Premixed vs diffusion flames

WHY this distinction even exists

WHY it matters for propulsion: an engine designer chooses the regime on purpose. Premixed gives fast, controllable heat release but risks flashback and knock. Diffusion is safe from flashback (can't burn back into unmixed fuel) but is mixing-limited and sooty. Rockets and gas-turbine afterburners live on this trade-off.

The controlling idea: rate-limiting step

This single line tells you everything else (speed, structure, soot, where it stabilizes).

Deriving the laminar premixed flame speed (from scratch)

We want $S_L$, the speed at which the flame eats into the fresh mixture. WHAT we need: a balance between heat generated by reaction and heat conducted ahead to preheat the next layer.

Step 1 — Set up the physics. Consider a 1-D flame, fresh gas approaching at speed $S_L$. The flame has a preheat zone of thickness $\delta$. Inside it, conduction must raise the gas from $T_u$ (unburned) to $T_b$ (burned). Why this step? A flame can only advance if it preheats fresh gas to ignition — conduction is the messenger.

Step 2 — Two characteristic times must match.

• Diffusion (thermal) time across the zone: $t_{diff}\sim \dfrac{\delta^2}{\alpha}$, where $\alpha = k/(\rho c_p)$ is thermal diffusivity.
• Chemical reaction time: $t_{chem}\sim \dfrac{1}{\dot\omega/\rho}$ (inverse of reaction rate per mass).

Why this step? The flame is steady only when "heat arrives" and "fuel burns" on the same timescale. If chemistry were slower, heat would outrun it and quench; if faster, the zone thins.

Step 3 — Relate speed and thickness. A wave crossing thickness $\delta$ in time $t_{diff}$ moves at $S_L \sim \frac{\delta}{t_{diff}} = \frac{\delta}{\delta^2/\alpha}=\frac{\alpha}{\delta}.$ Also the residence of gas in the zone is $t_{res}=\delta/S_L$, and combustion needs $t_{res}\sim t_{chem}$, so $\delta \sim S_L\, t_{chem}$.

Step 4 — Eliminate $\delta$. Substitute $\delta \sim \alpha/S_L$ into $\delta\sim S_L t_{chem}$: $\frac{\alpha}{S_L} \sim S_L\, t_{chem}\;\Rightarrow\; S_L^2 \sim \frac{\alpha}{t_{chem}}.$

Diffusion flame: where does it sit?

For a reaction $F + s\,O_2 \to$ products (s = mass of O₂ per mass of fuel), with oxidizer mass fraction $Y_{O,2}$ in the oxidizer stream and fuel mass fraction $Y_{F,1}$ in the fuel stream: $Z_{st}=\frac{Y_{O,2}/s}{Y_{F,1}+Y_{O,2}/s}.$ Why this step? At the flame, fuel and oxygen arrive in exactly the ratio they consume each other, so the local fuel/oxidizer balance defines a single surface in space.

A diffusion flame has no propagation speed — it doesn't travel into anything. Its "speed" is replaced by how fast mixing delivers reactants (the scalar dissipation rate $\chi$). Push mixing too hard ($\chi$ too large) and the flame extinguishes (blows out) because reactants are diluted faster than they release heat.

Worked examples

Common mistakes (steel-manned)

Active recall

Connections

• Laminar flame speed (S_L) — the quantity we derived; depends on $\phi$, $T_u$, pressure.
• Mixture fraction Z and conserved scalars — math backbone of diffusion flames.
• Flashback and blow-off limits — direct consequence of $S_L$ vs gas velocity.
• Soot formation and the rich premixed zone — why diffusion flames glow.
• Damköhler number — $Da=t_{flow}/t_{chem}$ formalizes "mixing- vs reaction-limited".
• Adiabatic flame temperature — sets $T_b$ used in the speed derivation.
• Diesel vs SI engine combustion — the propulsion payoff of this whole note.

Concept Map

Hinglish (regional understanding)