Flow Losses in Flexible Hose.
Flow Losses in Flexible Hose.
Flow Losses in Flexible Hose.
1967
Recommended Citation
Riley, Kenneth Lloyd, "Flow Losses in Flexible Hose." (1967). LSU Historical Dissertations and Theses. 1313.
http://digitalcommons.lsu.edu/gradschool_disstheses/1313
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This dissertation has boon
microfilmed exactly as received 6714,008
A Dissertation
in
by
Kenneth Lloyd Riley
B.S., Louisiana State University, 1963
M.S., Louisiana State University, 1965
May, 1967
ACKNOWLEDGEMENT
NAS 9-4630. I also want to thank the Dr. Charles E, Coates Memorial
ii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT 11
LIST OF TABLES vi
LIST OF ILLUSTRATIONS xi
NOMENCLATURE xii
ABSTRACT xvi
CHAPTER
I INTRODUCTION 1
Problem Definition 1
II LITERATURE SURVEY 6
Momentum Balance 7
iii
Ill EXPERIMENTAL CONSIDERATIONS 36
Flow Systems 40
Water System 40
Air System 44
Range of Measurements 48
IV RESULTS 50
V DISCUSSION OF RESULTS 92
iv
APPENDIX
VITA 166
v
LIST OF TABLES
Table Page
vi
LIST OF FIGURES
Figure Page
\
IV-31 Air System: fvs Re for NASA 51 (6 * 0) 76
ix
V-l f vs Re Showing Transition from Laminar to
Turbulent Flow 94
x
LIST OF ILLUSTRATIONS
Illustration Page
xi
NOMENCLATURE
2
A Characteristic area, ft
E+ Re (e/D0) /&T
F Force, lb-; Fg , force exerted by stationery fluid; F^, force
associated with moving fluid; F^ j, force associated with mov
ing fluid due to drag friction *
Gmax The mass velocity through the minimum free area of flow
perpindicular to the flow stream for a bank of tubes,
lb/(sec)(ft^)
xii
Nomenclature (cont'd)
T Absolute temperature, R
m Mass, lb.
p Wetted perimeter, ft
xiii
Nomenclature (contd)
Time, seconds
v/v*
(y) (v*)/v
Re(e/D) /F7T
Bend angle, degrees
2
v Kinematic viscosity y/p, ft /sec
xv
ABSTRACT
for predicting flow losses in flexible metal hose. Hoses with annu
lar and helical convolutions were used; their diameters ranged from
curved hose, relate the Fanning friction factor with Reynolds number
and hose geometry. For straight hose the correlation has the form:
1
4 log * if> (Re*)
F
tions are necessary: one for annular-type hose and another for helical-
type hose. These correlations are presented both graphically and mathe-
Reynolds numbers from 10,000 to 340,000. Data obtained from the litera
numbers from 2100 to 2,000,000. Data from this study also indicate that
xvi
For a given hose the flow behavior can be described by consider
olds numbers in the lower end of the turbulent flow regime (10,000
number and has a value of about 0.020. As the flow rate increases
as much as three times the value in the low range. A Flow model is
for both annular and helical hoses. The relationship for the ratio
of the friction factor for a curved hose to that for a straight hose
is:
0.17
b * 1.0 + 59,0 (Re)
f
sections of hose were studied with the bend angle varying from 0
xvii
CHAPTER I
INTRODUCTION
to minimize this invested capital and make the most effective use of
lations.
Problem Definition:
shape gives the lowest wetted surface area to volume ratio of any
conduit consumes less energy than would flow through a conduit of the
rigidly fixed to its supports it may bend, be torn loose, or even rupture.
In the process industries almost all metal pipes are used at temperatures
other than that at which they are installed. For this reason, provision
This hose has a corrugated (or "convoluted") inner tube of brass, monel,
strength. The flexible metal hose has three major advantages over con
allow for misalignment, and (3) it permits relative motion between two
rigid Jiydraulic lines. Short lengths are often used in piping systems
ment where the line must be moved frequently. Flexible metal hoses are
hose.
Perhaps the most significant design criterion for any conduit system
04
4
hoses. The flow loss through a given size flexible hose may be as high
This increased flow loss is due to the convoluted nature of the tube
wall which increases the surface to volume ratio and also creates sig
the process industries and space-oriented work, their selection and use
straight line installation, corrugated hose will produce three times the
pressure loss normally expected in pipe and interlocked hose double the
a flexible hose can give as much as 15 times the pressure loss as that
expected in pipe, it is obvious that this rule is not a very safe design
correlation.
of flexible metal hose, design methods must advance from the "rule of
and some work has already been done along these lines using the results
for flexible metal hoses. It is hoped that the end product is an accurate
systems.
CHAPTER II
LITERATURE SURVEY
This section deals with previous studies which form the basis
factor. The relationship between the Reynolds number and the friction
factor is then discussed for various cases. The first case considered
section. The fluid will exert a force F on the solid surface of the
conduit. This force may be split into two parts: Fs, that force which
would be exerted by the fluid even if it were stationery, and F^, that
factor:
6
7
F. * Akf (II-1)
K
Note that f is not defined until A and K are specified, With this
friction factor.
Momentum Balance:
then
Fk 2 (n -5)
r To-- sj0-f
8
gives:
_2
dP vdv" 4fv dx /TT ^
~ * 1 ; ' B5i7" 0 <n-6)
This equation can then be integrated to give the working equation for
straight forward.
dv" * 0 p constant
_2
therefore -AP a L. v (II 7)
0 5?c
f 1 D -APjfc (II-8)
4 L i _2 v
2 pV
data.
Et
constant (II- 9)
p p MIV
therefore
dv ,-dP (11-11)
v P
gives:
2 - p,2 = piv2Pi
1 ^2 14f TLT - 2 In
(11 -12 )
Introducing the Mach number M = v/c, the final working equation becomes
4f L
D
1
YMf - 2 In
(11-13)
16 (11-14)
Re
10
data of Stanton and PannellCl) and Senecal and Rothfus (2) show excel
2000.
is altered. This leads to a secondary flow in the pipe and hence the
frictional losses are greater than those in a straight pipe. Dean (3)
and Adler (4) have made theoretical calculations for the case of lami
Dean number:
De = 1_ (Re)
2
(11-15)
(11-16)
0.36 (11-17)
= 0.37 (De)
f
component normal to the axis of the conduit. For fully developed lami
v
v * max
2 (11-18)
v *v
max
equations.
12
to a Reynolds number of about 3300. From this point on, the friction
Reynolds number range 2100 - 3300 the flow is changing from a laminar
of about 3300. For the case of a smooth pipe of constant circular cross-
equation:
0.079
0.25
Re (11-20)
13
lumber of 100,000. At the time when Blasius made his study, data were
pipes for Reynolds numbers ranging from 4,000 to 3,240,000. The data
numbers above 100,000. It was shown that the Blasius equation pre
dicted a friction factor which was lower than that actually measured.
Using his own data and that of Stanton and Pannell, Nikuradse
Prandtl (10) led to the derivation of an equation with the same form as
(H-22)
14
very small.
that the density of the fluid was not constant along the length of
the pipe and that the velocity changed between the inlet and outlet of
the test section, he concluded that the friction factors are not mark
Keenan and Neumann (12) also indicates that the friction factor is
incompressible flow.
The friction factor plot based on this equation has been used exten
tions which make use of the analogy between the transfer of momentum
f * 0.046 (11-24)
0.20
Re
hold:
V vmax
T72T (11-25)
v vmax
1-f]*
Note that these relationships are based on empirical correlations and
vhere the exponent n varies with the Reynolds number. The value of
the exponent for the lowest Reynolds number studied (Re = 4000) is
can be used to express the relationship between the mean and the maxi
mum velocity:
Far a value of n = 6 , v/v max = 0.791: for n = 10, v/v max = 0.865.
sidering the fluid in a pipe as being divided into three separate zones:
zone, in which both laminar and turbulent effects are important, and a
this topic is given in Bird, Stewart, and Lightfoot (20) and Knudsen
1. Laminar sublayer:
v = y 0 <y+<5 (11-29)
2. Buffer zone:
greater. V."nite (22) has found that the friction factor for turbulent
fB = 1.0 + 0.075 Re
(11-32)
expressed as:
2 0.05
fB R e |'M
t W (11-33)
for Re (R/rg) > 6
0.25
^ 1/2 0.00725 + 0.076
(11-34)
for 300 > Re (P-/rB) >0.34
18
calculations on flow losses in turbulent flow have also been carried out
his studies. It should also be noted that the curvature of a pipe has a
The discussion thus far has been limited to smooth pipes, without
really defining smoothness. It has long been known that, for turbulent
flow, a rough pipe leads to a larger friction factor for a given Reynolds
reduction in the friction factor for a given Reynolds number. The pipe
some parameter which describes the roughness must be defined. The most
exact procedure is to describe the height, the spacing, and the orienta
larger value of the friction factor for the rough wall at a given
19
Reynolds number. Since the value of the friction factor plays a signifi
involves the matter of evaluating the degree of roughness and the extent
to which this increases the friction factor over that of smooth pipe.
choosing pipes of varying diameters and changing the size of the grain,
he was able to vary the relative roughness e/R from about 1/500 to 1/15.
