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Flow of Air Through Annular Channels

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non or ALR THROUGH AHHULAR CHAHMELB
A
tHesia
ptsBmxtD to m r im s r or
tb *
{Ibai >o a t e 8C h o o l or
CGJOIEU. OJTtTERSITy;FOB THE DEOHEE OF
DQCI08 or raiwBorax
By
Martin Jenkins Barnett
June, 1940
ProQuest N um ber: 10834570
All rights reserved
INFORMATION TO ALL USERS
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In the unlikely e v e n t that the a u th o r did not send a c o m p le te m a n u scrip t
and there are missing p a g e s, these will be n o te d . Also, if m a te ria l had to be re m o v e d ,
a n o te will in d ic a te the d e le tio n .
uest
P roQ uest 10834570
Published by ProQuest LLC(2018). C o p y rig h t of the Dissertation is held by the A uthor.
All rights reserved.
This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e
M icroform Edition © ProQuest LLC.
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,,is
fhe author was born in Meridian, Mississippi
on March 10, 19X5* He attended the public schools
of Meridian and was graduated ibik^*ittridlMhr'idk&'''
S
c
h
o
o
l
'-61 i;i^hiwad'tit
Bachelor'-of1Science1'ill‘Chemist^
College In Juhe, 1936.
of
f
Since that time he has
been registered in the graduate School of Cornell
University as a candidate for the degree of Doctor
of Philosophy*
Chemistry}
In 19S6-S? he was Assistant in
from 1937~4Q he was Assistant in
Chemical Engineering*
Acknowledgment
The author wishes to take this
opportunity
to #3tpress his sincere appreciation to Professor
F« 1.
hhodes, at whose suggestion this work was
undertaken, and under whose supervision it was
carried out*
fable of Contents
Page
Introduction ~ ** «* **.■* ~-> - -* .**..** - * ~ *> - *
1
Ptfvious Work
-
Z
Description of Apparatus ~ ~ - r ~ ~ ~ * .* * *■ ~
9
~ -\* -
~-. - - ^
^
Experimental Work
Calibration of Manometers
~
«*
Procedure
16
19
Data
.Individual 1urns - - .* ~ ~ ~.r ~
Fundaaental,Principles
Sample Computation - ~
$$
- -* » ~ *• *
68
~ ** ~ ~
~~
~~
Experimental Besulis - - - - - - - - - —
~ ~~
-
- —
Discussion of Errors ^
Summary
- - *» ~ ~ -r ~ ~ —
@9
- - -—
Summary of Velocity Traverses
Summary of Friction Factors
~- -
78
81
@7
108
.- - -.- <* —
Bibliography - - - - - ---- - — .- ~ -—
-* *
106
- -
108
Introduction
.-•A large amount of experimental work has been done on the
flow of air through cylindrical channels * We have very com­
plete information not only an to the value of the friction
factor under various conditions hut also as to the normal
distribution of velocities across the channel and as to the
ratio of the average to the maximum (mid-point) velocity*
these data enable us to estimate the rate of flow from a
single measurement of the velocity at the center of the duct*
Tery little work of a similar nature has been done with
channels of annular cross-section*
It Is the purpose of this
thesis to investigate the distribution of velocities in an­
nular channels, and to determine the accuracy of the compu­
tation of the frictional resistance as computed from the
average velocity and the so-called hydraulic radius of the
duct*
An attempt has also been made to develop a method for
the accurate estimation of the average velocity from a single
measurement of local velocity at a specific point in the
cross-section of the duct*
fne utility of such a method becomes evident when one
considers the methods available for measuring average velocity.
Orifice and Venturi meters, which give mean velocity directly,
are not applicable in annular channels*
Weight measurements
are often impractical, especially in measuring the
flow of
gases*
Dilution methods may he troublesome.
Accurate results
can he obtained fey making one or more series of traverses with
a deviee for measuring local velocities, such as a Fitot tube
or a hot-wlre aaeiBoaeter, but such measurements are trouble-
seme end ‘tedtewffi
The value of the Fanning friction factor for annular
ehaaaels was measured under various conditions and compared
with,the values obtained in cylindrical.channels*. The
validity of the hydraulic radius as a true , criterion of
diameter was tested experimentally*
Previous Work
lost of the experimental work on flow through annular
channele haafeeen directed toward an evaluation of the
factor, with the idea of correlation with
the-:fsctor for:circular'pipe*. tome; Investigators have dieeme§sd;'thw validity of the hydraulic radius*
inly a few '
httve e«sidered the distribution of velocity in annular
chajan41eir'
!:moet-of the work on this topic is of-'a theoretical
nabur#* 'hafortunatelyt 'auch of -the experiaental work has been
performed In the region ofwlseema flow under conditions such
that the results are m % readily applicable ,to the chemical
industry,
fo provide a background for the prosent discussion,
however,bhis work willfee briefly reviewed*'
r Atherton (1) determined the value of the friction factor
for the flow of air, water and oil in pipes of annular croieseetion*
fie used ah outer '-pipe %*Z inches in internal diame­
ter :and■inher pipes 0 #84, 1.040, and 1*88" inches ^in internal
diameter*
by substituting four-times the hydraulic radius of
the channels(that is,fourtime0 the cross-sectional area
divided by the wetted perimeter ) for the diameter term in the
ordinary form of :the Vanning equation, friction ■factors were
computed from'the'experiiinatai pressure drops#
The results,
both in the viscous and turbulent regions, were parallel to
and above values established in the past for
similar conditions in circular pipe*
flow under
if f is the friction
factor for circular pipe and $% and fa the Sectors for annular
pipe, Atherton found that '
f* * l%$$f
(turbulent)->> .
f* * 2 U M T
(vAeeous)■
'
Moody (1) objects to these conclusions, stating that
the only
proper basis of comparison forclreul&r andannularpipe
is
an ewperiaeai lawhiehthe two types of cross-sections have
tne aame mean hydraulic radius*
Krat2 , a<^uld, and MacIntyre (S) determined the relation
between f and the Meymoldslmsiber using/water and calciuu ■
chloride brine*
l i m m annular eross-seGtions and tv© circular
cross-sections were studied*
For the annular channels, a.
standard 2-iaeh wrought iron pipe was used as the: outer-,easing}
core# consisted of 11/4, 1, and &/4~ineh pipe respectively*
For the cylindrical channels, 1 i/4 m&- B**Inch pipes were, used*
A graph is given for the annular channels, all three channels
being represented by m single moan curve, although some ten­
dency for- a separation into throe sections- was noted*
s
with
of cros^sections,. the .channels, could no longer
he • considered -geometrically similar*^ Onder -m m conditions
the use of toh©*wea» hydraulic radius would m
longer toe valid*
ftoey conclude that m m r a wide range of conditions the- use of
the Etynolde Humber affords a means of correlating frietlon
factors*
■?■■■■
•■-■
Syateaatic experiments on the flow of air, vetoes end
steam through annular channels were made toy Becker (S)<* Me
eoneluded that the critical value of the Beynolds husaber
(tfemtfi, th«
which the flow shifts from the viscous
to the turbulent regies) is 2700#
Experiments by ihiller (3)
hsve iMicete^ that *th#' critical value ■for circular tubes -is SSS0* 'Therefore the error tnvoXvedbymsing heefcer’s value
*M^'iheili3Nlrettlle' <*a4i$e fe^aet greet!-■
'&e&a4ale (4) studied the flew of fester through vertical
anfiular ehaimels a M developM an empirical equatioa for pretl*t&«Mfpres*f»re4rep*4
ihe-veleetty at which viscous flew
ehanges^tO’turfeuleat -flow eg# -obtained graphically :for each
ahcui&r chaituiel fey plotting pressure gradients versus velvet*
ties oh .arithmetical paper and obeerving- the point at whibh
the curve first deviated from a straight'line^*
Iontdala
etates that it is nacesBaryte adept ‘Some empirical relation
to fetpress the varietion of critlcal velocity with the diameter
of the annulus sinee the linear dimensions of each systeEa eennot fee satisfactorily exprfessSdby a■single parameter *'•>■r
;
:, r Fiercy/aoopef,;and Winny (§) eemducted a thorough - 1
investigation on the offeet produced fey eccentricity of’the
inner pipe*
Ouch datoe-are important since small eccentri­
cities mayfee expected
inany experimentalaanuXias* due to
the mechanical difficulty of supporting a cere centrally
inside a pipe with accuracy except at isolated positions
along the length*
they give a graph shoving howtfee ratio
of the diechirge through an aanulu* with eccentric core to
the discharge through an annulus with central core at constant
pressure- differential: varies with the* degree of eccentricity*
them. the eccentricity 1* sera, the ratio is maltyf wh*& the
♦oesntrlcity laX (turner pipeto^ciiingthe outer pip*);the
ratio is 8ir£ f^r a diameter ratio of @#$*
Another graph shews
how tli# ratio of the resistance of a pipe with cor* to that
of the plpe withcmi cor* varies with the diameter retieof
the- anaul&sir^Wwo-the^iieatter rati© is aero,-.the ratio of
v#0i*$att«**>:lA mityj when the diaaeter ratiols ©.£> the
weeis%iMae**"te.’appro*i#»toi^^^^
si **<-*Poteeeot
'""■-
*WimNM*d*eil:much of the existing d&ta-her
H o w -throw#^ hew^ttrcuter <er*s*-s*£items *impah -effe*t¥tautest
*he^e*e*^lMrd*aia,4e •'retie* 4a~ tfc*' regies of
turbulent-flow*
Substituting the hydraulic radius for the
diameter tern iri the fanning equation, he computed friction
factors from the emperimentat values for pressure -drop end
velocity*
She result*'om rectangular sections show that' the
frletlon faster is the same as that obtained for circular
pipe*
this Is anticipated from the feet that the dimensions
wera v&rled and tfce Saiie swfsees kept In contact with the
fluid*
Because of similarity In shape, the roughness effect
?, ■
=•• *:>■ •>■■■
.V ■:-'
•
should compare in aiaghitude to that for Circular pipe* the
results on annular cross-seetions showed a wide variation in
friction factor*
Furthermore* the friction factor seemed to
increase slightly as the diameter
of the inner pipe was
liter*#**#* Peters explained this by assuming that the rela­
tive roughness of the atsnulus increased as the diameter of
the-fnaer pipe increased# me concluded that the friction
factor is higher for annular channela than for circular* the
magnitude of the increase depending upon the ratio of inner
pipe diameter to outer pipe diameter andon the roughness of
tte ^ptpO^vFUndaiaental work on the distribution of velocity in
tmbesof circular cross-section was performed by Mtanien (?)
aad by Stanton aad Paaaell (B>*
their graph of W ff / i
versus
t ^ T ^ has been widely used for obtaining average veloci­
ties*
Morrow (s)# filienradse (10) and Townend (11) have made
similar experiments*
Nikuradse (5) has also studied veleefe*
ty distributioh in triengular and rectangular sections*
Similar data for annular channels* however* are very meager
and incomplete*
he* (18) performed one experiment on the velocity dis­
tribution in an annular channel*
The velocity of water was
determined simultaneously at six points across an annulus
formed from a 5-inch cast iron pipe with a 5/6-inch core*
Me plotted r/R versus v/V where
r
R
v
?
