Cooling of an Injection Mold

Cooling is an important process in the production of injection molded plastics. First of all, the cooling time may well represent more than half of the production cycle time. Second, a homogeneous cooling process is desired to avoid defects in the manufactured parts. If plastic materials in the injection molding die are cooled down uniformly and slowly, residual stresses can be avoided, and thereby the risk of warps and cracks in the end product can be minimized.

As a consequence, the positioning and properties of the cooling channels become important aspects when designing the mold.

The simulation of heat transfer in molds of relatively complex geometries requires a 3D representation. Simulation of 3D flow and heat transfer inside the cooling channels are computationally expensive. An efficient short-cut alternative is to model the flow and heat transfer in the cooling channels with 1D pipe flow equations, and still model the surrounding mold and product in 3D.

This example shows how you can use the Nonisothermal Pipe Flow interface together with the Heat Transfer in Solids interface to model a mold cooling process. The equations describing the cooling channels are fully coupled to the heat transfer equations of the mold and the polyurethane part.

Figure 1: The steering wheel of a car, made from polyurethane.

Model Definition

Model Geometry and Process conditions

The polyurethane material used for a steering wheel is produced by several different molds. The part considered in this model is the top half of the wheel grip, shown in gray in Figure 2.

Figure 2: Polyurethane parts for a steering wheel. The top half of the grip is modeled in this example.

The mold consists of a 50-by-50-by-15 cm steel block. Two cooling channels, 1 cm in diameter, are machined into the block as illustrated in Figure 3.

Figure 3: Mold block and cooling channels.

The after injection of the polyurethane, the average temperature of the mold a the plastic material is 473 K. Water at room temperature is used as cooling fluid and flows through the channels at a rate of 10 liters/min. The model simulates a 10 min cooling process.

For numerical stability reasons, the model is set up with an initial water temperature of 473 K, which is ramped down to 288 K during the first few seconds.

Pipe Flow Equations

The momentum and mass conservation equations below describe the flow in the cooling channels:

(1)

Above, u is the cross section averaged fluid velocity (m/s) along the tangent of the centerline of a pipe. A (m2) is the cross section area of the pipe, ρ (kg/m3) is the density, and p (N/m2) is the pressure. For more information, refer to the section Theory for the Pipe Flow Interface in the Pipe Flow Module User’s Guide.

Expressions for the Darcy Friction Factor

The second term on the right-hand side of Equation 1 accounts for pressure drop due to viscous shear. The Pipe Flow physics uses the Churchill friction model (Ref. 1) to calculate fD. It is valid for laminar flow, turbulent flow, and the transitional region in between. The Churchill friction model is predefined in the Nonisothermal Pipe Flow interface and is given by:

where

As seen from the equations above, the friction factor depends on the surface roughness divided by diameter of the pipe, e/d. Surface roughness values can be selected from a list in the Pipe Properties feature or be entered as user-defined values.

In the Churchill equation, fD is also a function of the fluid properties, flow velocity and geometry, through the Reynolds number:

The physical properties of water as function of temperature are directly available from the software’s built-in material library.

Heat Transfer Equations

Cooling Channels

The energy equation for the cooling water inside the pipe is:

(2)

where Cp (J/(kg·K)) is the heat capacity at constant pressure, T is the cooling water temperature (K), and k (W/(m·K)) is the thermal conductivity. The second term on the right-hand side corresponds to heat dissipated due to internal friction in the fluid. It is negligible for the short channels considered here. Qwall (W/m) is a source term that accounts for the heat exchange with the surrounding mold block.

Mold Block and Polyurethane Part

Heat transfer in the solid steel mold block as well as the molded polyurethane part is governed by conduction:

(3)

Above, T2 is the temperature in the solids. The source term Qwall comes into play for the heat balance in Equation 3 through a line heat source where the pipe is situated. This coupling is automatically done by the Wall Heat Transfer feature in the Nonisothermal Pipe Flow interface.

