Physicochemical Properties of Gas

C H A P T E R

2.1 Physicochemical Properties of Gas Elise El Ahmar, Christophe Coquelet Mines ParisTech, PSL University CTP, Centre of Thermodynamics of Processes, Fontainebleau, France

List of Symbols A(T, v, n) fi G P R T Z x y v S H Hi

Helmholtz free energy [J mol
1] fugacity of compound i in the mixing [Pa] Gibbs free energy [J mol
1] pressure [Pa] gas constant [J/(mol K)] temperature [K] compressibility factor liquid mole fraction vapor mole fraction molar volume [m3 mol
1] entropy [J mol
1 K
1] enthalpy [J mol
1] Henry’s law constant [Pa/mol frac]

SUBSCRIPTS C cal exp i,j m v l

2.1.1 INTRODUCTION The goal of this chapter is to give the state of art concerning physicochemical properties of industrial gases used in agrofood processes. For example, these gases are CO2, N2, O3, and SO2. The chapter is divided into four parts. The first part concerns the fundamentals aspects in fluid thermodynamics (molecular interactions, chemical potential, and equilibrium conditions). The second part concerns the equation of state applied for these types of gas. The third part treats the gas solubility and Henry’s law constant correlations. The fourth part gives examples of phase diagrams (Pressure versus Temperature, Pressure versus Density, and Pressure versus Enthalpy) for pure

GREEK LETTERS ω acentric factor γi activity coefficient μi chemical potential of the component i in the phase [J mol
1]

SUPERSCRIPT E Id 0 ∞

excess property ideal mixing reference fluid infinite dilution

Gases in Agro-food Processes https://doi.org/10.1016/B978-0-12-812465-9.00004-9

critical property calculated property experimental property molecular species mixing vapor phase liquid phase

13

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14

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

gases and equilibrium properties of two mixtures. The chapter ends with a conclusion showing the utility of these properties for the design and the optimization of the agrofood processes.

2.1.2 FUNDAMENTAL ASPECTS IN FLUID THERMODYNAMICS: A REMINDER By definition, a phase diagram is a graphical representation of a physical or physical/chemical equilibrium and is a consequence of the chemical thermodynamics of the system. Because this equilibrium is dependent on the composition of the system, the pressure, and the temperature, a phase diagram should be able to tell us what phases are in equilibrium for any composition at any temperature and pressure of the system. Phase diagrams are essential in chemical engineering: a good understanding of phase diagrams leads to choosing or designing the best operating unit. There are many examples of diagram phase utility in chemical engineering. We can mention the distillation process. In fact, under heat effects, a fluid can dissociate into two phases or more. We also have liquid extraction based on the immiscibility between two liquids.

2.1.2.1 Molecular Interactions At the microscopic level, the two molecules can repulse and attract each other. These two classes of molecular interactions depend on the distance r between the cores of the molecules. The function linking force F(r) with the potential of interactions Γ(r) is expressed in Eq. (2.1.1): FðrÞ ¼

dΓ ðrÞ dr

2.1.2.1.2 Attractive Interactions According to the nature of the molecules (apolar, polar, ionic, etc.), there are many categories of attractive strengths. Let’s quote three categories: physical, quasichemical, and chemical (electrolyte). a. Physical interactions Between neutral (nonionic) molecules, van der Waals’ forces dominate the physical attractions. They are characterized by a potential of long-range low interaction. Van der Waals’ forces include three types: dispersion (London), dipole-dipole (Keesom), and induced dipole-dipole (Debye). • Dipole-dipole (Keesom) A polar molecule is electrically neutral but has a permanent electric dipole because of the difference in electronegativities of the atoms that form the molecule. The fact that the positive and negative poles attract each other causes an attractive force (see Fig. 2.1.1). The potential of interaction between two dipoles depends on the distance between two dipolar centers (r), their dipolar moments (μ), and the orientation of the dipolar moments. The potential of the average dipole-dipole interaction on all orientations can be expressed by Eq. (2.1.3):

(2.1.1)

where Γ(r) can be represented by the total of the repulsive and attractive contributions (Eq. 2.1.2): Γ ðrÞ ¼ Γ repulsion + Γ attraction

2.1.2.1.1 Repulsive Interactions When the distance between two molecules decreases, the latter ones repulse each other because of the presence of the electrons on their valence shell. The more the molecules approach, the more the potential for repulsive interactions increases. This potential approaches the infinite when the distance approaches zero.

(2.1.2)

Polar molecule

δ+ FIG. 2.1.1

δ−

Polar molecule

δ+

Dipole-dipole interaction.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

δ−

15

2.1.2 FUNDAMENTAL ASPECTS IN FLUID THERMODYNAMICS: A REMINDER

Polar molecule

δ+

δ−

δ+

δ−

FIG.

Apolar molecule

2.1.2

Induced

dipole-dipole

interaction.

δ+

δ−

δ+

δ−

Apolar polarizable molecule

Γ ij ¼


μ2i μ2j 1 3 kB T ð4πε0 Þ2 r6

Apolar molecule

(2.1.3)

where (
) indicates the attractive force, kB is Boltzmann’s constant, and ε0 the dielectric permittivity of the void.

δ−

• Induced dipole-dipole (Debye) All (polar or nonpolar) molecules are polarizable. A dipolar moment may happen when a molecule undergoes an electrical induction such as the presence of a polar molecule. The induced dipole is instantaneous and can attract another permanent dipole. Fig. 2.1.2 shows this type of interaction between a polar molecule and an apolar molecule. The average potential of interaction due to the induction of permanent dipoles can be expressed by Eq. (2.1.4): Γ ij ¼


αj μ2i + αi μ2j ð4πε0 Þ2 r6

(2.1.4)

The term “polarizability” (α) allows us to quantify the tendency of the formation of an instantaneous dipole within the (polar or apolar) molecule in the presence of a neighboring dipole. • Induced dipole-induced dipole (London) In an apolar molecule, the dipolar moment is statistically zero. But a temporary dipolar moment may happen through the instantaneously inhomogeneous distribution of electron density within a molecule (see Fig. 2.1.3).

δ+

δ−

δ+

Apolar molecule but temporally polarizable

FIG. 2.1.3

Induced dipole-induced dipole interaction

(dispersion).

The induced dipole-induced dipole interaction is generally called “dispersion.” The average potential of interaction of the dispersive interaction can be expressed by Eq. (2.1.5):
 Ii Ij 3 αi αj (2.1.5) Γ ij ¼
2 ð4πε0 Þ2 r6 Ii + Ij where I is the first molecule ionization energy (i.e., the energy required to remove an electron from the molecule). This type of interaction exists between all the molecules because the polarizability and the first ionization energy of a molecule are never zero. It is the only universal type of attractive interaction. In addition to van der Waals’ forces, there are also other types of physical forces associated with the quadrupole such as dipole-quadrupole and quadrupole-quadrupole. Their contributions are often low, but they can be highly

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

16

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

O O

R

O

H

R

H

O

H

H

R O

O H

R

H

R

R H

O H

O

H

R O

R R

FIG. 2.1.4

Hydrogen links in an alcohol.

significant in some cases (e.g., CO2, N2, etc.) (Kontogeorgis and Folas, 2010). b. Quasichemical interactions: hydrogen bonds The quasichemical forces refer to Lewis’ acidbase interactions. The transfer of electrical charges between the molecules leads to the formation of bonds (about 10 kJ mol
1) stronger than the physical forces (often below 1 kJ mol
1), but weaker than the chemical bonds (100 to 1000 kJ mol
1). The hydrogen bonds are a typical example of the quasichemical interactions. A hydrogen bond is established when a hydrogen atom bonded to a highly electronegative atom (e.g., oxygen, nitrogen, or fluorine) approaches another atom, also highly electronegative and carrying a free electric dipole. The hydrogen bonds are responsible for the formation of oligomers from monomers (see Fig. 2.1.4). This type of interaction is called “association” in thermodynamics. Depending on the nature of the molecular interactions, the fluid will adopt a behavior and propose several types of phase diagrams.

2.1.2.2 Phase Definition A phase is a homogeneous system composed of one pure component and/or a mixture of components. A system can be composed of one or several phases. Two kinds of systems

can be distinguished: monophasic homogeneous systems, for which composition and thermodynamics properties are identical in the whole space, and multiphase heterogeneous systems, for which thermodynamics properties changed brusquely at the interface. A phase is characterized by its temperature, its density, its pressure, and other thermodynamics properties (Gibbs energy, molar enthalpy and entropy, heat capacity, etc.). Phase is employed to distinguish the different states of matter. It exists mainly in three states: gas, liquid, and solid.