*
*
*
*
distance from center of pipe
radius of outer pipe
velocity at distance r
velocity at center of annulus
and proposed an equation of the type
The mean velocity occurred at 5/4 1 and equalled 0.88V*
In the region of streamline flow, the velocity distri­
bution in an annulus has been computed by Andrade (15)*
From
this equation an expression for the point of maximum velocity
in terms of the distensions of the annulus con he developed*
Andrade presente a chart showing how the point of maximum
velocity changes with a change in the ratio of inner to out­
er diameter*
Even when the ratio of outer to inner diameter
position of maximum velocity is noticeably
displaced from the wall of the inner-pipe*
Lea and Tadros
(14) present similar equations and. include experimental
results on
streamline .flow through tubes of very small,
dimensions« In the region of turbulent flow* complicating
factors- are Introduced and the problem of velocity distribu­
tion has not been solved on a theoretical basis#:
-0-
Description of Apparatus
The apparatus used in thee® experiments consisted
essentially of an annular channel made by supporting a rela­
tively small cylindrical pip® concentrically in a larger
outer pipe#
Air was blown at a controlled rate of flow
through the channel#
A Pitot tube was Installed so that
measurements of local velocities could be made at known
points along a traverse of the channelj pressure taps spaced
along the outer wall permitted the measurement of the
frictional resistance#
In some cases the channels were form­
ed from standard iron pipej in others standard steel tubing
was used-#
A sketch of the apparatus in which steel tubing was used
is shown in Figure 1(b)#
the blower discharged through a
control damper, set in a 6 5/8-inch line, to a galvanised
sheet metal reducer, 4 inches in diameter at the outlet#
Connection to the 4-ineh outer steel tubing was made by means
of a rubber connector taped in place and fixed securely with
a metal strap which could b© tightened against the connector#
tne same method was used to connect the two lengths of 4-inch
steel tubing, producing & straight channel 21 feet long#
Pressure taps were placed at 1 and I by drilling 15/04inch holes and soldering 1/Q-inch nipples in place#
The
static pressure tap at f for the Pitot tub® was formed in a
similar manner*
The inside of each opening was made smooth
figure 1
DIAGBM OF
k w blower
B w control damper
0 a* sheet metal reducer
D * standard flange and spider plate
1 * upstream pressure taps
F * static pressure tap for Pitot tube
0 * impact pressure, Pitot tube
h * Standard flange
1 * downstream pressure tap
3 * centering
device
K * rubber and strap metal connector
lb
FG
(TJ
i/)
D
5
UJ
rr
Z>
O
FIGURE
ir
£
CL
<
id =
oil
<
a:
O
<
Q
<
<
—XO—
whll of the 'p ip # *
and1fidsh with th«
fortbW-impact--*r« of the Pitot tube
the opening •at &
was made by drilling a
15/64~ineh hole and welding over this a short section into
which the i/2*intb
-of the fltot- tubi-ccmld be
ierewSi*' The Pilot tube could then be easily adjusted or
imiirfetjrremoved" fi»m-tWs'’ptf»#
^ -the”esmulervchattel was-fotttaSr- by suppowtlug •a length
of steel tubing within this cylindrical outer casing*
The
individual Icngthfe of inner- tubing' were eohneeted by iceotts of
solid' wdodew cylinders 'Which slipped inside"the■tube and sere
Ield:'ih'place-if%h wood Screws, countersunk toMnimise flow
diStUrbshOOik
:,f lTbe i m n w ^
J in the following manner*
three"' l/X6-inch' holes - were equally spaced around the circum­
ference of the outer plpe#
Through these-holes passed thin
wire which was:clamped around the circumference -of the inner
pipe and fasteaed on the outside to a bolt'with a long thread*
this-bolt, which was provided with a washer and nut
on its
Other end, was supported within a nipple which bore directly
on the outer wail*
By adjusting the nut which rested on the
hippie,*tension could
be applied to the wires and the inner
pipe could be rapidly and accurately centered*
This method
of centering, aside from its simplicity, has the further
advantage of introducing only a very slight disturbance to
flow conditions*
The very small wall thickness and consequent
light weight of the inner tube made the method applicable
lathis ease*
It
was mat entirely satisfactory ia the ease
of standard iron pipe*
- The arrangement of the apparatus for use with standard
iron pipe 1* shown in Figure i(&).
la M s
ease the adapter
radioed the diameter to $ iaehes, connection to the outer
pipe feeing made with d standard^ineh flange*
■ the two
length* of S~ineh pipe were likewise connected with standard
flanges*
Piezometer rings were used for the upstream and down'.t
stream pressure taps and for the static pressure,tap for the
Pitot tube,
These piesoaneter rings were,joade fey drilling four
holese^ualiy spacedon the circumference of the pipe and
tapping t& t 1/e-lndh nipples*
tees were attached to the ends
of.the nipples so that"the four openings could fee connected •
with mfeberthbing and ferought into a single channel fey means
of § glass f-tube. The opening for the Pitot tube was made
by Irilling and tapping a 1/2-inch hoi©*
When Inserting the
Pitpt tube* Ife was necessary to make certain that the threaded
brass plug carrying the impact tube was flush with the inner
wal| of the pipe* thus malting it possible to set th© tube at
the wall*
For centering the inner pipe, wires were again used at
the flange Junction near the Pitot tube*
of support was
Some sturdier means
required at the two ends, however, because of
the greater weight of the inner pipe*
This was provided by
open spider plates bolted between the flanges*
These plates
-12-
were tapped and threaded for the largest siae of inner pipe
and then provided with bushings for all smaller si2.es.
Cape
were screwed over the ends of the Inner pipe to prevent flow
through it.
the separate lengths of inner pipe were connect­
ed by standard threaded couplings*
The dimensions of 'the.various annular channels studied ‘
®re:'given in 'the following -tables."
Table 1
Standard Steel Tubing
■Outer lips..
Inner Pipe
fall*; ;
I.B#
(inches)
(inches)
4
3*334
0*033
0*30
0.049
3*334
4
. 3*334
0*033
1*000
0*040
1.334
4
,3*334
0*033
1*300
0.049
1*334
3*334
0*033
1*000
0*063
1*334
.3*334
0*033
3*000
0*063
0*334
,,
4
^
4 ,
Thickness
(inches)
Total length
0.1*
fall
. Diameter of
Thickness
Annulus
(inches) (inches)
0*1*
(inches)
•" $1 feet'
Distance between pressure taps® n « 0 s feet
■
-
.•
Standard
Iron Pipe
Inner Pipe
Outer Fipe
G.Jj.
(inches)
■’ ISiii
thickness
, X.D.
(inches)
(inches)
0*0.
(inches)
fall
Ihickaess
(inches)
liameter of
Annulus
(inches)
o.m
0.840
o.oaa
2.498
3*036
0*881
0.875
0.091
2*388
8*500
8*08§
0*881
0*840
0.109
2*198
3*500
8*088
o+m
1.05
0.118
1.988
8*800
8*088
0*831
1*515
0.188
1*728
8*800
' 8*080
3.500
total length
*
80 feet
distance between pressure tape « 88.0 feet
the Pitot tube (Figure 2) used throughout these expert-*
meats wee meg# in the shops of Cornell University.
It con­
sisted of a steel impact tube, 0*118 inches la external
diameter and 8*8 inches long, soldered to a brass radial arm,
0*190 laches in external diameter, which passed upward through
a threaded brass plug
and stuffing gland.
A hair-line
pointer, clamped to the upper portion of the radial arm and
operating over a scale graduated to 0*01 inch, was used to
indicate the position of the impact tube*
An additional
Figure 8
PltOf fhil
A ■« impaeb tub#
B * radial arm
C « threaded brass plug
B * packing gland
1 « steel scale
F » adjustable pointer
G » fixed pointer, aligned with impact tube*
n|©G G
15
-
-
T ^r
dbl
h3h2-
D
O
C O
B
PITOT
TUBE
—14pQinter,e©lder*dtQthetep of the radial arm and in line with
the impact tub#, assisted
$o
$m
the adjustment of the Pitot tube
^inpaot^ttSfe# pointed dlreetly upstream*
This im-
pdefstub*# ineon j\mctienwiththe static tub# described
abov*> was ussd to determin# the vslocity at various points ^
1.
Differential two-liquid manometer
The nature of this work required a manometer with a
range of approximately 1-12 centimeters of water and capable
of responding to pressure differences in the neighborhood of
0.01 centimeters wCwater*
From a practical point of view,
other important faster# ■were-ease and rapidity of taking ‘
read­
ings, reproduc ibillty and expense,
dfter- a consideration of
the$efftetors,&twe-liquiddifferentlalmano]&eterusing
’
ethyl' mleohoi^ -and '-petroleum bensine as the measuring liquids
was chosen.
These liquids ere readily obtainable and are-met
trouble.
Preliminary investi­
gation Shewed that the two liquids separated rapidly after
shakiGs tegether, glving anaarly flat menisens.
Tne two
Uquids showed -no tendency to stiefc to the side walls of the
emfSgriM^Ma^v'--'',
:
'-r
i
S
■
H-tmtbe msaometew wascoaetructed from ©-millimeter
Pyrex tubing and two
U
1/S—inch round bulbs.
Connection
between the bulb outlets and the ©-millimeter tubing was made
with rubber connectors to facilitate removal*
The lengths of
8-millimeter tubing were connected in a similar maimer at a
poXnt } ^ t^ w & r 4mm t ^
mBnom%iiT, providing a total length
of BgS eentiffieter** A short section of capillary tubing was
inserted at the feotiamof the B-twbe t# minimise fluctuations*
One stopcock was provided at the bottom for withdrawal and one
between the two
a#. a by*psas,
Two such manometers were
mounted on a hoard and connected to the Pitot tub© and pres­
sure taps with rubber tubing*
Ueber sticks, graduated be' :
1 millimeter, were mounted between the arms of the ft-tube for
reading the liquid height.
■
.
<\
-<.jKimrmMMmetwy-- .-:
.■■ti•■■■-For a few rm $ at extremely low velocities, a sensi­
tive mlcremamoaeter capable of reading to 8*01 millimeter of
water over:- a range CMS»S centimeters of water was, meed* .This
micromanometer was constructed at Cornell University and has
beeafully described by Winding and Shades (IS)*
The instru­
ment required no calibration, readings feeing given directly
in millimeters^.of water* a
The static pressure at thepttot tube, used In calcu­
lating.;,the pressure of the air, was obtained from an ordinary
U-tuba manometer filled with- water * All manometer connections
were made with new rubber tubing*
•16«*
Experimental Work
fhe multiplication Smmtm* for any differential twewliquM
manometer can bee&lenlated from the specific gravities of the
r,
^
;; *•*#* * (•***&)(&**&&)
where ?*«#*, * differential pressure*
g#«ge..*;':dlfferenow in specific gravity
of the two liquids*
'»•■'Weeding on manoiieter*
For accurate work, however, it Is advisable to determine the
relationship between F»-?i and h8~h* by ah
accurate direct
ediihratioh*^'•
Ordinary 9$ per cent ethyl alcohol was meed as the heavy
liquid! petroleum bensine (b*p# llG*~ltG*C) was used as the
light liquid#
the two liquids were nixed in approximately
equal amounts and agitated thoroughly*
A few drops of Congo
red solution were added to the mixture* Imparting an orange
color id the ethyl alcohol#
this made it possible to locate
the interface in the manometer easily#
fne liquids were
flowed tq remain in contact with each other overnight#
On
the following day they were separated and their specific
gravities measured at three different temperatures#
ny plat­
ting specific gravity against temperature, a line is obtained
whose slope indicates the rate of change of specific gravity
with - t*aperatura# A comparison of.these values far tha two
liquid* i#of importance In determining how the multiplication
factor change# with temperature#
assuming the relation given
above* the factor was ©alculatehfor -the -three M t im m t
temperettu*e»4 Hie weeult* are »uapiarlaed helow«
M
applies to
-
fable&
all w m s of Series I* Table 4
applies to all rune In herlea 11#
-
*
ip 7 " 1 7
v-WMmy*-..
~ *~
Temp*
Ethyl
0p#1^ ^
•C IlfWWHalcohol
Benzine
II^HUIUI.»|
J'-HXW.
IWI»||»B'L**<I||>|||HI
»II—
.xi^ii
'Oig^wwaii Factor'
20*
0.7896
3.7387
19.68
88*
0,7887
0.7 88 8
19.84
80*
0.7819
©,7817
19,98
Bp* G*
teap* ..Jttxl.
Bpo a.
#C •';; alcohol B e m im
A^ ^ K l e o t i 9 l * - * 00076
^
i
h e n z lm * ~ '° ° 070
factor.
to*
®4?au
Q.'twit
& i* l
88*
0*7809
0.7340
2 1 .3
80*
0,7777
0,7808
W ^ le o f a a * ~*0006e
*
^
&
f
H e n e ln * * — 60064
the irate:©T change of specific 'gfarity with ’temperature
la essentially the sane for the two liquids, so that the factor
is aimost constant over the range of temperatures covered*
the manometers
ware
only a single
%n t & ia range*.. . „.,; .
