Nuclear Science and Eng Lab Report
ProfEng Emanwanj-MatPhy
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Department of Nuclear Science and Engineering
Determination of the Drag Coefficient of a Sphere
Report from a laboratory experiment conducted on January 13, 2017
as part of ENGR 351 - Experimental Methods
Joe College Student
January 21, 2017
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
Abstract:
The drag force on a sphere in an air stream was measured at various free stream velocities below 100 ft/sec. This was done in a low speed wind tunnel using an integral balance system to measure the drag force and a pitot tube and venturi meter to measure the velocity.
The raw data were processed according to classical equations of fluid mechanics which define the Reynolds number and drag coefficient. An expression for the drag coefficient in terms of the Reynolds number was developed using a least squares curve fit to the experimental data.
The experimental results are compared to published results over the range tested.
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
Table of Contents
Abstract: 1 ........................................................................................................................................... Table of Contents 2 ............................................................................................................................. 1. Introduction and Background 1 ................................................................................................... 2. Theory 1 ....................................................................................................................................... 3. Description of Experimental Setup 3 .......................................................................................... 4. List of Equipment Used 3 ............................................................................................................ 5. Procedure 3 .................................................................................................................................. 6. Data 4 7. Analysis of Data 6 .......................................................................................................................
7.1. List of Variables: 6 ................................................................................................................. 7.2. Data Reduction Equations 6 ...................................................................................................
7.2.1.Calculation of air density 6 ................................................................................................ 7.2.2.Calculation of velocity pressure from manometer reading 6 ............................................. 7.2.3.Calculation of free stream air velocity 6 ............................................................................ 7.2.4.Calculation of Reynolds number 6 .................................................................................... 7.2.5.Calculation of drag coefficient, CD 7 ................................................................................
7.3. Calculated Results 7 ............................................................................................................... 7.4. Uncertainty Analysis 8 ...........................................................................................................
8. Discussion of Results 11 .............................................................................................................. 9. Conclusions 13 ............................................................................................................................ 10. References 13..............................................................................................................................
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
1. Introduction and Background
The drag coefficient is a well known parameter used to characterize the drag force a body immersed in a fluid experiences due to relative motion between the body and the fluid. Before using data collected from a wind tunnel experiment for complex shapes, the data for the drag force on a sphere should be analyzed and compared to results published in authoritative references. Published results are most often expressed in terms of a plot or mathematical correlation between the drag coefficient and the Reynolds number. If agreement between the experimental results for this simple shape and published data can be confirmed, then the apparatus can be used to determine the drag coefficients of other more complicated shapes with a greater degree of confidence.
2. Theory
It is known that any surface in contact with a flowing fluid is subject to a force exerted by the fluid. This force is commonly called a drag force. An expression relating to the drag force on a sphere immersed in a flowing fluid is easily derived by using dimensional analysis. (See Reference ‑ for more details concerning dimensional analysis.) 1
Consideration of the physical factors which influence the drag force leads to the listing of the following as principal variables:
FD the drag force on the sphere D the diameter of the sphere u∞ the free stream velocity of the fluid ρ the density of the fluid µ the viscosity of the fluid
Therefore, the following may be written:
! (1)
or, supplying some constants,
! (2)
Using the mass-length-time systems of units and substituting the proper dimensions,
! (3)
Since the dimensions must be the same on both sides of the equation, the exponents must be the same for each unit. Thus,
For M: !
),,,,( µρ∞= uDfFD
.dcbaD uCDF µρ∞=
.32 dcb
a
LT M
L M
T L
L T ML
⎟ ⎠
⎞ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛ ⎟ ⎠
⎞ ⎜ ⎝
⎛ =
dc+=1
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
For L: ! For T: !
Solving these equations in terms of d,
!
Thus,
! (4)
Now, grouping variables according to exponents,
! (5)
where ! is a dimensionless group called the Reynolds number. Regrouping, this equation can be rewritten in the general form
! (6)
that effectively reduces the number of variables to two dimensionless groups, which are, in turn, functions of density, viscosity, diameter, and velocity. By varying any one or more of these parameters, a correlation between the two groups can be formed.
An expression for the drag force on a body is usually given in the form
! (7)
where,
CD is a dimensionless drag coefficient, A is the frontal area of the body exposed to the flow (πD2/4 for a sphere),
gc is the gravitational constant which allows the left hand side to be expressed in units of force.
