Question

For an ideal gas turbine cycle, \(T_1\) and \(T_3\) are the compressor inlet temperature and turbine inlet temperature respectively. The ratio \(\dfrac{T_3}{T_1}\) 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. \(\; t^{\tfrac{2}{\gamma - 1}}\)
• B. \(\; t^{\tfrac{\gamma}{2(\gamma - 1)}}\)
• C. \(\; t^{\tfrac{\gamma}{(\gamma - 1)}}\)
• D. \(\; t^{\tfrac{\gamma - 1}{\gamma}}\)

Hint:

In Brayton cycle, maximum work output occurs when the pressure ratio is the geometric mean of temperature ratio \(t\). Remember formula: \[ r_p^{(\gamma - 1)/\gamma} = \sqrt{\frac{T_3}{T_1}} \]

Answer 1

The Correct Option is B

Step 1: Net work in Brayton cycle.
The specific work output is: \[ W_{net} = W_t - W_c \] where \(W_t\) = turbine work, \(W_c\) = compressor work.

Step 2: Temperature relations.
Compressor temperature rise: \[ T_2 = T_1 \, r_p^{(\gamma - 1)/\gamma} \] Turbine exit temperature: \[ T_4 = T_3 \, r_p^{-(\gamma - 1)/\gamma} \] where \(r_p\) = pressure ratio.

Step 3: Net work expression.
\[ W_{net} = c_p \left[(T_3 - T_4) - (T_2 - T_1)\right] \] \[ = c_p \left[ T_3 - T_1r_p^{(\gamma - 1)/\gamma} - T_3 r_p^{-(\gamma - 1)/\gamma} + T_1 \right] \]

Step 4: Optimization condition.
For maximum work, \[ \frac{dW_{net}}{dr_p} = 0 \] This yields: \[ r_p^{\tfrac{\gamma - 1}{\gamma}} = \sqrt{\frac{T_3}{T_1}} \]



Step 5: Final expression.
Since \(\dfrac{T_3}{T_1} = t\): \[ r_{p,opt} = t^{\tfrac{\gamma}{2(\gamma - 1)}} \] Final Answer:
\[ \boxed{r_{p,opt} = t^{\tfrac{\gamma}{2(\gamma - 1)}}} \]