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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?

The water transmission system in the figure is designed to supply water from a dam reservoir to a city and generate electricity through a turbine system. Calculate the design flows in each pipe while satisfying the velocity constraints and ensuring 6 MW of power generation.

This problem involves calculating the flow rates through pipes to meet velocity constraints while generating 6 MW of electricity from a dam's turbine. Using fluid dynamics concepts, this problem addresses pipe design, head loss, and turbine efficiency.

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