PRESSURE FLOW
through a
COILED TUBE
UTKU ALHUN
SALIM E. DARWICHE
ANDREA S. LEVINE
SURBHI PURI
BE – 309
BIOENGINEERING LAB III
FALL 2004
December 16, 2004
Department of Bioengineering
School of Engineering & Applied Science
UN IVERSITY of P ENNSY LVAN IA
ABSTRACT
The purpose of this investigation is to delve further into the pressure-flow experiments previously
conducted. This experiment was geared towards investigating different factors on fluid flow and
pressure. The variable factors considered were orientation of the coiled tubing, the radius of
curvature of the coils, and change in viscosity of the fluid used (comparing water and 10% sucrose
solution). Three hypotheses were tested. It was predicted that using the horizontal orientation of the
coils would give experimental results relatively similar to the theoretical values. It was also
hypothesized that the relationship between the experimental Dean’s number and the friction factor
obtained would be dependent of any variation in the radius of coil curvature. The final hypothesis to
be tested stated that the mathematically derived relationship of the Dean’s number and the friction
factor would be maintained by fluids of varying viscosities. It was found that the horizontal
orientation was, as per the hypothesis, is the preferred orientation in this experimental setup and this
was mainly attributed to the effect of gravity on the fluid flow varying with different orientations
(18.1% error). The second hypothesis was found to be true by the investigation. The results of the
experiment showed that the relationship between the Dean’s Number and the Friction Factor is
affected by the radius of coil curvature used. Finally the third hypothesis was supported by the data;
the mathematical Dean’s number proof can be implemented to both, the water and the 10% sucrose
solution. The sources of error (systemic and experimental) in this investigation give rise to the need
for further research to be done on pressure flow in coiled tubing. Suggestions include isolating
whether the coil number or the radius of coil curvature effect the relationship more and investigating
the effect of viscosity on the theoretical relationship trends and more experimental trends between
the Dean’s number and the friction factor, separately.
HYPOTHESIS
Due to the effect of gravity, both of the horizontal orientations (ascending and descending) would be
more similar to the theoretical curve than the vertical orientations. In addition, the radius of
curvature should affect the relationship between the Dean’s Number and the friction factor as it
appears in the coefficient of the power function relating De and f. Finally, the relationship which
Dean described mathematically should hold for fluids with different viscosities. Therefore, data
from both the 10% sucrose solution and water can be fit on the same curve and the relationship
between Dean’s Number and the friction factor will not be affected by the viscosity of the liquid.
BACKGROUND
Poiseuille’s law accurately describes the relationship between pressure-flow relationships in terms of
radius of the tube and viscosity of the Newtonian fluids for laminar flow. However, Poiseuille’s law
is not applicable to coiled tubes. This is due to the fact that the resistance to flow is greater in coiled
tubes than that in straight tubes. The more rapidly flowing central parts of the flow are forced
outwards by centrifugal action, while the slower parts along the walls are forced inwards, i.e.
towards the center of curvature. With laminar flow, the effects even of a very slight curvature are
dramatic when Reynolds’ number is not very small1. Since flow in curved tubes differs greatly from
Poiseuille’s law, W. R. Dean developed a theory to describe it.
The Reynolds’ number (Re), which is used to describe flow, is defined as:
U d
Re 
[1]

where, U is the mean fluid velocity, d is inner diameter of the tube, and ν is the kinematic viscosity.
The Dean’s number (De), a slight modification of the Reynolds’ number, is defined below for curved
tubes:
d U d
d
[2]
De  Re 


D

D
where, D is the diameter of curvature of the coil.
This experiment investigates pressure-flow relationship in a coiled tube wrapped around
cylinders of different radii. Head Loss is defined as:
2
P
L U
HL 
 12    
f
[3]
g
 D g
where, f is the friction factor and is a function of the Dean’s number.
