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