Simulation of gas-assisted injection mold-cooling process using line source model approach for gas channel

Int. Comm. HeatMass Transfer, Vol. 25, No. 7, pp. 989-998, 1998

Copyright © 1998 Elsevier Science Ltd Printed in the USA. All rightsreserved 0735-1933/98 $19.00 + .00

Pergamon

P I I S0735-1933(98)00090-6

S I M U L A T I O N O F GAS-ASSISTED I N J E C T I O N M O L D - C O O L I N G P R O C E S S USING LINE S O U R C E M O D E L A P P R O A C H F O R GAS C H A N N E L

Y. P. Chang, S. Y. Hu and S. C. Chen Mechanical Engineering Department Chung Yuan University Chung-Li, Taiwan 32023, R.O.C.

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT Whether it is feasible to perform an integrated simulation for process simulation based on a unified CAE model for gas-assisted injection molding (GAIM) is a great concern. In the present study, numerical algorithms based on the same CAE model used for process simulation regarding filling and packing stages were developed to simulate the cooling phase of GAIM using a cycle-averaged three-dimensional modified boundary element technique similar to that used for conventional injection molding. However, to use the current CAE model for analysis, gas channel was modeled by two-node elements using line source approach. It was found that this new modeling not only affects the mold wall temperature calculation very slightly but also reduces the computer time by 95% as compared with a full gas channel modeling required a lot of triangular elements on gas channel surface. This investigation indicates that it is feasible to achieve an integrated process simulation for GAIM under one CAE model resulting in great computational efficiency for industrial application. © 1998 Elsevier ScienceLtd

Introduction Gas-assisted injection molding (GAIM) process [1-6], being an innovative injection molding process, can substantially reduce production expenses through reduction in material cost, reduction in clamp tonnage and reduction in cycle time. In additions, part qualities can also be greatly improved by reduction in residual stress, warpage, sink marks and shrinkage. Although gas-assisted injection provides many advantages when compared with conventional injection molding (CIM), it also introduces new processing parameters in the process and makes the application more critical. One of the key factors is the design of gas channels which guide the gas flow to the desired locations. Due to the complexity of gas channel design and processing control, computer simulation is expected to become an important and a required tool to assist in part design, mold design and process evaluation in the coming age. 989

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Y.P. Chang, S.Y. Hu and S.C. Chen

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At the present stage, process simulation for the melt filling and gas-assisted filling stages [1-6] has been developed and some of them were incorporated into several commercial packages such as CGASFLOW and Moldflow/gas, etc.. Two key factors that affect the simulation accuracy most are the algorithm used for the calculation of skin melt thickness and the geometrical modeling approach used to represent gas channels. The most popular modeling for gas channel is to utilize a circular pipe of equivalent hydraulic diameter and a superimposition approach [1, 3-6] to represent the mixed I-D and 2D flow characteristics for melt and gas flow in the gas channel of non-circular cross section. Schematic of such modeling is shown in FIG. 1 for the semicircular gas channel. Such analysis approach has been verified for melt flow in thin cavity with channels of semicircular and quadrantal cross sections [4-5]. Simulation results on the secondary gas penetration by both C-GASFLOW and Moldflow/gas show only very rough accuracy. A recent study [6-7] suggests that to obtain good predictions for secondary gas penetration in gas-assisted packing stage, a new algorithm and a flow model of the shrinkage-induced origin may have to be introduced instead of just following the pressure-induced flow model used for tile post-filling simulation of CIM.

Equivalent Diameter "

Shape Factor "

!riD- + 8tD

2;

~)

S F. .

.

.

.

7~Oeq

FIG. 1 Schematic of the circular pipe to represent a semicircular gas channel. An equivalent diameter is calculated so that the melt flow area are the same for both models, Simulation of gas-assisted cooling is not available at this moment. Although the development for the simulation of gas-assisted cooling process is not too difficult, it does require to incorporate the geometry of the gas channels and the hollowed gas core into consideration. It does not know exactly how the geometrical simplification of gas channel will affect the accuracy of cooling simulation. Generally speaking, a complete CAE package for the entire phase of gas-assisted injection molding process is not available at the present time. That whether a unified finite element model can be used for process simulation (including melt-filling, gas-assisted melt-filling, gas-assisted packing and cooling stages). structural analysis and warpage calculation is a great ~oncern for CAE package development. Structural

