Dimensionless Niyama Criterion

You can use the dimensionless Niyama criterion to directly predict the amount of shrinkage porosity during solidification of metal alloy castings.

This criterion does not require any threshold value below which shrinkage porosity forms. You can use this criterion for pure thermal simulations without computing pressure or velocity fields. The criterion is computed for all Critical Temperatures. The implementation is based on [134].

Solid Fraction Curve Integral

First, the fraction solid curve is integrated:

Figure 1. EQUATION_DISPLAY

I (α_{l, c r}^{*}) = \int_{α_{l, c r}^{*}}^{1} 180 \cdot \frac{{(1 - α_{l}^{*})}^{2}}{α_{l}^{* 2}} \cdot \frac{\partial}{\partial α_{l}^{*}} (T^{*}) \cdot d α_{l}^{*}

(368)

where $α_{l}^{*}$ is the relative liquid volume fraction. $α_{s}^{*}$ is computed from the relative solid volume fraction $α_{s}^{*}$ as follows:

$α_{l}^{*} = 1 - α_{s}^{*}$

$α_{l, c r}^{*}$ is the critical liquid volume fraction at which the melt pressure decreases to the critical pressure value and porosity begins to form. $T^{*}$ is the normalized temperature and is defined as:

Figure 2. EQUATION_DISPLAY

T^{*} = \frac{T - T_{s o l i d u s}}{T_{l i q u i d u s} - T_{s o l i d u s}}

(369)

The integral in Eqn. (368) is evaluated numerically using at least the user-specified Min. Number of Integration Points (see Dimensionless Niyama Criterion Properties). The integral is calculated only once when the Dimensionless Niyama model is activated or when the Fraction Solid Curve material property changes.

If the fraction solid curve is specified as a function of pressure, a pressure of 1 atm is applied for its evaluation. Simcenter STAR-CCM+ assumes that the fraction solid curve is measured at this pressure.

Computation of the Dimensionless Niyama Criterion

At the end of every time step, the value of the Dimensionless Niyama criterion is computed as:

Figure 3. EQUATION_DISPLAY

{N y}^{*} = λ_{2} N y \sqrt{\frac{Δ P_{c r}}{μ_{l i q} β Δ T_{f}}}

(370)

where:


$Δ P_{c r}$	Critical pressure drop
$μ_{l i q}$	Dynamic viscosity at liquidus temperature
$β$	Solidification contraction
$Δ T_{f} = T_{l i q} - T_{s o l}$	Freezing temperature range
$λ_{2}$	Secondary Dendrite Arm Spacing from Eqn. (377)

The critical pressure drop is defined as:

Figure 4. EQUATION_DISPLAY

Δ P_{c r} = P_{l i q} - P_{c r}

(371)

where $P_{l i q}$ is the pressure at liquidus temperature. $P_{l i q}$ is the sum of the ambient pressure of the system and the local head pressure. $P_{c r}$ is the critical pressure and is calculated by considering the mechanical equilibrium that is necessary for a stable pore to exist. Here, $P_{c r}$ is an adjustable parameter.

The solidification contraction is defined as:

Figure 5. EQUATION_DISPLAY

β = \frac{ρ_{s o l} - ρ_{l i q}}{ρ_{l i q}}

(372)

The liquid and solid densities, $ρ_{l i q}$ and $ρ_{s o l}$ , are considered constant during solidification.

The original formulation of the Dimensionless Niyama criterion [134] uses the secondary dendrite arm spacing computed from the cooling rate (see Secondary Dendrite Arm Spacing (CR)).

Computation of the Solid Fraction Curve Integral Target Value

The target value of the solid fraction integral is given by:

Figure 6. EQUATION_DISPLAY

I (α_{l, c r}^{*}) = {N y}^{*}^{2}

(373)

Computation of the Pore Volume Fraction

Knowing the target value of the solid fraction integral, the critical liquid fraction ${α^{*}}_{l, c r}$ can be calculated. $α_{l, c r}^{*}$ is the integration limit of Eqn. (368).

Computation of the Absolute Pore Volume Fraction

The pore volume fraction is:

Figure 7. EQUATION_DISPLAY

α_{p} = β′ α_{l, c r}^{*} = \frac{β}{β + 1} α_{l, c r}^{*} = \frac{ρ_{s o l} - ρ_{l i q}}{ρ_{s o l}} α_{l, c r}^{*}

(374)

Dimensionless Niyama Criterion Properties


Dynamic Viscosity @ T_liquidus	The dynamic viscosity at liquidus temperature for the melt material.
Critical Pressure Drop	Defined as the difference between the pressure at liquidus temperature and the critical pressure (see Eqn. (371)).
Solidification Contraction	$β$ in Eqn. (372).
Min. Number of Integration Points	The minimum number of integration points used to numerically integrate Eqn. (368).