Vapour Power Systems

The question

Coal, nuclear, solar-thermal, geothermal and most biomass plants all generate electricity the same way: they boil water, spin a turbine with the steam, and condense it back. In 1900 such plants turned under a tenth of their fuel into electricity. Today's best manage nearly half. Nothing about the basic idea has changed. How do you squeeze 45% out of a pot of boiling water?

The Rankine cycle

You have already ridden it: the kilogram of water on the course home page went once around a . Its four steady-flow components each obey Lesson 4's energy equation:

Process	Device	Energy per kg
1 → 2	pump	$\cwork{w_p} = h_2 - h_1 \approx v_1(p_2 - p_1)$
2 → 3	boiler	$\cheat{q_\text{in}} = h_3 - h_2$
3 → 4	turbine	$\cwork{w_t} = h_3 - h_4$
4 → 1	condenser	$\cheat{q_\text{out}} = h_4 - h_1$

and the thermal efficiency is

$\eta = \frac{\cwork{w_t} - \cwork{w_p}}{\cheat{q_\text{in}}}.$

Why not simply run a Carnot cycle with steam? Because a Carnot cycle would need to compress a wet liquid–vapour mixture — a messy, inefficient business — and would limit the top temperature to the boiling point. The Rankine cycle instead condenses completely and pumps a liquid, which costs almost nothing: the back-work ratio of a steam plant is about 1%, against over 40% for a gas turbine. That single choice is the secret of steam.

For the ideal cycle of Moran and Shapiro's Example 8.1 — 80 bar saturated vapour, condensing at 0.08 bar — the course's steam tables give 37.1% efficiency and an exhaust quality of 67.5%, matching the book to three figures.

Three levers on efficiency

Lesson 5 said it: receive heat hot, reject it cold. For a cycle that receives heat over a range of temperatures, the right measure is the mean temperature of heat addition, $\bar T_\text{in} = q_\text{in}/\Delta s$ , and the efficiency is $\eta = 1 - T_\text{out}/\bar T_\text{in}$ . Every improvement in this lesson either raises $\bar T_\text{in}$ or lowers $T_\text{out}$ .

LabBuild a better steam cycle

Boiler pressure80 bar

Turbine inlet temperature480 °C

Condenser pressure0.080 bar

Turbine isentropic efficiency100%

Thermal efficiency39.6%Carnot limit 58%

Turbine exit quality79.3%too wet — blade erosion

Back-work ratio0.6%pump ÷ turbine work

Steam per MW0.80 kg/s

Challenges0/3

With an 85% turbine, keep the exhaust at least 90% dry at 150 bar or more.
Reach 40% with a realistic turbine (85% efficient or less).
Reach 45% with a realistic turbine.

The Rankine cycle computed with the course’s steam tables. Change the boiler and condenser conditions, the turbine’s isentropic efficiency, and add reheat (at 20% of boiler pressure) or an open feedwater heater fed by steam bled from the turbine. The red dashed line is the mean temperature at which heat is received — the single best predictor of efficiency.

Lower the condenser pressure. Condensing at 0.04 bar instead of 1 bar drops the rejection temperature from 100 °C to 29 °C. Plants condense in a deep vacuum, limited by the temperature of the cooling water.
Raise the boiler pressure. The water boils hotter, raising $\bar T_\text{in}$ . But the turbine exhaust gets wetter — and droplets travelling at hundreds of metres per second erode turbine blades. Exhaust quality should stay above about 90%.
Superheat. Hotter steam raises $\bar T_\text{in}$ and dries the exhaust. The limit is metallurgical: boiler tubes and turbine blades must survive at around 600 °C under enormous pressure.

Check your intuition

Raising the boiler pressure at fixed turbine inlet temperature increases efficiency. What is the main drawback?

Reheat and supercritical cycles

breaks the dilemma between high pressure and dry exhaust. The steam expands partway through a high-pressure turbine, returns to the boiler to be reheated to the top temperature, then expands through the low-pressure turbine. The exhaust is far drier, and since the reheat heat goes in at a high temperature, efficiency rises too.

