Prengle Rothfus, 1955 Transition Phenomena in Pipes and annular cross sections

Prévia do material em texto

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 ea- 
pacity and resistance to internal diffusion in addition to refine- 
ment of the apparatus. 
V = liquid rate, cc./min. 
2, x 
zo 
e = time, min. 
literature cited 
= = superficia1 distance along liquid length of exchange cm./min. tanks, cm. 
= end of exchange tank, 
Acknowledgment 
The aid of the United States Atomic Energy Commission in 
supplying the apparatus used, under Contract AT (30-1) 1108, 
is gratefully acknowledged. 
Nomenclature 
a = total exchange capacity, meq./gram sodium form (air 
C = copper concentration in solution, meq./l. 
Go = total cation concentration in solution, meq./l. 
C, = copper concentration in solution a t solid-liquid interface, 
C* = copper concentration in solution in equilibrium with 
E = exchanger rate, grams/min. 
KDS = over-all mass transfer coefficient, meq./(min.)(gram) 
~ D S = individual mass transfer coefficient, meq./(min.)(gram) 
dried) 
meq./l. 
solid a t concentration q, meq./l. 
(mes./l. 1 
(mes./l.) 
p 
qt 
= copper concentration in solid, meq./gram sodium form 
= copper concentration in solid a t solid-liquid interface, 
(air dried) 
meq./gram sodium form (air dried) 
(1) Bieber, E., Steidler, F. B., and Selke, W. A,, Chem. Eng 
Progr., Symposium Ser., No. 14, p. 17, 1954. 
(2) Crits, G. J., M.S. thesis, Columbia University, 1950. 
(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). 
posium Sw., No. 14, p. 87, 1954. 
(6) Higgins, I. R., and Roberts, J. T., Chem. Eng. Progr., Sym- 
(7) Jurpens. J. F., Reid, J. D., and Guthrie, J. D., Textile Research x , 18, 42 (1948). 
Progr. Symposium Ser., No. 14, p. 103, 1954. 
New York, 1947. 
404 (1953). 
1,722,938 (August 1929); 1,740,199 (December 1929). 
( 8 ) Koenig, W. W., Babb, A. D., and McCarthy, J. L., Chem. Eng. 
(9) Little, R. W., “Flameproofing Textile Fabrics,” Reinhold, 
(10) hIcCormack, R. H., and Howard, J. F., C h e m Eng. Progr., 49, 
(11) Nordell, C. H., U. S. Patents 1,608,861 (November 1926); 
(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). 
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. 
REAKDOWN of viscous motion in fluids flowing in conduits B 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 dis- 
tribution 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 pre- 
liminary dye filament experiments reported by Rothfus and 
Prengle (11). 
In 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 the 
still-laminar layer followed the simple relationship 
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 the effect of in- 
jector 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 com- 
plete 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 
I Present address, E. I. du Pont de Nemours & co., Buffalo, N. Y , Rothfus (is) and others, Ii is possible- that only large dis- 
March 1955 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 379 
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 
turbance eddies could be observed in the bentonite suspensions 
and that sinuous motion was consequently taken to 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 (11) 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 be- 
lieved 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. 
Pressure 
drop measurements by several investigators-e.g., Carpenter 
and coworkers (1)-have indicated that the friction factor 
Very little is known about transition flow in annuli 
rp - r1 I = (Y)(T) 
when plotted as a function of the Reynoldc: number, 2(r2-r l)T’p/p, 
does not “dip” in the transition zone as does the friction factor for 
a tube without a core. The limits of the transition zone have 
not, hoxever, been established with a very high degree of cer- 
tainty, even from the standpoint of pressure drop alone. 
Rothfus, Monrad, and Seneca1 (10) found that for fully turbu- 
lent flow, the pressure drop in annuli can be correlated with pipe 
data by applying the hydraulic radius concept to the portion of 
the fluid which lies outside the radius of ma\imum local velocity. 
That is, the friction factor a t the outsidesurface 
(3) 
is related to 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 de- 
pends 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 turbu- 
lence 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 
If the skin frictions, 71 and 7 2 , 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 princi- 
pally to check and refine Equation 1 for pipe flow and to deter- 
mine 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 aere determined 
in regions of the streams \Thearc the flo~v was viscous or only 
slightly sinuous. 
