Thermal characterisation of homogeneous materials

Quantitative InfraRed Thermography, 2014 http://dx.doi.org/10.1080/17686733.2014.955673

Thermal characterisation of homogeneous materials using a weak formulation technique Thierry Kouadioa*, Anissa Mezianea, Christophe Pradereb, Christophe Baconc, Jean-Christophe Batsaled and Christine Biateaub a

APY Department, University of Bordeaux, I2M, Talence, France; bCNRS, I2M, Talence, France; c Bordeaux INP, I2M, Talence, France; dArts et Metiers ParisTech, I2M, Talence, France (Received 13 February 2014; accepted 28 July 2014) A new method for the thermal characterisation of homogeneous materials is proposed. This method, specifically designed to measure the thermal diffusivity in the plane of thin specimens, is based on a low-order weak formulation (LOWF) method of the heat equation using a virtual test function. In this formulation, the spatial derivatives of temperature are coupled with an analytical expression less dependent on measurement noise. In this way, it is shown that the LOWF method leads to a robust method for the estimation of thermal diffusivity. The LOWF method is numerically tested on a two-dimensional model and validated on a homogeneous material using infrared thermography. Keywords: infrared thermography; thermal characterisation; weak formulation

1. Introduction In the field of thermal characterisation of materials, numerous methods have been developed. The first approach was proposed [1] for the measurement of the thermal diffusivity through the thickness of homogeneous and isotropic materials. Other methods were developed for in-plane measurements for anisotropic materials [2–5] or in-plane and through the thickness simultaneously.[6–8] Thanks to technological developments, temperature data can be recorded without contact and in several points of a surface simultaneously using an infrared camera. This type of camera gives better spatial resolutions and more data to process. Techniques such as integral transforms [9,10] or singular value decomposition also called principal components analysis or Karhunen–Loève Decomposition [11–16] reduce the amount of data to process. However, the low amount of data to be processed remains affected by measurement noise, causing errors in the estimation of thermal properties. In such a context, mathematical tools and techniques reducing noise effects are of great interest for material characterisation based on IRT. Two main approaches are usually used to estimate the thermal diffusivity. Modal approaches [9,17] such as the Fourier methods provide a global estimation over a relatively large domain and have the advantage of a good robustness to noise. The nodal approach,[17] on the other hand, allows local estimations on a pixel scale but with considerable sensitivity to measurement noise. This work proposes a new method for robust thermal characterisation *Corresponding author. Email: [email protected] © 2014 Taylor & Francis

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of homogeneous materials from noisy temperature data recorded by infrared thermography. The principle consists of a low-order weak formulation (LOWF) method of the heat equation using an analytical test function.[18,19] There are several methods for the thermal characterisation of materials using an analytical function. An approach using a Gaussian function [20] has been proposed for heat source reconstruction from noisy temperature fields. Also, the Fourier transform [3,4] is widely used to estimate the thermal properties of materials. An advantage of the Fourier transform is to provide a temporal differential equation that can be solved analytically, but requires to find a suited trigonometric test function according to the thermal boundary conditions of the problem. In practice, it can be difficult to define exactly these boundary conditions. In this context, the Fourier transform is more suited for global estimation of the thermal properties, i.e. on an entire domain, since the boundary conditions are better known at the edges. In the presented LOWF method, an approach based on the use of a polynomial test function is proposed. The goal is to allow, in first approach, a local estimation of the thermal diffusivity, i.e. on a subdomain whose boundary conditions are not exactly known. The LOWF method provides equations to solve numerically. In this paper, the method is tested numerically and experimentally in the case of a thermal Dirac excitation,[21] showing good results. The formalism of the method is first described and tested numerically on temperature data generated using a two-dimensional (2D) finite element (FE) model. A criterion is then proposed to optimise the accuracy of the method. Finally, the LOWF method is applied experimentally to estimate the thermal diffusivity of a thin homogeneous sample from temperature data recorded with an infrared camera. 2. LOWF of 2D heat transfer equation For a thin plate (Figure 1(a)), the 2D heat transfer equation without internal source is: @T ðx; y; tÞ @ 2 T ðx; y; tÞ @ 2 T ðx; y; tÞ 2h ¼ ax þ ay bT ðx; y; tÞ with b ¼ 2 @t @ x @2y u Cp e

(1)

where T (K) is the temperature, ax (m2 s−1) and ay (m2 s−1) are, respectively, the thermal diffusivity in the X- and Y-direction, h (W m−2 K−1) represents the convective heat transfer coefficient, u (kg m−3) is the density, Cp (J kg−1 K−1) is the specific heat capacity and e (m) is the thickness of the plate.

