To calculate the power produced by the turbine, we use the First Law of Thermodynamics for a control volume operating at steady state:

$\dot{Q} - \dot{W} = \dot{m} \left( h_2 - h_1 + \frac{v_2^2}{2} - \frac{v_1^2}{2} \right)$

Where:

• $\dot{Q}$ = Heat transfer rate (assumed negligible, $\dot{Q} = 0$)
• $\dot{W}$ = Power produced by the turbine
• $\dot{m}$ = Mass flow rate = $2\ \text{kg/s}$
• $h_1, h_2$ = Specific enthalpies at the inlet and outlet
• $v_1, v_2$ = Velocities at the inlet and outlet
• Neglecting potential energy changes.

The equation simplifies to: $\dot{W} = \dot{m} \left( h_1 - h_2 \right) + \dot{m} \left( \frac{v_1^2}{2} - \frac{v_2^2}{2} \right)$

Step 1: Determine specific enthalpies ($h_1$ and $h_2$)

• Inlet properties: From the steam tables, at $P_i = 3\ \text{MPa}$ and $T_i = 400^\circ\text{C}$: $h_1 = 3213.6\ \text{kJ/kg}$

• Outlet properties: At $P_o = 100\ \text{kPa}$ and quality $x = 0.9$, the specific enthalpy is: $h_2 = h_f + x \cdot h_{fg}$ From the steam tables at $P_o = 100\ \text{kPa}$: $h_f = 417.46\ \text{kJ/kg}, \quad h_{fg} = 2257.92\ \text{kJ/kg}$ $h_2 = 417.46 + 0.9 \cdot 2257.92 = 2459.57\ \text{kJ/kg}$

Step 2: Kinetic energy terms

The change in kinetic energy is: $\Delta KE = \frac{v_1^2}{2} - \frac{v_2^2}{2}$ Substituting $v_1 = 10\ \text{m/s}$ and $v_2 = 50\ \text{m/s}$: $\Delta KE = \frac{10^2}{2} - \frac{50^2}{2} = 50 - 1250 = -1200\ \text{m}^2/\text{s}^2$ Convert to $\text{kJ/kg}$ by dividing by 1000: $\Delta KE = -1.2\ \text{kJ/kg}$

Step 3: Power calculation

The total power output is: $\dot{W} = \dot{m} \left[ (h_1 - h_2) + \Delta KE \right]$ Substitute values: $\dot{W} = 2 \cdot \left[ (3213.6 - 2459.57) + (-1.2) \right]$ $\dot{W} = 2 \cdot \left[ 754.03 - 1.2 \right]$ $\dot{W} = 2 \cdot 752.83 = 1505.66\ \text{kW}$

Final Answer:

The power produced by the turbine is: $\boxed{1505.7\ \text{kW}}$

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Would you like further clarification or an explanation on any specific part?
Here are 5 related questions you might consider:

1. How is specific enthalpy determined for different states of steam?
2. What are the assumptions in applying the First Law of Thermodynamics in turbines?
3. How would the calculation change if heat transfer were not negligible?
4. How does steam quality impact turbine performance?
5. Why is the velocity change considered, and how significant is its effect?

Tip: Always refer to accurate steam tables for enthalpy and other thermodynamic property data at given conditions.