**3.1 Thermal diffusivity**

An open photoacoustic cell (OPC) was used to analyze the thermal diffusivity of samples [5]. Thicknesses and characteristics of samples are described in **Table 1**. In experimental setup the samples were placed on top of electret microphone, where a laser (450 nm wavelength) modulated was focused onto the sample surface. Frequency modulation range from 15 to 400 Hz was used. Photoacoustic technique was used to observe amplitude and phase signal behaviors in the sample [18, 19]. A thermal diffusion model was used to calculate the pressure at the photoacoustic gas


**65**

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method*

chamber [20], with the simplification for the thermally thick and optically opaque

where A is a constant that contains geometric parameters including factors as gas thermal properties, light beam intensity, and room temperature, *f* is the frequency scan, and *fc* is the cutoff which separates thick and thin regimens. The cutoff is related to thermal diffusivity α and the sample thickness *l* by the following

The thermal relaxation method was used to determine the heat capacity [8]. In the experimental setup, a thermocouple was connected to the back side of the sample. A laser light source of 450 nm wavelength was focused onto the sample surface with a continuous incidence and uniformly. Conduction and convection energy losses were reduced by a vacuum disposal, thereby mainly ensuring a radiation heat

> *P*0 <sup>η</sup> (1 <sup>−</sup> *<sup>e</sup>* −τ

and ε is the thermal emissivity, considered 1 in this case, and σ is the Stefan Boltzmann constant. T0 is the temperature, and lm is the thickness. The thermal

**Figure 5** shows the principle of PTR, where a beam of laser light is focused on a material. The radiation absorbed by the material penetrates a certain depth and becomes a wave of heat which propagates through the material. Using an infrared detector, it is possible to detect the signals coming from the thermal wave translated in amplitude and phase. In **Figure 6** the experimental setup of photothermal radiometry technique is shown. PTR technique was used to obtain thermal images of alpha brass samples. A high-power semiconductor laser (450 nm wavelength, 300 mW) was used. The laser beam was collimated, and then it was focused onto the surface of the sample with a 40 μm spot size using a GRADIUM lens. The modulated infrared radiation from the excited surface was collected and collimated by two off-axis paraboloid mirrors, and then, it was focused onto a Judson Model J15D12-M204 HgCdTe detector, which was cooled by liquid nitrogen. The detector signal was amplified by a low-noise preamplifier, and then, it was sent to a lock-in amplifier SRS-850 which was interfaced

conductivity is related to the heat capacity and thermal diffusivity by

3 \_\_\_\_\_ *lm*

*k* = *Cp* (8)

*A f e* − √ \_\_ *f* \_\_

*fc* (4)

<sup>2</sup> *fc* (5)

<sup>⁄</sup>*T*) (6)

(7)

*DOI: http://dx.doi.org/10.5772/intechopen.86924*

equation [21]:

**3.2 Heat capacity**

regime [21] according to the following equation:

*VOPC* = \_\_

α = *πl*

Δ*T*(*t*) = \_\_\_

where τ value is related to the heat capacity by

<sup>ρ</sup>*Cp* <sup>=</sup> <sup>8</sup>*T*<sup>0</sup>

**3.3 Photothermal radiometry (PTR)**

transfer [22]. The temperature variation was determined by

#### **Table 1.**

*Thermal properties, Vickers microhardness, and thermal diffusivity of Kunial brass samples.*

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.86924*

chamber [20], with the simplification for the thermally thick and optically opaque regime [21] according to the following equation:

$$\mathbf{V}\_{\rm OPC} = \frac{\mathbf{A}}{f} \mathbf{e}^{-\sqrt[\mathbf{\sqrt{f}}]{f}} \tag{4}$$

where A is a constant that contains geometric parameters including factors as gas thermal properties, light beam intensity, and room temperature, *f* is the frequency scan, and *fc* is the cutoff which separates thick and thin regimens. The cutoff is related to thermal diffusivity α and the sample thickness *l* by the following equation [21]:

$$
\mathfrak{\mathfrak{a}} = \pi l^2 f\_c \tag{5}
$$

#### **3.2 Heat capacity**

*Modern Spectroscopic Techniques and Applications*

**through thermal treatments**

solution, and 100 ml of distilled H2O.

due to its increase.

