A new molecular-thermodynamic model based on lattice fluid theory: Application to pure fluids and their mixtures

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Fluid Phase Equilibria 265 (2008) 112–121

A new molecular-thermodynamic model based on lattice fluid theory: Application to pure fluids and their mixtures Xiaochun Xu a , Honglai Liu a,∗ , Changjun Peng b,∗ , Ying Hu b a

State Key Laboratory of Chemical Engineering and Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China b Lab for Advanced Materials and Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China Received 21 November 2007; received in revised form 9 January 2008; accepted 15 January 2008 Available online 19 January 2008

Abstract In previous work, we have developed a close-packed lattice model for binary solutions of chain-like molecules. As a continuation, considering the effect of volume change, we present here a new molecular-thermodynamic model based on lattice fluid theory. The resulting equation of state shows good performance in describing thermodynamic properties such as pVT behavior, vapor pressure and liquid volume of pure normal fluids. Equation-of-state parameters are obtained by correlation of experimental data. The model is extended to calculate vapor–liquid equilibria of binary mixtures with only one binary interaction parameter. Comparison between the present model and other theories is also presented. © 2008 Elsevier B.V. All rights reserved. Keywords: Lattice fluid; Thermodynamic model; Equation of state; Vapor–liquid equilibrium

1. Introduction Various theories have been developed to describe the thermodynamic properties of species of chain-like molecules during the latter half of the last century. The most widely used and best known is the Flory–Huggins close-packed lattice theory [1,2] which illustrates the competition between the entropy of mixing and the attractive interaction energy in a simple way. However, the theory is based on a mean-field approach for which it is known that it cannot correctly describe the shape of the coexistence curves, especially near the critical point [3]. Some other lattice theories developed progressively, e.g. the lattice cluster theory [4–6] and the revised Freed’s theory [7–9]. Further, to account for volume effects, a lattice fluid model based on Flory–Huggins theory was first proposed by Sanchez and Lacombe [10–12] under the assumption of complete randomness in the distribution of molecules and holes on the lattice. Hu and co-workers [13] adopted a two-step mixing process to introduce holes to the revised Freed’s theory and established a lattice fluid model. Recently, Shin and Kim [14] extended a new quasi-

∗

Corresponding authors. Tel.: +86 21 64252921; fax: +86 21 64252921. E-mail addresses: [email protected] (H. Liu), [email protected] (C. Peng). 0378-3812/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2008.01.011

chemical nonrandom lattice fluid model to describe vapor–liquid equilibria of mixtures. Panayiotou et al. [15] developed a nonrandom hydrogen-bonding model, which was applicable to various systems, including nonpolar systems and highly nonideal systems with strong specific interactions, and Tsivintzelis et al. [16] derived new analytical expressions for the nonrandomness factor, which was used to describe nonrandomness in mixtures, and compared it with others corresponding expressions. Although the above mentioned theories or models indicate great progress, it is still necessary to develop more new lattice fluid models by using different theories or methods. Meanwhile, there are ever-increasing computer molecular simulation results based on lattice model appearing in the literatures, which provide a necessary to strictly testify the correctness of new theories. Recently, Yang et al. [17,18] developed a new close-packed lattice model for the binary mixtures, which is based on Zhou–Stell theory [19–21] and the study of Yan et al. [22] for the Ising system. By this model, the predicted critical temperatures and critical compositions, spinodals and coexistence curves as well as internal energies of mixing for systems with various chain lengths are in satisfactory agreement in comparison with computer simulation results and experimental data, indicating the superiority of this model. As a continuation, considering the volume effect, we have developed further a Helmholtz function model of mixing and the corresponding equation of state using

X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

113

Fig. 1. Process of two-step mixing.

the lattice fluid theory. The model was applied to pure fluids and mixtures. Comparisons with experimental data and other theories are also presented.

contact pairs and can be calculated by

2. The Model

where a and b are coefficients and can be obtained by fitting to the critical parameters from MC simulation data [23,24]. In this work, a = 0.1321, b = 0.5918 [18]. In the second step, the mixed close-packed solution is considered to be a pseudo-pure substance “a”, its average segment number ra and the energy parameter εaa are estimated by the following mixing rules:

2.1. Helmholtz function of mixing To extend a close-packed lattice model [17,18] to a lattice fluid, we adopt a two-step process [13] as shown in Fig. 1. In the first step, N1 molecules 1 with r1 segments and N2 molecules 2 with r2 segments are mixed to form a close-packed mixture. In the second step, the close-packed mixture, considered to be a pseudo-pure substance, is mixed with N0 holes to form an expanded real system with volume V at temperature T and pressure p. According the two-step process, the Helmholtz function of mixing can be expressed as Δmix A = Δmix AI + Δmix AII

(1)

