Extension of the CPA equation of state with dipolar theories to improve vapour–liquid-equilibria predictions

Fluid Phase Equilibria 312 (2011) 66–78

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Extension of the CPA equation of state with dipolar theories to improve vapour–liquid-equilibria predictions A.J. de Villiers, C.E. Schwarz, A.J. Burger ∗ Department of Process Engineering, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa

a r t i c l e

i n f o

Article history: Received 31 May 2011 Received in revised form 7 September 2011 Accepted 8 September 2011 Available online 17 September 2011 Keywords: CPA-JC CPA-GV Dipolar Vapour–liquid-equilibria

a b s t r a c t The Cubic-Plus-Association (CPA) equation of state (EOS) is extended with the dipolar theories of Jog and Chapman (JC) and Gross and Vrabec (GV). These new extended models are termed CPA-JC and CPAGV and each require four pure component model parameters for non-associating polar compounds and six parameters for associating compounds. Model parameters for selected ketones, aldehydes, esters, ethers and alcohols are presented for these two EOSs and were determined by including saturated vapour–pressure, liquid density and binary vapour–liquid-equilibria (VLE) data in the objective function. Furthermore, it is necessary to explicitly account for cross-association in binary mixtures where one component cross-associates, but does not self-associate. Major improvement in VLE predictions of polar/alkane and polar/polar systems are obtained with both CPA-GV and CPA-JC compared to normal CPA. In the majority of cases investigated, CPA-GV and CPA-JC require no or very small binary interaction parameters (kij ) to accurately represent VLE. The VLE of ternary mixtures is also predicted with good accuracy by both CPA-GV and CPA-JC. For the systems under investigation, no significant difference is observed between the performance of the CPA-JC and CPA-GV EOS models. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Simulation packages commonly use equation-of-state (EOS) models in the design of complex process units such as distillation columns. Often simple models with a lower numerical intensity are preferred over more complex models, because simulations can become rather time consuming. However, when mixtures with strong polar and associating forces are encountered in the system, simple models usually do not provide adequate representation of thermodynamic properties such as vapour–liquid-equilibria (VLE). Complex models such as PolarPC-SAFT [1–5], PCP-SAFT [4–7] and their sPC-SAFT counterparts [8] are able to model most mixtures with strong polar and hydrogen bonding forces accurately, but because these models are numerically intensive, it is useful to have a simplified model that is able to provide similar performance at lower computational cost. A possible approach is to modify the CubicPlus-Association (CPA) [9] EOS to explicitly account for strong dipolar interactions. CPA is essentially a combination of the SRK [10] EOS and the SAFT association term derived from Wertheim’s perturbation theory [11–14]. The SRK physical term of CPA is mathematically less complex compared to the physical term of

∗ Corresponding author. Tel.: +27 21 808 4494; fax: +27 21 808 2059. E-mail address: [email protected] (A.J. Burger). 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2011.09.007

PC-SAFT. Gani et al. [15] performed a mathematical and numerical analysis on several EOS, including CPA and PC-SAFT. From this study, good insight may be gained of the working equations required to perform phase equilibria calculations, and the difference in mathematical complexity between the EOS. In CPA, physical interactions are approximated with the cubic part and hydrogen bonding forces with the association term. Dipolar interactions are not explicitly treated in the original version of the model. A few approaches have been followed in an attempt to account for the effect of strong polar interactions with CPA. The most successful attempts seem to be the ‘pseudo-association’ approach where strongly polar components such as acetone are treated as if they are self-associating [16–20]. This approach seems to work well, but fairly large BIPs are still required to obtain acceptable results and the physical representation is not completely correct, because the molecules do not self-associate. Perfetti et al. [21,22] incorporated a polar term based on non-primitive MSA theory into the state function of CPA (CPAMSA) in order to explicitly account for polar interactions and modelled water–H2 S, water–methane and water–CO2 mixtures. As pointed out by Kontogeorgis and Folas [23], proper mixing rules and improvements are, however, still needed before the model can be considered fully functional. There are several other dipolar terms that can be incorporated into the state functions of EOS models, as reviewed by Tan et al. [24]. Two very successful dipolar theories are those developed by Jog and Chapman (JC) [1] (incorporated into polar-PC-SAFT [1–5]

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

and sPC-SAFT-JC [8]) and Gross and Vrabec (GV) [6] (incorporated into PCP-SAFT [4,6,7] and sPC-SAFT-GV [8]). These dipolar theories where primarily developed to be compatible with SAFT-type models which are based on a segment approach. Because the framework of CPA is significantly simpler compared to traditional SAFT-type models such as PC-SAFT [25], modifying the dipolar theories to be compatible with the framework of CPA might prove to be advantageous. Thus, the objective of this work is to evaluate and compare the performance of CPA when extended with the JC and GV dipolar theories. We present modified versions of the JC- and GV-polar theories that are slightly simplified and compatible with CPA. The newly developed models, CPA-JC and CPA-GV respectively, are then evaluated by investigating the VLE of several non-associating and associating polar systems.



Ai Bj = g exp g=

The CPA EOS was initially developed to extend the capabilities of cubic EOS to hydrogen bonding components [26]. As mentioned, the model uses the SRK EOS to represent physical interactions and the SAFT association term to model hydrogen bonding interactions. The CPA EOS is a pressure explicit equation and the dipolar terms of JC and GV are expressed in terms of the reduced Helmholtz energy. Therefore, it was decided to use the reduced residual Helmholtz energy as the state function with independent variables: temperature (T), total volume (V) and total moles (n) as presented below [26]: ArCPA (T, V, n) RT

=

ArSRK (T, V, n) RT

+

Aassoc (T, V, n) RT

(1)

The physical part, as represented by the SRK EOS, is expressed as follows [27]: ArSRK (T, V, n) D=

 i

B=





= −n ln 1 −

B V



−



D(T ) B ln 1 + RTB V



ni nj aij

(2) (3)

j

ni bi

(4)



ai aj (1 − kij )

εAi Bj =

1 AB (ε i i + εAj Bj ) 2

(10)

ˇAi Bi ˇAj Bj

(11)



2.2. Jog and Chapman’s dipolar term The JC-dipolar term is based on Wertheim’s thermodynamic perturbation theory [11–14] as explained elsewhere [1,2,4]. The dipolar term is expressed as a Pade approximate from Rushbrook et al. [29]: dipolar

A2 /RT Adipolar = dipolar dipolar RT 1 − A3 /A2



a(T ) = a0 1 + c1 1 −

 2

dipolar

  nc

ni

(ln XAi − XAi + 1)

Ai

i

−

RT

 1  ni nj XAi XBj Ai Bj 2 nc

nc

i

j

dipolar

A3

RT

=

i j 2 Na 1  ni nj xpi xpj I2,ij (∗) 9 V (kT )2 dij3

52 Na2 1 162 V 2 (kT )3

nc

nc

i

j

nc nc nc   

ni nj nk xpi xpj xpk j

2

2

2i 2j 2k dij djk dik

I3,ijk (∗)

(13)

(14)

k

(6)

where a0 , b and c1 are the pure component parameters for nonassociating components. Hydrogen bonding is approximated with the SAFT-association term [11–14] and requires two additional pure component parameters: the association energy (εAB /R) and association volume (ˇAB ). The Q-function of Michelsen and Hendriks [28] is used in this work to account for associations and is presented below: Aassoc (T, V, n) = Q (T, V, n) = RT

=−

i

Tr

(12)

The original expressions for the JC-dipolar term were developed to be compatible with the framework of SAFT-type models and some modifications are necessary in order to render the polar term compatible with the framework of CPA. CPA approximates molecules as single spheres. This implies that the polar term can be simplified accordingly. Furthermore, the expressions need to be transformed to be independent in (T, V, n) and the SAFT model parameters in the JC-term need to be related to the model parameters of CPA. The resulting expressions for the second- and third-order terms that were applied in this work are:

(5)

The ai parameter of CPA is calculated with the following equation [26]:



(9)

where g is the simplified radial distribution function, εAiBj and ˇAiBj are the respective cross-association energy and volume between site A on molecule i and site B on molecules j. The CR1 rule is applied to determine the cross-association energy and volume between two components as follows:

A2

i

aij =

(8)

The model described up to this point is known as the CPA EOS and requires five pure component model parameters to model associating components (a0 , b, c1 , εAB /R and ˇAB ).

