
μn(1) = μn(2) = ... = μn( π )
It says that at equilibrium, the temperature, pressure and chemical potentials of all species are
uniform over the whole system.
The chemical potential does not have an immediate equivalent in the physical world and it is
therefore desirable to express the chemical potential in terms of some auxiliary function that
might be more easily identified with physical reality. In the thermodynamic treatment of
phase equilibria, auxiliary thermodynamic functions such as fugacity coefficient and the
activity coefficient are often used. These functions are closely related to Gibbs energy.
1.2. Fugacity criteria for phase equilibria
The fugacity of component i in a mixture is defined as (Elliott, 1999):
RTd ln fi = d μi
at constant T
(1.8)
where fi is the fugacity of component i in a mixture and μi is the chemical potential of the
component.
The equality of chemical potentials at equilibrium, Eq (1.5), can easily be interpreted in terms
of fugacity. By integrating Eq (1.8) as a function of composition at fixed T from a state of
pure i to a mixed state, we find
μiV − μi0 = RT ln
f iV
fi 0
(1.9)
where μi0 and fi 0 are for the pure fluid at the system temperature. Writing an analogous
expression for the liquid phase, and equating the chemical potentials using Eq (1.5), we find
5
⎛ f iV ⎞⎟
μ − μ = RT ln ⎜⎜⎜ L ⎟⎟ = 0
⎝ f ⎠⎟
V
i
L
i
(1.10)
i
Then the condition for phase equilibria can be written as:
f iV = f i L
(1.11)
Eq (1.11) gives us a useful result. It tells us that the equilibrium condition in terms of
chemical potentials can be replaced without loss of generality by an equation in terms of
fugacities.
1.3. Fugacity coefficient
G.N. Lewis defined the fugacity of component i in a mixture is defined by:
dG = VdP ≡ RTd ln fi
at constant T
(1.12)
For a real fluid, the volume is given by V = ZRT / P , thus:
dG = RTZ
dP
P
(1.13)
for an ideal gas, we may substitute Z=1 into Eq. (1.13) and obtain
dG ig = RT
dP
= RTd ln P
P
(1.14)
Comparing Eqs (1.13) and (1.14) we see that
d (G − G ig ) / RT = d ln( f / P )
(1.15)
Integrating this equation at low pressure at constant temperature, we have for the left-hand
side:
1
RT
P
∫
0
1
d (G − G ) =
RT
ig
(G − G
⎡ G − G ig − G − G ig
(⎢⎣
)P (
) P=0 ⎤⎥⎦ = RT
ig
)
(1.16)
because (G-Gig) approaches zero at low pressure. Integrating the right-hand side of Eq.(1.12),
we have
⎛f⎞
⎛f⎞
ln ⎜⎜ ⎟⎟⎟ − ln ⎜⎜ ⎟⎟⎟
⎝⎜ P ⎠ P
⎝⎜ P ⎠ P=0
To complete the definition of fugacity, we define the low pressure limit,
⎛f⎞
lim ⎜⎜ ⎟⎟⎟ = 1
P→0 ⎜
⎝ P⎠
(1.17)
Here the ratio f/P is defined as the fugacity coefficient φ.
6
(G − G ig )
f
= ln = ln ϕ
RT
P
(1.18)
In practice, we evaluate the fugacity coefficient, and then calculate the fugacity by
f = ϕP
1.4. Calculation of fugacity
The fugacity of component i, fi, is related to its departure function chemical potential as:
RT ln ϕi = μir (T , P, n)
(1.19)
⎛ ∂G r , P ⎞⎟
⎟
μir , P (T , P, n) = ⎜⎜
⎜⎝ ∂ni ⎠⎟⎟
T , P , n j≠ì
(1.20)
where
It is simply another way of characterizing the Gibbs departure function at a fixed T, P. For an
ideal gas, the fugacity will equal the pressure, and the fugacity coefficient will be unity. For
representation of the P-V-T data in the form of Z=f(T,P), the fugacity coefficient is evaluated
using Eqs (1.13), (1.14) from:
(G − G ig )
RT
⎛f⎞
1
= ln ⎜⎜ ⎟⎟⎟ = ln ϕ =
⎜⎝ P ⎠
RT
P
P
⎛ Z −1⎞
ig
∫ (V −V ) dP = ∫ ⎝⎜⎜⎜ P ⎠⎟⎟⎟ dP
0
0
(1.21)
or the equivalent form for P-V-T data in the form Z=f(T,V):
(G − G ig )
RT
⎛f⎞
1
= ln ⎜⎜ ⎟⎟⎟ = ln ϕ =
⎜⎝ P ⎠
RT
ρ
∫
0
( Z −1)
ρ
d ρ + ( Z −1) − ln Z
(1.22)
which is the form used for the cubic equations of state.
The chemical potential of a component, μi, is the partial molar Gibbs energy, but it is also a
partial derivative of other properties:
⎛ ∂G ⎞
⎛ ∂A ⎞
⎛ ∂U ⎞⎟
⎛ ∂H ⎞⎟
⎟⎟
⎟⎟
= ⎜⎜⎜ ⎟⎟⎟
= ⎜⎜⎜
= ⎜⎜⎜
μi = ⎜⎜ ⎟⎟⎟
⎟
⎟
⎟
⎜⎝ ∂ni ⎠⎟
∂
∂
∂
n
n
n
⎝
⎠
⎝
⎠
⎝
⎠
i
i
i
T , P , n j≠ì
T ,V , n j≠ì
S ,V , n j≠ì
S , P , n j≠ì
(1.23)
So, when volume rather than pressure is fixed, the fugacity coefficients maybe also
derived from the total residual Helmholz energy, i.e. the Helmholz function of the mixture,
given as a function of temperature T, total volume V, and the vector of mixture mole numbers
n, minus that equivalent ideal gas mixture at (T, V, n). However, traditionally the definition of
the fugacity coefficient in Eq. (1.19) is not changed in the transition from G to A, it describes
7
the departure between the real fluid and ideal gas at given pressure, not at given volume. So,
we have to find Ar at fixed pressure. Using known expression (∂A / ∂V )T , n = −P and taking
into account that the pressure is by definition the same in the ideal gas and the real fluid, but
volumes are different, we can write:
Ar (T ,V ( P ), n) = A(T ,V ( P ), n) − Aig (T ,V ig ( P), n) =
V ( P)
=
∫
(π ig − π) dv −
∞
V ( P)
∫
V ( P)
π ig dv =
V ig ( P )
∫
∞
⎛ nRT
⎞
⎜⎜
− π ⎟⎟⎟dv − nRT ln Z
⎜⎝ v
⎠
(1.24)
Then
⎛ PV ⎞⎟ V
∂ ⎛⎜ Ar ⎞⎟
⎟⎟ = ∫
ln ϕi =
⎜ ⎟ − ln ⎜⎜⎜
∂ni ⎝⎜ RT ⎠⎟⎟T ,V
⎝ ni RT ⎠⎟ ∞
⎡
⎤
⎢ 1 ⎛⎜ ∂π ⎞⎟⎟
⎥
⎢ − ⎜⎜ ⎟⎟
⎥ − ln Z
⎢ v ⎝ ∂ni ⎠T ,V ,n j≠ì ⎥
⎣
⎦
(1.25)
The fundamental problem is to relate these fugacities to mixture composition. The fugacity of
a component is a mixture depends on the temperature, pressure, and composition of that
mixture. In principle any measure of composition can be used. For a mixture of ideal gases,
ϕi = 1 .
For the vapor phase, the composition is nearly always expressed by the mole fraction y. To
relate fiV to temperature, pressure, and mole fraction, it is useful to introduce the vapor-phase
fugacity coefficient ϕiV :
f iV
ϕ =
yi P
V
i
(1.26)
which can be calculated from vapor phase PVT-y data, usually given by an equation of state.
The fugacity coefficient depends on temperature and pressure and, in a multicomponent
mixture, on all mole fractions in the vapor phase, not just yi. The fugacity coefficient is, by
definition, normalized such that as P → 0 , ϕiV → 1 for all I (see Ch.1.1.1.b). At low pressure,
therefore, it is usually a good assumption to set ϕiV = 1 .
The fugacity of component i in the liquid phase is generally calculated by one of two
approaches: the equation of state approach or the activity coefficient approach.
8
Activity coefficient approach: The fugacity of component i in the liquid phase is related to
the composition of that phase through the activity coefficient γi . To develop expressions for
activity coefficients and their composition derivatives, we write the model expression for the
total excess Gibbs energy, nt g E , replacing mole fractions xi by mole numbers ni. Activity
coefficients are then found from
ln γi =
∂ ⎛⎜ nt g E ⎞⎟
⎟
⎜
∂ni ⎜⎝ RT ⎠⎟⎟T , P
(1.27)
Activity coefficient γ i is related to xi and to standard state fugacity f i 0 by
ai
fi L
ϕ (T , P, x1 , x,...)
