Estimation of Thermal Resistance Field in Layered Materials by Analytical Asymptotic Method

Round 1

Reviewer 1 Report

The work is dedicated to the quantitative assessment of thermal resistances in a multilayer sample investigated with a thermographic setup. This topic is interesting both for the infrared thermography and for the NDT scientific communities. However, the few following issues listed below should be addressed before publication.

Major remarks:

Section 2, specimen description: what about the emissivity of the specimen? Equation 4, check the indexing of time and layers (i) and disambiguate. Check for missing letters. Figure 5a is not clear, add in the legend the meaning of colors and explain in the text the 19.7s value. Figure 6, please provide a comment in the text about the unevenness of the temperature field (left side of the images) Figure 9 is not clear. Part a, explain in the text how the labelling is performed. Part b, it is hard to understand as ROI 3 and 4 seem to have a huge number of points outside the expected range. Please explain it inside the text (check the sentence “From the data table 1 and figure 9, one can see that the error is always lower than 5%”). Anyway, it would be probably better to analyze the ROI separately, having a histogram for each ROI and keeping the indication of the expected ranges. If you keep the figure, check y axis label.

 

Minor remarks:

Figure 2a and 2b, it would be better to rotate the image in order to have the same orientation of the thermal images (fig. 6 and 8) Figure 2 b, if you want to keep the thermal properties inside the figure, provide an image with better quality. Section 3.1, provide a better explanation for the subscripts e and s, maybe it could be integrated in figure 3. Section 3.1, for better clarity, extend with a textual description the thicknesses and thermal resistance ranges (such as [100 : 100 : 1000]), listing also the number of elements for each array. Alternatively report the ranges in a table. Section 3.1, with reference to figure 4, briefly describe in the text the -1/2 slope concept that is recalled in section 4 Figure 4 and 5a, mention in the caption or in the text that the axes are logarithmic Figure 4, legend, is Rth Rthg? Table 1, is the error missing the sign or do you consider the absolute value? Page 12, broken sentence: “and estimated values are reported in table 1 and illustrated” Check ref. 12

Author Response

Object: Response to reviewer                                                        February 2020

 

Dear Editor,

 

We would like to thank the reviewer for his pertinent comments. Therefore, we propose a corrected version of the article including all the remarks and comments given by the reviewer. And a second article with highlighted correction.

 

Below, a point by point reply to the reviewer’s comments is given.

 

Reviewer 1:

 

The work is dedicated to the quantitative assessment of thermal resistances in a multilayer sample investigated with a thermographic setup. This topic is interesting both for the infrared thermography and for the NDT scientific communities. However, the few following issues listed below should be addressed before publication.

 

Major remarks:

Section 2, specimen description: what about the emissivity of the specimen?

The sample is opaque with a uniform emissivity

 

Equation 4, check the indexing of time and layers (i) and disambiguate. Check for missing letters.

Thank you, the operator dt was missing in the left part of the Eq.4. The temperature of the measurement was rename T_m and not T_mi to avoid any ambiguity with the number I of the layers.

 

Figure 5a is not clear, add in the legend the meaning of colors and explain in the text the 19.7s value.

The value of 19.7s corresponds to the last time step of the measurements. The colors corresponds to the different Rth of the multilayer. We added the explanation in the legend caption: “the solid lines corresponds to the different R_th of the figure 4”

 

Figure 6, please provide a comment in the text about the unevenness of the temperature field (left side of the images)

We added: This normalization also avoid the the non-uniformity of the global temperature coming from the non-homogeneity of the flash lamp

 

Figure 9 is not clear. Part a, explain in the text how the labelling is performed.

In agreement with the remarks of you and other reviewers and to avoid any confusion due to the number outside the expected range we remove the figure 9.

Nevertheless to respond to your comment, the labelling was made by using a threshold based on Otsu method. The “huge number” of points are not so huge comparing to the number of “well” estimated one (average of 1500 points inside the error bar for 400 hundreds outside). The points outside come from the 3D effects that are not taking into account here. They correspond to the Rth gradient around the square (see figure 8.b).

