Flow of liquids in pipes of circular and annular cross-sections

HILLINOI SUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

PRODUCTION NOTE

University of Illinois atUrbana-Champaign Library

Large-scale Digitization Project, 2007.

UNIVERSITY OF ILLINOIS BULLETINISsUED W MBLT

Vol. XXVIII March 17, 1931 No. 29

[Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, underthe act of August 24, 1912. Acceptance for mailing at the special rate of postage provided

for m section 1103, Act of October 3, 1917, authorized July 31, 1918.]

FLOW OF LIQUIDS IN PIPES OF CIRCULARAND ANNULAR CROSS-SECTIONS

BY

ALONZO P. KRATZHORACE J. MACINTIRE

RICHARD E. GOULD

BULLETIN No. 222

ENGINEERING EXPERIMENT STATIONPoarIa BY T•x UNIxvBaITr or IuauoIs, UsBB

PaBIC: E'srnN ENTS

T HE Engineering Experiment Station was established by actof the Board of Trustees of the University of Illinois on De-cember 8, 1903. It is the purpose of the Station to conduct

investigations and make studies of importance to the engineering,manufacturing, railway, mining, and other industrial interests of the

State.

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THE ENGINEERING EXPERIMENT STATION,

SUNIVERSITY OF ILLINOIS,iURBANA, ILLINOIS

'**' * " ' ' " ' . , . ' ' ' * " * ^ .i ' ' '' , '

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UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

BULLETIN No. 222 MARCH, 1931


BY

ALONZO P. KRATZRESEARCH PROFESSOR, ENGINEERING EXPERIMENT STATION

HORACE J. MACINTIREASSOCIATE PROFESSOR, DEPARTMENT OF MECHANICAL ENGINEERING

RICHARD E. GOULDRESEARCH ASSOCIATE, ENGINEERING EXPERIMENT STATION

ENGINEERING EXPERIMENT STATIONPUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA

00010 UNIV ERS3IT1000 II 10 *'71 II PRESS Il

CONTENTS

I. INTRODUCTION . . . . .

1. Preliminary Statement2. Objects of Investigation3. Acknowledgments.

II. DESCRIPTION OF APPARATUS

4. Experimental Pipe Lines5. Annular Sections6. Elbows ..... .7. Pressure Measurement8. Temperature Measurement9. Velocity Measurement

PAGE

S .. . 5S . . . 5S . . . 6S . . . 6

S . . . 6

6S . 7

78

S . 9. .. . 10

III. METHOD OF CONDUCTING TESTS ..

10. Selection of Test Conditions .11. Control of Test Conditions

IV. CALCULATIONS . . . . . . . . .

12. Reynolds' Number . . . . .13. Friction Factor for Standard Pipe14. Friction Factor for Annular Sections15. Head Lost in Elbows . . . . . .

V. RESULTS OF TESTS . . . . . . .

16. Standard Wrought-Iron Pipe. .17. Channels with Annular Cross-Sections .18. Elbows . . . . . . . . . .

VI. APPLICATIONS OF DATA . . . . . . .

19. Statement of Problem. . . . . .20. Solution of Problem . . . . . .

VII. CONCLUSIONS . . . . . . . . . .21. Summary and Conclusions . . . .

10S 10

121213

131318

20

242424

2626

LIST OF FIGURESNO. PAGE

1. Schematic Plan of Apparatus for Friction in Standard Pipe . . . . . 62. Schematic Plan of Apparatus for Friction in Channels of Annular Cross-Section 83. Schematic Plan of Apparatus for Friction in 90-Deg. Elbows . . .. 94. Reciprocal of Kinematic Viscosity for Calcium Chloride Brine and Water . 115. Data and Curve Showing Variation of Friction Factor with Reynolds'

Number for Liquids Flowing in Wrought-Iron Pipe . . . . .. 146. Curve for Friction Factors for Liquids Flowing in Channels of Annular

Cross-Section . . . . . . . . . . . . . . . 197. Curves for Equivalent Length of l-in. and 2-in. Elbows . . . .. 238. Curves for Head Lost per Velocity Head in l-in. and 2-in. Elbows . . 23

LIST OF TABLES

1. Dimensions of Annular Sections . . . . . . . . . . . . 72. Principal Results of Tests on Flow of Liquids in 2-in. Wrought-Iron Pipe 153. Principal Results of Tests on Flow of Liquids in Annular Section No. 1 164. Principal Results of Tests on Flow of Liquids in Annular Section No. 2 175. Principal Results of Tests on Flow of Liquids in Annular Section No. 3 186. Principal Results of Tests on Flow of Liquids in 1 -in. Cast-Iron 90-Deg.

Elbows.. .. . . . . . . . . . . . . . . . . . 217. Principal Results of Tests on Flow of Liquids in 2-in. Cast-Iron 90-Deg.

Elbows . . . .. . ... ........ .. . 22


I. INTRODUCTION1. Preliminary Statement.-This bulletin constitutes a report on

the continuation of the work the results of which were published in En-gineering Experiment Station Bulletin No. 182.*

In Bulletin No. 182 a brief discussion of the general theory ofthe flow of fluids was given, and it was indicated that the loss ofhead produced by friction could be calculated from the Darcy formula,or some modification of the Darcy formula:

Iv2

h f J (1)2gm

in which h = loss of head due to friction, in feet of fluid flowingf= friction factor1 = effective length of pipe, in feetv = average fluid velocity, in feet per second

dm= mean hydraulic radius, in feet (m = when the fluidfills a pipe of circular cross-section)

d = inside diameter of pipe, in feetg = acceleration of gravity, in feet per second per second

The friction factor f, as defined by Equation (1), depends uponthe roughness of the rubbing surface of the pipe, and varies with thehydraulic mean radius, the velocity of flow, and the density and vis-cosity of the fluid. Hence in studies of friction loss in pipes it hasbecome customary to correlate the friction factor f with the dimension-less ratio designated as Reynolds' number, involving the various fac-tors influencing the numerical value of f, or

S dv 4mv (2)R . (2)v v

in which R = Reynolds' numberd = inside diameter of pipe, in feet

m = mean hydraulic radius, in feetv = average fluid velocity, in feet per secondv = kinematic viscosity (or absolute viscosity pt divided by

density p), in square feet per second.

