STATE OF THE ART OF EQUATIONS OF STATE Christophe Coquelet and Dominique Richon Laboratoire Thermodynamique et Equilibres entre Phases Centre Energétique et Procédés, Ecole Nationale supérieure des Mines de Paris CNRS FRE 2861 35, rue Saint Honoré 77305 Fontainebleau
[email protected] fax: 33164694968
[email protected] fax: 33164694968 Abstract Supercritical fluids (SFC) have unique advantages over gases and liquids. Many processes have been developed all around the world to benefit of possible SFC performances near critical conditions. Process optimization must rely on reliable models and unfortunately although thermodynamic properties are conveniently represented or even predicted in large ranges of conditions, the supercritical region caused drastic troubles. A quick review of modelling is done herein showing the various drawbacks pointed out regarding conventional data treatments. Finally, optimistic ways are examined to solve difficulties approaching critical points. INTRODUCTION Supercritical fluids (SF) exhibit very interesting properties, giving them great potential for industrial use. As the supercritical state is an intermediate state between a liquid and a gas, some of physical properties lie in between those of these two states, while the others show unique behavior around the critical point. Supercritical fluids are employed in many industrial applications, such as in the food and pharmaceutical industries, in the field biotechnologies, etc and for development of new materials. Typical examples of applications are: - Extraction of chemicals such as caffeine, aromas, nicotine, active principles for drug making, pesticides from food, … - Treatment of heavy hydrocarbons, polymer fractionation and extraction of monomers, - Regeneration of filters, absorbents and catalysts, - Separation, purification (isomer separation), - Use of supercritical CO as a green solvent or refrigerant, as a substitute of the forbidden CFCS, sterilization of food product with high pressure CO , … 2
2
- Chemical reactions in supercritical conditions - Supercritical fluid chromatography,etc. Supercritical fluid extraction (SFE) is definitely a more and more used technology. The basic steps of SFE processes are the following:
1
1- The material (solid or liquid) containing the substance (or substances) to be extracted is put in contact with a high-pressure supercritical fluid, and the substance (substances) is (are) partially dissolved by the supercritical phase 2-‐‑ The substance is recovered after changing the conditions of temperature and pressure. Two phases are created: the solvent which behaves as a gas, and the substance, which remains in the dense phase that is poor in solvent. Supercritical fluid extraction has several advantages: many substances are more soluble in high-‐‑pressure supercritical fluids than in liquids at ambient conditions. As diffusion coefficients are higher in supercritical fluids than in liquids so are mass transfer rates. The viscosity of SF is also lower than that of liquids. The most common compound used as solvent in SFE is carbon dioxide (CO2) which is a very interesting compound for several reasons: it is cheap, non-‐‑toxic, and its critical temperature (304.21 K) is close to ambient temperature, allowing reduced heating costs. Another advantage and not the least, CO2 is, at high pressures, a good sterilizer for food products. Very accurate thermodynamic models are required for both new designs and improvments of SFE separation processes. These thermodynamic models must be able to accurately predict both the densities and the phase equilibrium properties of systems involving supercritical and near-critical fluids. This issue is particularly important, as solubility of substances and selectivity in supercritical fluids depends greatly on the density of the system, which changes considerably even for very small pressure modifications. The use of high pressures in SFE processes represents unfortunately the largest part of the operating costs. Equations of State (EoS) allow developing the most convenient thermodynamic models to represent phase equilibrium properties when dealing with high pressures. The most widespread EoS in industry are probably the cubic equations of state, such as the Peng-Robinson and Patel-Teja EoS etc…, because these equations have a very simple analytical form, and they provide very accurate predictions of vapor-liquid equilibrium for mixtures of non polar molecules. Their great disadvantage is to provide inaccurate density representations, in the liquid and in the near critical regions. For pure compounds, it is possible to introduce temperature dependent volume translations in cubic EoS. For mixtures the main problem of most of proposed volume translations is their inadequacy to improve PVT properties on the whole range of pressures (improvements can be obtained in a given limited pressure range while deviation are higher in another limited pressure range…) For very non-ideal mixtures, volume translations are mostly unreliable and do not enable better representation of phase equilibria. During the last twenty years, several theoretical equations of state such as the SAFT EoS and its various versions (SAFT-VR, PC-SAFT, soft SAFT …) have been developed thanks to the progress in both statistical mechanics and computer simulation. Although these equations of state are more complex than cubic EoS, they are more and more used thanks to the increase of the computer speed; some of them have been implemented in process simulation softwares (for example, PC-SAFT in ASPEN + software from ASPEN Tech). The main advantages of these equations are their capabilities to treat very complex mixtures involving a large variety of compounds: associating fluids (molecules with hydrogen bonds), electrolytes, polymers, colloids… Such equations of state are usually more accurate than cubic EoS for the predictions of liquid densities, and require less binary parameters to represent phase equilibrium data. However, these EoS are unable to predict the thermodynamic properties in near critical regions, hence they have the same 2
drawback as cubic EoS. This is currently common problem to all classical EoS that do not take into account the large density fluctuations occurring close to the critical point. I – THERMODYNAMIC BACKGROUND I.1. Definition of the phase diagrams I.1.1. Pure compound The phase diagram of a pure compound is clearly defined by two points: the critical point and the triple point. At the triple point, solid, liquid and vapour phase are together coexisting. The critical point can be defined as a limit for the pure component vapour pressure. For temperature and pressure above the critical condition, there is no possibility to have vapour-liquid equilibrium. We can also consider the supercritical state as a “stable state” with no phase separation. For certain temperature and pressure conditions, phase separation appears leading to two phases in equilibrium (liquid and vapour). The critical point can be considered as a limit of stability of the supercritical phase. The first equation of state, which can describe the behaviour of a pure fluid, was developed by van der Waals [1]. Two types of interactions (repulsive and attractive interactions) are considered in his equation.
