Exergy

The question

Ask where the energy goes in a coal-fired power station and the answer is clear: more than half of it leaves through the condenser, into the cooling water. So the condenser is the plant's great waste? Strangely, no. Engineers who make plants more efficient spend their effort on the furnace and the boiler instead. Which losses really matter?

Energy is conserved. Useful energy is not.

A kilogram of water at 80 °C contains 230 kJ more internal energy than one at 25 °C. A litre of petrol contains about 32 000 kJ. Both are "energy", but you could run a car on one and not the other. The difference is the quality of the energy: how much of it could be turned into work.

That quality has a precise measure, (also called availability): the maximum theoretical work obtainable as a system is brought into equilibrium with its environment. The environment — air at about 25 °C and 1 atm, $T_0$ and $p_0$ — is the reference. A system in equilibrium with it is in the , and has zero exergy: nothing could ever be extracted from it.

For a closed system at a given state, the exergy is

$\cex{E} = (U - U_0) + p_0(V - V_0) - T_0(S - S_0) + KE + PE.$

The first terms are energy; the last, $T_0(S - S_0)$ , is the part of it that must be dumped into the environment as heat to satisfy the second law.

ExploreEnergy is not the same as useful energy

Temperature353 K · 80 °C

Environment T₀298 K · 25 °C

Energy above the dead state230 kJ

Exergy19 kJ8% of the energy

Challenges0/3

Find a state with negative energy (relative to the dead state) but positive exergy.
Find a state with more exergy than energy.
Make 1 kg of hot water hold under 5% of its energy as exergy.

The energy of a system relative to its surroundings (here the internal energy above the dead state, gold) and its exergy — the most work you could ever extract by bringing it into equilibrium with an environment at T₀ and 1 atm (pink). Hot water is full of energy and short of exergy; compressed air is the reverse.

The widget makes the point quickly. Hot water at 80 °C carries 230 kJ/kg of energy above the dead state, but only about 20 kJ/kg of exergy — 9%. Compressed air carries little internal energy at all (an ideal gas at room temperature has the same $u$ at any pressure), yet stores substantial exergy. And something colder than the environment, like ice, has negative energy relative to the dead state but positive exergy — you could run an engine between the warm air and the ice.

Exergy is destroyed

Here is the crucial difference from energy. Write an exergy balance for a closed system, and it looks like the energy balance with one extra term:

$\Delta \cex{E} = \underbrace{\int \left(1 - \frac{T_0}{T_b}\right)\delta \cheat{Q}}_{\text{exergy transfer with heat}} - \underbrace{\left[\cwork{W} - p_0\,\Delta V\right]}_{\text{with work}} - \cex{E}_d.$

The term $\cex{E}_d$ is , and it is never negative. It is tied directly to entropy generation by the Gouy–Stodola theorem:

$\cex{E}_d = T_0\,\cent{\sigma}.$

Every irreversibility — friction, mixing, heat transfer across a temperature difference, a free expansion, combustion — destroys exergy in exact proportion to the entropy it generates. Energy is never lost; work potential is lost all the time.

Notice the factor $1 - T_0/T_b$ on heat. It is a Carnot efficiency: heat transferred at temperature $T_b$ carries only that fraction of itself as exergy. Heat at 1500 K is 80% exergy; heat at 40 °C is barely 5%.

The power plant, audited twice

Now the question from the start. Here is a steam power plant with a coal or gas furnace, analysed both ways:

ExploreTwo ways to keep the books

Flame temperature2000 K

more excess air → cooler flame

Steam temperature540 °C

Boiler pressure120 bar

Turbine efficiency87%

Stack temperature147 °C

Plant efficiency33.7%energy = exergy here

Condenser, by energy59%by exergy 3.1%

Biggest exergy lossCombustion33% of the fuel

Challenges0/2

Get the plant above 40% efficiency.
Cut the boiler heat-transfer loss below 12% of the fuel. What happens to the combustion loss?

Where the fuel goes in a steam power plant, per 100 units of fuel. Top: an energy (first-law) balance. Bottom: an exergy (second-law) balance, where every irreversibility shows up as destroyed exergy. The plant model is the course’s Rankine cycle with an idealised furnace; fuel exergy is taken equal to its heating value.

By energy, the condenser is the villain: more than half of the fuel's energy leaves there. By exergy, it is a footnote of 3%. That condenser heat leaves at about 40 °C, just above the environment, so it was nearly worthless already: the damage was done earlier.

The exergy balance points to the real culprits:

Combustion — about a third of the fuel's exergy. Burning is a violently irreversible chemical reaction: highly ordered fuel molecules become a hot, chaotic gas.
Heat transfer in the boiler — another fifth or so. Gas at 2000 K heats water that boils at 325 °C. Heat crossing a 1400 K temperature gap is a textbook irreversibility.
Only then the turbine, the condenser and the stack.

This is the deep reason for the developments of Lessons 8 and 9: higher steam temperatures and pressures, reheat, supercritical boilers, and above all putting a gas turbine on top of the steam cycle, so the high-temperature heat does work directly before it is used to raise steam.

Check your intuition

A home's gas boiler heats water with 95% energy efficiency. Roughly what is its exergy efficiency, heating radiators at 60 °C with outside air at 0 °C?

Exergetic efficiency

The second-law, or exergetic efficiency, $\varepsilon$ , compares the exergy you got out with the exergy you put in. For a power plant, energy and exergy efficiencies are almost the same, because the product is work and the fuel's exergy is close to its heating value. For heating, cooling and heat recovery they can differ enormously, as the quiz showed. A turbine's exergetic efficiency, $\varepsilon = w/(e_{f1} - e_{f2})$ , is a little higher than its isentropic efficiency, because some of the "lost" work remains as exergy in the warmer exhaust.

Estimate first

Roughly what share of the world's primary energy supply ends up as useful exergy — work, electricity, useful heat at its true value?

110100

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

Going deeper
Going deeper: flow exergy and the exergy rate balance

For a stream flowing into a control volume, the exergy per unit mass is the flow exergy

$e_f = (h - h_0) - T_0(s - s_0) + \tfrac12 V^2 + gz,$

the flow analogue of the closed-system expression with $h$ replacing $u + p_0 v$ . The steady-state exergy rate balance for a control volume reads

$0 = \sum_j \left(1 - \frac{T_0}{T_j}\right)\dot Q_j - \dot W_\text{cv} + \sum_\text{in}\dot m e_f - \sum_\text{out}\dot m e_f - \dot E_d.$

Applied to each component of a plant in turn — exactly what the widget above does — it allocates every percent of the fuel's exergy to a product, a loss or a destruction, and the sum always closes. Moran and Shapiro use this exergy accounting to compare designs and to put a cost on each kilowatt of destruction: thermoeconomics.

Recap

What to carry forward

Exergy is the maximum work obtainable by bringing a system into equilibrium with its environment (the dead state, $T_0$ , $p_0$ ).
It is transferred with work, with heat (weighted by $1 - T_0/T$ ) and with flowing mass — and destroyed by every irreversibility: $E_d = T_0\sigma$ .
Low-temperature heat carries little exergy; compressed gases and cold bodies carry a lot relative to their energy.
In a steam plant, the energy loss is mostly at the condenser, but the exergy destruction is mostly in combustion and boiler heat transfer. That is where improvements pay.
Exergetic efficiency compares useful exergy out with exergy in.

Next: with energy, entropy and exergy in hand, we can take apart real power cycles. How do you squeeze 45% out of a pot of boiling water?