Following Ref. [13], the same model is applied for both steps. In the first step, the Helmholtz function of mixing Δmix AI is [17,18]:  Δmix AI φ2 z θ1 q1 φ1 ln φ1 + ln φ2 + ln φ1 = Nr kT r1 r2 2 r1 φ1  z q2 θ2 z 2 2 + + φ2 ln φ1 φ 2 − φ φ r2 φ2 2T ∗ 4T ∗2 1 2 2  z ri − 1 + λ i 2 2 2 2 φ φ (φ + φ ) − φi 1 2 1 2 ∗3 12T ri i=1   1 + φ2 (exp(1/T ∗ ) − 1) × ln 1 + φ1 φ2 (exp(1/T ∗ ) − 1)

−

(2)

where Nr = N1 r1 + N2 r2 is the total number of close pack sites, z is the coordination number, k is Boltzmann constant and T is temperature; T* = kT/εI is the reduced temperature for the first step, and εI = ε11 + ε22 − 2ε12 is the exchange energy between molecules 1 and molecules 2, and εij is the attractive energy of the i–j pair; φi and θ i are the volume fraction and surface fraction of component i, respectively, and can be determined as follows N i ri , i=1 Ni ri

φi = 

N i qi i=1 Ni qi

= 

z(ri − 1)(ri − 2) (ari + b) 6ri2

ra−1 =

(4)

φ1 φ2 + r1 r2

(5)

εaa = θ12 ε11 + 2θ1 θ2 ε12 + θ22 ε22 (6) ε12 , the interaction energy between a segment of species 1 and that of species 2, is assumed to be given by √ ε12 = (1 − κ12 ) ε11 ε22 (7) κij is an adjustable parameter, which can be obtained from experimental data. Then, N0 holes are mixed with “a”, and the Helmholtz function of mixing Δmix AII in this step is:  Δmix AII Nl ϕa ϕ0 ln ϕ0 + = ln ϕa Nr kT Nr ra   z qa z θ0 θa + + + ϕa ln ϕ0 ϕa ϕ0 ln 2 ϕ0 ra ϕa 2T0∗ z z − ∗ 2 ϕ02 ϕa2 − ϕ2 ϕ2 (ϕ2 + ϕa2 ) 4T0 12T0∗ 3 0 a 0   1 + ϕ0 (exp(1/T0∗ ) − 1) ra − 1 + λa ϕa ln − ra 1 + ϕ0 ϕa (exp(1/T0∗ ) − 1) (8) where Nl = Nr + N0 is the total number of sites for lattice fluid, T0∗ = kT/εII is the reduced temperature for the second step, εII is the exchange energy between pseudo-pure substance “a” and hole “0”, and the attractive energy around hole ε00 and ε0a are equal to zero. So εII = εaa + ε00 − 2ε0a = εaa

(9)

ϕi and θ i are the volume fraction and surface fraction of component i, respectively.

θi with zqi = ri (z − 2) + 2

λi =

(3)

ϕ0 =

λi in Eq. (3) is a parameter characterizing the long-range correlations between monomers in the same chain beyond the close

=

N0 , Nl N0 +

ϕa = 1 − ϕ0 = N0 

i=1 Ni qi

Nr , Nl

θ0

, θa = 1 − θ0 =

 N0 +

Ni qi i=1 

i=1 Ni qi

(10)

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X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

Fig. 2. Experimental (symbols) and calculated (line) vapor pressures of propane (squares), butane (triangles), pentane (diamonds), hexane (circles) and heptane (snowflakes).

Fig. 3. Experimental (symbols) and calculated (lines) saturated density of propane (squares), butane (triangles), pentane (diamonds), hexane (circles) and heptane (snowflakes).

λa is the parameter characterizing the long-range correlations between monomers in the pseudo-pure substance “a” beyond the close contact pairs and can be obtained by

expression for the equation of state for mixtures     2 1 z ˜ + ln − 1 ρ˜ + 1 p˜ = T˜ − ln(1 − ρ) 2 z ra

λa =

z(ra − 1)(ra − 2) (ara + b) 6ra2

(11)

z z1 z 1 − ρ˜ 2 − (10ρ˜ 6 − 24ρ˜ 5 (3ρ˜ 4 − 4ρ˜ 3 + ρ˜ 2 ) − ˜ 2 4T 12 T˜ 2

where a = 0.1321, b = 0.5918.

