Energy Balance at the Material Surface

Conducting an energy balance per unit area allows a closer analysis of the fuel surface. This implies the assumption that the energy is distributed homogeneously across the surface. This assumption is justifiable for most of the surface area of the fuel. The energy input to the surface is given by 

As it was done for ignition and flame spread, it is a common practice to simplify this term to a simple linearized total heat-transfer coefficient 

Finally, in-depth conduction is given by the temperature gradient at the surface 

This term varies significantly with the nature of the fuel. Liquid fuels are characterized by the presence of recirculation currents induced by buoyancy. These currents homogenize the temperature distributions, reducing in-depth conduction. Nevertheless, convective motion transfers heat to the interior of the pool leading to an additional term of in-depth convection. Convective heat transfer in polymer melts has a similar behavior and could potentially have an important effect on burning rates nevertheless, the impact of polymer melt rheology on the flammability of materials is still a matter of great controversy. 

Assuming that in-depth heat transfer is only by conduction, the energy balance at the surface is then given by 

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