EMPIRICAL METHOD FOR PREDICTION OF T H E CONCENTRATION DEPENDENCE OF MUTUAL DIFFUSIVITIES IN BINARY MIXTURES OF ASSOCIATED AND NONPOLAR LIQUIDS R O N A L D E. R A T H B U N A N D A L B E R T L. BABE Departments of Chemical Engineering and Nuclear Engineering, Unioersity of Washington, Seattle, Wash.
An empirical equation based on the principles of nonequilibrium thermodynamics is presented for predicting the concentration dependence of the mutual diffusion coefficient in binary liquid systems containing one component which is associated because of hydrogen bonding. The equation contains one empirical constant and pennits the prediction of the entire diffusivity-mole fraction relationship from only the diffusivities in the infinitely dilute solutions and activity coefficient data as a function of composition. With a different value of the empirical constant, the equation can be applied to binary systems with negative deviations from Raoult's law.
NE
of the primary problems of interest in the study of
0 diffusion in nonelectrolyte liquid systems is the concentration dependence of the mutual diffusion coefficient. I n recent years, the application of the principles of nonequilibrium thermodynamics to diffusion has yielded a general expression for the mutual diffusivity in a binary liquid system (77). For the volume-fixed refermence frame, this expression has the form
where L = a phenomenological coefficient, R = the gas constant, X = mole fraction, C = concentration, u = partial specific volume, M =- molecular weight, and a = thermodynamic activity. For a system which is thermodynamically ideal, the activity gradient term is unity and Equation 1 becomes
where Did is the difhsivity in a thermodynamically ideal system. If Equations 1 and i! are combined, one obtains DAB = Did(d In a / d In
X)T,p
(3)
A study (70)of the iconcentration dependence of the mutual diffusivity in three systems which were approximately thermodynamically ideal demonstrated that the diffusivity varied approximately linearly with mole fraction. O n the basis of this work, it has been assumed by several authors (72,77,79) that the diffusivity v x i e s linearly with mole fraction in a system which is thermodynamically ideal. Furthermore, as a consequence it has been assumed (72, 79) that the diffusivity which a nonideal system would have if it were ideal varies linearly with mole fraction between the experimental values of the diffusivity a t infinite dilution. Thus Did == DB'XA f DA'XB
(4)
where DB' is the mutual diffusivity in a n infinitely dilute solution of B in A and DA' is the mutual diffusivity in an infinitely dilute solution of A in B. If Equation 4 is combined with Equation 3, one obtains
Equation 5 has been applied to the systems ethanol-water and acetone-water, and the predicted values of the mutual diffusivity agreed well with the experimental values (79). However, when Equation 5 is applied to binary systems containing an associated component and a nonpolar component, the effect of the activity gradient is to "overcorrect" greatly the diffusivity-mole fraction curve. Since the shape of the curve predicted by Equation 5 was similar to that of the experimental curve, this suggested that the phenomenological coefficient, L, may vary as d In a / d In X raised to some power. I t is proposed, therefore, that a n empirical exponent be placed on the activity gradient so that Equation 5 has the form DAB = (DBOXA f DAoXB)(d In a / d In X ) 8 T , p
(6)
Results and Discussion
Equation 6 with s = 0.6 was applied to a number of binary liquid systems containing an associated component and a nonpolar component. These systems together with the maximum per cent difference between the diffusivity values predicted by Equation 6 and experimental values from the literature (3-5, 9, 74, 78, 20), and the mole fraction a t which the maximum deviation occurred, are listed in Table I. For the alcohol systems, this comparison was limited to alcohol mole fractions greater than 0.09. For smaller alcohol mole fractions, it was impossible to determine the activity gradient values accurately. A comparison between experimental and predicted diffusivities is given in Figure 1 for the ethanolcarbon tetrachloride system a t 25.0' C. and for the methanolcarbon tetrachloride system a t 55.0' C. in Figure 2. VOL. 5
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~i
2.
%
n..
0.5
"6
I!O
0:2 014 0:s 018 Mole fraction ethanol
Figure 1 . Diffusion coefficients for system ethanolcarbon tetrachloride at 25.0' C.
0
'
--
---
Hammond and Stokes ( 1 4 ) Anderson and Babb ( 4 ) Calculated from Equation 6 with s = 0.6 Calculated from Equation 6 with s = 1
Ob
--Table I. Comparison of Mutual Diffusivities Predicted by Equation 6 with Experimental Values for Binary Systems Containing One Associated Component s = 0.6
System
A CHIOH CHsOH CHxOH CH~OH CzHjOH CzHsOH CzHjOH CzH60H CH3COCH3 CH3COCH3 CH3COCHa CH3COCH3 CH3COCzHs CH3OH CHIOH CHsOH CiH60H CgH50H CzHsOH
Source of Data,
Ref.
