## 4 Results
### Pure Substances
#### 4.1.5 Practical Consequences. We have indicated in Figure 4 that, apart from the temperature range of the phase change, in the sensible zone, it is possible to determine the specific heat capacity by the formula:  
\[c=\frac{\Phi_{\text{sens}}}{m\beta} \tag{8}\]  
deduced from the models with homogeneous samples [1; 2; 3; 4; 6]. However, we can observe that the temperature inside the sample \(T_{\text{sens}}\) is almost uniform, the temperature differences being of the order of a few tenth of K.  
So, in this sensible zone, the specific enthalpy \(h\) is given by integration of this value of \(c\) (corrected to obtain the true temperature \(T_{\text{sens}}\) by the relation (10)). By extension certain authors define an artificial heat capacity, called equivalent capacity \(c^{\text{eq}}\), with a similar formula as Equation (8) but _even during the phase transform_. \[c^{\text{eq}}=\frac{\Phi}{m\beta} \tag{9}\]  
Moreover, the calculation of an \"apparent\" enthalpy \(h^{\text{app}}\) is proposed by an integration of \(c^{\text{eq}}\) versus \(T_{\text{plt}}\) and considering this plate's temperature as \"the\" temperature of the sample [1; 2; 3]. Given the preceding analysis and the corresponding conclusions, it is noteworthy to mention that this equivalent heat capacity representation, based on the integration of the heat flow-rate with respect to the plate's temperature, lead to a completely incorrect determination of the enthalpy function, for example even if we consider a supposed melting temperature range. Thus, in the present case of pure substances, such a definition leads to various functions for this equivalent heat capacity \(c^{\text{eq}}\), as depicted in Figure 11a and for the apparent enthalpy \(h^{app}\) in Figure 11b where different heating rates are considered. Figure 10: DSC curves at various masses for the heating at 5 K/min of a pure substance (water). (**a**) Basic thermogram; (**b**) Normalized thermogram. Firstly, it appears clearly (as it could have been inferred from the previous results) that the obtained functions \(h^{\rm APP}\) are not independent from the operational conditions i.e., in the present case, dependent on the heating rate. Yet, the enthalpy being a function of state, it cannot be dependent on the heating rates. In the present case of pure substances, it is known that this function is an Heaviside one. In other words, any method implying a dependency of the enthalpy function with the heating rate is totally inconsistent with the laws of thermodynamics. Moreover, we have shown [9; 10] that this method which enlarges the temperature range of melting is not representative of the true energy storage kinetics, the PCM seeming less efficient. Secondly, one can see that the dome zone of the equivalent heat capacity is narrower for lower values of the heating rate or the mass. However, the above results have shown that temperature gradients were still present even with low heating rates. Given that the true solution is a Dirac function, it is tempting to consider theses curves as different approximations of this Dirac function, having a result that should converge towards the correct value for infinitely low heating rate. Figure 11: Determination of the enthalpy by the equivalent capacity method. (**a**) Equivalent capacities calculated by Equation (9); (**b**) Enthalpies \(h^{\rm APP}\) given by an integration of \(c^{\rm eq}\). Unfortunately, such an approach is still imprecise. Indeed, when trying to further decrease the heating rate, one usually faces two drawbacks: (i) the ratio signal to noise is lowering and uncertainties consequently grow; (ii) generally, to maintain this ratio signal to noise to an acceptable value lowering the heating rate requires to use larger samples, that is to say to increase the mass, which leads to an increase of the width as shown before.