Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems
In this article, a general model for 1D thermal wave interference is derived for multi-layered coating systems (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-03-01
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| Series: | Applied Sciences |
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| Online Access: | https://www.mdpi.com/2076-3417/13/7/4185 |
| _version_ | 1797608368384245760 |
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| author | Dimitri Rothermel Thomas Schuster |
| author_facet | Dimitri Rothermel Thomas Schuster |
| author_sort | Dimitri Rothermel |
| collection | DOAJ |
| description | In this article, a general model for 1D thermal wave interference is derived for multi-layered coating systems (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> coating layers) applied on a thermally thick substrate. Such a model means the first step to building a non-contact photothermal measurement device that is able to determine the coating thickness of each layer. Test objects are to be illuminated on the surface using planar, sinusoidal excitation waves with fixed frequencies leading to the generation of thermal waves inside the object. Due to the multi-layered structure, each of these thermal waves is reflected and transmitted at layer interfaces. This process leads to infinitely many wave trains that need to be tracked to formulate the final surface temperature as a superposition of all waves. A mathematical and physical formulation of thermal wave interference is needed to model this process and relate the dependencies of the layer thicknesses, the materials, and the frequencies to the phase angle data, which then can be measured using, e.g., an infrared camera. In practice, the thermal properties of the layers might be unknown, which makes the process even more difficult. This article presents a concept to determine the thermal properties in advance. Finally, numerical experiments are presented that demonstrate the feasibility of the introduced layer thickness determination process. |
| first_indexed | 2024-03-11T05:42:30Z |
| format | Article |
| id | doaj.art-0cd60ac680474493b4016a11d7313a64 |
| institution | Directory Open Access Journal |
| issn | 2076-3417 |
| language | English |
| last_indexed | 2024-03-11T05:42:30Z |
| publishDate | 2023-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Applied Sciences |
| spelling | doaj.art-0cd60ac680474493b4016a11d7313a642023-11-17T16:16:45ZengMDPI AGApplied Sciences2076-34172023-03-01137418510.3390/app13074185Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating SystemsDimitri Rothermel0Thomas Schuster1Department of Mathematics, Saarland University, 66123 Saarbrücken, GermanyDepartment of Mathematics, Saarland University, 66123 Saarbrücken, GermanyIn this article, a general model for 1D thermal wave interference is derived for multi-layered coating systems (with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula> coating layers) applied on a thermally thick substrate. Such a model means the first step to building a non-contact photothermal measurement device that is able to determine the coating thickness of each layer. Test objects are to be illuminated on the surface using planar, sinusoidal excitation waves with fixed frequencies leading to the generation of thermal waves inside the object. Due to the multi-layered structure, each of these thermal waves is reflected and transmitted at layer interfaces. This process leads to infinitely many wave trains that need to be tracked to formulate the final surface temperature as a superposition of all waves. A mathematical and physical formulation of thermal wave interference is needed to model this process and relate the dependencies of the layer thicknesses, the materials, and the frequencies to the phase angle data, which then can be measured using, e.g., an infrared camera. In practice, the thermal properties of the layers might be unknown, which makes the process even more difficult. This article presents a concept to determine the thermal properties in advance. Finally, numerical experiments are presented that demonstrate the feasibility of the introduced layer thickness determination process.https://www.mdpi.com/2076-3417/13/7/4185photothermal measurementsinfrared thermographythermal wave interferenceparameter estimationlayer thickness determinationmulti-layered coating systems |
| spellingShingle | Dimitri Rothermel Thomas Schuster Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems Applied Sciences photothermal measurements infrared thermography thermal wave interference parameter estimation layer thickness determination multi-layered coating systems |
| title | Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems |
| title_full | Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems |
| title_fullStr | Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems |
| title_full_unstemmed | Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems |
| title_short | Development of a Generalized Photothermal Measurement Model for the Layer Thickness Determination of Multi-Layered Coating Systems |
| title_sort | development of a generalized photothermal measurement model for the layer thickness determination of multi layered coating systems |
| topic | photothermal measurements infrared thermography thermal wave interference parameter estimation layer thickness determination multi-layered coating systems |
| url | https://www.mdpi.com/2076-3417/13/7/4185 |
| work_keys_str_mv | AT dimitrirothermel developmentofageneralizedphotothermalmeasurementmodelforthelayerthicknessdeterminationofmultilayeredcoatingsystems AT thomasschuster developmentofageneralizedphotothermalmeasurementmodelforthelayerthicknessdeterminationofmultilayeredcoatingsystems |