A local composition model for multicomponent liquid mixture shear viscosity

Chemrcal Engineering Science Vol. 40. No. 3, pp. 40 I-408, Printed in Great Britain.

A LOCAL MULTICOMPONENT

1985

W9-2509/M $3.00 + .OO Q I985 Pergamon Press Ltd.

COMPOSITION MODEL FOR LIQUID MIXTURE SHEAR VISCOSITY

I. C. WE1 and R. L. ROWLEY* Department of Chemical Engineering,William Marsh Rice University, Houston, TX 77251, U.S.A. (Received

14 September

1983)

local composition model for multicomponent.nonaqueous,liquid mixtureshearviscosityhas been developed and tested. Only binary equilibriumthermodynamicinformation is used in the model in addition to pure component data. No mixtureshear viscositiesand no adjustableparametersare required. Predictionsbased on this model were compared to experimentaldata obtained from the literatureand in this laboratory, yielding an average absolute deviation of 1.1% for ln(qV) for 47 binary and 0.8% for seven ternary systems. Abstract-A

INTRODUCTION

There are currently increasing demands on the accuracy of liquid mixture shear viscosity prediction techniques for implementation in industrial property prediction computer routines. For these purposes, it is desirable to have equations that are predictive in nature, i.e. that do not require parameter fitting from experimental data for the desired property. Unfortunately, because of the system-specific nature of interactions in liquid mixtures, it appears that some kind of (at least) binary interaction information must always be included to make mixture predictions quite general for the variety of different interactions encountered. While a few nonparametric models[l4] have been developed for estimation of the composition dependence of liquid mixture viscosity, the results are not usually very accurate unless some viscosity data have been included to fit adjustable parameters. Many viscosity data correlations, requiring a varied number of adjustable parameters, have been utilized over the last twenty years[S12]. Because it appears that some binary interaction information is required to make accurate estimation of mixture properties, we propose here a method which utilizes pure component data and a limited amount of readily available binary mixture thermodynamics information to make accurate liquid mixture property predictions. We find the method to yield better results on a greater variety of systems than any existing predictive technique. Required binary data for this method are excess enthalpies and NRTL (nonrandom two-liquid) model parameters for the excess free energy. Although VLE experiments from which the excess free energy parameters are usually obtained are more difficult to accurately perform than viscosity measurements, there are compelling reasons for this approach; *Author to whom correspondenceshould be addressed. Presentaddress: Department of Chemical Engineering, 350

CB, Brigham Young

University, Provo, UT 84602, U.S.A.

401

namely: (1) Considerable VLE data have been obtained, compiled, published, and implemented in many property simulators. (2) The method is sensitive to the excess free energy not the model used. This allows information from computer data banks in the form of NRTL, UNIQUAC, UNIFAC, or Wilson parameters to be used to obtain the required NRTL parameters. (3) Group contribution techniques such as UNIFAC can be used to eliminate the need for any experimentally obtained NRTL parameters. (4) While many techniques require the use of data specific to that property, the work presented here is part of an overall program for predicting a variety of thermodynamic and transport properties from the same equilibrium thermodynamic input data. For example, we have previously shown that liquid mixture thermal conductivity can be accurately predicted using NRTL parameters [ 131. Thus a property prediction package can be set up based on simple pure component constants and the currently accessible binary thermodynamic GE model parameters. (5) There is currently no other way to make viscosity predictions with the same degree of accuracy and reliability for all nonaqueous systems, regardless of the nature of the constituents, without resorting to adjustable viscosity parameters. Most semi-empirical techniques for mixture tiscosity prediction have been based on either free volume theories for fluidity [ 141 or Eyring’s absolute rate theory [15]. Free volume theories have been quite successful in correlating the temperature dependence of pure component viscosities, but the composition dependence of the free volume is still unknown. Free volume theories do, however, make pure component data more accessible under a variety of conditions thereby enhancing the desirability of mixture models which require pure component data at the mixture conditions. McAllister [6] has developed an equation with two adjustable parameters which is based on the

