An Analytical Solution to the One-Dimensional Unsteady Temperature Field near the Newtonian Cooling Boundary

Round 1

Reviewer 1 Report

The authors proposed general theoretical solution using Fourier transform for one-dimensional heat conduction problems. The model was formulated and validated using numerical solution. The author made a great effort in writing the paper, presenting and discussing the results. However I have very minor comments:

   

1- In the abstract what is a, x, and λ?

2- out of 55 cited references, only 6 of them are published in 2022 and 2021.!!

3- References of some equations were not cited.

4- The meaning or definition of some symbols were not provided.

5- In figure 3, please explain why the difference between the numerical and analytical solution increase with increasing thermal diffusivity?

6- Lines 352 and 353 (This study adopts the 352 improved Morris screening method), please cite the reference

Author Response

Response to Reviewer 1 Comments

Dear reviewer,

 

We would like to express our sincere appreciation for your comments concerning our manuscript. Those comments are all extremely helpful and valuable for improving our paper. We have studied all the comments carefully and have made corresponding revisions using the track change function. The main corrections and the responses to the comments are listed as follows:

 

Reviewer 1 Comments

The authors proposed general theoretical solution using Fourier transform for one-dimensional heat conduction problems. The model was formulated and validated using numerical solution. The author made a great effort in writing the paper, presenting and discussing the results. However I have very minor comments:

 

Point 1: 1- In the abstract what is a, x, and λ?

 

Response 1:

In the abstract, we have supplement the corresponding explanation.

a is the thermal diffusivity, x is calculation distance, λ is the coefficient of cooling ratio.

Lines 20-21,

“The results show that T(x,t) is directly proportional to the thermal diffusivity (a) and is inversely proportional to calculation distance (x) and the coefficient of cooling ratio (λ).”

 

 

Point 2: 2- out of 55 cited references, only 6 of them are published in 2022 and 2021.

 

Response 2:

We supplement some related references in recent years , moreover, we summarize the recent relevant research work.

Lines 68-72,

“Through Newton's law of cooling and Fourier's law, combined with the relevant physical parameters, Tan et al established the heat conduction model of the fire suit and the model can improve the fitting accuracy, which is of great significance for the design and of and research into specialist fire clothing and equipment, and also has certain reference value for solving similar problems[11].”

 

Lines 95-112,

“Rosales et al used a generalized conformable differential operator and then a simulation of the well-known Newton’s law of cooling was made, which had an advantage with respect to ordinary derivatives[24]. Melo et al developed an active thermography algorithm capable of detecting defects in materials, based on the techniques of thermographic signal reconstruction, thermal contrast and the physical principles of heat transfer. Newton's law of cooling was used to store the normalized temperature data pixel-by-pixel over time and a compression ratio of 99% was obtained [25]. Konovalenko et al propose a novel method that extends the applicability of Newton's law of cooling to changeable ambient temperatures based on a set of temperature stability conditions and a sensor measurement error. In this method, an optimal number of measurements that characterize stable ambient temperatures and improve prediction reliability are selected[26]. Calvo-Schwarzwlder et al simulated the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment[27]. Herrera-Sánchez et al used Newton's Law of Cooling for heat transfer, which states that the rate of heat exchange between an object and its environment, to solve the problem of the packaging process when handling canned food[28].”

 

Lines 121-132,

“The physical law for describing the temporal temperature decrease has been dominated by Newton's law of cooling (NLC), which assumes that natural cooling occurs by following an exact exponential trend. However, several studies have questioned the broad validity of this law by arguing that cooling occurs following an approximate rather than an exact exponential trend. Silva introduced a new formulation of NLC based on generalized statistics that outperforms the classical NLC, and so demonstrates a new path to cooling analyses[48]. Yan et al. studied a discrete variable topology optimization method to solve the simplified convective heat transfer (SCHT) design optimization modeled by Newton's law of cooling. The discrete variable topology optimization was based on the proposed sequential approximate integer programming with trust-region, which could identify the convective boundary and carry out the optimization design.[49].”

 

 

Point 3: 3- References of some equations were not cited.

 

Response 3:

We have checked the whole paper carefully and cited the references of equations.

Please checked the revised version.

 

 

Point 4: 4- The meaning or definition of some symbols were not provided.

