Investment casting produces complex, thin-walled components with high dimensional accuracy and good surface finish, but the process chain begins with a wax pattern whose dimensions are transferred, with accumulated deviation, into the ceramic shell and finally into the metal casting. The filling stage is therefore not a minor preparatory step. It determines whether the die cavity is completely filled, whether the frozen skin is smooth, and whether the pattern contains entrapped gas, flow marks, sinks, or weld lines.
Wax injection is difficult to optimize by trial and error alone. Molten or softened wax is a shear-thinning, temperature-sensitive polymer-like material. Its viscosity changes by orders of magnitude across the filling path as shear rate and temperature vary, and the moving free surface interacts with a cold die wall. A useful simulation must therefore couple three-dimensional non-isothermal flow, a realistic rheological law, and free-surface tracking. Simplified models that assign a constant viscosity or ignore heat transfer can reproduce the general filling direction, but they are unreliable for thin sections, low injection pressures, and defect-sensitive geometries.
Rheology Sets the Structure of the Filling Model
Wax rheology is well represented by a Cross-WLF constitutive equation. The model combines a low-shear Newtonian plateau with a shear-thinning regime and includes temperature dependence; pressure dependence is retained in the general form but becomes inactive when the fitted pressure coefficient is zero. For the waxes discussed here, the viscosity law can be written as
$$
\eta(\dot{\gamma},T)=\frac{\eta_0(T)}{1+\left(\eta_0(T)\dot{\gamma}/\tau^{*}\right)^{1-n}},
\qquad
\eta_0(T)=D_1\exp\left[-\frac{A_1(T-T^{*})}{A_2+(T-T^{*})}\right],
$$
where \(\eta\) is the apparent viscosity, \(\dot{\gamma}\) is the shear rate, \(T\) is temperature, \(n\) is the power-law index, \(\tau^{*}\) is the critical stress marking the transition to shear thinning, and \(D_1\), \(A_1\), \(A_2\), and \(T^{*}\) are fitted constants. With \(D_3=0\), the reference temperature reduces to \(T^{*}=D_2\). This form is preferable to a simple power law because it captures the low-shear Newtonian region near the gate and the high-shear region near the advancing front. A second-order power law introduces unnecessary fitting constants and can produce nonphysical viscosity growth, while the Carreau model describes shear thinning but omits pressure effects.
| Parameter | 3-3-3-1 wax | K512 wax | Unit |
|---|---|---|---|
| \(n\) | 0.4213 | 0.3691 | – |
| \(\tau^{*}\) | 232.645 | 0.427887 | Pa |
| \(D_1\) | \(1.0\times10^{37}\) | \(2.18\times10^{36}\) | Pa·s |
| \(D_2\) | 263.15 | 263.15 | K |
| \(D_3\) | 0 | 0 | K/Pa |
| \(A_1\) | 150.37 | 123.92 | – |
| \(A_2\) | 51.6 | 51.6 | K |
Governing Equations and the Projection-VOF Strategy
The filling model treats the wax as an incompressible, isotropic, laminar fluid, assumes no slip at the die wall, and neglects inertial and gravitational terms relative to viscous resistance. The last assumption is an approximation: it is reasonable for the high-viscosity, low-Reynolds-number flows typical of wax injection, but it should be re-examined when injection rates are pushed upward. The die is also assumed to vent well, so the air phase is not explicitly tracked in the cavity.
With these assumptions, mass and momentum conservation reduce to
$$
\nabla\cdot\mathbf{U}=0,
\qquad
\frac{\partial\mathbf{U}}{\partial t}
=-\frac{1}{\rho}\nabla p+\nu\nabla^{2}\mathbf{U},
$$
where \(\mathbf{U}\) is the velocity vector, \(p\) is pressure, \(\rho\) is density, and \(\nu=\mu/\rho\) is the kinematic viscosity. The energy equation supplies the temperature field that feeds back into viscosity:
$$
\rho c_p\left(\frac{\partial T}{\partial t}+\mathbf{U}\cdot\nabla T\right)
=\lambda\nabla^{2}T,
$$
where \(c_p\) is specific heat and \(\lambda\) is thermal conductivity. In the numerical implementation, shear rate and temperature at time step \(n\) are used to evaluate the Cross-WLF viscosity, which is then lagged by one step when the viscous term is advanced. This explicit coupling is stable under the small time steps required by the filling Courant condition.
