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Journal of Molecular Liquid

Thermal conductivity of ternary liquid mixtures: Application of modified

Flory statistical theory and uniform gaseous model

Ranjan Dey, N.K. Soni, R.K. Mishra, V. Sanguri, J.D. Pandey *

Department of Chemistry, University of Allahabad, Allahabad-211 002, India

Received 2 June 2005; accepted 12 October 2005

Abstract

Modified Flory’s statistical theory and uniform gaseous model have been employed to evaluate thermal conductivity of six ternary liquid

mixtures viz. toluene+carbontetrachloride+chloroform (I), benzene+n-pentane+cyclohexane (II), benzene+chloroform+2-propanol (III),

toluene+benzene+n-pentane (IV), cyclohexane+carbontetrachloride+2-propanol (V) and n-pentane+ toluene+2-propanol (VI) at 298.15 K.

Theoretical results are found to be in fairly good agreement with the experimental results.

D 2005 Elsevier B.V. All rights reserved.

Keywords: Modified Flory theory; Gaseous model; Thermal conductivity; Ternary liquid mixture

1. Introduction

The present paper is a continuation of our studies [1] on the

thermodynamic properties of binary liquid mixtures. Thermal

conductivity of gaseous mixtures (binary and higher order) has

been analyzed on the basis of various models [2–5] using

mathematical theories of non-uniform gases [6]. Recently [7],

an attempt has been made to extend some of these gaseous

models for computing the thermal conductivity of few binary

liquid mixtures. In the present investigation, an extension of

these gaseous state models has been made to evaluate the

thermal conductivity of multicomponent mixtures. During

recent years, Flory’s statistical theory [8–10] has been used

to predict thermodynamic properties of binary and multicom-

ponent systems by various workers [11–15] for which the

characteristic and reduced parameters of pure and multicom-

ponent systems are required. In the present work, relationship

between thermal conductivity, ultrasonic velocity and ratio of

heat capacities have been developed in terms of characteristic

and reduced parameters using Flory’s statistical theory. As far

as our knowledge is concerned no such theoretical methods are

0167-7322/$ - see front matter D 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.molliq.2005.10.004

* Corresponding author.

E-mail addresses: drranjan@hotmail.com (R. Dey),

drsoni_nk@rediffmail.com (N.K. Soni).

available for the theoretical evaluation of thermal conductivity

of multicomponent systems.

2. Theoretical

Several empirical relations have been developed for

evaluating thermal conductivity of gaseous mixtures. In the

present work, theoretically justified Sutherland–Wassiljewa

equation [5] is applied to evaluate the thermal conductivity of

ternary liquid mixtures. The aforesaid equation for evaluating

the thermal conductivity of multicomponent liquid mixtures

can be expressed as,

k ¼X3i¼1

ki 1þXjmi

Aij

xj

xi

#" �1

ð1Þ

which in turn reduces to Eq. (2) for a ternary mixture,

k ¼ k1

1þ A12

x2

x1þ A13

x3

x1

þ k2

1þ A23

x3

x2þ A21

x1

x2

þ k3

1þ A31

x1

x3þ A32

x2

x3

ð2Þ

where k1, k2 and k3 are the thermal conductivities of the pure

components 1, 2 and 3 of the ternary mixture, respectively.

s 124 (2006) 102 – 105

www.elsev

R. Dey et al. / Journal of Molecular Liquids 124 (2006) 102–105 103

Sutherland, Wassiljewa and Hirschfelder obtained Eq. (1)

using quite different approaches. Sutherland expression was

based on the predictions of kinetic theory of gases, whereas

Wassiljewa derived it on the basis of mean free path.

Hirschfelder approximation was based on Chapman and

Enskog’s kinetic theory. Sutherland–Wassiljewa equation is

highly non-linear and flexible, and capable of explaining even

maxima or minima in any transport property composition

curve.

A new empirical form has been proposed for Wassiljewa

coefficient [7], Aij, to give the best possible results in the case

of liquids. It can be expressed as,

Aij ¼1

41þ ki

kj

�� 1=2Mi

Mj

�� 3=8#" 2

ð3Þ

where Mi and Mj represent molecular weights of components i

and j, respectively.

