Measurement of the external parameters in quantitative active thermography

Thermography is a non contact and non-destructive technique well suited for the fast detection of subsurface damages in heat conducting structures. Surface temperature maps u_k ($k=1, ..., N$), taken at discrete time instants by means of an infrared camera, are used to verify the integrity of the underlying object. The position, size and shape of the damage can also be identified.

Applications of thermographic nondestructive evaluation have been reported, for example, in the diagnostics of aircraft structures [2, 3], in the integrity analysis of pipelines [4], in the detection of cracks in steel bridges [5] and, in general, in the investigation of thick metallic structures [6, 7]. Thermography is also valuable for the diagnostics and monitoring of poorly conducting materials, like concrete, bricks and, in general, all kinds of masonry [8, 9]. A comprehensive list of the practical applications of thermography can be found in [10]. Interest in the measurement of cracks and defects in materials is continuously increasing, as testified by recent papers dealing with thermographic detection [19, 20, 21] and other measurement methods [22].

In this paper we assume that the specimen is a plate whose top surface is inaccessible to direct inspection and subject to the action of an aggressive environment. The damaged top side is a perturbed planar surface. On the other hand, we can stimulate the object from the bottom side by means of a controlled heat flux ϕ (active thermography). The same side is accessible to infrared vision.

Our approach to the detection and evaluation of inaccessible damage [12, 16] requires a thorough knowledge of both the 'internal' parameters of the problem—the specific heat capacity c, the density ρ and the thermal conductivity κ—and of the 'external' parameters relating to the measurement procedure and to the environmental conditions: the temperature maps u_k, the imposed heat flux ϕ, the heat exchange coefficient h, and the initial and ambient temperature U⁰.

We assume that c, ρ, κ and U⁰ are known a priori and that the temperature maps u_k are given with an error related to the accuracy of the camera. On the other hand, the heat flux ϕ, although related to the technical characteristics of the source and its positioning, remains essentially unknown. The same is true for h whose numerical value is strongly influenced by the environmental conditions close to the inaccessible side of the specimen.

A precise evaluation of ϕ and h is the main goal of the present paper. The importance of the knowledge of such parameters is often underestimated. As an example concerning the characterization of defects by pulsed thermography, although the surface temperature is shown to explicitly depend on the pulse energy ϕ [17], the solution is made independent of ϕ by computing the temperature contrast as the difference between that of the defective area and of a sound neighboring region. Apart from the difficulty of defining a non-defective region (where is it?), that choice implicitly assumes a constant ϕ, a condition usually not met in the real world. Such a difficulty has been highlighted in [18], where the necessity of exactly determining uneven heating is clearly addressed. The adopted 'substitution' technique, however, although valuable in a laboratory framework, is not applicable in real-world situations where a free-sample made having the characteristics of the material under investigation is barely feasible. Similar considerations can be made regarding the heat transfer coefficient h, whose importance is ofter undervalued.

Starting from the observation that ϕ is independent of the specimen, the heat flux is determined using two completely different methods (Fourier one-dimensional (1D) approximation and energy penetration depth) that validate each other, making our solution more robust (section 2). The heat transfer coefficient h is calculated by means of a simplified form of the thin plate approximation (see section 4 and [12]).

1.1. The experimental framework

A number of sources, usually lamps, heat the specimen from the accessible side. The other sides of the plate are subjected to natural or forced convection. The defect to be detected is located on the opposite inaccessible surface. Temperature maps are acquired by means of a thermographic camera.

Figure 1 shows the laboratory setup. Two high-power spotlights (1 kW each) are placed and focalized so as to produce a high heat flux density on the central region of the plate. The flux, as we will see in the following, has a parabolic shape. A thermographic image is acquired before the heating and for some hundreds of seconds after power on, at intervals of 5–6 s, by means of an FLIR B335 thermal camera.

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1.2. Details of the mathematical model

In mathematical terms, evaluating the damage means solving an inverse problem of the class of inverse heat conduction problems (see, for example, [1]).

