radial distance from origin of sphere, dimensionless particle Reynolds number, 2a U m p / p , dimensionless particle Schmidt number, p / p p , dimensionless local particle Sherwood number, 2aK,/P, dimensionless over-all particle Sherwood number, dimensionless mean stream velocity or superficial velocity, cm/ses velocity vector, dimensionless velocity vector, cm/sec velocity in radial direction, dimensionless velocity in angular direction, dimensionless 3x6 - 2Xe, dimenfunction of A, W = 2 - 3x sionless angle measured from front stagnation point, radians radial distance measured from surface of sphere, dimensionless p y , dimensionless step size of first step from surface of sphere in radial direction, dimensionless radial distance from surface of sphere a t grid point j , dimensionless void fraction or porosity in multiparticle systems, dimensionless function of c reaction rate parameter, dimensionless parameter used in lree surface model = (1 - e ) l i a , dimensionless stream function, dimensionless enhancement factor, dimensionless angle from front stagnation point, radians density of continuous phase, g/cc viscosity of continuous phase, poises
+
literature Cited
Acrivos, A,, Taylor, T. D., Phys. Fluids 5, 387 (1962). Baird, M. H. I., Hamielec, A. E., Can. J . Chem. Eng. 40, 119 (1962).
Carnahan, B., Luther, H. A., Wilkes, J. O., “Applied Numerical Methods,” T‘ols. I and 11, Wiley, New York, 1964. Chen, W. C., Ph.D. thesis, City University of New York, 1969. Friedlander, S. K., A.Z.Ch.E. J . 3 , 4 3 (1957). Friedlander, S. K., A.I.Ch.E. J . 7, 347 (1961). Froessling, N., Gerlands Beitr. Geophys. 52, 170 (1938). Garner, F. H., Grafton, R. W., Proc. Roy. SOC.A224, 64 (1954). Garner, F. H., Hoffman, J. M., A.1.Ch.E. J . 6, 579 (1960). Garner, F. H., Keey, R. B., Chem. Eng. Sci. 9, 119 (1958). Garner, F. H., Suckling, R. D., A.I.Ch.E. J . 4, 114, (1958). Goddard, J. D., Acrivos, A., Quart. J . hlech. Appl. Math. 20, 471 (1967).
Happel, J., A.Z.Ch.E. J . 4, 197 (1958). Houghton, W. T., Ph.D. thesis, McMaster University, Hamilton, Ontario. 1966. Johnson, A. I., Akehata, T., Can. J . Chem. Eng. 43, 10 (1965). Johnson, A. I., Hamielec, A. E., Houghton, W. T., -4.Z.Ch.E. J . 13, 379 (1967).
Levich, V. G., “Physicochemical Hydrodynamics,” PrenticeHall, Englewood Cliffs, N.J., 1962. Lochiel, A. C., Calderbank, P. H., Chem. Eng. Sci. 19, 471 (1964).
Peitzman, A., Pfeffer, R., Chem. Eng. Progr. Symp. Ser. 63, 49 (1967).
Pfeffer, R., IND. ENO.CHEM.FUNDAMENTALS 3, 380 (1964). Pfeffer, R., Happel, J., A.Z.Ch.E. J . 10, 605 (1964). Rhodes, J., Peebles, F. N., A.Z.Ch.E. J . 11,481 (1965). Rowe, P. N., Claxton, K. T., Lewis, J. B., Trans. Inst. Chem. Enorn. - 43. ~- T14 1196.5). 0
Rutland, L.: Pfeffer, R.; A.Z.Ch.E. J . 13, 182 (1967). Steinberger, R. L., Treybal, R. E., A.I.Ch.E. J . 6, 227 (1960). Yuge, T., Rept. Inst. High Speed Mech., Tokoku Univ. 57, 143 (1956).
RECEIVED for review August 9, 1968 ACCEPTED August 21, 1969
Finite-Amplitude, long Waves on Liquid Films Flowing Down a Plane William
B.
Krantzl and Simon 1. Goren
Department of Chemical Engineering, University of California, Berkeley, Calif. 94720
A theory for predicting the wave forms of finite-amplitude, long waves on liquid films flowing down a plane is presented. It is a power series expansion in wave number, retaining surface tension in the first-order terms. A major result is that equilibrium amplitudes for most highly amplified waves are expected to be a function of a single group (1.2 N R ~ cot W / N R , Nwe. New data for viscous oils are adequately correlated by this group, but the data of Kapitza and Kapitza for water and alcohol are not in agreement with those for the oils.