γi ≡ =
= i
0
xi xi fi
ϕi (T , P, xi = 1)
(1.28)
where ai is the activity of component i. The standard-state fugacity fi 0 is the fugacity of
component i at the same temperature of the system and at some arbitrary chosen pressure and
composition.
While there are some important exceptions, activity coefficients for most typical solutions of
non-electrolytes are based on a standard state where, for every component i, f i 0 is the fugacity
of pure liquid i at system temperature and pressure, i.e. the arbitrary chosen pressure is the
total pressure P, and the arbitrary chosen composition is xi = 1. Whenever the standard-state
fugacity is that of the pure liquid at system temperature and pressure, we obtain the limiting
relation that γi → 1 as xi → 1 (Raoult’s law standard state).
Equation of state approach: The liquid-phase fugacity coefficient, ϕiL , is calculated using Eq.
(1.25) just as for vapor. The only significant consideration is that the liquid compressibility
factor must be used.
When we extend the equation of state to mixtures, the basic form of the equations do not
change. The fluid properties of the mixture are written in terms of the same equation of state
parameters as fore the pure fluids; however, the equations of state parameters are functions of
compositions. The equations we use to incorporate compositional dependence into the
mixture constants are termed the mixing rules.
We will use the Equation of state approach in this work for methanol-water VLE calculation.
9
1.5. Fugacity of pure liquid
To calculate the fugacity for liquid, used in the Eq.(1.28), consider Fig. 1.1. Point A
represents a vapor state, point B – saturated vapor, point C – saturated liquid, and point D
represents a liquid. Calculation of the saturation fugacity may be carried out by any of the
methods for calculation of vapor fugacities. Methods differ slightly on how the fugacity is
calculated between points C and D. There are two primary methods for calculating this
fugacity change. They are Poynting method and the equation of state method.
The Poynting method applies Eq (1.12) between saturation (points B, C) and point D. The
integral is
PD
fD
RT ln sat
= ∫ VdP
f
P sat
D
Psat
C
T<Tc
Pressure
PD
(1.29)
B
A
sat L
sat V
Volume
Fig. 1.1. Schematic for calculation of Gibbs energy and fugacity changes
at constant temperature for a pure liquid (Elliott, 1999)
Since liquid are fairly incompressible for Tr<0.9, the volume is approximately constant over
the interval of integration, and may be removed from the integral, with the resulting Poynting
correction becoming
f
f sat
sat ⎞
⎛ L
⎜V ( P − P )⎟⎟
= exp ⎜⎜
⎟⎟
⎜⎜⎝
RT
⎠⎟
(1.30)
10
Then the fugacity for the liquid is calculated by
f
0, L
⎛V L ( P − P sat )⎞⎟
⎜
⎟⎟
= ϕ P exp ⎜⎜
⎟
⎜⎜⎝
RT
⎠⎟
sat
sat
(1.31)
Saturate volume can be estimated within a few percent error using the Rackett equation
V satL = Vc Z c(1−Tr )
0.2857
(1.32)
The Poynting correction is essentially unity for many compounds near the room T and P; thus,
it is frequently ignored.
f ≈ ϕ sat P sat
11
2. VAPOR-LIQUID EQUILIBRIA WITH EQUATIONS OF STATE
The volumetric properties of a pure fluid in a give state are commonly expressed with the
compressibility factor Z, which can be written as a function of T and P or of T and V:
Z≡
PV
= f P (T , P)
RT
(2.1)
= f v (T ,V )
(2.2)
where V is the molar volume, P is the absolute pressure, T is the absolute temperature and R
is so called universal gas constant. For an ideal gas, Z = 1 . For real gases, Z is somewhat less
than 1 except at high temperatures and pressures.
An algebraic relation between P, V and T is called an EoS. Many equations of state have been
proposed for engineering applications. From equation of state we get not only PVT
information but, from the interrelations provided by classical thermodynamics, departure
functions from ideal gas behaviour and phase equilibria can be calculated. The equations of
state may be classified as follows:
-
the virial equation;
-
semi theoretical EoS which are cubic or quadric in volume, and therefore whose
volumes can be found analytically from specified P and T;
-
non analytic equations
The virial equation can be derived from molecular theory, but is limited in its range of
applicability. It can represent modest deviations from ideal gas behaviour, but not liquid
properties. Semi theoretical EoS can represent both liquid and vapor behaviour over limited
ranges of temperature and pressure for many but not all substances. Finally, non analytic
equations are applicable over much broader ranges of P and T than are the analytic equations,
but they usually require many parameters that require fitting to large amount of data of
several properties. These models include semi theoretical models such as perturbation models,
chemical theory equations for strongly associating species, and crossover relations for a more
rigorous treatment of the critical region.
12
2.1. Virial equations of state
The virial equation of state, first proposed by Thiesen (in 1885), represents the volumetric
behaviour of a real fluid as a departure from the ideal gas equation. It is a polynomial series in
pressure or in inverse volume whose coefficients are functions only of T for a pure fluid. The
consistent form for the initial terms is:
2
⎛ P ⎞
⎛ P ⎞
B C
Z = 1 + B ⎜⎜ ⎟⎟⎟ + (C − B 2 )⎜⎜ ⎟⎟⎟ + ... = 1 + + 2 + ...
⎝⎜ RT ⎠
⎝⎜ RT ⎠
V V
(2.3)
where coefficients B, C,… are called the second, third,… virial coefficients. From statistical
mechanics, these coefficients are related to the forces between molecules; i.e. the second
virial coefficient represents the interaction between two molecules, the third virial coefficient
reflects the simultaneous interaction among three molecules, etc.
Despite its theoretical basis, the virial equation has not been widely used, mainly because
values of the virial coefficients are not known. Indeed, only the second virial coefficient has
been studied extensively for simple fluids and some light hydrocarbons, and less is known
about the third virial coefficient. As a result, in practice, the virial equation is used only for
vapours at pressures up to several atmospheres and away from the vapor-liquid transition
(Sandler, 1993). Extended virial equations of state are an important powerful tool for
calculating the VLE of non-polar mixtures. They are still preferred when volumetric and other
thermodynamic information of high accuracy are needed.
2.2. Analytical equations of state
An EoS used to describe both gases and liquids requires the form of Eq.(2.2), and it must be
at least cubic in V. Then, when T and P are specified, V can be found analytically rather than
only numerically.
Among analytical equations of state, cubic EoS are the most widespread and simple in form.
It is possible to formulate all possible cubic EoS in a single general form with a total of five
parameters. The general cubic form for P is
P=
Θ (V − η )
RT
−
V − b (V − b)(V 2 + δV + ε)
(2.4)
13
where, depending upon the model, Θ, b, η , δ , and ε may be constant, including zero, or they
may vary with T and/or composition. Relations among the Eq.(2.4) parameters for several
common cubic EoS are given in the Table 2.1.
Table 2.1. Equation (2.4) Parameters for Some Cubic EoS* (Poling, 2000)
EoS
δ
ε
Θ
Parameteres
Van der Waals (1890)
0
0
a
a, b
Y(Tc, Pc)
Redlich & Kwong (1949)
0
0
a / Tr0.5
a, b
Y(Tc, Pc)
Wilson (1964)
b
0
aα (Tr )
a, b, α(1)
Y(Tc, Pc, ω)
Soave (1972)
b
0
aα (Tr )
a, b, α(1)
Y(Tc, Pc, ω)
Peng & Robinson (1976)
2b
-b2
aα (Tr )
a, b, α(1)
Y(Tc, Pc, ω)
Soave (1979)
b
0
aα (Tr )
a, b, α(1)
N(2)
b+3c
2c2
aα (Tr )
a, b, c, α(1)
N(1)
2c
c2
aα (Tr )
a, b, c, α(1-2)
Y(Tc, Pc, ω), N(2)
4b+c
bc
aα (Tr )
a, b, c, α(3)
N(3)
Soave (1993)
b
0
aα (Tr )
a, b, α(1-2)
N(1-2)
Patel (1996)
b+c
-bc
aα (Tr )
a, b, c, α(3)
N(4)
2b
-b2
aα (Tr )
a, b, α(4-6)
N(3-6)
Peneoux, et al. (1982)
Soave (1984)
Twu et al. (1992)
Zabaloy & Vera (1996, 1998)
Generalized**
*Single letters (a, b, c, etc.) are substance specific parameters. Expressions such as α(T) are multiterm functions
of T containing from 1 to 3 parameters
** Y means that CSP(corresponding states principle) relations exist to connect all of the parameters a, b, c, etc.
to Tc, Pc, Zc, ω, etc. N means that at least some of the parameter values are found by data regression of liquid
densities and/or vapor pressures while others are critical properties or ω. The number of such fitted parameters is
in parenthesis.
