on catalytic wires of diameter cm < D < 10-l cm for gases other than hydrogen. Reaction rates up to 1021molecules cm+ s-' can be observed in reactions iiivolving H2 or its isotopes. In order to obtain rate data free of mass transfer resistances, reactions should be carried out under conditions such that no rates higher than 1019molecules cm-2 s-l (lozofor reactions involving H2) are obtained. Nomenclature c = total molar concentration, m0l/L3 YJ = binary diffusivity, L2/t D = wirediameter Da = Damkoeler number, k&YJ Gr = Grashof number, D3pg@AT/p2 h = heat transfer coefficient, M/t 3T k = thermal conductivity, ML/t 3T kH = reaction rate constant, t - l k, = mass transfer coefficient in a binary system, moleculesltL2 No = Avogadronumber NUH = Nusselt number for heat transfer, hD/k NUM = Nusselt number for mass transfer, k,D/cD P = pressure, M/Lt2 P r = Prandtl number, C p p / k Q = heat flux density, ML 2 / t3 R = gas constant, ML2/t2Tmol Re =Reynolds number, Dup/p r R = reaction rate, molecules/L2t Sc = Schmidt number, p / p B u = gas velocity, L/t x, = mole fraction of species j in gas bulk x,8 = mole fraction of species j in equilibrium with adsorbed j Greek Letters @ = thermal coefficient of volumetric expansion, T-'
6 = boundary layer thickness, L p = gas density, M/L3 p
= viscosity, M/Lt
Literature Cited Bird, R. B.. Stewart, W. E., Lightfoot, E.N., "Transport Phenomena," Wiley, New York, N.Y., 1960. Brown, F. T., Boo-Goh. Ong, Mason, D. M., J. Chem. Eng. Data, 15, 556 (1970). Cardoso, M. A. A.. Luss, D.. Chem. Eng. Sci., 24, 1699 (1969). Chim. Belg., Couper. A., Eley, D. D.. Hulatt, M. J., Rossington, D. R., Bull. SOC. 87, 343 (1958). Crank, J.. "The Mathematics of Diffusion," Clarendon Press, London, 1967. Hiam, L., Wise, H., Chikins, S..J. Cafal., 10, 272 (1968). Hinshelwood, C. N., Prichard, C. R., J. Chem. SOC. (London), 127, 327 (1925). Hinshelwood, C. N., Burk, R. E.,J. Chem. SOC.(London), 127, 1105 (1925). Hori, G. and Schmidt, L. D., J. Catal., 38, 335 (1975). Kyte, J. R., Madden, A. J., Piret, E. L., Chem. Eng. Prog., 49, 653 (1953). Langmuir. I., Phys. Rev., 34, 40 1 (19 12). Langmuir. I., Trans. Faraday SOC., 17, 621 (1921). Loffler, D. G., Schmidt, L. D., A.I.Ch.E. J., 21, 786(1975). Loffler, D. G., Schmidt, L. D., J. Catal., 41, 440 (1976a). Loffler, D. G.. Schmidt, L. D.. J. Cafal., 44, 244 (1976b). McAdams, W. H., "Heat Transmission," McGraw-Hill, New York, N.Y., 1954. Moss, R. L., Thomas, D. H., J. Catal., 8, 162 (1967). Nusselt, W. Z., 2. Ver. Deut. lng., 73, 1475 (1929). Patterson, W. R., Kembal, C., J. Cafal., 2, 465 (1963). Pignet, T. P., Schmidt, L. D., Chem. Eng. Sci., 29, 1123 (1974). Pignet. T. P., Schmidt, L. D., J. Catal., 40, 212 (1975). Rader, C. G., Ph.D. Thesis, State University of New York at Buffalo, 1974. Rader, C. G., Weller. S. W., A.l.Ch.E. J., 20, 515 (1974). Reid, R . C., Sherwood, T. K., "The Properties of Gases and Liquids, 2nd ed, McGraw-Hill, New York, N.Y., 1966. Schwab, G. M.. Schmidt, H., 2.Phys. Chem. (Leipzig), 38, 337 (1929).