a ⎞ ⎛ ⎜ P + 2 ⎟(v − b ) = RT v ⎠ ⎝
(1)
The stability conditions can be defined with the following equations:
⎛ ∂ 2 P ⎞ T ⎛ ∂P ⎞ >0 ⎜ ⎟ < 0, ⎜⎜ 2 ⎟⎟ < 0 and cv ⎝ ∂v ⎠ T ⎝ ∂v ⎠ T At the limit, these two previous conditions are equal to zero. ⎛ ∂ 2 P ⎞ T ⎛ ∂P ⎞ =0 ⎜ ⎟ = 0, ⎜⎜ 2 ⎟⎟ = 0 and cv ⎝ ∂v ⎠ T ⎝ ∂v ⎠ T It can be seen that at the critical point, the isochoric heat capacity has infinite value.
3
I.1.2. Mixture To illustrate this part, we just consider a binary mixture. Phase diagram concerning the mixture depend of the behaviour of the species. Van Konynenburg and Scott [2] have classified the mixture into VI types. Figure 1 presents the different types of phase diagrams.
Figure 1: Six types of phase behaviour 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 lines and hatching marks heterogeneous regions. Extract from J.M. Prausnitz, R. N. Lichtenthaler, E.G. de Azevedo, Molecular thermodynamics of fluid phase equilibria, 1999, Prentice Hall PTR, Upper Saddle river, New Jersey, USA. Type I phase behaviour It is the simplest type of phase diagram. The mixture critical point line starts at the first pure component critical point and finish at the second pure component critical point. It concerns mixtures where the two components are chemically similar or their critical properties are comparable. We can cite systems with CO and light hydrocarbons, systems with refrigerants HFC (figure 2), and benzene + toluene binary systems. 2
4
P /MPa
6.5 6.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 2.0 0.0
0.2
0.4
0.6
0.8
1.0
x1 , y1 Figure 2: Critical point line of R32 (1) + propane (2) (Coquelet et al. [3]) binary system calculated with cubic equation of state. Symbols: experimental VLE data at 343.26 K. Type II phase behaviour It is similar to type I but at low temperatures, the two components are not miscible in the liquid mixtures. Consequently a liquid - liquid equilibrium appears. The mixture critical point line for liquid - liquid equilibrium starts from UCEP (upper critical end point). At the UCEP, the two liquid phases merge into one liquid phase. We can cite systems with hydrocarbons and fluorinated fluorocarbons. Type III phase behaviour It concerns mixtures with very large immiscibility gaps. We can cite for example systems with water like hydrocarbons + water systems. A liquid – liquid - vapour curve appear and a first mixture critical point line starts from pure component 1 critical point and ends at the UCEP. The second one starts from the infinite (P-> ∞) and ends at the pure component 2 critical point, generally the solvent i.e. water. The slope of this second curve can be positive, negative or positive and negative. Concerning the positive curve, we have two phases at temperature larger than the critical temperature of the pure component two. Some example: Positive slope: helium + water binary system Negative slope: methane + toluene binary system Positive and negative slope: nitrogen + ammonia or ethane + methanol binary systems. Figures 3a, 3b and 3c present some example from the paper of Heidemann and Khalil [4]. It concerns CO + n-octane (a), CO + n-hexadecane (b) and CH + H S (c). 2
5
2
4
2
Figure 3a: Critical lines for CO + n-octane binary system. 2
Figure 3b: Critical lines for CO + n-hexadecane binary system. 2
Figure 3c: Critical lines for methane + H S binary system. 2
Type IV phase behaviour It is similar to type V. the vapour liquid critical point line starts at the critical point of component 2 and ends at the LCEP (lower critical end point). Vapour-liquidliquid equilibrium (VLLE) exists and is present in two parts. Ethane + n-propanol and CO + nitrobenzene binary systems behave like this. 2
Type V phase behaviour
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It is a modification of type III phase diagram. There are two vapour-liquid critical point lines. One goes from the pure component critical point 1 and ends at the UCEP. The other one starts at the pure component critical point 2 and ends at the LCEP. Contrary to type IV, below the LCEP the liquids are completely miscible. Ethylene + methanol binary system is classified as a type V system (see figure 4).