The following experimental items were studied in corisiderable 
detail: 
Reynolds number and position of the first observablc deviation 
Revnolds number and characteristics of the initial disturbance 
from truly viscous behavior 
eddy- 
the fluid 
Extent and characteristics of sinuous and turbulent regions in 
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 fol- 
A complete srt of the original data i3 available from the Ameri- 
lowing the breakdoxn of viscous motion 
can 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 to a heat es- 
changer 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 Borda- 
type 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‘nra- 
tion 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 
380 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 con- 
nected 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. 
peated. This was continued until the viscous motion was ob- 
served to break down at The condition 
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. In each case the entire 
procedure was repeated using injector needles of various diam- 
eters. 
Essentially the same procedure was used in 
measuring the thickness of the laminar film a t 
D I S C H A R G E higher Reynolds numbers. In 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 the filament to travel a known 
point in the fluid. 
C L E A R PLA 
OUTER T U B 
( U N D E R T E N S I O N 
Figure 1. 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 ordi- 
nary 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. 
In 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 magni- 
fying lens. Cross hairs above and below the conduit were aligned 
with the filament under observation in order to permit the ob- 
servation of very small deviations from viscous behavior. I t 
was estimated that a lateral movement of 0.01 inch could be 
detected in the filament under ordinary conditions. 
Procedure. City water from the supply tank mas circulated 
through the system and heated to room temperatureby the ex- 
changer. Thereafter, the operation was made essentially iso- 
thermal 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 tempera- 
tures of the fluid were determined from readings on a standard 
laboratory potentiometer. 
In 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 tra- 
versed by the injector. The dye filament was examined a t ap- 
proximately 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- 
* 
axial distance. Several complete velocity profiles 
were similarly obtained in the viscous flow region. 
In 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. In 
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. 
BOX 
Table I . Specifications of Tubes 
Tube Designation- 
P I Pz Ps 
Inside diameter, inches 1 620 1 .615 1 120 
Material Tennite Lucite Lucite 
Test section leApth inches 3 6 I / z 36'/z 34a/4 
Upstream calming iength inches 1703/s 1703/s 1701/s 
Downstream calming lenith, inches 1201/2 120l/z 1201/4 
Wall thickness inches '/I6 '/la 3/16 
Table II. Specifications of Annuli 
Annulus Designation ___ -__-- 
AI As As A4 As 
Core diameter, inches 
Core material 
Outer pipe 
Test section length, inches 
Upstream calming length, inches 
Downstream calming length, inches 
0.243 0.626 
Copper 
tubing 
Pa Pa 
343/4 30 
170'/s 137 
1201/4 49 
4 .61 1.789 
0 772 0.452 
Table 111. Specifications of Bent-Needle Injectors 
Needle Outside Inside 
1 0 0490 0 035 
2 0 0420 0 028 
3 0 0318 0 020 
4 0 0277 0 016 
5 0 0249 0 013 
6 0 0105 0 010 
No. Diam., Inch Diam., Inch 
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 agree- 
ment with the error estimated from the analysis of individual 
quantities and appeared to 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 
March 1955 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 381 
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 
which local velocities could be measured satisfactorily was limited 
by diffusion of dye from the 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 
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 
before any departure from laminar behavior could be observed 
vi th certainty. 
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 de- 
parture from truly viscous flow could be observed was called the 
wave point. The bulk Reynolds number associated n i th 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 the center of the injector which could 
be traversed without observing a deviation from viscous behavior 
in the dye filament. 
I300 
E 
2 
~ 1200 ..\ 
Lt B \. 
I300 
E 
2 
~ 1200 ..\ 
Lt B \. 
w 
z 
1000 
v) 
0 900 
W 
a 800 
I- 
630 
I I - 500 
.01 02 .03 .04 
N E E D L E D l A M / P I P E D I A M 
Figure 2. Effect of injector needle diameter on 
observed wave point Reynolds numbers in tubes 
Pipe Type of Entry Injector 
0 PI Borda Bent a P1 Square + screen Bent 
V PZ Bordo Inclined 
D p3 Borda Bent 
It was found that the diameter of the injectoi needle affected 
the observed value of the wave point Reynolds number in both 
pipes and annuli. The ratio of the needle diameter to some 
characteristic diameter of the conduit was chosen a8 the param- 
eter 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 to zero needle 
size. The extrapolated Reynolds number value thus obtained 
was checked experimentally by meam of the inrlined needle 
injector previously described. 