Observe that our plate-shaped structure has one dimension (thickness) significantly smaller than the other two. More precisely, let $\Omega_0$ represent the undamaged plate: $\Omega_0=\{(x, y, z) \in (-\frac{L}{2}, \frac{L}{2})\times (-\frac{L}{2}, \frac{L}{2}) \times (0, a)\}$ ($0< a \ll L$). Let $\rho c$ be the volumetric heat capacity. The specimen is assumed homogeneous and isotropic so that c, ρ and κ are positive scalar constants. We refer to [11] for a complete background regarding the classical theory of heat.

The accessible surface $S_B=\{(x, y, 0) x, y \in [-\frac{L}{2}, \frac{L}{2}] \}$ is heated by means of a device (e.g. the lamps) producing a heat flux $\phi(x, y)$, constant in time, through S_B.

The damaged specimen, that we denote by $\Omega_\epsilon$, consists of a 'deformed' version of $\Omega_0$ in which the top inaccessible surface $S_{T}=\{(x, y, a) x, y \in [-\frac{L}{2}, \frac{L}{2}] \}$ changes into $S_\epsilon=\{(x, y, a-\epsilon \theta(x, y)) x, y \in [-\frac{L}{2}, \frac{L}{2}] \}$, where is a scale factor of the 'shape' function $\theta(x, y)$ that describes the deviation from a planar surface.

The temperature maps of the bottom side of $\Omega_{\epsilon}$ are $u^\epsilon_k=u^\epsilon(x, y, 0, t_k)$ for $0<t_1<t_2<...<t_N\leqslant T_{\rm max}$. When we begin to collect the temperature maps, the temperature is U⁰, which is also the external environmental temperature.

Detecting the damage requires us to solve the following inverse problem IP:

Identify the function $\epsilon\theta$ from the knowledge of

• 1.  
  The thermal contrast $u^\epsilon(x, y, 0, t_k)-u^0(x, y, 0, t_k)$ for $0<t_1<t_2<...<t_N\leqslant T_{\rm max}$.
• 2.  
  The flux ϕ and the heat exchange coefficient h.
• 3.  
  The physical parameters U⁰, ρ, c and κ.

Actually, assuming that the physical parameters are known, the only available data consist of the sequence of the temperature maps $u^\epsilon(x, y, 0, t_k)$.

In this paper we show how to determine reliable approximated values of ϕ, h and, consequently, $u^0(x, y, z, t)$. We do this by exploiting typical features of stepped heating thermography (see [10], section 9.3). Once ϕ and h are known, it is easy to compute the background temperature of $\Omega_0$.

This inverse problem is framed in the following initial boundary value problem (IBVP) for the heat equation:

$$\begin{align} \displaystyle \label{eq:01} u_t=\alpha \Delta u \nonumber \end{align} \tag{ 1 }$$

in

$$\begin{align*} \displaystyle \Omega_{\epsilon}=\{(x,y,z): x,y \in (-\frac{L}{2},\frac{L}{2}) ; z<a-\epsilon\theta(x,y) \}\nonumber \end{align*}$$

with boundary conditions (called Robin or third kind) that account for energy exchanges between the specimen and the environment:

$$\begin{align} \displaystyle \label{eq:02} \kappa u_n(\sigma,t)+h(u(\sigma,t)-U^0)-\phi(\sigma)=0 \nonumber \end{align} \tag{ 2 }$$

where $\sigma \in \partial \Omega_\epsilon$ and initial data

$$\begin{align} \displaystyle \label{eq:03} u(x,y,z,0)=U_{{\rm in}}(x,y,z,0). \nonumber \end{align} \tag{ 3 }$$

Here, u_n is the outward normal derivative (normal with respect to the boundary $\partial\Omega_\epsilon$) and $\alpha=\frac{\kappa}{\rho c}$ is the diffusivity. The heat exchange coefficient h is related to the geometry of the specimen and to the environmental conditions close to $S_\epsilon$. We assume $h>0$ is constant at $S_\epsilon$; $h=0$ on the vertical sides of the plate. As for the heat flux, we assume that ϕ is concentrated on S_B (i.e. $\phi(\sigma)=0$ for $\sigma \in \partial\Omega_\epsilon - S_B$). As long as ϕ is constant in time we say that we are dealing with stepped heating thermography. Finally, we suppose that U⁰ is the same on S_B and on S_T or $S_\epsilon$ and that the initial temperature is $U_{{\rm in}}(x, y, z)=U^0$.

