Python Software Foundation - SfePy : GSoC 2013 : Week 10 & 11

The final problem accounting convection-diffusion along with phase change has been defined below along with the governing equations and validation case studies.

Problem
Heat transfer in the processing of materials involving solid-liquid phase transformations (melting and solidification) is commonplace in such fields as metallurgy, crystal growth from melts and solutions, purification of materials, and solidification of metals. The associated density gradients in a gravitational field can induce natural convection flows. Convection in the liquid phase influences the process in two different ways, one of which is beneficial and the other of which can be detrimental. During melting convection increases the overall transport rate and, hence, the growth rate of the new phase, which is desirable. On the other hand, during solidification convection decreases the growth of the new phase and also seems to affect the morphology of the solid-liquid interface adversely. The nature of the solid is largely determined by what occurs in the vicinity of the solid-liquid interface. The heat release (absorption), density change, and other processes that take place in the vicinity of the transformation front result in nonuniformities along the front that cause its shape to change. The resulting density gradients in the liquid generate buoyancy-driven convection that can affect the transport of heat, constituent chemicals, and the growth rate.
The physical domain considered is shown in the figure. The vertical side walls of the enclosure are maintained at uniform temperatures, while the connecting horizontal walls are adiabatic. The govering equations are written for the entire domain assuming constant thermophysical properties, Boussinesq approximation, laminar, incompressible, and Newtonian two-dimensional flow. The solid-liquid interface motion due to volume change upon melting or solidification is neglected through the assumption .

Nomenclature

	Density	kg/m
	Viscosity	Pa.s
	Specific heat capacity	J/kg.K
	Thermal conductivity	W/m.K
	Thermal expansion coefficient	1/K
	Latent heat of fusion	J/kg
	Enthalpy	J/kg
	Volume fraction
	Temperature	C
	Time	s
	Velocity	m/s
	Pressure	Pa
	Source term	m/s

Subscript

	Liquid
	Solid
	Reference
	Effective
	Hot
	Cold
	Initial
	Melting

Governing Equations

The continuum relations:









1. Continuity

2. Momentum

Now the key lies in modelling the source term. The coefficient which should tend to 0 as the liquid volume fraction approaches unity, and should become a large negative number to anhilate the motion in the fluid region at . Whereas these asymtotic conditions can be satisfied by several functions, we adopt the suggestion of Brent et al. :

where

2. Energy
Writing a general equation for conservation of thermal energy for all the zones in the domain is facilitated by focusing on an element undergoing phase change. Below are the energy equation of solid and liquid phases under the thermal equilibrium condition :
Solid

Liquid

where and are the interphase energy terms, having the same magnitude but being opposite in sign.
A single governing enthalpy equation results:

where

The latent heat content of the element is due to the fraction of liquid converted to, or from, the corresponding quantity of solid. Hence we write

where L is the latent heat of fusion.
The zones where , the entire element is in the liquid state and .
The zones where , the entire element is in the liquid state and .
It is the elements undergoing phase change at , where varies between 0 and 1. Substituting and in equation we get

where

Initial Conditions
everywhere
everywhere
everywhere.


Boundary Conditions
Left wall
Right wall
Top
Bottom
Left wall
Right wall
Walls


Validation Cases
Melting Gallium : Gau and Viskanta
Melting Gallium : Brent et al.
Melting Calcium chloride : Zivkovic and Fujii



References

1. Zivkovic, B., Fujii, I., 2001. “An Analysis of Isothermal Phase Change of Phase Change Material within Rectangular and Cylindrical Containers”. Solar Energy, 70, pp. 51-61.
2. Brent, A.D., Voller, V.R., Reid, K.J., 1988. “Enthalpy-porosity Technique for Modeling Convection-diffusion Phase Change: Application to the Melting of a Pure Metal”. Numerical Heat Transfer, 13(3), pp. 297-318.
3. Gau, C., Viskanta, R., 1986. “Melting and Solidification of a Pure Metal on a Vertical Wall”. Journal of Heat Transfer, 108(1), pp. 174-181.
4. Rajeev, K., Das, S., 2010. “A Numerical Study for Inward Solidification of a Liquid Contained in Cylindrical and Spherical Vessel”. Thermal Science, 14(2), pp. 365-372.
5. Vreeman, C. J., Krane, M. J. M., Incropera, F. P. , 2000. “The Effect of Free-Floating Dendrites and Convection on Macrosegregation in Direct Chill Cast Aluminum Alloys. Part 1: Model Development”. Int. J. Heat Mass Transfer, 43, pp. 677-686.
6. Flemings, M. C., 1974. Solidification Processing. McGraw-Hill, New York.
7. Kumar, A., Walker, M. J., Sundarraj, S., Dutta, P., 2011. “Grain Floatation During Equiaxed Solidification of an Al-Cu Alloy in a Side-Cooled Cavity: Part II-Numerical Studies”. Metallurgical and Materials Transactions, 42(B), pp. 783-799.
8. Voller, V.R., 2006. Handbook of Numerical Heat Transfer, 2nd ed.. Wiley, New York, NY, pp. 593-622.