In the region of laminar flow Nikuradse found that all rough pipes
had the same friction factor as a smooth pipe. The critical Reynolds
behavior of the friction factor was also observed above a Reynolds num
ber of about 2100. Again, as in the case of a smooth pipe, the friction
number of about 3000 was reached. In the turbulent region he found that
roughness behave in the same way as smooth pipes, that is, they follow
the relationship
this range and the friction factor depends on Reynolds number only.
as e/R decreases, the friction factor deviates from the smooth pipe rela
0 < < 5 f * g ( R e )
v
The size of the roughness is so small that all protrusions
2. Transition regime:
> 70 , f. g (e/R)
v
(11-35)
smooth to completely rough flow was established by Colebrook and White (27).
21
I1 - 4.67
(11-36)
(11-36) for the completely rough regime. In the transition region this
the transition region the friction factor is a function of both the rela
transition region. They reason that the flow in the turbulent core is
only affected by that part- of the roughness element which projects beyond
the sublayer. Thus one would expect the friction factor to correlate
better with e-5y+ than with e. The relationship which they developed
is as follows:
date the full range of Reynolds numbers and relative roughness which
develops for the friction factor at any roughness condition and Reynolds
4
number is:
22
vhere e+= Re /[)) /Tff . This equation is valid for e > 0,05.
Knudsen and Katz (21) present a friction factor chart for the
.pipes. The chart is derived from equation (11-14) for laminar flow,
for fully turbulent flow in rough pipes, and equation (11-36) for the
(28) state that the fundamental difference between the two types of
between the friction factor and the Reynolds number. For turbulent flow
passes through a minimum and then increases to its final asymptotic value.
They attribute this difference in behavior to the fact that the roughness
v- vM X . rj/n
1
(11-27)
value of n, for flow in tubes with roughness e/R = 0,1 and 0.2, changed
along the tube radius and had values from 2 to 4 (for Re - 10,000) and
from 2.5 to 4 (for Re * 135,000). This deviation from the power law
v+ * 2,5 In y + B
e (I1-39)
v*e/v. For the hydraulically smooth region it can readily be shown that
Nedderman and Shearer (28) also give a relationship for the velocity
(11-42)
Since the grains of sand were glued to the wall as closely to each
roughness e/D only. When this is the case, Schlichting (31) recommends
convenient when the flow is in the completely rough region and the fric
method involves correlating any given roughness with its equivalent sand
roughness and to define it as that value which gives the actual friction
had been made rough by cutting threads of various forms into them.
25
roughness, giving very large values of the friction factor, was dis
covered in a water duct in the valley of the F.cker. This pipe had a
diameter of 500 mm. and after a long period of usage it was noted that
the mass flow had decreased by more than 50 percent. Upon examination
it was found that the walls of the pipe were covered with a rib-like
deposit only 0.5 mm high, the ribs being at right angles to the flow
1/40 to 1/20, however, the actual geometric relative roughness had the
higher values of friction factor than sand roughness of the same abso
lute dimension.
For this system the friction factor is seen to be a function of two geo
tor and the two dimensionless groups is presented in the form of a chart.
at any point along the wall. The relationship for the friction factor in
2
0.079 + 8.2 (e/X)
f - (11-43)
0.25
Re
greater than 0.6. All values of E between 0.32 and 0.6 define an
f * 0.03075 (11-44)
[log (Dave/20]2
the wavelength.
inside a smooth tube. Koch also reports a tendency for the friction
increase in the drag coefficient of the plate to which the ribs were
Morris (39) proposed a concept of flow over rough pipe based upon
flow, and (c) quasi-smooth or skimming flow. Morris states that wake-
number.
in that the data presented primarily deals with straight hose. Bend
to a great extent. Also, the data that are available produce wide varia
C
28
maximum bore, 1.8 in. minimum bore, and 0.4 in. pitch of corrugations.
He observed that the loss of head was proportional to the mean velocity
argued that this would lead to the apparently paradoxical result than
and a pitch of 2/3 inch. Using these results and the data obtained by
They state that the friction factor was found to increase with increases
in the flow rate and water temperature. This result was found to occur
the words of the authors, "This unanticipated result was indicated quite
This trend for the friction factor to increase can also be seen in
cluded that the shape of the grooves was nearly as important as their
i
29
Finniecome (44). This comparison clearly shows that the friction factor
for a corrugated pipe does not tend to become constant until a higher
Reynolds number has been reached than would be the case for a pipe
tendency for the rising portion of the graph of f versus Reynolds number ,
smoother pipes. This effect was observed by Ifoeck (57) from many expe
minimum diameter, 0,813 inch maximum diameter and 0.104 inch pitch. For
olds number of 1700 - for flow in a smooth pipe the value is about 2100,
Also, he found that the index of the mean velocity v" in the equation
h CqV*1 (11-46)
approximately 2,31 over the upper portion of the velocity range. Alterna
the method of least squares. Allen's results clearly indicate that the
r
30
results also show that the influence of the corrugations may be decreased
water which takes no real part inthe general flow pattern. Some investi
under such conditions. The conclusion that Allen draws is that the effect
depth has been reached, because the disturbances are confined to the reg
ion adjacent to the crest of the corrugations, i.e., where the diameter
sure loss and calculated friction coefficients for annular and helical
type hoses. He indicates that the loss throug'h a given size flexible
size conventional pipe. Also, he indicates that the helical type hose
has a lower pressure loss than the annular type. Because his data were
taken at very high Reynolds numbers (above 500,000), Daniels found that
the friction factor was constant and not a function of Reynolds number.
Daniels and Fenton (47) present extensive data for both corrugated
hose and interlocked hose. They conclude from their data that the loss
factor for flexible hose elbows is normally higher than the value accepted
for smooth pipe elbows. A correlation for the friction factor is also
* 0.10 .e 1 *6
ND V
(11-47)
31
sure loss in both straight and bent flexible sections. Their plots show
The pressure drops reported are from 4 to 19 times the loss through an
of Reynolds number. The data were taken using metal hoses with diameters
a pressure loss coefficient for 90 bends for flexible metal hose with
rg/D = 0 to 36. Pressure loss correction factors for bends other than
equation from which the pressure losses for a gas flowing in flexible
(Aj)
1 = 3.48 - A0 In
1 A^*
(11-48)
(Re) 4
functions of the geometry of the hose and the Reynolds number. The
32
results obtained from this study also showed a sudden increase in the
which fit the available data on flexible hose quite well. One such cor
f = 0.01975 * (0.595) Y
0.2 3 (11-49)
D
vhere
6
Y = 1 , and X = 3.84 x 10
1,224
Re
for flow in rough pipes, was also given for flow in flexible hoses.
f * 2
2
A(E+) - 3.75 - 2.0 In 2c (11-50)
lowing relationships:
33
stated that flow losses are not induced in the valleys of the corruga
tions and therefore the relative roughness z/D is not a relevant vari
M
\)J
|i - L
[_
__E i
(p + 0.438Xj
In the study by Hawthorne and von Helms they assume that there is
been attacked on the basis of & study by Knudsen and Katz (34). They
turbulent flow there is at least one eddy observed in the region between
the fins. Tneir results can be analyzed by considering the ratio of the
fin spacing to the fin height. For values of this ratio between 1.15 and
0.73 the flow pattern is characterized by one circular eddy between the
fins, which becomes slightly elongated as the ratio nears the lower limit
of 0.73. Khen the ratio ranges from 0.51 to 0.45, two circular eddies
form between the fins, and they rotate in opposite directions. When
the ratio reaches a value of 0,31 a circular eddy forms at the outer
edge of the fin space, but in the space between this eddy and the tube
f * a (Re/ (11-52)
where a and 3 are functions of hose geometry. It was found that two
2
The quantity 3 is a function of the geometric parameter (oe/X ). The
of hose used - that is, it can be used for both helical and annular-
type hose.
3 * 0.299
N
/ac\ - 0.0313 (11-55)
The correlation for curved flexible hose sections was also found
tains all of the data used in this study. Also, Volume III of this
report contains the data reduction computer programs used for the data
4
CHAPTER III
EXPERIMENTAL CONSIDERATIONS
produced by flow in flexible metal hoses. Both air and water were
heavily on just one fluid system. Both straight and curved sections
hose and interlocked hose. Both of these types are available in awide
I. Annular-type hose
A. Close Pitch
B, Open Pitch
The results of this studv indicate that the annular-type hose need not
cations the hose tested had to cover a wide range of geometric variations.
The geometric variables for corrugated flexible hose can be seen in Illus
tration III1. Table A-lof the appendix gives the dimensions of the hoses
ILLUSTRATION III-l
r
38
Aside from the basic geometric linear variables there is also a shape
factor which must be considered. This shape factor describes the nature
volution.