2.1. CPA

RT

RT



− 1 bij ˇAi Bj

1 1 − 1.9

ˇAi Bj = 2. Theory

 εAi Bj 

67

Ai

Bj

The association strength is calculated with [26]:

(7)

where Na is Avogadro’s constant, k is Boltzmann’s constant, xp is the fraction of polar segments, d is the temperature dependent segment diameter,  is the dipole moment and I2,ij and I3,ijk are correlation functions. The temperature dependent segment diameter (d) of SAFT is easily related to the co-volume parameter (b) of CPA with [23]:

 d==

3

3b 2Na

(15)

The substitution assumes that the temperature dependent segment diameter is equal to the temperature independent segment diameter and may be calculated from the temperature independent co-volume parameter (b) of CPA. The following combining rule is used to calculated dij : dij = 0.5(di + dj )

(16)

68

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

The expressions for the correlation functions, I2,ij and I3,ijk , are as presented in the original publication [1] and the specific expression for the reduced density, *, is defined by: ∗ =

3B 2V

(17)

The polar term is somewhat simplified, because the derivatives of * are only dependent on V and n. This implies that the derivatives of I2,ij and I3,ijk , are also only dependent on V and n. The temperature derivatives of Eqs. (13) and (14) are also simplified, because the segment diameters are also no longer temperature dependent. When CPA is combined with the modified polar term of JC, the resulting EOS is termed CPA-JC. This is achieved by extending Eq. (1) with Eq. (12). For non-associating dipolar components, this EOS requires four pure component parameters (a0 , b, c1 and xp ) and for associating components, six parameters is required (a0 , b, c1 , xp , εAB /R and ˇAB ). Compared to the original CPA EOS, one additional parameter xp (fraction of polar segments) is required. 2.3. Gross and Vrabec’s dipolar term The GV-dipolar term is based on a third-order perturbation theory and more details are given elsewhere [4,6]. The dipolar term is also expressed in terms of a Pade approximant as given by Eq. (12). The resulting expressions for the second- and third-order terms, after similar modifications are made to render the term compatible with the framework of CPA, are presented below [24]: Add 2 RT

=−

nc nc 2i 2j Nav   n n · n n J dd i j pi pj V (kT )2 ij3 2,ij i

Add 3 RT

=−

(18)

j

nc nc nc 2 2 2i 2j 2k 4Nav 1  ni nj nk · npi npj npk J dd (19) 2 3 ij jk ik 3,ijk 3V (kT ) i

j

k

where  is the temperature independent segment diameter, np is the number of polar segments and J2,ij and J3,ijk indicate the respective integrals over the reference-fluid pair correlation function and over the three-body correlation function.  is related to the covolume parameter of CPA as shown in Eq. (15). The correlation functions used are provided below: dd = J2,ij

4 

[aw,ij + bw,ij ˛ij (T )]w

(20)

w=0

dd = J3,ijk

4 

cw,ij w

(21)

w=0

In the original publication by Gross and Vrabec [6], the coefficients in Eqs. (20) and (21) depend on chain length. In the framework of CPA, the coefficients are no longer functions of chain length, because all molecules are approximated as single spheres. Therefore, only 12 universal model constants are required compared to the 36 constants of the original expression. The reduced density, , is calculated with [26]: =

B 4V

(22)

In Eq. (20), the function ˛ij (T) is related to the dimensionless segment energy as follows: ˛ij (T ) = ˛i (T ) =

 ˛i (T )˛j (T )  a (T  ) i

(23)

4bi RT

when the modified dipolar term of GV is included in the state function of CPA, the resulting EOS is termed CPA-GV. This EOS

requires four pure component parameters (a0 , b, c1 and np ) for nonassociating polar components and six pure component parameters for associating components (a0 , b, c1 , np , εAB /R and ˇAB ). As for CPAJC, CPA-GV required one additional parameter, np (number of polar segments) compared to the original CPA EOS. 3. Determination of model parameters Pure component parameters of CPA-type models are usually fitted to saturated vapour pressure and liquid density data, but in this work, binary VLE data were also included in the parameter fitting procedure of CPA-JC and CPA-GV. This was motivated by the fact that similar problems were encountered during our initial fitting procedures as discussed in our previous work relating to sPC-SAFT-JC and sPC-SAFT-GV [8]. The discussion is therefore not repeated here. Essentially, including a binary VLE set in the regression procedure, where the polar component is in binary with a non-polar component, aids in determining the most appropriate contributions (physical, polar and association) to the state function. A similar strategy was followed by Dominik et al. [30]. Thus, in our parameter estimation, the Levenberg–Marquart algorithm with a least squares objective function was used. Saturated vapour pressure, liquid density and binary VLE data was included in the objective function. Thirty datapoints for each pure component property were generated from the DIPPR correlations [31] in the range 0.5 < Tr < 0.9. Binary VLE data was obtained from the literature. We recommend using good quality VLE data with measurements over the whole concentration range, if available. Furthermore, datasets with close boiling points that exhibit azeotropes usually provide good parameter sets. The values of the dipole moments were also obtained from the DIPPR database [31]. Alcohols were modelled with the 2B association scheme. Initially, high regression weights were assigned to the pure component data and a low regression weight to the binary data. Factors that influence the value of the regression weight assigned to the binary data include the number of measurements and temperature of the VLE set. We used higher regression weights for data sets with few measurements and at high temperatures (polar forces diminish with increase in temperature). The model parameters were then tested to establish if they provide the appropriate contribution from each term (physical, polar and association) to the state function. This was achieved by examining how well the pure component data and binary data included in the state function was predicted. If pure component data was not correlated with good accuracy, the regression weight of the binary data set was lowered and vice versa. We recognize that including binary data in the regression function is not the ideal method in determining pure component model parameters, but the significant improvement in the performance of CPA justifies the decision. The model parameters determined in this work are presented in Table 1 for CPA-JC and in Table 2 for CPA-GV. In Sections 4–6, comparisons are made to normal CPA and consequently model parameters had to be determined for some components that could not be found in the literature. The parameters used for CPA are presented in Table 3 and were fitted to saturated vapour pressure and liquid density with a least squares objective function. From Tables 1 and 2, it is clear that both CPA-JC and CPA-GV correlate the pure component properties with good accuracy and that no significant trade-off occurs as a result of including binary VLE data in the regression function. Common trends are observed with respects to the model parameters for CPA-JC, CPA-GV and CPA: the co-volume and energy parameters increase with molecular size. It is further noted that the fraction of polar segments (xp ) in CPA-JC and the number of polar segment in CPA-GV (np ) are larger than one for many of the components. Two contributing factors to this

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

69

Table 1 Model parameters for CPA-JC.

