Simulations of cyclic transient mold cavity surface temperatures in injection mold-cooling process

0735-1933/92 $5.00 +.00 Copyright © 1992 Pergamon Press Ltd.

INT. COMM. HEAT MASS TRANSFER Vol. 19, pp. 559-568, 1992 Printed in the United States

SIMULATIONS OF CYCLIC TRANSIENT MOLD CAVITY SURFACE TEMPERATURES IN INJECTION MOLD-COOLING PROCESS Shia-Chung Chen and Yung-Chien Chung Chung Yuan University Department of Mechanical Engineering Chung-Li 32023, Taiwan, Republic of China

(Communicated by J.P. Hartnett and W.J.

Minkowycz)

ABSTRACT In the present study, simulations of the injection mold-coollng process are developed by using a modified boundary element technique. The mold cavity surface temperature distr~utino is first calculated on a cycle-averaged basis. The temperature distr~ution and the temperature profile of the polymer melt and the corresponding transient heat flux on plastics-mold interface are then computed by finite difference method in a decoupled manner. Finally, the difference between cycle-averaged heat flux and transient heat flux is analyzed to obtain the cyclic transient mold cavity surface temperature. The analysis results for a box mold are illustrated and discussed. 1. I N T R O D U C T I O N Optimized cooling system design and cooling process conditions are of significant importance in injection molding industries because they affect both the part quality and process productivity [1-2]. An efficient cooling system design would reduce the cooling time and a uniform cooling would improve part quality by minimizing undesired defects. Basically, the cyclic cooling process is a three-dimensional time-dependent heat conduction problem with convective boundary conditions. Complicated boundary geometry is introduced by the cooling channel layout. Besides, there are many design parameters involved in the injection molding cooling process [3]. Although a complete analysis on the transient temperature variations of the mold and polymer melt simultaneously is possible in principle, the computational cost is too expensive to be implemented during the actual design process at the present time. In order to reduce the computing cost, a popular approach is to use the boundary element algorithm in which the heat flux is introduced on the plastics-mold interface on a cycled-averaged basis [4-9]. However, the cyclic transient behaviour of mold temperature of which the schematic diagram is shown in F I G . 1 may be 559

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very important especially when the mold cooling system is not efficient [10]. Hence, it is worthwile to simulate and study the periodic transient characterictics of mold temperature variation in details without increasing computational cost tremendously. In this paper, the cyclic transient mold cavity surface tempertures are calculated in two steps. The steady cycle-averaged temperature component is first calculated using modified boundary element method [9]. The time-varying temperature component is then evaluated by using transient plastics-mold interface heat flux obtained via finite difference method in a decoupled manner. An iterative procedure is implemented to obtain the final periodic transient temperature values using instantaneous and cycle-averaged heat flux as the boundary condition on cavity surface. The analysis algorithm is developed and applied to a box mold. The results are illustrated and discussed. 2. M E T H O D D E S C R I P T I O N The periodic transient mold temperature T,. can be seperated into two componentes: (1) a steady cycle-averaged component T,.... and (2) a time-varying component Tm,t within a typical cycle, i.e.

T,,,(z,y,z,t) = T,.,s(z,y,z) + T,.,t(x,y,z,t)

(1)

The steady temperature component T,.... can be obtained by the cycle-averaged boundary element analysis. The temperature field T,,,,s in the mold is expressed by the steady-state heat conduction equation of Lapalce type:

V2T = 0 fo;

-2ef~

(2)

with corresponding boundary conditions defined over the boundary surfaces of the mold as

• 0T

- K m ~ n =h~(T- Tc ) for

-ZeSc

- K m ~ n ' =hp(T- TM)

--2eSp

for

(3)

where S,, S, and Sp represent the exterior of mold boundary surface, cooling channel surface and melt-mold interface, respectively. The corresponding heat transfer coefficients for

