Unsteady contact meltingTim G. Myers

University of Cape Town

Contact melting configuration

Water droplet floatingabove hot steel: Leidenfrost effect

Applications: thermal storage, processmetallurgy, 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 equationsHeat 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 MethodClassic 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-meltingExact 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

HBIMStefan condition

Stage 3: More melting

Etc. etc.

where (from lubrication solution)

Force balance

Standard quasi-steady analysis

leads to without squeeze(Neumann solution)

Temperatureprofile

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

Evolution of liquid height for currentmodel 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 onN-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 models1. Modelling temperature in solid (using HBIM)2. Cooling condition at substrate3. Varying solid mass4. 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. 2008.2. 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.

3. 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.