402

I. C.

War and R. L. ROWLEY

Eyring model and works very well for binary systems[8]. His equation is based on three-body interactions and requires at least three adjustable parameters for ternary systems[S, 9, 161. Statistical mechanical methods have been developed which formally allow mixture viscosities to be computed from pair potential parameters [ 171, but currently unsolvable integrals over pair correlation functions are also involved. More recent techniques using corresponding states principles are severely hampered by generality problems. They, like the method proposed in this work, are predictive in nature using only readily available properties and thermodynamic concepts to estimate transport properties. Unfortunately, the basis of these techniques is strictly limited to a small group of systems, to those conformal to the reference fluid used in the calculation or to those that do not contain polar constituents. While some attempts have been made to extend corresponding states techniques to a broader application range, the accuracy of the predictions is quite often very poor unless an adjustable parameter is introduced to account for nonconformality. For example, the corresponding states method developed by Teja and Rice [IS], requires access to pure component shear vicosity data over a wide range of temperatures because the pure fluids are used as references at the mixture reduced temperature. Because the one-fluid theory is used, ad hoc mixing rules for the mixture critical properties are required. These contain an adjustable parameter which must be fit from viscosity data. The accent& factor is used as an interpolation parameter between the two pure component values. Similar procedures for equilibrium properties can only be used reliably, without recourse to an adjustable parameter, for nonpolar mixtures. Moreover, it is not yet clear what the accuracy of this method would be for multicomponent systems since only two of the components can serve as reference Auids. Presumably the method would be considerably less accurate for multicomponent mixtures. Ely and Hanley [ 19,201 have developed a corresponding states technique based on conformal solution theory. Their method is also predictive in nature because no adjustable parameters are required. Although this method requires less input data than our technique (no pure component shear viscosities or binary thermodynamic data), the method is not amenable to nonconformal fluids and cannot therefore be applied to a wide range of mixtures. Mixtures containing polar interactions or strong associations cannot be treated with this method. While corresponding states techniques do not have the versatility and wide constituent range that the method proposed herein has, they are applicable over a wider range of temperatures and pressures when strictly used on systems for which they were designed. We report here a method for prediction of nonaqueous liquid mixture viscosities based on a local composition model for the liquid which yields results that compare more favorably to experimental results

than the above mentioned techniques. It uses relatively readily available thermodynamic data for binary mixtures to evaluate local compositions in the multicomponent system. The mixture viscosity is then evaluated from the computed local compositions. The model requires neither mixture viscosity data nor system specific adjustable parameters to make it general. The local compositions used in the model contain information concerning the nonidealities and specific molecular interactions peculiar to each system. LOCAL

COMFOSlTION

Erying’s 1940 theory[l5]

MODEL

for shear viscosity,

WV = Nh exp (AG +/RT)

(1)

involves the activation free energy, AG +, required to remove molecules within the fluid from their most energetically favored state to the activated state. In eqn (I), 7 is hear viscosity, V is molar volume, N is Avagadro’s number, and h is Planck’s constant_ While eqn (1) was originally used for pure fluids, choice of proper mixing rules for the activation free energy allows it to be applied to mixtures. The free energy of activation can be thought of as a difference between the free energy of the activated state of a molecule moving through the interaction field of its nearest neighbors in response to an applied shear stress and the energetically more favored state characteristic of the equilibrium mixture without shear. A negative excess free energy of mixing contributes to the height of the activation barrier by lowering the equilibrium state relative to that of an ideal mixture. However, AG # for the mixture is an average of the activation energies of the individual molecular species and can therefore only be related to GE through a proportionality factor (because the nature of the average is unknown). Thus, AG+

= AG_$ -

mGE

(2)

where AG,$ represents the activation energy for a hypothetical ideal mixture (in the thermodynamic sense) of the components and u is a proportionality constant yet to be defined. Substitution of eqn (2) into eqn (1) and regrouping of GE in terms of SE and HE yields, qV = (‘IV),

exp @SE/R)

exp ( - aHE/RT),

(3)

where (q V), represents the volume-viscosity product for a hypothetical ideal mixture of the constituent components and has the following definition: (vV),~

= Nh exp (AG,$/RT).

