Phase-equilibrium calculations for n-alkane + alkanol systems using continuous thermodynamics

Fluid Phase Equilibria 217 (2004) 125–135

Phase-equilibrium calculations for n-alkane + alkanol systems using continuous thermodynamics D. Browarzik∗ Institute of Physical Chemistry, Martin-Luther-University, Halle-Wittenberg, 06217 Merseburg, Germany Received 27 July 2002; accepted 10 October 2002

Abstract A new association model based on continuous thermodynamics is introduced and applied to six systems of the type n-alkane (n-hexane, n-heptane, n-octane) + alkanol (methanol, ethanol). The alkanol is considered to be a mixture of chain associates with the composition described by a continuous distribution function. This distribution function is derived as an analytical expression from the mass action law applied to the association equilibrium. To consider the entropic contribution originating from the size differences of the molecules (associates) activity coefficients based on Flory–Huggins model are included in the mass action law. Unlike the molecular-mass distribution of a polymer the chain-length distribution of the associates depends on the temperature and on the mole fraction of the alkanol. The treatment of vapor–liquid equilibrium and liquid–liquid equilibrium is similar to that of an oil system or of a polymer solution using continuous thermodynamics. Different to other chemical models of association there is no additive split into a physical and a chemical contribution. The equilibrium constants of association were fitted to vapor-pressure data of methanol and ethanol. The model needs only one interaction parameter being independent of temperature and taking the same value for all systems studied. Considering the simplicity of the model, both the liquid–liquid equilibrium of the three methanol systems and the vapor–liquid equilibrium of all six systems are predicted with reasonable accuracy. © 2003 Elsevier B.V. All rights reserved. Keywords: Method of calculation; Liquid–liquid equilibrium; Vapor–liquid equilibrium; Continuous thermodynamics; Gibbs excess energy model; Associating systems

1. Introduction Alkanols form chain associates. Association strongly influences phase equilibria. So, many of the associates possess relatively high molar masses and, therefore, hardly contribute to the vapor pressure. For this reason alkanols possess much higher normal boiling-point temperatures than their homomorphs (non-associating substances with similar structure and molecular mass). Mixing alkanols and alkanes associates are degraded. Therefore, such binary mixtures show often maximum azeotropy in vapor pressure. Fitting Redlich–Kister constants to the liquid–liquid equilibrium of alkane–methanol systems the calculated vapor pressures are much too low. So, it is generally agreed that, in modeling Abbreviations: LLE, liquid–liquid equilibrium; VLE, vapor–liquid equilibrium ∗ Tel.: +49-3461-2133; fax: +49-3461-2129. E-mail address: [email protected] (D. Browarzik). 0378-3812/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2002.10.001

of phase equilibria of alkane + alkanol systems, H-bonding has to be specifically considered. In the past decades much progress has been achieved in understanding association as well as in describing excess properties and phase equilibria of associating systems. Two basic ways are known from literature. In the physical theory H-bonds are treated like strong physical interactions [1]. In the chemical theory mass action law is applied to the association equilibrium. Most of papers dealing with the chemical approach are based either on an Gibbs excess energy model [2–7] or on an equation of state (EoS) [8–15] (for overview see [8]). Recently equations of state from perturbation theory based on Wertheim’s theory are often applied (e.g. statistical association fluid theory (SAFT) [16–18], perturbed chain-statistical association fluid theory (PC-SAFT) [19]). Generally, an EoS should be favored in the high pressure region. However, often phase equilibria are to be calculated at low pressure. In this case Gibbs excess energy models can be applied with a similar accuracy as an EoS. Furthermore the treatment is much

