© A. Semerok et al., published by EDP Sciences, 2016

1 Introduction

Lock-in thermography is a non-destructive method which may be applied to test and to ensure remote control over materials in severe environment (e.g. nuclear installations) in a wide temperature range. The method is based on the laser heating of a sample with a modulated laser power at a given frequency f(Hz) followed by measurements of a thermal radiation emitted by the sample. The phase shifts Δϕ between the laser power and the thermal radiation measured at different modulated frequencies are then compared with those obtained with an analytical (3D + t) model developed at the LISL (DEN/DANS/DPC/SEARS) in case of the heating of a sample covered by a deposited layer [1,2]. Thus, it is possible to provide a tool to characterize some sample properties (thickness, thermal diffusivity, deposited layer/surface thermal contact resistance, characterization of under-surface defects and their evolution with time). The phase shift of heating temperature is presented in Figure 1.

Fig. 1 Phase shift of heating temperature.

2 Model for the heating of a plate

In a thermal model for homogeneous and isotropic plate with infinite dimensions, we supposed that:

• variations of the optical and thermal properties for a surface covered by a deposited layer during its heating are negligible;

• the surface roughness effect and heat exchange due to the sample surface/air convection are also negligible.

In the heating models [1,2] for a surface with a deposited layer, we supposed that the layer/surface thermal resistance (ℜ) and the layer optical thickness (αL) on laser wavelength are very high (ℜ → ∞ and αL >> 1). By applying the Fourier series analysis to the intensity of the laser beam and the temperature in the stationary regime of the laser heating [2,3], the complex temperature amplitude of the front face of a plate can be written as:(1)with for the Gaussian beam; ; ; ; ; ; ; , where: z and r, respectively, are the propagation direction of the laser beam and the radial distance from the center of the heated zone at the sample surface; C_v, k and L: the volumetric specific heat, the thermal conductivity and the thickness of the sample; α and R: the laser absorption coefficient and the reflectivity of the sample surface; r₀: the laser beam radius at 1/e intensity, I(t) and I₀: the intensity of the laser beam and its amplitude; f, t and i: the repetitive rate frequency of the laser, the time and the complex unity; ξ and τ: the variables of the integration; Φ_LP: the phase of the laser power.

The phase shift between the laser and the thermal power can be found by the expression:(2)

2.1 Environmental effect

The heat exchange with the environment by convection mechanism can be introduced by the conditions of limits at z = 0 and z = L [4]:(3)where: , h(Wm^–2 K^–1) is a coefficient of thermal exchange with environment by convection. The environment temperature T_a is supposed to be equal to the initial temperature of the plate, thus: ; . The losses by thermal emission are supposed to be negligible.

The solution of the heat equation:(4)on 0 ≤ z ≤ L with the initial condition was obtained for the nth harmonic of the laser repetition frequency:(5)where: ; ;

; ;

; ;

The phase shift can be written as:(6)

It depends on the parameters k, ρc, α, L, r₀, h. For thin metal plates, the environmental effect on the phase shift may be considered as negligible. For example, for SS 304L of 400 μm thickness, the phase shifts are not affected by the environment even for a material with h = 100 W/m² K (water) [4].

2.2 Simple analytical expressions

The numerical simulation of laser heating is used to fit the calculated phase shifts with the experimental ones by adjusting the material properties. A typical dependence of a phase shift on a laser modulation frequency is presented in Figure 2. For the laser beam with a diameter satisfying r₀/100 ≤ L ≤ r₀/2, one may observe a minimum on the phase shift curve with the corresponding values ϕ_min and f_min.

Multiparameter simulation of laser heating enables one to determine the effect of the interaction parameters and the material properties on the phase shifts [3–7]. Two analytical expressions were derived to relate laser parameters, sample properties, ϕ_min and f_min. This inter-relationship may provide rapid measurements of thickness L and diffusivity D of a sample with 99% accuracy:(7) (8)with ζ_ϕ and ζ_f calculated values (Fig. 3).

Fig. 2 Phase shift for SS 304L plate of 400 μm thickness.

Fig. 3 Calculated values of ζϕ and ζf parameters.

3 Experimental

The experimental setup with a compact fiber laser (low divergent near the Gaussian beam with M² = 1.1, beam radius r₀ ≅ 1 mm, 1060 nm wavelength) is simple in its arrangement (Fig. 4) and enables one to make remote measurements with a millimeter lateral resolution in a wide distance range (from some centimeters up to several meters).

Fig. 4 Scheme of the experimental setup.