*"Flow of Brine in Pipes," Univ. of Ill. Eng. Exp. Sta. Bul. 182, 1928.

ILLINOIS ENGINEERING EXPERIMENT STATION

FIG. 1. SCHEMATIC PLAN OF APPARATUS FOR FRICTION IN STANDARD PIPE

2. Objects of Investigation.-The objects of this investigation were:

(a) To determine the relation between the friction factor and

Reynolds' number for two sizes of standard wrought-iron pipe, with

two fluids, water and commercial calcium chloride brine.

(b) To determine the relation between the friction factor and

Reynolds' number for channels of annular cross-section, with waterand commercial calcium chloride brine.

(c) To determine the head loss resulting from the use of standardcast-iron elbows in pipe lines conveying commercial calcium chloride

brine.

3. Acknowledgments.-This investigation has been part of the work

of the Engineering Experiment Station of the University of Illinois, of

which DEAN M. S. KETCHUM is the director, and of the Department

of Mechanical Engineering of which PROF. A. C. WILLARD is the head.

Acknowledgment is also due to MR. M. I. LEVY for assistance rendered

in the tests on cast-iron elbows.

II. DESCRIPTION OF APPARATUS

4. Experimental Pipe Lines.-The pipe line used for the tests on

the 1¼/-in. pipe has been described in Bulletin No. 182. A similar

arrangement was used for the tests on the 2-in. pipe as shown in Fig.

1. The 2-in. pipe consisted of standard wrought-iron pipe having an

actual internal diameter of 2.08 in. The section through which the

friction pressure drop was measured consisted of two parallel lengths

of 53.5 ft. each, connected by a 2.12-ft. radius return bend of the

ICLVIII~ICil YTIUII~

FLOW OF LIQUIDS IN PIPES

same pipe and suspended in a horizontal plane, giving a total lengthof 113.7 ft. Corrections were made for the additional friction in thereturn bend as discussed in Section 13. The test section, includingthe entrance and exit lengths of pipe, was made up of 20-ft. lengths,the ends of which were sawed square. After the burrs were removed,the lengths were carefully aligned and the joints butt-welded. Theinner surface of the pipe at these joints was examined after the testswere completed, and it was found to be free from any exceptionalroughness. This examination also proved that the piezometer holeswere free from burrs and that the surface of the pipe was not appre-ciably corroded.

5. Annular Sections.-The apparatus used for the tests on the an-nular sections was similar to that used for the single pipe, except thatthe annular sections were substituted as shown in Fig. 2, and pro-vision was made for the use of a constant head tank. The sectionsthemselves were three in number, and the outer casing for each sectionconsisted of a standard 2-in. wrought-iron pipe, having a total lengthof 23.0 ft. The piezometer holes were placed in the outside casings,the distance between holes in each case being 10.0 ft. Each sectionhad an internal core with conical plugs at the ends. The core wasextended approximately 4.5 ft. beyond the piezometer holes, and wassupported concentrically with the casing by means of knife edges orset screws as shown in Fig. 2.

The sections were designated as Nos. 1, 2, and 3 and the cores con-sisted of standard 1¼-in., 1-in., and %-in. pipes, respectively. Thecomplete dimensions are given in Table 1.

6. Elbows.-The apparatus used for the tests on elbows was alsosimilar to that used for the single pipe except that ten 90-deg. elbowswere substituted as shown in Fig. 3. In the case of the 2-in. elbows,ten elbows were connected by lengths of 2-in. pipe and were spaced5 ft. between centers. The ten 1%-in. elbows were connected by

TABLE 1DIMENSIONS OF ANNULAR SECTIONS

Annular Section Number......................... 1 2 3

Inside Diameter of Casing, in ft.................. 0.1717 0.1723 0.1717

Outside Diameter of Core, in ft .................. 0.1383 0.1083 0.0883

Mean Hydraulic Radius, in ft ................... 0.00833 0.01604 0.02085


Cet/nergi Screw,

5ecf/,o7 A-A Secf/or B-B

Ar4271-141r Sclf/lw7 D9e/i/as

FIG. 2. SCHEMATIC PLAN OF APPARATUS FOR FRICTION IN CHANNELSOF ANNULAR CROSS-SECTION

lengths of 1%-in. pipe and were spaced 3 ft. between centers. Theends of the connecting pipes were sawed square and the burrs wereremoved. In making up the joints, all pipes were screwed equal dis-tances into the elbows.

The piezometers in each case were located as shown in Fig. 3, sothat the measured pressure loss included the loss in the elbows andthe connecting piping. The calculated loss in the straight pipe wasthen deducted as indicated in Section 15. The details of the elbowsare also shown in Fig. 3.

7. Pressure Measurement.-All pressure losses were measured bymeans of liquid manometers. Three types were used, adapting the


Center Ao Cen/er /fo'r--/'f"P/oe- 3'-"Z",'Pe- 5o"----411

IFaIce o Face g r-l"/pie-34"--

2",'oe- 5 --

Dete7/s of Con ect//Ž P/ves

P//71 of App0Ora;'fu ,-in. Casf /r-n7 E/bow

FIG. 3. SCHEMATIC PLAN OF APPARATUS FOR FRICTION IN 90-DEG. ELBOWS

liquids to the magnitude of the differential pressures to be measured,and all pressures were converted to terms of feet of the fluid flowingfor the final record. All manometers were connected to piezometerrings, the details of which are shown in Fig. 1, and were so arrangedthat any air which collected could be easily expelled.