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simpler than using an EoS. For this reason, here, we apply the chemical approach based on a Gibbs excess energy model. The simultaneous description of the liquid–liquid equilibrium (LLE) and the vapor–liquid equilibrium (VLE) is a long-standing problem, particularly, for associating systems. For example, the otherwise successful ERAS model [7] predicts the vapor pressure of the system n-hexane + methanol much too low, if the parameters were fitted to the LLE of the system. Deiters [15] combined the chemical approach with an EoS and could describe the VLE of this system reasonably well, but, the predicted immiscibility temperatures are much too high. According to so-called continuous association models [4,5,9,14], there are chain associates Ai with i = 1, 2, 3, . . . , ∞. Then a n-alkane + alkanol system is a polydisperse system similar to a solution of a polydisperse polymer. Such systems may be described very successfully by continuous thermodynamics [20,21]. So continuous thermodynamics should be convenient to describe associated systems too. Here, the concentrations of the associates are described by a continuous distribution function and, the concentration of the alkane is assigned a discrete value. So, the treatment is similar to that of a polymer solution or to that of an oil-solvent system. The most important difference is that the distribution function of a polymer or of a crude oil does not change, whereas, the association equilibrium and, therefore, the distribution function of the associated species depends on the temperature and on the mole fraction of the alkanol. The phase equilibria are influenced by interaction parameters (physical influence) and by the distribution function that has to be derived from the association equilibrium (chemical influence). However, there is no additive split into a physical and a chemical contribution. Both contributions are connected in a more sophisticated way. Because, there are chain associates the size differences of the associates result in a considerable entropic effect well known from polymer solutions (Flory–Huggins model). Especially because of this reason activity coefficients should be involved in the mass action law of the association equilibrium. Of course, this idea is not a new one. Already many years ago Brandani [4] as well as Kerber and Kehlen [5] considered activity coefficients including the Flory–Huggins contribution in the mass action law. However, phase equilibria were not predicted. This paper connects the ideas of these authors with continuous thermodynamics predicting LLE and VLE of alkanol–alkane systems.

2. Association equilibrium A liquid mixture of a n-alkane (A) and a alkanol B is considered. The alkanol forms chain associates governed by the following association equilibrium: Br−r + Br ⇔ Br

(1)

The subscript r is the association degree indicating how many monomers form the associate. Applying the mass action law to this association equilibrium continuous thermodynamics [20,21] results in KL =

xL W L (r)γB (r) xL W L (r − r  )γB (r − r  )xL W L (r  )γB (r  )

(2)

where KL is the equilibrium constant of association in liquid phase (L) depending on temperature. KL is assumed to be independent of the association degree r. The quantity xL is the total mole fraction of the associates and WL (r) is the corresponding continuous distribution function. xL W L (r) dr gives the mole fraction of all associates with values of r within the interval [r, +dr). The normalization condition of WL (r) reads
∞ W L (r) dr = 1 (3) 1

The quantity γ B (r) is the activity coefficient of the associate with the association degree r. The number average r¯B of the association degree is given by
∞ rWL (r) dr (4) r¯BL = 1

Knowing the number average of the association degree one can calculate the total mole fraction xL of the associates from L of the component B of the binary the true mole fraction xB mixture A + B. Then the quantity xL may be expressed by xL =

L xB L + (1 − xL )¯r L xB B B

(5)

Choosing the alkanol monomer as standard segment the associations degree r is equivalent to the segment number usually used in polymer thermodynamics. Neglecting the difference between O-atoms and C-atoms and assuming the segment number proportional to the number of C-atoms, the segment number rA of the alkane molecules is fixed too. For example, in the system n-hexane + methanol that means rA = 2. Then, the number average of the segment number of the mixture reads r¯ M = (1 − xL )rA + xL r¯BL

(6)

Evoking Flory–Huggins theory of polymer solutions the activity coefficients may be expressed by  r  r ln γB (r) = ln M + 1 − M + r ln γs,B (7) r¯ r¯ where γ s,B is the so-called segment-molar activity coefficient describing the intermolecular interactions. This quantity is assumed to be independent of the segment number r. Setting Eq. (7) into Eq. (2) one can find after some rearrangement KL =

rWL (r) r¯ M (r − r  )W L (r − r  )r  W L (r  ) xL e

(8)

D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

Introducing the segment-molar distribution function Ws (r) (practically, the mass distribution function) and the segment-mole fraction ψ (practically, the mass fraction) by Ws (r) =

rWL (r) , r¯BL

ψ=

xL r¯BL r¯ M

Eq. (8) may be rewritten as Ws (r) 1 KL =   Ws (r − r )Ws (r ) ψe

(9)

3. Calculation of the vapor–liquid equilibrium

Ws (r) = −λ exp[λ(r − 1)], r¯ L W L (r) = −λ B exp[λ(r − 1)] (11) r where the quantity λ is obtained solving the following equation in an iterative way: λ<0

(12a)

where with the aid of Eqs. (5), (6) and (9) the segment-mole L by fraction ψ may be related to the true mole fraction xB ψ=

L xB L + r (1 − xL ) xB A B

C = 0.57721567 is Euler’s constant. Usually, λ takes values between −0.01 and −0.1. According to that in Eq. (15) the series expansion can be truncated after few terms (n < 10). Based on Eqs. (11), (12) and (15) the distribution of associates is completely known.

