HuIgPHAS[ EQUIUBRIA

E L S E V I E R Fluid Phase Equilibria 137 (1997) 275-287

Viscosity prediction from a modified square well intermolecular potential model: polar and associating compounds

Wayne D. Monnery *, Anil K. Mehrotra, Wil l iam Y. Svrcek

Department of Chemical and Petroleum Engineering, The Unicersity of Calgary, 2500 University Driz~e NW, Calgar).', Alberta, Canada T2N 1N4

Received 30 May 1996; accepted 20 February 1997

Abstract

A theoretically based model for calculating both liquid and gas phase viscosities has been developed by modifying a statistical mechanics viscosity model based on the square well intermolecular potential. The original theory was corrected to account for the assumptions of only two-body interactions and molecular chaos for velocities and the inadequacy of the square well potential. In addition, the model was modified so that it approaches a consistent low density limit and to improve the dilute gas temperature dependence. The three model parameters are obtained from gas and liquid viscosity data and generalized with group contributions. With the resulting modified square well viscosity model, gas and liquid viscosities for a wide variety of over 100 polar and hydrogen bonding compounds are correlated with average deviations of 0.4% (gas) and 2.1% (liquid), and predicted with average deviations of 2.1% (gas) and 6.8% (liquid). © 1997 Elsevier Science B.V.

Keywords: Viscosity; Statistical mechanics; Model; Liquid; Vapour; Polar and associating

1. Introduction

In the chemical and process industries, viscosity is an important property in hydraulics, heat and mass transfer calculations for surface processing facilities, pipeline systems and porous media. With the increased popularity of process and reservoir simulators, there is a need for a reliable and analytical predictive model for viscosity calculations. Also, like some PVT data, viscosity data provide a macroscopic physical link to the intermolecular pair potential.

Recently, an extensive review of viscosity prediction and correlation methods was completed by

* Corresponding author. Tel.: 0014032205542. Fax: 0014032844852.

0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0378-3812(97)00090-3

276 W.D. Monnery et al. / Fluid Phase Equilibria 137 (1997) 275-287

Monnery et al. [1], which showed that, although there is an abundance of literature and many viscosity models are available, among the predictive methods, most have only been fit and /o r are restricted to a narrow range of compounds or mixtures with only a few empirical models widely generalized. Also, typically, many are applicable over limited ranges of temperature and pressure and relatively few are applicable continuously from the dilute gas to liquid phase. Of those, simple semi-theoretical methods based in statistical mechanics (such as corresponding states, hard sphere and square well theories) appear to be the most promising for engineering use. Their theoretical basis gives the correct qualitative trends with temperature and density, resulting in more confident extrapolation. Moreover, the parameters typically have some physical significance. However, more work is needed to apply these models to a wide variety of compounds and mixtures over wide ranges of temperature and pressure, generalizing parameters to keep the models predictive. This is especially true for polar and hydrogen bonding or associating fluids. For example, the modified hard sphere model of Chung et al. [2,3] has been generalized and applied to a variety of compounds and mixtures, although for associating fluids, it requires a parameter that has been determined and generalized for relatively few compounds (e.g. water, 1-alkanols). The extended corresponding states method is also predictive, but does not give particularly good results for fluids significantly different in structure from the reference fluid(s), and has not been extended to and generalized for polar and associating fluids. In addition, Monnery et al. [1] showed a need for a pure component model, since the best mixture viscosity models are functions of the pure component viscosities.

Previously, we presented a modified square well model and applied it to a wide variety of nonpolar fluids. Results were excellent with gas and liquid viscosities for 65 pure components correlated with average deviations of 0.7% (gas) and 1.3% (liquid), and predicted with average deviations of 2.2% (gas) and 4.7% (liquid) [4]. In this work, we apply the modified square well viscosity model to calculate liquid and gas phase viscosities for 100 organic polar and associating compounds and water.

