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Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1950
Velocity studies in a vertical pipe flowing full Velocity studies in a vertical pipe flowing full
Robert Franklin Tindall
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VELOCITY S'IUDI~
IN .A
VERTICAL PIPE
FWWING FULL
BY
ROBERT FRANKLIN TINDALL, Jr.
A
THESIS
submitted to the faculty of th
SCHOOL OF MINES AND METALlURGY OF THE UNIVERSITY OF MISSOORI
in partial fulfillment of the work required for th
Degre of
MASTER OF SCIENCE IN CIVIL ENGINEERING
Rolla, Mi sour!
1950
ACKNOWLEDGEMENT
For suggesting this study and offering valuable advice
the 'Writer wishes to express his sincere appreciation to
Professor V. A. C. Gevecker, and Professor E. W. Carlton,
both of whom are Professors of Civil Engineering, Missouri
School of Minea.
r am also grateful to Mr. Orlan King, who, as head of
the departmental shop, offered numerous helpful suggestions
on the construction and fabrication 0 f the equipment used in
these experiments.
11
iii
TABLE OF CONTENTS
Page
Acknowledgment............................................. 11List of Illustrations...................................... iv
List of Tables............................................. vi
Historical Sketch and Background........................... 1
Purpose and Object of Investigations....................... 5
Apparatus.................................................. 6
Testing Procedure.......................................... 15
Results.................................................... 18Conclusions................................................ 34
Nomenclature............................................... 38Bibliography............................................... 39
Vita. . • • •• • • • • • • . . • • • • • • • . . . • • • • • • • • •• • • • •• • •. • • .• • .•• •• ••• 42.
iv
LIST OF ILIlJSTRATIONS
Fig.No.
1
2
Picture of U-Tube Mercury Guagee •••••••••••••••••••••
Picture of Weighing Tank and Scales•••••••••••••••••••
PageNo.
7
3 Picture of U-Tube Mercury Guages, Weighing Tank
and Scales, and By-pass............................... 9
4 Picture of 6 Inch Centrifugal Pump•••••••••••••••••••• 10
5 Picture of 2 Inch Centrifugal Pump•••••••••••••••••••• 11
6 Picture of Stop Watch on Data Board, Micrometer,
Inside Caliphers, and Thermometer••••••••••••••••••••• 12
7 Picture of Intake from 2 Inch Feed Pipe to Test Pipe,
Also Number One Pressure Takeoff•••••••••••••••••••••• 13
8 Picture of Water nowing From Test Pipe to Weighing
Tar1lc. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • l.4.
9 Drawing of Flow of Wi ter for th Exper ent........... 17
10 Temperatu -visco i ty Curve for Some Common
Liqu.1ds••••••••••••••••••••••••••••••••••••.••••..•••• 19
II Temperature-viscosity and Density Cu~ s for ter.... 20
12 Reynold' Number-Friction Factor Chart................ 21
13 Relative Roughness Factors for New Clean Pipe •••••••• 22
14 Friction Factor IIfll and Reynolds' Number•••••••••••••• 23
15 Graph of Vele>city and Reynolds' Number, (Log-Log Plot) 27
16 Or ph of Velocity and Reynolds' Number, (Cooridinate
Graph Plot) ••••••••••••••••••••••••••••••••••••••••••• 28
17 Graph of Pressure Head and Velocity on each
Takeoff, (Log-Log Plot) ••••••••••••••••••••••••••••••• 30
18 Graph of Pressure Head and Reynolds' Number, (Log-Log
Plot)... . . . • .• • • • • • . . • . . . . • • • . . • . . . . . . . . . . . . . .. . . . . . •• 31
19 Graph of Friction Factor "f ll and Reynolds' Number,
(Log-Log Plot) ••••••••••••••••••••.••••••••••••••••••• 32
v
TableNo.
LIST OF TABLES
vi
PageNo.
1 Values of Darcy-Weisbach Friction Factor••••••••.••••• 24
2 Tabulation of Experimental Data••.•••••••••••••••••••• 26
HISTORICAL SKETCH AND BACKGRCUND
Although the origin of the 8 cience of hydraulics is not a
new discovery, the work done in this field was slow in pro
gress until about the seventeenth century.(l)
(1) Duba, J. G., Formulas for Flow of Fluids, MissouriSchool of Mines Thesis, 8)0, pp. 1-5, May, 1949.
In 1774 a new era in hydraulics was in the making, when
Turin and Bossut established as a fundamental principal that
1
formulae must be deduced from experiment. Bossut' s experimente
were among the first on the flow of water through pipes. Per-
haps the most famous engineer of that day was Antoine de
Che~ who in 1775 developed the basic formula,
v = c -VP'S \ (,1
an ression which carries his name, for the flow in pipe
and open channels.