Figure 1.

xy of the plate (b). Thin plate (a). Analytical function defined on a 2D subdomain X

Quantitative InfraRed Thermography xy (Figure 1(b)) defined by: Let us consider a rectangular subdomain X xy : x 2 ½x0 lx =2 ; x0 þ lx =2 ; y 2 y0 ly =2 ; y0 þ ly =2 X

3

(2)

where lx ; ly are dimensions of the plate in the X- and Y-direction, respectively. x0 ; y0 xy . For ax and ay assumed to be constant on X xy , are the coordinates of the centre of X the product of Equation (1) with an analytical test function f ðx; yÞ which belongs to xy Þ and integration over X xy gives: C 2 ðX Z Z @T ðx; y; tÞ @ 2 T ðx; y; tÞ f ðx; yÞdxdy ¼ ax f ðx; yÞdxdy @t @x2 xy xy X X Z @ 2 T ðx; y; tÞ þ ay f ðx; yÞdxdy @y2 Z X xy b T ðx; y; tÞf ðx; yÞdxdy (3) xy X

Applying double integration by parts, Equation (3) is written as: Z @T ðx; y; tÞ f ðx; yÞ dxdy @t xy X x0 þlx =2 @T ðx; y; tÞ @f ðx; yÞ x0 þlx =2 ¼ ax f ðx; yÞ T ðx; y; tÞ @x @x x0 lx =2 x0 lx =2 ! Z 2 @ f ðx; yÞ þ T ðx; y; tÞ dxdy @x2 Xxy y0 þly =2 @T ðx; y; tÞ @f ðx; yÞ y0 þly =2 f ðx; yÞ þ ay T ðx; y; tÞ @y @y y0 ly =2 y0 ly =2 ! Z Z 2 @ f ðx; yÞ þ T ðx; y; tÞ dxdy b T ðx; y; tÞf ðx; yÞ dxdy @y2 xy xy X X

(4)

The choice of an analytical function f ðx; yÞ verifying the following boundary conditions xy [18,19] (5) on X ( ( f ðx; yÞjy¼y0 ly =2 ¼ 0 f ðx; yÞjx¼x0 lx =2 ¼ 0 and (5) @f ðx;yÞ @f ðx;yÞ @x jx¼x0 lx =2 ¼ 0 @y jy¼y0 ly =2 ¼ 0 is a way to simplify Equation (3) into Z Z @T ðx; y; tÞ @ 2 f ðx; yÞ f ðx; yÞdxdy ¼ ax T ðx; y; tÞ dxdy @t @x2 xy X ZXxy @ 2 f ðx; yÞ þ ay T ðx; y; tÞ dxdy @y2 Z X xy T ðx; y; tÞf ðx; yÞdxdy b xy X

(6)

Equation (6) is a LOWF of Equation (1) in which the spatial derivatives are calculated by an analytical function f . The advantage of such a formulation is that the secondorder derivative is applied on the analytical function and not on the noisy measured temperature. In this way, errors occurring in the calculation of the higher order derivatives of noisy temperature data are reduced. In the following, the LOWF method

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consists of using Equation (6) to carry out robust estimations of thermal diffusivities ax and ay . The analytical function considered is defined by f ðx; yÞ ¼ ð1 x2 Þ2 ð1 y2 Þ2. For an isotropic material (ax ¼ ay ¼ a), Equation (6) is written as: xy ; tÞ ¼ a AðX xy ; tÞ b BðX where R

xy X

xy ; tÞ ¼ R BðX

xy X

@T ðx;y;tÞ @t

R

f ðx; yÞ dxdy

xy ; tÞ ¼ ; AðX

T ðx; y; tÞf ðx; yÞ dxdy

xy X

(7)