This expression can be related to equation 6 by solving for the drag coefficient:
! (8)
Therefore, the drag coefficient itself is a function of the Reynolds number.
dcba −−+= 31 .2 db−−=−
.1;2;2 dcdbda −=−=−=
.122 ddddD uCDF µρ −−
∞ −=
,22 d
D Du
uCDF −
∞ ∞ ⎟⎟
⎠
⎞ ⎜⎜ ⎝
⎛ =
µ
ρ ρ
µ
ρDu∞
( ),Re22 fuD FD ʹ=
∞ρ
c DD g
u ACF 2
2 ∞=
ρ
( ).Re82 222 fuD gF
uA gF
C cDcDD ʹ́=⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ==
∞∞ ρπρ
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
3. Description of Experimental Setup
! Figure 1 Low speed wind tunnel
A manually controlled variable speed wind tunnel similar to that shown in Figure 1 was used in this experiment. The wind tunnel was equipped with an integral force balance which measured both drag and lift forces and a multistation manometer tube bank to measure the velocity of the air stream. A separate pitot tube was used to verify the calibration of the built-in manometer. A mercury barometer was used to measure the atmospheric pressure and a thermometer was used to measure the air temperature.
4. List of Equipment Used
1. Flotek 250 wind tunnel located in the Mechanical Engineering Laboratory (S/N FT250-2784) 2. 2.5-inch diameter smooth calibration sphere wind tunnel accessory 3. Pitot tube and differential manometer (Property tag BSW365-22984) 4. Mercury barometer fixed to the wall near the wind tunnel. 5. Mercury thermometer (Sargent brand, no tag or serial number)
5. Procedure
Step 1: The atmospheric pressure and temperature were recorded at the beginning of data collection.
Step 2: The venturi meter velocity gage built in to the wind tunnel was calibrated by inserting the pitot tube at the center of the wind tunnel test section and varying the fan speed so as to produce 0.05-inch changes in the differential manometer attached to the pitot tube. At low speeds (less than 25 miles per hour) the built-in venturi meter was below the first scale reading, thus the
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
pitot tube was used to adjust the fan speed while taking drag measurements. At air speeds greater than 25 miles per hour, the venturi meter was very reliable, so it was used to adjust the fan speed setting during drag force measurements. A calibration table was made that correlated venturi air speed readings with desired pitot tube differential pressure values.
Step 3: The fan speed was set to produce a pitot differential pressure of 0.025 inches.
Step 4: The drag force of the specimen mounting stand was measured using the force balance and recorded on the data sheet. The 2.5-inch spherical test specimen was then mounted and the drag force was measured and recorded on the data sheet.
Step 5: The test specimen was removed from the test stand and the fan was shut off and the drag force indicator was checked to make sure it read zero.
Step 6: The fan was restarted and its speed was adjusted so as to produce a pitot differential pressure of 0.05 inches, then steps 4 and 5 were repeated. This process was continued, increasing the pitot differential pressure by 0.05 inches each run until the differential pressure reached 0.35 inches. At this point, the fan speed was adjusted by referring to the air speed calibration table that was made earlier. Measurements were made up to the maximum free stream air speed capability of the wind tunnel, which was 52 miles per hour (1.50 inches of pitot tube differential pressure).
Step 7: The entire data collection process was repeated in reverse, i.e., starting with the fan running at maximum speed, and lowering the speed by to match those used previously for each drag force reading.
Step 8: The atmospheric pressure and temperature were recorded at the conclusion of the last measurements.