8. Temperature Measurement.--The temperatures of the fluid en-tering into and discharging from the test section were obtained bymeans of two calibrated thermocouples. Each thermocouple was madeof No. 22 B. & S. gage copper-constantan wire immersed in oil in a1 -in. glass tube, which was inserted a distance of 4 in. directly in theflowing fluid at positions indicated in Figs. 1, 2, and 3. All pipe lines

z-i//2 C7'st /Iron E2 bow


were lagged with a layer of 11/2 in. of hair felt, in order to reduce the

temperature drop to a minimum.

9. Velocity Measurement.-The rate of flow of the fluid was deter-

mined by observing the time required to discharge a given weight of

fluid. The auxiliary apparatus used for this purpose is shown in Figs.

1, 2, and 3.

III. METHOD OF CONDUCTING TESTS

10. Selection of Test Conditions.-For the purpose of this investi-

gation the temperature and velocity of the fluid were regarded as in-

dependent variables, and were varied over as wide a range as was

found practical. The two fluids used were water and solutions of com-

mercial calcium chloride. A typical analysis, on the dry basis, of thecalcium chloride used is as follows:

CaCl ................. ........... . .. ............... . 98.09

NaCl .................. ................................ . 1.68

CaSO.................. ...... . ............... 0.08Ca (OH), . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . .. . .......... 0.09Insol. in HzO ....................................... 0.06

100.00

A test series consisted of a number of groups, each group run witha constant temperature and a number of different velocities. For anygiven test, the temperature and velocity were maintained constantduring the test.

11. Control of Test Conditions.-The fluid was maintained at aconstant temperature by means of a heater or cooler as the conditionsrequired. Since the pump was operated at a constant speed, the ve-locity of the fluid was controlled by means of a throttle valve in thedischarge pipe. The connections from the piezometer rings to themanometer were thoroughly purged of any entrapped air, and a periodof several minutes was always allowed for conditions of flow to becomeconstant before any observations were recorded.

IV. CALCULATIONS

12. Reynolds' Number.-Reynolds' number was, in all cases, cal-culated by substituting the numerical values in Equation (2), Sec-tion 1. In order to facilitate similar calculations the curves for the reci-procal of kinematic viscosity for water and calcium chloride brine areshown in Fig. 4.


2Te•pTe,-r'ae /,7 deg. /.FIG. 4. RECIPROCAL OF KINEMATIC VISCOSITY FOR CALCIUM

CHLORIDE BRINE AND WATER

The mean hydraulic radius in the case of the circular cross-sec-d

tion is numerically equal to 4. In the case of the annular cross-sec-

tions the mean hydraulic radius was calculated from the equation

7r(d - d2) 24 d, - d,m -d - - (3)

r (d, - d2) 4


in which m = mean hydraulic radius, in feetd, = internal diameter of the casing, in feetd, = external diameter of the core, in feet.

In the case of circular cross-sections, it has been customary toevaluate the Reynolds' number using the diameter d instead of themean hydraulic radius m. This practice gives values numericallyequal to those that would be obtained by using 4m. In the case ofgeometrically dissimilar cross-sections, it is doubtful whether anymathematical process ostensibly expressing the dimensions in termsof an equivalent diameter can be strictly justified. That is, while twocross-sections having the same mean hydraulic radius may be mathe-matically equivalent, they may not be physically equivalent owing tothe fact that a different velocity distribution results from the differ-ence in the shape of the channels, and the head lost under otherwisesimilar conditions is influenced by the velocity distribution as well asby the average fluid velocity. However, in order to obtain Reynolds'numbers of comparable magnitudes in the cases of both circular andannular cross-sections, the Reynolds' numbers for the annular cross-sections in this bulletin have been evaluated on the basis of 4m, asindicated in Equation (2).

13. Friction Factor for Standard Pipe.-Preliminary work indi-cated that the pressure loss around the return portion of the 2-in.pipe (see Fig. 1), which included 7.0 ft. of straight pipe and 6.7 ft.of curved pipe, was 14.6 per cent of the pressure loss through theentire test section of 113.7 ft. The equivalent length of the returnportion accordingly was

113.7 X 0.146 = 16.7 ft. of straight pipe.

Therefore, the effective length of the 2-in. test section was

113.7 - (7.0 + 6.7) + 16.7 = 116.7 ft.

For the 11 -in. pipe the effective length of the test section, as givenin Bulletin No. 182, was 108.0 ft. These effective lengths were sub-stituted in Equation (1), Section 1, in order to obtain the frictionfactor f.

14. Friction Factor for Annular Sections.-In the case of the annu-lar sections, the effective length was the distance between piezometerrings as shown in Fig. 2. The mean hydraulic radius was calculatedfrom Equation (3). These values for the length and mean hydraulicradius were substituted in Equation (1) in order to obtain the fric-tion factor f.


15. Head Lost in Elbows.-The measured pressure loss between thepiezometer rings shown in Fig. 3, expressed in feet of fluid flowing,included the loss in the elbows themselves and the loss in the connect-ing piping. The latter was calculated from the friction factors previ-ously determined for the straight pipe. For any given test on theelbows, the Reynolds' number was calculated from the known velocity,kinematic viscosity, and pipe diameter. The friction factor correspond-ing to this Reynolds' number was then determined from Fig. 5, and thehead lost was obtained by using this value of the factor f, togetherwith the proper length, in Equation (1). The total length of pipeused was 34.3 ft. for the 11-in. pipe, and 51.3 ft. for the 2-in. pipe.This calculated loss in the straight pipe was then deducted from themeasured loss and the result was divided by 10 in order to obtain theloss for a single elbow. For the purpose of plotting, this loss was con-verted to terms of equivalent length of straight pipe, and also to headlost per velocity head.

