DIFFUSION I N A HOMOGENEOUS FLUIDIZED BED
SIR: In principle two approaches may be followed for the analysis of diffusion in a fluidized bed: an (Eulerian) approach which has as its starting point equations of motion for the fluid and for the swarm of particles treated as a quasifluid, and a Lagrangian one which starts with either kinematic (Corrsin, 1962; Hinze, 1962; Taylor, 1921) or dynamical (Houghton, 1966; Ruckenstein, 1964) considerations. Though equations of motion for the two phases (fluid and particles) have been established and the instability of the reticulate structure to small perturbations has been examined by means of these equations (Pigford, 1965), we are not yet able to predict the stable state of a fluidized bed, which because of its unsteady character is analogous to that of a liquid in wave motion or of a turbulent fluid. Some attempts have been made recently in this direction (Ruckenstein, 1966) and the information obtained in this manner concerning the frequency of the density pulsations and their propagation velocity might be useful in the study of the mixing process. A Lagrangian kinematic description has been suggested for turbulent diffusion (Corrsin, 1962 ; Hinze, 1962 ; Taylor, 1921) and has been extended to a fluidized bed. Though valid for very restricted conditions, the Lagrangian descriptions used by Ruckenstein and by Houghton are based on dynamical considerations. The first (Ruckenstein, 1964) has as a starting point an equation of motion for a particle which is one of a swarm of particles interacting ivith a fluid (Ruckenstein, 1962). A similar equation, discussed by Houghton (1963), is used by Houghton (1966) as a basis for the extension to a fluidized bed of the equation established for the diffusion coefficient in the theory of Brownian motion. T h e aim of this communication is to show that the two dynamical approaches of Houghton and Ruckenstein refer to two extreme situations. The comparison will be made by means of the very simple method initiated by Langevin (Becker, 1954; Langevin, 1908) in his study of Brownian motion. The starting point will be the Langevin equation of motion dv dt
+ pv = A ( t )
\\.here pv is a dynamical friction experienced by the particle, while A ( t ) is a random function representing the action of the molecules. Let us multiply Equation 1 by x , the displacement of the solid particle. There results
I n the theory of the Broivnian motion it is essential that the average value of xA(t) taken over a large number of identical particles is nil. This ensemble average is nil, since the displacement of the particles and the action of the molecules are noncorrelated statistical quantities. I n this case, Equation 2 leads to
(3) Lrhich, if t
>>
Since
dx? -=2D dt one obtains
(5) Houghton (1966) writes an equation similar to Equation 1 for a particle in a fluidized bed and suggests, starting from it and using a more refined formalism (Way, 1954), the extension of Equation 5 to the diffusion of the solid particles in a fluidized bed. We remark that in the case of a fluidized bed in the composition of the quantity denoted by A ( t ) in Equation 1 there enter terms which contain the fluid velocity fluctuation (see Houghton’s Equations 9 and 11). Since the velocity fluctuation of the fluid and the displacements of the solid particles are in this case statistically dependent quantities, the ensemble average of the product is not zero. A similar comment may be made in connection \vith Houghton’s treatment of mixing in the liquid. The approach we proposed for mixing in the fluid (Ruckenstein, 1964) takes into account in a limiting case the correlation between the fluid velocity and the velocity of the particles. As in the case of the turbulent motion of a fluid, a general method of solving such problems is not available. For this reason we used dimensional considerations instead of the equation of motion of the turbulent interstitial fluid and some assumptions concerning the constancy of some coefficients (which are some kind of correlation coefficients) appearing in the expression of the average value of the square of the particle velocity. T h e two approaches therefore represent tlvo extreme approximations. Houghton shons that his theoretical equation does not agree with experiment and that the agreement may be improved by applying the stochastic approach to clusters of particles. I n this case too it would be necessary to take into account the statistical correlation between the motion of the clusters and that of the fluid. As shown by Kramers et al. (1962), particularly in a long liquid fluidized bed there exist clusters of particles which have oscillatory motions around some fixed points. Such a model has been used previously by us (Ruckenstein, 1964) assuming, however, one particle as a dynamical unit. Kramers’ observation suggests the consideration of a cluster of particles as a dynamical unit. However, the cluster is not formed as a result of adhesive forces, but as a result of “hydrodynamic forces;” unfortunately, we are not able to predict how their size depends on hydrodynamical parameters. Conclusion
I n the stochastic approach (Houghton, 1966) the statistical correlation between the motion of the particles and that of the fluid is not taken into account; in a limiting case, our previous approach (Ruckenstein, 1964) takes into account only the above-mentioned correlation.
becomes Nomenclature
(4)
618
I&EC FUNDAMENTALS
D t
= diffusion coefficient = time
= particle velocity = displacement of a particle = constant in Equation 1
U
x
E-
9,x2, x A ( t )
= ensemble average of
9,x2, and x A ( t )
Literature Cited
Becker, R., “Theorie der Warme,” Springer, Berlin, 1954. Corrsin, S., “Colloques internationaux du C.N.R.S., Mtcanique de la Turbulence,” p. 27, Editions du Centre National de la Recherche Scientifique, Paris, 1962. Hinze, I. O., “Colloques internationaux du C.N.R.S., Mecanique de la Turbulence,” p. 63, Editions du Centre National de la Recherche Scientifique, Paris, 1962. Houghton, G., IND.ENG.CHEM.FUNDAMENTALS 5,153 (1966). Houghton, G., Proc. Roy. Sot. A272,33 (1963).
Kramers, H., Westerman, M. D., de Groot, I. H., Dupont, F. A. A,, Proceedings of the Symposium on Interaction between Fluids and Particles, p. 114, London, June 1962. Langevin, P., Compt. Rend. 146, 530 (1908). Pigford, R. L., Baron, Th., IND.ENG.CHEM.FUNDAMENTALS 4, 81 (1965). Ruckenstein, E., INJ. ENG.CHEM.FUNDAMENTALS 3, 260 (1964). Ruckenstein, E., Rev. Phys. (Bucharest) 7, 137 (1962). Ruckenstein, E., Third Internatiorial Heat Transfer Conference, Paper 145, Chicago, August 1966. Taylor, G. I., Proc. London Math. Soc. A20, 196 (1921). Way, N., Ed., “Selected Papers on Noise and Stochastic Processes,” Dover Publications, Kew York, 1954.
E. Ruckenstein Polytechnical Institute Bucharest, Romania
Correctiori
Correction
THERMODYNAMICS OF SOLUTIONS. EQUATION OF STATE AND VAPOR PRESSURE
MOLECULAR THERMODYNAMICS OF SIMPLE LIQUIDS
In this article by Otto Redlich, F. J. Ackerman, R. D. Gunn, Max Jacobson, and Silvanus Lau [IND.ENG.CHEM.FUNDAMENTALS 4, 369 (1965)], Equation 12 should be replaced by:
+ [0.427481 P,T,-2.5
ZO3- Zo2
- 0.0866404 P7T,-l
x
(1 f 0.0866404 PTT,-’)]Zo- 0.0370371 Pr2TI-3.5 = 0
I n this article by Henri Renon, C. A. Eckert, and J. M. Prausnitz [IND. ENG. CHEM.FUNDAMENTALS 6 , 52 (1967)], some errors were found. O n page 54, left column, lines 7, 17, 20, and 26 from the bottom: fz = 1 (not zero), and on page 56, the captions for Figures 3 and 4 should be interchanged.
(12)
At the end of Equation 17, the term (E13
- T,2)
should be replaced by:
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