The expressions in Table 1.2 show how models have been developed to adjust density
dependence though different choices of δ and ε. Temperature dependence is mainly included
in α(T), though b, c, d, etc. may be varied with T.
2.2.1. Estimation of a, b and c parameters for the Soave-Redlich-Kwong EoS
To estimate parameters a and b for a pure fluid, for example, for the SRK EoS:
p=
RT
a(T )
−
v − b v (v + b)
(2.5)
14
The Eq. (2.5) is reshaped into a cubic polynomial:
v3 −
RT 2 ⎛⎜ a
RTb ⎞⎟
ab
v + ⎜ − b2 −
⎟⎟ v − = 0
⎜
⎟
p
p ⎠
p
⎝p
(2.6)
3
At the critical point this equation must fulfil (v − vc ) = 0 , that is v3 − 3v 2 vc + 3vvc2 − vc3 = 0 .
When compared term by term the two polynomials define a set of equations that can be
solved for a, b and vc:
3vc =
RTc
,
pc
3vc2 =
RT b
a
− b2 − c ,
pc
pc
vc3 =
ab
pc
(2.7)
3
The second equation is combined with the other two to yield 2vc3 − (vc + b) = 0 , which is
solved for the positive root b = (21/ 3 −1) vc . In terms of critical temperature and pressure, for
any component i, this is equivalent to:
bi = 0.08664
RTc ,i
ai = 0.042747
pc ,i
R 2Tc2
pc
(2.8)
At temperatures others than the critical
ai (T ) = ai α (T )
(2.9)
where αi (T ) is an non-dimensional factor which becomes unity at T=Tc. In SRK equation
(Soave, 1972):
αi0.5 = 1 + ci (1− Tr 0.5 )
(2.10)
Parameter ci can be connected directly with the acentric factor ωi of the related compounds by
ci = 0.480 + 1.547ωi − 0.176ωi2
(2.11)
Eqs. (2.9)-(2.11) yield the desired value of ai(T) of a given substance at any temperature:
(
(
ai (T ) = ai 1 + ci 1− Tr
))
2
(2.12)
2.2.2. Mixing rules
The greatest utility of cubic equations of state is for phase equilibrium calculations involving
mixtures. The assumption inherent in such calculations is that the same equation of state used
for pure fluids can be used for mixtures if we have a satisfactory way of obtaining the mixture
parameters. This had been done for decades using the simple van der Waals one-fluid mixing
rules with one or two binary parameters
15
a = ∑∑ xi x j aij
b = ∑∑ xi x j bij
(2.13)
In addition, combining rules are needed for the parameters aij and bij. The usual combining
rules are
aij = aii a jj (1− kij )
1
bij = (bii + b jj )(1− lij )
2
(2.14)
where kij and lij are the binary interaction parameters obtained by fitting equation of state
predictions to experimental VLE data for kij or VLE and density data for kij and lij. We can
find different expressions for the binary interaction parameter in the literature. For example,
Wong et al. (1992) suggest that the combining rule for kij be of the form:
1− kij =
2 (b − a / RT )ij
(bii − aii / RT ) + (b jj − a jj / RT )
(2.15)
Values for kij for various binary combinations are tabulated in the literature. In the absence of
the experimental data or literature values for kij, we may make a first-order approximation by
letting kij=0. For many mixtures lij is set equal to zero but there are situations where inclusion
of lij as a second interaction parameter leads to a better representation of VLE.
Nevertheless, this method was found to be satisfactory only for hydrocarbons, or
hydrocarbons and gases. It is only in recent years that new mixing and combining rules have
allowed the cubic equations of state to be used for accurate correlations, even for predictions
for more complicated mixtures involving organic chemicals.
2.3. Nonanalytical equations of state
The complexity of property behaviour cannot be described with high accuracy with the cubic
or quadric EoS that can be solved analytically for the volume, given T and P. Non analytical
equations of state include, for instance, strictly empirical BWR (Benedict-Webb-Rubin)
models and Wagner formulations, semi empirical formulations based on theory are
perturbation methods and chemical association models.
The technique of perturbation modelling uses reference values for systems that are similar
enough to the system of interest that good estimates of desired values can be made with small
16
corrections to the reference values. Perturbation terms, or those which take into account the
attraction between the molecules, have ranged from the very simple to extremely complex.
The simplest form is that of van der Waals which in terms of the Helmholz energy is
( vdW )
⎡ Ar (T , V ) / RT ⎤
⎣⎢
⎦⎥ Att
= −a / RTV
(2.16)
and which leads to an attractive contribution to the compressibility factor of
( vdW )
Z Att
= −a / RTV
(2.17)
This form would be appropriate for simple fluids though it has also been used with a variety
of reference expressions. The most complex expressions for normal substances are those used
in the BACK (Boublik-Alder-Chen-Kreglewski), PHCT (Perturbed Hard Chain Theory), and
SAFT (statistical associating fluid theory) EoS models. There general form is
Z
( BACK )
Att
n
m
= r ∑∑
i =1 j =1
⎡u ⎤
jDij ⎢ ⎥
⎢⎣ kT ⎥⎦
i
⎡η⎤
⎢ ⎥
⎢⎣ τ ⎥⎦
j
(2.18)
where the number of terms may vary, but generally n ~ 4-7 and m~10, the Dij coefficient and
τ are universal, and u and η are substance-dependent and may also be temperature dependent
as in the SAFT model.
2.3.1. Associating and polar fluids (Chemical theory EoS)
In many practical systems, the interactions between the molecules are quite strong due to
charge-transfer and hydrogen bonding. This occurs in pure components such as alcohol,
carboxylic acids, water and HF and leads to quite different behaviour of vapours of these
substances. Considering the interactions so strong that new “chemical species” are formed,
the thermodynamic treatment assumes that the properties deviate from an ideal gas mainly
due to the “speciation” plus some physical effect. It is assumed that all of the species are in
reaction equilibrium. Thus, their concentrations can be determined from equilibrium constants
having parameters such as enthalpies and entropies of reaction in addition to the usual
parameters for their physical interactions.
Associating (and solvating) species present a special problem with equations of state because
the occurrence of both weak chemical reactions and phase equilibrium. By proper coupling of
the contributions of the physical and chemical effects, the result is a closed form equation. A
similar formulation is made with the SAFT equation, where molecular level association is
17
taken into account by a reaction term that is added to the free energy term from reference,
dispersion, polarity, etc.
2.4. Summary on equations of state
To characterize small deviations from ideal gas behaviour the truncated virial equation with
either the second alone or the second and third coefficient should be used. Virial equations
should not be used for liquid phase.
For normal fluids, a generalized cubic EoS with volume translation should be used. All
models give equivalent and reliable results for saturated vapours except for the dimerizing
substances given above.
For polar and associating substances, a method based on four or more parameters should be
used. Cubic equations with volume translation can be quite satisfactory for small molecules,
though perturbation expressions are usually needed for polymers and chemical models for
carboxylic acid vapours.
18
3. THERMODYNAMIC PROPERTIES FROM VOLUMETRIC DATA
For any substance, regardless of whether it is pure or a mixture, most thermodynamic
properties of interest in phase equilibria can be calculated from thermal and volumetric
measurements. For a given phase, thermal measurements (heat capacities) give information
on how some thermodynamic properties vary with temperature, whereas volumetric
measurements give information on how thermodynamic properties vary with pressure or
density at constant temperature. Frequently it is useful to express a selected thermodynamic
function of a substance relative to that which the same substance has as an ideal gas at the
same temperature and composition and/or at some specified pressure of density. This relative
function is often called as residual function. The fugacity is a relative function because its
numerical value is always relative to that of an ideal gas at unit fugacity; in other words, the
standard-state fugacity fi 0 in Eq (1.28) is arbitrary set equal to some fixed value, usually 1
bar.
The thermodynamic function of our interest is the fugacity that is directly related to the
chemical potential. To obtain numerical values of the fugacity, we will find that an equation
of state is necessary. Since, such P, V, T, N relations are normally explicit in pressure, it will
be convenient to formulate the problem with T, V, N as the independent variables (Modell et
al., 1983). This conclusion suggests that a Legendre transform of the energy into T, V, n space
would be appropriate. Such a transform is the Helmholz energy, A:
A = U − TS
(3.1)
dA = −SdT − PdV + ∑ μi dN i
(3.2)
i
3.1. The Helmholz energy and its derivatives
Given an equation of state:
P = P(T , v, x)
(3.3)
where x is the vector of mixture mole fractions, the textbook approach to calculate mixture
fugacity coefficients is by mean of an integrals given by Eq (1.21) or (1.25). Using the Eq.
