# Numerical Integration of Weight Loss Curves for Kinetic Analysis

## Abstract

**:**

## 1. Introduction

_{0}+ HR·t) instead of the experimental relation T = f(t) can lead to major calculation errors if a strong exothermic or endothermic process is being produced [6]. A simple heating rate programme is not actual temperature, insofar as it is not possible to account for the sample’s temperature variations resulting from the reactions that are taking place. These variations can only be taken into account by performing a step-by-step numerical integration of the necessary differential equations.

_{0}= 6.1 × 10

^{11}s

^{−1}and E/R = 24,415 K, as well as with Avrami’s model, with n = m = 1.53, k

_{0}= 3.02 × 10

^{10}s

^{−1}and E/R = 21,273 K [9,10]. In these models, E/R is the activation energy (divided by the constant of the gases), k

_{0}is the pre-exponential factor and the values of ‘m’ and ‘n’ represent reaction and nucleation orders.

## 2. Numerical Methods for Simple Materials Based on One Process Only

#### 2.1. Theoretical Background

_{0}, E/R and n value, one can very simply integrate Equation (6) using a particular value of Δt and compare the calculated w-t curve values with that of the experimental one. Usually, the same t value as that of the experimental data is chosen, allowing to directly compare experimental and calculated data, and to use the actual temperature measured at the reaction point T = f(t).

#### 2.2. Application to Some Examples

_{1}= 10

^{6}s

^{−1}and k

_{2}= 10

^{7}s

^{−1}), and two activation energy values (E

_{1}/R = 12,500 K and E

_{2}/R = 15,000 K). Here, we can appreciate the effect of the change in both parameters. Many authors have found that a compensation effect exists between the preexponential factor and the activation energy, in such a way that different combinations of these parameters produce the same TG curve [24,25]. This is not completely true, as the compensation effect can be broken when considering different TG curves obtained under different experimental conditions and fitted with the same set of kinetic constants [7,14,26].

_{0}= 10

^{6}s

^{−1}, E/R = 15,000 K and reaction orders of 1, 2 and 0.5. Generally, low orders produce a sharp curve, whereas high values generate smoother curves.

_{0}= 10

^{6}s

^{−1}, E/R = 12,500 K, n = 1) shows a case where the curve moves appreciably at higher temperatures when increasing the heating rate, whereas the process in Figure 4 (simulation based on k

_{0}= 10

^{26}s

^{−1}, E/R = 48,100 K, n = 1) does not move to the same extent. Worthy of note, no other changes were introduced into the simulation, i.e., a kinetic triplet allows fitting the different behaviours of many materials, without any kinetic parameter modifications, whether in the degree of decomposition or heating rate.

^{cal}

_{j}and w

^{exp}

_{j}are the calculated and experimental values of the weight fraction, and the sub-index ‘j’ refers to a point in an experiment ‘j’. More complex forms taken by the OF can be used and have been discussed elsewhere [26]. To make the calculation more accurate, different runs are usually performed under different experimental conditions (generally a different temperature programme) simultaneously, using the same kinetic model and parameters. In this case, the formal expression of the OF would be:

_{0}= 1.7·10

^{−2}s

^{−1}, E/R = 13,200 K, n = 0.86. The spreadsheet used for this calculation is available in the Supplementary Material of this publication (Spreasheet2: PS_kinetics_fitting.odt).

## 3. Numerical Methods for Complex Materials

#### 3.1. Theoretical Background

_{i}the gases produced by this fraction, S

_{i}the solid residue, and s

_{i}the yield coefficient of the solid fraction, f

_{io}is the contribution of each fraction to the weight of the initial sample, and therefore:

_{i}= 0 for t = 0, for every fraction. The weight of each fraction t each time was calculated as follows:

_{i0}.

#### 3.2. Simulations and Application to Some Materials

_{01}= 10

^{7}s

^{−1}, E

_{1}/R = 12,500 K, n

_{1}= 1; k

_{02}= 10

^{6}s

^{−1}, E

_{2}/R = 12,500 K, n

_{1}= 1; k

_{03}= 10

^{5}s

^{−1}, E

_{1}/R = 12,500 K, n

_{3}= 1; f

_{i0}= 0.3; f

_{20}= 0.3.

_{i0}, f × (E

_{i}/R), f × n

_{i}, and (f − 1) values of f

_{i0}, i.e., a total of (4f − 1). This number is commonly within the 7–15 range, which is low compared to the usual number of points fitted using the method, implying that there would be no compensation effect.

## 4. Comparison with Other Methods

_{f}. This temperature is the one at which an equivalent (fixed) state of transformation is achieved at the different HRs. Several different explicit relations between the T

_{f}and the heating rate are proposed, but the vast majority present an equation similar to [40]:

_{f}at the different HRs would give a straight line with a slope equal to (−E/R).

