= Heat engine
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.<Thermodynamics and Propulsion/3.2 Generalized Representation of Thermodynamic Cycles>[§3.2, Figure 3.3: two reservoirs and net work output]{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:
$$
\Delta U=Q_h-Q_c-W=0,\qquad W=Q_h-Q_c.
$$
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.<University Physics Volume 2 by Samuel J. Ling and others/3.2 Work, Heat, and Internal Energy>[§3.2: heat, work and changes in stored internal energy]{ref}
For engine operation, $Q_h>Q_c$ and $W>0$. Its <thermal efficiency> is
$$
\eta=\frac{W}{Q_h}=1-\frac{Q_c}{Q_h}.
$$
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 $P_{\rm out}=\dot Q_h-\dot Q_c$ and $\eta=P_{\rm out}/\dot Q_h$, with the rates averaged over complete <thermodynamic cycles>.<University Physics Volume 2 by Samuel J. Ling and others/4.2 Heat Engines>[§4.2, equations 4.1–4.2: heat-engine balance and efficiency]{ref}
For a stipulated model at $T_h=600\,\mathrm K$ and $T_c=300\,\mathrm K$, let $Q_h=1000\,\mathrm J$ and $Q_c=600\,\mathrm J$ per <thermodynamic cycle>. Then $W=400\,\mathrm J$ and $\eta=2/5=40\%$. At two <thermodynamic cycles> per second, these become $2000\,\mathrm W$ incoming <heat transfer>, $1200\,\mathrm W$ outgoing <heat transfer>, and $800\,\mathrm W$ net <power>. These are constructed values, not measurements.
\Image[/heat-engine-cyclic-energy-flows.png]
{title=Two-<thermal reservoir> <heat engine>: $1000\,\mathrm J$ in, $400\,\mathrm J$ of net <work>, and $600\,\mathrm J$ rejected}
{description=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 $600\,\mathrm K$ and $300\,\mathrm K$.}
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> $W_{\rm in}$ and the balance $Q_h=Q_c+W_{\rm in}$. Their useful output is <heat transfer>, measured by a <coefficient of performance>. Finally, a one-time <isothermal expansion> can have $Q=W$ and $\Delta U=0$ while leaving the <volume (thermodynamics)> 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
$$
\eta\le\eta_C=1-\frac{T_c}{T_h}.
$$
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 $-1000/600+600/300=1/3\,\mathrm{J\,K^{-1}}$ per <thermodynamic cycle>.
\Image[/heat-engine-efficiency-temperature-model.png]
{title=<Thermal efficiency> versus hot <thermal reservoir> <temperature> at $T_c=300\,\mathrm K$}
{description=The upper curve is the reversible bound in <Carnot's theorem>. The lower curve is the stipulated model $\eta=0.8\eta_C$, not experimental data or a universal engine law. At $T_h=600\,\mathrm K$, the curves give $50\%$ and $40\%$, matching the <energy> budget above. The endpoint $T_h=T_c$ is a zero-output limit.}
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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