Received f o r reuiew November 12, 1976 Accepted April 25, 1977
This work was partially supported by the National Science Foundation under Grant No. ENG 75-01918.
Stability of Centrifugally Stratified Helical Couette Flow Larry A. Glasgow and Rlchard H. Luecke' Department of Chemical Engineering, University of Missouri, Columbia, Missouri 6520 1
Helical Couette flow, stabilized by viscous and centrifugal forces, undergoes catastrophic transition at particular combinations of axial pressure gradient and angular velocity of the outer cylinder. The dependence of the limit of stability upon flow parameters is predicted with an inviscid simplification of the method of small disturbances. Experimental evaluations of a laboratory-scale Couette device verify the disturbance equation within the limitations of the apparatus.
Introduction The stability of tangential flow in the annulus of rotating concentric cylinders has been considered both theoretically and experimentally by scores of investigators over the past 85 years. The flow field studied in this paper, however, represents a substantial difference from that case in that an axial pressure gradient was imposed upon the angular velocity of the fluid. Thus, the fluid in the annulus is subject simultaneously to both tangential and axial velocity components; the resultant motion is, of course, helical. The addition of the axial velocity alters the impact of centrifugal stratification upon stability. 366 Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
We became interested in this type of flow as part of studies of the influence of hydrodynamic shear upon floc. There have been a number of studies of particle coagulation dynamics in Couette devices but none have been reported that include an axial component of flow. This flow is important to our work in that large representative samples may be analyzed externally for particle size distribution. In general, the work described in this paper should prove useful to investigators requiring a well-defined shear field within the context of a recirculating system. This type of configuration is often a requisite for viscometry, lubrication research, and particle aggregation studies.
The initial experiments regarding stability without axial flow were performed by Couette (1890) and Mallock (1896) in the 19 century. Data were developed regarding conditions for which instability occurs but for years no theoretical predictions were available to compare with those data. By 1916, Lord Rayleigh had developed a stability criterion for inviscid fluids as a function of the variation of circulation with radius, but until Taylor’s work appeared in 1923, no one had successfully treated the viscous case. Taylor considered a variety of circular Couette flows and found that there was no limit for stability in the instance in which the outer cylinder rotates and the inner cylinder is a t rest. This result has been confirmed more recently by experiments performed by Schultz-Grunow (1959). Instabilities reported in previous investigations of this case are now thought to have resulted from finite disturbances caused by eccentricities or bearing imperfections. Few investigators have considered the stability of helical flow resulting when an axial pressure gradient is imposed upon circular Couette motion because of the experimental and theoretical difficulties inherent to this particular geometry. Goldstein (1977) has theoretically analyzed stability of helical flow produced by an axial pressure gradient imposed upon fluid motion resulting from rotation of the inner cylinder with the outer cylinder at rest using the method of small disturbances. He compared his analysis with the experimental work of Cornish (1933), whose apparatus consisted by brass cylinders with very small (0.011 to 0.023 cm) annular gaps. The results were in poor agreement; since Goldstein verified his equation with that of Taylor, it seems likely that the discrepancies with Cornish’s work are due to the nature of the equipment or the method of observation. Some facets of the design of the Cornish equipment are not very clear but several deficiencies are apparent. For example, a dc motor was used through direct coupling to drive the rotation of the inner cylinder. I t is nearly impossible to maintain a uniform speed of rotation with this arrangement. Concentricity in the device was obtained by adjustment of set screws with measurements made with a feeler-gauge. In practice, it is very difficult to eliminate eccentricities throughout the length of the device in this manner. Finally, Cornish depended upon pressure measurements for the detection of the onset of instability. Goldstein, in fact, noted that pressure-drop data probably relate to a phenomenon other than the first stages of transition and that therefore, direct observation of the flow could be vital. More recently, Ludwieg (1960) has examined helical flow generated by rotation and translation of the inner cylinder. Ludwieg predicted the limit of stability for an inviscid fluid using a modification of the Kelvin-Rayleigh circulation criterion (Rayleigh, 1916). Ludwieg’s stability condition was well supported by his experimental results. Joseph (1976) has summarized Ludwieg’s work as well as several other interesting examples of spiral Couette-Poiseuille flows. All of the investigations discussed above pertain to the case in which a stable secondary flow (Taylor vortices) results from the tendency of the fluid particle near the inner wall of the annulus to move centrifugally. At sufficiently high rates of rotation, these flows must become unstable. The difference between transitional processes in geometries common to the literature and in the configuration employed in the study reported in this paper has been illuminated by Coles (1965). Tangential motions dominated by rotation of the inner cylinder are characterized by an evolutionary process of transition-a process involving a succession of secondary flows beginning with the formation of Taylor vortices. The course of this type of transitional phenomena arises from the competing influence of two effects: viscous damping and centrifugal destabilization. However, the apparatus used in the present investigation
w
Figure 1. Modified Couette apparatus.