@ 2 f ðx;yÞ @2 x

R

xy X

þ@

2

f ðx;yÞ @2 y

T ðx; y; tÞ dxdy

T ðx; y; tÞf ðx; yÞ dxdy

Equation (7) is similar to the equation of a straight line whose slope corresponds to the thermal diffusivity a. For an anisotropic material (ax 6¼ ay ), an approach is proposed to estimate thermal diffusivities ax and ay separately. For this purpose, a one-dimensional (1D) form of Equation (7) is used. Let us estimate thermal diffusivity ax. The average form of Equation (1) in the Y-direction is written as: Z

Ly =2

@T ðx; y; tÞ 1 dy ¼ ax @t Ly Ly =2 ! Z Ly =2 T ðx; y; tÞdy b

1 Ly

Z

Ly =2

@ 2 T ðx; y; tÞ dy þ ay @x2 Ly =2

Z

Ly =2

@ 2 T ðx; y; tÞ dy @y2 Ly =2 (8)

Ly =2

which is also written as: @ T ðx; tÞ @ 2 T ðx; tÞ ay @T ðx; y; tÞ y¼Ly =2 ¼ ax þ b T ðx; tÞ (9) @x2 @t @y Ly y¼Ly =2 R L =2 where T ðx; tÞ ¼ L1y Ly y =2 T ðx; y; tÞdy is the average temperature field in the Y-direction. If one of the following conditions is verified, T ðx; y; tÞ is independent of variable y: the temperature field is 1D, that is T ðx; y; tÞ ¼ T ðx; tÞ, @T ðx;y;tÞ

(2) @y y¼L =2 ¼ 0: the thermal fluxes at Y-direction boundaries are equal to zero, y

@T ðx;y;tÞ

¼ @T ðx;y;tÞ : the thermal fluxes at Y-direction boundaries are equal, (3) @y

@y

(1)

Ly =2

Ly =2

then

@T ðx; y; tÞ y¼Ly =2 ¼0 @y y¼Ly =2

(10)

@ T ðx; tÞ @ 2 T ðx; tÞ ¼ ax b T ðx; tÞ @x2 @t

(11)

and Equation (9) is simplified in

T ðx; tÞ is a 1D temperature field. Hence, by analogy with Equation (6), the LOWF of x: Equation (11) is obtained using a 1D analytical function f ðxÞ and subdomain X

Quantitative InfraRed Thermography Z

@ T ðx; tÞ f ðxÞ dx ¼ ax @t x X

with f ðxÞ verifying

Z

(

@ 2 f ðxÞ T ðx; tÞ dx b @x2 x X

Z x X

5 T ðx; tÞ f ðxÞ dx

f ðxÞjx¼x0 lx =2 ¼ 0 @f ðxÞ @x jx¼x0 lx =2

¼0

(12)

(13)

x defined by and X x : fx 2 ½x0 lx =2 ; x0 þ lx =2 g X

(14)

Equation (12) can then be written as: x ; tÞ ¼ ax AðX x ; tÞ b BðX

(15)

R @ Tðx;tÞ R @ 2 f ðxÞ T ðx; tÞ dx f ðxÞ dx 2 @t x ; tÞ ¼ RXx x ; tÞ ¼ XRx @ x where BðX ; Að X x T ðx; tÞf ðxÞ dx x T ðx; tÞf ðxÞ dx X X Under condition (10), Equation (15) can be applied to an anisotropic material to estimate the thermal diffusivity ax in the X-direction. A similar demonstration can be carried out in the Y-direction to estimate the thermal diffusivity ay. In the following, the LOWF method is tested on numerical models for an estimation of thermal diffusivity for isotropic and anisotropic cases. Low values of diffusivity have been voluntarily used in the numerical models. In this way, the extension of the heat source is limited to a small area in order to reduce the calculation duration for a given mesh density. The presented results do not lose in generality. 3. Results and discussion 3.1. Case of an isotropic material (α = αx = αy ) A 2D model of an isotropic plate is built with a = 4.27 × 10−8 m2 s−1 using the FE code COMSOL Multiphysics®.[22] The geometric dimensions Lx = Ly = 3 cm of the plate are discretized into Nx = Ny = 300 nodes. The spatial pitch is then Dx = Dy = 100 μm, corresponding to the pixel size of a standard infrared camera. The initial temperature field is T0 ðx; yÞ ¼ 0. A heat pulse is imposed at the centre of the plate at t0 = 0 s. The temperature field is simulated for the temporal interval [t0 = 0 s, t120 = 120 s] discretized into Nt = 120 time steps with Dt = 1 s. The thermal boundary conditions in the X- and Y-direction are, respectively, as follows:

@T ðx; y; tÞ

@T ðx; y; tÞ

¼ 0 and ¼0 (16)

@x @y x¼Lx =2 y¼Ly =2 Heat losses are simulated perpendicularly to the plane (XY) with h = 5 W m−2 K−1. The simulated temperature field is written T ðxi ; yj ; tk Þ, with i 2 f1; 2; . . .; 300g, j 2 f1; 2; . . .; 300g and k 2 f1; 2; . . .; 120g. A Gaussian white noise dðxi ; yj ; tk Þ characterised by a standard deviation RN =S is added to the temperature data T ðxi ; yj ; tk Þ. RN=S is the representative of the noise-to-signal ratio and can be numerically adjusted to control noise intensity. The added noise dðxi ; yj ; tk Þ is previously normalised by division with its maximum value.

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Figure 2. FE isotropic model. Mapping of the simulated temperature field at t120 = 120 s with RN =S = 0%.

Figure 3.

xy . Scheme of the 2D plate Xxy and 2D subdomain X

xy ; tk Þ; yk ðX xy ; tk Þ and the fitted line (D) for Figure 4. FE isotropic model. Points Pk xkp ðX p RN =S = 0% (a) and RN =S = 10% (b).

Quantitative InfraRed Thermography

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Figure 2 shows the mapping of the simulated data T ðxi ; yj ; tk Þ at t120 = 120 s without noise (RN =S = 0%). xy . The LOWF method is Figure 3 shows the 2D plate X and the 2D subdomain X applied to the simulated temperature data in order to estimate the thermal diffusivity a. xy (Figure 3) is chosen arbitrarily with fx0 ¼ y0 ¼ 0g and The subdomain X lx¼ ly ¼ 150D . In accordance with Equation (7), points xy ; tk Þ; Bk ðX xy ; tk Þ are represented (Figure 4). The fitted line (D) is defined Pk Ak ðX xy ; tk Þ ¼ a^ Ak ðX xy ; tk Þ b, k ¼ 1; 2; . . .; 120. The slope a^ of (D) corresponds by Bk ðX to the estimated thermal diffusivity. In order to analyse the influence of noise on the accuracy of the LOWF method, data of good quality (RN =S = 0%) and medium quality (RN =S = 10%) are considered. The deviation e between the thermal diffusivity a and the estimated value a^ is calculated with e ¼ 100 jða a^Þ=aj. For RN =S = 0% (Figure 4(a)), there is no noise and points Pk are aligned. The slope of (D) provides the estimated thermal diffusivity a^ = 4.21 × 10−8 m2 s−1 corresponding to a deviation e = 1.47%. The fact that this deviation is not zero is essentially due to the spatial and temporal discretization of data. It has been verified that the use of a refined spatial mesh is a way to reduce this deviation value. For RN =S = 10% (Figure 4(b)), the added noise disturbs the alignment of points Pk . Indeed, the noise affects the accurate calculation of terms T ðx; y; tÞ and @t T ðx; y; tÞ in the weak formulation Equation (6). The slope of (D) gives a^ = 4.2 × 10−8 m2 s−1 corresponding to e = 1.54%. The deviation value e is unchanged for unnoisy (RN =S = 0%) and noisy temperature variation (RN =S = 10%). This result reflects a priori a good robustness of the LOWF method with respect to noise. To verify this, the method is tested for a different noise signal distribution with temperature data simulated for values of RN=S ranging from 0 to 10%. For each value of RN =S , 100 different distributions of the noise matrix dðxi ; yj ; tk Þ are simulated and 100 estimations a^n ðn ¼ 1; 2; . . .; 100Þ of thermal diffusivity are carried out. Figure 5 shows the minimum, mean and maximum deviation values as a function of noise intensity RN =S.

