k and k - 1 is capped probability that a given face between cells at k and k 1 is capped unfilled void volume in mercury penetration probability that a given cell belongs to a closed configuration surface tension
Nomenclature
wk’
= probability that a given face between cells at
C
wkv
=
W W,
= =
y
=
probability that a given cell is connected by a bridge of cells filled with liquid to the surface of the bed on the liquid side Ck = probability that a given cell located at k is connected d N / d s = experimentally determined distribution function of sphere separations F = fraction of void volume occupied by liquid Fk = probability that a given cell located a t k will be filled with liquid k = distance from the surface of the bed on the side of the displacing fluid m = thickness of bed in units of number of cells P = pressure across the bed of spheres p = probability that the opening on any given face of a cell will be open P R / y = reduced pressure q = I - p = radius of spheres used in packing R S = distance between sphere surfaces on the smaller diagonal of the face of a cell w = probability that a given face of a cell is capped = probability that a given face between cells both wk located at k is capped =
+
Literature Cited
Chizmadzhev, Yu. A., Soviet Electrochem. 2, 1 (1966). Dombrowski, H. S., Brownell, L. E., 2nd. Eng. Chem. 46, 1207 (1 954). Frevel, L. K., Kressley, L. J., Anal. Chem. 35, 1492 (1963). Harris, C. C., horrow, N. R., Nature 203,706 (1964). Iczkowski, R. P., IND.ENG.CHEM.FUNDAMENTALS 6,263 (1967). Ksenzhek, 0. S., Kalinovski, E. A., Tsiganok, L. P., Zh. Fiz. Khim. 38, 2587 (1964). Mason, G., private communication, 1967. Mason, G., Clark, W., Nature 207, 512 (1965). Mayer, R. P., Stowe, R. A.,J.ColloidSci. 20,893 (1965). RECEIVED for review November 13, 1967 ACCEPTED June 24, 1968
DEFORMATION AND BREAKUP OF LIQUID DROPLETS IN A SIMPLE SHEAR FIELD H. J.
KARAM AND J. C. BELLINGER
Plastics Fundamental Research, The Dow Chemical Go., Midland, Mich.
Theoretical factors involved in deformation and breakup of liquid droplets in a simple shear field are discussed. The theoretical formulas were checked experimentally; excellent agreement was obtained. Internal circulation within a drop was observed to make it more stable. The practical significance of the results is discussed in terms of the formation of suspensions.
HE
phenomena of deformation and breakup of liquid drop-
Tlets in a simple shear field are of utmost importance in
many chemical processes and unit operations. While actual cases are exceedingly complicated, a basic understanding of the breakup of liquid droplets under ideal conditions will probably give a better insight into the factors involved in the more complex cases. This paper discusses some of the theoretical aspects of drop formation and liquid droplet breakup and compares the results predicted by these theories with those from actual experimental data. Theory
Deformation of a Droplet in a Simple Shear Field. T h e theory of droplet deformation and breakup was formulated by Taylor (1932, 1934, 1954) and modified by Mason and coworkers (Bartok and Mason, 1959, 1961; Rumscheidt and Mason, 1961a,b; Travelyan and Mason, 1951) and by Cerf (1951). When one liquid is at rest in another immiscible liquid of the same density, it assumes the form of a spherical drop. Any movement of the outer fluid (apart from pure rotation or translation) will distort the drop because of the dynamic and viscous forces which act on its surface. Interfacial tension, however, will tend to keep the drop spherical. From experi576
I&EC FUNDAMENTALS
mental observations of these forces, Taylor formulated his theory. His basic assumptions are: The drops are so small that they remain nearly spherical when deformed. There is no slippage at the surface of the drop. The tangential stress parallel to the surface is continuous at the surface of the drop, so that any film which may exist between the two liquids merely transmits tangential stress from one fluid to the other. The normal stresses, on the other hand, are discontinuous at the drop interface and generate a pressure difference across this interface. The transmission of tangential stress across the film between the two fluids causes a system of gradients to be set up-Le., circulation within the drop. This circulation can result in the stabilization of the drop when it is subjected to shear. By the application of hydrodynamic theory, Taylor derived the following equation for the pressure difference across an interface :
H e further related the deformation of a drop to the physical properties of the fluid by the following equation:
I 1
-
I Figure 1.
..
surface tensLon forces ouuoslw " deformation are neelieible _ " compared with those due to viscosity. I n simule shear flaw the lim7irldroplet rotates and assumes a steady shape. The limiting de formation is given by the following equation:
Couette apparatus i 9 indmmdmt This eqLliltinn ~. ~...~.~ ...-.... nf .. the ....
4 will be along the direction of flow-Le., Mi lrop in a Simple Shear Field. Taylor furthcl ~ L ~ L C ~U ~ i d i a drop of minimum size, the disruptive forces due to viscosity tending to burst the drop are about equal to the forces due to interfacial tension tending to hold it together; the following relationship holds: 1 a = - 67
