drops across wavy surfaces becomes clear. The existence of flow reversal and a recirculation eddy is indicated at station C. Percentage turbulence intensities in the axial and radial directions appear in Figures 5 and 6, and the turbulent shear stress in Figure 7. The shear is normalized by the friction velocity, U,, calculated from measured pressure drop along the pipe a t the wall using the average radius. The calculated distribution, using the minimum and maximum radii, is shown as dotted lines. The importance of the secondary flows as a mechanism for momentum transfer for flow over wavy surface is particularly clear from Figure 6. r/R
Figure 7. Distribution of turbulent shear stress.
Application Secondary flows and turbulence were measured for flow of air in a corrugated pipe of 8.986-in. mean diameter and 23 f t long having a near sinusoidal wall shape with wave amplitude of 0.219 in. and length 2.751 in. Air temperature was 77OF and measurements were taken 87 wavelengths downstream of the inlet with this location followed by approximately 3 f t of conduit. Pressure was essentially atmospheric at the discharge. A special X array was mounted as shown in Figure 2 where the location of measuring planes along the wave is also indicated. The probe and its support are shown approximately scaled to the size of the wavy surface. It was thus possible to measure within the trough of the waves. A series of runs at several flow rates were executed and details are provided by Chen (1973). Results for one flow rate are presented in Figures 3 to 7. The distribution of time average velocity in the axial direction appears in Figure 3. This velocity is independent of position along the wave except for locations within 15-20% of the surface. Time average radial velocity distributions are shown in Figure 4. Note that significant radial flows can be detected at distances 30% from the wall. When one considers the large momentum flux associated with very small radial flows, the origin of the very high pressure
Nomenclature e = fluctuating voltage output from linearized hot wire anemometer ( e = 0) E = time average voltage output from linearized hot wire anemometer k = constant in eq 1 K = constant relating output of anemometer to effective cooling velocity rx = ratio of fluctuating to time average velocity, u/U rs = ratio of fluctuating to time average velocity, ufU R = ratio of time average velocities in the y and x directions, V fU u = fluctuating component of velocity in the x direction U = time average velocity in the x direction u = fluctuating component of velocity in the y direction V = time average velocity in the y direction w = fluctuating component of the effective cooling velocity W e = instantaneous effective cooling velocity W I = instantaneous velocity vector LY = angle between the normal to the cylinder and the x direction fl = angle between the normal to the cylinder and the instantaneous velocity vector
Literature Cited Chen, W. N.. M.S. Thesis, University of Houston, 1973. Champagne, F. H., Sieicher, C. H.,J. fluidMech., 28, 153 (1967a). Champagne, F. H.. Sleicher, C. H.,J. fluid Mech., 28, 177 (1967b). Hill, J. C., Sleicher, C. H.,Phys. fluids, 12, 1126 (1969). McCroskey, W. J., Durbin, E. J., ASME Paper 71-WA/FE-17 (1971).
Receiued for reuiew April 2, 1975 Accepted July 7,1975
COMMUNICATIONS
The Effect of Purely Sinusoidal Potentials on the Performance of Equilibrium Parapumps
The equilibrium theory of Pigford is used to analyze the effect of purely sinusoidal velocity on separation in a closed parametric pump. The results are compared with separations obtained using square wave potential. It is shown that a square velocity wave can produce separation factors increasingly larger than with a sinusoidal velocity. The equations describing the system characteristics when both velocity and temperature are purely sinusoidal are also presented.
The underlying mechanism for separation in closed parametric pumps has been nicely explained with the publication of the equilibrium theory of Pigford et al. (1969). Elegant in its simplicity, the equilibrium theory has been the nucleus around which extensions to the continuous mode (e.g., Chen and Manganaro (1974)) have been built. Disper362 Ind. Eng. Chem., Fundam., Vol. 14, No. 4, 1975
sive effects were ignored in Pigford's model, hence ultimate separation is unbounded. Nonetheless, the theory provides a useful basis for comparison. One aspect of the equilibrium theory which appears to have been overlooked is the type of periodicity the velocity and temperature potentials take. In this work, we take the
1.0
H
Figure 1. Characteristiccurves for purely sinusoidal velocity and square temperature wave.
square velocity and temperature wave structure used in the original equilibrium theory as the basis for comparison with purely sinusoidal potentials.
Transport Equation The conservation equation of Pigford et al. (1969) was taken as
where the first two terms account for accumulation in the fluid and solid phases, respectively, and the third term describes the uniform but periodic convection along the axis of the packed column. The equilibrium between fluid and solid phases was assumed to be linear s = 117 ( T ) y (2) Hence, solid phase composition ( x ) can be eliminated to give
In Pigford’s analysis, square wave potentials were used for m ( t ) and o ( t ) , i.e. r 0 ) = 2’0 sq(w/) (41 113 = 1110 - n sq(wt) From this point, we depart slightly from their work.
