SUBBCRIPTS refers to pipe axis = refers to the absence of transverse flow = refers t o particulate phase = refers t o a quantity at x / D = 0 =
C 0
P i
Literature Cited
Abramowitch, M., Stegun, I. A., Ed., “Handbook of Mathematical Functions,” Table 267, pp 978-983, Dover Publications. New York. N. Y.. 1967. Chai, 8. T., Oesteir. Ing.’Arch. 18, 7 (1964). Corsin, S., Lumley, J., Appl. Sci. Res., Sect. A 6, 114 (1956). Csanady, G. T., J . Atmos. Sci. 20,201 (1963). Friedlander. S. K.. A.I.Ch.E. J . 3. 381 (19573. Godson, W.’ L., Archiv. M e t e d . Geophys. Bioklimatol., Ser. A 10,
Lie, V. C., J . Meleorol. 13, 399 (1956). Lumley, J. L., Ph.D. Thesis, John Hopkins University, Baltimore. Md.. 1957. Norsetb, H. G., Mitchell, R. I., Ann. N . Y . Acad. Sci. 195, 88 11963). Rounds,‘W., Jr., Trans. Amer. Gwphys. Union 36, 395 (1955). Soo, S. L., Chem. Eng. Sci. 5,57 (1956): Soo, S. L., “Fluid Dvnamics of MultiDhase Systems,” Blaisdell Publishing Co., Waltham, Mass., 1967. Soo, S. L., Tien, C. L., J . Appl. Mech., Trans. A S M E 27, 5 (1960). Stukel, J. J., Soo, S . L., Powder Technol. 2, 278 (1969). Suueja, S. K., Ph.D. Thesis, Illinois Institute of Technology, Chicago, Ill., 1970. Tchen, C. RI., Dissertation, Delft, Martinus Nijhoft, The Hague, 1947.
305 (19.58).
Groenhof, H. C., Chem. Eng. Sci. 25, 1005 (1970). Hay, J. S., Pasquill, F., J . Fluid Mech. 2,299 (1957). Hinze, J. L., “Turbulence,” McGraw-Hill, New York, N. Y., 1959. Hughes, R. R., Gilliland, E. R., Chem. Eng. Progr. 48,497 (1952). Kada and Hanratty, A.1.Ch.E. J. 6 , 624 (1960).
RECEIVED for review February 22, 1971 ACCEPTEDSeptember 22, 1971 Financial support was provided by the Fine Particles Section of IIT Research Institute.
Theory of Coalescence by Flow through Porous Media Lloyd A. Spielman*l and Simon 1. Goren Department of Chemical Engineering, University of California, Berkeley, Calif, 9.4720
Our equations describing coalescence by flow through porous media are solved to yield specific predictions for oil-in-water dispersions. It i s shown that for small incoming volume fractions of suspended oil and oil-towater viscosity ratios which are not extremely large, the steady-state removal of droplets of a given size i s independent of the sizes and amounts of other droplets introduced; the aqueous pressure drop also is predicted to be independent of the incoming drop size distribution. Solutions are also obtained which are intended to describe the operation of semipermeable porous barriers. A correlation for the dimensionless filter coefficient as a function of a single dimensionless adhesion number i s suggested when capture i s by London forces in porous media of similar geometry and wettability. The theory i s largely supported by the fibrous mat coalescence experiments of the following paper.
Coalescence of liquid-liquid dispersions induced by flow through porous media has proved effective in a variety of industrial applications, including the treatment of aqueous wastes containing finely suspended oils. We recently summarized the most important applications and relevant literature (Spielman and Goren, 1970b). One of the chief barriers to greater use of porous coalescence was found to be the need for extensive pilot study because lack of understanding of this complex process has prevented the formulation of trustworthy general design equations. To overcome this, we combined concepts of two-phase flow through porous media with concepts of water and aerosol filtration to give a theoretical framework for predicting pressure drop, degree of phase separation, and some complicated capillary phenomena occurring in the separation of liquid-liquid dispersions by flow through porous media. Application of the theory requires the Division of Engineering and Applied Physics, Harvard University, Cambridge, Mass. 02138. 66
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 1 , 1972
capillary pressure, relative permeabilities, and filter coefficient be known as functions of the local oil saturation and other relevant parameters. I n this paper we obtain approximate solutions t o the equations presented earlier (Spielman and Goren, 1970b). Our aim is twofold: first t o provide interpretation of the fibrous mat coalescence experiments reported in the following paper (Spielman and Goren, 1972), and secondly to illustrate the use of the theoretical framework in establishing scaling criteria so that pressure drop and filtration data obtained from relatively small-scale apparatus may be used with confidence to design coalescers for specific industrial applications. Throughout the present treatment, oil is again arbitrarily taken as the suspended phase both for concreteness and because that was the actual situation in the experiments of the following paper, In most of the present work we assume the volume fraction of oil dispersed in the inlet aqueous phase is very small and the viscosity of the oil is not too large compared with that of the continuous phase. When these approximations
are valid as they are for our experimental conditions and for many (but certainly not all) practical applications, then the theoretical framework permits very great simplification, Conclusions of great practical significance are that the pressure drop is independent of the detailed size distribution of the dispersed phase, and the coalescence efficiency for any given size drop, while dependent upon that drop size, is also independent of the size distribution. It is shown that even when these approximations are not valid, the above conclusions still hold for sufficiently thick media likely to be used in practice. Summary of the Governing Equations
The equations for one-dimensional, steady-state coalescence of dispersed oil were given (Spielman and Goren, . 1970b) as capillary pressure pz
Table 1. Summary of Boundary Conditions (Spielman and Goren, 1970b) O i l dispersed in water'
Oil nonwetting Location
Inlet x =0
Zero backflow
Finite backflow
= 0 (above critical m, s 2 = SZC)
q20
sz =
Oil wetting" Zero backflow q20
szo
=
0
(above critical y10, Q20 = 0
Outlet Sz = SzC sz = s z c s 2 = 1 - 8 1, x = L a For oil wetting, finite backflow cannot occur. b For water dispersed in oil, interchange oil with water everywhere.
- PI = PO(8Z)
capillary conduction of aqueous phase ki(S2)
a = --pi
dpi dx
capillary conduction of coalesced oil with j given explicitly in terms of n(ap) through eq 7 . The boundary conditions on Sz, summarized in Table I, were discussed in detail (Spielman and Goren, 1970b). The quantity y20/y10fof r is the rate of deposited oil flowing back from the inlet face divided by the rate suspended oil is introduced into the mat. Because only finely suspended oil is inI0 for coalestroduced into the porous medium, -1 cence applications. Depending on physical conditions, I? is sometimes specified as zero but sometimes obtained as a n eigenvalue through solution of eq 9 with two saturation boundary conditions. Consistent with the outlet boundary condition, the coalesced oil globules released a t the outlet were predicted to have radius
deposition rate of suspended oil
)*d
dx
=
-A(&,