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GAS-ASSISTED INJECTION MOLD-COOLING PROCESS

991

analyses in the part design stage based on the same CAE finite model used for process simulation were also investigated [8-9]. All previous process simulations developed by different groups assume constant mold wall temperatures. This is not true as known by conventional injection molding [10-12]. Particularly, accurate mold cavity surface temperature and part temperature distributions calculation will not only be critical to the thermally induced residual stress calculation but also to the gas penetration simulation. As a result, only after non-constant mold temperature during the cooling phase of GAIM were obtained, the accurate prediction of gas penetration as well as warpage becomes possible for parts of complicate geometry. In order to reduce the computing cost, a popular approach used by conventional injection mold cooling analysis is to use the boundary element method (BEM) [10-12]. This paper investigates the mold cooling analysis of GAIM considering their influence from mold cooling system so that the cyclic, transient variations of mold cavity surface temperatures as well as part temperature distribution can be calculated for the first time following the same approach used in CIM. Analyses were applied to a 3 mm thick GAIM plate represented by a model with real semicircular gas channel geometry associated with hollowed core (FIG. 2a) and by the CAE model which converts the gas channel into and a line source model representing by two-node element (FIG. 2b). The later approach is a common CAE model approach used for simulations of melt filling, gas-assisted melt filling/holding phases [1,3-7] currently. Comparisons were also made in order to evaluate the differences in the predicted mold wall temperatures as well as their cyclic, transient variations due to the gas channel conversion and simplification.

cooling channel

~k

.~

~ I \channel ScX~ ~model

[\ut,,,,~,a,,,,~ \ \ [ ~ "--..~XS~ ~ \

~

Cavity side (Case A)

1O0 ~

R6.91

~

q

~

~

~

I 7~

Core side (Case A)

FIG. 2a BEM mesh model for mold exterior surface, cooling channel as well as part. Gas channel is modeled according to real geometry (Case A). Mesh models for both core side and cavity side required due to the asymmetry of the gas channel geometry.

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Y.P. Chang, S.Y. Hu and S.C. Chen

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gas chan

1

FIG. 2b BEM model for mold exterior surface, cooling channel as well as part. Gas channel is first converted into an equivalent circular gas channel then using line source approach so that it is finally modeled by twonode element (Case B) similar to that of cooling channel.

Theoretical Formulation

In previous studies [10-11] the periodic transient mold temperature has been separated into two components: (1) a steady cycle-averaged component. T,,,.~, and (2) a time-varying component, T.,,, within a typical cycle, i.e.

T,,,(7,t)

T,~(r)+ T.,,(z,O

~or

f

~

~

~l)

The steady temperature component, Tin..,, is obtained by the cycle-averaged boundary element analysis. 1"he temperature field in the mold is governed by a steady-state heat conduction equation of the Laplacian type: for

V2Tm.s = 0

? G ~')m

(2)

Corresponding boundary conditions defined over the boundary surface of the mold are described in the following -Kin

-

K m

On

' -

OTm.s

to,r)

- hc(Tm, s -

On

OTm,s -

Km

On

Ta,r)

for

E

for

r

E

for

? ~

Sc

(4)

S~p

(5)

-

qcp

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GAS-ASSISTED INJECTION MOLD-COOLING PROCESS

- K OTto ...... Jay

for

~ ~

993

Sp

(6)

" an where S~, S,., S,p, Sp and f2,, represent the mold exterior surface, the cooling channel surface, the clamping surface, melt-mold interface and the domain of the mold, respectively. The corresponding heat transfer coefficients for air and coolant are designated as ha~~ and hc ; qcp is the heat flux value defined at the mold clamping surface; ff,~ is the cycle-averaged heat flux transferred from the polymer melt into to cavity surface and is defined as follows. The ambient environment temperatures with respect to the mold, T~,r and Tc are defined for air and coolant correspondingly. K~, is the thermal conductivity of the mold. The cycle-averaged heat flux Jay is evaluated through numerical iterations. The time-varying component, T,,.~, within a typical cycle is approximated by one-dimensionaltransient heat conduction expressed as at

azk. m -

0z

)

for z ~ Sp

(7)

where 9,,, C,,, K,, are the density, specific heat and thermal conductivity of the mold, respectively. Within the mold cavity, the polymer melt also satisfies the transient conduction equation of

Ot = Ozk. p ~ z )

for ~ ~

Re

(8)

where 9p, Cp, Kp and ~p are the density, specific heat, thermal conductivity and domain of the polymer melt, respectively. On the mold cavity surface, Sp, compatible conditions must be fulfilled. That is,

K~ aTman= - Kp an

for

? ~ Sv

Tin: Tp

for

~ ~

Sp

(9) (10)