ILLUSTRATION III-2
All flexible metal hoses studied in this work had a "finger" shaped con
jinn
I L L U S T R A T I O N I I I -3
Che would expect that the "teardrop" shaped convolution would be more
annular, and helical. The sizes (nominal inside diameters) chosen were
1/2, 3/4, 1, 1 1/4, 1 1/2, 2, 2 1/2, and 3 inches. This gave a total of
twenty-four flexible metal hoses. All test hoses were 10 feet in length
with entrance and exit sections made of the same type flexible hose as
Special note should be made of the flanges used to connect the test
a flange section.
Df D
I L L U S T R A T I O N I I 1-4
There are two points which should be noted about this flange: (1) the
this is true for all hoses. Special care was also taken to see that
the flange was connected to the flexible hose at the crest of a convolu
tion.
IMMOVABLE
FLANGE
ILLUSTRATION III-5
40
Flow Systems:
The experimental equipment used in this study was designed and con
two units. The first was designed to measure the rate of flow of water
through corrugated hose and the corresponding pressure loss; the second
unit accomplishes the same objectives but with air as the flowing medium.
Water System:
III-2 and III-3 are photographs of the test system showing the actual
of equipment used.
sp.meter Thermometer
o
Venturi
Carbon Mercury
Tetrachloride Manometer
Manometer
D A T A P R O D U C T IO N SY ST E M FO R W ATER
FIGURE III-2
D A T A PR O D U C T IO N SY STE M FO R W A TER
SHOWING P U M P S AND CO N TR O LS
FIGURE III-3
44
Air System:
is a photograph showing the control system used to control the flow rate
r
AIR
Compressor
Storage Tanks
Auxiliary Compressor
Drain
VENT
Figure III-4
Schematic Diagram
Air System
VI
D A T A P R O D U C T IO N SY STE M FO R AIR
SHOW ING C O N TR O LS
FIGURE III-5
47
as follows:
follows:
4. This reading was then recorded along with the inlet tem
perature and pressure on the orifice section, and the
pressure drop across the orifice.
Range of Measurements:
Temperature - 40 to 80F
Temperature - 50 to 120F
The temperatures for the water system varied with the season, whereas for
temperature to change.
49
RESULTS
Chapter III. As noted there, the flow of air was investigated in one
This chapter will describe the results obtained from these experi
the chapter how the results from the two systems complement one another.
for hose NASA 62. These data were taken on the water system with the
Figures IV-2 through IV-9 show pressure drop versus Reynolds number
%
data for straight sections of all helical hoses tested. Note that except
for hose NASH1 a least squares analysis shows that all of the slopes are
-..wraTTStgi;
ope 1.92 T
Fipure IY-2
NASH 1 I
Water System
- REYNOLDS NUMBERM \l
LlU
_1 0 , 0 0 0 l+
r I r 1117
E M
I
Slone
Figure IV-3
Hose NASH_2
Water System
sshSSI
I S ii
rmrrr-m
Slope = 2.07
Figure IV-5
Hose NASH 4
Water System
.w^ooo
I
L'JJ-ffl.l.l IIJ1
REYNOLDS NUMBER
liii'ii'i1"
100.000
56
sT<
i S H g^ffg^ggpifnth-fafejss fraff sg5j~t;" F1
Slope = 2.14
Figure IV- 6
Hose NASH 5
= Water System
REYNOLDS NUMBER
10,000 ^ 100,000 ,
tttill '' 'ii''' m
57
Figure IV-7
Hose NASH 6
Water System
REYNOLDS NUMBER
it r t u
imTTnr:! |.,I
2.13
Figure IV- 8
Hose NASH 7
Water System
'OLDS UMBER
1U
M i n i
59
i Figure IV-9
J Hose NASH 8
Water System
_ T^i -u -u.i
| [
44-
L-4-U-u..uu . ^4
L ---- 44-4
;I ia iMp E h tr M
hi ^Reynolds vuMBEB^pr *7 ~ ::t.:r -
r itij :fB ;lE -f 1n I'liM t if te i 4 2
i l S s s M t v T i H i i E h i
r
60
slopes have different values - i.e., an average value would not ade
A value of 2,0 for the slope would indicate that the friction
value greater than 2 . 0 means that the friction factor would increase
from equation (II-7), noting that the Reynolds number is directly pro
Note that the abcissa is the mean velocity. Examination of these data
for hose NASH7 at angles of curvature of 0, 60, 120 and 180. These
data clearly show that for a given value of Reynolds number an increase
in bend angle increases the friction. Data for all other hoses follow
Figures IV-12 through IV-24 show the relationship between the Fan
ning friction factor and the Reynolds number for flow of water in a
Figure IV-10
Water System
Hose
NASH 1 a
NASH 2
NASH 4 A
NASH 6 +
MASH 8
F!'s-f :.r
inA ; 1 11 ifwfetTt~r
NASH 7
02121786
Figure IV-11
WATER SYSTEM
4+ff
-
63
=se?~
Figure IV-12
NASA 11
I'.ater System
REYNOLDS NUMBER
'M If M H IH
10.000 :.ioo,ooo
Figure IV-13
NASA 21
water System
REYNOLDS NUMBER
q i'tttiti rnii 11in it
,1 0 ,0 0 0 .,,1 0 0 , 0 0 0 ,TffHrr
by=r
Figure IV-14
NASA 31
Water System
-REYNOLDS NUN
T* il'lilllliJi
W-Wn.i
64
Figure IV-15
NASA 41
Vater System
= ?
_>.*^fp!
"
iTmTF
->- tWT
fiIffititH+i iIf
t
ijtnwutu
'Z^.ZJZREYNOLDS NUMBER
w g g ^ m m
Figure IV-16
NASA 51
lVater System
l a w s
:rr.
i t e 9 5 :.
REYNOLDS NUMBER
pps
^ Figure IV-18
m NASA 71
=E
fn.
tC
10,000 100,000
wm
Figure IV-20
NASA 32
Water System
,10,000 r
li * I IIIIFI
66
Pigure IV-21
MASH 3
Water System
RHYN.QL NUMBER
Figure IV-22
NASH 5
Water System
jTH iigfKigHTHttrfmf-
P.YNQLQS.muhbe .r-
i
3? Figure IV-24 !
gt NASH 7 |g
sff V.'ater System pi
number. The results from hoses NASA 31, NASA 61, and NASA 72 seem to
basis) for pressure loss and Reynolds number. However, Figures IV-12
the slopes of these curves. A slope of 2.0 on the pressure loss versus
tor versus Reynolds number curve, It is mucli easier to detect the dif
ence between 2.0 and 2.2. The reason the pressure loss versus Reynolds
number curve appears straight is that the percentage change in the slope
is very small.
friction factors. In the low Reynolds number region (<75,000) the magni
tude is about 0.020. For turbulent flow in a smooth pipe the Fanning
a rate four times greater for flow in flexible hose than for flow in a
smooth pipe. Furthermore, since the friction factor increases for higher
flow rates in flexible hose, the ratio of energy degradation becomes even
greater.
Experimental Results for the Air System:
drop and volumetric flow rate (SCFM) for hose NASA 51 (30 bend angle)
W/jT/Pj. Figure IV-26 shows the results for hose NASA 51 plotted in
this manner. Figure IV-27 is a plot of the data for hose NASA 32 and
Figure IV-28 is a plot of the data for hose NASH 4. Below a value of
(-AP)/Pj less than about 0.1 the relationship appears linear. Above
this value* the linear relationship breaks down and the line begins
Figure IV-29 shows the results obtained for four bend angles on
hose NASA 72. Because of the lack of compressor capacity it was impos
Note that the pressure drop data can be correlated for a given
Pi * C2 Pi/T (IV2)
N C3 P 1 VJ (IV-3)
| 4 rl
OF AIR 1
1
Figure IV-26
NASA 51
Air System. Jg
Figure IV-27
NASA 32
Air System
Figure Iv-28
NASH 4
;rj Air System
I' ^ w T g.
100 n
I i iiifTt fa&Wfl
74
22ifmn-U
Bend Angle
0 A
60 o
120 D
180 +
^00^.7,99
{
75
Figures IV-30 through IV-36 show data obtained for the flow of air
Note also that the data for hoses NASA21 and NASII2 indicate that the
Reynolds number.
data from both the air and water systems. From these data a general
a sigmoid-type curve, i.e., the data take the form of an elongated S when
plotted, the curve being characterized by a very small initial slope fol
val of nearly constant slope succeeded by a period when the rapidly decreas
ing slope approaches zero. At larger Reynolds numbers the friction factor
once again assumes a constant value which depends only on hose geometry.
The overall increase in the friction factor can be as much as 200 per cent.
previous sections of this chapter this is true for flow in flexible metal
C
MMimmm
Figure IV-30
NASA 21
Air System
m'tir-TU
m m m
10,000 1 00,000
fW-J-i4-M '
_ HS:~
Figure IV-31
NASA 51
Air System
Figure IV-32
NASA 32
Air System
}4i! 1
MYNOLDS NUMBER, &
77
Figure IV-33
NASA 42
Air_System
Figure IV-34
NASH 2
^ i r System
i_ i i 1 1 1 L i l i u t . s ' i n l i L
Figure IV-35
12781819 NASH 4
Air System
Figure IV-36
NASH 7
Air .-system
a
rrrrBKtW
OLDS NT
0 .0 0 0
l.i.TIu
Figure IV-39 Air +
NASA 32 JVater
REYNOLDS NUMBER
80
REYNOLDS n u m b e r -+Hup
Figure IV-41
NASH 2
^REYNOLDS NUMBER!