The pressure-velocity coupling uses a projection method rather than the iterative SOLA procedure commonly used in casting simulation. The momentum equation is first advanced without the pressure gradient to obtain an intermediate velocity,
$$
\mathbf{U}^{*}=\mathbf{U}^{n}+\Delta t\,\nu\nabla^{2}\mathbf{U}^{n}.
$$
Enforcing \(\nabla\cdot\mathbf{U}^{n+1}=0\) then gives a pressure Poisson equation,
$$
\nabla^{2}p^{n+1}=\frac{\rho}{\Delta t}\nabla\cdot\mathbf{U}^{*},
$$
and the velocity is corrected by
$$
\mathbf{U}^{n+1}=\mathbf{U}^{*}-\frac{\Delta t}{\rho}\nabla p^{n+1}.
$$
This split avoids repeated pressure-velocity iteration and is well suited to parallel computation. The free surface is tracked with a volume-of-fluid function \(F\),
$$
\frac{\partial F}{\partial t}+\nabla\cdot(F\mathbf{U})=0,
$$
where \(F=1\) denotes a cell filled with wax, \(F=0\) denotes an empty cell, and \(0<F<1\) marks an interfacial cell. The VOF approach preserves mass more strictly than interface-only level-set tracking, which is advantageous for the relatively smooth, viscosity-dominated fronts encountered in wax filling.
The equations are discretized on a uniform finite-difference grid. The time step is limited by the Courant condition,
$$
\Delta t<\min\left(\frac{\Delta x}{|u|_{\max}},\frac{\Delta y}{|v|_{\max}},\frac{\Delta z}{|w|_{\max}}\right),
$$
and is usually set to one quarter to one third of the critical value. Inlet boundaries may prescribe velocity or pressure; walls use no-slip velocity conditions and a convective thermal boundary; outlet boundaries impose zero normal velocity gradient and a static pressure. The solver is written in C++ and parallelized with OpenMP, with the three-dimensional cell loops distributed across shared-memory threads.
Validation on a Trigger-Shaped Wax Pattern
A trigger-shaped cavity component provides a demanding validation geometry: it is 270 mm × 60 mm × 45 mm, with concave, curved, inclined, and bifurcated features. The gate is a 15 mm long cylinder with an 8 mm diameter, oriented along the gravity direction. The pattern was meshed at a minimum cell size of 1.2 mm, giving 1,092,000 cells in a 70 × 65 × 240 grid. The wax was the 3-3-3-1 blend, composed of 30% paraffin, 30% stearic acid, 30% rosin, and 10% ozokerite.
| Quantity | Value |
|---|---|
| Pattern envelope | 270 mm × 60 mm × 45 mm |
| Gate | 8 mm diameter × 15 mm length |
| Mesh | 1.2 mm minimum cell size; 1,092,000 cells |
| Filling time | 1.2 s |
| Inlet velocity | 4.55 m/s |
| Injection temperature | 48 °C |
| Initial die and air temperature | 20 °C |
| Wax density | 1042 kg/m³ |
| Wax specific heat | 5000 J/(kg·K) |
| Wax thermal conductivity | 0.25 W/(m·K) |
| Wax-die heat transfer coefficient | 500 W/(m²·K) |
| Wax-air heat transfer coefficient | 20 W/(m²·K) |
Two simulations were compared. In the first, the viscosity was fixed at 3.28 Pa·s, corresponding to the average gate shear rate of about 1755 s⁻¹ and the injection temperature; using the gate diameter as the length scale, the Reynolds number was 11.56, so the flow remained nominally laminar. In the second, the Cross-WLF model updated viscosity locally from the computed shear rate and temperature.
The constant-viscosity case lost the strong viscosity contrast that stabilizes real wax filling. It predicted a less stable front, localized splashing-like patterns, and recirculation near the gate, and the filling pattern resembled a low-viscosity metal casting more than a wax pattern. The variable-viscosity case produced a smooth, laminar front. The wax first filled the raised boss around the gate, then split into two branches. The left branch, being closer to the gate, approached completion at about 0.75 s, while the front branch completed last at about 1.20 s. The experimental filling sequence and front shapes agreed with this prediction, although the simulated fill fraction was slightly ahead of the experiment at comparable stages; part of this offset is attributable to representing the experimental sprue as a simple cylinder.
The computed temperature field showed a maximum temperature decrease of only 0.85 °C during filling. This small drop reflects the low thermal conductivity and short filling time of the wax. The coolest regions were the concave sides of the raised boss, where boundary-layer wax contacted the die wall, lost heat, increased in viscosity, and became less mobile. The advancing front remained hotter because it contacted the wall over a limited area and because heat conduction within the wax is slow.
The pressure field developed smooth, nearly parallel bands from the gate to the front, with pressure approaching zero at the advancing surface. Completion of the left branch caused a rapid local pressure rise that propagated back toward the gate and toward the right branch. When the right branch opened, the pressure temporarily decreased as the flow area increased; final filling of the branch tips then generated a strong back pressure and a steep rise throughout the cavity. Point pressure histories at five locations reproduced these events, including the rise near 0.8 s and the temporary drop near 0.9 s.
