Gas power cycles
A car engine burns fuel, and the products of that burning are not the same gas that started the compression stroke. Analysing the actual chemistry, stroke by stroke, is possible and almost nobody does it, because a much cruder model gets within a few per cent of the right answer and is vastly easier to reason about. This topic is that model, and the three engines built from it.
Every idea needed to build these cycles has already been assembled: the ideal gas equation, the isentropic relations, and the first law applied to a closed system or a control volume as the device demands. What is new here is not another law but a deliberate simplification, applied on purpose and with its costs known in advance, that turns a messy combustion process into something four equations can describe.
Air-standard assumptions
A real petrol or diesel engine cycle is open: air and fuel enter, combustion products leave, and the working substance is never the same molecule twice. Modelling that honestly means tracking a changing mixture through a chemical reaction, which is exactly the territory of combustion as its own subject. Most of the time nobody needs that level of detail to understand why a compression ratio matters or how much efficiency a pressure ratio buys, and the air-standard cycle is the model built for exactly that level of need.
Four assumptions define it, and each one is doing specific work. The working fluid is air throughout, and only air: no fuel, no combustion products, no change in composition or molar mass at any point. The air behaves as an ideal gas, which is a good approximation at the temperatures and pressures involved even though the air-standard model already concedes it is not tracking the actual products. Every process in the cycle is internally reversible, so the isentropic relations from the previous topic apply without qualification to the compression and expansion strokes. And combustion is replaced by a heat-addition process supplied to the same air from an external source, with exhaust and intake replaced by a heat-rejection process that returns the air to its starting state, closing a loop that a real engine, breathing fresh charge every cycle, never actually closes.
That last substitution is the one worth dwelling on, because it is where the model’s honesty is easiest to lose track of. Nothing is burned in an air-standard cycle. A quantity of heat \(q_{\text{in}}\) simply appears, from nowhere in particular, exactly where combustion would have released it, and a quantity \(q_{\text{out}}\) is rejected exactly where exhaust and fresh intake would have exchanged the hot products for cold charge. The chemistry that fixes how much heat a given fuel-air mixture can actually deliver is not part of this model at all. It is supplied as a number, from a heating value look-up or a combustion calculation done separately.
The result is a specific and useful trade. The air-standard cycle gets thermal efficiency trends right (how efficiency responds to compression ratio, to pressure ratio, to the shape of the heat addition), because those trends are set by the ideal-gas and isentropic relations the model keeps faithfully. It gets absolute numbers wrong, typically overestimating real efficiency and especially real power output, because it has deleted every loss that does not fit inside “ideal gas, reversible, closed loop.” Used to compare two compression ratios against each other, it is trustworthy. Used to predict what a specific engine will actually deliver on a dynamometer, it is a starting point and nothing more, and the gap between the two uses is exactly the subject of the last section of this topic.
The real cycle breathes: an intake stroke in, an exhaust stroke out.
The Otto cycle
The Otto cycle is the air-standard idealisation of a spark-ignition engine, and it is built from four processes chosen to mirror what actually happens inside one, minus the losses. Air is compressed isentropically from state 1 to state 2, the piston rising fast enough that the process is treated as adiabatic and smooth enough to be reversible. Heat is added at constant volume, from 2 to 3: the spark ignites the charge so quickly compared with the piston’s motion that the volume is taken as fixed while the pressure and temperature jump. The hot gas expands isentropically from 3 to 4, delivering the cycle’s work, and heat is rejected at constant volume from 4 back to 1, closing the loop in place of the real exhaust and intake strokes.
