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Ind. Eng. Chem. Res. 2007, 46, 2628-2636
Nonrandom Hydrogen-Bonding Model of Fluids and Their Mixtures. 2. Multicomponent Mixtures Costas Panayiotou,*,† Ioannis Tsivintzelis,† and Ioannis G. Economou‡ Laboratory of Physical Chemistry, Department of Chemical Engineering, Aristotle UniVersity of Thessaloniki, GR-54124 Thessaloniki, Greece, and Molecular Thermodynamics and Modeling of Materials Laboratory, Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, GR-15310 Aghia ParaskeVi Attikis, Greece
Nonrandom hydrogen bonding (NRHB) lattice theory is extended here to multicomponent fluid mixtures rigorously. The model accounts for nonideal thermodynamic behavior of mixtures due to molecular connectivity of nonspherical molecules, weak van der Waals forces between first-neighbor molecular segments, and hydrogen-bonding interactions. The random distribution of molecules in the lattice is calculated using the generalized Staverman theory, while for the nonrandom correction, the Guggenheim quasi-chemical theory is adopted. Finally, the hydrogen-bonding contribution is based on Veytsman statistics as implemented in lattice fluid theory by Panayiotou and Sanchez. The equation of state is coupled with the mass action law due to hydrogen bonding, and both are solved simultaneously. The model is applied for the calculation of vaporliquid, liquid-liquid, and vapor-liquid-liquid equilibria at low and high pressures of binary mixtures of fluids with large molecular size differences and/or different types of interactions between unlike molecules. In addition, the model is applied to correlate low-pressure polymer-solvent data. Good agreement between experimental data and model predictions/correlations is obtained in all cases. Comparisons against mixture predictions from the model ignoring nonrandom contributions and from lattice-fluid-hydrogen-bonding model show a clear improvement of the NRHB model. Introduction The development of thermodynamic models for complex fluids applicable over a wide range of conditions is an active and fascinating research area. Recent advances in statistical thermodynamics and a better understanding of intra- and intermolecular interactions thanks to accurate experimental measurements and molecular simulations using realistic force fields have contributed significantly to this end. Thermodynamic models based on statistical mechanics can be classified into lattice models and nonlattice models. Lattice models are rooted to the pioneering work of Guggenheim1 and Flory2 for complex fluids, including polymers. These early works resulted in a number of successful models in subsequent years. The lattice fluid theory of Sanchez and Lacombe3,4 is probably one of the most widely used lattice models. Significant improvement in the performance of the abovementioned lattice models is obtained by accounting explicitly for the nonrandom distribution of free volume, and for highly specific forces between neighboring molecules resulting in hydrogen bonding.5-7 The resulting model known as the quasichemical hydrogen-bonding (QCHB) model was shown to be accurate for pure fluids and mixtures.8 More recently, QCHB was further modified by using a new combinatorial term based on the generalized Staverman model, and generalized nonrandomness factors were introduced. This latest model is known as the nonrandom hydrogen-bonding (NRHB) model.9 In this work, NRHB is generalized to multicomponent mixtures. The theoretical development of the model is presented while appropriate mixing rules are introduced. NRHB is applied to a wide range of mixtures that consist * To whom all correspondence should be addressed. Tel.: ++ 30 2310 996223. Fax: ++ 30 2310 996232. E-mail:
[email protected]. † Aristotle University of Thessaloniki. ‡ National Center for Scientific Research “Demokritos”.
