Page 173 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

TRANSIENT HEAT CONDUCTION ANALYSIS
where the subscript l denotes the liquid. Note that in the above equation, the convective
motion is neglected. For details of convection, the reader is referred to Chapter 7. Similarly,
the equation for the solid portion is written as 165
2
∂T ∂ T
ρ s c p s = k s 2 in
s (6.57)
∂t ∂x
where the subscript s represents the solid. The problem will be complete only if the initial
and boundary conditions and the interface conditions are given. The interface conditions are
T sl = T f (6.58)
and
∂T ds ∂T
−k s = ρ s L − k l on sl (6.59)
∂x s dt ∂x l
where sl represents the position of the interface, ds/dt represents the interface velocity
and T f is the phase change temperature. Equation 6.59 states that the heat transferred by
conduction in the solidified portion is equal to the heat entering the interface by latent heat
of liberation at the interface and the heat coming from the liquid by conduction. The main
complication in solving this classical problem lies in tracking the interface and applying
the interface conditions.

6.7.2 Enthalpy formulation

In the enthalpy method, one single equation is used to solve both the solid and liquid
domains of the problem. A single energy conservation equation is written for the whole
domain as
2
∂H ∂ T
= k in
(6.60)
∂t ∂x 2
where H is the enthalpy function, or the total heat content, which is defined for an isother-
mal phase change as
T
H(T ) = ρc s (T )dT if (T ≤ T f )
T r
T
T f
H(T ) = ρc s (T )dT + ρL + ρc l (T )dT if (T ≥ T l ) (6.61)
T r T f
and, for a phase change over an interval of temperature T s to T l , that is, the solidus and
the liquidus temperatures respectively, we have the following:

T
T s dL
H(T ) = ρc s (T )dT + ρ + ρc f (T ) dT (T s <T ≤ T l )
dT
T r T s
T s T l
T
H(T ) = ρc s (T )dT + ρL + ρc f (T )dT + ρc l (T )dT (T ≥ T l ) (6.62)
T r T s T l