(10)

The solution of this functional equation (for details see Appendix A) reads:

exp(λ) + λKL eψ = 0,

127

(12b)

Knowing the association constant KL , Eq. (12) permits the calculation of the quantity λ if the true mole fraction L of the associated component is given. The distribution xB functions of Eq. (11) continuously decreases with increasing r to give the value 0 for r → ∞, so that there is no maximum. Therefore, the occurrence of a higher associate is less probable than that of a lower one. For large values of the association constant KL with Eq. (12) λ takes very small negative values and, thus, according to Eq. (11) the associates are widely distributed. Otherwise, if KL is small λ takes strong negative values and, practically, there are only some lower associates. To calculate r¯BL the best way is to apply the relation
∞ 1 1 = Ws (r) dr (13) r¯BL 1 r Setting Eq. (11) into Eq. (13) and substituting λr = t it is easy to find
λ exp(t) 1 = λ exp(−λ) dt (14) L t r¯B −∞ The integral of Eq. (14) may be expressed by a series expansion given in usual mathematical handbooks. In this way, the reciprocal number average of the association degree reads 1 = λ exp(−λ) r¯BL   λ λ2 λ3 λn × C+ln|λ|+ + + +···+ 1 · 1! 2 · 2! 3 · 3! n · n! (15)

In the framework of continuous thermodynamics [20] the conditions for vapor–liquid equilibrium read p(1 − xV )ϕ˜ A = p∗A (1 − xL )γA

(16a)

pxV W V (r)ϕ˜ B = p∗ (r)xL W L (r)γB (r)

(16b)

Here, p is the vapor pressure of the mixture. p∗A is the vapor pressure and, γ A is the activity coefficient of component A, where r  rA A (17) ln γA = ln M + 1 − M + rA ln γs,A r¯ r¯ In this, γ s,A is the segment-molar activity coefficient of component A. p∗(r) is the fictive vapor pressure of the assoociates with the degree r (segment number). Furthermore, WV (r) is the molar distribution function and, xV is the total mole fraction of the associates in the vapor phase. The quantities ϕ˜ A , ϕ˜ B describe the real gas effects. For simplicity, we do not consider ϕ˜ B to be a function of the association degree r. Additionally, we assume the vapor phase to be an ideal mixture of real gases. So, we can write   BAA (p − p∗A ) (18a) ϕ˜ A = exp RT   BBB (p − p∗B ) (18b) ϕ˜ B = exp RT where BAA , BBB are the second virial coefficients of the pure components A and B disregarding association. Here, p∗B is the vapor pressure of component B considered to be a mixture of associates. For further treatment one has to know the vapor pressure p∗(r) as function of r and T. The simplest way to describe the vapor pressure in its dependence on temperature is given by the equation of Clausius and Clapeyron, here, expressed by   B(r) p∗ (r) = A(r) − (19) ln ; p+ = 101.325 kPa + p T Assuming A(r) and B(r) to be linear functions we obtain   B 0 + B1 r p∗ (r) = A0 + A1 r − (20) ln p+ T In terms of these parameters the normal boiling-point temperature reads (B0 + B1 r)/(A0 + A1 r) and, the vaporization enthalpy reads R(B0 + B1 r). Fitting the constants of Eq. (20) to the normal boiling-point temperatures of the n-alkanes (being homomorphs of the associates) [22] and to

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the vaporization enthalpy of n-hexane [23] (at the normal boiling-point temperature) the results are A0 = 7.12800,

B0 /K = 522.711,

A1 = 0.50401ncs ,

B1 /K = 491.189ncs

(21)

where λ∗ is the solution λ of Eq. (12a) for ψ = 1. Eq. (26) permits to fit the association constant KL (T) to experimental vapor pressure data of the associated component. The experimental vapor pressure data of methanol [18] and ethanol [18] result in

The quantity ncs denotes the number of carbon atoms the chosen standard segment is equivalent to (e.g. for methanol ncs = 2, for ethanol ncs = 3). Furthermore, the segment-molar activity coefficients are needed. The simplest way to calculate the segment-molar activity coefficients is to apply the Flory–Huggins theory of polymer solutions. Here, these quantities are given by

ln KL = −5.35852 +

2788.49 T

(methanol)

