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\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION \else STUDENT VERSION \fi\\WATER FALLING IN A CONE}





\markboth{Water Falling in a Cone}{Water Falling in a Cone}






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



\begin{abstract}
We offer an opportunity to model the height of a falling body of water in a right circular cone (funnel) and to estimate an appropriate parameter based on data collected from a video of the experiment found on YouTube. This is an application of Torricelli's Law. 
\end{abstract}


\section*{SCENARIO DESCRIPTION}



We have an inverted (point down) right circular conical tank  (top of funnel) with a top radius $R$ centimeters and a total height of $H$ centimeters, filled with water to a height of $h$ centimeters. There is a circular hole of radius $r_C$   centimeters at the very bottom of the cone from where water leaks. See Figure~\ref{fig:1-55-SetUp} of the apparatus used to  in the video \cite{sm2} from which we collected data which we offer below.


\begin{figure}[h]
\centering
\includegraphics[width=14cm]{1-55-SetUp.eps}

\vspace*{-11 true cm}
\parbox{12 true cm}{\caption{\label{fig:1-55-SetUp} Set up used to collect data on falling column of water from cone. Distance on ruler in cm and time on digital clock in minutes, seconds, and milliseconds.
}}
\end{figure}

We begin our analysis using data sampled from the movie of our experimental set up as found in SlowMoCapture-2  found at \cite{sm2}. See Appendix for a sample set of data taken by the author if the reader cannot collect the data from the videos \cite{sm1, sm2,c3}. These videos are also available for download on YouTube.


First we need to determine the governing equations for the height of a column of liquid in a cone with a hole in the bottom from which the liquid leaks out.  We refer to  Torricelli' s Law development as outlined in \cite[pp.~1-4]{Winkel2015}. 

Per this analysis, we calculate the flow out of the hole at the bottom of the inverted right circular cone in two different ways and set them equal. Here $h(t)$ is the height of the water from the base of the inverted cone and $A(h(t))$ is the cross sectional area of the surface of the column of water at height $h(t)$.

The left hand side (\ref{eq:LHS}) of our differential equation is given by a small element of height change, $h'(t)$, over a cross sectional area $A(h(t))$ at height $h(t)$:

\begin{equation}\label{eq:LHS}
A(h(t))*h'(t)\, . 
\end{equation}


The right hand side (\ref{eq:RHS}) of our differential equation, according to the application of the Conservation of Energy Principle derivation developed for Torricelli's Law in \cite{Winkel2015}, is
\begin{equation}\label{eq:RHS}
\alpha \pi (r_C)^2\sqrt{2 g h(t)}\, ,
\end{equation}
where $\alpha$ is the effective area or percentage of total  available area for fluid flow due to friction at the opening called the {\it discharge\/} or {\it contraction coefficient\/}) for fluid flow through the small opening at the bottom of the cone; $r_C$ is the radius of the hole on the bottom of our cone; and $g$ is the acceleration due to gravity.

\subsection*{Activity 1}

\begin{itemize}\item [(1)] Justify the differential equation (\ref{eq:de}) for the height  of the water, $h(t)$, in an inverted (point down) right circular cone  with total height $H$ and top radius $R$ is given by:
\begin{equation}\label{eq:de}
\pi \left(h(t)\frac{R}{H}\right)^2 h'(t)= \alpha \pi (r_C)^2\sqrt{2 g h(t)} \, .
\end{equation}
\end{itemize}


\subsection*{Activities 2-5}

Here are the main questions we need to address:

\begin{itemize}
  \item [(2)] Solve the differential equation model (\ref{eq:de}) for $h(t)$.
\item [(3)]
Explain what you have to do to find out how long  it takes the conical container to empty in terms of parameters? What might you do with this information to estimate the parameters $\alpha$?  Explain why this would be very difficult.
\item [(4)] Suppose we had data and measurements. Could we estimate the parameters from collected data?  In the Appendix we offer data collected from \cite{sm2} for use if the reader cannot collect  data.

\item [(5)] Further, what if the circular holes in the containers offer friction or resistance?
\end{itemize} 

\section*{Activities 6 - 8}

\begin{itemize}
  \item [(6)]
Create a mathematical model for a column of falling water based on Torricelli's Law as offered in \cite[pp.~2-4]{Winkel2015} when the container holding the water is an inverted right circular cone of height $H$ and base radius $R$. Here is the set of known working parameters and an unknown constant for our configuration as depicted in the  videos  on YouTube \cite{ sm1, sm2, c3}. These videos are also available for download on YouTube.

\begin{itemize}
\item  top radius of the right circular cone, $R = 4.$ cm;
\item  height of the right circular cone, $H = 9$ cm;
\item  radius of hole at bottom tip of the cone, $r_C = 0.4$ cm;
\item  acceleration due to gravity, $g = 980 {\rm cm}^2/{\rm s}$; and 
\item  the empirically determined resistance percentage, $\alpha$,  i.e.  the percent of the flow  rate of the liquid which actually goes through the hole in the {\it contraction coefficient\/}.
\end{itemize}

\item [(7)] Using the data from one of the videos  on YouTube \cite{sm1, sm2, c3} estimate the parameter $\alpha$.
\item [(8)] Validate your model by comparing your final model using the estimated parameter with your collected data. 
\end{itemize}

\section*{VIDEO RESOURCES}


We have placed three videos  on YouTube \cite{sm1, sm2, c3} which can be used to collect data on the height of a column of water (with blue food dye for visual ease) in a right circular cone (glass funnel) as it falls due to a hole in the bottom of the cone. The dimensions of the inner cone of this glass funnel used in this experiment is 4.95 cm for the radius of the top circular cross section of the cone and a height of 9 cm (roughly from visual inspection) from the top cross section to the ``point'' at the base of the inverted cone. The radius of the hole at the bottom of the cone is 0.4 cm. One can stop and start the video to collect data on the height of liquid as a function of time and build a mathematical model for this process. In this model (using Torricelli's Law as the guiding principle) one can then estimate the main parameter, namely the effective area (as a percent of the cross sectional area of the hole through which the water exits at the bottom of the cone) for fluid flow. That percent or fraction is called the {\it discharge\/} or {\it contraction coefficien\/}.

