Burning equations for a single particle

The combustion of a fine particle of solid fuel has many identical features to that of a liquid droplet but with important differences. One of the differences is that, in addition to no evaporation existing, the combustion of particles of solid fuel is frequently controlled by chemical kinetics, i.e., diffusion-governing is not the only possibility. 

Let us consider the burning of an ideal spherical particle in static gas. The oxidant diffuses to the surface of the particle to react with the carbon C + CO2, while the latter diffuses out from the surface of the particle. The combustion heat is transferred to the surrounding gas partially by convection and partially by radiation. The following assumptions were made in the modeling (1) The process is at a pseudo steady state. (2) The temperature the highest at the surface, and continuously drops down outwards from the surface of the particle and the concentration of oxidant is highest in the bulk 

The diffusion equation for the oxidant round the surface of the particle is written as 

Similarly, since M s = 0, there would be Ms = Mavs + M s = Mox s. The burning rate per unit surface area can be represented by 

The minus sign in the equation indicates that the motions of carbon and oxidant are in opposite directions. Using the continuity equation, Eq. (8.3), and Eq. (8.13) and with the known concentration of oxidant at the surface of the particle, mox s, the integration between r = rpo and r = x results in the equation for the flux of burning to be 

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