Figure 4: Vapour-liquid equilibrium pressure – composition diagram of ethylene + methanol binary system. Extract from J. M. Prausnitz, R. N. Lichtenthaler, E. G. de Azevedo, Molecular thermodynamics of fluid phase equilibria, 1999, Prentice Hall PTR, Upper Saddle river, New Jersey, USA. Type VI phase behaviour There are two critical point curves. The first one is similar to one presented with the type I diagram: a connection between the two pure component critical points. The second one connects the LCEP and thee UCEP. Between these two points, there are VLLE. The system “water + 2-butanol” is one typical example. I.2. Thermodynamic approach There are mainly two different approaches to model phase equilibrium properties. The two approaches are based on the fact that at thermodynamic equilibrium, fugacity values are equal in all phases, at isothermal conditions. For vapor-liquid equilibrium we have :
f iV (T , P, y ) = f i L (T , P, x )
7
(2)
I.2.1. gamma-Phi approach This approach is based on the use of an activity model for the liquid phase and an equation of state for the vapor phase, the γ – Φ approach. The equilibrium equation can be written:
Φ Vi (T , P, y )y i P = γ iL (T , x )xi f i 0 L (T , P )
(3)
because
f iV (T , P, x ) = γ iL (T , x )xi f i 0 L (T , P ) The vapor fugacity is calculated as follows: f iV (T , P, y ) = Φ Vi (T , P, y )y i P
(4)
The fugacity coefficient in the vapor phase is calculated using an equation of state or a virial equation. The fugacity of the pure compound, i, in the liquid phase can be written using the vapor pressure as:
(
⎛ v L P − Pi sat f i 0 L (T , P ) = Pi sat Φ i0 T , Pi sat exp⎜⎜ i RT ⎝
(
)
)⎞⎟ ⎟ ⎠
(5)
The exponential factor is known as the Poynting factor and viL is the liquid molar volume, at saturation, of the component i. But, the utilization of this approach is limited to systems studied at low pressures. And it is obvious that close to the mixture critical point, the calculation will not converge correctly as two different equations are used. The best way is to use similar equations for both vapour and liquid phase. This solution will be also used to describe supercritical states. 1.2.1. Activity coefficient models
γ iL (T , P, x ) is known as the activity coefficient. The activity is defined by the ratio between fugacity in the mixture and in the reference state. Activity coefficient is fi f introduced to characterise the non ideality of the mixture, γ i = ideal = *i . fi f i xi Several models can describe the behaviour of liquid mixtures. We van cite the empirical model based on Redlich Kister [5] equation. In general, models with theoretical origin are preferred for their better reliability when interpolating or extrapolating as necessary for process design. The most important one concerns the concept of local composition introduced by Wilson [6]. Models like NRTL [7], UNIQUAC [8] and predictive UNIFAC [9] model are based on this local composition concept. 1.2.2. Virial type equations These equations are used to accurately represent experimental properties of pure compounds. The extension from pure compounds to fluid mixtures is problematic, as 8
in theory a mixing rule is needed for each parameter. The virial equation of state is one of the most known of the large number of equations, which have been proposed. This equation is a development of the compressibility factor in series expanded in powers of the molar density with density-independent coefficients, B and C:
Z=
Pv B(T ) C (T ) = 1+ + 2 + .... RT v v
(6)
The density-independent coefficients are known as virial coefficients. (B called the second virial coefficient, C the third). Thus, the second virial coefficient depends on the interaction between two molecules, the third between three molecules, etc… Correlation based corresponding state principle are used to determine the different parameters (ref [10]). I.2.2. Phi-Phi approach This approach is more powerful as it can be used for low pressures and also for high pressures. In this approach the same model is used for the liquid and the vapour phase of either pure compounds or mixtures. Consequently, Eq. 2 becomes:
Φ Vi (T , P, y ) = Φ iL (T , P, x )
(7)
For the determination of fugacity coefficient, phi, we have to consider the following relation:
RTd ln( f i (T , P, z )) = vi dP
(8)
leading to : v ⎡ ⎤ RT ⎛ ∂P ⎞ ⎥dnv − RT ln(Z ) ⎟⎟ RT ln(Φ i ) = ∫ ⎢ − ⎜⎜ nv ∂ n ⎢ ⎝ i ⎠ T ,nv,n j ≠ i ⎥⎦ ∞ ⎣
(9)
It is obvious that a relation between all the variables T, P and v must be used. Equations of state correspond to the main way chosen in industry to describe and calculate fluids properties of mixtures.
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II. Description of models II.1. The van der Waals type model Van der Waals [1] develops the first equation of state which can describe vapor, liquid and supercritical states. He took into account the different interactions between the molecules. Several authors have proposed modifications of this equation and extended them to the representation of more sophisticated fluids (pure component or mixtures). II.1.1. Description of the equation Cubic equations of state can be described using this generalized form:
P=
RT a(T ) − 2 v − b v + ubv + wb 2
(10)
With parameters: a, b, u and w. a is the attractive term. It takes into account the attractive forces between the molecules. It is temperature dependant ( a(T ) = aCα (T ) ,α is called alpha function). b is designed as the molar co-volume. It takes into account the repulsive interactions. Table 1 gives the values of u and w for the most important cubic EoS. At the critical point, we have:
(v − vC )3 = v 3 − 3vC v 2 + 3vC2 v − vC3 = 0
(11)
Using the cubic equation of state at the critical point leads to equation (12).
⎡ RT ⎤ ⎡ RT a ⎤ ⎡ RT ab ⎤ v 3 − ⎢ C − (u − 1)b⎥v 2 − ⎢ C ub − (w − u )b 2 − ⎥v − ⎢ C wb 2 + wb 3 + ⎥ = 0 PC ⎦ ⎣ PC PC ⎦ ⎣ PC ⎦ ⎣ PC Where a = Ωa
(12)
R 2TC2 RTC , b = Ωb PC PC
Using values of u et w presented in Table 1, with c = Ω c
RTC the following system of PC
equations must be solved.