..:tty apr develop. m
to pe^it-e&XettXetion of the
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aquation
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where F*. « exporiment&lly determined factor at T*
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speoifie gravity of ethyl alcohol
% *
specific gravity of petroleum benzine
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In preparation for calibration, the two manometeri were
filled with the measuring liquids and carefully adjusted until
the'interface was near the middle of the O-tube*
The two
differential manometers, together with an ordinary water
-10-
manometer, were titm connected In parallel to the FItot tube*
Simultaneous readings of the three manometers at., different
velocities permitted calibration of the two manometers*
the
graph showing the variation of the readingon the differential
manometer (in centimeters) with the reading on the water
manometer (in centimeters) gave* for each manometer, a straight
line, the slope of which represented the multiplication factor*
The results of the calibrations for Series X and XX are given
in fable 3«
Calibration graphs are shown by Figure 5*
fable 5
Series X
Merles XX
©iff*
U-tube Oiff*
Mff*
©iff*
U-tube
Manometer He* X Manometer Ko* 8 Manometer Manometer
80* 1
50* 8
8o * 8
cm*
Ho« X
cm*
cm*
H*Q
cm*
H*0
cm*
cm.
10*4
88.7
m *m
04*1
40*0
49*8
66*0
04.4
04*0
110*4
100*7
146*4
104*4
0*7
1*45
1*58
8*07
8*78
8*90
8*97
5*07
5*65
6*58
7*88
8.87
8*97
temp*
17*0
85*4
58*0
45*1
51*9
68.7
80*4
96.4
101*8
188*4
150*6
140*8
158*8
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8*91
5,90
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5*60
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6*96
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54.0
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55*3
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187*3
151*5
166.6
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63.7
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y-tube
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3*30
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9*80
Temp*► * 7S.3*F
ix m s & m
fhe purpose of this work, as stated in the introduction,
was to make a rather thorough study of the distribution of
Figure 8
CAXJBBAfXOB OMFH FOB OXFmSXSXAb
MA80M8XBB6
Graph X
*
Meries
X - Manometer Mo* I
Greph 3
m
Merles
X «. Manometer Mo* U
Graph 5
m
Merles XX - Manometer Mo* I
Graph 4
*
Series XX .* Manometer Me* U
It
S liy is
i simis
y ilV M
aiivM
SH3131NI1N30
syiniAiiiNio
—BQ—
velocity in the turbulent flowwf air through annular channels,
The usual procedure In obtaining these data vas as follows.
An annular channel was built up by selecting one of the inner
pipes to be studied and carefully centering it within the outer
pipe.
The damper on the blower was then set at some fixed
point and, after.equilibrium had been attained, a traverse of
the channel was made with the PItot tube#
were, arrived atoim twe different ways*
Bitot tube settings
In one system, readings
were bake® at;certain' arbitrarily-selected; points which were
usually edmidiatout from each other and located on even
divisions of the scalef In the other'.system, the settings were .
calculated in such a way that each reading'-represented the
mean velocity through an equal portion of the total crosssectional area*
From a complete traverse carried outln this
manner, the average velocity could be computed In two different
waysi
(X) average the point velocities obtained at the calcu­
lated Bitot tube settings, ($) integrate graphically over the
cross*»e©tional area of the annulus# these
two methods will
be described in detail in the Sample Computations*11
BCfer# each rum, the following readings were taken*
Cl) Barometric Pressure
{2} Zero reading on both differential manometers
At each setting of the Pitot tube, the following readings were
usually takens
(1) Differential Pressure at Fitot tube
(Manometer Bo* 1)
(£) Static pressure at PItot tube
(g) Pressure dropdebvton ujastream and. downstream
pressure taps (Manometer So# 2}
r
(4) Dry bulbteaperuture
*
'-’
U ^*>.r iWb .iiuidi .t«skp«ri4tMr#.:^.-.>
.
;;Nv:C^
•.., .; .
iv- w.
After changing the setting of the Pitot tube, sufficient
time was allowed for the manometers to reach equilibrium under
the new conditions*
At the completion of a traverse, the
damper setting was changed and th e procedure repeated with the
new velocity*
This was continued
obtainable had been covered*
The
until the range of velocities
Inner pipe was then removed*
a pipe of different diameter substituted for it* and a new
group of runs completed*
At some convenient time during the experiments, similar
runs were performed with circular pipe in order
to obtain
certain information which could be used as a basis of compari­
son with the annular channels*
In some cases traverses were made across different radii
of the annulus, with a constant rate of flow of air#
It was
found that the several traverses gave results that were nearly
identic?*!*
For this reason, in most cases a single traverse
along a vertical radius was made*
-82-
■0■r ,•
1A1A
■BomeSalature'■*■
■-
A * differential head at Pitot tube, cm* manometer So* 1
B * differential head at Pitot tub©, em«
Ha0
C m point velocity at' indicated Pitot setting, ft ./sec*
D » g/jrp
B * radius to center of impact tube, feet
V « velocity at point B, ft*/sec*
I at barometric pressure, mm* Mg*
-^
P * static pressure at Pitot tube, cm. h*G
i * pressure drop between taps, cm*
Manometer Bo, 8
H » pressure drop, cm, laG
I • pressure drop, lb*/ft#*
J * aw*
dry bulb temperature •**
I a* av* wet bulb temperature *F
X» ^ temperature of manometer liquids *C
§ « setting of Pitot tube
Key
Series £i
Outer Pipe * 8" Standard Iron Pipe
A * 1/4* Inner Pipe
B » 8/8* Inner Pipe
C
» 3/8* Inner Pipe
D * 8/4* Inner Pipe
B « 1* Inner Pipe
0 * Circular Pipe
Buns 1,2,3, etc* * runs at varying velocities
for each annulus
-IS*
Series H i
Outer Fife * 4» Standard Steel Tubing
A * 1/B* Inner tubing
B » 1» Inner tubing
C * X#S* Inner tubing
'jb * S*0# Inner Tubing
g m.§*§” Inner tubing.
-0 * Circular Fife
Buns !*$*&# etc, « runs at varying velocities
lor each annulus
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90,36
98,08
07*01
100.77
100,03
111*43
108*33
Jj
I
jr
&
JC
730
4.9®
97*20
3*41
11*08
m
72
m
730
6.10
101*4
9*77
13*02
97,3
70
03
0
03.9
63,6
89.8
m i
«4.e
78.1
r#.i5
n.w
58.85
55.80
-59-
Serles IIS
fr*
.
8
A
S
5.00
4*90
4*30
4.70
4.60
4*50
4.40
4.30
4*80
4,196
4.335
4*034
4*333
4*433
4.838
88*0
107*4
188*5
154,0
148. X
144.5
153*0
126.95
103.4
103*4
33*8
lit,©
137.1
143*0
131.8
4*58
5*33
6.65
7.38
7*78
7,83
7.35
6.90
3*3©
5*77
4*83
3.38
7.45
7*88
6*58
*
■■-9":
a
I
I
JT
f
'*
8
?
83.6
103*85
113.36
118.83
188*40
183.48
181.04
116.73
106.86
105.88
06.76
111.33
180.8
183*13
118*01
T#0
V.&
M 5.0
8.10
16.54
#7
70
84.6
D
A
B
C
D
91*05
90.0
98*65
97.0
03*6
88.0
79.9
70*8
58*5
58*5
99.3
188.3
145*6
159.0
168,4
171*8
165*7
151*4
188.9
186*7
106.8
140*3
168*8
171*1
144*5
5.48
6*97
7.98
8,63
9.17
9,38
0,08
0.24
7.00
A QO
0*00
5*76
7.68
8.84
9*38
7.80
108*4
116.08
183.78
189.31
135,17
134*86
152*09
186,8
118*57
115,38
105,38
181*43
150,71
134.83
183.87
99.7
107.0
107.4
103,9
102*0
95.8
07*05
76.7
64*65
63.65
750
8.85
170.1
9.98
19.48
98.8
70
86.0
60Series IIS
<
Bua
ft
A
B
C
5.00
160.6
7.40
4.6ft
166*3
9.00
4.30
160*5
9.64
4.6ft
191.3 10.45
4.60
100.5 10.06
104*4 10.06
4.7ft
4.70 9 « h m
9.00
4.696
6.00
165.5
**
%
4#80
4.00
♦.t o
*
I
h
115*4
184.6
186.8
126*7
122.1
117.0
107.2
106.5
A
20*2
23*5
23*6
27*1
27.0
25.4
22.8
22.8
A
B
m*m
2*04
8*40
8*62
8.7ft
8.77
2*61
2*34
8*04
44.86
40.20
51.05
00*90
40*00
43*1
43*1
m
1.08
1.27
1.09
1.46
1.46
1*37
1*22
1.22
90
65
24
45.15
48.86
31.17
58.57
02.54
50.93
40.81
48.81
P
45.95
46.0
47.15
47.1
45.65
42.9
39.4
39.35
731
4.1
84
64
24
4
C
62.26
67.04
70.66
72*69
78.56
70.43
66*69
66.69
7.00
C
ft
0
Bon
5*00
4*88
110*7
131.0
167.0
141*15
140.47
168.66
131*24
130.6
P
751
96*6
100
70
94
F
..ft"
2
L
D
00.55
64.05
@5.0
65.15
@3.0
39*4
54.5
54.45
A
55.5
03*1
71.7
70*8
73*7
71.0
63*9
60.9
B
C
73*34
01*92
86*04
68.50
08.47
ft.ftft 80*94
3.47 81.09
0.47 61*09
3.02
3.34
3.90
4.12
4*18
751
10.16
69
67
20
P
70.5
77.55
79.4
79*03
70.9
78.40
66*0
@6.4
-61-
Series IIB
Hun
6
I
A
8.00
4.98
4.90
4.86
4.80
4.76
4.70
4.896
^
B
C
73.4
4.00
*6,6
96.4
4*78
100.1
100.8
94.8
84*9
84.9
6.44
Wn ^ i i w
6
y
^
67.2
94,78
99,84
101.7
101.79
96,95
@3.78
98,78
5,18
5.45
5.15
4.68
4.68
64.93
89.9
91.8
91.0
86.5
83.45
76.5
76.45
A
80.4
106.6
117.7
184,85
183.6
116.9
105.0
105.0
n ii i . , i ^ l . . 6 . i n w iw 4 w i-WW M .
I
K
L-
781
13.85
98.5
68
S3
Him
7
V
3
B
4.91
5,78
6.89
6.75
6.78
6.84
5.70
5.70
C
D
96.68 94,05
104,9 99,4
110.8 101.9
118.36 101.7
113.11 98.5
109.86 98.5
104.17 85.8
104.17 85.15
«■>' ■
*■■—
—
—
781
15.96
94.0
68.0
83.0
8
S
A
B
C
5.00
4.95
4.90
4.85
4.86
4.75
4.70
4.696
107.8
186,9
139.3
147.5
147.3
139.6
184.2
184.2
8*88
6.90
7.58
8.02
8*01
7.68
6.74
6.74
105.19
114.53
1*1.04
123.52
123.41
180.04
113.24
113.24
S
F
D
D
108,4
108.7
110.9
110.8
107.3
101,8
92,6
92.5
A
B
C
186.4
158.5
166.6
176.2
176,7
166*5
150.0
150.0
6.67
8.30
9.07
9.60
9*58
9*07
8.16
8.16
113.93
186.83
130.94
134,71
134.53
130.90
124.16
124.16
3
731
18.55
94
781
81.6
95
X
68
68
I
23
88
D
110.9
119.0
120.8
180.7
117.0
U0.3
101.7
101.4
nomenclature
ftuasasry of Velocity traverses
F * Static pressure Aft Fitot tub®, cm* ha0
1 * mmijmm velocity, ft*/®#®,
ft *. average velocity obtained by graphic integration,
O' #..|^rage ^iocity from calculated Fitot tub® setting®^
f
Q
* ratio of average to maximum velocity
setting of Fitot tub® at which maximum velocity was ob­
tained
B * setting of Fitot tub® at which average velocity (found by
graphic Integration) was obtained*
t * dietance from outer to inner wall, inches
P * distance from outer wall to point of maximum velocity
f « p/f a fraction of radius of annulus at which point of
maximum velocity is located, measured from outer wall
1 « difitanc® from outer wall to point of average velocity
X m f/f-fraction of radius of annulus at which point of
average yelocity is located, measured from outer wall
X *,.§$$ m $[,m fraction of distance to point of maximum velocity
at which point of average velocity is located
-63
Summary o f R es u lts
Series XA
%
Hun
S
ll*'
@7*1
£}<?•«•<7
59«@a
5.70
2*2
46.42 27.37
40.23 33.47
4*10
,305
•376
*301
.374
'w.wS®
4.67ft
1*249
.974
.781
*331
.805
•391
4
5
6
23*0
108.9
93*63
26.6
112.0
101*14
36.2
123*9
110*8
47.8
40*9
144.5 136.5
150.33 123.09
.906
4.20
4.735
1.131
.856
.788
•881
♦278
.875
.904
4.83
4*78
1.131
•386
*700
♦336
.234
•406
*904
4*25
4.78
1*131
.306
•032
•326
.276
•404
.902
,903
4,24
4.20
4.78 4*74
1.181 1.161
.816
•306
•690
•382
•356
.316
.267
.284
.416
.387
40.8
188.87
107.tx
%
$
1
f
.60S
.361
#305
*379
1
f
*781
•391
.313
•401
4*678
1.249
1.006
•803
*384
.303
•333
249
.803
•396
*317
*394
4#
1*
.936
.766
•331
.305
*393
Series IB
8
U
H
0
P
Q
u
V
1
X
X
f*48
,.