The direct IBVP (1)–(3) is well posed, whilst the inverse problem IP is not. It means, in practice, that a given temperature distribution on the surface S_B is 'compatible' with several shapes and scale parameters. A solution of the inverse problem can be obtained by a semi-explicit formula [12] or applying a suitable regularization strategy to treat ill-posedness using genetic algorithms, artificial neural networks, or other techniques to minimize an appropriate cost function ([7, 13, 14, 15] to cite only a few examples).

An easier approach to the 'on field' determination of heat flux consists of using a thermally insulating screen, for example, a plywood white-painted panel. That screen is placed in front of the surface of interest so as to receive exactly the same energy density. If the flux density $\phi=\phi_0$ is constant in space and time, the temperature $u^0=u^0(z, t)$ is independent of x and y. Indeed, it can be computed as a series dependent on z and t [11]:

$$\begin{align} \displaystyle u^0(z,t) = &\,U^0 + \frac{\phi}{h} \left[ 1 + \lambda \left.\Bigg(1 - \frac{z}{a} \right)\right. \nonumber \\ &\left.- \sum_{n=1}^\infty \frac{2 \lambda (\beta_n^2 + \lambda^2 ) {\rm cos}(\beta_n z/a)}{\beta_n^2 (\lambda + \lambda^2 +\beta_n^2 )} {\rm e}^{-\beta_n^2 \tau} \right] \label{unidim} \nonumber \end{align} \tag{ 4 }$$

where $\lambda = ah/\kappa$, $\beta_n$ are the real roots of $\beta {\rm tan}\beta = \lambda$ and $\tau = \alpha t/a^2$.

As the nth root $\beta_n$ belongs to the interval $[(n-1)\pi, (2n-1)\pi/2]$, a short time (a few seconds) yields very large exponents in (4) for $n > 1$, for the assumed values of the physical and geometrical parameters α, κ, h and a. Therefore, the first term of the series (4) provides a sufficiently good approximation of the background temperature.

For low values of κ (negligible longitudinal heat transfer) and t (negligible dependence on h before a time $t_a>0$), equation (4) can be used successfully for recovering ϕ from u⁰. It gives us complete quantitative information about the heat flux. It comes from formula (4) that a constant flux $\phi_0$ is determined by one temperature measurement taken in the first few seconds.

Assume that ϕ is a generic smooth positive function so that it can be approximated by the trigonometric polynomial

$$\begin{align} \displaystyle \phi(x) = {\phi_0}+ \sum_{k = 1}^N \left (\phi_{2k-1} \cos \frac{2 k \pi x}{L} + \phi_{2k} \sin \frac{2 k \pi x}{L} \right). \label{fourier_sum_fi} \nonumber \end{align} \tag{ 5 }$$

The coefficients $\phi_j$ for $j=0, 1, ..., 2N$ can be recovered from the knowledge of the temperature $u(x, 0, t)$ measured on the accessible surface $z = 0$ at suitable times.

Let $u_{j}(z, t)$ be the solution of the following two-dimensional (2D) IBVP:

$$\begin{align*} \displaystyle u_t=\alpha u_{zz} + \alpha u_{xx} \nonumber \end{align*}$$

$$\begin{align*} \displaystyle \kappa u_z(x,a,t)+h(u(x,a,t)-U^0)=0 \nonumber \end{align*}$$

$$\begin{align*} \displaystyle -\kappa u_z(x,0,t)=\phi_{j} \left \{\begin{array}{@{}rl@{}} \cos (2{\rm j} \pi x/L) & \nonumber \\ \sin (2{\rm j} \pi x/L) & \end{array} \right . \nonumber \end{align*}$$

$$\begin{align*} \displaystyle u(x,z,0)=U^0. \nonumber \end{align*}$$

We observe that the solution of the 1D problem (4) for a given constant ϕ allows us to closely estimate the solution in $z = 0$ of the above 2D problem for times sufficient to give a small 'Fourier number', i.e. $t \ll a^2/\alpha$.

