ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT ion exchange with phosphorylated cotton could a t least be brought to the level of operation obtainable with resins. The course of this development should include a study of the relation of the chemical structure and preparation of the cloth to its eapacity and resistance to internal diffusion in addition to refinement of the apparatus.
V
= liquid rate, cc./min.
x
= superficia1 distance along liquid length of exchange cm./min.tanks, cm. = end of exchange tank, = time, min.
2,
zo
e
literature cited
Acknowledgment
(1) Bieber, E., Steidler, F. B., and Selke, W. A,, Chem. Eng
The aid of the United States Atomic Energy Commission in supplying the apparatus used, under Contract AT (30-1) 1108, is gratefully acknowledged.
(2) Crits, G. J., M.S. thesis, Columbia University, 1950.
Progr.,
Nomenclature
a
= total exchange capacity, meq./gram sodium form (air
C Go C,
= copper concentration in solution, meq./l. = total cation concentration in solution, meq./l.
C*
= copper concentration in solution in equilibrium with solid a t concentration q, meq./l.
dried) = copper concentration in solution a t solid-liquid interface,
meq./l.
E = exchanger rate, grams/min. K D S = over-all mass transfer coefficient, meq./(min.)(gram) (mes./l. 1 ~ D = S individual mass transfer coefficient, meq./(min.)(gram) (mes./l.) = copper concentration in solid, meq./gram sodium form p (air dried) = copper concentration in solid a t solid-liquid interface, qt meq./gram sodium form (air dried)
Symposium Ser., No. 14, p. 17, 1954.
(3) Crumpler, R. B., ANAL.CHEM.. 19, 325 (1947). (4) Guthrie, J. D., IND.ENG.CHEM.,44, 2187 (1952). (5) Hiester, N. K., Phillips, R. C., Fields, E. F., Cohen, R. K., and Radding, S. B., Ibid., 45, 2402 (1953); Heister, N. K., Fields, E, F., Phillips, R. C., and Radding, S. B., Chem. Eng. Progr., 50, 139 (1954). (6) Higgins, I. R., and Roberts, J. T., Chem. Eng. Progr., Symposium Sw., No. 14, p. 87, 1954. (7) Jurpens. J. F., Reid, J. D., and Guthrie, J. D., Textile Research 18, 42 (1948). ( 8 ) Koenig, W. W., Babb, A. D., and McCarthy, J. L., Chem. Eng. Progr. Symposium Ser., No. 14, p. 103, 1954. (9) Little, R. W., “Flameproofing Textile Fabrics,” Reinhold, New Y o r k , 1947. (10) hIcCormack, R. H., and Howard, J. F., C h e m Eng. Progr., 49, 404 (1953). (11) Nordell, C. H., U. S. Patents 1,608,861 (November 1926); 1,722,938 (August 1929); 1,740,199 (December 1929). (12) Selke, W. A,, and Bliss, H., Chem. Eng. Prow., 47, 529 (1952). (13) Stanton, L. S., M.S. thesis, University of Washington, 1950. (14) Wilcox, A. I,.,U. 8. Patent 2,528,099 (October 1950).
x,
RECEIVED for review Scptemhpr 3, 1954.
ACCEPTED November 10, 1954.
Fluid Mechanics Studies
Transition Phenomena in Pipes Annular Cross Sections R. S. PRENGLE‘
AND
R. R. ROTHFUS
Carnegie lnsfifute of Technology, Piffsburgh, Pa.