The development of hydraulics from the seventeenth to
the nineteenth century grew in phenomenal proportions as
countless contributions were made in this field.
In about the middle of the nin teenth century a much
used pipe fomula came into use.
(2.)
Credit for its origin is given to Darcy, Weisbach, Fanning,
or Eytelwein by variQUS authors of the pres nt day. It is
widely known as the Darcy-Weisbach formula and will be so
called in this paper.
At about the same time the law of laminar flow was
first brought to light by Hagen. This work was almost
immediately confirmed by Poiseuille, who expressed his
findings in equation form. In terms of head 10s8, the
equation i ,
2
h - :3~..yL Vf - DJ. g P
(3)
Comparing this equation to the Darcy-Weisbach fonnula, it is
evident that the friction factor is,
(4-)
This relationship has since been Bubstantiated and is in
general use today.
A noticable split in the manner of treatment of hydrau-
lie pipe probl ha tak n pI c sine th 1 tt r part of th
nineteenth century.
JGhn R. Freeman and others were leaders in the detennin-
ation of friction factors and coefficients from experimental
d t. This type of data was widely ccept d and used by
practicing ngineer.
On the other hand, leaders suoh as Bakhm t if, and Rouse
favor a theoretical tr tment of hydraulics.
In the twentieth oentury men like Scobey, d Schoder,
did not continue experiments solely to determine the frieti~n
3
factor or Che~ coefficient. Instead they used their experi-
mental data as a basis for the development of the so called
"exact" or exponential type fonnula.
Since 1883 when Osborne Reynolds performed his cla.ssic
experiments, the parameter which carries hi name,
~ = V D P (5)A(
has proved a boon to the further development of pipe fiow
theory and practice.
Stanton and Pannell of the National Physical Laboratory
in London, ngland, utilized Reynolds number and put it in a
usable form. In 1914 they evolved the much used curve found
by plotting experimental data and corr lating Reynolds number
with the friction factor. other men quickly verified their
It was oon noticed by engineers that pipe roughness
al 0 affect d the friction factor det :nn1nation and new curves
wre plotted from exp rimental data, most of which approximately
par 11el the Stanton and Pann 11 curve in the turbulent flow
region.
Many experiments on th roughnes effect have been mad
since 1930. ikurad s th first to publi h hi finding
in 1933. He noted that th Reyn lds n r-friction f ctor
relation hip in th laminar flow region r main d unchanged,
but that an increase in the relative roughn 8 of pipe cau d
a corresponding increase in the friction factor n the tur-
bulent !.low region. V. L. Streeter published his findings in
1935, on artificially roughened pipe.
Meanwhile engineers interested primarily in the !.low of
water have continued to use long standing formulas such as the
Chezy, Kutter, Darcy-Weisbach, and Hazen-Williams with experi
mentally determined factors and coefficients.
PURPOSE AND OBJECT OF INVESTIGATIONS
The purpose of this investigation 1s to install a hard
dra'Wll copper pipe approximately i inch in diameter, and
approximately 35 feet long in vertical position.
The copper pipe is to have pressure takeoff points at
various distances along the length of the pipe. The purpose
of thee takeoffs i8 to measure the pressure of the water at
the point of takeoff in the pipe. The water i8 to be forced
into the pipe, and assisted by gravity, thereby causing a
pressur of some measurable quanity greater than absolute
zero pressure.
The discharge was to be by volumetric measurement, as
the pipe was flowing full, through the atmosphere into the
weighing tank.
The object of this invest1gatian is, to study the type
of flow in a v rtical pipe flowing full where gravity a8s1 ts
the lin pressu • The conditions xi ting within a vertical
pipe under free fall are at pre nt unknolm.
The effort of this study is to shed some light on this
unknown subject.
5
6
APPARATUS
The following Figures, 1 through 8 inclusive, show the
apparatus used in making the experimental tests.
Figure 1, page 7 is a picture of the U-Tube Mercury Guages,
numbered 1 to 4 for the 'Way the takeoffs are numbered from top to
bottom on the experimental teat pipe.
Figure 2, page 8 is the picture of the Weighing Tank and
Scales used in the volumetric measurement of water needed to
calculate the rate of discharge for each run.
Figure 3, page 9 is an overall picture of the U-Tube Mer
cur.y Guages, Weighing Tank and Scales, and By-pass for wat r
wen it ie not needed.