There is numerical evidence that for an arbitrary heat flux described by (5), an approximated solution could be obtained by superposition:

$$\begin{align} \displaystyle u(x,0,t) \approx u_0(t) + \sum_{k = 1}^N \left (u_{2k-1}(t) {\rm cos}\frac{2 k \pi x}{L} + u_{2k}(t) {\rm sin}\frac{2 k \pi x}{L} \right) \label{unidimensional_sum} \nonumber \end{align} \tag{ 6 }$$

with

$$\begin{align} \displaystyle u_0 &= \frac{\phi_0}{h} (1+ \lambda -\Sigma_N ) + U_0 \nonumber \\ u_j &= \frac{\phi_j}{h} (1+ \lambda -\Sigma_N ) \label{unidimensional_terms} \nonumber \end{align} \tag{ 7 }$$

for $j=0, 1, ..., 2N$, where λ, the $\beta_k$s and

$$\begin{align} \displaystyle \Sigma_N(t) = \sum_{k=1}^N \frac{2 \lambda (\beta_k^2 + \lambda^2)}{\beta_k^2(\lambda + \lambda^2 + \beta_k^2)}{\rm e}^{-\beta_k^2 \alpha t/a^2} \nonumber \end{align} \tag{ 8 }$$

(a finite truncation of the series appearing in the analytical solution of the unidimensional problem (4)).

For a plywood panel with a thickness of 10 mm, the Fourier number is lower than 0.2 for times up to about 200 s. In such conditions, for any N:

$$\begin{align} \displaystyle u(x,0,t) \approx U^0 + \left[ u^0(0,t) - U^0 \right] \left \{\begin{array}{@{}rl@{}} \cos (2 N \pi x/L) & \nonumber \\ \sin (2 N \pi x/L) & \end{array} \right. \!\!\! \!\!\!. \nonumber \end{align} \tag{ 9 }$$

Since we know $u(x, 0, t)$ from our thermal measurements, we can compute the u_js by means of the scalar products

$$\begin{align*} \displaystyle u_0(t) &= \frac{1}{L} \int_{-L/2}^{L/2} u(x,t) {\rm d}x \nonumber \\ u_{2k-1}(t) &= \frac{1}{L} \int_{-L/2}^{L/2} u(x,t) \cos \frac{2 k \pi x}{L} {\rm d}x \nonumber \\ u_{2k}(t) &= \frac{1}{L} \int_{-L/2}^{L/2} u(x,t) \sin \frac{2 k \pi x}{L} {\rm d}x. \nonumber \end{align*}$$

Finally, we invert relations (7) and determine the $\phi_j$s. In particular, we get

$$\begin{align*} \displaystyle \phi_0 &= \frac{(u_0(t) - U_0)h}{1+\lambda+\Sigma_N(t)} \nonumber \\ \phi_{j} &= \frac{u_j(t) h}{1+\lambda+\Sigma_N(t)}. \nonumber \end{align*}$$

We observe that the 'theoretical' $\phi_j$s are time-independent although they are infinite sums of the time-dependent quantities $u_j(t)$ and $\lim _{N \to \infty} \Sigma_N(t)$. We expect that the numerical approximations produced here are slightly dependent on $t$ as a consequence of both the truncation of the series and the noise/uncertainty on the temperature data.

An alternative way to compute the heat flux would therefore be appreciated as a validation tool. A simple estimate of ϕ can be obtained by measuring the temperature behavior on a thermal insulating plate subjected to the same heat flux density ϕ. As before, over short timescales the low thermal conductivity allows us to consider the problem as unidimensional because the lateral diffusion is negligible and the heating of the plate is nearly adiabatic. The thermal energy $E(t)$ in the specimen, per unit width of the plate, is straightforwardly related to the integral of the temperature at time t over the plate thickness:

$$\begin{align} \displaystyle E(t) = \rho c \int_0^a (u(z,t) - U_0) {\rm d}z \nonumber \end{align} \tag{ 10 }$$

and the heat flux is computed from the energies at two close times t₁ and t₂:

$$\begin{align} \displaystyle \phi = \frac{\Delta E}{\Delta t} = \frac{E(t_2) - E(t_1)}{t_2 - t_1}. \nonumber \end{align} \tag{ 11 }$$