 

Response 4:

The meaning or definition of symbols have been provided.

 

Line 254, 

“where F^-1 is the inverse conversion operator.”

Line 258, 

“where b is the intermediate variable, b = a (t - ξ).”

Line 260, 

“where ξ is the integral variable in time.”

 

Lines 345-348,

“where Δt is the time step, Δx is the space step length, (n,i) represents the position of a node in the time-space region and the corresponding temperature, recorded as T_nⁱ, T_nⁱ⁺¹ represents the temperature value of the point at x=n at time i+1; T_nⁱ, T_n+1ⁱ and T_n-1ⁱ represent the temperature values at times i, x=n, x=n+1 and x=n-1, respectively.”

 

 

Point 5: 5-In figure 3, please explain why the difference between the numerical and analytical solution increase with increasing thermal diffusivity?

 

Response 5:

The reviewer's opinion is very helpful for us to improve the quality of the paper. Indeed, the original text only pointed out this phenomenon, without necessary discussion.

In this paper, the explicit finite difference method is used for numerical verification, and the difference equation is Formula (27). It is well known that the numerical difference has some numerical oscillations. According to Formula (27), the difference value of temperature is positively related to the thermal diffusivity a, so the calculation error caused by the difference formula is also positively related to a. Therefore, the error between numerical solution and analytical solution will also increase with the increase of a under the same calculation conditions.

In the revised version, we have revised the original text as follows:

 

Lines 371-373,

“The difference value of temperature is positively related to the thermal diffusivity a, and the calculation error caused by the difference formula increases with the increase of a.”

 

 

Point 6: 6- Lines 352 and 353 (This study adopts the 352 improved Morris screening method), please cite the reference.

 

Response 6:

We gratefully appreciate for your valuable suggestion. We have cited the reference.

 

Lines 414-423,

“Sensitivity analysis research varies from field to field. It is a tool for evaluating the influence of input parameters on the model output. As a result, it not only aids in the construction and validation of models but also lessens uncertainty. This study adopted the improved Morris screening method[64] and sensitivity was expressed by a dimensionless index, which was calculated as the ratio of the relative change of the model output to that of the parameters. Another approach to defining parameter changes was taken into consideration in order to increase the sensitivity of the discrimination parameters. In this approach, the change in the parameters was defined as a fixed percentage of the effective parameter range rather than a fixed percentage of the initial value.[65, 66].”

 

 

We have studied all the comments carefully and have made corresponding revisions using the track change function. For other details, please refer to the revised paper.

 

 

Thanks again for your time and valuable comments.

 

 

Best regards,

All authors

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

REPORT on the article with title “Analytical Solution to the One-Dimensional Unsteady Temperature Field Near the Newtonian Cooling Boundary”

Submitted Journal: Axioms (ISSN 2075-1680)

Manuscript ID: axioms-2117279

Authors:  Honglei Ren , Yuezan Tao , Ting Wei , Bo Kang , Yucheng Li , Fei Lin

 

In this article, a general theoretical solution is established for the problem (I)  using Fourier transform, however, here  the boundary condition   φ(t) is not directly presented  in the transformation processes, φ(t) is substituted into the general theoretical solution to obtain the corresponding analytical solution, etc..  In addition, the analytical solution is applied for parameters calculation and verification in a case study, , especially for the study of related problems in the fields of fluid dynamics and peridynamics with the heat conduction equation.

The topic of the article is good and scientifically. A great effort has been to obtain the results of this article. However, the following items can be considered for the revision.

SUGGESTIONS:

-…-The sentences “For the one-dimensional heat conduction model in a semi-infinite domain, the forced convection in practice follows Newton’s law of cooling, but it is difficult to solve via conventional integral transformation methods when the boundary condition φ(t) is an exponential decay function” in “Introduction”.

-…-The similarity rate of the article is 26% (see, the PDF file). However, 6% and 4% of the article has been taken form the former papers of the of the “ authors”. These rates should be reduce lower rates.

-…- In entire article, there are numerous  typos and grammatical mistakes.  For example, for the equations in the article there is no  “.”,  “,”  and so on.

The article needs major revisions for that facts.

-…-The consider problem, in particular, the PDE (1) is linear and a simple mathematical model.

Why nonlinear model(s) have not been discussed. Satisfactory information can be given.