A similar procedure nas followed in eliminating the effect of 
needle diameter from the observed valum of the laminar film 
thickness. I n this case, the inclimd needle could not be used as 
a check because the position of the filament %*suing from it could 
not be controlled precisely enough Graphs of the observed 
laminar film thickness against bulk Reynolds number a t coiatant 
values of thc needle diameter function Rere made and smoothed. 
These were cross plotted to yield g r a p h of Reynolds number 
against diameter function at constant values of fEnr thicknews. 
Extrapolation to zero needle size was made a ; ~ for t h e wave point 
determination. 
Preliminary experiments indicated that either a sqrratreedged 
needles did, however, effect the behavior of their filament-. 
In pipes, sinuous motion and disturbance 
eddies appear at characteristic Reynolds numbers 
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 the bulk Reynolds number, t h e 
first observable indication of sinuous flow appeared a t the center 
line of the tube-i.e., a t the point of maximum local fluid velocity. 
Figure 2 shows the effect of inject'or needle diameter on the ob- 
served 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 thai the 
truc wave point Reynolds number is about 1225. 
The latter value is considerably higher than that first reported 
( 1 1 ) . However, when the needle effect is removed, the previous 
data also yield a wave point Reynolds number of about 1225. 
This valueis 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 nuIil- 
ber 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 to 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 in- 
creased in amplitude and iiitenait,y v i th increasing Reynolds 
number arid proxiiiiity to the center line. I n the range I500 
< X R ~ < 2100, the dye filainerit,s issuing from the bent needles 
took up a spiral motion, as previously reported by Rothfus 
and Prengle. The filaments issuing from the inclined needle, 
however, continued to Lvave haphazardly and dioxved lit,tle or 
no tendency to roll up in a spiral. It was therefore concluded 
that, the spirals were promoted by the wakes behind the bent 
needles and were not a truly stable mode of flow. 4 t a bulk 
Reynolds number of 2130 i 35 the dye filaments in the central 
part of the stream were broken into segments by the formation of 
t,hc first true disturbance eddy. 
The frequency with Ivhich the disturbance eddies were cast 
off increased with increasing Reynolds number in the range 2100 
< N R ~ < 3000. At higher Reynolds numbers than this, the nio- 
tion appeared to be entirely t'urbulent. There was Jome indica- 
tion that the disturbance eddies originated near the center of 
the stream, but the point remain8 in doubt. Photographs and 
a more complete description of the various flow configurations 
have been published (If). 
382 I N D U ST R TAL A KD E N G I N E E R I N G CHEMISTRY Vol. 47, No. 3 
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT 
Laminar Film Thickness. The local fluid velocity a t the edge 
of the laminar region was found to remain constant in a given 
pipe as the Reynolds number was varied between 1225 and 2200. 
The effect of variations in the fluid properties and pipe diameters 
could be correlated in the manner of Equation 1. Elimination of 
the injector diameter effect, however, yielded a different value 
of the right-hand side than previously reported. The corrected 
equation was found to be 
(9) 
Figure 3 shows the final experimental relationship between 
film thickness and Reynolds number corrected to zero injector 
diameter. The data from all three pipes showed r,/ro or its 
equivalent, the shear stress ratio, T J ~ T O , to be the same unique 
function of Reynolds number. An analytical expression for the 
curve in Figure 3 can be developed as follows: 
At any point within the laminar film, the Navier-Stokes equa- 
tion yields the local velocity 
But, by definition, 
0 01 0 2 0 3 04 0.5 06 07 0.8 
rt 1'0 
Figure 3. 
on laminar film thickness in tubes 
Effect of bulk Reynolds number 
Corrected to zero injector needle diameter 
Therefore, 
If Equation 12 is multiplied by Dp/fi and evaluated a t the edge 
of the film, Equation 9 can be substituted in the left-hand side 
to yield 
from which, by rearrangement, 
March 1955 
The friction factor defined in Equation 11 is not the loss per 
diameter of tube length expressed as a fraction of the velocity 
head, but one fourth of that amount. 
