Kinetic modeling of liquefied petroleum gas (LPG) reduction of titania in MATLAB

In the present study, reduction of Titania (TiO2) by liquefied petroleum gas (LPG)-hydrogen-argon gas mixture was investigated by experimental and kinetic modelling in MATLAB. The reduction experiments were carried out in the temperature range of 1100-1200°C with a reduction time from 1-3 hours and 10-20 minutes of LPG flowing time. A shrinking core model (SCM) was employed for the kinetic modelling in order to determine the rate and extent of reduction. The highest experimental extent of reduction of 38% occurred at a temperature of 1200°C with 3 hours reduction time and 20 minutes of LPG flowing time. The SCM gave a predicted extent of reduction of 82.1% due to assumptions made in the model. The deviation between SCM and experimental data was attributed to porosity, thermodynamic properties and minute thermal fluctuations within the sample. In general, the reduction rates increased with increasing reduction temperature and LPG flowing time.


Introduction
Titanium-containing compounds are very important in the advance technologies especially in the field of aerospace and biomedical. Commercially, titania was reduced into metallic titanium and titanium based compounds such as titanium carbide and nitride which have applications in catalyst, paint pigment or manufacture of composite materials [1]. The current technology used for production of metallic titanium through the Kroll process is by chlorination of titania at high temperature at about 800 o C-1100 o C [2]. According to Adipuri et al. [3] ,chlorination of titanium oxycarbide (TiOxC1-x) occurs at much lower temperature, around 400-500 o C than TiO2 in the production of titanium tetrachloride through the Kroll process. Synthesis of TiOxC1-x can be introduced through different technologies. It is mainly produced by reduction of titania with carbon in the temperature range of 1700 to 2000 o C [4]. Studies by Dewan et al. [5] have shown that carbothermal reduction of titania in different gas atmosphere was significant and it was found that hydrogen reduced titania and reacted with carbon to form methane gas. This methane gas promoted the formation of titanium sub oxides which was reduced into TiOxC1-x. For this paper, synthesis of TiOxC1-x by liquefied petroleum gas (LPG) could have few advantages such as the replacement of the coal as the reducing agent. Besides this, a low CO2 emission can be achieved

Experimental
TiO2 reduction data for the kinetic modelling work has been published by Yin et al. [9]. Besides this, experimental data by Sheikh Abdul Rezan et al. [11] was also used in the kinetic model. The reduction mechanism of TiO2 by H2 follows a shrinking core model (SCM) as proven by the latter studies. For methane (CH4) reduction of TiO2, studies by Guangqing Zhang et al. [12,13] have shown the SCM model gives a good representation of the kinetics.

Kinetic Modelling Methodology
The kinetic model was coded using MATLAB R2011b software based on the SCM. Details on the MATLAB model have been reported elsewhere [10,14]. In order to solve the complicated equations regarding CH4 reduction of TiO2 in the kinetic model, step by step modelling process was done based on its mass and thermal transport properties [15]. The kinetic model developed by MATLAB in this study has been successfully applied previously on isothermal and non-isothermal reduction of TiO2 and Fe2O3 in H2 gas atmosphere [10,14,22]. A flowchart on the computational methodology in the SCM is given in Figure 2.

Determination Rate of Transfer
Consider the chemical reaction in Eq. 1, it was assume TiO2 is a solid spherical reactant metallic oxide compound, CH4 is the reactant gas, TiC is the solid reaction product and CO with H2 is the evolved gaseous product. The evolved gas product was mainly H2 since CO content was less than 1 vol. % [9]. The overall reaction can be described by Eq. 2 with the symbol D designated as the product gas mixture of CO and H2. Since LPG gas concentration was 10 vol.% whereas H2 gas was 45 vol. %, the CO gas concentration could be ignored as it was less than 1 vol. % due to the small TiO2 pellet mass. If gaseous transport away from the pellet was defined to be positive magnitude in nature, the rates of mass transport for nA (CH4) and nB (H2) for ideal gas behavior was represented by Eq. 3-4.

