Combustion Fundamentals

Counting atoms: the equivalence ratio

Burning is bookkeeping. For a kerosene-like fuel, one idealised molecule needs a fixed amount of oxygen to become carbon dioxide and water:

$\mathrm{C_{12}H_{23}} + 17.75\,\mathrm{O_2} \;\longrightarrow\; 12\,\mathrm{CO_2} + 11.5\,\mathrm{H_2O}$

Because air is mostly nitrogen, that works out to a stoichiometricStoichiometricThe exact mixture in which every fuel molecule finds just enough oxygen to burn completely — about 1 kg of kerosene per 14.7 kg of air. fuel–air mass ratio of about 0.068, roughly one kilogram of fuel for every 14.7 kg of air. Any real mixture is described relative to that perfect ratio by the equivalence ratioEquivalence ratio φActual fuel–air ratio divided by the stoichiometric one. Below 1 is lean (spare air), above 1 is rich (spare fuel)., written in fuel gold throughout the course:

$\cfuel{\phi} = \frac{(F/A)_\text{actual}}{(F/A)_\text{stoichiometric}}$

• $\phi < 1$: lean. There is more oxygen than the fuel can use; the spare air just gets heated.
• $\phi = 1$: stoichiometric. Every atom finds a partner.
• $\phi > 1$: rich. Not enough oxygen, so some fuel leaves as CO, H₂ or soot.

That single number will be on the axis of almost every plot for the rest of the course.

The master curve

Now ask: if the mixture burns completely, with no heat lost, how hot does it get? That is the adiabatic flame temperatureAdiabatic flame temperatureThe temperature a mixture reaches if it burns completely with no heat lost. A bell-shaped function of φ, peaking slightly rich. $\cflame{T_{ad}}$. Here it is as a function of $\cfuel{\phi}$, and it is worth spending a few minutes playing with.

Three things to notice, because each one echoes through the whole course.

It's a bell, peaked just rich of stoichiometric. On the lean side, adding fuel adds heat to the same mass of gas, so the temperature climbs. On the rich side there isn't enough oxygen to release the extra fuel's energy, but the unburned fuel still has to be heated, so temperature falls. The peak sits slightly rich (φ ≈ 1.05), partly because of dissociation: at these temperatures some CO₂ and H₂O split apart and soak up energy.

The inlet temperature lifts the whole curve. A modern engine compresses air to 800–900 K before it ever meets fuel. That preheat adds almost directly to the flame temperature, which is why high-pressure-ratio engines are both more efficient and harder to keep clean.

Look at the purple curve. Thermal NOₓ formation spans many powers of ten across a range where the temperature changes by only a few hundred kelvin. That is not a quirk of NOₓ. It is what every chemical rate does, and the next widget shows why.

Arrhenius: why temperature is the master variable

Molecules in a gas are not all equally energetic. Their collision energies spread out in a distribution, and only the rare collisions in the far tail, those energetic enough to break the right bonds, actually react. The energy needed is the activation energyActivation energy EₐThe energy barrier a collision must overcome for a reaction to happen. $E_a$.

Heat the gas and the whole distribution stretches to the right. The bulk barely changes, but the tiny tail above $E_a$ fattens enormously:

The fraction of collisions above the barrier behaves, to an excellent approximation, like $e^{-E_a/RT}$. That is the Arrhenius lawArrhenius lawReaction rate ∝ exp(−Eₐ/RT): only molecules in the energetic tail can react, so rates grow exponentially with temperature., with the temperature in flame red:

$\dot{\omega} \;\propto\; \exp\!\left(-\frac{E_a}{R\,\cflame{T}}\right)$

The key quantity is $E_a/RT$: how many "thermal energies" tall the barrier is. When it is around 10, every 10% rise in temperature multiplies the rate by roughly $e^{1}\approx 2.5$. For the Zeldovich reaction that makes NO, $E_a/RT$ is closer to 35 at flame temperatures. That is why NOₓ, in the explorer above, lives on a logarithmic axis.

This one idea explains a surprising amount of combustor behaviour:

• Cold starts are hard: well-mixed but cold fuel and air simply don't react fast enough.
• Altitude relight is hard: thin, cold air lowers both temperature and collision rate.
• The primary zone must run hot: to finish the chemistry within a few milliseconds of residence time.
• NOₓ is almost purely a temperature problem: Lesson 9 is built on this.

Two kinds of flame

There is a second fundamental choice, and it shapes every low-emissions strategy in Lesson 9: are fuel and air mixed before they burn, or do they meet in the flame?

A premixed flamePremixed flameFuel and air are mixed before burning; the flame is a travelling wave whose temperature you choose via φ. is a travelling wave. It moves into the fresh mixture at the laminar flame speed $S_L$, a property of the mixture, so where the flow is faster than $S_L$ the flame must lean over. The Bunsen cone angle obeys $\sin\alpha = \cflame{S_L}/\cair{U}$: the cone is literally a flame-speed meter. You get to choose its temperature, because you choose φ. Run lean and it is cool and low in NOₓ. But it can flash backFlashbackA flame propagating upstream, against the flow, into hardware not designed to contain it. if the flow ever slows below $S_L$, and it is easy to shake into oscillation (Lesson 7).

A diffusion flameDiffusion flameFuel and air meet in the flame and burn where they reach stoichiometric proportions — robust, but always at the hottest surface. has no speed of its own. Fuel and air arrive separately and burn wherever they meet in stoichiometric proportion, so the flame sits on the hottest possible surface, whatever the overall mixture. It cannot flash back, it is extremely stable, and it is simple. That is why conventional aero combustors are diffusion-flame devices. The price is high peak temperatures (NOₓ) and fuel-rich pockets on the fuel side (soot, the yellow glow).

Chemistry-limited or mixing-limited?

One more distinction is worth making before we build a combustor around this flame. Combustion can be slow for two very different reasons:

• Chemically limited: fuel and air are mixed, but it is too cold (or too low in pressure) for the reactions to go fast. Think cold starts, idle, altitude. The fix is temperature, pressure or time.
• Mixing limited: the chemistry would be instant, but fuel and oxygen haven't met yet. This is typical of a diffusion flame at high power. The fix is better aerodynamics: more turbulence, better atomisation, smarter jets.

Diagnosing which regime you are in tells you which lever will actually help. Adding fuel to a mixing-limited flame just makes it longer and smokier.

The master trade-off

Everything that follows, from swirlers and fuel staging to cooling and RQL, is ultimately a strategy for navigating this table.

Next: the very first component the air meets. Before the flame can be anchored, the diffuser must slow the air from ~150 m/s without throwing away the pressure the turbine needs.