In addition, on the hollowed-core surface, Sgc, located within gas channel, boundary condition must satisfy

_xar~

P an = hN~(TP- TN')

for

r ~ Sg c

(11)

where hN~, TN~ are the heat transfer coefficient and the temperature of nitrogen injected during gasassisted filling and packing phases, respectively. Along the mold cavity surface, it is subjected to a periodic boundary condition specified by the time varying flux, .It = J(t) -Jay, at the cavity surface. During the analysis instant ti, the instantaneous heat flux J(ti) at the mold-plastic interface is computed from the polymer melt temperature when the mold is closed. When the mold is open, a convective boundary condition similar to equation (3) is defined for the cavity surface. The cycle-averaged heat flux j,~ is then obtained by

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Y.P. Chang, S.Y. Hu and S.C. Chen

~lav- ZJ(ti)At,

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(12)

t cycle

where At~ is the interval between two computations and

tcycle is the cycle time. Analysis as well as

iteration algorithms have been reported previously [ 10-12].

Numerical Methodology

Numerical methods for steady-state boundary element analysis on a cycle-averaged basis have been reported previously [10-12]. The finite difference method for polymer melt heat transfer is also introduced with the same criteria for the adjustment of analysis interval. The heat conduction equation (2) is then transformed into an integral equation using Green's second identity, the fundamental solution of Laplace equation:

C,T,: where T" =

~ T'qclF- ~q'TdF

(13)

1/4nr,q'=(8T*/On)and r is the distance between integral source point and field point. Eq. (13)

is a typical formulation used in the previous studies for part and mold. To avoid discretization of the circular channel along the circumference which requires a lot of small triangular shell elements and thus saves substantial amount of computer memory, the so-called line source (sink) approach developed by Rezayat and Burton [13] for cooling channels in 3-D parts was implemented. In this approach, the line source (sink) analog is applied to the temperature and heat flux on each segment of the circular channels. Schematic is shown in FIG. 2b. In such situation, for the elements on the axis of the jth cylindrical segment of the cooling lines, equation (13) can be further extended and expressed by [13] N

I, ~(T*q-q*T~Fdl(P)+ ~,~ I (T*q-q'T~ff~dl(P)=O

(14)

where P, lj, and N are the source point on the axis of the jth cylindrical segment, the axis of the jth segment and the total segment number of the cooling channels, respectively. Details have been reported [12-13]. When this method was applied to gas channel represented by two-node elements attributed with equivalent diameters, the gas channel behaves like a heat source in contrast with the heat sink behavior of cooling channels. The residual wall thickness (resulting from gas penetration) of the gas channel must be included for heat flux calculation. This approach not only minimize the using of tinny triangular elements on outer gas channel surfaces, but also avoids the aspect ratios issue [15] which will increase the large number of triangular elements required for the mold exterior surface during analysis. After discretization, equation (13) can be discretized into a set of linear equations

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GAS-ASSISTED INJECTION MOLD-COOLING PROCESS

IHI{T} = [Gl{q}

995

(15)

where {T} and {q} represent column matrices for temperature and heat flux, respectively. By introducing the boundary values, equation (15) can be rearranged to be the form of [A]{T} = {B}

(16)

Solving equation (16), the cycle-averaged mold wall temperatures including plastic-mold interface temperature are obtained. The mold cavity surface temperatures are then used to compute the polymer melt temperature profile and distribution by equation (8) using finite difference method. For gas channel, the heat conduction equation for hollowed cylinder must be used, discretized and solved. Numerical accuracy and stability of the presently developed BEM software have been tested in the previous studies [11, 14-15].

Simulated Results and Discussions

During the simulations, all the materials properties and the cooling operation conditions are listed in Table 1. Two BEM models of part with real semicircular gas channel geometry (Case A) as shown in FIG. 2a and by two-nodes elements (FIG. 2b) similar to that of cooling channel (Case B), respectively, were used. Basically, triangular elements were implemented on the mold exterior surface, parting surface, part surface (cavity surface) and surface of gas channel with real semicircular geometry The cooling channel mesh also reduced and represented by two-node elements. Due to the small triangular elements required for gas channel surface and the associated aspect ratio issue, total numbers of triangular elements during analyses for Case A is 4808. For Case B, the required number of triangular element is 156 plus 12 two-node elements for the gas channel. For both cases, each cooling channel needs 16 two-node elements. Because of asymmetry of gas channel shape, both core side and cavity side must be included in the BEM model for Case A. Simulated cycle-averaged mold cavity surface temperature distribution of the cavity side for GAIM plate with real semicircular gas channel geometry modeling (Case A) is depicted in FIG. 3a. Maximum temperatures are around gas channel laid out locations. Also, around plate center, the distance is most far away from mold exterior surface. Therefore, the mold temperatures show highest values. In Case B with gas channel modeled by two-node elements, the predicted cycle-averaged cavity surface temperature distribution is illustrated in FIG. 3b. The maximum and minimum values of the predicted temperatures are listed in Table 2. It can be seen that on the cavity side, the differences in these predicted temperatures using two models are only about l °C. On the core sides, predicted values of Case A are lower than others by about 3°C. However, the computer time has been reduced to only about 5% in Case B as compared with Case A. This does not include the additional time to rebuild the gas channel surface

