space-time distribution in filling a mold

Space-Time Distribution in Filling a Mold

I. MANAS-ZLOCZOWER’, J. W. BLAKE,** and C. W. MACOSKO

Department of Chemical Engineering and Materials Science University of Minnesota

Minneapolis, Minnesota 55455

The residence time of a fluid particle in mold filling i s total time spent in the mold. Displaying curves of constant residence time in the mold gives the space-time distribu- tion during the filling process. A simple method to calcu- late space-time distributions is presented. Applications to mold filling in reaction injection molding (RIM) are illus- trated.

INTRODUCTION olding operations in polymer processing in- M volve rapid injection of liquid systems into

cavities of different shapes. An understanding of the transport processes during mold filling is very helpful in establishing moldability criteria for these systems. There are numerous studies of the fluid mechanics of mold filling and all recognize the existence of two flow regimes: a main flow and a front flow. The problem was first considered by Spencer and Gilmore (1) in the early 1950s. Ballman and his co-workers (2, 3) conducted mold-filling experiments and recognized a specific flow pattern near the ad- vancing front. These authors saw a fairly flat front and stressed the importance of a trans- verse velocity in explaining this shape. The term “fountain effect” was coined and discussed by Rose (4) in an article about fluid displace- ment in capillary flow. Specifically, fluid enter- ing the fountain flow region decelerates in the direction of the flow as it acquires a velocity component in the perpendicular direction. Schmidt’s (5) experimental work with colored tracer particles supported a fountain effect in the front region when filling the mold. Tadmor (6) approximated the fountain flow by a steady elongational flow to explain semiquantitatively the commonly observed orientation distribu- tions in molded parts.

Obviously, this complex flow is difficult to describe mathematically. Bhattacharji and Savic (7) studied the displacement of a viscous liquid by a low viscosity fluid and derived a quasi-analytical expression for the stream function in the front flow. Castro (8) expanded this stream function in a power series to obtain velocity profiles for the flow front. He then com- pared his approximate analytical equations for the velocity profiles with Silliman’s (9) two-di-

* Present address: Case Western Reserve University, Department of Macromo- lecular Science. Cleveland, OH 44106. .* Present address: Ammo Chemical Co.. Naperville. IL 60566.

mensional finite element solution and found good agreement. Other methods also have been used to calculate the front flow. Huang (10). Manzione (1 1 ) and, recently, Kamal and co- workers (1 2) all employed marker-and-cell methods to describe the front flow. Domine (1 3) made use of a “demon” that computationally moves material from the central region of the front toward the front mold wall to simulate the fountain flow.

Past models of the filling process have fo- cused on solving the balance equations for mo- mentum, energy and, for reactive systems, mole in both the front flow and main flow regions. A flat flow front and a no-slip boundary condi- tion were assumed. Velocity, temperature and, where pertinent, conversion profiles at various times during filling were obtained. Such models are rather complex, and their solution requires a significant computational effort.

The goal of this work is to develop a simpler, yet useful model of the filling process involving the concept of space-time distribution. During mold filling, fluid particles entering the mold at the same time are distributed by the flow field into curves of constant residence time. Identi- fying curves of constant residence time in the mold gives the space-time distribution during the filling process. For fast polymerizations, as in reaction injection molding (RIM), thermal and mass diffusion often can be neglected. Then, space-time distributions also give directly tem- perature and conversion distributions within the mold.

FOUNTAIN FLOW

Obviously, the fountain flow has a dramatic effect on the space-time distribution. Consider an end-fed rectangular mold cavity of small thickness compared with the other two dimen- sions, as shown in Fig. 1 . Because of the as- sumed symmetry of the problem, it is sufficient

POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, YO/. 27, NO. 16 1229

I . Manas-Zloczower. J . Blake, and C. Macosko

to analyze only half of the mold. For conven- ience, the longitudinal position is referenced to the front position and the transverse location is referenced to half the mold thickness.

X v = -

Xf

c ; = - 2Y H

The contact line between the entering liquid and the departing gas is assumed to be perpen- dicular to the mold walls and moving with the average fluid velocity, uaUe. Taking a laminar unidirectional velocity profile for the main flow, u( [ ) . we define a dimensionless velocity:

(3)

For the unidirectional main flow, the line con- taining fluid particles moving at uaUe is desig- nated Ecr . Relative to a Lagrangian observer on the moving front line, fluid at tCr is stationary, fluid below tCr is entering the front region, and fluid above tCr is leaving the front region (Fig. 2). Idealizing the fountain flow as a single straight line coincident with the advancing front, there is a continuous transfer of fluid along this line from high-velocity regions of the main flow, 0 < ( < tC., to low-velocity regions of the main flow, tCr < [ < 1. Particles overtaking