3.1 Metal plates characterization

The lock-in thermography for a plate characterization was validated with a set of etalon samples. The obtained results on plate thickness and diffusivity measurements are presented in reference [3]. These results have demonstrated ≈90% accuracy of thickness and thermal diffusivity measurements. The measured thermal diffusivities along with the referenced values from literature are summarized in Figure 5.

Fig. 5 Comparison of the measured thermal diffusivities along with the referenced values from literature (presented as solid lines) [8]. The square points – Dm2 and the circle points – Dm1 (see Tab. 1 for Dm definition).

3.2 Zircaloy-4 cladding characterization

After the characterization of the metal plates by the above procedure, the method was used for studying Zircaloy-4 claddings (Fig. 6). The schematic comparison of the cladding diameter and the one of the heated zone (with the tested zone on it) is presented in Figure 6. As the heated zone diameter is smaller than the one of Zy4 cladding, the heated zone may be considered as a plane surface, and thus, the same procedure as the one for metal plates may be followed.

To study the effect of the oxide layer on the measured thermal diffusivity, some Zy4 claddings were artificially oxidized in a furnace at different regimes (temperature, environment, time) to obtain oxide layers of different thickness (5–15 μm). For 5 μm, 10 μm, and 15 μm oxide layer thickness, the regimes, respectively, were as follows: (500 °C, in air, for 37 hours), (550 °C, in air, for 23 hours), and (550 °C, in water vapor, for 51 hours). The surface of Zy4 with the oxide layer of 10 μm thickness has suffered nitriding effect, while the samples with 5 μm and 15 μm oxides were of a good quality.

The results on Zy4 claddings characterization are presented in Figures 7 and 8 and in Table 1. At low modulation frequency (f < 20 Hz), the phase shift is poorly affected by the presence of the oxide layer (Fig. 7). Thus, for Zircaloy-4 claddings, ϕ_min and f_min method may be used to determine thickness and thermal diffusivity (see Tab. 1). The measurement relative deviations were less than 10%.

At higher modulation frequency (f > 20 Hz), a clear effect of the oxide layer on the phase shifts is observed (Fig. 7). Due to the fact that ZrO₂ layers are semitransparent, the theoretical models for phase shift calculation [1,2] are not applicable in this case. However, there are all the reasons to suppose that further development of the thermal model of heating a semi-transparent layer on a metal plate will provide an adequate on-line in situ characterization of oxide formation.

Fig. 6 On the left: schematic comparison of the Zy4 cladding diameter and those of the heated zone (in red) and of the tested zone on it (in blue). On the right: the picture of Zy4 claddings. Some Zy4 claddings were artificially oxidized (on the right in the picture).

Fig. 7 Phase shifts for Zy4 claddings as a function of modulated frequency.

Fig. 8 Measured and reference values of thermal diffusivity as a function of temperature (°C) for Zy4 claddings.

4 Conclusions

The homemade thermal model of the local heating of a homogeneous and isotropic plate with infinite dimensions was developed and verified by characterizing different metal plates and Zircaloy-4 claddings. Two analytical expressions (7) and (8) for sample thickness L and thermal diffusivity D were derived. These expressions and the lock-in thermography measurements (the minimal phase shift ϕ_min and the corresponding minimal modulated frequency f_min [6,7]) are used to measure the thickness and the thermal diffusivity of the samples. The obtained results are in agreement with experimental data within an accuracy of 90%. The developed method may be applied for any material with a high absorption coefficient α, that is, for any plate with αL >> 1. Based on the results obtained, we may conclude that a rapid remote in situ control over components in nuclear installations may be ensured with a good spatial resolution (of the order of a laser beam diameter 2r₀).

Acknowledgments

Authors acknowledge DEN/DANS/DMN/SRMA/LC2M team for Zircaloy-4 claddings supply.

References

All Tables

All Figures

	Fig. 1 Phase shift of heating temperature.
In the text

	Fig. 2 Phase shift for SS 304L plate of 400 μm thickness.
In the text

	Fig. 3 Calculated values of ζϕ and ζf parameters.
In the text

	Fig. 4 Scheme of the experimental setup.
In the text

	Fig. 5 Comparison of the measured thermal diffusivities along with the referenced values from literature (presented as solid lines) [8]. The square points – Dm2 and the circle points – Dm1 (see Tab. 1 for Dm definition).
In the text

	Fig. 6 On the left: schematic comparison of the Zy4 cladding diameter and those of the heated zone (in red) and of the tested zone on it (in blue). On the right: the picture of Zy4 claddings. Some Zy4 claddings were artificially oxidized (on the right in the picture).
In the text

	Fig. 7 Phase shifts for Zy4 claddings as a function of modulated frequency.
In the text

	Fig. 8 Measured and reference values of thermal diffusivity as a function of temperature (°C) for Zy4 claddings.
In the text