Review Reports

Round 1

Reviewer 1 Report

The submitted paper was devoted to presenting analytical solutions using Adomian decomposition method for the problem of convection and radiation heat transfer in porous fin. Several questions need to answer to improve the quality of the paper:

1) State carefully the novelty in this work compare to others in literature including the authors' ones.

2) The quality of writing is very low. A lot of equations in the modeling section are not explained.

3) Most of the symbols are not defined and very poor nomenclature is provided at the end of the manuscript.

4) The model which is single nonlinear ODE can be solved in any other method such as the one given by Eqs. (23)-(24). Why do you select the Adomian decomposition method? 

5) Explain how did you build the model in Eq. (1). 

6) Explain why do you consider the thermal radiation mechanism with the convection one.

7) Why do you use the two particular formulas given in Eqs. (13), (14)? Are they work for porous fin thermal sink. Explain and give references.

8) What is the purpose of providing the solutions in Eqs. (23)-(24)?

9) discuss the efficiency, page 12, with some numeric examples.

Author Response

Response to Comment 1: Thank you very much for your comment. The significance of the present work is clearly stated as follow:

“The range of applications of FGM include nuclear, automobile structure, aerospace and optoelectronics. With the thermal capabilities, the use of FGM in heatsink fin design would serve as a viable cooling material. However, an in-depth review of existing works shows that research on the application of FGM for heatsink design is not exhaustive in the literature. Therefore, the present work is motivated from the capabilities of the FGM and several established thermal characteristics of porous fins. The present work focusses on the analysis of a porous heatsink of FGM operating under a convective-radiative environment for improved cooling low and high power electronic systems”.

Response to Comment 2: Thank you so much for your constructive criticism. The revised has been duly revised and checked for grammatical errors. Moreover, all the equations are explained along with the formulation with the associated assumptions clearly stated. However, the equations that are not the direct product of the present manuscript have been properly referenced.

Response to Comment 3: We have included all the definitions of all symbols used in the study as well as provide a complete nomenclature have been provided in the revised manuscript as requested.

Response to Comment 4: The analytical solutions obtained in Eq. (23) - (24) are the closed-form solutions of the linearized form of the nonlinear model of the present study. However, when the nonlinear term is incorporated, the developed analytical scheme (special functions) fails. This necessitates the need for an alternative analytical scheme as the numerical approach is employed for verification. ADM is employed in the present study because it transforms only the nonlinear terms into an Adomian function with all the linear terms preserved which increases the accuracy of the method. However, a strong limitation of ADM is the ability to obtain the right Adomian level, which when obtained, speeds up the convergence of the required solution.

Response to Comment 5: An explanation of the model in Eq. (1) has been included in the manuscript

Response to Comment 6: We consider the thermal radiation mechanism because of consideration for practical applications.

thermal conductivity may be used to enhance or reduce the rate of heat transfer. A reference to this idea is shown below:

28. Sobamowo, G, Oguntala, G. Yinusa, A. Nonlinear Transient Thermal Modeling and Analysis of a
Convective-Radiative Fin with Functionally Graded Material in a Magnetic Environment. Hindawi,
Modelling and Simulation in Engineering Article ID 7878564, 2019.

Response to Comment 8: The purpose is for validation at the linear level if required

Response to Comment 9: Thank you very much once again for the approaches employed towards improving the quality of our manuscript. The fin efficiency analysis is vital and has been discussed extensively based on the scope of the present study. However, the reviewer comment on numerical examples would be considered in our subsequent work.

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Author Response File: Author Response.pdf

Reviewer 2 Report

1) Validation of the present ADM results should be done with other published work by simplifying the work, if applicable. Currently, only numerical comparison is done in Table 1.

2) How many terms are considered in the ADM? Please mention it clearly.

3) Literature survey ignores many important works on the application of ADM and other semi-analytical and numerical techniques used in the heat transfer analysis of fins. Please see the following and discuss them to convincingly highlight the gaps within the existing studies and justify the importance of the present work.

"Analysis of transient heat transfer in a cylindrical pin fin." Journal of Thermophysics and heat transfer 12, no. 2 (1998): 281-283. 

"A transient 3-D inverse problem in imaging the time-dependent local heat transfer coefficients for plate fin." Applied Thermal Engineering 25, no. 14-15 (2005): 2478-2495.

"An iterative regularization method in estimating the base temperature for non-Fourier fins." International journal of heat and mass transfer 49, no. 25-26 (2006): 4893-4902.  

"An optimal fin design problem in estimating the shapes of longitudinal and spine fully wet fins." Computer Modeling in Engineering and Sciences (CMES) 44, no. 3 (2009): 249.

"An analytical prediction for performance and optimum design analysis of porous fins." International Journal of Refrigeration 34, no. 1 (2011): 337-352. 