(27a)

ln KL = −7.05090 +

3058.98 T

(ethanol)

(27b)

ln γs,A = χψ ;

H = −25.43 kJ mol

2

ln γs,B = χ(1 − ψ)

2

(22)

The χ-parameter is proportional to 1/T and to an energy describing the intermolecular interactions between A and B molecules related to the standard segment chosen. In rough estimation this interaction energy is proportional to the surface (of the standard segment) that is free available for intermolecular interaction. So, we assume χ=

˜w βA T

(23)

˜ w is the dimensionless van der Waals surface of where A the standard segment that may be calculated by the increment tables of Bondi [24]. β is the only parameter that has to be fitted to mixing data. This parameter should approximately take the same value for all n-alkane + alkanol systems. Now, the partial pressures pA , pB of the components A and B may be calculated. Applying Eqs. (16a), (17) and (18a) the partial pressure of the component A reads   p∗A (1 − ψ) rA pA = exp 1 − M + rA χψ2 (24a) ϕ˜ A r¯ Integrating Eq. (16b) with the aid of Eqs. (7), (18b) and (19)–(22) the following expression for the partial pressure of the associated component B is derived: pB =

p ∗ (1)ψλ exp[1 − (1/¯r M ) + χ(1 − ψ)2 ] (24b) ϕ˜ B [A1 − (B1 /T) + λ − (1/¯rM ) + χ(1 − ψ)2 ]

λ is obtained solving Eq. (12a) with the aid of Eq. (12b). The total vapor pressure of the mixture and the total mole fraction xV of the associates in the vapor phase are given by pB p = pA + p B ; (25) xV = p Setting ψ = 1; r¯ M = r¯BL ; ϕ˜ B = 1 the vapor pressure of the pure associated component B may be derived from Eq. (24b) as p∗B =

p ∗ λ∗ exp[1 − (1/¯rBL )] A1 − (B1 /T) + λ∗ − (1/¯rBL )

(26)

From these relations the standard enthalpy HL of the association reaction (1) may be easily derived. H L = −23.18 kJ mol−1 L

−1

(methanol)

(28a)

(ethanol)

(28b)

Practically, HL measures the strength of the H-bonds within the associates. The association enthalpies for methanol and ethanol are similar to the values given by Brandani [4] as well as by Kerber and Kehlen [5]. However, the association constants given by these authors are one order of magnitude bigger than calculated by Eq. (27). The reason is that Brandani introduced a modified association constant including the factor 2(z − 1). Here, z is the coordination number of a lattice originating from Staverman’s equation for the activity coefficients of an athermal mixture. The mentioned authors assumed z = 10. Knowing the association constants with the aid of Eqs. (11), (12), (15) and (27) the distribution functions Ws (r), WL (r) and the number average r¯BL of the association degree may be calculated. Especially, the values of r¯BL are interesting. These values strongly depend on temperature. So, r¯BL for ethanol at the normal boiling-point temperature accounts 6.5, but, at 25 ◦ C the value is 18.4. Asprion et al. [25] applied FT-IR and NMR spectroscopy to alcohol + hydrocarbon systems and evaluated the data with a UNIQUAC 1–2–n association model. The value n = 5 proved to be convenient in most cases. At least for higher temperatures the values of r¯BL are similar to that. Now, at given temperature and composition of the liquid phase the vapor pressure of the mixture can be calculated in an iterative procedure. In a first step the real gas corrections are neglected (ϕ˜ A = 1, ϕ˜ B = 1). Then the vapor pressure may be immediately calculated with the aid of Eqs. (24) and (25). After that, using Eqs. (18) and (26) real gas corrections are estimated. Introducing the values for ϕ˜ A , ϕ˜ B in Eq. (24) an improved vapor pressure value is obtained. Hence, Eq. (18) provides more accurate values for ϕ˜ A , ϕ˜ B . To obtain the final result this procedure has to be repeated only a few times. Additionally to the vapor pressure the vapor composition has to be determined. From Eq. (25) only the total mole fraction xV of all associates is known. However, we V of the associated need to know the true mole fraction xB component in the vapor phase. This quantity is related to

D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

xV by V = xB

xV r¯BV V xV r¯BV + 1 − xB

(29)

where r¯BV is the number average of the association degree (segment number) of the associates in the vapor phase. This quantity may be calculated by Eq. (4) replacing the superscript L by V. However, to do this the distribution function WV (r) of the associates in the vapor phase has to be known. According to Eq. (16b) we obtain W V (r) =

p∗ (r)xL W L (r)γB (r) pB ϕ˜ B

(30)