Acetone 2-Butanone 3-Pentanone MIPK MIBK Propanal Butanal Pentanal Methyl formate Ethyl formate Propyl formate Butyl formate Methyl acetate Ethyl acetate Propyl acetate Butyl acetate Propyl propionate Diethyl ether Dibutyl ether Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol

Sch

TC [K]

a0 /Rb [K]

b [L/mol]

c1

xp

 [D]

εAB /R [K]

ˇAB

VLE data

Psat [%]a

sat [%]a

– – – – – – – – – – – – – – – – – – – 2B 2B 2B 2B 2B 2B 2B 2B

508.2 535.5 560.9 553.4 574.6 504.4 537.2 566.1 487.2 508.4 538.0 559.0 506.6 523.3 549.7 575.4 568.6 466.7 584.1 512.5 514.0 536.8 563.1 588.1 611.3 632.3 652.3

2176.52 2413.61 2632.06 2451.56 2610.28 2188.46 2486.41 2575.66 1953.62 2159.87 2365.26 2588.32 2169.62 2313.77 2530.92 2803.26 2740.90 2299.72 2976.50 1573.83 1735.14 2140.71 2551.28 2714.60 2957.67 2960.60 3244.51

0.062685 0.078654 0.093605 0.093174 0.114107 0.061235 0.077191 0.094030 0.050851 0.067682 0.084000 0.102219 0.067180 0.084372 0.101395 0.118437 0.117960 0.083125 0.151493 0.031822 0.047843 0.064501 0.081741 0.097893 0.113694 0.135274 0.149121

0.73278 0.79914 0.86633 0.85124 0.88449 0.72867 0.75545 0.86640 0.73350 0.78554 0.85905 0.93402 0.83190 0.91598 0.96305 1.01500 1.00274 0.82606 1.06480 0.40024 0.82692 0.87654 0.08701 0.90702 0.85387 1.02245 0.94271

0.6700 0.8000 0.7851 0.9500 1.1982 0.7547 0.7120 1.0382 1.4500 1.4810 1.7773 1.7107 1.9551 2.0261 2.1700 1.8354 2.1425 2.0660 2.4182 0.3420 0.4694 0.7599 0.9501 1.4691 1.6776 2.5942 1.6155

2.88 2.76 2.82 2.76 2.67 2.52 2.72 2.57 1.77 1.93 1.91 2.02 1.68 1.78 1.79 1.84 1.79 1.15 1.17 1.70 1.70 1.68 1.67 1.70 1.65 1.74 1.65

– – – – – – – – – – – – – – – – – – – 2897.05 2643.03 2561.17 2650.10 2565.59 3058.02 3034.38 3237.25

– – – – – – – – – – – – – – – – – – – 0.01542 0.01459 0.00777 0.00305 0.00292 0.00086 0.00056 0.00064

n-Hexane [32] n-Heptane [33] n-Heptane [34] n-Octane [35] n-Octane [36] n-Pentane [39] n-Heptane [39] n-Heptane [40] n-Hexane [41] n-Hexane [41] n-Hexane [41] n-Hexane [41] n-Pentane [42] n-Hexane [43] n-Heptane [44] n-Hexane [45] n-Nonane [44] n-Pentane [46] n-Hexane [47] n-Hexane [48] n-Heptane [49] n-Heptane [50] n-Nonane [51] n-Heptane [52] n-Hexane [53] n-Decane [54] n-Decane [55]

0.70 0.47 0.24 0.99 2.35 0.47 0.74 0.43 0.29 0.80 1.16 0.68 0.39 0.82 1.14 0.84 1.10 0.74 0.46 0.82 0.36 0.10 0.09 0.06 0.08 0.65 0.18

1.64 1.68 1.17 0.54 2.78 0.97 0.61 0.84 0.77 0.66 0.61 1.54 0.88 0.84 0.79 0.78 0.85 0.56 0.66 0.46 0.38 0.52 0.74 0.69 0.67 1.75 1.08

Average a

n

The % absolute average deviation (%AAD) of the properties (saturated vapour pressure (Psat ), saturated liquid density (sat ), %AAD = (100/n) sat

n is the number of data points and X is P

,

sat

0.63

0.94 exp

i=1

|Xical − Xi

exp

|/Xi

, where

.

irregularity are that the effect of polar forces are underestimated as a result of the single sphere approach as discussed by Jog et al. [3], and that shortcomings within the JC and GV theories seem to underestimate the effect of dipolar forces, especially in components with small dipole moments, as previously shown by De Villiers et al. [8].

4. VLE of non-associating systems This section considers the predictive performance of CPA-JC and CPA-GV by investigating the VLE of non-associating polar/alkane systems. The influence of the dipolar terms can be thoroughly

Table 2 Model parameters for CPA-GV.

Acetone 2-Butanone 3-Pentanone MIPK MIBK Propanal Butanal Pentanal Methyl formate Ethyl formate Propyl formate Butyl formate Methyl acetate Ethyl acetate Propyl acetate Butyl acetate Propyl propionate Diethyl ether Dibutyl ether Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol

Sch

TC [K]

a0 /Rb [K]

b [L/mol]

c1

np

 [D]

εAB /R [K]

ˇAB

VLE data

Psat [%]a

sat [%]a

– – – – – – – – – – – – – – – – – – – 2B 2B 2B 2B 2B 2B 2B 2B

508.2 535.5 560.9 553.4 574.6 504.4 537.2 566.1 487.2 508.4 538.0 559.0 506.6 523.3 549.7 575.4 568.6 466.7 584.1 512.5 514.0 536.8 563.1 588.1 611.3 632.3 652.3

2342.74 2520.05 2701.31 2513.23 2700.91 2312.14 2562.04 2648.26 2151.77 2263.68 2455.02 2649.21 2291.67 2404.61 2597.21 2825.61 2776.06 2312.08 2988.11 1582.66 1785.72 2164.90 2568.84 2732.84 2967.17 2976.05 3229.32

0.058104 0.073725 0.089496 0.088731 0.105250 0.056988 0.073279 0.089087 0.046630 0.063081 0.078973 0.096705 0.062170 0.079119 0.096440 0.115049 0.1141000 0.081553 0.150845 0.031573 0.047423 0.063837 0.080979 0.096374 0.112193 0.132213 0.147919

0.69575 0.77373 0.85416 0.82519 0.85113 0.69885 0.74094 0.84755 0.68662 0.74903 0.83393 0.92012 0.79024 0.88067 0.93973 1.01055 0.99538 0.82466 1.06679 0.46917 0.73778 0.85852 0.86404 0.88379 0.83702 1.02715 0.94591

0.4541 0.5682 0.5750 0.7325 0.8838 0.5256 0.5195 0.7684 0.9500 1.0734 1.2883 1.2724 1.3785 1.4856 1.6321 1.4477 1.6563 1.6352 1.6604 0.2975 0.3655 0.6126 0.7579 1.1723 1.3829 2.0785 1.5799

2.88 2.76 2.82 2.76 2.67 2.52 2.72 2.57 1.77 1.93 1.91 2.02 1.68 1.78 1.79 1.84 1.79 1.15 1.17 1.70 1.70 1.68 1.67 1.70 1.65 1.74 1.65

– – – – – – – – – – – – – – – – – – – 2819.72 2751.72 2587.13 2665.97 2625.77 3099.88 3029.08 3231.03

– – – – – – – – – – – – – – – – – – – 0.01811 0.01105 0.00723 0.00291 0.00268 0.00083 0.00058 0.00064

n-Hexane [32] n-Heptane [33] n-Heptane [34] n-Octane [35] n-Octane [36] n-Pentane [39] n-Heptane [39] n-Heptane [40] n-Hexane [41] n-Hexane [41] n-Hexane [41] n-Hexane [41] n-Pentane [42] n-Hexane [43] n-Heptane [44] n-Hexane [45] n-Nonane [44] n-Pentane [46] n-Hexane [47] n-Hexane [48] n-Heptane [49] n-Heptane [50] n-Nonane [51] n-Heptane [52] n-Hexane [53] n-Decane [54] n-Decane [55]

0.73 0.42 0.28 2.02 1.87 0.44 0.83 0.43 0.53 0.22 1.25 0.52 0.38 0.70 1.13 0.96 0.97 0.83 0.50 0.77 0.35 0.09 0.09 0.05 0.11 0.69 0.21

1.63 1.60 1.02 1.78 0.82 0.98 0.62 0.81 0.91 1.14 0.74 0.80 0.96 0.86 0.77 0.73 0.75 0.67 0.67 0.39 0.35 0.56 0.74 0.71 0.74 1.38 1.01

Average a

n

The % absolute average deviation (%AAD) of the properties (saturated vapour pressure (Psat ), saturated liquid density (sat ), %AAD = (100/n)

n is the number of data points and X is Psat , sat .