of nonpolar, polar, and associating fluids as well as mixtures where one of the components is at supercritical conditions. Different types of phase equilibria are examined, including vapor-liquid (VLE), liquid-liquid (LLE), and vapor-liquidliquid (VLLE) equilibria. In all cases, model calculations are in good agreement with literature experimental data. Model Development In a molecular system, nonidealities appear as a result of the nonrandom distribution of molecular segments in the lattice as well as specific interactions between neighboring segments, such as hydrogen bonding. Consequently, the partition function can be factored into three contributions, according to the expression8
Q ) QRQNRQHB
(1)
where QR, QNR, and QHB account for the contribution due to the random distribution of molecular segments, the correction for the nonrandom segmental distribution, and the specific (hydrogen-bonding) interactions. Accurate expressions for QR, QNR, and QHB have been proposed over the years by Panayiotou and co-workers, resulting in different variations of the theory.5-10 In general, the molecular system consists of N1, N2, ..., Nt molecules of components 1, 2, ..., t, respectively, at a temperature T and an external pressure P. Let each component of type i be characterized by ri segments of segmental volumes Vi*. The molecules are assumed to be arranged on a quasi-lattice of coordination number z and of Nr sites, N0 of which are empty. The total number Nr of lattice sites is given by the expression
Nr ) N1r1 + N2r2 + ... + Ntrt + N0 ) rN + N0 ) N(x1r1 + x2r2 + ... + xtrt) + N0 (2) where N ) N1 + N2 + ... + Nt is the total number of molecules in the system and xi is the mole fraction of component i. The
10.1021/ie0612919 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/16/2007
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2629
average interaction energy per segment of molecule i is given by
i* ) (z/2)ii
(3)
where ii is the interaction energy per i-i contact. If zqi is the number of external contacts per molecule i, its surface-to-volume ratio, si, a geometric characteristic of molecule i, is given by
si ) qi/ri
(4)
An earlier version of the lattice theory, including QCHB,8 treated s as an adjustable parameter fitted to pure component thermodynamic data. In NRHB, we take advantage of the extensive databases available for the UNIQUAC model for the molecular volume and molecular area of different compounds. Furthermore, by using the widely accepted group contribution UNIFAC model, one may estimate si for a very large number of different compounds.11 In this way, si is no longer a fitted parameter but is calculated on the basis of well-established theories. This approach was shown to result in accurate calculations for pure fluids and will be used here for mixtures as well. In a mixture, parameters r and q are calculated through the following simple mixing rules: t
r)
∑i xiri
q)
∑i xiqi
isothermal-isobaric statistical ensemble, the partition function can be written as
(5)
t
(
Q(N,P,T) ) ΩRΩNR exp -
s ) q/r
(7)
t
ΩR )
i ) 1, 2, ..., t
θi )
qiNi t
∑k qkNk
)
qiNi qN
φisi
) t
∑k φksk
)
φisi s
i ) 1, 2, ..., t
(9)
The total number of contact sites in the system is
zNq ) zqN + zN0
where the same average segmental volume V* is assigned to an empty site and to an occupied site. Furthermore, it is assumed that two neighboring empty sites on the quasi-lattice remain discrete and do not coalesce. In earlier versions of the theory and in the NRHB model development for pure fluids, V* was assumed to be a pure component parameter adjusted to experimental data. In this work, V* is assumed to be constant for all fluids12 and is set equal to 9.75 cm3 mol-1. Let us turn now our attention to eq 1 by ignoring for the moment the contributions due to hydrogen bonding. In the
∏i ωiN
i
t
∏i Ni!
() Nq!
z/2
Nr!
z li ) (ri - qi) - (ri - 1) 2
(13)
(14)
For the nonrandom correction, Guggenheim’s quasi-chemical theory is used,1 as proposed in the original model:12
[( ) ]
0 Nr0 ! 2 QNR ) Nr0 Nrr!N00! ! 2
N0rr!N000!
2
[( ) ]
2
(15)
In this equation, Nrr is the number of external contacts between the segments belonging to molecules, N00 is the number of contacts between the empty sites, and Nr0 is the number of contacts between a molecular segment and an empty site. Superscript 0 refers to the case of randomly distributed empty sites. The Staverman’s combinatorial expression has been used also by Victorov and co-workers in their hole lattice model,14 which, in turn, was based on the early PV model.12 In the random case, N0rr takes the form
(10)
(11)
(12)
where ωi is a characteristic quantity for fluid i that accounts for the flexibility and symmetry of the molecule. In all applications of interest here, this quantity cancels out. Parameter li is calculated from the expression
qN z 1 ) qNθr N0rr ) zqN 2 N0 + qN 2
while the total volume of the system is given by the expression
V ) NrV* + N0V* ) NrV* ) V* + N0V*
i i
N0!