With the aid of Eqs. (7), (11), (20) and (24b) the distribution function of the vapor phase and the corresponding number average of the association degree are derived to be  1 r−1 V W (r) = V exp − V (31a) r¯B − 1 r¯B − 1 1 r¯BV = 1 − A1 − (B1 /T)+λ − (1/¯r M ) + χ(1 − ψ)2

(31b)

Hence, with Eqs. (25), (29) and (31b) the vapor composition is known. Finally, the model permits to connect the association constants of the vapor phase and of the liquid phase. Generally, the association constant of the vapor phase may be written as KpV

pxV W V (r)ϕ(r)/p+ = (pxV W V (r−r  )ϕ(r−r  )/p+ )(pxV W V (r  )ϕ(r  )/p+ ) (32)

Here, ϕ(r) is the fugacity coefficient of the associates consisting of r segments. Again considering the vapor phase as an ideal mixture of real gases and, assuming the fugacity coefficient to possess the same value for all associates independent of the chain length KpV may be simplified to

 BBB p W V (r) p+ exp − KpV = V V (33) px W (r − r  )W V (r  ) RT Applying Eqs. (2), (16b) and (18b) the association constant of the vapor phase reads

 BBB p∗B p+ p∗ (r) KpV = KL exp − (34) p∗ (r − r  )p(r  ) RT Introducing Eq. (20) the final relation is

 BBB p∗B B0 V L Kp = K exp −A0 + − T RT

(35)

Of course, the term related to the second virial coefficients is very small and, therefore, does not matter. Considering the parameters of Eq. (20) one can see from Eq. (35) that, indeed, the association constant of the vapor phase is essentially smaller than that of the liquid phase. Furthermore,

129

we assumed the association constant of the liquid phase to be independent of the size of the associates. In this case, Eq. (35) shows that also the corresponding association constant of the vapor phase does not depend on the chain length of the associates.

4. Calculation of liquid–liquid equilibrium The liquid–liquid equilibrium (LLE) of a system of the type non-associated component A + associated component B in the framework of continuous thermodynamics may be treated similar as the LLE of a polymer solution. The associated component, here, acts like a polymer. Continuous thermodynamics results in (for the fundamentals see [21]) the following equilibrium conditions between the phases I and II: (1 − ψII ) = (1 − ψI ) exp(rA ρA )

(36a)

ψII WsII (r) = ψI WsI (r) exp(rρB )

(36b)

Assuming the segment-molar activity coefficients to obey Eq. (22) the auxiliary quantities ρA , ρB are given by  1 I ψII ψI II (ψ − ψ ) + − I + χ[(ψI )2 − (ψII )2 ] ρA = rA r¯BII r¯B (37a)

temperature / K 320

310

300

290

280

270

260 0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 1. Liquid–liquid equilibrium of the system n-hexane + methanol; calculated: (—), experimental: (䉲) Goral et al. [26], ( ) Abbas et al. [27], (+) Savini et al. [28], (䊏) Lide and Kehiaian [29], (×) Kato et al. [30], (䉬) Radice and Knickle [31], (䉱) Hradetzky and Lempe [32].

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D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

1 ρB = (ψI − ψII ) + rA



ψII ψI − r¯BII r¯BI



+ χ[(1 − ψI )2 − (1 − ψII )2 ]

(37b)

Provided that the distribution function of the phase I is governed by Eq. (11) by integration of Eq. (36a) the distribution function of the phase II is obtained to be of the same type. Furthermore, in this way ψII = ψI

λI exp(ρB ) λII

(38)

λII = λI + ρB

(39)

may be found. The quantities λI , λII obey Eq. (12a). Of course, the association constant, given by Eq. (27), is the same for both liquid phases. For numerical solution it is convenient to form the difference of Eqs. (37a) and (37b) expressing ρA by Eq. (36a) and ρB by Eq. (39). The resulting equation reads

 1 1 − ψII λII − λI − ln − 2χ(ψII − ψI ) = 0 (40) rA 1 − ψI

Furthermore, combining Eqs. (36b) and (37a) the following condition is obtained: 

 1 1 − ψII 1 II ψII ψI I ln + (ψ − ψ ) − − I rA 1 − ψI rA r¯BII r¯B + χ[(ψII )2 − (ψI )2 ] = 0

(41)

Considering the segment-mole fraction of phase I to be given the numerical solutions of Eqs. (40) and (41) yield the unknowns ψII and T. The quantities λ and r¯B for both phases have to be calculated with the aid of Eqs. (12) and (15).

