Experiment 2: Fluid Dynamics

Introduction

The purpose of this experiment was to determine the diameter of the drain hole by using the concept of Bernoulli principle which states that an increase in velocity of the fluid is due to a decrease in pressure.  This statement is shown by the following equation in an incompressible fluid flow:

p₁ + ρgh₁ + 1/2ρv₁² = p₂ + ρgh₂ + 1/2ρv₂²

                This experiment was conducted by using a bucket with a small drain hole, 5 gallons of tap water, a 500ml beaker, a meter stick, and a stopwatch. First, the diameter of the drain hole was measured and recorded. Then, a bucket was filled with 3 inches height of water. In fact, this height was 2.3 inches above the drain hole. A beaker with 473 ml mark was put under the drain hole to let the water drain into the beaker. The time taken to empty 473 ml of water from the bucket was recorded. This procedure was repeated six times, and the average time taken was used to compare with the theoretical time taken.  Lastly, the actual diameter was computed based on the measured height of water and recorded time taken. Theoretical time taken and the actual diameter of the drain hole were computed by using the following equation:

V/t=A√2gh

in which V represents volume, t is time taken, A indicates area of the drain hole, g is the acceleration due to gravity, and h represents the height of water.

Figure 1: 3 inches height of water in the bucket with a drain hole 

Figure 2: Water from the bucket draining into the beaker

Data and Analysis
Uncertainties of each measured value were assumed to be large so that it took account of various errors in this experiment.

Table 1: Recorded measurements along with calculate values based on these measurements

Recorded measurements		Calculated values
Volume emptied (ml)	473 ± 5	Volume emptied (ft3)	0.0160 ± 0.0002
Diameter of drain hole (cm)	0.60 ± 0.01	Radius of drain hole (ft)	0.0098 ± 0.0016
Height of water (in)	2.30 ± 0.10	Area of drain hole (ft2)	3.0x10-4 ± 1.0x10-4
		Height of water (ft)	0.192 ± 0.008

Table-2: Comparison of experimental and theoretical time taken for 16 ounces (473ml) of fluid to flow out of the bucket

Trial	Time taken(s)	Average time taken (s)	Theoretical time taken (s)	% Error (%)
1	25.06 ± 1.00	25.30 ± 1.00	15.01 ± 4.93	68.55
2	24.85 ± 1.00			Smallest possible	Largest possible
3	25.24 ± 1.00			21.87	160.9
4	25.36 ± 1.00
5	25.17 ± 1.00
6	26.09 ± 1.00

Table-3: Comparison of the measured and calculated diameter/radius of the drain hole

	Measured	Calculated	% Error (%)
Diameter (cm)	0.60 ± 0.01	0.46 ± 0.01	29
Radius (ft)	0.0098 ± 0.0016	0.0076 ± 0.0002

Figure 3: Calculations of uncertainties 

Discussion

                According to table 2, the percent error of the time taken was 68.55%. The experimental and the theoretical time taken were not within uncertainties of each other as shown in table-2. This was because the equipment used in this experiment had high inaccuracy. For instance, the beaker which was marked at 473 ml could contribute larger error since the mark was only an approximation. In addition, using stopwatch to record the time fluid flow could create large deviation from the actual time taken. Since the starting and ending time happened in an instantaneous moment, it was difficult to record the time that matched with the actual beginning and ending of 16 ounces of fluid flow. Thus, there was deviation not only in measurement but also in recorded time. As shown in table-2, the uncertainty of theoretical time taken was almost one-third of its value. Since the theoretical time taken was computed based on several measured values such as height and diameter, uncertainty could be very high. This also contributed to larger percent error.
                However, as shown in table-3, there was smaller percent error, which was 29%, of the diameter of the drain hole. Again, the measured and the calculated values were not within the uncertainties of each other. This error was also contributed due to inaccuracy in the measurement and the recorded time. In addition, the tape that was used to cover the hole was not fully taken out; hence, it possibly prevented water from flowing out of the bucket. This led to longer time taken to empty 16 ounces of water.

Conclusion

                The experimental time taken for 16 ounces of water to flow out of the bucket was 25.30s on average while that of the theoretical value was 15.01s. This resulted in 68.55% error. Besides, the percent error of the diameter of the drain hole was 29% with actual diameter 0.46cm and the measured diameter 0.60cm.