uΩb = 1 + Ωb − 3Z C
(13)
Ω 3b + [(1 − 3Z C ) + (u + w)]Ω b2 + 3Z C2 Ω b − Z C3 = 0
(14)
Ω a = 1 − 3Z C (1 − Z C ) + 3(1 − 2Z C )Ω b + [2 − (u + w)]Ω b2 Equation of State u
(15)
10
w
Van der Waals Redlich and Kwong [11] Peng and Robinson [12] Patel and Teja [13]
0 1 2 1+ c
Harmens and Knapp [14]
c
b
0 0 -1 −c
b -(c-1)
Table 1 – u and w parameters of the main cubic equations of state. c is a third parameter. II.1.2. The alpha function Alpha functions are used to have the best representation of the pure component vapor pressure. They are empirical equation but their mathematics expressions are very important. α must tend towards zero at high temperatures (thermal agitation overrides the attractive interactions), α must be equal to 1 for TR = 1 (critical point) and must tend towards infinite values when the temperature tends towards 0 (the motionless molecules more strongly attract each other). All the alpha functions must take a similar form. In 1972, Soave [15] introduced an alpha function (see equations 16 and 17) in order to improve calculation of the vapor pressures (and of liquid and vapor volumes). TR is the reduced temperature and ω the acentric factor:
α (T ) = [1 + m(1 − TR1 2 )] m = 0.480 + 1.574ω − 0.175ω 2 2
(16) (17)
Other generalized alpha functions were developed according to the cubic type of equation. They were established starting from experimental data of pure substances between the boiling point under 1 atm and the critical point. The generalization of the coefficient m according to the acentric factor involves: For RKS : m = 0.47830 + 1.6337ω − 0.3170ω 2 + 0.760ω 3
(18)
For PR : m = 0.374640 + 1.542260ω − 0.26992ω 2
(19)
Mathias and Copeman [16] proposed a function with three parameters (c0, c1 and c2) adjustable on experimental data: 2 3 1 1 1 ⎡ ⎤ α (T ) = ⎢1 + c1 ⎛⎜1 − TR 2 ⎞⎟ + c2 ⎛⎜1 − TR 2 ⎞⎟ + c3 ⎛⎜1 − TR 2 ⎞⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣
2
(20)
When TR >1, one uses only the first term of the alpha function. 1
α (T ) = ⎡⎢1 + c1 ⎛⎜1 − TR 2 ⎞⎟⎤⎥ ⎣
11
⎝
⎠⎦
2
(21)
It should be noticed that this last expression makes it possible to have a representation much more precise of vapor pressures than with the alpha function of Soave. Nevertheless it is possible to choose other generalized alpha functions in order to obtain a better restitution of experimental vapor pressures and also for a better representation in supercritical conditions (use of temperature dependent alpha functions). Trebble and Bishnoi (1987) [17] proposed an expression different from that proposed by Soave: (22) α (T ) = exp(m(1 − TR )) Lastly, Twu et al. [18] developed an alpha function based on a different approach. They considered a linear function with respect to the acentric factor, ie:
α (T ) = α (0 ) (T ) + ω (α (1) (T ) − α (0 ) (T ))
(23)
with
α (T ) = TRN (M −1) [exp( L(1 − TRNM ))]
(24)
In the same style, Coquelet et al. [19] proposed a generalization of the Mathias and Copeman alpha function (equations 25 and 26) for the cubic equations of Redlich and Kwong and of Peng and Robinson, based on the use of 23 reference compounds. For the Redlich - Kwong equation: c1 = −0.1094ω 2 + 1.6054ω + 0.5178
c 2 = −0.4291ω + 0.3279
(25)
c 3 = 1.3506ω + 0.4866 For the Peng - Robinson equation:
c1 = −0.1094ω 2 + 1.6054ω + 0.5178 c2 = −0.4291ω + 0.3279
(26)
c3 = 1.3506ω + 0.4866 Coquelet et al. [19] also proposed another alpha function (see equations 27 and 28) which combines the advantages of the Mathias and Copeman and of the Trebble and Bishnoi alpha functions, and in particular for temperatures higher than the concerned critical temperature. 2 3 1 1 ⎡ ⎤ α (T ) = exp[c1 (1 − TR )]⎢1 + c2 ⎛⎜1 − TR 2 ⎞⎟ + c3 ⎛⎜1 − TR 2 ⎞⎟ ⎥ ⎝ ⎠ ⎝ ⎠ ⎦ ⎣
For the Peng - Robinson equation:
12
2
(27)
c1 = 0.1441ω 2 + 1.3838ω + 0.387 c2 = −2.5214ω 2 + 0.6939ω + 0.0325
(28)
c3 = 0.6225ω + 0.2236 II.1.2. Presentation of the mixing rules The mixing rules must be able to take into account the ideal and nonideal characters of the solutions. With the cubic equations containing two parameters, the objective consists in recalculating a and b parameters considering the mutual influence of the various compounds. The first set of mixing rules is that initially proposed by van der Waals, it corresponds to the so called "ʺclassical mixing rules"ʺ. On the basis of an equation of state, carrying out a development of the volume virial development and by applying statistical thermodynamics one leads to the following classical mixing rules: (29) a = ∑∑ xi x j aij i
j
and b = ∑ xi bi i
(
where aij = ai a j 1 − kij
)
kij is called binary interaction parameter. 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. Many more advanced mixing rules have been developed. Their authors took into account models based on the calculation of the activity coefficient (excess free enthalpy) and models involving equations of state. Indeed the first are adapted to the treatment of the polar and/or non polar bodies at low pressures and are in general not dependent of the pressure, while the equations of state give satisfactory results only for non-‐‑polar bodies but without pressure limitation.
gγE (T, P → ∞) = g EEoS (T, P → ∞) and v = b when P→∞. The first mixing rule (equations 30 and 31) of this type was presented by Huron and Vidal [20] in 1979.