0.64

i=1

0.89 exp

|Xical − Xi

exp

|/Xi

, where

70

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

Table 3 Model parameters for CPA.

Acetone 2-Butanone 3-Pentanone MIPK MIBK Propanal Butanal Pentanal Methyl formate Ethyl formate Propyl formate Butyl formate Methyl acetate Ethyl acetate Propyl acetate Butyl acetate Propyl propionate Diethyl ether Dibutyl ether Methanol Ethanol 1-Propanol 1-Butanol 1-Pentanol 1-Hexanol 1-Heptanol 1-Octanol

Sch

TC [K]

a0 /Rb [K]

b [L/mol]

c1

εAB /R [K]

ˇAB

Ref.

Psat [%]a

sat [%]a

– – – – – – – – – – – – – – – – – – – 2B 2B 2B 2B 2B 2B 2B 2B

508.2 535.5 560.9 553.4 574.6 504.4 537.2 566.1 487.2 508.4 538.0 559.0 506.6 523.3 549.7 575.4 568.6 466.7 584.1 512.5 514.0 536.8 563.1 588.1 611.3 632.3 652.3

2719.57 2824.42 2929.81 2850.80 3008.13 2626.38 2807.79 2923.67 2528.21 2633.77 2771.83 2904.73 2649.49 2723.51 2857.01 2991.73 2966.64 2391.73 3009.14 1573.71 2123.83 2234.52 2368.59 2808.75 2950.20 3273.13 3367.99

0.061900 0.077389 0.092444 0.091864 0.110108 0.060289 0.076192 0.092800 0.050060 0.066645 0.082904 0.099506 0.066250 0.083380 0.100189 0.117374 0.116578 0.082570 0.151181 0.030978 0.049110 0.064110 0.079700 0.097455 0.110800 0.131330 0.148500

0.80023 0.84707 0.90329 0.89382 0.90594 0.79227 0.80829 0.90619 0.79622 0.83753 0.89965 0.96345 0.87185 0.94265 0.98575 1.03340 1.02659 0.85074 1.07187 0.43102 0.73690 0.91709 0.97840 0.93580 0.98050 1.01096 1.14860

– – – – – – – – – – – – – – – – – – – 2957.78 2589.85 2525.86 2525.86 2525.86 2525.86 3159.97 3218.55

– – – – – – – – – – – – – – – – – – – 0.01610 0.00800 0.00810 0.00820 0.00360 0.00330 0.00030 0.00014

[56] This work This work This work This work This work This work This work [56] This work This work This work This work [56] This work This work This work This work This work [56] [56] [56] [56] [56] [56] [56] [56]

0.51 0.23 0.39 0.78 1.94 0.20 0.98 0.49 0.49 0.25 1.39 0.29 0.64 0.86 1.24 0.95 0.89 0.89 0.52 1.03 1.75 1.07 2.28 0.66 6.71 2.39 3.10

0.83 1.05 0.70 0.63 0.51 0.46 0.44 0.56 0.48 0.51 0.55 0.55 0.45 0.52 0.55 0.64 0.64 0.71 0.66 0.50 0.44 0.59 1.29 0.66 2.15 2.77 0.68

1.22

Average

n

The % absolute average deviation (%AAD) of the properties (saturated vapour pressure (Psat ), saturated liquid density (sat ), %AAD = (100/n) sat

n is the number of data points and X is P

,

sat

i=1

0.76 exp

|Xical − Xi

exp

|/Xi

, where

.

established by investigating polar systems where the association term is not active. Results for normal CPA without a dipolar term are also presented for comparative purposes. An overview of all non-associating polar/alkane systems investigated is presented in Table 4. It is clear that major improvements are obtained when a dipolar term is included in the state function of CPA when considering non-associating components. The pure component parameters presented in Tables 1 and 2 were fitted to pure component data and one binary VLE set, where the second component was nonpolar. These parameters do not only provide good predictions for the binary system included in the regression routine, but also for most other VLE systems investigated. This indicates that the model parameters determined in this work provide appropriate contributions (from the physical and polar terms) to the state function. In the following sub-sections it is shown that CPA-JC and CPA-GV provide very similar results. Generally, no distinct difference is observed between the performances of the two polar CPA variants. Unfortunately, a similar liquid–liquid demixing problem was encountered with CPA-GV as reported by Al-Saifi et al. [4] for PCPSAFT. CPA-GV predicted false liquid–liquid demixing for the methyl formate/n-heptane system at P = 1.013 bar. The error is easily corrected with a small BIP (CPA-GV kij = −0.009). False liquid–liquid demixing was not observed for any of the other systems with either CPA-JC or CPA-GV, but it is suspected that at lower temperatures, both models may predict false liquid–liquid splits. This problem seems to be common to the JC and GV theories, even when incorporated into PC-SAFT.

improved predictions compared to CPA. Unfortunately, a BIP is still required by both models to accurately predict the excess enthalpy. Furthermore, if model parameters are fitted to excess enthalpy and pure component data, poor VLE predictions are obtained (both models require the following BIPs to accurately describe the VLE when the parameters are fitted by including excess enthalpy data: CPA-JC kij = −0.024 and CPA-GV kij = −0.038). Similar results were found with sPC-SAFT-JC and sPC-SAFT-GV [8]. It is noted that kij values reported thus far are negative. A probable cause may be that the dipole moments in the vapour and liquid phases are not the same [66]. Both the GV and JC theories do not explicitly account for different dipoles in different mediums and this shortcoming should receive attention in future developments.

0.75

0.65

P [bar]

a

0.55

0.45

T =313.15 K CPA-GV CPA-JC CPA

4.1. Ketone/alkane systems 0.35

Fig. 1 shows the VLE predictions of the acetone/n-hexane system. Both CPA-JC and CPA-GV provide good VLE predictions without BIPs. Excess enthalpy results for the acetone/n-hexane system are shown in Fig. 2 and both CPA-JC and CPA-GV provide

0

0.2

0.4

0.6

0.8

1

mole fracon acetone Fig. 1. Isothermal VLE predictions of the acetone/n-hexane system at T = 313.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [32].

