Velocity Distribution and Fluid Friction in Smooth Concentric J
Annuli R. R. ROTHFUS, C. C. MONRAD, AND V. E. SENECAL Carnegie Institute of Technology, Pittaburgh, Pa.
T h e isothermal flow of air at room temperature through two smooth, horizontal, concentric annuli has been studied by measurement of point velocities and static pressure differentials. Data are presented for various Reynolds numbers in the viscous, transition, and lower turbulent flow ranges. Velocity distributions and pressure losses in viscous flow are in agreement with theoretical relationships. Both viscous and turbulent flow patterns are believed to exist simultaneously in the transition range. The radius of maximum velocity in fully developed turbulence appears to be predicted by viscous flow equa-
tions. Turbulent flow friction factors are affected by radius ratio and are higher than those calculated from hydraulic radius. The outer pipe common to both annuli has been operated without a core. Velocity profiles and pressure drops are in satisfactory agreement with data of other investigators. Friction factors in annuli have been correlated by the hydraulic radius concept applied to the portion of the fluid section outside the radiueof maximum velocity. Turbulent flow velocity profiles in the annuli have been correlated with open pipe data through a geometrical modification of the I&+, y + relationship.
UMEROUS investigations of fluid flow and transfer processes in annuli have been reported in the literature of the past half-century. Appreciable inconsistencies in the published data have made the suggested correlations uncertain, however, in spite of the wide variety of fluids and long ranges of operating variables covered by the investigations. One of the principal barriers to the analysis of the published data has been the almost complete absence of the velocity distribution measurements. Some general trends have been indicated in spite of specific differenced in the data of various investigators. The status of the literature is best summarized with respect to the type of flow existing in the annular space.
Lamb’s equation. Experimental friction factors obtained from pressure drop measurments show only rough agreement with the theoretical equation in the lower and middle diameter ratio ’ranges, and at larger values of rl/Tz the data are entirely inronsistent, (6, 22).
VISCOUS FLOW
Lamb (11) h a developed a theoretical expression for laminar flow velocity distribution under isothermal conditions:
The radius of maximum velocity computed from this equation i,q
TURBULENT F L O W
It can be shown from a simple force balance on the fluid that the pressure loss caused by fluid friction in any conduit is related to the skin friction by the equation
if the skin friction is the same at all points on the boundary surface, SL. For the case of annuli,
If the average skin friction weighted on the basis of peripheral areas is assumed to be exerted a t all points on the boundaries of an annular space, Apgo --
L
and the pressure drop caused by fluid friction is given by the equation
It seems reasonable to assume that the limiting conditiuns represented by such equations a t the extreme values of diameter ratio should be in agreement with flow in circular tubes at r l / n = 0 and flow between two flat plates a t r1ITZ 1. Fanning-type friction factors computed from Lamb’s equation actually reach their proper limiting values of 16~(/2 (TS r l ) Vp a t T ~ / T Z= 0 and 24p/2 ( T Z - r1) V p a t r1/r2 = 1. Friction factors in the diameter ratio range 0.8asingradius rat,io in the rarigr O.OBainfactors in the design and const,ruction of their experimental annuli. The principal points of variation are: Extreme differences in roughness of the inner and outer boundaries. Inadequate calming length upstream and downstream from the test section. Eccentricity of the cor? caused by improper centering or inadequate support. Excessive physical impediments in the fluid stream. The present apparatus was designed to minimize adverse effect,s of these factors.
The experimental apparatus was merely a series of horizontal conduits through which a measured quantity of room air could be made to flow under steady and e ntially isothermal conditions. The air was drawn into either of two Roots blowers, from which it was discharged to a sheet metal surge tank in the form of a 3foot cube. Leaving the tank, the air encountered a square-edged contractioii in passing to the annulus which consisted of an u p stream calming section 20 feet long, a Sfoot test section where vc.locities were measured, and a 10-foot downstream calming section. Between the annulus and the point of discharge from the system, the air passed successively through an lfl-inch radius ret,urn bend, a 21-foot calming length of 2-inch standard steel pipe,
TABLE I. Conduit Outer pipe Small core Large core Caljbration pipe Impact tubes
TITBE SPECIFICATIONS
Material Red brass pipe Yellow tubing brans Yellow brass tubing Red brass pipe Stainless steel hypodermic tubing
I.D., Inches 3,078
O.D.. Inches 3.500
Wall Thickness, Inch
Radius Ratio,
rdn ...
...
0.500
0.5
0.162
, . .
2.000
0.125
0.6*50
1 062
1.315
...
...
0.027
0.042
...
...
TABLE 11. CALMING LENGTHS Calming Section Annulus, upstream Annulus, downstream Orificd, upstream Orifice, downstream Calibration, upstream Calibration. downstream
Length, Feet 20 10
21 17 8 4
Equivalent Diameters Stnall core Large core Open pipe 223 78 93 112 39 46 ... 122 .. ... 95 , .
..
..
...
...