Part b, it is hard to understand as ROI 3 and 4 seem to have a huge number of points outside the expected range. Please explain it inside the text (check the sentence “From the data table 1 and figure 9, one can see that the error is always lower than 5%”). Anyway, it would be probably better to analyze the ROI separately, having a histogram for each ROI and keeping the indication of the expected ranges. If you keep the figure, check y axis label.

 

Minor remarks:

 

Figure 2a and 2b, it would be better to rotate the image in order to have the same orientation of the thermal images (fig. 6 and 8) Figure 2 b, if you want to keep the thermal properties inside the figure, provide an image with better quality.

We agreed the comment of the reviewer; we change the image resolution and color to improve the quality. Nevertheless we didn’t rotate the schema because the ROI were numbered overall the paper.

 

Section 3.1, provide a better explanation for the subscripts e and s, maybe it could be integrated in figure 3.

We agreed, we added the temperature and flux in the schema of the figure 3

 

Section 3.1, for better clarity, extend with a textual description the thicknesses and thermal resistance ranges (such as [100 : 100 : 1000]), listing also the number of elements for each array. Alternatively report the ranges in a table.

We change the notation by e_PC = [100,1000] µm (9 steps of 100 µm).

 

Section 3.1, with reference to figure 4, briefly describe in the text the -1/2 slope concept that is recalled in section 4 Figure 4 and 5a, mention in the caption or in the text that the axes are logarithmic Figure 4, legend, is Rth Rthg?

Yes the legend is the Rth. We also add the loglog plot in the figure caption, many thanks for this very important remark.

 

Table 1, is the error missing the sign or do you consider the absolute value?

In the last line of the table 1 the exact formulation was calculated. We just change the abs height

 

Page 12, broken sentence: “and estimated values are reported in table 1 and illustrated”

We do that

 

Check ref. 12

We check again the reference it seems ok.

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper is well structured and the english spell is fine.

However, it is not clear, for the reader, which is the research gap the work aims to bridge, and which is the novelty it introduces. The conclusions lack of discussion of both the aim of the paper and the results discussion. The paper cannot be published in this form.

Moreover, there are some errors in the unit of measurements that should be corrected.

 

 

In the following, some further suggestions:

PAGE 2. it is not nice to refer to the QIRT community.

Maybe authors can refer to the "research groups involved in quantitative thermography".

 

PAGE 2. it is unclear which is the final aim of the paper and, noreover, the research gap it bridges. This point must be solved

 

 

 

PAGE 3. Add more technical specification on the employed IR camera

 

PAGE 4. FIGURE 2B. Please use the SI way to express the units of measurement.

Please also add a black space to separate digits from unit of measurement (please revise accordingly the text and the figures of the manuscript - legends, axes, etc)

 

 

PAGE 6. Please avoid to use the dot to separate the unit of measurement. Revise all the text (es. page 9, y-axis of figure 5b)

 

PAGE 13. these sentences should be in the introduction.

the conclusion should summarize the main outcomes of the developed method and the approach followed, and discuss the results of experimental campaign (if any). Please revise this section.

 

 

PAGE 13. many relevant papers on this topic are not incuded in the reference list

Author Response

Object: Response to reviewer                                                        February 2020

 

Dear Editor,

 

We would like to thank the reviewer for his pertinent comments. Therefore, we propose a corrected version of the article including all the remarks and comments given by the reviewer. And a second article with highlighted correction.

 

Below, a point by point reply to the reviewer’s comments is given.

 

Reviewer 2:

 

The paper is well structured and the english spell is fine.

 

However, it is not clear, for the reader, which is the research gap the work aims to bridge, and which is the novelty it introduces. The conclusions lack of discussion of both the aim of the paper and the results discussion. The paper cannot be published in this form.

 

Moreover, there are some errors in the unit of measurements that should be corrected.

 

 

In the following, some further suggestions:

 

PAGE 2. it is not nice to refer to the QIRT community.