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

71

Table 4 VLE predictions of polar/n-alkane mixtures with CPA, CPA-GV and CPA-JC. Mixture

T or P

Acetone/n-pentane Acetone/n-hexane Acetone/n-heptane Acetone/n-octane 2-Butanone/n-hexane 2-Butanone/n-heptane 3-Pentanone/n-heptane MIPK/n-octane MIPK/n-cyclohexane MIBK/n-heptane MIBK/n-octane MIBK/cyclohexane Propanal/n-pentane Butanal/n-heptane Pentanal/n-heptane Methyl formate/n-hexane Methyl formate/n-heptanec Ethyl formate/n-hexane Ethyl formate/n-heptane Ethyl formate/n-octane Ethyl formate/n-nonane Ethyl formate/n-decame Propyl formare/n-hexane Propyl formate/n-heptane Propyl formate/n-nonane Butyl formate/n-hexane Methyl acetate/n-pentane Methyl acetate/n-heptane Ethyl acetate/n-hexane Ethyl acetate/n-heptane Ethyl acetate/n-heptane Ethyl acetate/n-octane Propyl acetate/n-heptane Propyl acetate/n-nonane Butyl acetate/n-hexane Propyl propionate/n-heptane Propyl propionate/n-nonane Diethyl ether/n-pentane Diethyl ether/n-hexane Dibutyl ether/n-hexane Average

n

298.15 K 313.15 K 313.15 K 313.15 K 333.15 K 318.15 K 353.15 K 1.013 bar 1.013 bar 343.15 K 0.94 bar 0.8 bar 313.15 K 318.15 K 348.15 K 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 298.15 1.013 bar 1.013 bar 323.15 K 343.15 K 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 1.013 bar 308.15 K

CPA

CPA-GV P (%)b / T (K)a

y (×102 )a

P (%)b / T(K)a

y (×102 )a

P (%)b / T (K)a

8.38 10.4 6.85 4.50 6.59 7.63 6.37 4.40 6.38 4.47 5.19 4.55 7.99 4.61 4.40 9.50 9.86 8.17 8.57 4.65 6.50 3.77 5.77 4.10 4.77 5.03 7.57 5.45 5.75 5.05 4.55 5.55 3.81 3.01 1.85 2.43 2.60 0.94 1.34 0.19

20.5 22.9 22.1 20.6 14.2 14.4 13.5 3.91 4.74 10.85 5.30 4.52 16.9 12.4 12.2 7.06 9.72 5.48 7.26 8.12 9.79 10.3 4.63 5.07 7.34 3.57 17.2 5.95 4.83 13.8 12.8 5.22 3.48 3.69 2.80 2.55 2.71 0.88 1.37 1.80

1.25 2.26 1.57 0.91 1.06 1.13 0.58 0.79 0.47 0.92 0.64 1.79 1.75 0.92 1.11 4.05 2.21 0.81 1.33 1.07 0.98 1.93 0.47 0.68 0.59 1.03 0.39 1.02 0.49 0.88 0.64 0.39 0.63 1.27 0.45 0.84 1.12 0.41 0.49 0.16

2.27 2.80 4.17 5.91 1.69 2.04 0.52 0.52 0.33 1.16 0.16 0.24 1.18 1.17 0.93 0.91 1.34 0.30 0.60 1.11 1.39 2.77 0.17 0.27 0.64 0.26 1.05 0.82 0.16 1.62 0.90 0.52 0.22 0.96 0.19 0.30 0.22 0.13 0.30 0.17

0.70 1.28 0.76 0.39 0.75 0.52 0.30 0.53 0.65 0.47 0.56 1.90 1.37 0.58 0.82 2.85 1.65 0.37 1.09 1.01 0.65 2.21 0.29 0.58 0.73 1.03 0.47 1.12 0.35 0.80 0.65 0.47 0.51 1.16 0.44 0.82 1.10 0.41 0.49 0.27

1.02 1.45 2.61 3.00 1.30 1.08 0.36 0.67 0.25 0.71 0.10 0.43 0.67 0.64 1.07 0.51 1.77 0.14 0.51 1.00 0.85 2.78 0.15 0.34 0.95 0.22 1.25 0.87 0.12 1.27 0.73 0.65 0.15 0.86 0.18 0.32 0.21 0.13 0.32 1.04

5.33

8.91

1.04

1.06

0.83

0.82

a

z = (1/n)

b

Deviations as %AAD. Results for CPA-GV were generated with kij = −0.009 to prevent false liquid-liquid demixing.

c

i

2500

−z

exp

n

Ref.

14 27 16 21 12 18 17 31 31 15 21 14 26 19 11 25 36 32 34 38 31 27 26 26 27 25 19 25 22 19 19 18 32 31 25 34 42 14 16 14

[57] [32] [32] [32] [58] [33] [34] [35] [35] [33] [36] [59] [39] [39] [40] [41] [60] [41] [61] [61] [61] [61] [41] [62] [62] [41] [42] [60] [43] [63] [63] [64] [44] [44] [45] [44] [44] [46] [46] [47]

|, where z represents y or T and n is the number of data points.

T = 293.2 K CPA-GV CPA-JC CPA

P = 1.013 bar CPA-GV CPA-JC CPA

395

T [K]

2000

HE [J.mol-1]

|z

cal

CPA-JC

y (×102 )a

1500

1000

385

375

500 365

0

0

0

0.2

0.4

0.6

0.8

1

mole fracon acetone Fig. 2. Excess enthalpy predictions of the acetone/n-hexane system at T = 293.2 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [65].

0.2

0.4

0.6

0.8

1

mole fracon MIPK Fig. 3. Isobaric VLE predictions of the MIPK (methyl isopropyl ketone)/n-octane system at P = 1.013 bar with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [35].

72

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78 T = 298.15 K CPA-GV CPA-JC CPA

0.5 1500

HE [J.mol-1]

P [bar]

0.45

0.4

1000

500

0.35

T = 323.15 K CPA-GV (kij = 0.0178) CPA-JC (kij = 0.0147) CPA (kij = 0.0797)

0.3

0

0.2

0.4

0.6

0.8

0

1

0

0.2

mole fracon 2-butanone

0.4

0.6

0.8

1

mole fracon butanal

Fig. 4. Isothermal VLE correlations of the 2-butanone/cyclohexane system at T = 323.15 K with CPA-GV (kij = 0.0147), CPA-JC (kij = 0.0178) and CPA (kij = 0.0797). Experimental data taken from Ref. [67].

Fig. 6. Excess enthalpy predictions of the butanal/n-heptane system at T = 298.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [69].

4.3. Ester/alkane systems Fig. 3 shows the VLE prediction for the MIPK (methyl isopropyl ketone)/n-octane system and both CPA-JC and CPA-GV provide accurate predictions of the data. These predictions are markedly superior to the predictions of CPA. Furthermore, the CPA-JC and CPA-GV model parameters for MIPK provide good VLE predictions of the cyclohexane/MIPK system (see Table 4). This indicates the model parameters are not system specific, but successfully capture the behaviour of the polar components. In some cases, small BIPs are still required by both CPA-JC and CPA-GV to give a very accurate description of the VLE e.g. the 2-butanone/cyclohexane system, as shown in Fig. 4. Nevertheless, the BIPs required by the polar models are significantly smaller than the BIPs required by CPA. 4.2. Aldehyde/alkane systems Both CPA-JC and CPA-GV predict the VLE of butanal and nheptane accurately, as shown in Fig. 5, while the prediction of CPA is not good. Fig. 6 shows the excess enthalpy of the butanal/n-heptane system and similar results are obtained compared to the acetone/nhexane systems (Fig. 2): the models cannot simultaneously predict the VLE and excess enthalpy with the same set of model parameters. It seems that more improvements to the theories are necessary before both properties will be predicted with good accuracy at the same time.

The VLE results for the methyl acetate/n-pentane system are shown in Fig. 7. Similar to the other results presented thus far, CPAJC and CPA-GV provide accurate predictions that coincide well with the experimental data. Several mixtures containing different esters were investigated (see Table 4) and both CPA-JC and CPA-GV seem to be able to account for the position of the ester functional group on the molecule. 4.4. Ether/alkane systems The small dipole moments of ethers imply that the influence of polar forces are very small, but not negligible, especially in systems such as diethyl ether and n-pentane, as shown in Fig. 8. Clearly, if the simulation of a distillation column depended on the VLE predictions of CPA, large errors would have been encountered in the process. Both CPA-JC and CPA-GV provide satisfactory VLE predictions of the system, therefore, improved simulation of process units becomes possible. Similar results were found with sPC-SAFT-JC and sPC-SAFT-GV [8]. 4.5. Non-associating polar/polar systems Generally, CPA provides good predictions of VLE for nonassociating polar/polar systems, because the like and unlike

0.4

0.8

0.65

P [bar]

P [bar]

0.3

0.5

0.2

0.35

T = 318.15 K CPA-GV CPA-JC CPA

0.1

0

0.2

0.4

0.6

0.8

1

mole fracon butanal Fig. 5. Isothermal VLE predictions of the butanal/n-heptane system at T = 318.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [39].