(4)

A nonzero excess entropy term is indicative of nonrandom mixing by the fluid components and can be grouped with (qV), to yield a local shear viscosity _ _. . including nonrandom mixing effects. It is this term

A local composition model for multicomponent liquid mixture shear viscosity for which the local composition model is applicable. The mixture viscosity is therefore composed of one contribution due to nonrandom mixing on the local level and another energetic portion related to the strength of intermolecular interactions which inhibit molecules from being removed from their most favorable equilibrium positions in the mixture. We label the former portion (q&Q,. Thus, (21v) = (tf VI,

exp ( - anE/RT)

allows local volume fractions to be computed from overall mole fractions and the equilibrium thermodynamic parameters Q and A,. Subject to the constraint that X1=, t#+= 1, eqns (8) and (9) yield,

where G, = 1 whenever i =_i. While many correlations have been based on mole fractions[l-5, 221, the same correlations have also been successfully used with volume fractions as the correlating composition variable. While it is not clear which variable should be used, we have chosen to work with volume fractions here because it produces a form for the binary interaction term consistent with that found for thermal conductivity[l3] and because additive equations give slightly better results in terms of volume than mole fractions (see Table 1). In the nonrandom two liquid model, deviations from a volume fraction average of the pure component tvalues are attributed to nonrandom mixing. Applying, the two fluid theory [ 13, 211 yields,

(5)

or defining a property, 5, as 5 = ln(77l% t = &

(6)

- ~TH~JIRT.

(7)

The nonrandom two-liquid theory (NRTL) as developed by Renon and Prausnitz[Zl] and extended to thermal conductivities by Rowley[l3] has been used to compute &. Local mole fractions can be defined as in the original NRTL model ?=:.exp(-

u

403

uAti/RT)=~Gg I

I

j=L

where xv represents the mole fraction of component i around a central molecule of type& G, is defined by eqn (8) and a! and A, are adjustable parameters obtainable from equilibrium data. Defining local volume fractions by

where cij equals &. yields

i-l

Combining

eqns (9) and (10)

Equation (12) contains viscous interaction terms which can be defined in a manner similar to that used

(9)

Table 1. Absolute average deviations (AAD) from experimentalmixture shear viscosity data using 5 and q for 47 binary (391 data points) and 7 ternary (63 data points) systems Equation

No. of Components

AAD

I

NRTL

l&

18b

18c

L8d

n

I

5

2

_-

1.1

__

4.4

3

__

0.8

-_

2.7

2

2.6

2.5

10.1

9.6

3

5.1

5.6

20.7

22.9

2

2.4

2.1

9.5

8.1

3

2.5

1.5

9.1

5.6

2

2.6

2.4

10.3

9.3

3

5.9

4.8

24.2

19.5

2

2.5

2.4

10.2

9.6

3

5.6

4.7

22.9

19.2

I. C. Wm and R. L. ROWLEY

404

by Rowley [ 131 for thermal conductive interactions_ In the pure component limit eqns (7) and (12) require that c =
x& exp G2d x :2exp (Cd x :1ew G d = ~2: exp (t23

(13)

where the x,: are determined entirely from the equilibrium thermodynamic interaction parameters and pure component values. In terms of specific volume fractions eqn (13) can be rewritten as

4: =

V V, &Z

4% exp (f2’/2)

exp (&O/2) + v1 K

exp (&O/2)

(14)

where the relationship between mole and volume fractions, eqn (8), and the fact that volume fractions must add to unity have been used. Likewise, local volume fractions at the point where eqn (13) is satisfied can be computed from eqns (10) and (14). At the * composition the binary &, is assumed to be equivalent to the cross interaction term !&,. It should be emphasized that in so doing no mixture data are used, but rather, elocis averaged or approximated by the interaction term <*, at this composition. This allows a direct solution for t2, from eqn ( 12),

(15) which is a quadratic average of pure component e values analogous to the weight fraction quadratic average used for mixture thermal conductivity interactions. Equations (7) and (12) can be combined to yield

- uHE/RT.

