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Thermal conductivity of binary liquid mixtures
CHEMICAL PHYSICS LETTERS
Volume 27, number 3
1 August 1974
THERMAL CONDUCTIVITY OF BINARY LIQUID MIXTURES M.P. SAKSENA and HARMINDER Physics Department,
University of Rajasthan, Jaipur-302004,
India
Received 17 April 1974 A semi-empirical method is developed for the prediction of the thermal conductivity of binary liquid mixtures. The proposed method is tested by calculating the thermal conductivity of twelve binary liquid mixtures and an excellent agreement between the observed and calculated values is obtained.
1. Introduction hm,
Recently Jamieson and Hastings [ 1] have measured the thermal conductivity of nearly sixty binary liquid mixtures, and have reviewed the existing methods for predicting the thermal conductivity. All the existing methods are either empirical or semi-empirical. Out of these methods the modified NEL equation, obtained by them empirically, is found to give best results. The NEL relation is h,,
= Olhl
+02x2
- alX2-X$[l
- (w2)1’21w2,
(1) where Wi and Xi represent the weight fraction and thermal conductivity of the ith species in the binary mixture. Eq. (1) contains an adjustable parameter (Y which has to be determined by one experimental value of the thermal conductivity at a given composition. Saksena and Harminder [2] have obtained a theoretical expression for the thermal conductivity of binary liquid mixtures by using the statistical mechanical theory of Bearman [3,4]. They have tested their relation by calculating the thermal conductivity of twenty-two binary liquid mixtures and have observed that their equation yields excellent results for all the binary mixtures except those which contain much heavier molecules like carbontetrachloride and tetrabromoethane. According to Saksena and Harminder [2] the thermal conductivity of a binary liquid mixture is given by
448
=
Xl 1-k (X2IX4A
x2 12 + 1+ (X1/X2)A21
’
(2)
where A,,A,,
= 1.
and Xi, respectively, denote the mole fraction and the thermal conductivity of the ith species. They have found the coefficients A I2 and A 21 to be dependent on the molar volume and the thermal conductivity of the comporients. We here intend to treat A 12 and Azl as unknown parameters, to be determined through one known value of X,, in conjunction with relations (2) and (3). Such a procedure will be useful for those mixtures for which the theoretical relation is unsuccessful in predicting the thermal conductivity data. Solving eqs. (2) and (3), we get
Xi
Thus knowing one X,, value, h, , X2, x1 and x2 the value of A, 2 and hence A,, can be determined.
2. Calculation
of X,i, and discussion of results
We have used the experimental values for the thermal conductivity of binary liquid mixtures reported by Jamieson and Hastings [l] to check the procedure
Volume
27, number
CHEMICAL
3
PHYSICS
Table Calculated
values of thermal
conductivity
Weight fraction Rinary
system
acetone-carbon (At* = 3.45;A*r
tetrachloride = 0.29)
108.2
118.3 (3.9)
121.6 (2.1)
129.5 (0.0)
135.5 (2.8)
126.8 (2.7)
139.6 (0.0)
149.0 (-1.3)
108.2
dichloromethane-carbon (Al2 = 3.72; AZ1 = 0.27)
108.2
122.8
toluene-carbon
tetrachloride
113.4
114.9
123.8
(3.9)
(0.0)
113.0
114.4
119.1
122.8
(0.7)
(0.9)
(0.0)
(0.3)
110.8
112.9 (-0.7)
(0.0)
108.2
(4.3) 108.2
(Al2 = 2.81; AZ1 = 0.35) benzene-tetrabromoethane (Al2 = 4.25; AZ1 = 0.23)
87.7
toluene-tetrabromoethane
87.7
(Al2 = 3.85; AZ1 = 0.26) methanol-benzene (A r2 = 3.69; Azl = 0.27)
152.0
methanol-toluene (A r2 = 4.69; AZ1 = 0.21)
140.0
acetone-toluene (A ,2 = 1.82; Azl
140.0
0.60
