Question

For an ideal gas turbine cycle, \( T_1 \) and \( T_3 \) are the compressor inlet temperature and turbine inlet temperature respectively. The ratio \( \frac{T_3}{T_1} \) is denoted by \( t \) and the ratio of specific heats is denoted by \( \gamma \). For any given \( t \), the optimum pressure ratio for the maximum specific work output is

• A. \( \frac{2}{t(\gamma - 1)} \)
• B. \( \frac{\gamma}{t^2(\gamma - 1)} \)
• C. \( \frac{\gamma}{t(\gamma - 1)} \)
• D. \( \frac{\gamma - 1}{t \gamma} \)

Hint:

In thermodynamic cycles, the optimum pressure ratio for maximum work output is often a function of the temperature ratio and the specific heat ratio. Higher temperatures and pressure ratios lead to better work extraction, but the efficiency depends on the system configuration.

Answer 1

The Correct Option is B

In an ideal gas turbine cycle, the specific work output depends on the pressure ratio and the temperatures at different points of the cycle. The optimum pressure ratio for maximum specific work output can be derived using thermodynamic relations.
The relationship between the compressor inlet temperature \( T_1 \) and turbine inlet temperature \( T_3 \) is given by \( \frac{T_3}{T_1} = t \), where \( t \) is a constant.
The general equation for specific work output in a gas turbine cycle is: \[ W_{{specific}} = \frac{R (T_3 - T_1)}{(\gamma - 1)} \] where \( R \) is the gas constant and \( \gamma \) is the specific heat ratio. The optimum pressure ratio, which maximizes the work output, is derived from the temperature and specific heat ratio relationships. The resulting equation for the optimum pressure ratio is: \[ P_{{opt}} = \frac{\gamma}{t^2(\gamma - 1)} \] Thus, the correct option is (B) \( \frac{\gamma}{t^2(\gamma - 1)} \).