+
(4; 74; i5j ( 7 . 75. 20)
+ +
CCla 25.0 C6Hs 1 1 . 0 C6Ha 27.06 C6Hs 4 0 . 0 CBHG 1 5 . 0 C6He 25.15 C ~ H B 39.98
- 9.01 $23.6 S1O.O +10.2 -12.5 +15.4 -15.4
0.0454 0.695 0.50321 0.2559 0.60 0.09574 0.5068
(3, 73) ( 9 , 78, 22) ( 9 , 22) ( 5 , 22) ( 78,24) (5,24) (5,24)
Table II. Comparison of Mutual Diffusivities Predicted by Equation 6 with Experimental Values for Systems with Negative Deviations from Raoult's l a w s = 0.3
M a x . Dev. of Pred. Values X A at Source System - ?mp., from Pot& of of Data, A B C. Exptl., % M a x . Dev. Ref. -2.66 (CzH6)zO CHCl3 2 5 . 0 0.404 (7,2) -2.36 CHsCOCH3 CHCI3 25.15 0.2081 ( 5 , 76, 27, 25 ) CHsCOCH3 CHCla 39.95 -4.57 0,3984 ( 5 , 76, 27, 25 )
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110
Figure 2. Diffusion coefficients for system methanolcarbon tetrachloride at 55.0' C.
0 --
M a x . Dev. of Pred. Values X, at - Tgmp., from Point of B C. Exptl., % M a x . Dev. -10.1 0.09955 CCla 1 0 . 0 $12.5 0.399 CCI4 2 5 . 0 0.8000 - 5.49 CCIa 4 0 . 0 0.09971 5.92 CCL 5 5 . 0 +23.6 0.1001 CCL 1 0 . 0 0.30 -12.3 CCla 25.0 0,7993 +11 . o CCla 4 0 . 0 0.4006 -11.4 CClr 55 .O 8 . 5 0 0.1997 CCla 1 0 . 0 0.2056 6.20 CC1a 2 5 . 0 0.2008 +10.4 CCla 4 0 . 0 - 2 . 1 0 0.5996 CClr 55 .o
0:2 014 0:s 018 Mole fraction methanol
PROCESS DESIGN A N D DEVELOPMENT
Rathbun (20) Calculated from Equa4ion 6 with s Calculated from Equation 6 with s
= 0.6 = 1
Slightly different values of the empirical constant, s, would have given better agreement for some systems. However, in the interest of determining an average constant, 0.6 was used for all the systems in Table I. The activity gradient values were determined from vapor-liquid equilibrium data available in the literature (6, 7, 8, 73, 75, 22, 23, 24). These values were interpolated or extrapolated to the required temperatures by assuming that the activity gradient varied linearly with reciprocal absolute temperature. Similarly, Equation 6 was applied to binary systems with negative deviations from Raoult's law, but a value of 0.3 was used for s. The results are tabulated in Table 11. Again the activity gradient values were determined from vapor-liquid equilibrium data available in the literature ( 7 , 76, 27, 25). I t is thus possible to predict satisfactorily the diffusivitymole fraction relationship for two general types of binary liquid systems from a knowledge of only the diffusivities in the infinitely dilute solutions and activity coefficient data as a function of composition. In addition, the good agreements between experimental diffusivity-mole fraction curves and those predicted by Equation 6 suggest the possibility of calculating activity-composition data for systems of the type used in this study for which diffusivity-mole fraction data happen to be available as suggested by Dullien and Shemilt (72). literature Cited (1) Anderson, D. K., Ph.D. dissertation, University of Washington, Seattle, Wash., 1960. (2) Anderson, D. K., Babb, A. L., J . Phys. Chem. 65, 1281 (1961). (3) Ibid., 66, 899 (1962). (4) Ibid., 67, 1362 (1963). (5) 62, 404 ~, Anderson. D. K.. Hall, J. K., Babb, A. L., Ibid., (1958). (6) Bachman, K. C., Simons, E. L., Znd. Eng. Chem. 44, 202 (1952). (7) Barker, J. A., Brown, I., Smith, F., Discussions Faraday SOC. 15, 142 (1953). (8) Brown, I., Smith, F., Australian J . Chem. 10, 423 (1957). ( 9 ) Caldwell, C. S., Babb, A. L., J . Phys. Chem. 59, 1113 (1955). (10) Ibid., 60, 51 (1956).