We REALLY apologize for the misunderstanding, we are also involved in the QIRT and the word community was not use to segregate. We include you proposal which is without any ambiguity.

Maybe authors can refer to the "research groups involved in quantitative thermography".

 

 

PAGE 2. it is unclear which is the final aim of the paper and, noreover, the research gap it bridges. This point must be solved

We improve the introduction by adding the following sentences:

“Nevertheless, in this challenging problem, the complete inverse processing of a multilayer analytical model is difficult due to the lack of sensitivity of some parameters (layer thickness, depth of thermal resistance, etc.) and the expansive computational iterative processing. For these reasons, the proposed strategy is to use a simple multilayer problem where only one resistive layer is estimated. Moreover, to simplify the inverse processing often based on iterative methods, an asymptotic development method is proposed. The main novelty comes from the calibrating method between the analytical calculation of a theoretical base and the estimated parameters of the asymptotic method. Then with the knowledge of this calibration curve, the estimated parameters based on the measured one can be turn to quantitative ones.”

 

PAGE 3. Add more technical specification on the employed IR camera

We added the wavelength range 9 to 11 µm

 

PAGE 4. FIGURE 2B. Please use the SI way to express the units of measurement.

Here we didn’t understand what are the SI way to express the units ??? Due to the size of the sample it is much more comprehensive to express the thickness and other in µm instead of x.1e-6 m.

 

Please also add a black space to separate digits from unit of measurement (please revise accordingly the text and the figures of the manuscript - legends, axes, etc)

We agree that this is not the final edited version linked with the template of Applied sciences. So this will be corrected by the editing step.

 

PAGE 6. Please avoid to use the dot to separate the unit of measurement. Revise all the text (es. page 9, y-axis of figure 5b)

Many thanks for this remark, We modify the units of the figure 5.b and also figure 4 to be in agreement with SI notations

We agree that this is not the final edited version linked with the template of Applied sciences. So this will be corrected by the editing step.

 

PAGE 13. these sentences should be in the introduction.

Which sentences? In agreement with the comment page 2 we already improve the introduction.

 

the conclusion should summarize the main outcomes of the developed method and the approach followed, and discuss the results of experimental campaign (if any). Please revise this section.

We understand the remarks because we only write on the main advantages and drawback of the method. So, we improve the conclusion by adding some comments on the experimental campaign.

“Nevertheless, the complete method applied on a heterogeneous bi-layer sample reveals a very good result with a complete estimation of the thermal resistance fields in fast time (complete calculation time lower than 7s) and accuracy with maximum thermal resistance errors around 5 %.”

 

PAGE 13. many relevant papers on this topic are not incuded in the reference list

We apologize for this mistake. Sometimes it is difficult to have non exhaustive bibliography study.

For this reason, if you have such missing paper, please specify it, we would include it in the reference and rewrite the introduction with this new part.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

This paper is entitled: Estimation of thermal resistance field in multilayer materials by analytical asymptotic method. From this, I expected a material made up of overlapping layers (more than two), instead the sample shown in Fig. 2 resembles mostly a coating of 1 mm polycarbonate over a 3 mm Aluminium with air bubbles that simulate debonding at the boundary between the two layers. The authors should modify the title or explain why a two-layer sample with defects can be considered as representative of a multilayer material.

Some other points

Fig.1 the term heat flux in the legend is not at all appropriate; the red arrows indicate radiated heat which is captured by the infrared sensor

Fig.2 is not clear since some symbols are not defined; they are described later in the next section. I suggest to define symbols sudden the first time they are introduced, alternatively make a note to refer the reader to the next section.

On p. 10 the paragraph after Fig.6 the term thermogram to refer to the temperature evolution in one pixel is not appropriate since thermogram is generally used as synonym of infrared image or thermal image. In my opinion it is better to use the term temperature plot.

Table 1 it is not clear why the maximum error is achieved for ROI 3. The authors should discuss this aspect and supply an explanation.