⎛ ⎛ a a = b⎜⎜ ∑ xi ⎜⎜ i ⎝ i ⎝ bi b = ∑ xi bi i
13
⎞ ⎞ ⎟⎟ + g PE=∞ × C ⎟ ⎟ ⎠ ⎠
(30) (31)
1 2 2 and C = − ln(2) ln 3 + 2 2 Robinson EoS respectively. With C = −
(
)
for the Redlich-Kwong and the Peng-
Wong and Sandler [21] (1992) preserved the classical mixing rules of mixtures obtained through virial development. However, as for the Huron-‐‑Vidal mixing rule, equality of the free excess enthalpies, according to the approaches by equation of state and activity coefficient makes it possible to obtain another relation between the a parameter of attraction and molar co-‐‑volume b. From there, one can also write starting from the Helmholtz molar free energy, A: E (T , P → ∞, x) = AγE (T , P → ∞, x) AEoS
(32)
On the other hand, the authors considered that the free energy is less pressure dependent than the free enthalpy. Thus, they wrote:
AγE (T , P → ∞, x) = AγE (T , P = 1 bar, x ) Indeed, the fundamental relation of thermodynamics is written:
g E (T , P, x) = A E (T , P, x ) + Pv E They considered that at low pressures the PvE product is negligible with respect to value of AE. Consequently, they obtained: E (T , P → ∞, x) = AγE (T , P → ∞, x) = AγE (T , P = 1 Bar, x) = gγE (T , P = 1 bar, x) AEoS Thus, after expressing the free energy of mixture, we obtain the Wong-‐‑Sandler mixing rule.
i
b=
b−
⎛
a ⎞
∑∑ x x ⎜⎝ b − RT ⎟⎠ i
j
j
ij
a ⎛ ⎞ ⎜ ∑ xi i ⎟ E bi g γ (T , P, x ) ⎟ i ⎜ 1− + ⎜ RT ⎟ CRT ⎜ ⎟ ⎝ ⎠
(33)
a a ⎞ a ⎞ 1 ⎡⎛ a ⎞ ⎛ a ⎞ ⎤ ⎛ ⎛ = ∑∑ xi x j ⎜ b − ⎟ with ⎜ b − ⎟ = ⎢⎜ b − ⎟ + ⎜ b − ⎟ ⎥(1 − kij ) (34) RT RT ⎠ ij RT ⎠ij 2 ⎢⎣⎝ RT ⎠i ⎝ RT ⎠ j ⎥⎦ ⎝ ⎝ i j
Michelsen [22] (1990) used the concept of the Huron-Vidal mixing rule, but he chose to calculate the a parameter by determining the free excess enthalpy at null pressure. For that, he applied it to the RKS cubic equation of state. Thus, it determined the molar excess free enthalpy of in the following form:
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gE ⎛ b ⎞ = ∑ xi Ln⎜ i ⎟ + q(α ) − ∑ xi q(α i ) RT ⎝ b ⎠ i i With α =
(35)
a bRT
We considered q(α) as q(α ) = q0 + q1α The corresponding modification leads to MHV1 mixing rules:
b = ∑ xi bi i
With
⎡ a RT a = b ⎢∑ xi i − bi q1 ⎣⎢ i
E ⎛ bi ⎞ gγ ⎤ ∑i xi Ln⎜⎝ b ⎟⎠ + q ⎥⎥ 1 ⎦
(36)
The numerical value of q1 is -‐‑0.593 for RKS EoS and -‐‑0.53 for the PR EoS. Another version of MHV1 was proposed by Dahl and Michelsen [23] (1990), it consists in choosing a second order q (α) function. (37) q(α ) = q0 + q1α + q2α 2 This leads to the second order equation which is the MHV2 mixing rule. (equation 38).
⎛ ⎞ ⎛ 2 GE ⎛ b ⎞ 2 ⎞ q1 ⎜ α − ∑ xiα i ⎟ + q2 ⎜ α − ∑ xiα i ⎟ = − ∑ xi Ln⎜ i ⎟ ⎝ b ⎠ i i i ⎝ ⎠ ⎝ ⎠ RT
(38)
a while for co-‐‑volume we still have bRT b = ∑ xi bi . The numerical values of q1 and q2 are respectively: -‐‑0.478 and -‐‑0.0047 for
Its resolution makes it possible to obtain α = i
RKS EoS and -‐‑0.4347 and -‐‑0.003654 for the PR EoS. In 1991, Holderbaum and Gmehling [24] developed a mixing rule by keeping the principle of MHV1 rule, but by taking the atmospheric pressure as reference pressure (Predictive Soave Redlich and Kwong, PSRK). The modification that they brought relates to the calculation of the excess free enthalpy. They use UNIFAC model to calculate it. On the other hand, the change of reference pressure involves a light modification of the MHV1 q1 parameter (-‐‑0.64663). The values of the UNIFAC parameters are different and directly adjusted on experimental data. This is why, we can speak about the PSRK model in the place of the PSRK mixing rule. II.1.3. one example
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With cubic equations of state it is possible to try to represent vapor-liquid equilibrium properties for any systems. Figure 5 presents phase envelops obtained with the RK equation of state using the MHV1 mixing rules and the NRTL model. This model appears as quite convenient in this case to represent data at temperatures where we still have the presence of an azeotrope. However at higher temperatures close to the mixture critical point (see T = 343.26 K), this model is really not able to determine correctly VLE data. 6 5
P /MPa
4 3 2 1 0 0.0
0.2
0.4
x1 , y1
0.6
0.8
1.0
Figure 5: R32 + propane binary system. Symbols: experimental data from Coquelet et al. [3], temperature from 278.10 K to 343.26 K. Lines: calculated with RK EoS involving the MHV1 mixing rules and NRTL G model. E
Figure 2, shown previously for its critical point line determination, concerns the same binary system but the model is changed: PR EoS with the Wong Sandler mixing rules instead of MHV1 mixing rules. II.2. The GC EoS The group contribution equation of state (GC EoS) is based on generalized van der Waals partition function Q. By definition, the free Helmholtz energy can be determined through equation (39).
