Question

An ideal Brayton cycle operates between maximum and minimum temperatures of \( T_3 \) and \( T_1 \), respectively. For constant values of \( T_3 \) and \( T_1 \), the pressure ratio \( r_p \) for maximum work output is:

• A. ( \left(\frac{T_3}{T_1}\right)^{\frac{\gamma}{\gamma-1}} \)
• B. ( \left(\frac{T_3}{T_1}\right)^{\frac{2\gamma}{\gamma-1}} \)
• C. ( \left(\frac{T_3}{T_1}\right)^{\frac{\gamma}{\gamma-1}} \)
• D. ( \left(\frac{T_3}{T_1}\right)^{\frac{2}{\gamma-1}} \)

Hint:

For maximum work output in a Brayton cycle, the pressure ratio is related to the temperature ratio raised to the power \( \frac{\gamma}{\gamma-1} \).

Answer 1

The Correct Option is C

In an ideal Brayton cycle, the maximum work output occurs at the optimum pressure ratio, which is related to the temperature ratio. The pressure ratio for maximum work output in a Brayton cycle is derived from the relationship between the temperatures and pressures during the isentropic compression and expansion processes. The formula for the pressure ratio \( r_p \) is given by the following relation:
\[ r_p = \left(\frac{T_3}{T_1}\right)^{\frac{\gamma}{\gamma-1}}, \] where \( \gamma \) is the specific heat ratio of the working fluid. This relationship results from the ideal gas law and the isentropic process conditions in the Brayton cycle.
Thus, the correct answer is option (C).