0.2

T = 298.15 K CPA-GV CPA-JC CPA

0

0.2

0.4

0.6

0.8

1

mole fracon n-pentane Fig. 7. Isothermal VLE predictions of the n-pentane/methyl acetate system at T = 298.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [42].

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

73

Table 5 VLE predictions of non-associating polar/polar mixtures with CPA, CPA-GV and CPA-JC. Mixture

T or P

Ethyl formate/2-butanone Methyl acetate/butanal Propyl acetate/butanal Methyl acetate/ethyl acetate Diethyl ether/acetone Dibutyl ether/acetone Acetone/2-butanone Acetone/3-pentanone Average

n

a

z = (1/n)

b

Deviations as %AAD.

i

313.15 K 323.15 K 323.15 K 353.15 K 303.15 K 1.013 bar 3.447 bar 1.013 bar

CPA

CPA-GV

CPA-JC

y (×102 )a

P (%)b / T (K)a

y (×102 )a

P (%)b / T (K)a

y (×102 )a

P (%)b / T (K)a

1.07 0.44 0.71 0.34 2.34 3.23 1.65 0.47

1.35 1.74 1.01 1.10 5.88 5.21 0.52 0.64

1.04 0.40 0.71 0.30 0.86 1.17 1.64 0.57

0.95 0.98 0.91 1.60 2.14 1.45 0.49 0.72

0.81 0.40 0.66 0.33 0.58 0.99 1.54 0.71

0.53 0.99 0.76 1.15 1.52 1.02 0.44 0.76

1.28

2.18

0.84

1.16

0.75

0.90

n

Ref.

9 17 14 11 13 17 14 12

[71] [72] [72] [73] [74] [75] [76] [77]

|z cal − z exp |, where z represents y or T and n is the number of data points.

moment of diethyl ether, which is the reason why CPA cannot describe the system accurately.

P = 1.013 bar CPA-GV CPA-JC CPA

309

T [K]

5. VLE of associating systems The associating systems considered in this study are limited to mixtures containing alcohols. The major sources of non-ideality present in alcohol-containing mixtures usually originate as a result of hydrogen bonding. Therefore, it is expected that the improvements gained by adding a dipolar term to the state function of CPA will be less pronounced, because hydrogen bonding is explicitly treated with the SAFT association term.

308

307

306

0

0.2

0.4

0.6

0.8

1

mole fracon diethyl ether Fig. 8. Isobaric VLE predictions of the diethyl ether/n-pentane system at P = 1.013 bar with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [46].

interactions are similar. However, this work shows that both CPAJC and CPA-GV, utilising the newly determined model parameters, offer predictions that are slightly superior to that of CPA (see Table 5). Fig. 9 shows the VLE of diethyl ether and acetone and both CPA-JC and CPA-GV accurately represent the experimental data. For this system, the unlike interactions are different, because the dipole moment of acetone is significantly stronger than the dipole

5.1. Alcohol/alkane systems Normal CPA seems to be able to correlate the VLE of alcohol/alkane systems accurately with a single BIP [26]. It is expected that CPA-GV and CPA-JC will require smaller or no BIP values to obtain accurate VLE predictions of these systems. A summary of alkane/alcohol VLE investigated in this study is presented in Table 6 and the results show that both CPA-GV and CPA-JC provide improved VLE predictions for the systems considered here, if compared to normal CPA. Although these improvements are slightly less pronounced compared to non-associating polar/alkane systems, they are nonetheless still significant. The VLE of nheptane/1-propanol is presented in Fig. 10 and VLE results for the n-hexane/1-hexanol systems are shown in Fig. 11. Related trends

0.95

0.45

0.8

P [bar]

P [bar]

0.35

0.65

0.25

0.5

0.35

T = 303.15 K CPA-GV CPA-JC CPA

0

0.2

0.4

0.6

0.8

1

mole fracon diethyl ether Fig. 9. Isothermal VLE predictions of the diethyl ether/acetone system at T = 303.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [74].

0.15

T = 333.15 K CPA-GV CPA-JC CPA

0

0.2

0.4

0.6

0.8

1

mole fracon 1-propanol Fig. 10. Isothermal VLE predictions of the 1-propanol/n-heptane system at T = 333.15 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [50].

74

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

1.2

0.9

0.6

0.3

0

P = 1.013 bar CPA-GV CPA-JC CPA

334

T [K]

P [bar]

339

T = 342.82 K CPA-GV CPA-JC CPA

329

324

0.2

0

0.4

0.6

0.8

319

1

0

0.2

mole fracon n-hexane

0.4

0.6

0.8

1

mole fracon acetone Fig. 12. Isobaric VLE predictions of the acetone/methanol system at P = 1.0132 bar with CPA-GV, CPA-JC and CPA when cross-association is not treated explicitly. Experimental data taken from Ref. [91].

Fig. 11. Isothermal VLE predictions of the n-hexane/1-hexanol system at T = 342.82 K with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [53].

indicate that it is advantageous to include a dipolar term in the state function when modelling alcohol/alkane VLE (Fig. 12).

have been followed by previous workers to treat this type of cross-association. Kleiner and Sadowski [7,88] suggested that the cross-association energy between the two components is equal to half the association energy of the associating component and that the cross-association volume is equal to the association volume of the associating component. With this approach, they managed to obtain a 10% reduction in %AAD values for the systems they considered. However, when Grenner et al. [89] applied this approach to other cross-associating systems, only marginal improvements were obtained. They then resorted to the approach proposed by Folas et al. [90], where the cross-association energy is also taken as half the association energy of the self-associating component, but with the association volume fitted to binary VLE data. In this work, we use a similar approach as proposed by Folas et al. [90]:

5.2. Associating polar/polar systems Two types of associating polar/polar systems are considered here. Type 1 systems are binary mixtures of alcohols. The VLE of these systems are generally predicted with good accuracy by normal CPA, because like and unlike interactions are similar and usually no BIPs are required. Type 2 systems involves binary mixtures where one component is an alcohol and the other a polar component that does not self-associate, but solvates (cross-associates) with the alcohol. This type of cross-association needs to be explicitly accounted for in order to obtain accurate VLE predictions and, in most cases, cannot be corrected with a BIP. A few approaches Table 6 VLE predictions of alcohol/n-alkane mixtures with CPA, CPA-GV and CPA-JC. Mixture

T or P

CPA y (×102 )a

Methanol/n-butane Methanol/n-pentane Methanol/n-hexane Methanol/n-hexane Ethanol/n-pentane Ethanol/n-hexane Ethanol/n-hexane Ethanol/n-heptane Ethanol/n-octane 1-Propanol/n-hexane 1-Propanol/n-heptane 1-Propanol/n-octane 1-Propanol/n-octane 1-Butanol/n-hexane 1-Butanol/n-heptane 1-Butanol/n-nonane 1-Butanol/n-decane 1-Butanol/n-decane 1-Pentanol/n-heptane 1-Pentanol/n-heptane 1-Hexanol/n-hexane 1-Heptanol/n-decane 1-Octanol/n-decane Average

n

a

z = (1/n)

b

Deviations as %AAD.

i

323.15 K 372.7 K 343.15 K 348.15 K 372.7 K 298.15 K 323.15 K 333.15 K 318.15 K 323.15 K 333.15 K 358.15 K 363.15 K 323.15 K 333.15 K 323.15 K 373.15 K 383.15 K 348.15 K 368.15 K 342.82 K 0.1359 bar 383.15 K