2.1.2.3 Phases Transitions A phase transition is commonly used to describe the transition between different states of matter. Table 2.1.1 illustrates all the phase transitions between the solid, liquid, and vapor phases. A phase transition can be obtained by a change of the composition of the system, a change of temperature and/or pressure, or by the application of external strength. Consequently, composition, density, molar internal energy, enthalpy, entropy, refractive index, and dielectric constant have different values in each phase. But, temperature and pressure are identical concerning multiphase systems regarding the thermodynamic principle. So when two phases (or more) exist, we can speak about phase equilibria.

TABLE 2.1.1

Phases Transitions

Phase 1

Phase 2

Transition 1 ! 2

Liquid

Vapor

Boiling

Liquid

Solid

Solidification

Vapor

Liquid

Liquefaction

Vapor

Solid

Condensation

Solid

Vapor

Sublimation

Solid

Liquid

Melting

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

2.1.2 FUNDAMENTAL ASPECTS IN FLUID THERMODYNAMICS: A REMINDER

We have two types of phase transition: First order: Concerning this type of phase transition, we have an important variation of the molar entropy due to an exothermic or endothermic effect on the system. The temperature will stay constant. Consequently, diverging exists concerning
heat capacity and susceptibil∂V ). In other words, the first ity (χ T ¼
V1 ∂P T derivative of Gibbs free energy with regard to thermodynamics variables is discontinuous. Second order: There is no discontinuity of entropy. The heat capacity and susceptibility diverge. In other words, the first derivative of Gibbs energy with regard to thermodynamics variables is continuous but not the second derivative, which is discontinuous.

2.1.2.4 Variance The variance F of the system is determined from the Gibbs phases rule (Eq. 2.1.6). It is the picture of the degree of freedom of the system. F ¼ C + 2
φ,

(2.1.6)

with F, the number of degree of freedom, C, the number of components, and Φ, the number of phases in presence. Table 2.1.2 determines the degree of freedom for one compound. TABLE 2.1.2 (C ¼ 1)

Degree of Freedom for One Compound

2.1.2.5 Phase Diagram for Binary Systems The phase diagram is clearly defined compared to current species so compared to the various molecular interactions. Indeed, molecular species 1 interacts with another molecular species 1 but also with a molecular species 2. The interactions between 1 and 2 can be of a different nature. There are several scenarios that may present themselves for liquid-vapor equilibrium. Table 2.1.3 shows the different cases. Van Konynenburg and Scott (1980) have classified the mixture into six types considering van der Waals EoS and quadratic mixing rules. Fig. 2.1.5 presents the different types of phase diagrams. The transition between each type of diagram can be explained by considering the effect of size and the repulsive interaction. Fig. 2.1.6 gives a view of the phase diagram transitions.

2.1.2.6 Chemical Potential The chemical potential is one of the most important thermodynamic variables in the context of phase equilibrium. If we consider one phase with volume V which content N components, at temperature T and pressure P, the chemical potential μi of the component i in the phase is defined by Eq. (2.1.7): μi ðP, T, n1 , n2 , …, nC Þ

Number of Phases

Degree of Freedom

Variables

Vapor pressure, or melting or sublimation curves

2

1

T or P

Liquid, vapor, or solid

1

2

P and T

Triple point

3

0

Everything is fixed

Critical point

2

1

TC or PC

Region of the Phase Diagram

17


 ∂GðP, T, n1 , n2 , …, nC Þ ¼ ∂ni T , P, n j6¼i

(2.1.7)

where G ¼ H-TS is the Gibbs free energy of the phase. The expression for the infinitesimal reversible change in Gibbs free energy is given by Eq. (2.1.8): X μi dni : (2.1.8) dG ¼ VdP
SdT + i

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18

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

Various Scenarios (Pressure Is Given With Arbitrary Unit Value)

7

40

6

35

Pressure

Pressure

TABLE 2.1.3

5 4 3 2

30 25 20 15 10

1

5

0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

x, y

x, y

Ideal mixture

Positive deviation from ideal mixture

1

30

40 35

25

Pressure

Pressure

30 25 20 15

20 15 10

10

5

5 0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

x, y

Negative deviation from ideal mixture

0.8

1

Strong negative deviation from ideality

1.4

2.5

1.2

2

Pressure

1 0.8 0.6

1.5 1

0.4 0.5

0.2 0

0 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6

0.8

x, y

x, y

Azeotropic mixture-Pressure minimum

Azeotropic mixture-Pressure maximum

2.5 2

Pressure

Pressure

0.6 x, y

1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

x, y

Heteroazeotropic mixture

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

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2.1.2 FUNDAMENTAL ASPECTS IN FLUID THERMODYNAMICS: A REMINDER

19

FIG. 2.1.5 Six types of phase behavior in binary fluid systems. C, Critical point; L, Liquid; V, Vapor; UCEP, Upper critical end point; LCEP, Lower critical end point. Dashed curve are critical.

Size effect (s)

TYPE V

TYPE I

TYPE IV

TYPE III S + MI

VdP
SdT +

X

ni dμi ¼ 0:

(2.1.9)

i

TYPE II Size effect (s)

Molecular interaction effect (MI)

Moreover, P c from the Euler theorem we can write G¼ N i¼1niμi. So, the Gibbs-Duhem (Eq. 2.1.9) equation can be given from Eqs. (2.1.7), (2.1.8):

Molecular interaction effect (MI)

2.1.2.7 Activity Coefficient and Fugacity

H bonding TYPE VI

FIG. 2.1.6

Evolution of phase diagrams.

The fugacity is defined from the variation of the chemical potential (Eq. 2.1.10); the mixing-free enthalpy can then be expressed with the help of the fugacities (see Eq. 2.1.11):

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

20

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

dμi ¼ RTd ln fi
 X fi M G ¼ RT Ni ln 0 f i i

(2.1.10) (2.1.11)

The quotient of the fugacities in a mixing and in the referential state is called “activity” and is given by Eq. (2.1.12). fi ai ¼ 0 fi

(2.1.12)

In 1923, Lewis defined the ideal solution with Eq. (2.1.13) (with T and P): fiid ¼ fi0 xi

(2.1.13)

where fi is the fugacity of compound i in the mixing, the exponent id the ideal mixing, and 0 the pure body. So, for an ideal mixing, the activity comes down to the molar fraction (Eq. 2.1.14). a i ¼ xi

(2.1.14)

The activity coefficient (γ i) allows us to measure the distance between a mixing and the ideality compared to an ideal mixing. Usually, it is suitable to choose γ i ¼ 1 when xi ! 1. Therefore, γi ¼ γ∞ i when xi ! 0. By developing Eq. (2.1.11), we can show the free-excess enthalpy in the expression of the mixing-free enthalpy (Eq. 2.1.15): X X Ni RT ln ai ¼ GE + Ni RT lnxi GM ¼ i

i

Gα +Gβ are the Gibbs free energy of the phases α and β, respectively. At equilibrium, the Gibbs free energy must be at the minimum, that is to say (dG ¼ 0) (Eq. 2.1.17): dG ¼ dGα + dGβ ¼

X i

μαi dnαi +

X

μβi dnβi ¼ 0

i

(2.1.17) In a closed system, we have dnβi ¼
dnαi for each component i between phases α and β. For each chemical species, we can write Eq. (2.1.18).     μαi P, T, nα1 , nα2 , …, nαNC ¼ μβi P, T, nβ1 , nβ2 , …, nβNC (2.1.18) The equilibrium conditions of a multiphase mixture are equality of temperature, pressure, and chemical potential μi of each component i in all the phases in equilibrium. Thermodynamic models are used to calculate chemical potentials. Several types of equations of states can be used to calculate the thermodynamic properties of gas.

2.1.3 EQUATIONS OF STATE In this section, three types of equations of state (EoS) will be presented: Fundamental EoS, cubic EoS, and molecular model.

