# Two Simplified Models of Premixed Laminar Flame Propagation

To get an appreciation of the physics of an industrial explosion, it is beneficial to understand the basics of flame dynamics and flame propagation under simplified conditions. This post briefly reviews two models of laminar flame propagation through a gas/air mixture to demonstrate the important system parameters. Future posts will look at the effect of non-premixed flame propagation and turbulence, as well industrial explosion of multiphase (solid/gas/air) mixtures. Readers interested in a more detailed explanation of the fundamental theories for laminar flame propagation and their mathematical derivations, are encouraged to see Chapter 5 in the textbook of Kuo [1] from which the following discussion was derived.

A flame front is a reaction zone which moves through a mixture of fuel and oxidizer, leaving burnt products in its wake. In most industrial cases involving flammable gas the oxidizer is the oxygen molecules from the air, and the gas is used as part of the processing application (e.g., methane, hydrogen, etc.). Inside the flame front the fuel is heated and reacts, providing energy which manifests itself as temperature and light. Once the flame temperature becomes high enough, it can heat the fuel ahead of it to ignition and the flame becomes self-propagating.

The self-propagating feature leads to a standing flame in the Bunsen burner apparatus shown above. The fresh fuel is supplied by a tube to the left of the image and reacts in the flame front. The color, shape, and height of the flame depends on the fuel/air mixture and the inflow velocity. Under industrial conditions a self-propagating flame may continue through the entire area containing the gas cloud, causing destructive temperature and overpressure.

## Thermal Theory

The first models of premixed flame propagation were based on the work of Mallard and Le Chatelier, who postulated that heat transfer (specifically thermal conductivity) is the driving mechanism. In their work the flame was divided into two regions: an inert Preheat Zone and a chemically active Reaction Zone. A one-dimensional schematic of this concept is shown in the figure below. In this figure the flame is propagating from right-to-left. Thermal conduction drives heat from the high-temperature reaction zone into the preheat zone. In the preheat zone the fresh fuel is brought to the fuel ignition temperature. At this temperature chemical reactions start, converting the fresh fuel to burnt products.

The goal of flame propagation models is generally to predict two parameters: the maximum flame temperature ($T_{\text{F}}$) and the laminar flame speed ($S_{\text{L}}$). The maximum flame temperature can be calculated directly from thermodynamics assuming equilibrium between the fuel and reaction products. The laminar flame speed depends on the overall dynamics of the fuel system and the interaction between the physical and chemical processes. From a mathematical description of the system above, Le Chatelier predicted that the flame speed is proportional to the square root of the chemical reaction rate $(RR, \text{1/s})$ and the thermal diffusivity of the gas $(\alpha, \text{m}^{2}\text{/s})$, giving the mathematic relation $S_{\text{L}} \propto \sqrt{RR\alpha}$.

## Diffusion Theory

A second group of flame propagation models were developed that assume that diffusion of gaseous species, and specifically the intermediate species of the chemical reactions, is the driving mechanism. For example in a hydrogen gas flame the global reaction $\text{H}_{2} + \text{O}_{2} \rightarrow 2\text{H}_{2}\text{O}$ occurs, but is actually broken into several intermediate processes including chain-branching and chain-terminating reactions (See Chapter 2 in Kuo [1] for a description of chemical kinetic mechanisms and their role in combustion). For hydrogen combustion some of these reactions include the intermediate steps: $\text{H}_{2} + \text{O}_{2} \rightarrow \text{H} + \text{H}\text{O}_{2}$ and $\text{H} + \text{O}_{2} \rightarrow \text{OH} + \text{O}$, for example.

The schematic below illustrates the diffusion theory for flame propagation. The y-axis in the figure represents species molar fraction (proportional to the mass of species per unit volume). It is plotted in log scale as some of the intermediate species/radicals are present in very small amounts (e.g., O). However, these intermediate species may have very high mass diffusion rates. Under diffusion theory it is predicted that the movement of these intermediates from the reaction zone to the area ahead of the flame is the driving mechanism. Under the diffusion theory the flame speed can be shown to be proportional to square root of the sum of the specie mass diffusivities $(\mathcal{D}_{i}, \text{m}^{2}\text{/s})$ multiplied by their partial pressure $(p_{i})$, giving the mathematic relation $S_{\text{L}} \propto \sqrt{\sum({p_{i}\mathcal{D}_{i})}}$.

Thermal and diffusion theories represent limiting conditions for flame propagation, and neither fully describes the whole process. In reality both mechanisms occur simultaneously, and several other features such as temperature dependent thermophysical properties, detailed chemistry kinetics, instability formation, and flame stretching may also be important. That being said, these simplified models demonstrate three of the major considerations for estimating laminar flame speed and demonstrate the importance of chemical reactivity, thermal diffusivity, and mass diffusivity of flammable gas hazards.

[1] K. Kuo, Principles of combustion, John Wiley & Sons, Inc., 2005.
[Bibtex]
@BOOK{Kuo2005,
title={Principles of Combustion},
author={Kuo, K.},
publisher={John Wiley \& Sons, Inc.},
}