B
REAKDOWN of viscous motion in fluids flowing in conduits of various shapes has been the subject of much speculation. The theoretical and experimental investigations of Meksyn ( 7 ) , Maurer (6),Schiller (I$’), Gibson ( 3 ) and others, however, have only partially clarified the physical picture of the phenomena that occur in the transition process. A recent study of velocity distribution and fluid friction in smooth tubes by Senecal and Rothfus ( I S ) has indicated that deviations from viscous behavior can be observed a t bulk Reynolds numbers as low as 1200 to 1300. This is in substantial agreement with the results of some preliminary dye filament experiments reported by Rothfus and Prengle (11). I n the latter investigation, thin filaments of aniline green dye were injected a t various points in the cross sections of two plastic tubes through which water was flowing. It was found that the first observable departure from laminar behavior occurred a t the center line of the tube and a t a bulk Reynolds number of 932. As the Reynolds number was increased above this value, the region of sinuous motion was observed to spread toward the tube walls a t such a rate that the velocity of the fluid a t the edge of t h e still-laminar layer followed the simple relationship I
Present address, E. I. du Pont de Nemours &
March 1955
co., Buffalo, N. Y ,
At Reynolds numbers between 1500 and 2100, there appeared to be a strong tendency to set up a stable spiral motion in the sinuous core of the fluid. At a Reynolds number of about 2100, the spiral motion was observed t o be replaced occasionally by a large disturbance eddy. The frequency with which the eddy was cast off increased with increasing Reynolds number until, a t a Reynolds number of about 3000, the eddy form became the stable one. The authors did not attempt to eliminate t h e effect of injector diameter on the flow characteristics. Therefore, the value of the right-hand side of Equation 1 was not firmly established, although the form of the expression appeared to be satisfactory. Lindgren ( 6 )recently studied the flow of birefringent bentonite suspensions in polished Plexiglas tubes and concluded that the basic flow was essentially laminar a t Reynolds numbers below 2900. Rare turbulent flashes were observed a t 2900 and complete turbulence was attained a t about 3600 Reynolds number. The bulk Reynolds numbers at which Lindgren reported changes in the flow regimes were somewhat too high to be consistent with the velocitv distribution and Dressure dror, data of Senecal and Rothfus (is) and others, Ii is possible- that only large dis-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
379
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT turbance eddies could be observed in the bentonite suspensions and that sinuous motion was consequently taken t o be truly laminar. It is also very possible that the particles in suspension may have had a calming effect on the flow. Lindgren believed the spiral flow mentioned by Rothfus and Prengle ( 1 1 )to be caused by the probe and the apparent thickness of the laminar sublayer to be the inserting depth at which the probe caused vorticity in the flow immediately downstream from it. By means of flash photographs, he showed the disturbances behind probes placed in the flowing suspensions. The presence of a turbulent wake a t low values of the tube Reynolds number was clearly indicated, but the flow pattern upstream from the probe exhibited the unlikely beharior described in the preceding paragraph. The present investigation, completed prior to the publication of Lindgren’s work, was also directed, in part, toward establishing the effect of the injector on the flow pattern. The authors believed that the preliminary data of Rothfus and Prengle ( 1 1 ) weie not sufficient to indicate the reality of spiral flow or the influence of injector diameter on local fluid motion. Very little is known about transition flow in annuli Pressure drop measurements by several investigators-e.g., Carpenter and coworkers (1)-have indicated that the friction factor
I=
( Y )rp(-Tr1 )
when plotted as a function of the Reynoldc: number, 2(r2-rl)T’p/p, does not “dip” in the transition zone as does the friction factor for a tube without a core. T h e limits of the transition zone have not, hoxever, been established with a very high degree of certainty, even from the standpoint of pressure drop alone. Rothfus, Monrad, and Seneca1 ( 1 0 )found that for fully turbulent flow, the pressure drop in annuli can be correlated with pipe data by applying the hydraulic radius concept t o the portion of the fluid which lies outside the radius of ma\imum local velocity. T h a t is, the friction factor a t the outside surface
(3) is related t o the Reynolds number
in the same manner that the friction factor in a tube is related to the bulk Reynolds number. I t should be noted that the function, JF, = 6, (;tXe,) is not unique in the viscous f l o ~range but depends on the radius ratio, r2/7 1. Therefore, it might be expected that corresponding friction factors in the transition zone should show some radius ratio effect unless a conipensating change occurs in the radius of maximum velocity. Velocity distributions obtained by Rothfus, RIonrad, and Seneca1 and by Knudsen and Katz ( 4 ) have indicated that the radius of masinium velocity is essentially the same in full turbulence as in truly viscous flon-,