**Table 1**.

**3.1 Thermal diffusivity**

**Sample PHT** 

**temperature °C**

**Thickness (μm)**

depth at the centerline illustrates the effect of the varying absorption coefficient

The Kunial alpha brass is a copper-zinc alloy plus nickel and corresponds to an alpha brass with chemical composition of 20% Znw, 6% Niw, 1.5% Alw, and the balance Cu. Six samples were characterized; one of them with solution heat treatment (SHT), as reference. Where SHT is the heating of an alloy to a temperature at which a particular constituent will enter into solid solution followed by cooling at a rate fast enough to prevent the dissolved constituent from precipitating. The other five samples were treated at different precipitation heat treatment (PHT) temperatures, as shown in **Table 1**. The thermal treatment by precipitation involves the addition of impurity particles to increase the strength of a material. PHT Samples were metallographically prepared with 240, 320, 400, 600, and 1000 grit sandpapers. Then, specimens were polished with 0.3 μm alumina and etched with 2 g of K2Cr2O7, 8 ml of H2SO4, 4 ml of NaCl saturated

To measure the hardness of the materials, 16 indentations were practiced on each polished brass sample, according to ASTM E92. One Leco Model LM300AT Vickers microhardness tester at 100 g load, according to ASTM-E70, was used [17]. The reported Vickers microhardness is shown in **Table 1**. Thermal diffusivity, conductivity, and heat capacity results of Kunial brass are shown in

An open photoacoustic cell (OPC) was used to analyze the thermal diffusivity of samples [5]. Thicknesses and characteristics of samples are described in **Table 1**. In experimental setup the samples were placed on top of electret microphone, where a laser (450 nm wavelength) modulated was focused onto the sample surface. Frequency modulation range from 15 to 400 Hz was used. Photoacoustic technique was used to observe amplitude and phase signal behaviors in the sample [18, 19]. A thermal diffusion model was used to calculate the pressure at the photoacoustic gas

> *α* **(cm2 /s)**

**ρC (J/m3 -K)**

*k* **(W/m**-**K)**

**Vickers microhardness (PTR technique)**

**Vickers microhardness [8]**

*Thermal properties, Vickers microhardness, and thermal diffusivity of Kunial brass samples.*

SHT NA 387 72 0.242 3.49 × 106 84.46 83.7 P300 300 392 82 0.463 2.60 × 106 120.46 92.1 P400 400 375 140 0.415 2.55 × 106 105.44 121.2 P500 500 392 201 0.339 2.77 × 106 93.32 200.6 P600 600 398 162 0.508 2.42 × 106 122.42 154.2 P700 700 401 135 0.587 2.70 × 106 158.63 133.5

**3. Experimental methods to characterize Kunial alpha brass** 

**64**

**Table 1.**

The thermal relaxation method was used to determine the heat capacity [8]. In the experimental setup, a thermocouple was connected to the back side of the sample. A laser light source of 450 nm wavelength was focused onto the sample surface with a continuous incidence and uniformly. Conduction and convection energy losses were reduced by a vacuum disposal, thereby mainly ensuring a radiation heat transfer [22]. The temperature variation was determined by

$$
\Delta T(t) = \frac{P\_0}{\eta} \{ \mathbf{1} - e^{-\gamma t} \} \tag{6}
$$

where τ value is related to the heat capacity by

$$
\rho \, \mathbf{C}\_p = \frac{8 \epsilon \sigma \, T\_0^3}{I\_m} \tag{7}
$$

and ε is the thermal emissivity, considered 1 in this case, and σ is the Stefan Boltzmann constant. T0 is the temperature, and lm is the thickness. The thermal conductivity is related to the heat capacity and thermal diffusivity by