Videos \cite{sm1, sm2} were taken on 24 February 2016 in Cornwall NY USA using the camera in an iPad 5 and then converted using an app called SloMo so that the user could more readily collect data. Video \cite{c3} was not converted.


\section*{APPENDIX}

In Table~\ref{table:data} we offer data (transformed for ease of use) from an experiment videoed using equipment as depicted in  configuration Figure~\ref{fig:1-55-SetUp} and shown in the YouTube video \cite{sm2}. Data such as this can be collected directly from the video and cleaned up for use in the modeling. However, in the event the user cannot gain access to the video we over a set of data which can serve. 

\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|}\hline
Time (s) & Height (cm)\\ \hline \hline
0.0 & 8.60\\ \hline
0.46 & 8.40\\ \hline
1.30 & 8.10\\ \hline
2.66 & 7.40\\ \hline
3.66 & 7.00\\ \hline
4.56 & 6.50\\ \hline
5.45 & 5.90\\ \hline
6.40 & 5.30\\ \hline
6.96 & 4.60\\ \hline
7.40 & 4.00\\ \hline
7.90 & 3.20 \\ \hline
\end{tabular}


\parbox{9 true cm}{\caption{\label{table:data} Data from \cite{sm2}.}}

\end{center}

\vspace*{-1 true cm}
\end{table} 


\teacher{
\section*{NOTES FOR TEACHERS}

We have given the essentials for the model based on Torricelli's Law derivation found in \cite{Winkel2015}. The key for students is to use some geometry (i.e. ratio or slope of the cone) to determine the element of volume loss per unit time as computed on the  left  hand  side of the differential equation (\ref{eq:de}).

Complete analysis for one set of data taken from \cite{sm2} is offered in the Mathematica notebook, 1-55-T-Mma-WaterFallingInCone.nb and an accompanying .pdf version of the same file.  We offer some convincing evidence from that file here from raw data collection (\ref{fig:1-55-SlowMoCapture2DataNEW}) to comparison of final model to data (\ref{fig:1-55-ModelOverData}). 



\begin{figure}[h]

\hspace*{3.8 true cm}\includegraphics[width=14cm]{1-55-SlowMoCapture2DataNEW.eps}

\vspace*{-13.9 true cm}
\centering
\parbox{11 true cm}{\caption{\label{fig:1-55-SlowMoCapture2DataNEW} Data taken from \cite{sm2}, before being converted and displayed in Table~\ref{table:data} above. Time is in minute:second plus millisecond format while Height is in cm as measured on adjacent meter stick as depicted in Figure~\ref{fig:1-55-SetUp}. \vspace*{-.5 true cm}
}}
\end{figure}

Figure~\ref{fig:1-55-SlowMoCapture2DataNEW} shows data  taken from \cite{sm2}. Time is in minute:second plus millisecond format while Height is in cm as measured on adjacent meter stick. To convert this raw data to usable form for the model offered one needs to subtract off the  time from the first data observation to zero out the time and subtract the estimated height of the bottom vortex of the inverted cone to get $h(t)$, the height of column of water in the cone at each time observation $t$. In our Mathematica notebook, 1-55-T-Mma-WaterFallingInCone.nb, we only entered the times as the number of milliseconds beyond the initial minute:second time recorded which was 1 minute and 36 seconds. 



In Figure~\ref{fig:1-55-ModelOverData} we show the data from our collection in comparison to our final model (\ref{eq:Model}), $h(t)$, for the height of the column of water in the cone obtained by minimizing the sum of square errors between data and model as a function of the parameter $\alpha$.  The final estimate of the parameter $\alpha$  obtained, i.e. $\alpha = 0.49$, is comparable to empirical values offered in the literature \cite{Neutrium2016}.

\begin{equation}\label{eq:Model}
h(t) = \sqrt[5]{813.443 t^2-13861.2 t+59049}
\end{equation}

\begin{figure}[h]
\centering
\includegraphics[width=11.3cm]{1-55-ModelOverData.eps}

\parbox{12 true cm}{\caption{\label{fig:1-55-ModelOverData} Plot of model (\ref{eq:Model}) compared to actual data.
}}
\end{figure}

}




\begin{thebibliography}{99}



\bibitem{sm1} SlowMoCapture 1. 2016. Video. \url{https://youtu.be/RV0reeXxfpw}. YouTube. Accessed 26 February 2016.   

\bibitem{sm2} SlowMoCapture 2. 2016. Video. \url{https://youtu.be/s9CZ-yR-j2Q}. YouTube. Accessed 26 February 2016. 


\bibitem{c3} Capture 3.  2016. Video. \url{https://youtu.be/9CpNqiMpHjU}. YouTube. Accessed 26 February 2016.  


\bibitem{Neutrium2016} Neutrium. 2016. Discharge Coefficient for Nozzles and Orifices. \url{https://neutrium.net/fluid_flow/discharge-coefficient-for-nozzles-and-orifices/}.  Accessed 27 February 2016.

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

\end{thebibliography}
 





\end{document}