Of course, we cannot measure the temperature profile $u(z, t)$ inside the plate. The temperature measured on the accessible surface $z = 0$, $u(0, t)$ can be used to determine a depth δ at which $E(t) = \rho c (u(0, t) - U_0) \delta$. It can be easily demonstrated that such a depth is related to the so-called thermal penetration depth:

$$\begin{align} \displaystyle \delta = \frac{1}{2} \sqrt{\pi \alpha t}. \nonumber \end{align} \tag{ 12 }$$

The above derivation is rigorously true when ϕ is constant, independent of x. At times t₁ and t₂ such that $\delta \ll a$, the heat flux is given by:

$$\begin{align} \displaystyle \phi = \frac{\sqrt{\pi}}{2} \left[ \frac{(u(0,t_2) - U^0) \sqrt{t_2} - (u(0,t_1) - U^0) \sqrt{t_1}}{t_2 - t_1} \right] \sqrt{\kappa \rho c} \label{approx_phi} \nonumber \end{align} \tag{ 13 }$$

the last square root term being the effusivity of the insulating plate. The formula (13) can also be used in the case of a variable flux, $\phi = \phi(x)$, at timescales short enough to neglect the heat diffusion. Figure 2 compares the ϕ obtained from the series (5) with $N = 10$ with the approximation (13) for a true heat flux $\phi(x) = 600 - 10^{4} (x - L/2){\hspace{0pt}}^2$. Temperature data have been obtained on a simulated wooden plate of width $L = 0.3$ m and thickness $a = 8$ mm, having thermal parameters $\kappa = 0.12 ~{\rm W~m}^{-1}~{\rm K}^{-1}$, $\rho = 500~$kg ${\rm m}^{-3}$, $c = 1760 ~{\rm J~kg}^{-1}~{\rm K}^{-1}$, and assuming $h = 5 ~{\rm W ~m}^{-2}~{\rm K}^{-1}$ on the top surface of the plate.

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In section 5 we will see how the above procedure is applied on real thermographic data.

The heat exchange coefficient h can be measured by means of the 'lumped-capacitance' method [23], briefly summarized in the following. An homogenous object of volume V and external surface $\partial V$ is initially at temperature U_i. From time $t = 0$ the object exchanges energy with the environment at temperature U_a. The heat balance equation for that system

$$\begin{align} \displaystyle \rho c V u_t + h S (u - U_a) = 0 \nonumber \end{align} \tag{ 14 }$$

is formally identical to the discharge equation for an electrical capacitor, and has the following solution:

$$\begin{align} \displaystyle u(t) = U_a + (U_i - U_a) {\rm e}^{-t/\tau} \nonumber \end{align} \tag{ 15 }$$

with the time constant τ given by

$$\begin{align} \displaystyle \tau = \frac{\rho c V}{h S} \label{tau}. \nonumber \end{align} \tag{ 16 }$$

If the volume is a parallelepiped of the base surface L² and height $d \ll L$, $\tau \approx 0.5 \rho c L/h$.

Experimentally, a small aluminum sheet (size $66 \times 44 \times 5$ mm) is heated by a heating plate up to a temperature of about 80 °C, with an ambient temperature of 21.9 °C. The temperature decay, measured by means of a thermocouple inserted in a small hole in the lateral surface, is recorded up to equilibrium conditions. Fitting an exponential to figure 3 and solving (16) for h, we obtain $h = 11\ {\rm W~m}^{-2}~{\rm K}^{-1}$.

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When the object under investigation is a metal sheet, e.g. an aluminum plate, the external parameters can be obtained simultaneously during the thermographic measurement. In that case none of the procedures outlined in section 2 can be applied because the Fourier number is too small. Instead, we can apply the thin-plate approximation (TPA) to the non-defective domain $\Omega_{0}$ of the specimen to obtain an expansion of ϕ and h in powers of the plate thickness a. In the 2D approximation, $\Omega_{0}$ represents a $y = {\rm const}$ plane 'far' from the defect.

