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Five Carnot engines in series between six reservoirs: the efficiency of each for a net efficiency of 211/243

Five Carnot engines in series, each feeding its rejected heat to the next: heat flows down the cascade, W/Q0 = 1 - (1 - eta)^5, and the run searches for the eta that gives 211/243 - eta = 1/3 = 0.33, with the reservoirs stepping down 1215, 810, 540, 360, 240, 160 K.

Sub-topics covered

• Carnot engine: efficiency eta = 1 - T_cold/T_hot = W/Q_absorbed
• Engines in series: Q_k = (1 - eta)^k Q0 and W = Q0 - Q_N
• Net efficiency 1 - (1 - eta)^N; efficiencies do not add
• Reversible engines in series equal one reversible engine between the extreme reservoirs

What it teaches

• Each engine keeps a fraction eta of its heat as work and passes (1 - eta) on, so the rejected fractions multiply
• W/Q0 = 1 - (1 - eta)^5 = 211/243 gives (1 - eta)^5 = 32/243 = (2/3)^5 and eta = 1/3
• For Carnot engines each reservoir is 2/3 of the one above: 1215, 810, 540, 360, 240, 160 K, and 1 - 160/1215 = 211/243
• The trap eta_net = 5 eta would exceed 1 for eta > 0.2

Simulation ID

• ADV-2026-P1-PHY-Q10 · revision 1

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