Figure 4 and 5, pages 10 and 11 respectively, show the 6
and 2 inch C ntrifugal Pumps used to furnisl1 'Water for the
experiment.
Figure 6, pag 12, i & pictur of the Stop W. tch on a
Data Board, Micrometer, Inside Caliphers, and Thermometer.
This equipment is used to measure the diameter of the test
pipe, temper ture of the air and water, and th numb r of
seconds needed to run th experiment for a certain number of
pounds of water.
Figure 7, pag 13, shows the entrance of th experimental
test pipe from th 2 inch fe d pipe, d also the numb r one
pressure takeoff.
Figure 8, page 14, shows the water flowing into the
weighing tank during a test run.
Figure 1
U-Tube Mercury Guages
7
Figure 2
Weighing Tank and Scales
8
Figure 3
U-Tube Mercury Guages, Weighing Tank and Scales, and By-pass
9
Figure 4-
6 Inch Centrifugal Pump
10
Figure 5
2 Inch Centrifugal Pump
11
Figure 6
Stop Wtch on D ta Board, Micrometer,
Inside Caliphers, and Thermometer
Figur 7
Intake From 2 Inch Feed Pipe to Test Pipe,
and Number One Pressure Takeoff
1.3
Figure 8
W ter Flowing From Test Pipe Into Weighing Tank
15
TESTING PROCEOORE
The equipment used in making the tests are illustrated
under the heading of apparatus.
The water used in the tests is pumped trom the sump in
the Fluid Mechanics Laboratory, by either the six or tw inch
centrifugal pumps.
There are three valves that can be used in combination
to regulate the flow of water furnished by the pump in use.
In making a test run from beginning to end, first the
diameter of the copper tubing is measured with t he inside
caliphers and micrometer and the measurement recorded. Then
by regulating the flow of the pump in use the reading of the
desired pressure is made wi. thin the range limit of the U-tube
mercury guages. The water furnished by the pumps is allowed
enough ti.me, to stabilize the flow 80 that any disturbance in
the flow will be eliminated. Then readings are recorded of the
differences in the mercury levels of all the pressure takeoffs,
also the height of water to the mercury in each tube connecting
the takeoff of the copper test pipe has to be recorded and sub
tracted from the pressure head reading, because these tubes are
always filled with water, thereby causing a greater pressure than
is possible at the point of takeoff. The air temperature is
recorded at the beginning of the run and as often as needed
during the experiment. The water temperature 1s recorded at
the end of the te t run.
16
The weighing of water is taken in the following manner.
There should be enough weights placed on the scale to ov rbalance
the weighing tank, the water is then passed from the by-pass into
the weighing tank, and the instant the water flowing in, balances
the scales the stop watch is started, then a known weight is placed
on the scales, when the water balances this known weight the stop
watch is stopped, then by using the by-pass the water is diverted
back to the sump for further use. From this method you have your
known amount of water in the number of seconds indicated by th
stop watch. This last procedure is run two or more times de
pending on the accuracy needed.
The readings of the U-tubes should be watched during the
run to see that there has been no change of reading duri g the
experiment.
The drawing Figure 9, following this page is a overall
drawing of the flow of water that is used in the experiment.
17
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18
RESULTS
The testing procedure gave the method of taking runs for
this experiment. The following results are a tabulation of 26
runs from the maximum pressure on the number one pressure take
off, to the minimum pressure of the test pipe, with the test
pipe still flowing full.
The pipe diameter was measured and found to be 0.525 inches,
or 0.04375 feet. The cross-sectional area in square inches is
0.216, or 0.0015 square feet.
The weight of water for all runs was 100 pounds. The weigh
ing procedure was followed twice for each run. The 100 pounds of
water was a constant figure, but the temperature ranged from 23.5
to 25 degrees Centigrade which gave the weight of water per cubic
foot a varying amount in the second decimal figure.
The distances of the pressure takeoffs 1 to 4, respectively
from the discharge opening are~
32.325 feet or 387.9 inches
24.325 feet or 291.9 inches
16.325 fe t or 195.9 inches
8.325 feet or 99.9 inches
The head of water that is always standing in the takeoff
tubes has to be subtracted from the mercury pressure, and in
the readings for the test runs the mercury heights were taken
from the disch rge opening as the reference datum.
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Densit.!;1 and Viscosity of Wo.ter- 67478 (,7 45
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. FT .v.>LuES or (VO) t.);1 WATER AT 60·f. (VE OCITY IN SEC x uiAMETEH IN I~ GHES)
20
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YN L S LJM8~R V V,nSCC,DlnFT,V InSEC
)
F,!vr" It-FrictIon FCClors fer An) Kind and SIze of Pipe
I II
43
e3
2
0201
05!--1-+-";'·
PIPE DIAMETER IN FEET, D.1 .2 ~ 4 S 6 .8 I 2 3 4 S6 8 10 2025
07, • 1 06,
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a:: 000,06 ,' I I' · I' , u..
