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Ideal Brayton cycle: diagrams and efficiency

Short answer

The ideal Brayton cycle has two isentropic and two constant‑pressure processes. Efficiency derivation: η = 1 − q_out/q_in = 1 − (T₄−T₁)/(T₃−T₂). Using isentropic ratios T₂/T₁ = T₃/T₄ = r_p^((γ−1)/γ) yields η = 1 − 1/r_p^((γ−1)/γ).

Sketch an ideal Brayton cycle on P-v and T-s diagrams. Also derive an expression for its efficiency in terms of pressure ratio.

The ideal Brayton cycle is the air-standard cycle of a gas turbine. It has two isentropic and two constant-pressure processes.

Diagrams

 P-v diagram                      T-s diagram
 P                                T
 |  2 ____________ 3              |               3
 |   \             \              |              /|
 |    \             \             |             / |
 |     '.            '.           |           /   |
 |       '.            '.         |    2 _.-'     |
 |         '-._          '-._     |    |          4
 |             '- 1 _________ 4   |    |       .-'
 |                                |    |   _.-'
 |                                |    1.-'
 +----------------------> v       +--------------------> s
 1-2 isentropic compression       2-3 constant-P heat addition
 3-4 isentropic expansion         4-1 constant-P heat rejection

Derivation of efficiency

Take air as an ideal gas with constant cp. Per kg of air:

  • Heat added (2-3, constant P): q_in = cp (T₃ − T₂)
  • Heat rejected (4-1, constant P): q_out = cp (T₄ − T₁)

Thermal efficiency:

η = w_net/q_in = 1 − q_out/q_in = 1 − (T₄ − T₁)/(T₃ − T₂)

For the isentropic processes 1-2 and 3-4, with P₂ = P₃ and P₁ = P₄, the pressure ratio is r_p = P₂/P₁ = P₃/P₄:

T₂/T₁ = (P₂/P₁)^((γ−1)/γ) = r_p^((γ−1)/γ) T₃/T₄ = (P₃/P₄)^((γ−1)/γ) = r_p^((γ−1)/γ)

Hence T₂/T₁ = T₃/T₄, which gives T₃/T₂ = T₄/T₁. Then

(T₄ − T₁)/(T₃ − T₂) = (T₁/T₂) × (T₄/T₁ − 1)/(T₃/T₂ − 1) = T₁/T₂

So

η = 1 − T₁/T₂

η_Brayton = 1 − 1/r_p^((γ−1)/γ)

The efficiency depends only on the pressure ratio and γ: it rises as r_p rises. For example, r_p = 10 and γ = 1.4 gives η = 1 − 1/1.931 = 48.2 %.

✓ Answer checked by Claude Opus 5.5 (high effort)

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