CPA-GV P (%)b / T (K)a

CPA-JC

y (×102 )a

P (%)b / T (K)a

y (×102 )a

P (%)b / T (K)a

1.14 4.02 2.90 2.94 2.82 1.20 2.74 2.41 1.74 1.77 2.45 2.52 2.04 0.46 1.74 1.34 2.60 2.34 2.94 2.99 1.36 6.05 3.37

7.80 7.72 7.00 7.54 7.45 4.22 6.03 4.05 3.64 7.77 5.39 8.46 6.34 5.52 5.90 3.24 6.53 6.02 7.10 6.96 10.3 2.65 10.6

0.75 2.74 1.93 2.60 1.66 0.18 1.56 0.71 0.99 0.53 0.98 0.92 0.84 0.43 0.77 0.92 1.67 1.40 0.99 1.18 0.24 1.17 0.60

3.17 4.56 2.24 2.89 3.94 1.01 2.82 0.71 2.73 1.28 1.12 2.75 1.11 3.46 2.72 1.36 3.38 2.74 1.54 2.16 2.43 0.22 0.98

0.76 2.71 1.86 2.69 1.57 0.34 1.41 0.62 1.11 0.54 0.95 0.89 0.80 0.45 0.74 0.93 1.72 1.44 1.03 1.22 0.24 1.17 0.67

3.03 4.40 1.98 2.64 3.75 1.80 2.41 0.63 3.24 1.29 1.08 2.78 1.11 3.40 2.53 1.42 3.41 2.79 1.67 2.29 2.50 0.23 1.48

2.43

6.44

1.12

2.23

1.12

2.25

|z cal − z exp |, where z represents y or T and n is the number of data points.

n

Ref.

11 11 24 24 10 9 20 16 17 22 33 25 24 10 19 15 22 22 19 20 22 15 14

[78] [79] [48] [53] [80] [81] [82] [49] [83] [50] [50] [84] [85] [86] [87] [51] [53] [53] [52] [52] [53] [54] [55]

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

75

Table 7 Cross-association volume values used for CPA, CPA-GV and CPA-JC. Component

CPA, ˇAB

CPA-GV, ˇAB

CPA-JC, ˇAB

VLE data included

Acetone 2-Butanone Methyl acetate Ethyl formate Ethyl acetate Propyl formate

0.15412 0.08261 0.05208 0.04676 0.03243 0.02089

0.20144 0.14154 0.06704 0.06493 0.06611 0.05386

0.21325 0.14623 0.07408 0.07452 0.07361 0.05703

Methanol [91] Ethanol [92] Methanol [93] Methanol [94] Ethanol [95] Ethanol [96]

339

P = 1.013 bar CPA-GV CPA-JC CPA

336

T [K]

the association energy parameter of the cross-associating component is set to zero and the association volume parameter is fitted to binary VLE data. When the CR1 rule is applied in the association term, it effectively implies that the cross-association energy between the two components is equal to half the association energy of the self-associating component and that the cross-association volume is fitted. Our reason for doing so is to verify if a single association volume parameter value can be used to describe the cross-associative behaviour of a component such as acetone in mixture with several alcohols. Presented in Table 7, are the crossassociation volume values used in this work for several components that do not self-associative, but do solvate. These cross-association values were determined from one binary VLE data set as indicated in Table 7. The components are modelled with the 2B scheme as proposed by Kleiner and Sadowski [7,88] (it should be kept in mind that, because the association energy is zero, no self association is induced). Table 8 shows VLE results for several Type 1 and Type 2 systems, indicating that very satisfactory VLE representations are obtained with CPA, CPA-GV and CPA-JC. The polar versions are marginally more accurate compared to normal CPA. For VLE predictions of the acetone/methanol system where cross-association is not treated explicitly (Fig. 13) CPA, CPA-JC and CPA-GV perform similarly. This indicates that the major source of error is not from the incorrect description of long-range polar forces, but from an incorrect description of the hydrogen bonding present in the system. Fig. 14 presents VLE representations of the same system when cross-association is described with the parameters presented in Table 7. Clearly, very accurate representations are obtained with

333

330

327

0

0.2

0.4 0.6 mole fraction acetone

0.8

1

Fig. 13. Isobaric VLE predictions of the acetone/methanol system at P = 1.0132 bar with CPA-GV, CPA-JC and CPA when cross-association is treated explicitly. Experimental data taken from Ref. [91].

Table 8 VLE predictions of associating polar/polar mixtures with CPA, CPA-GV and CPA-JC. Mixture

T or P

Methanol/ethanol Methanol/ethanol Methanol/1-propanol Ethanol/1-propanol Ethanol/1-butanol Ethanol/1-octanol 1-Propanol/1-pentanol Acetone/methanol Acetone/ethanol Acetone/1-propanol 2-Butanone/ethanol 2-Butanone/1-propanol Methyl acetate/methanol Methyl acetate/1-propanol Ethyl formate/methanol Ethyl acetate/methanol Ethyl acetate/ethanol Ethyl acetate/1-propanol Ethyl acetate/1-pentanol Propyl formate/ethanol Propyl formate/1-propanol Propyl formate/1-butanol Average

n

a

z = (1/n)

b

Deviations as %AAD.

i

298.15 K 373.15 K 333.35 K 333.15 K 343.15 K 1.013 bar 1.013 bar 1.013 bar 321.15 K 1.013 bar 1.013 bar 1.013 bar 318.15 K 1.013 bar 1.413 bar 1.413 bar 328.15 K 1.013 bar 1.013 bar 1.60 bar 1.60 bar 1.60 bar

CPA

CPA-GV

CPA-JC

y (×102 )a

P (%)b / T (K)a

y (×102 )a

P (%)b / T (K)a

y (×102 )a

P (%)b / T (K)a

0.25 0.27 0.59 1.22 0.80 1.59 0.58 0.44 3.38 2.46 0.96 0.84 1.15 1.80 0.89 2.46 1.31 1.44 2.03 0.88 1.33 1.57

0.36 0.79 1.58 4.02 1.56 1.36 0.29 0.14 7.31 1.59 0.21 0.50 1.06 1.43 0.45 1.36 0.89 0.32 2.12 0.38 0.54 0.75

0.52 0.29 0.61 1.52 0.87 3.44 0.57 0.36 0.94 1.19 0.74 0.67 0.78 0.64 0.64 1.19 0.98 1.08 2.07 0.80 1.21 1.50

0.55 0.62 0.76 4.43 1.40 2.50 0.36 0.09 1.87 0.51 0.16 0.36 0.76 0.36 0.32 0.49 0.64 0.22 2.12 0.31 0.55 0.72

0.58 0.29 0.61 1.51 0.83 3.34 0.57 0.41 0.75 0.94 0.58 0.59 0.48 0.48 0.79 1.01 0.77 1.04 2.17 1.05 1.59 2.44

0.59 0.56 0.77 4.34 1.30 2.34 0.40 0.08 1.95 0.52 0.10 0.31 0.41 0.41 0.30 0.50 0.49 0.22 2.17 0.34 0.54 1.24

1.28

1.32

1.02

0.91

1.03

0.90

|z cal − z exp |, where z represents y or T and n is the number of data points.

n

Ref.