A = −kT ln(Q)
(39)
⎛ E ⎞ Where, Q = ∑ exp⎜ − i ⎟ ⎝ kT ⎠ i The residual Helmholtz function (at constant volume and constant temperature) becomes fv
att
⎛ A R ⎞ ⎛ A R ⎞ ⎛ A R ⎞ ⎜⎜ ⎟⎟ ⎟⎟ ⎟⎟ = ⎜⎜ + ⎜⎜ ⎝ RT ⎠T ,V ,n ⎝ RT ⎠T ,V ,n ⎝ RT ⎠T ,V ,n
16
(40)
The repulsive term (free volume) corresponds to the expression given by Carnahan and Starling, equation for hard spheres). For mixtures, the free volume contribution is given by expression developed by Mansoori and Leland (1972) [25]. fv
⎛ A R ⎞ ⎛ λ3 ⎞ ⎜⎜ ⎟⎟ = 3(λ1λ2 / λ3 )(Y − 1) + ⎜ 2 2 ⎟ − Y + Y 2 − ln Y + n ln Y ⎝ λ3 ⎠ ⎝ RT ⎠T ,V ,n
(
)
(41)
With
Y = ⎛⎜1 − ⎝ And
πλ3
⎞ 6V ⎟⎠
−1
(42)
NC
λk = ∑ n j d kj
(43)
j
d is the hard sphere diameter for one mole of species j. j
The attractive part of the residual Helmoltz function is written as follow: att
⎛ A R ⎞ a ⎜⎜ ⎟⎟ =− V ⎝ RT ⎠T ,V ,n
(44)
For pure components, the parameter a has the following expression:
a=
z 2 gq 2
(45)
Where, g is the characteristic attractive energy parameter per segment and q is the surface segment area per mole as defined by the UNIFAC method or the number of surface segments assigned to molecule. z is the coordination number (set equal to 10). For mixtures, a two-fluid model is considered with local surface fractions (not local mole fractions). The following expression is similar to the NRTL model for the excess Helmholtz function.
R
⎛ A ⎜⎜ ⎝ RT
att
⎞ ⎟⎟ ⎠T ,V ,n
⎛ g kj q~τ kj θ k ⎜⎜ ⎛ z ⎞ NC NG i NG ⎝ RTV = −⎜ ⎟∑ ni ∑ν j q j ∑ NG ⎝ 2 ⎠ i j k θτ
∑ l
With
17
l lj
⎞ ⎟⎟ ⎠
(46)
q
θ j = ~j q
NC
∑ν
i j
qj
i
NC
NG
i
j
q~ = ∑ ni ∑ν ij q j
(47)
Δg ji = g ji − g ii
ν ij is the number of groups of type j in molecule i, θ j is the surface fraction of group j and q j is the number of surface segments assigned to group j.
⎛ α ji Δg ji q~ ⎞ ⎟⎟ τ ji = exp⎜⎜ ⎝ RTV ⎠
(48)
Skjold-Jorgensen [26] gives more details in his paper. More sophisticated equations of state where developed recently. They take into account associating interactions between molecules. II.3. SAFT approach Several version of SAFT (Statistical Association fluid theory) are currently available. The first one was developed by Huang and Radosz [27] in 1990. It is based on theory of associating fluids developed by Wertheim [28-31]. It expresses the residual Helmholtz energy as a sum of segment chains, and association terms. The general expression for the Helmholtz energy is given by:
a = a seg + a chain + a asso
(49)
In this paper, we just consider the PC-SAFT (Perturbation Chain) equation based on perturbation theory of Barker and Anderson [32]. Gross and Sadowski [33] considered a second order theory for the determination of dispersive term. Each molecule is regarded as a chain of spherical segments not being identified inevitably with an atom. The interaction potential, u(r), between segments of a chain corresponds to a modification of the square well potential (proposed by Chen and Kreglewski [34]). ⎧ ∞ r < (σ − s1 ) ⎪3ε (σ − s ) ≤ r < σ ⎪ 1 (50) u (r )⎨ − 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 potential well and λ the reduced width of the well (s1=0.12σ). The compressibility factor is the sum of three terms:
Z =1 + Z seg + Z chain + Z asso
18
(51)
The first term takes into account repulsive and attractive interactions. The repulsive interactions are considered using a model of hard sphere. Expressions of Boublick [35] and Mansoori et al. [36] are used (Eq. 53).