In order to determine the experimental relationship between the friction factor and the Dean’s
number, the equation is solved for f:
P  2  D
f 
[4]
 U 2  L
Using Dean’s work, the theoretical relationship between the Dean’s Number and the friction
factor is found to be:
64
d
0.36
0.64
f 
 0.37  De
or using [2] f  23.68 
De
[5]
Re
D
MATERIALS









Stand
Graduated Cylinder
4 Cylinders with different radii
Tank with flow cutoff valve
1/8” ID tubing 46’ in length
10% Sucrose Solution
Pressure Manometer
Stopwatch
Water
METHODS
The experimental set-up is illustrated in FIG 1, below. The water tank, supported on a stand, has a
needle valve which enables control over the flow rate. The 1/8” ID tubing, connected to a
manometer on the upstream, is coiled around a cylinder of known outer radius. The water is
collected in a graduated cylinder and the flow rate is measured using a stopwatch to compute the
volume of water flowing per unit time.
Legend
… Ma n o me te r
… Wa te r Ta n k
… Need le Va lv e
Descending horizontal
orientation of coiled tube
… Ru l e r
… S ta n d
S to p wa tch
Co i led Tu b e
Di rec tio n o f flo w
Mea su rin g
cyl in d e r
F IG
1. T HE
E X P ER IM E N TA L S E T - U P .
The first part of the experiment aimed to investigate the effects of different orientations of the coiled
tubing. The four different types of orientations were horizontal (descending/ascending coils) and
vertical (descending/ascending coils). The cylinder with an outer radius of 7.72 cm is used for all
the different orientations; the number of coils was 26, and kept constant.
The nomenclature used to define the orientation of coiling can be misleading; therefore, it is
worthwhile to explain the rationale behind it. For the horizontal orientation, the tubes are actually
coiled in a way that each individual coil is perpendicular to the plane of the table. On the other hand,
for the vertical orientation, each individual coil is parallel to the plane of the table. Basically,
horizontal and vertical are used to refer to the net direction of flow.
A
B
C
D
2. C O I L O R IE N TA TIO N S . A: Horizontal descending, B: Horizontal ascending, C: Vertical
descending, D: Vertical ascending.
F IG
For each orientation, four different flows of water were measured, approximately at equal intervals
of the height of water in the manometer. A maximum (approximately at 90 cm of water in the
manometer), a minimum (~ 20 cm of H20) and two intermediate flows (~70 and 40 cm H20) were
measured, three trials for each flow, for a total of 12 measurements.
After determining which orientation to use throughout the second part of the experiment, the
procedure was repeated using cylinders of different outer radii.
Radius
(cm)
Number
of coils
Thin white tube
1.37
121
Graduated cylinder
3.36
65
Bottle
7.72
26
Bucket
12.50
17
Cylinder
TAB LE 1.
R AD I I
NU MB ER O F C O ILS .
OF
C Y LIN D ER S
A ND
So as to investigate the effects of the radius of
curvature on pressure-flow measurements, four
different cylinders were used. The outer radii of
the cylinders and the number of coils around each
one are tabulated, on the right. The radius of
curvature is calculated by adding the radii of the
cylinder, and the tube wrapped around it.
Once again, the pressure readings were taken from the manometer, and the volumetric flow rate was
measured using a graduated cylinder and a stopwatch.
The third and the final part of the project consisted of using a 10% sucrose solution, instead
of water, to determine the possible effects of viscosity on fluid flow in a coiled tube. The procedure
outlined for the first part of the experiment was repeated.
R ES U LTS
The first part of the experiment tested the first hypothesis; namely, whether orientation of coiling
affects the relationship between Dean’s Number and the friction factor. Graph 1, below, shows the
Different directions on same coil
relationships for each orientation.
0.9
GR A P H
Experimental and Theoretical
f vs. De
for
Different Coil Orientations
0.8
0.7
Vertical orientation w/ coils t/b
Color coding is given as follows
Orientation w/ coils b/t
andVertical
is based
on the nomenclature
scheme
introduced
in the
Horizontal
Orientation
with Methods
coils
b/t
section.
0.6
Horizontal Orientation with coils
t/b
theoretical all
Yellow: [C]
Gold:
[D]
Power (Vertical orientation w/
coils t/b)
Orange:
[B]
Power
Red:
[A] (theoretical all)
f
0.5
f
1. f vs. De.
