Question

The figure below depicts two ideal gas turbine cycles, cycle 1-2-3-4-1 and cycle 1-2-3'-4'-1, on a \(T\)-\(s\) diagram. Which one of the following statements is FALSE?

• A. The thermal efficiency of the two cycles is the same
• B. The specific work of the two cycles is the same
• C. The processes 2-3 and 2-3' are isobaric
• D. The amount of heat added in the combustion process is greater for the cycle 1-2-3'-4'-1

Hint:

Efficiency depends only on the pressure ratio (shared compression leg), but net work depends on the peak temperature too.

Answer 1

The Correct Option is B

Step 1: Set up the ideal Brayton cycle relations.
Both cycles share the same compression process 1-2 (isentropic) and the same pressure levels, so both have the same pressure ratio \(r_p\).
For an isentropic process in an ideal gas,
\[ \frac{T_2}{T_1} = \frac{T_3}{T_4} = \frac{T_{3'}}{T_{4'}} = r_p^{(\gamma-1)/\gamma} = \tau \]
because 3-4 and 3'-4' are also isentropic expansions between the same two pressure levels as 1-2.

Step 2: Check the thermal efficiency (statement A).
The ideal Brayton cycle efficiency is
\[ \eta = 1 - \frac{1}{\tau} \]
which depends only on \(\tau\) (that is, only on the pressure ratio), not on the peak temperature \(T_3\) or \(T_{3'}\). Since \(\tau\) is the same for both cycles, \(\eta\) is the same for both. So statement (A) is TRUE.

Step 3: Check the specific work (statement B).
Net specific work is \(w = c_p(T_3 - T_4) - c_p(T_2 - T_1)\). Using \(T_4 = T_3/\tau\),
\[ w = c_p T_3\left(1 - \frac{1}{\tau}\right) - c_p(T_2 - T_1) \]
Since \(T_{3'} > T_3\) (point 3' is drawn higher on the T-s diagram), and every other term in the expression is identical for both cycles, \(w' > w\). So the specific work is NOT the same for the two cycles, statement (B) is FALSE.

Step 4: Check statements C and D.
Processes 2-3 and 2-3' both happen at the same, constant combustor pressure, so they are isobaric, statement (C) is TRUE.
Heat added is \(Q = c_p(T_3-T_2)\); since \(T_{3'}>T_3\), more heat is added for the 1-2-3'-4'-1 cycle, statement (D) is TRUE.

Final Answer:
Only statement (B) is false, the two cycles do not have the same specific work. \[ \boxed{\text{Option (B)}} \]