+ 21ρ˜ 4 − 8ρ˜ 3 + ρ˜ 2 )

2.2. Equation of state

+

Now, we can derive the equation of state from the Helmholtz function of mixing for the expanded solution. We defined ˜ and reduced density reduced temperature T˜ , reduced pressure P, ˜ ρ: T T˜ = = T0∗ , εaa /k

p pv∗ p˜ = ∗ = , p εaa

Nr v∗ ρ˜ = V

(12)

where v∗ is the hard-core volume of a site or a segment. The reduced density ρ˜ is related to the volume fraction by Nr ρ˜ = = ϕa , Nl

1 − ρ˜ = ϕ0

(13)

Using classical thermodynamics, we have  2 ∂(Δmix A/Nr kT ) ˜ p˜ = T ρ˜ ∂ρ˜ T˜ ,φ

(14)

However, Δmix AI /Nr kT for the first step is independent of density because of close-packing; therefore, there is no contribution to the equation of state. From Eqs. (8), (13) and (14), we obtain the

˜ 2−1 [1 + D1 (1 − ρ)] ra − 1 + λa ˜ 2 T ρ˜ ˜ − ρ)] ˜ ˜ [1 + D1 (1 − ρ)][1 ra + D1 ρ(1 (15)

D1 = exp(1/T˜ ) − 1

(16)

When the equation of state is applied to a pure fluid (pseudobinary system), εII = εaa = ε and ra = r. ε is the attractive energy of the neighbor monomers of the pure component, and r is the segment number. 2.3. Chemical potential For binary mixture, the chemical potentials of the components can be obtained by following equations,  μ1 − μ01 ∂(Δmix A/kT ) μ2 − μ02 = , kT ∂N1 kT T,V,N2  ∂(Δmix A/kT ) = (17) ∂N2 T,V,N1

Table 1 Comparison between SWCF, SAFT, PHSC, QLF, and present EOS in calculated saturated vapor pressure and liquid density Substances (Tmin − Tmax )

Hexane (282–492) Benzene (297–517) Acetone (283–463)

p (%)

ρ (%)

Present

SWCF

SAFT

PHSC

QLF

Present

SWCF

SAFT

PHSC

QLF

0.84 1.55 1.20

1.24 0.54 1.51

2.3 1.4 2.3

3.4 1.1 4.6

0.77 1.18 0.26

0.75 0.66 2.21

2.21 0.98 1.21

3.5 2.1 1.0

4.3 2.1 0.9

0.66 0.45 0.75

X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

115

Table 2 Parameters of pure fluids and the corrected results Components

T (K)

ε0 /k (K)

ε1 /k

r

p (%)

ρ (%)

Methane Ethane Propane Butane Pentane Hexane Heptane Octane Nonane Isobutane 2-Methylbutane 2,2-Dimethylpropane Ethene Propene 1-Butene 1-Pentene 1-Hexene 1-Octene trans-2-Butene cis-2-Butene Propyne 1-Butyne 2-Butyne 1,3-Butadiyne Propadiene 1,3-Butadiene 1,5-Hexadiene 2-Methyl-1,3-butadiene Acetone 2-Butanone 2-Pentanone 2-Heptanone 2-Nonanone 2-Undecanone Cyclopropane Cyclobutane Cyclopentane Cyclohexane Cycloheptane Cyclooctane Methylcyclopentane Methylcyclohexane Ethylcyclohexane Cyclopentene Cyclohexene Benzene Toluene Ethylbenzene Propylbenzene o-Xylene m-Xylene p-Xylene Styrene Tetrachloromethane Tetrafluoromethane Trichlorofluoromathane Chlorotrifluoromethane Carbon dioxide Overall average Δ

95–181 149–295 167–306 230–386 247–454 282–492 304–515 304–544 344–424 226–369 243–447 261–429 144–265 188–352 225–388 262–437 283–334 287–393 214–373 215–302 227–378 242–282 273–296 238–272 208–243 222–364 273–324 219–298 283–463 288–521 303–516 310–424 336–437 320–432 197–357 207–273 272–460 279–541 297–575 295–466 274–488 276–554 278–372 274–302 243–360 297–517 320–546 397–487 294–428 330–612 320–582 288–491 293–418 294–550 120–183 226–465 145–294 217–300

53.5864 80.5261 89.6839 95.5324 99.3591 102.1352 104.1338 105.9624 106.5516 91.1630 97.1265 90.3896 76.4120 90.4862 96.2312 99.8483 102.5820 105.9898 99.3462 101.8335 107.3208 107.6468 117.0228 117.5485 102.7331 100.5666 103.7618 105.8272 128.4890 123.2942 120.1438 119.7840 118.6053 118.4332 103.1990 110.0123 112.3436 113.8138 118.3600 120.0732 109.6412 110.7591 112.4261 115.2026 119.1344 124.5945 122.9239 122.1602 120.7868 125.1088 122.8928 122.3604 128.8144 118.1595 57.7248 105.8173 75.6933 91.9809

