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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Container Shape Falling Water}





\markboth{Container Shape Falling Water}{Container Shape Falling Water}






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


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\begin{abstract}
We examine many different physical situations to determine the time it takes a fixed volume of  water to flow out of different shape  containers through the same size exit hole at the bottom of the container.
\end{abstract}


\section*{SCENARIO DESCRIPTION}


\subsection*{Introduction}

We consider the question.  For a fixed volume of water, say $V = 1000 {\rm cm}^3$ and fixed exit hole at the bottom of a container does the shape of the column matter in determining the time of complete draining of the water from the container?

We examine many different  containers, each holding 1000 ${\rm cm}^3$ of water and each with an  exit hole for the water to fall out of the container of radius $r = 0.25$ cm.  If $\alpha$  is the effective area (percentage due to friction at the opening called the {\it discharge or contraction coefficient\/}) for water flowing through the exit hole then $\alpha = 0.70$.


First we need to recall the governing equations or model for the height of a column of liquid in a cone with a hole in the bottom from which the liquid leaks out.  


Per this analysis in \cite{Winkel2015}, we calculate the loss of water out of the container in two different ways and set them equal using Torricelli' s Law development as outlined in  \cite{Winkel2015}.

Here are the definitions and specifications for all the containers we shall work with:

\begin{itemize}
\item $h(t)$ is the height of the column of water in the container at time $t$.  Length units will be in centimeters and time units in seconds.
\item $A(h(t))$ is the cross sectional are of our container at height $h(t)$.
\item $r_c = 0.25$ cm  is the radius of the small bore hole through which the water exits our container.
\item $\alpha  = 0.70$  is the effective area (percentage of water that actually gets through the exit hole due to friction at the opening called the {\it discharge or contraction coefficient\/}.
\end{itemize}
 
\begin{itemize}
\item [(L)] At time $t$ a small element of height changes, $h'(t)$, over a cross sectional area $A(h(t))$. Hence we will have lost 
\begin{equation}\label{eq:slab}
A(h(t))\cdot h'(t)
\end{equation}
 volume of water. 
\item [(R)]  Since the water is flowing out of the opening with a velocity (according to Torricelli's Law) of $\sqrt{2\,  g\,  h(t)}$ m/s and our opening's cross sectional area is $a = \pi r_c^2$ m$^2$ then we are losing water at a rate of 
\begin{equation}\label{eq:opening}
\alpha\, a\, \sqrt{2\, g\, h(t)} = \alpha\, \pi (r_c)^2 \sqrt{2\, g\, h(t)}\, {\rm m}^3{\rm /s}\, ,
\end{equation} 
where we recall the effective area for fluid flow through the opening is only a fraction ($\alpha = .7$) of the area.  
\end{itemize}
Now if we equate the computations for the loss of water found in (\ref{eq:slab}) with that computed in (\ref{eq:opening}) we obtain the following differential equation, with our initial height $h(0) = h_0$ m as an initial condition:
\begin{equation}\label{eq:changevol}
A(h(t))\cdot h'(t)= \alpha\, \pi (r_c)^2 \sqrt{2\, g\, h(t)}\, ,\quad h(0) = h_0\, .
\end{equation}

Equation (\ref{eq:changevol}) is one form of Torricelli's Law for the height of falling column of water.



\subsection*{Activity 1}

\begin{itemize}
\item  [a)] Consider a right circular cylinder  of a fixed  volume $V = 1000\,  {\rm cm}^3$ of water in it and exit hole of radius$ r = 0.25$ cm, with  discharge or contraction coefficient $\alpha= 0.70$. Determine the time it takes to empty this container.
\item[b)] Now if we presume the volume of water in our cone is  $V = 1000\,  {\rm cm}^3$. Then we have a ratio between height and radius at every given height, for the volume of a cone of height $H$ and radius $R$ is $V = \pi R^2H$. This leads to a relationship between $R$ and $H$ when the volume is fixed at $V = 1000\,  {\rm cm}^3$.  For a range of heights, $0.01 \leq H \leq 100$ determine the time it takes to empty the tank and offer a plot of the time it takes to empty the tank for a given value of $H$ vs.  $H$ and comment on the shape of the plot.
\end{itemize}

\subsection*{Activity 2}

Consider a right circular cylinder of radius $R$ cm and height $H$ cm which contains $1000\,  {\rm cm}^3$. Note that $R$ and $H$ are related through the formula for the volume of a right circular cylinder of radius $R$ and height  $H$, i.e. $V = \pi R^2 H$. As such determine the time it takes to empty the container of water over a range of values of $H$, say $.01 \leq H \leq 100$. How do you think the time to empty the container changes (or does it?) as $H$ increases?  Plot time to empty the container as a function of height $H$. Comment on your plot and what it says physically.  



\subsection*{Activity 3}

Pick the dimensions of a right circular cone of radius $R$ and height $H$ that points downward which contains $1000\,  {\rm cm}^3$ and use the specifications outlined above. For $H = 20$ cm determine the time it takes to empty your container of water.


\subsection*{Activity 4}

\begin{itemize}
\item [a)] Pick the dimensions of a right circular cone of radius $R$ and height $H$ that points upward which contains $1000\,  {\rm cm}^3$ and use the specifications outlined above. Determine the time it takes to empty your container of water.
\item [b)] Note that $R$ and $H$ are related through the formula for the volume of a right circular cone of radius $R$ and height  $H$, i.e. $V = \frac{1}{3}\pi R^2 H$. As such determine the time it takes to empty the container of water over a range of values of $H$, say $.01 \leq H \leq 100$. How do you think the time to empty the container changes (or does it?) as $H$ increases?  Plot time to empty the container as a function of height $H$ and comment on your plot.  
\end{itemize}



\subsection*{Activity 5}

Consider a container in the shape of a sphere which has volume $V =1000\,  {\rm cm}^3$.  Determine the time it takes to empty the tank if initially filled with water.

\subsection*{Activity 6}

\begin{itemize}
\item [a)] Pick the dimensions of a right circular cone of radius $R$ and height $H$ that points downward which contains $1000\,  {\rm cm}^3$ and use the specifications outlined above. Determine the time it takes to empty your container of water.
\item [b)] Note that $R$ and $H$ are related through the formula for the volume of a right circular cone of radius $R$ and height  $H$, i.e. $V = \frac{1}{3}\pi R^2 H$. As such determine the time it takes to empty the container of water over a range of values of $H$, say $.01 \leq H \leq 100$. How do you think the time to empty the container changes (or does it?) as $H$ increases?  Plot time to empty the container as a function of height $H$ and comment on your plot.  
\end{itemize}

\subsection*{Activity 7}

Create a question of your own., and render an analysis?  E.g., other shapes, comparing emptying times for different shapes, for a given shape vary the exit hole radius.

\begin{thebibliography}{98}




\bibitem{Winkel2015} Winkel, Brian. 2015. 1-015-T-Torricelli.  


\end{thebibliography}

\teacher{NOTES FOR TEACHERS


This Modeling Scenario requires initial use of geometry to determine $A(h(t))$, the cross sectional area for various shape containers and follow through using Torricelli's Law. Further, it asks the students to compare various situations and do a reality check with their intuition on times to empty a tank give the different shapes.

Finally, we encourage students to ask questions of their own and carry out the analysis to address these questions.
}

\end{document}
