## 5 Analysis of Experimental Results
### Estimation of the Nondimensional Prefactor for Depth Evaluation  
Based on the theory of the one-dimensional propagation of heat, the observation time (\(T_{\rm max}\)) is a quadratic function of the depth of delamination (\(Z\)) [15; 40]. In this experimental study, this relationship is verified by plotting out the mean values of observation time in all cases of heating regimes with the real depth of delamination, as shown in Figure 14. The results of the statistical data analysis indicate that Equation (10) can be a quadratic function, with the coefficient of determination \(R^{2}\) of 0.993 proving high reliability of the dataset.  
\[T_{\rm max}=1.261\times Z^{2}, \tag{10}\]  
It is noted that even though the SNR for the delamination with the depth of 8 cm is always of negative value, the peak of absolute contrast is still possible to observe. Therefore, the case of depth 8 cm is considered in estimating the nondimensional prefactor \(k\).  
In addition, as demonstrated in Figure 14, the shallower the delamination is located, the smaller the standard deviation of observation time that is obtained. In particular, they vary from about 0.7 to 21.9 min for 1 to 8 cm depth, respectively.  
When the specimen is heated by a heat source, the transfer rate of heat of the specimen from the hot side to cold side is measured by the thermal diffusivity (\(\alpha\)), which is the thermal conductivity divided by the density and specific heat capacity at constant pressure [16]. Thus, the relationship between observation time (\(T_{\rm max}\)) and depth of defect (\(Z\)) can be expressed by the following equation [18].  
\[Z=k\sqrt{\alpha T_{\rm max}}, \tag{11}\]  
where, \(k\) is a nondimensional prefactor used to forecast the depth of delaminations, which is presented in Section 5.6. In this study, to estimate the \(k\) values corresponding to a wide range of the thermal diffusivity (\(\alpha\)), the \(\alpha\) values were divided into fifteen values ranging from (6.8 to 8.2) \(\times\) 10\({}^{-7}\) m\({}^{2}\) s\({}^{-1}\)[41] with the same interval of 0.1 \(\times\) 10\({}^{-7}\) m\({}^{2}\) s\({}^{-1}\). The relationship between \(Z\) and \(\sqrt{\alpha T_{\rm max}}\) in Equation (11) was analyzed, and the linear function was used to fit the data. Thus, the factor \(k\) could be estimated as the slope of the linear regression.  
The results showed that the factor \(k\) decreased from 1.414 to 1.288 when the thermal diffusivity increased from (6.8 to 8.2) \(\times\) 10\({}^{-7}\) m\({}^{2}\) s\({}^{-1}\). Figure 15 shows an example of the relationship mentioned in Equation (11) under the mean value of thermal diffusivity \(\alpha\) = 7.5 \(\times\) 10\({}^{-7}\) m\({}^{2}\) s\({}^{-1}\). The fitting linear regression function is shown in Equation (12), which has a very good fitness, with \(R^{2}\) = 0.992. The intercept is now set equal to zero to achieve a new equation, \(Z=1.347\times\sqrt{\alpha T_{\rm max}}\) and factor \(k\) can be obtained as 1.347. It should be noted that [PERSON] et al., in 2015, proposed nondimensional prefactor, \(k\) = 1.8 based on numerical simulation using \(\alpha\) = 7.5 \(\times\) 10\({}^{-7}\) m\({}^{2}\) s\({}^{-1}\). These results are different  
Figure 14: The relationship between the depth and observation time.  
with those from our experimental study which might be due to the assumption of uniform heating, homogenous material and without heat flux through the side faces of the specimen.  
\[Z=1.395\times\sqrt{\alpha T_{\text{max}}}-0.203, \tag{12}\]