(8)
and
∏i Nlr N
Nr!
t
Furthermore, segment fractions φi and surface (contact) fractions θi are defined as
riNi xiri φi ) ) rN r
)
where ΩR is the combinatorial term of the partition function for a hypothetical system with a random distribution of the empty sites and ΩNR is a correction term for the actual nonrandom distribution of the empty sites. For the random combinatorial term, earlier theories used the Flory expression2 resulting in the lattice fluid theory3,4 or the Guggenheim expression1 resulting in the Panayiotou and Vera (PV) model12 and QCHB model.8 The PV model formed the basis for the development of many other lattice models in recent years. In NRHB, the generalized Staverman expression is adopted, according to which13
(6)
and so
E + PV kT
(16)
where
θr ) 1 - θ 0 )
q/r q/r + V˜ - 1
(17)
and the reduced volume, V˜ , is defined as
V˜ )
V V*
)
1 F˜
)
1
∑i
(18) fi
where F˜ is the reduced density. The site fractions f0 and fi, for
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the empty sites and molecular segments of component i, respectively, are related by the expression
N0
f0 )
∑i riNi
Nr )
Nr
Nr
)1-
∑i fi
(19)
In the random case, the number of contacts between empty sites is given by the equation
N000
N0 z 1 ) N0z ) Nθ 2 Nq 2 0 0
(20)
and
ij ) xiijj (1 - kij)
where kij is a binary interaction parameter between species i and j and is fitted to binary experimental data. From statistical thermodynamics,15 it is known that in the isothermal-isobaric ensemble the partition function is related to the Gibbs free energy through the expression
G ) -RT ln Q(N,P,T)
N0 qN ) zN0 ) zqNθ0 ) zN0θr Nq Nq
(21)
The total number of intersegmental contacts is calculated as the sum of contributions between like molecules and between unlike molecules: t
t
(∂G∂F˜ )
T,P,N,N10,...,Nt0
[
li
t
P ˜ + T˜ ln(1 - F˜ ) - F˜
i
2 (22)
P ˜)
Nij ) N0ijΓij
i ) 1, ..., t
i ) 1, ..., t
T RT ) T* *
(31)
PV* P ) P* RT*
(32)
t
θiΓij ) 1 ∑ i)0
j ) 0, 1, ..., t
(25)
Finally, in the NRHB model, as well as in previous lattice models, it is assumed that only first-neighbor segment-segment interactions contribute to the potential energy E of the system. Consequently, for a mixture it is t
-E )
t
t
Niiii + ∑ ∑ Nijij ∑ i)1 i)1 j>i
(26)
t
∑ ∑ θiθjij* i)1 j)1
(33)
(34)
The 2t + t(t - 1)/2 + 1 different number of contacts Nij or, equivalently, the nonrandom factors Γij are calculated from the following set of minimization conditions:
)0
i ) 0, 1, ..., t and j ) i + 1, ..., t
(35)
T,P,N,F˜
which leads to the following set of t(t + 1)/2 equations:
ΓiiΓjj Similar expressions were also used by Victorov and co-workers in the hole-lattice model.14 The nonrandom factors Γ should obey the following material balance expressions:
]
lnΓ00 ) 0 (30)
ij* ) xi* j* (1 - kij)
∂G ∂Nij
(24)
+
and
( )
t>j>i
N00 ) N000Γ00 0 Ni0 ) Ni0 Γi0
* )
i * j (23)
In order to calculate the nonrandom distribution of molecular segments and empty sites, appropriate nonrandom factors Γ are introduced, as explained below. As a result, a total of 2t + t(t - 1)/2 + 1 contact value expressions become
]
where
t
qjNj ) zqiNi ) zqiNiθjθr ) zqjNjθiθr Nq
Nii )
[
q
z
