The first model computed through this work is called pSiO-T model. Its aim is to relate the amount of SiO(g) in the gas phase with the temperature gradient in the condensation zones. This model includes both Si-SiO2 and SiO2-SiC condensation reactions. The initial temperature of the gas is highest where the gas is produced. The temperature gradually decreases by going upwards from the gas production zone. At a certain position, the gas is not condensing anymore. The partial pressure of SiO(g) changes through the condensation zone, since condensates develop through a temperature gradient.
The initial gas composition is known from the amount of gas-generating materials consumed, and the final composition is known from the amount of condensates generated. The compositional change of the gas phase through the condensation zone is caused by the condensation reactions, which consume SiO and CO in different proportions. If the initial and final T and pSiO conditions are calculated, it is possible to track the partial pressure variation through the furnace temperature gradient. The pSiO-T curves will estimate the partial pressure of SiO(g) at any temperature inside the condensation chamber.
Procedure
The pSiO-T lines are traced for each experiment for which both the mass balance and temperature gradients were computed, and for each condensation reaction. To trace a pSiO-T line, one must interpolate between two points in a pSiO vs T diagram. The first point identifies the initial condition. i.e. the gas temperature and composition before condensation. The second point describes the final conditions, i.e. at the position where the last condensate is noticed. Hence, the edges of a pSiO-T lines have coordinates (TSiO,in ; pSiO,in) and (TSiO,out ; pSiO,out).
TSiO,in is the temperature at which the gas mixture of SiO(g) and CO(g) (and He/Ar when added) enters the condensation zone. In the large-scale setup, TSiO,in is assumed to be constant at 1890°C. In the small-scale setup, TSiO,in was measured to be 1815°C for all the experiments with a target temperature of 2000°C. TSiO,in was measured 20 mm below the condensation chamber for setup R1b, R2a and R4e. The inert gas enters the gas production chamber at room temperature. The inert gas is assumed to be heated up to 1815°C in a short time.
TSiO,out is the temperature at which the gas stops forming condensates of any kind. Above this point, the particles are not covered in any condensate.
Before calculating the partial pressures pSiO,in and pSiO,out, the total pressure inside the system must be calculated.
When it comes to the large-scale setup, the system is always exposed to atmospheric pressure, as the furnace is open. For the small-scale setup, the pressure is fixed at 1.5 atm. The overpressure is generated by the inert gas. The small-scale furnace is not equipped with a system recording the pressure continuously, so the pressure of the outgoing gas is measured in three different instants. In the small-scale setup, the pressure before the SiO(g) production reaction was 1.4 atm. Then the pressure increases up to 1.6 atm when the gas production reaches its maximum extent. Finally, the pressure stabilizes at around 1.5 atm few minutes after the CO(g) peak was reached. For simplicity, it was be assumed that the pressure of the system is constant at 1.5 atm through the whole condensation process.
pSiO,in is computed according to the pellets composition, the stoichiometries of Reaction (-1) or (-2) and the amount of inert gas injected. SiO2-SiC pellets will produce nSIO,in moles of SiO(g) and nCO,in moles of CO(g)
according to Reaction (-1), whereas Si-SiO2 pellets will follow Reaction (-2) and produce nSiO,in moles of SiO(g).
Both nSiO,in and nCO,in consider only the portion of the pellets charge that has reacted.
The volume of injected inert gas nIG is calculated by multiplying the gas production time tr and the injected gas flow. The amount of inert gas moles added nIG is computed with the ideal gas law. The gas phase contains ntot
moles of three compounds before condensation.
𝒏𝒕𝒐𝒕= 𝒏𝑰𝑮+ 𝒏𝑺𝒊𝑶,𝒊𝒏+ 𝒏𝑪𝑶,𝒊𝒏 Equation 7
In Si-SiO2 pellets large-scale experiments (IF5-8, IF10, IF12), the theoretical partial pressure of CO(g) should be pCO = 0. However, SiC-SiOx condensates form anyways in the setup, implying that a certain amount of CO(g) should be present in the gas phase before condensation. The graphite parts are believed to be responsible for CO production, as they are covered with a SiC layer (see Appendix A). It is assumed that CO(g) is generated by the interaction with the graphite wall (Reaction 3). nSiO,in should decrease by a number of moles corresponding to the CO moles generated in this way. To summarize, 0.1 mol CO are added to nCO,in, and 0.1 mol SiO are taken away from nSiO,in, in the mass balance of experiments IF5-8, IF10 and IF12.
