Padnos School of Engineering Grand Valley State University
EGR – 365 Fluid Mechanics Dr. Blekhman
Laboratory Report #10 Drag Coefficient of a Sphere
Matt Reimink
10/16/2013
Grading Rubric
Outline Purpose
The purpose of the laboratory exercise is to measure the drag coefficient of a sphere as a function of Reynolds number.
Experiment
Setup The setup for the wind tunnel is as shown.
Theory Definition of Drag Force from force balance. FW
−
FB
=
1 2
ρACD Vt
2
Definition of Reynolds Number R e
d
=
ρ Vd µ
Definition of error discrepancy %Error =
experimental − theoretical theoretical
×
100
Definition of propagated error % uncertainty in f ≅
Present Major Results
1 2
f
∂f 2 2 ∂f 2 2 u x1 + u x2 ∂x2 ∂x1
Mass (kg) Time 1 (s) Time 2 (s) Time 3 (s) Avg. Time (s) Avg. Velocity (m/s) Reynolds Number 0.0729 0.75 0.81 0.80 0.787 1.162 3.91E+04 0.0401 1.72 1.6 1.76 1.693 0.540 1.81E+04 0.0282
37.02
32.43
36.09
35.180
0.0260
8.73E+02
The experimental data was used to calculate the experimental drag coefficient. Mass (kg) 0.0729
CD Experimental 0.588
CD Theoretical 0.45
% Error 30.57%
0.0401 0.0282
0.738 7.839
0.44 0.5
67.75% 1467.77%
The theoretical and experimental values were then graphed against each other. 9.000 8.000 7.000 6.000 5.000 4.000 3.000 2.000 1.000 0.000 0.00E+00
1.00E+04
2.00E+04
3.00E+04
4.00E+04
5.00E+04
Re
Experimental
Theoretical
Error Discussion
The error was calculated using the same table as above. Mass (kg) 0.0729 0.0401 0.0282
CD Experimental 0.588 0.738 7.839
CD Theoretical 0.45 0.44 0.5
% Error 30.57% 67.75% 1467.77%
Conclusions
1.) The minimum experimental drag coefficient on the sphere was 0.588 and the maximum was 7.839. 2.) The minimum percent error for the experimental data was roughly 30% while the maximum was 1468%. 3.) Discrepancies were attributed to the wall effects, nonlinear descent, measurement errors, unsmooth surfaces and irregular compensation for helical effects.
Purpose
The purpose of the laboratory exercise is to measure the drag coefficient of a sphere as a function of Reynolds number.
Experimental Setup and Theory
As an object falls through a fluid, whether it be water or air, it will naturally experience a gravity force pulling it down, but less intuitive, it will experience a drag force and possibly a buoyant force opposing the downward motion. Both the gravity and buoyant force a constant, but the drag force is directly related to velocity. Figure 1 shows the force balance acting on the object as it falls.
Figure 1: Force balance of a falling object.
The experimental apparatus used to perform this laboratory exercise is shown below in Figure 2.
Figure 2: Experimental test apparatus.
Important dimensions, mainly those pertinent to calculations, can be seen in Table 1 along with other constants and variables needed. Table 1: Laboratory dimensions and constants.
Measurement Diameter of Sphere, (m) Volume of Sphere, (m3)
Value 0.03763 2.79E-05
Density of Water, (kg/m3) Viscosity (Ns/m2) Length of Fall, (m) Length before water, (m) Buoyant Force, (N)
1000 1.12x10-3 0.9144 0.4318 0.274
Each force acting on the body had to be calculated in order to determine the drag coefficient acting on the cylinder. First the buoyant force for a sphere was calculated using Equation (1). ρ =∀
FB
g
(1)
Next the force of the weight was calculated using Equation (2). FW
=
mg
(2)
Finally the drag force was evaluated. Equation (3) shows that the drag force is directly related to the velocity of the sphere. FD
=
1 2
ρACD V 2
(3)
From Figure 1 it is possible to notice that the weight force minus the buoyant force is equal to the drag force acting on the body. Therefore, Equation (3) can be rewritten to express the drag force in terms of the buoyant and weight forces, when the sphere has reached terminal velocity. FW
−
FB
1 =
2
ρACD Vt
2
(4)
The drag coefficient acting on the sphere can therefore be calculated once the terminal velocity is known. Since the drag coefficient is calculated for the sphere, it is also important to know the Reynolds number for the sphere so the data can be compared to theoretical published values. Equation (5) is the equation for calculating the Reynolds number. R e
Discussion of Results
d
=
ρ Vd µ
(5)
First the velocity was calculated as a function of time. Since the ball needs to be traveling at terminal speed in order to perform the correct force b alance, the velocity had to be determined. Equation (6) is the velocity in terms of time.
1 − e m V( t ) = ( FW − FB ) 1 2 ρACD m 1 + e
−2t 1 2 ρACD ( FW − FB )
− 2t 1 2 ρACD ( FW − FB )
(6)
The derivation of Equation (6) can be seen in Appendix A. Measurements should be taken at roughly 90% of terminal velocity; therefore Equation (6) was used to determine how far the ball had to fall in order for terminal velocity to be reached. Equation (7) was used to determine the length. l0.90Vt
=
mln(19)
(7)
ρACD
A drag coefficient of 0.45 was used for the calculation and the length was calculated for the heaviest sphere. The derivation can be seen in Appendix A. The length was calculated to be roughly 0.4289m or 16.89in. The length calculation is an overestimate of the actual path length because of the calculations. Since the distance left was 17in. it proves that the sphere was traveling at terminal velocity for the measurements. The length required could have also been determined experimentally by taking many data time and length readings to figure out where acceleration is zero.