Go further and you leave the dome behind. Supercritical plants run their boilers above 220.6 bar, where water never boils: it turns smoothly from liquid to vapour, and the constant-temperature boiling plateau — which dragged $\bar T_\text{in}$ down — disappears. Today's ultra-supercritical units run at about 280 bar and 600–620 °C, with one or two reheats.

Regeneration: feedwater heating

The coldest heat a steam boiler receives is spent warming the feedwater from condenser temperature. That low-temperature heat addition is what pulls $\bar T_\text{in}$ down most. attacks it from inside the cycle: bleed some steam from part-way down the turbine and use it to preheat the feedwater.

In an open feedwater heater, the bled steam mixes directly with the feedwater; in a closed one, it condenses in tubes and passes its heat through the walls. The bled steam does less work in the turbine, but the heat it gives up stays inside the cycle instead of being bought from the furnace at a large temperature difference — the exergy destruction of Lesson 7 shrinks. Large plants use seven or eight feedwater heaters and preheat the water to nearly 300 °C before it reaches the boiler.

A century of steam

Put the levers together, era by era:

HistoryA century of steam

Era1905 · Early turbine station

Steam conditions12 bar · 250 °Cno reheat

Cycle efficiency15.8%Low pressure, modest superheat, a poor vacuum and crude turbines.

Fuel per kWh0.86 kg coalat 29.3 MJ/kg, 90% boiler

Challenges0/1

Step through all seven eras.

Seven generations of steam power plant, each computed with the same simplified cycle model (one feedwater heater at most, idealised boiler). Real plants used more feedwater heaters and reached somewhat higher cycle efficiencies; their overall plant efficiencies, after boiler and auxiliary losses, ran roughly ten percentage points lower than the cycle figures here.

Estimate first

A 1 GW coal plant at 45% efficiency burns coal at 29 MJ/kg. How many tonnes of coal per hour?

1101001,00010,000100,000

Drag along the line to place your guess — each tick is ten times the last.

Going deeper
Going deeper: cogeneration, other fluids and carbon capture

A condensing steam plant rejects most of its fuel energy at about 35 °C, too cool to be useful. A cogeneration (combined heat and power) plant condenses at a higher pressure — 100 °C or more — and uses the rejected heat for district heating or industrial processes. Its electrical efficiency falls a little, but 80–90% of the fuel energy ends up useful.

For low-temperature heat sources — geothermal brine, waste heat, solar ponds — water is a poor working fluid, because its saturation pressure at 100 °C is so low that turbines become huge. Organic Rankine cycles use refrigerants or hydrocarbons whose vapour dome suits the temperatures available. At the other extreme, supercritical CO₂ cycles (actually a closed Brayton cycle, Lesson 9) promise compact, efficient turbomachinery for nuclear and concentrated solar heat.

Fossil steam plants with carbon capture divert a large stream of low-pressure steam to regenerate the solvent that absorbs CO₂ from the flue gas, costing 8–12 percentage points of efficiency. The second-law analysis of where that steam is drawn from is an active design problem.

Recap

What to carry forward

The Rankine cycle: pump a liquid, boil and superheat it, expand it in a turbine, condense it. Pumping a liquid makes the back-work ratio tiny.
$\eta = (w_t - w_p)/q_\text{in}$ ; think of it as $1 - T_\text{out}/\bar T_\text{in}$ .
Lower condenser pressure, higher boiler pressure and superheat raise efficiency; the exhaust must stay above about 90% quality.
Reheat allows high pressures with a dry exhaust; supercritical cycles remove the boiling plateau.
Feedwater heating with bled steam raises $\bar T_\text{in}$ and cuts exergy destruction in the boiler.

Next: replace water with air and the boiler with a flame inside the engine. Why do jet engines and car engines work so differently?