90 4.5
December 1950
INDUSTRIAL AND ENGINEERING CHEMISTRY
a standard flange-tap orifice meter, and a second calming section
17 feet long. For the calibration of the. impact tubes, an additional 12-foot length of 1-inch brass with necessary accessories was fitted to the discharge end o ? ! c standard apparatus. Important dimensions of the conduits are summarized in Tables I and 11. The rate of air flow to the annulus was controlled by a standard 2-inch brass globe valve supplemented by a bleed line to the a t mosphere containing a 1-inch by-pass control valve. The annuli were constructed in sections to facilitate centering of the cores, the section ends beinq alternately recessed and shouldered to reserve alignment and internal dimensions. As shown in Tab{ I, the same outer pi e was used for both annuli, but the core diameter was varied)fourfold. Preliminary calculations indicated that core supports were nccefrsary at about 3.5-foot intervals to prevent appreciable sagging. Consequently, three sets of supports were installed in each 10-foot calming section and two sets in the test section. Each set of su ports consisted of three separately adjustable members s p a c e f a t 120" intervals on the periphery of the outer pipe. In order to determine the effect of core support size on pressure drop measurements, supports having diameters of 0.036, 0.063,and 0.094 inch were used. The sup rts were made of steel drill rod which was shouldered on one endPand pressed into brass machine screws threaded into the outer pipe. The cores were ccwtered by means of precisely machined brass cradles which could be slipped into the outer pipe during assembly of the annulus. Average fluid temperatures in the annulus were estimated from the readings of calibrated thermocouples situated in the main stream a t each end of the annular section. Static ressure taps wwe situated at intervals along the outer pipe to apow preliminary measurement of entrance and exit effects on the pressure gradient. Midway through the downstream calming section, four taps installed in the same cros9 plane allowed measurement of transverse differences in static pressure. Air flow rates were measured by a standard, flange-tap, Foxh r o orifice meter installed between calming sections in the standard %inch steel pipe. Two standard orifice lates, one having a O.5OO-inch bore and the other a 1.250-inch were used in order to maintain a practical differential head over the flow range. Most of the pressure differences across the orifice were measured by simple U-tubc mercur or red oil manometers having millimeter scales. Small pressure Jfferentials across the orifice as well &s the point velocity pressures and friction heads were measured by a tilting-type micromanometer. The motion of the assembly necessary to re-establish an air-water interface a t a predetermined zero point after application of a pressure difference was measured by a micrometer. Pressure differences of O.OOO1 inch of water could be read directly on the instrument and differences of O.ooOo5 inch of water could be estimated with reasonable certainty. Three impact tubes (see Table I) were used to traverse the romplete annular section. Main stream velocities were obtained with a bent tube of the usual type, but velocities close to the walls were measured by separate straight tubes having rectangular side-wall openings. The tube used for measurement of velocities close to the core wall had a side o enin 0.015 inch wide and 0.029 inch long, while the openin in t i e tuge used for measuring velocities near the outer ipe wafi was 0.015 inch wide and 0.024 inch long. The ends of t i e straight impact tubes were sea1ed.b-y caps of aluminum foil 0.001 inch thick. Brass plugs were driven into the interior of the cores to revent leakage around the impact tubes and to aid in guiding tEe tubes through the cores. Details of the tubes and plugs are illustrated in Figure l. The traverse of the fluid section was accomplished by means of a micrometer feed mechanism which allowed the im act tube openings to be positioned with a precision of *0.001 in&.
kre,
EXPERIMENTAL PROCEDURE
.r
Impact tube calibrations were made in a 12-foot length of 1inch standard brass pipe installed in series with the usual flow system. Use of a 1-inch pipe allowed viscous flow of air to be attained a t velocities high enough for accurate measurement wit,hout undue effect of pipe wall curvature on the impact tube behavior. Calibrations of all thrce tubes were made a t two Reynolds numbers in the viscous flow range to establish whether the calibration curves were actually independent of Reynolds number as reported by Stanton, Marshall, and Bryant. Preliminary runs were also made to determine which of the static pressure taps were free from entrance and exit effects.
2513
After completion of the preliminary and calibration runs, the outer pipe was operated without a core. There were three principal reasons for this procedure: (1) to test for constant error in the apparatus by checking viscous flow drita against the parabolic velocity distribution law; (2) to ascertain the extent to which the calibrations obtained in viscous flow were i n d i d a t e d by turbulence; and (3)to establish a basis of comparison for data obtained in the annuli. Velocity profiles were determined a t two Reynolds numbers in the lower turbulent range and one Reynolds number in the viscous flow range. Pressure drop data were obtained over as long a Reynolds number range as possible with the availahle blowers and manometers. FOIL CAP 7
0942' O.D. DETAIL OF STRAIGHT IMPACT TUBES
FRONT
SIDE
-
ASSEMBLY SHOWING CORE PLUG AND IMPACT TUBES IN POSITION
NOT TO SCALE m,
Figure 1.