has a distinction in that the tangential motion of the fluid results from the rotation of the outer cylinder. Both viscous and centrifugal effects exert stabilizing influences on this system. Thus the evolutionary process involving secondary flows does not develop. Instead the tangential flow in combination with the axial flow component results in catastrophic transition.
Description of the Apparatus The modified Couette-type device employed in this work consisted of two transparent cylinders (one cast acrylic and the other glass) carefully centered on a 1.905-cm stainless steel shaft. The smaller cast acrylic cylinder had an outer diameter of 7.62 cm and was rigidly attached to the shaft. The larger glass cylinder had an inner diameter of 10.16 cm (.z0.013 cm) and was free to rotate about the axis on two sealed, flange ball bearings. Oil-impregnated powder bronze bearings had been used initially, but were found to produce undesirable vibrations a t high speeds. The annular gap provided by this geometry was 1.27 cm while the length of the plain annular region was 47 cm. The length-to-gap ratio was, therefore, 37. A ratio of about 20 has been commonly used to avoid undesirable end effects (Van Duuren, 1968) in Couette devices. Centering of the cylinders on the shaft was assured within f0.003 cm through use of the available machine tools. Provision for the axial component of flow was made by drilling 1.27-cm diameter holes into both ends of the shaft and milling entrance and exit slots (diameter = 0.80 cm) from the outer surface to points coincident with the extreme depth of the centered, shaft holes. Considerable effort was devoted to the minimization of the effects of the peripheral eddies associated with the sudden expansion of the jet through the entrance orifice. A tapered and highly polished aluminum cone was fit on the inner cylinder at the entrance end of the apparatus. An end plug (and bearing mount) with a complementary taper was fit on the entrance end of the large cylinder. A cross-section drawing of the apparatus appears in Figure 1. The outer cylinder was belt driven and normally operated in a range of angular velocities from 1 to 65 rad/s. The axial flow rate was commonly in the range of 0.1 to 8 L/min. Angular velocities were determined with a General Radio Company Strobotac and the volumetric flow rate was measured with a calibrated Fischer & Porter Stabl-Vis Rotameter. Flow stability was to be evaluated by observation of dye filaments. Provision was made for the injection of a neutraldensity dye into the center of the annulus a t a point 21.6 cm downstream from the entrance. Teflon capillary tubing transmits the dye from outside the device through the inlet flow passage and into a small (0.318-cm) hole which emerges Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
367
from the shaft inside the cylinder. The capillary tubing is a t that point attached to a 26-gauge needle (diameter = 0.046 cm) which runs through the wall of the inner cylinder into the center of the annulus. The needle was bent so that its orientation conformed to the expected helical flow pattern (Figure 1).The laminar velocity profile was expected to be fully developed at the point of dye injection due to the large distance (several hundred centimeters) traveled in a helix (with an approximate diameter of 8.9 cm). The bulk of the experimental data was derived from experiments in which water filled the annular region of the experimental apparatus. T o extend the range of Reynolds and Taylor numbers over which data could be taken, however, aqueous solutions of glycerine, sucrose, and ethylene glycol were also used. The range of absolute viscosities of the fluids employed was approximately 1to 25 cP. Viscous heating effects were neglected. Characteristics of t h e Mean Flow: Velocity a n d S h e a r Distributions Under laminar-flow conditions, the tangential component of the fluid velocity is produced solely by the rotation of the outer cylinder, The velocity profile of the tangential fluid motion is obtained by integrating the appropriately simplified Navier-Stokes equation twice to obtain B Vo = Ar t r The constants A and B are determined from the boundary conditions and substituted into eq 1to yield
