Thermodynamics of Carbothermic Reduction and Impurity Partitioning

01Thermodynamic Phase Equilibria in the Si-O-C System

The carbothermic reduction of silica ( $SiO_{2}$ ) is fundamentally constrained by the phase equilibria of the ternary Si-O-C system. Unlike simple metal reductions (such as iron smelting), silicon reduction cannot proceed directly from oxide to metal without passing through condensed intermediate compounds ( $SiC$ ) and volatile gaseous phases ( $SiO$ ).

[ Sio2 (Solid/Liquid) ]Schematic · source diagram

Free Energy Dependencies & Ellingham Analysis

The thermodynamic feasibility of reactions within the furnace shaft is governed by the standard Gibbs free energy change ( $Δ G^{\circ}$ ) as a function of temperature (T in Kelvin):

Formation of Silicon Carbide ( $SiC$ ):

SiO_{2} (s) + 3 C (s) \to SiC (s) + 2 CO (g)

Δ G^{\circ} = 607, 000 - 347.7 \cdot T (J/mol)

Equilibrium temperature ( $T_{e q}$ where $Δ G^{\circ} = 0$ at $P_{CO} = 1 atm$ ): $\approx 1746 K (147 3^{\circ} C)$ .

Primary Reduction to Liquid Silicon ( $Si$ ):

SiO_{2} (l) + 2 SiC (s) \to 3 Si (l) + 2 CO (g)

Δ G^{\circ} = 688, 000 - 358.1 \cdot T (J/mol)

Equilibrium temperature ( $T_{e q}$ at $P_{CO} = 1 atm$ ): $\approx 1921 K (164 8^{\circ} C)$ .

Volatile Silicon Monoxide Generation:

SiO_{2} (l) + Si (l) ⇌ 2 SiO (g)

Δ G^{\circ} = 521, 000 - 336.2 \cdot T (J/mol)

02Volatilization Dynamics and Silicon Yield Equations

The total silicon yield ( $η_{Si}$ ) of a submerged arc furnace is dictated by the fraction of silicon lost as volatile $SiO$ gas escaping through the top charge shaft.

[ Total Si Input (Sio2) ]Schematic · source diagram

The thermodynamic minimum partial pressure of $SiO$ gas inside the arc zone is defined by the equilibrium constant $K_{p}$ :

P_{SiO} = K_{p} \cdot a_{SiO_{2}} \cdot a_{Si}

Where $a_{SiO_{2}}$ and $a_{Si}$ represent the thermodynamic activities of liquid silica and liquid silicon, respectively. If the upper bed temperature drops below $140 0^{\circ} C$ or lacks reactive carbon surface area, $SiO$ gas cannot react according to $SiO + 2 C \to SiC + CO$ , resulting in massive silicon yield degradation. Overall silicon recovery is modeled by:

η_{Si} = 1 - \frac{n ˙ _{SiO, exhaust}}{n ˙ _{Si, input}}

Modern high-efficiency furnaces achieve $η_{Si} = 85% - 92%$ , with remaining losses manifesting as silica fume ( $SiO_{2}$ micro-dust) in the gas abatement baghouses.

*FIG. 1*A high-precision photorealistic laboratory molten metal testing setup

03Impurity Partitioning Thermodynamics

During carbothermic reduction, trace metallic and non-metallic elements present in quartzite and carbon reductants partition between the liquid silicon phase, slag phase, and off-gas stream.

Raw Feedstock Impurities (Fe, Al, Ca, B, P, Ti)Schematic · source diagram

Thermodynamics of Trace Impurity Elements:

Impurity Element	Thermodynamic Behavior & Partition Coefficient ( $k_{p}$ )	Primary Source	Fate / Removal Strategy
Iron ( $Fe$ )	$k_{p} \approx 1.0$ ; Nearly $100%$ retains in liquid silicon.	Quartzite, Steel tools	Cannot be removed by ladle refining; requires chemical chlorination.
Aluminum ( $Al$ )	$k_{p} \approx 0.7 - 0.9$ ; High oxide affinity ( $Al_{2} O_{3}$ ).	Ash in coal, Clay minerals	Oxidized via ladle oxygen bubbling ( $Al \to Al_{2} O_{3}$ slag).
Calcium ( $Ca$ )	$k_{p} \approx 0.5 - 0.8$ ; Volatile at arc temperatures.	Woodchips, Limestone	Oxidized via oxygen/chlorine gas bubbling into floating slag.
Titanium ( $Ti$ )	$k_{p} \approx 0.95$ ; Forms stable titanium silicides ( $TiSi_{2}$ ).	Heavy minerals in quartzite	Retained in liquid metal; requires chemical distillation downstream.
Boron ( $B$ )	$k_{p} \approx 0.8 - 0.9$ ; Extremely stable in molten Si.	Coal ash	Extremely difficult to remove; strictly controlled at feedstock intake.
Phosphorus ( $P$ )	$k_{p} \approx 0.6 - 0.8$ ; Highly volatile gas species.	Coal, Charcoal	Partially volatilizes; remainder retained in silicon matrix.

*FIG. 2*A thermodynamic Ellingham-style free energy diagram ( G^ vs Temperature in Kelvin) specifically showing the equilibrium lines for SiO_2 + 3C SiC + 2CO, SiO_2 + 2SiC 3Si + 2CO, and SiO_2 + Si 2SiO

04Mathematical Model of Ladle Slag Refining Kinematics

The oxidative removal rate of calcium and aluminum from liquid silicon in the refining ladle follows first-order mass-transfer-controlled kinetics:

\frac{d [ % Me ]}{d t} = - k_{m} \cdot \frac{A}{V} \cdot ([% Me] - [% Me]_{eq})

Where:

$[% Me]$ is the weight percentage of impurity metal ( $Al$ or $Ca$ ) in molten silicon.
$k_{m}$ is the overall mass transfer coefficient ( $m/s$ ).
A/V is the interfacial reaction area per unit melt volume ( $m^{- 1}$ ).
$[% Me]_{eq}$ is the thermodynamic equilibrium concentration at the slag-metal interface.

By injecting gas fine bubbles ( $O_{2} / Cl_{2}$ ) through bottom porous plugs, A/V is maximized, reducing $Al$ and $Ca$ levels from $> 0.5%$ down to $< 0.02%$ in under 45 minutes.

*FIG. 3*A detailed matrix infographic titled 'Impurity Partitioning in Silicon Smelting'

→Prerequisites & continue reading

Prerequisites: Advanced thermodynamics (Ellingham diagrams, Gibbs free energy, phase equilibria)

Also in Topic 1.1, Quartzite Extraction & Carbothermic Reduction:

Tier 1 · From Sand to Silicon: The Journey of Quartzite Reduc…Tier 2 · Inside the Submerged Arc Furnace: Thermal & Electric…