(2.1.15) The activity coefficient is linked to the freeexcess enthalpy with Eq. (2.1.16): X GE ðT, P, xÞ ¼ xi RTLnðγ i Þ (2.1.16)

2.1.2.8 Equilibrium Conditions We consider a multicomponent system in equilibrium between two phases (α and β), at temperature T and pressure P. The Gibbs free energy of this system is G ¼ Gα +Gβ, where

2.1.3.1 Fundamental Equations of State It is well known from Helmholtz energy that all thermodynamic properties can be calculated. Table 2.1.4 presents the thermodynamic properties obtained from Helmholtz free energy A(T, v, n). 2.1.3.1.1 Pure Fluids Fundamental equations of state are explained in terms of reduced molar Helmholtz free energy (Eq. 2.1.19).

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

21

2.1.3 EQUATIONS OF STATE

TABLE 2.1.4

Properties Calculation From Helmholtz Free Energy

Property

Relation
 ∂A μi ¼ ∂ni V , T, nj

Chemical potential


 ∂A P¼
∂V T  ! ∂ A= U ¼ 1 T  ∂ =T V

Pressure Internal energy

H ¼ A + TS
 ∂A S¼
∂T V

Enthalpy Entropy

G ¼ A
pV
 ∂H CP ¼ ∂T p

Gibbs energy Isobaric heat capacity Cp


 ∂U ∂T V

Isochoric heat capacity Cv

CV ¼

Fugacity coefficient

ln φ ¼ Z
1
ln Z + ART
res  ∂A ðT, V, nÞ res
RT lnZ + ART ln φi ¼ ∂ni T, v, nj

res

Partial molar fugacity coefficient of component i in mixture

AðTr , ρr Þ Aid ðTr , ρr Þ Ares ðTr , ρr Þ ¼ + RT RT RT id ¼ a ðTr , ρr Þ + ares ðTr , ρr Þ

(2.1.19)

id

concerns ideal gas contribution and res concerns residual contribution. ρr and Tr are the reduced variables. Tillner-Roth and Baehr (1994) have proposed an expression concerning the Helmholtz free energy model (Eq. 2.1.20). Temperature and density are expressed in the dimensionless variables δ ¼ ρρ and τ ¼ TTr .

For some fluids, it is more convenient to use a virial type equation for state (Eq. 2.1.21) such as the Benedict Webb Rubin (Benedict et al., 1949, Eq. 2.1.22) or the modified BWR (Eq. 2.1.23) equations of state. Z ¼ 1 + Bρ + Cρ2 + Dρ3 + …

(2.1.21)

P (Z ¼ ρRT )

Z is the compressibility factor and is equal to one for an ideal gas; B, C, and D are the second, third, and fourth virial coefficients respectively. vr is the reduced molar volume.

r

X X   AðTr , ρr Þ ¼ ln ðδÞ + αi τti + αk τtk δdk exp
γδlk RT i k (2.1.20)

Eq. (2.1.20) contains several adjustable parameters (αi and αk) and several adjustable exponents (tk, dk, and lk). Data are required to adjust these parame6 0. ters. Moreover, if lk ¼ 0, γ ¼ 0 and γ ¼ 1 if lk ¼

BWR:



 B C D c4 γ γ Z ¼ 1 + + 2 + 5 + 3 2 β + 2 exp
2 vr vr v r T r v r vr vr (2.1.22)    with B ¼ b1
b2 Tr
b3 T2
b4 T3 , r r  C ¼ c1
c2 =Tr + c3 =Tr3 , and D ¼ d1 + d2 Tr .

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

22

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

mBWR: P¼

9 X

"
αn ρn + exp

 # 15 ρ 2 X ρC

n¼1

MR n°1: αn ρ2n
17

X

A0 ¼

n¼10

Eq. (2.1.22) contains several parameters (12) β, γ, bi, ci and dj (where i ¼ 1–4 and j ¼ 1–2). Eq. (2.1.23) also contains several adjustable parameters (αn). All parameters must be adjusted on experimental data (equilibrium properties or densities) using a specific algorithm (objective function to minimize, numerical methods). Using these equations of state, it is possible to calculate the Helmholtz free energy. Considering Eq. (2.1.19), the residual term can be calculated and is given in Eq. (2.1.24). ares ðT, vÞ ¼

1 RT



v¼∞

RT
P v

X

C0 ¼

1

=2

E0 ¼

X

b¼

1

xi b i

d¼



1

xi d i

=3

i

dv

(2.1.24)

X

γ¼

T

1

xi γ i

=2

X

1

xi a i

=3

=3

1

xi c i

X

,α ¼

1

xi αi

1
R

ðT

cid p T

,

!2 ,

,

=3

!2 ,

i

!2

i

The determination of ideal term requires the calculation of the isobaric heat capacity (Eq. 2.1.25). hid sid ρT 1 aid ðT, vÞ ¼ 0
0
1 + ln + RT R ρ0 T0 RT

!2

!2

i

!2

=2

xi D0i

i

,c ¼

i

X

X

!2

=3

1

i

,a ¼

x i E0 i

i

X

, D0 ¼

!2

=2

x i B0 i ,

i

!2

xi C0i 1

X

,B0 ¼

xi A 0 i

i

X

!2

=2

i

(2.1.23)

ðv

1

(2.1.27) MR n°2:

ðT cid p dT T0

XX

A0 ¼

i

dT

B0 ¼



2.1.3.1.2 Mixtures Helmholtz free energy can also be split into ideal and residual terms (Eq. 2.1.26). A ð Tr , ρ r , x Þ ¼ a ð Tr , ρ r , x Þ RT ¼ aid ðTr , ρr , xÞ + ares ðTr , ρr , xÞ (2.1.26) For the mixture, one possibility is to directly use the previous equations of state, but mixing rules have to be considered. For example, with the BWR equations of state, the following mixing rules (MR) can be used (Eqs. 2.1.27, 2.1.28).

D0 ¼

xi B0i , C0 ¼

XX i

E0 ¼

1

1

j

X X

1

xi a i 1

xi c i

=3

α¼

 1
kij ,

i

2

 2

=3

1

1

,b ¼

xi α i

 2

1



xi xj C0i C0j





2

2

3 1
lij ,

2

4 1
mij ,

5 1
nij ,

X

1

xi b i

=3

i

!2

=3

1

j

1

!2

i

X

i



xi xj E0i E0j

i

c¼

2

XX

xi xj D0i D0j

j

XX i

a¼



xi xj A0i A0j

i

(2.1.25)

1

2

j

X

T0

1

,d ¼

X

1

xi d i

=3

!2 , !2

i

!2 ,γ ¼

X

1

xi γ i

=2

, !2

i

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

(2.1.28)

23

2.1.3 EQUATIONS OF STATE

Generally, in order to have the best prediction of thermodynamic properties, it is useful to consider binary interaction parameters (kij, lij, mij, and nij). Also, the determination of these interaction parameters requires numerous experiments for the mixture (equilibrium properties and densities). The second possibility is to consider the multifluid approximation. This approach was introduced by Tillner-Roth (1993). It applies mixing rules to the Helmholtz free energy of the mixture components (Eq. 2.1.29).  X  id res x a ð T , ρ , x Þ + a ð T , ρ , x Þ aðTr , ρr , xÞ ¼ j r r r r j j j + xj lnxj +

X X

xp xq Fpq aEpq

(2.1.29)

p¼1 q¼p + 1

P P E Δares is called the pq ¼ p¼1 q¼p + 1 xp xq Fpq apq departure function from ideal solution. p and q are the component index. It is an empirical function fitted to experimental binary mixture data. In this departure function, the Fpq parameters take into account the behavior of one binary pair with another. If only vapor liquid equilibrium properties are available, aEpq is considered to be equal to zero (Mac Linden and Klein, 1996). With the multifluid approximation, it is important to calculate the new critical properties that correspond to the mixture studied as reducing parameters are used. Eqs. (2.1.30), (2.1.31) detail one type of mixing rule. TCmel ¼

XX

 0:5 kT, pq xp xq TCp TCq

(2.1.30)

p¼1 p¼1

VCmel ¼

XX p¼1 p¼1

kv, pq xp xq

1=3  1=3 3 1  VCp + VCq 8 (2.1.31)

kT, pq and kv, pq are adjustable parameters. Kunz and Wagner (2012) proposed different mixing rules (Eqs. 2.1.32, 2.1.33).