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Y.P. Chang, S.Y. Hu and S.C. Chen

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and mesh model required for analyses. The calculated cyclic, transient variations of cavity wall temperatures at different locations of the plate and gas channel for all two cases are also shown in FIG. 4. Comparison of cyclic, transient cavity wall temperatures from two BEM models result in predicted values of less than 7% difference. This indicates that the current CAE model used for the simulations of filling and packing phases can be used also for cooling simulation. Only that the cooling channel mesh as well as mold exterior mesh must be added in order to count for the influence of mold cooling system. It is feasible to achieve an integrated process simulation for GAIM process under one CAE model resulting in great computational efficiency for industrial application.

6'1 6 ~

~

'

~

0

67.5

~

~

9"L9

~

.

~i-~9

b~,

6~"

.-

FIG. 3a Cycle-averaged mold wall temperature distribution of cavity side for the GAIM part with real semicircular gas channel modeling (Case A).

FIG. 3b Cycle-averaged mold wall temperature distribution of cavity or core side for the GAIM part using twonode element to represent the real gas channel (Case B).

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GAS-ASSISTED INJECTION MOLD-COOLING PROCESS

997

80

t..) 75

8 E ~ 70 0

no 0

E 65

C

~ 6o

~

0

10

20

Time

30

40

(sec)

50

FIG. 4 Variations of cyclic, transient mold wall temperatures for Case A and Case B at different gas channel locations designed as G1 and G2 and part locations labeled as P1 and P2.

TABLE l Materials Properties and Processing Parameters 1. Thermal properties of polymer melt and mold: Kv= 0.15 W/m.K, Cv= 2.1 kJ/kg.K, and p = 1040 kg/m 3 K,, = 36.5 W/m.K, Cm = 0.46 kJ/kg'K, and Pm "~ 7820 kg/m 3 2. Flow rate of coolant: I0 #min, Diameter of cooling channel: 10 mm 3. Ambient air temperature: 25°C, Heat transfer coefficient of air and nitrogen: 10 W/m2-K 4. Thickness of plate for the GAIM part: 3 mm 5. Initial polymer melt temperature: 230°C 6. Coolant temperature: 60°C, Nitrogen temperature: 25°C 7. Mold open time: 5 seconds, Cooling time: 45 seconds

TABLE 2 Maximum and Minimum Cycle-Averaged Mold Cavity Temperature Maximum / Minimum (Cavity side)

Maximum / Minimum ( Core side )

CASE A

70.97°C / 64.61°C

68.97°C / 65.42°C

CASE B

71.57°C /65.64°C

71.57°C /65.64°C

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Y.P. Chang, S.Y. Hu and S.C. Chen

Vol. 25, No. 7

Acknowledgments This work was supported by the National Science Council under NSC grant 87-2216-E033-007 and Chung Yuan University under distinguished research funding.

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L.S. Turng, SPE Tech. Papers, 38, 452 (1992).

2.

A. Lanvers and W. Michaeli, SPE Tech. Papers, 38, 1796 (1992)

3.

G. Sherbelis, SPE Tech. Papers, 40, 411 (1994).

4.

S.C. Chen, N. T. Cheng and K. S. Hsu, Int. J. Mech. Sci., 39, 335 (1996).

5.

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6.

S.C. Chen, N. T. Cheng and K. S. Hsu, Int. Comm. Heat Mass Transfer, 22, 319 (1995).

7.

S.C. Chen, N. T. Cheng and S. Y. Hu, to appear in 3~ of Applied Polym. Sci. (1998).

8.

S.C. Chen, S. Y. Hu, R. D. Chien and J. S. Huang, accepted by Polym. Eng. Sci. (1998).

9.

S.C. Chen, S. Y. Hu, R. D. Chien, and J. S. Huang, to appear in J. of Applied Polym Sci. (1998).

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Received May 14, 1998