2. Modified square well viscosity model development

The square well intermolecular potential is the simplest potential model that accounts for both repulsion and attraction, albeit crudely. However, as stated in Ref. [5], the resulting viscosity model is an excellent compromise between realism and ease of calculation. As in dense fluid hard sphere theory, the two important assumptions in deriving the model are: (1) only 2-body interactions are important and (2) molecular chaos for velocities (but not positions), i.e., no statistical correlation between succeeding velocities of particles. Also analogous to dense hard sphere theory, a modified Boltzmann equation was derived accounting for positional correlation at higher density through the radial distribution function. By also accounting for the different types of interactions or collisions for a square well particle and solving the resulting modified Boltzmann equation, Davis et al. [6] and Davis and Luks [7] derived the following model:

r/ [1 +0.4bp(g(o' l ) + R3g(Ro',)~bl)] 2

I g(o-,) + 6 kTJ

48 (bp)2[g(crl)+R4g(Ro_)~2 ] (1) + 25----~

W.D. Monnery et al. / Fluid Phase Equilibria 137 (1997) 275-287 277

where parameters q'~ and qt 2 are defined as:

0, = 1 - e ~/kr + ( E / 2 k T ) 1 + ( 4 / e ~/kr e-X-x2dx (2)

o 0

02 = e ' / * r - e / 2 k T - 2 fo x2( x 2 + e / kT ) l / 2e - ' 2dx (3)

For a given component at a given density and temperature, this model requires five parameters, namely o- 1 (or b) e (or e/k) , R, g(o- 1) and g(Rol ) , as well as values of the parameters qtj and qt 2.

2.1. Model parameters

To calculate qt I and qt 2, the integrals in Eqs. (2) and (3) have been approximated by the following algebraic equations that reproduce the values of g t and g'2 within 0.8% for 0 < E / k T < 4 and extrapolate smoothly:

1(0~) = 1.0/[2.2519 + 0.46614(E/kT) + 1.4843(E/kT) 2

- 0 . 4 4 1 1 7 ( E / k T ) 3 + 0.18849(E/kT) 4] (4)

I( 02 ) = [0.50419 + 0.28304( e / k T ) ] / [ 1 . 0 + 0.15360( E/kT)] (5)

Although slightly more complicated, Monnery et al. [4] showed that the above expressions are considerably more accurate than those given by Du and Guo [8], who also modified the Davis et al. [6], Davis and Luks [7] theory.

The well width parameter, R, was set to 1.5, which is a typical value shown to provide a good compromise between fitting second virial coefficient data, and critical temperature and volume for real fluids [9]. According to Benavides et al. [10], with this value, the phase behavior of a square well fluid is similar to a Lennard-Jones 6-12 fluid. Furthermore, with a constant value of R = 1.5, calculation of radial distribution functions is considerably simpler, resulting in a more practical model.

The radial distribution function values are determined by the EXP approximation of the WCA perturbation theory [11,12], as proposed by Vimalchand et al. [13]:

g(cr ,) = gHs(O'I) exp[ c~(o-,)E/kT] (6)

g(R° '1) = gHs(R°' ,) exp[ a(Ro ' , )E / kT] (7)

where gHs(°'l) = the hard sphere value of the radial distribution function at o- l, determined from the Carnahan and Starling [14] equation, and g ns(R°~) = the hard sphere value of the radial distribution function at Ro-~ determined from a correlation fitted to the Monte Carlo simulation results of Barker and Henderson [15].

gHs(° l ) = [1.0 - 0 . 5 ( b p / 4 ) ] / [ 1 . 0 - (bp /4 ) ] 3 (8)

gHs(Ro',) = 0.99948 + 0.20601 bp - 0.21686(bp) 2 (9)

Eq. (9) matches the results of Barker and Henderson [15] with an overall AAD of 0.35% over a reduced density range of 0.2 < 0 o-3 < 0.9 (0.419 _< bp < 1.885).

278 W.D. Monnery et al. / FIuM Phase Equilibria 137 (1997) 275-287

The advantage of the EXP approximation is that it reduces correctly to the theoretical dilute gas limit when c~ is a function of density and approaches unity. However, in this work, a is taken as a constant for simplicity. The values of c~ (o- l) and c~ (R o- l) were determined to be - 0.2 and - 0.5, respectively, by minimizing the deviations between the predicted values from Eqs. (6)-(9) and the Monte Carlo simulation values of Henderson et al. [16], matching these values within 5% [17].

With the well width parameter set and the radial distribution function values determined, the square well model is reduced to two parameters, b and e/k.