−0.019659 −0.004793 0.003806 0.026658 0.033622 0.039250 0.043155 0.044053 0.046626 0.029750 0.036962 0.043272 −0.015389 0.007707 0.023312 0.032750 0.034912 0.041417 0.017755 0.005398 −0.022987 −0.016893 −0.009809 −0.025850 −0.046013 0.014665 0.036878 0.010626 −0.006203 0.009766 0.019367 0.027662 0.035247 0.036456 −0.004463 −0.001565 0.027770 0.037758 0.042333 0.044447 0.037860 0.046978 0.042752 0.011008 0.020249 0.023585 0.033508 0.035131 0.037281 0.037164 0.036653 0.035279 0.026709 0.034731 −0.042472 0.023044 −0.004746 −0.093547

3.846 5.388 6.952 8.580 10.145 11.653 13.173 14.679 16.373 8.652 10.036 10.201 4.910 6.539 8.115 9.658 11.226 14.365 8.066 7.845 5.826 7.331 7.199 6.341 5.654 7.593 10.676 8.936 7.330 8.937 10.496 13.386 16.515 19.626 5.851 7.232 8.737 10.108 11.321 12.781 10.325 11.692 13.285 8.158 9.524 8.645 10.255 11.760 13.329 11.628 11.755 11.843 11.222 9.237 5.202 8.382 6.444 3.765

1.73 2.14 2.55 1.13 0.94 0.84 0.74 1.01 0.18 1.16 0.50 0.32 2.16 1.77 0.94 0.60 0.17 0.74 1.41 1.02 1.89 0.25 0.09 0.37 0.21 1.48 0.36 3.11 1.20 1.56 1.22 0.88 0.53 1.30 2.64 0.66 1.14 1.44 0.95 1.43 0.94 1.10 0.68 0.17 1.23 1.55 0.83 1.35 1.21 0.82 0.74 1.50 0.87 1.06 1.51 2.01 2.57 1.21 1.14

3.14 2.56 1.45 0.36 0.44 0.75 0.96 1.34 0.30 0.29 0.96 0.73 2.69 1.89 1.01 0.25 0.08 0.16 0.94 0.91 2.26 0.43 0.29 0.65 0.78 1.09 0.09 0.59 2.21 – – 0.22 0.09 0.13 1.12 1.40 0.40 – 1.55 0.33 0.43 1.17 0.15 0.23 0.67 0.66 0.31 0.32 0.22 0.64 0.49 0.34 0.73 1.09 1.75 1.22 1.98 4.16 0.93

p (%) =

100 Nm



pexp −pcal

pexp , ρ (%) =

100 Nm



ρexp −ρcal

ρexp , and Nm is the number of data.

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X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

Fig. 4. Experimental (symbols) and calculated (line) vapor pressures of ethanol (squares), 1-propanol (triangles), 1-butanol (diamonds) and pentanol (circles).

Combining Eq. (17) with Eqs. (1), (2) and (8), we can obtain. μ1 − μ01

=

μ1,I + μ1,II − μ01

(18)

  μ1,I − μ01 ∂(Δmix AI /kT ) = ln φ1 = ∂N1 kT T,V,N2

ϕ02 (D1 + 1) ra − 1 + λa r1 D2 ra φ1 (1 + ϕ0 D1 )(1 + ϕ0 ϕa D1 )

D1 = exp(1/T˜ ) − 1,

 D2 = 2θ1 θ2

θ1 θ2 − θ1 θ2 + − T˜ 1 T˜ 12 T˜ 2



z/6(3a − b) + 1 z/6(6b − 4a) zb D3 = + − 3 ra ra2 ra

(22-b)

where T˜ 1 , T˜ 2 and T˜ 12 are defined by kT , T˜ 1 = ε11

kT T˜ 2 = , ε22

kT T˜ 12 = ε12

(23)

The chemical potential of the component 2 can be also obtained from Eqs. (18)–(22) by interchanging the subscripts 1 and 2. 3. Result and discussion 3.1. Application to pure substances The new equation of state, Eq. (15), can be applied to describe thermodynamic properties such as pVT behavior and phase equi-



μ1,II ∂(Δmix AII /kT ) ˜ II + r1 p˜ = = r1 A kT ∂N1 ρ˜ T˜ T,V,N2   r1 1 z r1 + 1− ln θa + ϕ0 D 2 1 − ϕ 0 ϕ a ra 2 φ1 T˜  1 − ϕ0 ϕa (ϕ02 + ϕa2 ) − (r1 − ra )D3 2T˜ 2  1 + D 1 ϕ0 × ln 1 + D 1 ϕ 0 ϕa −

Di s are calculated by

(22-a)

z θ1 z z + q1 ln + r1 φ22 − ∗2 r1 φ1 φ22 (2φ2 −φ1 ) ∗ 2 φ1 2T 4T z − r1 φ1 φ22 [φ12 (4φ2 − φ1 ) + φ22 (2φ2 − 3φ1 )] 12T ∗3   1 C3 −(r1 − 1 + λ1 ) ln + 1− r1 C4 C4  r1 − 1 + λ 1 r2 − 1 + λ 2 φ2 − r1 r2   1 r1 − 1 + λ 1 1 r2 − 1 + λ 2 + (19) − C3 r1 C2 r2 

Fig. 5. Experimental (symbols) and calculated (lines) saturated density of ethanol (squares), 1-propanol (triangles), 1-butanol (diamonds) and pentanol (circles).