5. Results To prove the accuracy of the model LLE and VLE of the systems n-hexane + methanol, n-heptane + methanol, n-octane + methanol and VLE of the systems n-hexane + ethanol, n-heptane + ethanol, n-octane + ethanol are to be predicted. Generally, the reference segment is the monomer considered. For the methanol systems, the size of the reference segment is similar to that of an ethyl group. So, for example, in the system n-hexane + methanol rA = 3 was assumed. For the ethanol systems the size of the reference segment is similar to that of a propyl group. So, in the sys-

pressure / kPa 160 temperature / K 340

140

330

120

333.15 K 100

320

80

310

60

300

313.15 K 40

290

20

298.15 K

280

0

270

0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 2. Vapor pressure and vapor composition at 298.15, 313.15, 333.15 K for the system n-hexane + methanol; calculated: (—), experimental: (×) Goral et al. [26], (䊏) Hongo et al. [33], (䉬) Sörensen and Arlt [35].

0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 3. Liquid–liquid equilibrium of the system n-heptane + methanol; calculated: (—), experimental: (䉱) Marino et al. [36], (+) Sörensen and Arlt [37], ( ) Orge et al. [38].

D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

tem n-hexane + ethanol rA = 2 was assumed. Generally, for the methanol systems rA = n/2 and, for the ethanol systems rA = n/3 were chosen. Here, n indicates the number of carbon atoms within the n-alkane molecules considered. The model developed includes three parameters. The association constant KL of the liquid phase was fitted to VLE data of the pure alkanols giving Eq. (27). The mixing parameter β involved in Eq. (23) was adjusted to LLE data of the system n-hexane + methanol. All other calculations are real predictions. The comparison with corresponding experimental data serves to evaluate the model. Based on the dimensionless van der Waals surface Aw = 3.58 for methanol (see increment tables of Bondi [24]) the parameter fit for β yields β = 25.14 K

(42)

In Fig. 1 the calculated miscibility gap is compared with numerous experimental LLE data [26–32]. The agreement with the experimental data is reasonably well only for higher methanol concentrations the description is not quite perfect. To calculate the VLE for n-alkane + alkanol systems both the vapor pressures of the n-alkanes as well as the second virial coefficients of the n-alkanes and of the alkanols are

pressure / kPa

131

needed as functions of temperaure. The temperature dependence of the vapor pressures of the pure n-alkanes (n-hexane, n-heptane, n-octane), here, is obtained by Antoine’s equation with the parameters given by Hongo et al. [33]. The second virial coefficients of n-hexane, n-heptane, n-octane, methanol and ethanol were taken from the data collection of Smith and Srivastava [34]. The functions describing the temperature dependence of the second virial coefficients are given in Appendix B. In Fig. 2 the predicted VLE of the system n-hexane + methanol at three temperatures compared with experimental data [27,33,35] is presented. The calculated vapor pressures are somewhat lower than the experimental ones. In the vicinity of the azeotropic point the errors are between 4% and 5% at all three temperatures considered. Perhaps Eq. (20) should be improved. Particularly, for the lower associates the vapor pressure as function of temperature is important to know. The calculated azeotropic mole fraction of methanol is too large, especially, at 333.15 K. Probably, the reason is an overestimation of the association degree of the vapor phase. Eq. (31b) yields values between 1.4 and 1.5. Replacing these values by 1.1 or 1.2 the azeotropic concentration would be predicted correct. The system n-hexane+methanol was often studied by other authors using different association models. For comparison, here, three papers are to be discussed. Bender and Heintz [7] applied the ERAS model.

30

temperature / K

395

25

375

20 355

15 335

10 315

5 295

0

275

0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 4. Vapor pressure and vapor composition at 298.15 K for the system n-heptane + methanol; calculated: (—), experimental: (䊏) Hongo et al. [33].

0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 5. Vapor–liquid–liquid equilibrium at normal pressure for the system n-octane + methanol; calculated: (—), experimental: (䉱) Sörensen and Arlt [35], (+) Sörensen and Arlt [37], (䊏) Orge et al. [38].

