ters of gas and argon molecules as defined by eq 17, dimensionless Re,, = ratio of logarithms of Ostwald coefficients as defined by eq 18, dimensionless r = molecular separation distance, 8, rt, = radius of solute molecule as determined from van der Waals equation of state, 8, r, = radius of cavity in solvent, cm; crystallographic radius of solute molecules, A r, = molecular separation distance a t minimum potential energy of interaction, A ro = radius of solute molecule derived from LennardJones (12-6) potential energy function, A A S ' = entropy of solution for the process identified by eq 6, eu A s p = entropy of isothermal expansion of gas as defined by eq 9, eu T = absolute temperature, OK Greek Letters y = "microscopic" surface tension of solvent, dyn/cm t = absolute magnitude of the potential energy mini-
mum, ergs = solvent density, g/cm3 = separation distance a t which the molecular potential energy of interaction is zero, A @ ( r ) = potential energy of interaction at the separation distance r, ergs
Literature Cited Blander, M., Grimes. W. R., Smith, N. V., Watson, G. M..J. Phys. Chem., 63,1164 (1959). Briggs, R. B., Reactor Techno/., 14, 335 (1971-1972). Cantor, S.,U. S. Atomic Energy Commission, Report ORNL-4396, 174 (1969), Field, P. E.,Green, W. J., J. Phys. Chem., 75, 821 (1971). Grimes, W. R . , Cantor, S.,"The Chemistry of Fusion Technology," D. M . Gruen, Ed., p 161, Plenum Press, New York. N. Y., 1972. Hirschfelder, J. O., Curtiss, C. F.. Bird. R . B.. "Molecular Theory of Gases and Liquids," pp 250, 1110, Wiley, New York. N. Y., 1954. Huheey, J. E., J. Chem. Educ., 45, 791 (1968). Lee, A. K. K., Johnson, E. F.. Ind. Eng. Chem., Fundam., 8, 726 (1969). Malinauskas, A. P., Richardson, D. M.,Savolainen, J. E., Shaffer, J. H., Ind. Eng. Chem., Fundam., 11,584 (1972). Paniccia, F.. Zambonin, P. G., J. Chem. SOC.,Faraday Trans. 7, 11, 2083 (1972). Romberger, K. A., Braunstein. J., Thoma, R. E., J. Phys. Chem., 76, 1154 (1972). Sisskind, B., Kasarnowsky, I., 2.Anorg. Allg. Chem., 214, 385 (1933). Uhlig, H. H.,J. Phys. Chem., 41, 1215 (1937). Veleckis, E., Dhar, S. K., Cafasso, F. A., Feder, H. M.. J. Phys. Chem.. 75,2832 (1971). Watson, G. M., Evans. R. B., Grimes, W. R.. Smith, N. V., J. Chem. €ng. Data, 7, 285 (1962),
Receiuedfor reuiew November 1,1973 Accepted February 21,1974
p u
This work was supported by the U. S. Atomic Energy Commission under contract with the Union Carbide Corporation.
Thermal Conductivity of Binary Liquid Mixtures Mahesh P. Saksena* and Harrninder Physics Deparfmenf, University of Rajasfhan, Jaipur-302004, lndia
Assuming a quasi-lattice model for liquid mixtures, a theoretical expression for the cross coefficients is given, following the treatment of Horrocks and McLaughlin for a pure liquid. The appropriateness of the mixing rule and the proposed relation has been tested, with success, by calculating the thermal conductivity of five binary mixtures.
Introduction The subject of the thermal conductivity of liquids is of considerable interest due to its technological applications as well as the understanding it provides for the molecular processes. In recent years several review articles have appeared on the subject (Eyring and John, 1969; McLaughlin, 1964, 1969). Comparatively little work has been done on the thermal conductivity of liquid mixtures, which are equally important. Recently, Jamieson and Hastings (1969) have reported binary mixture data at 0°C for about 60 systems and this extensive work provides an opportunity to carry out further investigations in this field. These authors have also reviewed the existing methods of estimating the thermal conductivity of the mixtures and have shown that the NEL modified equation (NEL stands for National Engineering Laboratory, Glasgow, Scotland) and the modified Filippov equation (Filippov and Novoselova, 1955) are in best agreement with the experimental data. Both of these equations contain a constant whose value has to be empirically determined for the system under investigation. McLaughlin (1964) has also suggested a quadratic mixing law of the form
where x refers to the mole fraction, hl and X Z are the values of the thermal conductivity for the pure components, and A' is the effective cross coefficient. This equation appears to be most logical and has been found to be in satisfactory agreement with experimental data by the empirical adjustment of the cross coefficient (Jamieson and Hastings, 1969). We here intend to rewrite McLaughlin's equation in a more appropriate form and suggest a theoretical procedure for calculating the cross coefficient, using the analysis of Horrocks and McLaughlin (1960) for pure liquids. Development of Theory Horrocks and McLaughlin (1960), assuming a lattice structure for the liquid, derived a relation for the thermal conductivity of a liquid in which the excess energy due to the temperature gradient is transferred mainly by means of the vibrations of the molecules. There is also a convective contribution to the thermal conductivity, but it is found to be usually less than 1% of the total value. The resulting equations are Ind. Eng. C h e m . , F u n d a m . , Vol. 13, No. 3 , 1974
245
Table I Liquid Acetone Benzene Carbon tetrachloride Methanol Toluene
A,
elk,
Mlv,,*2)}1’2
mW m-1 “(2-1
O K
u, A
158.4 440.0
5.485 5.630
327.0 194.9 377.0