(20)

where coefficients Ci are calculated by C1 = exp(1/T0∗ ) − 1, C3 = 1 + C1 φ2 ,

C 2 = 1 + C 1 φ1 C4 = 1 + C1 φ1 φ2

(21-a) (21-b)

Fig. 6. Segment number r as linear functions of molar mass, for n-alkanes (squares), 1-alkenes (triangles), cycloalkanes (diamonds), n-alkylbenzenes (circles), 2-alkanones (snowflakes) and n-alkanols (crosses).

X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

Fig. 7. Molecular interaction energy parameter (ε0 /k) as smooth but nonlinear function of molar mass, for n-alkanes (squares), 1-alkenes (triangles), cycloalkanes (diamonds), n-alkylbenzenes (circles), 2-alkanones (snowflakes) and n-alkanols (crosses).

librium of the fluids and their mixtures. In this model, each pure fluid is fully characterized by four molecular parameters: the coordination number (z), the segmental volume (v∗ ), the segment number (r) and the interaction energy (ε). For lattice fluid theories, following Refs. [14,25], we set z = 10. The hard-core volume per segment, v∗ , is considered to be the same for all fluids and equals 9.75 cm3 /mol as suggested by Panayiotou and Vera [25]. We further consider that the interaction energy ε is a function of temperature given by ε = ε0 + (T − 273.15)ε1

(24)

Therefore, the present model contains three temperatureindependent parameters, r, ε0 , ε1 , which should be obtained by simultaneously correlating experimental vapor pressure and saturated liquid density [26,27] of real fluids. Fig. 2 shows the comparison between the calculated and experimental saturated vapor pressures for propane, n-butane, n-pentane, n-hexane, and n-heptane, respectively. Fig. 3 compares the saturated density for propane, n-butane, n-pentane, n-hexane, and n-heptane with corresponding calculated results by this model, respectively. The top part of each curve represents the saturated liquid density

117

Fig. 8. Molecular interaction energy parameter (ε1 /k) as smooth but nonlinear function of molar mass, for n-alkanes (squares), 1-alkenes (triangles), cycloalkanes (diamonds), n-alkylbenzenes (circles), 2-alkanones (snowflakes) and n-alkanols (crosses).

of the substance, and the bottom part represents the saturated vapor density. As shown in Fig. 3, the present lattice fluid EOS describes the saturated density of these substances successfully over a large range of temperatures, but fails in the critical region. Generally, the present EOS overestimates the critical temperature, but underestimates the critical density. The main reason for this is that essentially, the present lattice model is still based on mean-field theory. For several representative substances, Table 1 compares the average absolute deviations for vapor pressure and liquid density calculated by present EOS and other EOSs, where the SWCFEOS [28], the SAFT-EOS [29], and the PHSC-EOS [30] are all derived from models based on free volume theory, and QLF [14] is a quasi-chemical nonrandom lattice fluid model. As observed, the present EOS is as accurate as these EOSs in calculating the saturated vapor pressure and liquid density of pure substances, and moreover, the present EOS generally shows slightly better results than the SWCF-EOS, the SAFT-EOS, and PHSC-EOS. Table 2 lists temperature range, r, ε0 /k, ε1 /k and the average absolute deviation (%) of vapor pressure and saturated density for the selected systems. The total average absolute deviation

Table 3 Parameters of pure fluids containing hydrogen-bond and the corrected results for vapor pressures and the saturated densities Components

T (K)

ε0 /k (K)

ε1 /k

ε2 /k × 104 (K−1 )

r

p (%)

ρ (%)

Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol 2-Propanol 2-Butanol 2-Methyl-1-propanol 2-Methyl-2-propanol 2-Pentanol 3-Pentanol Overall average Δ

257–482 271–495 297–517 295–531 278–508 310–428 343–445 296–549 279–494 281–488 300–526 304–443 287–381 297–382

194.1525 166.5842 155.4809 149.2762 143.7471 139.9623 137.7364 135.8915 146.8115 141.1195 143.9267 132.3171 136.6987 135.5490

−0.219421 −0.178635 −0.150168 −0.130815 −0.098442 −0.085860 −0.078452 −0.067686 −0.155920 −0.136195 −0.125524 −0.133798 −0.109046 −0.108581

1.9970 2.0899 2.3010 2.4812 2.0375 2.0207 2.1482 2.0229 2.6289 3.0839 2.4635 3.3096 2.6724 3.1342