Flow front position I

. Gate

L

f Dimensionless variables: q = x l x

c - y11i12

Fig. 1. Schematic representation of an end-fed rectan- gular mold.

main flow veloclty proflle veioclty profile relatlve to the v( c 1 lront velocity

Fig. 2. Eulerian and Lagrangian representations of the uelocity profiile in mold fi l l ing.

the front at position [ should turn back through the front at position [' according to the mass conservation law (incompressible fluid is as- sumed):

Equation 4 emphasizes the cumulative effect of the fountain flow. Notice that the fountain flow is completely defined in terms of the main flow velocity profile.

In the case of a Newtonian fluid, the main flow velocity profile is (1 4)

3 u* = - (5) (1 - t2) for which

1 c; =- cr Jz

Substituting Eq 5 into Eq 4 and integrating gives

"3 - 4' + 4 - (3 = 0. (7) Similarly, for a power-law fluid (14)

s + 2 s + l

u* = - (1 - ? + I )

where s is the reciprocal of the power-law model parameter. In this case

- 1 - tcr = (s + 2)"+1 (9)

and integration of Eq 4 gives

0. (10) " S + 2 - (' + [ - y+2 =

Given that a fluid particle entered the front flow at [, Eqs 7 and 10 provide the position [' where that particle reenters the main flow, for Newtonian and power-law fluids, respectively. Equation 7 has an analytic solution (1 5) and Eq 10 was solved using a Newton-Raphson scheme

To test our idealized model of the fountain flow, we compared it with the two-dimensional front flow velocity profiles used by Castro (8). Castro's velocity profiles are

u,' = (0.5 - 1.5 t2)[1 - 1.45e-5(f'-"1)

(16)-

(1 1) .sin(0.76 + 2(t* - 77 + l))] + 0.53(1 - 5[4)e-5(t'-1+11

.sin(2(t* - 7 + 1)) + 1.0

u i = 0.5[(1 - [2)[3.63e-5(t0-q+1)

-sin(0.76 + 2(t* - v + 1))

- 1 .45e-5~t'-n+1~cos(0.76

+ 2(t* - 4 + l))]

- [(l - t4)[1.315 . e-5't'-"+l'sin(2(t* -

(12)

11 + 1)) - 0 .53e -5 ( t ' -~+1)~~~(2 ( t* - 4 + I))]

1230 POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, Vol. 27, NO. 16

Space-Time Distribution in Filling a Mold

E q u a t i o n s 1 1 and 12 were integrated in time for several points on the starting material line using a high order backward difference method (1 7) to achieve the solid curves shown in Fig. 3. Similarly, in our model E q 5 was integrated in time to obtain the dashed curves in Fig. 3. When fluid particles encountered the front line, E q 7 was used to determine the particle’s new loca- tion, and then the time integration of E q 5 resumed. Residence time on the front line in our model was ignored. This, coupled with our simplified representation of the front flow, ac- counts for the poor agreement in particle loca- tions while they are in Castro’s front flow (i.e., t = 3H/uaue). However, the good agreement shown in Fig. 3 between particle locations en- tering and exiting the front flow lends support to the notion that a simple model ( E q s 5 and 7 ) describes accurately enough the cumulative ef- fect of the fountain flow.

SPACE-TIME DISTRIBUTIONS To determine lines of constant residence time

in the mold, one has to know when material particles entered the mold and their flow history up to the present. I t is convenient to distinguish three groups of fluid particles: fluid particles first to enter the mold; particles that entered the mold later and did not reach the flow front; and particles that entered later and did reach the flow front. Obviously, the fluid elements that were at the front line when the filling process started (at time t = 0) were the first to enter the mold. These first fluid elements will

L _ _ _ _ \ l _ _ _ _ _ i = H / v

&----_-__----____----------I- t=8H/?

Fig. 3. Schematic representation of the motion for a ma- terial line entering the mold near the center line (0.05 < [ < 0.5). Solid curves are results obtained using Castro’s model (Eqs 1 1 and 12) and dashed curves are results obtained using the present model (Eqs 5 and 71.

form an envelope of maximum residence time in the mold. For convenience, the residence time is referenced to the current filling time:

t* = tre,/t. (13 ) Therefore, at any time during the filling pro-

cess, t , points on this envelope of maximum residence time will have a dimensionless resi- dence time, t* = 1 . In accordance with E q 1 , the dimensionless longitudinal positions for these points will be given by

ve = $’ U*(EeW* = u*(Ee) l c r < te < 1 . (14 )

Figure 4 shows this envelope of maximum residence time and the regions where the other two groups of fluid particles mentioned above can be found for a Newtonian fluid. Fluid ele- ments below this envelope never reached the front line, whereas above the envelope, all fluid elements have experienced the fountain flow. Pearson ( 1 8 ) presented a similar concept by defining a n interface between regions of mate- rial in mold flow that have and have not suf- fered mixing at the front line (assuming mixing through fountaining).