0.4
0.3
Power (Horizontal Orientation
coilsblue
b/t) line represents the
Thewith
navy
Power (Horizontal Orientation
expected
theoretical
curve.
with coils
t/b)
0.2
Power (Vertical Orientation w/
coils b/t)
0.1
0
0
20
40
60
80
100
The red [A] and orange [B] trends
are the closest to the theoretical
trend.
De
Both the experimental and the theoretical data were fit to power curves of the following form
y=kx^n (Graph 1 in the Appendix). The Coefficients and R2 values are listed in the table below:
Orientation
K
n
R2
Vertical Descending
30.198
-1.1929
.987
Vertical Ascending
1.108
-.482
.864
Horizontal Descending 4.534
-.7881
.962
Horizontal Ascending
3.740
-.7491
.991
Theoretical
3.166
-.6251
.997
2
Table 2: Coefficients and R values of power curves fit to the orientation data.
These coefficients fit the analysis above. Both A and B for horizontal orientations both are the
closest numerically to their respective theoretical coefficients. The percent deviation from the
theoretical for Horizontal Ascending is (k: 18.1% and n:19.8% ) and for Horizontal Descending is
(k: 43.2% and n: 26.1% ) For Vertical Descending (VD) and Ascending (VA), these deviations are
much higher (VD: k: 853% n: 90.8%, VA: k: 65.0%, n: 22.8% )
For the next part of the experiment, the relationship between the Dean’s Number and friction factor
was plotted for different radii of curvature; only horizontal ascending loops were used. Both the
experimental data and the theoretical data (friction factor calculated from Dean’s Numbers using the
mathematical relationship) were plotted.
Graph 2 combines the theoretical and experimental friction factor (f) data as it relates to De for all
coil sizes and both 10% sucrose and water solutions.
The theoretical data points were fit to a power function which differs with coil size; the lighter the
blue color is, the larger the coil size. The power function curve flattens and shifts to the right as the
coil decreases. The experimental data follows the same trend. Dashes mark theoretical data points
while triangles mark experimental data points. No graphical distinction is made between water and
sucrose data since the data distributions seem to overlap and follow the same trend (see appendix on
data distribution).
Experimental and Theoretical f vs. De f or dif f erent coil sizes
1.60E+00
f
Experimental and Theoretical
f vs. De
for
Different Radii of Curvature
1.40E+00
\\
1.20E+00
f
1.00E+00
8.00E-01
6.00E-01
4.00E-01
2.00E-01
0.00E+00
0
20
40
60
80
100
120
140
D
e
De
Small
Medium
Large
X-Large
Small exp
Medium exp
Large exp
X-Large exp
Pow er (Small)
Pow er (Medium)
Pow er (Large)
Pow er (X-Large)
GR A P H
2. f vs. De for different radii of curvature.
The next part of the experiment of the experiment further explored the how well the experimental
data matched the theoretical data for the relationship between Dean’s Number and friction factor for
the different radii of curvature. It incorporates both sucrose and water data.
3b - Medium Theoretical & Exp f vs. De
3a - Small Theoretical & Exp f vs. De
1.4
1.2
1.2
1
1
0.8
f
f
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
50
De
100
0
150
De
100
150
3d - X- Large Theoretical & Exp f vs. De
3c - Large Theoretical & Exp f vs De
f
50
1.6
1.6
1.4
1.4
1.2
1.2
1
0.8
0.8
f
1
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0
20
40
De
60
80
20
40
60
80
De
GRAPHS 3a, 3b, 3c and 3d show the theoretical and experimental f vs. De for small (a), medium (b),
large (c) and x-large (d) coil sizes. The plots combine both sucrose and water data. Red points mark
the experimental friction factor values (f) obtained for every De while navy points mark the
theoretical values of f for the same De values.
Table 3, below, shows the curve fits and exponential for both the theoretical and experimental
y=kx^n curves for different radii:
K
n
R2
Theoretical Experimental Theoretical Experimental Theoretical Experimental
Small
6.79
1.87
-.612
-.373
.998
.601
Medium
5.70
4.59
-.670
-.721
.998
.974
Large
4.98
4.37
-.745
-.789
.981
.982
X-Large
4.33
3.42
-.774
-.777
.976
.985
TABLE 3: Coefficients of power fits of both the experimental and theoretical data for each different
radius of curvature.