6)

Mixture component viscosities can be computed from the NRTL model by first computing r$: from eqn (14) and equilibrium thermodynamic data in the form of G, and Gji for each of the possible binary pairs. Binary interactions are then computed from

eqn (15) for each possible pair within the multicomponent mixture. Using the obtained viscous binary interaction terms, the excess enthalpy of mixing, and a value for the proportionality or mixing factor, Q, the mixture viscosity can be calculated from eqn (16) and r! = exp (0/v

(17)

where Y is the molar volume of the mixture at the composition in question. At this time there is no theoretical value for u. However, testing has shown that for nonaqueous systems a constant of 0.25 represents a reasonable choice. The objective of this work has been development of a nonparametric liquid mixture predictive equation; eqns (15H17) represent such an equation when d is chosen in this manner. If correlation prowess rather than predictive capability is desired, B can be left as an adjustable parameter with excellent results for a single parameter equation. However, all of the calculations presented in this paper are predictive in nature using a value of 0.25 for the mixing factor and no adjustable mixture viscosity parameters. RESULTS The NRTL viscosity model has been tested on 47 nonaqueous binary liquid mixtures consisting of, as much as possible, systems with a variety of interactions including polar-polar, polar-nonpolar, and nonpolar-nonpolar. Thermodynamic data used in the predictions were obtained from the literature[23-251 for both VLE and LLE data. If u, A 21, and A,, were not reported directly in the literature, values were obtained from the reported equilibrium data using a nonlinear least squares fitting routine. Excess enthalpies were obtained exclusively from Christensen et al. [26] using a NRTL model with temperature dependent parameters for temperature interpolation. Experimental shear viscosities for comparison purposes were obtained from the literature[5,8,27-331 where available and were measured in this laboratory where incomplete or unavailable. The experimental apparatus, error analysis, and data obtained for those systems measured in this laboratory are reported elsewhere[34]. Results are shown in Table 1 along with a comparison of corresponding results obtained from nonparametric additive equations [2-4, 81:

tl

(18b)

In t] = 5 zi In vi i=l

(18~)

Inv = 2 zilnvi i= 1

(18d)

A local composition model for multicomponent liquid mixture shear viscosity

0.3

0

0.2

0.4

0.6

0.8

Ix)

01 and experimental shear viscosity as Fig. 1. Predicted (-) a function of volume fraction for the binary systems triethylamineand acctone-cyclohexane [34] (0) chloroform[34] [I) at 298 K.

where v is the kinematic viscosity and z can be either mole fraction or volume fraction. The average absolute deviation for each system was obtained over the entire composition range using generally 7-10 binary points per system. Typical results for q are shown in Fig. 1 for two systems that show very different composition behaviors. The results shown in Fig. 1 are representative of the effective system-specific nature which incorporation of equilibrium thermodynamic data imparts to the NRTL viscosity model without the use of adjustable mixture viscosity parameters. Ternary mixture shear viscosities have also been computed and compared to experimental data obtained primarily in this laboratory. Experimental details and data obtained can be found elsewhere[34]. While eqns (14X16) were derived in general for an n-component system, only binary interaction parameters appear in the equations. The NRTL model requires only binary thermodynamic data and pure component shear viscosity data. Consequently, testing ternary predictions provides a more stringent test of the model. Table 1 also shows the average absolute deviations obtained from a comparison of both the NRTL and the additive nonparameteric models to experimental ternary viscosities. The results show agreement with experiment similar to that found for binaries. Heat of mixing data for the ternary systems were obtained by fitting binary data to a ternary NRTL equation for WE with temperature dependent parameters for extrapolation and interpolation to the mixture viscosity conditions. DISCUSSION

Correlation of multiple properties based on parameters obtained from a particular property, while very attractive from a theoretical and practical viewpoint, can often be inaccurate. The success of the local composition or NRTL model in predicting transport properties such as thermal conductivity [ 131 and viscosity is somewhat remarkable in this sense. It appears that the value obtained for the binary interaction transport parameter plays a more important