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CENTRIFUGALLY-SPUN 01CEMENT AND BITUMINUSLININGS, TRANSITE, ETC.'
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60 100 300PIPE DIAMETER IN INCHES 0",
FIe.. 1.3 RELATIVE ROUGHNESS FACTORS FORNEW CLEAN PIPES
,J:'X
ABSCISSA SCALE
~=Kinem~ticViscosity
QG=Discharge:n U.S·9P·mQB =- Di5-charge In b:'1. ::Je r :'r.R -:: Reynold's Ncn:'er
UNITS FOR MAIN
Din: Diameter in inchesV ::Velocityinfl.persec.
P =Specific Gravity
p :: Viscosity in poises
To change from R In :nefnc :J{':;.Js,,:e. J) In em. Vln em. per$ec Jot. 1:7 poise:;,(mdp=derwfy; foR /nordlnar.unils, i.e. DIn in inches, V:';7 ff.per seandp:=specific grcmfy. dl;'ldeby 774
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Figut'e 14 Friction factor 'If" and Relj VlO Id0'
'''::>I.
~ per c' .:..-,.n u m bey
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Table 1 (2)
L V2.Valu 8 of f in the Darcy-Wei bach Fonnula, h - ft - 0 2.9
For water flowing in straight smooth pipe
Dia.of Pipe Mean Velocity (V) in Feet per SecondIn Inches
0.5 1.0 2.0 3.0 4.0 5.0 10.0 15.0 20.0
1/2 0.042 0.038 0.034 0.032 0.030 0.029 0.025 0.024 0.023
3/4 .041 .037 .033 .031 .029 .028 .025 .024 .023
1 .040 .035 .032 .030 .028 .027 .024 .023 .023
1 1/2 .038 .034 .031 .029 .028 .027 .024 .023 .023
.2 .036 .033 .030 .028 .027 .026 .024 .023 .022
.3 .035 .032 .029 .027 .026 .025 .023 .022 .022
4 .034 .031 .028 .026 .026 .025 .023 .022 .021
5 .033 .030 .027 .026 .025 .024 .022 .022 .021
6
8
.032 .029 .026 .025 .024 .024· .022 .021 .021
.030 .028 .025 .024 .023 .023 .021 .021 .020
10 .028 .026 .024 .023 .022 .022 .021 .020 .020
12 .027 .025 .023 .022 .022 .021 .020 .020 .019
14 .026 .024 .022 .022 .021 .021 .020 .019 .019
16 .024 .02; .022 .021 .020 .020 .019 .019 .018
18 .024 .022 .021 .020 .020 .020 .019 .018 .018
(2) King, H. W., Wisler, c. 0., and Woodburn, J. G. Hydraulic.5th Ed. N. Y., John Wil y and Sons, Inc., 1948, p. 184.
25
The openings in the wall of the test pipe for the pressure
takeoffs was approxiJnately five sixty fourths of an inch.
Figures 10 and 11, pages 19 and 20 respectively were used in
calculating ReYnOlds I number for all t he runs.
Table 1, page 24, and Figures 12, 13, and 14, pages 21, 22,
and 23, respectively, are relationship curves of Reynolds'
number-friction factor, friction factor f, and the relation of
relative roughness.
Tabl 2, page 26 is the tabulation of the experimental
data, a.nd gives the Temperature in degrees Centigrade, Weight
of water per cubic foot for that temperature, Run number, Average
time for each run, the calculated Discbarge, Velocity, Reynolds'
number and pressure for each takeoff. The Reynolds I number was
calculated by using Figures 10 and 11, density and viscosity
charts.
Figures 15 and 16, pages 27 and 28 respectively, are both
graphs of Reynolds' number and Velocity. The plottil18 of the
graphs was done on log-log, and coordinate graph paper. The
dat~ for these graphs was taken from Table 2, page 26.
The explanation and proof is on page 29, and it clearly shows
that the experiment is following the mathematical equation form of
y =mx + b, wh re m is the slope of the line, and b is the intercept.
Assuming the friction factor f to be 0.02. The head loss is
then found by using equatio (2), page 1 in terms of velocity. The
velocity is then foun for the point of takeoff by using Bernou.ll:i'
theorem.