It was impossible to check Equation 14 by means of dye experi- 
ments a t Reynolds numbers above 2200 because the viscous layer 
was frequently penetrated by strong crosscurrents. The velocity 
profiles of Senecal and Rothfus ( I S ) and of Rothfus, Monrad, 
and Senecal (IO) could, however, be extrapolated into the region 
near the wall with reasonable accuracy. These investigations 
also included friction data and thus permitted Equation 14 to 
be examined up to a Reynolds number of 24,500. 
I O 
09 
0 8 
07 
0 6 
,o 0 5 
c 0 4 
03 
0 2 
0 1 
0 0 
. 
100 200 400 600 1000 2000 
Figure 4. Correlation of laminar film thick- 
ness in tubes at bulk Reynolds numbers 
between 1225 and 24,500 
Equation 14 was confirmed, within the accuracy of the present 
experiments and the above extrapolations, over the entire range 
of Reynolds numbers between 1225 and 24,500. Figure 4 shows 
a good compromise between Equation 14 and the actual support- 
ing data. Friction factors calculated from Figure 4 deviate by 
less than 2% from those of the previous work (IO, I S ) . The graph 
of rf/rO against N R ~ Z / ~ . is completely smooth even though the 
friction factor-Reynolds number curve dips in the transition 
range. 
Caution should be exercised in using Equation 14 a t high tur- 
bulent Reynolds numbers. It must be remembered that Equa- 
tions 9 and 14 are based on the concept of a laminar film which is 
never penetrated by cross-current fluctuations. The existence of 
such a layer a t Reynolds numbers above 2100 has not been shown 
experimentally. It is likely that Equation 14 fails to represent 
even the average situation satisfactorily a t sufficiently high 
Reynolds numbers. The very small film thickness indicated by 
the equation are, however, consistent with the observations of 
Gazely ( a ) and others. 
Correlation for laminar film thickness in 
annuli i s similar to that obtained for pipes 
Viscous Flow. Several annular velocity profiles obtained in 
completely viscous flow are shown in Figure 5. Comparison with 
Lamb's equation shows that the assumption of zero slip at the 
boundaries seems to be a good one (6). If any slip occurs, i t 
exerts negligible influence on the profile in the main portion of 
the stream. The results also confirm the fact that precise veloc- 
ity data can be obtained from dye filament studies a t flow rates 
too small for ordinary impact tubes. 
Wave Point. The first deviation from viscous flow appeared 
a t the point of maximum local velocity in all five annuli-i.e., at 
the radius given by Equation 5. I n this respect, the behavior 
was like that observed in pipes without cores. 
In the annuli, however, there were several bulk Reynolds 
numbers which could be used almost equally well as correlating 
I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 383 
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 
I .o 
0.9 
0 8 
0 7 
0 6 
0 5 
0 4 
0 3 
0 2 
0 1 
n v 
0 0. I 0.2 0 3 0 4 0.5 0.6 07 0.8 0 9 1.0 
Figure 5. Viscous-flow velocity distribution 
in annuli, showing comparison with 
theoretical relationship 
r*/n 
24.80 
18 .35 
12.72 
4 .61 
1 . 7 9 
NRe2 
470 
387 
281 
31 5 
266 
parameters. 
the Reynolds number 
After the raw experimentaldata were examined 
which had been found to correlate outer-wall friction factors in 
fully turbulent flow was used. Table IT summarizes the wave 
point Reynolds numbers which were obtained with the inclined 
needle injector. Although there was some variation in the ex- 
perimental values, no clear trend with changing radius ratio was 
evident. Similarly, extrapolations on NRe2 versus D,/4RH, 
coordinates yielded a single value of 700 a t zero needle diameter 
within the limits of experimental precision. 