405

role in the accuracy of the generated data than the actual values of G,, and G,,. While the interaction terms are themselves determined from the G, values, they depend strongly on the pure component values and the form of the cross interaction mixing rule. Nevertheless, it is rather remarkable and significant that both thermal conductivity and viscosity yield quite good results when an equation of the same form is used to calculate the binary interaction terms from equilibrium-property-based local compositions; especially since we have found that a relatively small change in the magnitude of the interaction term can lead to quite unacceptable differences between calculated and predicted values. As a more effective means of comparison of the NRTL model to experimental data, an excess 5 can be defined as the difference between the measured and volume fraction average of pure component values, n (191 or from eqn (16),

- oH E/RT.

(20)

Figure 2 shows both the calculated and the experimental c E obtained for five systems which show quite different behaviors. The NRTL model accurately reproduces the composition profiles for these systems although the magnitude ranges from negative to positive and the shape may be symmetric or skewed. The methanol
0.50

0.25

-w, 0.25

_ I

0

02

0.4

0.6

0.8

I

pr,

Fig. 2. Predicted () and experimentalc6 as a function of volume fraction for binary systems: triethylamine-

chloroform [34] (0) at 298 K, aceton~hloroform [34] (0) at 298 K, methanol
I. C. War and R. L. ROWLEY

Fig. 3. Plot of cE for the chlorofonn(l)+thanol(2~methanol(3) ternary system at 298 K. While the main mechanisms for thermal conductive heat transfer in liquid mixtures are through intermolecular interactions in the form of vibrational and rotational energy transfers, molecular mobilities become extremely important in shear viscosity calculations. Thus, the NRTL model applied to thermal conductivity involves only a nonrandom mixing effect. Corresponding deviations from a bulk mole fraction average of pure component S-values must include an enthalpy term for translational interactions as in eqn (20). Equation (20) therefore shows not only an entropic or random mixing term but also an excess enthalpic term proportional to the enhanced or diminished strength of interactions within the mixture relative to pure components. The importance of the excess enthalpy term can be seen in Fig. 4 where experimental values for q are compared to those calculated from the NRTL model and those computed from eqn (16) with NE set equal to zero.

Fig. 4. Shear viscosity for the aceton~hlorofonn [34] system at 298 K as predicted by the NRTL model (-), measured (I), and predicted without the excess enthalpy term (---).

While the local compositions are quite significant in reproducing the correct shape for the composition dependence of the shear viscosity, the excess enthalpy contributes dramatically to the magnitude and shape of mixture viscosities. Several other authors have recognized the importance of the excess enthalpy term15, 35-371. It is instructive to compare the type of results and accuracy of predictions obtained in this work to those obtained from current corresponding states approaches. While Teja and Rice have achieved good results for binary systems with a three parameter corresponding states model1 181, the price paid for this accuracy was the need for pure component shear viscosities over a large temperature range and an adjustable parameter that must be fit from mixture viscosity data. In contrast, our approach was development of an entirely predictive equation that uses easily obtained equilibrium thermodynamic properties as a source of interaction information instead of an adjustable parameter. The real significance of this work is that thermodynamic properties can be used to predict transport properties, i.e. interaction terms transcend the individual property, without adjustable parameters. Use of a value determined from VLE data in place of the adjustable parameter in the mixing rule of the Teja-Rice equation to eliminate need for viscosity data would be comparable to the approach taken in this work. Present indications are that the adjustable parameter in the mixing rule is not a true physical interaction term and must take on different values when applied to each different property. To illustrate the advantage of the local composition model over the corresponding states techniques for mixtures containing polar fluids when no adjustable parameters are fit from experimental mixture viscosity data, we have calculated the average