There is some indication, not well established, however, that the radius of maximum velocity shifts toward the core in transition flow. Since a simple force balance on an annular element of fluid yields the shear stress distribution
the ratio of the skin frictions a t the inner and outer boundaries of the annulus must be
380
If the skin frictions,
71
and
72,
happen to be equal,
rather than the value obtained from Equation 5 , Consequently, if an inxaard shift of r , actually occurs in the transition zone, the phenomenon is consistent with a tendency to equalize the skin frictions. By analogy with transition phenomena in pipes, i t might be expected that the first olxervable departure from viscous flow in annuli should occur a t the point of maximum local velocity Furthermore, it would be reasonablr to suppose that the main stream of the fluid might exhibit a pinuous-turbulent progression siinilai to that in a pipe. Scope of the investigation
The experiments described in this paper were performed principally to check and refine Equation 1 for pipe flow and to determine the general nature of transit,ion flow in concentric annuli of different radius ratios. The investigation was limitcd to the study of dye filaments injected into water flowing horizontally in three smooth tubes and five sinooth concentric annuli. The radius ratios, rpll.1, in the annuli varied from 1.79 to 24.8. The flox was esserit,ially isothermal in every case and the over-all range of bulk Reynolds numbers investigated Tvas roughly 200 to 2400. Above t,he latter value, it was not possible to study the dye behavior successfully. By terminating the filament issuing from the eject,or, local fluid velocities a e r e determined in regions of the streams \Thearc the flo~vwas viscous or only slightly sinuous. The following experimental items were studied in corisiderable detail: Reynolds number and position of the first observablc deviation from truly viscous behavior Revnolds number and characteristics of the initial disturbance eddyExtent and characteristics of sinuous and turbulent regions in the fluid Laminar film thickness as a function of Reynolds number Local velocity a t the edge of the laminar region Viscous-flow velocity profiles, especially in the annuli Behavior of the radius of maximum velocity in the annuli following the breakdoxn of viscous motion
A complete srt of the original data i3 available from the American Docunientation Institute. Experimental technique includes study of behavior of dye filament injected into flowing water
A diagram of the esperiniental apparatus is shown in Figure 1 and pertinent details of the tubee and annuli are summarized in Tables I and 11. Details of the external piping not shoivn in Figure 1 are available (0). The test fluid was pumped from a supply tank t o a heat eschanger fitted with a by-pasa and thence t’o a surge tank before i t entered the inlet box shown in Figure 1. The poeition of the box could be varied to produce either a square-edged or Bordatype entrance to the experimental conduit as desired. Inlet and outlet water temperatures were measured by means of copper-constantan t#hermocouples. The water leaving the test conduit passed through a swivel arni Ivhich permitted it t o be recirculated or measured volumetrically in an. one of three calibrated tanks. Each tank was fitted with a glass side arm on which t,he reference level x-as scribed, and the volumetric cali‘nration was est,ablished to an accuracy of 0.1%. The annulus cores passed completely through the test length and calniing sections, as shown in Figure 1. They were centered by means of carefully machined packing glands. The three smallest cores were prevented from sagging in the horizontal outer tube by tension applied through an externally situated screw. The t x o largest cores were supported by tight wires which extended through their interiorj. The alignment of the
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Vol. 41,No. 3
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT various sections of the plastic outer tube was maintained by foam rubber supports. The sections within the test length were connected by dry butt sleeves in order to preserve constant inner dimensions. The entire experimental conduit was mounted on a heavy iron frame to minimize external disturbances. It was believed, however, that more useful data could be obtained if no spectacular measures were taken to prevent small vibrations of the unit as a whole.
DISCHARGE
CLEAR OUTER
PLA
TUB
(UNDER
Figure 1.
TENSION
Annular section
The injectors were formed from stainless steel hypodermic tubes with the diameters shown in Table I11 and mounted on traversing mechanisms. Most of the injectors were of the ordinary bent-needle design shown in Figure 1. The length of the bent portion was 0.875 inch in each case. I n order to determine the effect of the wake behind the bent tubes an inclined injector was also used. This was simply a straight length of hypodermic tubing inserted through the outer tube wall a t an angle to the flow. The filament from the inclined injector was "fired" across the fluid stream to a predetermined position in the cross section. I n some of the runs, two of the bent needles were inserted in the same pla'ne a few inches apart along the length of the conduit. This arrangement permitted the wake from the upstream tube to be studied through the behavior of the downstream filament.
