Thermal Wave Scattering and Temperature Concentration around the Opening in Platinum–Rhodium Leaky Plates

Round 1

Reviewer 1 Report

On the basis of the "non-Fourier" heat conduction law, the Authors study the heat propagation problem in a Pt-Rh GFLP with a circular opening. The topic is well developed and potentially interesting for readers. However, I have minor criticisms:

- In the “Numerical examples” Section, it should be specified which computing environments the Authors used;

- Some typos must be corrected; see for example line 274;

- Character, line spacing and alignment must be homogeneous;

- Some sentences are not clear and need to be reworded; see for example lines 67-69 or 226-228;

- The text must be corrected by an English native speaker.

Author Response

Please see the attachment.

 

Author Response File: Author Response.pdf

Reviewer 2 Report

Brief Description of the Work

The objective of this work is to analyze the heat propagation problem in a platinum-rhodium glass fibre leaky plate with a circular opening. The experimental device is the following: a laser pulse beam with a (modulation) frequency is irradiated on the outer surface of the leaky plate. The temperature distribution in the z -direction is considered to remain uniform so, the authors examined the wave process in the two-dimensional plane x-y. One of the main results obtained by the authors is that when the laser pulse frequency is taken in the range (0Hz-5Hz) , the temperature at the centre of the circular opening of the Pt-Rh GFLP increases with the increase of frequency. In the range (5Hz-300Hz) the temperature at the centre of the circular opening gradually decreases with the increase of frequency and tends to be stable.

Questions/Suggestions

Q1) In general in physics and mathematics, the heat equation is a P.D.E. describing how the distribution of heat evolves over time in a solid medium, as it spontaneously flows from places where it is higher towards places where it is lower. It is kindly asked to the authors to explain, from the physical point of view, why they decided to use the non-Fourier heat conduction equation, rather than the standard heat P.D.E., for studying the temperature behaviour in the solid medium. This P.D.E. simply pops up in the manuscript and the authors refer to reference [15] (Jia L. Advanced heat transfer. Peking: Higher Education Press, 2003) without any explanations.

Q2) The starting point of the authors' study is the P.D.E. (1). However, the solution of this differential equation requires the specification of the boundary conditions, which have to be set up on the base of the experiments that the authors are performing. The question is: "why does expressions (2) and (3) reflect the experimental conditions to which the material is subject ?". The authors only state "we assume the expression of the temperature field is Eq. (2)" and “the general solution of the thermal wave scattering field of an opening in a Pt-Rh GFLP is determined by Eq. (3)”. This requires further explanations, since, once again, it is not clear for which reasons these expressions reflect the appropriate boundary conditions.

Q3) Please specify since from the beginning the boundary conditions related to the circular opening in the leaky plate, which generates the scattering waves. I know, that this is mentioned from Eq. (8) onwards, but the boundary conditions should be specified in clear way since from the beginning and, possibly, first of all in the coordinates x-y.

Q4) The authors have obtained, numerically, a series of results, three of which, in particular, are quite predictable (obtainable without having to solve any P.D.E.). In my opinion, the most relevant result is the dependence of the temperature at the centre of the circular opening of the Pt-Rh GFLP on the modulation frequency. More specifically, the authors obtained that when the laser pulse frequency is taken in the range (0Hz-5Hz), the temperature at the centre of the circular opening of the Pt-Rh GFLP increases with the increase of frequency. However, in the range (5Hz-300Hz) the temperature at this place gradually decreases with the increase of frequency and tends to be stable. Unfortunately, the authors give no interpretation of this phenomenon/result. It is therefore required that:

Q4a) the authors provide an interpretation, at least qualitatively, of the dependence of the temperature at the centre of the circular opening of the Pt-Rh GFLP on the frequency and

Q4b) the interpret, at least qualitatively, the reason why in the range (5Hz-300Hz) the temperature tends to stabilize. Have the authors studied, numerically, the stability of the solution (3) with respect to frequency?

Conclusions

The theoretical work of the authors starts from a P.D.E. non-standard (non-Fourier heat conduction equation) without any explanations and without specifying in a clear way the boundary conditions (at the first sight, the expressions (2) and (3) seem to be only mere assumptions). Written in this form, the risk is that the reader will remain skeptical about the work and the manuscript will be ignored. In conclusion, it is my opinion that if the ultimate objective of this work is to provide theoretical support and engineering reference for the research on the temperature field of Pt-Rh GFLP, the authors should make an extra effort and take into account the suggestions above mentioned.