(11) DeGroot, S. R., Mazur, P., “Non-Equilibrium Thermodynamics,” pp. 245-52, Interscience, New York, 1962. (12) Dullien, F. A. L., Shemilt, L. W., Nature190, 526 (1961). (13) Fowler, R. T., Norris, G. S., J . Appl. Chem. (London) 5, 266 (1955). (14) Hammond, B. R., Stokes, R. H., Trans. Faraday Sac. 52, 781 (1956). (15) Ishikawa, F., Yamaguchi, T., Bull. Znst. Chem. Res. ( T o k y o ) 17, 246 (1938). (16) Karr, A. E., Scheilsel, E. G., Bowes, W. M., Othmer, D. F., Ind. Eng. Chem. 43,961 (1951). (17) Laity, R. LV., J . Pkys. Chem. 63,80 (1959). (18) Lemonde, H., Ann. Phys. 9 , 539 (1938). (19) LisnyanskiY, L. I., Vuks, M. F., Russian J . Phys. Chem. 38, 339 (1964).
(20) Rathbun, R. E., Ph.D. dissertation, University of Washington, Seattle, Wash., 1965. (21) Rosanoff, M. A,, Easley, C. W., J . Am. Chem. Sac. 31, 953 (1909). (22) Scatchard, G*’ Wood, E‘a Mochel, M*3 68, 1957 (1946). (23) Zbid.3 P. l960. (24) Udovenko, V. V., Fatkulina, L. G., Russian J . Phys. Chem. 26, 719 (1952). (25) Zawidski, J. V., Z. Physik. Chem. 35, 129 (1900).
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RECEIVED for review September 7, 1965 ACCEPTEDMarch 24, 1966
SIMPLE ASYMPTOTIC RELATIONS FOR HEAT TRANSFER T O UNIFORMLY OR NONUNIFORMLY SIZED CLOUDS OF PA R T IC 1, ES GODFREY C. GARDNER Central E1ectricity:Research Laboratories, Leatherhead, Surrey, England
The heat exchange process between a cloud of uniformly or nonuniformly sized particles and a concurrent or countercurrent stream of gas or liquid is studied. Simple expressions are developed which describe the asymptotically approached temperature distributions within and between particles and the asymptotically approached heat transfer rate. These expressions may often b e applied with negligible error, as is demonstrated by cornparison with the exact solutions for uniformly sized particles of Munro and Amundson. In any case they teind to overdesign rather than underdesign the size of equipment. The parameters included in the analysis--thermal conductivity of the particles, heat transfer coefficient from the surfaces of the particles to the fluid, velocities of the particles relative to the walls of the equipment, and ratio of the heat capacity of the fluid stream to that of the liquid stream-are the same as those studied b y Munro and Amundson, but extended to include particle size distribution.
industrial processes involve the transfer of heat between a cloud of nonuniformly sized particles and a concurrent or countrrcurrent gas or iiquid stream. The boundary condition ai. the particle inlet to the system is normally that all particles have the same temperature and that the temperature within particles is uniform. T h e problem is usually to determine the size of equipment necessary to perform a given heat transfer cluty, knowing the mass flow rates of the fluid and particle streams and their average inlet and outlet temperatures Interest, however, may also attend the details of the temperature distribution that develops both within and between particles For example, the process may involve the freezing of a liquid sp-ay when the temperature of the largest or hottest particle is inportant, or it may be desired to limit the vaporization of particles and it will be necessary to limit the range of particle temperature a t any location in the system. This paper obtains expressions, which are accurate within the assumptions stated later, determining the heat transfer rate and the temperature (distributions, both between and within particles, which are asymptotically approached as the particles move away from their inlet boundary condition. As a n approximation it may be assumed that these asymptotic exANY
pressions prevail throughout the equipment and this will often result in negligible error, especially for engineering purposes. Moreover, if the particle inlet conditions are the usual ones noted above, the determined size of the heat exchanger will be conservatively large. This is important, because Munro and Amundson ( 3 ) noted with respect to clouds of uniformly sized particles that the numerical methods employed by Love11 and Karnofsky (2) gave errors which underdesigned. Exact solutions are available for the problem involving uniform particles when the temperature is initially uniform within a particle. The solution of Gurney and Lurie ( 7 ) is well known in its graphical form and is applicable to the case where the mass flow rate of the fluid stream is infinite compared with that of the particle stream. Munro and Amundson (3) solved the problem more generally to include finite mass flow rate ratios. Their solution is in the form of a rapidly convergent series and they state that a competent calculator can determine the size of a heat exchanger in 4 hours. T h e asymptotic solution is much simpler and in many cases is adequately accurate, as a comparison will show. For nonuniformly sized particles the asymptotic solution is the only one available without recourse to tedious numerical methods. VOL. 5
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