A Shortcut Method to Solve for a 1D Heat Conduction Model under Complicated Boundary Conditions

Round 1

Reviewer 1 Report

The paper describes a simplified way to solve Fourier's heat conduction equation for a complicated boundary condition T = f(t). The paper is interesting and good but requires a few modifications to make it acceptable for publication. 

* Please elaborate further on the motivation of the paper in the introduction. For example, you mention in lines 208-212 an important application is for developing heat exchangers for geothermal applications. Place that information in the introduction. And you mention one of the main goals of the paper is to calculate the thermal diffusivity. Is the thermal diffusivity of soil in reference text books not sufficient for the design calculations required for geothermal heat exchangers? 

* With today's ubiquitous computing power, what is the reason for developing an exact solution for the heat equation, as opposed to solving the equation numerically using Runge-Kutta or other discretization methods? Is the level of accuracy not sufficient for analyzing temperature data? 

* Figure 4: Remove the symbols for the curves (a = 0.03 --> 0.033) so that they don't overlap with the collected data points. 

* You use variables like "a" and "beta" in the abstract. Replace "a" with "thermal diffusivity" as well as "beta" and t_g. 

* Line 55-56 is not a complete sentence. 

* Please add a Nomenclature section to clearly define all variables as well as units. 

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

In this paper, the authors studied the calculation problem of one-dimensional heat conduction equation defined on a semi-real axis (semi-infinite space) region with boundary measurements f(t). Firstly, they deduced the theoretical solution by Fourier transform method. To obtain a calculation result based on the measurement, the authors applied the piecewise linear interpolation to discretize f(t) and obtained a theoretical solution that can be used to solve quickly the practical problem. Due to the variation of boundary conditions, the authors introduced the inflection-point method and curve-fitting method for calculating the parameter of the model. This result is very interesting. I recommend it for publication after some corrections. 

1) There are lots of symbols appear in the text and it is better to add a glossary of nomenclature.

2) Condition 3.3.3 seems to be a special case of 3.3.2, here combing them to discuss is better.

3) Add some literature to illustrate the inflection-point method and curve-fitting method.

4) Figure 4 is not clear enough and it should be improved. 

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

The paper can be accepted.

Author Response

Dear reviewer,

We are very excited and honored that you have given such a positive review for our manuscript.

Thank you again for your time and kind consideration. Have a nice day!

Best regards,

All authors

Reviewer 4 Report

The paper under the title "A Shortcut Method to Solve 1D Heat Conduction Model Under 2 Complicated Boundary Conditions" tried to present a new solution for PDE. The manuscript has serious issues such as quality of text, poor discussion of methodology, and results. I am also not sure about the novelty of this method. The authors present complicated boundary conditions, but the paper solves them in a simple form. In addition, the calculation cost of this method seems high. Compared to the standard method, it's interesting to have a comparison between these methods. Also, the auhors didn't mention the time. I'm eager to see the response letter.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

The authors didn't pay attention to my concerns in this paper. None of my comments have any effect on the manuscript.