Isentropic processes and efficiency

Every quantity in this topic is a comparison. Nothing here is measured directly off a real machine; it is computed for an imagined one and then used as the yardstick the real machine is held against. That is a familiar move: the whole of the second law was built the same way, with the Carnot engine standing in for the best any cycle could do. Here the comparison is not a cycle but a single device, and the standard it is held to has a name of its own.

What isentropic means

A process is isentropic when the entropy of the substance undergoing it does not change. That is a narrower condition than it first sounds, and it is worth taking apart into the two things that have to be true at once, because both are load-bearing. The entropy balance for a closed system reads \(\Delta s = \int (\delta q/T)_{\text{rev}} + s_{\text{gen}}\), and \(\Delta s\) vanishes only when both terms on the right do. The first requires the process to be adiabatic, \(q = 0\), so there is no transfer term to integrate. The second requires it to be internally reversible, so that \(s_{\text{gen}} = 0\). Drop either condition and the entropy is free to change; keep both and it cannot.

Students who have just met the T–s diagram tend to remember the picture and forget the argument: a vertical line on a T–s plot, full stop. The picture is correct. Reversible and adiabatic together force a straight drop, exactly as the previous topic showed for the sides of a Carnot rectangle, but the picture is a consequence, not the definition. An adiabatic process that is not reversible is not isentropic, and it does not draw as a vertical line even approximately; it leans to the right, because entropy is still being generated even though none is being transferred. The two words adiabatic and isentropic get used almost interchangeably in casual speech and they are not synonyms. Every isentropic process is adiabatic. Very few adiabatic processes are isentropic.

What makes this worth a whole topic rather than a footnote is what the isentropic process is used for. No turbine, compressor or nozzle is built without friction, without leakage, without some departure from equilibrium in the passages the fluid moves through, so no real device is ever isentropic in fact. The isentropic process is the idealisation of that device: same inlet state, same exit pressure, but with the irreversibility switched off. It plays exactly the role the reversible engine played in the Carnot argument: not a thing anyone expects to see, but the fixed point that lets a real thing be scored.

It is worth being precise about which processes this applies to. Nothing here says every process worth analysing should be adiabatic. A boiler is deliberately not adiabatic; heat transfer is the entire point of it. Isentropic analysis is reached for specifically in steady-flow devices where heat transfer is small by construction (the fluid passes through too quickly, or the casing is insulated on purpose) and where the quantity of interest is work or velocity rather than heat. Turbines, compressors, pumps and nozzles are exactly that list, and they are the subject of the rest of this topic.

One more piece of vocabulary earns its keep here. Because the two conditions are independent, it is useful to name them separately: a process is internally reversible if no irreversibility occurs within the substance itself, whether or not heat crosses the boundary. An isentropic process is the special case of an internally reversible process that also happens to be adiabatic. A reversible isothermal process, by contrast, is internally reversible but very much not isentropic: it is one of the two straight legs of the Carnot cycle, and its whole job is to move entropy, not to conserve it.

The isentropic relations

For an ideal gas with constant specific heats, the isentropic condition can be turned into closed-form relations between the properties at the two end states, and the fastest route to them is the entropy-change formula already derived in the previous topic. Set it to zero:

Rearranging and using \(R/c_p = (c_p - c_v)/c_p = 1 - 1/k = (k-1)/k\), with \(k = c_p/c_v\) as before, gives the temperature–pressure form:

The companion relation, built from \(0 = c_v \ln(T_2/T_1) + R \ln(v_2/v_1)\) in exactly the same way, gives the temperature–volume form:

Both are the same statement in different clothes, and eliminating \(T\) between them gives the form most often quoted and least often understood:

It is worth seeing why this curve sits where it does relative to the isothermal \(Pv = \text{constant}\) through the same starting point. Expand the gas along either curve to the same final volume and compare the two pressures reached. Along the isotherm the temperature has not moved, so the pressure is set by \(RT_1/v\) alone. Along the isentrope the temperature has fallen as well, so the pressure is \(RT_2/v\) with the smaller \(T_2\), and it must therefore land lower. The isentrope is the steeper of the two curves on a P–v plot precisely because it is doing two things at once: losing pressure to the expanding volume and losing pressure again to the falling temperature, while the isotherm only does the first.

These relations hold only where the assumption behind them holds: specific heats taken as constant, which is a reasonable approximation over a modest temperature range and a poor one across several hundred kelvin, for the same reasons set out earlier. Where the swing is large (a gas turbine combustor, a compression stroke that gets genuinely hot), the constant-\(k\) relations are replaced by tabulated functions of temperature alone, \(P_r(T)\) and \(v_r(T)\), defined so that \(P_2/P_1 = P_{r2}/P_{r1}\) and \(v_2/v_1 = v_{r2}/v_{r1}\) along an isentrope. They are built once from the same entropy function \(s^{\circ}(T)\) that appeared in the general entropy relation, so nothing new is being assumed: only the constant-specific-heat shortcut is being given up.

