I
=
= = Jo , ju: j H = g
k kC
m NU n
P
r Pr
R Re 40
sc St t
T U+ V V, Vmax
Y Y-
= = = = = = = = = = = = = = =
current, amperes integrand defined in Table I11 diffusional mass flux a t wall, positive if out of fluid Chilton-Colburn j factors for mass, heat, transfer thermal conductivity mass-transfer coefficient, defined by Equation 4 constant (Equation 24) Nusselt number, hD,/k for heat and kcD,/D, for mass constant in vrlocity profile (Equation 17) pressure radial coordinate for cylindrical geometry Prandtl number, E D p / k tube radius for circular tube Reynolds number, De < v > / v \\.all heat flux, positive if out of fluid Schmidt number, u / a ) t m Stanton number, Nu/Re .i time temperatur:
= u/d/7o/p = velocity = axial velocity component = flow-average velocity = maximum velocity = distance from wall into fluid = (Y/Y)d\/7O/p
Y++
=
(yvmax/v)(f/2)
Y
= =
height of duct axial distance
2
=
(
y+ df72
GREEKLETTERS a 6 A p V
c p T~
Q $ u
thermal diffusivity, k / p t p thickness of concentration boundary layer = Schmidt or Prandtl number = viscosity = p ’ p = kinematic viscosity =
=
= = = = =
2t/Y
density wall shear stress, positive dimensionless velocity, temperature, or concentration asymptotic shape of velocity, temperature, or concentration profiiles = constant in Deissler model for v ( L ) (Table IV)
literature Cited
Analysis,” 2nd ed., Harper and Row, New York, 1963. (3) Brennan, W. C., Trass, O., 52nd Meeting, A. I. Ch. E., Memphis, Tenn., February 1964. (4) Colburn, A. P., Trans. A . I. Ch. E . 29, 174 (1933). (5) Deissler, R. G., Natl. Aeronaut. Space Admin., NACA Rept. 1210 (1955). (6) Deissler, R. G., Taylor, M. F., Zbid., NASA TR 4-31 (1959). ( 7 ) Eisenberg, M., Tobias, C. W., Wilke, C. R., Chem. Eng. Progr. Symp. Ser. 51, No. 16, 1 (1955). (8) Eisenberg, M., Tobias, C. W., Wilke, C. R., J . Electrochem. Soc. 103, 413 (1956). ( 9 ) Zbid., 101, 306 (1954). (IO) Hamilton, R. M., Ph.D. thesis, Cornel1 University, 1963. (11) Hubbard. D. W.,Ph.D. thesis. University of Wisconsin, ’ 1964. (12) Laitinen, H. A , , Kolthoff, I. M., J . Am. Chem. Sac. 61, 3344 (1939). (13 ) Lebich, V., “Physicochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N. J., 1962. (14) Lin, C. S., Denton, E. B., Gaskill, H. S.,Putnam, G. L., Znd. Eng. Chem. 43, 2136 (1951). (15) Lin, C. S., Moulton, R. FV., Putnam, G. L., Ibid., 45, 636 (19531. (16) Ludwieg, H., 2.Flugwiss. 4, 73-81 (1956). (17) Opfell, J. B., Sage, B. H., Aduan. Chem. Eng. 1 , 241 (1956). (18) Page, F., Jr., Corcoran, W. H., Schlinger, W. G., Sage, B. H., Ind. Eng. Chem. 44, 419 (1952). (19) Page, F., Jr., Schlinger, W.G., Breaux, D. K., Sage, B. H., Ibid., 44,424 (1952). (20) Reiss, L. P., M.S. thesis, University of Illinois, 1960. (21) Reiss, L. P., Hanratty, T., A . Z. Ch. E . J . 9, 154 (1963). (22) Schlichting, H., “Boundary Layer Theory,” 4th ed., p. 516, McGraw-Hill, New York, 1960. (23) Ibid., Par. 19. g. (24) Sherwood, T. K., Chem. Eng. Progr. Symp. Ser. 55, No. 25, 71 (1959). (25) Sherwood, T. K., Ryan, J. M., “Recent Advances in Heat and Mass Transfer,” J. P. Harnett, ed., McGraw-Hill, New York. .. .. . 1961 -~ 1
(26) Siegel, R., Sparrow, E. M., ( Trans. A.S.M.E., Sect. C ) J . Heat Transfer 82, 152 (May 1960). (27) Siegel, R., Sparrow, E. M., Hallman, T. M., Appl. Sci. Res. A7.386 (19581. (28) Sleicher, C: A., Trans. A S M E 80,693 (1958). (29) Sparrow, E. M., Hallman, T. M., Siegel, R., Appl. Sci. Res. A7,37 (1957). (30) Van Shaw, P., Reiss, L. P., Hanratty, T. J., A . I . Ch. E . J . 9, 362 (1963). (31) Vieth, W. R., Porter, J. H., Sherwood, T. K., IND.END. CHEM.FUNDAMENTALS 2 , l (1963). (32) Walker, J. E., Ph.D. thesis, Carnegie Institute of Technology, 1957. f 33) Whan, G. A , , Ph.D. thesis. Carnegie Institute of Technology, 1956.