TCmel ¼

X

x2i TCi +

XX

2xp xq βTpq γ Tpq

p¼1 p¼1

i

0:5 xp + xq  TCp TCq 2 βTpq xp + xq VCmel ¼

X i

x2i VCi +

XX

(2.1.32) xp xq βvpq γ vpq

p¼1 p¼1

1=3  1=3 3 xp + xq 1   V + V Cp Cq β2vpq xp + xq 8

(2.1.33)

βT,pq, βv,pq, γ T,pq and γ v,pq are adjustable parameters. Using these different equations, the thermodynamic properties are calculated with a high degree of accuracy, particularly if the composition approaches a mole fraction of 1, as high accuracy equations of state for the components are used. The main disadvantage of this model is the number of parameters to be adjusted. So, the number of experimental data requires to fit them.

2.1.3.2 Cubic Equations of State The first tests done in the 19th century were related to the study of the gases. The first thermodynamic model that allowed us to understand the behavior of the gases is the model of the perfect gas (Eq. 2.1.34). Pv ¼ RT

(2.1.34)

The determination of this state equation is based on the kinetic theory whose assumptions are:





The gas is assimilated to a monoatomic gas. Speed isotropy. Uniform molar density. No interaction between the molecules (the pressures are relatively low).

This model is similar to an EoS because it connects the various intensive variables (P, T, v). Unfortunately, it has a limited gas use. Later, Van der Waals (1873) attempted to take into account the interactions between molecules.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

24

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

Considering the repulsive and attractive forces, he proposed changing the kinetic pressure with (attractive a negative molecular pressure
a v2 interactions between molecules, where a is the energy parameter). The expression of the molecular pressure results from the expression of the potential of interaction between the molecules. With the perfect gas equation, the volume becomes zero when the pressure gets very high. Therefore, he also took into consideration the repulsive interactions via the molar covolume b. The equation for the pure substance is given by Eq. (2.1.35):  a (2.1.35) P + 2 ðv
bÞ ¼ RT v where a is the attraction parameter (called energy parameter) and b the molar covolume. The determination of a and b is done at the critical point where (see Eq. 2.1.36):
2 
 ∂P ∂ P ¼ ¼0 (2.1.36) ∂v T ∂v2 T We then can find: a¼

27 R2 TC2 1 RT C PC v C ; b¼ ; ZC ¼ ¼ 0:375 RT C 64 PC 8 PC

This equation was the first to reveal the existence of a liquid-vapor phase transition and to report on the existence of a critical point. It represents the properties of the liquid phase less than those of the vapor phase. In order to better show the thermodynamic properties of the fluids (vapor tensions and volumes), other researchers have proposed to modify the Van der Waals equation (VdW). The cubic equations are only improvements of the VdW equation (more particularly of the expression of the molecular pressure that comprises the attractive parameter a). In 1949, Redlich and Kwong proposed a first modification. In 1972, Soave modified the expression of the attractive term and used a temperaturedependent function. Soave, Redlich, and Kwong’s equation (SRK), which applies for

apolar (or slightly polar) compounds, is given by Eq. (2.1.37):
 aðT Þ ðv
bÞ ¼ RT (2.1.37) P+ ðv + bÞv The determination of the a and b parameters of this equation is done the same way we do for the Van Der Waals one at the critical point. R2 TC2 RT C ; Ωa ¼ 0:42748; b ¼ Ωb ; PC PC 1 Ωb ¼ 0:086640; Zc ¼ 3 a ¼ Ωa

This equation allows us to better correlate the experimental data in a wider field and to improve the representation of the critical zone. In 1976, Peng and Robinson proposed another modification of the attractive term. This equation is generally used for polar compounds (also used for hydrocarbons) and gives results that are closer to experimental results (mainly for volumetric properties) than those of the SRK equation. The expression of Peng and Robinson’s equation (PR) is given by Eq. (2.1.38):
 aðT Þ ðv
bÞ ¼ RT (2.1.38) P+ 2 ðv + 2bv
b2 Þ As for a and b, we get: a ¼ Ωa

R2 TC2 RT C ; Ωa ¼ 0:47236; b ¼ Ωb ; PC PC Ωb ¼ 0:07780; ZC ¼ 0:3074

Other cubic EoS have been developed; we can cite Patel and Teja (1982), Trebble and Bishnoi (1987), and Coquelet et al. (2016). In 1986, Trebble and Bishnoi made a comparative study of the two cubic EoS (SRK and PR). Table 2.1.5 show the comparisons about the pressures and the liquid and vapor volumes in absolute relative mean value (ΔY ¼ j(Yexp
Ycal)/Yexp j). We can note that the SRK cubic equation does not allow us to make a very precise

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

25

2.1.3 EQUATIONS OF STATE

TABLE 2.1.5 Ranges on the Calculation of Vapor Pressures and of Liquid and Vapor Molar Volumes Authors

Δ(psat)/%

Δ(vliq)/%

Δ(vvap)/%

Soave (1972)

1.5

17.2

3.1

Peng and Robinson (1976)

1.3

8.2

2.7

calculation of the liquid volumes. The vapor pressures are generally well calculated because the researchers have developed alpha functions that make the attractive term a vary with temperature. In 1972, Soave added the following alpha function (Eqs. 2.1.39, 2.1.40) in order to improve the calculation of the vapor pressures (and of the liquid and vapor volumes). TR is the reduced temperature and ω the acentric factor: h  i2 1=2 αðT Þ ¼ 1 + m 1
TR (2.1.39) m ¼ 0:480 + 1:574ω
0:175ω2

(2.1.40)

Other alpha functions have been developed; we can cite Stryjek and Vera (1976), Mathias and Copeman (1983), Twu et al. (1995a,b), and Coquelet et al. (2004). Recently Jaubert and coworkers [Le Guennec et al. (2016, 2017)] proposed a new method to characterize alpha function and defined new parameters for the Twu et al. alpha function. 2.1.3.2.1 Mixing Rules The mixing rules must be able to take into account the ideal and nonideal characteristics of the solutions. With the two-parameter cubic equations, the objective is to calculate again the a and b parameters, considering the mutual influence of the various compounds. The first set of mixing rules is that of Van der Waals, which corresponds to what is commonly called “classical mixing rules.” Starting from the state equation, developing the virial about the volume and applying the statistical thermodynamics, we have Eq. (2.1.41):

a¼

XX i

xi xj aij

(2.1.41)

j

 P pffiffiffiffiffiffiffi where aij ¼ ai aj 1
kij ;b ¼ i xi bi kij is called the binary interaction parameter of the decoupling constant. This parameter must take into account the fact that the attractive interactions between compounds i and j are different from those between i and i and j and j. Several mixing rules have been developed. We will present two of them (Huron-Vidal and Wong-Sandler mixing rules). Their authors have taken into account models based on the calculation of the activity coefficient (excess-free enthalpy) and models by state equation. Indeed, the first ones are adapted to the low-pressure treatment of polar and/or nonpolar bodies while the state equations give satisfying results only for apolar bodies but without any pressure limitation. Thus, they have written (Eq. 2.1.42): gEγ ðT, P ! ∞Þ ¼ gEEoS ðT, P ! ∞Þ

(2.1.42)

and v ¼ b when P ! ∞. This mixing rule has been presented by Huron and Vidal (1979): ! X X
ai  E + gP¼∞  C ; b ¼ xi xi b i ; a¼b bi i i r1
r2  C¼
1
r1 Ln 1
r2 where r1 and r2 depend on the chosen state equation. Wong and Sandler (1992) have kept the classical mixing rules obtained with the development of the virial. However, as for the Huron-Vidal mixing rule, the equality of excess-free enthalpies, according to the processing with state equation and activity coefficient, allows us to obtain another relation between the attraction parameter a and the molar covolume b. Starting from that, we can also write from Helmholtz free energy (Eq. 2.1.43):

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

26

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

aEEOS ðT, P ! ∞, xÞ ¼ aEγ ðT, P ! ∞, xÞ

(2.1.43)

On the other hand, the authors have considered that the free energy depends less from the pressure than from the free enthalpy. Thus, they have written Eq. (2.1.44): aEγ ðT, P ! ∞, xÞ ¼ aEγ ðT, P ¼ 1 Bar, xÞ

(2.1.44)

In fact, the fundamental relation of the thermodynamics is written (Eq. 2.1.45): g ðT, P, xÞ ¼ a ðT, P, xÞ + Pv E

E

E

(2.1.45)