 

"A decomposition analysis on convecting–radiating rectangular plate fins for variable thermal conductivity and heat transfer coefficient." Journal of the Franklin Institute 349, no. 3 (2012): 966-984.

"Analytical solution for convective–radiative continuously moving fin with temperature-dependent thermal conductivity." International Journal of Thermophysics 33, no. 5 (2012): 924-941.  

"Semi-analytical method for solving nonlinear equation arising in natural convection porous fin." Therm. Sci 16, no. 5 (2012): 1303-1308.

"Determining the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method." International Journal of Numerical Methods for Heat & Fluid Flow 22, no. 2 (2012): 263-272.

"A model on the basis of analytics for computing maximum heat transfer in porous fins." International Journal of Heat and Mass Transfer 55, no. 25-26 (2012): 7611-7622.  

"Application of Adomian decomposition method and inverse solution for a fin with variable thermal conductivity and heat generation." International Journal of Heat and Mass Transfer 66 (2013): 496-506.

"Approximate analytical method for porous stepped fins with temperature-dependent heat transfer parameters." Journal of Thermophysics and Heat Transfer (2016): 661-672.

"Application of simulated annealing in a rectangular fin with variable heat transfer coefficient." Inverse Problems in Science and Engineering 21, no. 8 (2013): 1352-1367.

"A comparative study of longitudinal fins of rectangular, trapezoidal and concave parabolic profiles with multiple nonlinearities." Energy 51 (2013): 243-256.

"Analytical investigation of nonlinear model arising in heat transfer through the porous fin." Thermal science 18, no. 2 (2014): 409-417.  

"Direct and inverse approaches for analysis and optimization of fins under sensible and latent heat load." International Journal of Heat and Mass Transfer 124 (2018): 331-343.

"Heat transfer improvement of a wet fin under transient response with a unique design arrangement aspect." International Journal of Heat and Mass Transfer 127 (2018): 1239-1251.

4) Why an insulated tip/adiabatic boundary condition is considered in equation 10? In reality, it should be at-least convective.

5) Please provide suitable referencing to various expressions written in equations 1 to 9. Equations which are not the direct outcomes of the present study must be referred.

6) What type of fluid is considered here? Is it air or any liquid?

7) Mention clearly various assumptions used in the study. In particular, kindly indicate the conditions where a 1-dimensional model as used in the present study holds good.

8) Why ADM is used here? In the literature, there are many semi-analytical techniques. So, what is the importance of ADM with respect to other techniques must be highlighted.

9) It is better to study the variations of fin efficiency for some cases of FGM.

Author Response

Response to Comment 1: Thank you very much for the good review work. The result of the present study has been compared with other works as presented in Table 1 of the revised manuscript.

Response to Comment 2: 15 Adomian terms were considered for each of the nonlinear parts of the model as clearly stated in the revised manuscript.

Response to Comment 3: Thank you very much the authors have considered all related works and integrated into the revised appropriately.

Response to Comment 4:  For most practical applications, the fin thickness at the end is relatively small that the heat transfer from the fin tip can be neglected and the solution can be obtained assuming fin is insulated at the tip. Therefore, in this work, we assumed that the tip of the fin is adiabatic/insulated.

Response to Comment 5: The equations were derived in the present study. However, the equations that are not the direct product of the present manuscript have been properly referenced.

Response to Comment 6: The main fluid is air but other fluids may be considered by simply supplying their thermos-physical values.

Response to Comment 7: The reviewer’s comment on this has been considered and effected in the revised paper.

Response to Comment 8: The analytical solutions obtained in Eq. (23) - (24) are the closed-form solutions of the linearized form of the nonlinear model of the present study. However, when the nonlinear term is incorporated, the developed analytical scheme (special functions) fails. This necessitates the need for an alternative analytical scheme as the numerical approach is employed for verification. ADM is employed in the present study because it transforms only the nonlinear terms into an Adomian function with all the linear terms preserved which increases the accuracy of the method. However, a strong limitation of ADM is the ability to obtain the right Adomian level, which when obtained, speeds up the convergence of the required solution.

Response to Comment 9: Thank you very much once again for the approaches employed towards improving the quality of our manuscript. The fin efficiency analysis is vital and has been discussed extensively based on the scope of the present study. However, the reviewer comment on numerical examples would be considered in our subsequent work.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have addressed my comments and the paper now is in an acceptable form.

Author Response

Thank you so much for your comment. The whole manuscript has been duly revised and all errors corrected accordingly. 

Reviewer 2 Report

Revised paper seems satisfactory.

Author Response

Thank you so much for your recommendation of acceptance. The whole manuscript is revised and all errors have been duly corrected.