Application to a Thin-Walled Skeleton Pattern
The second case is a large, thin-walled skeleton pattern with a trapezoidal outer frame, a central block, and intersecting ribs. Its upper and lower frame widths are 70 mm and 233 mm, its height is 396 mm, and its maximum depth is 71 mm. The frame walls are 13 mm and 7.5 mm thick, the central block reaches 24 mm, and the curved and straight reinforcing ribs are only 3 mm thick. The pattern is symmetric about the OXY plane, so the parting plane is placed on that symmetry plane and the die opens vertically. This arrangement reduces the risk of damaging the fragile pattern during ejection and allows a more balanced fill in the vertical direction.
The pattern material was K512, a medium-temperature pattern wax with a nominal melting point of 84 °C and linear shrinkage of 0.9–1.1%. Differential scanning calorimetry over 30–70 °C at 5 °C/min gave a transition temperature of 49 °C. Thermal conductivity was measured by the transient plane source method, separately for solid and liquid samples, and specific heat was obtained from the same DSC program. Both specific heat and thermal conductivity varied non-monotonically with temperature over the measured range, first rising and then falling. Rheology was measured with a rotational rheometer over 1–1000 s⁻¹ at 50, 54, 58, 61, and 64 °C. The wax showed clear shear thinning, and viscosity decreased substantially as temperature increased.
The measured K512 data were fitted to the Cross-WLF model, producing the parameters listed above. For the skeleton simulation, the thermal properties were read as temperature-dependent tables rather than fixed constants, while the density was 1042 kg/m³, the air density was 1.29 kg/m³, the latent heat was 122.7 J/g, and the wax-die heat transfer coefficient was 100 W/(m²·K).
The skeleton mesh used 1.5 mm cells and approximately 2.9 million cells in a 274 × 182 × 58 grid. Single-gate simulations were first run with a 9 mm diameter, 25 mm long cylindrical gate, a filling time of 10 s, an inlet velocity of 0.94 m/s, an injection temperature of 60 °C, and a die temperature of 20 °C. The purpose was not only to compare gate positions but also to examine whether the thin ribs could be filled before the front temperature fell too far.
| Metric | Central gate in thick block | Side gate near thin frame |
|---|---|---|
| Minimum front temperature | 59.30 °C | 59.03 °C |
| Front temperature drop | 0.70 °C | 0.97 °C |
| Filling stability | Smooth front; limited splashing | Smooth front; limited splashing |
| Injection pressure requirement | Lower and relatively stable | Higher, with pronounced oscillation |
| Thin-wall filling | Ribs fill late; far right frame is last | Right-side ribs fill earlier, but final thin sections remain difficult |
| Overall suitability | Better balance and lower pressure demand | Local benefit, but limited by high resistance |
Both single-gate cases kept the front temperature drop within the commonly accepted 2–5 °C window, so neither would be expected to produce a gross short shot under the stated conditions. The central gate was nevertheless superior. Placing the gate in the 24 mm thick block reduced the inlet pressure needed to maintain the prescribed velocity, and the fill remained more uniform. The side gate sat on a thin frame section, so the incoming wax had to overcome a high local resistance; the inlet pressure was much larger, and its history showed clear oscillations as the flow encountered alternating thick and thin regions. The side gate did improve early filling of the right-hand thin structure, but this advantage faded in the final stage because the 3 mm ribs produced severe hesitation once the melt reached a thick-to-thin transition. The flow velocity dropped sharply at those transitions, leaving the last rib sections to fill under a very high resistance.
Dual-Gate Design and Pressure-Controlled Filling
To combine the low resistance of the central block with the improved reach of a right-side gate, a dual-gate system was designed. Gate a enters the central block; gate b enters the right side of the trapezoidal frame. The runners were arranged so that both gates are at equal distance from the sprue, and the circular side gates taper from 10 mm at the start to 8 mm at the end. The sprue has a 15 mm start diameter, a 20 mm end diameter, and a 40 mm length, so its cross-section increases in the flow direction and reduces inlet resistance. The runner diameter is 16 mm. The complete system was simulated with a pressure inlet of 0.1 MPa.
The dual-gate pattern filled in 9.75 s. The fill-time contours were smooth and broadly spaced in the thick central block and frame, indicating faster flow, while they were closely spaced in the thin ribs, indicating hesitation. The two gates began to admit wax at nearly the same time, around 0.4 s, which confirms that the runner lengths were well balanced. Gate a nevertheless delivered a slightly higher velocity than gate b. Part of the difference comes from geometry: flow into gate a continues in the sprue direction, whereas flow into gate b makes an additional turn and loses more momentum. The thinner frame section adjacent to gate b also raises the local resistance.
The minimum front temperature in the dual-gate case was 59.66 °C, higher than the 59.30 °C of the central single gate, and the area of low front temperature was smaller. This is the main functional benefit of the second gate: it shortens the flow path to the right-side thin structure and reduces cooling before the last ribs are filled. The dual-gate system is not perfectly balanced, however. Because gate b suffers greater entry loss and feeds a thinner adjacent section, shortening the runner to gate b or slightly enlarging its gate would help equalize the two streams.