(1.24)
19
V
⎛
nRT ⎞⎟
A (T , V , n) = ∫ ⎜⎜ P −
⎟ dV
⎜⎝
V ⎠⎟
∞
r
(3.4)
where Ar is the residual Helmholz function. Ar is a homogeneous function of degree 1 in the
extensive variables (V,n) and, given an expression for Ar, all other properties can be derived
solely by differentiation. The pressure equation itself, normally used to define the “equation
of state”, is actually just one of these derivatives given by
P =−
∂Ar nRT
+
∂V
V
(3.5)
The expression for the residual Helmholz energy is thus the key equation in equilibrium
thermodynamics, where all other residual properties are calculated as partial derivatives in the
independent variables T, V, and n (Mollerup et al., 1992). It is important to note that mole
numbers rather than mole fractions are independent variables. Derivatives with respect to
mole fractions are best avoided, as they require a definition of the “dependent” mole fraction
and in addition lead to more complex expressions missing many important symmetry
properties.
3.2. Calculation of the derivatives of the Helmholz energy
As said above, when a particular mathematical model is chosen for the Helmholz energy,
derived properties such as fugacity coefficients, enthalpy, heat capacity etc. are obtained as
partial derivatives with respect to the independent variables T, V and n. Although not
necessary, it is more convenient to work with the reduced Helmholz energy, defined by
F=
Ar
RT
Then the equation for the fugacity coefficient, for example, is as follows
⎛ ∂F ⎞
ln ϕi = ⎜⎜ ⎟⎟⎟ − ln Z
⎜⎝ ∂ni ⎠⎟T ,V
(3.6)
(3.7)
Normally model provides an expressions for F in terms of T, V, total moles n, and one or
more “mixture parameters”, a, b, …, e.g.:
F = F (T ,V , n, b, a)
(3.8)
The calculation of the derivatives of F is better performed as a two-step procedure. In the first
step, F is differentiated with respect to its primary variables, e.i. T, V, n, and the mixture
parameters, and in the second step the derivatives of the mixture parameters are evaluated
(Mollerup et al., 1992).
20
The partial derivatives of F needed for calculation of thermodynamic properties are:
⎛ ∂F ⎞⎟
⎜⎜ ⎟ = ∂F + ∂F ∂b + ∂F ∂a = FT + Fb bT + Fa aT
⎜⎝ ∂T ⎠⎟V ,n ∂T ∂b ∂T ∂a ∂T
(3.9)
⎛ ∂F ⎞⎟
⎜⎜ ⎟ = ∂F + ∂F ∂b + ∂F ∂a = FV + FbbV + Fa aV
⎜⎝ ∂V ⎠⎟T , n ∂V ∂b ∂V ∂a ∂V
(3.10)
⎛ ∂F ⎞⎟
⎜⎜ ⎟
⎜⎝ ∂n ⎠⎟⎟
(3.11)
i V ,T
=
∂F ∂F ∂b ∂F ∂a
+
+
= Fn + Fb bi + Fa ai
∂ni ∂b ∂ni ∂a ∂ni
where bi and ai are abbreviated notations for the composition derivatives of b and a.
For example, in case of the generalized cubic equation of state (2.4), we will have an
expression for the residual Helmholz energy as follows:
V ⎛
⎞⎟
⎛ NRT
Θ (v − η )
⎜ NRT NRT
cubic ⎞
⎟⎟ dv
⎟
⎜
⎜
=∫⎜
− P ⎟⎟ dv = ∫ ⎜
−
−
2
⎟
⎜⎝ v
⎜⎜⎝ v
⎠
−
v
b
−
+
+
v
b
v
δ
v
ε
(
)
(
)
⎠⎟
∞
∞
V
A
r ,V , cubic
(3.12)
V
−ΘΦ
V −B
(3.13)
∂Ar ,V ,cubic
V
= NR ln
−ΘT ΦT
∂T
V −B
(3.14)
Ar ,V ,cubic = NRT ln
⎛
⎞⎟
(v − η )
⎜
⎟⎟ dv
where Φ = ∫ ⎜⎜
⎜⎜⎝(v − b)(v 2 + δv + ε)⎠⎟⎟
∞
V
Then the corresponding derivatives are:
where ΘT and ΦT are temperature derivatives of Θ and Φ, B = ∑ bi ni
i
∂Ar ,V ,cubic
B
Θ(V − η )
= −NRT
−
∂V
V (V − B) (V − B )(V 2 + δV + ε)
(3.15)
∂Ar ,V ,cubic
V
= RT ln
−ΘiΦi
∂ni
V −B
(3.16)
21
where Θi and Φi are composition derivatives of Θ and Φ.
In case of η = b , we will have :
Ar ,v ,cubic = nRT ln
V
+
V −b
2
⎡
⎢V + δ −
2
⎢
Θ
ln ⎣
2
⎡
δ
−ε ⎢V + δ +
2
2
⎢
⎣
( )
⎤
( δ 2) −ε ⎥⎥
2
⎦
⎤
δ
−ε ⎥
2
⎥
⎦
( )
2
⎡
⎢v + δ −
2
⎡ ∂Ar ,v ,cubic ⎤
(∂Θ ∂T )v ,n ⎢⎣
v
⎢
⎥ = nRT ln
ln
+
2
⎢
⎥
⎡
v −b
⎣ ∂T ⎦ v , n
2 δ
−ε ⎢v + δ +
2
2
⎢
⎣
( )
⎤
( δ 2) −ε ⎥⎥
2
⎦
⎤
δ
−ε ⎥
2
⎥
⎦
( )
2
⎡ ∂Ar ,v ,cubic ⎤
b
Θ
⎢
⎥ = nRT ln
+ 2
⎢ ∂v ⎥
v ( v − b ) v + δv + ε
⎣
⎦ T ,n
⎛
⎞
⎜⎜∂b ⎟⎟
⎡ ∂Ar ,v ,cubic ⎤
∂ni ⎠
v
⎝
⎢
⎥
= RT ln
+ nRT ln
+
⎢ ∂ni ⎥
v −b
v −b
⎣
⎦ T ,v ,n j≠i
⎡⎛
⎤ ⎡
⎞
⎛
⎞
⎛
⎞
⎢v + δ −
Θ δ ⎜⎜∂δ ⎟⎟
+ ⎜⎜∂ε ⎟⎟
⎢ ⎜⎜∂Θ ∂n ⎟⎟
⎥
4 ⎝ ∂ni ⎠T ,v ,n
2
⎝ ∂ni ⎠T ,v ,n j≠i ⎥ ⎢
i ⎠T , v , n j≠i
1 ⎢⎝
j≠i
⎢
⎥ ln ⎣
−
2
2
2
⎥ ⎡
⎛
⎞
2⎢
δ
⎜ δ
⎢
⎥ ⎢v + δ +
−ε
−ε⎟⎟ δ
−ε
⎜
2
2
2
⎝ 2
⎠
⎢⎣
⎥⎦ ⎢
⎣
⎛
⎞
Θ ⎜⎜∂δ ⎟⎟
⎡
⎤
⎝ ∂ni ⎠T ,v ,n j≠i ⎢ δ 2 v + ε⎥
⎣
⎦
⎛ δ 2
⎞⎟ ⎡ v 2 + δv + ε⎤
−ε⎟ ⎣⎢
2 ⎜⎜
⎦⎥
⎝ 2
⎠
( )
( )
( )
( )
2
⎦−
⎤
δ
−ε ⎥
2
⎥
⎦
( )
2
( )
( )
In case of the Soave-Redlich-Kwong equation (2.5) η = b ; δ = b ; ε = 0 and Θ = a (T , n) , the
residual Helmholz energy for the fluid will be
V
A
r ,V , SRK
⎛ NRT
⎞
⎛ V ⎞⎟ A ⎛ V ⎞⎟
= ∫ ⎜⎜
− p SRK ⎟⎟⎟dv = NRT ln ⎜⎜
+ ln ⎜
⎜⎝ v
⎜V − B ⎠⎟⎟ B ⎝⎜⎜V + B ⎠⎟⎟
⎠
⎝
∞
⎤
( δ 2) −ε ⎥⎥
(3.17)
and the corresponding first derivatives are:
⎛ r ,V , SRK ⎞⎟
⎛ V ⎞⎟ AT ⎛ V ⎞⎟
⎜⎜ ∂A
⎟⎟ = NR ln ⎜⎜
+ ln ⎜
⎜⎝V − B ⎠⎟⎟ B ⎜⎝⎜V + B ⎠⎟⎟
⎜⎝ ∂T ⎠⎟V ,n
⎛ ∂Ar ,V , SRK ⎞⎟
B
A
⎟ = NRT
−
⎜⎜⎜
⎟
⎟
V (V − B) V (V + B )
⎝ ∂V ⎠T ,n
22
⎛ ∂Ar ,V , SRK ⎞⎟
⎜⎜
⎟
⎜⎝ ∂N ⎠⎟⎟
i
T ,V , N j≠ `k
⎛ V ⎞⎟
bi
Ab ⎞ ⎛ V ⎞⎟
Abi
1⎛
= RT ln ⎜⎜
+ NRT
+ ⎜⎜ Ai − i ⎟⎟⎟ ln ⎜⎜
⎟⎟ −
⎜⎝V − B ⎠⎟⎟
⎜
⎜
V −B B⎝
B ⎠ ⎝V + B ⎠ B(V + B)
where coefficients are defined as follows:
B = ∑ bi ni
(3.18)
i
A = ∑∑
i
Ai =
ai a j ni n j (1− kij )
(3.19)
j
∂A
= ∑∑ 2 ai a j n j (1− kij )
∂ni
i
j
(3.20)
23
4. THE CPA EQUATION OF STATE
Species forming hydrogen bonds often exhibit unusual thermodynamic behaviour. The strong
attractive interactions between molecules of the same species (self-association) or between
molecules of different species (cross-association). These interactions may strongly affect the
thermodynamic properties of the fluids. Thus, the chemical equilibria between clusters should
be taken into account in order to develop a reliable thermodynamic model.
