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.