Isentropic efficiency of turbines and compressors

A turbine takes a fluid in at state 1 and expands it to a lower pressure \(P_2\). If that expansion were isentropic, the fluid would leave at a specific state \(2s\), fixed the moment \(P_2\) is fixed, by the relations of the previous section or by a table, and the work per unit mass, from the steady-flow energy equation with heat and kinetic terms dropped, would be

The real turbine reaches the same pressure \(P_2\) but not the same state; friction in the blade passages and leakage past the tips generate entropy, so the actual exit state \(2a\) sits to the right of \(2s\) on a T–s or h–s diagram, at a higher enthalpy. The real work is correspondingly smaller,

and the isentropic efficiency of the turbine is the ratio of what was delivered to what was possible:

Well-built industrial turbines reach 0.85 to 0.92; small or badly-matched ones can fall well below that. Note what the definition assumes without saying so: the comparison is always between two states at the same exit pressure. Comparing a real exit state against an isentropic one at a different pressure answers a different question and is not what \(\eta_T\) means.

A compressor runs the same picture with the roles exchanged. Work goes in rather than out, so the ideal case is the smallest input that would reach the target pressure, and the real machine always needs more than that. The efficiency is built so that it is still a number no greater than one, which means the ratio has to be inverted:

Mixing up the two definitions (dividing the wrong way round for a compressor) is one of the most common slips in this topic, and it is caught immediately by the sanity check that an efficiency has to come out below one. A pump is a compressor for a liquid, and because a liquid’s specific volume is nearly constant, its isentropic work reduces to \(w_s \approx v_1(P_2 - P_1)\), exactly as in the steady-flow topic, with the actual work still found from the same ratio.

Isentropic efficiency of nozzles

A nozzle has no shaft, so it earns no work term at all, and its entire purpose is to convert enthalpy into kinetic energy. With heat transfer negligible and the shaft term absent, the steady-flow energy equation for a nozzle collapses to

Taking the inlet velocity as small enough to neglect (usually fair, since a nozzle exists precisely to make the exit velocity large by comparison), the isentropic exit velocity comes from the full enthalpy drop being spent on speed:

Friction against the walls of a real nozzle, and the small amount of turbulence that comes with any finite flow rate, generate entropy exactly as they do in a turbine, so the real exit state \(2a\) again sits to the right of \(2s\) at the same exit pressure, with a smaller enthalpy drop available to convert:

The isentropic efficiency of a nozzle is defined on kinetic energy rather than on work, because kinetic energy is the product the device sells:

The two ratios in that equation are equal only because \(V_1\) was dropped from both numerator and denominator; keep it and the enthalpy form is still exact while the bare velocity ratio picks up a correction. Well-designed nozzles are efficient by the standards of this topic (figures of 0.90 to 0.99 are typical), because the flow path is short, smooth and does not give friction much distance to act over, unlike the long, twisting passage a fluid takes through a turbine stage.

A diffuser is the same duct run in reverse, decelerating a fluid and raising its pressure, and it earns exactly the same treatment with the inequality signs flipped: the ideal case recovers the most pressure rise for a given drop in kinetic energy, and the real one recovers less. Aircraft intakes and the inlet to a centrifugal compressor are both diffusers, and both are scored the same way.

Why real machines fall short

Every efficiency defined in this topic is a ratio built around the same fixed point, and it is worth stating plainly why that point can never be reached. The isentropic exit state is not simply the best state anyone happens to have found; it is the state that corresponds to zero entropy generation, and the increase-of-entropy principle forbids \(S_{\text{gen}}\) from ever being negative. There is no direction in which a real, adiabatic device can improve past it. It is a ceiling in the strict sense: a bound that can be approached as closely as engineering allows and never crossed.

Adiabatic devices are worth calling out separately here because the entropy balance simplifies for them in a way it does not for a boiler or a condenser. With \(q = 0\), the transfer term in the entropy balance vanishes entirely, leaving

which means that for a turbine, compressor or nozzle the entropy rise is the entropy generation, directly, with no reservoir bookkeeping needed to extract it. That is a genuine convenience: it is why entropy generation is so often read straight off the horizontal distance on a T–s or h–s diagram for these devices, and nowhere near as simply for a heat exchanger, where the transfer term has to be accounted for as well.

The mechanisms responsible are the same short list from the second law: friction in the boundary layer along every blade and wall, separation and turbulence where the flow cannot follow the passage shape, leakage past tip clearances and seals that bypasses the intended path entirely, and shock waves in any transonic or supersonic stage. None of these is a design failure in the sense of being avoidable in principle. They can be reduced (better aerodynamics, tighter clearances, more stages doing less work each), and every one of those reductions costs money, which is why isentropic efficiency is as much an economic quantity as a physical one. A compressor that improves from 0.80 to 0.85 has not changed what is possible; it has bought back some of the gap between the possible and the actual.

One consequence is worth stating because it surprises people who have only just met it. Comparing a real device’s performance against another real device of a different design tells you which is better, but it does not tell you how much room for improvement either one has left, because neither number carries the reversible limit inside it. That limit (how much better a real process could conceivably become, and how to price the gap between where it sits and where it could sit) needs a quantity that has not been introduced yet, one built directly out of the entropy generated rather than the enthalpy consumed. That is the next topic.