I
R. M. CHRISTIANSEN’ and A. NORMAN HIXSON University of Pennsylvania, Philadelphia, Pa.
Breakup of a Liquid Jet in a Denser Liquid
I
C o r r e l a t i o n of D r o p Size a n d Inff uence of Interfacial Tension
THE
disruption of a liquid jet in extraction columns produces drops, creating extensive interfacial area. Because the rate of mass transfer between two liquid phases depends directly upon the extent of the area separating those phases, the precise correlation of mass transfer coefficients requires an accurate measure of interfacial area. Despite this fundamental importance, few authors ( 5 , 7) have attempted to correlate interfacial area generated by jet breakup with the properties of the system and nozzle size. However, numerous correlations have related capacity coefficients to flow rates in jetting systems. In many cases the correlations probably reflect changes in area rather than in the coefficients themselves. The prime objective of this investigation was to develop a theoretical basis for correlating drop sizes from the breakup of liquid jets. A second objective was to characterize the anomalous effects of certain surface active solutes on jet disruption in terms of a nonequilibrium interfacial tension existing at the time of breakup.
Present address, Research Laboratorv, Owens Corning Fiberglas Corp., Newark; Ohio.
Previous Work The first studies directed toward understanding jet breakup were concerned with the readily observable maximum jet length that occurs at some critical flow velocity. Smith and Moss (76) found this velocity to be proportional to d~/pd, and independent of viscosity. Tyler and Watkin (78) found that viscosity had no effect on jet breakup and correlated flow a t maximum jet length by relating two dimensionless groups,
V,
(e) ’/’,
the square root of the nozzle
Weber number, and
(2) ”’
the ratio
of nozzle Reynolds number, to this modified nozzle Weber number. Ohnesorge (77) used the latter dimensionless ratio and the nozzle Reynolds number to define jet breakup over wide ranges of flow rates and showed that maximum jet length corresponded to a transition between two different mechanisms of jet disintegration. Merrington and Richardson (70) investigated the transition region and based the correlation of its occurrence with flow rate on nozzle diameter. Hayworth and Treybal (5) analyzed jet breakup in terms of an energy balance around a drop breaking away from a nozzle and developed a semitheoretical equation limited to jet velocities below 10 cm. per second, which incorporated the Harkins and Brown (4)
volume correction factor. Again nozzle size and jet size were assumed equal and viscosity appeared to have no effect. Keith and Hixson (7) found that as nozzle flow velocity increased the total surface area formed by the drops reached a maximum a t a flow rate slightly below that for maximum jet length. The velocity a t which the maximum area occurred was reproducible and varied with nozzle diameter and properties of the phases for two-component two-phase systems. Correlation of the flow rate for maximum area was based upon a dimensional analysis. As in previous work, the fundamental length unit was chosen as the nozzle diameter and no provision was made for contraction of the jet after it leaves the nozzle. Because of this it was necessary to treat large and small nozzles and systems with high and low interfacial tensions separately. Both large nozzles and low interfacial tension systems lead to jet sizes that differ appreciably from the nozzle size. Keith and Hixson also observed that a t the flow rate for maximum area, or minimum surface average drop size, the drops were surprisingly uniform. This uniformity suggested that some regular, periodic disturbance occurring on the jet may influence jet breakup. Hence it was felt that an analysis of jet breakup from the standpoint of jet stability could provide a theoretical VOL. 49, NO. 6
0
JUNE 1957
1017
C o n stont heid tank
ters into computation of the most unstable wave length and therefore (ka), must be evaluated for each system. For a physical interpretation it is helpful to examine the general equation given by Lamb (8) for the displacement, q’, at any point on the jet under the restriction of axial symmetry: 7
, = --AkIA(ka) cos(kz)sin ( q t
-+ e ) -
P
(5)
When q takes on an imaginary value, i i q l , the displacement grows exponentially with time according to ,I’
Camera 17‘
qotarneter
Diagram of apparatus
basis for correlating the variables affecting jet behavior at the maximum area point.