qiNi z z ) qiNi ) qiNiθiθr 2 Nq 2
N0iiΓii
z
φi - ln 1 - F˜ + F˜ ∑ r 2 r i)1
T˜ )
N0ij
(29)
which leads to the equation of state:
where
N0ii
)0
t
N0ij ∑i N0ii + ∑i ∑ j>i
N0rr )
(28)
At equilibrium, the reduced density of the system is obtained from the following minimization condition:
while the number of contacts between a segment and an empty site is given by 0 ) zqN Nr0
(27)
Γij2
) exp
( ) ∆ij RT
i ) 0, 1, ..., t and j ) i + 1, ..., t (36)
where
∆ij ) i + j - 2(1 - kij)xij
(37)
and 0 ) 0. Equations 25 and 36 form a system of 2t + t(t 1)/2 + 1 nonlinear algebraic equations which is solved analytically for pure fluids and numerically for the case of multicomponent mixtures. In this work, the algorithm proposed by Abusleme and Vera16 based on the generalized NewtonRaphson method is used. For phase equilibrium calculations, the chemical potential of each component i in the mixture is needed. It is obtained from the following expression:
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2631
µi )
( ) ∂G ∂Ni
(38)
T,P,Nj,j*i,N
where, νH is the average per segment number of hydrogen bonds in the system, and is given by
10,...,Nt0,F˜
m
By making the appropriate substitutions, eq 38 results in the following expression for a non-hydrogen-bonding component i:
µi RT
) ln
φi ωiri
- ri
∑j
φjlj rj
[
+ ln F˜ + ri(V˜ - 1) ln(1 - F˜ )
][
[
]
The expression for the chemical potential of a pure component, µoi , can be obtained from eq 39 by setting φi ) θi ) 1 and the number of components in the summations equal to 1. According to Flory-Huggins theory, the excess chemical potential of the solvent is related to the χ parameter according to the expression
( )
r1 µ1 - µo1 ) ln φ1 - 1 - φ2 + χφ22 RT r2
[
(∑
P ˜ + T˜ ln (1 - F˜ ) - F˜
li
) [
q
]
φi - νH - ln 1 - F˜ + F˜ + ri 2 r i)1 2
∑R ∑β
m
n
∑R ∑β
νRβ )
NRβ,H (42)
rN
RT
m
) riνH -
∑
diR ln
νRd
n
-
νR0
R)1
∑
aiβ ln
β)1
νβa ν0β
(43)
where m
NRd
νRd )
)
∑ dkRNk k)1
rN
(44)
rN
and n
νβa )
Nβa
)
akβNk ∑ k)1
rN
(45)
rN
while
z
z
µi,H
(40)
By combining eqs 39 and 40, one may calculate the χ parameter from the equations of state. Hydrogen-Bonding Contribution. In NRHB, as well as in previous quasi-chemical theories where hydrogen-bonding interactions are calculated explicitly such as QCHB and LFHB, it is postulated that intermolecular forces can be divided into physical (repulsion and dispersion) and chemical (such as hydrogen bonding) forces.8-10 This is a major difference between these models and plain quasi-chemical models.12,14 A direct implication of the postulate made here is that the partition function can be factored out into a product of terms, as shown in eq 1. Furthermore, using eq 28 and standard thermodynamic equations, the distinct contribution of hydrogen-bonding interactions to the Gibbs free energy, chemical potential, enthalpy, and so forth can be calculated. Such contributions will be denoted using index (either subscript or superscript) H. The formalism that will be used here for hydrogen bonding was proposed by Panayiotou and Sanchez10 for the lattice fluid theory using Veytsman statistics.17 Only the key equations of the model are presented here. As proposed previously, we consider that there are m types of proton donors and n types of proton acceptors in the mixture. Let dkR be the number of donor groups of type R in each molecule of type k and akβ be the number of acceptor groups of type β in each molecule of type k. Let NRβ,H be the total number of hydrogen bonds between a donor of type R and an acceptor of type β in the system. Using the NRHB procedure, we obtain the following expression for the equation of state of the mixture: t