5.881 108.2 111.7 (3.2) 4.082 209.6 188.7(-9.9) 5.930 140.0 121.1(-13.5)
Obsd
Calcda
where
171.1 178.2(4.1) 152.0 1 4 1 . 9 ( - 6 . 6 )
‘,I*
SO
where the frequency of the vibration is
with =( r , / ~ ) ~
(5)
Here m, KO, a, c,, and 2 represent the mass of the molecule, the frequency of the diffusive displacement, the nearest-neighbor distance, the specific heat a t constant volume, and the number of the nearest neighbors, respectively. c and ro are the potential parameters. L1 and MI are lattice summation constants, as defined by Hirschfelder, et ul. (1967). In order to apply the above theory to liquid mixtures, the quasi-lattice model for a liquid mixture is assumed. The choice of such a model has two advantages: first, the Horrocks and McLaughlin (1960) theory can be directly applied, and secondly this model can be easily used to understand the behavior of complex liquid mixtures (Henderson and Leonard, 1971). For such a model one has to consider the interactions between the dissimilar molecules. We define two cross coefficients X l z and X z l instead of one as done by McLaughlin. Here A 1 2 will correspond to the vibration of a molecule of species 2 in the cell formed by the molecules of species 1 and similarly will correspond to the vibration of a molecule of species 1 in the cell formed by the molecules of species 2. With this picture, the McLaughlin mixing rule has to be written as Further, extending the Lennard-Jones-Devonshire model (Hirschfelder, et ul., 1967) to the dissimilar interactions, one can write the corresponding frequency for the vibration of the molecule of the j t h species in the cell formed by the molecules of the ith species, in analogy to eq 4,as Table 11. Calculated
Amix
=
(roL]/al])3
that hi] would be equal to
In relations 7-9 the Lorentz-Berthelot combination rules are to be used for the determination of tij, Uij, and roij, i. e.
The percentage errors are given in parentheses.
E*
(7)
Equation 9 reflects that X I J and XJ1 are in general not equal as they are dependent upon the masses of the species j and i, respectively. At, + A,, has an inherent property that its value remains the same on interchanging the species i a n d j . Hence for a particular mixture ALJ + A J L is a constant and is equivalent to W‘ (eq 1). However, now the values of X,I and XJ1 can be theoretically predicted while that of A’ (eq 1)has to be obtained empirically.
Calculation of Amix and Discussion of Results We have calculated the thermal conductivity of five binary mixtures, the choice of which was governed by the availability of the experimental thermal conductivity data and appropriate potential parameters. The distances between the nearest neighbors “a” for different liquids are calculated from the density of these liquids, which are taken from the work of Timmermans (1950). For the determination of the frequency of vibrations vil from relations 7 and 10 the Lennard-Jones (12-6) potential parameters were used. Several sets of such parameters are often available and we have chosen the ones which have given the best results for the thermal conductivity of the pure liquids using relation 2. These parameters along with the calculated and experimental thermal conductivity values for the liquids studied are given in Table I. For the specific heat at constant volume, per molecule, a constant value 3k is used, where k is the Boltzmann constant. Finally, hmix is calculated from relation 6 in conjunction with relation 9. The calculated and observed thermal conductivity values of the binary mixtures methanol-benzene, benzene-toluene, toluene-carbon tetrachloride, methanolacetone, and acetone-toluene along with the percentage errors are given in Table I1 as a function of the weight fraction of the liquid having the higher X value. In these calculations the experimental values of the thermal conductivity of the pure liquids (Jamieson and Hastings, 1969) have been used. This has been done to check the appropriateness of the mixing rule and the theoretical
Values (mW m-l OC-I) Weight fraction“
Liquid mixture Methanol-benzene
0.0
0.20
0.25
0.40
0.50
0.75
0.80
1.00
152.0 140.0
182.0 (7.3)
Toluene-carbon tetrachloride Methanol-acetone
171.1
Acetone-toluene
140.0
187.9 (9.1) 139.2 (-3.4) 124.9 (4 ’ 0 ) 198.1 (4.6) 154.8 (-0.2)
200.1 (1’ 3) 144.3 (-1.5) 132.8 (2.2) 204.8 (3.7) 163.1 (0.9)
202.2 (4.5) 145.7 (-2.4)
108.2
171.9 (3.9) 137.4 (-3.9) 116.4 (5.2) 187.8 (4.3) 146.7 (-0.7)
209.6
Benzene-toluene
168.2 (3 ’ 3)
a
246
The percentage errors are given in parentheses. Ind. Eng. Chem., Fundam., Vol. 13,No. 3, 1974
152.0 140.0 209.6 171.1
Table 111. The A Values in mW m-l "C-'
Mixture Methanol-benzene Benzenetoluene Toluenecarbon tetrachloride Methanol-acetone Acetone-toluene
Empirical best fit value of 2 A', rel. 1
Theoretical value of
261.5 282.4
337.7 262.6
213.7 365.0 291.9
234.5 390.0 292.9
AI2
+
the thermal conductivity of liquids using the relations given here has to be treated with reserve. However, regarding the appropriateness of the suggested mixing rule and the relation for X i j , Table II reveals that the deviations of the calculated values from the observed ones are almost always well within the experimental uncertainties. The average absolute percentage deviation for the five binary systems CH30H-CsH6, CI&-C~H~CH~, C6H5CH3CC14, CH30H-(CH3)2CO, and (CH3)2CO-C.&CH3 are 5.1, 2.8, 3.8, 4.2, and0.6, respectively.
