A pump delivers water through two pipes laid in parallel. One pipe is \(100 \mathrm{~mm}\) diameter and \(45 \mathrm{~m}\) long and discharges to atmosphere at a level \(6 \mathrm{~m}\) above the pump outlet. The other pipe, \(150 \mathrm{~mm}\) diameter and \(60 \mathrm{~m}\) long, discharges to atmosphere at a level \(8 \mathrm{~m}\) above the pump outlet. The two pipes are connected to a junction immediately adjacent to the pump and both have \(f=0.008 .\) The inlet to the pump is \(600 \mathrm{~mm}\) below the level of the outlet. Taking the datum level as that of the pump inlet, determine the total head at the pump outlet if the flow rate through it is \(0.037 \mathrm{~m}^{3} \cdot \mathrm{s}^{-1}\). Losses at the pipe junction may be neglected.

The Darcy-Weisbach equation helps us determine the frictional head loss in a pipe due to the flow of fluid. This is essential in fluid mechanics to understand energy losses in systems like pipelines. The equation is given by: \[ h_f = f \times \frac{L}{D} \times \frac{v^2}{2g} \] Where:

• h_f is the head loss due to friction (m)
• f is the Darcy friction factor (dimensionless)
• L is the length of the pipe (m)
• D is the diameter of the pipe (m)
• v is the velocity of the fluid (m/s)
• g is the acceleration due to gravity (\(9.81 \text{ m/s}^2\))

In this exercise, we used the Darcy-Weisbach equation to calculate the head loss due to friction for each of the two pipes in the system. This is a critical step in determining the total head required at the pump outlet.

Volumetric flow rate is a measure of the volume of fluid passing through a cross-section of a pipe per unit time. It is denoted by \(Q\) and expressed in cubic meters per second (\(m^3/s\)). The relationship between volumetric flow rate, cross-sectional area \(A\), and velocity \(v\) is given by: \[ Q = A \times v \] In this exercise, we have a total flow rate of 0.037 \( \text{m}^3/\text{s} \). We used this together with the cross-sectional areas of the two pipes to find the individual flow rates in each pipe. The cross-sectional area \(A\) for a circular pipe is calculated as: \[ A = \frac{\text{π} D^2}{4} \] Where \(D\) is the diameter of the pipe. By splitting the total flow rate into each pipe using the areas, we determined how much water flows through each.

The continuity equation is a fundamental principle in fluid mechanics, stating that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another, assuming incompressible flow. Mathematically, it expresses: \[ Q = Q_1 + Q_2 \] Where:

• Q is the total flow rate
• Q_1 and Q_2 are the flow rates in each individual pipe

In this problem, we used the continuity equation to ensure that the total flow rate of 0.037 \( \text{m}^3/\text{s} \) was split correctly between the two pipes based on their respective areas. This allows for the calculation of the individual velocities and subsequent head losses due to friction.

Head loss due to friction is the loss of energy (or pressure) in a pipe due to the friction between the moving fluid and the pipe walls. This is an essential factor to consider in fluid mechanics systems because it affects the required pump energy. The Darcy-Weisbach equation helps calculate this as previously explained. In our exercise, the calculated head loss due to friction for the 100 mm diameter pipe was 0.296 m, and for the 150 mm diameter pipe, it was 0.375 m. Summarizing, understanding head loss due to friction enables us to determine the total head requirements at the pump outlet by combining these with elevation heads and other losses, if any, to ensure efficient fluid distribution.