Flow Resistance due to Solidification

The melting and solidification model simulates only the effect of phase change on the energy equation; it does not modify the momentum equations. A further model is required to account for the change in material properties from the liquid state to the solid state.

Slurry Viscosity Model

In the early stages of solidification, small crystals nucleate and grow in the melt. These crystals move with the liquid but do not interact with each other. The mixture of the solid and liquid phases is called a slurry. However, the presence of the solid crystals in the liquid increases the viscosity of the two-phase mixture, which increases the flow resistance. This increased resistance is simulated using the Slurry Viscosity model.

The Slurry Viscosity model is used for low values of the solid volume fraction, ${\alpha }_s^{*}\le {\alpha }_{\mathrm{cr}}^{*}$ , where the critical solid fraction ${\alpha }_{cr}$ defines the upper limit of applicability.

The method of Metzner ([594]) expresses the variation in viscosity of the mixture as a function of the solid volume fraction ${\alpha }_s^{*}$ :

$$\mu \text{*}={\mu }_l\left[{(1-\frac{F_{\mu }{\alpha }_s^{*}}{A})}^{-2}-1\right]$$

(2713)

where:

• ${\mu }^{*}$ is the effective viscosity of the solid-liquid mixture
• ${\mu }_l$ is the dynamic viscosity of the liquid
• $F_{\mu }$ is a switching function that is given by Eqn. (2719)
• $A$ is a crystal constant which depends on the aspect ratio and surface roughness of the crystal.

Mushy Zone Permeability Model

For high solid volume fractions, equiaxed grains grow and start to agglomerate and to form dendritic regions. Dendrites also grow from cooled solid surfaces or the solidification front into the melt. A partially solidified stationary region permeated with dendrites is called a “mushy zone”.

A common engineering application for this model is high-pressure die casting where melt is pressed through ducts with wall temperatures below solidus temperature at high velocities. The melt solidifies at the walls and dendrites grow from the solidification front into the melt. This partially solidified region causes an additional pressure loss in the duct that influences the melt mass flow. This pressure loss is simulated using the Mushy Zone Permeability model.

The Carman-Kozeny Mushy Zone Permeability method provides a mushy zone for the flow resistance, eliminating the need to set up porous media. In addition, the method updates the velocity field in the melting and solidification process. The flow resistance is active only in cells that have a solid volume fraction greater than zero and is applied to all regions in the physics continuum.

The flow resistance in a mushy zone can be modeled similar to an isotropic porous medium. Therefore, to quantify the permeability, the Carman-Kozeny equation [583] for flow through a porous medium is used:

$$K=\frac{{\left(1-{\alpha }_s^{\text{*}}\right)}^3}{{\alpha }_s^{\text{*2}}{F}_K{c}_s}$$

(2714)

where:

• $K\left[m^2\right]$ is the permeability
• ${\alpha \text{*}}_s$ is the relative solid volume fraction
• $F_K$ is a non-dimensional switching function that is given by Eqn. (2720).
• $c_s$ is the shape factor:

$$c_s=c/d^2$$

(2715)

where $c$ is a shape constant with a default value of 180 and $d$ is the secondary dendrite arm spacing.

The contribution of the mushy zone permeability to the momentum equation Eqn. (2588) enters the porous medium resistance force ${\mathbf{\text{f}}}_p$:

$${\mathbf{\text{f}}}_p=-{\mathbf{\text{P}}}_{\text{v, Mushy Zone}}\cdot \mathbf{\text{v}}$$

(2716)

where $\mathbf{\text{P}}$ is the porous resistance tensor, which is given by:

$${\mathbf{\text{P}}}_{\mathrm{MushyZone}}=(\frac{{\mu }_l}{K}+\frac{C_F{\rho }_l{\alpha }_l^{*}}{\sqrt{K}}\Vert \mathbf{\text{v}}\Vert )$$

(2717)

${\mathbf{\text{P}}}_{\text{v}}$ and ${\mathbf{\text{P}}}_{\text{i}}$ are the viscous (linear) and the inertial (quadratic) resistance tensors, respectively.

