Comparison

Torricelli's Law and Fluid Dynamics
MATH 2413 Research Project 2002

Comparison

Experiment 1

Pipe BB drains in 13.3 seconds.
Pipe BS drains in 50.4 seconds.
Pipe SB drains in 3.8 seconds.
Pipe SS drains in 13.1 seconds.

According to the Analysis section, .

Using that information:
Pipe BB should drain in 14.4 seconds (7.6% error).
Pipe BS should drain in 59.4 seconds (15.2% error).
Pipe SB should drain in 3.7 seconds (2.7% error).
Pipe SS should drain in 14.5 seconds (9.7% error).

Except for Pipe SB, all pipe drain times were shorter than expected. We anticipate that a higher value of alpha should have been used.

Experiment 2

We see that the data is linear; that is, each equation is of the form v(t) = -bt + a. We use this to verify Torricelli's Law. Recall from the Analysis page:

Using our known values for a and b, we can examine the slope fields (from dh/dt) and the graphs of h(t) from the equations above. Note that they are essentially identical. Note that the slope fields should stop when they hit 0.

We now continue to examine our equation for h(t) to prove Torricelli's Law by reinserting v:

We now point out that when v = 0, h = 0:

Torricelli's Law is derived here for a specific value of alpha. Given the data for this experiment:
Pipe BB 1.07
Pipe BS 1.17
Pipe SB 1.02
Pipe SS 1.09
Average 1.08
This will be the value for alpha used in all following experiments. It is higher than the predicted value of 0.84.

A comparison of the experimental h(t) graphs to the theoretical h(t) graphs (in order of BB, BS, SB, SS):

We note that the experimental data is always lower, which can be accounted for by the theoretical alpha value of 0.84, lower than the value found through experimentation.