Rankine Cycle.

The Rankine cycle takes its working fluid through 4 processes and returns it to the state it began in. In order, those are:

1. Feed pump
2. Boiler and superheater
3. Turbine expansion
4. Condenser

You will find it in coal, gas, nuclear and solar-thermal power stations. At the standard set-up above — boiler pressure 6,000 kPa, turbine inlet temperature 600 °C, condenser pressure 10 kPa, turbine efficiency 1 — it reaches a thermal efficiency of 40.24%, against a Carnot limit of 63.47% for the same temperature span. The net work is 1,393 kJ/kg, from 3,461 kJ/kg of heat in and 2,068 kJ/kg rejected.

Its efficiency has a closed form: η = (w_turbine − w_pump) / q_boiler. The simulator above does not use it — it solves the state points and divides net work by heat in — but the two agree to machine precision, and the test suite behind this page checks that on every build.

Switch the diagram pane between P–v and T–s and watch the same cycle change costume. On the pressure–volume plot the area inside the loop is the net work per kilogram of working fluid, because work is the integral of P dv. On the temperature–entropy plot the area inside the loop is the net heat, because heat in a reversible process is the integral of T ds.

Those two areas are the same number. That is the first law applied to a closed loop: the fluid ends where it started, so its internal energy has not changed, so everything that went in as heat came out as work. Two different pictures, one arithmetic.

The T–s view is the more revealing of the two once you are comfortable with it, because it shows the temperature at which heat crossed the boundary — and that, not the amount, is what decides how much of it can become work.

William Rankine, a Scottish engineer and one of the founders of thermodynamics as a discipline, gave his name to the cycle that describes a steam power plant. It is the only cycle on this page whose working fluid changes phase, and that single fact is what makes it the cycle most of the world’s electricity is generated by — from coal, gas, nuclear and concentrated solar alike.

The reason is the pump. Compressing a gas is expensive; pumping a liquid is nearly free, because liquid water barely changes volume under pressure. So a steam plant spends almost nothing getting its working fluid up to boiler pressure, and keeps almost all of what the turbine produces.

Raising boiler pressure raises the average temperature at which heat is added, which raises efficiency. It also drives the turbine exhaust wetter. Water droplets travelling at turbine tip speeds erode the last-stage blades, so plants hold exhaust quality above roughly 88% dry.

Superheat is the usual counter, and it helps. But sweep the turbine inlet temperature on the panel above and watch: no realistic simple-Rankine setting gets a 16 MPa cycle back above 88%. That is physics, not a limitation of this model — and it is precisely why real stations do not simply superheat harder. They expand partway, take the steam back to the boiler to be reheated, and expand again.

A large subcritical coal station quotes something near 40% thermal efficiency, and the ideal cycle above gives about the same at realistic steam conditions — which looks like agreement and is partly luck. The ideal cycle is optimistic; the real plant has losses the model does not carry; and they roughly cancel at these conditions.

Drag the turbine efficiency slider from 1 down to 0.88 and you will see the largest single one. Real expansion is not isentropic; the steam leaves with more entropy and less enthalpy drop than the model assumes. Beyond that: boiler losses up the stack, condenser pressure that depends on the temperature of the river or the air, feed pump power, and everything the auxiliaries consume.