A stationary spherical particle with a diameter of $10$ mm$,$ consisting of solid carbon, is burning in air in the form of an ember.

a$)$ Derive the conversion time expression by establishing shell balances for oxygen and carbon.

b$)$ How long does it take for the particle to completely burn and disappear?

Given:

Bed porosity: $0\mathrm{.}5$

L$.585$

c$)$ What are the percentage proportions of the controlling resistances during combustion?

Given:

Temperature $($K$)$: $873$

Air viscosity $($kg$/$m$.$s$)$: $3\mathrm{.}897$ x $10^-8$

Diffusion coefficient of oxygen in air $($m$/$s$)$: $1\mathrm{.}528$ x $10^-4$

Air density $($kg$/$m$)$: $0\mathrm{.}4036$

Activation energy $($kcal$/$kgmol$)$: $35700$

Frequency factor $($kg$/$m$.$s$.$atm$)$: $87100$

Oxygen concentration in air $($kgmol$/$m$)$: $2\mathrm{.}643$ x $10^-3$