Problem 3 (35 points)
Here is a power extraction system from a water tower with a depth of 1000 ft . The upper surface of the water tower is a free surface. There are two circular pipes: one connecting the water tower to the turbine, with a diameter of 0.1 ft and a length of 100 ft . The other one (with a diameter of 0.2 ft and a length of 200 ft ) connects the turbine and a leaky piston moving at a speed \( \boldsymbol{U}_{\boldsymbol{p}}=\mathbf{1} \boldsymbol{f} \boldsymbol{t} \boldsymbol{/} \boldsymbol{s} \). The hole diameter is 0.05 ft . A force is exerted on the piston to keep it moving at a constant speed. The turbine is connected to a power grid with efficiency \( \boldsymbol{\eta}=\mathbf{5 0} \% \), and the power grid has an electrical power head \( \boldsymbol{h}_{\text {electric }}=\mathbf{2 f t} \). The pipes' roughness \( \boldsymbol{\varepsilon}=\mathbf{0 . 0 0 2} \boldsymbol{f} \). The flow within the pipe has a volumetric flow rate of \( \dot{\boldsymbol{V}}=\mathbf{0 . 2} \boldsymbol{f t}^{3} / \boldsymbol{s} \). The minor losses \( \boldsymbol{K}_{\boldsymbol{L}} \) are listed in the figure.
1. Determine \( P_{1}, P_{2}, P_{3}, P_{4}, P_{5} \), in psi. Pressure is measured right before their corresponding minor
\( -25 \) loss at the locations. [3 points each; 15 points in total]
2. Determine the leaky flow speed on the piston \( U_{\text {hole }} \) in \( f t / s \). [5 points]
3. Determine where the Bernoulli's equation can be applied in this system. [ 5 points]
4. Determine the force exerted on the piston in lbf. [10 points]
Hint: no iteration is required to solve these problems. The water tower can be regarded as infinitely wide, such that the depth \( H \) does not change with time. The system can be regarded as in a steady state. Water can be regarded as incompressible. Gravitational acceleration and viscosity can be regarded as constants. You may assume correction factors \( \alpha=\beta=1 \) for both pipes. Use gauge pressure only.