Installation of Impact Tubes
Complete velocity traverses were made at one. transition and two turbulent Reynolds numbers in the annulus with the smaller core. I n addition to two velocity runs in turbulent flow, one completely viscous flow profile was obtained in the annulus with the larger core to check the validity of Lamb's equation. One complete traverse waa repeated to check the reproducibility of the experimental results. Friction data were obtained over the entire accessible Reynolds number range in both annuli. Separate pressure drop runs were made over the entire range for each core support size. I n all cams the distance between the statio pressure taps was 16.59 feet. SOURCES OF DATA
Orifice coefficients for orifice Reynolds numbers above 10,OOO were obtained from the American Society of Mechanical Engineers (8). In lower ranges of flow, coefficients were taken from the best data of Ambrosius and Spink (1). Air viscosities were obtained from Keenan and Kaye (IO). Densities of air were calculated from the perfect gas law. The densities of liquid water were taken from a handbook (8). RESULTS
Impact Tube Calibrations. Main stream velocities were measlured with the bent impact tube a t Reynolds numbers of 1540 and 1750 in the standard 1-inch brass calibration pipe. Both velocity profiles could be extrapolated to zero velocity a t a point lying 0.006inch outside the observed interior pipe wall. Because no precedent could be found in the literature for the w u m p t i o n that wall effect was important in the range traversed by the bent tube, it was concluded that the error lay in the observed position of the impact tube opening. The velocity profiles were consequently translated 0.006 inch toward the center of the pipe, and the corrected data were compared with the parabolic distribution based on the same average fluid velocity. As indicated in Figure 2, the observed and theoretical velocities a t corresponding points in the fluid section were in close agreement. The bent impact
INDUSTRIAL AND ENGINEERING CHEMISTRY
2514
Vol. 42, No. 12
tube a t corresponding points. Because the main stream velocities were in substantial agreement with those of Stanton (16) and Nikuradse (15), a portion of the straight tube distribution w s raised to meet the main-stream points. It was assumed that thc straight impact tube, calibrated in viscous flow, retained it,s calibration in the region close to the wall where an approximately linear velocity profile was obtained. Therefore, only the data between this region and the outermost main-stream point werc shifted to smooth the profile. The straight tube points farthest from the pipe wall were raised to extrapolate the mainstream data in the region where the slopes on logarithmic coordinates w ( w similar. The remaining velocities, lying in the narrow huffer zone, were given an arbitrary correction approximately linear in the distance from the outer boundary of the uncorrected regioii near the wall. The data thus affectcrd lip in the region l O < y + < 3 0 on Figure 4.
0.02 I
0:
Ee
0.01
B
F INCHES FROM
PIPE
it
WALL
Figure 2. Comparison of Streamline FlowVelocity Data i n 1-Inch Standard Brass Pipe with Parabolic Profiles Based on Same Average Fluid Velocities
=
1
-
(;)*
l00,000
10*000
REYNOLDS NUMBER -N, Figure 3. Fanning Friction Factors i n 3.078-Inch Inside Diameter Brass Tube Operating without Core
tube was therefore assumed to have a coefficient o f unity in viscous flow. The straight impact tubes were calibrated in the manner of Stanton, Marshall, and Bryant. A series of point velocities was obtained with each tube at Reynolds numbers of about 1560and 1770 in the 1-inch pipe. From the parabolic distribution U Urn
0.003
(9)
the distance from the wall corresponding to each observed velocity was calculated. Thie was taken to he the “effective distance’] of the center of the impact tube opening from the wall. The calibration was summarized in a fmoothed plot of the effective distance against the observed distance from the wall to the center of the impact opening. Very close to the pipe wall, thcb effective distances greatly exceeded the observed distances, hut a t positions farther than 0.06 inch from the wall the corrections werc negligible. The straight tube and bent tube data were in agreement in the region where overlapping occurred. -4sindicated by Stanton, the effect of Reynolds number on the cahbrations wax negligible in the investigated range. Open Pipe Data. Pressure drop data were obtained between Reynolds numbers of 1300 and 70,000 in the 3-inch outer pipr operated without a core. As shown in Figure 3, Fanning friction factors computed from the experimental data were in agreement with the best line through Stanton’s smooth tube data for air in the turbulent range and Poiseuille’s law in the viscous range. The limits of the transition zone were clearly defined a t 2100 and 3700 Reynolds numbers, which was in agreement with numerous investigations of flow in smooth tubes. No constant error of sufficient magnitude to affect friction factor calculations appeared to exist in the apparatus] and the outer pipe acted as a smooth tube over the rather limited Reynolds number range investigated. The velocity profiles shown in Figure 4 were obtained a t Reynolds numbers of 6210 and 24,500 based on observed averagr linear velocities. At distances from the pipe wall greater than 0.1 inch, the straight impact tube appeared to be influenced by turbulence in such a way that the point velocities measured bv the tube lay somewhat be!ow the velocities measured b y the bent
Illustrates comparison with Poiseuille’s law and turbulent flow air curve of Stanton and Pannell (18)