The Cubic-Plus-Association (CPA) model is an equation of state that combines the cubic
SRK equation of state and an association (chemical) term described below. In terms of the
compressibility factor Z it has an appearance:
Z = Z SRK + Z assoc |
(4.1)
The compressibility factor contribution from the SRK equation of state is:
Z SRK =
Vm
a (T )
−
Vm − b RT (Vm + b)
(4.2)
and the contribution from the association term is given by:
⎡⎛ 1
1 ⎞⎟ ∂X Ai ⎤⎥
− ⎟⎟
Z assoc = ∑ xi ∑ ρi ∑ ⎢⎢⎜⎜⎜
⎜X
2 ⎠⎟⎟ ∂ρi ⎥⎥
i
i
Ai ⎢⎝
⎣ Ai
⎦
(4.3)
where Vm is the molar volume, X Ai is the mole fraction of the molecule i not bonded at site A,
i.e. the monomer fraction, and xi is the superficial (apparent) mole fraction of component i.
The small letters i and j are used to index the molecules, and capital letters A and B are used
to index the bonding sites on a given molecule.
While the SRK model accounts for the physical interaction contribution between the species,
the association term in CPA takes into account the specific site-site interaction due to
hydrogen bonding. The association term employed in CPA is identical with the one used in
SAFT.
Before we describe the model, let’s give definitions of “sites” and “site-related” parameters
used in CPA and SAFT models.
24
4.1. Association energy and volume parameters
The key features of the hydrogen-bonds are their strength, short range, and high degree of
localization. In Fig. 4.1. it is shown a simple example of prototype spheres, or spherical
segments, with one associating site, A. Such spheres can only form an AA-bonded dimer
when both distance and orientation are favourable.
σ
A
A
A
A
A
Wrong distance
Wrong orientation
Site-site attraction
A
0
εAA
εAA Interaction energy
κAA Interaction volume
corresponding to rAA
rAA
Fig. 4.1. Model of hard spheres with a single associating site A illustrating a simple case of molecular
association due to short-distance, highly orientational, site-site attraction (Chapman et al., 1990)
The associating bond strength is quantified by a square-well potential, which, in turn, is
characterized by two parameters. The parameter εAA characterizes the association energy (well
depth), and the parameter κAA characterizes the association volume (corresponds to the well
width rAA). In general, the number of association sites on a single molecule is not constrained
and they are labelled with capital letters A, B, C, etc. Each association site is assumed to have
a different interaction with the various sites on another molecule. Examples of two associating
sites molecules are given in the Fig. 4.2.
Thus, for each pure component we need three molecular parameters, σ, ε/k, and m, which are
the temperature independent segment diameter in angstroms, the Lennard-Jones interaction
energy in Kelvins, and the number of segments per chain molecule, respectively. In addition
we need two association parameters, association energy, εAiBj/k in Kelvins and volume κAiBj
25
(dimensionless), for each site-site interaction. The usual method for deriving the σ, ε and m
values is to fit vapor pressure and density data for pure components. The association
parameters εAiBj/k and κAiBj can be fitted to bulk phase equilibrium data.
A
Model monomer (methanol)
B
A
1
2
3
m
Model m-mer (alkanol)
B
Fig. 4.2. Models of hard sphere (monomer) and chain (m-mer) molecules with two associating sites A
and B; the chain molecule represent nonspherical molecule (Chapman et al., 1990).
4.2. Association term in CPA EoS
For pure components, the association term is defined in terms of the residual Helmholz
energy ar per mole, defined as
a r (T ,V , n) = a (T , V , n) − a ig (T ,V , n)
(4.4)
where a and aig are the total Helmholz energy per mole and the ideal gas Helmholz energy per
mole at the same temperature and density. The residual Helmholz energy is a sum of three
terms representing contributions from different intermolecular forces: segment-segment
interaction, covalent chain-forming bonds, and site-site specific interactions among the
segments, for example, hydrogen-bonding interactions:
a r = a seg + a chain + a assoc
(4.5)
The extension of the CPA EoS to mixtures requires mixing rules only for the parameters of
the SRK-part, while the extension of the association term to mixtures is straightforward. The
mixing and combining rules for a and b are the classical van der Waals (Chapter 2.2.2).
The mixture Helmholz energy for the association term is linear with respect to mole fractions,
ni:
⎛
Aassoc
1
1⎞
= ∑ ni ∑ ⎜⎜ln X Ai − X Ai + ⎟⎟⎟
⎜
RT
2
2⎠
i
Ai ⎝
(4.6)
26
Here, A is used to index bonding sites on a given molecule, and X Ai denotes the fraction of Asites on molecule i that do not form bonds with other active sites, and
∑
represents a sum
A
over all associating sites. Examples for molecules with two attractive sites and one attractive
site are given as follows (Chapman et al., 1990):
X
X
Aassoc
= ln X A − A + ln X B − B + 1
RT
2
2
X
1
Aassoc
= ln X A − A +
2
2
RT
(2 sites)
(1 site)
(4.7)
(4.8)
4.2.1. Fraction of nonbonded associating molecules, XA
Since the mixture contains not only monomer species but also associated clusters, we need to
define the mole fraction (X) for the total components and their monomers. The mole fraction
of all the molecules of component i is Xi. The mole fraction of (chain) molecules i that are
NOT bonded at site A is XAi, and hence 1-XAi is the mole fraction of molecules i that are
bonded at site A. This definition applies to both pure self-associated compounds and to
mixture components and is give in terms of mole numbers.
The site fractions in Eq.(4.6), X Ai , is related to the association strength between site A on
molecule i and site B on molecule j, Δ
Ai B j
, and the fractions XB of all other kind of association
sites B by:
X Ai =
1
1+ ρ∑ n j ∑ X Bj Δ
j
Ai B j
(4.9)
Bj
where ρ is the molar density of the fluid , and nj – mole fraction of substance j.
So, the key quantity in CPA and SAFT EoS is the association strength Δ. In SAFT it is
approximated by the equation
Δ AB = g (d ) seg ⎡⎢ exp (ε AB / kT ) −1⎤⎥ (σ 3κ AB )
⎣
⎦
hs
g (d ) seg ≈ g (d ) =
2−η
2(1− η )3
(4.10)
(4.11)
27
Since CPA is a molecular based (not a segment-based) EoS, Kontogeorgis et al. (1996)
proposed to calculate the reduced fluid density by
η=
b
4V
(4.12)
where b = 2π N AV d 3 / 3 , and substituted the product σ 3κ AB in Eq. (4.10) by equivalent bβ. So,
in CPA, Δ
Ai B j
, the association (binding) strength between site A on molecule i and site B on
molecule j is given by:
Δ
where ε
Ai B j
and β
Ai B j
Ai B j
= g (ρ )
ref
⎡
⎛ Ai B j ⎞ ⎤
⎢ exp ⎜⎜ ε ⎟⎟ −1⎥ b β Ai B j
⎢
⎜⎜ RT ⎟⎟ ⎥ ij
⎝
⎠ ⎦⎥
⎣⎢
(4.13)
are the association energy and volume of interaction between site A of
molecule i and site B of molecule j, respectively, and g (ρ ) ref is the radial distribution
function for the reference fluid.
