Solving the Inverse Heat Conduction Problem in Using Long Square Pulse Thermography to Estimate Coating Thickness by Using SVR Models Based on Restored Pseudo Heat Flux (RPHF) In-Plane Profile

Abstract

This research develops a method to estimate opaque coating thickness based on time-and-space-resolved thermography. Thermography is a viable technique for measuring coating thickness. However, time-resolved thermography becomes unreliable when uncontrolled constant thermal stimulation amplitude appears. Therefore, a time- and space-resolved thermography technique for coating thickness measurement called restored pseudo heat flux (RPHF) has been developed by using Fourier–Hankel transform. A non-dimensional analysis was conducted and the results show RPHF curves with different coating thicknesses converging as the thermal diffusivity ratio or thermal conductivity ratio goes to one. For large thermal conductivity ratio values, the RPHF curves have two inflection points along the non-dimensional radius. Fifty-nine samples were tested using the proposed method. Support vector regression (SVR) models were constructed with the in-plane distribution of RPHF and the temporal distribution of the measured surface temperature as inputs. To avoid overfitting, cross validation was applied to all the models. Later, another twenty-eight samples were tested to validate the SVR models. The results suggest that a support vector regression model with in-plane profiles of RPHF handles uncontrolled heat flux variation better and yields a better performance in coating thickness measurement than the temporal profile does. With in-plane RPHF profile as input, the SVR model can evaluate coating thickness with a relative root mean square error at 25.3% even when heat amplitude varies 50%.

Appendices

Appendix 1: Derivation of RPHF and Dimensional Analysis

See Figs. 8 and 9.

Fig. 8

Coating thickness of the training samples

Fig. 9

Coating thickness of testing samples

To provide background about the underlying theory in this paper, the derivation of RPHF is shown below. First, by applying Fourier transform with respect to t, r, and θ respectively, Eqs. (1–5) can be transformed into frequency domain as :

$$ \frac{{{\text{i}\upomega}}}{{{{\upalpha }}_{1} }}v_{1} + {\xi^{2}} v_{1} = \frac{{\partial^{2} v_{1} }}{{\partial z^{2} }}, $$

(12)

$$ \frac{{{\text{i}\upomega}}}{{{{\upalpha }}_{1} }}v_{2} + {\xi^{2}} v_{2} = \frac{{\partial^{2} v_{2} }}{{\partial z^{2} }}. $$

(13)

With boundary conditions set as:

$$\begin{aligned} & - k_{1} \frac{{\partial v_{1} }}{\partial z}\left( {z = z_{0} } \right) \\ &\quad = \bar{q}\left( {\omega ,\xi } \right) = A sinc\left( {\frac{\omega }{{\omega_{0} }}} \right)\exp \left( { - \frac{{{\xi^{2}} B^{2} }}{8}} \right)/4\pi R^{2} ,\end{aligned} $$

(14)

$$ v_{1} \left( 0 \right) = v_{2} \left( 0 \right), $$

(15)

$$ k_{1} \frac{{\partial v_{1} }}{\partial z}\left( 0 \right) = k_{2} \frac{{\partial v_{2} }}{\partial z}\left( 0 \right), $$

(16)

$$ k_{2} \frac{{\partial v_{2} }}{\partial z}\left( { - \infty } \right) = 0. $$

(17)

Equations (12) and (13) are solved using:

$$ v_{1} = A\exp \left( { - z\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} } \right) + B\exp \left( {z\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} } \right), $$

(18)

$$ v_{2} = B_{2} \exp \left( { - z\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} } \right) + A_{2} \exp \left( {z\sqrt {{\xi^{2}} + \frac{{{\text{i }\upomega}}}{{\alpha_{1} }}} } \right), $$

(19)

According to (17):

$$ B_{2} = 0. $$

(20)

By adding (18) (19) and (20) to Eqs. (14) (15) and (16), the following equations are obtained:

$$ A\exp \left( { - d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right) - B\exp \left( {d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right) = \frac{{\bar{q}\left( {\omega ,\xi } \right)}}{{k\sqrt {{\xi^{2}} + \frac{i\omega }{{\alpha_{1} }}} }}, $$