They have considered that, at low pressure, the PvE product is insignificant compared to aE. Consequently, they have obtained (Eq. 2.1.46): aEEOS ðT, P ! ∞, xÞ ¼ aEγ ðT, P ! ∞, xÞ ¼ aEγ ðT, P ¼ 1 Bar, xÞ (2.1.46) ¼ gEγ ðT, P ¼ 1 Bar, xÞ Thus, after having written the mixing-free energy, we come to Wong-Sandler’s mixing rule (see Eqs. 2.1.47, 2.1.48, and 2.1.49):  XX a  xi xj b
RT ij i j (2.1.47) b¼ 0X a i 1 xi bi gEγ ðT, P, xÞC B i C 1
B @ RT + CRT A and


the NRTL model: A local composition model based on the lattice theory.
the UNIQUAC model: Also based on the concept of local composition.
the UNIFAC model: a group contribution model. The gE can also be calculated by the Redlich and Kister (1948) form given by Eq. (2.1.50): gE ¼ x1 ð1
x1 Þ

2 X

ðRTGn ð2x1
1Þn Þ

where Gn(n ¼ 0–2) are the adjustable parameters. 2.1.3.2.3 NRTL Model (Nonrandom Two Liquids) Proposed in 1968 by Renon and Prausnitz, this model is based on mixing internal energy according to the local compositions. The expressions of the activity coefficients and of the free-excess enthalpy are given by Eqs. (2.1.51), (2.1.52), respectively: X τj, i Gj, i xj j

Lnðγ i Þ ¼ X

Gj, i xj

j

+

0

X

(2.1.48)

with

  a  1  a   a   1
kij b
¼ + b
b
RT ij 2 RT i RT j (2.1.49) 2.1.3.2.2 Activity Coefficient Models We need activity coefficient models to calculate Gibbs energy. For this reason, different models have been developed, called “activity coefficient models” or “GE models.” We can cite:

Gk, j τk, j xk

1

X Gi, j xj B C Bτi, j
kX C X @ A G x G x k, j k k, j k j k

 XX a a  b
xi xj b
¼ RT RT ij i j

(2.1.50)

n¼1

Ci , j ; τj, i ¼ RT

k



(2.1.51)  ; Ci, i ¼ 0 

C
αj, i RTj, i

Gi, j ¼ Exp
Cj, i X X xj Exp
αj, i RT
 Cj, i (2.1.52) gE ¼ xi X Ck, i i j xk Exp
αk, i RT k

with

We can notice that we have six parameters for a binary mixing αi, j and Ci, j which are likely to be adjusted from experimental data. Generally, the parameters αij are set (0.2 or 0.3 and even 0.5, which corresponds to mixture families). Other researchers have created the UNIQUAC model, which has fewer parameters.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

27

2.1.3 EQUATIONS OF STATE

2.1.3.2.4 The UNIQUAC Model (UNIversal QUAsi Chemical) Abrams and Prausnitz (1975) developed this model. As for the NRTL model, the UNIQUAC model is based on the concept of local composition. They have considered that each constituent could be decomposed into segments (parameter ri) and that the interactions depend on the external surface of the constituents (surface parameter qi). Thus, by writing the mixing internal energy, they have shown two excess-free enthalpies that take into account the interactions between the constituents (residual excess-free enthalpy) and size parameters (ri and qi) of each constituent (combinatory excess-free enthalpy). The expressions of the activity coefficients are given by Eqs. (2.1.53, 2.1.54, and 2.1.55): Lnðγ i Þ ¼ Lnðγ i Þcombinatory + Lnðγ i Þ Residual (2.1.53) with Lnðγ i Þ

Combinatory



 Φi z Θi + qi Ln + li ¼ Ln xi Φi 2 Φi X
xj lj xi j (2.1.54)

where

li ¼ ðri
qi Þ 2z
ðri
1Þ; Θi ¼ Xi

x qi

Moreover,

xj qj

xi ri ; Φi ¼ X ; z ¼ 10 x j rj

j

2

j

0

3

1

X Θj τij 7 X 6 7 @ X Lnðγ i Þ Residual ¼ qi 6 Θj τji A
5 41
Ln Θk τkj j j k

(2.1.55)

The expression of the excess-free enthalpy is given by Eqs. (2.1.56), (2.1.57), and (2.1.58): g ¼g E

E,combinatory

gE, combinatory ¼ RT

"

X i

+g


xi Ln

E, residual

(2.1.56)


Φi zX Θi xi qi Ln + xi Φi 2 i

#

0 0 11 X X E, residual g ¼
RT @ qi xi Ln@ Θj τji AA (2.1.58) i

j

Thus, the UNIQUAC model requires knowledge of only two parameters for each binary. The volume and surface parameters (Van der Waals surface and volume) have been determined from the volumes and the surfaces of the molecules. Despite everything, if we do not have any experimental data, we cannot make predictive calculations. From the UNIQUAC model, the researchers have developed predictive models based on the contributions from groups. One of the first predictive models is the UNIFAC model. 2.1.3.2.5 The UNIFAC Model (UNIquac Fonctional Group Activity Coefficient) The UNIFAC model was proposed by Fredenslund et al. in 1975. Its principle is founded on that of the UNIQUAC model, that is to say the excess-free enthalpy can be decomposed into two free enthalpies, combinatory and residual. However, the authors have taken into account interactions between groups instead of taking into account interactions between constituents, knowing that a constituent is an assembly of these groups. Thus, in the framework of a binary, for example, we must not consider a two-compound solution anymore but a group solution. The whole difficulty is in the UNIFAC decomposition of the molecule. The values of the binary interaction parameters are generally available in the literature and they are constantly updated. The expression of the residual activity coefficient is Eqs. (2.1.59, 2.1.60, and 2.1.61):  i   X ðiÞ h ðiÞ ¼ υ Ln ð Γ Þ
Ln Γk Ln γ residual k i k k

(2.1.59) 3

2

! X X Θm Ψkm 7 6 X LnðΓk Þ ¼ Qk 41
Ln Θm Ψmk
5 Θn Ψnm m m

(2.1.57) 2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

n

(2.1.60)

28

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

where: Xm Qm Θm ¼ X Xn Qn

(2.1.61)

n

and Xm is the molar fraction of the group in the mixing and υkis the number of k under-groups present in the mixing.  a  mn Ψmn ¼ Exp
(2.1.62) T where amn is the interaction parameter between the various undergroups. As for the combinatory term, ri and qi are calculated by simple additivity rules: X X ri ¼ υm, i Rm ;qi ¼ υm, i Qm m

m

where Rm and Qm are the volume and surface parameters of each undergroup.

2.1.3.3 Molecular Model Based on Wertheim’s (1984) statistical theory of associative fluids, Chapman et al. (1989, 1990) developed the first EoS SAFT (Statistical Associating Fluid Theory) called SAFT-0. Many versions exist today, such as LJ-SAFT (1994), SAFT-VR (1997), Soft-SAFT (1997), PC-SAFT (2001), etc. The various versions differ mainly in the choice of the reference fluid, the radial

Segment of hard spheres

distribution function, and explicit expressions of the terms of disruption. The PC-SAFT (Gross and Sadowski, 2001, Perturbed-Chain Statistical Associating Fluid Theory) EoS will be presented in this section. Each molecule is considered as a chain of spherical segments that is not inevitably identified as an atom. The u(r) interaction potential between the segments of a chain corresponds to a modification of the potential of square wells (proposed by Chen and Kreglewski, 1977) (Eq. 2.1.63). 8 ∞ r < ð σ
s1 Þ > > > < 3ε ðσ
s1 Þ  r < σ (2.1.63) uðrÞ >
ε σ  r < λσ > > : 0 r  λσ where r is the distance between two segments, σ the diameter of the segment, ε the depth of the potential trough, and λ the reduced width of the well (s1 ¼ 0.12σ). The compressibility factor is the total of the three terms (Eq. 2.1.64). Z ¼ 1 + Zseg + Zchain + Zasso

The first term takes into account the repulsive and attractive interactions. Fig. 2.1.7 shows the PC-SAFT different contributions. The repulsive interactions are studied with a hard sphere model. The Boublik (1970) and Association

Polarity Dispersion

Chain’s formation

FIG. 2.1.7

(2.1.64)

Illustration of the various PC-SAFT contributions. 2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

29

2.1.3 EQUATIONS OF STATE

Mansoori et al. (1971) expressions are used (Eqs. 2.1.65, 2.1.66); Zseg ¼ mZhc + Zdisp

(2.1.65)