Tube
Note that if the data points encircled above in the small coil graph are taken out (they all refer to the
measurement of the lowest flow in the small coils for sucrose which yielded a very unsteady
pressure reading in the monometer) the correlation factor (R2) becomes 0.987.
D IS C U S S IO N
1 summarizes the first part of the experiment, where the orientation of the coiled
tubing was varied to determine the optimum direction of flow. The orientation was shown to affect
the relationship between the friction factor and Dean’s Number as predicted by our hypothesis. The
main difference between the orientations of the coiled tubing is the effect of gravity on the direction
of fluid flow. For the horizontal ascending and descending orientations, the fluid flows in a
perpendicular fashion through each individual coil. Therefore, to a good approximation, the number
of ascending coils and the number of descending coils are equal to cancel out the effect of gravity.
However, for the vertical ascending and descending orientations, this is not the case. For the
ascending one, the fluid moves against the force of gravity; this greatly hinders the flow. For the
descending one, the fluid moves along the force of gravity; this enhances the flow. Therefore we
chose to use the horizontal ascending position as it yielded the least % error from the theoretical
curve (k: 18.1% and n: 19.8%).
G R A P H S 3 a, b, c, d show the distribution of water and glucose data superimposed on one
graph for each radius of curvature, separately. As shown by the graphs, the relationship between f
and the Dean’s Number was not affected by the viscosity. Referring to Equations [1] and [2], it can
be seen that Reynolds number is a function of viscosity, and Dean’s number is a function of the
Reynolds’ number, therefore Dean’s number is a function of the viscosity of the fluid. However, the
number of fluids with different viscosities tested should be increased to determine whether this
observation will hold true. The values for the Dean’s number are also found to be lower for the
more viscous 10% sucrose solution, occupying a broader range.
According to the theoretical and experimental data, there is a distinct relationship between
the Dean’s Number and the friction factor for each cylinder. Theoretically, this is expected as from
equation [5] the coefficient multiplying De contains the coil diameter. Indeed, it was found, by
plotting both experimental and theoretical curves, that as the radius of curvature increased, the data
was shifted down to the left and the largest cylinders yielded more precise data points. However,
note that as coil size increased, coil number increased as well. While coil number is not accounted
for in the theoretical equation and therefore should not matter, the fact that it was not maintained
constant might suggest that increased coil number also has an effect on the friction factor values.
Unfortunately, since both parameters varied together we cannot conclusively say that coil number
does not affect the friction factor. In addition, since f proportional to De0.36 (according to Equation
[5]), then an increase in the radius of curvature will increase the diameter of curvature and will
therefore decrease the Dean’s Number. The decrease in Dean’s Number will lead to a lower
theoretical f, causing a lower friction factor which explains the curves’ shift downwards and to the
left for increasing radii. While it would make sense that radius of curvature would influence the
Dean’s Number, as it is accounted for in the equation, further work should be done to explore this
relationship and the relationship between coil number, Dean’s Number, and friction factor.
Sucrose and water data are shown to lie on the same curve (for each cylinder separately).
This demonstrates our hypothesis that the relationship between Dean’s Number and friction factor
do not vary based on viscosity. As can been seen in Graph 4, more sucrose values could be obtained
for a lower Dean’s number for both theoretical and experimental cases. Indeed, the higher viscosity
for approximately the same flow rates would decrease the Dean’s Number. Graph 3 suggests that at
lower f values (as well as higher Dean’s values), the theoretical and experimental Dean values do not
differentiate drastically. Indeed, flow being higher in these cases, generally the volume and time
measurements are more accurate. Additionally, it seems that with increasing coil size the
GRAPH
repeatability and accuracy of the readings increase. With smaller coil sizes, it is harder to simulate an
efficient flow system; hence, some aberrant data with decreasing coil size and number is shown to
exist; also, pressure seems to fluctuate more at low flows for small coil sizes. In general, all
comparisons of sucrose and water data to the calculated theoretical values, showed good correlations
between the data sets.
