Kinetics, Flammability and Flame Stability

Arrhenius rates and the law of mass action, chemical against residence time, ignition delay, flammability limits, flame speed and stabilisation, and the relight envelope

Speedbird 9 — 24 June 1982

British Airways Flight 9, a Boeing 747 bound for Auckland, flies by night into a cloud of volcanic ash from Mount Galunggung, about 180 km south-east of Jakarta. One after another, all four engines flame out. From 37,000 feet the 747 can glide for about 23 minutes. The crew divert towards Jakarta and run the restart drill again and again, far above 28,000 feet, the recommended ceiling for starting the engines in flight. Nothing lights. At 13,500 feet, close to the height at which they would have to turn out to sea and ditch, engine four starts, then the other three. The 747 lands safely in Jakarta, and all 263 people on board survive (British Airways Flight 009). A flame in a combustor is a balance between how fast chemistry can burn fuel and how fast the gas carries it away — and that balance is why engines have a ceiling for relighting at all.

Combustion proceeds through hundreds of elementary reactions between molecules and radicals. Each runs at a rate set by the law of mass action — proportional to the concentrations of the colliding species — and a rate constant that climbs steeply with temperature, the Arrhenius law. Because concentrations scale with pressure, a reaction between two molecules runs faster at high pressure in proportion to

p^2

, and the time it needs falls as

1/p

k = A\,T^n\,e^{-E_a/(\mathcal R T)} = A\,T^n\,e^{-T_a/T}

The Arrhenius rate constant, with activation energy Ea or activation temperature Ta = Ea/ℛ.

Derivation

Why rates climb so steeply

A reaction needs a collision energetic enough to break bonds — more than

E_a

. In a gas at temperature

T

the fraction of collisions with that much energy follows the Boltzmann factor

e^{-E_a/(\mathcal R T)}

. Raising

T

\Delta T

multiplies the rate by about

\frac{k(T + \Delta T)}{k(T)} \approx e^{T_a\left(\frac{1}{T} - \frac{1}{T + \Delta T}\right)} \approx e^{T_a \Delta T/T^2},

so the rate doubles every

\Delta T = \ln 2\; T^2/T_a

. For fuel oxidation,

T_a \approx 15{,}000

K, it doubles every 104 K at 1500 K (Lab 1). For the reaction that makes thermal NO,

T_a = 38{,}370

K, every 41 K.

💡Two clocks

A flame survives only if its chemistry keeps up with the flow. Compare the chemical time

\tau_{chem}

with the residence time

\tau_{res}

the gas spends in the hot zone — a few milliseconds in a combustor. Their ratio is the Damköhler number:

\mathrm{Da} = \frac{\tau_{res}}{\tau_{chem}}.

Large Da: a robust flame. Da near one: the flame is on the edge, and a gust, a throttle chop or water in the inlet blows it out. Chemistry slows in cold, thin gas — exactly what a windmilling engine at high altitude holds.

📐Ignition delay

How long does a fuel–air mixture take to ignite on its own? With detailed chemistry (GRI-Mech 3.0, methane as a stand-in for kerosene, Lab 3):

At 20 bar: 8.1 ms at 1100 K, 0.70 ms at 1300 K, 0.10 ms at 1500 K — 80 times faster for 400 K.
At 1300 K: 2.9 ms at 5 bar, 0.70 ms at 20 bar, 0.33 ms at 40 bar.

Hot, dense gas lights almost instantly; cool, thin gas may never light within a combustor's few milliseconds.

📐A relight envelope

A windmilling engine at altitude holds cold air at low pressure. In a simple time-scale model (Lab 2) — chemistry slowing as

1/p

and with the Arrhenius factor — relight is possible only below a ceiling that rises with flight speed, because ram compression raises the pressure and temperature: 3.9 km at Mach 0.4, 7.3 km at Mach 0.8. Make the chemistry 1.5 times slower and the Mach 0.7 ceiling drops from 6.3 to 6.0 km. Real relight envelopes are measured in altitude test cells and printed in every pilot's handbook.

💡Flammability and flame stabilisation

Flammability limits: too lean or too rich and a flame cannot propagate. At room conditions methane burns in air only between 5 and 14.3 % by volume ( $\phi \approx 0.5$ – $1.6$ ), kerosene vapour between about 0.6 and 5 % ( $\phi \approx 0.5$ to over 4). A combustor's overall $\phi$ of 0.36 is below that lean limit: hot, compressed air widens the limits, but a flame that lean is weak and easily blown out — the first reason combustors burn a richer primary zone.
Flame speed: a laminar hydrocarbon flame at room conditions moves at about 0.4 m/s; turbulence wrinkles and multiplies it, but combustor air arrives far faster.
Stabilisation: the flame must be held where the flow is slow enough. A swirler sets up a recirculation zone that carries hot products back to the fuel injector, continuously relighting the incoming mixture — the flame holder of every modern combustor (Lesson 7-3).

Ladies and gentlemen, this is your captain speaking. We have a small problem. All four engines have stopped. We are doing our damnedest to get them going again. I trust you are not in too much distress.

– Captain Eric Moody, British Airways Flight 9, 24 June 1982 (Wikipedia)

🕰What the ash did

The ash cloud was dry, so it never showed on the weather radar, which looks for moisture. Inside Speedbird 9's engines the ash melted in the combustion chambers and stuck to the inside of the engines. As they cooled and the 747 descended out of the cloud, the ash solidified and broke off, letting air flow through again. Nineteen days later a Singapore Airlines 747 had to shut down three engines in the same area, and only then was the airspace closed for good. In December 1989 KLM Flight 867, a new 747-400, lost all four engines in ash from Mount Redoubt in Alaska; after descending more than 14,000 feet its crew restarted two and landed safely (KLM Flight 867).

💡Where you will use this

Lesson 7-3 builds the swirl-stabilised zones into a combustor. Lesson 7-4 uses the same Arrhenius steepness to explain why a flame 200 K cooler makes several times less NO. Lesson 8-3 asks how fast an engine can decelerate before its flame blows out.

📚Further reading

Stephen R. Turns, An Introduction to Combustion (3rd ed., McGraw-Hill, 2012), on chemical kinetics and flame speed.
Arthur H. Lefebvre and Dilip R. Ballal, Gas Turbine Combustion (3rd ed., CRC Press, 2010), on ignition, stability and relight.
British Airways Flight 009 on Wikipedia.

↓But there is a problem

A stable flame wants a hot, nearly stoichiometric zone with products recirculating through it. That is exactly where carbon monoxide is made, where soot forms, and where nitrogen starts to burn. The combustor has to give the flame what it needs and still deliver an even, turbine-safe temperature a few centimetres downstream.