A tank of plan area \(5 \mathrm{~m}^{2}\), open to atmosphere at the top, is supplied through a pipe \(50 \mathrm{~mm}\) diameter and \(40 \mathrm{~m}\) long which enters the base of the tank. A pump, providing a constant gauge pressure of \(500 \mathrm{kPa}\), feeds water to the inlet of the pipe which is at a level \(3 \mathrm{~m}\) below that of the base of the tank. Taking \(f\) for the pipe as \(0.008\), calculate the time required to increase the depth of water in the tank from \(0.2 \mathrm{~m}\) to \(2.5 \mathrm{~m}\). If the combined overall efficiency of pump and driving motor is \(52 \%\), determine (in \(\mathrm{kW} \cdot \mathrm{h})\) the total amount of electricity used in this pumping operation.

The Darcy-Weisbach equation is vital in fluid mechanics for estimating head loss due to friction in pipes. It is represented as: \[ h_f = f \frac{L}{D} \frac{v^2}{2g} \] where:
- \( h_f \): Head loss due to friction (m or ft)
- \( f \): Friction factor (dimensionless)
- \( L \): Length of the pipe (m or ft)
- \( D \): Diameter of the pipe (m or ft)
- \( v \): Flow velocity of the fluid (m/s or ft/s)
- \( g \): Acceleration due to gravity (9.81 m/s² or 32.2 ft/s²)
To calculate the head loss, you need the pipe's diameter, length, fluid velocity, and friction factor. The friction factor \( f \) is often determined experimentally or found using the Moody chart, depending on the Reynolds number and pipe roughness. For the problem at hand, knowing these parameters allows the calculation of head loss, which impacts the system's total head.

Pump efficiency is a crucial metric that determines how well a pump converts electrical energy into hydraulic energy. Mathematically, it can be written as: \[ \text{Efficiency} (\boldsymbol{\text{η}}) = \frac{\text{Hydraulic Power Output}}{\text{Electrical Power Input}} \times 100% \]
For example, if a pump has an efficiency of 52%, it means that only 52% of the electrical energy is converted into useful work in lifting water, while the rest is lost to factors like heat and friction. In the exercise, calculating the efficiency of the pumping system helps determine the total energy consumption. This efficiency metric is critical for designing energy-efficient systems and estimating operational costs accurately.

Flow rate is a measure of the volume of fluid passing through a cross-section of a pipe per unit time. It is usually denoted by \( Q \) and can be calculated using the formula: \[ Q = A \times v \] where:
- \( Q \): Flow rate (m³/s or ft³/s)
- \( A \): Cross-sectional area of the pipe (m² or ft²)
- \( v \): Flow velocity (m/s or ft/s)
In the given problem, calculating the flow rate requires finding the area first using the diameter of the pipe: \[ A = \frac{\boldsymbol{\text{π}}}{4} D^2 \]
Then, we use this area to find the velocity \( v \), which in turn helps determine the flow rate. Accurate flow rate calculation is indispensable for estimating the time required to fill the tank from one depth to another.

Head loss refers to the energy lost due to friction and other resistance factors as fluid flows through a pipe. It is a key consideration in designing piping systems and pumps. Factors contributing to head loss include pipe length, diameter, flow speed, and the fluid's viscosity.
There are two main types of head loss:

• **Major Head Loss:** Caused by friction within the pipe's walls, primarily determined by the Darcy-Weisbach equation.
• **Minor Head Loss:** Caused by fittings, bends, valves, and other pipeline components.

In the example, calculating head loss due to friction allows us to understand the total head rise needed. Adding this to the height difference and the pump’s head helps determine the effective energy conversion required to pump water to the desired level in the tank. Properly accounting for head loss ensures that the system operates efficiently, avoiding under or over-sizing of pumps and pipes.