Calculate the time required to burn to complete spherical particles of graphite (radius 12 mm, bulk density 2.4 g/cc) in a 12% oxygen stream at 900°C and 1 atm. Assume gas film resistance to be negligible. Surface reaction rate constant = k” = 25 cm/s.

Solution :

In burning of particle of graphite there is no ash formation. The reacting particle shrinking in size as reaction progresses. As per data k" is only used in Chemical reaction control

Formulae for Chemical reaction control

$$ \tau=\frac{\rho_{\mathrm{B}} \mathrm{R}_{0}}{\mathrm{bk}^{\prime \prime} \mathrm{C}_{\mathrm{Ag}}} $$

Combustion Reaction :

$$ \begin{array}{ll} & \mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{CO}_{2}(\mathrm{g}) \end{array}$$

compare with $$\mathrm{A}(\mathrm{g})+\mathrm{bB}(\mathrm{s}) \rightarrow \text { fluid product }$$

$ \therefore \mathrm{b}=1 \text { (stoichiometric coefficient of C }: \text { Carbon) } $

$ \mathrm{R}_{0}=5 \mathrm{~mm}=0.5 \mathrm{~cm} $

Density of graphite $=\rho_{\mathrm{B}}=2.2 \mathrm{~g} / \mathrm{cm}^{3}$

rate constant $=\mathrm{k}^{\prime \prime}=20 \mathrm{~cm} / \mathrm{s}$

Evaluate $\mathrm {C_{A g}}$ : bulk gas phase concentration of $\mathrm {A}$.

Here $\mathrm{A}$ is $\mathrm{O}_{2}$.

$$ \mathrm {C_{A g}=\frac{p_{A g}}{R_{T}}} $$

A in gas stream $-8 \%$ by volume, total pressure $=1$ atm.

For an ideal gas : volume $\% \mathrm {A}=$ pressure of $\mathrm {A}$

$$\begin {aligned} \text { pressure } \% \mathrm{~A} &=8=\frac{\mathrm{P}_{\mathrm{Ag}}}{\mathrm{P}} \times 100 \\ \therefore \quad \mathrm{P}_{\mathrm{Ag}} &=\frac{8 \times 1}{100}=0.08 \mathrm{~atm} \end{aligned}$$

$$ \begin{aligned} T &=900^{\circ} \mathrm{C}=1173 \mathrm{~K} \\ \mathrm{R} &=82.06\left(\mathrm{~cm}^{3} \cdot \mathrm{atm}\right) / \mathrm{mol} . \mathrm{K} \\ \mathrm{C}_{\mathrm{A}} &=\frac{0.08}{82.06 \times 1173}=8.31 \times 10^{-7} \mathrm{~mol} / \mathrm{cm}^{3} \end{aligned} $$

$\rho_{B} =2.2 \mathrm{~g} / \mathrm{cm}^{3} $ convert this mass density into molar density mol/cm³

$$\begin {aligned} \rho_{B} &=\frac{2.2}{12}=0.183 \mathrm{~mol} / \mathrm{cm}^{3},(\mathrm{At}.\text { Wt. } \mathrm{C}=12) \\ \tau &=\frac{0.183 \times 0.50}{1 \times 20 \times 8.31 \times 10^{-7}}=5505.4 \mathrm{~s} \end{aligned} $$

Time required for complete burning of particles of graphite $=5505.45$ or $\sim 1.53 \mathrm{~h}$