Fluid Dynamics

The purpose of this Fluid Dynamics lab is to utilize Bernoulli's equation and the volume flow rate in estimating a theoretical time for a pre-determined volume of fluid to be ejected from a container given the dimensions of the setup.  This theoretical time will be compared with experimental data to evaluate its accuracy and consider factors that may contribute to any deviations observed.

The Setup:

Water is ejected from the small hole near the base of the container

• A hole has been drilled near the bottom of a container
• Tape is applied to the outside of the hole
• The container is filled with water to a depth of 3 inches (7.62 cm)
• The tape is removed and water is ejected into a beaker
• Using a stop watch, the amount of time for the beaker to fill to 500 mL is measured
• The trial is performed 6 times

Measurements and Calculations:

• We start with the special form of Bernoulli's equation which relates the velocity of the ejected water from the hold to the he above it (v=sqrt(2gh))
• We determine h to be the distance from the hole to the surface of the water in the bucket
  (3 in. - 0.7 in = 2.3 in. = 5.842 cm = 0.05482 m)
• We measure the diamter of the hole to be 8.5mm, which gives a radius of 4.25mm
• The area, A, of the hole is calculated as 56.7mm^2 = 0.0000567 m^2, (A=pi*r^2)
• We convert 500 mL volume, V, emptied to be 0.0005 m^3
• We combine Bernoulli's equation along with the volume flow rate and solve for the theoretical time in which 500mL would be emptied:
  
  t = V/(A*sqrt(2gh)) = 8.23 seconds
• 6 trials were performed:
  Trial Time (seconds)
  
1 29.70
2 29.21
3 29.54
4 29.56
5 29.50
6 29.20

Results:

We see that there is about a 72% difference, which is a considerably large deviation from the theoretical time.  There are, however, some factors that surely could have contributed to this difference:

• There is uncertainly propagated from the rulers used to measure the hole
• The area of the hole was calculated using the area of a circle and could only serve as an approximation to the actual area of the hole - although this is a fair approximation
• Bernoulli's equation is based on the ideal situation where the velocity of the ejected fluid is constant - This was not the case in our situation as the velocity decreased as the height, h, of water above the hole decreased.  This adds a significant amount of time

Using our theoretical time, we could reverse our calculation to determine what size of hole would be required to meet the time expectation:

diameter = 4*sqrt(V/(pi*t*sqrt(2gh))) = 17 mm = 0.017 m

This is a difference of 50%.  This tells us that Bernoulli's equation, based on an idealized model with constant velocity of ejected fluid, may not be suitable for a shallow initial height of liquid.