ξ 3ξ1 ξ2 3ξ32
ξ3 ξ32 + (2.1.66) Z ¼ 3 + 1
ξ3 ξ0 ð1
ξ3 Þ2 ξ0 ð1
ξ3 Þ3 P where ξn ¼ π6 ρ i xi mi dni and d is the collision diameter corresponding to a chain segment while m is the number of segments for each chain. The chain term is calculated with the help of Eq. (2.1.67): hc

Z

chain

¼


X i

∂ ln ghc ii xi ðmi
1Þρ ∂ρ

(2.1.67)

ghc ii is the function of radial distribution for the segments of the compound i in a system of hard spheres. As for the dispersive part, Barker and Henderson’s (1967) theory is used while taking as a reference the chain of hard sphere segments. The dispersive interactions are only a disturbance in the reference state. They are applied to molecules having several segment chains. This allows the PC-SAFT model to be used for the study of the polymers. We have Eq. (2.1.68): Adisp A1 A2 ¼ + NkT NkT NkT

(2.1.68)

where A1 and A2 are the first and second range contributions, k Boltzmann’s constant. They are determined from the following relations (Eqs. 2.1.69, 2.1.70), which can be applied to any interaction potential.
 ∞
 ð A1 σ 2 ε eðxÞghc m; x x2 dx σ3 u ¼
2πρm kT d NkT

  eðxÞghc m; x σd x2 dx eðxÞ ¼ uðεxÞ; I1 ¼ u with x ¼ σr , u 1 " # ∞   Ð ∂ hc σ 2 eðxÞg m; x d x dx . and I2 ¼ ∂ρ u Ð∞

1

ghc is the distribution function allowing us to know the number of molecules in a certain volume element. In the PC-SAFT theory, I1 and I2 can be estimated from weighted sums. Finally, the compressibility factor of the dispersive part is written by Eq. (2.1.71): ∂ðηI1 Þ 2 3 m εσ ∂η

∂ðηI2 Þ
πρm C1 + C2 ηI2 m2 ε2 σ 3 (2.1.71) ∂η

1 ∂Zhc hc 1 2 3 with C1 ¼ 1 + Z + ρ ; C2 ¼ ∂C ∂η ; m εσ ∂ρ ε  XX ij ¼ xi xj mi mj σ 3ij m2 ε2 σ 3 kT i j  ε 2 XX ij ¼ xi xj m i m j σ 3ij : kT i j Zdisp ¼
2πρ

The associative term is directly deduced from Wertheim’s expressions. If we take into account two spherical segments having an association site A, the associative bond can occur only when the distance and orientation are favorable. The association is modeled by a square well interaction potential centered on site A. Two parameters are required: parameter εasso, which corresponds to the depth of the well, and parameter κ asso, which characterizes the association volume (linked to the range of the interaction). These parameters allow us to calculate XA, a molar part of the molecules that is not associated with site A.

1

(2.1.69)

1 A2 ∂Zhc ¼
πρm 1 + Zhc + ρ NkT ∂ρ 3 2 ∞
 ð σ 7 6 eðxÞghc m; x x2 dx5 ∂4 u d
2 ε 1 m2 σ3 ∂ρ kT

Z

asso

¼

X i

2

3 "

# X X
∂XAj  1 xi 4 ρj
0:5 5 ∂ρi T, ρk6¼i XAj A j j

(2.1.72) (2.1.70)

This model allows us to calculate, for each kind of associative molecule, the thermodynamic properties. Parameters of binary interactions

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

30

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

allowing us to calculate m2 εσ 3 and m2 ε2 σ 3 can be adjusted from experimental data.

2.1.4 GAS SOLUBILITY The solubility of a gas in a liquid is often proportional to its partial pressure in the gas phase. The equation that describes this observation is commonly known as Henry’s law. At high pressure, the effect of the pressure is not negligible and it is necessary to consider how Henry’s constant depends on pressure. In 1935, Krichevsky and Kasarnovsky showed how to explain the correct Henry’s law (H) (Eq. 2.1.73).
V     f ðT, P, yH2 O Þ ¼ ln H T, Psat ln solute solvent xsolute v∞ + solute Psolute RT (2.1.73) v∞ solute is the molar volume at infinite dilution of the solute. It can be obtained with Eq. (2.1.74):  2
3 ∂P 6 ∂nsolute 7 6
 T, V, nsolvent 7 v∞ ¼
(2.1.74) 6 7 solute 4 ∂P 5 ∂v T, nsolute , nsolvent n ¼0 solute

It is common to estimate the solubility of a species in a solvent mixture by using the solubility of the same species in each of the pure solvents that comprises the mixture. The global Henry’s coefficient is determined by using a relation that includes each Henry’s coefficient. The unsymmetrical normalized activity coefficient and Gibbs’ energy are used to determine the following relation (Prausnitz et al., 1999) (for this example a ternary mixture is used with the solvent, composed of 1 and 3, and the solute, 2) (Eq. 2.1.75): lnðH2, mixture Þ ¼ x1 lnðH2,1 Þ + x3 lnðH2,3 Þ
a13 x1 x3 (2.1.75)

a13 is a parameter that can be estimated from the vapor–liquid equilibrium data for the solvent mixture. Consequently, it is sometimes customary to define an excess Henry’s constant E (Eq. 2.1.76): Hi,mixture m X  E    xj ln Hi, j ¼ lnðHi, mixture Þ
ln Hi,mixture j¼1

(2.1.76) Concerning the Henry’s law constant at solvent vapor pressure, it is necessary to consider a temperature dependency. Several correlations can be found in the open literature (see Carroll, 1999). For example, an exponential expression (Eq. 2.1.77) can be considered. It is inspired from the expression used by Yaws et al. (1990) or the Design Institute for Physical Properties (DIPPR) equation n°101, which was obtained from Dauber et al. (2000)   B ln Hicor ¼ A + + ClnðTÞ + DT T

(2.1.77)

Harvey also proposed another correlation to estimate Henry’s law constant coefficient in a large range of temperature for pure water (Eq. 2.1.78)     Harvey ¼ ln Psat ln Hi water +

A Bð1
Tr Þ0:355 + + Ceð1
Tr Þ Tr
0:41 Tr Tr (2.1.78)

In 1999, Yaws et al. (1999) proposed a correlation to predict the Henry’s law constant at atmospheric pressure (Eq. 2.1.79). Parameters for CO2, N2, and SO2 are presented in Table 2.1.6. Fig. 2.1.8 presents the evolution of the Henry’s law constant as a function of temperature for the three molecules.   B log HiYaws ¼ A + + ClogðT Þ + DT T

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

(2.1.79)

31

2.1.5 APPLICATION TO INDUSTRIAL FLUIDS

TABLE 2.1.6 mol Fraction

Henry’s Law Constant for CO2, N2, and SO2 in Pure Water According to Yaws et al. (1999), Hi is in atm/

Name

A

B

C

D

Tmin /K

Tmax/ K

Carbon dioxide

69.4237


3796.46


21.6694

0.000478857

273.15

353.15

Nitrogen

78.8622


3744.98


24.7981

0

273.15

350.15

Sulfur dioxide

22.3423


1987.11


5.6854

0

283.15

323.15

FIG. 2.1.8

Evolution of Henry’s law constant of CO2, N2, and SO2 as a function of temperature.

2.1.5 APPLICATION TO INDUSTRIAL FLUIDS In the following section, three phase diagrams (Pressure versus Temperature, Pressure versus Density and Pressure versus Enthalpy) of some gases used in the food industry (CO2, N2, O3 and SO2) and their uses (Girardon, 2004) will be presented. In order to illustrate the application of thermodynamic to the representation of equilibrium properties of mixtures, two binary systems have been investigated. The first one concerns the binary system CO2 + water and the second one concerns the binary system CO2 + ethanol.

2.1.5.1 CO2 Carbon dioxide is used in food processing, preservation, and packaging. Its main applications are the carbonation of soft drinks and the monitoring of temperatures in deep freezing and transport (due to its ability to be stored in snow or ice forms). Figs. 2.1.9 and 2.1.10 present two phase diagrams of pure CO2. A fundamental EoS is used (REFPROP v10.0 software from the National Institute of Standards and Technology). Fig. 2.1.11 gives the pressure enthalpy phase diagram. Its critical properties are 304.13 K and 73.77 bar.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

32

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

8.0 7.0

Pressure/MPa

6.0 5.0 4.0 3.0 2.0 1.0 0.0 225

FIG. 2.1.9

235

245

255

265 275 Temperature /K

285

295

305

Pressure temperature phase diagram of CO2.

