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\begin{document}

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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\Toricelli Box}



\markboth{Column of Water Flow with Box in Water}{Column of Water Flow with Box in Water}

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

\makeStitlePDFLaTex


\begin{abstract}
The time it takes a column of water to empty and the time it takes the same volume of column of water with a box (various sizes) submerged in the column of water are compared through modeling with Torricelli's Law.
\end{abstract}


\section*{SCENARIO DESCRIPTION}

\subsection*{Torricelli's Law}

We study the  emptying a column of water through a small bore hole at the bottom of the column of water.   We use Torricelli's Law \cite{Winkel2015} in which we have a differential equation (\ref{eq:torr}) we can use to solve for the height, $h(t)$ m,  of the water in the column at time $t$ as the water is emptying the column through a small bore hole at the bottom of the column of water:

\begin{equation}\label{eq:torr}
	A(h(t)) h'(t) = - \alpha a \sqrt{2 g h(t)},
\end{equation}

\noindent where $a$ is the area of the small bore hole at the bottom of the cylinder through which the water exits the cylinder, $\alpha$ is the restriction coefficient or percent of the water which could flow out which actually does flow out, and $A(h(t))$ is the cross sectional area of the water in the cylinder at height $h$.


\subsection*{Emptying Column of Water with Boxes of Varying Size at Base}


We consider the problem of emptying a column of water through a small bore hole at the bottom of the column of water with a box on the bottom of the column, displacing water of volume equal to the volume of the box. Thus the water in the column will rise accordingly.  We keep the Length ($L$) and Width ($W$) of the box the same in all cases, but vary the height, $s$, of the box.

We ask the question, ``As the height of the box, $s$, increases how does the time to empty the column of water change?''  Does the time to empty the column of water get more or less than when there is no box in the column of water?


Consider a right circular cylinder partially filled with a column of water to a height of $h_i$ m.  Let us suppose the cross-sectional shape of the column of water is a circle of radius $r$ m. Place a rectangular solid (solid box) of dimensions $L$ m long, $W$ m wide, and of s m high, into the column of water so it settles on the bottom. It therefore displaces $L \cdot W\cdot s$ m$^3$ of water. 

This box causes the water level in the cylinder to rise, but by how much? Call this height change $h_c$ m. The volume of water displaced by the  box in the water is $L\cdot W\cdot s$ m$^3$. Thus we have the original volume of water, $V_i = \pi r^2 h_i$, and we add the volume of displacement $L\cdot W\cdot s$ to get a total volume of material in the cylinder of radius $r$. This means we can determine the change in the height of the  water, $h_c$  by solving for $h_c$ in $\pi r^2 h_c = L W s$, where  $\pi r^2 h_c$ is the cylindrical volume of water displaced by the box.



Use Torricelli's Law appropriately and address these questions.  As the height of the box, $s$, increases how does the time to empty the column of water change?  Does the time to empty the column of water become more or less than when there is no box in the column of water?

Use the following variables:  $\alpha =  0.70$,  $a = \pi (.02)^2$ m$^2$, $r =  0.1$ m, $L = 0.06$ m, 
   $W = 0.06$ m, $g = 9.8$ m/s$^2$, and $h_i= 0.6$ m.
   
   Examine the cases where the height of the submerged box, $s$, is $s = 	0.1, 0.2, 0.3, 0.4, 0.5,$ and $0.6$ as well as the maximum height of the box where the top of the box is level with the surface of the water, for beyond that height a portion of the box is out of the water and that portion of the box which is out of the water does not alter our model. In each case, determine the time it takes to empty the column of water and compare that time to the time it takes to empty the column of water with no box in it.
   
   
   
\subsection*{Modeling the Emptying Column of Water with Box at Base Using Video Data}

We provide two videos 1-064-Torricelli'sLawWithNoBox.mp4 and 1-064-Torricelli'sLawWithBox.mp4  in which we show the fall of the height of a column of water in a graduates cylinder, the first with no box at the base of the column of water and the second with a box at the base of the column of water.

We apply Torricelli' s Law for a Falling Column of Water to model the height of the column of water for a fixed graduated cylinder in two cases:

\begin{enumerate}
\item  the graduated cylinder is filled to 500 ml (marked as a height of 23.6 cm) and water exits a small bore hole at the bottom which has a diameter of 5/32" or radius of 0.198438 cm
\item  the graduated cylinder is initially filled to 500 ml and a configuration of four small blocks is placed at the bottom of the cylinder, thus raising the height of the column of water, now marked as a height of 26.25 cm. The  water exits a small bore hole at the bottom which has a diameter of 5/32" or radius of 0.198438 cm.
\end{enumerate}

The inner radius of the graduated cylinder in the videos is 2.6 cm.

Model both situations for the height of the column of water as a function of time using the differential equation (\ref{eq:torr}) for  Torricelli's Law and  ascertain which column of water empties first, the column with the blocks or without the blocks.


\begin{thebibliography}{98}




\bibitem{Winkel2015} Winkel, B. 2015.  1-015-S-Torricelli. 


\end{thebibliography}

\teacher{
\section*{NOTES FOR TEACHERS}


This are examples of building a model from Torricelli's Law which is given or from  first principles (Conservation of Energy) leading to a derivation of Torricelli's Law and then applying it in two situations, when the cross sectional area of the surface of the column of water is just the circular cross section of the column itself and when the cross section of the box needs to be subtracted from the larger cross sectional area.


In the section ``Emptying Column of Water with Boxes of Varying Size at Base'' we offer a Mathematica  notebook, 1-064-T-Mma-TorricelliBox-TeacherVersion.nb and a pdf version of this file for those who do not use Mathematica. We discuss several box configurations, i.e. we vary the height of the box in the water keeping all other variables the same. We also offer an animation, 1-064-T-Mma-TorricelliBox-Animation.mp4, to illustrate the comparison between the falling column of water with and without the box at the base of the column of water.  


With regard to the issue of which takes longer to empy, the column of water with or without the box at the base of the column, it is well worth asking students first what they think, because if they understand the principle behind Torricelli's Law, namely due to Conservation of Energy the exit velocity of the water through the bore hole at the bottom of the column of water is higher when the column of water is higher, then they should believe adding boxes to the column of water  decreases the time to empty the tank.  However, modeling this phenomena is convincing and we have created the animation to illustrate this, through comparison of the two situations.


In  the section ``Modeling the Emptying Column of Water with Box at Base Using Video Data'' we offer a Mathematica  notebook, 1-064-T-Mma-TorricelliBox-With-WithoutBox-TeacherVersion.nb and  a pdf version of this file for those who do not use Mathematica.  

We offer a complete analysis in which we first determine $\alpha$, the discharge coefficient, from the initial time interval in the video 1-064-Torricelli'sLawWithBox.mp4 in which the top of the column of water stays above the level of the top of the box.  From here we use this $\alpha$ value to model the entire process and compare our model predictions with the actual data, quite favorably, it would appear. 




}

\end{document}
