Fuel Injection

The d² law: why size is everything

A droplet evaporates from its surface. Work through the heat and mass transfer around a small evaporating sphere and a beautifully simple result drops out: the droplet's surface area shrinks at a constant rate.

$\cfuel{d}^2(t) = \cfuel{d_0}^2 - \cflame{K}\,t \qquad\Longrightarrow\qquad t_\text{life} = \frac{\cfuel{d_0}^2}{\cflame{K}}$

The evaporation constant $K$ depends on the fuel and on how hot and fast the surrounding gas is. Under combustor conditions it is of order 1 mm²/s. The consequence hides in that square: double the diameter and the lifetime quadruples. Let three droplets race through a combustor with a 5 ms residence time:

A 20 μm droplet is gone within half a millisecond, well inside the primary zone. A 70 μm droplet lives for ~6 ms: it escapes the primary zone, drifts through the cooler dilution air, and is still partly liquid when it reaches the turbine. There it leaves as unburned hydrocarbons, CO and smoke. The injector's job is to make sure almost no fuel ends up in the big droplets.

The right way to average a spray

A real spray contains droplets from a few microns to a few hundred. How do we summarise it in one number? The obvious average, the mean diameter, is the wrong one. Evaporation happens at the surface, but the fuel (the energy) is stored in the volume. The number that captures that balance is the Sauter mean diameter (SMD)Sauter mean diameterD₃₂ = Σd³/Σd²: the droplet size with the same volume-to-surface ratio as the whole spray — the right average for evaporation.:

$\text{SMD} = D_{32} = \frac{\sum_i n_i d_i^3}{\sum_i n_i d_i^2}$

It is the diameter of a single droplet with the same volume-to-surface ratio as the entire spray. A few large droplets drag the SMD up hard, because they hold a disproportionate share of the fuel. Modern atomizers aim for SMDs of roughly 15–40 μm at operating conditions; a human hair is about 70 μm across.

Three ways to shatter a liquid

All atomizers do the same thing at heart: they spread the liquid into a thin sheet or jet and let aerodynamic forces tear it into ligaments and then droplets. The Weber number, the ratio of disruptive aerodynamic force to the restoring force of surface tension ($\rho_A U^2 d/\sigma$), decides who wins. They differ in where the energy for shattering comes from.

Pressure-swirl (simplex). Fuel is forced at high pressure through tangential slots into a swirl chamber and leaves through an orifice as a thin, hollow, spinning cone that breaks up. The energy comes from the fuel pressure. Lefebvre's correlations show

$\text{SMD} \;\propto\; \sigma^{0.25}\,\mu_L^{0.25}\,\dot m_L^{0.25}\,\Delta P_L^{-0.5}\,\rho_A^{-0.25}$

and here is the catch. For a fixed orifice, flow scales with $\sqrt{\Delta P_L}$, so the pressure drop scales with the square of fuel flow. At idle, fuel flow falls several-fold, the pressure drop falls by the square of that, and the spray coarsens dramatically. That is the idle problem.

Duplex (dual-orifice). Two circuits: a small primary orifice sized to keep a high pressure drop at low flows, and a larger secondary that opens at higher power. It fixes the idle problem, at the cost of a coarse "hiccup" as the secondary circuit opens.

Airblast.Airblast atomizerAn injector that spreads fuel into a thin film and shatters it with fast air, rather than with fuel pressure. Rather than pressurising the fuel, spread it into a thin film and blast it with high-velocity air on both sides. The energy comes from the air, and the scaling changes character:

$\text{SMD} \;\propto\; \left(1 + \frac{\dot m_L}{\dot m_A}\right)\left(\frac{\sigma}{\rho_A\, U_A^2}\right)^{0.5} + \ldots$

Because the air velocity is set by the liner pressure drop, which stays a roughly fixed fraction of the rising combustor pressure, atomization stays good across the operating range, and improves at high power. Better still, the fuel is already mixed into the air as it atomizes, so there are fewer rich pockets and far less soot.

Sweep the power down to idle with each atomizer. The simplex falls apart at low flow, the duplex rescues it, and the airblast holds steady, until engine start, where there is barely any air velocity to do the work. Production aero engines therefore combine an airblast main circuit with a small pressure-atomising pilot for starting and low power.

Beyond SMD

A good spray is not just fine. It must also be placed correctly:

• Cone angle: too narrow and the fuel is concentrated on the axis (rich core, soot); too wide and droplets hit the liner wall before evaporating (carbon deposits, hot spots). Typical full cone angles are 60–90°.
• Penetration: large droplets carry their momentum further; small ones follow the gas almost instantly.
• Circumferential uniformity: an injector that sprays unevenly around its circumference creates local rich and lean zones, which appear downstream as hot streaks and a poorer pattern factor (recall the clogged injector in Lesson 4).

Viscosity: the quiet enemy

Lefebvre returns again and again to viscosity. It damps the instabilities that break a liquid sheet apart and resists the swirling flow inside a simplex nozzle. Cold fuel is more viscous, and at high altitude after a long cold soak, fuel can be far more viscous than on the ground. SMD rises, efficiency falls, and relight (Lesson 5) becomes harder still. This is one reason fuel is often warmed by using it as a heat sink for engine oil.

The flame is now fed, anchored and burning. Next: what happens when that flame starts talking to the sound waves in the chamber around it, and the combustor begins to sing.