Foundations — Premixed vs diffusion flames

This page assumes you know nothing. We build every letter the parent note used, one at a time, each with a picture and a reason it exists. When you finish, go read the parent topic and every symbol will already be an old friend.

0. What is a flame, in pictures?

Before any symbol, fix the mental image. Fuel (something that wants to give away electrons, like the carbon and hydrogen in ${\text{CH}}_4$) and an oxidizer (something that wants to grab them, usually the ${\text{O}}_2$ in air) sit near each other. When they are hot enough, they collide and rearrange into new molecules (like ${\text{CO}}_2$ and water), releasing heat. That heat keeps the next batch hot. A flame is that self-feeding hot reaction zone.

1. Concentration: mass fraction $Y$

The parent note writes things like $Y_{O,2}$ and $Y_{F,1}$. What is a $Y$?

The picture: imagine a jar of 100 marbles. If 23 are blue (oxygen) and 77 are grey (nitrogen), then $Y_{\text{blue}}=0.23$. That is exactly why the parent uses $Y_{O,2}=0.233$: in ordinary air, 23.3% of the mass is oxygen (the rest is mostly nitrogen).

Why the topic needs it: to say "fuel and oxidizer meet in the right ratio", you must first be able to count "how much fuel" and "how much oxidizer" — $Y$ is that count.

The subscripts just say which substance and which stream:

• $Y_{F,1}$ :: fuel ($F$) in the fuel stream (stream $1$).
• $Y_{O,2}$ :: oxidizer ($O$) in the oxidizer stream (stream $2$).

2. Stoichiometry: the ratio $s$

How to get it from a reaction equation. For methane: ${\text{CH}}_4+2{\text{O}}_2\to {\text{CO}}_2+2{\text{H}}_2\text{O}$ One molecule of ${\text{CH}}_4$ (mass 16) uses two molecules of ${\text{O}}_2$ (mass $2\times 32=64$). So per kg of fuel we need $64/16=4$ kg of oxygen: $s=\frac{2\times 32}{16}=4.$

The picture: four oxygen-marbles must arrive for every one fuel-marble, or someone goes hungry.

Why the topic needs it: the whole idea of a diffusion flame is "the flame sits where reactants arrive stoichiometrically". Without $s$ you can't say what "the right ratio" means.

3. Where the flame sits: mixture fraction $Z$ and $Z_{st}$

Now imagine two separate streams flowing toward each other: pure fuel from the left, air from the right. Somewhere in the middle they blend.

The picture: stand anywhere in the mixing region and ask "if I trace every molecule back, how much of my mass was born in the fuel tank?" That number is $Z$. It slides smoothly from 1 on the fuel side to 0 on the air side.

Using $s$, $Y_{F,1}$ and $Y_{O,2}$ from before: $Z_{st}=\frac{Y_{O,2}/s}{Y_{F,1}+Y_{O,2}/s}.$

Why the topic needs it: a diffusion flame doesn't travel; it just parks at $Z=Z_{st}$. This one number tells you where in space the flame will be. For methane in air it comes out to about $0.055$ — meaning the flame sits deep on the air side (see the parent's worked example).

4. How heat spreads: thermal diffusivity $\alpha$ (and $k$, $\rho$, $c_p$)

The parent's flame-speed derivation lives or dies on one idea: a flame advances by warming the gas just ahead of it. To measure "how fast warmth spreads", we need $\alpha$.

Three helper quantities first:

The picture: drop dye in still water and watch the blot grow. In time $t$ the blot spreads a distance of roughly $\sqrt{\alpha t}$. Heat does the same thing with temperature instead of dye.

Why the topic needs it: the flame preheats fresh gas by conduction. $\alpha$ is the number that says how quickly that preheating outruns into the cold gas — which sets both flame speed and thickness.

5. The flame's fingerprints: thickness $\delta$, speed $S_L$

The picture: stand in the frame of the fresh gas. The flame wall marches toward you at $S_L$. On a burner, the gas flows out at $S_L$ while the flame tries to come in at $S_L$ — they cancel and the flame looks frozen in place. That balance is the whole story of flashback (Flashback and blow-off limits) and blow-off.

Why the topic needs it: $S_L$ is the number that characterizes a premixed flame. A diffusion flame has none (it isn't travelling into anything), which is precisely why the two families are different.

6. How fast the chemistry runs: reaction rate $\dot{\omega }$ and $t_{chem}$

The picture: $\dot{\omega }$ is the flow of a waterfall; $t_{chem}$ is how long to empty the bucket. Fast waterfall (big $\dot{\omega }$) empties fast (small $t_{chem}$).

Why the topic needs it: a premixed flame is reaction-limited. Its speed is a race between how fast heat spreads ($\alpha$) and how fast fuel burns ($1/t_{chem}$). That race gives the parent's headline result $S_L\sim \sqrt{\alpha /t_{chem}}$ — the geometric mean of transport and chemistry. See also Laminar flame speed (S_L).

7. The "$\sim$" sign and why we scale, not solve

The parent uses $\sim$ ("scales like / of the same order as"), not $=$.

Why the topic needs it: the honest job of the derivation is to reveal dependence — that doubling $\alpha$ raises $S_L$ by $\sqrt{2}$, that thinner flames come from faster chemistry. A full $=$ needs messy constants; the physics lives in the $\sim$.

8. How it all connects

The left branch (colours, recipe, position) builds the diffusion side; the right branch (heat spread + chemistry race) builds the premixed side. Both feed the parent topic.

9. Two quick self-checks

Equipment checklist

Cover the right side and answer out loud. If any fails, re-read that section before the parent note.