Z seg = mZ hc + Z disp
(52)
ξ3 3ξ 23 − ξ 3ξ 23 3ξ1ξ 2 + + 1 − ξ 3 ξ 0 (1 − ξ 3 )2 ξ 0 (1 − ξ 3 )3 π where ξ n = ρ ∑ xi mi d in 6 i Z hc =
(53)
d is the diameter of collision corresponding to a segment of chain, m is the number of segments per chain. The chain term is calculated by considering the following relation:
∂ ln g iihc (54) ∂ρ i g iihc is the radial distribution function for the segments of compound i in a system of hard spheres. Z chain = −∑ xi (mi − 1)ρ
Concerning the dispersive part, the theory of Barker and Anderson is used regarding the chain of segments of hard spheres as a reference. The dispersive interactions are only one perturbation of the reference state. They are applied to molecules comprising several chains of segments. It is what makes it possible PC-‐‑SAFT model to be able to be used for the study of polymers. One has:
A A A disp = 1 + 2 NkT NkT NkT
(55)
where A1 and A2 are the first and second order contributions, and k the Boltzmann constant. A1 and A2 are obtained through the following relations which can be applied to any potential of interaction. ∞ A1 σ ⎞ ⎛ 2 ⎛ ε ⎞ 3 ~ = −2πρm ⎜ ⎟σ ∫ u (x )g hc ⎜ m; x ⎟ x 2 dx NkT d ⎠ ⎝ kT ⎠ 1 ⎝
⎡∞ ~ ⎤ σ ⎞ ⎛ ∂ ⎢ ∫ u (x )g hc ⎜ m; x ⎟ x 2 dx ⎥ 2 hc −1 d ⎠ ⎛ ⎝ A2 ∂Z ⎞ ⎛ ε ⎞ ⎦ ⎟⎟ m 2 ⎜ ⎟ σ 3 ⎣ 1 = −πρm⎜⎜1 + Z hc + ρ NkT ∂ρ ⎠ ∂ρ ⎝ kT ⎠ ⎝ with x =
19
r ~ u (x ) and , u (x ) =
σ
ε
(56)
(57)
∞
σ ⎞ ⎛ I 1 = ∫ u~ (x )g hc ⎜ m; x ⎟ x 2 dx d ⎠ ⎝ 1 ∞ ∂ ⎡ ~ σ ⎞ 2 ⎤ hc ⎛ I2 = ⎢∫ u (x )g ⎜ m; x ⎟ x dx ⎥ ∂ρ ⎣ 1 d ⎠ ⎝ ⎦
(58) (59)
ghc is the distribution function making it possible to know the number of molecules in a certain element of volume. In PC-‐‑SAFT theory, I1 and I2 can be estimated through weighted sums. The associative term is deduced directly from the expressions of Wertheim. If one considers two spherical segments provided with a site of association A, the associative connection can be formed only when the distance and the orientation are favourable. Association is modelled by a square well potential of interaction centered on site A. Two parameters are necessary: the parameter ε
asso
which corresponds to
asso
the depth of the well and the parameter κ which characterizes the volume of association (related to the range of the interaction). These parameters make it possible to calculate XA, the molar fraction of molecules not associated with site A.
Z
asso
⎡ ⎡⎛ ∂X A j ⎢ = ∑ xi ∑ ρ j ∑ ⎢⎜⎜ ⎢ j i A j ⎢⎝ ∂ρ i ⎣ ⎣
⎤ ⎤ ⎞ 1 ⎡ ⎤ ⎟ ⎥ ⎥ ⎢⎣ X A j − 0.5⎥⎦ ⎥ ⎥ ⎟ ⎠T , ρk ≠i ⎦ ⎦
(60)
This model makes it possible to calculate for any type of associative molecules the thermodynamic properties. Binary interaction parameters, which allow the calculation of m 2εσ 3 and m 2ε 2σ 3 (see equations 61 and 62), can be adjusted directly on experimental data.
⎛ ε ij ⎞ m 2εσ 3 = ∑∑ xi x j mi m j ⎜⎜ ⎟⎟σ ij3 i j ⎝ kT ⎠
(61)
2
⎛ ε ij ⎞ m ε σ = ∑∑ xi x j mi m j ⎜⎜ ⎟⎟ σ ij3 i j ⎝ kT ⎠ 2
2
3
(62)
II.4. Limitation of these models Cubic or SAFT equations of state are limited to represent correctly the properties close to the critical point. The critical point is a particular point where the densities of the vapor and the liquid become identical. To describe correctly thermodynamic properties very close to critical points another theory must be used, based on statistical physics. III. The crossover EoS III.1. General presentation Both original cubic and non cubic EoS are unable to represent the densities and other properties over the full fluid range, because they do not satisfy the asymptotic laws 20
that are observed near the critical point. For instance, it is experimentally observed that the saturated liquid density ρ scales as ρ L − ρ C ∝ (TC − T )β where ρ and T are the critical density and temperature, and β one of the critical exponents. The experimental value of β is universal and is about 0.326, while any classical equation of state predicts that β = 0.5. This is why all cubic EoS underestimate the liquid densities. Classical EOS is unable to represent the divergence of the isochoric heat capacity and isothermal compressibility at the critical point. Other expressions can be written (for example): L
P − PC ρ − ρC ∝ PC ρC
C
C
δ
for the pressure along the coexistence VLE curve
T − TC TC The exponents δ, α and Δ are called critical exponents. It exist other critical exponents: β , γ , ν and η . There are relations between them that were determined from the renormalization group theory: for instance, α + 2β + γ = 2. The experimental and classical values of these exponents are reported in table 2.
(
)
−α Δ CV ∝ kα 0 + kα1 (Δτ ) 1 + kα1 (Δτ ) 1 + ... for ρ ρ with Δτ = =
C
1
Critical exponents α β γ δ ν η Δ 1
Best estimate values 0.110 0.3255 1.239 4.800 0.630 0.033 0.510
Values from a classical EoS 0 0.5 1 3 0.5 0 0
Table 2: Critical exponents Asymptotic scaled equations of state have been developed from the renormalization group to take density fluctuations into account. Such EoS satisfies the universal scaling laws observed experimentally, but are inapplicable far away from the critical point. It is possible to combine such an asymptotic and non-analytical model to a classical EoS that is valid far away from the critical point, to obtain a global EoS applicable over the entire fluid range. Such approaches are called crossover equations of state (see the good review by Sengers in “Equations of State for Fluids and fluid mixtures”, Elsevier, Amsterdam, 2000]). In this paper, we will limit our discussion to pure fluids. III.2. Crossover Equation for pure fluids A simple method for incorporating scaling laws into an analytical equation of state has been developed by Kiselev [37]. This approach is based on renormalization group results in the critical region. First, we start with the Landau expansion for the pure fluid at the critical point gives the value of the thermodynamic potential.