Below the envelope of maximum residence time, the residence time of a fluid element cur- rently at position (7, t ) is simply

t2elow = v / ~ * ( t ) . (15 ) For fluid elements above the envelope of max-

imum residence time, the residence time in the mold is the sum of the time spent reaching the front line, t?, the time spent in the front line, t;, and the time passed after experiencing the fountain effect, t?.

t:booe = t ; + tz’ + t;. (16 ) For a particle currently above the envelope at

position (77’. t ’ ) , the time spent reaching the front line is given by

where E is related to [’ through E q 4 and v1

.. I

EWPERIENCEO F D W T A I N FLOY

Y NEVER REACHED FLOY FRCUT

0 1 OIMENSIDNLESS LCNGTHn

Fig. 4. Envelope of maximum residence time in the mold for a Newtonianfluid.

POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, Vol. 27, No. 16 1231

I. Manas-Zloczower, J . Blake, and C. Macosko

represents the position of the front line when the particle overtook it, referenced to the cur- rent front line position.

As mentioned earlier, in this paper the resi- dence time of particles on the front line is ne- glected; i.e., t$ = 0. This assumption is valid for molds with a large length-to-thickness ratio.

The residence time elapsed since the particle experienced the front flow can be calculated in terms of the distance currently separating the particle and the front line, 1 - q' , and the particle's current velocity relative to the front velocity, 1 - u*( [ ' ) :

1 - q'

1 - u"(4')' t; =

Knowing t ; enables us to calculate v 1 . the lo- cation of the front line when the particle en- countered it:

q1 = q' - u*(")t$. (191 Combining Eqs 17, 17, and 19 into Eq 16

yields the total residence time of a fluid particle currently located above the envelope at (7'. 4 ' ) .

where 4 and 4' are related through Eq 4. Equations 4 , 14, 15, and 20 constitute the

general equations necessary for generating space-time distributions during mold filling. For the specific case of a Newtonian fluid, with velocity profile given by Eq 5 and a n idealized front flow given by Eq 7, E q s 14, 15, and 20 become

and

The space-time distribution in filling a mold with a Newtonian fluid is shown in Fig. 5. It was obtained by plotting the envelope with Eq 21 and then Eqs 22 and 23 for constant values of the dimensionless residence time, t*.

A similar analysis can be carried out for a power-law fluid. With the velocity profile given by Eq 8 and a n idealized front flow given by Eq 10, Eqs 14, 15, and 20 become

s + 2 s + l 77e = - (1 - Ed") (24)

and (s + l)q'[(s + 2)F+1 - 1 )

(26) + (s + 2)(s + l)(['S+l - Ffl)

(s + 2)(1 - ?'l)[(s + 2)4'S+l - 11' t L u e =

The space-time distribution for mold filling with a power-law fluid (s = 2.5) is shown in Fig. 6.

APPLICATION OF SPACE-TIME DISTRIBUTION TO RIM

RIM is the high-speed production of polymeric parts by quick injection of low-viscosity reactive monomers or prepolymers into a mold. Dur- ing mold filling, temperature and conversion changes influence the viscosity of the material. Mold filling is typically a fast step (1 to 5 s) , and the aforementioned changes combine to give a nearly constant viscosity. Under the assump- tion of constant viscosity, space-time distribu- tions as described above can be generated.

Considering a phenomenological nth order ki- netic expression with Arrhenius temperature dependence and neglecting heat conduction in the main flow direction, one can write the ap- propriate dimensionless energy and mole bal- ances: DT* 1 2xfd2T* Da 2xf Dt* P e H at2 P e H

- + - - k*(l - c*)" (27)

k*(l - C* 1 (28) Dc* Da 2xJ Dt* P e H

-

0 DIMENSIONLESS LENCTH.r)

1

Fig. 5. Space-time distribution for mold filling with a Newtonianfluid.

0 DIMENSIONLESS LENC1H.r)

Fig. 6. Space-time distribution for mold filling with a power-lawfluid (s = 2.5).