6. Data
Temperature at start of experiment: 77° F
Barometric pressure at start of experiment: 29.80 inches of mercury
Pitot Tube Differential Pressure-Δh
(inches)
Venturi Meter
Reading
(miles/hour)
Going Up Coming Down
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
0.025 N/A 0.00 0.010 0.00 0.007
0.05 N/A 0.002 0.010 0.002 0.010
0.10 N/A 0.003 0.015 0.003 0.017
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
Temperature at end of experiment: 77° F
0.15 N/A 0.004 0.020 0.004 0.020
0.20 N/A 0.005 0.020 0.005 0.030
0.25 N/A 0.007 0.030 0.007 0.037
0.30 N/A 0.009 0.035 0.009 0.040
0.35 25.0 0.011 0.045 0.011 0.045
0.40 26.8 0.013 0.050 0.013 0.050
0.45 29.0 0.015 0.060 0.015 0.060
0.50 30.6 0.018 0.070 0.018 0.070
0.55 31.2 0.021 0.075 0.021 0.075
0.60 33.6 0.024 0.080 0.024 0.080
0.65 34.8 0.027 0.080 0.027 0.085
0.70 35.2 0.030 0.085 0.030 0.085
0.75 36.5 0.033 0.090 0.033 0.090
0.80 37.5 0.036 0.095 0.036 0.100
0.90 40.0 0.039 0.105 0.039 0.105
1.00 43.0 0.042 0.120 0.042 0.120
1.10 45.2 0.045 0.130 0.045 0.135
1.20 47.8 0.050 0.145 0.050 0.145
1.30 49.0 0.055 0.150 0.055 0.150
1.40 50.6 0.060 0.150 0.060 0.160
1.50 52.5 0.065 0.165 0.065 0.160
Pitot Tube Differential Pressure-Δh
(inches)
Venturi Meter
Reading
(miles/hour)
Going Up Coming Down
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
Mounting Stand Drag
(lbf)
Total Drag
(lbf)
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
Barometric pressure at end of experiment: 29.80 inches of mercury
7. Analysis of Data
7.1. List of Variables: FD - drag force in lbf CD - drag coefficient Re - Reynolds number D - diameter of sphere in inches ρ - density of air in lbm/ft3 u∞ - velocity of air stream in ft/sec2 P - atmospheric pressure in lbf/ft2 Δp - pressure difference in manometer in lbf/ft2 Δh - difference in heights of liquid in manometer in inches T - atmospheric temperature in °R µ - viscosity of air in lbm/ft-hr ρo - density of oil in manometer in lbm/ft3
7.2. Data Reduction Equations The following equations were used to reduce the raw data. Sample calculations for each equation are given in the Appendix.
7.2.1.Calculation of air density Assuming ideal gas conditions, the density of air can be calculated using
! (9)
where R is the gas constant for air, ! .
7.2.2.Calculation of velocity pressure from manometer reading
! (10)
7.2.3.Calculation of free stream air velocity By neglecting compressibility effects, the free stream air velocity can be derived from the Bernoulli equation as
! (11)
7.2.4.Calculation of Reynolds number The Reynolds number based on the sphere diameter is defined by the equation
RT P
=ρ
Rlb lbft
34.53 m
f
° =R
. c
o g g
hP Δ=Δ ρ
. 2 ρ
Pg u c
Δ =∞
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
! (12)
7.2.5.Calculation of drag coefficient, CD The drag coefficient can be calculated using equation 8,
! (8)
where the average net drag force (total drag force – mounting stand drag force) is used for each data point.
7.3. Calculated Results The following values were used to compute the values in the data reduction equations:
! ! , ! Applying equation 9, the
density is calculated as The calculated values of free stream air velocity (u∞), Reynolds number (Re), and drag coefficient (CD) are given in the table below.
.Re µ
ρDu∞=
, 8
22 ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ =
∞uD gF
C cDD ρπ
R,537F77 °=°=T fthr lb
0444.0 m ⋅
=µ . ft lb
2105Hgin802.29 2 f=⋅=P
. ft lb
07349.0 3 m
Pitot Tube Differential Pressure-Δh
(inches)
Free Stream Air Velocity-
u∞ (ft/sec)
Reynolds Number-Re
Drag Coefficient-CD
0.025 9.70 12,302 2.223
0.050 13.72 17,398 1.046
0.100 19.40 24,604 0.8501
0.150 23.76 30,134 0.6975
0.200 27.43 34,795 0.6539
0.250 30.67 38,902 0.6932
0.300 33.60 42,616 0.6212
0.350 36.29 46,030 0.6352
0.400 38.80 49,209 0.6049
0.450 41.15 52,193 0.6539
0.500 43.38 55,017 0.6801
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
The calculated values for drag coefficient versus Reynolds number are plotted on the next page. A least-squares best fit logarithmic equation for the experimental data was found to be
! (13)
7.4. Uncertainty Analysis The uncertainty associated with each of the measured variables is given in the table below. These values were chosen based on the stated accuracy of the instrument, if available. Otherwise, they are reasonable estimates based on values typically reported.
