FeO Reduction by Carbon via a Gas-Halo Intermediate: 21-Mat-A2 Materials Transport Phenomena, December 2013, Q2

Question 2: FeO Reduction by Carbon via a Gas-Halo Intermediate (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Reaction pair: at the droplet/char surface, $C_{(droplet)} + CO_2 \rightarrow 2CO$; at the slag-halo interface, $CO + FeO \rightarrow Fe + CO_2$ — net $C + FeO \rightarrow Fe + CO$, with CO₂ shuttling across the halo as the diffusing intermediate. $T=1600\,{}^{\circ}\text{C}=1873\,\text{K}$, gas pressure in halo $P=1.2$ atm, halo thickness $\delta=120\,\mu\text{m}$, $D_{CO}=D_{CO_2}=1.0\times10^{-3}\ \text{m}^2/\text{s}$, bulk (slag-side) $y_{CO_2}=6\%$, droplet-side $C_{CO_2}\approx0$ (fast interfacial reaction). Fe smelting rate $=4$ t/hr, slag $=40$ t at 6 wt% FeO, $\rho_{slag}=3300\ \text{kg/m}^3$, average particle diameter $d_p=1$ cm.

Find. Total char + droplet surface area $A$ sustaining the observed smelting rate, and the resulting dispersed-phase volumetric loading in the slag.

CO₂/CO diffuse across the thin (120 μm) gas halo separating the char/droplet surface from the slag; CO₂ is the species consumed at the droplet and regenerated at the slag interface.

Approach. Since $\delta\ll d_p$, treat the halo as a flat stagnant film: Fick’s law gives the CO₂ flux from the slag-side (6%) to the droplet-side ($\approx0$) interface; this flux fixes the molar rate of FeO reduction per unit area (1:1 stoichiometry through the coupled reactions), which combined with the observed Fe production rate gives the required total interfacial area, then the particle count and dispersed volume.

Total gas concentration and CO₂ driving force in the halo. $C_{tot}=\dfrac{P}{RT}=\dfrac{1.2\times101325}{8.314\times1873}=7.808\ \text{mol/m}^3$. Bulk (slag-side) $C_{CO_2}=0.06\times7.808=0.4685\ \text{mol/m}^3$; droplet-side $\approx0$.
Fick’s-law flux across the thin film. $N_{CO_2}=\dfrac{D\,\Delta C}{\delta}=\dfrac{(1.0\times10^{-3})(0.4685)}{120\times10^{-6}}$ $$\boxed{N_{CO_2}=3.904\ \text{mol/(m}^2\text{s)}}$$ This flux equals the molar rate of both carbon consumption at the droplet and FeO reduction at the slag interface per unit area (each mole of CO₂ delivered drives one net mole of FeO→Fe).
Observed molar Fe production rate. $\dot n_{Fe}=\dfrac{4000\ \text{kg/hr}\times1000}{56\ \text{g/mol}\times3600\ \text{s/hr}}=19.84\ \text{mol/s}$.
Required total interfacial area. Substituting into $\dot n_{Fe}=N_{CO_2}\,A$: $$A=\dfrac{19.84}{3.904}\Rightarrow\boxed{A=5.08\ \text{m}^2}$$ of combined char + droplet surface must be present in the slag at any instant.
Particle count and dispersed volume. Per 1 cm-diameter sphere: $a_p=\pi d_p^2=3.142\times10^{-4}\ \text{m}^2$, $v_p=\tfrac{\pi}{6}d_p^3=5.236\times10^{-7}\ \text{m}^3$. $N=A/a_p=5.08/3.142\times10^{-4}=1.62\times10^4$ particles; $V_p=N\,v_p=1.62\times10^4\times5.236\times10^{-7}=8.47\times10^{-3}\ \text{m}^3$.
Volumetric loading. $V_{slag}=\dfrac{40{,}000}{3300}=12.12\ \text{m}^3$. $$\boxed{\%\,\text{loading}=\dfrac{V_p}{V_{slag}}\times100=\dfrac{8.47\times10^{-3}}{12.12}\times100=0.070\%}$$ — a very dilute dispersion, consistent with a foaming slag carrying suspended reacting particles rather than a packed bed.

Question 2 — final results
Quantity	Value
CO₂ flux across halo	3.90 mol/(m²·s)
Required char + droplet area	5.08 m²
Particle count (1 cm dia.)	≈ 1.62 × 10⁴
Dispersed volumetric loading	0.070 vol%

Topic: Gas-film-controlled indirect reduction (carbon–gas–slag mechanism).

Key relations: Fick’s first law across a thin stagnant film, $N_A=D\Delta C/\delta$; ideal-gas concentration $C=P/RT$; sphere surface/volume $A=\pi d^2$, $V=\pi d^3/6$.

Why this works: Because the two interfacial reactions ($C+CO_2\to2CO$ at the droplet, $CO+FeO\to Fe+CO_2$ at the slag) are fast relative to gas-phase diffusion, the halo behaves as a simple diffusion resistance in series with negligible reaction resistance, so a linear concentration profile and steady flat-film Fick’s law are valid despite the curved geometry, because $\delta\ll d_p$.

Common pitfall: Forgetting that CO₂ (not CO) is the species whose flux directly paces FeO reduction, or applying Fick’s law with the wrong driving-force side (using total P instead of the CO₂ partial concentration difference).

Source: Szekely & Themelis, Rate Phenomena in Process Metallurgy, gas-film mass-transfer chapters; Turkdogan, Fundamentals of Steelmaking, indirect-reduction mechanism.

Question 2 of 9: FeO Reduction by Carbon via a Gas-Halo Intermediate

Question 2: FeO Reduction by Carbon via a Gas-Halo Intermediate (20 marks)