The only nonzero stress tensor component for this flow is rro, which may be found by differentiation of the velocity profile
The axial component of flow in the experimental apparatus is produced by the pressure gradient, dPldz. The appropriate form of the Navier-Stokes equation may be integrated twice subject to the applicable boundary conditions to obtain
V ( r )= (Vg2
+ V,2)1/2
(8)
Theoretical Analysis of Stability For Couette flows in which the inner cylinder rotates and the outer cylinder is fixed, the tendency of the more rapidly moving fluid near the inner surface to move centrifugally promotes a laminar secondary flow (Taylor vortices) at relatively low rotational velocities. Increasing the rate of rotation of the inner cylinder produces a succession of secondary states and ultimately, complete turbulence. In contrast, for the circumstance in which the outer cylinder is rotating and the inner cylinder is a t rest, fluid particles with the greatest angular velocity are restrained from centrifugal motion by the wall of the outer cylinder. Similarly, throughout the fluid volume particles with lower angular velocities are restrained from centrifugal motion by the faster moving outer fluid. In this manner, the flow is stabilized by “centrifugal forces.’’ It is now known that for this type of Couette flow without an axial velocity component, no limit to stability can exist (Schlichting, 1968) for infinitesimal disturbances. The apparatus described in this paper, however, constitutes a significantly different case because of the imposition of the axial pressure gradient upon the rotational motion of the fluid. With this geometry, the influence of large axial, inertial forces can overcome the effect of centrifugal stabilization. A principal technique employed in studies of the stability of laminar flows is the method of small disturbances. Such stability analyses for two-dimensional paralled flows often begin with the well-known intermediate result, the OrrSommerfeld equation. The point-of-inflection stability criterion was first demonstrated by Raleigh by integrating as inviscid simplification of the Orr-Sommerfeld equation; the criterion was later strengthened from a necessary to a sufficient condition by Tollmien (1936). It is initially tempting to apply the Rayleigh-Tollmien inflectional stability criterion to the flow configuration described in this paper since it is apparent that certain combinations of the component flows will produce points of inflection in the point velocity distribution. However, the onset of instability can be but crudely predicted in this manner for this type of helical Couette flow since the Orr-Sommerfeld equation does not account for centrifugal stabilization. To develop an applicable analysis for the case of interest, therefore, it is necessary to begin with the three NavierStokes equations and the equation of continuity in cylindrical coordinates. A three-dimensional disturbance is assumed such that each component of the total fluid motion is written as the sum of the mean and fluctuating velocities
vr = v,’ ( V , = 0) =
Ve + v,’
v, =
v,+ v,!
v, where rmaxis the radial position corresponding to maximum axial velocity and for the pressure (5) The stress tensor component applicable to axial flow in plain annuli is T , ~ ;the shear distribution is again found by differentiation of the velocity profile
and
The actual point velocity in the annulus is the resultant of the axial and tangential components, Le. 388
Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
p=P+p’ These relations are substituted into the Navier-Stokes equations and the equation of continuity. The resulting four partial differential equations are simplified by: (1) the assumption that the mean flow terms satisfy the equations separately; (2) linearization with respect to quadratic terms in the fluctuating velocity components and their derivatives; (3) elimination of pressure terms (appearing twice in four equations) by cross-differentiation and subtraction; and (4) the assumption that the disturbance is rotationally symmetric. This last assumption has been widely used in stability investigations of rotating flows, e.g., Taylor (1923), Goldstein (1937), and others. Thus, the disturbance has the form of
