Lab10 - Drag coefficient of Sphere PDF

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Drag coefficient of Sphere...

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Drag coefficient of Sphere By Quang Nguyen Lab Partners: Huynh Anh, Aldo Fenalli, Kendall Shaw

Laboratory 10 EGR 365 – Fluid Dynamics

July 17, 2017

1. Introduction

Quang Nguyen EGR 365 Lab 7 Summary Report

The purposes of this laboratory was to measure the drag coefficient of a sphere as a function of Reynolds number. At the beginning of this lab, fundamental equations to calculate drag coefficient of a sphere were derived. From the force balance on the sphere, the equation to determine velocity as a function of time was:

[

√

(

−2 t

) √0.5 ρA C D (F w− F b)

(F w −F b ) 1−e m V= −2t ) ( 0.5 ρA C D √0.5 ρAC 1+e m Fw

where

was gravity force,

Fb

D

(Fw −Fb )

]

(1)

was viscous drag force,

ρ

was the fluid

C D was the drag coefficient, A was the frontal area, V was the speed, m was

density,

mass of the sphere, and t was the time to reach V. The time to reach 0.99 V t t 0.99V = t

was determined using equation:

m ln(199) (2) ρ V t A CD

and the path length to 0.99 V t l0.99 V = t

where

is:

m ln(199) (3) ρA C D

ρ

was the fluid density,

C D was the drag coefficient, A was the frontal area,

V was the speed, m was mass of the sphere, l was path length and t was the time to reach V. After deriving equation, the masses and diameters of five balls were collected before letting these balls to shrink in water tank. The distance that each ball travel was also measured, and the time for each ball reach the bottom of the tank was recorded. All of these data was employed to calculate experimental drag coefficient of the sphere using equation: 2 F w −Fb =0.5 ρA C D V t (4)

where Vt

ρ

was the fluid density,

was the speed, F w

C D was the drag coefficient, A was the frontal area,

was gravity force, and Fb

was viscous drag force.

Quang Nguyen EGR 365 Lab 7 Summary Report

These experimental values of drag coefficient were then compare with published values, which were obtained from graph in Figure 1.

Figure 1: Published graph of sphere drag coefficient versus Reynolds number The drag coefficient obtained experimentally and the published drag coefficient did not match well. The highest percent difference was 86.2% when the weight of the ball was 32g and the lowest percent difference was 5.4% when the weight of the ball was 36g. The possible sources of error were uncertainty in laboratory equipment, propagated uncertainty in calculation, human error in measurement, and non-straight path of the balls. 2. Raw Data Table 1: Balls’ weight and time to reach bottom of the tank Mass (g) 32 ± 0.0005 36 ± 0.0005 40 ± 0.0005

Time (sec) 4.51 ± 0.005 5.15 ± 0.005 2.94 ± 0.005

Quang Nguyen EGR 365 Lab 7 Summary Report

42 ± 0.0005 60 ± 0.0005 66 ± 0.0005

3.41 ± 0.005 1.94 ± 0.005 1.71 ± 0.005

Table 2: Experiment Parameters Diameter (m) 0.0390 ± 0.00005

Distance traveled (m) 1.599 ± 0.0005

3. Results and Discussion Table 3: Experimental results Mass (g) 32 36 40 42 60 66

l0.99 V (m) 0.158 0.177 0.197 0.207 0.296 0.325

t

velocity (m/s) 0.355 0.311 0.544 0.469 0.824 0.935

Re 12338.14 10804.86 18926.87 16318.18 28682.99 32540.94

The path length to reach 0.99 V t

Fb (N) 0.305 0.305 0.305 0.305 0.305 0.305

Fw (N) 0.314 0.353 0.392 0.412 0.589 0.647

CD CD experimental published 0.062 0.45 0.422 0.4 0.248 0.49 0.409 0.48 0.350 0.495 0.328 0.496

was calculated using

C D = 0.45. According

to table 3, as the mass of the ball increase the path length to reach 0.99 V t also increased. The lowest path length to reach 0.99 V t

was 0.158m when the mass was

32g, while the highest path length to reach 0.99 V t

was 0.325m when the mass was

66g. Figure 2 showed the graph of experimental drag coefficient versus Reynolds number. Comparing to the graph of published drag coefficient values versus Reynolds number in Figure 1, the experimental results did not match well, especially at the points whose Reynolds number was 12338.14. This was also the point that associated with 32g

Quang Nguyen EGR 365 Lab 7 Summary Report

ball and had the highest percent error in compare with published value which was 86.23%. The possible reasons for this high percentage error might be human’s measurement skill when measuring the time to reach 0.99 V t

and the fact that the

travel path of the ball was not straight line. Moreover, the propagated uncertainty in calculation caused the error. The lowest propagated uncertainty was 0.0108 when the weight was 66g, while the highest propagated uncertainty was 0.08801 when the weight was 36g. On the other hand, the lowest percent difference was 5.4% when the weight of the ball was 36g. The source of error in the experiment were human error in measurement, uncertainty in equipment, propagated uncertainty in calculation, and zigzag path of the ball when it shrank.

Drag Coefficient vs Reynoids number 0.450 0.400 0.350 0.300

Cd

0.250 Cd experieemtal 0.200 0.150 0.100 0.050 0.000 5000.00

10000.00 15000.00 20000.00 25000.00 30000.00 35000.00

Re

Figure 2: Experimental drag coefficient versus Reynolds number.

4. Conclusion

Quang Nguyen EGR 365 Lab 7 Summary Report

The experimental drag coefficient and published drag coefficient did not match well with high percentage error. The highest percent difference was 86.2% when the weight of the ball was 32g, while the lowest percent difference was 5.4% when the weight of the ball was 36g. The possible sources of error in this experiment were uncertainty in laboratory equipment, propagated uncertainty in calculation, human error in measurement, and non-straight path of the balls....