$$k = a\rho \,\mathrm{C}\_p \tag{8}$$

### **3.3 Photothermal radiometry (PTR)**

**Figure 5** shows the principle of PTR, where a beam of laser light is focused on a material. The radiation absorbed by the material penetrates a certain depth and becomes a wave of heat which propagates through the material. Using an infrared detector, it is possible to detect the signals coming from the thermal wave translated in amplitude and phase. In **Figure 6** the experimental setup of photothermal radiometry technique is shown. PTR technique was used to obtain thermal images of alpha brass samples. A high-power semiconductor laser (450 nm wavelength, 300 mW) was used. The laser beam was collimated, and then it was focused onto the surface of the sample with a 40 μm spot size using a GRADIUM lens. The modulated infrared radiation from the excited surface was collected and collimated by two off-axis paraboloid mirrors, and then, it was focused onto a Judson Model J15D12-M204 HgCdTe detector, which was cooled by liquid nitrogen. The detector signal was amplified by a low-noise preamplifier, and then, it was sent to a lock-in amplifier SRS-850 which was interfaced

with a PC. XYZ microstages were used to obtain PTR amplitude and phase images [23, 24].

An area of 2×2 mm in each sample was scanned to obtain the thermal images. The PTR technique comprises the optical excitation of the sample with a modulated laser light source and the detection of the recombination-induced infrared emission. PTR technique covers the thermal infrared emission band from 2 to 12 μm. In PTR the amplitude and phase signals parameters were obtained from a highly focused laser beam, with a waist of 40 μm.

The thermal wave generated in the sample becomes attenuated at a distance μ, thermal length. Only information due to changes in the thermal properties of the surface of the sample is obtained. The thermal length is defined by

$$
\mu = \sqrt{\frac{\alpha}{\pi f}} \tag{9}
$$

**67**

aging.

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method*

√ \_\_ <sup>α</sup> <sup>=</sup> <sup>√</sup>

are shown in **Figures 7a** and **b**. In **Figure 7c** the highest heat capacity value was for the SHT sample. The noticed behavior was that as diffusivity and conductivity increased, the heat capacity trended to decrease. This can be explained by the thermal treatment; due to the temperature increased, the grain size increased, and this had an effect on diffusivity and conductivity. The heat capacity decreased due to the PHT effect because intermetallic precipitates migrated to grain boundaries

\_\_\_\_

where k is the thermal conductivity, ρ is the material density, and c is the specific

Lowest diffusivity and conductivity values that corresponded to the SHT sample

Vickers microhardness results are shown in **Figure 8a**. The lowest value of Vickers microhardness corresponds to the SHT sample and the highest value to the P500 sample. Due to recrystallization and precipitation phenomena, as the PHT temperature increased, the Vickers microhardness decreased. The precipitation had a little effect from 300 to 400°C. However, the P500 sample reached the highest Vickers microhardness value at 500°C. In P600 y P700 samples, the PHT temperature increased, and the microhardness decreased due to over

increased. As the Vickers microhardness decreased, the crystallinity quality

increase as the PHT temperature increased. The crystalline quality improved as the PHT temperature increased. In spite of the highest Vickers microhardness value being reached at 500°C (P500), higher temperatures caused that the Vickers

k*c* (11)

increased as the PHT temperature

value was at 400°C, and then, it trended to

*DOI: http://dx.doi.org/10.5772/intechopen.86924*

<sup>ε</sup> <sup>=</sup> \_\_*<sup>k</sup>*

heat at a constant volume.

*Experimental setup of PTR technique.*

**Figure 6.**

**3.4 Thermal properties**

affecting the heat capacity.

**3.5 Vickers microhardness and FWHM<sup>−</sup><sup>1</sup>**

In **Figure 8b** is shown that the FWHM<sup>−</sup><sup>1</sup>

increased. The lowest FWHM<sup>−</sup><sup>1</sup>

where α is the thermal diffusivity of the sample and *f* = ω/2*π*.

The PTR amplitude signal generated in the sample due to the absorption of modulated laser can be described by the following equation (8):

$$T(\mathbf{x},t) = \frac{I\_0}{2\varepsilon\sqrt{\alpha}} \exp\left(-\frac{\mathbf{x}}{\mu}\right) \cos\left(\alpha t - \frac{\mathbf{x}}{\mu} + \frac{\pi}{4}\right) \tag{10}$$

where ω is the angular frequency, I0 is the laser intensity, *x* is the sample thickness, and ε is the thermal effusivity. The pre-factor in Eq. (10) is constant for a fixed modulation frequency *f* = ω/2*π*.

