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\begin{document}

\newpage
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\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\Three Hole Column}



\markboth{Three Spigot (Hole) Column of Water}{Three Spigot (Hole) Column of Water}

\author{Brian Winkel, Director
SIMIODE\\ 
Chardon OH 44024 USA\\
BrianWinkel@simiode.org
}

\makeStitlePDFLaTex


\begin{abstract}
We consider a column of water with three holes or spigots through which water can exit and ask students to model the height of the column of water over time.
\end{abstract}

\section*{SCENARIO DESCRIPTION}

Consider a cylinder containing a falling column of water as water exits through three spigots (see Figure~\ref{fig:1-063-ColumnPhoto}).

			
\begin{figure}[h!]
\begin{center}
\includegraphics[width=6.4cm]{1-063-ColumnPhoto}



\caption{\label{fig:1-063-ColumnPhoto} Cylinder with column of water and three spigots.\cite{NC2018}}
\end{center}



\end{figure}

This cylinder has three exit holes or spigots. We create a cylinder and offer dimensions, perhaps different from the one depicted in \cite{NC2018}, and ask you to  model the height of the water in the cylindrical column as a function of time using Torricelli's Law\cite{Winkel2015s}.
Wewill consider two cases. First we will vary the radii of the spigot holes and second we will keep the radii of the spigot holes the same.

\subsection*{Case 1 - varied  radii of the spigot holes}

The radius of the cylinder is $R = 2.5$ cm and the radius of the lower spigot hole  is $r_3 = 0.8$ cm, the radius of the middle spigot   is  $r_2 = 0.5$ cm, and the radius of the top spigot hole  is $r_1= 0.2$ cm. 
  
The middle spigot hole is 12 cm above the bottom spigot hole (measured from center of the hole to center of hole) while the top the spigot hole  is 10 cm above the middle spigot whole (measured from center of the spigot hole to center of the spigot hole).

Identify the spigot holes as bottom is spigot hole 3, middle spigot hole is hole 2, and top spigot hole is hole 1. 


The water level initially is at $h_0k = 14$ cm above the top spigot  hole (measured from center of the spigot  hole to the water level).  The  top  spigot hole  is $h_1k = 10$ cm above the middle spigot hole (measured from center of the spigot hole to center of the other spigot hole) while the middle spigot hole is $h_2k = 12$ cm above the bottom spigot hole (measured from center of the spigot hole to center of the other spigot  hole). The bottom spigot  hole is $h_3k = 5$ cm above the base of the container.


\subsection*{Case 2 - identical  radii of the spigot holes}

The radius of the cylinder is $R = 2.5$ cm and the radius of the lower spigot  hole  is $r_3 = 0.3$ cm, the radius of the middle spigot hole  is  $r_2 = 0.3$ cm, and the radius of the top spigot hole  is $r_1= 0.3$ cm. 
  
The middle spigot  hole is 12 cm above the bottom spigot  hole (measured from center of the spigot  hole to center of spigot  hole) while the top spigot   hole  is 10 cm above the middle spigot  hole (measured from center of the spigot  hole to center of the spigot  hole).

Identify the spigot  holes as bottom is spigot  hole 3, middle spigot  hole is spigot  hole 2, and top spigot  hole is spigot  hole 1. 


The water level initially is at $h_0k = 14$ cm above the top spigot  hole (measured from center of the spigot  hole to the water level).  The  top  spigot  hole  is $h_1k = 10$ cm above the middle spigot  whole (measured from center of the spigot  hole to center of the other spigot  hole) while the middle spigot  hole is $h_2k = 12$ cm above the bottom spigot  hole (measured from center of the spigot  hole to center of the other spigot   hole). The bottom  spigot  hole is $h_3k = 5$ cm above the base of the container.

\subsection*{Analysis}

Recall from \cite{Winkel2015s} that the exit velocity of the stream of water exiting a column of water at a depth of $h$ units below the surface of the column of water is given by $v = \sqrt{2 g h}$, where $g$ is the acceleration due to gravity in the appropriate units.

Using this fact you should be able to build a set of differential equations for the  falling column of water when the surface is in the three regions: (i) above spigot 1, (ii) between spigot 1 and spigot 2, and (iii) below spigot 2 and above spigot 3. In each case you can compute the rate at which the volume is changing from the outflow of the spigots and set that equal to the volume loss, $\pi R^2 h'(t)$ where $h(t)$ is the height of the column of water above a spigot hole, $h'(t)$ is the rate at which the height of the column of water is falling, and $\pi R^2$ is the constant cross sectional area of the cylinder.

One other thing to consider is the fact that through any spigot hole about only 70\% of the possible flow actually occurs due to friction on the spigot wall as an example of loss.  So we need to employ some lessening of the flow rate by multiplying the theoretical or maximum flow rate by some constant $\alpha = 0.70$, where $\alpha$  is an empirically observed number which indicates the percentage of the maximum flow rate which actually gets through the small spigot hole. $\alpha$  is called {\it the discharge or contraction coefficient\/}\cite{Wiki2018}.

You should produce plots of the height of the column of water over the several sections of the column of water above and between spigots and in  Case 2 with identical  radii of the spigot holes you should comment on the rate at which the height of the column of water decreases in each section.

\begin{thebibliography}{98}

 

\bibitem{NC2018} North Carolina State University, Physics Department DemoRoom. 2018. {\it Velocity of Efflux Three Hole Can Experiment.\/}. Raleigh NC:  North Carolina State University DemoRoom. \url{https://www.youtube.com/watch?v=0EsUiBhBIX8}. Accessed 15 December 2018.

\bibitem{Wiki2018}Wikipedia contributors. 2018.  Discharge coefficient.  Wikipedia, The Free Encyclopedia. \url{https://en.wikipedia.org/w/index.php?title=Discharge_coefficient&oldid=839574186}. Accessed 17 December 2018.

\bibitem{Winkel2015s} Winkel, B.  2015.  1-015-S-Torricelli.


\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}

This is an opportunity to apply Torricelli's Law which can be found in a number of sources and which we have developed elsewhere\cite{Winkel2015t}.  Torricelli's Law says that the exit velocity of a stream of water through a spigot placed in a column of water at a depth of $h$ units below the surface of the water is $\sqrt{2 g h}$, where $g$ is the acceleration of gravity, in the appropriate units.

The difficulty in this situation is the ``bookkeeping'' of stopping the flow out of a spigot hole once the water level in the column falls below that level.

We offer two situations, one in which there are fixed distances between the spigot holes, but the spigot holes are of different radii and the other in which  the same fixed distances between the spigot holes occur, but the spigot holes are of the same radii.
  

We offer a Mathematica notebook (1-063-T-Mma-ThreeHoleColumn-TeacherVersion) and a pdf version for those we do not read Mathematica files of our analysis. Further, we render an animation of our model in the case that all three spigots have the same radius. These files are  placed in the Supporting Docs for this Modeling Scenario. The animation comes close to modeling the behavior of the video rendered in \cite{NC2018} in that during the last stage, i.e.~when the column of water is below the second of three spigot holes, the rate at which the column of water falls really slows down. This is reassuring!



\begin{thebibliography}{99}



\bibitem{NC2018} North Carolina State University, Physics Department DemoRoom. 2018. {\it Velocity of Efflux Three Hole Can Experiment.\/}. Raleigh NC:  North Carolina State University DemoRoom. \href{https://www.youtube.com/watch?v=0EsUiBhBIX8}{https://www.youtube.com/watch?v=0EsUiBhBIX8}. 

\bibitem{Winkel2015t} Winkel, B.  2015.  1-015-T-Torricelli.


\end{thebibliography}
}





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