Question

The air standard efficiency of closed gas turbine cycle is given by: Where r = Pressure ratio for compression and turbine; and $\gamma$ = Isentropic index of air.

• A. $\eta = \{ 1 - \frac{1}{r^{(\frac{\gamma - 1}{\gamma})}} \}$
• B. $\eta = \{ 1 - \frac{1}{r^{(\gamma - 1)}} \}$
• C. $\eta = \{ 1 - r^{(\gamma - 1)} \}$
• D. $\eta = \{ r^{(\gamma - 1)} - 1 \}$

Hint:

Don't confuse the Brayton cycle (Gas Turbine) with the Otto cycle (Petrol Engine).

• Otto Cycle: Uses Volume Ratio ($r$) and the power is $(\gamma - 1)$.

• Brayton Cycle: Uses Pressure Ratio ($r$) and the power is $(\frac{\gamma - 1}{\gamma})$.
Just remember that pressure ratios always need that extra "$\gamma$" in the denominator of the exponent.

Answer 1

The Correct Option is A

The closed gas turbine cycle operates on the Brayton cycle (also known as the Joule cycle). It consists of two isentropic processes (compression and expansion) and two isobaric processes (heat addition and rejection). 1. Efficiency Derivation: The thermal efficiency ($\eta$) of a Brayton cycle is defined as: $$\eta = \frac{\text{Net Work}}{\text{Heat Added}} = 1 - \frac{\text{Heat Rejected}}{\text{Heat Added}}$$ For constant pressure heat addition ($Q_{in} = c_p(T_3 - T_2)$) and rejection ($Q_{out} = c_p(T_4 - T_1)$): $$\eta = 1 - \frac{T_4 - T_1}{T_3 - T_2}$$

2. Relation to Pressure Ratio ($r$): Using the isentropic relations for the compressor ($1 \to 2$) and turbine ($3 \to 4$): $$\frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{\gamma-1}{\gamma}} = r^{\frac{\gamma-1}{\gamma}} \quad \text{and} \quad \frac{T_3}{T_4} = \left(\frac{P_3}{P_4}\right)^{\frac{\gamma-1}{\gamma}} = r^{\frac{\gamma-1}{\gamma}}$$ Substituting these into the efficiency equation and simplifying leads to the final expression: $$\eta = 1 - \frac{1}{r^{\left(\frac{\gamma - 1}{\gamma}\right)}}$$

3. Significance: This formula shows that the efficiency of a gas turbine depends primarily on the

pressure ratio. Increasing the pressure ratio ($r$) or using a gas with a higher isentropic index ($\gamma$) will increase the theoretical efficiency of the cycle.