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However, I'm not going to use the stagnant gas film model as shown in the figure, since I lack data for the film thickness, and I want to evaluate the problem numerically. Instead, I'm going to use spherical coordinates with boundaries r = R, the radius of the carbon particle, and r = ∞. I'm going to assign A to O

_{2}and B to CO

_{2}for the mathematical models. I will be using molar fractions because the global molar density of the system can be considered to be constant. The diffusion equation, concentration profile and molar flux in terms of A are given by

[tex]\frac{d}{dr} \left( r^2 \frac{dx_A}{dr} \right) = 0[/tex]

[tex]x_A = x_{A \infty} \left( 1 - \frac{R}{r} \right)[/tex]

[tex]N_{Ar} = - \frac{c \ D_{AB} \ x_{A \infty} R}{r^2}[/tex]

Where x

_{A∞}is the molar fraction of oxygen at r = ∞. The negative sign in the molar flux expression is there because oxygen is diffusing towards the carbon particle. My problem lies within giving a numerical value to x

_{A∞}. The statement says the particle burns in presence of

**air**, so first I let x

_{A∞}= 0.21, but that creates a problem. For this scenario, [itex]N_{Ar} = - N_{Br}[/itex], and if I make the analysis in terms of B, I obtain the following expressions

[tex]x_B = x_{BR} \frac{R}{r}[/tex]

[tex]N_{Br} = \frac{c \ D_{AB} \ x_{BR} R}{r^2}[/tex]

The problem is that x

_{BR}, the molar fraction of CO

_{2}in the surface of the carbon particle, is always 1. Then I'm forced to assign x

_{A∞}= 1, in order for [itex]N_{Ar} = - N_{Br}[/itex] to be satisfied. The problem is that the particle is burning in air, not oxygen, so what about the presence of nitrogen? Do we just consider species A and B and neglect any other species present in the system? Do we ignore nitrogen because it is not participating in the mass transfer process? The process is diffusion-controlled, so I ignored the kinetics of the combustion and setted [itex]x_A |_{r=R} = 0[/itex] and [itex]x_{BR} = x_B |_{r=R} = 1[/itex].

My ultimate goal is to calculate the time it takes for the carbon particle to consume completely, using a quasi-steady state analysis. I have enough data to calculate this, but I want to sort this boundary condition issue out first.

The data I have are:

Carbon particle diameter: 0.05 in

Carbon particle density: 85 lb ft

^{-3}

Pressure: 1 atm

Temperature: 2500 °F

Diffusivity: 8 ft

^{2}hr

^{-1}

Thanks in advance for any input!