2.2. Model modifications and corrections

In the ideal gas limit, the square well model reduces to r /= rlo/D(e/kT) where:

D( E/kT) : g( o-l; p --+ O) + R2 g( Rtr, ;p --+ O)f( e/kT) (10)

which is the denominator of the first term of Eq. (1) and:

f ( e / k T ) = ~2 + ( 1 / 6 ) ( e / k T ) 2 (11)

According to Chapman-Enskog theory, the dilute gas viscosity for a fluid interacting with the square well intermolecular potential is written as:

2f~ (2,2) ~ - - ~ ( ~ "~)* r/o,sw = (5/16rrl/2)(mkT)'/2/( cr sw ) = r /o / sz~" (12)

This implies that in the ideal gas limit, D(e/kT) should be equivalent to 0(2'2)* ~ s w , as long as the radial distribution functions approach the correct ideal gas limit, e x p ( - crP/kT). As shown previously [4], this is not the case. Furthermore, when the perturbation theory is used for determining the radial distribution function values, D(e/kT) does not match 0{2"2)* ~-sw at e/kT> 0.8, due to the constant empirical coefficients.

To ensure that the proper ideal gas limit is reached with the radial distribution function values determined from the perturbation theory, f ( e / k T ) has been modified and is determined from the following expression [4]:

f ( e / k T ) = 0.811 + 0.788 exp(e /2 .055kT) (13)

As previously stated, the assumptions made in the derivation of the square well model are essentially the same as those made in the derivation of the dense hard sphere model of Enskog. As such, it likely suffers from the same shortcomings. For dense hard spheres, by comparing the results of the Enskog model to the results of molecular dynamics simulations, Alder et al. [18] showed that the assumption of molecular chaos breaks down at sufficiently high densities, resulting in significant underprediction of viscosity, and they provided approximate corrections for this. Similarly, Michels and Trappeniers [ 19] have compared viscosity results of the square well model to molecular dynamics simulation results that show significant underprediction of the model at high densities and low temperatures. Based on these results, a multiplicative correction term, C, was formulated as a function of reduced density, bp, such that the model matched the simulation results at higher densities and tended to unity in the ideal gas limit, where no correction is required:

rt = r/swC (14)

C = 1.0 + ks(bp) k4 (15)

W.D. Monnerk" et al. / Fluid Phase Equilibria 137 (1997) 275-287 279

where ~/sw is the original square well model from Davis et al. [6], Davis and Luks [7]. Parameter k 5 was found to be a function of E / k T and expressed as:

k 5 = k I + k 2 ( e / k T ) k3 (16)

With the values of kl to k 4 set to match the square well model viscosity results to the molecular dynamics results of Michels and Trappeniers [19], values of b and e , / k were regressed in an attempt to simultaneously match liquid and gas phase viscosity data for real fluids. The model could not match gas viscosities well and liquid viscosities were only matched for a few hydrocarbon families. Although both gas and liquid phase data were used, sensitivity analysis of the regressions showed that liquid viscosity tended to control the optimum values of b and that e l k did not have enough 'correlating power' to match gas viscosity. This is the well-known problem of parameters regressed from a gas phase property giving poor results in the liquid phase and vice versa, which indicates inadequacy of the intermolecular potential and resulting model [20,21 ]. In addition, the square well model does not represent the temperature dependence of dilute gas viscosity as well as the Lennard-Jones 6-12 potential. Based on these facts, the dilute gas part of the correction was modified by multiplying by a power function, which approximated the ratio of the reduced collision integrals of the square well and Lennard-Jones potentials, such that the dilute gas square well model has similar temperature dependence to a Lennard-Jones model. This ultimately resulted in the correction being expressed as:

C = k o ( k T / e ) ° z s + [k, + k2(e /kT)k3] (bp) k4 (17)

Physically, this correction accounts for the breakdown of the model assumptions at high densities and low temperatures and the inadequacy of the square well potential. The correction is perturbative with the first term accounting for the inadequacy of the square well potential where k 0 is a structural correction and ( k T / e ) °zs corrects the temperature dependence. This term dominates at low densities and is negligible at high densities. The second or perturbation term accounts for three-body or higher interactions and correlations of velocities between successive collisions. This term dominates at high densities but is negligible at low densities.

