*3.2.2.1 Calculation of T0*

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:

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

*<sup>∂</sup><sup>t</sup>* � *Ae<sup>α</sup> <sup>z</sup>* <sup>1</sup> <sup>þ</sup> *<sup>e</sup> <sup>j</sup>ω<sup>t</sup> if* � *ls* <sup>≤</sup>*z*<sup>≤</sup> <sup>0</sup>

Indeed, the application of the boundary conditions of temperature and heat flux

We will study in this paragraph the variations of the photothermal signal as a function of the square root of the frequency in the case of a bulk sample. **Figure 2** represents the theoretical variations normalized amplitude and phase of the photothermal signal as a function of the square root of the frequency for three values of the thermal conductivity (k = 0.1, 2.0 and 4.0 W/m K) of a bulk sample.

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

*square root of modulation frequency for three numerical values of thermal conductivity.*

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

at the different interfaces allows us to determine the expression of the periodic

*=* ð Þ 1 þ *g* ð Þ 1 þ *b e*

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

*<sup>σ</sup><sup>s</sup> ls* � ð Þ <sup>1</sup> � *<sup>g</sup>* ð Þ <sup>1</sup> � *<sup>b</sup> <sup>e</sup>*

(5)

�*σ<sup>s</sup> ls*

(4)

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

*Different medium browsed by the heat.*

**Figure 1.**

**Figure 2.**

**126**

*∂*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*

temperature T0 at the sample surface:

þ 2ð Þ *r* � *b e*

*3.2.1.2 Illustration of theoretical model*

coefficient of the sample.

*∂T <sup>f</sup>*

*Zero-Energy Buildings - New Approaches and Technologies*

*∂Ts*

*∂Tb*

*<sup>T</sup>*<sup>0</sup> ¼ �*<sup>E</sup>* ð Þ <sup>1</sup> � *<sup>r</sup>* ð Þ <sup>1</sup> <sup>þ</sup> *<sup>b</sup> <sup>e</sup><sup>σ</sup><sup>s</sup> ls* � ð Þ <sup>1</sup> <sup>þ</sup> *<sup>r</sup>* ð Þ <sup>1</sup> � *<sup>b</sup> <sup>e</sup>*�*σ<sup>s</sup> ls*

�*α ls* 

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:

$$\begin{aligned} \frac{\partial^2 T\_f}{\partial x^2} &= \frac{1}{D\_f} \frac{\partial T\_f}{\partial t} & \text{if } 0 \le z \le l\_f\\ \frac{\partial^2 T\_c}{\partial x^2} &= \frac{1}{D\_c} \frac{\partial T\_c}{\partial t} - A\_c e^{a\_c z} \left(1 + e^{j\_{wt}}\right) & \text{if } -l\_c \le z \le 0\\ \frac{\partial^2 T\_s}{\partial x^2} &= \frac{1}{D\_s} \frac{\partial T\_s}{\partial t} & \text{if } -l\_c - l\_s \le z \le -l\_s\\ \frac{\partial^2 T\_b}{\partial x^2} &= \frac{1}{D\_b} \frac{\partial T\_b}{\partial t} & \text{if } -l\_c - l\_s - l\_b \le z \le -l\_c - l\_s \end{aligned} \tag{6}$$

**Figure 3.**

*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.*

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

**4.1 Thermal diffusivity of Sorel cement without PVAc**

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

three values of the thermal diffusivity 0.2, 0.4 and 5.0 m<sup>2</sup>

frequency for magnesium oxychloride cement.

**Figure 6.**

*conductivity.*

**Figure 7.**

**129**

the experimental curve is obtained for 0.4 10<sup>7</sup> m<sup>2</sup> s

**4.2 Thermal diffusivity of Sorel cement with PVAc**

**Figure 6** presents the experimental curves of normalized amplitude (a) and phase (b) of the photothermal signal and the corresponding theoretical ones for

modulation frequency for the reference sample (MOC without PVAc). Furthermore, **Figure 6** also presents the experimental normalized amplitude and phase of the photothermal signal and the corresponding theoretical ones for three values of the thermal conductivity 0.01, 0.3 and 0.7 w/mk versus the square root modulation

As noted in the previous section, the PTD signal is insensitive to thermal con-

1 .

ductivity. We can only determine the values of the thermal diffusivity of the magnesium oxychloride cement. The theoretical curve which coincides best with

We will proceed the same way for the rest of the magnesium oxychloride cement samples to which PVAc has been added with different percentages using the same method. **Figure 7** shows the normalized amplitude and phase of experimental

*Experimental and theoretical variation of the normalized amplitude (a) and phase (b) of the photothermal signal with the square root of modulation frequency for three values of thermal diffusivity and thermal*

*The normalized amplitude (a) and phase (b) of experimental photothermal signal versus square root of modulation frequency of the magnesium oxychloride cement with PVAc fitted with theoretical curves (line).*

/s versus square root

#### **Figure 5.**

*Theoretical variation of the normalized amplitude (a) and phase (b) of the photothermal signal with the square root of modulation frequency for three values of thermal diffusivity and thermal conductivity.*

where Ac = αcI0/2kc are constant numbers and α<sup>c</sup> is the optical absorption coefficient of the black layer.

By applying the continuity conditions of temperature and heat flow at the different interfaces allows us to determine the expression of the periodic temperature T0 at the sample surface:

$$\begin{split} T\_{0} &= E\_{c}[(1-b)e^{-\sigma\_{l}l\_{\star}}](1-r\_{c})(1-c)e^{\sigma\_{l}l\_{\star}} + (1+r\_{c})(1+c)e^{-\sigma\_{l}l\_{\star}} \\ &- 2(1+c\,r\_{c})e^{-a\_{l}l\_{\star}}] - (1+b)e^{\sigma\_{l}l\_{\star}}[(1-r\_{c})(1+c)e^{\sigma\_{l}l\_{\star}} + (1+r\_{c})(1-c)e^{-\sigma\_{l}l\_{\star}} \\ &- 2(1-c\,r\_{c})e^{-a\_{l}l\_{\star}}] \, / [(1+b)e^{\sigma\_{l}l\_{\star}} \left[ (1+c)e^{\sigma\_{l}l\_{\star}} + \left(1-\frac{\mathcal{g}}{c}\right)(1-c)e^{-\sigma\_{l}l\_{\star}} \right] \\ &- (1-b)e^{-a\_{l}l\_{\star}} \left[ (1+\frac{\mathcal{g}}{c})(1-c)e^{\sigma\_{l}l\_{\star}} + \left(1-\frac{\mathcal{g}}{c}\right)(1+c)e^{-\sigma\_{l}l\_{\star}} \right] \end{split} \tag{7}$$

where Ec = Ac/(α<sup>c</sup> <sup>2</sup> � <sup>σ</sup><sup>c</sup> 2 ), rc = αc/σc,b=kbσb/ksσs,c=kcσc/ksσs, and g = kfσf/ks.

#### *3.2.2.2 Illustration of theoretical model*

**Figure 5** shows the variations of normalized amplitude and phase for three values of thermal conductivity with equal thermal diffusivity D = 0.4 10�<sup>7</sup> m<sup>2</sup> s �1 and well-defined values of the properties of the ink layer [11]. In addition, **Figure 5** also shows the variations normalized amplitude and phase for different values of thermal diffusivity with k = 1 Wm�<sup>1</sup> K�<sup>1</sup> .

We notice that the PTD signal is sensitive to both the conductivity and the thermal diffusivity of sample (substrate). Since the value of thermal diffusivity is determined in the previous case (sample without layer), the value of thermal conductivity can be determined with great precision.