n
The full expression of the chemical potential of component i in the mixture is calculated by adding to eq 39 the following hydrogen-bonding contribution:
]
qi
q ln 1 - F˜ + F˜ - ri V˜ - 1 + 2 ri r zqi ri P ˜ V˜ qi + ln Γii + (V˜ - 1) ln Γ00 + ri - (39) 2 qi T˜ T˜ i z
νH )
]
ln Γ00 ) 0 (41)
νR0 ) νRd -
n
∑ νRβ
(46)
β)1
and similarly m
ν0β ) νβa -
∑ νRβ
(47)
R)1
As discussed previously,10 the νRβ’s satisfy the minimization conditions:
( )
H -GRβ νRβ ) F˜ exp νR0ν0β RT
for all (R,β)
(48)
Equations 48 are a set of (m × n) quadratic equations that are solved simultaneously with the equation of state for reduced density. H In eq 48, GRβ is the free enthalpy of formation of the hydrogen bond of type R-β and is given in terms of the energy (E), volume (V), and entropy (S) of hydrogen-bond formation by the equation H H H H GRβ ) ERβ + PVRβ - TSRβ
(49)
The formalism presented here is general and sufficient for solving phase equilibrium problems in systems of hydrogenbonded fluids of any number of donor and acceptor groups. Results and Discussion Pure Components. In the formulation of the NRHB model for pure fluids,9 non-hydrogen-bonding fluids had three pure component characteristic parameters adjusted to experimental thermodynamic data for vapor pressure and saturated liquid and vapor density. These parameters were the average intersegmental
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Table 1. NRHB Scaling Parameters for Pure Fluids and Deviation between NRHB Correlation and Experimental Vapor Pressure and Liquid Density Values in the Temperature Range Indicated
methane ethane propane n-pentane n-decane n-octacosane cyclohexane dichloromethane carbon dioxide methanol ethanol 1-propanol water polystyrenea poly(dimethyl siloxane)a a
h* (J mol -1)
s* (J mol-1 K-1)
Vsp,0* (cm3 g-1)
s
T (K)
% AAD in Psat
% AAD in Fliq
1956.2 2997.4 3409.2 3841.6 4199.4 4469.4 4469.2 5163.3 3468.4 4202.3 4144.3 4231.0 6614.7 5341.5 3872.6
-0.9181 -0.3018 0.5624 1.4279 2.0254 2.3140 1.8391 -1.3305 -4.5855 1.5269 1.0622 1.1751 -6.5100 4.5361 5.1720
2.12519 1.58559 1.44141 1.31072 1.21463 1.15461 1.19596 0.68800 0.79641 1.15899 1.12571 1.11492 0.98440 0.9027 0.9219
0.961 0.941 0.903 0.867 0.836 0.814 0.801 0.881 0.909 0.941 0.903 0.881 0.861 0.667 0.744
95.54-185.79 136.48-298.59 171.75-361.32 231.17-458.16 339.36-600.56 551.14-803.69 292.74-538.85 239.80-498.09 218.00-301.00 265.86-499.29 278.41-499.88 295.56-521.85 273.15-643.15 373.45-513.35 314.75-579.85
1.1 1.9 1.3 0.7 1.1 3.8 0.6 4.5 1.0 2.2 1.0 0.8 0.9
0.9 1.1 0.9 0.6 0.4 0.4 1.9 0.9 1.7 2.6 0.5 0.5 2.2 0.1 0.3
For polymers, Vsp* ) Vsp,0* + (T - 298.15)Vsp,1* - 0.135 × 10-3 P where T is in K and P in MPa.
