*3.2.1.1 Calculation of T0*

The sample consists of only one layer (**Figure 1**) of thermal conductivity ks, thermal diffusivity Ds and thickness ls. It is fixed at backing of kb, Db, lb respectively thermal conductivity, thermal diffusivity and thickness. The sample and backing are in a fluid (air) of thermal conductivity kf, thermal diffusivity Df and thickness lf. To determine

We see that the normalized amplitude and the phase of the PTD signal are insensitive to variations of k (the three curves are coincident), which means that in this

**Figure 3** represents the theoretical variations normalized amplitude and phase curves for different values of thermal diffusivity D. These curves show a remarkable sensitivity of the PTD signal to variations in thermal diffusivity, which allows

The sample is composed of a layer deposed on a substrate (**Figure 4**). Fernelius [10] was developed the theoretical model of a layer deposed on a substrate by writing the heat equations in the four medium: fluid (Kf, Df, lf), sample (Ks, Ds, ls), black layer (Kc, Dc, lc) and backing (Kb, Db, lb). Assuming that all the light is absorbed only

by the black layer, the heat equations in the different media are given by:

*<sup>∂</sup><sup>t</sup> if* <sup>0</sup><sup>≤</sup> *<sup>z</sup>*≤*<sup>l</sup> <sup>f</sup>*

*<sup>α</sup><sup>c</sup> <sup>z</sup>* <sup>1</sup> <sup>þ</sup> *<sup>e</sup> <sup>j</sup>ω<sup>t</sup> if* � *lc* <sup>≤</sup>*z*≤<sup>0</sup>

*<sup>∂</sup><sup>t</sup> if* � *lc* � *ls* <sup>≤</sup>*z*<sup>≤</sup> � *ls*

*Theoretical variation of the normalized amplitude (a) and phase (b) of the photothermal signal with the*

*square root of modulation frequency for different values of thermal diffusivity.*

*<sup>∂</sup><sup>t</sup> if* � *lc* � *ls* � *lb* <sup>≤</sup>*z*<sup>≤</sup> � *lc* � *ls*

(6)

case we cannot determine the thermal conductivity of the sample.

determining thermal diffusivity with great precision.

*Improvement of the Thermal Properties of Sorel Cements DOI: http://dx.doi.org/10.5772/intechopen.91774*

*3.2.2 Sample composed (layer deposed on a substrate)*

*∂T <sup>f</sup>*

*∂Tc <sup>∂</sup><sup>t</sup>* � *Ac <sup>e</sup>*

*∂Ts*

*∂Tb*

*3.2.2.1 Calculation of T0*

*∂*2 *T f <sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *Df*

*∂*2 *Tc <sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *Dc*

*∂*2 *Ts <sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *Ds*

*∂*2 *Tb <sup>∂</sup>z*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *Db*

**Figure 3.**

**Figure 4.** *Different medium.*

**127**

#### **Figure 1.** *Different medium browsed by the heat.*

the periodic temperature T0, we solve the one-dimensional heat diffusion equation in the different media backing, sample and fluid. Assuming that only the sample is absorbing the pump light beam, the heat equations in the different media are given by:

$$\begin{aligned} \frac{\partial^2 T\_f}{\partial x^2} &= \frac{1}{D\_f} \frac{\partial T\_f}{\partial t} & \text{if } \ 0 \le z \le l\_f\\ \frac{\partial^2 T\_s}{\partial x^2} &= \frac{1}{D\_s} \frac{\partial T\_s}{\partial t} - A e^{ax} \left(1 + e^{j\alpha t}\right) & \text{if } -l\_s \le z \le 0\\ \frac{\partial^2 T\_b}{\partial x^2} &= \frac{1}{D\_b} \frac{\partial T\_b}{\partial t} & \text{if } -l\_s - l\_b \le x \le -l\_s \end{aligned} \tag{4}$$

where A = αsI0/2ks are constant numbers and α<sup>s</sup> is the optical absorption coefficient of the sample.

Indeed, the application of the boundary conditions of temperature and heat flux at the different interfaces allows us to determine the expression of the periodic temperature T0 at the sample surface:

$$\begin{aligned} T\_0 &= -E\left[ (\mathbf{1} - r)(\mathbf{1} + b) \, e^{\sigma\_l \, l\_t} - (\mathbf{1} + r)(\mathbf{1} - b) \, e^{-\sigma\_l \, l\_t} \\ &+ 2(r - b) \, e^{-a \, l\_t} \right] / \left[ (\mathbf{1} + \mathbf{g}) \left( \mathbf{1} + b \right) \, e^{\sigma\_l \, l\_t} - (\mathbf{1} - \mathbf{g})(\mathbf{1} - b) \, e^{-\sigma\_l \, l\_t} \right] \end{aligned} \tag{5}$$