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

This paper obtains some particular solutions of the evolution of the temperature profile in a plate of a given material (platinum-rhodium in this case, but the mathematical results could be applied to other materials as well) with a longitudinal cylindrical defect (behaving as a leak) in the plane of the layer, when the layer is suddenly heated. The original aspects of the paper are: the use of the Cattaneo-Vernotte equation instead of the classical Fourier law (this is logical because the heating is assumed to be very fast, with a rate comparable to the inverse of the relaxation time of the heat flux), and the illustration that this equations leads to a transient concentration of the temperature increase around the defect line. The influence of geometrical and physical features is explored. Since this concentration of the temperature increase may be deleterious for the system, this study is well-motivated and of practical interest, besides its purely theoretical interest. Furthermore, these results are of interest for internal flaw detection in the layer and for the measurement of material properties by means of a fast periodic heating. The mathematical analysis is sound, the paper is well explained, and the previous bibliography on particular solutions of the Cattaneo-Vernotte equation is well references in the introduction. The paper has interesting original results deserving publication, but it is not acceptable in the present form, as it in not sufficiently clear and it is confusing at several points.

 

Before acceptation, I have a considerable number of questions and remarks asking for deep changes:

• Page 2, section 2, first line (line 65): the authors refer to a “circular opening”, but in Fig. 1 the opening has an ellipsoidal profile instead of a circular profile. This is confusing for the reader. In line 119, the authors refer to “circular openings of any shape” (what does mean “circular of any shape”?). In line 128 it is eventually clarified where the elliptical shape comes.
• Page 3, caption figure 1: the caption should be more explicative; for instance, it would be convenient explaining what are the ellipses above and below the plane (as they refer to the mathematical technique used in the paper, but not to the physical system itself); also the words “pulse heating” are misleading, as here one is showing the geometry of the system, but not the pulse; instead of GFPL, the complete words shoul be used, to be more explicit; it could be said that this is a transversal cross section of the system being considered, and that the central ellipse is the leak defect
• Page 2, line 78: It is surprising that the thermal conductivity is the same for 5% and 10 % composition of the platinum-rhodium alloy, in contrast to the mass density, which is perceptibly different. In many occasions, the thermal conductivity depends strongly on the composition of the alloys, at least for dilute alloys (for instance, in Si-Ge alloys, the thermal conductivity changes very much for Ge with small concentrations of Si and for Si with small concentration of Ge). Could the authors comment this point? The authors should include the references from which these values of the physical parameters have been taken.
• Which are the physical properties of the defect, i.e. of the material inside the circle or the ellipse? What are the transfer properties of heat between the material of the plane and that of the defect? This is an essential point of the paper, but it is not explained. One would intuitively guess that if the material of the defect has a higher thermal diffusivity than that of the plane, the transient temperature inside the defect could be higher than that of the surrounding region of the plane, as in Figs 2, and viceversa. If the wall between the plane and the defect is adiabàtic and heat cannot enter inside the defect, what is the meaning of temperature inside the defect? (it should not be affected by the heat flowing along the plate)
• Page 2, line 55: it should be explained that the frequency refers to the periodic repetition of laser pulses; otherwise, one has the impression that one is considering a pulse of a laser of a given frequency (seeing later the low valui of the frequencies being considered it is clear that this is not so)
• Page 3, in expression (2) it would be useful writing T(x,y) and Greek theta(x,y), in order that the space dependence is clearly visible; the authors should explain in this page how the system is heated: is it heated from the upper boundary plane?, from the lower boundary plane?, from both the upper and lower planes?, from the center of the system? From the left or the right ends of the plate in the x direction? Only in line 134 it is said for the first time how the plate is heated, but it is not yet said that the upper and lower plane boundaries are assumed to be adiabatic (this is said in line 154, in a not very clear form, as it is said “under the adiabatic condition”, without specifying which boundaries are adiabàtic: the upper and lower boundary planes?, the perimeter of the defect?).
• Page 3, line 100, I would suggest writing “kappa is the complex wave number” rather than writing “kappa is the wave number of the complex variable”
• Page 5, expression (11) is inconsistent with expression (2): the term in exp -iwt should not be inside theta, as in (2) it is multiplying theta.
• Lines 175-177: again, one is referring to elliptical opening in line 175, but to a circle of radius a in line 177
• Line 181: the frequency of repetition of pulse is indicated, but not the amplitude of the pulse; the authors should comment on it
• In pages 9 and 10, the y direction is called longitudinal, but the heat is flowing along the x direction. Thus, I would say that x is the longitudinal direction and y the transversal direction.
• Conclusions, point 2: the authors refer to the heat transfer between the plate and the surrounding air. How and where has been described such a transfer in the paper? This is an essential point but, for instance, the heat transfer coeficient between the plate and the air has not been discussed. Instead, and “adiabatic condition” has been mentioned in line 154.

For future works dealing with thin plates, heat tranfer equations more general than the Cattaneo-Vernotte equation should be considered. It would be of interest taking this into account in some future paper. See for instance A. Sellitto, V. A. Cimmelli and D. Jou, Mesoscopic theories of heat transport in nanosystems, Springer, Berlin, 2016, or H. Machrafi, Extended Non-Equilibrium Thermodynamics: From Principles to Applications in Nanosystems.

