The effect of a moving boundary on droplet heating and evaporation

1. The effect of a moving boundary on droplet heating and evaporation I.G. Gusev Sir Harry Ricardo Laboratories, University of Brighton, UK 13th January 2012

2. Plan • Introduction • The droplet radius is a linear function of time • The droplet is an arbitrary function of time • Results

3. Receding radius Linear model Conventional model Rd Rd = Rd0(1+αt) Rd = const t

4. Temperature equation

5. Mass fraction equation “Speed of evaporation”: where: Dv is the binary diffusion coefficient of vapour in air and BM is the Spalding mass transfer number defined as Yvis the total vapour mass fraction

6. Mass fraction equation “Speed of evaporation”: Droplet radius receding during the time step where

7. Analytical solution

8. Results

9. Results

10. Mass fraction equation Mass fraction equation inside the droplet: Boundary condition: where

11. Newsolution

13. The plots of the ethanol mass fraction Yeth versus ξ = R/Rd, as predicted by the conventional model (dashed) and the new model, taking into account the effect of the moving boundary, (solid) for times 0.001 s, 0.01 s and 0.03 s.

14. The plots of the ethanol mass fraction Yeth(ξ = 1), as predicted by the conventional model (dashed) and the new model, taking into account the effect of the moving boundary, (solid).

15. The plots of the droplet radius Rd versus time, as predicted by the conventional model (dashed) and the new model, taking into account the effect of the moving boundary, (solid) .

16. The plots of the Spalding mass transfer number BM versus time, as predicted by the conventional model (dashed) and the new model, taking into account the effect of the moving boundary, (solid).

17. Receding radius Integral model Rd t

18. Integral model

19. Integral model Algorithm: Calculate Rd(t) using conventional model Calculate T(R,t) using Rd(t) Calculate Rd(t) using T(Rd,t)

20. Integral model - Results

21. Integral model - Results

22. Integral model - Results

23. Integral model - Results

24. Numerical Solution • S.L. Mitchell, M. Vynnycky, I.G. Gusev, S.S. Sazhin, An accurate numerical solution for the transient heating of an evaporating spherical droplet, Applied Mathematics and Computation,217 (2011) 9219-9233.

25. Numerical Results

26. Numerical Results

27. Results • Conventional approach takes  3586 seconds to calculate  1191 steps. • Linear approach takes  2773 seconds to calculate  893 steps. • Integral approach takes  453 seconds to calculate  15 iterations + time to perform calculations for initial radius function.

28. Selected Journal Papers • S.S. Sazhin, P.A. Krutitskii, I.G. Gusev, M.R. Heikal, Transient heating of an evaporating droplet, Int. J. Heat and Mass Transfer 53 (2010) 2826-2836. • S.S. Sazhin, P.A. Krutitskii, I.G. Gusev, M.R. Heikal, Transient heating of an evaporating droplet with presumed time evolution of its radius, Int. J. Heat and Mass Transfer 54 (2011) 1278–1288. • I.G. Gusev , P.A. Krutitskii, S.S. Sazhin and A.E. Elwardany, New solutions to the species diffusion equation inside droplets in the presence of the moving boundary, Int. J. of Thermal Sciences, (in press) • S.L. Mitchell, M. Vynnycky, I.G. Gusev, S.S. Sazhin, An accurate numerical solution for the transient heating of an evaporating spherical droplet, Applied Mathematics and Computation,217 (2011) 9219-9233.

29. Conference Papers • S.S. Sazhin, I.N. Shishkova, I.G. Gusev and M.R. Heikal, Hydrodynamic and kinetic models for monocomponent droplet heating and evaporation: recent developments, The 21st International Symposium on Transport Phenomena 2-5 November, 2010, Kaohsiung City, Taiwan • S.S. Sazhin, A.E. Elwardany, I.G. Gusev, I.N. Shishkova and M.R. Heikal, Modelling of fuel droplet heating and evaporation: recent results and unsolved problems, Anniversary volume honoring Amalia and MicklosIvanyi, Pollack Mihaly Faculty of Engineering, University of Pecs, Oct 25-26, 2010, pp. B:197-B:209 • S.S. Sazhin, I.N. Shishkova, I.G. Gusev, A.E. Elwardany, P.A .Krutitskii and M.R. Heikal, Fuel droplet heating and evaporation: new hydrodynamic and Kinetic models, Proceedings of the ASME 2010 International Heat Transfer Conferences,IHTC-14, August 8-13, 2010, Washington, USA • S.S. Sazhin, I.N. Shishkova, I.G. Gusev, A.E. Elwardany and M.R. Heikal, Modelling of droplet heating and evaporation: recent results and unsolved problems. Journal of Physics, Conference Series, 012026. International Workshop on Multi-Rate Processes & Hysteresis in Mathematics, Physics and Information Sciences. University of Pecs, Hungary, May 31 -- June 3, 2010. Published by Institute of Physics (UK). • S.S. Sazhin, I.G. Gusev, J.-F. Xie, A.E. Elwardany, A.Yu. Snegirev and M.R. Heikal, New approaches to modelling droplet heating and evaporation. Proceedings of DIPSI Workshop 2011 on Droplet Impact Phenomena \& Spray Investigation, May 27, 2011, Bergamo, Italy. Editors E Cossali and S Tonini, Bergamo University Press • S.S Sazhin, I.G. Gusev, M.R. Heikal, P.A. Krutitskii, Modelling of liquid droplet heating and evaporation taking into account the effects of the moving boundary. Proceedings of the Asian Symposium on Computational Heat Transfer and Fluid Flow - 2011, 22--26 September 2011, Kyoto, Japan, paper 029.

30. Acknowledgements The authors are grateful to the European Regional Development Fund Franco-British INTERREG IVA (Project C5, Reference 4005) for financial support of the work on this project.

31. Thank you for your attentionYour questions are more than welcomed

32. The effect of a moving boundary on droplet heating and evaporation I.G. Gusev Sir Harry Ricardo Laboratories, University of Brighton, UK 13th January 2012