The hard-sphere radial distribution is further simplified by Kontogeorgis et al.(1999) to:
g (ρ ) =
1
1−1.9η
(4.14)
1
η = bρ
4
Also Yakoumis et al.(2001) proposed a much simpler general expression for the association
term instead of Eq.(4.3):
∂ ln g ⎞⎟
1⎛
⎟ ∑ ni ∑ (1− X Ai )
Z assoc = − ⎜⎜1 + ρ
∂ρ ⎠⎟⎟ i
2 ⎜⎝
Ai
(4.15)
⎛ 1 ⎞⎟
∂ ln ⎜⎜
⎟
b
⎜⎝1−1.9η ⎠⎟
∂ ln g
b ⎞⎟ 1.9 4
−2 ⎛
⎜
where
=
= (1−1.9η)(−1)(1−1.9η) ⎜−1.9 ⎟⎟ =
⎜⎝
∂ρ
∂ρ
4 ⎠ 1−1.9η
( )
ρ
∂ ln g
1.9η
=
.
∂ρ
1−1.9η
In term of Volume, we have the result for sCPA and CPA, respectively:
ρ
∂ ln g
0.475B
=
∂ρ
V − 0.475B
(4.16)
28
ρ
∂ ln g
(10V − B)
= 2B
∂ρ
(8V − B)(4V − B)
(4.17)
The resulting EoS is referred to as simplified CPA (sCPA).
All phase equilibria calculations performed in this work are based on the simplified CPA
model.
4.2.2. Association schemes
As seen from Eq. (4.15), the contribution of the association compressibility factor in CPA
depends on the choice of association scheme, i.e. number and type of association sites for the
associating compound.
Huang and Radosz (1990) have classified eight different association schemes, which can be
applied to different molecules depending on the number and type of associating sites.
Examples of one-, two-, three-, and four-site molecules for real associating fluids are given in
Fig. 4.3. According to them, for example, for alkanols, each hydroxylic group (OH) has three
association sites, labelled A, B on oxygen and C on hydrogen. The association strength Δ due
to the like, oxygen-oxygen or hydrogen-hydrogen (AA, AB, BB, CC) interactions is assumed
to be equal to zero (since two lone pairs electrons on protons cannot attract each other). The
attraction can only occur between a lone pair electron and proton, i.e. the only non-zero Δ is
due to the unlike (AC and BC) interactions, which moreover are considered to be equivalent.
Another approximation is to allow only one site of oxygen (A) and one site of hydrogen (B).
In case of self association, the association scheme for alkanols is 2B.
29
A
OB
CH
Alkanol
B
Water
Amines:
A
O HC
HD
A
tertiary
N
secondary
N
HB
primary
A
C
N HB
HA
Fig. 4.3. Types of bonding in real associating fluids (Huang et al., 1990)
30
The 2B association scheme (Huang et al., 1990):A
Δ AA = Δ BB = 0
(4.18)
Δ AB ≠ 0
XA = XB =
−1 + 1 + 4ρΔ AB
2ρΔ AB
(4.19)
The 4C association scheme is used for water:
Δ AA = Δ AB = Δ BB = ΔCC = ΔCD = Δ DD = 0
(4.20)
Δ AC = Δ AD = Δ BC = Δ BD ≠ 0
X A = X B = XC = X D =
−1 + 1 + 8ρΔ AC
4ρΔ AC
(4.21)
These schemes are in agreement with the accepted physical picture that alcohols form linear
oligomers and water three-dimensional structures.
When CPA is used for the cross-associating mixture, e.g. alcohols-water, combining rules are
needed for the cross-association energy and volume parameters (ε
AiBj
, β
AiBj
) or for the
association strength ΔAiBj. Examples for the selection of the combining rule are given by Fu
and Sandler (1995). According to them, in water-alcohol mixture, water has three association
sites and an alcohol has two, but only the unbonded electron pair can form a hydrogen bond
with a hydrogen atom thereafter Eq. (4.22) can be described to all sites in methanol-water
system:
B
C
O
C
A
H
Alkanol (1)
H O
B
D A
H
Water (2)
The scheme of self association : A1B1, A2C2 , B2C2 and the scheme of cross association
A1C2 , A2B1 , B2B1 (Kraska,1998) then we rewrite Eq. (4.9) as:
X A1 =
X B1 =
X C2 =
1
1 + ρ(n1 X B1 Δ
A1B1
1 + ρ(n1 X A1 Δ
A1B1
1 + ρ(n2 X A2 Δ
+ n2 X C2 Δ A1C2 )
1
+ n2 X B2 Δ B1B2 + n2 X A2 Δ B1 A2 )
C2 A2
1
+ n2 X B2 ΔC2 B2 + n1 X A1 ΔC2 A1 )
31
1
X B2 =
1 + ρ(n2 X C2 Δ
B2C2
1 + ρ(n2 X C2 Δ
A2C2
+ n1 X B1 Δ B2 B1 )
1
X A2 =
+ n1 X B1 Δ A2 B1 )
If we set Δ A2C2 = Δ B2C2 , X A2 = X B2 , we have:
1
X A1 =
1 + ρ(n1 X B1 Δ
A1B1
+ n2 X C2 Δ A1C2 )
1
1 + ρ(n1 X A1 Δ + 2n2 X B2 Δ B1 A2 )
X B1 =
A1B1
X C2 =
1
1 + ρ(2n2 X A2 Δ
X A2 = X B2 =
C2 A2
+ n1 X A1 ΔC2 A1 )
1
1 + ρ(n2 X C2 Δ
B2C2
+ n1 X B1 Δ B2 B1 )
There are four non-linear equations with four variables and we can solve them
simultaneously using Newton-Raphson method with objective function (for all sites) :
⎛all sites
⎞⎟
⎛
⎞
⎜⎜
F ⎜ ∑ X Ai ⎟⎟⎟ = X A1 ⎜⎜⎜1 + ρ∑ n j ∑ X B j ⎟⎟⎟ −1 ≈ 0
⎜⎝ X A
⎝⎜
⎠⎟
j
Bi
⎠⎟
i
In order to simplify the problem further, all the cross-association energy and volume
parameters are taken to be equal and are estimated as follows:
ε A1B2 = ε B1B2 = εC1 A2 = ε A1C1 ε A2 B2
β
A1B2
=β
B1B2
=β
C1 A2
=
(β
A1C1
+ β A2 B2 )
2
(4.23)
(4.24)
According to Derawi (2002), following mixing rules for the energy parameters shows good
correlation with the experimental data on methanol-water system:
CR-1:
ε
Ai B j
=
ε A1B1 + ε A2 B2
;
2
β
Ai B j
= β A1B1 + β A2 B2 ;
(4.25)
CR-3:
ε
Ai B j
= ε A1B1 + ε A2 B2 ;
β
Ai B j
= β A1B1 + β A2 B2 ;
(4.26)
The Elliott rule:
Δ
Ai B j
= Δ A1B1 + Δ A2 B2
(4.27)
32
4.3. Chemical potential and fugacity coefficient from CPA EoS
In the calculation of the fugacity coefficient in phase equilibria calculation, the NewtonRaphson iteration method is applied to calculate the volume from the CPA equation of state.
This method needs the first and second derivatives of X Ai with respect to the density, and as
seen in Eq. (4.3) this calculation is not quite straightforward especially for the second
derivative. Michelsen and Hendriks ( 2001) proposed a much simpler general expression for
the association term:
∂ ln g ⎞⎟
1⎛
⎟ ∑ xi ∑ (1− X Ai )
Z assoc = − ⎜⎜1 + ρ
2 ⎜⎝
∂ρ ⎠⎟⎟ i
Ai
(4.28)
It is evident from Eq. (4.28) that for non-associating compounds the association term is zero,
and the SRK model is retained.
Derived properties, e.g.
∂ ⎜⎛ Aassoc ⎞⎟
P assoc
⎟
=−
⎜
RT
∂V ⎜⎝ RT ⎠⎟⎟
(4.29)
are determined by differentiation of (4.6). That gives:
⎛ 1
P assoc
1 ⎞⎟ ∂X Ai
= ∑ ni ∑ ⎜⎜⎜
− ⎟⎟
⎜ X Ai 2 ⎠⎟⎟ ∂V
RT
i
Ai ⎝
(4.30)
and thus requires that the volume derivatives of the solution of Eq.(4.30) are calculated.