0.75
0.80
1.00
141.9
152.1 (-1.5)
155.6 (-5.0)
171.1
(0.7) 159.1 (-1.0)
176.0 (-5.0)
182.1 (-2.2)
209.6
136.2 (-1.8)
139.2 (-2.5)
154.5
114.7
113.4
120.0
(2.5)
(0.0)
104.3
120.5
(4.7)
(0.0)
100.5
113.5
(4.7)
(0.0)
134.6 (-1.1)
152.7
121.4 (-3.3)
135.5
128.5 (-1.1)
140.0
136.4 (-1.7)
152.0
126.7 (-1.8)
140.0
160.1 (-1.6)
162.4 (-1.9)
169.6
174.9
(0.0)
(1.6)
180.7 (-0.9)
190.3 (-3.7)
149.2
151.8 (-0.2)
160.2 (0.0)
166.5 (-1.3)
177.3 (2.3)
185.1 (-0.7)
(0.0)
= 0.55)
methanol-nitrobenzene (A r2 = 4.16; A2, = 0.24)
X_
(3.1)
(Al2 = 3.58; All
in mW M-’ “C-l
a) 0.40
butan-1-ol-carbon tetrachloride (A r2 = 4.08; AZ, = 0.24)
ether-carbon tetrachloride = 0.26)
liquid mixtures
0.25
(3.7)
151.0
147.0 (-0.4)
154.5
162.5
(0.0)
(0.9)
164.8
179.1
(0.0)
(0.6)
194.1 (-0.7)
a) The weight fractions are of the component having the higher thermal conductivity, written first in the mixture. errors are given in parentheses. All the binary mixture thermal conductivity values are at 0°C.
developed here. By using relations (2) and (4) the thermal conductivity of twelve binary mixtures has been calculated. The calculated values are compared with the observed data in table 1. We have chosen the binary mixtures acetone-carbon tetrachloride, methanol-carbon tetrachloride, butan-l-olcarbon tetrachloride, dichloromethane-carbon tetrachloride, di-n-ethyl ether-carbon tetrachloride, toluene-carbon tetrachloride, benzene-tetrabromoethane and toluene-tetrabromoethane, for which Saksena and Harminder [2] found their theoretical relation relatively unsuccessful. As a further check for this relation we have also included four mixtures, methanolbenzene, methanol-toluene, acetone-toluene, and
1974
1
of binary
0.20
108.2
di-n-ethyl
1 August
0.0
methanol-carbon tetrachloride (A r2 = 7.14; AZ1 = 0.14)
tetrachloride
LETTERS
193.8
209.6
(0.1) 189.4
209.6
(1.0) 171.1
209.6
The percentage
methanol-nitrobenzene, for which good agreement obtained with the theoretical relation. For the determination of the coefficients A I2 and A,, , the experimental X,, value, at a composition near to equimolar one, was used. All these coefficients are also listed in table 1. It may be mentioned that the coefficients A r2 and AZ1 will be found to have different values, if the hix value at different compositions is used in their calculation. However, the increase in one is always associated with a corresponding decrease in the other. In our calculations we have treated these coefficients as composition independent and the good agreement obtained between calculated and experimental values supports this procedure. Was
449
Volume
27,
number 3
CHEMICAL PHYSICS LETTERS
For the first nine binary mixtures the average absolute percentage deviations are 2.6, 2.3, 2.5,0.6, 2.2, 1.2,2.1 and 2.2, respectively, when calculated by the semi-empirical method developed here, while the corresponding average absolute percentage deviations using the exact equation [2] are 5.4,6.5, 7.6, 3.7, 7.1, 5.2,8.8 and 8.4, respectively. For the remaining four binary mixtures the average absolute percentage deviations are 1.4,0.8, 0.4 and 0.4 when calculated by the present method as compared to 1.5, 1.6,0.3 and 1 .l%, respectively, obtained by the exact equation [2].
450
1 August 1974
References 111 D.T. Jamieson and E.H. Hastings, in: Proceedings of VIII International Conference of Thermal conductivity, ed. C.Y. Ho (Plenum Press, New York, 1969). [21 M.P. Saksena and Harminder, Chem. Phys. Letters 25 (1974) 445. 131 R.J. Bearman, J. Chem. Phys. 29 (1958) 1278. [41 J.K. Horrocks and E. McLaughlin, Trans. Faraday Sot. 58 (1962)
1357.