9.0 8.0 7.0

Pressure/MPa

6.0 5.0 4.0 3.0 2.0 1.0 0.0 0

5

10

15

20

25

30

Density/mol.dm–3

FIG. 2.1.10

Pressure density phase diagram of CO2. At 243.3, 304.13 and 310.21 K.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

33

2.1.5 APPLICATION TO INDUSTRIAL FLUIDS

8.0

7.0

6.0

Pressure/MPa

5.0

4.0

3.0

2.0

1.0

0.0 0

5000

10000

15000

20000

25000

Enthalpy/Jmol–1

FIG. 2.1.11

Pressure enthalpy phase diagram of CO2 at 243.3, 304.13 and 310.21 K. Reference: h ¼ 0 at ¼0 K.

2.1.5.2 N2 Nitrogen is proving to be a beneficial ingredient for the prepared foods sector, with a range of creative applications that includes freezing, packaging, mixing, coating, and grinding. Nitrogen can be used to process a wide variety of foods including fruits, vegetables, pasta, dairy products, baked goods, and prepared meals. Figs. 2.1.12 and 2.1.13 present two phase diagrams of pure nitrogen. A fundamental equation of state is used (REFPROP v10.0). Fig. 2.1.14 gives the pressure enthalpy phase diagram. Its critical properties are 126.19 K and 33.96 bar.

2.1.5.3 O3 Ozone is particularly suited to the food industry because of its ability to disinfect microorganisms without adding additional chemicals to the

treated food, or to the water used or the atmosphere in which the food is stored. Figs. 2.1.15 and 2.1.16 present two phase diagrams of pure SO2. A cubic equation of state (Peng Robinson Equation of state implemented in Simulis Thermodynamics software from Prosim, France) is used. Fig. 2.1.17 gives the pressure enthalpy phase diagram. Its critical properties are 261 K and 55.7 bar.

2.1.5.4 SO2 Sulfur dioxide is the most widely used additive in winemaking. It is also used as an additive in the food industry in general (dry fruits, mustard, prepackaged food preparation, shellfish, cereals). Its main functions are to inhibit or kill unwanted yeasts and bacteria, and to protect wine from oxidation. Figs. 2.1.18 and 2.1.19

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

4.0

3.5

Pressure/MPa

3.0

2.5

2.0

1.5

1.0

0.5

0.0 60

FIG. 2.1.12

70

80

90 100 Temperature/K

110

120

130

Pressure temperature phase diagram of N2.

8.0

7.0

6.0

Pressure/MPa

5.0

4.0

3.0

2.0

1.0

0.0 0

FIG. 2.1.13

5

10

20 15 Density/mol.dm–3

25

30

35

Pressure density phase diagram of N2 at 88.33, 126.19 and 128.72 K.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

8.0

7.0

6.0

Pressure/MPa

5.0

4.0

3.0

2.0

1.0

0.0 –5000 –4000 –3000 –2000 –1000

0

1000

2000

3000

4000

5000

Enthalpy/Jmol–1

FIG. 2.1.14

Pressure enthalpy phase diagram of N2 at 88.33, 126.19 and 128.72 K. Reference: h ¼ 0 at ¼0 K.

6.0

5.0

Pressure/MPa

4.0

3.0

2.0

1.0

0.0 190

200

210

220

230

240

250

260

270

280

Temperature/K

FIG. 2.1.15

Pressure temperature phase diagram of O3.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

8.0

7.0

Pressure/MPa

6.0

5.0

4.0

3.0

2.0

1.0

0.0

FIG. 2.1.16

0

5000

10000

15000 20000 Density/mol.dm–3

25000

30000

Pressure density phase diagram of O3 at 208.80, 261 and 266.22 K.

8.0

7.0

Pressure/MPa

6.0

5.0

4.0

3.0

2.0

1.0

0.0 –25000

–20000

–15000

–10000

–5000

0

Enthalpy/Jmol–1

FIG. 2.1.17

Pressure enthalpy phase diagram of O3 at 208.80, 261 and 266.22 K. Reference: h ¼ 0 at T ¼ 298.15 K, ideal gas.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

9.0 8.0 7.0

Pressure/MPa

6.0 5.0 4.0 3.0 2.0 1.0 0.0 200

250

300

350

400

450

Temperature/K

FIG. 2.1.18

Pressure temperature phase diagram of SO2.

10.0 9.0 8.0

Pressure/MPa

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0

0

5

10

15

20

25

30

Density/mol.dm–3

FIG. 2.1.19

Pressure density phase diagram of SO2 at 301.45, 430.64 and 439.25 K.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

38

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

10.0 9.0 8.0

Pressure/MPa

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 –10000 –5000

FIG. 2.1.20

0

5000

10000 15000 20000 25000 30000 35000 Enthalpy/Jmol–1

Pressure enthalpy phase diagram of SO2 at 301.45, 430.64 and 439.25 K. Reference: h ¼ 0 at ¼0 K.

present two phase diagrams of pure SO2. A fundamental equation of state is used (REFPROP v10.0). Fig. 2.1.20 gives the pressure enthalpy phase diagram. Its critical properties are 430.64 K and 78.84 bar.

2.1.5.5 Binary Systems 2.1.5.5.1 CO2-H2O Fig. 2.1.21 presents the phase diagram at 298.28 K. The CO2 + water binary system is classified as type II according to Van Konynenburg and Scott (1980). The Peng Robinson equation of state associated with the modified Huron Vidal Mixing rules and NRTL activity coefficient model is used to correlate the experimental data. Experimental data are from Nakayama et al. (1987) and Valtz et al. (2004). As we can see, the solubility of CO2 is perfectly predicted but not the water content in the liquid-liquid region.

2.1.5.5.2 CO2-C2H5OH Fig. 2.1.22 presents the phase diagram of the CO2 + ethanol binary system at 293.15 K. The same equation of state was used. Experimental data are from Secuianu et al. (2008). As we can see, the model is in good agreement with the experimental data and so the ethanol content can be easily predicted.

2.1.6 CONCLUSION Throughout this chapter, we presented the fundamentals aspects in fluid thermodynamics (molecular interactions, chemical potential, and equilibrium conditions) and various thermodynamic models (fundamental EoS, cubic EoS, and molecular model). These models are at the heart of all developments and optimization of industrial processes and energetic

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

39

10,000

10,000

9000

9000

8000

8000

Pressure/kPa

Pressure/KPa

REFERENCES

7000 6000 5000 4000 3000 2000

7000 6000 5000 4000 3000 2000

1000

1000

0 0

0.2

0.4

0.6

0

0.8

1

0.98

0.985

0.99

(B)

X1, Y1

(A)

0.995

1

X1, Y1

FIG. 2.1.21 Vapor liquid equilibrium of CO2 (1) + Water (2) binary system at 298.28 K. (A) complete phase diagram; (B) zoom around pure CO2 vapor pressure. (Δ): Valtz et al., (◊): Nakayama et al., solid line: Peng Robinson equation of state prediction.

7000 6000

Pressure/kPa

5000 4000 3000 2000 1000 0

0

0.2

0.4

0.6

0.8

1

X1, Y1

FIG. 2.1.22

Vapor liquid equilibrium properties of the CO2 (1) + Ethanol (2) binary system at 293.15 K. (Δ): Secuianu et al. (2008), solid line: Peng Robinson equation of state prediction.

efficiency. Phase diagrams help to define the best process and to show how one can articulate the unit operations according to the temperature, pressure, and composition conditions. The study of the phase diagrams also helps to understand the physicochemical phenomena in order to improve the functioning of the separation units: knowledge of the molecular aspects helps in the development of tomorrow’s processes.

