Water flows from left to right in a pipe. On the left the diameter is \SI{8}{\centi\metre}, the water comes in with a uniform velocity of \SI{1,5}{\metre\per\second}. The diameter increases gently until it reaches \SI{16}{\centi\metre}; the expansion is smooth, so that losses (specifically, energy losses due to wall friction and flow separation) are negligible.

\begin{enumerate}

\item What are the mass and volume flows across the pipe?

\item What is the average velocity of the water at the right end of the expansion?

\item What is the pressure change in the water across the expansion?

\end{enumerate}

The volume flow of water in the pipe is now doubled.

\begin{enumerate}

\shift{3}

\item What is the new pressure drop?

\end{enumerate}

The water is drained from the pipe, and instead, air with density \SI{1,225}{\kilogram\per\metre\squared} is flowed in the pipe, incoming with a uniform velocity of \SI{1,5}{\metre\per\second}.

\begin{enumerate}

\shift{4}

\item What is the new pressure drop?

\end{enumerate}

%%%%

\subsubsection{Pipe flow with losses}

\wherefrom{\cczero\oc}

\label{exo_pipe_with_losses}

Water flows in a long pipe which has constant diameter; a valve is installed in the middle of the pipe length. Water comes in the pipe with a uniform velocity of \SI{1,5}{\metre\per\second} and the pipe diameter is \SI{250}{\milli\metre}. The heat capacity of the water is \SI{1}{\kilo\joule\per\kilogram\per\kelvin}.

The pipe itself and the valve, together, induce a pressure loss which can be quantified using the \vocab{loss coefficient}$K_\text{valve}$ (we will study this as eq.~\ref{eq_def_loss_coeff} p.\pageref{eq_def_loss_coeff}). With this tool, the pressure loss is related to the average incoming speed $V_\text{incoming}$ as:

\item What is the pressure drop of the water across the pipe?

\item What is the power required to pump the water across the pipe?

\item If the heat losses of the pipe and valve are negligible, what is the temperature increase of the water?

\end{enumerate}

%%%%

\subsubsection{Combustor from a jet engine}

\wherefrom{\cczero\oc}

\label{exo_combustor}

A jet engine is equipped with several \vocab{combustors} (sometimes also called \vocab{combustion chambers}). We are interested in fluid flow through one such combustor. Air from the compressor enters the combustor, is mixed with fuel, and combustion occurs, which greatly increases the temperature and specific volume of the mix, before it is run through the turbine.

The conditions at inlet are as follows:

\begin{itemize}

\item Air mass flow: \SI{0,2}{\kilogram\per\second};

\item Air properties: \SI{20}{\bar}, \SI{240}{\degreeCelsius}, \SI{12}{\metre\per\second}

\item Fuel mass flow: \SI{1}{\milli\gram\per\second}.

\end{itemize}

At the outlet the conditions are as follows:

\begin{itemize}

\item Gas properties: \SI{20}{\bar}, \SI{1800}{\degreeCelsius}

\end{itemize}

We consider that the air and gas keep the same thermodynamic properties throughout ($c_\text{p}=\SI{1050}{\joule\per\kilogram}$)

\begin{enumerate}

\item What is the volume flow at inlet and at outlet?

\item What is the flow velocity at outlet?

\item What is the power provided to the flow as heat?

\item What is the net force exerted on the gas as it travels through the combustor?

\end{enumerate}

%%%%

\subsubsection{Water jet on a wall}

\wherefrom{White \smallcite{white2008} Ex3.9}

\label{exo_water_jet}

A horizontal water jet hits a vertical wall and is split in two symmetrical vertical flows (\cref{fig_water_wall}). The water nozzle has a~\SI{3}{\centi\metre\squared} cross-sectional area, and the water speed at the nozzle outlet is $V_\text{jet}=\SI{20}{\metre\per\second}$.\\

\begin{figure}

\begin{center}

\includegraphics[width=7.5cm]{nozzle_plate}

\vspace{-0.5cm}

\end{center}

\supercaption{A water jet flowing out of a nozzle (left), and impacting a vertical wall on the right.}{\wcfile{Nozzle flow case 1.svg}{Figure}\cczero\oc}

\label{fig_water_wall}

\end{figure}

\begin{enumerate}

\item What is the net force exerted on the water by the wall?

\item What is the net force exerted on the water by the wall?

\end{enumerate}

Now, the wall moves longitudinally in the same direction as the water jet, with a speed $V_\text{wall}=\SI{15}{\metre\per\second}$. This conceptual setup allows us to approach the case where water acts on the blades of a turbine.

\begin{enumerate}

\shift{1}

\item What is the new force exerted by the water on the wall?