Volume 27, number 3
1 August 1974
THERMAL CONDUCTIVITY OF BINARY LIQUID MIXTURES M.P. SAKSENA and HARMINDER Physics Department,
University of Rajasthan, Jaipur-302004,
India
Received 17 April 1974 A semi-empirical method is developed for the prediction of the thermal conductivity of binary liquid mixtures. The proposed method is tested by calculating the thermal conductivity of twelve binary liquid mixtures and an excellent agreement between the observed and calculated values is obtained.
1. Introduction hm,
Recently Jamieson and Hastings [ 1] have measured the thermal conductivity of nearly sixty binary liquid mixtures, and have reviewed the existing methods for predicting the thermal conductivity. All the existing methods are either empirical or semi-empirical. Out of these methods the modified NEL equation, obtained by them empirically, is found to give best results. The NEL relation is h,,
= Olhl
+02x2
- alX2-X$[l
- (w2)1’21w2,
(1) where Wi and Xi represent the weight fraction and thermal conductivity of the ith species in the binary mixture. Eq. (1) contains an adjustable parameter (Y which has to be determined by one experimental value of the thermal conductivity at a given composition. Saksena and Harminder [2] have obtained a theoretical expression for the thermal conductivity of binary liquid mixtures by using the statistical mechanical theory of Bearman [3,4]. They have tested their relation by calculating the thermal conductivity of twenty-two binary liquid mixtures and have observed that their equation yields excellent results for all the binary mixtures except those which contain much heavier molecules like carbontetrachloride and tetrabromoethane. According to Saksena and Harminder [2] the thermal conductivity of a binary liquid mixture is given by
448
=
Xl 1-k (X2IX4A
x2 12 + 1+ (X1/X2)A21
’
(2)
where A,,A,,
= 1.
and Xi, respectively, denote the mole fraction and the thermal conductivity of the ith species. They have found the coefficients A I2 and A 21 to be dependent on the molar volume and the thermal conductivity of the comporients. We here intend to treat A 12 and Azl as unknown parameters, to be determined through one known value of X,, in conjunction with relations (2) and (3). Such a procedure will be useful for those mixtures for which the theoretical relation is unsuccessful in predicting the thermal conductivity data. Solving eqs. (2) and (3), we get
Xi
Thus knowing one X,, value, h, , X2, x1 and x2 the value of A, 2 and hence A,, can be determined.
2. Calculation
of X,i, and discussion of results
We have used the experimental values for the thermal conductivity of binary liquid mixtures reported by Jamieson and Hastings [l] to check the procedure
Volume
27, number
CHEMICAL
3
PHYSICS
Table Calculated
values of thermal
conductivity
Weight fraction Rinary
system
acetone-carbon (At* = 3.45;A*r
tetrachloride = 0.29)
108.2
118.3 (3.9)
121.6 (2.1)
129.5 (0.0)
135.5 (2.8)
126.8 (2.7)
139.6 (0.0)
149.0 (-1.3)
108.2
dichloromethane-carbon (Al2 = 3.72; AZ1 = 0.27)
108.2
122.8
toluene-carbon
tetrachloride
113.4
114.9
123.8
(3.9)
(0.0)
113.0
114.4
119.1
122.8
(0.7)
(0.9)
(0.0)
(0.3)
110.8
112.9 (-0.7)
(0.0)
108.2
(4.3) 108.2
(Al2 = 2.81; AZ1 = 0.35) benzene-tetrabromoethane (Al2 = 4.25; AZ1 = 0.23)
87.7
toluene-tetrabromoethane
87.7
(Al2 = 3.85; AZ1 = 0.26) methanol-benzene (A r2 = 3.69; Azl = 0.27)
152.0
methanol-toluene (A r2 = 4.69; AZ1 = 0.21)
140.0
acetone-toluene (A ,2 = 1.82; Azl
140.0
0.60
0.75
0.80
1.00
141.9
152.1 (-1.5)
155.6 (-5.0)
171.1