TABLE 2
TABt~TION OF EXPERIME.7AL DATA
Run Temperature Weight e! Water Average Time DiSCharge, Q, in Velocity, V, in Reynolds' Pressure in Feet of Water for each Takeoff
Number Degree Centigrade per Cubic Feot in Seeond8 Cubic Feet per Seeend Feet per Second Number Number 1 Number 2 Number 3 Number 4
1
2
3
4
5
6
7 (Mcu)
8
9
10
11
12
13
15
16
17
18
19
20
21
22
23
2425
26 (Min-)
2J.O
24.0
25.0
24.5
24.5
25.
25.
25.0
25.0
25.
25.0
25.
25.0
25.025.0
25.
62.24
62.24
62.23
62.22
62.21
62.22
62.22
62.22
62.22
62.21
62.21
62.21
62.2162 0 21
62.21
111.4
138.0
85.0
91.8
74.8
64.6
65.5
68.3
70.2
80.3
81.9
85.7
90.5
105.8
115.3136.1
166 0 6
0.014
0.017
Q.012
0.019
0.018
0.021
0.025
0.025
0.024
0.023"
0.023
0.021
0.021
0.020
n.n20
0.020
0.019
0.019
0.018
0.018
0.017
0.016
0.015
O.OIL
0.012
O. 10
9.33
1 .33
8.00
12.67
12.00
14.00
16.67
16.67
16.00
15.33
15.33
14.00
14.00
13.33
13.3'3
13.33
12.67
12.00
12.00
1 .33
10.67
10.00
9.33
8.00
39,810
47,933
33,845
54,f.9f:
52,338
61,665
72,706
72,706
69,784
66,861
65,511
59,827
61,061
58,714
58,714-
5 ,714
55,807
55,8CY7
52,855
52,855
49,904
46,997
44,0/J:J
41,095
35,237
29,379
- 11.98
- 3.4-1 - 0.91
- 17.96 - 12.10
+ 1.96 + 27.65
- 1.44 ... 0.58
... 11.95 of. 9.86
~ 26.62 + 19.53
• 24.99 + 18.30
+ 21.J1 + 15.71
... 18.03 + 13.40
;- 13. '53 9 .71
+ 12.16 8.90
... 10.65 + 7.81
+ 9.15 + 6.58
+ 7.79 + 5.63
+ 6.43 + 4.53
... 4.66 + 3.30
+ 3.55 + 2.35
+ 1.66 ... 1.26
... 0.56 + 0.17
- 1.35 - 0.93
- 4.49 - 3.51
- 7.48 - 5.29
- , 0.90 - ~.28
- 15.81 - 11.83
- 20.34 - 14.01
- 0.51
+ 1.94
... 4.44
... 6.72
i- 13.52
t 12.44
+ 10.53
+ 9.04
+ 6.72
... 6.03
... 5.21
l' 4.54
... 3.85
+ 3.17
l' 2.35
T loll?,
l' 0.85
+ 0.1?
- 0.79
- 0.29
- 4.25
+ 0.94
+ 0.12
+ 3.26
to 6.81
+ 5.45
... 4.63
+ 3.26
+ 2.98
i- 2.58
... 1.76
of" 1.61
of' ] .21
t 0.80
t 0.39
- 0.02
- 0.83
- 2.22
- 2.M
- 3.98
- 4.94
100 I .
80
+0
I 0 -I'-~---+-ri-
~"-
~
2
II .IIO~
"I
~
n urn he I" (g)
28
,._-I
-_Q_~Ii) . - ---t::::s
.c
- ---- Q ,-....,....~-crr
""-~ll)
-Q~
-CJ-~II) ~
- ---.- jI
I
I.- .I
III
I
JII
tI
0- _1..
J'~ft-~~-L~:I~I (I\) 1l-lIJ OI<l/\ I
1 I II I
II
----~-I
~_ . slope:::m
1
lj=mX+b
Velocit~ YS. ReljnoJds ' number
R = Vd (J = V de- l' -r
Coor-dinate Gr-Qph Plot
V= R -'V V- d P
when V= 0 B =0
UJr,e n V::. 3.. R :: Id (0
A(dP= sloF e
o = ,'Yl t e Ii' C e pt 0.tor "9i nof coord'lnQtes,
29
Log- L09 Gro.ph Plot
LogV = L09 R +Logif
s J 0 Fe ~ 1: 1
y = V = cl1' VAn9le tanned:: 45°
11. 0
J} I
~'l I,.-fniH oj;q tt-..,..I~~:-~+H--t4
'" ~O - - SO 4-0 .so ~O·~ 'f0 60 0
Ve' D,C ;t~
f-.
.'fl,
I
f-,j .., •._- ~
--
10
'107 -.-'
,t - -
ir
of
, _. t
I.!
