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Unsteady contact melting. Tim G. Myers University of Cape Town. Water droplet floating a bove hot steel: Leidenfrost effect. Contact melting configuration. Applications: thermal storage, process metallurgy, geology, nuclear technology, Leidenfrost , ice skating ….
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Unsteady contact melting Tim G. Myers University of Cape Town
Water droplet floating above hot steel: Leidenfrost effect Contact melting configuration Applications: thermal storage, process metallurgy, geology, nuclear technology, Leidenfrost, ice skating …
Three stages of melting for block with insulated sides and top surface
Governing equations Heat equations in liquid and solid Navier-Stokes equation and incompressibility condition Mass balance Stefan condition
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
Non-dimensionalisation Navier-Stokes equation and incompressibility condition
Similarly Governing equations Boundary conditions Thermal problem Stage 2 Stage 1
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
Neumann’s solution Stefan condition
Integrate heat equation … Couple to Stefan condition … i.e. two equations for two unknowns; before melting have single first order ODE to solve
Application to contact melting Three stages of melting for block with insulated sides and top surface Stage 1: pre-melting Exact solution HBIM solution
Stage 2: Melting HBIM Stefan condition where (from lubrication solution) Stage 3: More melting Etc. etc.
Force balance Standard quasi-steady analysis leads to without 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
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
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.