Air +
niL'.Vater <>
Figure IV-4 2
NASH 4
REYNOLDS NUMBER
mttvtnt
01 J U
81
hoses. This study found that two correlations are necessary to adequately
describe the data obtained on all the flexible hoses. One correlation must
(Re*) = 4 log
F " W
0 (IV-5)
This correlation can be divided into three parts. The first part corre
S = log j 20 $ [ (IV-9)
M o g ( 1 0 0 - *)f
* x 0,094
0.090 0.539 (x) (IV-10)
r
Figure IV-43
Annular-type
hose
HIMHi
IE
HI
Re* = Re/f
10,000 100,000
1 * 1 M l I
tI\I IiIIItHl
83
The third part of the correlation corresponds to the high Reynolds num
Figure IV-44. Again, the correlation can be divided into three parts:
for x S 0.58
x ~ 0.116______
0. 153 + 0.4375 (x) (IV-14)
5 * X - 0.116 (IV-15)
b.Oati + 0.600 (x)
Equation (IV-9) is applicable for this figure but equations (TV-7) and
(IV-8 ) are not. The corresponding equations for Figure IV-44 are:
n
1
.t|(Re*;
4ffi-
5
Figure IV-44
Helical-type
Hose
I
II
Re* = Re/f
I lI m m Li00,000
00
A
85
that in this region the friction factor might correlate better with X/o
than with i)/X. This being the case, the following correlations have been
developed:
1. Region I
a. Annular-type hose
1_ _ 4 log X. - 6 , 3 4 (IV-19)
b. Helical-type hose
and IV-44 have been put into more useful forms. Figure IV-45 shows
However, note also that this effect tends to diminish as the Reynolds
n , i 1 i l i. j
* |4 H l l t i 1 i j f i l i illr Z?, r 1" ^ T j C ' ^ 'T 1 ;~ " T '
j jjjlj|R p Y N j ) ^ ?jtl|MREft
..!'. .11ii!APP9fPfl9iiiiii ml I
0 7 0 0 1 2 a
00
ON
! ! 'j mnniiimf tlillMn
FigUTe IV-46 . i II !>!
HTiTmm
-T
.Helical
I11 i <hi n
- rrrrTFmi'fT! n'TTfn
iiiliilipillH H
X/D = 0.35
% n i 30
TIB*""1- '
mat;
I:;1 !hi
iiii
lii'lf;;! ajj
H
t<
,
1 000,000
00
^1
0
7 8 9
wwmm Inr ttti
|TTT * - iMMi TTTTHi
, j r i.
t
ep^-ooee<sr|?
REYNOLDS NUMBER , 1
MiiilliiiH
t it.. 1 . 0 0 0
:1 PQpy(?;n j *i!
!: :- ii
;.i '-r->Li-4'Hi-n
Figure IV-48
NASA 72
Water System
Bend angle
-Ui'QrH QDI8
p -JT.J*
XXX
<a-Q
!I,REYNOLDS
" iiii i<i( <NUMBER
' ' ;Is;
fi<
-I!, 100,000 ' 1 ,000,000
! I :t:r : 1 . :l *4 1 I
I I1
6 7 8 9 1
I |.'JJ U t i l ' ; J u t r t
T T T m m m i-vulii-i; - i f f ] T w :U ;
IV-49
NASH 6
,<t'
Water System Hr Bend angle ~i
REYNOLDS NUMBER
i ^ ii
to
o
91
Note that fg is the friction factor for a curved section and f is the
DISCUSSION OF RESULTS
ble hoses. It is then shown how this model leads to certain conclusions
as to which geometric parameters are important for the three flow regions
Any proposed flow model must take into account the observed flow
behavior. For flow in flexible metal hoses the flow behavior appears
used to explain the behavior for these systems can be used to help
The flow behavior for flexible hose will be described using the
For the Reynolds number range used in this study there appears to be
three distinct flow regions. At the low end of the turbulent regime the
study was about 10,000. However, the work of Allen (45) indicates
that this region extends to Reynolds numbers much lower than this.
Figure V-l shows data obtained by Allen in the Reynolds number range
350 to 54,000. These data were taken on a flexible pipe of 0.S inch
minimum diameter, 0.813 inch maximum diameter and 0.104 inch pitch.
Note that the critical Reynolds number for the transition from lami
hose geometry. It has not been established if the lower limit is simi
the behavior of the friction factor. Whereas initially the friction fac
tor was independent of the Reynolds number, it now becomes a strong func
number.
and Daniels and Cleveland (51) found it for flow in flexible metal hoses.
gations of the friction loss for flow across tube banks have been made.
These data also are usually presented in the form of a plot of friction
geometry between the two flow systems, the friction factors and Reynolds
Kays, London, and Lo (55) for 4 different banks of tubes. The defining
4G2 max L
Re d'Gmax (V-2)
for flow in flexible hoses. This curve represents data taken for tubes
in an in-line arrangement.
ently from a staggered arrangement may provide a clue to the flow mech
ll-
>i4i.
Staggered
Staggered
In-Line
i-H -H
i!;i ;
HIM*
Figure V-2
W l l
0 .0 0 0 ;
to
O'
97
the flov; patterns which occurred as the water flowed through the tube
bank. Both the in-line arrangement and the staggered tube arrangement
IN-LINE STAGGERED
ARRANGEMENT ARRANGEMENT
o o o o o
o o o o o
o o o o 0
FLOW Ocf/Vtf
w FLOW O
ILLUSTRATION V-l
\
98
first. At low flow rates the pattern around the tube is similar to
that observed for flow around a single circular cylinder. The sepa
ration of the boundary layer and the turbulent wake behind each tube
was quite evident. As the flow rate increased the turbulent wake
transverse row, and only a very thin boundary layer formed on that
tube. The spaces between the tubes in all transverse rows contained
For the staggered tube arrangement in which the tubes are widely
since the next tube was two transverse rows Away, this turbulent wake
did not reach the next tube. It was found that a boundary layer was
formed on the forward part of each tube in the bundle and subsequently
these spacings the tubes are not in the turbulent wake of the tubes
The only place where there was a large turbulent wake was behind the
Since the flow behavior of the in-line tube arrangement was simi
lar to that for flexible hose, it appears that similar mechanisms are at
<
99
friction and (b) form friction. Using this concept equation (V-4) can
tion, both skin and form friction may be active in varying degrees.
the total friction is largely form friction and skin friction is unim
portant.
The following discussion deals with the proposed flow model for the
f
100
the convolutions. The flow model for this region assumes that a stable
vortex (swirling eddy) is present in this valley and that this is the
is at the crest of the convolution. The flow model assumes that on the
ant assumption for this low Reynolds number range is that the wake has
e H f s + Hf d s - ^ ^ l <v"6)
p * * p o
where -PS is the pressure drop due to skin friction and is the
ate the latter two terms in equation (V-6 ) in order to estimate the
Fk * A K f (V-7)
The assumption is now made that the convolutions behave as if they were
be less than the average velocity in the flexible hose. Using these
(V-8 )
and not f. The term (L/X) has been introduced to take into account the
(V1 0 )
calculate the energy required to maintain the eddy motion in the valleys
between the convolutions and of length equal to the flexible hose per
The energy of flow per unit time through any concentric cylindrical
{
102
dm = ppvrdr (V-12)
dt
E pp vr 3 dr (V-13)
8c
To solve for the energy per unit time required to maintain vortex flow
* ppsv0 3 (V-15)
16 gc
(V-17)
{
(bmbining equations (II-8 ), (V17), and (V-18) an expression for the
f * (V-19)
and the velocity ratio (vQ/v). For a given hose o and X are constant
and (v0 /v). The pertinent question now becomes, assuming that the
and (v0 /v) become independent of the Reynolds number and hence cause
The only thing which can be done at the present time to answer
this question is to take systems for which data are available and
assume that they are approximations to the real system. First, con
range from 100 to 200,000. It' appears reasonable to assume that for
104
v * 2.5 In M + B
1 1 (V-20)
v0 = 2.5 W y \ + B (V-21)
v \ r j
vhere y^
'o is some small distance measured from the crest of the con-
volution. From the definitions of the friction factor and v* the
v * _j (V-22)
v* /TTT
Using equations (V-21) and V-22) to solve for vn/v^
vo - 17 12.5 In Xo + B| (V-23)
7 ~ )
This result indicates that for a given hose the ratio vQ/v depends on
the value of the friction factor, however, since the friction fa.ctor
remains constant for turbulent flow in the low Peynolds number region,
number.
105
flow through hose NASH 4. For this hose a/X 0.125/0.250 0.50
. 0.50
v \2
vo\ = 0.04
The experimental values for the friction factors in the low Reynolds num
ber range are all of the order of magnitude of 0,020. Note that equation
friction factor is very small. For the assumed case just discussed
skin friction contributed only a little more than 2 % of the total fric
tion. This is in general agreement with the statement that when bound
ary layer separation and wake formation is present the total friction is
the geometric ratio o/X. This correlation has been developed for
1. Annular-type hose
(V-24)
This relationship has been tested for values of Pe/F/(P/X) from 170
to 1400.