- _I _. _I
r' _. I..
t.,O
~e~ no Ids'
. ..... - ....._~.
- .'
- --_. -- ,--..~--:-
-+-"---~-------
. ~ -'- -I
--_.--J--------
t-'-'~-+-+--+..I' I -E-~-~' -1 ._.
I ' •f-.................· -L-.-+-ro+--- -~----II- f 1 I ~___._-=~f:---.-~~-._-_-.-_--.-_--,,-+---.....-----l-,-.------r--+-..,-JH-........--~::--+.f_--'~-'_,_-_-It-~_'_i.__--:--+-+4__..er..,...a:--,,4
~_._-
.. '-_ ..I
----,r----- - . _.,.-
__ J
.--
,- ---_. -....,.-- ---~.-
--~-' ------"---------f----.'-'L..-...:...--::;...
;r--'--~' -'
~-----,.-- - .
-- ~
10'0
I'- -- t----_.. -_.-,
~--~---- .. ---- -t-.-'I'f""'" -
---'-" ------+----j""f-.,..--:ll,.,..--+~rl_•..11,
~ 4 ..-.------:--....:......----:---J----~-----
tU+(j
34o
Q)
~
:)Cf)
<J)
~~r----+------------1~------.......-.-.;~-----+-..--...1-1+--~-~---r----+:.,4-t-4-~~-+-~.;..,j~..,.- .....+-+.....+-...,......;.~1o-'-,....-r--H-O"";-+.-.--i!--.:......+--.+ .........r+--+--~n..
50 .sl-'-------,.~---_+_---::-----:;=--'----__r-t-_+_---:--~>_lr___+__--+__r_+_-.....---+_+_-----J-__,.~_+_~
I~
I .
-.. , '.08
,. ··.. ,07t l'
:-.- .09
. . j0'
C
~I.'. I··..~,
01·~-IVI
i dl'.03
I II
I· I, I '1
I
.- -,.0., I
I ' ' I
, Il' I
. - i·f)5, i
80 90 100.J
80 90
\. III
,_ J
I I
i I
I ,
I ,
I. ,
II
" -
\
70
.1
, ,
-.1 ..III
I'1
. I
I'I,
I:I
: , '1I I ._ . !. I - .. I,
e I'1,
: J. 1'; I
. ~ : - !~-l---.------ - --4--+--r' "I'" ,- .. r
I'... . I : 1
, , ~_ G
: .... _... " ,_ la-'"--•. •:::. tI '
"I'" ..
.: I· . '. 'j , .. L;
I ,.I' .. - ~._---~,.. . .
.. !. I"
-- - ~---~
I'
._. .. :!
, t .. ·..-, ,
. .,.... , •• " P.
, .j •
. ',' .. ..-- ~ .. -.., . : t ' ....
• • • .J
: '1.......
I
..
.. '
- ...- _. -- -~-- ~-
1- . :. ,. I.-: ;- -~'7' - -. ~ .
06 ;- --------+~--'-I-'__'•
.07'
10 20., r
__--:.!._-: _._'_.:-ll.~~~-; ..0'; . I.
.05
~." I0+'-"-------'---+------11--- - - ..~--__T_-~-I~10'I"i--
~03 ~.------'''''''-a-..,..--~.
I ~ .
10'........
I ' . I I
I· ";' ,.. : ..I . I' I. , ... j -, . 'I":"j,'" 1,1
"'!i I • r i . I..
j; . I • i. I,Oll...---'-- -.L_........l.-.:~__......L__ ____i'__L ....L..._ ___!__L___I_ __L_'__J..........__l._~·___ll ~ _
10 20 30 +0 .50 60Re~nolds' number (E)I '/n ,a:s un'lts
33
Figures 17, 18, and 19, pages 30, 31, and 32, respectively
are graphs of Velocity versus Pressure, Pressure versus Reynolds'
number, and Friction f ctor versus Reynolds' number. The data
for all of the listed above graphs are taken from Table 2, page
26, ldth the exception of the Friction factor, which was calculat-
ed by using Bernoulli t s equation and the Darcy-Weisbach equation.
Example:
Bernoulli's equation:
2Pl + V1 Sl- -+2g
=
Pl' P2' W, g, V1 , V2, i!'l' and *2 are all known, from this hL can
be calculated. Th n by using th Darcy-Weisbach quation:
~= f ~ y2D 2g
re ~, L, D, V, and g are known, f can be calculated.
34
CONCIlJSIONS
Reference is made to Figures 15 and 16, pages 27 and 28,
r spectively. These graphs of Velocity versus Reynolds'
number are the result of plotting experimental data, and follow
the analytical proof shown on page 29.