(0.7) 159.1 (-1.0)
176.0 (-5.0)
182.1 (-2.2)
209.6
136.2 (-1.8)
139.2 (-2.5)
154.5
114.7
113.4
120.0
(2.5)
(0.0)
104.3
120.5
(4.7)
(0.0)
100.5
113.5
(4.7)
(0.0)
134.6 (-1.1)
152.7
121.4 (-3.3)
135.5
128.5 (-1.1)
140.0
136.4 (-1.7)
152.0
126.7 (-1.8)
140.0
160.1 (-1.6)
162.4 (-1.9)
169.6
174.9
(0.0)
(1.6)
180.7 (-0.9)
190.3 (-3.7)
149.2
151.8 (-0.2)
160.2 (0.0)
166.5 (-1.3)
177.3 (2.3)
185.1 (-0.7)
(0.0)
= 0.55)
methanol-nitrobenzene (A r2 = 4.16; A2, = 0.24)
X_
(3.1)
(Al2 = 3.58; All
in mW M-’ “C-l
a) 0.40
butan-1-ol-carbon tetrachloride (A r2 = 4.08; AZ, = 0.24)
ether-carbon tetrachloride = 0.26)
liquid mixtures
0.25
(3.7)
151.0
147.0 (-0.4)
154.5
162.5
(0.0)
(0.9)
164.8
179.1
(0.0)
(0.6)
194.1 (-0.7)
a) The weight fractions are of the component having the higher thermal conductivity, written first in the mixture. errors are given in parentheses. All the binary mixture thermal conductivity values are at 0°C.
developed here. By using relations (2) and (4) the thermal conductivity of twelve binary mixtures has been calculated. The calculated values are compared with the observed data in table 1. We have chosen the binary mixtures acetone-carbon tetrachloride, methanol-carbon tetrachloride, butan-l-olcarbon tetrachloride, dichloromethane-carbon tetrachloride, di-n-ethyl ether-carbon tetrachloride, toluene-carbon tetrachloride, benzene-tetrabromoethane and toluene-tetrabromoethane, for which Saksena and Harminder [2] found their theoretical relation relatively unsuccessful. As a further check for this relation we have also included four mixtures, methanolbenzene, methanol-toluene, acetone-toluene, and
1974
1
of binary
0.20
108.2
di-n-ethyl
1 August
0.0
methanol-carbon tetrachloride (A r2 = 7.14; AZ1 = 0.14)
tetrachloride
LETTERS
193.8
209.6
(0.1) 189.4
209.6
(1.0) 171.1
209.6
The percentage
methanol-nitrobenzene, for which good agreement obtained with the theoretical relation. For the determination of the coefficients A I2 and A,, , the experimental X,, value, at a composition near to equimolar one, was used. All these coefficients are also listed in table 1. It may be mentioned that the coefficients A r2 and AZ1 will be found to have different values, if the hix value at different compositions is used in their calculation. However, the increase in one is always associated with a corresponding decrease in the other. In our calculations we have treated these coefficients as composition independent and the good agreement obtained between calculated and experimental values supports this procedure. Was
449
Volume
27,
number 3
CHEMICAL PHYSICS LETTERS
For the first nine binary mixtures the average absolute percentage deviations are 2.6, 2.3, 2.5,0.6, 2.2, 1.2,2.1 and 2.2, respectively, when calculated by the semi-empirical method developed here, while the corresponding average absolute percentage deviations using the exact equation [2] are 5.4,6.5, 7.6, 3.7, 7.1, 5.2,8.8 and 8.4, respectively. For the remaining four binary mixtures the average absolute percentage deviations are 1.4,0.8, 0.4 and 0.4 when calculated by the present method as compared to 1.5, 1.6,0.3 and 1 .l%, respectively, obtained by the exact equation [2].
450
1 August 1974
References 111 D.T. Jamieson and E.H. Hastings, in: Proceedings of VIII International Conference of Thermal conductivity, ed. C.Y. Ho (Plenum Press, New York, 1969). [21 M.P. Saksena and Harminder, Chem. Phys. Letters 25 (1974) 445. 131 R.J. Bearman, J. Chem. Phys. 29 (1958) 1278. [41 J.K. Horrocks and E. McLaughlin, Trans. Faraday Sot. 58 (1962)
1357.