 

Conclusions

It seems there is contradiction between two statements:

The main advantages of this method are as follows: (i) the possibility to address any type of multilayer (high numbers of layers, etc.),

The main drawbacks of the proposed methods are (i) the 1D character of the model and (ii) the sensitivity to only one thermal resistive layer.

 

Author Response

Object: Response to reviewer                                                        February 2020

 

Dear Editor,

 

We would like to thank the reviewer for his pertinent comments. Therefore, we propose a corrected version of the article including all the remarks and comments given by the reviewer. And a second article with highlighted correction.

 

Below, a point by point reply to the reviewer’s comments is given.

 

Reviewer 3:

 

This paper is entitled: Estimation of thermal resistance field in multilayer materials by analytical asymptotic method. From this, I expected a material made up of overlapping layers (more than two), instead the sample shown in Fig. 2 resembles mostly a coating of 1 mm polycarbonate over a 3 mm Aluminium with air bubbles that simulate debonding at the boundary between the two layers. The authors should modify the title or explain why a two-layer sample with defects can be considered as representative of a multilayer material.

We totally agreed with the comment of the reviewer. Here it is important to note that whatever the number of layer we should have one “high contrast” layer. To determine this contrast the sensitivity study (not presented here but extracted from analytical calculation software) should be done. To point out this, we decide to only make a bilayer sample with this high contrast.

If we understand this question is related to the last one of the conclusion.

Perhaps our explanations are not clear enough. This new method is independent of the number of layer. In fact, this only depends on the quadrupole calculation. THEN, for any multilayer assembly, IF one resistive interface varied or should be estimated, the proposed method can be used. WE never claim to be able to estimate two or three resistive layers estimations. A next paper will deal with such problem.

 

Some other points

 

Fig.1 the term heat flux in the legend is not at all appropriate; the red arrows indicate radiated heat which is captured by the infrared sensor

We totally agreed with the reviewer comment, the figure was updated.

 

Fig.2 is not clear since some symbols are not defined; they are described later in the next section. I suggest to define symbols sudden the first time they are introduced, alternatively make a note to refer the reader to the next section.

We totally agreed with the reviewer comment, we added the description in the sentence of the symbols included in the figure 2.

 

On p. 10 the paragraph after Fig.6 the term thermogram to refer to the temperature evolution in one pixel is not appropriate since thermogram is generally used as synonym of infrared image or thermal image. In my opinion it is better to use the term temperature plot.

We are partially agreed with the reviewer comment. In fact the term thermogram was introduced before the FPA sensor, so they can also be used also for one pixel. Nevertheless we use the term you have proposed.

 

Table 1 it is not clear why the maximum error is achieved for ROI 3. The authors should discuss this aspect and supply an explanation.

Perhaps that during machining the error was bigger. We have no explanation about why the ROI3. We can just say that the maximum gap between is around 5%.

 

 

Conclusions

 

It seems there is contradiction between two statements:

 

The main advantages of this method are as follows: (i) the possibility to address any type of multilayer (high numbers of layers, etc.),

The main drawbacks of the proposed methods are (i) the 1D character of the model and (ii) the sensitivity to only one thermal resistive layer.

No the asymptotic method is sensitive to only one resistive layer variation. But this resistive layer can be anywhere in any multilayer assembly.

Author Response File: Author Response.pdf

Reviewer 4 Report

Introduction - selection of quadrupole method clear, but not the autoregressive asymptotic method.  Why did you select this method?  What other methods could be applicable.

Experimental 

There is a focus on the resolution of the camera and the machining of the layer - why is this important in your study?  Please expand.

Can you discuss the assumptions you are making with a single layer experiment and how it affects the multilayer analysis?  Your design in Fig 2 does not simulate Fig 3.

The photos in Figure 2 are not clear.  2a has a shadow, and the windows are difficult to see.  In Fig 2b, you do not define the variables (e, rth).  Where did you get the thermal properties?  I do not see a reference.