2. Helical-type hose
(V-2S)
This expression has been tested in the range of Re>^?/(D/X) from 180
to 2 ,0 0 0 .
As the flow rate through the flexible hose increases the wake
with tne boundary layer or the next convolution. The proposed flow
model for flexible hose assumes that this is the point where the fric
result follows from the assumption that the flow mechanism in flexible
roughened fcy densely-packed uniform sand grains, and the correlation devel
oped for annular hose from the results of this study. Nikuradse's correla
D/e, However, it is evident that D/X is numerically equal to D/e for his
pipes and hence either parameter can be used without affecting the numeri-
cal results. It should also be noted that Morris' correlation was derived
Experimental data for very high Reynolds numbers were not obtained
in this study. This was due to capacity limitations in the pumps and
that the friction factor would approach a constant value at very high
riments at very high Reynolds numbers and their results clearly indicate
Morris (39) concluded from his study that this limiting valueis appli
\
i
Figure
i!' Hi
IP'
!l'
Annular hose
correlation
Morris
Nikuradse
fit
73
fl
e* = Re/f
fiTITT
liliiiU O O O H n u ll*
109
is a good visual flow experiment. At the present time this has not
and more often than not this essential information has been provided
velocity profiles in the main stream of the hoses (i.e., the center
relations.
Accuracy of Correlations:
tions for the data produced in this study. The appendix contains a list
ing of the computer program used to obtain the results given in this
table. For each hose (straight sections only) the average error ( % ) 1
N\SA 11 9.4 30.1 7.2 29.4 -17.6 40.1 259.1 244.9 146.9 140.1
NASA 12 29.7 61.9 38.5 52.9 24.9 64.4 175.1 147.6 -1 . 2 49.5
NASA 21 7.5 35.8 1.1 27.4 -5.0 34.3 270.0 235.0 141.4 124.0
NASA 22 -23.6 39.4 -5.6 21.7 -33.2 48.6 165.2 146.0 56.8 52.7
NASA 31 0.8 35.0 -8 . 2 33.5 -18.7 33.4 189.3 166.8 87.8 80.4
NASA 32 32.5 52.0 34.7 35.3 33.8 47.6 336.9 163.3 152.8 70.8
NASA 41 1.4 25,9 -17.4 33.0 -12.3 29.9 180.5 161.0 84.8 77.7
NASA 42 -7.0 31.6 5.2 23.2 -2 0 . 6 38.0 168.5 144.1 41.0 40.4
NASA 51 11.5 24,0 -6 . 2 22.1 -15.2 30.0 182.2 163.0 68.5 62.7
NASA 52 10.7 20.9 38.7 53.8 -24.4 31.8 149.8 148.7 -18.0 26.3
110
Table V 1 (cont'd)
NASA 61 9.8 28.3 -16.5 33.2 -7.6 32.6 161.3 146.2 59.9 57.6
NASA 62 4,6 29.3 23.4 40.9 -16.4 38.4 129.1 120.1 -33.1 47.0
NASA 71 25.4 50.6 -5.7 54.5 -7.5 53.3 174.8 148.9 54.7 60.3
NASA 72 5.4 11.1 18.7 32.1 -14.2 17.0 168.0 174.3 10.3 13.8
NASA 81 45.1 49.6 13.8 26.0 2.6 9.0 250.9 259.6 102.9 108.0
NASA 82 13.7 17.8 22.9 30.4 -1 2 . 2 21.7 202.8 205.7 38.9 41.6
NASH 1 16.7 40.8 20.7 37.0 -13.6 51.1 240.9 185.6 33.9 42.4
NASI! 2 -20.5 38.4 -3.6 25.7 -18.0 41.6 197.9 177.9 61.7 57.9
NASH 3 -15.6 29.0 -0.4 15.4 -27.6 38.3 176.7 165.5 49.7 48.6
NASI! 4 - 3.8 26.8 3.3 19.2 - 9.1 28.6 194.3 173.8 56.0 52.2
NASH 5 - 4.5 19.3 3.7 23.1 -23.3 31.8 180,4 174.2 49.2 49.1
NASI! 6 - 4.4 19.5 -7.8 20.7 -23.7 31.0 149.8 144.6 30.4 33.5
111
NASH 7 15.8 17.4 -3.3 10.7 -2.3 15.0 206.7 204.5 65.5 65.4
NASH 8 29.3 46.7 2.1 4S.7 13.4 41.4 244.2 218.3 90.5 84.5
f
Table V-l (cont'd)
NASA 21 9.0 19.8 25.0 50.6 11.0 31.1 41.4 50.7 121.1 107. 7
NASA 22 -4.3 20.8 25.5 53.7 -11.7 28.9 88.8 30.7 58.2 54. 4
NASA 31 4.8 26.7 8.1 39.1 0.2 31.7 6.5 40.0 68.1 65. 8
NASA 32 35.9 29.9 23.3 30.0 37.8 41.8 75.8 52.8 153.8 71. 8
NASA 41 -2 . 0 23.8 4.1 47.7 2.2 25.6 4.0 37.3 57.8 56. 8
NASA 42 4.7 22.6 2.8 34.7 1.5 34.7 8.0 3.1 49.6 46. 0
NASA 51 4.5 19.8 5.3 37.1 4.6 19.6 0.9 30.4 56.3 53. 1
NASA 52 -4.8 20.6 9.0 51.7 -4.3 19.5 35.2 36.6 35.4 47. 5
NASA 61 -3.6 28.3 16.2 51.0 0.5 26.8 -14.0 37.7 41.5 44. 4
NASA 62 -12.4 32.1 -7.4 41.7 -12.5 32.3 22.3 31.6 20.3 31. 1
Table V-l (cont'd)
NASA 71 0.9 52.1 16.1 64.8 7.7 51.5 -6 . 1 52.8 47.5 57. 4
NASA 72 -3.1 15.0 3.5 42.7 2.1 13.6 16.9 19.2 43.2 46. 7
NASA 81 19.2 28.5 21 .0 56,4 32.7 41.4 19.7 21.6 89.4 95. 4
NASA 82 6.1 11.8 4.5 29.5 16.6 20.6 23.1 27,0 62.6 6 6 ,0
NASI! 2 4.9 36.9 21.4 63.3 -3.2 24.6 30.4 36.6 78.1 72. 1
NASH 3 13.0 32.1 6.8 39.7 0.05 14.9 20.2 27.7 64.0 61. 4
NASH 4 10.8 26,0 21.2 51.4 12.1 20.5 21.6 31.1 66.9 61. 2
NASH 5 14.6 35.5 10.y 52.3 9.6 23.2 17.1 24.0 60,3 59. 6
NASH 6 0.07 20.1 8.8 41.4 4.0 18.9 -1 . 2 21.2 37.9 39. 8
NASH 7 6.5 14.5 3.3 36.9 16.2 20.5 14.3 19.6 67.2 67. 5
NASH 8 12.2 46.2 9.6 53.2 27.6 48.5 21.7 42.5 85.7 81. 3
113
Table V-l.(cont'd)
114
Table V-l (cont'd)
115
116
2
and average standard deviation (%) are given. The following is a list
FN - Equations (11-45)
able for flow in flexible hose. Also, it appears at first that correlation
correlation FP1 was tested only for data in the lower Reynolds number reg
ion where the friction factor is constant. Correlation FR, on the other
2 Standard deviation
\
117
hand, was tested over the entire range of Reynolds numbers. The con
correlation FR1 can predict friction factors in the low Reynolds num
is the region where the lowest pressure drops were measured. At low
nificant per cent of the total readings and hence the accuracy and
precision decreases.
Another important fact should be brought out about some of the cor
FHVH, FNK1, FNK2, and FCW do not predict the correct behavior observed
for flow in flexible hose. FNASA assumes that there is a straight line
and Reynolds number data from the water system. As can be seen from
and lower flow rates would have been available the nonlinear nature of
the curve would have been more apparent. However, with the data avail
very high flow rates the data does indicate a nonlinear relationship,
Correlations FN, FDF, FNK1, FNK2, and FHVH indicate that the fric
flexible hose.
