unsteady contact melting
DESCRIPTION
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 …. - PowerPoint PPT PresentationTRANSCRIPT
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Unsteady contact meltingTim G. Myers
University of Cape Town
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Contact melting configuration
Water droplet floatingabove hot steel: Leidenfrost effect
Applications: thermal storage, processmetallurgy, geology, nuclear technology,Leidenfrost, ice skating …
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Three stages of melting for block with insulated sides and top surface
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Navier-Stokes equation and incompressibility condition
Governing equationsHeat equations in liquid and solid
Mass balance
Stefan condition
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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
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Non-dimensionalisation
Navier-Stokes equation and incompressibility condition
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Governing equations
Boundary conditions
Thermal problem Stage 1 Stage 2
Similarly
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Heat Balance Integral MethodClassic heat flow problem …
Heat balance formulation – replace BC at infinity
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Heat Balance Integral
Optimal n method
Where n = 2.233
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Classical Stefan problem
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Neumann’s solution
Stefan condition
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HBIM solution
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Integrate heat equation …
Couple to Stefan condition …
i.e. two equations for two unknowns;before melting have single first order ODE to solve
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Stage 1: pre-meltingExact solution
HBIM solution
Three stages of melting for block with insulated sides and top surface
Application to contact melting
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Temperature at end of Stage 1
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Stage 2: Melting
HBIMStefan condition
Stage 3: More melting
Etc. etc.
where (from lubrication solution)
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Force balance
Standard quasi-steady analysis
leads to without squeeze(Neumann solution)
Temperatureprofile
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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
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Temperature in solid and liquid half-way through melting process
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Maximum value of neglected terms for HTC of 855 and 5000
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Comparison of solid thickness with experiments onN-octadecane, current method (solid), current with infinite HTC (dotted) and Moallemi et al (1986)theory (dash-dot)
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Leidenfrost effect Now must calculate shape of droplet as well
Young-Laplace equation
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Constant volume droplet
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Unsteady calculation
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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.