The results of this investigation support the hypothesis that Dean’s mathematical proof will
apply for varying viscosities. Graph 4 shows both sucrose and water data (with all other factors kept
constant), confirms that the data of the two solutions of different viscosities can be integrated on a
curve and the trends between Dean’s Number and the friction factor will remain unaffected.
Although care was taken to minimize experimental error (including having the same people
perform the same jobs in order to minimize errors due to inaccuracies in measuring equipment),
there are sources of error. Inherent in this experiment is error due to timing and measuring volumes
and pressures. In addition, a negative systematic error was found. This could be due to slowly
fluctuating pressures, which may have been read before they rose to the final constant pressure. This
could be confounded in slow flows, as the pressure rise was gradual; resulting in thinking the
pressure was constant before it actually was. Finally, the sucrose solution may not have been fully
mixed or dissolved, which would introduce error into the viscosity used.
Flow in curved pipes has not been studied extensively as implied by the lack of the Dean’s
number in scientific literature. However, “Flow in Curved Pipes” by S. A. Berger, Talbot, and L. S.
Yao [Annual Reviews of Fluid Mechanics, 1983, 15:461-512] is an extensive study in this particular
area of fluid dynamics. The study introduces the use of the toroidal coordinate system and a series
of complex differential equations in investigating about the subject. In short, the Dean’s number is
defined as a function of the inertia, and centrifugal and viscous forces. Together with another
parameter dependent upon the geometry of the tube, the Dean’s number is found to characterize the
flow in curved pipes. The review stressed on finding the velocities of the fluid at different points in
the cross section and discussed the diverse definitions of Dean’s number and their boundary limits.
However, it did not stress or elaborate on the pressure-flow relationship as it relates to Dean’s
number, rather it focused on other consequences of the secondary flow due to centrifugal forces. It
was also mentioned that critical Re numbers for curved pipes were 3 to 5 folds higher than that for
straight tubes which basically says that the onset of turbulence occurs at higher flow rates in coiled
tubes.
CONCLUSION
Future projects should focus on determining both how the number of coils and the radii of
curvature in isolation affect the Dean’s Number-friction factor relationship. This will allow us to
determine which factor contributes more to the variation in the relationship. Also, a greater number
of fluids with a broad range of viscosities should be used to further determine whether the
relationship between Dean’s Number and friction holds for fluids with different viscosities. Two
fluids are not enough to fully determine whether viscosity affects this relationship; however, it does
indicate that this relationship does not vary with viscosity. Furthermore, non-Newtonian fluids can
be used to investigate further into the subject.
APPENDIX:
Different directions on same coil
0.9
Vertical
Descending
0.8
Vertical
Ascending
-1.1929
y = 30.198x
R2 = 0.9865
0.7
0.6
y = 3.1663x-0.6251
0.5
R = 0.9971
Horizontal
Descending
2
f
Horizontal
Ascending
-0.7881
0.4y = 4.5337x
2
R = 0.9618
0.3
y = 3.7395x-0.7491
0.2
2
R = 0.9909
theo all
Power
(Vertical
Descending)
Power (theo
all)
-0.482
0.1
y = 1.1075x
R2 = 0.8643
0
Power
(Horizontal
0
20
40
60
80
100
Descending)
De
Power
The above graph shows the equations for the trend lines to which the data (Horizontal
points were fit. This
Ascending)
relates to the data taken for water at different orientation of coiling. The below
Powergraph shows the
(Vertical
experimental data distribution as it varies from water to sucrose.
Ascending)
Sucrose and Water De and experimental f distribution
1.4
1.2
1
0.8
f
Water
Sucrose
0.6
0.4
0.2
0
0
50
100
De
150
REFERENCES:
1
W. H. Dean and C. M. White
“Flow in Curved Pipes” by S. A. Berger, Talbot, and L. S. Yao [Annual Reviews of Fluid
Mechanics, 1983, 15:461-512]
3
Project report spring 2004
2