interaction energy *, the segment volume V*, and the closepacked specific volume Vsp*. As already mentioned, in this work, V* is assumed to have a constant value for all fluids12 equal to 9.75 cm3 mol-1. Furthermore, in order to increase the model accuracy over a wide temperature range, * and Vsp* are allowed to vary linearly with temperature as follows:
* ) h* + (T - 298.15)s*
(50)
Vsp* ) Vsp,0* + (T - 298.15)Vsp,1*
(51)
Subscripts h and s in eq 50 denote an “enthalpic” and an “entropic” contribution to the interaction energy parameter, respectively, reminiscent of Flory’s χ parameter contributions. In eq 51, parameter Vsp,1* is treated as a characteristic parameter for different homologous series. In this work, Vsp,1* assumes a constant value of -0.412 × 10-3 cm3 g-1 K-1 for nonaromatic hydrocarbons, -0.310 × 10-3 cm3 g-1 K-1 for alcohols, and 0.150 × 10-3 cm3 g-1 K-1 for all other fluids. Consequently, the model has three pure component adjustable parameters, which are h*, s*, and Vsp,0*. These parameters are fitted to the experimental saturated liquid density and vapor pressure of pure components.18 Polymers have practically no vapor pressure. As a result, model parameters are fitted to PVT data of the melt over a wide
Figure 1. Methane-ethane VLE. Experimental data20 at 172.04 K (O) and 199.92 K (b), NRHB predictions (s), and predictions ignoring nonrandomness (----).
temperature and pressure range. Often, the pressure range of available data is extended from ambient pressure up to, or even higher than, 200 MPa. In order to correlate the experimental data over the full pressure range, we have used a small “compressibility” correction to the scaling constant Vsp*. Of course, this compressibility correction is not needed in applications at low to moderate pressures (say, below 10 MPa). In Table 1, model parameters are shown for the fluids, including polymers, examined in this work. In ref 19, parameters for an extensive list of components can be found. For hydrogen-bonding fluids, NRHB contains three additional pure component parameters, which are the energy EHi , volume VHi , and entropy SHi , of hydrogen bonding. These parameters can be calculated on the basis of experimental spectroscopic data or accurate ab initio quantum mechanics calculations. In this work, these parameters were set constant for the alcohols examined and equal to the values proposed previously for NRHB:9 EHi ) -25 100 J mol-1, VHi ) 0.0, and SHi ) -26.5 J K-1 mol-1. Furthermore, for the hydrogen bond between water molecules, these parameters are EHi ) -15 000 J mol-1, VHi ) 0.0, and SHi ) -15.2 J K-1 mol-1. Fluid Mixtures. In this work, a representative set of binary fluid mixtures was examined. An effort was made to examine mixtures of components that differ in molecular size, polarity, and/or the ability to hydrogen bond. Furthermore, different types of phase equilibria such as VLE, LLE, and VLLE were examined. In the following paragraphs, detailed results for these families of mixtures are presented. Methane-Hydrocarbon Mixtures. Methane-hydrocarbon mixture phase equilibria are important for the oil and gas industry. In Figure 1, experimental data and NRHB predictions (kij ) 0) are shown for the methane-ethane VLE. It is a relatively simple mixture of two nonpolar components of similar size. Model predictions are in excellent agreement with experiments. For comparison, predictions from the model ignoring nonrandomness are shown. For this simple system, the effect of the latter on the phase equilibria is small. For higher hydrocarbons, the nonideality of the thermodynamic properties increases and the accurate prediction of phase equilibria is a challenging task. In Figure 2, the solubility of methane in n-octacosane at two different temperatures (348.2 and 423.2 K) is shown. Model prediction is very satisfactory. Cubic equations of state typically require a binary interaction parameter adjustment to correlate experimental data for such mixtures.21 In
Ind. Eng. Chem. Res., Vol. 46, No. 8, 2007 2633
Figure 2. Methane solubility in n-octacosane: experimental data21 at 348.2 K (b) and 423.2 K (O), NRHB predictions (s), and predictions ignoring nonrandomness (----). Figure 5. Carbon dioxide-n-pentane VLE: experimental data24 at 310.15 K (O) and 333.15 K (b), NRHB correlation (s) with kij) 0.093, and LFHB correlation (----) with kij ) -0.011.
