laminar-Turbulent Transition Criterion for Certain Stratified Two-Phase Flows Hanks' stability criterion has been extended to stratified two-phase flows. For an oil-water system of viscosity ratio it predicted the less viscous water phase to undergo transition to turbulence first. However, the predicted transition agreed with experimental observations only at those input ratios of the two fluids where the maximum velocity prior to transition always appeared in the less viscous phase,
5.58:l
I n the study of the laminar-turbulent transition, criteria are examined for single-phase and two-phase flow. Transition Criterion for Single-phase Flow
The onset of instability and turbulence is usually described in terms of some critical Reynolds number, Re, = d G p / p . The numerical value of this depends on the flow geometry as well as on the choice of the characteristic length d and velocity u. Hanks (1963) has proposed a generalized stability parameter which is independent of the flow system geometry and he incorporated an earlier stability parameter by Ryan and Johnson (1959) as a special case. Hanks based his parameter on the premise that fluid motion will be unstable to certain types of disturbances if the magnitude of the acceleration forces acting on the fluid becomes a certain multiple of the magnitude of the viscous forces. The ratio of the forces becomes the stability parameter K . For rectilinear flows in the steady state, this parameter was expressed as
The parameter K , like that of Ryan and Johnson, is a function of position. It will vanish a t the solid boundaries and a t the position of maximum velocity. Somewhere in the flow field within the above limits, the parameter reaches a maximum, K,,,. When this K,,, becomes equal to or exceeds a critical value K,, instability of the flow is to be expected. Based on the transition Reynolds number of 2100 in fully developed pipe flow, this critical stability parameter can be shown to be numerically equal to 404. Hanks (1963) also claimed that this parameter remains constant and applies equally well to parallel plates and other flow geometries. This has been substantiated a t least in part by Hanks and coworkers (1966, 1967,1970) and by XcEachern (1969). Using the velocity profile for plane Poiseuille flow along with an equivalent diameter de equal to four times the hydraulic radius h, eq 1 for parallel plates becomes
for computation of K in each phase separately. Laminar velocity profiles can easily be obtained for the fully developed, concurrent, stratified flow of two immiscible, incompressible Newtonian fluids between parallel plates assuming no slip a t the smooth interface (Bird, et al., 1960). The stability calculations for the above two-phase system were based on a hypothetical plate separation. A plate separation can be found such that the actual velocity profile in each phase corresponds to part of a hypothetical, single-phase velocity profile. An example of the hypothetical plate separations and velocity profiles is illustrated in Figure Ib. Using this method, it is possible to compute the transition parameter K in the two-phase system by treating each fluid layer as a separate single phase. If the velocity profiles for a given interfacial position are substituted into eq 1, expressions for the point values of K parameters for each phase become
for (-hz
I y I h2)
Since only the maximum value of Ka (i = 1, 2) is meaningful as a transition criterion, the position where this maximum occurs must be determined for each interface position which in turn corresponds to a different hypothetical plate separation. Kl,maxand K S , ~can ~ now , be computed in terms of pressure gradients from eq 3a and 3b for those portions of the velocity profiles actually exhibited by the two fluids. Pressure gradient a t which transition to turbulence would appear in a given phase will therefore be
p Az
=-
!t,transition
404
Kt,rnax
(4)
The parameter K has its maximum value a t y/h = d1/3, which a t the critical point would have the numerical value of 404. This would yield a critical Reynolds number Re, = 2800 for parallel plates, which is in good agreement with the experimental findings of Davies and White (1928).
The above procedure assumes no inhence of one phase on the transition in the other. Since the pressure gradients for the two phases are necessarily identical in a fully developed laminar flow, the phase with the lower transition pressure gradient will undergo transition first. Volumetric flow rates can be computed by integrating the laminar velocity profiles just before transition a t the appropriate pressure gradient from eq 4 and the initially chosen interface position.