A velocity traverse was also made a t a Reynolds number of 1740. Experimental precision in the viscous flow range was not high a t any point in the fluid cross section and consistent readings could not be obtained a t points closrr to the pipe wall than 0.3 inch. Data obtained farther out in the air stream indicated agreement with the parabolic distribution to *5%, Annular Data. Over-all friction factors of the Fanning type computed from pressure drop measurements in the annuli are shown in Figures 5 and 6. The friction factors have been plotted in the conventional manner against a Reynolds number containing the equivalent diameter based on total wetted perimeter. The hydraulic radius prediction] obtained directly from open pipe data, and the theoretical relationship for streamline flow have been included for purposes of comparison. The experimental friction factors in viscous flow were observed to approach agreement with the theoretical values a t low Rry25
20
* = l
I
a W &-
y
15
i2 2! to
5
0
I
IO FRICTION DISTANCE
io0 PARAMETER- y’
Figure 4. Generalized Correlation of Turbulent FlowVelocity Data i n 3.078-Inch Inside Diameter Tube Comparison with generally accepted relationships
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1950
2515
TABLE 111. RADIUSOF MAXIMUM VELOCITY ( m
NR. 21,600 5,670 1,820 1,390 i4,400 3,800 1,250
n/rr
0.162 0.162 0.162 0.162 0.650 0.660 0.650
- ri)/(n
Observed
- Lamb n)
0.427 0.424 0.365 0.408 0,482 0.472 0.480
0.424 0.424 0.424 0.424 0.480 0.480 0.480
TABLEIV. COMPARISONOF OBSERVED FRICTION FACTORS WITH T H O SCOMPUTED ~ FROM VELOCITY GRADIENTS OVERALL REYNOLDS NUMBEA-NR,
ri/n = 0.162
nolds numbers. The theoretical friction factor line was attained within experimental precision by the annulus with the 2-inch core. The “dip” characteristic of open pipes in the transition region waa not obtained in either annulus. The turbulent flow friction factors were considerably higher than predicted by the hydraulic radius concept, the deviation being greater in the annulus with the 0.5-inch core. The effect of core support diameter was not significant in the investigated flow range. Complete velocity profiles were obtained at Reynolds numbers of 1250,3800, and 14,400 in the annulus with the 2-inch core and a t 1820, 5670, and 21,600 in the annulus with the 0.5-inch core. In addition, a main-stream traverse was obtained in the latter annulus a t a Reynolds number of 1390. All Reynolds numbers were brtsed on the total equivalent diameter and the observed average velocities. The experimental profiles, adjusted in the same manner M the open pipe data, are presented in Figures 7 to 10. As shown in Table 111, the profiles obtained in completely tur-
bulent flow exhibited, within experimental accuracy, the same radius of maximum velocity aa calculated from Lamb’s velocity distribution in completely viscous flow. The profile obtained a t 1820 Reynolds number in the annulus with the 0.5-inch core, however, showed a radius of maximum velocity considerably less than the viscous flow value. The viscous flow traverse in the annulus with the %inch core indicated excdlent agreement with Lamb’s theoretical distribution. Table IV shows that it wa8 impossible to predict skin friction from the turbulent flow velocity data on the assumption of negligible eddy viscosity near the walls. DISCUSSION OF RESULTS
*
Table V shows a comparison of aver: age velocities measured by the orifice with those obtained by integration of the adjusted velocity profiles. The integrated velocities were generally higher than the observed values by amounts roughly proportional to the average. velocities. A maximum kxprimental error of about 2% was possible in the observed average velocities. The balance of the deviation probably resulted from (1) the effect of turbulence on mainstream velocity measurements and (2) overcorrection of the buffer layer data. It was not possible to weigh the relative importance of the two effeck because a r exact, method of impact tube calibration was not available. I n this regard, it was noted that agreement of the turbulent main-stream velocities in the open pipe
N R ~ 24,500 6,210 21,600 5,670 14,400 3.800
n/rr 0.000 O.Oo0 0.162 0.162 0.650 0 650
Figure 5. Over-all Fanning Friction Factors in Annulus with 0.5-Inch Core
TABLEV. ri/n
0.000 0.000 0.162 0.162 0.162 0.650 0.650 0.650
f (Obsd.)
f (Calcd.) 0 0062
0.0063 0.0091 0.0041 0.0106 0.0041 0.0088
0.0108 0 0075 0.0107 0.0078 0.0104
RELATIONSHIP BETWEEN OBSERVED ANIJ
INTEGRATED AVERAGE VEWCITIES
N R (Obsd.) ~ 24,500 6,210 21,600 5,670 1,820 14,400 3,809 1,260
V (Obsd.)
16.67 4 10 17.43 4.49 1.48 27.45 6.90 2.35
V (Int.) 17.17 4.15 18.22 4.70 1.48 28.40 7.30
yo Deviation
..
2.9 1.2 4.3
4.5 0.0 3.3 5.5 Negl.
w t h the data of other investigators (8, 16, 16) did not indicate that the measured velocities were necessarily correct. The point velocities close to the wall of the open pipe scattered considerably less than expected, in view of the low impact pressures in this region. Approximate agreement with the u+, y + rdationship near the wall indicated that the distance from the wall was a reaeonable parameter to use in the impact tube calibrations. The reason for failure to predict skin friction from the velocity gradient near the wall was not determined. Stanton, Marshal4 and Bryant also failed to obtain the proper skin frictions from their velocity measurements. The deviation appeared to be a direct function of the fluid velocity and therefore might have been caused by an effect of turbulence on the impact tube calibration or by the presence of appreciable eddy viscosity in the region of the pipe wall. The radii of maximum velocity in the annular spaces remained
0.04
’ 8 0.02 Z
Q
p:
0.006 600
8000
I,OOO
l0,000
20,000
OVERALL REYNOLDS NUMBER-N, Figure 6.