V. RESULTS OF TESTS

16. Standard Wrought-Iron Pipe.-The results of the tests on thestandard 2-in. pipe are given in Table 2. The points representing theseresults have been plotted in Fig. 5, which also includes the results ofthe tests on the 11-in. pipe previously reported in Bulletin No. 182.The band of points shown in Fig. 5 therefore represents results ob-tained with two sizes of pipe, and with two fluids, over a wide rangeof specific gravity, temperature, and viscosity. This band of pointsmay be represented in the region of turbulent flow by a mean line, asshown in Fig. 5, such that the maximum deviation of any point fromthe line does not represent a deviation of more than 9 per cent in thevalue of the friction factor. Furthermore, the distribution of points issuch that it is not possible to draw two distinct lines thus separatingthe performance of the 11-in. and 2-in. pipes, or to distinguish be-tween the fluids used. Therefore, in cases where geometrically similarcross-sections are involved, the use of the Reynolds' number affordsa means of correlating the friction factors over a wide range of condi-tions, and the difference in roughness ordinarily encountered in com-mercial pipes does not introduce errors of sufficient magnitude to betaken into serious consideration. The difference between turbulent andviscous flow has been discussed in Bulletin No. 182.

14 ILLINOIS ENGINEERING EXPERIMENT STATION

m

z

ozp

0

r4

z

'a

u oa<

fe &d% 0


TABLE 2

PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS IN2-IN. WROUGHT-IRON PIPE

Head Lossft. of fluidTest

No.

Friction Reynolds'Factor Number

f R

SpecificGravity

2

1.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1571.1561.1561.1551.1551.1541.1521.1521.1521.1521.1521.1521.1521.1521.1521.1521.1521.1521.152

AverageTemp.

deg. F.

3

61.061.062.062.062.062.062.062.062.062.063.063.063.064.064.076.576.576.577.077.077.077.077.077.077.077.077.077.577.577.578.078.078.015.017.017.519.019.021.024.024.023.022.022.023.023.024.024.025.025.025.025.0

Reciprocalof Kinematic

Viscositysec. per sq. ft.

1/x

4

84 20084 20084 20085 30085 30085 30085 30085 30085 30085 30086 50086 50086 50087 60087 60066 50066 50066 50066 80066 80066 80066 80066 80066 80066 80066 80066 80067 10067 10067 10067 50067 50067 50026 40027 50027 80028 60028 60029 90031 60031 60031 00030 50030 50031 00031 00031 60031 60032 30032 30032 30032 300

7 8

0.0062 67 5000.00598 93 6000.00615 83 7000.00614 78 4000.00647 57 8000.00671 46 0000.00688 40 8000.00699 37 8000.00714 34 7000.00738 29 4000.00773 22 9000.00810 18 6000.00877 14 3000.00870 12 0500.00980 9 4200.00630 65 6000.00640 60 5000.00658 47 1000.00657 45 4000.00674 38 4000.00671 38 8000.00681 34 3000.00702 30 3000.00690 26 4000.00735 24 5000.00769 20 4000.00792 17 8000.00830 14 9000.00868 11 9000.00903 9 8800.00966 8 4200.00992 7 0100.01050 5 7200.00720 26 4000.00723 25 4000.00730 23 4000.0154 2 2800.0111 4 2000.0139 3 0600.01005 6 8300.00845 7 6200.00902 9 0200.00883 10 2000.00882 12 0800.00795 15 0000.00776 17 1000.00755 20 0500.00731 22 1000.00728 24 8000.00720 28 2000.00694 32 4000.0113 36 200

6

5.5510.308.487.214.142.732.191.911.651.220.7600.5190.3340.2290.1588.567.444.624.253.113.182.512.031.581.391.000.7850.5680.3830.2770.2110.1500.106

10.08.607.250.1360.3340.2020.6520.6951.061.3851.792.603.274.235.036.037.629.750.198

AverageVelocity

ft. per see.

v

5

4.626.415.735.303.913.112.762.562.351.991.531.240.9530.7940.6205.695.254.093.923.323.352.962.622.282.121.761.541.281.020.850.720.600.495.765.324.860.4600.8480.5891.251.391.681.932.282.793.183.664.044.445.035.800.646

------------


TABLE 3PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS IN

ANNULAR SECTION NO. 1

TestNo.

1

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647

SpecificGravity

2

1.0001.0001.0001.0001.0001.0001.0001.0001.0001.1481.1481.1481.1481.1481.1481.1481.1481.1531.1531.1531.1531.1531.1531.1531.1531.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.150

AverageFluid

Temp.

deg. F.

3

67.068.573.069.068.067.567.067.067.071.071.071.071.071.071.071.071.034.034.034.034.034.034.034.535.067.067.067.067.066.566.566.066.065.565.562.563.063.063.063.063.063.064.064.564.564.064.0



1/l

4

91 00092 80097 50093 50092 20091 60091 00091 00091 00062 50062 50062 50062 50062 50062 50062 50062 50038 00038 00038 00038 00038 00038 00038 20038 50091 10091 10091 10091 10090 50090 50090 00090 00089 40089 40057 00057 30057 30057 30057 30057 30057 30058 00058 40058 40058 00058 000

AverageVelocityft. per sec.