The two constant-volume legs mean no boundary work crosses them, so the two heat transfers are the simplest possible energy balances for a closed system at fixed volume:
and thermal efficiency, exactly as it was defined for any heat engine in the second law, is one minus their ratio:
What makes the Otto cycle worth a closed-form result rather than a case-by-case calculation is that the two isentropic legs relate every temperature to the compression ratio, \(r = v_1/v_2\), and nothing else. Since \(v_4 = v_1\) and \(v_3 = v_2\), both isentropic relations from the previous topic pick up the same ratio \(r\) raised to the same power, and it factors cleanly out of the efficiency expression:
For air at \(k = 1.4\) and a compression ratio of \(r = 8\), typical of a petrol engine, \(\eta_{\text{th}} = 1 - 1/8^{0.4} = 0.565\), or 56.5 per cent: a ceiling, in the air-standard sense, that no real petrol engine gets close to, for reasons the last section returns to. The formula also explains, on its own, why compression ratio is the single number engine designers fight hardest to raise: it is the only lever this model gives them. It cannot be raised without limit in a real spark-ignition engine, because a high enough compression ratio ignites the fuel-air mixture on its own, ahead of the spark (knock), which is a chemistry and materials limit the air-standard model has no way to represent, since it never models combustion in the first place.
Isentropic compression squeezes the air before ignition.
The Diesel cycle
The Diesel cycle changes exactly one process from the Otto cycle and inherits the rest unchanged, which makes the comparison between them unusually clean. Diesel engines have no spark and no premixed charge: air alone is compressed, hard enough that its temperature exceeds the fuel’s ignition point, and fuel is then injected gradually as the piston nears top dead centre. Because the injection and burn take a finite time (long compared with a spark discharge, short compared with the whole stroke), the idealisation treats this heat addition as happening at constant pressure rather than constant volume, with the piston moving outward to accommodate it.
Compression, 1 to 2, is isentropic exactly as before, and typically runs to a much higher ratio than a petrol engine dares, since there is no premixed charge left to knock. Heat addition, 2 to 3, is now at constant pressure, so this leg does boundary work and the energy balance uses \(c_p\) rather than \(c_v\):
The volume ratio over which the heat is added gets its own name, the cutoff ratio, \(r_c = v_3/v_2\), and it measures how long injection lasts relative to the compressed volume. Expansion, 3 to 4, is isentropic once more, and heat rejection, 4 to 1, is at constant volume, unchanged from Otto. Carrying the same algebra through with the new heat-addition leg gives
The bracket is always greater than one for \(r_c > 1\), so at a given compression ratio a Diesel cycle is always less efficient than an Otto cycle: spreading the heat addition over a finite volume rather than adding it instantaneously at top dead centre loses some of the advantage of a sharp, high-pressure burn. As \(r_c \to 1\) the bracket itself tends to one and the Diesel expression collapses onto the Otto formula exactly, which is the useful check that nothing was smuggled into the derivation: instantaneous injectionis the Otto cycle.
What makes real diesel engines more efficient than real petrol ones despite that formula, not less, is the compression ratio each is permitted to run. With no premixed knock limit, a diesel engine can compress to \(r = 18\) or higher, where a petrol engine is confined to something near 8 to 11. Take \(r = 18\) and a modest cutoff ratio \(r_c = 2\), and the formula gives \(\eta_{\text{th}} = 1 - (1/18^{0.4})[(2^{1.4}-1)/(1.4 \times 1)] = 0.632\), comfortably above the Otto figure at \(r = 8\). The two engines are not competing at the same compression ratio; they are each running at the ratio their own ignition method allows, and the Diesel cycle’s formula explains why that ratio is worth chasing even though the constant-pressure heat addition itself is a slight penalty.
A higher compression ratio, with no spark needed — the air alone ignites the fuel.
The Brayton cycle
Otto and Diesel are both piston engines: one working chamber, a fixed mass of air, four strokes repeated in time. A gas turbine is built differently: air flows continuously through a compressor, a combustor and a turbine in series, each doing its job on whatever fluid happens to be passing through it at that instant. The air-standard model built for it, the Brayton cycle, is written in the language of steady-flow devices rather than a closed system undergoing strokes.
