A reservoir \(A\), the free water surface of which is at an elevation of \(275 \mathrm{~m}\), supplies water to reservoirs \(B\) and \(C\) with water surfaces at \(180 \mathrm{~m}\) and \(150 \mathrm{~m}\) elevation respectively. From \(A\) to junction \(D\) there is a common pipe \(300 \mathrm{~mm}\) diameter and \(16 \mathrm{~km}\) long. The pipe from \(D\) to \(B\) is \(200 \mathrm{~mm}\) diameter and \(9.5 \mathrm{~km}\) long while that from \(D\) to \(C\) is \(150 \mathrm{~mm}\) diameter and \(8 \mathrm{~km}\) long. The ends of all pipes are submerged. Calculate the rates of flow to \(B\) and \(C\), neglecting losses other than pipe friction and taking \(f=0.01\) for all pipes.

Reservoir dynamics involves understanding how water moves from one reservoir to another. In this problem, water is supplied from a higher elevation reservoir (A) to lower elevation reservoirs (B) and (C). Water flows due to the difference in elevation, which provides the driving force for the flow. The elevations of 275 m, 180 m, and 150 m for reservoirs A, B, and C respectively create an imbalance that drives the water to flow downhill. It's like water naturally following the path of least resistance, seeking to level itself out.

The Darcy-Weisbach equation is a fundamental formula used in fluid mechanics to calculate the head loss due to friction in a pipe. The equation is given by: \[ h_{f} = f \frac{L}{D} \frac{V^2}{2g} \]where \( h_{f} \) is the head loss, \( f \) is the friction factor, \( L \) is the length of the pipe, \( D \) is the diameter of the pipe, \( V \) is the flow velocity, and \( g \) is the acceleration due to gravity (9.81 \( m/s^2 \)). Head loss represents the energy lost as water flows through the pipes due to friction against the pipe walls. By applying the Darcy-Weisbach equation to each pipe section, one can determine how much energy is lost before the water reaches junction D.

The continuity equation is a principle of fluid dynamics that states that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another, as long as there are no additions or losses within the system. It can be expressed as: \[ Q = A \times V \]where \( Q \) is the volumetric flow rate, \( A \) is the cross-sectional area of the pipe, and \( V \) is the flow velocity. In this problem, at junction D, the continuity equation is applied to state that the flow rate from reservoir A (\

Head loss calculation is an essential part of determining the efficiency and effectiveness of a fluid transport system. It accounts for the reduction in total head (or energy) due to frictional forces within the pipe system. Applying the Darcy-Weisbach equation, the head loss for each pipe section is calculated based on the friction factor, the length and diameter of the pipe, and the velocity of the fluid flow. By knowing the overall head loss, engineers can ensure the system is designed to deliver adequate pressure and flow rates where needed. This problem involved calculating the head losses to understand how much energy is lost in transit from reservoir A to reservoirs B and C.