_{f}at w = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 and 0.1 were extracted from the simulated data presented in Figure 1 corresponding to k

_{1}and E

_{1}/R (equal to 12,500 K). In this case, an Octave programme was used for simplicity. Worthy of note, the experimental errors were avoided because the data was simulated by numerical integration of the differential equation. Figure 8a shows the lines obtained for the different values of fixed ‘w’. As can be observed, the lines are straight, and the slope of the plots give an average value of E/R = 12,790 K, that is, an error of +2.32% was calculated. Let us try other values of constants, for example those corresponding to Figure 4, i.e., k

_{0}= 10

^{26}s

^{−1}, E/R = 48,100 K, n = 1. Figure 8b shows the resulting plot. For these parameters, a value of E/R = 51,188 K is obtained, representing an error of +6.48%.

_{0}is overlooked. Naturally, the (previously mentioned) methods, that do not even consider curves obtained at different HRs, are even worse.

_{1}/R = 17,673 K, E

_{2}/R = 11,919 K and E

_{3}/R = 12,566 K were found using numerical integration and optimisation. A considerable error is committed when using simplified methods. Figure 10 shows all activation energy (E/R) values calculated, applying the py-isoconversion method [40] and using Equation (13). For comparison purposes, we plotted the values of E/R obtained by numerical integration and optimisation. As we can observe, the E/R values are similar, but it is not possible to distinguish, using graphical methods, the different organic fractions contained in the husks.

## 5. Conclusions

## Supplementary Materials

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Effect of activation energy and the preexponential factor on the position and shape of the decomposition curve at a fixed heating rate of 10 K/min. Values used in the simulation: k

_{1}= 10

^{6}s

^{−1}, k

_{2}= 10

^{7}s

^{−1}, E

_{1}/R = 12,500 K, E

_{2}/R = 15,000 K.

**Figure 2.**Effect of the reaction order on the position and shape of the decomposition curve at a fixed heating rate of 10 K/min.

**Figure 3.**Effect of the heating rate (HR) on the decomposition curve’s position without varying the kinetic constants. In this case, the curve moves appreciably to the right when the HR increases (in K/min, indicated in the graphics).

**Figure 4.**Effect of the heating rate (HR) on the decomposition curve’s position without varying the kinetic constants. In this case, the curve does NOT move appreciably to the right when the HR increases (in K/min, indicated in the graphics).

**Figure 6.**Evolution of the decomposition of three different initial organic fractions (F1, F2 and F3) and total weight loss (wtotal) for the simulation of a run performed at 10 K/min. The values of the kinetic parameters are shown in the main text.

**Figure 7.**Thermal decomposition of coffee husks at three different HR. Experimental and calculated curves (the kinetic parameters values are: k

_{01}= 10.6 s

^{−1}, E

_{1}/R = 17,673 K, k

_{02}= 7.26 s

^{−1}, E

_{2}/R = 11,919 K, k

_{03}= 5.1 s

^{−1}, E

_{3}/R = 12,566 K, f

_{10}= 0.26, f

_{20}= 0.31, all orders were considered equal to 1).

**Figure 8.**Plots corresponding to the different values of T

_{f}calculated based on the data in Figure 1 and Figure 4, using as fixed decomposition state of transformation: w = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 and 0.1 (the line moves to the right as the level of decomposition increases). (

**a**) simulated with values of k

_{1}and E

_{1}of Figure 1, (

**b**) simulated with data of Figure 4.

**Figure 9.**Plots corresponding to the different T

_{f}values calculated for the decomposition of coffee husks (Figure 7), using as fixed decomposition state of transformation w = 0.9, 0.8, 0.7, 0.6, 0.5, 0.4, 0.3, 0.2 and 0.1.

**Figure 10.**Activation energy (E/R) calculated using the py-isoconversional method linearizing differential equation. The E/R values obtained by numerical integration and optimisation were plotted for comparison purposes.

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Conesa, J.A.
Numerical Integration of Weight Loss Curves for Kinetic Analysis. *Thermo* **2021**, *1*, 32-44.
https://doi.org/10.3390/thermo1010003

**AMA Style**

Conesa JA.
Numerical Integration of Weight Loss Curves for Kinetic Analysis. *Thermo*. 2021; 1(1):32-44.
https://doi.org/10.3390/thermo1010003

**Chicago/Turabian Style**

Conesa, Juan A.
2021. "Numerical Integration of Weight Loss Curves for Kinetic Analysis" *Thermo* 1, no. 1: 32-44.
https://doi.org/10.3390/thermo1010003