Computer Vision Analysis of a Melting Interface Problem with Natural Convection - Visual Computing: Scientific Visualization and Imaging Systems - page 209

12.2 The Phase Change Problem

This study deals with melting of a vertical ice slab upon a gravitational field in a
rectangular enclosure due to a horizontal thermal gradient. The process is driven
by thermally induced natural convection in the liquid phase.

12.2.1 Mathematical Model

The physical problem at time t * = 0 is shown in Fig. 12.1 . The asterisk superscript
(*) is used to indicate the dimensional variables. In the initial condition considered
half of the material volume is in the solid state, while the other half is in the liquid
state. Initially, all the volume of the testing substance is set at its fusion temper-
ature, i.e., T 0 = T Fus . The melting process begins when the temperature of one of
the vertical walls of the rectangular cavity, represented by T H , is increased. H is the
cavity's height and L is the width. The right vertical wall is kept isothermal at T 0
and the horizontal walls are adiabatic. The entire process is controlled by the
natural convection in the liquid phase. The position of the interface at time t* and
level z* is defined by its distance from de hot wall c ð z ; t Þ .
The hypotheses below were assumed in order to formulate the equations that
manage the problem:

• The flow is laminar and two-dimensional.
• The liquid material is Newtonian and incompressible.
• The fluid's physical properties are constant, except for density in the buoyancy
force term.
• The viscous dissipation is negligible.
• The density change of the material upon melting is neglected.
• It is assumed that the velocity of propagation of the melting front is several
orders of magnitude smaller than the fluid velocities in the boundary layers on
the vertical walls. This suggests that it is possible to divide the process in a
number of quasi-static steps, separating, therefore, the melting front motion
calculations from the natural convective calculations.

The coordinate system adopted, the time and the melting front were made
dimensionless in the following way:

y ¼ y = H; z ¼ z = H

ð 12 : 1 Þ

t ¼ t t = H 2 ; c ð z ; t Þ ¼c ð z ; t Þ= H ð 12 : 2 Þ

Dirichlet thermal boundary conditions are taken on the vertical wall and at the
interface, and the horizontal walls are adiabatic. In the liquid cavity, zero velocity
dynamic boundary conditions are considered at the four walls.