5

0.8871.4622.1102.6403.4604.0604.6205.1105.7805.3704.7204.0803.4702.7602.2401.7901.2205.2404.6504.0003.8203.2302.5901.9201.3500.5090.6070.7110.8350.8940.9401.0321.0901.2101.3030.1290.2430.3080.4090.4790.6080.7640.9651.0301.1341.3341.522

Head Loss Frictionft. of fluid Factorflowing

h f

6 7

0.160 0.010940.370 0.009270.700 0.008451.070 0.008221.730 0.007722.350 0.007622.990 0.007483.580 0.007354.500 0.007204.200 0.007903.310 0.007952.530 0.008111.890 0.008301.230 0.008680.850 0.009080.570 0.009550.300 0.010804.360 0.008533.480 0.008642.670 0.008952.470 0.009061.840 0.009481.230 0.009830.750 0.010900.385 0.011300.059 0.012300.088 0.012800.113 0.011900.135 0.010600.156 0.010450.177 0.010780.206 0.010300.230 0.010400.272 0.009900.310 0.009780.025 0.081500.042 0.038300.053 0.029900.067 0.021600.079 0.018500.103 0.015000.134 0.012300.208 0.012000.239 0.012100.279 0.011700.365 0.011000.460 0.01060

Reynolds'Number

R

8

2 6924 5206 8488 24010 64012 40014 00015 48017 52011 1909 8308 5007 2605 7504 6603 7202 5406 6305 8805 0604 8304 0903 2802 4401 7301 5451 8432 1602 5002 6902 8303 0903 2603 6003 860

243464588781915

1 1601 4601 8602 0002 2002 5802 940

i


TABLE 4

PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS INANNULAR SECTION NO. 2

SpecificGravity

TestNo.

1

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

AverageFluid

Temp.

deg. F.

3

71.572.571.572.571.572.571.571.571.571.563.063.062.061.561.060.560.060.060.060.058.558.558.559.559.559.059.059.059.059.560.060.060.060.060.064.064.064.064.064.064.064.064.564.564.564.565.065.065.065.0



1/C

4

97 50098 60097 50098 60097 50098 60097 50097 50097 50097 50086 50086 50085 40084 80084 20083 70083 00083 00083 00083 00054 20054 20054 20054 90054 90054 50054 50054 50054 50054 90055 20055 20055 20055 20055 20058 00058 00058 00058 00058 00058 00058 00058 20058 20058 20058 20058 50058 50058 50058 500

AverageVelocity

ft. per see.

v

5

0.6210.8461.3501.6152.0102.2002.5202.7502.9603.2700.3310.4820.6170.8090.9551.0881.2200.8200.6670.3910.1850.2970.4370.6100.7100.8880.6370.4820.3070.9020.9811.0731.2321.8562.4303.6803.4803.2503.0802.8302.5702.3402.1401.8101.5401.2401.0900.9260.7110.481

Head Lossft. of fluid

flowing

h

6

0.0400.0720.1590.2250.3260.3880.4930.5830.6700.8010.0180.0280.0390.0590.0810.1060.1310.0670.0480.0250.0140.0200.0300.0480.0610.0850.0540.0390.0250.0850.1000.1160.1480.3160.5281.1000.9990.8910.7950.6820.5820.4850.4010.2990.2260.1540.1220.0930.0630.037

FrictionFactor

f

0.010650.010320.009030.008920.008300.008280.008050.007950.007900.007740.017100.012300.010500.009300.009130.009260.009090.010400.011200.017200.043400.024000.016300.013400.012600.011200.013800.017200.027800.010850.010800.010400.010100.009560.009210.008450.008520.008690.008670.008810.009070.009150.009010.009440.009850.010300.010600.011300.01300.01660

1.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1511.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1501.1491.1491.1491.149

Reynolds'Number

R

3 8905 3508 450

10 20012 60013 90015 80017 20018 60020 5001 8352 6703 3804 4105 1605 8406 5004 3703 5502 080

6441 0331 5202 1502 5003 1002 2301 6801 0703 1803 4703 8004 3606 5608 610

13 70012 90012 10011 50010 5009 5508 7008 0006 7505 7504 6304 0903 4802 6701 810


TABLE 5

PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS INANNULAR SECTION No. 3

TestNo.

1

123456789

1011121314151617181920

SpecificGravity

2

1.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001.1511.1511.1511.1511.1511.1511.1511.1511.1511.151

AverageFluid

Temp.

deg. F.

3

69.068.568.568.068.067.067.067.073.069.050.551.051.051.051.052.052.552.553.552.5

AverageVelocity

ft. per sec.

v



1/P

4

93 50092 80092 80092 20092 20091 00091 00091 00097 50094 00048 80049 10049 10049 10049 10049 90050 00050 00050 80050 000

Head Lossft. of fluid

flowing

h

6

0.0570.0880.1600.2390.3600.4520.6030.7700.1130.4810.0350.0560.0830.1260.3780.0340.0590.1240.3900.395

FrictionFactor

I

0.009440.008330.007180.006810.006530.006380.006150.006040.006900.005990.010120.008850.008200.007680.007130.012030.009600.007830.006890.00703

17. Channels with Annulartests on channels with annular

Cross-Sections.-The results of thecross-sections are given in Tables 3,

4, and 5. These results have been plotted in Fig. 6. The points ob-tained from all three sections have been represented by a single meancurve. In this case, considerable deviations from the mean line maybe observed, the maximum deviation being approximately 14 per cent.There is also some tendency for the points to separate into distinctlines for the three sections. A consideration of the principles of di-mensional homogeneity may serve to explain these deviations.