1st edition, Taylor & Francis Group, London (2019); D. Jou, J. Casas-Vazquez, G Lebon.: Extended Irreversible Thermodynamics. Springer, New York (2010); V A Cimmelli, Different thermodynamic theories and different heat conduction laws, Journal of Non-Equilibrium Thermodynamics 34 (2009), 299-333;

 

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Second Report

Unfortunately, the authors responded ver hastily and, in my opinion, not exhaustively, to the questions raised in my first report. So, I am obliged to answer, once again, may be in a more precise way, the same questions, that is:

1) It is known that the classical Fourier equation admits infinite speed of propagation of heat signals within the continuum field. So, the speed of information propagation is faster than the speed of light in vacuum, which is inadmissible within the framework of relativity. To overcome this contradiction, workers such as Cattaneo [1], Vernotte [2], Chester [3], and others [4] proposed that Fourier equation should be upgraded from the parabolic to a hyperbolic form.

However, in this manuscript the authors justifiy the use of the Cattaneo et al. wave equation by stating "The use of the Cattaneo-Vernotte equation instead of the classical Fourier law is logical because the heating is assumed to be very fast, with a rate comparable to the inverse of the relaxation time of the heat flux".

The authors are kindly invited to justify in a less hasty way, the reason why they are proposing the equation of Cattaneo et al. also in the field of classical theory. Furthermore, what would be obtained if instead of using a hyperbolic equation (i.e. Cattaneo's P.D.E.) they had used a parabolic equation (i.e. Fourier's P.D.E.)?

[1] Cattaneo, C. R., "Sur une forme de l'équation de la chaleur éliminant le paradoxe d'une propagation instantanée". Comptes Rendus. 247 (4), 431 (1958).

[2] Vernotte, P., "Les paradoxes de la theorie continue de l'équation de la chaleur". Comptes Rendus. 246 (22), 3154 (1958).

[3] Chester, M., "Second sound in solids". Physical Review. 131 (15): 2013–2015. Bibcode:1963, PhRv..131.2013C. (1963) doi:10.1103/ PhysRev.131.2013.

[4] Morse P. M. and Feshbach H.,  Methods of Theoretical Physics. New York: McGraw-Hill (1953).

2) The boundary conditions have been more fully specified (although the boundary conditions for the left and right ends of the Pt-Rh GFLP, which are not specified in the manuscript, will "turn up the noses" of several readers).

3) The authors did not answer to the question 4a) mentioned in my previous report: “4a) the authors are kindly asked to provide an interpretation, at least qualitatively, of the dependence of the temperature at the centre of the circular opening of the Pt-Rh GFLP on the frequency”.

Also in this case, very hastily, the authors replied:

“frequency is one of the important factors affecting temperature. The influence of frequency on temperature is closely related to the thermal diffusivity of materials. Only in a certain range of thermal diffusivity can the influence of frequency on temperature work, and when it exceeds this range, it tends to be stable”.

The authors are kindly invited to explain, at least qualitatively, from the physical point of view, the following issues:

3a) The dependence of thermal diffusivity to frequency (please, at least provide a reference);

3b) " Only in a certain range of thermal diffusivity can the influence of frequency on temperature work". Why ? Please explain.(or, please, provide a reference):

3c) "when it exceeds this range, it tends to be stable". Why ? Have the author tested the stability (mathematically or numerically) the stability of the solution. If not, at least qualitatively, the authors should explain, the existence of this range of frequencies and the reason why beyond this range the solution tends to be stable (or, please, at least provide a reference).

In conclusion, it is still my opinion that if the ultimate objective of this work is to provide theoretical support and engineering reference for the research on the temperature field of Pt-Rh GFLP, the authors should make a real extra effort and take into account the suggestions mentioned in my two reports.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The authors have clarified the points I was asking for. They have done a good work and now I see more clearly and directly the several points I was confused or uncertain about. I recommend publication.

I have still four minor suggestions:

line 94 of the paper they could introduce the values of the thermal conductivity of the 5% and of the 10% materials, and comment that the difference is very small, instead of considering for it a single value withou any further discussion. In view of their explanation of this point in the cover letter, their assumption is very reasonable, but a comment on it would take only three lines and the reader would not be wondering about this point.

In line 141, instead of “openings of triangle, openings of rectangle” it would be preferable: “opening of triangular or rectangular cross section”

In Figure 5, the authors study materials with mass fraction changing between 0 % and 30 %. However, in line 91-94 they have only given the vàlues of density, specific heat and thermal conductivity for mass fraction 5 % and 10 %. The readers wonders how the changes in physical parameters of the material between mass fractions from 0 % to 30 % have been modelized. A very brief comment would be suficient.

In line 319, it could be added: “The analysis of rapid heating requires going beyond Fourier’s law and taking into consideration the non-vanishing relaxation time of the heat flux, as it has been done in the present paper”. (In this way, the reader is reminded again that the use of the hyperbolic heat transfer law in this paper is well motivated ad necessary)

Author Response

Please see the attachment.

Author Response File: Author Response.pdf