(1) Bird, R. B., Stewart, kV. E., Lightfoot, E. N., “Transport
RECEIVED for review February 1, 1966 ACCEPTEDMay 2, 1965
Phenomena,” \.Viley, New York, 1960. ( 2 ) Blaedel, W. J., Meloche, V. W., “Elementary Quantitative
GRAVITA,TIONAL THINNING OF FILMS Efect of Surface IViscosip and Surface Elasticity STEPHEN W H I T A K E R Department of Chemical Engineering, University of California, Davis, Cal;f.
THE role of surface arctive agents in stabilizing foams and emulsions is known i:o be complex (2, 6, 77, 75). Surface viscosity, surface elasticity, electrical forces, evaporation, and marginal regeneration are all governed by the adsorption of surface active agents a t the gas-liquid interface, and each is capable of playing a n influential role in stabilizing a foam. I n this paper the effect of surface viscosity and surface elasticity on the drainage rate of a single free film is analyzed in the absence of electrical repulsive forces between the two surfaces, marginal regeneration, and evaporation. The calculated drainage rates cannot be valid for the latter
stages of film thinning where the electrical properties of the surface provide the major stabilizing force ( 6 ) ,and the neglect of marginal regeneration [which enhances the film thinning process by “pumping” fluid to the plateau border (75)] and evaporation requires that the calculated results be considered as a lower bound on real film-thinning rates. Discussion
T h e analysis of the drainage of a single free film such as that indicated in Figure 1 requires application of: the laws of mechanics for both the bulk and surface fluids, conservation of VOL. 5
NO. 3
AUGUST 1966
379
The effect of surface viscosity and surface elasticity on the thinning of soap films has been analyzed mathematically. Two special cases of interfacial mass transfer were considered: kinetically controlled transfer between the surface and a substrate of constant concentration, and equilibrium between the surface and a substrate of varying concentration due to convective transport. The first case illustrates the role of the Marangoni effect in foam stabilization, while the second case illustrates the Gibbs effect. Both effects give rise to an “elastic” type of behavior, and the calculated drainage rates indicate that the Gibbs effect is by far the more important.
inverted funnel from a solution containing a surface active agent. The edge of the funnel supports the film, and permits adjustment of the pressure inside the film to produce vertical walls (75). Since the diameter of the film is much larger than the thickness, the flow may be analyzed in rectangular coordinates.
,SUPPORT
T
Bulk Fluid Equations of Motion
For two-dimensional incompressible flow, the equations of motion for an isothermal Newtonian fluid are
.($ + vz 2+ vu
L
1
Figure 1 .
GRAVITY
1
and the continuity equation is
av, -+ ax
Film thinning by gravity
mass for both the bulk and surface fluids, the thermodynamics of surfaces (the surface tension-surface density relationship), and kinetics (rate of mass transfer between surface and substrate). Some insight into the physics of the problem may be gained by examining the equations governing mass and momentum transport for the bulk and surface fluids. Illustrative versions of these equations are shown in Figure 2. I n this work, the surface is treated as a two-dimensional Newtonian fluid and the “elasticity” under consideration is compositional, as opposed to structural, in nature. Under these conditions the surface stress may depart from the equilibrium stress (or surface tension as discussed in thermodynamics) because of surface viscosity effects which are proportional to the rate of deformation. In addition, the equilibrium stress may change with time and space because of transient and spatial variations of the surface density. If the solubility of the surface active agent is comparatively low, a large fraction is adsorbed a t the surface and variations in surface density of the adsorbed species are nearly proportional to the deformation. This gives rise to a true elastic behavior, which is designated as the Gibbs effect ( 7 7 ) . If the solubility is comparatively high, only a small fraction of the surface active agent present in the system is adsorbed a t the surface. Under these conditions, departures of the surface density from the equilibrium value are only temporary since the surface is replenished by adsorption from the bulk phase, and the effect on the surface stress is comparable to a viscoelastic type of behavior. The film shown in Figure 1 may be formed by pulling an 380
- = bv, o
l&EC FUNDAMENTALS
(3)
ay
where vz and uy are the components of the velocity vector in the x and y directions, respectively. If the Reynolds number, N R e , defined as NRs
= V,d/v
(4)
where
vo
= us, y =
0, x = L
d = q,t = 0,x = L is much less than unity, an order of magnitude analysis indicates that the convective inertial terms may be neglected, and Equation 1 reduces to
We need not be concerned with the y-direction momentum equation, for a solution for u Z ( x , y , t ) will provide us with the drainage rate. O u r analysis will be restricted to films having a length much greater than the thickness, and the continuity equation indicates
v,