Parameters k o to k 4 were also made adjustable to observe the optimum values and trends, resulting in 7 adjustable parameters in the model. As expected, the model now matched both gas and liquid viscosities very well ( < 1% error). Analysis of regressed values of the parameters indicated definite trends for k l, k z and k 4 and relatively low sensitivity of calculated viscosities to these parameters. To minimize adjustable parameters, these were generalized as functions of molar mass and acentric factor, reducing to their square well values as the acentric factor approaches zero:

k 1 = 0.03072 + 0 .00128Mw 1/4 (18)

k 2 = 0.6270 exp( - 0.00242M w'/Z) (19)

k 4 = 1.0 + e x p ( - 0 . 0 3 8 9 5 M w '/2) (20)

Further analysis indicated that typically, 0.6 < E / k T c < 0.7 and could be generalized with critical temperature,

~ / k = 0.65T~ (21)

leaving three adjustable parameters, k o, k 3 and b, which were then regressed again.

280 W.D. Monneo' et al. / Fluid Phase Equilibria 137 (1997) 275-287

3. Database

The density and viscosity data used to regress the parameters and test the model were obtained from the DIPPR database [22]. The polar compounds were selected to represent a variety of organic families and are summarized in Table 1. The temperature range of the gas and liquid viscosity and density data was, typically, from a reduced temperature of 0.48 to 0.70-0.80. However, ultimately, bounds were set in accordance with the recommended temperature ranges for the data as stated in the database along with judgement based on consistency checks.

Note that, in the database, inevitably there are some gaps in available experimental data that have been filled with predicted values. An attempt was made to minimize the use of predicted data, only using it where necessary to cover the desired range of compounds in a given family. Based on quality indicators that are stated in the database, for these polar compounds, predicted gas and liquid viscosity data are typically considered to be accurate within 5-10%. Overall, density data were considered accurate to < 3% and the accuracy of the viscosity data were considered to be 5-15%, based on database quality indicators and comparison with DECHEMA [23] and Thermodynamics Research Center data. This appears to corroborate earlier estimations of overall data accuracy, such as by Ely and Hanley [24].

4. Model results

Adjustable parameters k o, k 3 and b were regressed and the modified square well model simultaneously correlated gas and liquid viscosities excellently, as shown in Table 2. Except for alcohols, phenols and glycols, the model can typically correlate liquid viscosities within about 2% and gas viscosities within about 1%, which is well within experimental error. Although the liquid viscosity data for these exceptions is experimental, perhaps the existing literature data should be reevaluated or new data measured and the database modified and the model refitted. For example, for 1,2- and 1,3-propanediol, the model fits the data of van Velzen et al. [25] and Bleazard et al. [26], respectively, considerably better. (AADs are 50% of the values given in Table 2.)

Table 1 Database compounds

Family No. of compounds

Ethers 15 Ketones 12 Aldehydes 5 Esters 16 Alcohols 14 Phenols 4 Diols 3 Acids 9 Amines 17 Nitriles 5 Water 1 Total 101

W.D. Monnery et al. / Fluid Phase Equilibria 137 (1997) 275-287

Table 2

Results of square well viscosity model correlation and prediction

281

Family Correlation Prediction

% AADL % AADG % AADL % AADG

Ethers 0.70 0.60 7.34 3.50

Ketones 1.13 0.34 4.16 1.39

Aldehydes 0.88 0.43 5.37 1.46

Esters 1.16 0.48 4.05 2.98

Alcohols 3.47 0.59 7.49 1.79

Phenols 3.97 0.17 7.35 1.44

Diols 6.29 0.33 11.85 3.11

Acids 0.54 0.11 9.21 1.81

Amines 1.26 0.41 5.24 2.18

Nitriles 1.84 0.36 5.70 1.20

Overall 2.12 0.38 6.78 2.09

Water 3.46 3.10

Analysis of the regressed optimum values of k 0, k 3 and b showed that they could not be adequately generalized with corresponding state relationships but showed trends within a given family of compounds, as shown in Fig. 1. This suggested that these parameters were related to the structure of the molecule and could be generalized with the well-known concept of group contributions.

As pointed out by Constantinou and Gani [27], the most common groups used for determining pure component properties are different from those used for determining mixture properties. For example, the groups used to determine critical properties of pure components given in Ref. [28] are not the same as the groups used in the UNIFAC activity coefficient model, also given in Ref. [28]. In addition, there is no firm theoretical basis for most sets of groups [28]. In an attempt to provide some basis for group definitions, Wu and Sandier [29] performed quantum mechanical calculations to determine the net charges of specific groups. Groups were established based on having approximately the same charge to make the same contribution to a property, independent of the remainder of the

90o

8o0

700

60o

500

400

"~ 300

200

100

• n-paraffins v olefins [] cy¢lopentanes

cyclohe×anes 0 aromatics 0 alcohols

s~ ~v 0 . . . . i . . . . i . . . .

