Further work in progress aims to continue these studies using both theoretical and experimental approaches. It is believed that this microscopic approach to the problem of fluid-solids contacting adds to knowledge available by the contrasting statistical or empirical procedures often used. Acknowledgment
The authors are grateful to the Research Corp. and the National Science Foundation for providing support for this study.
Motion of Spheres and Fluid in
Cylindrical Tubes
Nomenclature a = radius of spherical particle
T H I S theoretical investigation continues previously published studies of a fundamental solution for hydrodynamic relationships underlying low Reynolds number phenomena involving particles suspended in a fluid. As in earlier studies, so-called creeping motion equations are employed in conjunction with spherical particles in a cylindrical tube. Attention is confined to purely hydrodynamic aspects of these systems. Behavior of a single sphere, free to occupy any position in a tube and move in an axial direction, is first developed. For this situation, the first approximation which neglects the effect of a/R,-Le., assumes a small ratio of sphere to cylinder diameter-results in the following expressions for drag on a sphere, W , and pressure drop increase caused by its presence, AP:
Relationships developed for a single sphere can be applied to the behavior of more than one sphere, employing the same mathematical techniques involved in obtaining Equations 1 and 2 . To the approximation employed here, the fluid velocity field consists of the original undisturbed Poiseuille flow. The drag exerted on each particle is given by Stokes' law. The pressure drop caused by each particle depends solely on its location and velocity, independent of other spheres. Three cases are developed. I n the first, the particles do not move relative to each other and are uniformly distributed over the cylinder cross section; in the second, the particles remain uniformly distributed in space but are free to move relative to each other; and in the third, radial distribution of the
particles is not uniform and they are free to move relative to each other. I n all cases it is assumed that, except at the ends of the tube, motion is confined to the axial direction. From the practical standpoint, the first case is realized in sedimentation. The second case may be approximated where both fluid and particles are introduced into a system simultaneously as in pneumatic conveying and pumping of slurries or when fluid velocity is relatively low as in particulate fluidization phenomena. The third case is believed especially applicable to the behavior of aggregative fluidization. I n this case, the conclusion is reached that a build-up of particles near the walls of the tube takes place, Pressure drops are greatest relative to bed weight for fixed assemblages. Recirculation of a uniformly dispersed suspension lowers pressure drop. If redistribution as well as recirculation occurs, still lower pressure drops are attainable but a n unstable condition results.
b = distance of sphere center from cylinder axis n, = number of spheres per unit volume in outer annular area where particles are moving down n, = number of particles per unit volume in inner cylindrical space where particles are moving up AP = pressure drop caused by sphere R = perpendicular distance from longitudinal axis of cylinder R, = radius of cylinder U = velocity of sphere in an axial direction with respect to an origin fixed with respect to cylinder wall U, = velocity of fluid with respect to cylinder wall, at the axis of the cylinder at sufficient distance from the sphere for the pattern to be parabolic
JOHN HAPPEL and HOWARD BRENNER College of Engineering, New York University, New York, N. Y.
The complete paper is b ing reviewed by the A.I.Ch.E. Journal a& the Journal of Fluid Mechanics for possible publication.
PARTICLE CONCENTRATION,
n 0
In aggregative fluidization, particles build up near the walls of the tube VOL. 49,
NO. 6
JUNE 1957
969