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TEMPERATURES IN INJECTION MOLD-COOLING PROCESS

561

air, coolant and polymer melt are designated as hair, he and hp. The ambient enviroment temperatures with respect to the mold, Tair, Tc and TM are defined for air, coolant and ploymer melt, correspondingly. K,,, is the thermal conductivity of mold. The cycle-averaged heat transfer coefficient hp for the polymer melt is defined from the total heat transported into the mold in one cyle per degree C in temperature difference. Therefore, hp can be matlmmatically formulated as q~ = h p A T = Q -Q tcyc tcool-I- t2

(4)

where q~ represents the cycle-averaged heat flux, tc~c is the cycle time, t¢oot is the cooling time, t2 = t¢~ - t~oot= t fin (filling time) + top~,, (mold open time), Q is the total heat removed from the polymer melt by the mold and A T is the temperature difference between the polymer melt and the mold cavity wall. Heat released during the filling stage has been neglected in the present sinmlation. For a plastic part of thickness H, density p , specific heat Cp, initial melt temperature TM, ejection temperature TE and cycle-averaged cavity wall temperature Tw, hp is given by hp _ p c , ( H / 2 ) TM -- TE tcvc [TM Tw ]

(5)

where A T = TM - Tw and Q = pCv(H/2 ) (TM - TE). The cooling time tcoot, being the period of time for the average melt temperature to reach the ejection temperature TE, can be approximately described by [9]

tcoo, =

H2

in

.8 TM-Tw : ]

(6)

where ap is the thermal diffusivity of the polymer melt. It may noted that the cooling time tcoot and hp are dependent on the steady cyclic mold wall temperature Tw which is not knwon a priori. Therefore, to start the simulation, an initial value of Tw is assumed. Then the analysis is carried out to provide the new values of Tw. In the next iteration, hp is evaluated from equation (5) based on the new wall temperature Tw. Once the steady cycle-averaged heat flux on cavity surface is obtained during later stage of the analysis, hp can also be calculated directly from equation (4).

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The heat conduction equation (2) is then transformed into an integralequation using Green's second identity [11] and fundamental solution of Laplace equation:

C,T, = fr Tq'dr + fr qT'dF

(7)

where T* = ~ ln(~), q* = (~,~*), C=1/2 for boundary points and C = I for interior points [11]. After discretization over the boundary surface of the mold, equation (2) can be applied to each element to obtain a set of linear equation [HI{T} =[a]{ 0 ~ }

(8)

where T represents the temperature value and { Y;fn} aT is the associated normal derivative on the boundary element. By introducing the boundary values and solving equation (8), the cycle-average mold wall temperatures Tm,s or

Tw including plastics-mold interface

temperature are obtained. The steady mold cavity surface temperatures are then used to compute the polymer melt temperature profile and distribution by the transient heat conduction equation:

pc,

cgT

O( . OT

=Vz I':P-8fz)

(9)

where p, Cp and Kp are the density, specific heat and thermal conductivity of the polymer melt, respectively. Equation (9) is solved by finite difference method using steady cavity wall temperature as the boundary condition. Equation (9) is also solved for the period when mold is open by replacing the thermal properties of polymer melt by mold. In such situation, the ambient air temperature is used as the boundary condition. During the analysis, the instantaneous heat flux and the cycle-averaged heat flux are also computed. In typical meshes for part geometries the element lengths are greater than the distance travelled by the thermal pulse in the plannar direction. This implies that Tm(x,y,z,t) = T,n(z,t) and the time-varying temperature component ca~l be approximated by onedimensional transient heat conduction equation for semi-infilfite solid [12]. Therefore, the difference between instantaneous heat flux

qt and steady cyclic heat flux qj can be used

as the boundary condition for equation (9) to obtain the cyclic transient temperature fluctuation Tm,t. The summation of T,.... and

T,~,t gives the periodic transient cavity wall

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TEMPERATURES IN INJECTION MOLD-COOLING PROCESS

563

temperature T,,,. An iterative procedure is implemented to obtain stable values of both

T,n,t and T,n,e. Usually, only two or three iterations are required for a stable calculation. 3. R E S U L T S A N D D I S C U S S I O N During the simulations, all the material properties and the cooling operation conditions are listed in Table I. Table I