0 100 200

Molar Mass, M (g/tool)

Fig. 1. Parameter b versus molar mass, M.

0

300

282 W.D. Monnery et al . / Fluid Phase Equilibria 137 (1997) 275-287

Table 3 Parameter group contributions for square well viscosity model

Group A b A k o 2~ k 3

Alcohols

CH 3OH 0.6285 0.7970 3.5090 CH 3CH 2OH 1.0371 0.5168 6.0770

CH2OH 0.7739 ./~glyb 0.5168 ~f~.tck,, fg,yk, 6.4610 falck3 Jglyks CHOH 0.7919 fglyb 0'1697./'~,,~k~ fglyk~ 8.3901 fglyk3 c o n 0.9105 - 0.2351 14.9816

Phenols

aCOH 0.4110 0.1565 2.8160facon

Ketones

CH 3COCH 3 1.0240 - 0.6470 1.9220 CH 2COCH 2 0.8124 - 0.0408 1.6220 CH3COCH 2 0.9363 0.3204 1.6838 CH3COaC 0.7922 0.1268 - 0.4560

Aldehydes HCHO 0.4567 0.7250 2.2000 CH 3CHO 0.7213 0.6410 2.3770 CH 2CHO 0.6802 (/.3287 1.6268 aCCHO 0.7288 0.1595 - 1.5860

Esters

HCOOCH 3 0.8590 0.6890 2.4900 HCOOCH 2 0.8175 0.3716 2.0403 CH3COOCH 3 1.1414 0.6580 2.1650 CH 2COOCH 2 0.9970 - 0.0207 2.0739 CH 3COOCH: 1.0776 0.3346 2.1724 CH 3COOaC 0.7289 (/.0505 ! .5935

Acids

HCOOH 0.6173 0.6870 4.7170 CH 3COOH 0.9445 0.7470 3.4540 CH2COOH 0.9075 0.3746 2.7872 faci CHCOOH 0.8501 0.0523 2.7937 aCCOOH 0.0101 - 0.0950 11.6910

Ethers

CH 3OCH 3 0.7260 0.6130 0.6710 CH2OCH2 0.6640 0.0026 1.4871 CH 3OCH 2 0.6845 0.3256 1.3645

cCH 2OCH 2 0.6605 0.5070 1.0260 fethc aCO aC 0.4093 0.4760 - 2.0100 CH 3 ° aC 0.5655 0.0915 0.3230 CH3OC 0.6471 - 0.3071 1.2216

W.D. Monnery et al./ Fluid Phase Equilibria 137 (1997) 275 287 283

Table 3 (continued)

Group A b A k o A k 3 Amines

CH 3 NH z 0.5951 0.6600 2.6270 CH 2 NH 2 0.5432 0.3386 3.1088 CHNH 2 0.5801 0.0362 1.2230 (CH 3)2 NH 0.8831 0.7690 1.9091 (CH z)e NH 0.7495 0.0202 2.2677 (CH3)3N 1.2119 0.6850 1.4880 (CH 2)3 N 0.8671 - 0.3043 1.1770 aCNH2 1.5230 0.7220 3.8910

Nitriles CH 3CN 0.7725 0.6251 0.1940 CH eCN 0.6730 0.2906 1.3692 aCCN 1.5975 0.6530 2.3060

For corrections, see Appendix C.

molecule. A table of proposed groups, which showed many groups that differed from the UNIFAC definitions, was given.

In this work, for convenience and consistency, it was attempted to utilize existing groups and not develop new, customized ones although, initially it was not known which groups would work the best. However, as work progressed, it was found that in most cases, the groups proposed by Wu and Sandler [29] worked very well in correlating the model parameters along with the following additive equations:

b i / l O 0 = Y'~ n i j A b j + 0.06617 j = l

g

ko,i = E nijAko4 j = l

g

k3, i = 1.0 + Y'~ nijAk3, j j = l

g )2 EnijAbj (22)

j = l

(23)

(24)

The group contributions were obtained using one family at a time, starting with n-paraffins to determine CH 3 and CH e groups and continuing in this manner to determine the remaining groups. The resulting groups and contributions for nonpolar compounds are given in Ref. [4] and those for the polar and associating compounds in this work, are given in Table 3. As expected, the first members of a homologous series (methanol etc.) did not trend well with higher members, and to overcome this problem, each of these compounds is treated as its own group. In addition, water was considered as a unique component with the values of its parameters as: k 0 = 0.584, k 3 = 4.356, b = 21.43 cm3/mol.

