Question

A coal particle burns in air at 1145 K and atmospheric pressure. The process is limited by diffusion of the oxygen counterflow to the CO₂ and CO that are formed at the particle surface. Twice as much CO is formed as CO₂. Assume that the coal is pure carbon with a density of 1280 kg/m³ and that the particle is spherical with an initial diameter of 0.015 cm. Under the conditions of the combustion process, the diffusivity of oxygen in the gas mixture may be assumed to be 10^-4 m²/s. If the combustion is instantaneous:

Find an algebraic equation for the combustion rate of oxygen, W_A. State all assumption and boundary conditions.

Determine the time that is necessary to reduce the diameter to 0.005 cm.

mass transfer course

Answer 1

When carbon combines with oxygen from the air it given by

We have the diffusion of oxygen (A) toward the surface and diffusion of carbon dioxide (B) away from the surface. The molar flux of oxygen is given by

r is the radial distance from the center of the carbon so N _A,r = - N _{B, r}

The molar flux is not independant of r since the area of mass transfer is not a constant. Using quasi steady state assumption, the mole transfer rateis assumed to be independent of r at any instant of time

At the surface of the coal particle, the reaction rate is much faster than the diffusion rate to the surface so that the oxygen concentration can be considered to be zero: y _A,R = 0. Sepaating the variables and integrating gives

Since one mole of carbon will disappear for each mole of oxygen consumed at the surface

Making a carbon balance gives

Separating the variables and integrating from t = 0 to t gives

Know the total gas concentration can be obtained from the ideal gas law

where R = 0.08206 m³.atm/kmol.^oK

The time necessary to reduce the diameter of the carbon particle from 0.015 cm to 0.005 cm is then

convert cm to meter then 0.015 to 1.5 * 10 ^-4 m and 0.005 cm to 5 * 10^-5 m

Initial radius = 7.5x10^-5 m

final radius = 2.5 x 10 ^-5 m

then substituting the value from data in the following equation

we get