Equilibrium Simulation

Each experiment was simulated in an equilibrium model developed by Tang et al. [30], using the ChemSheet (GTT-Technologies [107]) add-on in Microsoft Excel [108]. The model uses ChemSheet to calculate equilibrium phases and compositions at minimum Gibbs free energy based on thermodynamic properties of phases and compounds in a database. The model includes equilibrium reactions at the interface and not the gas-phase reaction (Equation (3.4)), nor convection and diffusion.

H₂O + SiO = SiO2(s) + H₂ (3.4)

An extract of the database for thermodynamic properties of solar grade silicon materials by Tang et al. [39] is used for the melt, with Redlich-Kister polynomials for boron, oxygen and hydrogen impurities. The activity coefficient for boron corresponds to that in COST 507 [40] Thermochemical Database for Light Metal Alloys. Intermediate compounds, ox-ides and other solids in the H-B-O-Si system are included as pure phases. A liquid slag of silica and B₂O₃ is modeled with the quasi-chemical model for oxide solutions. The gas phase is an ideal mixture of gases in the H-B-O-Si system with data from JANAF [44]

Thermochemical Tables, except the standard enthalpy of formation of HBO is modified to Δ_fHHBO^−◦ (298 K) =−251 kJ/mol as recommended by Page [51]. The Gibbs free energy at standard state for selected compounds in the equilibrium model are in Figure 3.15 plotted over a relevant temperature for the melt surface. The enthalpy is referenced to the most stable state of elements at 1 bar and 298 K. The standard state is pure compound at 1 bar. The

values plotted in Figure 3.15 are calculated from modeling results of molar Gibbs free energy (Gj) and the activity (aj) of the compounds asGj= G⁻j^◦+ RT ln(aj).

Figure 3.15: Gibbs free energy at standard state for selected compounds in the equilibrium model. SiO₂(c) is cristobalite and both SiO₂(l) and B₂O₃(l) are components of a slag.

Inputs to the model are the initial boron concentration, temperature, pressure, gas compo-sition and flow rate, which were taken from experimental data in Section 3.3. ChemSheet uses the amount of substance for the equilibrium calculations, which was calculated from

ideal gas law according to Equation (3.5) for each input gas compound “j”. The gas flow rate (Q) is input in [lN/min] with the gas constantR = 0.082 lN· atm/mol/K and temperature TQ= 293 K. The equilibrium simulates the course of refining by iterative equilibrium calculations for ev-ery minute of gas blowing with steam. The input mass of silicon is the mass weighted into the crucible and that of boron is calculated from ICP-MS analysis of the boron content in the first sample. The input mass of oxygen in the melt is set close to saturation for silica formation.

The amount of gas calculated by Equation (3.5) with Δt = 1 min and the whole of the melt are input to ChemSheet for calculating the resulting equilibrium compositions. Remaining amounts in the melt is transferred to the next minute iteration and re-equilibrated with the next minute of gas supply, while the equilibrated gas amounts output from ChemSheet are discarded. Table 3.21 presents an excerpt from equilibrium simulation of experimentQ_16b.

Table 3.21: Excerpt from iterative equilibrium simulation of experimentQ_16b. The amount of gases in line Input is used for every iteration (Δt = 1 min), together with T = 1500^◦C for the melt andp = 1.12 bar measured in the furnace. Input masses in the melt are used in the first equilibrium calculation and the second iteration uses the masses in condensed phases (mj) from the first output, while the output gas amounts (Δnj) are discarded. Selected simu-lation outputs are listed for the first, second and 69th equilibrium calcusimu-lation (tr= 69.1 min forQ_16b), from which silicon weightloss compares to the weightloss of the crucible during the experiment. Additional parameters for calculating inputs: 79 ppmw boron in first sample, m[O]= 20.8 · 10⁻⁶m[Si],pH₂O= 0.011 bar, pH₂= 1.11 bar, Q = 16.00 lN/min.