Some of the group contributions are corrected for iso position, attached chain lengths, number of side chains and occurrence number. These types of corrections are not unreasonable because, as pointed out by Makita [30], liquid viscosity is remarkably sensitive to structure. For example, as a paraffinic side chain length increased, it was observed that ring structures tended to behave more

284 W.D. Monnery et al. / Fluid Phase Equilibria 137 (1997) 275-287

Table 4 Recommended viscosity prediction methods results

Family % AADG % AADL

Reichenberg [31-33] Chung et al. [2,3] van Velzen et al. [34] Chung et al. [2,3]

Ethers 1.91 5.38 15.54 18.28 Ketones 1.66 3.33 3.39 30.17 Aldehydes 1.06 6.01 3.47 N / A Esters 4.08 5.29 6.81 19.20 Alcohols 2.31 2.93 12.87 39.36 Phenols 0.71 14.69 31.24 43.88 Diols 2.61 4.83 10.78 74.37 Acids 1.12 10.22 16.30 37.91 Amines 0.97 8.79 36.25 29.58 Nitriles 4.43 24.09 N / A N / A Overall 2.09 8.56 14.24 32.66

paraffinic with the ring dominance decreasing. As such, chain length corrections have been expressed to model this behavior. The number of side chains on ring structures also appeared to have an effect.

As for other examples, it is not expected that the contribution of CH 2 groups on different cyclic rings would be identical and it appeared that more than one occurrence of a group, as in dienes, has an effect. These effects have also been accounted for in the work of Constantinou and Gani [27] as second-order corrections. However, as can be seen, the corrections in this work are very simple and do not complicate the calculation procedure.

The results of predicted viscosities using the group contributions are also shown in Table 2. The results of the viscosity prediction methods recommended in Ref. [28], applied to the database used in this work, are shown in Table 4. Note that N / A (not applicable) in Table 4 means either a group contribution method did not have the required group or that a method had some kind of problem dealing with the particular compounds, but was not penalized for it. For the most part, these results are consistent with results reported in Ref. [28]. It is beyond the scope of this work to discuss the details of the results from Table 4. These methods have been reviewed in [i]. However, it should be pointed out that the method of Chung et al. [2,3] performs much better when the association factor in the model is available.

The important point to be made is that the results in Tables 2 and 4 show that the predictive modified square well method compares very favorably with the recommended methods in Ref. [28], which was also the case for nonpolar compounds as shown in [4]. In addition, calculations indicate that the modified square well model can be considered accurate within 15% from a minimum temperature of 0.26 + Ttpr/2 up to a b o u t T r = 0.90 [17]. An example to illustrate the viscosity calculation procedure is given in Appendix A.

5. Conclusion

A theoretically based viscosity model suitable for engineering calculations and applied to a wide variety of polar and associating compounds, has been presented. It typically correlates gas and liquid

W.D. Monnery et al. / Fluid Phase Equilibria 137 (1997) 275-287 285

viscosity data within 0.4 and 2.1%, respectively. The model has been generalized and typically predicts gas and liquid viscosities with average deviations of 2.1 and 6.8%, respectively. Previous work showed that the model gives similar results for a wide variety of nonpolar compounds which compare very favorably with currently recommended methods for calculating gas and liquid viscosi- ties.

Appendix A. Example of viscosity calculation procedure

1. Estimate the viscosity of liquid and gas 2-propanol at 270 K.

2. Data: M = 88.11 g / m o l , T c = 523.30 K, w = 0.3664; liquid molar volume at 270 K = 74.51 c m 3 / m o l ; gas molar volume at 270 K = 2.53323 × 1 0 6 cm 3 /m o l .

3. Groups: 1 × H C H O H and 2 × H CH 3.

4. From Eqs. (18)- (20) : k 1 = 0.100, k 2 = 0.557, k 4 = 1.148.

5. From Eqs. (22) - (24) , Table 3, [4]: k o = 0.873, k 3 = 10.322, b = 165.98 cm 3 /m o l .

6. From Eq. (21): Elk = 330.39 K; E/kT= 1.224.

7. From Eqs. (2), (3) and (5): ~1 = B0.7803, 1/ t 2 = 1.3559.

8. From the definition of the hard sphere dilute gas viscosity (in practical units): r/0 = 3.1163 × 10 -3 (MT)l/2/b2/3 = 0.01314 mPa s.