A21,
rel. 9
Literature Cited relation for the cross coefficients. In Table 111, a comparison of the values of W' (eq 1) obtained by the best fit of the experimental data and the theoretical values of A 1 2 A 2 1 calculated from eq 9 is presented. The two sets of values are in good agreement for d l the systems except methanol-benzene. All these results are a t 0°C. It is observed in these calculations that any variation in the value of (= 2t1/%-o) affects the frequency of vibrations and hence the thermal conductivity value considerably. The potential parameters available in the literature have such a large scatter that the values of thermal conductivity predicted through them vary considerably from the observed data. It would therefore be worthwhile to determine these parameters afresh using some data on liquids. In the absence of such parameters the prediction of
+
Eyring, H., John, M. S., "Significant Liquid Structure," pp 92-95, Wiiey, New York, N. Y., 1969. Filippov. L. P., Novoselova. N. S., Vestn. Mosk. Univ. Ser. F / z . - M a t . . 10, 37 (1955). Henderson, D., Leonard, P. J., "Physical Chemistry, An Advanced Treatise," Vol. 8B, D. Henderson, Ed., pp 435-41, Academic Press, Ne w York, N. Y., 1971. Hirschfelder, J. O., Curtiss, C. F . , Bird, R . B., "Molecular Theory of Gases and Liquids," pp 293-98, Wiiey, New York, N. Y.. 1967. Horrocks, T. K., McLaughlin, E., Trans. FaradaySoc., 56, 206 (1960). Jamieson, D. T., Hastings, E. H., "The Thermal Conductivity of Binary Liquid Mixtures," Proceedings of V I i i International Conference of Thermal Conductivity, C. Y. Ho, Ed., Plenum Press, New York, N. Y . , 1969. McLaughlin. E., Chem. Rev.. 64, 389 (1964). McLaughlin, E., "Thermal Conductivity," Vol. 1 1 , R . P. Tye, Ed., pp 1-62. Academic Press, New York. N. Y., 1969. Timmermans, T., "Physico-Chemical Constants of Pure Compounds," Elsevier Publication Co.. New York, N. Y . , 1950.
Received f o r review September 6, 1973 Accepted M a r c h 11, 1974
Fluid Mechanical Description of Fluidized Beds. Convective Instabilities in Bounded Beds James Medlin, Hin-Wai Wong,' and Roy Jackson* Rice University, Houston, Texas 7700 1
A linearized stability analysis based on previously developed dynamical equations is used to examine the behavior of small perturbations in uniformly fluidized beds of finite depth. lt is shown that convective instabilities may develop, generating a particle motion analogous to the well-known motion of a fluid heated from below. It is also shown that this instability can be suppressed by increasing the pressure drop across the bed support or by reducing the width of the bed.
Introduction Over the past decade or so a dynamical theory of fluidized suspensions, modeled as two interpenetrating and interacting fluid continua, has had some success in accounting for the principal features of bubble motion (Davidson, 1961; Jackson, 196313; Murray, 1965b) and suggesting the mechanical origins of the contrasting behavior of typical gas and liquid fluidized beds in terms of a hydrodynamic stability analysis of the state of uniform fluidization (Anderson and Jackson, 1968; Jackson, 1963a; Murray, 1965a; Pigford and Baron, 1965).
' Present address, Gulf General Atomic, San Diego, Calif
The stability analyses have shown that a uniformly fluidized suspension of infinite extent is unstable, the instability taking the form of growing waves of voidage fluctuation rising through the bed, and that the rate of growth, though very modest in typical liquid fluidized beds, is two decimal orders of magnitude larger in similar beds fluidized by air. The generation of rapidly intensifying voidage fluctuations in gas fluidized beds is suggestive of bubble formation, though of course a linearized perturbation analysis cannot pursue the development of the voidage fluctuation into anything resembling a fully developed bubble. Analyses such as these can be criticized on the grounds that real fluidized beds are not of infinite extent, and that some account should therefore be taken of boundary conditions at the upper and lower surfaces of a bed of finite Ind. Eng. Chern., Fundarn.. Vol. 13, No. 3, 1 9 7 4
247