Cooling Operation Condition

Thermal properties of polymer melt: Kp = 0.25 w/(m*°K),

Cp =

lS00 J / ( k g . ° K ) and p = 938 k g / m 3

Thermal properties of mold: K m = 36.5 w/(m*°K), C,, = 4260 J/(kg*OK) and p,, = 7670 k g / m ~ Part thickness: 2 mm. Coolant temperature of core side and cavity side: 300C Ambient air temperature: 300C Heat transfer coefficient of core coolant and cavity coolnat: 1825 w/rn 2 .° C Heat transfer coefficient of air: 77w/rn 2 .° C Initial polymer melt temperature: 240°C Polymer melt frozen temperature: 130°C Mold open time topen: 4 seconds. Mold filling time t fin: 2 seconds. Mold cooling time tc : 4 seconds.

A simple box mold with cooling channel arrangements as indicated in F I G . 2 , is analyzed. The cycle-averaged mold cavity surface temperatures calculated using full-gapboundary approach [9] axe illustrated in F I G . 3 . The cyclic transient temperature for element # 2 7 located in the center part of the box on the cavity side is shown in F I G . 4 . The obtained results from the first three iterations axe also indicated. The first iteration almost provides the stable temperature values. The periodic transient cavity surface tem-

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Vol. 19, No. 4

peratures at different locations on both core and cavity sides are depticted in F I G . 5 . For a specified location the transient temperature may vary as high as 5°C in a cycle. Also, the minimum cavity surface temperature is higher than coolant temperature by about 15°C. The simulated results show reasonable description on the transient charcateristics of the cyclic mold cavity surface variations as those observed [10]. 4. C O N C L U S I O N In the present study, the cyclic transient cavity surface temperatures of the mold cooling process are calculated using iterative BEM/FDM in decoupled sequence. First, the steady cycle-averaged temperature componet is computed using modified boundary element technique. The instantanous heat flux qt and the cycle-averaged heat flux q8 on the cavity surface are then evaluated from polymer melt temperature profile using finite difference method. Finally, the cyclic time-varying temperature component is analyzed from the difference between qs and qt. The analysis algorithm provides reasonable description on the cyclic transient characterictics of the mold cavity su~fface temperature variation in an injection cycle. Since only two or three iterations are required to obtain stable results, the computational cost is greatly reduced as compared with standard three-dimensional transient heat transfer analysis. 5. A C K N O W L E D G E M E N T S This work was supported by National Science Council under NSC grand 80-0405E033-04. 6. N O M E N C L A T U R E Cp

specific heat of polymer melt

[G]

square matrix for coefficients associated with heat flux

[H]

square matrix for coefficients associated with temperature

H

thickness of the cavity

hair heat transfer coefficient between the exterior mold boundary and the ambient air he

heat transfer coefficient between the cooling channel wall and the the coolant

hp

defined cycle-averaged heat transfer coefficient between the polymer melt and

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TEMPERATURES IN INJECTION MOLD-COOLING PROCESS

the cavity wall Km

thermal conductivity of mold

K,

thermal conductivity of polymer melt

n

normal direction of mold boundary

Q

total heat removed from the polymer melt by mold in one cycle

q

heat flux at mold boundary

q8

steady cycly-averaged heat flux at mold boundary

qt

insta~xeous heat flux at mold boundary

q*

heat flux at mold boundary defined from fundamental solution T*

r

distance between the source point and the field point

T

temperature field

T*

temperature represented by fundamental solution formulation

T~ir ambient air temperature

T,

coolant temperature

Te

averaged gapwise temperature of polymer melt upon ejection

TM

polymer melt temperature

Tm

cyclic transient mold temperature

Tra,s steady cyclic component of mold temperature Tm,t cyclic time-varying component of mold temperature Tw