Heat Exchange

The heat exchange term Qwall (W/m) couple the two energy balances given by Equation 2 and Equation 3:

The heat transfer through the pipe wall is given by

(4)

In Equation 4 Z (m) is the perimeter of the pipe, h (W/(m2·K)) a heat transfer coefficient and Text (K) the external temperature outside of the pipe. Qwall appears as a source term in the pipe heat transfer equation.

The Wall heat transfer feature requires the external temperature and at least an internal film resistance.

Text can be a constant, parameter, expression, or given by a temperature field computed by another physics interface, typically a 3D Heat Transfer interface. h is automatically calculated through film resistances and wall layers that are added as subnodes. For details, refer to the Theory for the Heat Transfer in Pipes Interface in the Pipe Flow Module User’s Guide.

In this model example, Text is given as the temperature field computed by a 3D heat transfer interface, and automatic heat transfer coupling is done to the 3D physics side as a line source. The temperature coupling between the pipe and the surrounding domain is implemented as a line heat source in the 3D domain. The source strength is proportional to the temperature difference between the pipe fluid and the surrounding domain.

The Wall Heat Transfer feature is added to the Nonisothermal Pipe Flow interface, and the is set to the temperature of the Heat Transfer in solids interface.

Figure 4: In the Wall Heat Transfer feature, set the External temperature to the temperature field computed by the Heat Transfer in Solids interface.

The heat transfer coefficient, h, depends on the physical properties of water and the nature of the flow and is calculated from the Nusselt number:

where k is the thermal conductivity of the material, and Nu is the Nusselt number. dh is the hydraulic diameter of the pipe.

COMSOL detects if the flow is laminar or turbulent. For the laminar flow regime, an analytic solution is available that gives Nu = 3.66 for circular tubes (Ref. 2). For turbulent flow inside channels of circular cross sections the following Nusselt correlation is used (Ref. 3):

(5)

where Pr is the Prandtl number:

Note that Equation 5 is a function of the friction factor, fD, and therefore that the radial heat transfer increases with the surface roughness of the channels.

All the correlations discussed above are automatically used by the Wall Heat Transfer feature in the Pipe Flow Module, and it detects if the flow is laminar or turbulent for automatic selection of the correct correlation.

Results and Discussion

The mold and polyurethane part, initially at 473 K, are cooled for 10 minutes by water at room temperature. Figures below show sample results when flow rate of the cooling water is 10 liters/minute and the surface roughness of the channels is 46 μm. After two minutes of cooling, the hottest and coldest parts of the polyurethane part differ by approximately 40 K (Figure 5).

Figure 5: Temperature distribution in the polyurethane part and the cooling channels after 2 minutes of cooling.

Figure 6 shows the temperature distribution in the steel mold after 2 minutes. The temperature footprint of the cooling channels is clearly visible.

Figure 6: Temperature distribution in the steel mold block after 2 minutes of cooling.

After 10 minutes of cooling, the temperature in the mold block is more uniform, with a temperature at the center of approximately 333 K (Figure 7). Still, the faces with cooling channel inlets and outlets are more than 20 K hotter.

Figure 7: Temperature distribution in the steel mold block after 10 minutes of cooling.

To evaluate the influence of factors affecting the cooling time, use Material and Parametric sweeps. Figure 8 shows the average temperature of the polyurethane part as function of the cooling time for the several flow rates of the cooling water, the surface roughness of the cooling channels, and the mold materials.

Figure 8: Average temperature of the polyurethane part as function of time and cooling conditions.

References

1. S.W. Churchill, “Friction factor equations span all fluid-flow regimes,” Chem. Eng., vol. 84, no. 24, p.91, 1997.

2. F.P. Incropera, D.P. DeWitt. Fundamentals of Heat and Mass Transfer, 5th ed., John Wiley & Sons, pp. 486–487, 2002.

3. V. Gnielinski, “New Equation for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” Int. Chem. Eng. vol. 16, pp. 359–368, 1976.

Pipe_Flow_Module/Heat_Transfer/mold_cooling

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
T_init_mold	473.15[K]	473.15 K	Initial temperature, mold
T_coolant	288.15[K]	288.15 K	Steady-state inlet temperature, coolant
Qw	10[l/min]	1.6667E-4 m³/s	Coolant flow rate
e	0.046[mm]	4.6E-5 m	Surface roughness

Step 1 (step1)

Create a smooth step function to decrease the coolant temperature at the beginning of the process.

