4. References

Kennedy and Zheng [200] discussed the difficulty of obtaining material data for use in the the-oretical models. To overcome this problem, they proposed a semi-empirical model that in-troduces some correction factors to tune the theoretical predicted results and used measured shrinkage data on simple geometrical molded specimens to determine the correction factors for different materials by the multi-variable regression technique. This approach is known as the CRIMS (Corrected Residual In-Mold Stress) method in commercial software.

5.10 The 2.5D Approximation for Runners

Runners are frequently trapezoidal in cross-section to ensure easy ejection from the mold.

However most 2.5D mold-filling simulation software uses only runners of circular cross-section. While the user of a simulation program may choose various cross-sections, the soft-ware will generally reduce this to an equivalent circular cross-section. This may not be a sat-isfactory approximation, however, and can introduce error in shear heating calculations and convection of the melt around corners. Nevertheless, we ignore this for now and return to the problem in Section 6.4.

Just as for the cavity, the flow in the runner system is determined by the conservation equations of mass, momentum, and energy. However, simplifications are obtained using the material ap-proximations discussed in Section 5.2 and the assumption of symmetry about the runner axis.

Figure 5.10 shows the geometry of the circular runner and its associated coordinate system. We

Figure 5.10 Geometry and coordinate system for runners

assume that the temperature and flow fields are symmetrical about the longitudinal axis of the runner. Consequently no quantity will depend onθ. In particular, it means that the interface between melt and frozen layer depends only on r and x.

The most significant aspect in considering runner systems is the adoption of a cylindrical co-ordinate system for the conservation equations of mass, momentum and energy. We provide these here and then look at further approximations later in this chapter and Chapter 6.

5.10.1 Conservation of Mass for Runners

The conservation of mass equation, Equation 4.13, may be expressed in cylindrical coordinates [35, 260] as

Due to symmetry about the axis of the runner, terms involving derivatives ofθ are zero and Equation 5.98 becomes:

The densityρ depends on both the temperature and pressure. Therefore, using the chain rule for differentiation, and the definitions of expansivity and compressibility given by Equations 3.23 and 3.26 we have:

Substituting these expressions into Equation 5.99 and rearranging gives:

κ³∂p It is usual to ignore the pressure variation in the radial direction, as well as the pressure con-vection term and so,

Ignoring pressure dependence in the radial direction is analogous to ignoring the pressure dependence in the thickness direction of the cavity. It introduces errors at the flow front where convection of melt from the centerline is convected out to the runner wall (fountain flow).

However, as in the cavity, these errors reduce after the flow front passes.

The assumption of temperature symmetry about the runner axis, while apparently reasonable, introduces errors in convection of temperature fields when the runner feeds a number of sim-ilar cavities. These errors depend on processing conditions and runner configuration. If there is substantial shear heating in the runner system, the error becomes more prevalent and can lead to variations in filling times and properties of the molded components. We discuss this in more detail in Section 6.4.

5.10 The 2.5D Approximation for Runners 93

5.10.2 Conservation of Momentum for Runners

In cylindrical coordinates, the momentum equation, Equation 5.12, may be written as three scalar equations [35, 260]: Ignoring body forces, assuming symmetry about the center axis of the runner centerline and employing the dimensional analysis techniques of Section 5.6.2 to determine the magnitude of terms, the momentum equations may be reduced to

∂p

5.10.3 Conservation of Energy for Runners

Using the gradient and Laplacian operators in cylindrical coordinates [35, 260], the energy equation, Equation 5.13, may be written in cylindrical coordinates as:

ρcp

Assuming the runner is circular, and therefore the temperature and pressure fields are inde-pendent ofθ, Equation 5.109 becomes:

ρcp

Further simplification is obtained by ignoring the pressure variation in the radial direction

∂p/∂r and the conduction term in the x direction ∂²T /∂x². The former is unjustified at the flow front and is analogous to ignoring the velocity vzin the cavity. Moreover, this assumption is unjustified at bends in the runner system. This can be important and is discussed in Section 6.4. Ignoring the temperature conduction along the x direction is reasonable and can be es-tablished by dimensional analysis. Finally we ignore the pressure convection term, vx∂p/∂x.

In view of these further assumptions, we arrive at the final form for the 2.5D approximation for the energy equation:

5.10.4 Integration of the Momentum Equation for Runners

In Section 5.6.4 we integrated the momentum equations for the cavity to obtain expressions for the velocities v_xand v_y. A similar treatment may be given to the momentum equations for runners given by Equation 5.108. The nontrivial momentum equation, Equation 5.108,

∂p

can be multiplied on both sides by r and then integrated with respect to r to obtain

∂p

Dividing both sides of Equation 5.114 by rη and integrating again with respect to r , 1

Let r⁺denote the value of r at the interface of the frozen layer and the melt as shown in Figure 5.10. By definition, the velocity vxis zero in the frozen layer, and so vx(r⁺) = 0. Then from

5.10 The 2.5D Approximation for Runners 95

An average velocity in the x direction v_x, at a point x, may be defined as the total flow rate through the runner, at x, divided by the area A_c, of the melt channel at x. That is,

¯

From Figure 5.10, and since we have assumed a circular cross-section, we have A = πr²and A_c= 2πr⁺². So Equation 5.118 becomes,

Equation 5.117 defines v_x(r ) and may be substituted into Equation 5.119:

¯

The double integral on the right-hand side of Equation 5.120 may be evaluated using integra-tion by parts as discussed in Secintegra-tion 4.2.4. To this end, set

M =

Then the double integral on the right-hand side of Equation 5.120 may be written:

Z r⁺

Substituting this result into Equation 5.119 gives,

where the fluidity S1, is defined by

S1= 1

Note that the fluidity subscript of 1 is to distinguish it from the fluidity S2 for the cavity in Section 5.6.4.

5.10.5 Integration of the Continuity Equation for Runners

The previous section developed an equation that related average velocity to the pressure drop along the runner. In order to combine the momentum and continuity equations, we integrate the continuity equation and use the average velocity to develop a single equation for pressure in the runner system.

We rearrange the energy equation, Equation 5.112, to get

∂T

Substituting this into the continuity equation, Equation 5.104, we obtain, 0 = κ∂p

Using dimensional analysis, the quantityβ²T /ρcp is much smaller thanκ and so may be ig-nored. Consequently, Equation 5.126 becomes

Taking A_c= πR²as the cross-sectional area of the runner, an average value of compressibility coefficient ¯κ may be defined as:

κ¯c= 1

5.10 The 2.5D Approximation for Runners 97

Noting that d p/d t is independent of x, and using the average compressibility coefficient, Equation 5.127 may be integrated over the cross-section of the runner to get

0 = 2π

Consider now the last term on the right-hand side of Equation 5.129,

2πZ r⁺

where, in the last two steps, we have used the average velocity, Equation 5.123. Substituting Equation 5.130 into Equation 5.129 gives;

0 = ¯κcR²∂p

This concludes the 2.5D treatment for runners.