Table IV. W a v e Point Reynolds Numbers in Annuli Obtained 
with Inclined Injector 
Annulua r2/1.1 n'Re2 at TVave Point 
Ai 
A2 
A3 
A& 
As 
24.80 
18.35 
12.72 
4.61 
1 . 7 3 
750 
677 
690 
714 
653 
KO explanation has been found for the fact that the wave 
point Reynolds numbers in the annuli are lower than thoPe in 
the pipes. It seems reasonable to expect a progression toward 
the pipe value with increasing radius ratio, n/n. If such exists, 
however, it is too small a variation to be picked up by the present 
procedure in the range of radius ratios covered in this investiga- 
tion. I n defense of the experimental facts, it should be noted 
that there is a discontinuity in the velocity gradient and rate of 
energy dissipation a t the center line of a pipe when an infini- 
tesimal concentric core is inserted. Thus there remains the pos- 
sibility that the wave point Reynolds number is also discontinu- 
ous. I n any case, a wave point Reynolds number of about 700 
seems to be the proper choice for annuli of ordinary dimensions. 
Development of Turbulence. The turbulent patterns in the 
annuli developed ainiilarly to those in pipes. The thickness of 
the laminar regions in the fluid decreased steadily with increasing 
Reynolds number and progressively stronger disturbances were 
apparent in the centra! portion of the stream. The dye filament3 
in the latter region wavered irregularly but remained intact unt,il 
broken by the first disturbance eddies a t Reynolds numbers, 
ATRel, of 2200 to 2300. The condition under which the initial 
disturbance eddies were cast off in the annuli was not, as well 
defined as in the case of pipes. The average deviation of the 
break point Reynolds numbers was 5.17. for annuli compared 
with 1.7% for pipes. 
The friction data reported previously (10, 23) in the form of 
friction factor v e r s u ~ Reynolds number curves indicate that' the 
casting off of the firat, disturbance eddy was accompanied by a 
marked change in tthe slope of the curves. This was true for 
both annuli and pipes, the change being more sharply defined in 
the latter case. The precision with which the break point could 
he determined froin pressure drop measurements was the same 
as the precision obtained in the present experiments. 
1-iscous flow velocity profiles, 
such as those shown in Figure 3, indicated that the radius of 
maximum velocity could he calculated by means of Equation 5 up 
t,o the wave point Reynolds number, NRea, of 700. Once this 
Reynolds number was exceeded, however, the whole top of the 
velocity curve suddenly flattened, making it difficult to ascertain 
whether the maximum point had shift,ed. Close examination of 
data obtained just above the wave point led to the conclusion 
that the niasinium ve!ocity had actually moved toward tlie radius 
indicated by Equat,ion 8. 
Radius of Maximum Velocity. 
Rn, ' R H 2 
Figure 6. Effect of Reynolds number 
on laminar f i lm thickness at outer walls 
of annuli 
Corrected to zero injector needle diameter 
To check this concluaion, careful explorations were made in 
the region of the maxiinurn point a t Reynolds numbers between 
the Tvave point and the appearance of the first disturbance eddy. 
Only bent-needle injectors were used and the position of the first 
spiral formation in the dye filament was determined as closely as 
possible. Since the spiral formation \vas not apparent a t Reyn- 
olds numbers just above the wave point, i t a a s rrasonable to 
assume that the onset of such flow would occur a t the point of 
384 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 41, No. 3 
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 
maximum local velocity. As shown in Table V, the first spirals 
were obiervable a t radii much closer to the values calculated from 
Equation 8 than from Equation 5. I t was therefore concluded 
that the radius of maximum velocity actually shifts toward the 
core in transition flow and probably does not return to the viscous 
evidence of normal as well as parallel flow. Such behavior em- 
phasizes the difficulty of obtaining satisfactory data in the vicinity 
of the inner boundary. 
Summary 
flow position until thezntjre Ytream is in fully turbulent motion. 
The extent of the shift is undoubtedly a function of the Reynolds 
number. As a first approximation, however, it can be assumed 
that the maximum velocity attains the position necessary to 
equalize the skin frictions on the inner and outer boundaries of 
the annular space. This, in effect, makes the annulus equivalent 
to two flat plates, which is ronsistent with the transition-zone 
friction data of Rothfus, Monrad, and Senecal. 
