The purpose of this lab is to explode fluid dynamics as well as the Bernoulli equation. The Bernoulli Principle is a series of relationships with fluid flow, pressure and height of the fluid. The model is simplified to ignore loss of energy and internal flow. This lab will attempt to verify the model by relating volume flow, potential energy, and time.
If the fluid is incompressible and the stream is continuous with no internal flow, then the volume moved at point 1 has to be the same as the volume at point 2. The conserve energy pressure, height, and velocity become dependent on one another. The result is the Bernoulli Equation:
In this lab the goal is to explore fluid dynamics and to possibly verify the Bernoulli model. The method used to study fluid dynamics in this lab is to measure a volume flow over time. By using measurements, in conjunction with the Bernoulli Principle, it will be possible to juxtapose a theoretical value of time for a given volume to flow versus actual measured time values.
The equipment used in this lab is as follows:
- A medium sized bucket
- Water
- Beaker
- Ruler
- Calipers
- Stopwatch
Procedures
A hole was put in the bucket near the bottom. The bucket is then filled with water and a set volume will drain out of the hole. The beaker used is a 200 ml beaker, this was the target volume and the measurement was the time it took to fill the 200 ml beaker. After every trial, the water was filled back up in the bucket of 144.5 mm above the exit hole to insure consistency. The values of interest are as follows:
Theoretical Volume = 200 ml = 200 cubic centimeters
Height of Water = 144.5 mm = +/-0.05mm
Area of Hole = 0.0000385 meters squared +/-0.00000005 meters squared
Before discussing trials, the math involved will be addressed. Since the hole was the point of zero potential and the top of the water was the point of highest potential with no velocity, the Bernoulli equation simplifies to:
Since we are interested in time we can break down velocity as h/t. After that, both sides can be multiplied by the area of the hole. When that is done, solving for t yields:
This equation has all the variable measured in the procedures. It will be possible to use the know values of the equipment and the set parameters to obtain a theoretical value for time. With the theoretical time, it will be possible to verify the model used in this lab by comparing the experimental time values.
Trials
- t = 3.82s V = 204.0 ml
- t = 4.75s V = 218.0 ml
- t = 4.35s V = 198.5 ml
- t = 4.38s V = 197.5 ml
- t = 4.35s V = 210.0 ml
- t = 4.52s V = 211.5 ml
After obtain the data, a time average was calculated with a standard deviation. Furthermore, A theoretical time was calculated with propagated error.
Average Experimental Time With SD = 4.35s +/- 0.31s
Theoretical Time With Uncertainty = 3.48s +/-0.39s
Percent Error = 25% +/-23.2%
Looking at the average time with standard deviation and theoretical time with uncertainty, the potential difference is 0.17 seconds. This is an acceptable validation of the model when you consider the rigor and pit falls within the experiment. Some of those issues are:
- Variable volume with the experimental time
- Slight variances in the drilled hole
- The table not being completely level
- Unaccounted fudge factors in using a stop watch
- Repeatability
By using the average value for the experimental time it is possible to solve for the diameter of the hole in the bucket. The measured value of the hole used in the previous calculation was:
D = 7.0 mm
The equation for time rearranged for diameter:
The result of experimental values used in the equation yield a diameter of:
D = 5.99 mm
The % difference between the measured diameter and the experimental diameter is 14%. This is to be expected when you consider the hole was made with a drill and never properly finished. This would create variations in stream diameter due to roughness around the hole.
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