Figure 5. of RN =S .

FE isotropic model. Minimum, mean and maximum deviation e curves as a function

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The mean deviation remains stable at 1.47% (Figure 5). The non-null deviation value obtained even for RN =S = 0% is due to the discretization process in COMSOL Multiphysics®. Indeed, complementary verifications shown that this value decreases with refined meshes. The gap between the maximum and minimum deviation curves increases with the noise intensity RN =S, but remains low with a maximum value of dispersion equal to ±0.08% for RN =S = 10%. This result shows the ability of the LOWF method to robustly estimate thermal diffusivity from noisy temperature data. 3.2. Case of an anisotropic material (αx 6¼ αy ) The previous FE model is used to simulate temperature data with ax = 4.27 × 10−8 m2 s−1 and ay = 2 ax = 8.54 × 10−8 m2 s−1. All other parameters of the model are unchanged. Figure 6 shows the mapping of the simulated data T ðxi ; yj ; tk Þ at t120 = 120 s without noise (RN =S = 0%). The boundary conditions (16) of the FE anisotropic model allow the averaging LOWF approach. Thus, the averaged 1D temperature fields T ðxi ; tk Þ and T ðyj ; tk Þ are considered to estimate ax and ay , respectively. 3.2.1. Estimation of αx

x is defined The analytical function f ðxÞ ¼ ð1 x2 Þ2 is considered. The subdomain X arbitrarily with x0 ¼ 0 and lx ¼ 150D x (Figure 7). x ; tk Þ; Bk ðX x ; tk Þ , k ¼ 1; 2; . . .; 120 and the fitted line (D) are Points Pk Ak ðX represented for RN =S = 0% (Figure 8(a)) and RN =S = 10% (Figure 8(b)). For RN =S = 0% (Figure 8(a)), points Pk are perfectly aligned along (D). The slope of (D) gives the estimated thermal diffusivity a^x = 4.25 × 10−8 m2 s−1 corresponding to a deviation e = 0.51%. The low dispersion of estimations is characterised by a standard deviation r = 7 ×10−24 m2 s−1. For RN =S = 10% (Figure 8(b)), the alignment of points Pk is more disturbed. The estimated thermal diffusivity is a^x = 4.24 × 10−8 m2 s−1 corresponding to a deviation

Figure 6. FE anisotropic model. Mapping of the simulated temperature field at t120 = 120 s with RN =S = 0%.


Figure 7.

9

x with x0 ¼ 0 and lx ¼ 150Dx . Scheme of the 1D subdomain X

x ; tk Þ; Bk ðX x ; tk Þ and the fitted line (D) for Figure 8. FE anisotropic model. Points Pk Ak ðX RN =S = 0% (a) and RN =S = 10% (b).

x ; tk Þ; Bk ðX x ; tk Þ and the fitted line (D) for Figure 9. FE anisotropic model. Points Pk Ak ðX RN =S = 0% (a) and RN =S = 10% (b).

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e = 0.83% with r = 1 × 10−10 m2 s−1. The LOWF method has then provided an accurate estimation of ax from unnoisy data or data corrupted with noise. 3.2.2. Estimation of αy