The PTR amplitude signal is proportional to the reciprocal of the thermal effusivity, while the PTR phase lag is proportional to the *x*/*μ* term. The thermal effusivity and the thermal diffusivity are dependent parameters from the thermal wave propagation which determines the material inertia. The thermal effusivity is a significant heating periodic surface and a heat transport parameter. It is representing the dissipated heat energy in the solid material depending on the temperature change at the beginning of the periodic warming process. The thermal effusivity is related to Eq. (9) by the diffusivity coefficient (α), as shown in the following equation:

**Figure 5.** *Principle of PTR technique for characterization of a material.*

*Analysis of the Absorption Phenomenon through the Use of Finite Element Method DOI: http://dx.doi.org/10.5772/intechopen.86924*

**Figure 6.** *Experimental setup of PTR technique.*

*Modern Spectroscopic Techniques and Applications*

focused laser beam, with a waist of 40 μm.

*<sup>μ</sup>* <sup>=</sup> <sup>√</sup>

*<sup>T</sup>*(*x*,*t*) <sup>=</sup> \_\_\_\_ *<sup>I</sup>*<sup>0</sup>

modulation frequency *f* = ω/2*π*.

in the following equation:

images [23, 24].

with a PC. XYZ microstages were used to obtain PTR amplitude and phase

An area of 2×2 mm in each sample was scanned to obtain the thermal images. The PTR technique comprises the optical excitation of the sample with a modulated laser light source and the detection of the recombination-induced infrared emission. PTR technique covers the thermal infrared emission band from 2 to 12 μm. In PTR the amplitude and phase signals parameters were obtained from a highly

The thermal wave generated in the sample becomes attenuated at a distance μ, thermal length. Only information due to changes in the thermal properties of the

The PTR amplitude signal generated in the sample due to the absorption of

where ω is the angular frequency, I0 is the laser intensity, *x* is the sample thickness, and ε is the thermal effusivity. The pre-factor in Eq. (10) is constant for a fixed

The PTR amplitude signal is proportional to the reciprocal of the thermal effusivity, while the PTR phase lag is proportional to the *x*/*μ* term. The thermal effusivity and the thermal diffusivity are dependent parameters from the thermal wave propagation which determines the material inertia. The thermal effusivity is a significant heating periodic surface and a heat transport parameter. It is representing the dissipated heat energy in the solid material depending on the temperature change at the beginning of the periodic warming process. The thermal effusivity is related to Eq. (9) by the diffusivity coefficient (α), as shown

\_\_\_ \_\_α

*<sup>μ</sup>*) *cos*(*t* − \_\_

*x <sup>μ</sup>* + \_\_ *π*

*<sup>f</sup>* (9)

<sup>4</sup>) (10)

surface of the sample is obtained. The thermal length is defined by

where α is the thermal diffusivity of the sample and *f* = ω/2*π*.

2ε √ \_\_ <sup>ω</sup> *exp*(−\_\_ *x*

modulated laser can be described by the following equation (8):

**66**

**Figure 5.**

*Principle of PTR technique for characterization of a material.*

$$
\mathbf{e} = \frac{k}{\sqrt{\alpha}} = \sqrt{\mathbf{k}\rho c} \tag{11}
$$

where k is the thermal conductivity, ρ is the material density, and c is the specific heat at a constant volume.

## **3.4 Thermal properties**

Lowest diffusivity and conductivity values that corresponded to the SHT sample are shown in **Figures 7a** and **b**. In **Figure 7c** the highest heat capacity value was for the SHT sample. The noticed behavior was that as diffusivity and conductivity increased, the heat capacity trended to decrease. This can be explained by the thermal treatment; due to the temperature increased, the grain size increased, and this had an effect on diffusivity and conductivity. The heat capacity decreased due to the PHT effect because intermetallic precipitates migrated to grain boundaries affecting the heat capacity.