two-dimensional Tollmien-Schlichting waves traveling parallel to the axis. A discussion of this simplification (in connection with parallel flows) has been given by Squire (1933). After dropping terms related to viscous forces, the simplified equations are obtained a2uzJ+ -2 a U r f au -
atar
+ ur'2 a2u + z au au z
ar ar
ar2
ar az 2ugaugi a2uZ' a2urf + uz ----u azar dtaz r az
a%, ' z 2
-
az
-0
-
(10)
Table I. Calculated Neutral Disturbance Reynolds Numbers for Various Combinations of Taylor Number and Wavelength" Ta 644 1288 2576 3864 9660
Axial Reynolds number, Re, X 3.83 cm X 10.24 cm X 31.92 cm 460 969 1992 3018 7834
777 1690 3671 5402 14020
926 2057 4354 6662 16480
Note: calculations were based on the kinematic viscosity of water at room temperature. and
-auri + L u+ L au =o
atr=Rl,&=O (12)
dr r az The form of the disturbance is, as stated, assumed to be that of a wave, symmetric about the axis, and propagated along it ur' = 4 1 ( r ) e i ( d + ~ ~ ) ug' = 42(r)ei(ot+az)
(13)
v,' = $3(r)ei(ot+az)
where a is real and u, in general, is complex (the analysis will be confined to consideration of the temporal growth of disturbances). The relations for the disturbance (13) are substituted into eq 10-12 along with further simplifications arising from the known character of the base flow (see eq 2 and 4)
+ B-r
ug = Ar
u, = Clr2 - C2
c4 + C3 In r
(14)
After considerable rearrangement, and introduction of /3 for u/a,a single equation is obtained
Expressions involving Taylor and Reynolds numbers can be substituted for A, B, C3, and uz in eq 15 since uTa
A=-
(16)
R21/2(Rz- R1)3/2 uTa
B=
R~I/Z(R ~R1)3/2
1 - 1 -
(Rz2 -2uRe,
c3 = ( R ~-
2
~~2
(20)
As a result of the omission of viscous effects, it is not possible to treat eq 15 as a classic characteristic value problem and obtain critical values for the parameters Ta, Re,, and A. However, it is possible to determine whether or not a particular laminar flow (at large Reynolds numbers) is neutrally stable. Equation (15) was investigated numerically using a Runge-Kutta scheme similar to that described by White (1974). A systematic search procedure was applied in conjunction with the integration to identify the parametric values that would result in compliance with the boundary condition ($1 = 0 at r = Rz) for cases of neutral disturbances (that is, the amplification factor, pi, equals zero and the neutral solution is assumed to be steady such that the principle of exchange of stabilities holds and Pr can be taken as zero). These results appear in Table I. Additional discussion of the principle of exchange of stabilities has been given by Jeffreys (1928), Lin (1955), and Joseph (1976). Experimental Results a n d Discussion The basic strategy of the experimental procedure was to inject dye into the center of the annulus a t various combinations of pressure gradient, angular velocity, and viscosity and then photographically to observe the degree of mixing, i.e., the dispersion of the dye filament. An example of a photographic record appears in Figure 2. A correlation of all of the experimental observations made in the course of this investigation is shown in Figure 3. Turbulent conditions are represented by open circles; laminar flow by solid circles. Classification of data near the transition curve was sometimes difficult. The tangential component of flow (the abscissa in Figure 3) is expressed in the form of the Taylor number, Ta, where
(17)
R12)
F ) (+R -~ ~
atr=Rz,q51=0
(18)
2rmax2)
and
As discussed previously, in the absence of an axial pressure gradient, centrifugally stratified Couette flow may be expected to remain stable for infinitesimal disturbances at all values of the Taylor number. The axial component of flow is represented in Figure 3 by the Reynolds number, Re,, where 4rdJ~ Re, = c1
and the hydraulic radius, r H , is based upon the point of maximum velocity The boundary conditions for eq 15 result directly from the requirement that urf must vanish at the solid surfaces a t each side of the annular gap
(23)
This is the same Reynolds number as that used by Prengle and Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
369
. ....