The value 0.1 mol was estimated from the amount of SiC-SiOx condensates found in these experiments, assuming that all the CO(g) generated from Reaction 3 is used for Reaction 1. The same correction was applied for the SiO2-SiC pellets, and only for the large-scale setup experiments. After the correction, the number of moles is scaled to partial pressures (with respect to the total pressure of each setup), and pSiO,in is found.
The value of pSiO,out is found from the weight of Si-SiO2 and SiO2-SiC condensates collected from each experiment.
It is possible to calculate the SiO(g) and CO(g) moles used during condensation, either by Reaction 1 (SiC-SiOx
condensates) or Reaction 2 (Si-SiO2 condensate). These moles are called nR1cond,SiO, nR1cond,CO and nR2cond,SiO, where the superscript specifies the condensation reaction, and the subscript the gas species considered. At this point, the mole balance is carried out by the equations below. nIG remains constant through the experiment.
𝒏𝑺𝒊𝑶,𝒊𝒏− 𝒏𝑺𝒊𝑶,𝒄𝒐𝒏𝒅𝑹𝟏 − 𝒏𝑺𝒊𝑶,𝒄𝒐𝒏𝒅𝑹𝟐 = 𝒏𝑺𝒊𝑶,𝒐𝒖𝒕 Equation 8
𝒏𝑪𝑶,𝒊𝒏− 𝒏𝑪𝑶,𝒄𝒐𝒏𝒅𝑹𝟏 = 𝒏𝑪𝑶,𝒐𝒖𝒕 Equation 9
Finally, nIG, nSiO,out and nCO,out are proportional to the partial pressures at the point where condensation ends.
The value of pSiO,out can be found, with respect to the total amount of moles in each setup. In particular, for the large-scale setup, pCO is assumed to be (1 - pSiO), as no other species are assumed to be present in the gas phase.
Once the two points (TSiO,in ; pSiO,in) and (TSiO,out ; pSiO,out) are found for every experiment, a linear regression gives a pSiO-T curve with the form 𝑝𝑆𝑖𝑂= 𝛾𝑇 + 𝛿. The graphs should be read from right to left, therefore the partial pressure of SiO(g) decreases linearly through the condensation chamber, according to the model. The points are united by a line, as shown in Figure 182.
Reactions 1 or 2 take place in the temperature interval [TSiO,R1,start ; TSiO,R1,stop] (or with R2 subscript, in case Reaction 2 is considered). These temperatures can be related to the corresponding pSiO on the linear interpolation, after having found 𝛾 and 𝛿. In this way, the corresponding partial pressures pSiO,R1,start and pSiO,R1,stop
can be calculated.
Figure 182 shows how the pSiO-T curve looks like for experiment R1b, considering Reaction 2. The interval between (TSiO,R2,start ; pSiO,R2,start) and (TSiO,R2,stop; pSiO,R2,stop) is continuous, and represents the interval where the condensate was found. The dashed line represents the zone where the reaction is not occurring.
All the values of T, pSiO, 𝛾 and 𝛿 discussed in this section, for all the experiments, are collected in Appendix D.
193 Figure 182: Example of pSiO vs T diagram for Reaction 2, experiment R1b.
Results
This section shows all the pSiO-T curves computed for the small- and large-scale setup experiments. The slope of the curves is very similar between experiments in different conditions and setups, whereas the intercept 𝛿 shifts to lower pSiO by increasing the inert gas flow added. The partial pressure does not decrease dramatically through the condensation setup, according to the mass balance calculations performed.
In Figure 183 and Figure 184, experiments at the same inert gas flow have been collected in groups of lines with similar colors. A higher inert gas flow gives a lower pSiO,in and shifts condensation to temperature intervals located at colder positions. pSiO,out and TSiO,out decrease by adding a higher inert gas flow, whereas TSiO,start follows the same trend for Reaction 2 only. Experiments at longer holding times have wider gaps between TSiO,start and TSiO,stop . For example, Experiments R3a, R4 and R5a have respectively 30, 60- and 240-min holding times, as well as R14a, R17a and R19-21.
In the large-scale setup (Figure 185 and Figure 186), every experiment has a different color. Experiments at shorter holding times (IF10, IF11, IF12) tend to have a higher TSiO,stop compared to their respective experiments at longer holding time (IF1b, IF1b and 5b). A broader particle size distribution has the opposite effect, as it happens between IF1b - IF2a – IF4 – IF3a and IF5b – IF8 – IF7b (each group is ordered by increasing particle size distribution broadness).
Figure 183: pSiO-T curves for SiO2-SiC condensation, small scale setup.
Figure 184: pSiO-T curves for Si-SiO2 condensation, small scale setup.