01The question
Q.10 As shown in the figure, five Carnot engines, each with efficiency η and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider Q0 to be the amount of heat absorbed per cycle by the first engine and W as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be ηnet = W/Q0 = 211/243. The value of η is:
02The experiment
Answer—
Press START: heat flows down the cascade, and the run tunes η until W/Q₀ = 211/243.
Ready.
Engines
Speed
03Detailed solution
Answer: η = — (exactly 1/3; the accepted range is 0.32 to 0.34)
- Heat passed down. Each engine turns a fraction η of its input into work and passes (1 − η) of it on:
Q1 = (1 − η)Q0, Q2 = (1 − η)²Q0, … , Q5 = (1 − η)⁵Q0.
- Total work. Energy is conserved across the whole stack, so W = Q0 − Q5:
ηnet = W/Q0 = 1 − (1 − η)⁵ = 211/243.
- Solve. (1 − η)⁵ = 32/243 = (2/3)⁵, so 1 − η = 2/3 and η = 1/3 ≈ 0.33.
- The Carnot picture. A Carnot engine has η = 1 − Tcold/Thot, so each step lowers the temperature by the factor 2/3. For example: 1215 K → 810 → 540 → 360 → 240 → 160 K. The whole cascade is then exactly one Carnot engine between 1215 K and 160 K: 1 − 160/1215 = 211/243.
Understand the concept
In a series of engines, the heat each one rejects feeds the next. The efficiencies do not add; the rejected fractions multiply: 1 − ηnet = (1 − η₁)(1 − η₂)…. For Carnot engines this is the statement that temperature ratios multiply, so a chain of reversible engines behaves like one reversible engine between the first and last reservoirs. A common trap is to write ηnet = 5η. That would exceed 1 for η > 0.2.
Key takeaways
- Engines in series: 1 − ηnet = Π(1 − ηi). For five equal engines, 1 − ηnet = (1 − η)⁵.
- 243 = 3⁵ and 32 = 2⁵, so spotting the powers gives 1 − η = 2/3.
- Reversible engines in series are equivalent to one reversible engine between the extreme temperatures.
Verification: an independent derivation, a sympy route, a cycle-by-cycle energy bookkeeping of five Carnot engines with explicit reservoir temperatures and entropy checks, and the page's own engine, run by tests/verify_p1phyq10.py. The official key (0.32 to 0.34) and the MathonGo solution (0.33) agree. Status: script-verified by the PrayogX simulation factory; no review by a person has been recorded. The reservoir temperatures shown are one consistent example: only their ratios matter.
How to use this simulation
- Press ▶ START EXPERIMENT. Heat pours down the cascade, and each engine sends its share out as work.
- Watch the pipes get thinner from top to bottom: each passes on (1 − η) of what it received.
- The run tries values of η and plots W/Q₀ against η until it hits 211/243 = 0.868. It lands on η = 1/3.
- Drag Try η to test any value, or change Engines to 3 or 7 to see 1 − (1 − η)N.
- Replay runs it again. Reset clears it. Classroom fills the screen for a projector.