Page 172 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

TRANSIENT HEAT CONDUCTION ANALYSIS
164
show the temperature variation at the centre point of both the two- and three-dimensional
geometries with respect to time. It may be seen that both the results are identical. It should
be noted that the temperature increases rapidly and reaches a value of 200.4 at about four
seconds and thereafter remains constant.
6.7 Phase Change Problems—Solidification and Melting
Materials processing, metallurgy, purification of metals, growth of pure crystals from melts
and solutions, solidification of casting and ingots, welding, electroslag melting, zone melt-
ing, thermal energy storage using phase change materials, and so forth, involve melting
and solidification. These phase change processes are accompanied by either absorption or
release of thermal energy. A moving boundary exists, which separates the two thermo-
physical states in which the thermal energy is either absorbed or liberated. If we consider
the solidification of a casting, or ingot, the super heat in the melt and the latent heat liber-
ated at the solid–liquid interface are transferred across the solidified metal interface and the
mould, encountering at each of these stages a certain thermal barrier. In addition, the metal
shrinks as it solidifies and an air gap is formed between the metal and the mould. Thus,
additional thermal resistance is encountered. The heat transfer processes that occur are
complex. The cooling rates employed range from 10 −5 to 10 10 K/s and the corresponding
solidification systems extend from depths of several metres to a few micrometres. These
various cooling rates produce different microstructures and hence a variety of thermo-
mechanical properties. During the solidification of binary and multi-component alloys, the
physical phenomena become more complicated owing to phase transformation taking place
over a range of temperatures. During the solidification of an alloy, the concentrations vary
locally from the original mixture, as material may have been preferentially incorporated,
or rejected, at the solidification front. This process is called macro-segregation. The mate-
rial between the solidus and the liquidus temperatures is partly solid and partly liquid and
resembles a porous medium and is referred to as a mushy zone.
A complete understanding of the phase change phenomenon involves an analysis of
the various processes that accompany it. The most important of these processes, from a
macroscopic point of view, is the heat transfer process. This is complicated by the release,
or absorption, of the latent heat of fusion at the solid–liquid interface. Several methods have
been used to take into account the liberation of latent heat. The following subsections give
a brief account of commonly employed methods that deal with transient heat conduction
during a phase change.

6.7.1 The governing equations

The classical problem involves considering the conservation of energy in the domain,
,
by dividing this into two distinct domains,
l (liquid) and
s (solid), where
l +
s =
.
The energy conservation equation for the one-dimensional case is
2
∂T ∂ T
ρ l c p l = k l 2 in
l (6.56)
∂t ∂x