Figure 3. Carbon dioxide-ethane VLE: experimental data22 at 230 K (O) and 270 K (b), NRHB correlation (s) with kij ) 0.11, and LFHB correlation (----) with kij ) 0.086.
Figure 6. Carbon dioxide-n-decane VLE at 344.3 K: experimental data25 (b), NRHB correlation (s) with kij) 0.201, and LFHB correlation (----) with kij) -0.092.
Figure 4. Carbon dioxide-propane VLE: experimental data23 at 230 K (O) and 270 K (b), NRHB correlation (s) with kij ) 0.117, and LFHB correlation (----) with kij ) 0.072.
addition, model predictions ignoring nonrandomness (dashed lines) result in substantial deviations from experimental results. Carbon Dioxide Mixtures. Carbon dioxide is a widely used component in many chemical processes, mostly as part of a mixture, and accurate knowledge of the phase equilibria is required. In this work, mixtures of carbon dioxide with different n-alkanes were examined. In Figures 3-6, representative results are shown for carbon dioxide-ethane, -propane, -n-pentane, and -n-decane mixtures at different temperatures. The ethane mixture exhibits an azeotrope at both temperatures examined (Figure 3). Furthermore, in the case of n-pentane (Figure 5) and n-decane (Figure 6) mixtures, carbon dioxide is at super-
critical conditions. NRHB correlates the experimental data very accurately using a temperature-independent binary interaction parameter. This binary parameter increases from approximately 0.1 for the lower n-alkanes to approximately 0.2 for the n-decane mixture. This may be attributed to the larger molecular size difference for the latter mixture. Correlations using the latticefluid hydrogen-bonding (LFHB) model, an earlier lattice model,10 are also shown in Figures 3-6 (dashed lines). In all cases, the new model consists of an improvement over this previous model. A mixture of carbon dioxide with a polar chlorinated hydrocarbon, namely, dichloromethane, with a dipole moment of 1.8 D was examined. A binary interaction parameter fitted to the coexisting phase compositions (Figure 7a) was used subsequently for the prediction of density of liquid and vapor phases (Figure 7b). Model predictions are in very good agreement with experimental data for both types of calculations. Furthermore, carbon dioxide mixtures with lower alcohols that strongly hydrogen-bond were examined. In Figure 8, carbon dioxide-ethanol VLEs at 313.2 and 328.2 K are shown. NRHB correlates accurately the phase equilibria with a single binary
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Figure 7. Carbon dioxide-dichloromethane VLE at 318.2 K. Experimental data26 (b) and NRHB calculations (s) with kij ) 0.036. (a) Coexisting phase compositions and (b) coexisting phase densities.
interaction parameter (Figure 8a). In addition, the model predicts accurately the coexisting densities with the parameter optimized to phase equilibria (Figure 8b). A carbon dioxide-1-propanol mixture VLE was also examined. NRHB predictions (kij ) 0) are in very good agreement with the experimental data at high pressures (doted lines in Figure 9), but some deviations are observed at lower pressures. If one uses the kij value of 0.05 optimized for the carbon dioxide-1-propanol mixture, then an excellent prediction of the entire pressure range is obtained (solid lines in Figure 9). In addition, LFHB correlation significantly overpredicts the composition pressure in both temperatures (dashed lines). Hydrogen Bonding-Nonpolar Mixtures. Finally, two representative binary mixtures were examined, where one of the components hydrogen-bonds strongly and the other component is a nonpolar fluid. Such considerable difference in intermolecular interactions results in complex phase equilibria with a large range of partial miscibility of the two components. An extensive examination of the model’s ability for these mixtures will be examined in the future, and only some characteristic results are shown here. In Figure 10, the methanolethane mixture is examined at 298.15 K. At low pressures, the mixture exhibits VLE which terminates at 4 MPa with the appearance of a second liquid phase (Figure 10a). At higher pressures, LLE is observed. NRHB correlates the entire phase diagram very accurately using a single binary interaction parameter (Figure 10a). Furthermore, the model predicts very accurately the coexisting densities at different pressures (Figure 10b). Water-hydrocarbon mixtures are highly nonideal. As a result, their mutual solubility is most of the times very low. In Figure 11, the solubility of methane in water over a wide temperature
Figure 8. Carbon dioxide-ethanol VLE. Experimental data26 at 313.2 K (b) and 328.2 K (O), and NRHB calculations with kij ) 0.05. (a) Coexisting phase compositions and (b) coexisting phase densities.