U*,v'
i.94
*o
4.
4#
1.
•
*
*
18.15
78*0
70*74
18
**
•901
4,20
4.78
1.181
Q1S.&
«O0W
*785
*
*
8m
*876
.331
7
6
»64*
$#rim 1C
Run
i
f
k
H
0
f
1
I
f
!
w
X
*
1QX, 5
m m
m M
4,54
*k*7
X.099
*7X6
*isi
*866
.E61
*
84.0
90,4
68.69
88*88
♦9X0
4.88
4,76
1*099
.786
.070
.990
.970
.402
9
10.5
80.3
73.14
70.76
*0X9
4.35
4.77
1*099
*700
*044
.900
*000
.405
4
S
12.25
48.0
09*4
119.0
99.00 108.8
99*91 109.87
•914
*9X44*94
4.82
4.740
4.77
1.099
1.099
1*4ir>
*7919
.66E
.070
.£86
.311
.861
.903
•880
•493
0
49.1
107*9
115*99
110.79
.911
4.31
4*703
1.090
:SS
.301
.074'
.408
7
6
7*3
3.5
49.9
34.7
40.38 31*30
•
*
*910
4.33
4*773
1.099
*706
.644
.093
.809
*401
.900
4,35
4.79
1*009
.706
.644
*366
•843
.377
7
8
8©ri#s 10
Bun
F
I
S
f
F
«
i
f
u
V
f
1
f
1
8
42.6
40.75
135.S
188.6
183.18 117.18
184*36 118*64
*911
-#ixi4450
-4*80
4*838
4*833
*994
*984
*556
*366
•560
*300
*224
,228
.825
,225
*401
.403
3
4
88*6
24,8
100.9
113*8
105*44 02,00
108.78 93*18
*213
•911
4*50
4.50
4,831
4*835
•994
.904
•536
.536
*560
.825
.223
*227
.224
•404
*401
5
15*9
79,9
78*86
75.62
.905
4.505
4.848
•994
•351
•568
.208
.209
.577
6
11,9
69,2
62.82
63.56
.907
4.49
4.33
•094
•566
.570
.826
.223
•399
7.40
88.7
48.32
49,39
*904
4*50
4.34
.994
.556
.560
.816
.217
.338
4,00
33.8
34.54
m
,891
4*50
4*87
.994
*356
•560
•186
.187
.335
ieriee IS
Stto
w
17.8©
i
8
Q
f
5
4
’*0
i0
'*6
101*45 1XX.07
*004
«I
4*03
4*
4,
12*3
79*8
7i;95
ST
4,60
4*34
.36X5
4*03
i
t
a
53*6
154*4
123.5
184.,i'
»0?3
«!
i
%
*<
5*3
43*30
33*61
08*30
*9X4
4*30
4*300
*36X0
*406
*383
#JS03
.886
*646
X.TO
36*63
34.15
34*47
*903
4.60
4*33
*36X6
*466
*689
*176
*304
*036
Series IXA
X
4*70
f
142.7
M
S . 127*22
187*33
0
» 0*398
■0*7$
■.•4^i60
1
f
0
7
I
t
0
1*876
0*703
0*466
0*874
0.558
2
1*06
60*5
08*7
62*34
0*378
0*79
4*673
1*667
1*286
0*76
0*431
0*839
0*03
0
1*56
73*6
69.01
69.51
0.877
0*79
4*30
1*86?
1*833
0*73
0*466
0*874
0*831
4
5
3
2*45
99*16
87.46
88*17
0*881
0*80
4.61
1*337
1.256
0*764
0*443
0*838
0.556
4.0
185*45
118*52
1X9.00
0*883
8*775
4.61
1.867
1*331
0*77
0*446
0*238
0*848
8*3
124*3
1X1*06
110*93
0*301
3.79
4.59
1.667
1.286
0.76
0*466
0*28
0*363
7
3*0
118*5
99*85
100.17
0*383
5*79
4*80
1.867
1*366
0*76
0*456
0*374
0*361
8
2 .0
36.4
77.98
78.18
0.864
3*60
4*605
1.667
1.856
0.764
0*465
0*272
0*561
IXB
Kxm
F
I
1
0
f
4
1
*
0
f
f
1
!l
....
"
1 "
5*05
144*4
150*1
ISO.08
0*90$
:?4*3U»
4*57
1*417
0.956
0*661
0.566
0*67$
0.415
”
.
'9
$
0*60
49*44
44*09
44*$7
0*094
4*4#
4*67
1.417
0.966
0.676
0.53$
0.971
0.401
X*70
71*9
64.15
64.81
0*694
■ 4*10
4.68
1*417
0.956
0*676
0.576
0.266
0*594
4
2*50
90*0
81.15
81 #46
0.901
4.10
4*63
1.417
0*956
0.676
0*379
0*068
0*597
5
5*55
106*5
33.04
95.81
0,801
4*13
4.66
1*417
0.926
0*654
0*376
0*366
0.406
7
8
4.65
137.0
113.93
114.53
0*895
4.14
4*60
1*417
0.916
0.847
0.576
0*366
0.411
5*35
157*06
103*35
105*99
0.898
4*10
4.67
1.417
0.936
0*861
0*584
0.071
0.410
6
4*00
115.43
104.13
104*6$
0.903
4*14
4*68
1*417
0*916
0.647
0.376
0.366
0.411
aeries IIC
p
$
8
0
P
0
I
f
8
7
f
X
t
1
8
$
4
7i;l5
140*1
107.64
108*11
0*610
4.35
4*75
1*107
0*706
0.606
0.509
0.365
0*487
1*40
87.4
51*83
51.99
0*905
4*64
4,76
1*187
0.716
0,614
0*800
0*357
0.419
3,00
74*38
67.05
67.40
0*904
4,84
4*76
1,187
0,706
0.614
0*09$
0.353
0*410
3.1
88.05
60,19
60.06
0*910
4,85
4*76
1*167
0,700
0.606
0*809
0*365
0.437
5
3*95
100*43
90*96
31,45
0.905
4.35
4*76
1.107
0*766
0,606
0*398
0*356
0*403
6
4.78
110.7
101,17
101*01
0.915
4.54
4.75
1*167
0*716
0*614
0*306
0*963
0*439
7
5*55
130.46
109.44
109*98
0*909
4.35
4.76
1*167
0*706
0.606
0*396
0*354
0*419
6
6.55
151*6
119*15
130*09
0*907
4,54
4.76
1*187
0.716
0.614
0.291
0*250
0.407
-67*
Series IID
Bun
t
M
N
0
P
Q
R
f
13
f
W
X
X
'
I
8
3
9*45
141*3,
1*40
8*55
69*84
40*1
43.69
63.04
i m **
188*82 45*87
65.15
0*910
0*909
0*910
4.50
4*30
;4*81
4*08
4*88
4 m
0*917
0.917
0*917
0*656
0*556
0*546
.606 - *595
♦606
*054
*856
*887
*865
*857
*859
*458
*487
*400
4
5
6
7
3*75
86*9
78*97
79*31
0*910
4*31
4*08
0*917
0*346
•595
•236
*857
•452
4*90
90*9
90*44
90.48
0*915
,4*51
4*82
0*917
0.546
•395
•239
*261
•439
6.10
111.6
101*07
102*16
0*912
4*31
4*82
0*917
0.546
•595
*256
*857
•432
7*8
123*4
112*76
112*09
0.918
4*50
4*62
0.917
0.550
*006
.840
*262
*432
8
8.35
134.3
125*42
123.03
0 *910
4.51
4*01
0*917
0*346
*595
•246
•268
•450
Series III
Hun
1
f
23*8
141*2
130*3
1
B
0
P
0
M
t
U
7
f
X
X
m
•983
4*04
4.987
*417
0*211
•506
*093
*287
♦807
S
4*1
82*6
48*87
—
♦92©
4.83
4,982
.417
*226
*848
*104
*249
*400
•
3
4
7*30
72*8
67*84
4»
*930
4*84
4*948
*417
•221
*530
*103
.259
*489
10*18
03*6
81.04
**•
*924
4*83
4*952
.417
.221
*530
*104
*249
*470
3
6
13*25
101.9
94*45
15.95
113*4
105*11
♦926
4*32
4*958
*417
•236
•565
.103
*247
.437
•926
4*84
4*931
.417
•221
*530
.105
•252
*475
7
13*55
123*7
114•34
—
•925
4*83
4.951
*417
*226
•542
.105
.252
•465
8
21.6
134.9
124*12
•921
4.68
4.955
•417
•886
*548
•101
•242
• 446
Someaelature
limwiiiary of Friction Factors
A*A
m diameter of aanulus, foot
8*1
« average. velocity, foot/see*
0-C ■ « overage
density, ibs*/ft**
IMD
® overage viscosity * 10*, lbs./ft*,sec*
1*1
m Beynolds Somber »
F-F
*
®*G
® Fanning friction factor, computed
from F~P
1*31
pressure drop, Xb»*/ft**
» Fanning friction factor for circular
pipe, at same Eeynolds lumber as
B-E
Bvmmry of Friction Factors
Bus
A-**'
3*#
&&
10*68
*<&
A 1
2
5
i
s
3
4
5
070
187
46,700
18*85
18*78
80.18
70*74
98*63
101*14
U8*8
18*78
18*65
18*66
18*78
18*85
12.66
18*55
18*55
18*48
0 I
%
%
4
78*14
18
18*68
7
18*65
15 J
0
0
4
5
t
$
0706
IS.40
5
6
7
8
0708
I486
.1486
.1436
36.61
24.13
46.800
~-
52
71.600
4d»*Sw
15.70
12.70
IS.*7
12.68
18,47
18,45
*00341
*00391
*00556
.00356
*00676
•00431
*00395
.00396
*00597
*0040
.00423
—
*0057$
*00382
*00584
*00385
*0038$
*0099$
•00391
.0042
*00396
*00596
*00399
•0040
•00599
•00395
•0048
•004$
98*48
$6*03
16*99
12*44
9*40
5*90
6.97
.00595
•0039$
•00386
•0041
.00415
.00467
*00467
•00395
.00397
•00398
.0040
•00408
•00489
•00475
20.09
14*5
8*78
87*55
33*96
40.26
4*56
1.96
•00366
.00411
.00419
•00407
.00414
.00598
.00461
97,900 17.51
$7,100 14*04
76,600 11*0
66,000
7*97
114,700 £5*9
195,@00 29.5
4.29
43,300
£.15
51,600
18*67 116,800
18*15 U S , 800
18*37 100,400
84,200
18*70
67.800
18.58
88,400
18*57
89*18
71*91
1 I
*00394
6*80 *005$$ *00403
63,750
67,000 11*78 *00375 .00997
76,500
9*01 *00370 •00399
106,500 16*5
*00347 •00395
111,600 17.86 *00343 •00395
150,000 23*8$ *00554 *00994
145,500 29.3
•00591 *00593
164,000 £6*5$ *0039 •00394
0788
§
4
*»«!.**1
19*01
JUJ.SO
12.15
12.70
i
s
b
H-M
0-0
89.800
48,700
78.800
89,400
97.600
51,000
19,420
•004
•004
•0043
.00398
•00396
.00396
.00490
.00506
-70-*
Bun
A-A
8-B
11
*2339
47,45
•2939
97.9
61.1
9 •2958
, 4 *8532
72*0
m •8358
79.9
6 .2332
84*09
7 •2558
89*7
83 *2532
99.fi
m *2939 108.fi
10 *2939 116*2
113 •2532 188*2
12 '*2532 22*0
30.4
-■153 *2592
41.8
143' *®§0®
.-
"rV
C-C
F-F
0-0
H-M
0—0
B-E
*0793
•0708
12*85
12*37
12.4
IB.42
12*45
12*48
12*47
18,48
18*50
12.61
12*58
12*45
18*42
12*40
60,600
84,600
91,900
104,309
114,500
181,000
189,000
142,900
158,500
186,800
174,700
31,500
43,800
8*87
8*05
5*99
7.79
10*18
11*05
18*28
15*18
18.0
21*48
88*1
0.899
1*47
8*70
•00386
*00391
*•0059
•00591
.0048
*00407
*00899
*00461
*004
•00418
•00403
*00486
*00418
.00464
197,800
9*86
8.0*
,0033
*0040
♦00375
*00354
•00339
*0034
*00346
*00368
*0717
*0719
*0712
*0711
*9710
*0708
*0797
*0706
*9798
.0706
•0799
♦0708
69,500
^
H
A 1
8."
