Modern Physics: Fluid Dynamics

Introduction:
Using Bernoulli's equation we may obtain a fairly accurate formula for calculating the exiting speed of fluid from a large container.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

The left side of the equation models the top of our large container while the right models the exiting path at the bottom of a large container.
Since
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

We find

With this exiting speed, the rate at which some volume exits the the container can be found.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

It follows that we may find the time needed for some amount of volume to empty out of the container.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

This value is taken as the theoretical time, ttheoretical,needed to empty a volume V.In this experiment, we will explore the how well this formula reflects experimental values. By emptying some amount of water V from a large cylindrical bucket on successive trial runs we will acquire actual time data. An analysis of the error between ttheoretical and tactual will follow. Additional parameters from our time equation will also be compared to experimental values.

Steps:
Six trials runs will be taken to acquire a range of actual values that will be compared the the theoretical value calculated later. A simple setup and a timer is used to measure the time taken for V = 300mL of water to empty into a beaker. The hole at the bottom of our bucket is initially covered with tape then opened, at which point we begin the timer until we fill the beaker to the 300mL mark.

Releasing, timing, and watching the amount of water flow is a two-man job as seen in this attempt to measure the time need to fill our beaker to 300mL.

The times acquired are tabulated below:

	1^st Run	2^nd Run	3^rd Run	4^th Run	5^th Run	6^th Run
Time to empty 300 mL t_actual (s)	8.59	9.87	9.59	8.12	8.57	9.06

tactual = tavg ± (tamax-tamin)/2 = 8.97 ± .74s

As the equation above suggest, we require parameters V,A and h to calculate ttheoretical .

Calipers were used to measure the diameter d.

As suggested by my lab partner, James, we scaled the diameter of the exit hole by outlining the edge created by an inserted pencil.

The height h was obtained, simply, with a meter stick lined from the center of the whole to the edge of the water level. The measurements were as follows:
V = 300mL ± 15mL
d = 6.0 ± 1mm
h = 14.0 ± .2cm

Calculations are depicted below (SI units of volume, length, and time are acounted for):

Without accounting for uncertainties we observe the theoretical time to be
ttheoretical = 6.40 s
To calculate the percent error for each trial run we use the equation
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

	1^st Run	2^nd Run	3^rd Run	4^th Run	5^th Run	6^th Run
%Error_t	34.2	54.2	49.8	26.9	33.9	41.6

Uncertainty from our theoretical result is obtained by calculating
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

As seen in the above diagram, our theoretical result is only certain within
ttheoretical = 7.11 ± 2.66 s

Thus, the time values agree within uncertainty, with the exception of our maximum value on trial 2.

Assuming the diameter measured is inaccurate, we may solve for the "actual" diameter by rearranging the time-to-drain equation.

Since A = π(d/2)2

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

dcalculated = 5.7 mm
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Errord = 5.4%

Questions/Conclusions:

The consistencies of these measurements really comes down the reliability of the whole apparatus, including the lab participants who determine an appropriate time to clock the timer. Refilling the bucket container to the appropriate height after each height is another concern.

The actual time values taken from our trial runs range from a minimum value of 8.12s to a maximum of 9.87s. Comparing
tactual = 8.97 ± .74s
ttheoretical = 7.11 ± 2.66 s

Our results fall within uncertainty ranges, but with errors still greater than 26% from the theoretical value. The accumulation of such a great uncertainty is expected for the reason that many measurements were required to obtain ttheoretical.

Modern Physics

Tuesday, March 5, 2013

Fluid Dynamics

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