9. From Eqs. (6) and (8) ( a = - 0 . 5 ) : g(0-~) = 6.4947 (L), 0.7829 (G).

10. From Eqs. (7) and (9) ( a = - 0 . 2 ) : g(Ro- I) = 0.2073 (L), 0.5421 (G).

11. From Eq. (17): C = 12.292 (L), 0.8246 (G).

12. From Eq. (1) with f(E/kT) f rom Eq. (13): %red = 4.825 (L) mPa s, TIpre d = 0.00701 (G) mPa s.

13. Liquid experimental: r/exp = 5.001 mPa s, error = - 3 . 5 % .

14. Gas experimental: T~exp : 0.00689 mPa s, error = + 1.8%.

15. Overall error for 2-propanol: AADL = 6.5%, A A D G = 1.20%.

Appendix B

List of symbols a

b C

g(0-1) g( R 0-1) k

= aromatic (Table 3) = covolume = (2/3)'rrN A o-13, c m 3 / m o l = correction factor = radial distribution function at repulsion diameter = radial distribution function at attraction diameter = Boltzmann constant

286 W.D. Monneo, et al. / Fluid Phase Equilibria 137 (1997) 275-287

k o . . . k 4 = parameters in correction term m --par t ic le mass, g M = molar mass, g / t o o l N A = Avogadro ' s Number n i j : number o f jth groups in compound i R = well width parameter T = temperature, K T~ = critical temperature, K T r = reduced temperature Tip r = reduced triple point temperature D = density, m o l / c m 3

Greek letters ot = empirical coefficient in radial distribution function perturbation theory A Pi = group contribution o f jth group for parameter P where P = b, k 0 or k 3 • = energy well depth

= viscosity, mPa s ~/o = hard sphere di lute gas v i scos i ty = ( 5 / 1 6 7 r l / 2 ) ( m k T ) l / 2 / o - i 2

lO-3(MT)l/Z/b2/3, mPa s

o-~ = repulsion diameter (R o-~ = attraction diameter) to = acentric factor

(2,2)" = reduced collision integral

= 3 .11630 ×

Acronyms A A D L = average absolute deviation (liquid) A A D G = average absolute deviation (gas) WCA = Weeks Chandler Andersen

Appendix C. Corrections for Table 3

Iso: fcH~b = exp( - -0 .16ncH ~ - 0 . 0 6 n c n - 0 . 1 6 n , , c n - - 0.24nCH~COCH2 -- 0.06n(CH2)2NH), fCH3k3 = exp( --0.10riCH (riCH + n C) -- 3.()(ncn~b -- 1)(nCH2COCH, q- n(c H ) NH)).

Diene: if(nc-'H,=CH q- nCH=CH q- n c t i _ C) > l, fdio -" ().8. 22 Cyclic ring si~te chain length: f c c , _G e x p ( - 0.20ncno), fcc6 = e x p ( - 0.15ncn2). Aromatic ring side chain length and number: fben = exp( -- 0.11 nCH2), faC,b ~- (0 .613 + 0.387(n~c '

q- n aCO. )), f a C , ~'3 = 1.0/(n~c ' + n aCOH)" Alcohol chain length: fdlck,, = e x p ( - 0.03riCH ) , Jalck~ = e x p ( - 0 . 0 0 9 5 9 (n c . a - 3 ) 2 q- 0 . 2 8 r i C H ) .

Diol correction: if (ncHao H + nCHOH) > 1, fg~yb = 0"32nCHEOH + 0"52nortOn + 0.06ncH2, fglygo = 0-40ncH~OH + 0"36nCHOH, fglyk~ = 0.27ncmoH + 0"35nCHOH + 0-09nc.2"

Phenol side chain number: f,,COH = naCOH + 0'33n,,c, '

Cyclic ether: fethc = ncCH2OCH:" Carboxylic acid chain length: f~ci = 1.0 + ln(1.0 + 0.10riCH2).

W.D. Monnery et a l . / Fluid Phase Equilibria 137 (1997) 275 287 287

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