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 Incoming links: Thermodynamics and Propulsion / 3.2 Generalized Representation of Thermodynamic Cycles

Heat engine 2026-10-14

A heat engine uses a repeating thermodynamic cycle to turn part of an incoming heat transfer into net outgoing work. In the two-thermal reservoir model, the hot thermal reservoir supplies heat

Q_{h} > 0

at fixed temperature

T_{h}

, and the cold thermal reservoir receives heat of magnitude

Q_{c} > 0

at fixed temperature

T_{c} < T_{h}

. A Stirling engine makes the repeated motion visible; the abstract energy budget applies independently of its particular mechanism.^[ref]

Choose the entire cyclic device as a closed system, with no net change in bulk kinetic energy or potential energy. Count heat entering as positive and net work delivered as positive, as in the first law of thermodynamics. Thus

Q_{h}

enters with a plus sign and

Q_{c}

, although a positive magnitude, enters with a minus sign. Assume these two heat transfers and the net work are the only transfers across the boundary. Over one complete thermodynamic cycle, all stored energy, including internal energy

U

, returns to its starting value:

Δ U = Q_{h} - Q_{c} - W = 0, W = Q_{h} - Q_{c} .

(1)

The device does not accumulate heat from cycle to cycle. More precisely, heat is a transfer, while internal energy is what can be stored; even a component that temporarily stores internal energy must return to its initial state for this boundary to be cyclic.^[ref]

For engine operation,

Q_{h} > Q_{c}

and

W > 0

. Its thermal efficiency is

η = \frac{W}{Q _{h}} = 1 - \frac{Q _{c}}{Q _{h}} .

(2)

Use net work, after subtracting any work supplied during compression or required by auxiliaries included in the chosen device. For periodic operation, average power satisfies

P_{out} = \dot{Q}_{h} - \dot{Q}_{c}

and

η = P_{out} / \dot{Q}_{h}

, with the rates averaged over complete thermodynamic cycles.^[ref]

For a stipulated model at

T_{h} = 600 K

and

T_{c} = 300 K

, let

Q_{h} = 1000 J

and

Q_{c} = 600 J

per thermodynamic cycle. Then

W = 400 J

and

η = 2/5 = 40%

. At two thermodynamic cycles per second, these become

2000 W

incoming heat transfer,

1200 W

outgoing heat transfer, and

800 W

net power. These are constructed values, not measurements.

Figure 1.
Two-thermal reservoir heat engine: $1000 J$ in, $400 J$ of net work, and $600 J$ rejected
. Illustrative transfers per complete thermodynamic cycle, with positive magnitudes labeled on the arrows. The closed system returns to its initial state. Arrow widths represent the energy split; the thermal reservoirs remain at $600 K$ and $300 K$ . **Details**.

Different signs describe different tasks. With the same directions of heat transfer,

Q_{h} = Q_{c}

gives

W = 0

: a device transmitting heat without net work output. If

Q_{c} > Q_{h}

, net work is supplied to that device. A refrigerator or heat pump instead extracts heat from the cold side and delivers heat to the hot side, using positive input work

W_{in}

and the balance

Q_{h} = Q_{c} + W_{in}

. Their useful output is heat transfer, measured by a coefficient of performance. Finally, a one-time isothermal expansion can have

Q = W

and

Δ U = 0

while leaving the volume changed: it is not a complete thermodynamic cycle. A warming device also fails the cyclic assumption, and a fuel-consuming device requires an accounting of matter crossing its boundary.

For two fixed thermal reservoirs at positive thermodynamic temperatures, the Second law of thermodynamics strengthens the energy balance to

η \leq η_{C} = 1 - \frac{T _{c}}{T _{h}} .

(3)

Equality requires the overall operation to be a reversible thermodynamic process, including its heat transfer to the thermal reservoirs. The Carnot cycle explains how the reversible benchmark is constructed, and Carnot's theorem explains its universality and the entropy production penalty. Use kelvin values, never ratios of degree Celsius readings. The example above lies below the

50%

bound: its total entropy production is

- 1000/600 + 600/300 = 1/3 J K^{- 1}

per thermodynamic cycle.

Figure 2.
Thermal efficiency versus hot thermal reservoir temperature at $T_{c} = 300 K$
. The upper curve is the reversible bound in Carnot's theorem. The lower curve is the stipulated model $η = 0.8 η_{C}$ , not experimental data or a universal engine law. At $T_{h} = 600 K$ , the curves give $50%$ and $40%$ , matching the energy budget above. The endpoint $T_{h} = T_{c}$ is a zero-output limit. **Details**.

This is useful before building a machine: warmer cooling surroundings reduce the possible thermal efficiency, even if the mechanism is unchanged. A small observed thermal efficiency does not by itself test the Second law of thermodynamics; heat leaks, friction, incomplete recovery of work, and the measurement boundary can all reduce the reported ratio. A meaningful comparison requires the actual thermal reservoir temperatures, matched heat and net work measurements, and the hypotheses of Carnot's theorem. Finite bodies whose temperatures change are treated separately there.

The practical question already appears in the 1824 Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance by Sadi Carnot: pp. 6–8 ask whether machines driven by heat have a fundamental limit and whether changing the working substance can improve their output. Pages 9–11 describe heating and cooling in real machines, but use the historical caloric theory, rather than the modern first law of thermodynamics balance. The historical discussion under Carnot's theorem explains that distinction.

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