The following statement is made by Stanton and Pannell:* "Simi-

larity of motion in fluids at constant values of the variable d- willv

exist, provided the surfaces relative to which the fluids move are geo-metrically similar, which similarity, as Lord Raleigh has pointed out,must extend to those irregularities in the surfaces which constituteroughness."

Since the same sized outside casing was used for all three of thesections tested, and the diameter of the core was varied, the sections

*Stanton and Pannell, "On Similarity of Motion in Relation to the Surface Friction ofFluids," Phil. Trans. Roy. Soc. A, Vol. 214, 1914, p. 199.

0.9021.1881.7252.1702.7203.0803.6204.1301.4753.2800.6830.9221.1611.4822.7000.6160.9111.4522.7502.740

Reynolds'Number

R

7 0409 160

13 36016 68020 92023 36027 48031 32012 00024 160

2 8003 8104 8106 140

11 8002 5903 8306 110

11 75011 520

FLOW OF LIQUIDS IN PIPES 19

z

0

0

z

z

.)rf.ci

IO


were not strictly geometrically similar. Furthermore, they were notsimilar to the circular sections. As indicated in Section 14, by evalu-ating the Reynolds' number on the basis of the mean hydraulic radius,cross-sections having the same mean hydraulic radius could be re-garded as being mathematically equivalent, thus making it possible toplot all points to the same scale. However, mathematical equivalencyin this case may not necessarily constitute physical equivalency, sincethe friction pressure loss is influenced by the velocity distributionacross the cross-section, as well as by the average velocity. Cross-sections having the same mean hydraulic radius do not necessarilyhave the same velocity distribution, and different friction pressurelosses may be obtained even when the average velocities and viscosi-ties are the same. Therefore, it is not unreasonable to find that greaterdeviations from the mean curve occur when the three different annularcross-sections are considered than those occurring in the case of thepipes with circular cross-sections. It is also reasonable to expect thatthe mean curve for the annular sections should not coincide with thecurve for the circular pipes. This lack of coincidence is shown inFig. 6, in which the curve for the circular pipes has been reproducedfor the purpose of comparison. It is also possible that the results fromthe three annular cross-sections should be represented by three sepa-rate curves, but since variations in roughness seem to introduce anelement of uncertainty that might lead to inconsistency, it has beenconsidered best to represent the results from all three of the sectionsby a single mean curve. This procedure may be justified by the limitedrange of cross-sections investigated, but it is doubtful whether asingle curve can be regarded as applicable to a wide range of cross-sections.

18. Elbows.-The results of the tests on elbows are given in Tables6 and 7. The relation between Reynolds' number and the friction lossin the 1%-in. elbow, expressed in terms of the equivalent length of11/-in. pipe, is shown in Fig. 7. A similar relation for the 2-in.elbow, expressed in terms of the equivalent length of 2-in. pipe, isalso shown in Fig. 7, and for the purpose of comparison, a similarcurve given by McAdams* has been reproduced.

The results must apparently all be represented by separate anddistinct curves. The tests on elbows reported in this bulletin wereall made with a solution of calcium chloride, and apparently a greater

*W. H. McAdams, "Flow of Fluids," Mass. Inst. Tech. Serial No. 121.


TABLE 6

PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS IN1Y 4 -IN. CAST-IRON 90-DEG. ELBOWS

TestNo.

1

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960

SpecificGravity

2

1.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2001.2101.2101.2101.2091.2091.2081.2081.2081.2071.2061.2061.2061.2051.2051.2051.2051.0971.0971.0971.0971.0971.0971.0971.0971.0971.097

AverageTemp.

deg. F.

3

67.067.067.067.067.568.068.068.068.062.062.062.063.063.063.064.064.564.565.065.065.065.058.559.559.560.060.060.061.061.061.061.061.061.032.032.032.533.033.535.035.535.536.037.037.037.539.039.541.039.072.572.573.073.073.573.573.574.074.575.0

Reciprocalof Kine-

maticViscositysec. persq. ft.

1/c

4

49 40049 40049 40049 40049 70050 00050 00050 00050 00046 50046 50046 50047 10047 10047 10047 60047 90047 90048 20048 20048 20048 20044 60045 10045 10045 40045 40045 40046 00046 00046 00046 00046 00046 00030 00030 00030 50030 60031 00031 80032 00032 00032 30032 80032 80033 00033 90034 00035 00033 90075 90075 90076 20076 20076 60076 60076 60077 00077 60078 300

AverageVelocity

in ft.per sec.

5

6.566.356.035.694.964.594.013.382.616.525.845.083.573.132.482.301.712.251.401.100.8040.6310.7140.7380.5960.77.90.8420.9380.8880.5751.294.323.196.435.936.494.933.652.583.342.842.411.971.581.410.9201.270.3050.7803.266.325.904.683.263.322.802.432.031.210.776

Reynolds'Number

R

6 '

37 20036 00034 20032 30028 40026 40023 10019 40015 00034 80031 20027 10019 30017 00013 40012 600

9 41012 4007 7506 1004 4603 5003 6603 8303 1004 0604 4004 9004 7003 0406 790

22 80016 85034 00020 50022 40017 30012 8809 200

12 20010 4608 8607 3205 9605 3203 4904 9501 1913 14012 70055 10051 50041 00026 90028 30024 60021 40018 00010 760

7 000

HeadLoss in

Pipe andElbows

ft. of brine

h

7

12.4011.5810.438.997.066.164.753.392.11

12.7210.237.743.872.981.901.561.051.600.6480.4250.2470.1582.082.101.512.362.743.282.851.445.925.603.07