\item How would the force be modified if the volume flow was kept constant, but the diameter of the nozzle was reduced? (briefly justify your answer, e.g. in 30 words or less)

\end{enumerate}

%%%%

\subsubsection{High-speed gas flow}

\wherefrom{\cczero\oc}

\label{exo_high_speed_gas_flow}

Scientists build a very high speed wind tunnel. For this, they build a large compressed air tank. Air escapes from the tank into a pipe which decreasing cross-section. The pipe diameter reaches a minimum (at the tunnel \vocab{throat}), and then it expands again, before discharging into the atmosphere.

For simplicity, we assume that heat losses through the tunnel walls are negligible, and that the fluid has uniformly-distributed velocity in cross-sections of the pipe.

In the tank, the air is stationary, with pressure \SI{12}{\bar} and temperature \SI{200}{\degreeCelsius}.

At the throat, the pressure and temperature have dropped to \SI{2,1}{\bar} and \SI{-10}{\degreeCelsius}. The throat cross-section is \SI{0,04}{\metre\squared}.

\begin{enumerate}

\item What is the mass flow through the tunnel?

\item What is the Mach number in the tank and at the throat?

\item What is the net force exerted on the fluid between the tank and the throat?

\item What is the kinetic energy per unit mass of the air at the throat?

\end{enumerate}

Downstream of the throat, the pressure keeps dropping. By the time it reaches a point A where the cross-section is \SI{0,08}{\metre\squared}, the air has seen its pressure and temperature drop to \SI{0,8}{\bar} and \SI{-45}{\degreeCelsius}.

\begin{enumerate}

\shift{4}

\item What is the fluid velocity at point A?

\item What is the Mach number at point A?

\item What is the net force exerted on the fluid between the throat and point A?

\item What is the kinetic energy per unit mass of the air at point A?

\end{enumerate}

Once it has passed point A, the air undergoes complex loss-inducing evolutions (including going through a \vocab{shock wave}, where its properties change very suddenly), before it exits to the atmosphere with pressure \SI{1}{\bar}. As it exits to the atmosphere, the tunnel cross-section is \SI{0,09}{\metre\squared} and the tunnel air temperature is \SI{12}{\degreeCelsius}.

\begin{enumerate}

\shift{8}

\item What is the fluid velocity at outlet?

\item What is the Mach number at outlet?

\item What is the net force exerted on the fluid between section A and the outlet?

\item What is the kinetic energy per unit mass of the air at the outlet?

\end{enumerate}

\begin{comment}

\item [\ref{exo_water_jet}]%

\tab 1) $F_\text{net on water}=\SI{-120}{\newton}$;

\tab 2) $\vec F_\text{water/wall}=-\vec F_\text{net on water}$;

\tab 3) $F_\text{net on water}=\SI{-7,5}{\newton}$;

A horizontal water jet hits a vertical wall and is split in two symmetrical vertical flows (\cref{fig_water_wall}). The water nozzle has a~\SI{3}{\centi\metre\squared} cross-sectional area, and the water speed at the nozzle outlet is $V_\text{jet}=\SI{20}{\metre\per\second}$.\\

In this exercise, we consider the simplest possible case, with the effects of gravity and viscosity neglected. In that case, the water is perfectly split into two vertical jets of identical mass flow, the speed is identical everywhere, and the velocity distributions are uniform.

\begin{figure}

\begin{center}

\includegraphics[width=7.5cm]{nozzle_plate}

\vspace{-0.5cm}

\end{center}

\supercaption{A water jet flowing out of a nozzle (left), and impacting a vertical wall on the right.}{\wcfile{Nozzle flow case 1.svg}{Figure}\cczero\oc}

\label{fig_water_wall}

\end{figure}

\begin{enumerate}

\item What is the velocity of each water flow as it leaves the wall?

\item What is the force exerted by the water on the wall?

\end{enumerate}

Now, the wall moves longitudinally in the same direction as the water jet, with a speed $V_\text{wall}=\SI{15}{\metre\per\second}$.

\begin{enumerate}

\shift{2}

\item What is the new force exerted by the water on the wall?

\item What is the velocity of each water jet as it leaves the wall, relative to the nozzle?

\item How would the force be modified if the volume flow was kept constant, but the diameter of the nozzle was reduced? (briefly justify your answer, e.g. in 30 words or less)

\end{enumerate}

%%%%

...

...

@@ -406,10 +379,6 @@ Change in angular momentum:

\tab 1) $\dot m =\SI{1,0603}{\kilogram\per\second}$