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\lhead{CE 3105 -- Fluid Mechanics Laboratory}
\rhead{SPRING 2024}
\lfoot{CE 3105 -- Cleveland}
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\rfoot{REVISED: 22 APR 2024}
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\begin{document}
\section*{\center{ { CE 3105 -- Fluid Mechanics Laboratory} {Final Exam} } }
\begin{itemize}
\item Please read through the entire exam before you begin - notice there are blank pages to show your work, so make it legible and follow a logical problem solving protocol.
\item Write your name on \textbf{each} sheet before beginning to work the exam.
\end{itemize}
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\section*{Question 1}
What are the names of your laboratory team members:
\begin{enumerate}
\item TEAM MEMBER 0 (YOU): $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item TEAM MEMBER 1: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item TEAM MEMBER 2: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item TEAM MEMBER 3: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item TEAM MEMBER 4: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item TEAM MEMBER 5: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\end{enumerate}
\clearpage
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\section*{Question 2}
Before each laboratory you received a safety briefing.  What were three safety reminders before each laboratory?
\begin{enumerate}
\item REMINDER 1: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item REMINDER 2: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\item REMINDER 3: $\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}$
\end{enumerate}
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\clearpage
\section*{Question 3}
A metal sphere measured using a dial-caliper (in inches) is pictured. The same sphere is weighed using a digital scale (in grams). 
\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.75in]{sphere-dimension.png} 
   \caption{Sphere in dial-caliper instrument reading diameter in inches}
   \label{fig:sphere-dimension}
\end{figure}

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=2.75in]{sphere-mass.png} 
   \caption{Sphere on scale reading mass in grams}
   \label{fig:sphere-mass}
\end{figure}
\newpage
Determine:
\begin{enumerate}
\item The volume of the sphere in cubic meters.
\item The density of the sphere in $\frac{kg}{m^3}$
\end{enumerate}
\clearpage
Show work here
\clearpage
\section*{Question 4}
The sphere (color coated for visibility) is used to estimate viscosity for an unknown liquid as depicted in the photographs below. A 50 ml sample of the light amber liquid has a mass of 71.5 grams at 20$^o$C. The time required for the sphere to traverse 127 mm was observed to be 3.9 seconds.
\begin{figure}[h!]
\centering
\begin{subfigure}{.3\textwidth}
  \centering
  \includegraphics[width=.4\linewidth]{drop.png}
  \caption{Dropping sphere}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.3\textwidth}
  \centering
  \includegraphics[width=.4\linewidth]{begin.png}
  \caption{Time = 0 sec.}
  \label{fig:sub2}
\end{subfigure}
\begin{subfigure}{.3\textwidth}
  \centering
  \includegraphics[width=.4\linewidth]{end.png}
  \caption{Time = 3.9 sec.}
  \label{fig:sub2}
\end{subfigure}
\caption{Spherical object for viscosity measurement by Stokes Law (Laboratory 1)}
\label{fig:test}
\end{figure}
Determine:
\begin{enumerate}
\item The density of the unknown liquid (in $\frac{g}{ml}$)
\item The viscosity of the unknown liquid (in Pa$\cdot$s)
\end{enumerate}
\clearpage
Show work here
\clearpage
\section*{Question 5}
The following discharge data and head change were obtained using the apparatus in the photograph below:

\begin{table}[h!]
    \centering
    \caption{Pipe head loss apparatus (Laboratory 4)}
    \begin{tabular}{p{1in} p{1in} p{1in}}
    Volume (mL) & Time (s) & $\Delta h$ (m) \\
    \hline
    122&8.36&1.34\\
    138&6.67&2.56\\
    180&7.9&3.09\\
    205&8.03&3.98\\
    217&8.26&3.92\\
    \end{tabular}
    \label{tab:table1}
\end{table}

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=4in]{labsetup.png} 
   \caption{Pipe head loss apparatus (Laboratory 4)}
   \label{fig:labsetup}
\end{figure}

Determine:
\begin{enumerate}
\item The discharge rate for each of the 5 measurements.
\item The flow velocity in the 508 mm long, 3-mm diameter, brass tube for each of the 5 measurements.
\item The Reynolds number for each of the 5 measurements.
\item The Darcy-Weisbach friction factor for each of the 5 measurements.
\item The flow regime (laminar, transitional, or turbulent) in each experiment?
\item Plot (sketch) the friction factor versus Reynolds number for these data.
\end{enumerate}
\clearpage
Show work here
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Show work here
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\section*{Question 6}
A small piece of volcanic ejecta from Amboy Crater is at the front of
the classroom

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
   \centering
\includegraphics[width=2in]{amboy.png}
\end{figure}

The mass of the object is 23.93 grams. The porosity of typical pumice is
\(\eta = 64–85\%\) by volume
\url{https://en.wikipedia.org/wiki/Pumice}

The ellipsoid method to approximate volume uses 3 measurements
\(A\),\(B\), and \(C\), called semi-axes.

\begin{figure}[htbp] %  figure placement: here, top, bottom, or page
\centering
\includegraphics[width=2in]{ellipsoid.png}
\end{figure}

\(V = \frac{4}{3} * \pi * A * B * C\)

The porosity of typical pumice is \(\eta = 64–85\%\) by volume. The porosity can
be used to approximate the solids volume from the expression:

\(V_{total} \cdot (1-\eta) \approx V_{solids}\)

Determine: 
\begin{enumerate}
\item An estimate of the volume of the irregular shaped object (in milliliters). 
\item An estimate of the solids volume, based on a porosity of \(64\%\) 
\item If the object will float in air.
\item If the object will float in water.
\item If the object will float in glycerine.
\item Write an experimental procedure to obtain definitive answers to the three previous (will it float in ...) questions.  
\end{enumerate}
\clearpage
Show work here
\clearpage
Show work here
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\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{sphere-dimension.png} 
   \caption{Sphere in dial-caliper instrument reading diameter in inches}
   \label{fig:sphere-dimension}
\end{figure}
\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{sphere-mass.png} 
   \caption{Sphere on scale reading mass in grams}
   \label{fig:sphere-mass}
\end{figure}

\begin{figure}
\centering
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.4\linewidth]{sphere-dimension.png}
  \caption{Sphere in dial-caliper instrument reading diameter in inches}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
  \centering
  \includegraphics[width=.4\linewidth]{sphere-mass.png}
  \caption{Sphere on scale reading mass in grams}
  \label{fig:sub2}
\end{subfigure}
\caption{Photographs of spherical object for viscosity measurement}
\label{fig:test}
\end{figure}

\begin{figure}[h!] %  figure placement: here, top, bottom, or page
   \centering
   \includegraphics[width=3in]{circlegate.png} 
   \caption{}
   \label{fig:circlegate}
\end{figure}