The compressor raises the air’s pressure isentropically, 1 to 2, consuming work. The combustor adds heat at constant pressure, 2 to 3, in place of fuel burning continuously in the airstream, the same substitution the piston cycles make, applied to a flow rather than a fixed mass. The turbine expands the hot gas isentropically, 3 to 4, delivering work, part of which is taken straight back to drive the compressor on a shared shaft, with the remainder available as the engine’s net output. In a real gas turbine the cycle is open: state 4 is simply exhausted to the atmosphere, and a fresh charge of air enters at 1 rather than the same air returning, but the air-standard idealisation closes it with a notional constant-pressure heat rejection, exactly as intake and exhaust were closed into a loop for Otto and Diesel.
Because both the compression and expansion legs are once again isentropic and both heat transfers are at constant pressure, the same style of algebra that produced the Otto result produces a Brayton efficiency in terms of a single ratio, not the compression ratio this time, but the pressure ratio, \(r_p = P_2/P_1\):
At \(r_p = 8\) and \(k = 1.4\), this gives \(\eta_{\text{th}} = 1 - 1/8^{0.2857} = 0.448\), and the formula says the same thing the Otto result said about compression ratio: efficiency climbs monotonically with \(r_p\), and pressure ratio is the single lever a simple Brayton cycle has to pull. Modern jet and industrial gas turbines run pressure ratios well into the twenties and thirties precisely because of this equation, tempered, as with every formula in this topic, by real limits the air-standard model does not see, chiefly the temperature the turbine blades can survive at state 3.
A compressor raises the air's pressure, taking work in.
Why real engines fall short
Every efficiency in this topic was computed for a cycle that does not exist, and the gap between the air-standard number and a real engine’s measured number has several separate causes, each traceable to one of the four assumptions being quietly false. Compression and expansion are not isentropic: piston-ring friction, boundary-layer drag on the cylinder wall, and turbulence in a real gas turbine’s blade passages all generate entropy exactly as the previous topic described, and the isentropic efficiencies defined there apply to the compressor and turbine legs of a real Brayton cycle without modification.
Combustion is not instantaneous and not adiabatic. A spark-ignition burn takes a few crank-angle degrees rather than zero, which rounds off the sharp corner at state 3 that the constant-volume idealisation assumes, and every real combustion chamber loses some heat straight through its walls to the coolant rather than keeping it all in the working gas. Both effects shrink the pressure the real cycle reaches at the top of the loop, which is exactly the quantity the ideal cycle is leaning on for work. Draw the real pressure-volume trace measured off a running engine, the indicator diagram, against the air-standard loop computed for the same compression ratio, and the real trace sits visibly inside the ideal one at every point; the area between the two is work that the ideal cycle promised and the real cycle never delivered.
The working fluid is not air either, in the end. Real combustion products have a different molar mass and different specific heats from the air that entered, and those specific heats are not constant. They climb with temperature, for the same reason raised in ideal and real gases, and the effect is largest exactly where the cycle runs hottest, right after combustion. A more careful cold-air-standard versus air-standard distinction exists for this reason: properties evaluated at room temperature against properties allowed to vary with \(T\), and the gap between the two is a fair estimate of how much the constant-specific-heat shortcut alone is costing the prediction, separate from every other loss in this list.
Gas turbines carry one further penalty that piston engines barely feel. The back-work ratio, the fraction of the turbine’s gross output spent driving the compressor on the same shaft, runs to somewhere between 40 and 80 per cent of \(w_{\text{turb}}\), strikingly high compared with a Rankine cycle’s feed pump, which the next topic shows costs almost nothing by comparison, because compressing a gas takes far more work per unit pressure rise than compressing an almost-incompressible liquid. Any inefficiency in the compressor therefore does double damage in a gas turbine: it wastes work directly, the way it would in any device, and it enlarges the slice of the turbine’s own output that has to be spent covering for it. That coupling, one component’s loss inflating another component’s burden, is invisible to an air-standard analysis that treats compressor and turbine work as two numbers to be subtracted, and it is one clear reason gas turbine efficiency improved for decades chiefly through compressor aerodynamics rather than through any change to the cycle itself.