References Abrams, D.S., Prausnitz, J.M., 1975. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AICHE J. 21, 116–128. Barker, J.A., Henderson, D., 1967. Perturbation theory and equation of state for fluids: the square-well potential. J. Chem. Phys. 47 (8), 2856–2861. Benedict, M., Webb, G.B., Rubin, L.C., 1949. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures. J. Chem. Phys. 8, 334–344.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

40

2.1. PHYSICOCHEMICAL PROPERTIES OF GAS

Boublik, T., 1970. Hard sphere equation of state. J. Chem. Phys. 53, 471–478. Carroll, J.J., 1999. Henry’s law revisited. Chem. Eng. Progr. 95, 49–56. Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M., 1989. SAFT: equation-of-state solution model for associating fluids. Fluid Phase Equilib. 52, 31–38. Chapman, W.G., Gubbins, K.E., Jackson, G., Radosz, M., 1990. New reference equation of state for associating liquids. Ind. Eng. Chem. Res. 29 (8), 1709–1721. Chen, S.S., Kreglewski, A., 1977. Applications of the augmented van der Waals theory of fluids. I. Pure fluids. Ber. Bunsen-Ges. 81, 1048–1052. Coquelet, C., Chapoy, A., Richon, D., 2004. Development of a new alpha function for the Peng–Robinson equation of state: comparative study of alpha function models for pure gases (natural gas components) and water-gas systems. Int. J. Thermophys. 25 (1), 133–158. Coquelet, C., El Abbadi, J., Houriez, C., 2016. Prediction of thermodynamic properties of refrigerant fluids with a new three-parameter cubic equation of state. Int. J. Refrig. 69, 418–436. Dauber, T.E., Danner, R.P., Sibu, H.M.I., Stebbins, C.C., Oscarson, J.L., 2000. Physical and Thermodynamic Properties of Pure Chemicals. DIPPR, Taylor & Francis. Girardon, P., 2004. Utilisation des gaz industriels en agroalimentaire. Techniques de l’ing
enieur. Agroalimentaire 2, 1–22. F1275. Gross, J., Sadowski, G., 2001. Perturbed-chain SAFT: An equation of state based on perturbation theory for chain molecules. Ind. Eng. Chem. Res. 40, 1244–1260. Huron, M.J., Vidal, J., 1979. New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non ideal mixtures. Fluid Phase Equilib. 3, 255–271. Kontogeorgis, G.M., Folas, G.K., 2010. Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories. Wiley, Chichester, UK. Kunz, O., Wagner, W., 2012. The GERG-2008 wide range equation of state for natural gases and other mixtures: an expansion of GERG-2004. J. Chem. Eng. Data 57, 3032–3091. Le Guennec, Y., Lasala, S., Privat, R., Jaubert, J.N., 2016. A consistency test for α-functions of cubic equations of state. Fluid Phase Equilib. 427, 513–538. Le Guennec, Y., Privat, R., Lasala, S., Jaubert, J.N., 2017. On the imperative need to use a consistent α-function for the prediction of pure-compound supercritical properties with a cubic equation of state. Fluid Phase Equilib. 445, 45–53. Mac Linden, M.O., Klein, S.A., 1996. A next generation refrigerant properties database. In: International Refrigeration and Air Conditioning Conference Paper 357, pp. 409–414.

Mansoori, G.A., Carnahan, N.F., Starling, K.E., Leland, T.W., 1971. Equilibrium thermodynamics properties of the mixture of hard spheres. J. Chem. Phys. 54, 1523. Mathias, P.M., Copeman, T.W., 1983. Extension of the Peng Robinson equation of state to complex mixtures: evaluation of the various forms of the local composition concept. Fluid Phase Equilib. 13, 91–108. Nakayama, T., Sagara, H., Arai, K., Saito, S., 1987. High pressure liquid-liquid equilibria for the system of water, ethanol and 1,1-difluoroethane at 323.2 K. Fluid Phase Equilib. 38, 109–127. Patel, N.C., Teja, A.S., 1982. A new cubic equation of state for fluids and fluid mixtures. Chem. Eng. Sci. 37, 463–473. Peng, D.Y., Robinson, D.B., 1976. A new two parameters equation of state. Ind. Eng. Chem. Fundam. 15, 59–64. Prausnitz, J.M., Lichtenthaler, R.N., Gomes de Azevedo, E., 1999. Molecular Thermodynamics of Fluid-Phase Equilibria. Prentice Hall PTR, New Jersey. ISBN: 0-13977745-8. Redlich, O., Kister, A.T., 1948. Thermodynamics of nonelectrolyte solutions – x-y-t relations in a binary system. Ind. Eng. Chem. 40 (2), 341–345. Secuianu, C., Feroiu, V., Geancentsa, D., 2008. Phase behavior for carbon dioxide + ethanol system: experimental measurements and modeling with a cubic equation of state. J. Supercrit. Fluids 47, 109–116. Soave, G., 1972. Equilibrium constants for modified RedlichKwong equation of state. Chem. Eng. Sci. 4, 1197–1203. Stryjek, R., Vera, J.H., 1976. PRSV: an improved PengRobinson equation of state for pure compounds and mixtures. Can. J. Chem. Fundam. 15, 59–64. Tillner-Roth R., 1993 PhD. University of Hannover, Germany. Tillner-Roth, R., Baehr, H.D., 1994. J. Phys. Chem. Ref. Data 23, 657–729. Trebble, M.A., Bishnoi, P.R., 1987. Development of a new four-parameter equation of state. Fluid Phase Equilib. 35, 1–18. Twu, C.H., Coon, J.E., Cunningham, J.R., 1995a. A new generalized alpha function for a cubic equation of state. Part 1. Peng Robinson equation. Fluid Phase Equilib. 105, 49–59. Twu, C.H., Coon, J.E., Cunningham, J.R., 1995b. A new generalized alpha function for a cubic equation of state. Part 2. Redlich-Kwong equation. Fluid Phase Equilib. 105, 61–69. Valtz, A., Chapoy, A., Coquelet, C., Paricaud, P., Richon, D., 2004. Vapour-liquid equilibria in the carbon dioxidewater system, measurement and modelling from 278.2 to 318.2K. Fluid Phase Equilib. 226, 333–344. Van der Waals, J.D., 1873. Over de Continuiteit van den € Gas- en Vloestoftoestand. (Uber die Kontinuitt€at des Gas- und Fl€ ussigkeitszustands). Dissertation, Universit€at Leiden, Niederlande, deutsche € Ubersetzung, Leipzig.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS

FURTHER READING

Van Konynenburg, P.H., Scott, R.L., 1980. Critical lines and phase equilibria in binary van der Waals mixtures. Philos. Trans. R. Soc. 298, 495. Wertheim, M.S., 1984. Fluids with highly directional attractive forces. I. Statistical thermodynamics. J. Stat. Phys. 35, 19–34. Wong, D.S.H., Sandler, S.I., 1992. A theoretically correct mixing rule for cubic equation of state. AIChE 38, 671–680. Yaws, C.L., Hopper, J.R., Wang, X., Rathinsamy, A.K., Pike, R.W., Hansen, K.C., 1999. Calculating solubility and Henry’s law constant for gases in water. Chem. Eng., 102–105. Yaws, C.L., Yang, H.C., Hopper, J.R., Hansen, K.C., 1990. Organic chemicals: 168 water solubility data; keep these values at your fingertips for engineering and environmental impact studies. Chem. Eng., 115–118.

Further Reading Blas, F.J., Vega, L.F., 1997. Thermodynamic behaviour of Homonuclear and Heteronuclear Lennard-Jones chains

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with association sites from simulation and theory. Mol. Phys. 92 (1), 135–150. Fredenslund, A., Jones, R.L., Prausnitz, J.M., 1975. Group contribution estimation of activity coefficients in non ideal-liquid mixtures. AICHE J. 21, 1086–1099. Gil-Villegas, A., Galindo, A., Whitehead, P.J., Mills, S.J., Jackson, G., Burgess, A.N., 1997. Statistical associating fluid theory for chain molecules with attractive potentials of variable range. J. Chem. Phys. 106 (10), 4168–4186. Krichevsky, I.R., Kasarnovsky, J.S., 1935. Thermodynamical calculations of solubilities of nitrogen and hydrogen in water at high pressures. J. Am. Chem. Soc. 57, 2168–2171. Redlich, O., Kwong, J.N.S., 1949. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev. 44, 233–244. Renon, H., Prausnitz, J.M., 1968. Local composition in thermodynamic excess function for liquid mixtures. AICHE J. 14, 135–144. Trebble, M.A., Bishnoi, P.R., 1986. Accuracy and consistency comparison of ten cubic equations of state for polar and non polar compounds. Fluid Phase Equilib. 29, 465–474.

2. CHEMICAL AND PHYSICAL GASES PROPERTIES, GASES PRODUCTION PROCESS, UNITS