PHYSICAL ASPECTS OF FOAM 3O7 DRAINAGE After a standing column of foam has been draining for some time, the liquid content will be low at the top and high at the bottom. For many foams, the average liquid content can be approximated in terms of an exponential time decay, or in other ways. However, most of the analyses of drainage which have been offered so far require the use of empirical constants because the approaches were in large measure empirical. A more fundamental attack (11) recognizes that, at least for dry foams, the drainage takes place primarily through the Plateau borders, which essentially are randomly oriented and of curved triangular cross section (as shown in Fig. 1) with walls that are subject to a finite surface viscosity. Regular dodecahedra of equal size were taken as the model for the bub- bles. By geometry, this gives 1.952X/r 2 of PB length per unit volume of foam, where X is the volumetric fraction of gas in the foam, and r is the radius of the sphere equal in volume to a dodecahedron. From the randomness of the PB orientation, the number of PB's intercepted by a unit of cross-sectional area is half, or 0.976x/r 2 (23). The differential equation of momentum conservation for incompress- ible rectilinear steady flow was solved for a general PB, and the resulting local velocities were combined vectorially with the upward movement (if any) of the foam as a whole. The resultant velocities were then inte- grated with respect to the horizontal cross section of the vertical column to give the rate of steady drainage for a stationary foam (or the net rate of steady liquid upflow if the foam is rising). The analysis does not in- clude coalescence as such, so the results are limited to stable foams unless the extent of coalescence is available from other information. Steady drainage of a stationary foam in a column can be achieved by feeding liquid steadily to the top of the foam. For such a system, the theory relates the drainage rate to the liquid content of the foam (24). In principle, through combination with an unsteady-state mass balance over a horizontal element of differential thickness, this theory also can be applied to the unsteady drainage of a stable foam. The theory also relates the rate of foam overflow to the various in- dependent parameters by means of two dimensionless groups, oe and •. Group o is defined as 4QAgt•re/Get• or its equivalent, 41togre/lto%. Q is the volumetric rate of foam overflow on a gas-free basis (in other words, just the liquid), A is the horizontal cross-sectional area of the column, g is the acceleration of gravity, G is the volumetric rate of gas flow, t• is the dynamic liquid viscosity, uo is the superficial linear velocity of collapsed

308 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS (gas-free) foam overflow, uo is the superficial linear velocity of gas, and v is the kinematic liquid viscosity, t•/p. Group 0t is defined as t•aG/t•82gpA or its equivalent uova/gv?, where •8 is the kinematic surface viscosity, Figure 3 shows the theoretical relationship (9) between oe and • for very dry foam. For wetter foam, the theoretical results are more com- plicated, but can be well approximated by simply multiplying Q ob- tained from Fig. 3 by (1 + 3Q/G) to give a revised Q. For variation in bubble size, r should be replaced by ra.•. For columns of nonuniform cross section, the theoretical development must be modified. 20O 4O ioc 8o 60 2o IC I0 -5 I0 -4 I0 -'• IC• 2 IC[ I I I0 Figure 3. Theoretical relationship (9) between oe and • [or very dry loam The theory also predicts that a very stable foam in a vertical column of uniform cross section under conditions of steady drainage or steady overflow will show a liquid content that does not vary appreciably with level (height). This interesting result has been supported by experi- ment (3, 9, 10). The quantitative predictions of liquid content and overflow rate have been fairly well confirmed too, provided a proper effective t•8 is employed for the surfactant involved (3). In particular, the values of pts ) 104 dyne sec/cm for saponin, bovine serum albumin, and Triton X-100 in water are 4.2, 2.6, and 1.0, respectively (based on the bubble size distributions observed at the column wall, without further correc- tion). Such low effective values for saponin and albumin in comparison to those obtained by surface viscosimetry may reflect, at least in part, the thixotropic or other non-Newtonian behavior of such otherwise rigid surfaces--the average transit time for flow through a PB (from mixing cell to mixing cell) being less than one second.