Table V. Position of Initial Spiral Formation within Annular 
Cross Sections 
Annulus 
A i Ai A2 
Nus2 1668 1565 1470 
Injector diameter, inch 0.0249 0.0277 0.0277 
Radius, r , a t which first spiral observed, inch 0,095-0.266 0.160 0.193 
r m from Equation 5, inch 0.319 0.319 0.334 
l'm from Equation 8, inch 0.162 0.162 0.189 
Laminar Film Thickness. It was found that the local fluid 
velocity a t the edge of the laminar region remained constant in 
a given annulus as the Reynolds number, NRen, was varied be- 
tween 700 and 2300. The effects of annulus size and fluid prop- 
erties were adequately accounted for by the general relationship 
An experimental study has been made of the behavior of dye 
filaments injected into water flowing isothermally through three 
pipes and five concentric horizontal annuli. 
At bulk Reynolds numbers less than about 1200 in pipes only 
viscous characteristics are detectable by means of such expel i- 
ments. The first observable stable deviation from viscous flow 
in pipes is an irregular waver occurring on the center line of the 
pipe-Le., the position of maximum local velocity-at a bulk 
Reynolds number of 1225 i 40. 
At Reynolds numbers between 1225 and 2100, the fluid in the 
central portion of a pipe takes up a sinuous motion while the 
fluid near the wall remains laminar. The thickness of the laminar 
film decreases with increased Reynolds number. At 2100 Reyn- 
olds number, the first disturbance eddy is cast off. The fre- 
quency with which disturbance eddies are formed increases with 
the Reynolds number. A t Reynolds numbers greater than about 
3000, the disturbance eddies appear to be stable. 
In the Reynolds number range from 1225 to 25,000, the local 
fluid velocity a t the '&&vaf the laminar film in a smooth pipe 
appears to obey the simple relationship 
tJ 
The corresponding distance, r!, from the center of the pipe to the 
edge of the laminar film is given by the equation 
which is the counterpart of Equation 9. 
(7) ~ is the viscous-flow quantity 
The velocity ratio, The 2xistence of a laminar film a t high turbulent Reynolds num- 
bers remains in doubt. Caution must therefore be used in 
extrapolating these equations to Reynolds numbers greater than 
25.000. 
4 R H 3 T 7 P . 
(lG) It is convenient to use the Reynolds number, - 1 xl 
At values of this 
tJ 
r; - r: - 27-2, In '2 
rm 
ri + rf - 2r~ 
dealing with flow through annular spaces. 
in which rm is calculated by means of Equation 5. Most of the 
data supporting Equation 15 were obtained in the region between 
the outer wall and theradius of maximum velocity. It was pos- 
sible to study only a few positions inside the maximum point, 
but the same relationship appeared to be valid over the entire 
section investigated. 
After correction to zero injector needle diameter, the laminar 
film thicknesses in the outer portion of the annuli were found to 
be represented satisfactorily by the function shown in Figure 6. 
The abscissa was chosen to be R,/RH, because the shearing stress 
distribution, 7 / 7 2 , is linear in this quantity. In unidirectional 
flow the local shearing stress is always linear in the hydraulic 
radius formed on that section of the fluid lying between the posi- 
tion of maximum velocity a t the point in question. 
Figure 6, like Figure 3, is, therefore, equivalent to a graph of 
bulk Reynolds number against the ratio of shearing stresses a t 
the film edge and the fluid boundary. The actual value of the 
radius of maximum velocity could not be measured accurately 
in the Reynolds number range under consideration. Conse- 
quently, i t was necessary to form Figure G on an arbitrary basis. 
Fortunately, when the value of rn obtained from Equation 5 was 
used in calculating R x f and RH$, the data were drawn into the 
single line shown on the graph. A final appraisal of Figure 6 
must await further experimental information about the radius, rm. 
I n the annuli with fine wire cores, i t was noted that once a 
disturbance eddy was cast off, crosscurrent motions occasionally 
carried the dye to the other side of the core. This observation 
was consistent with the finding of Mueller (8) , who concluded 
that turbulent-flow heat transfer coefficients a t fine cores showed 
parameter less than about 700, only viscous motion is observable 
by means of the present technique in annuli having radius ratios 
between 1.79 and 24.8. The first observable stable deviation 
from viscous behavior in such annuli takes the form of a waver 
occurring on the radius of maximum local fluid velocity a t a 
Reynolds number, of 700 i 50. KO reason has been 
found to explain the difference between the wave point Reynolds 
numbers in pipes and annuli. 