0.550 45.49 57,702 0.6421
0.600 47.52 60,268 0.6103
0.650 49.46 62,729 0.5584
0.700 51.32 65,097 0.5138
0.750 53.13 67,382 0.4970
0.800 54.87 69,592 0.5027
0.900 58.20 73,813 0.4796
1.000 61.34 77,806 0.5101
1.100 64.34 81,603 0.5202
1.200 67.20 85,232 0.5177
1.300 69.94 88,712 0.4779
1.400 72.58 92,061 0.4437
1.500 75.13 95,292 0.4251
.Re87.407 5978.0−=DC
Uncertainty Description Symbol Numerical Value
Pitot tube differential pressure 0.05 inch
Drag force 0.005 lbf
Sphere diameter 0.01 inch
Atmospheric pressure 0.005 inch Hg = 0.353 lbf/in2 ! DU
! hUΔ
! PU
! DFU
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student January 21, 2017 ENGR 351 – Experimental Methods
The detailed calculations of the uncertainty associated with the calculated variables (in accordance with Reference ‑ ) are given in the Appendix. The table below summarizes the results 2 of these calculations.
Temperature 1°F
Manometer oil density 3.12 lbm/ft3
Viscosity of air
! oUρ
lbm/hr-ft 5104.6 −×! µU
! TU
Uncertainty Description Symbol Numerical Value
Air density
Manometer pressure difference
Free stream air velocity 2.52 ft/sec
Reynolds number 3181
Drag coefficient 0.131
! ReU
lbm/ft3 410883.1 −×
! ∞uU
! PUΔ
! DCU
! psi 310774.1 −×
! ρU
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College of Engineering !
Determination of the Drag Coefficient of a Sphere
Joe College Student March 21, 2002 ENGR 351 – Experimental Methods
! Figure 2 Plot of Drag Coefficient versus Reynolds Number
Drag Coefficient vs. Reynolds Number for a Sphere
D ra
g C
oe ff
ic ie
nt (C
D )
0.1
1
10
Reynolds Number (Re) 10000
Experimental Data CD=exp(-.597*ln(Re)+6.01)
! 10
!
Determination of the Drag Coefficient of a Sphere
Joe College Student March 21, 2002 ENGR 351 – Experimental Methods
8. Discussion of Results
The results of this experiment are best depicted in Figure 2. Over the air speed range tested, the drag coefficient generally decreases as the Reynolds number increases. Similar results are reported in Reference ‑ (shown in Figure 3). Owing to the limitations of the low speed wind 3 tunnel used in this experiment, comparison of results is possible over only a single decade
! The results calculated using the present experimental data compare very favorably at the upper end of this region, where both curves show ! At the lower end of the region, the present experimental data yielded values of CD near unity, while those reported in Reference 3 remained considerably less than unity.
An examination of the uncertainty in the values of the drag coefficient provides some interesting information. At the lowest velocity the uncertainty calculation yields a probable error of 2.625, which exceeds the calculated value. At the highest velocity the probable error is 0.044, or only approximately 10%. This is so because the uncertainty in the velocity changes relatively little even though the velocity itself increases greatly. The principal factor contributing to this, and therefore to the greater reliability of the values of drag coefficient at higher velocities was the uncertainty in the reading of the manometer tube, which did not vary with velocity. Therefore, the uncertainty due to this factor represented a greater percentage of the velocity and thus the drag coefficient at low velocities that at the higher velocities. For the same reasons the uncertainties in the higher values of the Reynolds number would be less on a percentage basis that those at lower values.
).10Re10( 65 ≤≤ .4.0Re ≈
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!
Determination of the Drag Coefficient of a Sphere
Joe College Student March 21, 2002 ENGR 351 – Experimental Methods
! Figure 3 Experimental Values of Drag Coefficient vs. Reynolds Number for a Sphere (from Reference 3)
Drag Coefficient vs. Reynolds Number for a Sphere
D ra
g C
oe ff
ic ie
nt (C
D )
0.01
0.1
1
10
100
1000
Reynolds Number (Re)
0 0.1 1 10 100 1000 10000 100000
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!
Determination of the Drag Coefficient of a Sphere
Joe College Student March 21, 2002 ENGR 351 – Experimental Methods
9. Conclusions
The results of this experiment show that the drag coefficient for a sphere can be calculated reasonably accurately using a low speed wind tunnel. The results obtained agree with other published results at the higher range of velocities used. More accurate results at lower velocities would probably require a more sensitive force balance and air speed indicator.
10. References ! R. L. Daugherty and J. B. Franzini, Fluid Mechanics with Engineering Applications (McGraw-Hill, New York, 1965).1
! H.W. Coleman and W.G. Steele, Experimentation and Uncertainty Analysis for Engineers (John Wiley & Sons, Inc., New 2 York, 1989).
! H. E. Donley, The Drag Force on a Sphere (1991). Available at http://www.ma.iup.edu/projects/CalcDEMma/drag/3 drag.html (21 March 2002).
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