Fluid Dynamics (Flow of water out of bucket)

Objective:

The objective of this lab was to document how well the measured values of time agree with uncertainty, in order to properly prove the Bernoulli equation.

Method and Results:

A bucket that contained a drain hole on the bottom vertical wall was obtained and sealed with tape.  This Bucket was then filled with tap water up to a constant height.  Next a measurement was taken from the drain hole to the surface of the water (h). 

h= 0.110 m  +/-  0.0012 m

The diameter of the drain hole was given and then measured for accuracy.  That value was then divided by two to determine the radius of the drain hole (r). 

R*=  3.175e-3 m 

r = 3.675e-3  m +/-  1.5e-3 m

* given value

This value was used to determine the cross sectional area of the drain hole (A). 

A= (pi)r^2= 4.24e-5 m  +/- 3.00e-3 m

A Graduated cylinder was placed under the drain hole to measure the volume of water that was emptied from the bucket (V). 

V= 5.0e-4 m^3 +/- 1.5e-6 m^3

When the tape was removed from the drain hole a stopwatch was used to determine the amount of time (t) it took for the water to drain down the complete height (h). 

                  Time to empty (s)

1^st Run         20.30

2^nd Run         19.74

3^rd Run         19.87

4^th Run          20.30

5^th Run          20.89

6^th Run          20.21

The acceleration due to gravity (g) was taken as a constant 9.8 m/s^2.  These values were used to calculate the theoretical value of time (T).

T= 8.03 s  +/-  5.40 s

Conclusion and Analysis:

                                                               Time (s)

Avg                                                                20.22                                   

Range                                                      1.15

Standard deviation                                    0.404        

Error %                                                      60 %                                   

Measured Uncertainty                           5.40

Calculated uncertainty                           4.82

Although the uncertainty of the measured values is high, its predicted range of uncertainty is not adequate to explain the large gap between the predicted value of 8.03 s and the actual average of 20.22 s.  Reasons for error could be an underestimation of the actual intrinsic error involved with lab measurements.  One source of error that seems to stick out is the diameter of the drain hole, which was given as a ½ in which in turn was calculated to be 6.35e-3 m, this value was measured and found to be 3.68e-3, which is a large variance at that level.  Other reasons for the error may be that the height (h) of the water displaced was too large of a value.  With a smaller value the effects of the water draining may have been more accurately modeled by the equation for time derived from Bernoulli’s equation.  Interesting results though seem to be that there is only a slight variance between the measured uncertainty and the calculated uncertainty, which leads me to believe that there may have been a minor error in the recorded values of the calculated time.