Prediction of Convective Mass Transfer Coefficients
The convective mass transfer coefficient between a gas and a particle with spherical shape was calculated from Eq. 5-6, where Sherwood number, Sh is a function of the mass Reynolds number (Re), Schmidt number (Sc), and binary diffusivity (DAB). AB was the reductant and product gas defined as CH4-H2.

Prediction of Binary Diffusivity
The method was based on kinetic theory and gives predictions with an average deviation of 4 pct. (%) with a maximum deviation of 16% [16]. The binary diffusivity (cm 2 /sec)prediction was made using Eq. 7.

Prediction of Convective Heat Transfer Coefficients
Equations 8-9 below enables accurate prediction of the convective heat transfer coefficient between a sphere and flowing gas using dimensionless parameter called Nusselt (Nu) number. where Re =Reynolds number, Pr = Prandtl number, hc = convective heat transfer coefficient (cal/cm 2sec-°K) and Kf = thermal conductivity of the gas film (cal/sec-cm-°K).

Determination Rate of Reaction
The location of the reaction interface on the spherical particle determines the reaction points E and F in Eq. 10. These points enable the calculation of R' (% Reduction) or extent of reduction. This, along with the bulk gas composition, evaluation of transport mass coefficients, and the temperatures of the reaction interface and gas stream enables calculations of the instantaneous reaction rate (nC) at any given time during the reaction. The extent of conversion of TiO2 to TiC was proportional to the magnitude of nC.

Determination of Spherical Pellet Temperature
Prediction of the variation of spherical pellet temperature change during reaction was more difficult than the determination of the gaseous mass transport rate. The reasoning behind this argument was that heat transfer at the reactant core consisting of TiO2 occurred simultaneously with heat transfer through the accumulated solid reaction product (TiC) and the bulk gas. To simplify the heat transfer calculations in MATLAB, a few assumptions was made. The first assumption, the accumulation of heat in reaction zone was negligible. The 2 nd assumption, heat transfer was 1-dimensional and by conduction, convection and radiation. The 3 rd assumption, the average pellet core temperature of TC was defined to account for the overall heat transfer away the pellet. This allowed the simplification of the temperature calculations by Eq. 11-14. where Tp= Average temperature of the particle (°K) Cpp= heat capacity of solid product (calorie/moleo K), CpC = heat capacity of solid reactant core (calorie/mole-o K), Cpg= heat capacity of reducing gas (calorie/mole-o K).
The overall chemical reaction for the system was represented by Eq. 2 from SCM [17] and the reaction rate can be predicted by the simultaneous solution of Eq.10-15 done in MATLAB. Lastly, after the reaction rate (nc) and time of reaction (t) have been determined, the R' can be calculated based on Eq. 15-16:

Results and discussion
The experimental data for extent of reduction (R') versus time at different temperatures is shown in Figure 3. Table 1 tabulates the final R' calculated from the weight loss of samples. The R' was increasing with reduction temperature and time because a higher amount of TiO2 was reduced into Ti2O3 and TiC. As shown in Figure 3, there was significant difference in R' between higher and lower reduction temperature. At 1200 o C and 180 min of reduction time with 20 min LPG flow time, the R' was the highest at 37.9%. The titania sample reduced at latter conditions contained 3.68 wt.% carbon when analyzed by elemental carbon analysis.  During the reduction process, thermal cracking of propane and butane took place, which is the major component in LPG; propane was cracked into ethane, ethylene, methane, and hydrogen [18,19]. The final stable state was methane (CH4). Deposition of solid carbon occurred at the end of the cracking process, as shown in Eq. (17). The presence of carbon monoxide (CO) gas was due to the chemical reaction in Eq. (1).

CH4 (g) → C (s) + 2H2 (g)
The cracking process enhances the rate of reduction as more carbon is available for reduction. Furthermore, similar work done by Itao, G.B. and N.M. Anacleto [6] showed the thermodynamic activity of C3H8 and C4H10 was much higher than CO and H2. This higher activity allowed a greater degree of cracking into solid carbon. The longer flowing time of LPG was able to produce more free carbon and thus increase the reduction rate. Free solid carbon can reduce and carburize titania into TiC but the reduction was slow due to the low reduction temperature studied in this work.