3
4;:
5
8
7
8:-
*8778 127*22
38*7
*2778
*2778
88.01
87*46
*8778
*8778 116*92
*2778 111.03
99.85
*8778
77.88
*8778
*0704
*0788
*0708
•0704
•0686
*0869
•8682
•0888
12*38
12*38
12*58
12*56
12*73
12*68
12*62
12*80
B 1
2
5
4.::
5
e
7
8
.2368
*8368
•2562
•2508
*2362
•8568
*2368
*2349
130*5
44.09
64*15
81*13
95*24
104.13
113*92
123*23
.0622
•0891
*8697
.0692
*8888
•0389
•0389
•0833
12*84
12*49
12,48
12*88
12,80
12*69
12*70
12*70
C 1
8
3
4
5
6
7
3
*1945
.1945
*1943
•1945
,1945
•1985
•1945
•1943
127,84
51*83
67,05
80*12
90.96
101*17
109*44
119.15
*0705
♦0702
*070
,0706
•0706
*0706
,070
*0692
lO^TOO
130.000
177.000
167,400
6*85
uejooo
6*16
4*04
•90985
.00422
00868
00978
00810
00946
00960
00388
12.38
1*78
3*49
5*88
7*20
8*87
9.88
11.16
00561
00442
00401
ftftM*
00358
00360
00379
00366
*0054
*00458
*0040
*0058
•00362
*00357
*00358
*00544
12*60 139,100 15*67
3*13
12,80 56,650
72,600
4*93
12*85
6*80
12.57 87,700
8*58
99,350
12*53
12*58 110,600 10*87
12*70 117.300 21*99
12.80 135.300 13*97
00360
0047
00444
00484
00416
00408
00404
00401
*00353
•00468
.00481
*00396
.00384
•00373
*00370
•0036
170.000
57*000
u 9m
100.300
104.300
133.900
145.900
137.900
7*59
m m
.0066
*0070
,0078
,00395
-72-
.
P tndftM U l P ria o lp lii
Bitot tubes are of two general types*
(a) those having
static'pressure'%eningsincorporated in'the sides of the
ihStrWbt, (b) thote without static' pressure openings*
tube haid1W
shonld he
the
thSi&e^^iierimeBt's was of the latter 'type and
oalled, more
exactly, an impact tube.
bihc&*;'with'1the'taH'Static'1ophing,'sui ^
doctd chpahle of indicating'' point:velocities*
When comwas pro-
the simple
Pltot tube formula
-v, •
.:='r ■-.?■..■
s
'vdiere ? ’•»■ v e lo c ity , f t * /s e c . •
g * fo rc e o f g r a v ity , f t * /s e c .*
■" S g ^
has been
a feet head ©f fluil flowing
used throughout th is worfc, thereb y assuming an
In stru m en t constant o f u n ity *
B ather c a re fu l and e x te n s iv e
in v e s tig a tio n s (1 6 ,1 7 ,1 8 ) have shown th a t the dimensions and
g e n e ra l shape o f the Im pact tube a re o f l i t t l e
th is type o f tub e*
im portance in
th e constant w i l l be u n ity i f a c o rre c t
s t a t ic pressure is obtained#
th e fo llo w in g a re o f im portance
in o b ta in in g c o rre c t read in g s!
( I ) read ing s should be made a t a s e c tio n approxim ately
SO diam eters from any p o in t o f d is tu rb a n c e , so th a t norm al
flo w c o n d itio n s a re c lo s e ly approached*
fs) static openings shouldbe perpei^icular to the axis
(g) aU,;hart#-"ehhhM be rewewedfrom the inside of the
fJLgwi... -.
- ) (4 ) the
of the Impact tube should he parallel to
bfce^.ii*is:,ef the pipe# Mo*ta (l^) state* that m U b m U q b by
•ye 10 sufficient else© angles of yaw.of a few degrees give
m appreciable, difference in readings*
states that
Mother authority tie)
the instrument may be as much as 11* out of
allghlleht hefore the error exceeds 1 per .cent# :
<
Measurements made.with.the fitot tube .give: point veloci-
tlesaadare therefore useful in exploring a duet to determine
how the velocity changes frompoini to point.*
to find theaverage veloeity In c y lin d r ic a l
In attempting
channels from
p o in t measureaents, the following methods have been used with
w s£?yiiii-'ieg^^
(i) Messnr© the v«iocIty at saBje designateO poi-nt suoh
m
the;-center- of the stream and multiply thie by anexperideterw M ed; factor* --■
(t) Measure the velocity at- some point at which it haw
been found:that M e *local v ^ o c i ^ is the same as the average
welooitr over the entire cross-sectional area.
(8) Measure the loc&l velocities at a series of points
end,
by
a proper system of averaging or integrating# compute
the average velocity.
—74~
bfeawldely used.
Stanton and
Parnell :{ e ) . aadeextenaive,experiments.and, from their data,
published a graph showing the relation between the ratio
7av/?max and the Eeynolds Humber.
hater work by Hikur&dse
(10) has substantiated and extended the earlier experiments.
Towend (11) has published similar graphs for circular and
ifuare cress-sections*
Ho data are given for annular channels.
th e second method has n o t been used to any g re a t e x te n t
due to th e r a th e r -la r g e e rro rs which may r e s u lt from i t s
a p p lic a tio n .
From an in s p e c tio n o f a flo w -fr o n t fo r c y lin d r i­
c a l p ip e , i t can be seen th a t th e v e lo c ity increases very ra p id ­
ly in th e neighborhood o f th e average v e lo c ity .
e s p e c ia lly tru e in tu rb u le n t flo w .
T h is is
Consequently s lig h t e rro rs
in th e s e ttin g o f the P it o t tube r e s u lt in la rg e errors in the
average v e lo c ity .
There are no corresponding experim en tal
d ata fo r an n u lar ch an n els.
The th ir d m ethods M a t o f p ro p e rly averaging s e v e ra l
p o in t v e lo c itie s * is th e most g e n e ra lly a p p lic a b le to a l l types
o f e ro s s -s e c tio n s and w i l l be used throughout th is work*
th e d a ta obtain ed in t h is manner, Method® l-an & U
illu s t r a t e d *
used
W ith
w i l l be
Two d if f e r e n t systems o f averaging have been
in a l l ru n s , p ro v id in g a check on th e average v e lo c ity
obtain ed and f in is h in g * basis fo r comparison o f th e two
methods*
Method I t
Measure the lo c a l v e lo c itie s a t p o in ts which
a re lo c a te d ia such a way th a t each measurement rep re s e n ts the
velocity over m •emeX
■e# the cyoss-teettenai area*
In a circular or annular duct^ the loeetion* of the
joints of measurement may be computed as followfi*
Let
U@
«
o u ter d iam eter
of
_
stream
* in n e r diam eter o f stream * I 6 0
* in n e r d iam eter o f m y ann ular p o rtio n o f stream
'T''‘ •
’'
•'
«••" •
'
"!-’'
•V *5
» in n e r diam eter o f next inn er- segment
I
** number o f an n u lar segments, o f equal a re a , in to
e tre a m -d iL te v b e ,d iv id e d *.,
vi
-v,.%
■■:
distance from inside o f-o u te r w a ll ..
C ro s s -s e c tio n o f * e n tir e stream *
C ro s s -s e c tio n o f any segment •
.r '" :■."" '
1 -
^ 4 X/B
“ y/4 <4-l -«/>
&Ut . - #0;,..*,-1^ ..
I::,
* V-** V » (l-l*) D**
,
di • [l - J/B
’V*t1* <*“**J*'*®
°
*( *
$m: c y lin d r ic a l pipe^'IC '
*.:■■■ U ,
*
^
and -
<* * U * X/B)0’* %
& * « * / » (8«e4»)
*>r-<o,&a.< .I/P ,£#m>«
:
., ®bc. ■
-76-
''
Ih order tO iaake measnreaents at 6 points along the
tadimi ofam •es«riiljar:' d ^
measurement representing an
equal" Ifcea, tbe^erocs^seetionai dram is divided into 10 equal
segments*
Then Measurements are made at the lines of contact
between.,,segments X and #, 3 and 4, 5 and 6, 7 and a, and 9 and
10*
Complete cbaiptltations for this method will be given later
in this report* :
This
method is generally satisfactory for annular ducts
of reasonable diameter*
For channels of
very small diameter,
however, the diameter of the Pitot tube may make it impossible
to ©bi&iar those readings'nearest the inner and outer walls*
Furthermore, with the method outlined above, only five read­
ings are obtained across the duct} if one of these is incor'Xi,■.^'*_.■
' v ..
. './ ,
- -
.
rect, ^a’eMsiderabie sxror is introduced*
Mit& the normal
velocity distribution In a circular pipe a single 10-point
travef|0 -^ives •■-aTWwMm;i
felocity theoretically 0*30 per cent
high and a MO-polat traverse gives a result 0*1 per cent high.
It is reasonable to assume that these errors are duplicated
in annular channels.
The number of readings
can be in­
creased, of course, but this results in a large number of
readings near the p i p e h . J a t additional disadvantage is
the fact that computations usually indicate a Fitot tube
setting at some uneven fraction of the scale.
The uncertainty
in setting introduces small errors*
:'
^ Measure local velocities at points which are
evenly Spaced across the radius of the annulus and located
AklSSL9ggO!RMr
.fa|«»;a^iM*«EMrl.*Mlt scale.
Proper integration
across 1he channel gives- the ty to X rate of flow, from which .
the average velocity mm fee tompmte<i# fhe methQd of integra­
tion
be developed*.r:
l»et ftm mid«p#int ratine of any very narrow annular
segment, fiet* :,- ;
da * width of annular segment# feet
V * mid-point velocity through this segment, ft ./sec,
to • Inner radius of outer pipe
* outer radius of inner pipe
Gross- sectional area * 77TC& ♦ X/t
of segment
^ ■
%;■ ,,
*
*
m *m
*..
l/M 4*)*7
•■
r'
:U-:.
Volume of air flowing
!!.*■«*.!*.
through segment
• *V$V<m
Volume rate of flow
through entire eross^-: <w
section
1
i^lTdl,
*
The integral can fee evaluated by computing values of
fe^lt'end'^
*M> cdrrefpdnding values of E.
Graphic integration of the area under the curve from E* to
E0 gives the value of the integral*
fae method will be
Illustrated in the Sample Computations*
fiftifrulation of Pressure Proas
the FanMng Equation is generally used for the calculation
of pressure drop of liquids flowing through cylindrical pipe.
It is given by the expression
■viher#v..df :m.;jrese^e drop# lb*/ft**
.- U * length# feet.
v/?«.average .density of fluid, iWft#*
&%
-m c«wage velocity, ft*/see»
g * force of gravity * &&ȣ fU/sec*#
.firs* diameter ©fplpe, feet
/ < m average viscosity of fluid, ib./ft*,«ec.