Table V-2 gives the average en*or (%) and average standard deviation (%)
Figure V-4 shows data obtained by Daniels and Celveland for hose
DCA4. These data clearly demonstrate that the friction factor assumes a
The curved hose correlation was developed from data covering a range
-0.17
(IV-18)
Note the comparison between this expression and one given by V.Tiite (22)
1/2 1/4
fB 1.0 + 0.075 Re (11-32)
\
TABLF. V-2
DCA1 7.4 17.4 115.7 107.0 -17.7 36.7 -5.7 15.5 18.6 25.5
DCA2 -14.5 34.8 .94.0 71.7 13.2 36.4 9.2 14.1 -13.6 29.3
DCA3 17.4 24.0 152.0 137.0 26.9 30.6 20.3 25.1 19.0 25.5
DCA4 -5.8 15.3 104.1 98.8 19.8 23.2 5.9 12.1 13.4 15.5
DCA5 -23.9 29.6 81.3 82.0 35.4 37.1 -4.3 14.2 11.6 16.5
DCA1 7.5 27.4 18.5 29.9 46.6 47.2 13.0 28.5 0.3 16.8
DCA2 -2 1 . 0 50.2 14.1 36.6 7.8 37.8 8.1 37.8 -20.3 45.5
DCA3 2.5 25.8 39.0 3S.6 42.3 41.0 32.5 34.2 8.0 20.4
DCA4 -26.5 35.0 12.7 19.6 4.5 17.9 7.4 18.1 -13.5 21.0
120
tixiiiitl::r:I n t Hnnij
Figure V-4
Hose DCA 4 .] I U lM ji,- l!j Jij *- - J l f tt
L (1$- REYNOLDS,NUNBERtttft hij.iiji*
Water
. Hmitt iiJlfl ra 11,000,000 |W j.:: I
122
than for common pipe. However, note that this effect decreases as
flexible hose. The results reported in this table are (1) the average
O o O o o o
Hose 30 60 90 120 150 180
CD (2 ) (1 ) (2 ) CD (2 ) CD C2) (D (2 ) CD C2 )
NASA 11 12.9 28.2 -13.6 20.9 -14.5 21.5 -1 2 . 0 23.9 -15.2 30.2 2.1 24.9
NASA 12 34.4 70.8 37.1 68.8 33.4 59.0 34.5 54.6 34.1 54.5 36.7 56.3
NASA 21 6.1 9.8 6.6 11.0 4.4 8,8 0.4 11.8 2.3 13.9 -0.9 13.9
NASA 22 13.2 25.3 -4.0 24.2 2.5 35.5 7.6 14.5 2.4 17.4 -0.3 20.8
NASA 31 -6 . 6 16.0 -7.6 14.0 -11.9 18.7 -21.9 27.7 -26.9 36.0 -24.4 29.8
NASA 32 2.4 12.0 0.3 14.5 11.6 33.2 -8 . 8 23.8 -1 0 . 2 27.4 -14.5 28.2
NASA 41 -8 . 6 12.7 -1 1 . 2 15.5 -18.1 2*2.5 -24.5 30.6 -19.9 27.2 -28.2 33.}
NASA 42 3.2 21.5 1.3 IS.5 -5.7 16.9 -6 . 1 24.3 -6 . 6 16.4 -7.0 22.8
NASA 51 -9.3 17.8 -13.7 19.4 -15.8 24.0 -21.4 81.2 -16.7 23.9 -23.4 . 30.4
NASA 52 45.4 C9.7 51.0 76.5 52.2 74.2 50.1 72.1 49.0 71.1 53.2 75.7
NASA 61 -6 . 8 14.3 -10.9 23.0 -8.9 14.4 -13.6 18.4 -1 1 . 8 21.0 -13.8 24.4
NASA 62 16.0 29.3 13.8 27.2 14.0 25.3 11.2 28,9 7.2 23.6 8.7 23.7
123
(1 ) Averag e error ()
t
Table V-3 (contd)
0 0 0 o
Hose 30 60 90 120 150 180
Cl) (2 ) CD (2 ) CD (2 ) CD (2 ) CD (2 ) CD (2 )
NASA 71 9.5 16.3 -0.7 7.6 -7.8 11.7 -8.7 13.4 -6 .7 16.9 -12.4 18.1
NASA 72 1.3 44.9 7.6 28.3 2.1 36.3 -2.5 30.0 6 .2 25.4 4.1 24.8
NASA 81 -1 1 . 1 17.7 -8.7 16.0 -17.0 22.3 -24.5 33.9 -14. 5 20.9 -19.8 39.3
NASH 1 15.1 21.9 16.8 24.5 16.6 23.9 19.3 27.8 2 2 .8 31.3 16.7 27.5
NASH 2 1.2 15.6 1.5 11.3 1.2 12.7 0.5 17.2 S. 1 13.9 -0.5 11.0
NASH 3 3.1 14.7 5.3 14.2 2.7 17.9 1.6 14.2 3. 5 11.1 0.7 12.6
NASH 4 4.3 16.9 6.3 16.0 -0.9 22.0 -3.6 20.5 -3. 9 18.1 -3.9 36.4
NASH 5 4.7 32.4 11.0 25.6 9.6 23.1 5.0 23.2 2 .9 19.2 0.4 22.5
NASH 6 -7.9 34.8 -10.5 24.3 -1 2 . 0 17.9 -20.3 38.5 -2 2 .3 55.4 -14.5 20.2
NASH 7 -7.2 IS. 2 -7.2 12.7 -1 0 . 2 15.0 -19.4 27.7 -14. 8 17.9 -22.7 ' 37,8
NASH 8 -1.3 7.8 -6 . 2 12.5 -1 1 . 8 15.1 -18.9 22.3 -13. 0 18.0 -2 1 . 6 40.6
124
CHAPTER VI
CONCLUSIONS
and
RECOMMENDATIONS
Conclusions:
for flow in flexible metal hose than for flow in a smooth tube of the
Reynolds number increases above this regime the friction factor begins
Reynolds number.
125
126
D/X.
Re* ^ 11,000
i - 4log /-1- 2.28 (VI-3)
If
D
- 4 log * 2.28 (VI-6 )
/F X
is the plot for annular hose and Figure (IV-46) the plot for helical
hose.
applicable to both annular and helical hoses. The ratio of the fric
tion factor for a curved hose to that for a straight hose is:
Recommendat ions:
in smooth rigid pipe. The behavior found in the present study was
just concluded has pointed out as being most needed. For example,
\
128
studies of the velocity profiles in the main stream of the tubes and
improved. Work in this area might take the form of developing a corre
{
LITERATURE CITED
(1) Stanton, T, E. and J. F. Pannell. Phil. Trans. Roy. Soc. A., 214,
(1914), p. 199.
129
130
}
(31) Schlicting, H. boundary Layer Theory, 4th ed., McGraw Hill Book
Co., Inc., New York, 196u.
(36) Nunner, '-V. "Heat transfer and pressure drop in rough pipes",
VDI, Forschungsheft, (1958) p. 455.
(37) Koch, R. "Pressure drop and heat transfer for flow through empty,
baffled, and packed tubes", VDI, Forschungsheft, (1558), p. 469.
(55) Kays, .V. M., A. L. London, and P. K. Lo. "Heat Transfer and Fric
tion Characteristics for Gas Flow Normal to Tube Banks - Use of
a Transient-Test Technique", Trans. ASME, 76:387 (1934).
<
APPENDIX A
133
134
All flexible metal hoses used in this study are denoted by the
helical hose. The number following these four letters is used with
Table A-l to define the internal geometry of the flexible hose, e.g.,
APPENDIX A
TABLE A-l
D X e 0
Annular: (inches) (inches) (inches) (inches)
helical:
{
136
TABLE A-2
D X e
Hose (inches) (inches) (inches)
137
138
APPENDIX B
Many sigmoid curves, both normal and skewed, can be fitted sat
5 * x - xj (B-l)
a + bx
vhere
Illustration Bl:
tionship between -^(Re*) and Re* for helical-type hose has a sigmoid
shape in the range of Re* from 2000 to 16,000. The data in this
<
139
Pe* ip(Re*)
2000 4.28
2250 4.25
2500 4.20
3000 4.10
3400 4.00
3800 3.90
4500 3.75
4800 3.60
5600 3.40
6000 3.30
6900 3.10
8000 2.90
8600 2.80
9300 2.70
10000 2.61
12000 2.45
13000 2.40
14000 2.35
15000 2.30
16000 2.28
for x 0.58
S x - 0.116
0.153 + 0.4375 (x) (B-5)
f
140
Figure B-l
Figure B-2
141
S * x - 0.116 (B-6 )
'0.059 + 0.6U0 (x)
143
C THIS PROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 7040 COMPUTER.
C THE PROGRAM CALCULATES PREDICTED FRICTION FACTORS FROM VARIOUS
C MODELS AND COMPARES THEM TO OBSERVED FRICTION FACTORS. THE
C OUTPUT FROM THE PROGRAM IS ERR O R (PER CENT) AND STANDARD DEVIATION
C (PER CENT).