Over-all Fanning Friction Factors in Annulus with &Inch Core n/rr
0.650
2516
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. 42, No. 12
terns in the transition zone is believed to explain the lack of a dip in the plot of over-all friction factor against Reynolds number. Turbulent momentum exchange is also thought to account for the discrepancy between observed and theoretical friction factors in the upper viscous Reynolds number region. It is reasonable to postulate that the transition zone in the annulus with the 0.5inch core should be more extensive than that in the annulus with the larger core because of the larger difference in skin frictions a t the inner and outer boundaries. The experimental friction factors substantiate this, agreement with the theoretical viscous flow relationship being attained at a much higher Reynolds number in the annulus with the larger core. I
0.I
I
I
0.2 INCHES FROM
0.3 GORE
1 0.4 WALL
0.5
-
Figure 7. Comparison of Streamline FlowVelocity Data with Lamb’s Equation Based on Same Average Fluid Velocity Annulus with 2-inch core.
r1/r2
= 0.650
CORRELATIONS Experimental data on velocity distributions and friction head losses in two smooth annuli have been reported above, Correlations of the turbulent flow velocity profiles and friction factors are presented here. I n view of the relatively large numbers of smooth tube data available, open pipe relationships were used as the basis for correlating data obtained in t,he annuli. OPEN PIPE RELATIONSHIPS
Turbulent flow velocity distributions in smooth tubes of circular cross section have been correlated in t e r m of two parameters (3): 1L __
+ I-
u+ =
P
and
,.-
y+ =
Y
da
-- Y
If a laminar film actually exists in thc region ot the pipe wall, U f
INCHES
Figure 8.
FROM
CORE
at any point within the film, provided the film is so thin that the variation of shear stress through it can be neglected. Combination of the Prandtl form of the velocity deficiency law with the data of Nikuradse (16) yields the main-stream distribution
WALL
Turbulent Flow-Velocity Data
Annulus with 0.5-inch core, r i / m = 0.162
the same in turbulent and viscous flow within experimental accuracy. The only exception was the profile measured at 1820 Reynolds number in the annulus with the 0.5-inch core, Figure 10 shows that the maximum point lay nearer the core than the position predicted by the streamline flow equation. The velocities outside the maximum point agreed, however, with Lamb’s distribution calculated on the basis of the observed radius of maximum velocity. The profile inside the maximum point, on the other hand, showed no such agreement. It was postulated, therefore, that some degree of turbulence existed near the core where the shear stress was high and that viscous flow was simultaneously present in the outer part of the annular section. The maximum point of the profile a t 5670 Reynolds number in the same annulus was not disturlied from its position in fully developed turbulence, but the velocity pattern near the outer tube wall was flattened. It was believed that this traverse represented the u p p r limit of the transition zone and that the velocity distribution in the outer part of the annular space was tending toward the viscous form. The run was repeated and the point velocities wwe found to be reproducible, indicating that the flow pattern was stable. The simultaneous presence of viscous and turbulent flow pat-
(12)
= yf
u+ =
5.5
+ 2.5 In
(13)
$+
40
g 30
15
d
v)
\
c U’
\ ln
::20 8
10
4 2! 5
? r0 9 W
10
5!E
4
0 0
0.1
0.2 INCHES
Figure 9.
FROM
a3 GORE
0.4
0.5
WALL
Turbulent Flow-Velocity Data
Annulus with 2-inch core, n / r > = 0.650
0
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1950
25 17
Nikuradse's points show a Reynolds number effect on uf, y + coordinabs, and Equation 13 represents a n average line through data obtained over his whole Reynolds number range. The data a t any one Reynolds number generally lie above the average line near the center of the tube but cross it before the buffer layer is reached. The data of both Fage (7') and Stanton (16)show a similar effect. Suhtitution of Equations 10 and 11 in Equation 13 yields
$ -
=
5.5
+ 2.5 In
+i .
which can be rearranged to give
1
4%
-=z
=
4.07loglo
[ ( y4 )51-
0.40 0
But the von K k m h friction factor expression (5), which correlates smooth tube date extremely well, is 1
~ - 0.40 77 = 4.0 logio ( N R d)
0.4 INCHES
a6 FROM CORE
0.8 WALL
1.0
1.2
Figure 10. Transition Flow-Velocity Profile Annulus with 0.5-inch core, r l / m = 0.162 Illustrates shift in m a x i m u m point and comparison with Lamb's equation
(16)
It is seen, therefore, that the usual plot of Fanning friction factor against Reynolds number can be used to predict main-stream velocity profiles iff is replaced by 2rO/pu2and N Rby ~ 8yu/v. Equation 12 indicates t h a t if a laminar film exists,
I n t e r m of an average skin friction applied over the entire wchtted perimeter of an annulus, Equation 19 can be written
. r
By rearrangement, Equation 17 becomes 23 =
E
pu2
8p4
which is analogous to f = 16/Nylk,the relationship for streamline flow. Figure 11 shows the experimental open pipe data plotted in the suggested manner. FLUID FRICTION IN ANNULI
When applied to the annular section outside the radius of maximum velocity, Equation 19 becomes
and by combination of Equation 20 and 21, 72
=
r,2 - r: &r2 rl> low
-
In terms of the corresponding Fanning friction factors based on the over-all average velocity, Equations 15 and 16 suggest that turbulent flow velocity distributions in conduits of noncircular cross section might be correlated by 0.6 similar relationships. To make this possible, the open pipe friction factor correlation would have to apply over some portion of the boundary area. h pointed out previously, observed over-all friction factors in the exQI perimental annuli were considerably "Q : higher than predicted by the hy\ draulic radius based on the total r-0 01 wetted perimeter. Hydraulic radius might, however, be expected to apply over any portion of ai1 annulus acted on by a uniform skin 0.01 friction. The hydraulic radius concept was consequently applied to the part of I I I , t , I I I111111 I I11111 I IIllrppl 0.00320 I I I I I 111 the fluid cross section lying outside 100 w WQ0 woo0 the radius of maximum velocity. For 8PP'P any section influenced by a uniFigure 11. Correlation of Point Velocities in 3.078-Inch Inside Diameter form skin friction, a force balance Brass Tube by Modified Friction Factor Plot yields Illustrates comparison with Poineuills'n law and friction curve o f S t a n t o n a n d Pannell (le)
proximate agreement with the predicted skin friction ratios (Table VI). The absolute values of the gradients, however, could not be obtained from pressure drop measurements by application of the! laminar film hypothesis.