Three measurements were taken for each of the three spheres, and the average velocity was calculated and tabulated. Table 2 is the tabulation of the experimental results. Table 2: Experimental data.
Mass (kg) Time 1 (s) Time 2 (s) Time 3 (s) Avg. Time (s) Avg. Velocity (m/s) Reynolds Number 0.0729 0.75 0.81 0.80 0.787 1.162 3.91E+04 0.0401 1.72 1.6 1.76 1.693 0.540 1.81E+04 0.0282
37.02
32.43
36.09
35.180
0.0260
8.73E+02
Using the data in Table 2 it was possible to calculate the experimental and theoretical drag coefficient. The Reynolds number was used along with Figure 9.21 from Fundamentals of Fluid Mechanics, 4th Edition. The values were calculated and tabulated and can be found in Table 3.
Table 3: Comparison of experimental and theoretical values.
Mass (kg) 0.0729 0.0401
CD Experimental 0.588 0.738
CD Theoretical 0.45 0.44
% Error 30.57% 67.75%
0.0282
7.839
0.5
1467.77%
The theoretical and experimental values were then plotted against the Reynolds number. The graphical representation of the data is shown in Figure 3. 9.000 8.000 7.000 6.000 D C
5.000 4.000 3.000 2.000 1.000 0.000 0.00E+00
1.00E+04
2.00E+04
3.00E+04
4.00E+04
5.00E+04
Re
Experimental
Theoretical
Figure 3: Drag coefficient versus Reynolds number for theoretical and experimental data.
For the most part, the theoretical values held together, while the experimental values varied by quite a bit. For our range of Reynolds numbers, the graph should have represented the linear region as shown in Figure 4.
Figure 4: CD versus Reynolds number.
Since the drag coefficient relies so heavily on the Reynolds number, it was important to try and model sphere that would yield Reynolds number in the linear region between 103 and 104.
Error Discussion
For this particular laboratory exercise the error will be a discrepancy between the theoretical values calculated and the experimentally measured data and the propagated error in calculating the velocity. This percent discrepancy can be found using the equation: %Error =
experimental − theoretical theoretical
×
100
(8)
Table 4 is a tabulation of the error data, both for individual values. Sample calculations can be seen in Appendix B.
The reason for the insanely high error for the mass of 0.0282kg is the fact that the weight force and the buoyant force almost cancel each other out. Therefore, the sphere barely sank down in the tube. Also, due to the fact that the balls had to be filled with BB’s, they were no longer smooth on the outside, but rather had irregular rough spots.
This had two main effects on the
drag coefficient calculations. First, the drag coefficient was desired for a smooth sphere. Second, the irregular rough spots caused the ball to move in a helical pattern. This increased both the time of the descent and the length travel by the ball. However, the time value was the only one that was compensated, which ended up skewing the velocity calculations.
Another smaller effect was the ball bumping into the wall on the descent. These wall effects change many of the properties of the flow, mainly the viscous effects.
The way to deal with error propagation is to determine the uncertainty in every variable used when solving for another variable. If f was a function of x1 and x2, f (x1, x2) the uncertainty of f would be % uncertainty in f ≅
1 f 2
∂f 2 2 ∂f 2 2 u x1 + u x2 ∂x2 ∂x1
(9)
Using Equation (9) it will be possible to determine the propag ated error in all our measurements.
Conclusions
1.) The minimum experimental drag coefficient on the sphere was 0.588 and the maximum was 7.839. 2.) The minimum percent error for the experimental data was roughly 30% while the maximum was 1468%. 3.) Discrepancies were attributed to the wall effects, nonlinear descent, measurement errors, unsmooth surfaces and irregular compensation for helical effects.
Appendix A Derivation of Equations
Appendix B Sample Calculations
Appendix C Experimental results.
Measurement Pressure Reading
-Cp
% of Chord Location
dc
Riemann Sum
1 2 3
-0.06 -0.115 -0.125
0.3 0.575 0.625
80 70 60
2.80 2.45 2.10
0.3506 0.3506 0.3506
0.1052 0.2016 0.2191
4 5 6 7 8 9 11 12 13 14 15 16 17 18 10
-0.145 -0.164 -0.169 -0.188 -0.172 -0.135 -0.16 -0.08 -0.015 -0.01 -0.005 0 0.003 0.006 0.188
0.725 0.82 0.845 0.94 0.86 0.675 0.8 0.4 0.075 0.05 0.025 0 -0.015 -0.03 -0.94
50 40 30 20 10 7.5 7.5 10 20 30 40 50 60 70 0
1.75 1.40 1.05 0.70 0.35 0.26 0.26 0.35 0.70 1.05 1.40 1.75 2.10 2.45 0
0.3506 0.3506 0.3506 0.3506 0.08765 0.26295 -0.08765 -0.3506 -0.3506 -0.3506 -0.3506 -0.3506 -0.3506 -0.3506 -0.26295
0.2542 0.2875 0.2963 0.3296 0.0754 0.1775 -0.0701 -0.1402 -0.0263 -0.0175 -0.0088 0.0000 0.0053 0.0105 0.2472
CL L (lbf)
0.5551 1.747
Appendix D Theoretical lift coefficient values for a Clark Y-14 airfoil1.
1
Compliments of the University of Tennessee website. http://www.engr.utk.edu/~rbond/airfoil.html
Appendix E Dimensioned test apparatus.