Transition Criterion for Two-Phase Flow
Results
Hanks' stability parameter is directly related to the velocity distribution within the fluid. To adapt this criterion to a twophase flow system, laminar velocity profiles must be available
Loci of the critical superficial Reynolds numbers were calculated as a function of interface position following the method outlined for parallel plates. The stability parameter K was
-
416
Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
m; u,=o;
2h2
+
I
I,'
I/
,' VELOCITY
Figure 1 . Model of stratified two-phase flow: a, terminology for parallel plates; b, concept of hypothetical plate separation. Note: both velocity profile maxima (real and hypothetical) are at the same value of y
-
LAMINAR
IO+ ' '
I
'
than their amplitudes, no careful calibration of the anemometer response was attempted. Absolute turbulence intensity measurements therefore could not be made. More detailed descriptions of the experimental system and the techniques used have been reported elsewhere by Charles (1963), Charles and Lilleleht (1965a), and Stellmach (1967). The results for an oil to water viscosity ratio of 5.58:1 are illustrated in Figure 2, which is a plot of the calculated superficial oil Reynolds number us. the calculated water Reynolds number assuming: (a) transition in water, the phase with oil remaining laminar and (b) transition in oil, the phase with water still laminar. Results are shown for parallel plates as well as the rectangular conduit of 7.95: 1 aspect ratio. Also plotted in Figure 2 are the experimental conditions a t which the water phase had reached 0.5 intermittency, Le., 50% of the time laminar and 50% of the time turbulent patches. The agreement between the calculated and experimental transition points for the water phase is good for interface positions greater than 0.4. As expected, agreement is better for the rectangular conduit than for parallel plates. Note that the calculated transition Reynolds numbers correspond to the lower critical Reynolds numbers, whereas the experimental values correspond to the point where the disturbances have already grown to occupy half the time a t a fixed location (y = 0.5). It should be further noted that according to the stability criterion used to predict the transition for 5.58: 1 viscosity ratio fluids, the less viscous fluid will always become turbulent first. Hanks' stability parameter thus appears to be successful in predicting laminar-turbulent transition in a two-phase system only if the less viscous phase occupies a sufficiently large fraction of the conduit flow area for the maximum velocity to appear in that phase. Since the transition parameter K , is based on a ratio of the acceleration forces to the viscous forces within a flow system, this criterion can therefore be expected to predict transition only in those two-phase systems where these forces represent the major agents of instability. Nomenclature
"'
l0,OOO
1000
l00,000
half width of rectangular conduit, L defined as ( d Z 2 - pZb1*)/(wbz p 2 b J , L depth of fluid i, L diameter or characteristic length, L equivalent diameter equal to four times hydraulic radius, L body forces per unit mass, j / X gravitational constant, J I L / ( f )( t ) 2 half separation of parallel plates, L hypothetical separation of parallel plates t o match the velocity for phase i, L actual parallel plate separation = hl hp, L stability parameter, defined by eq 1, dimensionless transition parameter, K , = 404, dimensionless pressure, f / L 2 volumetric flow rate of fluid i, L3/t superficial Reynolds number of phase i = d e f i Z p Z / p 2 , dimensionless point velocity, L/t superficial average velocity of phase i = Q , / ( 2 a H ) ,
+
RE WATER
Figure 2. Flow regime boundaries computed from the modified stability criterion: ---, flow between parallel , flow in rectangular conduit; 0 , experimental plates; data for rectangular conduit at y = 0.5
--
also computed for a rectangular conduit of 7.95: 1 aspect ratio using the laminar velocity profiles given by Charles (1963) and Charles and Lilleleht (1965b). The experimental observations of the laminar-turbulent transition were made in a horizontal Lucite channel with a nominal inside width of 8 in., height of 1 in., and an overall length of 37.5 ft. The structure of the flow field and the transition from laminar flow was monitored by a constant-temperature anemometer using a hot film probe located a t approximately 160 equivalent diameters from the channel entry. The hot film element was mounted on the 30' conical end of a 4-mm diameter, 10-cm long glass probe which could be moved vertically across the channel. Both the ac and de components of the anemometer output were recorded permitting the detection of turbulent patches and the evaluation of the intermittency factor y.Since this study was primarily interested in detecting the first appearance of velocity fluctuations, rather
+
L/t
rectangular coordinates defined in Figure 1 GREEKLETTERS
p
=
interface position
=
bl/H
y
= intermittency factor:
time during which flow is turbulent divided by total time of observation viscosity, X / L t density, X / L 3
P P
= =
Ind. Eng. Chem. Fundarn., Vol. 1 1 , No. 3, 1972
419
Hanks, R. W., Ruo, H-C., IND.&a. CHEM.,FUNDAM. 5, 558 (1966). Hanks, R. W., Song, D. S., IND. ENQ.CHEM.,FUNDAM. 6 , 472 (1967). McEachern, D. W., A.I.Ch.E. J . 15, 885 (1969). Ryan, N. W., Johnson, M. M., A.I.Ch.E. J . 5,433 (1959). Gtellmach, H. S., “Laminar-Turbulent Transition in Co-Current Stratified Oil-Water Flow,” M.Sc. Thesis, Department of Chemical and Petroleum Engineering, University of Alberta, Edmonton, Alberta, Canada, 1967.