0 04
-c1
0.162 0.162 0.650 0.650
2 0.005 600
p00
3poo 2[(r2*-r-*)/rp]
l0,Ooo
31,600 5,670 14,400
1.85 2.05 1.02 1.02
3.800
2.03 2.03 1.16 1.16
20,Ooo
(vp/p)
Figure 12. Correlation of Friction Factors,fi, in Annuli By hydraulic rsdius concept applied outaide radius of m a x i m u m velocity
The hydraulic radius concept indicates that f i should be related to the Reynolds number
by the usual smooth tube correlation in the turbulent range. Figure 12 shows the friction factors a t the outer surface to be in close agreement with the hydraulic radius concept. Limited data obtained from heat transfer by Monrad and Pelton (IS)a p pear to confirm this result. It seemed reasonable, therefore, to compare annular velocity profiles with those in a pipe of radius (r; r i ) / r 2 operating a t a skin friction equal to r2 with the same fluid. By analogy with Equation 15, the parameters of the velocity correlation were choRen to be
-
It appeared in the light of these observations that the distance parameter relating an annulus and its equivalent pipe in full turbulence might be developed from a consideration of viscous flow relationships. Lamb's equation can be written in the form'
The annular distribution is parabolic with respect to the group
If this group is designated by R, it is seen that the velocity a t radius r in the annulus is equal to the velocity a t distance R from the center of a pipe of radius ( r i r i ) / r 2operating a t the same preasure gradient and with a fluid of the same viscosity. By analogy with the open pipe, a distance parameter, Y , can be defined such that
-
This quantity is related to the distance variable, y, in an open pipe and possesses similar properties. For example, if a modified shear stress, TR, is defhed as where Y was a linear unit developed in the manner outlined in the ollowing section. VELOCITY DISTRIBUTION IN ANNULI
du 7R =
in viscous flow, i t is easily shown that
Any value of the point velocity, except the maximum, occurs a t two different radii in an annular section. It was observed that equal velocities in the experimental turbulent flow profiles occurred a t the same radii as predicted by Lamb's viscous flow distribution, within the limits of experimental accuracy. This was true not only in the main stream but also close to the boundary surfaces. The distribution of shearing stress in an annulus, regardless of the type of flow, is given by the equation
The modified shear pattern is therefore linear in both R and Y by virtue of Equation 28. The value of TR at. the boundaries of the annulus can be found by evaluating R a t r = r l and r = r2. The resultant modified skin friction, T ~ R ,has the same value at both limits of r arid is given by the equation
(25)
and the corresponding ratio of skin frictions a t the inner and outer walls is
It has been reported in the previous section that the experimental velocity profiles in entirely turbulent flow exhibited the same radii of maximum velocity as Lamb's viscous flow distributions. The wholly turbulent skin friction ratios were predictable, therefore, from the viscous flow relationship. It was observed that the ratios of the velocity gradients a t the walls were in a p
It is apparent that the modified skin friction is equal to that in a pipe of radius (rg - r i ) / r 2 operating a t the same pressure gradient as the annulus and with a fluid of the same viscosity. The velocity and shear patterns in the annulus have therefore been transformed to those of an equivalent pipe in viscous flow through the use of the modified distance variable, Y . However, du/dr in the annulus reaches zero a t r = rm but du/dY retains a value greater than zero a t this radius, The modified annulus therefore corresponds to only the outer portion of the equivalent pipe. On the modified basis, both the core and outer tube of the annulus correspond to the wall of the equivalent pipe, and the radius of maximum velocity corresponds to radius R evaluated a t
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1950
20
E
I5
5
q/5=0.162
OBSERVED N,
-21,WO
‘ a
2519
influenced by wall proximity a4 a r s l l as velocity, the agreement with open pipe data is not as closc. Lack of smooth tube data in the buffer region at low turbulent Reynolds numbers makes the comparison somewhat uncertaiu. There is also considerable question as to whether the differericev we real or apparent, because the accuracy of the impact tutw calibration may be low in the buffer layer. A real difference is indicated, however, by the fact that the velocity data on either side of the maximum point in earh itnnulus yield the same line on the modified coordinates. It can be concluded that main-stream velocities in annuli can be predicted accurately from the smooth tube u+,y relationships if the coordinates are modified as suggested. The buffer layer, which occupies only about 10%of the fluid cross section, is not aa well correlated. Provided the flow is fully turbulent, velocities inside and outside the radius of maximum velocity follow a single relationship over the entire fluid section. The recommended correlation should be used with caution a t Reynolds nurfibers and radius ratios outside the experimrntal range until verified by more clstensive data. +
0
0
Do
50
Y’
Figure 13.
