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?

Question

The difference of temperature in a material can convert heat energy into electrical energy. To harvest the heat energy, the material should have:

Answer 1

On a T-s diagram, the area enclosed by a cycle equals the net specific work of that cycle, and the shape of the constant-pressure lines (2-3 and 4-1) tells you about heat transfer. Use these two facts to check each statement instead of writing out the full algebra.

Thermal efficiency is the same: efficiency of an ideal Brayton cycle depends only on the pressure ratio across the compressor and turbine, and both cycles share exactly the same compression leg 1-2, so the same pressure ratio. That means the same $T_2/T_1$ ratio, hence the same efficiency for both cycles, regardless of how high point 3 or 3' goes. This statement is TRUE.
Specific work is the same: net work is the area enclosed by the cycle on the T-s diagram. Cycle 1-2-3'-4'-1 reaches a higher peak temperature at 3', so its top edge sits higher and its enclosed loop is visibly larger than that of cycle 1-2-3-4-1. A bigger enclosed area means more net work, so the two cycles cannot have the same specific work. This statement is FALSE, this is the one we are looking for.
2-3 and 2-3' are isobaric: both segments represent heat addition in the combustor, which happens at constant pressure in the ideal cycle, so they are isobaric. TRUE.
Heat added is greater for 1-2-3'-4'-1: heat added is proportional to the temperature rise $T_3-T_2$ (or $T_{3'}-T_2$) along the constant pressure line. Since 3' sits higher than 3 on the diagram, the temperature rise, and so the heat added, is larger for the primed cycle. TRUE.

The only false statement is that the specific work of the two cycles is the same, because raising the turbine inlet temperature to $T_{3'}$ enlarges the cycle's enclosed area and increases its net work output, even though it leaves the efficiency unchanged.

Let's summarize:

Efficiency of the ideal Brayton cycle is set by the pressure ratio alone.
Net work is set by both the pressure ratio and how high the peak (turbine inlet) temperature goes.

So the false statement is (B), the specific work of the two cycles is not the same.

Answer 2

On a T-s diagram, the area enclosed by a cycle equals the net specific work of that cycle, and the shape of the constant-pressure lines (2-3 and 4-1) tells you about heat transfer. Use these two facts to check each statement instead of writing out the full algebra.

Thermal efficiency is the same: efficiency of an ideal Brayton cycle depends only on the pressure ratio across the compressor and turbine, and both cycles share exactly the same compression leg 1-2, so the same pressure ratio. That means the same $T_2/T_1$ ratio, hence the same efficiency for both cycles, regardless of how high point 3 or 3' goes. This statement is TRUE.
Specific work is the same: net work is the area enclosed by the cycle on the T-s diagram. Cycle 1-2-3'-4'-1 reaches a higher peak temperature at 3', so its top edge sits higher and its enclosed loop is visibly larger than that of cycle 1-2-3-4-1. A bigger enclosed area means more net work, so the two cycles cannot have the same specific work. This statement is FALSE, this is the one we are looking for.
2-3 and 2-3' are isobaric: both segments represent heat addition in the combustor, which happens at constant pressure in the ideal cycle, so they are isobaric. TRUE.
Heat added is greater for 1-2-3'-4'-1: heat added is proportional to the temperature rise $T_3-T_2$ (or $T_{3'}-T_2$) along the constant pressure line. Since 3' sits higher than 3 on the diagram, the temperature rise, and so the heat added, is larger for the primed cycle. TRUE.

The only false statement is that the specific work of the two cycles is the same, because raising the turbine inlet temperature to $T_{3'}$ enlarges the cycle's enclosed area and increases its net work output, even though it leaves the efficiency unchanged.

Let's summarize:

Efficiency of the ideal Brayton cycle is set by the pressure ratio alone.
Net work is set by both the pressure ratio and how high the peak (turbine inlet) temperature goes.

So the false statement is (B), the specific work of the two cycles is not the same.

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?

Show Hint

The Correct Option is B

Solution and Explanation

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