 

Methods

Perhaps change wording to "...the complete stack can be overlaid onto a substrate with different thermal properties..."  "For example, a foam..."  What does imposed temperature relate to?  Do you have a constant temperature on the surface with a gradient, or does the metal have a constant temperature throughout.  Not clear.

For Fig 3 - you should spend more time describing the schema and how it relates to your experiment.  Rth(i) is not described in the text.

"A matlab code was developed to generate the general solution and to conduct a sensitivity analysis..."  

Describe M in more detail - is this the transformation matrix for the quadrupole method?  Be clear what you are modeling.  It would be beneficial to show how you are translating a complex analytical model, but that has been shown in other works.

Clarify where the convection is taking place - which boundaries?

I'm not certain that Stehfest algorithm is common knowledge - you may want to describe it quickly.

Fig 4 - are these the results you expected?

Was the sensitivity analysis conducted?

Can you hypothesize what the results would be if you did not use the autoregression model?

You should refer to an equation when discussing the principle of SVD - there are many variables, but no equation to put them in context.

Results 

Can you explain the difference in variation in your results? (Fig 7)

Can you explain how you converge to the thickness in Fig 9b?  

Can you explain the results in Fig 9?  There is little supporting documentation in the text.

 

 

Aside - what do you suggest to improve the model to incorporate >1D problems?

Author Response

Object: Response to reviewer                                                        February 2020

 

Dear Editor,

 

We would like to thank the reviewer for his pertinent comments. Therefore, we propose a corrected version of the article including all the remarks and comments given by the reviewer. And a second article with highlighted correction.

 

Below, a point by point reply to the reviewer’s comments is given.

 

Reviewer 4:

 

Introduction - selection of quadrupole method clear, but not the autoregressive asymptotic method.  Why did you select this method?  What other methods could be applicable.

We propose an autoregressive method to try to optimize the sensitivity of the parameters as function of time.

 

Experimental

 

There is a focus on the resolution of the camera and the machining of the layer - why is this important in your study?  Please expand.

We agreed with the reviewer in our case the camera resolution has no influence in the paper. Moreover, the machining resolution is important to explain the uncertainties of the estimated parameters which are not due to the method sensitivity.

 

Can you discuss the assumptions you are making with a single layer experiment and how it affects the multilayer analysis?  Your design in Fig 2 does not simulate Fig 3.

We agreed with the reviewer, the fig 2 is a particular case of the figure 3.

 

The photos in Figure 2 are not clear.  2a has a shadow, and the windows are difficult to see.  In Fig 2b, you do not define the variables (e, rth).  Where did you get the thermal properties?  I do not see a reference.

We agreed with the reviewer, we made a new photography, we define the variables in the text and we also introduce a reference for the thermal properties.

 

Methods

 

Perhaps change wording to "...the complete stack can be overlaid onto a substrate with different thermal properties..."  "For example, a foam..."  What does imposed temperature relate to?  Do you have a constant temperature on the surface with a gradient, or does the metal have a constant temperature throughout.  Not clear.

We agreed we changed the sentence. Moreover the foam blocked the heat flux transfer whereas the metallic sample acts as a heat sink.

 

For Fig 3 - you should spend more time describing the schema and how it relates to your experiment.  Rth(i) is not described in the text.

We understand the comment. The Rth was described in the text.

 

"A matlab code was developed to generate the general solution and to conduct a sensitivity analysis..." 

Describe M in more detail - is this the transformation matrix for the quadrupole method?  Be clear what you are modeling.  It would be beneficial to show how you are translating a complex analytical model, but that has been shown in other works.

We added the link to upload the matlab code which is a guide to realize both the direct calculation of any assembly as well as the sensitivity analysis.

 

Clarify where the convection is taking place - which boundaries?

We have natural convection taking place at the front face of the sample.

 

I'm not certain that Stehfest algorithm is common knowledge - you may want to describe it quickly.

The Stehfest algorithm is very common in the quadrupole formalism. The algorithm is written in the quadrupole book, it is inside the reference [15].