The mushy zone permeability contributes to ${\mathbf{\text{P}}}_{\text{v}}$ through:

$${\mathbf{\text{P}}}_{\text{v, Mushy Zone}}=(\frac{{\mu }_l}{K}+\frac{C_F{\rho }_l{{\alpha }^{*}}_l}{\sqrt{K}}\Vert \mathbf{\text{v}}\Vert )$$

(2718)

where:

• $\frac{{\mu }_l}{K}$ is Darcy’s term
• $\frac{C_F{\rho }_l{{\alpha }^{*}}_l}{\sqrt{K}}\Vert \mathbf{\text{v}}\Vert$ is Forchheimer’s term

where:

• $C_F$ is the Ergun coefficient
• ${\rho }_l$ is the liquid phase density
• ${\alpha }_l^{*}$ is the liquid fraction that is given by ${\alpha }_l^{*}=\left(1-{\alpha }_s^{*}\right)$.

${\mathbf{\text{P}}}_{\text{v,MushyZone}}$ can be considered a drag coefficient. The drag coefficient is given by the Darcy’s and Forchheimer’s terms, which are often referred to as the “viscous-drag” and “form-drag” terms.

The Forchheimer’s term is optional and its contribution to the mushy zone permeability can be deactivated. The Forchheimer’s term must be considered when relatively high interdendritic fluid-flow velocities are expected. Such high velocities can be important during the intensification phase in die or squeeze casting.

Switching Function for Metzner Slurry Viscosity and Carman-Kozeny Mushy Zone Permeability Model

The Metzner slurry viscosity model and the Carman-Kozeny mushy zone permeability model are typically used in combination. Both models cover different ranges of solidification state. The applicability of each model depends on the extent to which the liquid has solidified.

The state of solidification is expressed using the relative solid volume fraction ${\alpha \text{*}}_s$. The critical relative solid fraction ${\alpha }_{\mathrm{cr}}^{*}$ separates the applicability ranges of the two models.

${\alpha }_s^{*}\le {\alpha }_{\mathrm{cr}}^{*}$	Metzner Slurry Viscosity Model
${\alpha }_{\mathrm{cr}}^{*}<{\alpha }_s^{*}$	Carman-Kozeny Mushy Zone Permeability Model

For the Metzner slurry viscosity model and the Carman-Kozeny mushy zone model, a switching function ensures that each model is applied within its appropriate range of solid volume fraction. The switching function is applied as a multiplicative factor to the respective model equation (see Eqn. (2713) and Eqn. (2714)).

The switching function for the slurry viscosity model is:

$$F_{\mu }\left({\alpha \text{*}}_s\right)=0.5-\frac{\text{arctan}\left[s\left({\alpha \text{*}}_s-{\alpha \text{*}}_{\mathrm{cr}}\right)\right]}{\pi }$$

(2719)

where $s$ is a switching function constant.

The switching function for the mushy zone permeability model is given by:

$$F_K\left({\alpha \text{*}}_s\right)=1-F_{\mu }\left({\alpha \text{*}}_s\right)=0.5+\frac{\text{arctan}\left[s\left({\alpha \text{*}}_s-{\alpha \text{*}}_{\mathrm{cr}}\right)\right]}{\pi }$$

(2720)

The switching function has a smooth transition near the critical relative solid volume fraction ${\alpha \text{*}}_{cr}$(see the figure below) where solid grains start to agglomerate to a significant extent:

The default value of the non-dimensional constant $s$ is 100 because the shape is rather insensitive for values from 80 through 120. The default value for the critical solid volume fraction ${\alpha \text{*}}_{cr}$ is 0.27 according to Baeckerud and others [579] for Aluminum alloy A201.

Oldenburger and Spera [600] initially suggested this switching function. The switching function that is implemented in Simcenter STAR-CCM+ is a slightly modified formulation that is based on the model of Chang and Stefanescu [585].

The following figure depicts the switching function $F_K$ for critical solid volume fraction ${\alpha }_{cr}^{\text{*}}=0.27$ (default value).