A local composition model for multicomponent liquid mixture shear viscosity absolute deviation (AAD) for six binary systems each of which have an alcohol as one of the components. The adjustable parameter in the Teja-Rice equation was set equal to unity as suggested by the authors. The resultant AAD was 7.55% using the Teja-Rice equation as opposed to 2.66% using the local composition method. A graphic illustration of the need to include the thermodynamic information inherent in the local composition model is shown in Fig. 5. Although we would like to compare out predictions to those obtained using the method of Hanley and Ely, their method cannot yet be used for polar components. However, two of the systems that they used to test their predictions were also used in our composition model. They tests of the local report[l9,20] AAD’s of 6.02% and 6.09% for the toluene-n-heptane and the benzene-n-hexane systems, respectively; this compares with AAD’s of 0.80% and 0.97%, respectively, using the local composition model. It is also important to note the difference between the two properties used throughout this work. While we have found a local compositional model based on < to be much more effective than trying to use ‘1, it is the mixture viscosity that is usually desired. Many correlations involving adjustable parameters are also based on < [6,8, 151 because of the large deviations from linear composition averages that are exhibited by q, for example that displayed in Fig. 1. Nevertheless, the two properties are related by eqn (17). As can be seen from Table 1, the local composition model yields excellent predictions for 5. Errors due to inaccuracies in the thermodynamic properties, local compositions, and the model itself are amplified when e is exponentiated to obtain q_ The errors in 4 are generally, therefore, perceptibly larger than those in < upon which the NRTL model was based. This fact is evident in Table 1. To date this model is restricted to nonaqueous systems. Further work is in progress aimed at its extension to aqueous systems. It may be used, however, as an accurate data correlation for aqueous

0

0.2

0.4

0.6

0.8

Fig. 5. Shear viscosity vs mole fraction for the acetone-ethanol system at 298.15 K and ambient pressure: measured,(); predictedby the NRTL local composition model, (m): and predicted by the work of Teja and Rice[l8], (0).

407

systems in its existing form if 0 is treated as an adjustable parameter. CONCLUSIONS

A model based on the NRTL theory has been developed which allows reasonably accurate predictions of nonaqueous liquid mixture shear viscosity from pure component viscosities and binary equilibrium thermodynamic data. The specific nature of the interactions associated with mixtures of different components is contained within the thermodynamic data used to compute local compositions and the interaction excess enthalpy characteristic of strengths. The thermodynamic data effectively adjust the model to each specific system without recourse to any adjustable viscosity parameters. While an unknown mixing factor, tr, appears in conjunction with the excess enthalpy term, very good results have been obtained when Q is assigned a value of 0.25. This value is recommended unless an accurate single parameter data correlating equation is desired. The developed model has been tested on 47 binary and seven ternary systems for which the average absolute deviation for < and r] are 1.1% and 4.4% for the binary and 0.8% and 2.7% for the ternary systems, respectively. While other mixture viscosity equations of comparable accuracy contain at least one adjustable parameter for binary and additional parameters for multicomponent systems (for those models extendible to multicomponent mixtures), the NRTL model developed and tested in this work can be used for any number of components without adjustable parameters. Acknowledgement-This

work was supported under DOE

contract number DE-AS05-82ER13008. NOTATION

AAD

average absolute deviation NRTL parameter A, AG+ free energy of activation free energy of activation for ideal mixture AG,:, GE excess free energy Planck’s constant 2 excess enthalpy N Avagadro’s number R Gas constant T absolute temperature V molar volume xi mole fraction of component i x/I local mole fraction of component i around central molecule i mole fraction when 5 21-- c 1.X local mole fraction when 5 *a-- < hX arbitrary composition variable

Greek symbols

a 11 v 5 r 21

NRTL nonrandomness parameter shear viscosity kinematic viscosity ln(rlV) intermolecular viscous interaction term

I. C. WEI and R. L. ROWLEY pure component 5 value < based only on local compositions excess < based on eqn (19) free energy mixing parameter volume fraction volume fraction when czI = & local volume fraction ofj around component locAl volume fraction when &, = El,

