Unsteady contact melting

1. Unsteady contact melting Tim G. Myers University of Cape Town

2. Water droplet floating above hot steel: Leidenfrost effect Contact melting configuration Applications: thermal storage, process metallurgy, geology, nuclear technology, Leidenfrost, ice skating …

3. Three stages of melting for block with insulated sides and top surface

4. Governing equations Heat equations in liquid and solid Navier-Stokes equation and incompressibility condition Mass balance Stefan condition

5. Standard assumptions: • The temperature of the solid remains at the melting temperature, throughout the process. • The melting process is in a quasi-steady state, i.e. h(t)=constant. • Heat transfer in the liquid is dominated by conduction across the film. • The lubrication approximation holds in the liquid layer, so the flow is primarily parallel to the solid surface and driven by the pressure gradient. The pressure variation across the film is negligible. • The amount of melted fluid is small compared to that of the initial solid. • There is perfect thermal contact between the liquid and substrate or there is a constant heat flux, Now develop a model without invoking 1, 2, 5, 6

6. Non-dimensionalisation Navier-Stokes equation and incompressibility condition

7. Similarly Governing equations Boundary conditions Thermal problem Stage 2 Stage 1

8. Heat Balance Integral Method Classic heat flow problem … Heat balance formulation – replace BC at infinity

9. Heat Balance Integral Optimal n method Where n = 2.233

11. Classical Stefan problem

12. Neumann’s solution Stefan condition

13. HBIM solution

14. Integrate heat equation … Couple to Stefan condition … i.e. two equations for two unknowns; before melting have single first order ODE to solve

16. Application to contact melting Three stages of melting for block with insulated sides and top surface Stage 1: pre-melting Exact solution HBIM solution

17. Temperature at end of Stage 1

18. Stage 2: Melting HBIM Stefan condition where (from lubrication solution) Stage 3: More melting Etc. etc.

19. Force balance Standard quasi-steady analysis leads to without squeeze (Neumann solution) Temperature profile

20. Evolution of melted thickness for current model and quasi-steady solutions for infinite HTC and HTC=855 Evolution of liquid height for current model and quasi-steady solutions for infinite HTC and HTC=855

21. Temperature in solid and liquid half-way through melting process

22. Maximum value of neglected terms for HTC of 855 and 5000

23. Comparison of solid thickness with experiments on N-octadecane, current method (solid), current with infinite HTC (dotted) and Moallemi et al (1986) theory (dash-dot)

24. Leidenfrost effect Now must calculate shape of droplet as well Young-Laplace equation

25. Constant volume droplet

26. Unsteady calculation

27. Conclusions • Difference with standard models • Modelling temperature in solid (using HBIM) • Cooling condition at substrate • Varying solid mass • Unsteady • Can match contact melting experiments almost exactly • (really should be error due to 3D), v. close to Leidenfrost results • Extensions: 3D, include convection in liquid/vapour • Related publications: • Myers T.G. & Charpin J.P.F. A mathematical model of the Leidenfrost effect on an axisymmetric droplet. Submitted to Phys. Fluids Aug. 2008. • Myers T.G., Mitchell S.L. & Muchatibaya G. Unsteady contact melting of a rectangular cross-section phase change material. Phys. Fluids 20 103101 2008, DOI: 10.1063/12990751. • Myers T.G. Optimizing the exponent in the Heat Balance and Refined Integral Methods. Int. Commun. Heat Mass Transf. 2008, DOI:10.1016/j.icheatmasstransfer. 2008.10.013.