Laplace transform isotherm .pdf - University of Hertfordshire ...

and
(uc − uf)
u2 = uc − �
erfc
µ (α1/α2) 1
2
�erfc
�
x
2(α2t) 1
2
where µ is a constant to be determined, uf is the solidifying point of the
material and α1 and α2 are the thermal constants associated with the solid
and liquid phases respectively.
Lightfoot (1929) used an integral method in which he assumed that the
thermal properties of the solid and liquid were the same. He considered the
surface of solidification which was moving and liberating heat and this led
to an integral equation for the temperature.
2.3.2 The heat-balance integral method
Goodman (1958) integrated the one-dimensional heat flow equation with re-
spect to x and inserted boundary conditions to produce an integral equation
which expressed the overall heat balance of the system. Goodman says that
although the solution was approximate it provided good accuracy and the
problem was reduced from that of solving a partial differential equation to
one of solving an ordinary differential equation. Poots (1962) extended the
heat-balance integral method to study the movement of a two-dimensional
solidification front in a liquid contained in a uniform prism.
2.3.3 Front tracking methods
These are methods which compute the position of the moving boundary at
each step in time. If we use a fixed grid in space-time, then in general,
the position of the moving boundary will fall between two grid points. To
resolve this, we either have to use special formulae which allow for unequal
space intervals or we have to deform the grid in some way so that the moving
boundary is always on a gridline. Several numerical solutions based on the
finite difference method have been proposed. Their approach to the grid
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