# Dynamic Character of Thermal Analysis Where Thermal Inertia Is a Real and Not Negligible Effect Influencing the Evaluation of Non-Isothermal Kinetics: A Review

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Thermal Inertia and Newton’s Law of Cooling

_{0}exp (−t/τ) where ΔT(t)

_{0}and ΔT(t) are the initial and actual temperature differences, with the latter decaying exponentially as a function of time, t, where τ takes on a value known as a thermal inertia function.

_{true}) is correct only when the controlled temperature is added by the difference (ΔT) due to the reaction detected between the sample and its inertia that is read correctly as (T + ΔT), cf. Figure 1 left.

## 3. Historical Kinetics by Thermal Analysis

## 4. Physical Meaning of the Phenomenon Called Thermal Inertia and Reaction Kinetics by Thermal Analysis

_{p}ΔT/dt, where ΔT is the experimentally detected difference between the sample and reference temperatures.

## 5. Impact of Thermal Inertia in Thermal Analysis and Calorimetry

_{p}ΔT/dt. Its significance is indisputable especially in determining the so-called calorimetric cooling constants clearly showing that the heat transfer to the heated/cooled sample is not instantaneous and that its description requires introduction of an explicit time indicating an increase or decrease in temperature in the exponential manner.

**Figure 2.**Historical replica of a calibrating pulse of heat transferred from Tian’s 1933 publication compared with rectangular heat pulse artificially inserted recently into the sample by either method: electrically by resistant heating inside the sample carried out under steady heating and (j) external heat irradiation onto the sample surface during the isothermal regime. Both pulses are normalized on the <ΔT vs. t> axis as to fine-tuning the same shape regarding the inserted rectangular pulse (thin solid). The as-measured DTA response peak (dashed line, resistant heating) was mathematically corrected on the heat inertia effect by the differential method to yield the rectified peak (full line) [11,48]. The as-measured DTA feedback on the externally applied heat-pulse (small-circle line) was corrected by the standard Netzsch instrumental software [49] based on integral method providing a rectified peak (small-triangles line). Both rectifications encompass the same character of the necessary adjustment showing the necessity of corrections for kinetic interpretations due to radical changes in the ratio of partial areas used to determine the degree of conversion. It is clear that the effect of thermal inertia is non-linear (s-shaped) and therefore cannot be compensated [12,14] for a kinetic analysis of a singular peak by a single multiple coefficient often known as calorimetric constant, τ, derived from the fading part of a thermal record [46]. Shaded area of rectangular misfit on the right curves associates to unenclosed effects of temperature gradients in the sample body.

## 6. DTA Equation and Thermal Inertia Effect in Kinetics

_{P}dΔT + K.ΔT dt

_{P}(dΔT/dt) + KΔT)/(K(A − a) − C

_{P}ΔT)

_{0}and this correction is included in auxiliary equations {12} to {15} taken again over from Reference [38].

_{P}dΔT/dt) and C

_{P}ΔT are usually an order of magnitude smaller than the quantities to which they are added and subtracted”. They results show, however, that term “C

_{P}dΔT/dt varies from 0.634... to −2.70” while the term KΔT “varies from 4.67 to 13.1 going through a maximum of 28.1”. For that reason the above neglecting seems to be rather erroneous because the heat inertia term shows a more significant influence as being asymmetrical on the level approaching the curve inflection points, i.e., differing at least 20% from the original signal. The shape of the kinetic curve and the derived kinetic parameters become thus extremely sensitive to this heat inertia consequence especially given by the s-shaped peak background.

_{mr}

^{2}) against −1/RT

_{mr}where T

_{mr}is substituted by temperature T

_{m}

_{Δ}at which an extreme of the peak on DTA curve is reached. It means the temperature difference ΔT = T

_{S}− T

_{R}between the sample under study (S) and the reference sample (R) attained its extreme value under the constant heating rate, β, for which the condition is valid as dΔ

_{m}

_{Δ}T/dt = 0. However, Kissinger’s assumption [59] that the temperature T

_{m}

_{Δ}(in the point where temperature difference ΔT reaches the extreme value = Δ

_{m}

_{Δ}T) is identical with temperature T

_{mr}where the reaction rate r = dα/dt reaches its maximum is not correct at all. This identity can be assumed justifiable only for curves obtained by compensating DSC (Perkin–Elmer) method [28]. The same inequality T

_{mr}≠ T

_{m}

_{Δ}is also relevant to the spontaneous heat flux DSC (calibrated DTA where the spontaneous heat flux q is given as q = K ΔT). The correct equation for a DTA curve ΔT

_{(t,TR)}in the simplest form (after subtracting of the baseline B

_{L}:ΔT

_{S}= ΔT − B

_{L}) is as:

_{S}(dΔT/dt) − ΔH(dα/dt)

or ΔT = R

_{t}[C

_{S}(dΔT/dt) − ΔH(dα/dt)]

_{s}(J/K) the heat capacity of the sample under study including its holder, ΔH is integral enthalpy (J) of the process under study and R

_{t}≡ 1/K is called thermal resistance.

## 7. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic view of any (thermo-analytically) placed sample studied inside a heated furnace showing possible internal and external flows which thus forms the basis of the methodology of thermal analysis studying heat processes (middle). According to the method of the given measurement, i.e., detection of the measured quantity, we can distinguish two types of thermal analysis. In the classical design, the furnace temperature is regulated and the degree of sample conversion is measured, while in the second less common mode, the sample conversion is kept constant by changing the ambient temperature, which is also measured as the indicated response temperature. Both methods actually deal with heat transfer and the inclusion and description of this transfer in the standard thermo-analytical literature is missing. The main simplification of the standard method of daily thermal analysis (left) is the unification of the temperature of the external source (heated furnace, inert sample) with the actual reaction temperature of the studied sample, which is considered usable and rationally true despite its inaccuracy. The (left) shown curves trace true-life measurements of temperature where the shaded area represents the deviation of programed temperature from that factually measured. This shows the magnitude of the discrepancy between the programed temperature (which is traditionally used as representative) and the actual value (which is, in fact, mostly ignored via simplification [13]).

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Šesták, J.
Dynamic Character of Thermal Analysis Where Thermal Inertia Is a Real and Not Negligible Effect Influencing the Evaluation of Non-Isothermal Kinetics: A Review. *Thermo* **2021**, *1*, 220-231.
https://doi.org/10.3390/thermo1020015

**AMA Style**

Šesták J.
Dynamic Character of Thermal Analysis Where Thermal Inertia Is a Real and Not Negligible Effect Influencing the Evaluation of Non-Isothermal Kinetics: A Review. *Thermo*. 2021; 1(2):220-231.
https://doi.org/10.3390/thermo1020015

**Chicago/Turabian Style**

Šesták, Jaroslav.
2021. "Dynamic Character of Thermal Analysis Where Thermal Inertia Is a Real and Not Negligible Effect Influencing the Evaluation of Non-Isothermal Kinetics: A Review" *Thermo* 1, no. 2: 220-231.
https://doi.org/10.3390/thermo1020015