Comisiones del CIEMI

As outlined in Section 2.5.2, Vittee (2012) presented a complete adaptation of the RPM to the algorithm developed by Scott et al. (2006(a)). Some of the important steps carried out during the DAE based model application are discussed in the sections that follow. As explained in Section 2.4.3, the first order and RPM reaction models will be considered.

4.1. Data collection via simulation

TGA data can be simulated via various ODE solvers and the use of a semi-analytical method. Vittee (2012) used two ODE solvers, ODE45 and ODE15s for simulation of kinetic data of the same kinetic triplet. The author applied Scott‟s algorithm to the simulated data and showed the variance in accuracy for the different solvers. According to Vittee (2012), the semi-analytical method of data simulation is most suitable and accurate as proven by the successful application of Scott‟s algorithm.

4.1.1. The semi analytical method of data simulation

This method involves the derivation of a solution for the mass fraction remaining. Equation [2-1] may be presented as shown below:

[2-1]

Separating variables and integrating:

∫

∫ ⁄ [4-1]

The right hand term (the temperature integral) of the above expression has no analytical solution, and must be numerically evaluated, hence the name semi-analytical. An in-built numerical integration function in Mat Lab is applied to solve the integral in this study. The function approximates the integral using recursive adaptive Simpson quadrature. The left hand side can however be analytically evaluated, and the mass fraction remaining is solved for. A

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matrix of is then generated, with the number of columns varying as per number of specified reactions. Multiplication of the mass fraction remaining during each reaction by the specified fractional components normalizes the respective reactions to their specified initial masses (Vittee 2012). This method of data simulation was used in this study.

4.2. Statistical methods for analysing the quality of fit

The model accuracy was measured using two methods: the correlation coefficient (Vittee, 2012), also known as the R2 statistic, and the root mean square value of the differences between the simulation and the experimental plot (Sima-Ella et al., 2005). The simulation in this case is obtained by the use of the kinetics determined by the DAE based model for the particular conversion. The two parameters are obtained by the application of regression analysis on the experimental data and the simulated curve. The correlation coefficient is defined by the Equation [4-2] (Vittee, 2012; Draper and Smith, 1981).

∑ ̂

∑ ̅ [4-2]

Here, ̂ is the model predicted dependant variable, is the experimentaly determined dependant variable and ̅ is the mean of the experimental values. The root mean square (RMS), error value is defined as

√ ∑ ̂ [4-3]

On application of the model, the kinetics are determined by the use of conversion data from two different heating rates. The conversion data from the third heating rate is normally used for evaluation of the quality of fit (Vittee 2012). This is carried out by using the kinetics determined by the first two heating rates to predict the reaction progression at the third heating rate. The two methods of evaluating the quality of fit in this study were not only applied on the heating rate being predicted, but instead, the quality of fit was evaluated for all three heating rate conversion data. The average values of these were then given out by the model. This allowed an evaluation of the determined kinetics based on its suitability for predicting the reaction progression, not only at one heating rate, but at three different heating rates.

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Figure 4-1: Statistical parameters for a diagonal line through an RPM conversion

The statistical parameters were measured for a straight diagonal line through the conversion of a given random pore reaction. The specified kinetic triplet for the random pore reaction and the statistical parameter results are displayed on Figure 4-1. The R2 statistic for the diagonal is reported as 0.771 and the RMS error value is reported as 0.212. In this study, an accuracy of 0.0100 and 0.999, RMS error value and R2 statistic respectively will be considered of high accuracy. Clearly, a perfectly accurate plot must give an R2 value as close to 1 as possible, and a RMS error value as close to 0 as possible. The RMS value obtained when comparing the diagonal line to the random pore reaction represents an increase in the relative error greater than 2000% from the value being considered of high accuracy in the current study. The R2 value obtained when the diagonal line is compared to the random pore reaction conversion represents an increase in the relative error of only 129% from the value of 0.999. Even though the R2 statistic is an important tool for evaluating the quality of the fits obtained, it is less sensitive to changes in the relative errors as compared to the RMS error values. The use of both parameters will therefore be employed in this study.

4.2.1. RPM DAE based model evaluation

At this stage a set of kinetics is specified to simulate typical conversion vs. temperature data. The simulated data is fed onto the DAE based model for kinetics determination. The DAE based model determined kinetics are then compared to the initial specified kinetics and the percentage error is then evaluated. Not only does this process evaluate the model accuracy, it ensures the

900 950 1000 1050 1100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 T emperature (K) M a ss F ra ction R e ma ining ( 1 -x)

Statistical parameters sensitivity analysis

Random pore reaction Diagonal line Relative error measured E (Kj/mol) A (s-1m-1) f 𝝋 R2 RMS error 250 10^10 1 4 0.771 0.212

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elimination of errors in the coding. The code has also been tested on simulated data for multiple reactions, these are not discussed here.

Table 4-1: Model evaluation by simulation (RPM) (kJ/mol) (s-1 m-1) 𝝋 R2 RMS Specified Kinetics 200 1.0E+12 1.00 4 0.99999 0.0011 Model determined kinetics 199.85 1.02E+10 0.998 4.1366 Relative Error 0.07% 2% 0.2% 3%

The relative errors for all the kinetic parameters are desirably low. As observed in Table 4-1, the overall fit accuracy is maintained at an excellent value of R2= 0.99999 and RMS=0.0011. It can then be concluded that no errors in the coding exist.

4.2.2. First order DAE based model evaluation

A similar analysis is carried out for the first order DAE based model. Table 4-2 displays the outcome obtained.

Table 4-2: Model evaluation by simulation (First order) (kJ/mol) (sec-1 ) R2 RMS Specified Kinetics 200 10E+10 1.00 1 5.76e-006 Model determined kinetics 199.99 9.98E+09 1.00004 Relative Error 0.0% 0.2% 0.0%

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The results displayed show almost exact replicability of the simulated curve by the DAE based model. The quality of the fit is measured at a remarkable R2 value of 1 and a corresponding RMS value of 5.76e-006. The DAE based first order model has therefore been successfully validated and it can be concluded that no coding errors exist. The code is working successfully as described in Section 2.5.2.

4.3. Conclusion

The semi analytical method described in this chapter is used in the study for simulation of conversion data. The two statistical parameters, the correlation coefficient and the root mean square value, were used for the assessment of the quality of the fits obtained. The two models have been successfully validated by simulation and it is concluded that no coding errors exist. The model is ready for application on experimental data. The preliminary analysis follows in Chapter 5.

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