S'1
* Reynolds Bumber
_
,
o : .. v
f(I^) «, Fanaihg friction factor
In order to extend the range of applioabiilty of this
equation to channels of non-circular cross-section, a term
called the hydraulic radius is substituted for the diameter
v.f, # ijrdrgulid Radius * * * - wetted perimeter
tor annular 'channels*
a *
u^iiT
ffl *
"TT"
-»79-
The ra&aisi
therefore becomes
For gases th e 4rop in pressure along the duct causes a
decrease la density and Increase in linear Telocity#
the
equation given above applies only to a differential length of
pipe in which the density is constant*
For the isothermal
flow of a gas in a duet of uniform cross-section, Ferry (80)
gives the following equation*
p
ji
?*-p*
*^Es»
where
U
*
1 * 4*® ft x°* ft
w
P* * upstream and downstream absolute
static pressures, lWaq*ft#
P v * density of gas at temperature in
pipe and
at pressure
* 1/8(1* ♦ P«)
0 * mass velocity, lb#/ft*, sec*
When the pressure drop is small (less than 10# of the initial
pressure), the factor £1 * 4*6
log Px/P.] differs little
from unity and may he omitted#. Hence for ordinary pressure
gradients, the equation becomes identical in form with the
equation for liquids*
Actual flow for gases may deviate from
isofcherm&l conditions but unless
experimental determination
of friction factors is available, use of the isothermal flow
equation is satisfactory*
-80Experimental pressure drops were substituted in the
Fanning equation and friction factors calculated*
©ample Computation
Series IXC,
1*
Bun 4
Computation of Pitot tube spacings
Inner diameter of outer pip® « 3*834*
Outer diameter of inner pip® *■ lig©#"
X * X#800/3#884 * 0*391
S * 10
P 38 distance from outer wall to center of Pliot tub#
• t, -
0.056
6 « setting
m
PItot tabs * 5.000 - P
4a * [l ? V S
%
* 8.668
$.854 - 3*660
ft
* 0*08$ - 0*08© * 0*08?
$g
*■■ 8*000 <* 0.08?
1
1
8
5
7
8
5.868
3.311
8.911
8.446
1.669
P
* 4*97$
mi l l i iinnp ini u i"
0 .0 8 3
0.8818
0.4615
0.694
0.9688
*1
0.08?
0.808
0.405
0.686
0.986
8
ill I'w i<l n r i« iij i m i i i i,
4.97S
4.798
4.695
4.868
4.074
n.m~
rttok Beading at
Inner Sail - 6.00 *
+ 0.118
• 8.946
Caleulatioza of point velocities
tm ‘
Water fortMi* n » > » flee* @a page 53
Barometric Pressure * 749 n * Mg*
S ta tie Pressure
a t’ P lto t tube
« 3*1 m * M,Q
fetal Pressure art Pitot tube (p) * 749 4
* 701*3 mm* Eg*
Brgr bulb temperature (t) * 91*p
li
Wetbulb temperature
^WtiSeXute Humidity (H)
*
*
$X*f
®#®®4Si W l b #
* w «* (£||) (i|$§£)2|& * ^i(,i»^|?.)2sa.
.7»ot ♦ aao.?)2!2^
▼ • (0.780
♦
* 81 ♦ 886
» 14*18 cu. ft ./lb. dry air ■*■ water vapor
Seeding of PItot tube at setting * 69.9 cm. differential
manometer Ho. 1
of 4.80
m
3.265 om. Hs0
(Calibration Graph)
Temp, of H&noaeter Liquids
* 80®C
Density of Water <80*C) « 0.9988 gm./cc.
* 68.58 lbs./ft.*
T%, head of air « -SaSSfi.
■ 94.01
T »
• \Tfi X 58.8 X 94.01
• 77.76 ft./sec.
Foist velocities for all pitot tube settings war#
calculated in this manner and plotted as a function of the
Pitot setting*
this produced a typical flow-front graph
which is shown in Fig* 4*
$*
Calculation of average velocity.
A.
Point velocities obtained at predetermined points
were added and divided by five to give the average velocity*
ICIMM
4.375
4.795
4*505
aa m
4*074
47*50
78*05
84*70
88*05
81*88
Average Velocity * 80*06 ft*/see*
Figure 4
m o em
M s m w r a o i is as a f f o u b c h a h s e i
Outer Pipe « 4*
Inner Pipe « 1*8*
88
86
SEC.)
80
POINT
VELOCITY
82
(F T . PER
84
78
76
74
72
5.0
4.8
4 .6
PITO T
TUBE
4 .4
SETTING
4 .2
(IN C H E S )
4 .0
3 .8
-84-
B*
Graphic Integration
Begins of Outer Xube
* X.9I7#
Maximum radius to Center * 1.917 - 0.056
of Pitot tube
• 1.88X#
Kadius at any setting * 1*861 - (5.000-s)
• S-3*139
Radius at setting of 4*3 « 4*800-5.139
* 1*861*
Radius * 1.661/IE * 0*1384*
8VTBY * 8 * 3*1416 x 0*1364 x 77.78
» 67*8
Values of %1fWl for all Pitot tube settings (not including
the
five calculated settings) were computed and plotted as a
function of R*
The graph is shown in Fig* 3.
The limits of
the area under the curve are set by the inner and outer radii
of the anaulus*
Bo * 3.884/3 at 18 * 0.1898*
%
m 1*800/3 X 12 * 0.0688*
Integration of the area with a.planiaeter gave a value
of 340.3 units.
Value of 1 square « .004 x 4
Q «
340*3 x *016
a 8*448 cu* ft./sec*
* .016 cu.ft./sec.
70
60
50
2TTRV
40
30
OUTER
WALL
20
INNER
0 .1 7 0
0 .1 5 0
0 .1 3 0
0.110
RA D IU S } FE E T
WALL
0 .0 9 0
0 .0 7 0
Cross-sectional area
of aimulus
« 9*679
■M
■/-j
l*■>
Average velocity «
Q/A
< 1*448
■■ * W w m
« 80*13 ft•/sac*
Observed press\sre drop * 59*8 cm* differential Manometer
No* 2
am* MgO (Calibration Graph)
Pressure drop, lto./ft** •
* *®*®*
* 6.SO
Reynolds Number *
/“
4m *
fo r
rn
• 0*1945 ft.
l/v « X/ld.lS * 0*708 lba./ft.*
*av * 80*19 ft./sec.
/* * 19*87 at K T # lbs ./ft., sec.
All viscosities were calculated from Sutherlands
formula (Perry Chemical Engineers handbook, p. 876)*
-86-
4mY/
~
/f
0*1845
x
60.19 x Q.70S
a
1Q~*
** r'r]i|Br
18*57 s
* 10J.f>J V..' I-....
• 87,700
I« ,. * :diataace D*t*esa pr«»aur« :t«p*
m U » P tL tU r,
ft..* ii.0 » .;;;,*U
0.004iE4
,s
7“
-87-
of
Aesults
By mea#urlrig fch* velocity at different points in a duct
and piottini theee veioeities against the distance from the
pipe wall, a £1#*-front such as that shown in Figure 4 can
he obtained# , these curves
of velocity distribution become
tangent to the pipe wall, indicating a velocity approaching
aero at the wall of the pipe*
At small distances from the
wall there la a film of fluid moving- in viscous flow although
the flow in the mainhody of the stream is turbulent#
In
cylindrical,pipe,this velocity increases uniformly with the
distance from the pipe wall, reaching- a maximum at the
center, .and decreasing to aero at the opposite wall:*
In
streamline flow, the resulting flow-front is a true parabola,
sharply pointed, in .the middle and tangent-,,to the walls of...the
pipe*
the average .velocity la 0*8 times the -maximum*
In ,
turbulent flow, the curve is paraboloid hut is not a true.,
parabola# It la somewhat flattened, in the middle*
average velocity can be
the
found from a graph of W f i / * . versus
%vAsaax.*
these relations hold only for straight sections
of
cylindrical pipe* .XXk .annular channels the distribution of
velocity is affected by tb# presence of the inner pipe* the
maximum velocity is never, found at the center of the annulus
but is always displaced toward the inner pipe*
The magnitude
of this displacwaent is determined by the dimensions of the
annulua*
fhia
em
be explained by
&
consideration of the
effect ©f the drag produeed by the wells*
Consider a point at the middle- of the eanulus*
For any
infinitesimal segment, the drag on the point (transmitted ■
•
:
.
■
through the fluid) is the same for both walla, provided the
ef the two wallr «^e: idaiitleal*' however,. since the
outer pipe etirwesioward the point while the Inner pipe curves
away from
it, Integration over the entire turfae© producing
drag shows that- the outer wail has* ireater retardlzig effect*
Heace the maximum velocity will hot’be attained at this feint*
It ls a reasonable assumption that'-the drag "from any ’
influencing surface will be proportlonalto the area of that
surface element and inversely proportions! tosoae function of
the distfence of that element from the point*
As the point la
moved inward from -the middle of the anhulu© * ' the' effective'
drag oftheoutarpipe decreaaesvhile that of the Inner pipe
increases*
Finally seme point Is reached at which the two
effects arei;etual| this Is the point of maximum velocity*
FroeoodiM$ further inward, -the
--ehe*-Jjftfcee-'pipe-increases
more-rapidly thaw that'Of the outer pipe decrease* and the
velobltF diiiiinlshea* -;,:
- Aa the diameter of the inner pipe IS thcreased, the total
surface effective for drag rises and the point of maximum
velocity Should Shift toward the center of the atmulucj that
is, toward the outer pipe*
As the diameter of the iisaev pipe
approaches that of the outer pipe, the ease approaches that
of flow between parallel plates and the maximum velocity is
at the center of the annulus*
is the diameter of the inner
pipe approaches zero, the conditions approach those of flow
through cylindrical pipe and the maximum velocity is at the
wall of the inner pipef, that is, at the center of the pipe*
Between these
two theoretical limits, the point of maximum
velocity shifts with the dimensions of the annulus,
tm m
theoretical considerations are illustrated by the
graphs in Figure
Fire runs, representative of the fire
different annul! studied in the flow through steel tubing,
are selected from the data*
Approximately the same velocity
range is selected in each ease for purposes
of comparison*
to show the trend of the change, the percentage of the total
radius of the annulus is computed for each Pitot tub© setting
and plotted against the corresponding velocity* the scale of
ordinates is displaced in order to separate the individual
runs*
The lowest curve is for the largest inner pipe; each
higher curve represents a smaller inner pipe* Iven with the
largest inner pipe, the point of maximum velocity is definite­
ly displaced from the center
of the amnulus*
This point
shifts uniformly toward the inner pip© as the diameter of the
inner pipe is decreased, reaching a value of approximately 60
per cent with the 1/2-ineh tube#
Figure 6, while typical, does not represent average
values*
In Table 6 a complete summary of this information is
figure 6
effect of raratftxosft of m m im on
mint of maximum TOLoem
Graph 1
Merles 11 A
jtun 8
Graph B
Maries 11 ft
M an 4
Graph $
Series XX €
Bun 4
Graph 4
Series XX
ft
Eu& 4
Graph 5
Series XX I
lun 4
V E L O C IT Y
POINT
O
20
40
PER CENT
60
80
RADIUS OFANNULUS
IOO
§ im n tm ,:
»*
Xftbie ® (a)
Standard Iron Pipe
lories X
M
A
i.IiiiiVi.
*
3
a.osa
a.ooa
a.aga s;«88 - 8. om
0*640
0.67B
9*840
l.O B
1*318
0*178
0.888
0*878
0*348
0*433
0*877
0*902
0 .9 1 8
0*898
0,018
0*701
0*797
0.8 01
0*888
0.881
0 *8 0 8 <
0*898
0*888
0*883 -
0 .8 8 8
0 *8 8 1
0*889
Hi!