Figure 2. Laminar flow helix in Couette apparatus.
a-
1
4
101
x
Rothfus (1955) in their study of transition in plain annuli. From their data, one would expect transition to occur (with no tangential velocity component) in the range of Reynolds numbers
700 < Re,
< 2200
< 1400
(25)
In Figure 3. the enwlope of the experimentally detvrmined limit ofstahility is indicated by thedashed line; the result of the numerical analysis of the theoretical eq 15 is represented by the solid line. One very curious outcome may he immediately noted. At moderate rates of rotation of the outer cylinder (Ta 300), instability was observed a t Reynolds numbers as low as Re, ^I 350, which is well below the axial flow rate producing transition in the case of the plain annulus. It is not apparent whether this result is due to deficiencies of the apparatus or is a normal consequence of the added inertia provided by rotation (at angular velocities too low to produce significant centrifugal stabilization). Note that Re, is characteristic of only the net axial flow and not of the true resultant flow velocity. In general, increased rates of rotation (larger Ta's) provided stable flows at elevated axial flow rates. This is an inherent result of the tendency of the fluid toward centrifugal motion: the greater the rate of rotation, the greater the stabilizing influence. Results of the numerical integration of the inviscid stability equation presented in Figure 3 show that there is general agreement in trend between the theoretical and the experimental results in the range 200 < Re, < 6000. Since the linearized stability equation does not account for viscous damping of small disturbances at low velocities, one cannot expect to obtain meaningful theoretical results for small Taylor 6nd Reynolds numbers. For moderate values of the Taylor sumher, however, the stabilizing influence of viscosity i i much lrs tliilrr that ariiing fnm rentriruqal effww. In thi'se cases. omission of viscous forces does nut greatly . mmair . rhi, accuracy of the prediction of neutral stahkty. At Taylor numbers of ahout 500% however, the exPerimental results show a general onset of turbulence a t all Reynolds numbers. There can he little doubt that this outcome stems directly from eccentricities, vibrations, or other mechanical factors serving in the amplification of disturbances. 370
Ind. Eng. Chern., Fundam.. Vol. 16, No. 3.1977
8
' ' , ~ ~ ~ ' 100
I
1.000
I 10.000
TaYLOR NUMBER. la
Figure 3. Experimentally determined stability envelope and theoretical neutral stability curve for inviscid water at wavelength h 3.83 cm.
(24)
During evaluation of axial flow in the experimental device, transition was observed in the range of Reynolds numbers
800 < Re,
IO
Summary Experimental data have generally verified theoretical predictions regarding the stability of centrifugally stratified-helical Couette flow, For plain annular flow (with both of' cylinl]ers SfRll,,nars) t r a n s i t i ~ , n was obsrrved i n a Revnoids in,,,, 80" tO 1,+00. \Vith the ,,uter " - ~ ~ cylinder rotating at low angular velocities (w S= 10 rpm) transition was observed at au axial Reynolds number of about 350. Increasing the rotational speed of the outer cylinder tended, as expected, to stabilize the system. At a Taylor number of 1000, transition was observed a t an axial Reynolds number of about 550. At large Taylor numbers (Ta t 5000), however, transition was observed for Reynolds numbers less than ~~~~
~
~
~
~
~
~
100. A theoretical analysis utilizing the method of small disturbances for the inviscid case indicates the expected trend, Le., stability with respect to the axial flow (Re,) is exhanced by increased angular velocities of the outer cylinder over a wide range of wavelengths (Figure 3 and Table I). Failure of correspondence between data and theory at large Taylor numbers (Ta t 5000) undoubtedly results from the amplification of disturbances by eccentricities or other mechanical deficiencies of the apparatus. It should be noted, however, that at no experimental condition was significant equipment vibration discernible.
Nomenclature A = constant defined by eq 16 B = constant defined by eq 17 C, = constants appearing in eq 14 g, = gEvitational constant = -1 = pr.9sure r = radius rH = hydraulic radius defined by eq 23 = radial position of maximum axial velocity defined by
'
On
R
r.
