To solve this problem, we need to calculate the flow rate in pipe 2 ($Q_2$) for a system driven by gravity with turbulent flow. Here’s the breakdown of the approach:

Key Steps:

1. Determine the Head Loss in Pipes (Using Energy Balance): Since gravity drives the flow between reservoirs, there’s a potential energy difference based on elevation heights ($z_1, z_2, z_3$). The head loss due to friction must equal the energy difference between reservoirs.

2. Colebrook–White Equation for Friction Factor: To find the friction factor $f_d$ in each pipe, we’ll use the Colebrook–White equation: $\frac{1}{\sqrt{f_d}} = -2 \log_{10} \left( \frac{\epsilon}{3.7D} + \frac{2.51}{\text{Re} \sqrt{f_d}} \right)$ This equation is implicit, meaning $f_d$ needs to be solved iteratively, as it appears on both sides.

3. Reynolds Number Calculation: For each pipe, calculate the Reynolds number $\text{Re} = \frac{\rho V D}{\mu}$ where $V$ is the flow velocity, $D$ is the diameter, $\rho$ is the fluid density, and $\mu$ is the dynamic viscosity. Assuming the fluid properties are known, this will help us in the iterative solution for $f_d$.

4. Flow Balance at Junction $A$: Since we have three pipes connected at junction $A$, the sum of inflows must equal the sum of outflows. If we denote flows in pipes as $Q_1, Q_2, Q_3$, then: $Q_1 + Q_3 = Q_2$ Here, we will determine flow directions based on positive or negative flow rates relative to the junction.

5. Iterative Solution for $Q_2$: With the friction factors calculated for each pipe, use the energy and continuity equations to solve for $Q_2$ iteratively until it converges to a stable value.

Given Parameters:

• Lengths $L_1 = 125 \, \text{m}, L_2 = 25 \, \text{m}, L_3 = 125 \, \text{m}$
• Elevations $z_1 = 1275 \, \text{m}, z_2 = 1000 \, \text{m}, z_3 = 100 \, \text{m}$
• Diameters $D_1 = 0.2 \, \text{m}, D_2 = 0.2 \, \text{m}, D_3 = 0.15 \, \text{m}$
• Roughness $\epsilon_1 = 1 \, \text{mm}, \epsilon_2 = 0.25 \, \text{mm}, \epsilon_3 = 0.5 \, \text{mm}$

Next Steps:

1. Assume an initial guess for $f_d$ and $Q_2$ to start the iteration.
2. Use the Colebrook–White equation to refine the friction factors based on the guessed flow rate.
3. Apply the energy balance and continuity conditions to update the flow rates.
4. Iterate until convergence for $Q_2$ to three significant figures.

Would you like a more detailed walkthrough of any specific part of this process, or have any questions?

Here are some further questions related to the topic:

1. How would you use the Moody chart if the Colebrook–White equation wasn't provided?
2. What are the implications of assuming all reservoirs are "wide" in this scenario?
3. How does the choice of initial guess affect the convergence of iterative methods?
4. What are some common methods to numerically solve implicit equations like the Colebrook–White equation?
5. Why do we ignore minor losses in this analysis, and how would including them affect the result?

Tip: When iterating to find the friction factor, use the previous iteration's value as the starting point for the next. This often speeds up convergence.