The temperature of $\Omega_{0}$ for $t \in (0, T_{\rm max}]$, fulfills the heat equation

$$\begin{align} \displaystyle u^0_t=\alpha \Delta u^0 \nonumber \end{align} \tag{ 17 }$$

with boundary conditions

$$\begin{align} \displaystyle \kappa u^0_n+ah^{\prime}(u^0-U^0)-a\phi^{\prime}=0 \nonumber \end{align} \tag{ 18 }$$

and initial data

$$\begin{align} \displaystyle u^0(x,z,0)=U_{\rm in}(x,z,0)=U^0. \nonumber \end{align} \tag{ 19 }$$

As usual we scale the thickness variable: $\zeta=\frac{z}{a}$. We have also scaled the heat flux $\phi=\frac{\phi^{\prime}}{a}$ and the heat transfer coefficient $h=\frac{h^{\prime}}{a}$ because in this way we get an expansion in even powers only. More precisely, the IBVP becomes

$$\begin{align} \displaystyle \label{eq:011} \frac{a^2}{\alpha} v_t= a^2 v_{xx}+v_{\zeta\zeta} \nonumber \end{align} \tag{ 20 }$$

with boundary conditions

$$\begin{align} \displaystyle \label{eq:021} \kappa v_\zeta(x,1,t)+a^2h^{\prime}(v(x,1,t)-U^0)=0 \nonumber \end{align} \tag{ 21 }$$

$$\begin{align} \displaystyle \label{eq:031} -\kappa v_\zeta(x,0,t)=a^2\phi^{\prime} \nonumber \end{align} \tag{ 22 }$$

and initial data

$$\begin{align} \displaystyle \label{eq:041} v(x,z,0)=U_{\rm initial}(x,z,0)=U_0. \nonumber \end{align} \tag{ 23 }$$

Since v₀ is independent of ζ, if we take the temperature maps at two instants $t_1 \neq t_2$, we have the linear system

$$\begin{align*} \displaystyle h^{\prime}(v_0(x,t_1)-U^0)-\phi^{\prime}+\rho c {v_0}_t(x,t_1) -\kappa {v_0}_{xx}(x,t_1)=0\nonumber \end{align*}$$

$$\begin{align*} \displaystyle h^{\prime}(v_0(x,t_2)-U^0)-\phi^{\prime}+\rho c {v_0}_t(x,t_2) -\kappa {v_0}_{xx}(x,t_2)=0\nonumber \end{align*}$$

whose solution comes from a straightforward calculation:

$$\begin{align} \displaystyle h^{\prime}=-\rho c \frac{{v_0}_t(t_1)-{v_0}_t(t_2)}{{v_0}(t_1)-{v_0}(t_2)}+ \kappa \frac{{v_0}_{xx}(t_1)-{v_0}_{xx}(t_2)}{{v_0}(t_1)-{v_0}(t_2)} \label{h_primo} \nonumber \end{align} \tag{ 24 }$$

and

$$\begin{align} \displaystyle \phi^{\prime} = h^{\prime} ({v_0}(t_1)-U^0)+\rho c {v_0}_t(t_1)-\kappa {v_0}_{xx}(t_1) \label{fi_primo} \nonumber \end{align} \tag{ 25 }$$

once the physical parameters κ, ρ and c, typical of the material at hand, are known.

The desired values of ϕ and h are computed by an iterative procedure, computing an initial value of h by means of (24) and alternatively applying equation (25) to the computation of ϕ and h until convergence is reached in both parameters. A simulation run on a 2D aluminum plate covered with a high-emissivity material (as in the experimental conditions) on the bottom surface, subjected to a 'cylindrical' heat flux identical to that plotted in figure 2, and exchanging heat at a rate $h = 20~{\rm W~m}^{-2}~{\rm K}^{-1}$ at the top surface (the reason the value is so 'high' will be clear in the following), allows us to test the iterative procedure. Figures 4 and 5 show the convergence of $\phi_0$ (the maximum of ϕ) and h, to values nearly identical to the true ones: $\phi_0 = 600.2$, $h = 19.99$.

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Figure 6 shows the heat flux error dependence on the X coordinate: the comparison with that obtained on a thermally-insulating panel (see figure 2) is very satisfying, confirming the reliability of the procedure.