Intermolecular energy (Eo) can be expressed as,

� Eo

rN¼ s

2Vh1g11 þ h2g22 þ h3g33 � h1h2Dg1 � h2h3Dg2½

� h3h1Dg3� ð4Þ

where N is the number of particles comprising the system. The

characteristic pressure for the pure components are given by

g11 ¼2P4

1VT2

s1; g22 ¼

2PT2V

T2

s2; g33 ¼

2PT3V

T2

s3ð5Þ

and also

X12 ¼S1Dg12V T2 ; X23 ¼

S2Dg22V T2 ; X31 ¼

S3Dg32V T2 ; X13 ¼

S1Dg12V T2 ;

X21 ¼S2Dg22V T2 ; X32 ¼

S3Dg32V T2

Adding consecutive expressions and simplifying, we obtain

similar expressions for Dg1, Dg2, and Dg3

Dg1 ¼X12 þ X13ð ÞV T2

s1; Dg2 ¼

X23 þ X21ð ÞV T2

s2;

Dg3 ¼X31 þ X32ð ÞV T2

s3ð6Þ

Putting required values from Eqs. (5) and (6) in Eq. (4) one

gets

� Eo

rN¼ s

2Vh1

2PT1V

42

s1þ h2

2PT2V

T2

s2þ h3

2PT3V

T2

s3

�

� h1h2X12 þ X13ð ÞV T2

s1� h2h3

X23 þ X21ð ÞV T2

s2

� h3h1X31 þ X32ð ÞV T2

s3

�ð7Þ

The molecular element or segment is defined in the investiga-

tion for the component species such that r1, r2 and r3 shall

be in the ratio of the respective molar core volumes v1T, v2T and

v3T. s1, s2 and s3 are the ratio of the molecular surface area of

contact per segment.

Random mixing of the species is assumed and as an

approximation, we take the expectation that a species of kind i

neighbors of any given site to be equal to its site fraction hidefined as

hi ¼siriNi

srNð8Þ

r ¼X

riNi

Nð9Þ

s ¼X

siriNi

rNð10Þ

N ¼X

Ni ð11Þ

The segment fraction is defined as

wi ¼riNi

rNð12Þ

where

1

r¼X

wisi ð13Þ

and therefore the site fractions are defined as

h1 ¼w1s1

s; h2 ¼

w2s2

s; h3 ¼

w3s3

sð14Þ

Putting these values in Eq. (7) and simplifying one gets,

� Eo

rN¼ V T2

Vw1P

T1 þ w2P

T2 þ w3P

T13 � w1h2

X12 þ X13ð Þ2

�

� w1h2X23 þ X21ð Þ

2� h3h1

X31 þ X32ð ÞV T2

2

�ð15Þ

by analogy with the energy of pure component, defined earlier

[16],

� Eo

rN¼ PTV T

VVð16Þ

comparing Eqs. (15) and (16),

PT ¼ w1PT1 þ w2P

T2 þ w3P

T3 � w1h2

X12 þ X13ð Þ2

�

� w1h2X23 þ X21ð Þ

2� w3h1

X31 þ X32ð ÞV T2

2

�ð17Þ

which is the modified expression of the characteristic pressure

of ternary liquid mixtures. In the case of multicomponent

system, the interaction parameter is defined as [17]

Xij ¼ PTi 1�

PTj

PTi

! 1=2V Tj

V Ti

! 1=635

24

2

ð18Þ

Table 1

Thermal conductivity of ternary liquid mixtures (in mW/(m-K)) at 298.15 K

x1 x2 k k k

(Exp) (Eq. (2)) (Eq. (28)

Toluene (x1)+carbon tetrachloride (x2)+chloroform (x3)

0.6864 0.1370 116.16 121.56 127.42

0.2802 0.5035 102.98 108.96 112.29

0.2555 0.1530 108.20 113.34 119.93

0.5038 0.3018 108.99 115.24 119.23

0.4828 0.1446 111.77 117.43 122.49

0.2673 0.3202 105.73 111.15 115.12

0.4177 0.2502 107.76 114.57 118.30

APD �5.35% �9.60%

Benzene (x1)+n-pentane (x2)+cyclohexane (x3)