radius ofcylinder(s) Re, = axial Reynolds number defined by eq 22 s = annular gap t = time I
T a = Taylor number defined by eq 21 U = bulk velocity u = total fluid velocity u' = fluctuating velocity V = meanvelocity
Greek Letters wavenumber ratio of u to wave number wavelength absolute viscosity kinematic viscosity fluid density u = circular frequency plus amplification factor T = shear stress tensor component 4 = amplitude function w = angular velocity
a = p = X = p = v = p =
Subscripts = radial direction = axial direction = tangential direction = inner wall (radial position) = outer wall (radial position) = amplitude function for radial component of disturbance = amplitude function for tangential component of disturbance
3 = amplitude function for axial component of disturbance Literature Cited Coles, D., J. FluidMech., 21, 385 (1965). Cornish, R . J., froc. Roy. SOC.London, Ser. A, 140, 227 (1933). couette, M., Ann. Chim. fhys., 21, 433 (1890). Goldstein, S . , froc. Cambridge Phil. SOC., 33, 41 (1937). Jeffreys, H., froc. Roy. SOC.London, Ser. A, 118, 195 (1928). Joseph, D. D., "Stability of Fluid Motions. I," Chapters 2 and 6, p 27, SpringerVerlag. New York, N.Y., 1976. Lin. C. C., "The Theory of Hydrodynamic Stability," Chapter 7, p 109, University Press, Cambridge, -1955. Lord Rayleigh, "On the Dynamics of Revolving Fluids," "Scientific Papers," Volume VI, 1916. Ludwieg, H.,2.Flugwiss. 8, 135 (1960). Mallock, A., Phil. Trans. Roy. SOC.London, Ser. A, 187, 41 (1896). Prengle, R. S.,Rothfus, R. R.. lnd. Eng. Chem., 47, 379 (1955). Schlichtina. H., "Boundary-Layer . . Theory," ChaDter 2, D 27, McGraw-Hill. New York, NTY., 1968. Schultz-Grunow, Von F., Z.Angew. Math. Mech., 39, 101 (1959). Squire, H. B.. Proc. Roy. SOC.London, Ser. A, 118, 621 (1933). Taylor, G. I., Phil. Trans. Roy. SOC.London, Ser. A, 223 (1923). Tollmien, W., NACA, Technical Memorandum, 792 (1936). van Duuren. F. A,, J. Sank Eng. Div., ASCE, 671 (1968). White, F. M., "Viscous Fluid Flow," Chapter 5,p 399, McGraw-Hill, New York, N.Y., 1974.
Received f o r review October 19, 1976 Accepted M a y 11,1977 T h e authors are grateful t o t h e N a t i o n a l Science F o u n d a t i o n a n d t o t h e Office o f Water Research a n d Technology of t h e D e p a r t m e n t o f the I n t e r i o r for financial support.
EXPERIMENTA1 TECHNIQUES
Breakage and Coalescence Processes in an Agitated Dispersion. Experimental System and Data Reduction F. H. Verhoff,'' S. L. Ross, and R. L. Curl Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48 104
A method for the rapid and accurate determination of the bivariate distributions of drop volume and a tracer dye concentration in a continuous flow agitated dispersion is described. The method involves extracting a sample of the dispersion from a vessel, protecting the sampled drops with a surfactant, and pumping the sample at a constant flow rate through a capillary smaller than the smallest drop of interest; light transmission gives dye concentration, and time-of-passage of a drop gives drop volume simultaneously. The method yields information about the rate processes of breakage and coalescence in an agitated dispersion.
If a mixture of two immiscible liquids is agitated in a batch or continuous flow system, a dispersion of droplets may be formed whose characteristics are functions of the geometry and size of the mixer and its materials of construction, the intensity of agitation, the phase fraction, and fluid properties including viscosities, densities, surface energies, and the concentration of dissolved solutes. These factors act to proAddress correspondence t o t h i s author a t t h e D e p a r t m e n t of Chemical Engineering, W e s t V i r g i n i a University, Morgantown, W. Va. 26506.
duce the directly observable drop size distribution as determined by breakage and coalescence. These two rate processes are called dispersed phase mixing because they mix substances between droplets in the dispersed phase. These processes may be important in the design of liquid-liquid contacting equipment. Although such properties as interfacial area and mean drop size have been measured and correlated with system parameters, a better understanding of droplet breakage and coalescence in an agitated system would permit the treatment of problems involving mass transfer and chemical reaction, Ind. Eng. Chem., Fundam., Vol. 16, No. 3, 1977
371