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D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

Fitting the parameters to LLE data the VLE was predicted. The predicted vapor pressures are essentially too low, but the predicted azeotropic concentration agrees with the experimental one. Deiters [15] combined an EoS with the chemical association theory. Fitting the parameters to VLE data the LLE was predicted very unsatisfactory. Although above 309 K the system is completely miscible at 323.15 K phase decomposition is predicted. The SAFT model seems to be more successful. Kahl and Enders [18] applied SAFT to LLE and VLE of n-hexane + methanol and achieved satisfactory predictions. However, SAFT is much more complicated than this model. Besides, SAFT is based on many parameters, this model needs an only one. In Fig. 3 the prediction of LLE for the system n-heptane+ methanol is compared to experimental data [36–38]. The calculated demixing curve is too narrow, but, as a true prediction the result is satisfactory. Obviously, the parameter β fitted to the system n-hexane + methanol is also suitable to the system n-heptane + methanol. Fig. 4 shows the prediction of vapor pressure and vapor composition for the system n-heptane + methanol at 298.15 K compared with experimental data of Hongo et al. [33]. The deviations from the experimental data are similar as in Fig. 2.

Fig. 5 presents the phase diagram of the system n-octane+ methanol at normal pressure. The comparison between the calculated VLE and the experimental data [35] reveals somewhat too high equilibrium temperatures and, again a too large azeotropic mole fraction of methanol. The miscibility gap compared to the experimental LLE data [37,38] is too narrow. However, the predictions are approximately correct and, so the parameter β given by Eq. (42) is also suitable to the system n-octane + methanol. The next step is to prove whether the model is suitable for n-alkane + ethanol systems too. With the relative van der Waals surface Aw = 4.93 for ethanol [24] and Eq. (42) for the parameter β, VLE predictions were performed for the systems n-hexane+ethanol, n-heptane+ethanol, n-octane+ ethanol. Figs. 6–8 show the results at some temperatures. Surprisingly, the agreement between the calculated curves and the experimental data is better than for the methanol systems. The azeotropic mole fraction of ethanol is again too large but, the deviation from the experimental azeotropic point is smaller than for the methanol systems. Obviously, the parameter β obeys Eq. (42) too, if methanol is replaced by ethanol.

pressure / kPa pressure / kPa

100

90 80

80 70 60

343.15 K

60

323.15 K 50 40

40

30

323.15 K 20

20

298.15 K 10

303.27 K 0

0 0

0.2

0.4

0.6

0.8

1

mole fraction ethanol Fig. 6. Vapor pressure and vapor composition at 298.15 and 323.15 K for the system n-hexane + ethanol; calculated: (—), experimental: (䊏) Hongo et al. [33], (䉱) Sörensen and Arlt [35].

0

0.2

0.4

0.6

0.8

1

mole fraction ethanol Fig. 7. Vapor pressure and vapor composition at 303.27, 323.15 and 343.15 K for the system n-heptane + ethanol; calculated: (—), experimental: (䉱, 䉬) Busch et al. [39], (䊏) Berro et al. [40].

D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

on the chain length of the associates meaning, a function KL (r, r ) would have to be considered. The generalization of the treatment towards this goal could be an interesting subject for forthcoming papers.

pressure /kPa 100

80

60

348.15 K

40

20

318.15 K

0 0

0.2

133

0.4

0.6

0.8

1

mole fraction ethanol Fig. 8. Vapor pressure and vapor composition at 318.15 and 348.15 K for the system n-octane + ethanol; calculated: (—), experimental: (䊏, 䉱) Goral et al. [41].

6. Conclusions The developed model based on continuous thermodynamics needs only the mixing parameter β to describe simultaneously liquid–liquid equilibrium and vapor–liquid equilibrium of the six n-alkane–alkanol systems considered. According to Eqs. (23) and (42) the χ-parameter is very small. Since the χ-parameter measures the physical interactions association effects (chemical contribution) seem to be dominant. On principle, the model is suitable, but, in detail there are deficiencies. Firstly, miscibility gaps are predicted too narrow. However, this is not a specific problem of associated systems, but rather a general difficulty. Secondly, the vapor pressure of the mixture is predicted somewhat too low. Thirdly, the calculated azeotropic mole fraction of the alkanol is too large (particularly, for the methanol systems). This lack originates from the too large association degrees predicted for the vapor phase. So, for methanol values between 1.4 and 1.5 are found. Replacing these values by 1.1 or 1.2 the azeotropic concentration would be predicted nearly correct. Some improvement seems to be possible introducing better vapor pressure equations than Eq. (20). Besides, the association constant KL should be permitted to depend