11 10 26 9 8 25 19 22 14 10 21 21 32 34 28 27 14 33 16 30 34 28

[97] [98] [99] [100] [101] [102] [103] [91] [104] [85] [92] [92] [93] [105] [94] [94] [95] [106] [85] [96] [96] [96]

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A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

Table 9 VLE predictions of ternary mixtures with CPA-GV, CPA-JC and CPA at P = 1.013 bar. x1 (×102 )a

Mixture/model

x2 (×102 )a

multi-component systems in a satisfactory manner without using x3 (×102 )a

y1 (×102 )a

y2 (×102 )a

y3 (×102 )a

Acetone(1)/diethyl ether(2)/n-hexane(3) CPA-GV (all kij = 0) CPA-JC (all kij = 0) CPA (k12 = 0.025, k13 = 0.08, k23 = 0.025)

0.00 0.00 0.00

3.36 2.68 2.30

3.36 2.68 2.25

2.60 2.18 1.95

2.19 1.79 2.32

1.66 1.57 1.28

Ethyl acetate(1)/n-hexane(2)/acetone(3) CPA-GV (all kij = 0) CPA-JC (all kij = 0) CPA (k12 = 0.069, k13 = 0.011, k23 = 0.08)

0.00 0.00 0.23

3.10 3.07 4.35

3.10 3.07 4.11

1.36 1.23 1.67

3.24 2.86 5.03

3.95 3.58 4.84

Cyclohexane(1)/MIPK(2)/n-octane(3) CPA-GV (all kij = 0) CPA-JC (all kij = 0) CPA (k12 = 0.067, k13 = 0, k23 = 0.064)

0.00 0.00 0.01

1.37 1.46 1.54

1.37 1.46 1.52

0.95 0.92 1.26

1.08 1.12 1.25

0.74 0.82 0.62

Ethyl acetate(1)/acetone(2)/ethanol(3) CPA-GV CPA-JC CPA

0.08 0.03 0.08

3.84 3.74 4.38

3.79 3.72 4.31

2.55 2.54 4.81

1.23 1.25 1.33

2.65 2.48 4.61

Acetone(1)/methanol(2)/1-propanol CPA-GV CPA-JC CPA

0.00 0.00 0.00

1.37 1.60 2.24

1.37 1.60 2.24

0.94 0.97 2.11

1.48 1.57 2.69

0.72 0.89 1.08

a

z = (1/n)

n i

n

Ref.

15

[108]

106

[109]

77

[35]

59

[110]

15

[111]

|z cal − z exp |, where z represents y or x and n is the number of data

points.

all three models. We generally found that the cross-association volume values presented in Table 7 can be used to describe the cross-association in other alcohol systems as well. The association volume parameter values for acetone were determined from acetone/methanol VLE data. These association volume values also provide very good VLE predictions of the acetone/ethanol and acetone/1-propanol systems (see Table 8), especially for CPA-GV and CPA-JC. In some cases, the prediction of CPA was less accurate compared to CPA-JC and CPA-GV, as indicated in Fig. 14 for the methanol/ethyl acetate system (the association volume parameter ethyl acetate were determined form ethanol/ethyl acetate VLE). 6. VLE of multi-component systems The results presented in Table 9 indicate how the polar CPAGV and CPA-JC models are able to predict the VLE of some

362

P = 1.413 bar CPA-GV CPA-JC CPA

T [K]

357

352

347

BIPs. Therefore, the polar CPA-variants appear to be powerful models that may be used to model complex multi-component systems. CPA without the polar terms can, of course, also represent such systems satisfactorily, but with BIPs. 7. Conclusions In this work, the JC and GV polar theories are modified and incorporated into the CPA EOS. The resulting EOS, CPA-JC and CPA-GV respectively, each require four pure component model parameters to model non-hydrogen bonding polar components and six pure component parameters for hydrogen bonding components. Pure component parameters were regressed for several polar components by including saturated vapour pressure, liquid density and binary VLE data in the objective function. Compared to normal CPA, both CPA-JC and CPA-GV offer considerable improvement in the prediction of VLE data for ketone/alkane, aldehyde/alkane, ester/alkane, ether/alkane and alcohol/alkane binary mixtures. No or very small BIPs are required by CPA-GV and CPA-JC to obtain accurate VLE representations. However, at very low temperatures, both CPA-GV and CPA-JC are prone to predict false LLE splits. Also, the VLE and excess enthalpy still cannot be predicted simultaneously with the same set of model parameters. Very good predictions are possible with the two polar models when describing the VLE of binary cross-associating systems, where one component solvates but does not self-associate. In addition, the VLE of multi-component systems are predicted with good accuracy. CPA-JC and CPA-GV perform similarly, with neither model being superior to the other and both prove to be powerful models that are numerically less intensive than their PC-SAFT counterparts. List of symbols

342

0

0.2

0.4

0.6

0.8

1

mole fraction methanol Fig. 14. Isobaric VLE predictions of the methanol/ethyl acetate system at P = 1.413 bar with CPA-GV, CPA-JC and CPA. Experimental data taken from Ref. [94].

A P T Q V

Helmholtz free energy (J) pressure (bar) temperature (K) function used to describe hydrogen bonding (mol) total volume (L)

A.J. de Villiers et al. / Fluid Phase Equilibria 312 (2011) 66–78

a0 energy parameter in CPA (bar L2 /mol2 ) b co-volume parameter in CPA (L/mol) ˚ temperature dependent segment diameter (A) d g radial distribution function k Boltzmann constant (J/K) binary interaction parameter kij m segment number in SAFT-type models n number of moles (mol); number of data points np number of polar segments parameter in GV-term fraction of polar segments in JC-term xp I2,ij and I3,ijk correlations functions in GV-term J2,ij and J3,ijk correlations functions in JC-term Greek letters ˇAB association volume parameter ε/k dispersion energy parameter in SAFT models (K) εAB /R association energy parameter (K)  reduced density reduced density in JC-term * ˚  segment diameter (A) dipole moment (D)  Superscripts calc calculated exp experimental nc number of components sat saturated Abbreviations %AAD absolute average deviation BIP binary interaction parameter Cubic-Plus-Association CPA CPA-GV CPA extended with the GV dipolar term CPA-JC CPA extended with the JC dipolar term EOS equation of state Gross and Vrabec dipolar term GV JC Jog and Chapman dipolar term PCP-SAFT PC-SAFT extended with the GV dipolar term Polar PC-SAFT PC-SAFT extended with the JC dipolar term sPC-SAFT simplified Perturbed-Chain-Statistical-AssociatingFluids-Theory vapour–liquid-equilibria VLE Acknowledgements The financial assistance of Sasol Technology (Pty) Ltd and the Department of Trade and Industry (DTI) of South Africa through the Technology and Human Resources for Industry Programme (THRIP) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the sponsors. References [1] P.K. Jog, W.G. Chapman, Mol. Phys. 97 (1999) 307–319. [2] S.G. Sauer, W.C. Chapman, Ind. Eng. Chem. Res. 42 (2003) 5687–5696. [3] P.K. Jog, S.G. Sauer, J. Blaesing, W.G. Chapman, Ind. Eng. Chem. Res. 40 (2001) 4641–4648. [4] N.M. Al-Saifi, E.Z. Hamad, P. Englezos, Fluid Phase Equilib. 271 (2008) 82–93. [5] F. Tumakaka, G. Sadowski, Fluid Phase Equilib. 217 (2004) 233–239. [6] J. Gross, J. Vrabec, AIChE J. 52 (2006) 1194–1204. [7] M. Kleiner, G. Sadowski, J. Phys. Chem. C 111 (2007) 15544–15553. [8] A.J. de Villiers, C.E. Schwarz, A.J. Burger, Fluid Phase Equilib. 305 (2011) 174–184. [9] G. Kontogeorgis, E. Voutsas, I. Yakoumis, D. Tassios, Ind. Eng. Chem. Res. 35 (1996) 4310–4318. [10] G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. [11] M.S. Wertheim, J. Stat. Phys. 35 (1984) 19–34. [12] M.S. Wertheim, J. Stat. Phys. 35 (1984) 35–47.

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