Air at 3.5 MPa and 500°C is expanded in an adiabatic gas turbine to 0.2 MPa. Calculate the
maximum work that this turbine can produce, in kJ/kg.

To calculate the maximum work that can be produced by an adiabatic gas turbine, we assume the process to be **isentropic**, meaning no heat transfer and entropy remains constant. Here's how to solve it:

### Given Data:
- Initial pressure, \( P_1 = 3.5 \, \text{MPa} = 3500 \, \text{kPa} \)
- Initial temperature, \( T_1 = 500^\circ \text{C} = 773 \, \text{K} \) (since \( T_1 = 500 + 273 \))
- Final pressure, \( P_2 = 0.2 \, \text{MPa} = 200 \, \text{kPa} \)
- Air behaves as an ideal gas, and the process is adiabatic and isentropic.

### Step-by-Step Solution:
#### 1. Assume air behaves like an ideal gas, so we use the isentropic relations for an ideal gas:

\[
\frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\frac{\gamma - 1}{\gamma}}
\]

where:
- \( \gamma \) is the ratio of specific heats for air. For air, \( \gamma = 1.4 \).

#### 2. Calculate the temperature at the exit, \( T_2 \):

\[
T_2 = T_1 \times \left( \frac{P_2}{P_1} \right)^{\frac{\gamma - 1}{\gamma}}
\]

Substituting values:

\[
T_2 = 773 \times \left( \frac{200}{3500} \right)^{\frac{1.4 - 1}{1.4}} = 773 \times \left( \frac{200}{3500} \right)^{0.286}
\]

Let’s calculate this.

The exit temperature of the air, \( T_2 \), is approximately 341.21 K.

#### 3. Calculate the work produced by the turbine using the specific heat at constant pressure \( c_p \):

The work done per unit mass in an isentropic process for an ideal gas is given by:

\[
W = c_p (T_1 - T_2)
\]

For air, the specific heat at constant pressure is approximately \( c_p = 1.005 \, \text{kJ/kg·K} \).

Substituting the values:

\[
W = 1.005 \times (773 - 341.21) \, \text{kJ/kg}
\]

Let’s compute the work output.

The maximum work that the turbine can produce is approximately **433.95 kJ/kg**.

Would you like further details on the process, or have any other questions?

Here are 5 related questions:
1. How does the efficiency of a real turbine compare to an ideal isentropic one?
2. What would happen if the turbine process wasn't adiabatic?
3. How does changing the pressure ratio affect the work output of a gas turbine?
4. What role does the specific heat ratio \( \gamma \) play in turbine performance?
5. How does an increase in inlet temperature \( T_1 \) impact the work produced?

**Tip**: In many thermodynamic processes, assuming ideal gas behavior simplifies calculations, but for more accuracy, use real gas properties depending on pressure and temperature ranges.

Air at 3.5 MPa and 500°C is expanded in an adiabatic gas turbine to 0.2 MPa. Calculate the maximum work that this turbine can produce, in kJ/kg.

This solution explains how to calculate the maximum work produced by an adiabatic gas turbine expanding air from 3.5 MPa and 500°C to 0.2 MPa. It involves key thermodynamic concepts, including isentropic processes and the ideal gas law. Ideal for university-level engineering students.

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