y is defined The analytical function considered is f ðyÞ ¼ ð1 y2 Þ2 . The subdomain X x ; tk Þ; Bk ðX x ; tk Þ , arbitrarily with y0 ¼ 0 and ly ¼ 150Dy . Points Pk Ak ðX k ¼ 1; 2; . . .; 120 and the fitted line (D) are represented for RN =S = 0% (Figure 9(a)) and RN =S = 10% (Figure 9(b)). For RN =S = 0% (Figure 9(a)), points Pk are aligned along (D). The estimated value of thermal diffusivity is a^y = 8.5 × 10−8 m2 s−1 corresponding to e = 0.5% with r = 7 × 10−23 m2 s−1. For RN=S = 10% (Figure 9(b)), as previously, the alignment of points Pk is disturbed. The estimated thermal diffusivity is a^y = 8.5 × 10−8 m2 s−1 corresponding to e = 0.54% with r = 6 × 10−11 m2 s−1. Thermal diffusivities ax and ay have been estimated with good precision. This result shows that, from a 2D temperature field verifying condition (10), the LOWF method can be applied to estimate the thermal diffusivity of an anisotropic material in the principal directions of the plan. It is possible by applying the method to the 1D averaged temperature field calculated in each principal direction. In the numerical examples presented, the subdomain size has been chosen arbitrarily. A criterion is proposed for an optimal sizing of the subdomain in order to improve the accuracy of the LOWF method. 4. Criterion for an optimal sizing of the subdomain In this section, the optimal size of the subdomain is searched for the case of a heat source lowly extended. However, it can be noted that more generally, the size of the heat source probably also determines the optimal size of the subdomain.

x with x0 ¼ 0 and lx ¼ 10Dx , lx ¼ 100Dx and Figure 10. Scheme of the 1D subdomain X lx ¼ 200Dx .


11

x ; tk Þ; Bk ðX x ; tk Þ and the Figure 11. FE anisotropic model with RN =S = 10%. Points Pk Ak ðX fitted line (D) for 1D subdomain Xx with x0 ¼ 0 and lx ¼ 10Dx (a), lx ¼ 100Dx (b) and lx ¼ 200Dx (c).

12 Table 1.

T. Kouadio et al. x with RN =S = 10%. Estimations for three sizes lx of subdomain X 10 Δx

lx 2 −1

a^x (lx) 3r (m s ) Deviation e (%)

−8

100 Δx −11

5.4 × 10 ± 2.7 × 10 27.1

−8

4.3 × 10 ± 6 × 10 0.24

200 Δx −11

4.2 × 10−8 ± 1.2 × 10−9 1.3

x centred in x0 ¼ 0. In order to analyse Let us optimise the size of a subdomain X the influence of the subdomain size on the accuracy of the LOWF method, the thermal x : lx ¼ 10Dx , diffusivity ax is estimated considering three different sizes for X lx ¼ 100Dx and lx ¼ 200Dx (Figure 10). Temperature data are simulated using FE anisotropic model with the previous x ; tk Þ; Bk ðX x ; tk Þ , k ¼ 1; 2; . . .; 120 and RN =S = 10%. Figure 11 shows points Pk Ak ðX the fitted line (D) for lx ¼ 10Dx, lx ¼ 100Dx and lx ¼ 200Dx . Results are summarised in x. Table 1 for sizes lx ¼ 10Dx , lx ¼ 100Dx and lx ¼ 200Dx of the subdomain X k For lx ¼ 10Dx (Figure 11(a)), points P show a low dispersion around (D). The high deviation value e = 27.1% can be explained by the fact that the small amount of data taken into account when considering a small size of subdomain is not sufficient to estimate the thermal diffusivity accurately. For lx ¼ 100Dx (Figure 11(b)), a dispersion of points Pk around (D) occurs, but the estimation is more precise with a deviation e = 0.24%. For lx ¼ 200Dx (Figure 11(c)), the dispersion of points Pk around (D) is consider x is large, more temperature data with a high RN =S ratio are taken into able. Because X account and involves a larger dispersion of estimations. Here, the interval 3r even represents a dispersion larger than the deviation e = 1.3%. The results of these three estimations show that there is an optimal subdomain size, somewhere between small and large. A criterion is proposed to find the optimal subdomain size. This criterion is based on the fact that the dispersion of points Pk around (D) is related to the size of the subdomain (Figure 11). The dispersion can be associated to the value of the linear correlation coefficient K 2 ½1 1 of points Pk defined by PNt k k k¼1 ðA AÞ ðB BÞ ffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (17) PNt P Nt k AÞ k BÞ 2 2 ðA ðB k¼1 k¼1 P Nt k P Nt k 1 1 ¼ with A k¼1 A and B ¼ Nt k¼1 B . Nt The coefficient K is a statistical tool to evaluate the linearity of variables Ak and k k k B . For values of K close to −1 or 1, the alignment of points P A ; B is good: the dispersion of points Pk around (D) is low, increasing the accuracy of the estimation. For values of jKj close to 0, variables Ak and Bk are linearly independent: the dispersion of points Pk around (D) is high, decreasing the accuracy of the estimation. The proposed criterion consists of defining the optimal subdomain size as that which correspond to the middle of the jKj-curve section where maximum values are stabilised. In order to test this criterion, the value of K is calculated for different sizes x , ranging from smallest to largest. For each size lx , the deviation e is calculated. lx of X The link between the coefficient K and the deviation e is then discussed. Figure 12 shows K-curve and e-curve as a function of subdomain size nx , where lx ¼ ðnx 1ÞDx . The evolution of the two curves (Figure 12) is in concordance with the previous x results (Figure 11). The lowest deviation values are observed for intermediate sizes of X (50 ≤ n ≤ 170) and correspond to the highest values of the linear correlation coefficient K. k