(5)
Sinusoidal Velocity Instead of a square velocity wave, we now consider a purely sinusoidal velocity I , ( / ) = zs0 sin ( w / ) (6) where uo denotes the fluid amplitude, frequency product, Am. There are strong practical reasons for using sine waves, not the least of which is ease of generation from rotating machinery. Intuitively, one would not expect dramatic differences in separation from the square wave situation, but further analysis will show this is not the case. The method of characteristics (see for example, Hildebrand (1965)) can be used to solve the first-order hyperbolic partial differential equation (3) to form the pair of ordinary equations
dZ--dl
1
?’(/) A
/I/,,
- (1
sin (of) sq(of)- 1 - h sq(w/) (7)
where we allow the temperature to take a square wave form as in Pigford’s work. Aris (1969) has shown that the particular pictorial representation of Pigford was in fact a special case, the special character of which implies that the bed length must be
L =
2 [ l-2h
7TZl
-
-1 + 1
1
(9)
11
For the purposes of comparison, we shall continue to pursue this special case and to simplify matters we use (9) to express the characteristic equation (7) in dimensionless form, thus
Equation 8 shows that along a characteristic path, y ( {,O) [ 1 - b sq(B)] is constant, while eq 10 shows that the characteristic curves are not straight and during a cycle follow the path
+
constant
(11)
This somewhat complicates the analysis. However, by sketching a single cycle using (11) a template can be constructed to generate the remaining characteristic curves shown in Figure 1. The value of b = 0.185 used in this figure was deduced from Pigford’s Figure 3 by taking the ratio of his straight characteristic lines during hot and cold periods. The shaded region in Figure 1 denotes the passage of the original contents in the column; the straight characteristics of Pigford are drawn for comparison. For the square velocit y , the special case Pigford presented showed that the initial composition of the column does not affect the solution after the second full cycle. The construction of generalized finite-difference equations for any cycle then required only top and bottom compositions a t the second cycle ( ( Y T ) ~ , ( Y B ) ~ )as initial conditions. However, for a sinusoidal velocity of the same amplitude as the square wave, Figure l shows that five full cycles must elapse for the initial composition to no longer affect the final solution. Furthermore, there are in general three concentration fronts that contribute to the well-mixed tops composition. The analysis now proceeds exactly as for Pigford’s case, those second-cycle characteristics arriving a t the top along the line ac’ are enriched by the amount ( ~ ~ ) 1+(b1) / ( l b ) = yo(l b ) / ( l - b ) and those arriving along c’d originate from the initial composition. The calculation of the fractions uc’lc’d can be deduced from eq 11 to be
+
Thus, the top and bottom compositions up to the fifth cycle are taken as Ind. Eng. Chem., Fundarn., Vol. 14, No. 4, 1975
363
In general, the form of the forcing function in this finite difference equation depends on the selection of b. Applying the initial condition (YT)~, and letting (1 b ) / ( l - b ) = r, there results
+
(n
>
5 ) (18)
and
The separation factors, defined as ( Y T ) , / ( V B ) ~ for the square and sine velocities are compared in Figure 2 for b = 0.185. There is a significant difference. A calculation not shown in Figure 2 for n = 50 cycles gives separation factors of 5 X lo6 and 3.1 X los for sine and square velocities, respectively; clearly there is an enormous difference in potential separation. Both modes of pulsing produce unbounded separations (proportional to ekn) owing to the absence of dissipative effects in the equilibrium model. In a laboratory situation, what these theoretical results mean is that purely sinusoidal pulsing is expected to take considerably longer to reach the ultimate separation than square wave pulsing.
NUMBfR W CYCLES, n
Figure 2. Comparison of separation factors for b = 0.185.
and 1-11
(?'B)n
= 3'0
(E)
(14)
as before. Now, the top composition a t the fifth cycle receives material originating from ( Y T ) 4 along line ab, from yo along line bc' and from ( Y B ) ~along line c'd. I t can easily be shown using eq 11 that the net movement upward per cycle (change in () of a characteristic curve can be calculated to be ( 4 b / ~ ) / ( l 3 b ) . Hence a characteristic beginning a t the origin would have a position a t the end of the fourth cycle of (4 = ( 1 6 b / ~ ) / ( 1 3 b ) . When this information is applied to eq 11,the fraction ae'lad is determined to be
+
+
nc'lnd =
CY
= cos-1
[ (17b
1) - 7i(l 1 - b
+
3b)
so that the fifth cycle composition is calculated from /?T)5
=
One can generalize this result by inserting the successive values of ( Y T ) ~ but , this is tedious and such work is eminently suited to the computer. We can now write a general finite-difference expression to find (YT),, for the specific value of the parameter b used in Figure 1. In the manner analogous to Pigford, there results 364
Ind. Eng. Chem., Fundarn., Vol. 14, No. 4, 1975
Sinusoidal Velocity and Temperature Under the conditions of fast pulsing, heat transfer resistance in the parapump distorts the square temperature wave applied through the jacket surrounding the bed. The bed then sees, on the average, a temperature field that is more sinusoidal than square. Thus, if the previous analysis were undertaken with a sinusoidal temperature field, it would simulate in some sense a dispersive thermal effect. Under such conditions, eq 7 and 8 became d- z_ sin w t (20) df - 1 - 11 sin w/ d In (1 - 11 sin o f ) dt dt The latter equation shows that along a characteristic y ( t , t ) [ l - b sin u t ] is constant, which means during a halfcycle, y depends as 1/(1 - b sin u t ) and is not a simple constant as before. Equation 20 can be integrated to obtain the characteristic curves thusly
_ _d _In y
+ IA
dt csc (wf) -
[-;
+
(tan - b2 t1
2 tan-' wt 1
ot) - b ) ]
- 112
(22)
The analysis can proceed as before but is somewhat intractable.