(21)

$$ A + B = A_{2} , $$

(22)

$$ \left( {B - A} \right) = \frac{{k_{2} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{2} }}} }}{{k_{1 } \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} }} A_{2} , $$

(23)

When Eq. (22) is added to Eq. (23), A and B are expressed as:

$$ B = \left( {1 + \frac{{k_{2} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{2} }}} }}{{k_{1 } \sqrt {{\xi^{2}} + \frac{i\omega }{{\alpha_{1} }}} }}} \right)A_{2}\bigg/2. $$

(24)

$$ A = \left( {1 - \frac{{k_{2} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{2} }}} }}{{k_{1 } \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} }}} \right)A_{2} \bigg/2. $$

(25)

Define $R\left( {\xi ,{\text{i}}}\omega \right)$ as the ratio between A and B:

$$ R(\xi) = \frac{A}{B} = \frac{{\sqrt {{{\xi^{2}}} + \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} - \chi \sqrt {{{\xi^{2}}} + n^{2} \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} }}{{\sqrt{{{\xi^{2}}} + \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} + \chi \sqrt{{{\xi^{2}}} + {n^{2}} \frac{{{\text{i}\upomega }}}{{\alpha_{1} }}} }}, $$

(26)

where $\chi = \frac{{k_{2} }}{{k_{1} }}$ and $n = \sqrt {\frac{{\alpha_{1} }}{{\alpha_{2} }}} .$

Add Eqs. (25) (24) and (26) to Eq. (21):

$$ A\left( {1/R - \exp \left( { - 2z_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right) = \frac{{\bar{q}\left( {\omega ,\xi } \right)\exp \left( { - d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} }}, $$

(27)

$$ A = \frac{{\bar{q}\left( {\omega ,\xi } \right){\text{Rexp}}\left( { - d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} \left( {1 - R\exp \left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right)}}, $$

(28)

$$ B = \frac{{\bar{q}\left( {{{\upomega }},\xi } \right)\exp \left( { - d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} \left( {1 - R\exp \left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right)}}, $$

(29)

$$\begin{aligned} v_{1} & = \frac{{\bar{q}\left( {\omega ,\xi } \right){\text{Rexp}}\left( { - \left( {z + d_{0} } \right)\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} \left( {1 - R\exp \left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right)}} \\ &\quad + \frac{{\bar{q}\left( {{{\upomega }},\xi } \right)\exp \left( {\left( {z - d_{0} } \right)\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} \left( {1 - R\exp \left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right)}}. \end{aligned} $$

(30)

At the surface of coating z = − d₀, one can obtained:

$$ v_{1} = \frac{{\bar{q}\left( {\omega ,\xi } \right)\left( {1 + {\text{Rexp}}\left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right) } \right)}}{{k\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} \left( {1 - R\exp \left( { - 2d_{0} \sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}}}{{\alpha_{1} }}} } \right)} \right)}}, $$

(31)

A dimensional analysis should be applied to Eq. (31) in order to test whether the derivation is consistent with previous studies. The unit of v₁ can be calculated from Eq. (31) by knowing that

$$ \bar{q}\left( {\omega ,\xi } \right)\sim \left[ {\frac{\text{J}}{{{\text{m}}^{2} {\text{s}}}}} \right], $$

$$ k\sim \left[ {\frac{J}{m \cdot K \cdot s}} \right], $$

$$ {{\upxi }}\sim {\text{m}}^{ - 1} , $$

$$ {{\upomega }}\sim {\text{s}}^{ - 1} , $$

$$ {{\upalpha }}\sim \frac{{{\text{m}}^{2} }}{\text{s}} . $$

The dimension of the term R, $\sqrt {{\xi^{2}} + \frac{i\omega }{\alpha }}$ can be derived:

$$ {\text{R}}\sim \frac{{\sqrt {\left[ {{\text{m}}^{ - 1} } \right]^{2} + \frac{{\left[ {{\text{s}}^{ - 1} } \right]}}{{\left[ {\frac{{{\text{m}}^{2} }}{\text{s}}} \right]}}} - \left[ 1 \right]\sqrt {\left[ {{\text{m}}^{ - 1} } \right]^{2} + \left[ 1 \right]\frac{{\left[ {{\text{s}}^{ - 1} } \right]}}{{\left[ {\frac{{{\text{m}}^{2} }}{\text{s}}} \right]}}} }}{{\sqrt {\left[ {{\text{m}}^{ - 1} } \right]^{2} + \frac{{\left[ {{\text{s}}^{ - 1} } \right]}}{{\left[ {\frac{{{\text{m}}^{2} }}{\text{s}}} \right]}}} + \left[ 1 \right]\sqrt {\left[ {{\text{m}}^{ - 1} } \right]^{2} + \left[ 1 \right]\frac{{\left[ {{\text{s}}^{ - 1} } \right]}}{{\left[ {\frac{{{\text{m}}^{2} }}{\text{s}}} \right]}}} }}\sim \left[ 1 \right] $$

$$ \sqrt {{\xi^{2}} + \frac{i\omega }{\alpha }} \sim \sqrt {\left[ {{\text{m}}^{ - 1} } \right]^{2} + \left[ 1 \right]\frac{{\left[ {{\text{s}}^{ - 1} } \right]}}{{\left[ {\frac{{{\text{m}}^{2} }}{\text{s}}} \right]}}} \sim \left[ {m^{ - 1} } \right] $$

The dimension of v₁ should be

$$ v_{1} \sim \frac{{\left[ {\frac{\text{J}}{{{\text{m}}^{2} {\text{s}}}}} \right]}}{{\left[ {\frac{J}{m \cdot K \cdot s}} \right] \cdot \left[ {m^{ - 1} } \right]}} \cdot \frac{{\left( {1 + \left[ 1 \right]\exp \left( {\left[ m \right] \cdot \left[ {{\text{m}}^{ - 1} } \right]} \right)} \right)}}{{\left( {1 - \left[ 1 \right]\exp \left( {\left[ m \right] \cdot \left[ {{\text{m}}^{ - 1} } \right]} \right)} \right)}} \sim \left[ K \right] $$

Besides, as the thermal effusivity is defined as $e = \sqrt {\rho ck}$ while thermal diffusivity is defined as $\alpha = \frac{k}{\rho c},$ the Eq. (31) can be rewritten as:

$$ v_{1} = \frac{{\bar{q}\left( {\omega ,\xi } \right)\left( {1 + {\text{Rexp}}\left( { - \frac{{2z_{0} }}{{k_{1 } }}\sqrt {k_{1}^{2} {\xi^{2}} + {\text{i}\upomega}e_{1}^{2} } } \right) } \right)}}{{\sqrt {k_{1} {\xi^{2}} + {\text{i}\upomega}e_{1}^{2} } \left( {1 - R\exp \left( { - \frac{{2z_{0} }}{{k_{1} }}\sqrt {{\xi^{2}} + {\text{i}\upomega}e_{1}^{2} } } \right)} \right)}} , $$

with R defined as:

$$ R = \frac{{\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}e_{1}^{2} }}{{k_{1}^{2} }}} - \chi \sqrt {{\xi^{2}} + n^{{{\prime }2}} \frac{{{\text{i}\upomega}e_{1}^{2} }}{{k_{1}^{2} }}} }}{{\sqrt {{\xi^{2}} + \frac{{{\text{i}\upomega}e_{1}^{2} }}{{k_{1}^{2} }}} + \chi \sqrt {{\xi^{2}} + n^{{{\prime }2}} \frac{{{\text{i}\upomega}e_{1}^{2} }}{{k_{1}^{2} }}} }}. $$

$$ n^{{\prime }} = \frac{{e_{2} }}{{e_{1} }} \cdot \frac{{k_{1} }}{{k_{2} }}. $$

Appendix 2: Measured Thickness for Each Sample

The coating thickness δ_c is determined from measured sample thickness δ_s and foil thickness δ_f as

$$ \delta_{c} = \delta_{s} - \delta_{f} $$

(32)

Therefore, the standard deviation of coating thickness δ_c as

$$ Std_{c} = \sqrt {Std_{s}^{2} + Std_{f}^{2} } $$

(33)