The gate-velocity histories also reveal a general feature of pressure-controlled wax injection. Under a nearly constant inlet pressure, the velocity first rises rapidly as the initial resistance is low, then grows slowly as the cavity fills, and finally declines once the combined flow resistance exceeds the available pressure. In the dual-gate case, this sequence occupied roughly 0–1.2 s, 1.2–4.3 s, and 4.3–9.75 s. A constant inlet pressure therefore does not produce a constant filling rate. To keep the front speed stable, the injection-pressure profile should rise with the evolving flow resistance; otherwise the early stage may be too fast and the late stage too slow.
Process Parameters and Filling Stability
The baseline dual-gate case was compared with three single-parameter variations: lower injection pressure, higher injection temperature, and higher die temperature. All other thermal and material settings were held constant.
| Parameter | Baseline | Lower pressure | Higher wax temperature | Higher die temperature |
|---|---|---|---|---|
| Injection pressure | 0.1 MPa | 0.09 MPa | 0.1 MPa | 0.1 MPa |
| Injection temperature | 60 °C | 60 °C | 64 °C | 60 °C |
| Die temperature | 20 °C | 20 °C | 20 °C | 25 °C |
| Filling time | 9.75 s | 10.12 s | 5.79 s | 8.82 s |
| Maximum front-temperature drop | 0.34 °C | 0.48 °C | 0.17 °C | 0.24 °C |
Lowering the injection pressure reduced the driving force, lengthened the fill, and allowed more heat loss before the last thin sections were reached. Raising the wax temperature reduced viscosity, shortened the fill to 5.79 s, and reduced the front-temperature drop, but it also pushed gate velocities to 2–3 m/s. At that speed the front became less stable, and at 89% fill the thin-wall region contained isolated cells with \(F<1\). The present solver does not track the air phase and assumes good venting, so these cells disappear numerically in the next step; in a real die they represent locations where gas may be trapped and porosity may form. Raising the die temperature from 20 °C to 25 °C reduced the wax-die temperature difference, slowed cooling, lowered viscosity, and shortened the fill to 8.82 s with a 0.24 °C maximum front-temperature drop.
The process trends are consistent with the rheology. Injection pressure controls the driving force and therefore the filling speed, but excessive pressure can destabilize the front. Injection temperature changes viscosity exponentially and strongly affects both fill time and flow stability; a high temperature may solve a filling problem while creating an air-entrapment problem. Die temperature acts through the cooling rate and the resulting viscosity: a very cold die raises viscosity and flow resistance, while a warmer die improves fillability but lengthens the cooling portion of the cycle.
Implications for Wax Injection Practice
The validated trigger case and the skeleton case point to a practical design sequence. First, characterize the wax with temperature-dependent thermal properties and a Cross-WLF fit rather than a single room-temperature viscosity. Second, place the gate in a thick section when possible, but check whether a second gate is needed to reach isolated thin ribs. Third, evaluate the proposed runner and gate system with a pressure-controlled filling simulation, because a velocity-controlled simulation can hide the pressure demand and the natural decline in filling speed. Fourth, choose the injection temperature and die temperature together: the objective is not the shortest fill time but a stable front that stays above the transition temperature without exceeding the velocity at which air entrapment becomes likely.
The dual-gate skeleton design meets these criteria better than either single-gate concept. It reduces the inlet pressure compared with the side gate, keeps the front temperature higher than the central gate alone, and fills the thick block and the thin ribs in a more coordinated sequence. The remaining imbalance between gates a and b can be reduced by shortening the runner to the right-side gate or by enlarging that gate slightly.
Limitations and Outlook
The model still contains assumptions that matter for production accuracy. The wax-die interfacial heat-transfer coefficient was prescribed rather than measured in situ, and the skeleton case was evaluated numerically without a corresponding filling experiment. Air entrapment, weld-line formation, and surface sink are not predicted by a dedicated criterion; the VOF field can indicate where gas may be trapped, but a realistic venting model or a gas-phase solver would be needed to quantify the risk. These are appropriate next steps: inverse estimation of the interfacial heat-transfer coefficient from measured wax temperatures, experimental validation on thin-walled skeleton patterns, and defect criteria built on the computed front history and volume-fraction field.
Within those limits, the combination of Cross-WLF rheology, three-dimensional non-isothermal flow, Projection-VOF free-surface tracking, and shared-memory parallelization provides a quantitative way to compare gate designs and injection schedules before tooling is cut. For investment casting, where dimensional error accumulates from the wax pattern onward, that capability directly supports the goal of producing thin-walled wax patterns with fewer filling defects and more predictable dimensions.