A new solution to a weakly non-linear heat conduction equation in a spherical droplet: basic idea and applications

Dmitrii Antonov, E. Shchepakina, Vladimir Sobolev, Elena Starinskaya, Vladimir Terekhov, Pavel Strizhak, Sergei Sazhin

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A new analytical solution to a non-linear heat transfer equation in a spherically-symmetric droplet is suggested. All thermophysical properties inside the droplet are considered to be close to their average values. This allows us to consider the non-linearity of this equation as weak. The solution is presented as T=T_0+T_1, where T_0 is the solution to a linear heat conduction equation, and T_1 << T_0. The equation for T_1 is presented as a linear heat conduction equation with a source term depending on the distribution of T_0 and its spatial derivatives inside the droplet. The latter equation is solved analytically alongside the linear equation for T_0, and the final solution is presented as T=T_0+T_1. The predictions of the numerical code in which this solution was implemented are verified based on a comparison of those predictions with the predictions of COMSOL Multiphysics code using input parameter values that are typical for nanofluid (water and SiO_2 nanoparticles) droplet evaporation in atmospheric conditions. It is demonstrated that for these experiments T_1 << T_0 which justifies the applicability of the linear heat conduction equation used for the analysis of this process. Small differences in the temperatures predicted by both non-linear and linear models lead to a much more noticeable difference in integral characteristics such as time before the start of the formation of the cenosphere when the mass fraction of nanoparticles at the droplet surface reaches about 40%
Original languageEnglish
Article number124880
Number of pages10
JournalInternational Journal of Heat and Mass Transfer
Publication statusPublished - 6 Nov 2023

Bibliographical note

Funding Information:
The authors would like to thank the Ministry of Science and Higher Education of the Russian Federation (Grant 075-15-2021-575 ) and the Russian Science Foundation (Grant 21-19-00876 , https://rscf.ru/en/project/21-19-00876/ ) for their financial support. Grant 075-15-2021-575 supported the contributions by D Antonov (who contributed mainly to the development of the numerical code and formal analysis), E Starinskaya and V Terekhov (who contributed mainly to planning, the performing of experiments and formal analysis) and S Sazhin (who coordinated the work on the paper, and contributed to formal analysis, and writing and editing the paper). Grant 21-19-00876 supported the contributions by E Shchepakina and V Sobolev (who were mainly responsible for the derivation of the analytical solution which was implemented in the numerical code used in the analysis), and S Sazhin (who coordinated the work on the paper, and contributed to formal analysis and writing and editing the paper). The research presented in this paper was initiated during work on a project supported by the Royal Society (United Kingdom) (Grant IEC 192007 ).


  • Evaporation
  • Heating
  • Mathematical model
  • Nano-fluid droplets
  • Non-linear heat conduction equation


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