t m[Si] m[B] m[H] m[O] mSiO2(c) ΔnH2 Δn^sH₂O ΔnSiO ΔnHBO

[min] [g] [g] [g] [g] [g] [mmol] [mmol] [mmol] [mmol]

Input 200.23 14.8 0.000 4.16 0 658 6.8

1 200.05 12.6 0.265 4.49 0 665 0.011 6.58 0.201

2 199.86 10.8 0.264 4.52 0 665 0.011 6.63 0.173

69 187.05 1.31 · 10⁻⁴ 0.247 4.34 0 665 0.011 6.80 2.30 · 10⁻⁶

Regression of ln_C^C[B]

[B](t=0)to_V^t providesAckeq, called the equilibrium rate coefficient. Loss of the silicon melt by oxidation is taken into account by adding ^Δt_V successively for each itera-tion to calculate _V^t used for regression. The reaction area does not take part in equilibrium modeling and does not affect the equilibrium rate of refining. The mass transfer coefficient for equilibrium simulation of experiments (keq) is found by dividing the equilibrium rate co-efficient over the available crucible cross-section area like in regression for kt from boron contents in samples. If the boron content in simulations become too low, like fort = 69 min

in Table 3.21, it start to deviate from the first order rate law used for regression. Concen-trations far below the detection limit for ICP-MS measurements (∼ 1 ppmw boron) are not included in regressions of mass transfer coefficients for comparison to experiment.

Boron removal in experiments is compared to the equilibrium simulations in order to estimate the fraction of equilibrium that is achieved in the experiments. The ratio _k^kt

eq is called gas uti-lization when the feed gas is input to the equilibrium model, as it estimates the fraction of the feed gas that is utilized completely to equilibrium in the interface reactions (see Section 5.2). Almost all of the utilized gas reacts with silicon for equilibrium at the interface and a minor fraction reacts with boron impurities in the melt as shown in Section 2.1. The gas utilization thus approximates the fraction of the feed gas that reacts with the melt in order to reproduce the experimental removal rate of boron for equilibrium at the interface. The frac-tion of the feed gas that reacts with boron is orders of magnitude less than the gas utilizafrac-tion of experiments.

Also the weightloss in experiments is compared to the weightloss of silicon due to SiO for-mation in the equilibrium model (−Δmeq) for the experimental time of blowing steam in the feed gas. The comparison estimates the fraction of steam in the feed gas that has reacted at the melt interphase in the experiments. In accordance with considerations by Ratto et al.

[78] and the conclusion by Næss et al. [31] that oxygen supply to the interface is the sole rate determining step for active oxidation of silicon, local equilibrium is assumed at the high-temperature melt surface. Equilibrium modeling predicts that essentially all of the supplied oxygen is consumed (see discussion above Reaction (5.34)). The weightloss comparison is thus used to estimate the fraction of steam in the feed gas (with partial pressurepH2O) that is supplied to the interface (p^sH₂O), which is called the steam supply fraction (^p_p^sH2O

H₂O in Equa-tion (3.6).) Since the partial pressures relates to the mole fracEqua-tion through the total pressure, which may be assumed constant in the open system, the steam supply fraction also expresses the fraction of steam supply on a molar basis. The steam supply fraction accounts for loss of steam by fuming Reaction (2.72) and diffusion resistance through the gas boundary layer, which are not included in the equilibrium model.

p^s_H2O

pH2O = −Δm

−Δmeq

(3.6)

Equation (3.6) is used to estimate actual supply of steam to the interface reactions in exper-iments and the equilibrium model was run a second time for each experiment withp^sH₂O =

−Δm−ΔmeqpH2O as input for steam (pH2O is in tables in Section 3.3). The resulting simulation estimates the equilibrium rate limit for boron removal for the actual supply of steam to the interface reactions (keq(p^sH₂O)), which is used as a measure of the equilibrium rate limit for boron removal (Equation (2.60)). A new comparison to the experimental mass transfer co-efficient (_k ^kt

eq(p^s_H2O)) estimates resistances to transfer of boron in Section 5.2, as separate from the resistances to steam supply.