13

Figure 12. FE anisotropic model with RN =S = 10%. K-curve and e-curve as a function of subdomain size nx for the averaged temperature field T ðxi ; tk Þ.

x is defined by nopt = 67, that is lopt ¼ 66Dx . The estimated The optimal size of X thermal diffusivity is then a^x = 4.26 × 10−8 m2 s−1, corresponding to a deviation e = 0.14%. y . Figure 13 shows the The criterion is also tested for the sizing of subdomain X K-curve and the e-curve as a function of subdomain size ny where ly ¼ ðny 1ÞDy . As y is defined by previously, the two curves are in good agreement. The optimal size of X nopt = 79, that is lopt ¼ 78Dy and the estimated thermal diffusivity is a^y = 8.49 × 10−8 m2 s−1 corresponding to e = 0.6%. The proposed criterion based on the value of K to optimise the size of the subdomain is numerically validated, thanks to the accurate estimations of ax and ay . In a similar way, a sensitivity analysis has been performed [23] to define the optimal position x0 of the centre of the subdomain. Considering a subdomain with a fixed size at different positions x0 , this analysis shows that the estimation accuracy is the best when x0 coincides with the point of maximal temperature, i.e. where the RN =S ratio is the lowest.

Figure 13. FE anisotropic model with RN =S = 10%. K-curve and e-curve as a function of subdomain size ny for the averaged temperature field T ðyj ; tk Þ.

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Figure 14. View of the experimental device. Thermal excitation of a Titanium thin plate with a laser diode. Recording temperature field during thermal relaxation using an infrared camera.

5. Experimental estimation of thermal diffusivity A thin Titanium plate with dimensions L = 50 mm, l = 5 mm and e = 0.5 mm is used as a sample. The thermal diffusivity value a = 9.3 × 10−6 m2 s−1 provided in the literature for Titanium is considered as the nominal [24] thermal diffusivity of the plate. A 50 mJ burst of energy is deposited at the surface of the sample with a laser diode pulse. In order to minimise heat transfers through the thickness, the sample is placed on an insulating plate in Skamol. The temperature field is recorded during the thermal relaxation with an IR camera FLIR SC-7000 (Figure 14). Temperature data are recorded as frames of 100 × 35 pixels with Dx ¼ Dy = 143 μm. The acquisition frequency is 40 Hz, corresponding to Dt = 25 ms. The recorded temperature data are written as T ðxi ; yj ; tk Þ, with i 2 f1; 2; . . .; 100g, j 2 f1; 2; . . .; 35g and k 2 f1; 2; . . .; 80g. The temperature fields at t1 ¼ 1Dt, t2 ¼ 2Dt and t5 ¼ 5Dt are represented in Digital Level (Figure 15). Digital level is a unit proportional to the thermal radiation received by the camera. The calibration curve of the infrared camera lens enables the Digital Level to be considered as a unit proportional to the temperature. Nevertheless, in such estimation processes, absolute temperature is not required. The surface of the sample has been depolished to avoid optical reflections. At t1 (Figure 15(a)), the plate shows a 3D thermal behaviour: heat diffusion occurs along the three directions X, Y and Z of the plate. The 3D heat transfer is relatively short since e=L 1; e=l 1: At t2 (Figure 15(b)), the characteristic diffusion time in the thickness s ¼ e2 a is exceeded, i.e. the Fourier number Fo ¼ a t [ 1: heat diffusion occurs in the plane (XY). During this 2D diffusion, the thermal fluxes can be considered as null at the up and down edges of the sample. The temperature field then verifies one of the three conditions allowing the use of the averaging LOWF approach in a specific direction. The 2D diffusion continues until about t5 . At t5 (Figure 15(c)), the up and down edges of the sample are reached and the isotherms of the temperature field are almost perpendicular to the X-direction. That is the