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 2.5.

In the text field, type 1.

In the text field, type 0.

Click to expand the section. In the text field, type 5.

Optionally, you can inspect the shape of the step function:

Click

Variables 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
T_inlet	T_coolant+(T_init_mold-T_coolant)*step1(t[1/s])	K	Ramped inlet temperature, coolant

Geometry 1

Import 1 (imp1)

First, import the steering wheel part from a CAD design file.

In the toolbar, click

In the window for , locate the section.

Click .

Browse to the model’s Application Libraries folder and double-click the file mold_cooling_top.mphbin.

Click .

Locate the section. Find the subsection. Click .

In the dialog box, type Wheel in the text field.

Click .

Second, draw the mold and cooling channels. To simplify this step, insert a prepared geometry sequence from file. After insertion you can study each geometry step in the sequence.

Click the

button in the toolbar.

In the toolbar, click

Browse to the model’s Application Libraries folder and double-click the file mold_cooling_geom_sequence.mph.

In the toolbar, click

Click the

button in the toolbar.

Click the

button in the toolbar.

Work Plane 1 (wp1)

Create the selections to simplify the model specification.

In the window, click .

In the window for , locate the section.

Find the subsection. Click .

In the dialog box, type Cooling channels in the text field.

Click .

Materials

The next step is to specify material properties for the model. First, select water from the built-in materials database.

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

Materials

Water, liquid (mat1)

In the window for , locate the section.

From the list, choose .

Material Switch 1 (sw1)

Define the mold materials to switch between during a solver sweep.

In the toolbar, click

and choose .

Select Domain 1 only.

Add Material

Go to the window.

In the tree, select .

Click

In the tree, select .

Click

In the toolbar, click

to close the window.

Materials

Polyurethane

Next, create a material with the properties of polyurethane.

In the window, under right-click and choose .

In the window for , type Polyurethane in the text field.

Locate the section. From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Thermal conductivity	k_iso ; kii = k_iso, kij = 0	0.32	W/(m·K)	Basic
Density	rho	1250	kg/m³	Basic
Heat capacity at constant pressure	Cp	1540	J/(kg·K)	Basic

Nonisothermal Pipe Flow (nipfl)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Pipe Properties 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the di text field, type 1[cm].

Locate the section. From the list, choose .

In the text field, type e.

Temperature 1

In the window, click .

In the window for , locate the section.

In the Tin text field, type T_inlet.

Inlet 1

In the toolbar, click

and choose .

Select Points 3 and 4 only.

In the window for , locate the section.

From the list, choose .

In the qv,0 text field, type Qw.

Heat Outflow 1

In the toolbar, click

and choose .

Select Points 269 and 270 only.

Wall Heat Transfer 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the Text list, choose .

Internal Film Resistance 1

In the toolbar, click

and choose .

Initial Values 1

In the window, click .

In the window for , locate the section.

In the u text field, type 0.1.

In the T text field, type T_init_mold.

Heat Transfer in Solids (ht)

Initial Values 1

In the window, under click .

In the window for , locate the section.

In the T2 text field, type T_init_mold.

Heat Flux 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

In the h text field, type 2.

Mesh 1

Edge 1

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Size 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Free Tetrahedral 1

In the window, right-click and choose .

Even with a maximum element size of 0.003 m, the mesh contains some collapsed elements, resulting in solver errors. Trial and error gives that lowering the curvature factor somewhat will create a mesh with good quality.

Size

In the window, click .

In the window for , locate the section.

Click the button.

Locate the section. In the text field, type 0.003.

In the text field, type 0.55.

In the window, right-click and choose .

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,2,28) range(30,30,600).

Definitions

To evaluate the average temperature of the polyurethane part for different conditions and mold materials (Figure 8), perform parametric and material sweeps. To avoid accumulating a lot of data while solving, keep only last solution and save the average wheel temperature in a table. For this purpose, add a global probe to the model.

Domain Probe 1 (dom1)

In the toolbar, click