From the conditions of the experiment, the coordinate graph
plots on a slope of ~ , and passes through the origin ofd~
coordina.tes. The log-log plot of the data forms a 45 degree angle
with the horizontal. The intercept is not the origin of the axis
as it is in the preceeding case.
The proof clear~v shows that the data follows the mathe-
matioal equation form. of y = mx+ b. In this case of the coordi-
nate graph plot the intercept b is 0, the slopem of the line is
~. In the case of the log-log graph plot the velocity ha acAPvalue ofd~' when the Reynolds' number has the value ot 1.
Example:
Given: A Velocity
To find: The Pressure
(A)
R ~'-V=X.B =f(V)
35
R
(B)
R= f (P)
This equation will give a series of curves.
f(p) = f(v)
The result of a combination of (A) and (B), which are
empirical and analytical, will produce a graph (C), which will also
be a series of curves.
(e)
v
The result of t he combination of (A) and (B), will produce
a series of curveeon the graph (C). In the range of the experi
mental data contained herein the Reynolds' number is practically
a straight line. For this reason, in fitting the curves to the
experimental data straight lines were used.
The graphs of Pressure versus Velocity, and Pressure versus
Reynolds' number a1 0 indicated that there is no shock existing
in the reversal of flow from negative to positive pressure.
36
The intersection of the curves on the plots of Pressure
versus Reynolds' number and Pressure versus Velocity indicate
the point at which the transition from negative to positive
pressure takes place. The crossing of all the lines at one point
would suggest that the Reynolds' number for all the takeoffs
would be constant at that particular point.' Therefore, the
Reynolds' number is constant at the point of reversal of flow
from negative to positive pressure. This graph indicates that
there is only one velocity, at which the pressure is t e same for
all the takeoffs on the test pipe. This pressure is zero guage
pressure. Any velocity greater than this critical velocity, a
positive pressure exists within the entire length of the te t
pipe. Any velocity less than the critical velocity a negative
pressur xists within the ntire length of the test pipe.
Figure 19, page 32 is a log-log plot of Friction factor
versus Reynolds' number. The calculated Reynolds' numbers of the
experiment showed that the flow was in the turbulent region.
There has been no opportunity to examine the interior of the
test pipe to determine if pitting has occured. It is believed
that any change of the interior of the pipe is due to corrosion
alone.
Practical application applied to a vertical pipe flowing full
in a f ctory building where on t\'o'O or more floors there are active
takeoffs of a liquid.
The diameter, roughness, and the various other factors in
volved held to a minimum, the flow of the liquid can be regulated
so that the pressure c an be equalled on every floor, and that a
37
positive pressure maintained at all takeoffs. The elimination of
negative pressures at various points on the line prevent a back
flow, or siphon action on the operating equipment.
The writer woul.d like to recommend that in future experiments
in this line to vary the diameter of the test pipe, both larger
and smaller diameters used. To use liquids of different viscosi
ties. Also, to increase the length of the test pipe, within the
limits of the building space used for experimentation.
From a combination of data of the above variations, additional
information could be detennined and used in design problems of
other pipes of different diameters, and different viscosities with
the utmost confidence.
NO NCLA'lURE
---------- e of cross section
D Diameter in feet
38
d
f
g
H
h
Diameter in inches
Darcy-Weisbach friction factor
Acceleration due to gravity
Total head
Head
hf ---------- Head loss due to friction
~ ---------- Pressure head
bv --------- Velocity head
LengthL
P Tot pr ssur
p ------ Unit pressure
Pf ---- Pressure drop in psi
Q ----- Discharge in cfs
Reynolds' numb r
T ---- Temperature in degrees centrigrade
t Time in seconds
v --------- Velocity in teet per second
UT, 2S ----- Specific weight
A( ---- Absolute viscosity
11 ----- Kinematic viscosity
~ ------- Density
39
BIBLIOGRAPHY ON THE FLOW OF
INCOMPRESSIBLE FllJIDS IN PIPES
This bibliography includes only publications in the
English language pertaining directly to the subject for the
period from 1926 to date. For earlier publications the reader
is referred to the fourth listing below.
(1) Bakbmeteff, B. A. Mechanics of Turbul nt Flow.
Princeton University Pr ss, 1936.
(2) Bard 1 1', C. E. Historical ketch of Flow of Fluids
Through Pip • Publication #44, Oklahoma A. & M. Engineering
Experiment tation, tillwater, Oklahoma, April 1940.