Similarly, the association contribution to the chemical potentials are calculated from
⎛
⎛
μiassoc
∂ ⎛⎜ Aassoc ⎞⎟
1
1⎞
1 ⎞⎟ ∂X Aj
⎜ 1
⎟⎟ = ∑ ⎜⎜ln X Ai − X Ai + ⎟⎟⎟ + ∑ n j ∑ ⎜⎜
=
− ⎟⎟⎟
⎜
⎜
⎜⎝ X A 2 ⎠⎟ ∂ni
RT
2
2⎠
∂ni ⎜⎝ RT ⎠⎟
Ai ⎝
j
Aj ⎜
j
(4.31)
This requires that the solution of Eq. (4.31) is differentiated with respect to all composition
variables.
When pressure rather than total volume is specified in the property calculation, V must be
determined iteratively, typically by means of some variant of Newton’s method. This requires
calculation of ∂P / ∂V , and use of Eq.(4.29) would require evaluation of the second
derivatives of the unbonded site fractions with respect to volume. The calculation of derived
properties becomes much simpler when we take advantage of the fact that the association
contribution to the Helmholz energy is in itself the result of minimization. As a result,
33
Michelsen & Hendricks (2001) give simplified equation for the contribution to the chemical
potential in CPA as follows:
μiassoc
h ∂ ln g
= ∑ ln X Ai −
2 ∂ni
RT
Ai
Where h = ∑ ni ∑ (1− X Ai ) and
i
Ai
(4.32)
⎛b ⎞
∂ ln g
= 1.9 g ⎜⎜ i ⎟⎟⎟ ρ
⎜⎝ 4 ⎠
∂ni
In term of Volume, we have the result for sCPA and CPA, respectively:
0.475bi
∂ ln g
=
∂ni
V − 0.475 B
(4.33)
∂ ln g
(10V − B)
= −2bi
∂ni
(8V − B)(4V − B)
(4.34)
Then the residual chemical potential from CPA EoS will be
μi r ,CPA = μi r , SRK + μi r ,ass
(4.35)
and the fugacity coefficient can be calculated using:
ln ϕi =
μir ,CPA
− ln Z CPA
RT
(4.36)
4.3.1. Volume from CPA EoS
For the determination of the total volume using Newton-Raphson iteration, we need good
estimates for the start volumes. Volume from SRK part of CPA could be such estimates. For
this purpose, we can solve the SRK equation for volume. However, solution of the equation of
state for Z is greatly preferred over solution for V. The value of Z often falls between 0 and 1,
V often varies from 50-100cm3/mol for liquids to near infinity for gases as P approaches zero.
It is much easier to solve for roots over the smaller variable range using the compressibility
factor Z.
The standard method for solution to cubic equations is as follows (Elliott, 1999). The equation
can be made dimensionless prior to application on the solution method. By noting that:
bρ ≡ B / Z
(4.37)
B ≡ bP / RT
(4.38)
Z ≡ P / ρ RT
(4.39)
34
A ≡ aP / R 2T 2
(4.40)
the Soave-Redlich-Kwong equation of state becomes
Z=
1
A
−
1− B / Z Z (1 + B / Z )
(4.41)
Rearranging the dimensionless SRK equation yields a cubic function in Z that must be solved
for vapor, liquid or fluid roots:
Z 3 − Z 2 + ( A + B − B 2 ) Z − AB = 0
(4.42)
The larger root from this equation will be the vapor root and considering the case when
P=Psat, it will be the value of Z for saturated vapor. The smallest root will be the liquid root
and will be the value of Z for saturated liquid. The middle root corresponds to a condition that
violates thermodynamic stability, and cannot be found experimentally; the derivative of
volume with respect to pressure must always be negative in a real system, and this root
violates that condition. Below the critical temperature, when P>Psat, the fluid will be a
superheated vapor and the liquid root is less stable. Then T>Tc, we have a supercritical fluid
which can only have a single root but it may vary continuously between a “vapor-like” or
“liquid-like” densities and compressibility factors.
4.3.2. Newton-Raphson technique in calculating total volume for CPA EOS
Starting from The Taylor series expansion of f ( x ) around point x = x0 :
2
(
f ( x ) = f ( x0 ) + ( x − x0 ) f '( x0 ) + (1 2)( x − x0 ) f ''( x0 ) + O x − x0
3
)
(4.43)
and setting the quadratic and higher terms to zero, we solve the linier approximation of f ( x ) ,
which gives for x:
x = x0 −
f ( x0 )
f '( x0 )
(4.44)
Subsequent iterations are defined in a similar manner as
xn+1 = xn −
f ( xn )
f '( xn )
(4.45)
35
In this calculation, starting from a function P = f(v) at constant temperature and composition,
to calculate the next value of iteration, we used equation:
f (v) = f (v0 ) + f '(v)(v − v0 )
v = v0 −
f (v0 )
f '(v0 )
(4.46)
(4.47)
The first derivative of equation is needed. Because it is not easy to find the first derivative of
CPA EOS analytically, numerical method is used. There are three methods in numerical
differentiation: forward, backward and central difference formula. The central difference is
the best method and it can be described shortly as follows:
df
f ( v + e) − f ( v − e)
f ( v + e) − f ( v − e)
= f '(v) = lim
= lim
e
→
0
e
→
0
dv
( v + e) − ( v − e )
2e
(4.48)
The Newton-Raphson method requires only one initial value as the initial guess for the root.
To calculate the total volumes for CPA EoS, it is needed initial value from liquid or vapour
volumes which is calculated from SRK EoS .
In CPA EoS, when pressure (P), temperature (T) and composition are known, to solve the
equation, we can rearrange CPA EoS as an objective function.
CPA EOS :
P=
∂ ln g ⎤
RT
a
1 RT ⎡
−
−
1+ ρ
xi ∑ (1 − X iA )
∑
⎢
⎥
∂ρ ⎦ i
v − b v (v + b) 2 v ⎣
Ai
(4.49)
Objective Function :
f (v ) =
∂ ln g ⎤
RT
a
1 RT ⎡
−
−
1+ ρ
xi ∑ (1 − X iA ) − P
∑
⎢
⎥
∂ρ ⎦ i
v − b v (v + b) 2 v ⎣
Ai
(4.50)
The detailed procedure for calculating both liquid and vapor volumes from CPA EOS using
Newton-Raphson method is given in Appendix 2.
To check the results of the first derivative using numerical method, we can take the first
derivative from SRK equation (2.5) :
f ' (v ) =
dP
RT
a
a
=−
+ 2 −
<0
2
dv
(v − b ) bv b(v + b) 2
36
It will be correct when the value of f ' (v ) from central differentiation is negative and it can be
checked from the Fig. 4.4. There are three roots and the one is discarded, actually in
calculation we should be aware from this root, because it seem to be so easy the value of
liquid volume jump to this value and will get a positive f ' (v ) value or this has a meaning that
the result is lay on meta-stable volume.
P
Liquid volume
Vapor volume
Psat
Metastable Volume
vl
vv
v
Fig. 4.4. PV-diagram
4.4. Phase split calculation
Using fugacity coefficients obtained from CPA model, the phase split may be calculated by
Rachford-Rice method, known also as K-value method. According to the method, the
equilibrium relations are solved as a set of K-value problems on the form
yi = Ki xi
(4.51)
where yi and xi are the mole fractions of component i in the vapor and liquid phases
respectively.
In Equation of state method for calculating liquid phase fugacity coefficients, we determine
fugacities for both phases as:
fiV = ϕiV yi P
fi L = ϕiL xi P
(4.52)
37
Then, using the condition for thermodynamic phase equilibrium(1.11), we can write:
ϕiL
Ki = V
ϕi
(4.53)
V, yi
Liquid feed:
P, T
F, zi
L, xi
Fig. 4.5. Isothermal flash calculation
For the system shown in the Fig. 4.5.:
1. Overall material balance: F = V + L
2. Component material balance: Fzi = Vyi + Lxi
3. Elimination of V produces: Fzi = ( F − L) yi + Lxi
4. Using: z L =
L
;
F
zV =
V
, substitution of yi (using Eq.(4.51)) yields:
F
xi =
Fzi
z
= L i V
L + K i ( F − L) z + K i z
K i zi
z + K i zV
yi =
n
5. By definition,
∑ xi = 1 and
1
(4.54)
(4.55)
L
n
∑ y = 1 , we can find zL and
i
z V = 1− z L
1
n
6. Using symmetric condition: f ( z L ) = ∑ ( xi − yi ) = 0 , we have
i =1
n
(1− Ki ) zi
i =1
z L + K i zV
f (zL ) = ∑
n
= ∑ f i zi = 0
i =1
Which is solved with respect to zL using a Newton-Raphson iteration, and need first
derivative of f ( z L ) respect with zL.. We can rearrange f ( z L ) in term zL at denominator,
as:
n
(1− Ki ) zi
i =1
z L (1− K i ) + K i
f (zL ) = ∑
n
= ∑ f i zi = 0
i =1
38
2
n
n
(1− Ki ) zi
∂f ( z L )
L
=
'(
)
=
−
=
−
f
z
f 2 i zi
∑
∑
2
L
L
∂z
i =1 ( z (1− K ) + K )
i =1
i
−1
−1
z
L , k +1
=z
L ,k
⎛ ∂f ⎞
− ⎜⎜ L ⎟⎟⎟
⎜⎝ ∂z ⎠
f =z
L ,k
⎛ n
⎞
+ ⎜⎜∑ f i 2 zi ⎟⎟⎟
⎜⎝ i=1
⎠
f
(4.56)
Substitution of the obtained zL into Eqs. (4.54) and (4.55) produce new values of xi and yi.