φ (P, T ,η ) = φ0 (P, T ) + Aʹ′τη 2 + bη 4 − ηhV
21
(63)
With h, the intensity of the external field in Pa ( h = P − PC ), τ =
T v − 1 and η = − 1. TC vC
For constant T, P and η, the Landau expansion reduces to:
dφ = ΔA(τ ,η ) = Aʹ′τη 2 + bη 4
(64)
Considering the Helmholtz energy, the Taylor series expansion at the critical point, v where v r = = 1, gives (fourth order development): vC
⎛ 4 ⎞ ⎛ ∂A ⎞ ⎟⎟ (vr − 1) + ⎜⎜ ∂ A4 ⎟⎟ (vr − 1)4 + ... A(Tr , vr ) = A(Tr ,1) + ⎜⎜ ⎝ ∂vr ⎠Tr ,vr =1 ⎝ ∂vr ⎠Tr ,vr =1
(65)
The Landau expansion is inserted to the Taylor expansion:
⎛ ∂A ⎞ ⎟⎟ (vr − 1) + ΔA(τ ,η ) A(Tr , vr ) = A(Tr ,1) + ⎜⎜ ⎝ ∂vr ⎠ Tr ,vr =1
(66)
The development of this equation leads to
A (Tr , vr ) =
A(Tr , vr ) v = ΔA (τ ,η ) − P 0 +µ0 RT vC
(67)
III.3. Crossover function In order to satisfy the correct critical behavior in the vicinity of the critical point, i.e. satisfy the deviation of the thermodynamic properties like heat capacity with the critical exponents, the Landau expansion had to be corrected. The renormalization group theory is applied to renormalize the dimensionless variable τ and η (Nicoll et al. [38-40], Chen et al. [41-42]). It has been developed to deal with the effect of critical fluctuations. Variables are renormalized into:
τ = τ ×Y
−α 2Δ1
(68)
γ −2 β
η =η ×Y
4 Δ1
(69) Chen et al.[41-42] added a new term which is called the Kern term in order to provide correct critical isochoric specific heat capacity.
( )
ΔA(τ ,η ) = Aʹ′τ η 2 + bη 4 − K τ 2
( )
Where K τ
2
−α ⎞ 1 2 ⎛⎜ Δ1 = cτ Y − 1⎟ . ⎜ ⎟ 2 ⎝ ⎠
The crossover function Y can be written in the parametric form: 22
(70)
Δ1
⎡ q 2 ⎤ Y (q ) = ⎢ ⎥ ⎣ R(q )⎦ q is the measure of the distance from the critical point. Kiselev [37] uses the following crossover function. ⎡ q 2 ⎤ R(q ) = ⎢ + 1⎥ ⎣1 + q ⎦
(71)
2
(72)
It can be seen that close to the critical point (q=0), R(q) = 1 and Y (q ) = q 2Δ1 , far from the critical point, q → ∞ R(q ) = q 2 and Y (q ) = 1. The distance q is given by:
⎛ η + d1τ + d 2τ 2 ⎞ ⎟⎟Y q = + b ⎜⎜ β G G ⎝ ⎠ 2
τ
2 lm
1− 2 β Δ1
(73)
With blm2 = 1.361 and G, the Ginsburg number which was introduced to find the 1 distance where the fluctuations become negligible. Moreover, q = gr where g ∝ G and r is the distance from critical point. Finally, the resulting expression for the pressure (pure fluid) is:
P(T , v ) =
RT v0C
⎡ v0C ⎢− ⎢⎣ vC
⎛ ∂ΔA ⎞ v ⎜⎜ ⎟⎟ + P 0 + 0C vC ⎝ ∂η ⎠T
⎛ ∂K ⎞ ⎤ ⎜⎜ ⎟⎟ ⎥ ⎝ ∂η ⎠T ⎥⎦
(74)
Figure 6 presents the results obtained with CO for temperature higher than the critical temperature of CO . It can be seen that in modeling the renormalization theory can be used with cubic equation of state in order to improve the representation of the PVT properties in the critical region. Unfortunately, this model has too many parameters and so it is very difficult to easily apply this equation in industry and also to adjust parameters from experimental data. 2
2
23
60
Pressure /MPa
50 40 30 20 10 0 0
5000
10000
15000
20000
25000
30000
-3
Molar density/mol.m
Figure 5: Comparison of crossover Patel-Teja equation of state (in red) and classical Patel-Teja equation of state (in black) calculation of density for CO at 313 K. (Δ): Calculated with the Span and Wagner equation for CO2 [43]. 2
CONCLUSION Industrial applications need various models depending on the operating conditions. Thus for long decades, modelling of thermodynamic properties has been approached through several ways. At low pressures dissymmetric methods with equation of state for vapour phase and GE models for liquid phase are preferred and more again when dealing with very non ideal mixtures. At medium and high pressures, symmetric methods implying one equation of state for all phases are preferred. The most used equations in industrial world are the cubic equation of state even if PVT properties are not well represented. If VLE and PVT properties are of interest volume translations in cubic EoS can give satisfactory results but only on small pressure ranges. For non ideal mixtures we have seen that G models can be used with success, even in the symmetric approach, through the mixing rules. Near the critical point crossover corrections give very good results. Convenient tools implying combinations of means are now available for full representation of thermodynamic properties and for accurate development of SFC processes. E
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