4.351 6.043 7.571 9.094 10.625 12.230 13.740 15.216 7.667 9.057 9.135 9.127 10.583 10.427

0.32 0.19 0.41 0.49 0.57 0.32 0.71 0.71 0.39 0.32 0.45 0.12 0.06 0.15 0.37

4.51 2.42 1.23 0.41 0.42 0.21 0.40 1.25 1.80 0.70 0.80 0.17 0.28 0.35 1.07

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X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

Table 4 Parameter r as function of molecular mass

Table 6 Parameter ε1 /k as function of molecular mass

Systems

Mm range

A(1)

A(2)

Systems

Mm range

A(1)

A(2)

A(3)

n-Alkanes 1-Alkenes Cycloalkanes n-Alkylbenzene 2-Alkanone n-Alkanols

16–128 28–112 42–112 78–120 58–170 32–130

2.0584 1.7747 1.7689 0.0209 1.0386 0.8322

0.1113 0.1125 0.0982 0.1107 0.1090 0.1113

n-Alkanes 1-Alkenes Cycloalkanes n-Alkylbenzene 2-Alkanone n-Alkanols

16–128 28–112 42–112 78–120 58–170 32–130

0.0616 0.0452 0.0564 0.0450 0.0410 0.0530

1.8354 3.5582 3.2576 3.8931 5.2986 6.2516

0.017896 0.034705 0.026261 0.027323 0.026402 0.008118

is 1.14% for the saturated vapor pressures and 0.93% for the saturated liquid densities. From Table 2, one can observe that the results using the equation of state with three-parameter are in excellent agreement with the experimental data. For the substances with specific interactions such as hydrogen-bonding between molecules, the contribution due to such specific force should be included in the model. The doublelattice model (DLM) [7] and the lattice fluid hydrogen-bonding model (LFHB) [31,32] may be the best choice to account for the orientation effect of hydrogen-bonding. In this work, we extend simply the present model to hydrogen-bonding systems, by writing the interaction energy ε as follows: ε = ε0 + (T − 273.15)ε1 + (T − 273.15)2 ε2

(25)

In this case, the EOS contains four parameters. Figs. 4 and 5 compare the experimental [26] and calculated vapor pressures and saturated density based on this model for ethanol, 1propanol, 1-butanol and 1-pentanol, respectively. As shown, the calculated results are also in satisfactory agreement with experimental data except in the critical region. The average deviations for vapor pressure are 0.19% for ethanol, 0.41% for 1-propanol, 0.49% for 1-butanol and 0.57% for 1-pentanol, respectively. The corresponding deviations are 0.86%, 0.16%, 0.23% and 0.32% for the SAFT-EOS [29]. When the SWCF-EOS [28] is used, the deviations are 0.48%, 0.38%, 0.57% and 0.40%, respectively. It shows that when a three-parameter equation is applied simply to express interaction energy, the present lattice fluid EOS can describe appropriately systems containing hydrogen-bonding as well. Of course, our future work will also focus on developing the hydrogen-bonding model with strict physical meaning. The estimated parameters, temperature range and average absolute deviation of vapor and density for the self-associating fluids are listed in Table 3. It can be found that the value of ε for hydrogenbonding systems is greater than that of other systems, indicating that the specific interaction between molecules is included in parameter ε.

Fig. 9. Experimental (symbols) and calculated (lines) vapor pressures and vapor compositions of the mixture of methane (1) + tetrafluoromethane (2) at 159.61 K (triangles), 161.58 K (circles), 169.38 K (squares), 173.90 K (snowflakes) and 178.93 K (crosses).

Now we focus on the parameter behavior according to molar mass. This is very important because the key of future challenge lies in estimating the equation-of-state parameters for polydispersed, poorly defined pseudocomponents of real fluid mixtures, rather than in fine-tuning precise predictions (however important) for well-defined pure components [29]. The segment number r is plotted versus molar mass for n-alkanes, 1-alkenes, n-cycloalkanes, n-alkylbenzenes, 2-alkanone and nalkanol in Fig. 6. It is observed that, the segment number r practically increases linearly with increasing molar mass within

Table 5 Parameter ε0 /k as function of molecular mass Systems

Mm range

A(1)

A(2)

n-Alkanes 1-Alkenes Cycloalkanes n-Alkylbenzene 2-Alkanone n-Alkanols

16–128 28–112 42–112 78–120 58–170 32–130

105.1187 105.0418 118.5250 120.4447 118.2983 132.3826

0.0449 0.0461 0.0466 0.0427 0.0432 0.0276

Fig. 10. Experimental (symbols) and calculated (lines) vapor pressures and vapor compositions of the mixture of acetone (1) + hexane (2) at 308.15 K (triangles), 318.15 K (circles) and 328.15 K (squares).