I50 200 (=YVlqnh)
250
aoo
Correlation of Turbulent Flow-Velocity Data in Annuli
B y modified generalized correlation for amooth tuber
r = rm. The modified velocity distributions both inside and outside rmin the annulus follow a common parabolic profile in the equivalent pipe. On the assumption that the Y developed for viscous flow might prove to be a satisfactory parameter for the correlation of turbulent velocity distributions, a modified u+, y + relationship as constructed as suggested above. Figure 13 shows the correlation thus obtained using values of average velocity based on the integration of the measured point velocities over the entire crosF, section. I~ISCUSSION
It might be expected that the hydraulic radius concept applied inside the radius of maximum velocity would yield a proper friction factor, f i . It is readily seen, however, that such is not the case, in spite of uniform skin frirtion over the entire periphery of the core. Agreement of the outside friction factor, fi, with hydraulic radius in that part of the fluid cross section must, therefore, be viewed only as an experimentally determined fact. As previously mentioned, the smooth tube data of other investigators show a Reynolds number effect on the generalized u + , y + correlation. The magnitude of the variation appears to be greater a t low turbulent Reynolds numbers than at very high ones; consequently, the Reynolds number range of the present investigation is particularlv affeckd. The deviations of the annular, main-stream data from Nikuradse’s average line in Figure 13 are in the same direction as those obtained by Fagr at comparable Reynolds numbers in smooth tubes. The shear stress at any point in a turbulent fluid can be related to the velocity gradient through an eddy viscosity, e, defined by the equation 7
= -(p
+
€)%
For an annulus, combination of Equations 25 and 32 gives (33)
The variables can be separated if E is a function of u only, Apgo/2L being constant for a particular velocity profile. Upon integration between the limits of r2 and r, the right-hand side of Equation 33 yields the radius function of Equation 27. Because Ap/L is the same in the annulus and equivalent pipe, agreement between annular and smooth tube data on the modified u+, g+ coordinates might be expected in regions of the fluid where the eddy viscosities are similar functions of velocity alone. This appears to be the case in the main stream, where turbulence is relatively unrestricted and very closc to the walls where eddy viscosity is small. ‘In the buffer layer where eddy viscosity may be
SUMMARY The iwthermal flow of air a t room temprature through two smooth, horizontal annuli of widely different radius ratios has been investigated with regard to velocity distribution and static pressure drop. Point velocities were measured by small impact tubes calibrated in streamline flow. Adjustments were made to offset the effect of turbulence on the calibrations. Preliminary experimenb in the outer pipe common to both annuli showed the pressure drop to be in agreement with published smooth tube data. Velocit? data followed generally accepted correlations in the lower turbulent range and the paraholic law in streamline flow. Lamb’s equation for isothermal streamline flow-velocity distribution in annuli was confirmed with experimental accuracy by velocity data. The radius of maximum velocity in fully developed turbulent flow appeared to be the same ILR predicted by Lamb’s equation for entirely viscous flow. In the transition zone, the maximum point shifted toward the core, Velocity profiles in the transition range indicated that some degree o f turbulence existed inside the radius of maximum velocity, even though the rest of the fluid was in viscous motion at the same time. This was thought to explain the absence of a pronounced dip in the curve of over-all Fanning friction factor against over-all rteynolds number. The extent of the transition region was a function of the radius ratio, being longer in the annulus with the smaller rl/re value. The onset of turbulence near the core was’thought to be related to the ratio of skin frictions a t the inner and outer walls. Over-all friction factors computed from turbulent flow pressure drop measurements were appreciably higher then those predicted by the hydraulic radius concept. The larger deviation was obtained in the annulus with the smaller rl/rz value. It has been established that the main-stream and “film” velocities in smooth tubes of circular cross section under fully turbulent conditions can be predicted from ordinary plot8 of Fanning friction factor against Reynolds number by using coordinates 2r,/pu2 and 8yu/v in place of 5 and N R ~ . The static pressure drops caused by fluid friction in the annuli have been correlated with smooth tube pressure losses by applying the hydraulic radius concept between the radius of maximum velocity and the wall of the outer tube. The friction factor, 2 r ~ / p V 2has , been shown to be related to the Reynolds number, 2(r; r:)V/rp, by the smooth tube correlation off and NR*in fully turbulent flow. By analogy, the annuli have h e n modified through a distance parameter to equivalence with a pipe of radius (rg r;):)T2
-
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
2520 operating a t the skin friction distarice parameter
TL
with the same fluid.
Using the
y
= distance from boundary surface, feet
y+
=
e p
=
= =
Y
a modified u + , I / + relationship has kweti evolved to correlate the groups
Fully turbulent main-stream velocities in the annuli can be predicted satisfactorily from open pipe relationships on the modified u+,y + correlation. Velocities in the narrow buffer layer are somewhat uncertain, however, because of limited smooth tube data in this region. Velocities on both sides of the radius of maximum velocity are equally well correlated over the entire fluid section. Until further data are obtained, caution should b e exercised in extrapolating the velocity correlation beyond the present experimental ranges of Reynolds number and radius ratio,
The authors wish to thank Robert Ritzmann for his aid in the early phases of the investigation and the Allied Chemical and Dye Corporation for fellowship assistance. NOMENCLATURE
f go
SeC.)