 

Fig 4 - are these the results you expected?

Yes, yes yes. These results demonstrate the coupled effect of the thermal resistance depth and thickness.

 

Was the sensitivity analysis conducted?

Yes but we didn’t introduce it in this paper.

 

Can you hypothesize what the results would be if you did not use the autoregression model?

It is what we have done by calculated the calibration curve at a fixed time.

 

You should refer to an equation when discussing the principle of SVD - there are many variables, but no equation to put them in context.

We agreed, we added the eq.4. But the SVD method is referenced [20-23] and we just explain that the SVD is calculated for the matrix X of the equation 4.

 

Results

 

Can you explain the difference in variation in your results? (Fig 7)

Yes, the following sentence was added

“In the figure 7, it can be clearly observed that for the deepest ROI the temperature evolution quit the -1/2 slope further. For higher thermal resistance, the behavior is closer to adiabatic case with variable inflexion tendencies.”

 

Can you explain how you converge to the thickness in Fig 9b? 

In agreement with the remark of other reviewer the figure 9 was deleted

Can you explain the results in Fig 9?  There is little supporting documentation in the text.

 

 

Aside - what do you suggest to improve the model to incorporate >1D problems?

Write a 3D model.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

After the thorough revision provided by the authors, only few minor remarks are left. The online publication of the code adds great value to the work. This is a publication matter, but it would be nice to have the plain text code as an appendix of the paper.

 

 

Section 3.1, provide a better explanation for the subscripts e and s, maybe it could be integrated in figure 3.

We agreed, we added the temperature and flux in the schema of the figure 3

- You added a new symbol (psi), describe it inside the text

 

 

Figure 6, please provide a comment in the text about the unevenness of the temperature field (left side of the images)

We added: This normalization also avoid the the non-uniformity of the global temperature coming from the non-homogeneity of the flash lamp

- Ok, remove the double the

 

 

Check ref. 12

We check again the reference it seems ok.

- I apologize, the incomplete reference is #10. Oswald-Tranta, B., Maier, A. & Schledjewski, R. Defect depth determination in a cfrp structure using tsr technique. The 12 (2014). 2

 

 

Figure 9 is not clear. Part a, explain in the text how the labelling is performed.

In agreement with the remarks of you and other reviewers and to avoid any confusion due to the number outside the expected range we remove the figure 9.

Nevertheless to respond to your comment, the labelling was made by using a threshold based on Otsu method. The “huge number” of points are not so huge comparing to the number of “well” estimated one (average of 1500 points inside the error bar for 400 hundreds outside). The points outside come from the 3D effects that are not taking into account here. They correspond to the Rth gradient around the square (see figure 8.b).

Part b, it is hard to understand as ROI 3 and 4 seem to have a huge number of points outside the expected range. Please explain it inside the text (check the sentence “From the data table 1 and figure 9, one can see that the error is always lower than 5%”). Anyway, it would be probably better to analyze the ROI separately, having a histogram for each ROI and keeping the indication of the expected ranges. If you keep the figure, check y axis label.

 

- It is ok to remove the figure, but you should keep in the text the fact that you have a strong (not huge, figure 9 was misleading, but 400 over 1900 points is not negligible) gradient around the borders. It is already shown in the figure 8 but it is impossible to assess it quantitatively from the image.

 

 

 

 

New remarks:

- page 9 spatial resolution of 300 μm per pixel

- be consistent between the spatial resolution listed in page 3 and in page 9

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

The Authors have modified the manuscript according to the comments. It is now worthy of publication

Author Response

Thank you for your comments.

Reviewer 3 Report

 

My question is still about the title which refers to  multilayer materials while in reality the authors tested a bilayer sample. I suggest to change the title as:

 

Estimation of thermal resistance field in a bilayer material by analytical asymptotic method

 

In the conclusion they can explain that their method can be extended to multilayer materials but by overcoming the existing problems when approaching a sample composed of multiple overlapping layers.

Author Response

please see the attachment.

Author Response File: Author Response.pdf