The development of full turbulence in annuli follows the same 
course as in pipes. Sinuous motion occurs in the main stream 
a t Reynolds numbers between 700 and 2200 to 2300, and the lam- 
inar film thickness decreases with increased Reynolds number. 
The first disturbance eddy is cast off a t 2200 to 2300 Reynolds 
number and the progression to full turbulence is characterized by 
more and more frequent formation of disturbance eddies, as in the 
case of the pipe. 
The edges of the laminar regions in annuli are marked by local 
fluid velocities, ut, given by the equation 
 RH VP 
P 
where ?$) is the ratio of maximum to bulk average velocities 
in fully viscous motion. The laminar film thickness in the vicin- 
ity of the outer wall can be correlated by simple empirical means. 
The present data are not sufficient to permit correlation of film 
thicknesses a t the core. 
At Reynolds numbers between $00 and 2300, the point of 
March 1955 I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y 385 
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT 
maximum velocity in an annulus appears to shift inward from its 
position in fully viscous and fully turbulent flow. The amount '' = average fts/sec. 
of the shift may well be enough to equalize the skin frictions 
on the inner and outer boundaries. The exact position of the 
maximum point in transition flow is not yet known precisely. 
served a t the position of maximum local velocity does not neces- 
sarily mean that initial turbulence is generated at that point. 
Nomenclature 
urn = maximum local fluid velocity, ft./sec. 
= fluid viscosity, lb./(sec.)(ft,) 
= fluid densit,!., lb,/cu. ft. 
= local shearing stress, poundals/pq. ft. 
= skin friction a t wall of pipe, poundalsjsq. ft. 
T 
TO 
7 2 = skin friction a t out,er boundary of annular space, The fact that the first deviation from viscous behavior is ob- 
TI skiil friction a t core of a n n u b PoundaWsq. f t . 
poundalsjsq. ft. 
Literature cited 
D, D, = diameter of pipe, ft. 
D, 
fL 
ATRe 
KRet 
= diameter of needle, ft. 
= Fanning friction factor, dimensionless 
= length of conduit over which pressure drop is measured, 
= bulk Reynolds number for pipes = ---> diinerisioriless 
4 R .va T i P 
= special bulk Reynolds number for annuli = -, 
Et. 
D V p 
P 
Ll 
dimensionless 
in fluid, ft. 
4pgo 
r 
= pressure drop due to fluid friction, poundals/sq. ft. 
= radius from geometrical center to point of measurement 
ro 
T, 
r2 
ri 
= inner radius of pipe, ft. 
= inner radius of annular space, ft. 
= outer radius of annular spx< 
= radius from geometrical center'^^ edge of laminar film, 
c i 
lb. 
rnL 
RH 
= radius from geometrical center to point of maximum 
= hydraulic radius of that portion of fluid lying between 
local fluid velocity, f t . 
T,, and r . f t . 
?2 - 7 2 
2r 
=- In in annuli 
= r / 2 in pipes 
= hydraulic radius of that portion of fluid lying between RHp 
r m and r2, f t . 
ri - r; 
2 ~ 2 
=- in annuli 
= ro/2 in pipes 
= local fluid velocity, ft.jsec. 
= local fluid velocity a t edge of laminar film, ft./sec. 
u 
u, 
(I) Carpenter, F. G., Colburn, AL P., Schoenborn, E. AI . , and 
Wurster, A, Trans. Am. Inst . Chem. Engrs., 42, I65 (1946). 
(2) Gazely, C., Jr . , Ph.D. thesis, TJniversity of Delaware, 1948; 
Dukler. -1. E., and Bergelin, 0. P., Chem. Eng. Progr. , 48, 
557 (1952). 
(3) Gibson, -1. IT.. Phil. illaa., 7, 15 (1933). 