-…-The formulations of the article can be more regular. Most of them are not regular.

-…- The following papers can be added to the references of this article to rich  them:

-…-Constructions of the optical solitons and others soliton to the conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Journal of Taibah University for Science. 14 (2020), no.1, 94–100.   

-…-Heat Transport Exploration of Free Convection Flow inside Enclosure Having Vertical Wavy Walls. J. Appl. Comput. Mech. 7(2) (2021) 520-527.

I would like to suggest the acceptation of the article  in “Axioms “ after revision is done.

Comments for author File: Comments.pdf

Author Response

Response to Reviewer 2 Comments

Dear reviewer,

 

We would like to express our sincere appreciation for your comments concerning our manuscript. Those comments are all extremely helpful and valuable for improving our paper. We have studied all the comments carefully and have made corresponding revisions using the track change function. The main corrections and the responses to the comments are listed as follows:

 

Reviewer 2 Comments

In this article, a general theoretical solution is established for the problem (I) using Fourier transform, however, here the boundary condition φ(t) is not directly presented in the transformation processes, φ(t) is substituted into the general theoretical solution to obtain the corresponding analytical solution, etc.. In addition, the analytical solution is applied for parameters calculation and verification in a case study, especially for the study of related problems in the fields of fluid dynamics and peridynamics with the heat conduction equation.

 

The topic of the article is good and scientifically. A great effort has been to obtain the results of this article. However, the following items can be considered for the revision.

 

I would like to suggest the acceptation of the article in “Axioms “ after revision is done.

 

 

Point 1: -The sentences “For the one-dimensional heat conduction model in a semi-infinite domain, the forced convection in practice follows Newton’s law of cooling, but it is difficult to solve via conventional integral transformation methods when the boundary condition φ(t) is an exponential decay function” in “Introduction”.

 

Response 1:

We have revised the relevant statements.

 

Lines 12-14,

“For the one-dimensional heat conduction model in a semi-infinite domain, although forced convection obeys Newton's law of cooling, it is challenging to solve using standard integral transformation methods when the boundary condition φ(t) is an exponential decay function.”

 

Lines 113-116,

“In practice, even if the immediate increase in the boundary temperature and subsequent decline are consistent with Newton's law of cooling, the heat transfer issue is challenging to solve directly by integral transformation when the boundary conditions are the exponential decay function ∆T₀ e^−λt.”

 

Point 2: -The similarity rate of the article is 26% (see, the PDF file). However, 6% and 4% of the article has been taken form the former papers of the of the “ authors”. These rates should be reduce lower rates.

 

Response 2:

For the similarity rate of the article, we have reduced the rates carefully.

Please check the revised version.

 

 

Point 3: - In entire article, there are numerous typos and grammatical mistakes. For example, for the equations in the article there is no “.”, “,” and so on.

The article needs major revisions for that facts.

 

Response 3:

For the typos and grammatical mistakes, we have checked the whole paper and revised the corresponding content.

The revised manuscript undergo extensive English revisions with MDPI's Author Services.

The following is the English-Editing-Certificate.

Please check the revised version.

 

 

 

 

 

Point 4: -The consider problem, in particular, the PDE (1) is linear and a simple mathematical model.

Why nonlinear model(s) have not been discussed. Satisfactory information can be given.

 

Response 4:

The opinion raised by the reviewers is accurate and forward-looking, which is also an issue we are further exploring.

Although the universal definite equation (1) in model (I), that is, the universal definite equation (5) in model (II) is the simplest form of this kind of problem, the boundary condition (3) in model(I), and the boundary condition (7) in model (II) is an exponential function, φ(t) = ∆T₀e^-λt, the solution of model (I) and (II) is relatively complex.

When the universal equation is a complex nonlinear equation, such as Boussinesq equation in the unsteady phreatic flow, convection diffusion equation of contaminant diffusion in the aquifer or atmosphere combined with exponential function boundary, the model solution is also relatively complex.

In view of the universality of the fact that the boundary conditions are exponential decay functions, so we will take the simplest universal definite equation (1) as an example to discuss the problem of model solution under such boundary conditions.

In the revised paper, we made a relevant discussion in the conclusion part of the paper.