X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

119

Table 7 Binary interaction parameters of mixtures and the corrected results Systems

T (K)

κij

p (%)

y (%)

Methane–ethane Methane–propane Methane–tetrafluoromethane 1,3-Butadiene–butane 1,3-Butadiene–benzene 1,3-Butadiene–dichloromethane 1,3-Butadiene–tetrachloromethane Cyclopentane–tetrachloromethane Cyclopentane–trichloromethane Pentane–benzene Pentane–hexane Benzene–cyclohexane Benzene–2-butanone Acetone–benzene Acetone–cyclohexane Acetone–hexane Acetone–toluene Acetone–heptane 2-Butanone–toluene 2-Butanone–heptane 2-Butanone–ethylbenzene 2-Butanone–octane Tetrachloromethane–cyclohexane Tetrachloromethane–2-butanone Hexane–2-butanone 2-Pentanone–toluene Heptane–2-pentanone

158.15–172.04 172.04 159.61–178.93 324.82–338.82 298.15 298.15 298.15 298.15 298.15 308.15–313.15 298.15 333.15 313.15–333.15 298.15–323.15 298.15–328.15 308.15–328.15 308.15–328.15 323.15–338.15 328.15–348.15 323.15 338.15–348.15 338.15 333.15 323.15 333.15–338.15 323.15 363.15

−0.004724 0.0033 0.079536 0.006864 0.002749 −0.002503 0.005036 0.000490 0.018191 0.017189 −0.001418 0.018361 0.007241 0.019882 0.073197 0.069078 0.024351 0.069536 0.011101 0.049132 0.016166 0.053108 0.005632 0.026045 0.044854 0.003791 0.042719

2.00 1.88 1.89 0.19 0.37 0.36 0.57 0.65 0.22 0.86 0.37 0.63 1.26 1.25 3.56 2.90 1.58 4.95 1.06 3.52 0.96 2.33 0.70 0.77 2.95 0.64 1.89

0.43 0.17 1.42 0.46 0.83 1.01 0.07 1.33 1.67 0.34 0.33 1.18 1.52 1.62 3.08 4.32 1.23 3.43 1.00 5.23 0.90 2.00 1.21 4.30 3.65 3.83 4.53

1.49

1.89

Overall average Δ



yexp −ycal

y (%) = 100

yexp . Nm

each homologous series. As shown in Figs. 7 and 8, a similar molar mass correlation can be developed for the molecular interaction energy parameter ε0 /k, which indicates the interaction energy in 273.15 K, and for ε1 /k, which describes the temperature influence on the interaction energy. Unlike r, ε0 /k and ε1 /k are nonlinear with respect to the molar mass. However, for selfassociating fluids, the parameter ε2 /k, which is difficult to be correlated by a simple function, usually has a small and similar value for a homologous series. For example, the ε2 /k values of n-alkanols almost all equal to 2 × 10−4 K−1 . In this work, the parameters r, ε0 /k and ε1 /k have been regressed as simple functions of the molar mass (Mm ) for many homologous series. For example, r = A(1) + A(2) Mm for all series ⎧ (1) A (1 − exp(−A(2) Mm )) ⎪ ⎪ ⎪ ⎨ for n-alkanes, 1-alkenes, n-cycloalkanes ε0 = ⎪ k A(1) (1 + exp(−A(2) Mm )) ⎪ ⎪ ⎩ for n-alkylbenzenes, 2-alkanone, n-alkanol ε1 = A(1) (1 + A(2) exp(−A(3) Mm )) k

for all series

(26)

(27)

the equation-of-state parameters for non-associating pure fluids when no accurate data are available. Observed the regression coefficients A(1) and A(2) of function 25 in Table 4, it is found that although the values of A(1) vary for each series, the values of A(2) almost equal to 0.11. In essence, the value of A(1) is relative to the functional groups of heterogeneous series, while the value of A(2) is dependent of the contribution of adding the –CH2 – group to r. Back to the original purposes of developing this model, we have expected that values of A(1) are different but values of A(2) are the same for different hydrocarbon series. That idea is consistent to the results of regressing model parameters, which were obtained by correlating experimental data. 3.2. Application to mixtures Let us now apply the model to mixtures. We randomly select some representative binary mixtures for vapor–liquid equilibrium calculations. The equilibrium condition can be written as follows (α)

(β)

μi (p, T, xi ) = μi (p, T, yi ) (28)

Values for the regression coefficients A(i) are reported in Tables 4–6. Now we have a useful method for estimating all

(29)

Combining with Eqs. (19) and (20), the vapor–liquid equilibrium of a binary mixture can be calculated. The pure substance parameters could be obtained by the method introduced in the former section. So, when applied to binary vapor–liquid equi-

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X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

Table 8 Comparison between NRHB model, NLF modela and present model in correlation of vapor–liquid equilibria (VLE)b Systems