(4) Knudsen, J. G., and Katz, D. L., Proc. Midwestern Conf. on 
Fluid Dynamics, 1st Conf., No. 2, 175 (1950). 
(5) Lamb. H.. "Hydrodynamics," 5th ed., p. 555, Cambridge Uni- 
versity Press, London, 1924; Lindgren, E. R., A p p l . Sci. Re- 
senrch, Sect. A , 4, KO. 4, 313 (1954). 
(6) Maurer, E., 2. Phvsik. 126, 522 (1939). 
(7) AIelrsyn, D.. and Stuart, J. T., Proc. Roy. SOC. ( L o n d o n ) , 208A, 
(8) I\Iueller, A. C., Trans. Am. Inat. C h e m Engrs., 38, 613 (1942). 
(9) Prengle, K. S., Ph.D. dissertation in chemical engineering, 
(10) Rothfus, R. R.. LIonrad, C. C., and Senecal, T. E., ISD. EXG. 
517 (1951). 
Carnegie Institute of Technology, Ma37 
CHEM., 42, 2511 (1950). 
(11) Rothfus, B. R.. and Prengle, R. S., Ibid. , 44, 1653 (1952). 
(12) Schiller, L., Proc. Intern. Congr. Appl. Mech., 3rd Congr., 
Stockholm, 1, 226 (1930). 
(13) Senecal, V. E., and Rothfus, R. R., C i ~ e m . E T I ~ . Progr., 49, 533 
(1953) 
RECEITED for review July 30, 1934. ACCEPTED November 8, 1964. 
Submitted by R. S. Prengle in partial fulfillinent of the requirements 
for the degree of doctor of science a t Carnegie Institute of Technology. 
Material supplementary t o this article has been deposited a3 Document No. 
4434 with the AD1 Auxiliary Publications Project, Photoduplication Service, 
Library of Congress, Washington 2 5 . D. C. A copy may be secured by citing 
the document number and by remitting 517.50 for photoprints or R5.60 for 
35-mm. microfilm. Ad\-ance payment is required. Make checks or money 
orders payable to Chief, Photoduplication Service, Library of Congress. 
M e t a l 4 1 m esistance Ther eters 
for Measuring Surface 
C. C. WINDING, L. TOPPER', AND B. v. BAUS2 
Cornell Universify, Ifhaca, N. Y. 
CCURATE average surface temperatures ate frequently re- A quired in heat transfer investigations. LIeasurement of true 
surface temperature is particularly necessary in determining film 
coefficients when the driving force is very small. For horizontal 
tubular exchangers where condensation or boiling occurs, surface 
temperatures generally vary both longitudinally and around the 
circumference,as vel1 as a i t h tube position in the bundle. I n 
such a case, point values of surface temperature may be inade- 
quate for the calculation of film coefficients. Nuclear boiling and 
dropwise condensation further reduce the accuracy of point 
values because of temperature variations a t the surface. 
Two techniques now in use for measuring true surface tempera- 
tures are radiation pyrometry and the optical methods of E. 
Schmidt (schlieren photography) and R. B. Kennard (interferom- 
1 Present address, The Johns Hopkins University, Baltimore 18, Md. 
3 Present address, E. I. du Pont de Kernours & Go., Inc., Wilmington, 
Del. 
etry) These method.: require that the observer see the surface 
and so they are not applicable to many heat transfer studies. 
Colburn and Hougen (6) have reviewed the conventional 
methods for surface temperature measurements. Most of them 
involve the use of embedded wire thermocouples which measure 
point values and are seldom positioned exactly at the surface. 
Flow patterns are frequently disturbed, and heat may be con- 
ducted t o or from the junction by lead wires. Different methods 
of inetalling the thermocouples do not give consistent results. 
Surfaces, such as condensers, which have a wide variatioii in 
temperatures, require the installation of many thermocouples to 
obtain a useful average value. 
Bendersky (4) used a nickel film deposited 011 a steel probe as a 
thermocouple. Xear the turn of the century, Callendar suggested 
that the average temperature of a metal tube might be derived 
from a measurement of its electrical resistance. Jeffrey ( 7 ) re- 
examined this proposal and %-as able to derive a relation between 
I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y Vol. 47, No. 3 386