Predicted Reduction Kinetics of Titania
The reduction of titania has been one of the most important and complex gas-solid reactions in kinetic study by various researchers [9,12,13,22]. The complexity was due to the formation of titania suboxides such as Ti4O9, Ti3O5 and Ti2O3. These sub-oxides affects the reaction rate and differ greatly in terms of its physical and material properties. A shrinking core model (SCM) developed by Szekely et al. [20] was employed to determine the reaction rate. From the mass and heat transport equations listed above, the transport-limited reaction rate for reduction of titania was calculated in MATLAB. results are significant. All the simulated plots are similar in which the predicted non-isothermal R' data was closer to the experimental results compared with isothermal conditions. Even though the deviation was significant, it can be observed that the R' increased with temperature for both simulated and experimental. The source of the deviation will be discussed below. The comparison of temperature variation with respect to reaction rate was shown in Figure 6. From the figure, it shows the deviation between predicted and experimental measured results. On the other hand, the experimental reaction rate changed linearly with reduction time, which showed the reaction was chemically controlled. In this case, temperature was a key parameter in determining the overall reaction rate. The graphical trend of reaction rate for experimental and predicted results should be similar. An explanation for the deviations observed was due to the SCM model assumptions that were made upon developing the kinetic model. Those assumptions were that the porosity and pellet size of sample was constant during the reduction. Firstly, the sample was assumed to be porous without close pores for gas transport in and out during the reaction time. However, Ronald Charles Gower [15] have suggested that this situation was ideal to take place because of the presence of open and closed pores. Moreover, the porosity throughout the sample also varied and was not constant. These pores formed within the product and restricted the movement of the CH4 gas. Therefore, until the product layer cracks to allow free movement of CH4, the diffusion of oxygen and carbon atoms in the solid phase or gaseous reactants can only contribute to the reaction kinetics. Delay in this cracking of intermediate product layers of Ti2O3 also slows down the reaction. Furthermore, Ti2O3 does not convert directly to TiC but as another intermediate phase of TiOxC1-x. Studies by Dewan et al. and Guangqing Zhang et al. [5,13] have shown that the end product to be TiOxC1-x instead of TiC at the reduction temperature used. The presence of these intermediates phases also was difficult to model by SCM. The slow reaction for intermediate product to transform into TiOxC1-x caused the build up in non-porous Ti2O3. As a result, mass transfer for carbon and CH4 to penetrate the Ti2O3 layer was difficult.
Pure TiC was employed in developing the kinetic model instead of using the actual end product of the reaction, which was a solid solution titanium oxycarbide (TiOxC1-x). There was insufficient thermodynamic and physical properties for TiOxC1-x phases to be used for the kinetic model compared with TiC. Added to this, it was impossible to reduce titania into pure TiC at this low temperatures. Instead of producing pure TiC, a solid solution of TiOxC1-x with similar cubic crystalline structure as TiC was produced during reduction [21]. Due to these supporting evidence, the predicted and experimental results deviation was observed. Even though there were deviations between the predicted and experimental results, the model still shows the kinetic trends that was expected from SCM. The trend indicated the increasing in rate and extent of reduction when temperature increases when governed by chemically controlled reaction rate.

Conclusions
Titania was reduced to TiOxC1-x in a temperature range of 1100 -1200 o C by liquefied petroleum gas (LPG)-hydrogen-argon gas mixtures. The variation in the LPG flowing time and reduction time showed a significant effect on the extent of reduction. The results obtained confirmed that the reduction rate was accelerated by the longer flow time of LPG. The highest experimental extent of reduction of about 37.9% occurred at a temperature of 1200 o C with 3 hours reduction time and 20 min of LPG flowing time. The reduction rates increased with increasing reduction temperature and LPG flowing time. The SCM was implemented in MATLAB that modelled isothermal and nonisothermal reduction. For the nonisothermal reduction at 1200°C at 3 hours with 20 min of LPG flowing time, the R' calculated by the kinetic model was about 82%. There were differences between the predicted and experimental results, but the overall kinetic trend of reduction rate was similar. The differences was due to limitation of SCM to predict changes in porosity, thermodynamic properties of TiOxC1-x and minute changes in temperature in the concentric shells during reduction. Future models will address this limitations.

Acknowledgement
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