12.5010.9012.687.434.092.133.612.691.891.360.9160.7280.3370.5980.9430.2543.71

12.1310.206.453.223.392.431.821.370.4780.215

EquivalentLength ofElbow inpipe di-ameters

8

40.238.137.634.836.136.235.233.632.640.639.438.134.633.230.527.631.530.926.926.226.324.425.024.125.625.626.626.824.427.129.735.332.041.036.635.834.129.727.632.132.127.929.429.127.024.526.591.024.837.846.542.541.538.640.638.737.439.431.530.0

Head Lossper Elbowdivided by

VelocityHead

9

1.0811.0401.0300.9611.0101.0351.0281.0141.0421.1131.0931.0801.0431.0301.0000.9151.1201.0301.0101.0461.1451.1501.2001.1081.2601.1581.1711.1371.0431.3341.1531.0360.9961.1231.0951.0501.0500.9850.9911.0791.1171.0121.1151.1631.1121.1551.1214.9101.2101.2521.1901.1051.1071.0971.1351.1201.1051.2101.0901.150

-------------------


TABLE 7

PRINCIPAL RESULTS OF TESTS ON FLOW OF LIQUIDS IN

2-IN. CAST-IRON 90-DEG. ELBOWS

Test SpecificNo. Gravity

1 2

Reciprocalof Kine-

Average maticTemp. Viscosity

sec. p-sq. ft.

deg. F. 1/c

3 4

63.0 47 00064.0 47 50064.0 47 50064.0 47 50064.0 47 50065.0 48 20067.0 49 30065.0 48 20065.0 48 20065.0 48 20065.0 48 20065.0 48 20065.0 48 20066.0 48 70066.0 48 70067.0 49 30033.5 30 00034.0 30 00035.0 30 50035.0 30 50035.5 30 80036.0 31 00036.5 32 00037.5 32 00038.0 32 50038.5 32 70039.0 32 80039.5 33 00040.0 33 30040.5 34 00041.5 34 50041.0 34 20042.0 34 50071.0 75 00071.0 75 00072.0 75 70072.0 75 70072.0 75 70072.0 75 70072.0 75 70072.0 75 70072.5 76 30073.0 76 50073.0 76 500

AverageVelocity

in ft.per sec.

5

0.430.600.801.011.290.261.652.032.422.202.683.053.542.112.031.933.423.112.952.652.422.311.981.631.471.271.110.8610.7060.5560.4830.4310.4073.563.202.332.722.941.851.641.340.8940.7190.560

Reynolds'Number

R

6

3 5004 8806 5608 270

10 5502 160

14 00016 90020 10018 30022 30025 40029 50017 70017 00016 40017 69016 10015 50013 92012 88012 35010 950

8 9908 2107 1306 2804 9004 0503 2602 8702 5402 420

46 00041 00030 40035 60038 50024 20021 40017 41011 780

9 4807 400

EquivalentLength ofElbow inpipe di-ameters

8

Head Lossin Pipe

andElbows

ft. of brine

h

7

0.0690.1200.2030.3130.4900.0300.8001.2001.7401.4202.0902.6903.5701.2901.1901.0903.4802.8202.5702.1101.7301.5501.1900.8100.6780.5070.3980.2520.1770.1240.0880.0770.0663.4402.8001.5852.1502.5000.9860.7840.5260.2450.1580.092

1.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2021.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.2071.0971.0971.0971.0971.0971.0971.0971.0971.0971.0971.097

Head Lossper Elbowdivided by

VelocityHead

9

1.0080.9350.8780.8480.8281.3400.9110.9521.0170.9760.9891.0621.0000.9520.9470.9281.0050.9450.9610.9820.9230.890

.0.9380.8950.9530.9180.9180.9480.9901.2301.0201.0721.0651.0020.9690.9981.0751.0580.9900.9920.9930.9840.8970.776


gbo

60

.4'----

^ S

Inn lI I I I I I I I I Illr~7

I- l l I I , I---- Univ. of ////ois, /K C .E/bow---- /Uni/ of ///no, e" C/ E/bow----- MA/doams, Proposea' Carv for

*SAte/ /a r/ F/hni/ I

0 00co ° o>o^*

__„ ° ^^s2^5S'^o

N

I

-. a

^

N1

000 4000 6000 0000 Z 0000 40000 /0000

FIG. 7.

Reyno/a's' Number, R= v

CURVES FOR EQUIVALENT LENGTH OF 11

4-IN. AND 2-IN. ELBOWS

I I * I I i I I I I Iil

Reyno/ds' Number, R= v

FIG. 8. CURVES FOR HEAD LOST PER VELOCITY HEAD IN 1/4-IN. AND 2-IN. ELBOWS

o

I Ilrllillll


head loss occurred in the 1¼-in. than in the 2-in. elbows. The testsreported by McAdams were run with oil and water, and includedelbows of from % in. to 6 in. No assurance can be had that the elbowsused were exactly comparable with the ones used in this investigation.

The pressure loss in an elbow results largely from the action ofeddies and cross currents rather than from surface friction alone.The exact nature of these eddies depends to a large extent on thecharacter of the elbow and the amount of pipe thread included, ratherthan on the size of the elbow. Therefore, it is doubtful whether anymethod of dimensional analysis is applicable in the case of elbows.Each elbow is probably an individual case, and coincidence of thecurves is hardly to be expected. Plotting the friction loss against theReynolds' number, however, affords a convenient means of correlatingthe data and of making comparisons provided that unwarranted con-clusions are not drawn.

A second method of expressing the friction loss in elbows is oftenused in practice. By this method the loss is expressed in terms of thehead lost in the elbow per velocity head in the pipe. For convenience,the results have been plotted on this basis and are shown in Fig. 8.