Ethane–methanol Ethane–ethanol Ethane–propanol CO2 –ethane CO2 –propane CO2 –butane CO2 –pentane CO2 –ethanol CO2 –propanol CO2 –butanol CO2 –pentanol Overall average Δ a b

T (K)

298.15 313.4–333.4 313.4–333.4 230.0–270.0 230.0–270.0 319.3–344.3 310.2–363.2 313.2–328.2 313.4–333.4 314.8–337.4 314.8–337.5

NRHB

NLF

Present model

κij

p (%)

y1 (%)

κij

p (%)

y1 (%)

κij

p (%)

y1 (%)

−0.046 −0.038 −0.053 0.097 0.088 0.078 0.035 −0.036 −0.044 −0.059 −0.079

3.5 8.7 5.6 1.2 3.7 3.8 3.0 2.5 4.1 7.6 9.2

1.5 1.9 2.7 2.8 3.0 9.6 8.9 1.0 0.9 0.6 0.5

0.012 0.012 0.003 0.108 0.120 0.069 0.094 0.040 0.039 0.042 0.052

2.8 6.3 3.3 2.0 6.0 2.0 5.3 4.9 4.9 7.0 6.9

1.3 1.7 0.3 3.3 3.1 5.1 4.2 1.1 0.9 0.7 0.6

-0.010 0.020 0.014 0.108 0.118 0.063 0.091 0.041 0.043 0.055 0.061

5.9 4.6 3.5 1.6 5.7 3.8 8.2 2.0 4.5 4.5 5.0

0.6 1.8 0.2 2.3 2.7 4.9 3.7 0.9 0.6 0.6 0.5

4.8

3.0

4.7

2.0

4.5

1.7

The value of κij and Δ of NRHB model and NLF model are from Ref. [16]. High-pressure VLE data are from Refs. [36–44].

librium, the present model has only one adjustable parameter, the binary interaction parameter κij . The values of κij have been fitted to experimental vapor–liquid data of the mixtures and are summarized in Table 7. The average absolute deviation (Δ%) of pressure and vapor compositions is also included in the same table. The vapor–liquid equilibria data are taken from handbooks [33,34] and Refs. [35–44]. As shown in Table 7, the present model can be used to correlate satisfactorily the experimental data with only one interaction parameter. As a representative result, Fig. 9 compares the experimental [35] and calculated vapor–liquid equilibria (VLE) for methane (1) + tetrafluoromethane (2) at five temperatures. In this example, only one value for the interaction parameter κij was used at five temperatures, but the agreement is rather good. An analogous comparison is made in Fig. 10, where the experimental [34] and the calculated VLE for the acetone (1) + hexane (2) at 308.15, 318.15 and 328.15 are compared. The agreement is again rather satisfactory. As shown in Table 8, the present model is compared with the nonrandom hydrogen-bonding model (NRHB) [15], which accounts for nonrandomness through the quasi-chemical approximation, and a nonrandom lattice fluid model (NLF), proposed recently by Tsivinzelis et al. [16]. It is found the present model is at least as good as the NRHB model and the NLF model. 4. Conclusion A new lattice fluid model is proposed based on our previous work. Corresponding equation of state and chemical potentials are derived from the Helmholtz function of mixing. It was found that the present EOS can describe satisfactorily the vapor pressure and the saturated density of pure normal fluids and can be extended simply to hydrogen-bonding systems. The parameters in the equation of state can be expressed as simple functions of the molar mass for many homologous series. We also have tested the present model on vapor–liquid equilibrium data sets for mixtures. It was found that the present model can describe quantitatively the vapor–liquid equilibria

of binary mixtures with only one binary interaction parameter. List of symbols A Helmholtz energy (J) Δmix A Helmholtz energy of mixing (J) defined by Eqs. (26)–(28) A(i) C1 , C2 , C3 , C4 defined by Eq. (21) D1 , D2 , D3 defined by Eq. (22) k Boltzmann constant ≈1.38 × 10−23 J/K M molecular weight N number of molecule Nl number of total sites of lattice fluid model number of total sites of close-packed lattice model Nr p pressure (MPa) p* characteristic pressure (MPa) q surface area parameter r number of segments per molecule ra average segment number T temperature (K) T* reduced temperature for the first step (K) reduced temperature for the second step (K) T0∗ v∗ hard-core volume of one segment V volume (cm3 ) z coordination number Greek letters ε interaction energy (J) εI exchange energy of the first mixing step (J) εII exchange energy of the second mixing step (J) interaction energy of i–j pair (J) εij φ volume fraction based on close-packed lattice model ϕ volume fraction based on lattice fluid model κ adjustable parameter λ parameter charactering the long-range correlations μ chemical potential θ surface fraction

X. Xu et al. / Fluid Phase Equilibria 265 (2008) 112–121

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