*
Either aniline green or Congo red dyes were supplied to the injectors from pressurized reservoirs. The rate of dye flow vias finely controlled by needle valves in the supply line. The dye filaments were observed through a 10-power magnifying lens. Cross hairs above and below the conduit were aligned with the filament under observation in order to permit the observation of very small deviations from viscous behavior. I t was estimated that a lateral movement of 0.01 inch could be detected in t h e filament under ordinary conditions. Procedure. City water from the supply tank mas circulated through the system and heated to room temperature by the exchanger. Thereafter, the operation was made essentially isothermal by passing a small fraction of the recirculated stream through the exchanger which now used cooling water instead of steam in order to remove the heat added by the action of the pumps. The conduit alignment was checked and runs were begun when the thermocouples on each end of the test conduit differed by less than 0.25" F. Only a few runs could be made on each filling of the system because of the accumulation of dye in the test fluid. The flow rate was determined by allowing the discharge from the test conduit to flow into one of the calibrated metering tanks for a measured period of time. The additional (small) amount of fluid necessary to fill the tank to the reference mark on the glass side arm was then determined and the flow rate calculated. Each time a flow rate mas taken, the inlet and outlet temperatures of the fluid were determined from readings on a standard laboratory potentiometer. I n order to determine the point of first departure from viscous flow, the rate of discharge was set a t a sufficiently low value to ensure viscous behavior and the cross section of the fluid was traversed by the injector. The dye filament was examined a t approximately 0.025-inch intervals across the entire fluid section. If no transverse motion of the filament was noted a t any point, the flow rate was increased slightly and the procedure was re-
March 1955
peated. This was continued until the viscous motion was obpoint in the fluid. The condition served to break down at of the initial breakdown was then approached from the other , direction-Le., from transition flow-and the last point to be sinuous rather than laminar was noted. I n each case the entire procedure was repeated using injector needles of various diameters. Essentially the same procedure was used in measuring the thickness of the laminar film a t higher Reynolds numbers. I n this case, the edge of the film was approached from both directions in the cross section aa well as from both higher and lower flow rates. The local velocity a t the film edge was obtained by shutting off the dye stream completely and measuring the time necessary for the terminus of t h e filament to travel a known axial distance. Several complete velocity profiles BOX were similarly obtained in the viscous flow region. I n this case, the pipe data were used to check the performance of the equipment and the annular data were used to verify further the theoretical velocity equation of Lamb ( 5 ) which assumes no slip a t either wall. Analysis of Errors. An analysis of the errors involved in the measurement of individual experimental quantities indicated a maximum error of 1.2% in the value of the lowest Reynolds number a t which viscous flow was observed to break down. I n addition, however, an indeterminate error was associated with the act of observing the behavior of the dye filament. The data for any one injector diameter showed an average deviation of 1.9% which indicated an uncertainty of from 20 to 40 on the Reynolds number scale.
Table I.
Specifications of Tubes Tube DesignationPz Ps 1 620 1.615 1 120 Tennite Lucite Lucite P I
Inside diameter, inches Material Wall thickness inches Test section leApth inches Upstream calming iength inches Downstream calming lenith, inches
Table II.
'/I6
Core diameter, inches Core material Outer pipe Test section length, inches Upstream calming length, inches Downstream calming length, inches
Needle No. 1 2
3
4 5 6
36'/z 1703/s 120l/z
34a/4
1701/s
1201/4
Specifications of Annuli AI
Table 111.
3/16
'/la
36I/z 1703/s 1201/2
___
Annulus Designation -__-As As A4
As
0.243 0.626 Copper tubing Pa Pa 343/4 30 170'/s 137 1201/4 49 1.789 4.61 0 772 0.452
Specifications of Bent-Needle Injectors Outside Diam., Inch 0 0490 0 0420 0 0318 0 0277 0 0249 0 0105
Inside Diam., Inch 0 035 0 028 0 020 0 016 0 013
0 010
The maximum error in reproducing a flow rate a t a fixed setting of all valves was actually about 0.4%. This was in exact agreement with the error estimated from the analysis of individual quantities and appeared t o mean that very little error was caused by minor fluctuations in the flow. The maximum error in measuring local fluid velocities was estimated to be about 1.3%. The lowest Reynolds number a t
INDUSTRIAL AND ENGINEERING CHEMISTRY
381
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT which local velocities could be measured satisfactorily was limited by diffusion of dye from t h e filament which made it difficult to observe the position of the terminus. The measurement of laminar film thickness was, of course, complicated by the fact that sinuous deviat,ions of 0.01-inch amplitude were necessary before any departure from laminar behavior could be observed v i t h certainty.
