§ 23. Ideal Heat Engine & Carnot Cycle in 3D (Grade 10)
Conversion of heater heat Q₁ into mechanical piston work A and heat rejection Q₂ to the cold sink. No heat engine can convert all heat from a hot reservoir into work without rejecting heat to a colder sink: maximum efficiency is set by T₁ and T₂.
Theory Summary & Scientific Mechanism
A heat engine operates cyclically between a hot reservoir at absolute temperature T₁ and a cold sink at T₂ 0).
In 1824, Sadi Carnot proved that the maximum possible efficiency between temperatures T₁ and T₂ is achieved by a reversible Carnot cycle comprising two isotherms (T₁ = const, T₂ = const) and two adiabats (δ Q = 0, PV^γ = const). For an ideal gas, net cycle work is W = ν R (T₁ - T₂)ln (V₂)/(V₁), and the thermal efficiency is η_Carnot = 1 - (T₂)/(T₁).
Carnot’s theorem establishes an universal thermodynamic bound: maximum efficiency is independent of the working substance and depends solely on the absolute temperature ratio T₂/T₁. Raising turbine inlet temperature T₁ and lowering exhaust temperature T₂ is the primary route to higher efficiency in power engineering.
Everyday Analogy
A car engine or power plant runs on a temperature difference, just like a waterwheel runs on a height drop. No engine can convert 100% of heat into work, some heat must always be exhausted to the cold radiator!
Interactive Question & Derivation
An ideal Carnot heat engine operates between a hot reservoir at T₁ = 600 K and a cold sink at T₂ = 300 K, absorbing Q₁ = 120 J of heat per cycle. What is the maximum mechanical work A the piston can perform in one cycle?
- A: All 120 J of heat is converted into mechanical work (η = 100%)
- B: The Carnot efficiency is η = 1 - T₂/T₁ = 50%, so A = 60 J becomes useful work while Q₂ = 60 J must be rejected to the cold sink
- C: The work equals the temperature difference 600 - 300 = 300 J
Study Hint: From Carnot’s efficiency formula η = 1 - T_C / T_H: how does raising hot reservoir temperature T_H (or lowering T_C) affect efficiency η?
Proof & Explanation: By Carnot's theorem, no heat engine operating between temperatures T₁ and T₂ can exceed efficiency η_Carnot = 1 - (T₂)/(T₁). For T₁ = 600 K and T₂ = 300 K, η = 1 - 300/600 = 0.5 (50%). Out of Q₁ = 120 J, the piston performs A = η Q₁ = 60 J of work and transfers Q₂ = 60 J to the cold reservoir.
Key Equations & 3D Simulation Parameters
η_Carnot = 1 - (T₂)/(T₁), A = η_Carnot Q₁, Q₂ = Q₁ - A
- Hot reservoir temperature T₁: 450 … 900 K (default: 600 K)
- Cold sink temperature T₂: 200 … 400 K (default: 300 K)
- Heat absorbed per cycle Q₁: 60 … 200 J (default: 120 J)
- Thermodynamic cycle freq. ν: 0.8 … 3.2 Hz (default: 1.8 Hz)