21-Mat-A2 Materials Transport Phenomena · December 2015
Question 3 of 8: Direct Steelmaking — Gas-Film-Controlled FeO Reduction
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
National Exams — December 2015 — 10-Met-A2 Metallurgical Rate Phenomena. Three-hour, open-book exam; any non-communicating calculator permitted. Candidates answer Question 1 (compulsory) plus any four of Questions 2–8; the five answered count equally (20 marks each). All eight are solved below for completeness. Candidates were told to state any interpretive assumptions where doubt exists.
Reference texts: Geiger, G. H. & Poirier, D. R., Transport Phenomena in Materials Processing (TMS) — the primary reference for the momentum-, heat- and mass-transfer analyses in Questions 3, 6, 7 and 8; Szekely, J., Evans, J. W. & Sohn, H. Y., Rate Phenomena in Process Metallurgy — reactor-network and interfacial mass-transfer modelling (Questions 2, 3); Welty, J. R. et al., Fundamentals of Momentum, Heat and Mass Transfer — dimensional analysis and pipe-friction data (Questions 4, 7); Gaskell, D. R., Introduction to the Thermodynamics of Materials — Fe–C phase relations (Question 1h).
Question 3: Direct Steelmaking — Gas-Film-Controlled FeO Reduction (20 marks)
Fig. 3 — iron droplet / char particle surrounded by a 120 µm gas halo; CO diffuses out, CO₂ diffuses in.
Given.
Quantity
Symbol
Value
Fe smelting rate
$\dot m_{Fe}$
4 tonnes/hr
Slag FeO content
—
6 wt%
Slag weight
$m_{slag}$
40 tonnes
Gas film thickness
$\delta$
120 µm
Diffusivity, $D_{CO}=D_{CO_2}$
$D$
$1.0\times10^{-3}$ m$^2$/s
Halo gas pressure
$P$
1.2 atm
Halo gas temperature
$T$
1873 K
CO$_2$ mole fraction at slag interface
$y_{CO_2,s}$
0.06
Droplet/char diameter
$d_p$
1 cm
Slag density
$\rho_{slag}$
3300 kg/m$^3$
Find. Total interfacial (droplet + char) surface area required, and the resulting dispersed-phase volumetric loading of the slag.
Approach. Treat the 120-µm halo as a stagnant film supporting equimolar counter-diffusion of CO (outward, generated at the carbon surface) and CO$_2$ (inward, consumed there); use Fick's law for the film flux, match it to the observed Fe production rate via the slag-interface stoichiometry ($1\,\text{mol CO}\to1\,\text{mol Fe}$), then convert the resulting area to a particle count and dispersed volume.
Total gas concentration in the halo (ideal gas).
$$C_{tot}=\dfrac{P}{RT}=\dfrac{(1.2)(101{,}325)}{(8.314)(1873)}=\boxed{7.81\ \text{mol/m}^3}$$
Driving force for CO across the film. The two interfacial reactions add to the overall reaction $FeO+\underline{C}\to Fe+CO$ (carbon-boat/CO intermediate reduction), so CO is essentially exhausted (consumed by FeO) at the outer, slag-side edge of the halo ($y_{CO,s}=1-y_{CO_2,s}=0.94$) while it is essentially pure at the inner, droplet/char surface ($y_{CO,d}\approx1$, since CO$_2$ there is consumed as fast as it arrives). Under the equimolar counter-diffusion (stagnant-film) approximation,
$$N_{CO}=\dfrac{D\,C_{tot}(y_{CO,d}-y_{CO,s})}{\delta}=\dfrac{(1.0\times10^{-3})(7.81)(1-0.94)}{120\times10^{-6}}=\boxed{3.90\ \text{mol/(m}^2\text{s)}}$$
Fe production rate, molar basis.
$$\dot n_{Fe}=\dfrac{\dot m_{Fe}}{M_{Fe}}=\dfrac{4000\ \text{kg/hr}}{3600\ \text{s/hr}}\div0.056\ \text{kg/mol}=\boxed{19.8\ \text{mol/s}}$$
Required interfacial area. Stoichiometry ($CO+FeO\to Fe+CO_2$) needs 1 mol CO per mol Fe, so
$$A_{tot}=\dfrac{\dot n_{Fe}}{N_{CO}}=\dfrac{19.8}{3.90}=\boxed{5.08\ \text{m}^2}$$
Particle count and dispersed volume. With $d_p=1$ cm spheres, area per particle $=\pi d_p^2=3.14\times10^{-4}$ m$^2$, so
$$n_p=\dfrac{A_{tot}}{\pi d_p^2}=\dfrac{5.08}{3.14\times10^{-4}}\approx1.62\times10^4\ \text{particles}$$
$$V_{p}=n_p\left(\dfrac{\pi}{6}d_p^3\right)=8.47\times10^{-3}\ \text{m}^3$$
Volumetric loading. Slag volume $V_{slag}=m_{slag}/\rho_{slag}=40{,}000/3300=12.1$ m$^3$, so
$$\boxed{\text{loading}=\dfrac{V_p}{V_{slag}}\times100\approx0.070\%}$$
Quantity
Value
CO film flux, $N_{CO}$
3.90 mol/(m²·s)
Total droplet+char surface area, $A_{tot}$
5.08 m²
Equivalent particle count (1 cm spheres)
≈ 1.62×10&sup4;
Dispersed-phase volumetric loading
≈ 0.070% of slag volume
Check
Assumes (i) equimolar counter-diffusion of CO/CO$_2$ across the film — the real halo has a net outward molar flow (2 mol CO generated per 1 mol CO$_2$ consumed at the droplet), so this is a simplifying, exam-level approximation of a genuinely Stefan-flow problem; (ii) $y_{CO}\approx1$ at the droplet/char surface (complete local consumption of incoming CO$_2$); (iii) the outer (slag-side) halo area is close enough to the inner (particle) area to use $\pi d_p^2$ directly, valid since $\delta=120\ \mu$m $\ll d_p=1$ cm; (iv) droplets and char are lumped into one dispersed population sharing the computed total area, since the trial data do not split the smelting rate between the two solid types.