SUBSCRIPTS
1
=
2
= = = =
i
C
max
less viscous bottom phase or water more viscous top phase or oil lor2 critical or a t transition point maximum
literature Cited
Bird, R. B;: Stewart, W. E., Lightfoot, E. N., “Transport Phenomena, pp 54-56, Wiley, New York, N. Y., 1960. Charles, 11. E., “The Co-Current Stratified Flow of Two Immiscible Liquids in a Rectangular Conduit,” Ph.D. Thesis, DeDartment of Chemical and Petroleum Enrineerinr. Univer’sity of Alberta, Edmonton, Alberta, CanaJa, 19637 Charles, 31.E., Lilleleht, L. U., J . Fluid Mech. 22, 217 (1965a). Charles, JI. E., Lilleleht, L. U., Can. J . Chein. Ena. - 43,. 110 ( 1965b ). Davies, S. J., White, C. M., Proc. Roy.Soc.,Ser. A 119,92 (1928). Hanks, R. W., A.1.Ch.E. J . 9 , 4 5 (1963). Hanks, R. W., Cope, R. C., A.Z.Ch.E.J. 16, 258 (1970).
University of Alberta Edmonton, Alberta, Canada University of Virginia Charlottesville, Vu. 22901
HARRY S. STELLMACHI LEMBIT U. LILLELEHT*
Present address, Shell Canada Ltd., Box 100, Calgary, Alberta, Canada. RECEIVED for review August 13, 1971 ACCEPTED May 31, 1972
Prediction of Binary Diffusion Coefficients for Polar Gas Mixtures Three diffusivity correlations (those of Fuller, et a/., of Saksena and Saxena, and a combination of these two) are tested with the data of Chakraborti and Gray for polar-polar binary combinations of dimethyl ether, methyl chloride, and sulfur dioxide. The results suggest that a correlation of all binary diffusion data might be achieved with as few as two adjusted parameters.
V a r i o u s methods have been proposed for correlation and prediction of binary diffusivities in gases. Relative merits of these methods have been discussed and reviewed recently by Fuller, et al. (1966), and by Saksena and Saxena (1966). The common feature in the development of these empirical formulas has been the replacement of the collision diameter, U , and the depth of the potential-energy minimum, e, of the rigorous Chapman-Enskog expression for the diffusion coefficient by some experimentally measurable physical quantity (cube root of the critical volume or LeBas atomic volume, boiling temperature, critical temperature, etc.). Fuller, et al. (1966), have determined such suitable quantities (called “special diffusion volumes”) by minimizing a n error discriminant. Their correlative equation is
DI= ~
0.001T’ .75 (1/’Mi
+ l/Xz) ’/’
p[(CvP+ (C~,)’/T
(1)
where D12is the binary diffusion coefficient (cm2/sec), T is temperature (OK), P is pressure (atm), M I and Mz are molecular weights, and the 21, are the empirical diffusion volumes of Fuller, et al. (1969), which are summed over atoms, groups, and special structural features for each diffusing species. The formula suggested by Saksena and Saxena (1966) can be written as
D l2
+
B ~ T ~ J ( ~ /I /MM~~1/2-) V,21/3)2(l B,T,,,/T)
- P(V,,’/8
+
+
(2)
where V,,and V,, are critical volumes (cm3/mole), T,, and T,, are critical temperatures, with T,,, = (T,,TCJ1’*,and BI and BPare adjustable constants. 420
Ind. Eng. Chem. Fundarn., Vol. 1 1 , No. 3, 1972
The form given by Fuller, Schettler, and Giddings (FSG) has been used with polar-polar gas pairs, but that of Saksena and Saxena (SS) has not. We make such calculations here, testing the two procedures and giving a third method based on the formula
where B is a n adjustable parameter. Chakraborti and Gray (1966) have measured diffusion coefficients a t 1 a t m for different binary combinations of dimethyl ether, methyl chloride, and sulfur dioxide. The accuracy of their measurements is abuot 1.5%. I n this note we have attempted to reproduce their data by using SS and FSG correlations as well as that of eq 3. Different sets of the calculated values along with the experimental results are reported in Table I. Column 3 of Table I represents Chakraborti and Gray’s correlation of their experimental data by means of the Stockmayer potential. Each binary system was treated as a pure gas, and two independent parameters were determined for each of the three systems by least-squares analysis, thus requiring a total of six adjustable parameters for the entire data set. Column 4 is also based on Chakraborti and Gray’s work. Here parameters of the Stockmayer potential were estimated from literature data on other transport and equilibrium measurements for the pure components, and calculations were based on mean values of the parameters obtained by the usual combining rules. For column 4, then, there were effectively no adjustable parameters.