(3) Beale, S. L., and Docks y, p. Flow of Fluids in
Pipe. Journal of the In titute of Petroleum. Technologist.
Volume 14, pages 236-262, pril 1928.
(4) Butler, J. B. Descriptive Bibliography on Oil and
Fluid Flow and Heat Transfer in Pipes. Volume 9 Number 4, 1926.
Technical Series Bulletin, Missouri School of Mines and
Metallurgy. 62 pages.
(5) Colebrook, C. G. Turbulent now in Pipes, with
Particular Reference to the Transition Region between Smooth
and Rough Pipe Laws. Journal of the Institute of Civil
Engineers, Volume 12, Number 4, February 1939, pages 133-156.
(6) DaUgherty, R. L. Hydraulics. McGraw-Hill, 1937.
Chapter 8.
(7) Di Tirro, D. A. Fluid Pressure Drop Losses Through
oth Straight Tubing. Product Engineering, Volume 19,
September 1948, pages 117-120.
(8) Dodge, R. A., and Thompson, M. J. Fluid Mechanics.
McGraw-Hill, 1937. Chapter 8, and 9.
(9) Heltzel, W. G. Fluid Flow and Friction in Pipe
Line. Oil and Gas Journal. Vol. 29, No.3, June 5, 1930,
page 203.
(10) Kemler, E. Study of Data. on Flow of Fluids in Pipes.
Transa.ctions of A.S.M.E., Vol. 55, Hydraulics, 1933, pp 7-32.
(11) King, H. W., Wisler, C. 0., and Woodburn, J. G.
Hydraulics, 5th ed., 1948. John Wiley & Sons. Chapter 7.
(12) Lea, F. C. Hydraulics. dward Arnold & Co.,
London, 1938. Chapter 5.
(13) Lee, C. H. The Flow of Vi cous Liquids Th ough
Pipe. ngineeri • Volume 125, No. 25, pp 498-499, pril 27,
1928.
(14) MOody, L. F., Friction Factor for Pipe Flow. A. S. M. E.
Transactions, 1944, Vol. 66, pp 671-684.
(15) Moody, L. F. Approximate Formula for Pipe Friction
Factors. Mechanical Engineering, Vol. 69, December 1947, pp 1005
1006.
(16) Nikuradse, J. Laws of Fluid Flow in Rough Pipes.
(A translation from an article published in German in 1933)
Petroleum ngineer, Vol. 11, March pp 164-166; &3", PP 75,78,80,
82; June, pp 124, 127-128, 130; July, PP 38, 40, 42; August
1940, PP 83-84, 87.
(17) Nissan, A. H. Flow of Liquids under Critical Condi 10ns.
Journal of the Institute of Petroleum, Vol. 28, PP 257-273, November
1942.
(18) Pigott, R. J. •
41
Flow of Fluids in Closed Conduiis•
Mechanical Engineering, Vol. 55, August 1933, pp 497-501.
(19) Towl, F. M. f. The Pipe Line Flow Factor in the
Hydraulic Flow Formula (Darcy-Weisbach) and Its Relation to
Density and Viscosity. 26 Broadway, N. Y.
(20) Vennard, J. K. Elementary Fluid Mecha.nics. John
Wiley & Sons, 2nd ed., 1947. Chapter 8.
(21) Pipe Friction-Tentative Standards of Hydraulic
Institute. Hydraulic Institute, N. Y., 1948, 82 pages.
VITA
Robert Franklin Tindall, Jr. was born on September 4,
1925 at Sa! t Joseph, Missouri, the son of Robert and
Virginia Tindall.
He received his grade school education in the public school
at DeKalb, Missouri. Further education was received in the
pUblic school at Saint Joseph, Missouri. In June 1943 he
graduated from Benton High School, then w:>rked for the summer
until September, 1943.
t that time he enlisted in the Un1ted States Marine Corps
Reserve and served as a Field Artillery Fire Control Operator
lxnti1 his honorable discharge in December, 1945.
In January, 1946 he enrolled at aint Joseph Junior College
and gr du ted from there with a Sixty Hour (60) Certificate in
Science in June, 1947.
In June, 1947 he enrolled at Missouri School of Mines and
Metallurgy and graduated from there with the degree of Bachelor
of Science in Civil Engineering in Yi&Y, 1949.
He was married to Lorene Kathryn Beger, daughter of Mr. and
Mrs. H. E. Beger of Saint Joseph, Missouri, in August, 1948.
Upon graduation he accepted a tour of ctive Duty with the
United States ~.
In September, 1949 he enrolled in the Graduate School of
the Missouri School of Mines and Metallurgy and has been in that
capacity to date.