4.4.1. Bubble-Point Pressure Calculation
Starting from Eq (1.11) and (4.52), we can write :
ϕiL xi PiV = ϕiV yi P
(4.57)
The relation between ϕi and the residual chemical potential of component i can be found
from Eq (1.25),. Then the phase equilibrium constant, Ki can be obtained from Eq (4.53),
which can be rewritten as follows:
⎡⎛ μ res ⎞L ⎛ μ res ⎞V ⎤
Zv
(4.58)
exp ⎢⎢⎜⎜ i ⎟⎟⎟ − ⎜⎜ i ⎟⎟⎟ ⎥⎥
Ki =
ZL
⎢⎣⎜⎝ RT ⎠⎟ ⎝⎜ RT ⎠⎟ ⎥⎦
In Bubble-Point Pressure calculation we calculate vapour phase fraction of each component
until the sum of vapour phase fractions is equal to 1 (less than tolerance).
C
C
i
i
∑ yi =∑ Ki xi = 1
4.5. Results and discussion
Vapor-liquid phase equilibria calculation has been performed for the binary cross-associating
mixture of methanol and water. The algorithm for the Matlab code is given in Appendix 1.
The CPA pure-compound parameters, used for the calculations, have been obtained by
Kontogeorgis et al. (1999), and are listed in the Table 4.1.
39
Table 4.1. CPA pure-compound parameters for water and methanol
3
b, dm /mol
6
2
a0, bar.dm /mol
c1
ε, bar.dm3/mol
Β
1999
1996
Other**
Methanol
0.030978
0.0330*
0.045587
Water
0.014515
0.0152
0.021127
Methanol
4.0531
4.7052
9.58644
Water
1.2277
2.5547
5.608392
Methanol
0.43102
0.9037
0.5536
Water
0.67359
0.7654
0.65445
Methanol
245.91
183.36
Water
166.55
174.03
Methanol
0.0161
0.0449
Water
0.0692
0.0595
* b=1.52Vw
** calculated using critical parameters Pc and Tc, and SRK acentric factor
Vapor pressure vs volume (PV) diagrams for the pure components are presented in the
Fig.4.6.
To test the performance of the CPA and of the SRK equations of state, they were tested on the
correlation of the saturated vapor pressures of pure water and methanol. The results compared
with the reference data in Fig. 4.7. As we can see in this figure, CPA model gives better
correlation than original SRK. At the higher temperatures (vapor phase), SRK and CPA
models give similar result for both methanol and water, but at the lower temperatures (liquid
phase) SRK have a significant error.
Pressure and chemical potential vs volume diagrams for SRK and CPA EoS at 333.15K are
given in Fig. 4.8. As can be seen from these diagrams, SRK EoS gives higher values of
chemical potential. Starting from this calculation (at equilibrium condition, we have equal
chemical potentials for both phase), we have compared liquid and vapour densities from these
models (Fig.4.9). We see that SRK failed to give a good fitting in liquid phase. Summarized
results on the average absolute error are given in the Table 4.2. Generally, we can say that
CPA model gives better correlation of the experimental data than SRK EoS for associating
compounds.
40
a) methanol
b) water
Fig. 4.6. PV diagrams for pure methanol and water using CPA and SRK EoS (T=333.15K)
41
Fig. 4.7. P-T diagrams for pure water and methanol using experimental data
and calculated using SRK and original CPA equations of state
Table 4.2. Average error calculated from SRK and CPA Model
Average Error (%)
SRK
CPA
Methanol
Water
Methanol
Water
Saturated Pressure
6.09
6.73
1.11
7.07
Liquid Density
28.56
29.97
2.77
2.57
Vapor Density
8.96
10.19
8.88
9.03
42
a) methanol
b) water
Fig. 4.8. μ-P diagrams for pure Methanol and Water using CPA and SRK EoS ( T=333.15K)
43
Fig. 4.9. P-Density (ρ) diagrams for pure methanol and water using experimental data
and calculated using CPA and SRK EoS
44
We have tested also the performances of CPA model, both simplified (sCPA) and original
(CPA), for different systems. P-x,y diagram for methanol (self-associating) – n-pentane
(inert ) system is given in the Fig. 4.10. Our calculation results obtained using original CPA
EoS are similar to those given in the reference paper (Yakoumis, et. al., 1997) if we compare
the average absolute error in bubble-point pressure and average absolute deviation in vapor
phase fraction (2.0 % and 0.0172, respectively). However, simplified sCPA gave worse fitting
than original CPA.
Fig. 4.10. P-x,y diagrams for Methanol / n-Pentane system from experimental data
and calculated using CPA EoS
45
P-x,y diagram for the methanol-water system is given in the Fig.4.11. This diagram is
obtained using simplified CPA EoS not taking into account the cross-association (two selfassociating compounds). The average error in bubble-point pressure calculation in this case is
9.48 % and the average absolute deviation in vapor phase fraction is 0.021. We used 2B
model for Methanol and 4C model for water.
Fig. 4.11. P-xy diagrams for methanol -Water system experimental data
and calculated using CPA EoS
46
As a result of our work, we have good correlation for pure components (methanol and water)
and system consisting of one self-associating and one inert compound (methanol – pentane).
The correlation of the experimental data for the methanol-water system was not that good,
because we did not take into account cross-association that takes place between methanol and
water molecules. We have tried to implement cross-association using schemes proposed by
Kraska (1998) but did not get any better results. So, further study should be done on the
implementation of the cross-association into sCPA model, that should give better fitting of the
experimental data.
47
LIST OF SYMBOLS
General Symbols
A
G
K
P
R
T
V
XA
Z
a
a0
b
c1
f
g
kij,lij
n
x
y
Helmholz free energy
Gibbs free energy
equilibrium constant, K-value
pressure, bar
gas constant, 0.08314 bar dm3/mol-K
temperature, K
molar volume, dm3
mole fraction of the molecule not bonded at site A
compressibility factor
activity
parameter in the energy term, bar-dm6/mol2
covolume parameter, dm3/mol
parameter in the energy term (acentric factor), dimensionless
fugacity
radial distribution function
binary interaction parameters
mole number
mole fraction in the liquid phase
mole fraction in the gaseous phase
Greek Symbols
Δ
Θ,η,ε
α
β
γ
ε
κ
σ
μ
ρ
φ
ω
association strength
coefficients in generalized cubic EoS
coefficient in generalized cubic EoS
association volume parameter, dimensionless
activity coefficient of component in the liquid phase
association (interaction) energy parameter, bar-dm3/mol
interaction volume
segment diameter
chemical potential
molar density, mol/dm3
fugacity coefficient
acentric factor, dimensionless
Subscripts/ Superscripts
0
ig
i,j
m
r
vap
standard state
ideal gas
components in a mixture
molar property
residual function
vapor
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50
APPENDIX 1
ALGORITHM FOR VOLUME CALCULATION FROM CPA EOS
USING NEWTON-RAPHSON ITERATION METHOD.
1. The objective function f (v) ≈ 0 .
2. At the first calculation, put the liquid ( vl ) or vapor volume ( vν ) from SRK EOS as
initial value ( v 0 ).
3. From this initial guess ( v 0 ), we calculate f (v0 ) .
4. To calculate f ' (v 0 ) , take a small value of ( e ≈ 10 −5 ) and then calculate
f (v0 + e) and f (v0 − e) , respectively.
5. The first derivative is calculated from f ' (v0 ) =
f (v 0 + e) − f (v 0 − e)
2e
6. The next value of iteration is calculated from v new = v 0 −
f (v 0 )
f ' (v 0 )
7. The procedure is repeated until convergence is obtained and the iteration will stop
when the value of
f (v 0 )
< Tolerance of convergence (i.e 10 −12 ) , it means that
f ' (v 0 )
f (v 0 )
≅ 0 or f (v) ≈ 0 and v new = v0 .
f ' (v 0 )
51
APPENDIX 2
ALGORITHM FLASH CALCULATION FOR CPA MODEL
52
ALGORITHM BUBBLE-POINT PRESSURE CALCULATION FOR CPA MODEL
53