0.5987 0.2161 125.33 130.81 137.49

0.1932 0.6275 117.51 119.07 117.96

0.2055 0.2224 118.01 122.37 129.17

0.3927 0.4251 121.39 124.73 128.03

0.4050 0.2192 120.83 126.53 133.56

0.1991 0.4312 112.42 120.68 123.67

0.3291 0.3563 118.22 123.90 128.51

APD �4.15% �7.75%

Benzene (x1)+chloroform (x2)+2-propanol (x3)

0.6056 0.1321 131.73 135.11 145.60

0.2846 0.4605 118.34 123.38 134.10

0.1801 0.1178 131.22 133.42 148.48

0.4340 0.2840 125.89 129.25 139.97

0.3837 0.1245 131.98 134.21 146.90

0.2038 0.2666 124.13 128.43 140.91

0.3341 0.2186 126.64 130.70 142.70

APD �4.79% �12.24%

Toluene (x1)+benzene (x2)+n-pentane (x3)

0.5498 0.2162 126.21 129.31 133.4

0.1719 0.6085 130.09 133.26 136.81

0.1664 0.1962 118.18 121.12 118.62

0.3548 0.4186 128.43 131.28 135.44

0.3488 0.2057 123.02 125.02 126.36

0.1691 0.3990 123.82 126.96 127.35

0.2861 0.3375 124.87 127.53 129.46

APD �2.27% �3.61%

Cyclohexane (x1)+carbon tetrachloride (x2)+2-propanol (x3)

0.6064 0.1106 115.72 120.07 129.49

0.2474 0.4061 105.28 112.21 116.88

0.1740 0.0952 122.46 127.70 129.86

0.4450 0.2435 110.15 116.19 123.10

0.3740 0.1023 118.49 123.87 135.04

0.2043 0.2236 113.79 120.04 128.09

03345 0.1830 113.99 120.24 129.02

APD �5.09% �11.49%

n-pentane (xl)+ toluene (x2)+2-propanol (x3)

0.6020 0.1571 119.77 121.37 116.51

0.2198 0.5163 128.22 129.05 131.73

0.1857 0.1454 131.32 132.43 139.89

0.4196 0.3286 123.09 125.04 122.70

0.3858 0.1510 124.43 126.83 125.74

0.2013 0.3153 129.92 130.73 134.97

0.3311 0.2593 125.26 127.62 126.93

APD �1.28% �2.67%

R. Dey et al. / Journal of Molecular Liquids 124 (2006) 102–105104

The characteristic temperature is given by [18, 19]

TT ¼ PT

X3i¼1

wiPTi

Ti

ð19Þ

Ideal reduced volume and the ideal reduced temperature for a

multicomponent system are expressed as [10],

VV - ¼X3i¼1

xiVVi ð20Þ

TT - ¼ VV -1=3 � 1

VV -4=3ð21Þ

For the evaluation of reduced parameters, the following

relations are used:

TT ¼ T

TT ¼ VV 1=3 � 1

VV 4=3ð22Þ

VV ¼ aT3 1þ aTð Þ

��þ 1

�� 3

ð23Þ

The excess reduced volume is given by [10,19]

VV E ¼ VV -7=34

3� VV -1=3

�� �1

TT � TT -� �

ð24Þ

The reduced volume for the mixtures can be expressed as

VV ¼ VV E þ VV - ð25Þ

V T ¼ V

VV: ð26Þ

Assuming the contribution of translational and rotational

degrees of freedom in the transference of energy, the thermal

conductivity of liquid is related to the velocity of sound by [16]

k ¼ 2:8kn2=3c�1=2c ð27Þ

where k is the Boltzmann constant, n the number of moles per

unit volume, c the heat capacities ratio and c the ultrasonic

velocity. For multicomponent systems, Eq. (27) can be

expressed in terms of characteristic and reduced parameters

obtained by Flory’s statistical theory,

k ¼ 2:8kXi

ni/i

! 2=3P4V 4 1� 3 VV 1=3 � 1

� �� �3VV

Xi

XiMi

! VV 1=3 � 1� �

377775

266664

1=2

ð28Þ

3. Results and discussion

In the present work thermal conductivity of six ternary

liquid systems namely toluene+carbontetrachloride+chloro-

form (I), benzene+n-pentane+cyclohexane (II), benzene+

)

