# On the Use of Infrared Thermography for the Estimation of Melting Enthalpy
## 2 Materials and Methods
### Principle of the IRT-Based Method for the Estimation of Melting Enthalpy
In this work, this is assumed to be the case, and therefore, it is possible to simply express the temperature in function of the infrared signal as in Equations (3)-(5), which are, respectively, the Stefan-Boltzmann, Wien and Rayleigh-Jeans formulations. Those three formulations are common approximations of [PERSON]'s law for specific wavelength ranges. A demonstration is made in Appendix A detailing the origin of the expression of Equations (3)-(5). \[T\;=\;f(T)\;\times\;s(T)^{\frac{1}{4}} \tag{3}\]  
\[T\;=\;f(T)\;\times\;\ln(s(T)) \tag{4}\]  
\[T\;=\;f(T)\;\times\;s(T) \tag{5}\]  
where \(f\) is a temperature-dependent emissivity-controlled function. The function mostly depends on the evolution of the emissivity with temperature. The knowledge of the emissivity is, therefore, the key challenge in getting an accurate temperature measurement of the sample from the infrared signal. This is also an important issue in thermal imaging [1; 3; 21], as it constitutes an important technological lever towards quantitative infrared measurements. Amongst the different methods available to evaluate emissivity and its variation, the contact method [3] is used in this work. It is the simplest method available and requires the use of a thermocouple to continuously monitor the actual temperature of the samples. However, using thermocouples directly in the samples is incompatible with the method, as it would interfere with the signals of the sample and  
Figure 1: Scheme of the experimental setup. as it would require a large number of thermocouples. Therefore, it is assumed that, by setting a low enough heating rate (1 \({}^{\circ}\)C/min), the temperature in the samples is equal to the temperature of the aluminum supporting plate. Hence, the signal of a sample with no ongoing phase change (i.e., either completely solid or completely liquid) can be associated to a linear temperature evolution matching the plate temperature. The uncertainty and uncontrollable aspect of the method is being able to follow the evolution of emissivity upon phase change. If the emissivity drastically changes between the solid and liquid state, then the interpolation of the emissivity between the beginning and the end of the melting process will likely have a significant impact on the results. This aspect has already been mentioned when studying the influence of the sample shape on the raw infrared signal for different types of materials [22]. It was shown that the infrared signal for fatty acids is rather continuous between the solid and liquid state and that, therefore, the interpolation of the emissivity during the melting process should not have a significant impact. In a similar fashion to the expression of the latent heat of fusion in DTA measurements, the melting enthalpy can be calculated with the IRT method using the formula in Equation (6) once the emissivity-dependent function \(f\) has been determined. Given that the infrared camera detects signals between 1.5 and 5.5 \(\upmu\)m, the Wien approximation appears to be best suited. \[\Delta H_{i}\ =\ \frac{S_{c}}{m_{i}}\int_{t_{1}}^{t_{2}}\Bigl{(}T_{plate}\ -\ f(T)\ \times\ \ln(s(T))\Bigr{)}dt \tag{6}\]