 

Lines 540-543,

“(6) For the above problem, in particular, the PDE is linear and a simple mathematical model. There is a very hot field in mathematics and physics that has been studying nonlinear PDE for a long time. So, in the future research, the related nonlinear PDE equations could be studied further.”

Point 5: -The formulations of the article can be more regular. Most of them are not regular.

Response 5:

For the formulations of the article, we have checked the whole paper and revised the corresponding content. It is expected that the formula format can be more standardized.

Please check the revised version.

 

 

Point 6: - The following papers can be added to the references of this article to rich them:

-Constructions of the optical solitons and others soliton to the conformable fractional Zakharov-Kuznetsov equation with power law nonlinearity. Journal of Taibah University for Science. 14 (2020), no.1, 94–100.

-Heat Transport Exploration of Free Convection Flow inside Enclosure Having Vertical Wavy Walls. J. Appl. Comput. Mech. 7(2) (2021) 520-527.

 

 

Response 6:

We have added the related references. For detail, in the revised version.

 

Lines 44-50,

“Fayz et al expresses a numerical study of flow features and heat transport inside an enclosure. Governing equations are discretized by a finite-element process with a collected variable arrangement. Streamlines and isotherm lines are utilized to show the corresponding flow and thermal field inside a cavity. Velocity and temperature profiles are displayed for some selected positions inside an enclosure for a better perception of the flow and thermal field[3].”

 

Lines 200-205,

“For the above problem, in particular, the PDE is linear and a simple mathematical model. There is a very hot field in mathematics and physics that has been studying nonlinear PDE for a long time. For example, Md and Cemil examined the modified (G′/G)-expansion process for generating closed-form wave answers of the conformable fractional ZK equation, including power law nonlinearity[52]. So, in the future research, the related nonlinear PDE equations could be studied further.”

 

 

We have studied all the comments carefully and have made corresponding revisions using the track change function. For other details, please refer to the revised paper.

 

 

Thanks again for your time and valuable comments.

 

 

Best regards,

All authors

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

Report on the REVISED article with title “Analytical Solution to the One-Dimensional Unsteady Temperature Field Near the Newtonian Cooling Boundary”

 

Submitted Journal: Axioms (ISSN 2075-1680)

 

Manuscript ID:axioms-2117279

 

Authors: Honglei Ren , Yuezan Tao , Ting Wei , Bo Kang , Yucheng Li , Fei Lin

 

Previous Comments and Present Responses

1-Comment: The sentences “For the one-dimensional heat conduction model in a semi-infinite domain, the forced convection in practice follows Newton’s law of cooling, but it is difficult to solve via conventional integral transformation methods when the boundary condition φ(t) is an exponential decay function” in “Introduction”.

Response 1: It was done accordingly.

Lines 12-14,

“For the one-dimensional heat conduction model in a semi-infinite domain, although forced convection obeys Newton's law of cooling, it is challenging to solve using standard integral transformation methods when the boundary condition φ(t) is an exponential decay function.”

Lines 113-116,

“In practice, even if the immediate increase in the boundary temperature and subsequent decline are consistent with Newton's law of cooling, the heat transfer issue is challenging to solve directly by integral transformation when the boundary conditions are the exponential decay function ∆T₀ e^−λt.”

2-Comment: The similarity rate of the article is 26% (see, the PDF file). However, 6% and 4% of the article has been taken form the former papers of the of the “ authors”. These rates should be reduce lower rates.

Response 2:

The similarity rate of the article is now 30%. It is higher than before. However, the similarity rates 6% and 4% of the article were reduced to 5% and 3%, respectively.  See the attached similarity report on PDF file.

3-Comment:  In entire article, there are numerous typos and grammatical mistakes. For example, for the equations in the article there is no “.”, “,” and so on.

The article needs major revisions for that facts.

Response 3:  There are still some typos in the article. It should be revised during the galley proofs.

4-Comment: The consider problem, in particular, the PDE (1) is linear and a simple mathematical model.

Why nonlinear model(s) have not been discussed. Satisfactory information can be given.

Response 4: Satisfactory information has been given.

5-Comment: The formulations of the article can be more regular. Most of them are not regular.

Response 5:Some  suitable arrangements have  been done.

6-Comment: The following papers can be added to the references of this article to rich them.

Response: It was added accordingly.

I would like to suggest the acceptation of the REVISED article in “Axioms “.

Comments for author File: Comments.pdf