specified cavity wall temperature

{Z}

column matrix for temperature variables

t

time

tcyc

cycle time

tcool cooling time t2

tc~c - tcool

Z

direction across cavity thickness thermal diffusivity of polymer melt

p

density of polymer melt

AT

temperature difference between the polymer melt and the cavity wail

565

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S.-C. Chen and Y.-C. Chung

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7. R E F E R E N C E i. K. J. Singh, in "Computer-Aided Engineering for InjectionMolding", E. C. Bernard (ed.),Hanser Publisher, Munich, 1983. 2. L. T. M~,~z~one,(ed.) "Application of Computer-Aided Engineering in InjedtionMolding", Hanser Publisher, Munich, 1987. 3. S. C. Chen, S. Y. Hu and A. DavidoiT, Chung Yuan Journal, 19, 82, (1990). 4. C. Austin, SPE Techanical Papers,31, 764, (1985). 5. T. H. Kown, S. F. Shen, and K. K. Wang, SPE Technical Papers,32, 110, (1986). 6. K. ~ima.sekhar, C. A. Hieber, and K. K. Wang, SPE TechnicalPapers, 34,352, (1988). 7. S. C. Chen, S. M. W a n g and Y. L. Chang, SPE Technical Papers, 36, 1090, (1990). 8. L. S. Turng and K. K. Wang, J. Eng. Indus.,Trans. ASME, i12,161, (1990). 9. S. C. Chen, S. Y. Hu, Int. Comm. Heat Mass Transfer,18, 823, (1991). I0. W. I. Pearson, M. R. l(amal a~d F. Gao, SPE Technical Papers, 36, 227, (1990). II. C. A. Brebbia, J. C. F. Tellesand L. C. Wrobel, "Boundary Element Technique", pp. 60-71, Springer-Verlag,New York (1984). 12. M. Rezay,~t and T. Burton, in Symposium of Advanced Bounda-,'y Element Methods : Application in Solids and Fluids, Springer-Verlag,New York (1987).

Steady cyclic period

Cycle time

~u

"A-7~-A-7~-A-~ FA-7~-A-7~-A-7~""

t Cyclic

V V V V V V V ~ / V v V V----i-transient

E E-"U o

I

=

Operation starts

Time PIG.1 The schematic diagram of mold temperature variationversus injectioncycles.

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TEMPERATURES IN INJECTION MOLD-COOLING PROCESS

567

180.0

e l e m e n L JlE

¢ ¢ element. #1

element #105

FIG.2 The box mold together wi~h cooling system configuration.

60.0

elementi~90 56.0 w

52.0

,~'ementH11ernenti~27

1 0 , / e l e m e.enlement t # 8~0

F5

Q_ i,i

48.0

44.0

elementH16

40.0 ,,] 1 , O.Oq

,ul

I = l S = l l l ; l ~ l l l ~ g l | l t t ,

20.00

lilll

I l O t 6 1 , , * l l l l l g

i i I

4-0.00 6 0 . 0 0 80.OO 100.00 POSION ALONG THE MOLD

F I G . 3 Cycle-averaged cavity surface temperature distribution calculated by full-gapboundary element analysis.

568

S.-C. Chen and Y.-C. Chung 53.0

Vol. 19, No. 4

ELEMENT 27 . . . . . cyclic-averaged temperature =:=-*-~ transient temperature

52.5

~ 52.0

~

51.5 51.0

(~ 50.5 O3

50.0

~49.5 49.0

48.5 A t 8 . 0

,

i

i

i

,

,

i

0.0(

i

i

I

i

,

I 0.00

,

i

i

,

,

i

i

I

i

i

,

i

20.00 TIME(sec)

i

i

i

i

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30.00

FIG.4 Variation of cycic transient cavity surface temperature with iteration numbers for element #27.

6°°i/L

58.0

56.0 <

w54.0 ~" ~-uJ

~

: ~-"~-" element #80

.-'1 J/q~

-'-'~: element ~90 (core side)

~. . . . . element #105

co 52.0 m 50.0 >p-

...............

g 48.0 -~ -~ 46.0 l ~ ..o7

/

/'~

~ ~

...................

0.00

2.00

n

..... t

element ~1 element #16 (cavity side)l #27 I

, .......

4.00

II17111111711C17[

6.00 TIME(sec)

8.00

10.00

FIG.5 Vaziation of cycic transient cavity surface temperature for specified locations indicated in FIG. 2. Received April 30, 1992