In pipes, sinuous motion and disturbance eddies appear at characteristic Reynolds numbers
Wave point Reynolds number marking local departure from viscous flow i s used in correlations
The set of operating conditions under which the first local departure from truly viscous flow could be observed was called the wave point. The bulk Reynolds number associated n i t h this set of conditions was called the wave point Reynolds number. The thickness of the laminar film was taken to be the longest distance from the wall to t h e center of the injector which could be traversed without observing a deviation from viscous behavior in the dye filament. I300 E
2 ~
B ..\\.
1200
Lt
w
z 1000 v)
900
0 W
800
a I-
630
-I
I
500
.01 NEEDLE
02 DlAM /PIPE
.03
.04
DIAM
Figure 2. Effect of injector needle diameter on observed wave point Reynolds numbers in tubes Pipe
Type of Entry
0
PI P1
V
PZ
D
p3
Borda Square screen Bordo Borda
a
+
Injector Bent Bent Inclined Bent
It was found t h a t t h e diameter of the injectoi needle affected the observed value of the wave point Reynolds number in both pipes and annuli. T h e ratio of t h e needle diameter to some characteristic diameter of the conduit was chosen a8 the parameter through which t o express the needle effect in each case. The observed wave point Revnolds numbers were plotted against the diameter function8 and extrapolated t o zero needle size. The extrapolated Reynolds number value thus obtained was checked experimentally by meam of t h e inrlined needle injector previously described. A similar procedure n a s followed in eliminating t h e effect of needle diameter from t h e observed valum of the laminar film thickness. I n this case, t h e inclimd needle could not be used as a check because the position of t h e filament %*suingfrom it could not be controlled precisely enough Graphs of t h e observed laminar film thickness against bulk Reynolds number a t coiatant values of t h c needle diameter function Rere made and smoothed. These were cross plotted to yield g r a p h of Reynolds number against diameter function a t constant values of fEnr thicknews. Extrapolation t o zero needle size was made a ; for ~ t h e wave point determination. Preliminary experiments indicated t h a t either a sqrratreedged
382
or Borda-type entrance could be used without influencing the f l o ~characteristics. I n addition, it was established that the presence of an injector needle exactly in line m-ith and upstreain from another needle appeared to have little effect on the filairierit issuing from the latter. The wake behind the belit iiiiec5tor needles did, however, effect the behavior of their filament-.
Wave Point. The stable modes of transition flow in pipes were found to be cssentially those previously reported by the authors (11). At a particular value of t h e bulk Reynolds number, t h e first observable indication of sinuous flow appeared a t the center line of t h e tube-i.e., a t the point of maximum local fluid velocity. Figure 2 shows t h e effect of inject'or needle diameter on the observed wave point Reynolds number. The points on the axis DJD, = 0 n-ere obtained with the inclined needle. Both t,hese data and the extrapolation of the ot.lier points indicat,e t h a i the truc wave point Reynolds number is about 1225. The latter value is considerably higher than that first reported ( 1 1 ) . However, when t h e needle effect is removed, the previous data also yield a wave point Reynolds number of about 1225. This value is likewise in agreement with friction data reported by Seneca1 and Rothfus ( I S ) . I n the experiments reported (If), a high level of turbulence !vas promoted a t th'e entrance by means of a globe valve installed in the h b e . It was found that the wave point Reynolds nuIilber was independent of the value setting. The pressure drop and velocity distribution data ( I S ) were obtained in tubes having true square-edged entrances. I n neither case were any unusual steps taken t o eliminate normal amounts of vibratioh. Since the prwent data agree closely with those previously reported ( I f , I S ) , it appears likely that the observed wave point Reynolds number of 1225 is the characteristic value for ordinarily high levels of vibration and initial disturbance. No attempt was made to establish the threshold conditions precisely. The wave point Reynolds number reported in this paper is of courpe only an indication of the conditions under which lyaves of a t least 0.01-inch amplitude were evident a t the center line of the tube. I t should not be construed to be a criterion for the origin of turbulence without further evidence; nor should it be concluded that the turbulence necessarily originates a t the axis of the tube. Present experiments can do no more than point, out where first departure from laminar behavior was observable. Development of Turbulence. At Reynolds numbers above the wave point value, the region of fully viscous motion retreated steadily toward ihe tube x-alls as the Reynolds number increased. The crosscurrent motion in the central portion of the stream increased in amplitude and iiitenait,y v i t h increasing Reynolds number arid proxiiiiity to the center line. I n the range I500 < X R
