Classical thermodynamics, heat transfer Definition

Premixed flame

Builds on Combustion theory Traveling wave Reaction–diffusion system

Definition

Take the reacting flow model of combustion theory in one space variable $x$ , for a gas at rest. Let $T(x, t)$ be the temperature, $Y(x, t)$ the fuel mass fraction and $\omega(T, Y)$ the reaction rate. Let $(T_u, Y_u)$ with $Y_u > 0$ be the state of the fresh mixture, and $(T_b, 0)$ the state of the burnt gas, with $T_b = T_u + q Y_u / c_p$ . A planar premixed flame is a travelling wave solution $T(x, t) = \Theta(x + S_L t), \qquad Y(x, t) = \Phi(x + S_L t)$ with speed $S_L > 0$ such that:

$(\Theta(\xi), \Phi(\xi)) \to (T_u, Y_u)$ as $\xi \to -\infty$ ;
$(\Theta(\xi), \Phi(\xi)) \to (T_b, 0)$ as $\xi \to +\infty$ .

The flame moves towards $x = -\infty$ , into the fresh mixture, and $S_L$ is the laminar burning velocity. A premixed flame in general is a reaction zone separating fresh premixed gas from burnt gas and moving into the fresh gas.

Remarks

The Arrhenius rate is positive at $T_u$ , so the fresh state is not exactly at rest and no exact travelling wave exists. The standard fix sets $\omega = 0$ below an ignition temperature between $T_u$ and $T_b$ . With this cut-off, the travelling wave exists and its speed is unique, as for other ignition-type reaction–diffusion systems. Scaling: if $\kappa$ is the thermal diffusivity and $\tau$ the reaction time, then $S_L$ is of order $\sqrt{\kappa / \tau}$ and the flame thickness is of order $\kappa / S_L$ . Check of the burnt state: when $\kappa = D$ , the quantity $H = T + qY/c_p$ solves the heat equation with no source. A bounded travelling wave keeps $H$ equal to its value $T_u + q Y_u / c_p = T_b$ in the fresh gas, so $Y = 0$ gives $T = T_b$ . Non-example: in a diffusion flame, fuel and oxidiser arrive from opposite sides, and the flame does not propagate into a fresh mixture.

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Tag	`RWZ4JG3RE1`
URI	`m2://RWZ4JG3RE1`
Kind	definition
Writing	W0 stub
MSC	80A25, 76V05
Defines	Premixed flame
Uses	Combustion theory, Traveling wave, Reaction–diffusion system
Changed	3 November 2026

Premixed flame

Definition

Remarks

Reading path

Further reading

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