An \(800 \mathrm{~mm}\) diameter pipe \(1200 \mathrm{~m}\) long supplies water at \(1.1 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) to a power plant. A simple, open, cylindrical surge tank \(2.5 \mathrm{~m}\) in diameter is connected \(120 \mathrm{~m}\) upstream of the turbines. The base of the surge tank is \(15 \mathrm{~m}\) below the water level in the reservoir. Neglecting entrance and other losses in the surge tank and taking \(f\) constant at \(0.007\), estimate the height of tank required to cope with instantaneous complete shut-down of the turbines without overflowing.

Fluid mechanics is a branch of physics that studies the behavior of fluids (liquids and gases) and the forces acting on them. It encompasses both fluid statics (the study of fluids at rest) and fluid dynamics (the study of fluids in motion). Principles of fluid mechanics are essential in many fields, including hydraulic engineering, which deals with the flow and conveyance of fluids. Understanding how water moves through pipes, how pressure changes, and how fluids can be controlled is key to solving problems like the surge tank design mentioned in this exercise.
In the given scenario, fluid mechanics principles help us understand the movement of water through the pipeline and the effect of a sudden shutdown on the turbine. Concepts like flow rate, fluid velocity, and pressure differences are foundational to determining how high the surge tank needs to be.

The Darcy-Weisbach equation is a fundamental equation in fluid mechanics used to calculate the frictional head loss due to fluid flow in a pipe. The equation is:
\[ h_f = f \frac{L}{D} \frac{V^2}{2g} \]
Where:

• \( h_f \) = frictional head loss
• \( f \) = friction factor
• \( L \) = length of the pipe
• \( D \) = diameter of the pipe
• \( V \) = velocity of the fluid
• \( g \) = acceleration due to gravity

The friction factor (\( f \)) depends on the pipe's material and the flow regime (laminar or turbulent). Typically, it is determined through empirical correlations or by using the Moody chart.
In our exercise, we used the Darcy-Weisbach equation to calculate the frictional head loss (\( h_f \)) in the pipeline, with known values for the friction factor, pipe length, diameter, and velocity. This value is crucial for determining the height of the surge tank needed to manage the instant energy change upon turbine shutdown.

Frictional head loss occurs when a fluid flows through a pipe and frictional forces between the fluid and the pipe's walls resist the flow. This results in a loss of pressure and a reduction in hydraulic head (the height of fluid energy).
We calculate the frictional head loss using the Darcy-Weisbach equation mentioned earlier, which factors in the length and diameter of the pipe, the velocity of the fluid, and the friction factor. This loss is the energy the fluid 'loses' while overcoming the frictional forces inside the pipeline.
In our given problem, the frictional head loss calculated was approximately 3.43 meters. This value is added to the initial elevation difference to determine the total height required for the surge tank. The calculation shows that the surge tank must account for the energy dissipated due to friction within the pipeline.

Hydraulic engineering focuses on the flow and conveyance of fluids, primarily water and sewage. It includes the design and management of structures such as pipelines, water supply networks, and surge tanks.
Surge tanks are crucial in hydraulic systems to manage changes in pressure. They absorb sudden rises in pressure and provide a backup supply of water. In power plants, surge tanks prevent pipeline damage by coping with rapid shutdowns. The height of the surge tank must be designed to handle the maximum pressure changes without overflowing.
In the exercise, we calculated that the surge tank height needed to cope with an instantaneous shutdown was about 18.43 meters. This height ensures the tank can accommodate the pressure surges and water volumes without causing structural or operational issues. Designing such systems requires a deep understanding of both fluid dynamics and structural principles.