>888
0.8 08 ' 0 ,4 8 8
i.rv
table 8 (8)
fta a O a r* S te e l tubing
Series
.,;•% i
8.6348.984
,■
0.600
Bi/%
0.1804 8.8806 0.8818
0 .8 64
0*898
0 .7 8 9
:
..,8,....■
1*080
3.684
1.888
8.8 84
8.080
8.884
8 .8 8 4
8.000
0
0.8818
0*7888
0
0*890
0.0 18
0.8 86
0.087
0.688
0,610
0.888
0.840
0.880
0 .8 7 8
0,868
0.888
0*800
9.860
0 .3 6 8
0.400
0*484
0.488
0 .4 0 8
For nomenclature, see page 62
The proper interpretation
of these data and an Intelligent
recognition of their limitations provide a method of obtain**
ing average velocities in annular channels*
The development
of such a method was one of the main objects of this investi­
gation.
The curvet obtained by plotting the
ratio of inner to
outer diameter against the fraction of the radius of the
annulus at which the maximum velocity is located are shown
in Figure.7*
Two separate curves are shown, one for standard
iron pipe and one for standard steel tubing*
This seems
to
be justified both by the data, which define two separate
curves, and by theoretical considerations*
Any factor which
affects velocity distribution will effect the position of
maximum velocity and therefore change the shape of the curves
in Figure ?«
toughness of the walls of the channel is known
to be a factor, although its quantitative
thoroughly determined*
effect has not been
Standard iron pipe is noticeably
rougher than the standard steel tubing which was used*
additional less important
An
factor Is the fact that the die*
tance from the blower discharge to the Fitot tube was slightly
less in the experiments made with the steel tubing*
The points at Di/B0 ** 0 and 1 are of course theoretical
and are plotted merely for completeness*
They serve to
Indicate the probable shape of the curve but values taken
from the graph in this extrapolated region may be subject to
large error*
Five experimental values are Included for each
series of runs*
The rather large range not covered by actual
Figure 7
t m u n m .m
» i « of maximum m o c m
W1TI T i l M T I0 OF x m m TO Q TO ii iX A iifim
II
» d ie t * from o u te r w e ll to p o in t o f maximum
v e lo c ity
f
* d is ta n c e from o u te r to in n e r w e ll
J*l * o u te r d ia m e te r o f
in n e r
p ip e
£0 * in n e r d ia m e te r o f
o u te r
p ip e
graph 1
*
S e rie s
I
*
Ir o n P ip e
Qreph B
«
S e rie s
II
*
S te e l Tubing
o q
LO
ID
empeyiment was caused pertly hy experimental limitations and
partly toy a consideration e l U i applicability of such data#
Witts ■m outer.tuba of.
,soma reasonable dimension, such as,4
inches, it is .difficult to insert and support any tube larger
then $ inebts# f^ich gives a diameter ratio of only
F#rtbf?i*dfe.the clearance :becomes so small that appreciable.,
errors are intreduced by,the presence of the fltot tube and
ethfrvSiaUierrorf-mre.magnified# , In tlbe other direction, any
tub«,4#«»^t^i^rl/#wi|i©h in outer diameter is difficult to.. .,
dappert' ;because of.:e»fi#ai?e beading under- its earn weight*
41a#-further:,shffte.;fn the position of maximum, velocity
become to#,.snail to be -accurate^ measured#
Finally, these
data-are intended for appUeaUoa ^industry* extremely
large or s#all-clearances .are not ordinarily,encountered in
a o e h \ # # r ^ ■■/.
^b» point of m&ximun velocity moves rapidly toward the
eenterof the anmilus until a ratio of diameters of approxi­
mately 0 , 4 i s reachedj beyond this the point of maximum
felacity:
.chaiigfs only slowly with further, increase, in the
ratio of the diameter of the inner pipe to that of the outer
r fha point of maximum,v#l©oity for any given aanulua
is
constant for variations in the rate of flow within the range
of rates studied#
data#
this is shown clearly in the summarised
lush constancy is implied in Figure 7 by plotting
average values for the point of maximum velocity# da increase
In the rate of flow affects all retarding
surfaces equally,
-94-
producing no change In the shape of the flow front*
Keferenee to the summarised data on velocity traverses
shows that the ratio of average to maximum velocity is sub­
stantially constant for all values of the Beyaolds Humber
studied,
This can he explained hy analogy to the situation
in cylindrical channels*
was approximately 80,000*
The lowest &eyholds Humber obtained
the majority were much higher*
At
this value in cylindrical pipe, the graph of Vfiv/Vaax versus
Beynolds Humber becomes flat and shows practically no change
for higher Beyaolds numbers*
it appears that this relation
Is at least approximately duplicated in annular channels*
The ratio of average
to maximum velocity does change,
however, with changes in the dimensions of the annulus*
The
magnitude of the variation is shown in Figure 8* The ratio
is at a minimum with the smallest inner pipe and gradually
Increases as the diameter of the inner pipe is increased*
The total change is not very great, being 4*89 per cent in
Series I and 4.75 per cent in Series IX*
This can be explain­
ed by a consideration of the dragging effect of the two walls*
As the Inner diameter is Increased, the drag of the inner pipe
Increases and the drags of the two walls approach each other,
allowing the flow-front to assume a more
form*
How
nearly symmetrical
consider two flow fronts across the same annulus,
each having the same maximum velocity#
The area under the
symmetrical curve is greater than that under the flattened
curve*
Therefore its volume rate of flow and average velocity
Figure 0
SFfSCT OF THE JftlimXO&t Of fit m m W B
m
tie ba tio of r n m i m to
mmxm
m o c m
Jjl • outer diameter, inner pipe
0o * inner diameter, outer pipe
v
« average velocity
? m x M maximum velocity
Graph 1
*
Graph 8
m Series II -
Series I
<* Iron Pip#
Steel Tubing
XEUU
( i lAdno)
ABi
GO
CM
CD
in
'Mo
A
GO.
00
)
0i\
in
oo
O
o
O
co
o
(CURVE
CM
Ld
>
in
D
O
O
^
O
cc
C5\
CM
°\
cm
<
O
Lf)
o
o
o
in
O
G)
CM
CD
(2
O
CD
3Am o )
XPUU
CD'
CO
GO
GO
A
96-
toto elae be greater.
the average
Thus, u
to* inner diameter increases,
velocity increases for constant values or fib*. -
maximum velocity duet© the shift la for* of the flow-front
and tits ratio
increase*.
A mefchod v i U now Peoutlined for determining to* average
velocity in annular channels froaa singiePitot tuba measure­
■■■‘i
ment.
■ ■ * . :.li; ;,n
— ; , o:.-
....
(1) 0al*uls»tft-.'.tto. rattoifig/B* jfort-the annulue used,
(a> Obtain oorred* value ■for-fraction afyadlua -.of.•
annulus a t which otto maximum velocity is
7«
located from Figure
Vs* Graph 1 fer iron pig# end Graph 8 for steel tubing.
,(g) Calculate the e*rr*eV**itl»g from the-.yalue of this
fSMtleh end the toewa radius of tto annulus.
,, (d) insert.a Pitot tube at this point end, measure the
maximum velocity*
(6) Obtain the ratio of average te maximum velocity from
FigUrOof#; ;.
(6)
.-0:-,,
Calculate the average velocity.
.
fhto tottod* used with proper limitations, should give
reasonably aecurate results*
the following precautions are
nodessary».....
(X) the roughness of the channel should approximate that
used in these experiments.
(2) The inner pipe should be accurately centered.
(I) The aises of the
outer and lamer pipes should
deviate widely fron the sises used in these experiments
though the value of the ratio »j/D0 remains the same,
not
even
the
-96-
relations probe,bly do hold t or larger ©r smaller pipe, since
the #hap*» eve ge«^etrically similar* but this point has not
be#m pyeved mp%$imehbslly*
Another method for determining average velocity fro®
Ajpejl# measurement I* suggested by;tfe* data# -..to# value of
Pi/% Is compute# *s-before and the fraction of the radius of
the armulugs at which the average velocity is located la taken
directly:from fable 6*
Insertion of the Pitot tube at this
point them gives the average velocity,
This method la not
recommended* fh© Value of this fraction is not es reliable
an.'Indloatiom-of velocity as is the fraction to the point of
maximum velocity*
plotted*
Per this reason* the values were not
It the point of average velocity# the velocity is
changing rapidly with increasing distance from the wall* a
slight error in-the'setting of the Pitot tube may result in
a-very large:.error- id She velocity*- dm the other hand#'at
the
point of maximum velocity the flow-front is flat for a
short -distance* -flight errors in setting the Pitot tube
produce very little error in the velocity measured*
Discussioa of Pre$tur<? D rw
Pressure drop measurements and the computed values of
the
Fanning friction factor are summarised on pages 69 to 71 .
The
method of computation is given on page 85*
In order to
provide a true basis for comparison, similar experiments were
carried out for cylindrical channels*
summarised on pages 70 -to 71.
These results are
-97-
A graph of the Reynolds numbers as abscissas and the
friction factors as ordinates gives a smooth curve similar to
the corresponding graph for
circular pipe (figures 9 and
10)*
Some tendency for separation of the data into sections for each
annulus is noted, especially in Series 1,
However the devi­
ations are not very marked so that one mean curve has been
given for all runs*
this tendency to separate was noticed
also by Erats, Gould, and MacIntyre (£)♦
A similar graph was plotted for circular pipe*
Values of
E-S on pages 69 to 71 were obtained by taking Beyaolds numbers
for annular channels and reading off the corresponding value of
the friction factor from the curve for circular pipe*
These
fictitious friction factors were then compared with the actual
friction factors and the percentage deviations calculated*
The results for steel tubing are consistent among themselves and agree to within 3 per cent with the values obtained
for circular pipe*
The friction factors for annular channels,
with a very few exceptions, are slightly higher than those
for circular pipe*
The results for iron pipe are not as consistent as those
for steel tubing*
The percentage deviation from circular
pipe varies widely for the different annuli, although for
each annulus the results are reasonably consistent*
This is
a manifestation of the tendency to separate into sections
which was noticed when plotting Beyaolds numbers versus
friction factors.
The average deviation of the friction
Figure 9
VARIATION IN fSI FAiiNIHfl FRICTION FACTOR
WITH
«HIQjW3S BtitfSUt FOB I8QH PIPS OF
ABSBMB CB08B-SECTI0H
O
C\J
To
00
x
NUMBER
O
REYNOLDS
CD
C\J
O
GO
CO
p O I x dOXOVJ
Tt
N O IIO IU J
C\J
9N IN N V J
Figure 10
TAKIAIIOI 18 TH£ FAiiHIHG FRICTIOH FACTOB
7IXB BEXBOUiS BOMBER FOB fSMBS, ZUMUW OF
AHBUUfi C£iG8S-SECII0ii
NUMBER x IO
REYNOLDS
CO,
CD CD Is*
CD
LT)
Ol x H 01DVJ
CO
NOI1DIUJ
ONINNVJ
■ facto r from th a t f o r c ir c u la r p ip e Is ? per c e n t*
In th is
;$& ***, however, th e re s u lts a re low er w ith the a n n u la r channels ♦
:¥h$ v a r ia tio n In f r ic t io n fa c to rs may he caused by th e
i i f f ©reiit.. arrangem ent o f th e apparatus in th e two cases#
th e
In
apparatus using iro n p ip e , th e d is ta n c e from th e blower
o u tle t to th e upstream pressure ta p was o n ly 48 Inches*
th is
d is ta n c e is probably in s u ffic ie n t to s e t up a s ta b le v e lo c ity
d is tr ib u tio n ; th e s t a t ic pressure a t th is p o in t may th e re fo re
be s u b je c t to flu c tu a tio n s #
In th e apparatus; using s te e l
tu b in g , th e ,d is ta n c e from blower o u tle t to upstream pressure
ta p was 10E inches and appears to be s u ffic ie n t*
Throughout these experim ents
I t has been assumed th a t by -
s u b s titu tin g .. th e h y d ra u lic ra d iu s f o r th e d iam eter the o rd in a ry
Fanning equatio n used fo r c ir c u la r cro ss^sectio ns could be
used w ith eq u al v a lid it y fo r computing f r ic t io n fa c to rs in
a n n u la r 'Channels* , T h is may no t be
(4 )(S l)-«
e n t ir e ly J u s tifie d ( 8 } f
S ince th e same o u ter pipe was used and the d iam eter ,
o f th e in n e r pipe v a rie d , th e s e c tio n s were n o t s t r i c t l y
g e o m e tric a lly s im ila r *
For such s e c tio n s i t is d o u b tfu l
whether' any m athem atical process expressing dim ensions in terms
o f an e q u iv a le n t d iam eter can be f u l l y J u s tifie d *.