The results of these tests as shown in Figs. 7 and 8 are indicativeof the fallacy that exists in adopting a single value for either thelength of pipe equivalent to an elbow, or the ratio of head lost tovelocity head in the pipe, and in using such adopted values over awide range of conditions. The curves show varying values for bothof these quantities ordinarily used as constants in estimating thefriction loss in pipe lines.

VI. APPLICATION OF DATA

19. Statement of Problem.-Assume that the total pressure lossin a 2-in. pipe line containing 200 feet of straight pipe and 5 cast-iron 90-deg. elbows is required. The weight of brine circulated isto be 250 lb. per min. The average temperature of the brine is 25deg. F. The specific gravity of the calcium chloride brine to be usedis 1.15.

20. Solution of Problem.-The first step in the solution of theproblem is the determination of the velocity v and the reciprocal

of the viscosity- for the brine.v


v = (4)60XkXsXA

in which v = velocity of brine in feet per secondw = weight of brine circulated in pounds per minutek = density of water in pounds per cubic foot = 62.4s = specific gravity of brine

A = cross-sectional area of pipe in square feet.

Since the nominal diameter of 2-in. pipe is 2.08 in., the value of Ato be substituted in Equation (4) is

7rX (2.08) 2A X 0.0236 sq. ft.4 X 144

Hence the velocity is

250v= = 2.46 ft. per sec.60 X 62.4 X 1.15 X 0.0236

The reciprocal of the viscosity, from Fig. 4, at a temperature of25 deg. F. is 32 000.

Therefore, Reynolds' number from Equation (2) is

dv 2.08 X 2.46 X 32 000 3R 13 650.

v 12

From Fig. 7, the equivalent length of one 2-in. elbow, for a Reynolds'number of 13 650, is 28 pipe diameters. Therefore, the total equiva-

5 X 28 X 2.08lent length of the 5 elbows is 2 24.3 feet. The total12

effective length of pipe to be used in Equation (1) then becomes200 + 24.3 = 224.3 feet.

Using the same value of Reynolds' number, 13 650, from Fig. 5,the friction factor f is found to be 0.0082. The mean hydraulic

d 2.08radius is 12 = 0.0433. Hence the pressure loss as determined

4 12 X 4from Equation (1) is

0.0082 X 224.3 X (2.46)2h = 3.99 feet of brine.2 X 32.2 X 0.0433

Expressed in pounds per square inch this becomes

3.99 X 62.4 X 1.15- = 1.99 lb. per sq. in.144


VII. CONCLUSIONS

21. Summary and Conclusions.-As a result of this investigationthe following conclusions may be drawn:

(1) The use of the Reynolds' number affords a means of correlatingfriction factors over a wide range of conditions and is independent ofthe fluid used, in cases in which channels with geometrically similarcross-sections are involved.

(2) In the case of the two commercial pipes used in this investiga-tion the difference in the roughness of the rubbing surfaces did not in-troduce deviations in the friction factor of sufficient magnitude to betaken into serious consideration when a single mean curve was usedto represent all of the data.

(3) The use of the Reynolds' number for correlating friction fac-tors in cases of channels in which the cross-sections are not geometri-cally similar is not strictly justifiable, although for any particularsection it serves to correlate the friction factors with the temperature,velocity, and viscosity for all fluids.

(4) The Reynolds' number is not applicable for correlating thehead lost in elbows, but does afford a convenient means for express-ing and plotting the data.

(5) The loss in head in an elbow is largely influenced by thepresence of eddies and cross currents, rather than by surface friction.

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UNIVERSITY OF ILLINOIS

THE STATE UNIVERSITY

URBANA

HARRY WOODBURN CHASE, Ph.D., LL.D., President

The University Includes the Following Departments:

The Graduate School

The College of Liberal Arts and Sciences (Curricula: General with majors inthe Humanities and the Sciences; Chemistry and Chemical Engineering;Pre-legal, Pre-medical, and Pre-dental; Pre-journalism, Home Economics,Economic Entomology, and Applied Optics)

The College of Commerce and Business Administration (Curricula: GeneralBusiness, Banking and Finance, Insurance, Accountancy, Railway Adminis-tration, Railway Transportation, Industrial Administration, Foreign Com-merce, Commercial Teachers, Trade and Civic Secretarial Service, PublicUtilities, Commerce and Law)

The College of Engineering (Curricula: Architecture, Ceramics; Architectural,Ceramic, Civil, Electrical, Gas, General, Mechanical, Mining, and RailwayEngineering; Engineering Physics)

The College of Agriculture (Curricula: General Agriculture; Floriculture; Home,Economics; Landscape Architecture; Smith-Hughes-in conjunction withthe College of Education)

The College of Education (Curricula: Two year, prescribing junior standing foradmission-General Education, Smith-Hughes Agriculture, Smith-HughesHome Economics, Public School Music; Four year, admitting from the highschool-Industrial Education, Athletic Coaching, Physical Education. TheUniversity High School is the practice school of the College of Education)

The School of iusic (four-year curriculum)

The College of Law (three-year curriculum based on a college degree, or threeyears of college work at the University of Illinois)

The Library School (two-year curriculum for college graduates)

The School of Journalism (two-year curriculum based on two years of collegework)

The College of Medicine (in Chicago)

The College of Dentistry (in Chicago)

The School of Pharmacy (in Chicago)

The Summer Session (eight weeks)

Experiment Stations and Scientific Bureaus: U. S. Agricultural ExperimentStation; Engineerintg Experiment Station; State Natural History Survey;State Water Survey; State Geological Survey; Bureau of EducationalResearch. -

The Library collections contain (May 1, 1930) 836,496 volumes and 221,800pamphlets.

For catalogs and information address

THE REGISTRARUrbana, Illinois

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Transcript of Flow of liquids in pipes of circular and annular cross-sections