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Figure 15. Two-dimensional temperature fields recorded with the infrared camera at t1 ¼ 1Dt (a), t2 ¼ 2Dt (b) and t5 ¼ 5Dt (c).

end of heat diffusion along the Y-direction. The temperature field has a 1D behaviour for a long time: heat diffusion occurs along the X-direction. During the 1D thermal behaviour, the temperature field verifies one of the three conditions allowing the use of the averaging LOWF approach in a specific direction. In the following, the LOWF method is used to estimate thermal diffusivity ax of the plate in the X-direction. For this purpose, the averaged thermal field T ðxi ; tk Þ is considered and the optimal sizing criterion is applied. The first image corresponding to t ¼ t1 is not taken into account in the estimation process, since the 2D heat transfer starts at x is t ¼ t2 . The analytical function f ðxÞ ¼ ð1 x2 Þ2 is used and the subdomain X centred on the temperature field with x0 ¼ 50 (Figure 15). Figure 16 shows the K-curve and the e-curve as a function of subdomain size nx and points Pk corresponding to the optimised subdomain size. The optimal size of sub x is defined by nopt = 17 (Figure 16(a)), that is lopt ¼ 16Dx corresponding to domain X the linear correlation coefficient K = 0.77. The slope of (D) gives an estimated thermal diffusivity a^x = 9.2 × 10−6 m2 s−1 corresponding to a deviation e = 0.8% (Figure 16(b)). An analysis of uncertainty on the slope of the linear regression (D) in the vicinity of

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Figure 16. Experimental temperature data. K-curve and e-curve as a function of the subdomain size nx (a), points Pk and the fitted line (D) for the optimised subdomain defined by x0 ¼ 50 and lopt ¼ 16Dx .

the optimal size nopt (12 n 22) provides a dispersion on the estimated diffusivity r = 3 × 10−7 m2 s−1. The thermal diffusivity of the Titanium thin plate is estimated with a good accuracy in the X-direction. This result confirms the experimental validation of the LOWF method. 6. Conclusion A new method for the thermal characterisation of homogeneous materials has been used to determine the thermal diffusivity of a titanium plate. The LOWF method proposed is based on a weak formulation of heat transfer equation in association with InfraRed Thermography experiments carried out on thin samples. An analytical test function is used for a robust estimation of thermal diffusivity from noisy data. The formalism of the LOWF method is presented for the case of an isotropic material. Numerical tests on simulated temperature data show a good accuracy of the method when data are corrupted with noise. An extension of this formalism is proposed for an anisotropic material and also provides robust estimations. Another advantage of the LOWF method is that thermal diffusivity can be estimated on a subdomain. For optimal accuracy of the method, it has been shown that the size of the subdomain should not be chosen arbitrarily. A criterion for an optimal sizing of the subdomain is then proposed and tested with conclusive results obtained numerically and experimentally. With further developments, the LOWF method could be an interesting alternative for a multiscale thermal characterisation of materials, between point-by-point and global scales. Funding This work was supported by: Fondation de Recherche pour l’Aéronautique et l’Espace (FRAE).

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