REFERENCES

Ill Reid R. C., Prausnitz J. M. and Sherwood T. K., The Prouerties of Gases and Licwids. 3rd Edn. McGrawHill, New York 1977. _ PI Bingham E. C., Fluidity und Plasricity. McGraw-Hill, New York 1922. 131 Kendall J. and Monroe K. P., J. Am. Chem. Sot. 1977 39 1787. [41 Coronauer D. C., Rothfus R. R. ,and Kermore R. I., J. Chem. Engng Data 1965 10 131. PI Katti P. K. and Chaudhri M. M., J. Chem. Engng Data 1964 9 442. McAllister R. A., A.I.Ch.E.J. 1960 6 427. F! Grunberg L. and Nissan A. H., Nature 1949 164 799. 181 Heric E. L., J. Chem. Engng Data 1966 11 66, Heric E. L. and Brewer J. G., J. Chem. Engng Data 1967 12 574. 191 Kallidas R. and Laddha G. S., J. Chem. Engng Data 1964 9 142. [lOI Reed T. M. and Taylor T. E., J. Phys. C&m. 1959 63 58. Cl11 Tamura M. and Kurata M., Bull. Chem. Sot. Japan 1952 25 32. [121 Cullinan H. T., Jr., Znd. Engng Chem. Fundls 1966 5 281. Rowlev R. L.. Chem. EmmE Sci. 1982 37 897. :::; Hildedrand J. ‘H., Viscosit> &d Dt#iiivity: A Predictive Treatment. Wiley, New York 1977. [I51 Glasstone S., Laidler K. J. and Eyring H., The Theory of Rafe Processes. McGraw-Hill, New York 1941.

Noda K., Ohashi M. and Ishida K., J. Chem. Engng Data 1982 27 326. Bearman R. J. and Jones P. F., J. Chem. Phys. 1961 33 1432. Teja A. S. and Rice P., Ind. Engng Chem. Fun&s 1981 20 77. v91 Ely J. F., J. Res. N. B. S. 1981 86 597; Ely, J. F. and Hartley, H. J. M., Ind. Engng Chem. Fun& 198120 323. PO1 Hanley H. J. M., Cryogenics 1976 16 643. Renon H. and Prausnitz J. M., A.Z.Ch.E.J. 1968 14 135. s;; Irving J. B., NEL Report No. 631, National Engineering Laboratory, East Kilbride, Scotland 1977a. Nagata I., J. Chem. Engng Japan 1973 6 18. Grnehling J. and Onken U., Vapor-Liquid-Equilibrium Data Collection; Dachema Chem. Data Series, Vol. 1, Part 2a. Verlag and Druckerel Friedrich Bischoff,

Frankfurt 1977.

WI Nagata I., Ohta T. and Uchiyama Y., J. Chem. Engng Data 1973 18 54.

WI Christensen J. J., Hawks

W. and Izatt R. M., Wiley, New York 1982. WI Mussche M. J. and Verhoeye L. A., J. Chem. Engng Data 1975 20 46. P81 Dizechi M. and Marschall E., .J. Chem. Engng Data 1982 27 358. r291 Kumar A., Prakash 0. And Prakash S., J. Chem. Engng Data 1981 26 64. (301 Katti P. K., Chaudhri M. M. and Prakash O., J. Chem. Engng Data 1966 11 593. r311 Katti P. K. and Prakash O., J. Chem. Engng Data 1966 11 46. 1321 Howard K. S. and Pike F. P., J. Chem. Engng Data 1959 4 331. 1331 Hammond L. W., Howard K. S. and McAllister R. A., J. Phys. Chem. 1958 62 637. [34] Wei I. C. and Rowley R. L., J. Chem. Engng - - Data 1983 29 332; Wei I. C. and Rowley R. L., J. Chem. Engng Data 1983 29 336. 1351 Raman G. K., Naidu P. R. and Krishnan V. R., Aust. _ _ J. Chem. 1968 21 2717. [36] Bloomfield V. A. and Dewan R. K., J. Phys. Chem. 1971 75 3113. [371 Nath J. and Dubey S. N., J. Phys. Chem. 1981 85 886. Handbook

R.

of Heats of Mixing