List of symbols A0 , A1 constants of vapor pressure Eq. (20) Aw dimensionless van der Waals surface of the monomer of the associates B0 , B1 constants of vapor pressure Eq. (20) BAA , BBB second virial coefficient of component A or B standard enthalpy of the association HL reaction (1) association constant of the association KL reaction (1) for the liquid phase association constant of the association KpV reaction (1) for the vapor phase ncs number of carbon atoms the chosen standard segment is equivalent to (e.g. for methanol ncs = 2, for ethanol ncs = 3) p pressure p+ normal pressure p∗A , p∗B vapor pressure of pure component A or, B p∗ (r) fictive vapor pressure of the pure associates consisting of r monomers p A , pB partial pressure of component A or B r number of monomers in an associate (association degree) and segment number rA segment number of the non-associating molecules r¯B number average of the association degree number average of the segment number of r¯ M the liquid mixture R universal gas constant T temperature W(r) molar distribution function of the associates Ws (r) segment-molar distribution function x totale mole fraction of the associates xB true mole fraction of component B (alkanol) Greek letters β parameter of Eq. (23) χ interaction parameter of Eqs. (22) and (23) γA activity coefficient of component A γ B (r) activity coefficient of the associates consisting of r monomers γ s,A , γ s,B segment-molar activity coefficients of component A or B ϕ˜ A , ϕ˜ B real gas correction for component A or B, defined by Eq. (18) λ parameter that describes the width of the associate distribution of the liquid phase ρA , ρB auxiliary quantitiy of the component A or B of Eq. (37) ψ total segment-mole fraction of the associates

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D. Browarzik / Fluid Phase Equilibria 217 (2004) 125–135

Subscripts A B s *

Table B.1 Constants of Eq. (B.1) describing the temperature dependence of the second virial coefficient of the pure substances

non-associating component (n-alkane) associating component (alkanol) segment-molar quantity pure component

Superscripts L liquid phase V vapor phase I, II liquid phases

Appendix A Introducing the abbreviation K∗ = KL eψ into Eq. (10) the following functional equation has to be solved: K∗ Ws (r − r  )Ws (r  ) = Ws (r)

(A.1)

Integration of Eq. (A.1) over r in the limits r + 1 and ∞ yields with the aid of Eq. (3)
∞ K∗ Ws (r  ) = Ws (r) dr (A.2)

Substance i

α0i

α1i

α2i

n-Hexane n-Heptane n-Octane Methanol Ethanol

635.8 920.5 1304.2 326.4 461.0

142,3 348.3 829.4 6493.7 9274.8

15.68 17.11 18.57 27.58 27.59

n-hexane, n-heptane, n-octane, methanol, ethanol have to be known as function of temperature. To describe the temperature dependence of the second virial coefficient the following expression proves to be suitable. 

 T Bii (cm3 mol−1 ) = − α0i + 1000α1i exp −α2i 1000 K (B.1) The constants α0i , α1i , α2i , are listed in Table 1.

r +1

References

Differentiation with respect to r provides K∗

dWs (r  ) = −Ws (r  + 1) dr 

(A.3)

By series expansion of Ws (r  + 1) and by replacing of r by r one can easily find the equation dWs (r) 1 d2 Ws (r) (1 + K∗ ) + dr 2! dr 2 3 1 d Ws (r) 1 dn Ws (r) + + · · · + =0 3! dr 3 n! dr n

Ws (r) +

(A.4)

The general solution of this differential equation is Ws (r) = c exp(λr)

(A.5)

where c and λ are constants with respect to r. Applying the normalization condition (3) to Eq. (A.5) the constant c results in c = −λ exp(−λ) and with the aid of Eqs. (9) and (A.5) the final result is given by Eq. (11). Introducing the general solution (A.5) in Eq. (A.4) we obtain   1 1 1 c exp(λr) 1+λ(1+K∗ )+ λ2 + λ3 + · · · + λn = 0 2! 3! n! (A.6) Using the series expansion of the function exp(λ) Eq. (A.6) with K∗ = KL eψ results in Eq. (12a). If KL is known this equation permits to calculate the quantity λ in an iterative way.

Appendix B To calculate the VLE for the systems considered in this paper the second virial coefficients of the pure substances

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