Evaluation of smooth reaction rate of noisy experimental data using Legendre series expansion

Abstract

The accurate calculation of reaction rates from experimental data is crucial for understanding and characterizing chemical processes. However, the presence of noise in experimental data can introduce errors in rate calculations. In this study, we introduced a novel approach that utilizes the Legendre series expansion method to directly derive smooth reaction rates from noisy experimental data, eliminating the need for numerical differentiation methods. This approach proves to be highly effective in handling noisy thermogravimetric analysis (TGA) data obtained from the thermal decomposition of specific polymers. We demonstrated the robustness and reliability of this method and provided Gnu Octave codes as a free alternative to MATLAB, making the implementation more accessible. Furthermore, the smooth reaction rates obtained were used to evaluate the activation energy using the Friedman isoconversional method. The results showed excellent agreement with those obtained using the Vyazovkin integral method. Additionally, the proposed method can be applied to obtain smooth derivative thermogravimetric (DTG) curves using noisy TGA data set.

Appendices

Appendix A

1.1 Differential isoconversional method

To investigate the correlation between activation energy (E), pre-exponential factor (A), and the conversion degree ($\alpha$), denoted as $E_{\alpha }$ and $A_{\alpha }$ respectively, isoconversional methods are utilized. One notable approach is the model-free differential isoconversional procedure introduced by Friedman [18]. This technique is applicable to any arbitrary temperature program:

$$\begin{aligned} \ln \left( \frac{d\alpha }{dt} \right) _{\alpha ,i} = \ln {A_{\alpha }f(\alpha )} - \left( \frac{E_{\alpha }}{RT_{\alpha ,i}} \right) \end{aligned}$$

(A.1)

where the subscript i indicates the ordinal number assigned to a specific experiment, while $T_{\alpha ,i}$ corresponds to the temperature at which the desired extent of conversion is achieved in that experiment. The application of the Friedman method can be broadened to encompass non-isothermal conditions, where multiple experiments are conducted at various constant heating rates denoted as $\beta _{i}$. In such scenarios, the equation associated with this methodology is expressed as follows:

$$\begin{aligned} \ln \left[{\beta _{i}\left( \frac{d\alpha }{dt} \right) }_{\alpha ,i} \right]= \ln {A_{\alpha }f(\alpha )} - \left( \frac{E_{\alpha }}{RT_{\alpha ,i}} \right) \end{aligned}$$

(A.2)

Therefore, the slope of plotting $\ln \left( {d\alpha }/{dt} \right) _{\alpha ,i}$ or $\ln \left[\beta _{i}\left( {d\alpha }/{dT} \right) _{\alpha ,i} \right]$ against ${1/T}_{\alpha ,i}$ will be $\left( - E_{\alpha }/R \right)$ and the activation energy $E_{\alpha }$ can be obtained from the slope.

1.1.1 Incremental integral isoconversional method

The incremental integral isoconversional methods are based on the assumption that the kinetic parameters, $A_{\alpha }$ and $E_{\alpha }$, remain constant within a specified conversion range ${\Delta }\alpha$. This range is defined by lower bound $\alpha _{-}$ and upper bound $\alpha _{+}$, with ${\Delta }\alpha = \alpha _{+} - \alpha _{-}$. In our study, we set $\alpha _{+} = \left( \alpha + {\Delta }\alpha /2 \right)$ and $\alpha _{-} = \left( \alpha - {\Delta }\alpha /2\right)$.

Vyazovkin [19] has proposed an advanced isoconversional method that introduces an incremental approach. By minimizing the function $\Phi (E_{\alpha })$ within the interval $[\alpha _{-}, \alpha _{+}]$, the activation energy $E_{\alpha }$ value can be determined:

$$\begin{aligned} \Phi (E_{\alpha }) = \sum _{i = 1}^{n}{\sum _{j \ne i}^{n}\frac{J\left[E_{\alpha },T_{i}\left( t_{\alpha } \right) \right]}{J\left[E_{\alpha },T_{j}\left( t_{\alpha } \right) \right]}} \end{aligned}$$

(A.3)

where the function $J\left[ E_{\alpha },T\left( t_{\alpha }\right) \right]$ is:

$$\begin{aligned} J\left[E_{\alpha },T\left( t_{\alpha } \right) \right]= \int _{t_{\alpha _{-}}}^{t_{\alpha _{+}}}{{\exp }\left[- \frac{E_{\alpha }}{RT(t)} \right]dt} \end{aligned}$$

(A.4)

Table 2 The first few Legendre polynomials and their derivatives

Similarly, for a series of runs performed at different constant heating rates, the function $\Phi (E_{\alpha })$ has the following form:

$$\begin{aligned} \Phi (E_{\alpha }) = \sum _{i = 1}^{n}{\sum _{j \ne i}^{n}\frac{I\left( E_{\alpha },T_{\alpha ,i} \right) \beta _{j}}{I\left( E_{\alpha },T_{\alpha ,j} \right) \beta _{i}}} \end{aligned}$$

(A.5)

where the function $I\left( E_{\alpha },T_{\alpha }\right)$ is:

$$\begin{aligned} I\left( E_{\alpha },T_{\alpha } \right) = \int _{T_{\alpha _{-}}}^{T_{\alpha _{+}}}{{\exp }\left( - \frac{E_{\alpha }}{RT} \right) dT} \end{aligned}$$

(A.6)

Appendix B

1.1 Legendre polynomials

The Legendre polynomial $P_{n} (x)$ is a solution to the Legendre differential equation:

$$\begin{aligned} \left( 1-x^2\right) y''-2xy'+ n \left( n+1 \right) y=0\;,\;\;\;n=0,1,2,\dots \end{aligned}$$

(B.1)

The first few Legendre polynomials and their first derivatives are listed in Table 2. The Legendre polynomials and their first derivatives obey the recurrence relations. This study employed the following relations:

$$\begin{aligned}{} & {} \left( n+1\right) P_{n+1}\left( x\right) =\left( 2n+1\right) xP_n\left( x\right) -nP_{n-1}\left( x\right) \end{aligned}$$

(B.2)

$$\begin{aligned}{} & {} P_{n+1}^{'}\left( x\right) =\left( 2n+1\right) P_n\left( x\right) +P_{n-1}^{'}\left( x\right) \end{aligned}$$

(B.3)