R. Dey et al. / Journal of Molecular Liquids 124 (2006) 102–105 105

chloroform+2-propanol (III), toluene+benzene+n-pentane

(IV), cyclohexane+carbontetrachloride+2-propanol (V) and

n-pentane+ toluene+2-propanol (VI) have been computed at

298.15 K and ambient pressures over the complete mole

fraction range using Wassiljewa equation and Eq. (27). The

experimental data needed for thermal conductivity values of

pure components have been taken from earlier papers [15,16].

Various workers [3,4,6] have applied Wassiljewa equation to

study the transport properties of binary and multicomponent

systems of gases. In the present work an extension of this

equation to ternary liquid mixtures for computing the thermal

conductivities has been done. Interaction parameters Al2, A21,

A13, A31 A23 and A32 have been evaluated using Eq. (2).

Thermal conductivities of pure components and previously

mentioned interaction parameters have been utilized to compute

the thermal conductivities of aforementioned ternary systems

using Sutherland–Wassiljewa approach given in Eq. (2), which

is completely based on non-uniform gaseous model. Results,

thus obtained, are in close agreement with the maximum and

minimum average percentage deviations (APD) of 5.35% and

1.28% for systems I and VI, respectively, as reported in column

four of Table 1. The order of average percentage deviation for

six ternary systems under the present investigation with

Wassiljewa equation has the following sequence:

system I > system V > system III > system II > system IV

> system VI:

In liquid mixtures interactions are stronger in comparison to

gases. Due to the lack of considerations of these interactions in

the Wassiljewa equation there is slight difference between the

theoretical and experimental values of thermal conductivities.

Flory’s statistical theory [8–10] has also been employed to

compute the thermodynamic properties of binary and multi-

component systems by various workers [11–15]. Characteristic

and reduced parameters of pure components are evaluated by

the procedure adopted by Pandey and Sanguri [11] and Pandey

et a1. [17]. Characteristic and reduced parameters of ternary

liquid mixtures have been computed utilizing Eqs. (17), (25)

and (26).

A relationship between thermal conductivity, ultrasonic

velocity and heat capacity ratio, based on Eyring’s assumptions

where only the rotational and translational degrees of freedom

are assumed to be effective in transferring energy, can be

expressed in terms of characteristic pressure, characteristic

volume and reduced volume of ternary liquid systems, and is

given by Eq. (28). This equation applies well for a large

number of liquids. Eq. (28) in conjunction with Eqs. (17), (25)

and (26) has been used to compute thermal conductivity of

ternary systems under the present investigation. With the

reference of this equation, theoretical values of thermal

conductivity show maximum and minimum APD of 12.24%

and 2.67% for system III and system VI, respectively, as

reported in column five of Table 1. However, trend of APD for

all the six ternary systems is given as follows:

system III > system V > system I > system II > system IV

> system VI:

4. Conclusion

Computation of thermal conductivity has been carried out

by employing the Sutherland–Wassiljewa relation as well as

the Flory statistical theory for six ternary liquid mixtures at

298.15 K. A comparative study reveals that the Sutherland–

Wassiljewa relation holds superior to the Flory theory,

although the results obtained from Flory theory are found to

be in good agreement. The reason for the slight discrepancy

arising from Flory theory is due to the fact that only two

body interactions are taken into consideration while the three

body interactions are ignored to avoid the computational

complications.

Acknowledgements

One of the authors (J.D.P.) is grateful to U.G.C. for the

award of Emeritus Fellow and N.K.S. and V.S. are thankful to

CSIR, New Delhi, for the award of Research Associateship.

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