C
C K L RILEY 1303-4C060 MODEL COMPARISON FOR FLEX-HOSE
C DIMENSION VARIABLES
DIMENSION R E (200 )FA(2C0)FP(2G0)R(7),FAA(200)
DIMENSION DIAM(3,8)PITCH(3,8),DEPTH(3,8),AXIAL (3,8)
C READ INTERNAL GEOMETRIC PARAMETERS
DO 1 1= 1 3
D01K= 1,8
REAC2,DIAMl I,K),PITCH!I,K),DEPTH!I,K),AXIAL (I,K)
2 FORMAT(4F1C.0)
1 CONTINUE
C READ fiENC RADII
REAC51,(R( I),1= 1,7)
51 F O R M A T (7F1C.0)
C REAC HEADER CARD
C N=NUMBR CF DATA POINTS
C NN=INDICATES TYPE OF HOSE, IF 1 OR 2 ANNULAR-TYPE, IF 3 HELICAL
C 0 = INS ICE CIAMETER, INCHES
C LANG=REND ANGLE, DEGREES
78 REAC4 3NNN,D,LANG
43 F O R M A T (I10,10X,110,F1C.0,110)
C REAC CBSERVEC DATA POINTS
C R E (I)=REYNOLCS NUMBER
C FA(I)=CBSERVEO FRICTION FACTOR
REAC5C (R E ( I),F A (I ), I= 1,N )
50 FORMAT(2F20.0)
PRINT64,NN,D,N,LANG
64 FO RM A T (I1C*F10.32I10)
C SET SUBSCRIPTS FOR GEOMETRIC VARIABLES
144
IFtO.GT.0.500)NG=1
IF(C.GT.0.750)NG=2
IF(D.CT.1.000)NG=3
IF(D.GT.1.250)NG=4
IFIC.CT.1.480)NG=5
IFtC.GT,2.000)NG=6
IF(C.CT.2.500)NG=7
IFID.GT.2.SOO)NG=8
IF(LANG.EC.O)NA = 7
IF(LANG.EG.30)NA=1
IFtLANG.EC.60)NA=2
IF(LANG.EC.90)NA=3
IF(LANG.EC.12C)NA=4
IF(LANG.E0.150)NA=5
IFILANG.EC.180)NA=6
C CALCULATE PRECICTED FRICTION FACTOR, F P U )
00 3CC I = 1 N
F P U ) = FNASA(NN,RE(I ) ,DIAM(NN,NG> ,DEPTH1N N , N G )PIT C H {N N , N G ) ,
AXIAL (NN,NG)R(NA)L A N G )
300 CONTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCER = ERROR(N , F A , F P )
PCSER = DEVR(N,FA FP,LANG)
PRINT 32,PCER,PCSER
32 FORMAT! /1C-X,5HFNASA,2X,E20.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, FPlI)
DG500 I= 1,N
F P U )=FR(NN,RE(I ) ,DIAM(NN,NG) ,P ITCHt NN, NG ) ,LANG,R( NA ) )
500 CONTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCER=ERRCR(NFA,FP)
PCSER=DEVRfN,FA,FP,LANG)
PRINT52,PCER,PCSER
52 FORMAT(/1GX,2HFR,5X,E20.8,E20.8)
IF!LANG)416,416,78
416 J=0
C CALCULATE P R E d C T E O FRICTION FACTOR, FP (I )
00 44C 1= 1,N
RRSsOIAM(NNNG)/PITCH(NN*NG)
RSS=PITCH(NN,NG)/AXIAL(NN,NG)
R6S= (R E ( I)FA ( D / R R S )
IF(NN-2)411,411,412
411 IF(RES-14C0.0)413,413,440
413 J=J + 1
F A A (J J= F A ( I)
FRES=t.34
GO TC 415
412 IF(RES-2C00.0)414,414,440
414 J=J+1
F A A (J )= F A ( I)
F RES =5 .77
415 FP(J)=1.0/((FRES+4.C*AL0G10(RSS))**2)
440 CONTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCEr.-ERROR!J,FAA,FP)
PCSR=0E3!J,FAA,FP)
PRINT42,PCER,PCSER
42 F O RM AT (/10X,3HFR1,4 X ,E 20.8,E2 0.8)
C CALCULATE PREDICTED FRICTION FACTOR, FP(I)
417 D06CCI = 1, N
FP!I)=FN(DEPTH(NN,NG),DIAM(NN,NG))
600 CONTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCER=ERROR!N,FA,FP)
PCSER=DE3(N,FA,FP)
PRINT53,PCER,PCSER
53 FORMAT!/1CX,2HFN,5X,E20.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
146
t
)
DG70CI = 1 N
FPtI)=FDF(PITCH(NN^NG),DIAM(NN,NG),D E P T H (N N , N G ))
700 CCNTINUE
G CALCULATE ERROR AND STANDARD DEVIATION
PCER=ERRCR(N,FA,FP)
PCSER=DE4(N,FA, FP)
PR INT54, PCER,PCSER
54 FORMAT </1CX,3HFCF,4X,E 2 0 .8,E 2 0 .8>
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
D080CI= I ,N
FPI I )= F D C 1 (R E (I),DIAM<NN,NG),DEPTH(NN,NG))
8C0 CCNTINUE
C CALCULATE ERRCR AND STANDARD DEVIATION
PCER=ERROR(N,FA,FP)
PCSER=DEV(NFA,FP)
PRINT55,PCER,PCSER
55 FCRMAT(/1CX,4HFDC1,3X,620.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
DC90CI = I *N
F P U I= F M S U (R E (I ),01 AM(NN,NG),DEPTH(NN,NG),PITCH(NN,NG),AXIAL(NN,NG
*) )
900 CCNTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCER=ERROR(NFA,FP)
PCSER=DE7(N,FA,FP)
PRINT56,PCER,PCSER
56 FCRMAT(/10X,4HFMSU,3X,E20.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
DDIOOC 1= 1,N
FP( I)=FDC2(RE(I),D E P T H {N N ,N G ),DI A M (N N ,N G ) )
1000 CCNTINUE
C CALCULATE ERRCR AND STANDARD DEVIATION
PCER=?ERRCRIN,FA,FP)
PCSER=DEV(N,FA,FP)
147
POINTS7, PC E R -PCSER
57 FORMAT(/10X,4HFDC2,3X,E20.8,E20.8)
C CALCULATE PRECICTED FRICTION FACTOR, F P f I )
DC1 20 C1= 1, N
F P U ) = F H V H ( C I A M I N N , N G ) ,P I T C H (N N , NG )J
1200 CCNTINUE
C CALCULATE ERRCR AND STANCARD DEVIATION
PCER=ERROR(N,FA,FP)
PCSER=DE3(N*FA,FP)
PRINT59,PCER,PCSER
59 FORMATI/10X,4HFFVH,3X,E20.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
D0130CI=1N
F P U )=FNK1 IRE!I),DIAM<NN,NG),DEPTH(NN,NG))
1300 CCNTINUE
C CALCULATE ERROR AND STANDARD DEVIATION
PCER=ERRCRIN,FA,FP)
PCSER=DEV(N,FAFP)
PRINT60,PCERrPCSER
60 FORMAT(/10X,4HFNK1,3X,E20.8,E20.8)
C CALCULATE PREDICTED FRICTION FACTOR, F P U )
DO 1ACC 1= 1,N
F P (I)= F N K 2 (R E ( I ),DI A M (N N , N G ),PITCHINN,NG))
1400 CCNTINUE
C CALCULATE ERRCR AND STANDARD DEVIATION
PCER=ERRCR(NFA,FP)
PCSER=DEV(NFA,FP)
PRINTfcl,PCER,PCSER
61 FORMAT </lCX,4HFNK2,3X,E20.8,E20.8)
C CALCULATE PRECICTED FRICTION FACTOR, FP(I)
D015CC 1= 1,N
F P I I )= FCWIREII),D1AM(NN,NG),DEPTHINN,NG))
1500 CCNTINUE
C CALCULATE ERRCR AND STANDARD DEVIATION
148
PCER=ERRORtN,FA,FP)
PCSER=DEV(NFAFP)
PRINT62,PCER,PCSER
62 FORPATI/10X,3HFCW,AX,E 2 0 .8,E20.8 J
C CALCULATE PREDICTED FRICTION FACTOR, FP CI)
D0160C I= IN
FP(I) = FM{RE( I ),CIAM(NN,NG),PITCH{NN,NG)J
16C0 CCNTINUE
C CALCULATE ERRCR AND STANDARD DEVIATION
PCER=ERRORIN,FA,FP)
PCSER=DEV(N,FA,FP)
PRINT63,PCER,PCSER
63 F O R M A T (/1CX,2HFM,5 X ,E20.8, E 20.8>
PR INT 1450
1450 F O R M A T (1H1 )
G0T078
END
149
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 704C COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTION FACTORS FROM MODEL FNASA.
C
FUNCTION FN*SA(NN,REN,DI,EPS*PH,StRB,LANG)
GECMI = CSEPS)/(PH**2)
B 1 = 2 .9866834E-01*GE0Ml-3. 1293821E-02
GECM2 = (PH-SJ/EPS
IF(NN.EC.3)GO TO 720
BO * 1.5882587E-02*GE0M2-2.IA825I1E-03
GC TC 703
720 B0S2.9 1 56283E 02*GEUM2 8 8 6 16C99E 03
703 FFM = B 0 (REN*B I)
IF(LANG)701,701,702
701 FNASA = FFM
RETURN
702 GECM3 DI/{12.0*RB)
RFF = 1.0 + 7.897679*(GE0M3**.8956039)
FNASA = RFFFFM
RETURN
END
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 70AC COMPUTER.
C THIS SUBPROGRAM IS USCD BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTIGN FACTORS FROM MODEL FR.