the more general problem without the restriction of axial symmetry was solved hy Christiansen ( 3 ) and the phase velocity found to be
A k f : ( k a ) cos(kz) sinh (1q1f
Id
+
E)
(6)
Thus the wave length of greatest instability may be interpreted as one whose amplitude increases more rapidly than all other wave forms that may be present on the jet; hence, it is this wave form which most influences jet breakup. I t can be reasoned from Equation 6 that the wave form is not periodic with time but only with jet length, z . For every value of z there is a fixed displacement which groxvs with time, according to sinh ( 1 q t 4- E ) . Physically, this means that as the jet velocity increases the jet length increases: as the breakup time is related to jet diameter. When the restriction of axial symmetry is not imposed on Equation 3-i.e., s = 1, 2, 3 , . . .. etc.-the phase velocity for the fundamental mode of vibration (J = 1) is real for all values of (ka) and Equation 3 describes a real wave form moving out on the jet with a celerity of q / k where q is given by
Theoretical Studies of Jet Stability Rayleigh (72, 73) analyzed the breakup of a jet in terms of the phase velocity of a critical disturbance on the jet, assuming axial symmetry and neglecting viscosity. When only the jet was considered to possess inertia, the phase velocity was given by
where a is the local jet radius and k the period of oscillation. When only the phase into which the jet issued was considered to possess inertia. the phase velocity was given by
Equations 1 and 2 are special cases of the more general solution for the phase velocity of a disturbance on a jet when the inertial effects of both jet and surrounding medium are considered. Following the velocity potential method used by Lamb ( 8 ) for these special cases,
1 0 18
For the restriction of axial symmetry, s = 0, and when (ka) is less than 1, q 2 is negative and q therefore imaginary. The value of (ka) which maximizes - q 2 defines a disturbance which exercises the most influence on jet breakup. Rayleigh termed it the disturbance of greatest instability. From the relationship between wave celerity, c, and phase velocity-Le., c = q h / 2 n = q/k-the wave length of greatest instability, ,A, can be computed; A, = 2na/(ka),. Rayleigh found A, = 4.508 X 2a for a jet having inertia and A, = 6.48 X 2 a for a noninertial jet issuing into a phase having inertia. When both jet and surrounding phase possess inertia, the wave length of greatest instability can be obtained from Equation 3 under the restriction of axial symmetry by maximizing -q2 in the equation
Clearly, the density of both phases en-
INDUSTRIAL AND ENGINEERING CHEMISTRY
This analysis of jet breakup neglects the effect of phase viscosity. Weber (79) and Tomotika (77) considered both viscous and inertial forces, but other limitations in the analyses prevented application to liquid-liquid systems. However, based upon the findings of other workers (5: 76, 78): viscosities in the range of most common fluids appear to have little effect on jet breakup and thus may be neglected. Equation 3 indicates that interfacial tension plays an important role in jet breakup. Keith ( 6 ) presented some evidence that in the case of AlkatergeC, the interfacial tension-depressing effect of this solute was only partially realized at the time of jet breakup. This phenomenon should be expected, for its counterpart in liquid into air jetting has been known since 1879, when Rayleigh (74) devised a method of estimating effective surface tension of a solution using the vibrating jet technique. Addison ( 7 ) later employed this techique to investigate the migration time of various solutes to the surface of a water in air jet.
FLUID MECHANICS Addison's migration times are equivalent to the time necessary for the solute to depress surface tension to equilibrium conditions. Addison's data (listed in Table I) were used as a guide in this study of the effective interfacial tensions of liquid into liquid jets. Clearly, the delay time for the establishment of surface equilibrium a t an air-water interface differs from the delay time a t a liquid-liquid interface. In the latter case both liquids contain an equilibrium amount of solute determined by the distribution coefficient. Presumably migration to the surface from both phases shortens the time required to reach surface equilibrium. As expected, the size of the solute molecule affects migration times in liquid-liquid systems as it does in liquid-gas systems. Keith's results indicated that with the smaller sized ben-
Table t. Pre-equilibrium Times for Solutes Migrating in Water [Data of Addison (1)] Concn.,
Wt. %, t o Reach u = 50
Solute
Dynes/Cm.
Methanol Ethyl alcohol n-Propyl alcohol n-Butyl alcohol Isoamyl alcohol n-Amyl alcohol n-Hexyl alcohol n-Heptyl alcohol sec-Octyl alcohol n-Octyl alcohol n-Decanoic acid
16.0 8.2 3.0 1.2 0.45 0.40 0.15 0.047 0.030 0.015 0.0023
Preequilibrium Time, Sec.
< < <