KRAXTZ
and Goren (1970) recently reviewed the literature concerning small-amplitude waves on liquid films flowing down an inclined plane; the geometry of this flow is shown in Figure 1. These authors presented the first measurements of wavelengths, wave velocities, and wave amplification rates for waves resulting from imposed disturbances of controlled amplitude and frequency. Their data and data from the literature on wavelengths and wave velocities of waves resulting from room disturbances (most highly amplified waves) were in good semiquantitative agreement with the various 1 Present address, Depart,ment of Chemical Engineering, University of Colorado, Boulder, Colo. 80302.
linear stability analyses. The data and theories show that the stability of thin film flow a t moderate or low Reynolds numbers is governed by disturbances having long wavelengths compared to the film thickness. For example, for water flowing on vertical columns the wave number (for maximum growth rate) , a = 2nho/h, is less than 0.2 when the Reynolds number, NR*= tiLho/v, is less than -170, and less than 0.1 when the Reynolds number is less than -10. Although the wave number is small, the product a2Nnre, where Nw. = a/photi2 is the Weber number, is of order unity, a fact of considerable importance in the theory presented. This paper presents a theory describing finite-amplitude, VOL. 9 NO. 1 FEBRUARY 1970
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107
Figure 1.
Geometry for rippled film flow
two-dimensional, equilibrium waves on thin films flowing down an inclined plane. I n accord with the comments above, as a first but practical approximation attention is restricted to waves of very small wave number but with aZNweof order unity. Wave amplitudes may be comparable to the film thickness. Measurements of equilibrium wave amplitudes for viscous oils flowing on vertical and inclined columns are reported and correlated as suggested by the theory. Previous Work
With the exception of the theory of Anshus (1965), nonlinear differential equations for the wave amplitude have been derived from the kinematic surface condition relating the wave form to the velocity components a t the surface. The velocity components are eliminated from this equation by solving the unsteady-state, nonlinear, two-dimensional equations of motion. Among the techniques for solution are the momentum integral approach of Kapitza (1948) and subsequent workers, the power series solutions in wave number of Benney (1966), and the power series solutions in y of &lei (1966) and Nakaya and Takaki (1967). Kapitaa (1948) solved boundary-layer type equations for long, traveling waves using a momentum integral approach. The resulting velocities were substituted into the surface kinematic condition to obtain a nonlinear total differential equation for the wave form. A solution of this equation obtained by the method of successive approximation reveals that in the second approximation the wave velocity and mean film thickness are functions of the wave amplitude. The equilibrium amplitude, however, is not obtained by directly solving this nonlinear equation. Rather, Kapitaa obtains it by invoking the assumption that the most stable periodic flow corresponds to that having the minimum mean film thickness subject to the constraint that the energy dissipation is equal to the work done by the body forces. The energy dissipation is calculated using the steady-state, periodic solution to the linearized equation for the wave form. This analysis yielded a wave velocity of 2.4 and a wave amplitude (one half peakto-peak) of 0.46 independent of the wave number, Reynolds number, and surface tension. Bushmanov (1961), carrying out a similar analysis including the cross-stream convection term which Kapitaa neglected, found a wave velocity of 2.0 and a wave amplitude of 0.58. Lu (1965) considered the effect of varying angles of inclination and solved the amplitude equation to a higher order of approximation to obtain the Reynolds number and column-angle dependence of the wave velocity and amplitude. Also using Kapitza’s approach, Massot, Irani, and Lightfoot (1966) included the axial diffusion term in the equation of motion and considered higher order terms in the viscous dissipation function to arrive a t a 108
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NO. 1 FEBRUARY 1970
Weber number dependence of the equilibrium amplitude. This work is questionable, however, since the new terms these authors incorporated into their analysis are of the same order as other terms which they omitted. They predict that the equilibrium amplitude will increase with increasing Weber number (u/ph$), but the converse is observed experimentally. Benney (1966) solved the nonlinear equations of motion in a manner similar to Yih’s (1963) power series in wave number solution for the linear stability problem. Each term in this series can be evaluated in terms of the unknown film thickness. A nonlinear differential equation is then obtained for the film thickness by applying the kinematic surface condition. For the special case of small perturbations around the undisturbed film thickness Benney obtains Yih’s linear long wave results to first order in wave number, except that the term involving the surface tension has been omitted. Surface tension does not appear because it occurs as an ~ * N term; w ~ Yih (1963) argues that it should be considered along with the terms of order CY, since it is the only term involving surface tension. Benney’s analysis is correct if the Weber number is of order unity. However, this is often not the case. For mater, the Weber number is large for moderate or small Reynolds numbers (Nw.,> 10 when N R ~ 10 when N R
