Unsteady contact melting Tim G. Myers University of Cape Town

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

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

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

Standard assumptions: 1.The temperature of the solid remains at the melting temperature, throughout the process. 2.The melting process is in a quasi-steady state, i.e. h(t)=constant. 3. Heat transfer in the liquid is dominated by conduction across the film. 4.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. 5. The amount of melted fluid is small compared to that of the initial solid. 6.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

Non-dimensionalisation Navier-Stokes equation and incompressibility condition

Governing equations Boundary conditions Thermal problem Stage 1 Stage 2 Similarly

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

Heat Balance Integral Optimal n method Where n = 2.233

Classical Stefan problem

Neumann’s solution Stefan condition

HBIM solution

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

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

Temperature at end of Stage 1

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

Force balance Standard quasi-steady analysis leads towithout squeeze (Neumann solution) Temperature profile

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

Temperature in solid and liquid half-way through melting process

Maximum value of neglected terms for HTC of 855 and 5000

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)

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

Constant volume droplet

Unsteady calculation

Conclusions Difference with standard models 1.Modelling temperature in solid (using HBIM) 2.Cooling condition at substrate 3.Varying solid mass 4.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: 1.Myers T.G. & Charpin J.P.F. A mathematical model of the Leidenfrost effect on an axisymmetric droplet. Submitted to Phys. Fluids Aug Myers T.G., Mitchell S.L. & Muchatibaya G. Unsteady contact melting of a rectangular cross- section phase change material. Phys. Fluids , DOI: / Myers T.G. Optimizing the exponent in the Heat Balance and Refined Integral Methods. Int. Commun. Heat Mass Transf. 2008, DOI: /j.icheatmasstransfer