Melting and Solidification

The melting-solidification model does not track the liquid-solid interface explicitly. Instead, the model uses an enthalpy formulation to determine the distribution of the solidified portion of the liquid-solid film.

The enthalpy of the liquid-solid film $h_{ls} *$ includes the latent heat of fusion $h_{fusion}$ :

Figure 1. EQUATION_DISPLAY

h_{ls} * = h_{ls} + (1 - Y_{s}^{*}) h_{f u s i o n}

(2827)

where $h_{ls}$ is the sensible enthalpy. The relative solid mass fraction $Y_{s}^{*}$ is defined as the portion of the mass of the liquid-solid film which the solid state occupies. In the enthalpy model, the relative solid mass fraction $Y_{s}^{*}$ is a function of temperature:

Figure 2. EQUATION_DISPLAY

Y_{s}^{*} = {\begin{matrix} 1 & i f & T *< 0 \\ f (T *) & i f & 0 < T *< 1 \\ 0 & i f & 1 < T * \end{matrix}

(2828)

where $T *$ is the normalized temperature that is defined as:

Figure 3. EQUATION_DISPLAY

T * = \frac{T - T_{solidus}}{T_{liquidus} - T_{solidus}}

(2829)

The function $f (T *)$ is called the fraction solid curve. For a linear dependence between $Y_{s}^{*}$ and $T *$ , the solidification path is defined as:

Figure 4. EQUATION_DISPLAY

f (T *) = 1 - T *

(2830)

If melting and solidification take place at one temperature (that is, $T_{solidus} = T_{liquidus}$ ), a linear solidification path is assumed and a small temperature interval of 0.002 K is automatically introduced.

Solidified Mass Removal and Morphing

Simcenter STAR-CCM+ provides an option to remove the solidified mass from the film. To accurately represent the geometry in combination with the solidified layer that forms on it, you use morphing motion to adjust the surface of the solidified volume when the solidified mass is removed. The removed solid mass determines the distance the morpher displaces the boundary to model the thickness of the solidified layer. This option is typically used to model icing.

During every time-step, a solid film thickness increment $Δ h_{s}$ is computed, updated at every inner iteration during the segregated process. This value represents the increase in solidified thickness during the time-step. $Δ h_{s}$ is then used to construct the sources and sinks into the various transport equations.

The updates (at every inner iteration) are computed as:

Figure 5. EQUATION_DISPLAY

{Δ h}_{s}' = {\begin{matrix} - \frac{Δ H (T) - Δ H_{liq}}{Δ H_{fus}} (h + {Δ h}_{s}) & i f & T \geq T_{liq} \\ h \cdot α_{s} & if & T < T_{liq} \end{matrix}

(2831)

with $Δ h_{s}'$ the correction to $Δ h_{s}$ . This expression implies that (within each time-step) the amount of solidified mass increases when the temperature is below the liquidus temperature, but is reduced if the temperature is above the liquidus temperature. However, the present implementation has a lower limit of $Δ h_{s} \geq 0$ . Therefore, this feature only allows mass to solidify; it does not allow melting. Once the solidified mass has been removed, it leaves the system and cannot be retrieved.

The solid film thickness increment can be related to a mass sink $\frac{k g}{m^{2} s}$ :

Figure 6. EQUATION_DISPLAY

\dot{m} = \frac{ρ_{l i q} Δ h_{s}}{Δ t}

(2832)

which is used as a base for corresponding sources in the continuity, momentum, energy, and species transport equations in the film.

For each time-step, the solidification is computed as follows:

A mass sink is applied to the liquid film, corresponding to how much liquid is turning into solid mass.
This mass sink is used to calculate the velocity of the solid front (that is, the velocity that the interface between solid and liquid is moving), also taking into account the density of the solid mass.
This velocity is used to calculate the solid thickness increment $Δ h_{s}$ for that time-step; simply $Δ h_{s} = V \cdot d t$ .

Typically, this solidified mass accumulates on the wall: it remains in place (it does not flow any further) and deforms the surface as seen by the gas and the remaining film. This effect is modeled by feeding the displacement $Δ h_{s}$ into the morpher.

For each time-step, the displacement of every face is:

Figure 7. EQUATION_DISPLAY

d_{face} = Δ h_{s} a

(2833)

where $a$ is a unit vector, normal to the film surface and pointing into the gas (that is, the growth is assumed to happen in the direction normal to the surface). The displacement $d_{face}$ is then interpolated to displacements $d_{vert}$ on the vertices.

Simcenter STAR-CCM+ scales the solid front velocity by a factor $K$ as $V^{*} = K \cdot V$ . Consequently, within each time step, the solid thickness increment is increased by a factor $K$ . For example, each 1kg/s of solidifying liquid is turned into $K$ kg/s of solid. This can be considered as a mass source, although there are no transport equations solved for the solid: the solid is removed from the simulation and its effect is included only through the morphing of the boundary. The total physical time should be reduced by a factor $K$ to get the same physical result.

When the vertices are displaced according to the algorithm that is described above, the displacement can lead to roughness in the deformation, forming a zig-zag pattern. This roughness can cause instability issues in the fluid film transport.

To resolve this issue, an optional smoother is available. The smoother recomputes the displacement as a weighted average, including the displacements at the nearest neighbors. This smoothing can be repeated multiple times.