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:
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
Similarly
Governing equations
Boundary conditions
Thermal problem
Stage 1
Stage 2
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
Application to contact melting
Three stages of melting for block with insulated sides and top surface
Stage 1: pre-melting
Exact solution
HBIM solution
Temperature at end of Stage 1
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
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. 2008.
2. Myers T.G., Mitchell S.L. & Muchatibaya G. Unsteady contact melting of a rectangular crosssection 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.