A reservoir \(A\) feeds two lower reservoirs \(B\) and \(C\) through a single pipe \(10 \mathrm{~km}\) long, \(750 \mathrm{~mm}\) diameter, having a downward slope of \(2.2 \times 10^{-3}\). This pipe then divides into two branch pipes, one \(5.5 \mathrm{~km}\) long laid with a downward slope of \(2.75 \times\) \(10^{-3}\) (going to \(\left.B\right)\), the other \(3 \mathrm{~km}\) long having a downward slope of \(3.2 \times 10^{-3}\) (going to \(\left.C\right)\). Calculate the necessary diameters of the branch pipes so that the steady flow rate in each shall be \(0.24 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\) when the level in each reservoir is \(3 \mathrm{~m}\) above the end of the corresponding pipe. Neglect all losses except pipe friction and take \(f=0.006\) throughout.

The Darcy-Weisbach equation is fundamental in fluid mechanics for calculating head loss in a pipe. Head loss is the reduction in the total head (sum of potential energy, pressure energy, and kinetic energy per unit weight of fluid). The equation is given by: \ \[ h_f = f \frac{L}{D} \frac{V^2}{2g} \] ewline Here, \textbf{f} is the friction factor, \textbf{L} is the length of the pipe, \textbf{D} is the diameter, \textbf{V} is the flow velocity, and \textbf{g} is the acceleration due to gravity. To use this equation, the velocity \( V \) must be calculated using the flow rate \( Q \) and the cross-sectional area \( A \) of the pipe, where \ \[ A = \frac{\pi D^2}{4} \]. ewline Once you have all these variables, you can substitute them into the Darcy-Weisbach equation to find the head loss.

Head loss in a pipe system due to friction can significantly impact the design and efficiency of fluid transport systems. It is necessary to calculate the head loss to ensure proper flow rates and sufficient pressure. For instance, in the problem of calculating the necessary diameters for the pipes leading from reservoir A to reservoirs B and C, head loss had to be considered to meet specific flow requirements. The necessary steps involve: \ 1. Determining the flow velocity using the flow rate and the pipe's cross-sectional area. ewline2. Applying the Darcy-Weisbach equation: \ \[ h_f = f \frac{L}{D} \frac{V^2}{2g} \] ewline The goal is to find a balance where the head loss does not exceed the pressure head, ensuring steady flow to the reservoirs.

Pipe friction refers to the resistance to fluid flow within a pipe, caused by the interactions between the fluid and the pipe's inner surface. This friction results in energy loss, often expressed as head loss. The friction factor \( f \) in the Darcy-Weisbach equation accounts for this resistance and depends on factors like: ewline \begin{itemize} \item Pipe material and roughness - rougher pipes increase friction ewline \item Flow rate and velocity of fluid - higher flow rates increase turbulence and friction ewline \item Pipe diameter - smaller diameters lead to higher friction \end{itemize} Understanding pipe friction is crucial for designing efficient piping systems as it helps in selecting appropriate pipe sizes and materials to minimize energy losses while maintaining required flow rates.