That is ,
w h ile two cro ss-sectio n ® having the same mean h y d ra u lic ra d iu s
may be m ath em atically e q u iv a le n t, they may no t b®
e q u iv a le n t, since a
p h y s ic a lly
d if f e r e n t v e lo c ity d is tr ib u tio n re s u lts
from
the d iffe re n c e in the shape o f th e channels and the head
lo s t
is in flu e n c e d
by v e lo c ity d is tr ib u tio n as w e ll as by
th e average f lu id v e lo c ity *
S h ille r ( 2 2 ) , on th e o th e r hand,
beliefs that- ihr1hydraulic radius is 'valid:f o r ' thfe5turbulent
regiomi ■-
■':
P m to the absence o f any better method, these pressure
drops heir© been evaluated on the basis of the mean hydraulic
radius*
two sesumpiiaaa are present rif:the methodi
(1) the mom hydraulic is an accurate measure of the
diameter for annular channels#
•, ■ ■■
'■■ ’ ■ (8) friction factors forfmnul&r channeii vary with the
Heyuolds Kmber in the same manner .as thosevim cyliiadrlcal ^
channeI© >and can be, eomskmbmd. from;the Fanning:equation* ..;.
fb© good agreeaaeut obtained from ..individual -runs and v
from a ■eompertsom with the values for. circular pipe indicates
that ;these
-at least reasosteb&y good mpwemmk-
..^.uta ^ai^roatoa^e fens of .the .Fanning equation, which i*.~
aufflolemtly accurate for many purposes* is developed befam
if/ii *
'*# #0SS7
wh eye
f (ae)
fs$*/**
, |, » rate of. flow, gaist/mlm.
dp 3* pressure drop, lbs./sq*in*
s;« density, gms*/cc.
4. » diameter, inches
•IOQ
* $Bt th e graph
In the region %s/ds ® 1*0 to ©bout %s/ds
of f versus <|s/d& Is m approximately straight line with a
slope of ~G*&„
Thus
*
when
• * cf£r*
0e/ds « 1, t - .018
t » *oi* ( W d * ) “0**
W t - ■* * *
In order to test
« £ / “ * “
-
this equation for annular channels, the
average velocity was platted on logarithmic paper as a function
of pressure drop*
these graphs are shown la Figures 11 and 12*
A separate straight line was obtained for each annulus, the
intercept increasing uniformly as the diameter of the inner
pipe increased*
fhls is confirmed by the approximate equation
given above, since for a given rate of flow the pressure drop
varies inversely as some power of the diameter*
and intercepts of these graphs are
the slopes
given in the following table*
fable
Outer tips m $e
mr lisa Siaea irta,ssfiBl
0
l/4»
8/8*
1/8*
8/4*
1*
1 .9 8
1.89
1.988
1.8»
1.98
1.84
,00177
*00292
.0084
.00882
.0084
.0086
Outer fube * 4"
Inner J?afe» Mai*
0
X/2«
1*
1.5«
fi.0«
8.0"
1*79
1.77
1*30
1*80
1*80
1*66
Intercept
*00134
*0019
.00197
*0024
*0035
*00145
Figure 11
VARIATION OF PEESBUBE DBG? WlfH AVERAGE m O C I f l
i i xmn pipe o f &umh<m c rq s s -s e c tio k
Graph 1 - 3-Inch
Graph
circular pipe
2 - 1/4-inchinner pipe
Graph 5
- 3/8-inchinner pipe
Graph 4
-
Graph 5
- 3/4~inehinner pipe
Graph 6 - i-inch
inner pipe
inner pipe
( 'l a 'tos/'Qi ) doaa lanssiad
40
20
AVERAGE.
40
VELOCITY
60
IOO
( F T ./S E C .)
80
200
Figure
IB
rm uuon of piiasoii mo? with &vehaoe m o c i H
XM STEEL XUBIMG Of AJfSULAK €tOS®-#ECTIOl
Graph X - 4 -inch c ir c u la r tubing
Graph 2
*■l/S -in c h in n e r tube
Graph 5
«• X -lnch
Graph 4 -
X
in n e r tube
l/& ~ ln c h in n e r tube
Graph 5
8-inc.h
in n e r tube
Graph 6
* Scinch
in n e r tube
O U1
CD
CJ.J'vs/■<ai)
doaa
ianss3Hd
O
xoi-
These slopes ere fairly consistent for each series and
check the approximate equation reasonably well#
The values
for thr series usia# fe^inch pipe are high, which is in line
with the ■
friction factors obtained#
data obtained
%rpi'
0
\ cr cit
'■
0
in' this Manner,; equations of
...... v.
v:,.,.gfrigrjf* ;■^
;
the
--V. ‘■■I ■-■
V
Mhere-intercept
T * velocity, ft./sec*
:;h'*» slope"
Can 'be- developed'■Thus, for '!~inch tublng ;tila 4*lfcch "tube
■ :
•
•*-■■■■'
'Witch equations arehsefhl when: applied under Identical experi­
mental eonditions bbt ere'hob' reeoasmended for use under other
C onditions!
;
;iV:-;
v ; ‘" r-:v;
-AU8-
Discussion of Byyora
1*
Eccentricity of Fife
the pife wee accurately centered at three
points along
its length, one of which was located within a foot of the
Fitot tube#
t*01 inch*
fh© possible eccentricity at the Pilot tube was"
Between the points of support, the inner pip©
sagged under its own weight, the amount of sag
the sis© and weight of the pipe*
varying with
fh© amount of sag, although
The effect
of uncertain magnitude, was small in all eases*
of ecoentriolty Is not.fully. raftgte&tood, but it. may become
very important in narrow annuli*
One authority (S3) states
that with a given .differential pressure, the flow obtained
with a core touching the outer wall is U l/ B times as much as
with the core'cbhcdhtfio*'' "'**
2*
Fitot. tub©
a)
fh© seal© was .graduated to 0*Ql inch*
even division could be made to gO*0Gl inches*
Bettings of
Calculated
settings, used in obtaining the average velocity directly,
occurred at uneven divisions and were subject to an error of
approximately &«099 inches*
b) The M t o t tube was aligned with the axis of the pip©
by eye, aided by an outside pointer.
been iriejptfar *& degrees* Ho**
this alignment may have
(19) state* the angle of jaw
aaybeseveral degrees without noticeable effect*
108-
c) th e assumption of an instrument constant of unity
should introduce no appreciable error since the design of the
tube adhered!Cloeely-to conditions
set dom by the Aero­
nautical Besear«^,C^»iaittee (17)*
0*
Manometers
The differential manometer* were graduated to 1 milli­
meter * 0ue- to-elight- fluetuatioas 1n ' theline, the probable
accuracy, is $8 millimeters*
a) Iffeet. en^felecity Mfltrlbutlea
• *rsr
t
:''A Probable accuracy of manometer « 0*4 centjieter;~~
■MV
m #0 i ©a. water
-
lowest p f t * re tTf } k * i . , , h * I cm* h *0
Irror in velocity *
» l/M e|&
• 0 *1#
Highest rat© g? floi|i,^.h » .18. cp* M^Q
Error in weioeity ,w ^
m
m 0*04#
fh© 4$ffeyenf&al,..aanoaietprs were calibrated with a
simple water manometer* sc©»tte to 1 millimeter*
Therefore,
the absolute values obtained from a single reading are no
more accurate tbefe'the water manometer * Increments of
104
■
-
pressure, however, can be read with the accuracy Illustrated
above*
b) Effect on Pressure drop
m 'gfc.-;
' ,
l
' Z
,..,.lowest pressure drop*
..::
1
h,«.1*00 m ? 5*0
*.$ 9 $
Error in pressure drop*
'
Bluest rate'-of:
Hewt
h * 55
cm* 5®0
Error la pressure drop •
* Q*d£
c) Manometers were calibrated at 7®* P and this calibra­
tion used at all subsequent temperatures*
The magnitude of
the error produced by a 10* variation in temperature will
mow be computed*
*» where
ptlbijF*
Pa * factor at 74* P
Pi « factor at 66* P
X« « *#007 s 10 » **007
£
It
76* Pi
50 cm*
n
f* *
* **000753 x 10 « *#00753
* 3*01 cm* 5*0
« | ^ | - ie.61
1 - 1,007 -161o07S8) 18.81
■ 16.48
005-
**
At 66* F*
% error m
50 cm*
»
5#038 m * H3Q
% 100
* 0*85
Shi variation i n temperature was usually only a few
degrees ad that the average error was much lower*
for any one t m
constant factor*
Furthermore,
the temperature was constaut, providing a
although the absolute value might he slightly
in error* variations in pressure could be read with e^ual
aeeuraey at all temperatures*
106**
Summery
The factors influencing velocity distribution in annular
channels have been investigated for the flow of air in the
turbulent region* The following conclusions
may be drawn:
(1) the point of maximum velocity shifts toward the
center of the annulus as the diameter of the inner pipe in­
creases*
(8)
the point of maximum velocity does not shift with
Increase in the rate of flow*
(3) the ratio of average
to maximum velocity is inde­
pendent of the Reynolds Slumber for all values greater than
80*000*
(4) the ratio of average to maximum velocity increases
slightly as the ratio of inner to outer pipe increases*
(5) the magnitude ©f the shift in maximum velocity is
dependent upon the roughness of the channel*
by a graphical application of the data used in obtain­
ing the conclusions given above, a method is presented for
obtaining average velocities from one Pitot tube measurement
and a knowledge of the dimensions of the annulus.
Friction factors have been determined for annular
channels and compared with the values obtained for cylindri­
cal channels#
The use of one
graph of T N f fa
versus f for
-107-
both cross-sections will probably give results accurate to
10 per cent*
The roughnesses of the two channels must be
comparable*
Ovbr the range of cross-sections studied, the mean
hydraulic radius appears to be a reliable measure of the
diameter*:- ■
-103-
Bifcliogr&phy
(1) Atherton, Steeh* Eng. 18, 1112 (1926).
(S) irate, Gould and Ssclntyre, Univ. 111. Eng. Exp. Station,
Bulletin Bo* 222p March,
(«)
Murneghan, and m temaf1 Mrodyaiiiei* National
Ha®oareh Council, 1932*
Lonsdale, Phil* lag* (6) J§, 161 (1923)*
(5) tier@y, hooper, and Wiamy, Phil. lag. jyfc, 64? (1935).
(6) Kemier, Trans. Am. ice. lech. Eng* RIB 55-2, SO (1933)*
(7) Stanton, Froc. Hoy* Soc* (Load©a) A S£> 366 (1911).
Stanton and Pannell, Trans. Hoy* hoe. (London) A 114*
199 (1914)*
(9) Morrow, Proc* Boy* Soc. (London) A 76* 205 (1905) .
10) Biknradse, Fors chungarbeiten V.D.X. 5ft6f 34 (1933)*
11) Tomend, Proc. Boy* So©. 145* 100 (1954).
IE) Lea, Hydraulics. 5 ed*, Longman®, Green and Co., flew York;,
1950*
15) Andrade, Trans* Faraday Soc* ||£, SOI (1951).
14) Lea and Tadros, Phil. Mag# H , 1235
15) Winding and Rhodes, Ind. Eng. Chem. Anal. Id* Ifi, 60S
(1930).
16) Gregory and Sehoder, Trans. Am. Soc. Mech. Eng. 50,
(1900).
351
17) Reports and Memo. Ho* 71 of Aero. Research Comm*, Pec.
1912*
10) Reports and Memo* Ho. 901 of Aero. Research Comm.,
August, 1925.
19) Moss, Trans. Am. Poe.
Mech. Eng. j&&, 761 (1916)*
Perry, Chemical Engineers handbook, p. 725, McGraw-hill,
1934*
-*109-**
(HI) Spitsglass, frans. Am.
(X9S0).
Soc. Mech. Eng. MXD 5H-7. Ill
(H8) Schiller, Forschungarbelten 7er. deut* lag., Vol. S48,
p. 16
(HS) Botes by Editor, Engineering 1HH. Ill
(19H6)*
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