Comments and Conclusions We have analyzed and compared the situations when a sinusoidal velocity of equal frequency and amplitude as a square velocity wave is used to drive a closed parametric pump; the motivation for the current comparative study arises from recent efforts (Rice, 1973) to predict ultimate separation using a sine representation for velocity, rather than the complete Fourier series representation of the square velocity potential.
Perhaps a more instructive comparison would be to compare sine and square potentials of equal displacement. In relation to the present study, this would mean that the sine velocity is everywhere (except in the length expression, eq 9) multiplied by x/2. If this is done, i t can be easily shown that the square and sine characteristic lines begin and terminate at the same points in the region 0 < ( < 1. However, a t the points of intersection along { = 1 (which determines the rate of enrichment) the sine and square velocities produce characteristic lines which intersect in general, at different points. If eq 9 still holds, the ratio of the sine wave interaction (27r + a d ) to the square wave intersection (27r ac) along ( = 1,and 27r < % < 37r, can be shown to be
+
and for the specific value of b = 0.185 used earlier, this relation shows that nc'/nc = 0.6225/0.687 = 0.906
(24)
Hence for this case, the sinusoidal velocity of displacement equal to the square velocity will always produce the slower enrichment. In general, discussions relating to the best selection of velocity wave form will obviously depend on the value of the parameter b, even under conditions of equal displacement; for the specific case cited in eq 24, the rate of enrichment will be slightly, but distinctly, different. Acknowledgments The author gratefully acknowledges the constructive comments of Professor Philip C. Wankat, who reviewed the manuscript.
L = totalbedlength M ( 7') = temperature-dependent equilibrium constant m = p,(l - c)M(T)/pp rno = steadyvalueof rn n = number ofcycles uo = Vo/(l + mol u = interstitial velocity Vo = velocity amplitude x = solids concentration y = fluid concentration z = axial position in bed Greek Letters a , = separation factor, ( Y T ) n / ( y B ) n (Y = fraction defined by eq 15 @ = fraction defined by eq 12 c = bed void fraction p = phasedensity w = cycle frequency % = dimensionless time, wt ( = dimensionless distance, z/L Subscripts B = bottom (lean) reservoir n = cyclenumber f = fluid phase s = solid phase T = top (rich) reservoir L i t e r a t u r e Cited Aris, R., Ind. Eng. Chem., fundam., 8 , 603 (1969). Chen, H.T.. Manganaro, J. A,. AIChEJ., 20, 1020 (1974). Hildebrand. F. B., "Advanced Calculus for Applications", pp 379-388, Prentice-Hall, Englewood Cliffs, N.J.,1965. Pigford, R. L., Baker, B., Blum, E. E., Ind. Eng. Chem., fundam., 8, 144 (1969). Rice, R. G.,Ind. Eng. Chem., Fundam., 12,406 (1973).
Department of Chemical Engineering University of Queensland Brisbane, Australia 4067
Nomenclature a = change in equilibrium constant b = a / ( l ma)
Richard G . Rice
Received for reuiew December 2,1974 Accepted June 5,1975
+
Mass Transfer from Spherical Drops at High Reynolds Numbers
The mass transfer resistance external to a spherical drop at high Reynolds and Peclet numbers is obtained using the thin concentration boundary layer assumption and the interfacial velocity of Harper and Moore (1968). Earlier treatments of the same problem are in error because an incorrect expression was used for the interfacial velocity. The available experimental data agree well with the present analysis.
-
The resistance to mass transfer external to a spherical drop for Pe 00 is given by
-
where ui is the velocity in the 8 direction a t the interface. For Re the flow field external to the drop is given by potential flow with ui 3 . - = - sin 8
u 2 substitution yields the well-known Boussinesq equation
(3)
A number of authors (Lochiel and Calderbank, 1964; Winnikow, 1967; Cheh and Tobias, 1968) attempted to extend the analysis to finite Reynolds numbers using the velocity boundary layer approach of Moore (1963). In all cases the velocities they used were in error. The error, which consisted of neglecting the circulation of vorticity inside the drop, is described by Harper and Moore (1968) and Harper (1972). The correct interfacial velocity is given by Ind. Eng. Chem., Fundam., Vol. 14, No. 4. 1975
365