L = axial length of conduit over which A p is measured, feet N R ~= Reynolds number = 4RHV/Y, diniensionless A p = static pre,ssure drop caused b y fluid friction, Ib. force per sq. foot Apgo = static pressure drop caused by fluid friction, poundals/ sq. foot R = distance parnnieter =
RH
=
r
=
rm
=
g =-
U+ = u
=
;r:>2
fiz,
Urn
=
=
Y y+
= distance parameter =
v
=
=
(91’
+ +
- r,’ r2 2rz In 12, feet hydraulic radius, feet radial distance froni geomrtrical center to a point in fluid stream, feet radius of maximum velocity i n an annulus, feet inner radius of pipe, feet peripheral area of conduit, sq. fpet diInensionless modified velocity parameter = u/ local fluid velocity, feet per second maximum local fluid velocity, feet persecond velocity parameter for pipes = U / ~ T ~ / dimensionless P , averagp fluid velocity, feet per second
u+
modified friction dimensionless
T T ~ .
T~
=
TR
=
T ~ R
=
6
=
+
Subscripts 1 designates core wall or region inside radius of mafirnun1 velocity. 2 designates outer tube wall or region outside radius of maximum velocity. LITERATURE CITED
Ambrosius, E. E., and Spink, L. K., Tram. Am. SOC.Mech. Am. SOC.Mech. Engrs., A.S.M.E. Research Committee, “Fluid Meters,” 4th ed., Part 11, New York, 1937. Bakhmeteff, B. A., “Mechanics of Turbulent Flow,” Princeton, Princeton University Press, 1936. Carpenter, F. G.,Colburn, A. P., Schoenborn, E. M., and Wurster, A., Trans. A m . Inst. Chem. Engrs., 42, 165 (1946). Chen, C. Y.,Hawkins, G. A., and Solberg, H. L., Trans. Am. SOC.Mech. Engrs., 68,99 (1946).
= area of fluid cross section, sq. feet = Fanning-type friction factor = 2 l p g 0 R a / p V 2 L , dimensionless = conversion factor = 32.2(1b. mass)(ft.)/(lb. force)(sq.
[(G
= = =
p
friction distance parameter for pipes = Y ~ T ~ / u dimensionless eddy viscosit Ib. mass/(sec.)(feet) absolute fluiptiscosity, Ib. mass/(sec.)(feet) kinematic viscosity = b / p , sq. feet per second fliiid densit), Ih. mass per cu. foot local shear stress in fluid, poundals per sq. foot average skin friction weighted on basis of peripheral areas, poundals per sq. foot skin friction in open pipe, poundals per sq. foot du modified local shear stress = ( p E) dppoundals per sq. foot modified skin frict>ion,poundals per sq. foot function
Engrs., 69,8 , 805 (1947).
ACKNOWLEDGMENT
A
Vol. 42, No. 19
- R, feet (‘ : i “> distance parameter --
=
Y~T&/u,
Davis, E. S.,Zbid., 65,755 (1943). Fage, A., Phil. Mag., 21,80 (1936). Hodgman, C. D.,and Holmes, H. L., “Handbook of Chemistry and Physics,” 22nd ed., p. 1226, Cleveland, Chemical Rubber Publishing Co., 1937. Jordan, H. P., Proc. Znst. Mech. Engrs. ( L c m h ) , 73, 1317 (1909).
Keenan, J. H., and Kayc, J., “Thermodynamic Properties of Air,” New York, John Wiley & Sons, 1945. Lamb, H., “Hydrodynamics,” 5th ed., p. 555, London, Cambridge University Press, 1924. Mikrjukov, V., J. Tech. Phys. (U.S.S.R.), 4, 961 (1937). Monrad, C. C., and Pelton, J. F., Trans. Am. Inst. Chem. Engrs., 38,593 (1942).
Mueller, A. C., Zbid., 38,613 (1942). Nikuradse, J., 8.D. I . Forschungsheft, 356, 1 (1932). Stanton, T.E.,Proc. Roy.SOC.(London),A85, 366 (1911). Stanton, T. E.,Marshall, D., and Bryant, C . N., Zbid., A97, 413 (1920).
Stanton, T. E., and Pannell, J. R., Trans. Roy. SOC.(London), A214, 199 (1944).
Tomotika, S., and Imai, I., Rept. Aeronaut. Research Znst.,Tokyo Z m p . Univ., 180,14,299 (1939). Wiegand, J. H., Trans. Am. Inst. Chem. Engrs., 41, 147 (1945). Wiegand, J. H.. and Baker, E. M., Ibid., 38,569 (1942). RECEIVED M a y 18, 1950. Submitted by R. R . Rothfus in partial fulfillment of the requirements for the degree of doctor of science a t Carnegie Institute of Technology. Original data and calibrations presentea in thesia. “Velocity Distribution a n d Fluid Friotion in Concentric Annuli.” by’ R. R. Rothfus available on interlibrary loan from Carnenie Institute of Technology, Pittsburgh 13, Pa.
* * * * * Numerous contributions toward solution of some of the urgent problems of air pollution will be made on December 28 and 29, 1950, when the A.C.S. Division of Industrial and Engineering Chemistry meets for its 17th Christmas Symposium-subject this year is “dispersions in gases.” The symposium will include papers on the composition and generation of smogs, particle size distribution, the transfer and collection of aerosols, and the performance characteristics of filters. The program will be presented a t Johns IIopkins University in Baltimore, Md.