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The computation based on the TPA assumes a 2D geometric domain. That essentially means that the true heat flux ϕ, having an almost parabolic shape in both coordinates x and y, is assumed to have a cylindrical shape (uniform along y). This, in turn, means that no heat dispersion is assumed along y. Therefore the h value is over-estimated by the above procedure because all heat is assumed to escape from the line $z = a$. We call h_eff such an 'effective' h-value. A comparison among a 3D realistic model of the plate under measurement and the 2D model used in the TPA allows us to obtain the relation between h and h_eff. Figure 7 shows that such a relation is linear, the slope of the straight line only depending on $\phi(x, y)$.

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The linear relationship shown in figure 7 completes the procedure described in section 4:

• The iterative computation gives $\phi(x, 0, 0)$ and h_eff.
• A similar procedure can be applied to obtain $\phi(0, y, 0)$, or this last can be estimated by a simple proportion between the X and Y dependence of the temperature.
• Once $\phi(x, y, 0)$ is known, the linear relation among the true (unknown) heat exchange h and h_eff is obtained by a numerical simulation, allowing us to obtain the correct h.

Two problems arise when trying to heat a bare metallic surface [12]:

• (i)  
  The very low emissivity (e.g. lower than 0.1 for a polished aluminum surface) prevents the material from being significantly heated.
• (ii)  
  The high reflectivity makes it impossible to take thermographic images of the heated surface because any object (including the heating lamps) is reflected into the thermographic camera.

The above well-known problems [10] are usually solved by covering the surface with a high-emissivity paint. Such an approach, while suitable in the laboratory, appears to be rather unpractical and usually too invasive for a real wall. Looking at the published emissivity tables of common materials (e.g. [10], table 8.1), matt paper appears to have an emissivity greater than 0.9. That suggests to us the possibility of using an adhesive paper sheet to overcome the emissivity/reflectivity problem.

Figure 8 schematically shows the geometry of the heated plate.

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The influence of the paper sheet on the temperature distribution can be described in terms of its thermal resistance. Actually, in steady-state conditions, the thermal conductivity of the paper-covered plate is obtained by an electric-equivalent circuit as a series of two conductances (paper sheet and aluminum plate, respectively). In transient conditions, the effect of the thermal resistance of the paper sheet can be taken into account by developing the temperature in a Maclaurin series of its thickness d. Thanks to the very small value of d, we can assume the first term as dominant, therefore the temperature of the interface between the paper and the aluminum is given by

$$\begin{align} \displaystyle u(x,d,t) = u(x,0,t) - \frac{d}{k_{\rm p}} \phi(x) \label{mclaurin} \nonumber \end{align} \tag{ 26 }$$

k_p being the paper conductivity.

Figure 9 shows the experimental heat flux measured on a plywood panel for a given placement of the heating lamps during an experimental session.

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After replacing the plywood panel with the aluminum plate, equation (26) allows us to compute the temperature on the paper/aluminum interface from the thermographic images taken during heating. Figure 10 shows the effect of the correction for the paper thermal resistance at three time values.

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The effective heat exchange coefficient is computed from the corrected temperature maps according to the procedure outlined in the previous section. The value obtained is $h_{\rm eff} = 20.6~{\rm W~m}^{-2}~{\rm K}^{-1}$, corresponding to a true $h = 10~{\rm W}$${\rm ~m}^{-2}~{\rm K}^{-1}$, in good agreement with that obtained by the lumped method in section 3.

Knowledge of the heat flux and the heat exchange coefficient is mandatory for a correct application of active thermography to the non-destructive testing of 'thin' plate structures. The procedures developed in this paper allow the evaluation of both parameters. The reliability of such procedures has been tested on numerical models simulating the true experimental conditions, and then subsequently applied to an experiment. In particular, an approach based on the thin plate approximation allowed us to obtain an accurate estimate of the heat exchange coefficient (as compared to the values obtained by the lumped-constant method). The same approach also allows us to estimate the heat flux intensity and spatial distribution, although that parameter is accurately and robustly measured by means of a non-conducting panel placed in front of the accessible surface of the metal plate under evaluation.