$(25\%)$ The steady flow hydroelectric turbine shown here converts the potential and kinetic energies of the dam into electricity. The surface of the dam and the turbine exit flow are at atmospheric conditions.

$\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{t}}{\displaystyle \underset{cv}{\overset{\phantom{}}{}}}edY+{\displaystyle \underset{cx}{\overset{\phantom{}}{}}}((u^{})+\frac{p}{}+\frac{V^2}{2}+gz)V*hat(n)dA=Q_{\{\begin{array}{ccc}\text{:}[& \text{n}\text{a}\text{t}& ]\\ & in\end{array}}^{}+w_{\{\begin{array}{ccc}\text{:}[& \text{s}\text{h}\text{a}\text{f}& ]\\ & \text{n}\text{e}\text{t}& in\end{array}}^{}$

$4($a$)(15\%)$ Starting with the energy equation provided here, show that for this system, the turbine work can be expressed as:

$h_T=h-\frac{v_2^2}{2g}-h_l$

$h_T$ is the turbine head in $m$

$h$ is the depth of the water to the center of the turbine inlet pipe in $m$

$h_L$ is the head loss in $m$

$V_2$ is the turbine exit flow velocity in $\frac{m}{s}$

Provide a neat and well$-$labeled sketch of a suitable control volume for the problem. State all assumptions made and show the step$-$by$-$step approach to your solution; no short cuts!

$4($b$)(10\%)$ Determine the turbine power, $W^{}$ for $h=10m$ and $d=0\mathrm{.}5m.$ Volume flow rate is $1\frac{m^3}{s}.$ Assume ideal turbine operation. State all assumptions and apply units and unit conversion ratios.

$5(25\%)$ A thin rectangular plate having a width $w$ and a height $h$ is located normal to a moving stream of fluid. Assume the drag $D$ that the fluid exerts on the plate is a function of $w$ and $h,$ fluid viscosity $,$ fluid density $p,$ and the velocity $V$ of the fluid approaching the plate.

$5($a$)(5\%)$ Provide a neat and well$-$labeled diagram for this problem.

$5($b$)(15\%)$ Using the M$-$L$-$T system of basic dimensions, determine a suitable set of dimensionless paramete for the flow problem. Show the step$-$by$-$step approach to your solution; no short cuts.

$5($c$)(5\%)$ Closely inspect the formulated pi$-$terms and state the fluid parameter that each of the pi$-$term represents, such as ratios, numbers, coefficients, etc.