Figure 9. Carbon dioxide-1-propanol VLE. Experimental data27 at 310.15 K (O) and 333.15 K (b). NRHB calculations (s) with kij ) 0.05 and (‚‚‚‚) with kij ) 0.0 and LFHB calculations (red ---) with kij ) -0.108.
and pressure range is shown. Even with no binary parameter adjustment, the model predicts accurately the experimental data. A temperature-independent kij value results in excellent agreement between experiment and theory. Polymer-Solvent Mixtures. Modeling of high polymer systems is a challenging task. In this work, a few representative results are shown. A manuscript devoted to this subject is underway. In Figure 12, sorption of carbon dioxide in polystyrene at 80 and 100 °C is presented. NRHB correlation is in excellent agreement with the experimental data over a wide pressure range. Experimental data are often used to calculate the FloryHuggins χ parameter, which accounts mainly for energetic
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Figure 12. Carbon dioxide sorption isotherms in polystyrene. Experimental data30 (b and O) and NRHB correlation (s) with kij ) 0.013.
Figure 13. Flory-Huggins χ parameter for cyclohexane (l)-PDMS (2) mixture at 25 °C. Experimental data31,32 (b) and NRHB correlation (s) with kij ) 0.016. Figure 10. Methanol-ethane phase equilibria at 298.15 K. Experimental data28 (b), and NRHB correlation (solid lines, kij)0.021). (a) Coexisting phase compositions and (b) coexisting phase densities.
Figure 11. Methane solubility in water. Experimental data29 at 423 K (b), 523 K (O), and 603 K (2) and NRHB calculations: solid lines (kij ) 0.0) and dashed lines (kij ) 0.161).
but also entropic interactions between the polymer and solvent. According to the Flory-Huggins model, χ is composition-independent. However, this is not the case for many real mixtures. In Figure 13, experimental data for a cyclohexanepolydimethylsiloxane (PDMS) mixture at 25 °C are shown. NRHB correlation is in excellent agreement with experimental data, predicting a significant variation of χ with composition.
Conclusions NRHB lattice theory was successfully generalized to mixtures. For the case of hydrogen-bonding mixtures, the equation of state needs to be solved together with nonlinear algebraic equations originated from the mass action law. VLE, LLE, and VLLE of many highly nonideal binary mixtures were correlated accurately, in many cases without any binary parameter adjustment. Representative results for solvent-polymer mixtures were presented. A wide range of temperature and pressure conditions were examined. In all of the cases examined, the new model was shown to be more accurate than LFHB, an earlier version of the model. Furthermore, it was shown that explicit account for the fluid nonrandomness results in improvement in model predictions. Future work will focus on extensive model applications to specific types of mixtures, including mixtures of multiple associating fluids, polymer mixtures, and so forth. In parallel, efforts will be made to simplify the coupled nonlinear equations that describe mixture nonrandomness (eq 36), as well as hydrogen bonding between different components (eq 48) in order to speed up calculations and to make the model useful for routine calculations, even as part of a process simulation. Acknowledgment Financial support for this work has been provided by the Greek General Secretariat of Research and Technology through the PENED 2001 program. I.G.E. acknowledges a visiting professorship at IVC-SEP, Department of Chemical Engineering,
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ReceiVed for reView October 10, 2006 ReVised manuscript receiVed January 19, 2007 Accepted February 1, 2007 IE0612919