C
FUNCT ION FRI NN, REN ,D I PH,L*ANG BR )
DIMENSION Y A (L O O )* YH (100),F(1C0)
RRS=CI/PF
F (1 )= C . 0 2 0
D020CJ = 1 * ICO
JJ=J+1
RES=(RENMF(J)*0.5)J/RRS
I F (N N - 2)7 71 8
7 IF{RES-1A00.0)A 5,5
A FRES = A . 35
GC TO 1A 1
5 IFtRES-llGOO.O)6,9,9
6 XA = L0G101RS)-3.1A6128
IF(XA-.5A)11,12,12
11 S = (XA-0.09A)/(0.090+0.539*XA)
Y A (1) = 0.5
GO TO 1A2
12 Y A ( 1) = 1.6
S * (XA-0.C9A)/(0.0395+0.633*XA)
142 DO 30C L = 1,100
LL = L + 1
YA< LL )= M 1 0 . 0 * * S ) / 2 0 . 0 ) *( AL OG 10 ( l OO .O -Y A( L )))
IF(ABS(YA(LL)-YA(L))-C.0001)13,13,300
13 FRES = A.35 - YA(LL)
GO TG 1A 1
300 CCNTINUE
9 FRES = 2.28
GC TO 1AI
8 IF(RES-2C0C.0)1A,15,15
1A FRES =5 A.28
GO TO 141
15 IFIRES-160C0.0)16,17,17
16 XH = AL0G10(RES)-3.30103
IFIXH-.58)18^18*19
18 S * (XH-0.1l6)/t0.153+0.4375*XH)
YHfl) = 0.5
GC TC 143
19 Y H (1) = 1.6
S * (XH-C.116)/(0.059+0.600*XH)
143 00 4CC M
= 1,100
MM = M + 1
Y H (M M )={(10.0**S)/20.C)*(ALOG10(100.0-YH(M)))
I FtABS(Yh( MM )-YH(M>.)-0. 0005)21,21, 400
21 FRES = 4.28-YHIMM)
GC TC 141
400 CCNTINUE
17 FRES = 2.28
141 F (JJ ) = l.C/((FRES+4.0*ALOG10(RRS))**2)
IFtABS(F(JJ)-F(J n-0.0015)22,22.200
200 CCNTINUE
22 IFILANG)23,23,24
24 CCNTINUE
R F F = 1.0+59.0*1(01/12.0)/BR)*(REN**(-0.17))
FFB = RF F * F (J J )
FR*FFE
RETURN
23 CCNTINUE
F R = F (J J )
RETURN
END
in
K>
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 7040 COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTION FACTORS FROM MODEL FN.
C
FUNCTION FN(EPS,DI)
FN*C.16*(EPS/DI ).* * 0 5
RETURN
END
153
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 704C COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM CALCULATE PREDICTED
C FRICTICN FACTORS FROM MODEL FMSU.
C
FUNCTION FMSUIREN,DIEPS,PH,S)
0 IMENS ION F(IOO)
RA*S/2 .0
B1*<2CO.OO*RA*EPS)/{DI*2)
C=17.C*(RA*EPS/(PH*0I)J-0.3
82=0.S 6 8 / B 1
Cl=3.6 3E13*ttRA*EPS)/{DI**21)**(-3.71)
H = 0 . 1415*{1C00.0*((RA*EPS)/(DI**2)))**3.5
A=(C*B2)/(1.0+(D1/{REN**4>))
F ( 1 )= C . 0 2 0
DC101J=1,ICO
JJ=*J+1
F(JJ) = 1.0/1(3.48 B1*A LO G(A f H / (REN*SORT(F tJ )))))**2)
IFtABS(F(J J ) F tJ))-0.CC01) 102,102,101
101 CCNTINUE
102 FMSU= F (J J )
RE T U R N
END
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 7040 COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTION FACTORS FROM MODEL FDC2.
C
FUNCTION FCC2(REN,EPS,DI)
DIMENSION Ft 100)
RR = EPS/t DI+2.0*EPS)
Ft 1 )=C.020
DC201J=1,100
JJ=J+1
ES*REN*SQRTtF<J)/2.0)*RR
IFtES-lOCC.0)202,202,203
202 AES= 11 .0
GCTC241
203 IFtES-lOCCO.0)205,206,206
205 AES=-3.0*ALOG10tES)-i20.0
GCTC241
206 AES=8.0
241 F (JJ)* 2. 0 / t(AES-3.75 2.0*ALOGt2.0*RR))**2)
IFtABS(FtJJ)-F(J))-0.CC01)204,204,201
201 CCNTINUE
204 F C G 2 = F (J J )
RE T U R N
END
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 7040 COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTICN FACTORS FROM MODEL FNK1.
C
FUNCTION FNK1(REN? D I E P S )
CIMENSION F(100>tYY(100)
RRR = EP S/CI
F ( 1)= C 0 20
D 0 3 C 1J = 1100
JJ=J+l
RES=REN*SCRT(FlJ))*(RRR)
RESL=AL0G10(RES>
IFIRESL-0.6 5)302,302,303
302 FRES=4.Q*RESL 0.4C
GCTC341
303 IFtRESL-l.20)304,304,305
304 F R ES = 2 .20+0.75*{1.0-EXP(-5.34*1RESL-O.65)))
GCT034 I
305 IFIRESL-2.10)306,307,307
306 X X = R E S L - 120
IF(XX-O.47)308,308^309
308 S=(XX-0.20)/<0.14+0.49*XX)
Y Y { 1) = 0 . 2 5
G0TG34 2
309 S=(XX-0.2)/(-0.06*0.93*XX)
YY{ 1)=0.68
342 D0350L=1,1CO
LL=L+ 1
Y Y I L L )={(1C.0**S)/20.0)*(ALOG10tlOO.O-YY(L)))
IFtABS(YY(LL)-YY(L))-0.0001)310,310,350
310 FRES=2.95-YY(LL)
G0TC341
350 CCNTINUE
307 FRES=2.15
156
341 FI JJ)?1.0/( (FRES-i4.0*AL0G10(1.0/RRR) )**2)
IFtABS(FIJJI-FIJ))-0.CGO 1)361361*301
301 CCNTINUE
361 FNK1=F(JJ)
RETURN
ENC
C T H I S S U B P R O G R A M IS W R I T T E N IN F O R T R A N IV FO R AN IBM 7 0 A C C O M P U T E R .
C T H I S S U B P R O G R A M IS U S E D BY THE M A I N P R O G R A M TO C A L C U L A T E P R E D I C T E D
C F R I C T I O N F A C T O R S F R O M M O D E L FHVH.
C
FUNCTION FHVH(DIfPH)
RRS = C I / P H
F H V H = C . 2 5 * R R S * I I 10 C1 . 0 / 1 1 . 0 + 0 . A 3 8 / R R S ) ) * * 2 ) * * 2 >
RET U R N
ENC
C THIS SUBPROGRAM IS WRITTEN IN FORTRAN IV FOR AN IBM 7040 COMPUTER.
C THIS SUBPROGRAM IS USED BY THE MAIN PROGRAM TO CALCULATE PREDICTED
C FRICTION FACTORS FROM MODEL FNK2.
C
FUNCTION FNK2(REN,DI,PH) *
DIMENSION F<100),Y Y { 100)
RRS=CI/PH
F 11 )=C.020
DC401J=I1 CO
JJ=J+1
R ES=REN*SCRT{F(J))*(1.0/RRS)
R E SL =*LOG10(RES)
IFCRESL-O.65>402,402,403
402 FRES=4.0*RESL-0.40
GGTC441
403 IFIRESL 1.20)404*404,405
404 FRES = 2.20 + 0 . 7 5 * I 1.O - E X P (-5.34*{R ESL-0.65))>
G0T044 I
405 IFIRESL-2.10)406,407,4G7
406 XX*RESL 1.20
IF(XX-O.47>408,408,409
408 S=(XX-0.20)/(0.14+0.49*XX)
Y Y ( 1)=0.25
GCTG442
409 S=IXX-0.2)/( 0.06+0.93*XX)
Y Y (1)=0.68
*42 D045CL = 1*100
LL=L+1
Y Y (L L )- IClC.0**S)/20.0)*(AL0G10(100.0-YYIL)>)
IF IABS(YY(LL)-YY(L))-C.0001)410,410,450
410 FRES=2.95-YYILL>
GCT0441
450 CONTINUE
407 FRES = 2 15
441 FtJJl=1.0/t(FRES+4.0*ALQG10tRRS))**2)
IF(ABStFtJJ)-FtJ))-0.COO 1)461,461,401
401 CONTINUE
461 FNK2=F(JJ)
RETURN
ENO
162
IF(LANG)1901.19Q1*1902
1901 SSDEV=SUMSF/(AN-5.0)
GCTC1S03
1902 SSCEV t SUMSF/1AN 6.0)
1903 DEVR=(SQRT(SSDEV)/AVEFF) *1*00.0
RETURN
ENO
164
SSDE<V*SUMSF/ (AN-4.0)
D E4=<SCRT(SSDEV)/AVEFF)*100.0
RETURN
END
he was a member of Phi Eta Sigma, Tau Beta Pi, Phi Kappa Phi, and
Engineering.
166
EXAMINATION AND THESIS REPORT
Approved:
EXAMI N I N G COMMITTEE:
^ fi.,
jt, gTt/r-r
Date of Examination:
May 8 , 1967