The problem you have provided deals with designing a water transmission system that satisfies two objectives: supplying water downstream and generating 6 MW of electricity through a turbine system. The flow rates in the pipes must be determined to satisfy given velocity constraints and also ensure the turbine produces the required power.

To summarize:

Given Data:

• Water characteristics:
  • Density $\rho = 998 \, \text{kg/m}^3$
  • Dynamic viscosity $\mu = 1 \times 10^{-3} \, \text{Pa·s}$
• Turbine power requirement:
  • $P = 6 \, \text{MW} = 6 \times 10^6 \, \text{W}$
  • Efficiency $\eta = 0.95$
  • Elevation of dam $Z_1 = 1069 \, \text{m}$
  • Elevation of reservoir 2 $Z_2 = 150 \, \text{m}$
  • Base reference $Z_3 = 0 \, \text{m}$
• Velocity constraints:
  • Minimum velocity $V_{\text{min}} = 0.75 \, \text{m/s}$
  • Maximum velocity $V_{\text{max}} = 2 \, \text{m/s}$

Pipe Data:

1. Pipe 1: $L = 3092 \, \text{m}$, $D = 1520 \, \text{mm}$, Material: Cast iron, Roughness $\epsilon = 0.26 \, \text{mm}$
2. Pipe 2: $L = 1551 \, \text{m}$, $D = 919 \, \text{mm}$, Material: Galvanized iron, Roughness $\epsilon = 0.15 \, \text{mm}$
3. Pipe 3: $L = 1915 \, \text{m}$, $D = 1607 \, \text{mm}$, Material: Welded steel, Roughness $\epsilon = 0.045 \, \text{mm}$

Formulas to use:

1. Turbine power equation: $P = \eta \cdot \gamma \cdot h_T \cdot Q$ where:

   • $P$ is the power (6 MW),
   • $\eta = 0.95$ (efficiency),
   • $\gamma = \rho \cdot g$ (specific weight of water),
   • $h_T = Z_1 - Z_2$ (gained head by the turbine),
   • $Q$ is the flow rate to be determined.

2. Flow rate ($Q$) related to velocity ($V$): $Q = V \cdot A = V \cdot \left(\frac{\pi D^2}{4}\right)$ where $A$ is the pipe's cross-sectional area.

3. Reynolds number: $\text{Re} = \frac{\rho V D}{\mu}$ to check for turbulent or laminar flow.

4. Friction losses (Darcy-Weisbach): $h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$ where $f$ is the Darcy friction factor, and it can be estimated using the Moody chart or Colebrook equation.

Steps to solve:

1. Calculate the total head gained by the turbine: $h_T = Z_1 - Z_2 = 1069 \, \text{m} - 150 \, \text{m} = 919 \, \text{m}$.

2. Determine the required flow rate $Q$ using the turbine power equation: $P = \eta \cdot \gamma \cdot h_T \cdot Q$ and solve for $Q$.

3. Check velocity constraints: Ensure that the flow rates in each pipe satisfy the given velocity limits ($V_{\text{min}}$ and $V_{\text{max}}$).

4. Check friction losses using the Darcy-Weisbach equation and calculate the head loss for each pipe.

Let me proceed with the first few steps to calculate the required flow rate $Q$. I'll start by determining $Q$ from the turbine power equation.The required flow rate $Q$ to generate 6 MW of power is approximately $0.702 \, \text{m}^3/\text{s}$.

Next, we need to distribute this flow across the three pipes while ensuring the velocity constraints of $V_{\text{min}} = 0.75 \, \text{m/s}$ and $V_{\text{max}} = 2 \, \text{m/s}$ are satisfied for each pipe.

Would you like me to proceed with the velocity calculations in each pipe, considering their diameters and flow rates?