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Space-Time Distribution in Filling a Mold

with UaveH

2a P e = - (29)

The Peclet number describes the relative im- portance of convective heart transfer to con- ductive heat transfer. The Damkoehler number can be interpreted as the ratio of heat genera- tion due to chemical reaction to heat conduc- tion. For Peclet numbers much larger than un- ity, heat conduction can be neglected in Eq 27. With only source terms remaining on the right hand side of E q s 27 and 28, a fluid particle's temperature and conversion depend only on its residence time in the mold. Under these condi- tions, lines of constant residence time in the mold are also lines of constant temperature and conversion.

For RIM polyurethane systems, Castro and Macosko (1 9) reported the thermal and kinetics parameters presented in T a b l e 1.

In a typical RIM process, with u,,, = 0.2-0.4 m/s, H = 3.2 X m, and 2xflH = 300-600 at the end of filling, the Peclet number varies between 3500 and 7000, and conductive heat transfer can be considered negligible. For ex- ample, given an initial material temperature of 50°C and a total temperature rise of 20 to 40°C during the filling stage, the dimensionless group ( o a k " ) ranges from 5 to 18, which justifies re- taining the source term in E q s 27 and 28. Under these circumstances, lines of constant resi- dence time represent lines of constant temper- ature and conversion as well.

Figures 7 and 8 show the space-conversion and space-temperature distributions at the end of filling (2.45 s) for the RIM 2200 polyurethane system, using To = 50°C.

From Figs. 7 and 8, conversion and temper- ature at different locations in the mold after filling can be easily seen. Figure 9 shows con- version profiles at different axial locations as taken from Fig. 7. Also shown in Fig. 9 are points calculated with the model for the RIM process developed by Castro and Macosko (20). for which the momentum, energy, and mole balances were integrated numerically using ex- plicit finite differences. In Castro's model, vis- cosity was allowed to vary with conversion and temperature, and the wall was assumed to be isothermal. Figure 10 compares temperature

Table 1. Thermal and Kinetic Parameters

Par am e t e r Model System-RIM 2200

a (m2/s) 9.1 x 10-8

co (mol/rn3) 2.4 x 103

n 2 ko (m3/mol s) 10560 E (kJ/mol) 53.2

ATad (4 126

W

r'

2 I Y)

w A

z 5 T

0

0 1 OIHENSIONLESS LENG1H.r)

Fig. 7 . Space-conversion distribution at the end of filling for a RIM 2200 polyurethane system with To = 50°C and

= 2.45 s.

r 3

I Y

YI

6 .. P

0

0 1 DIMENSIONLESS LENC1H.r)

Fig. 8. Space-temperature distribution at the end of fill- ing for a RIM 2200 polyurethane system with To = 50°C and t f = 2.45 s.

W +. 2 I YI

Y

6 H .z Y)

0

0 .5 CONVERSION

Fig. 9. Conversion profiles at the end of filling for the system in Fig. 7 at several longitudinal positions in the mold. Symbols are results obtained using Castro's model with an isothermal wall at 65°C.

profiles at different axial locations from Fig. 8 at the end of filling with points calculated by Castro's model. Differences between the model results are due to our idealization of the front flow, our assumption of constant viscosity, and

POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, Vol. 27, NO. 16 1233

I . Manas-Zloczower, J. Blake, and C. Macosko

- ,--- \-?--I

a q-0.25

i q -0.5

.3 100

TEHPERATURE. C 50

Fig. 10. Temperature profiles a t the end of filling for the system in Fig. 8 at several longitudinal positions in the mold. Symbols are results obtained using Castro’s model with an isothermal wall at 65’C.

the fact that we neglected heat conduction. The disagreement at the wall is due primarily to Castro’s isothermal boundary condition.

CONCLUSIONS In this work, a simple representation for the

fountain flow is presented. I t is followed by a general approach to calculate space-time distri- butions during mold filling. Results are pre- sented for Newtonian and power-law fluids. Space-time distribution is particularly applica- ble to typical RIM processes, where filling is a fast step and conductive heat transfer can be neglected. Under these circumstances, the space-time distribution also gives the tempera- ture and conversion distributions. The predic- tions of this simple model compare well with results obtained using a numerical solution of the momentum, energy, and mole balances for the RIM process developed by Castro and Ma- cosko (20). The conversion and temperature dis- tributions are useful for process control and optimization. They give guidelines for prevent- ing premature gelling or thermal degradation during filling and represent initial conditions for the curing step.

ACKNOWLEDGMENTS The authors gratefully acknowledge Mr. Rob-

ert Secor for many helpful comments and dis- cussions. They also thank the Productivity Cen- ter-Institute of Technology, University of Min- nesota, and the National Science Foundation for partial financial support.

NOMENCLATURE c = concentration of reactive species at time

c, = initial concentration of reactive species. c* = extent of reaction (conversion): (c, - c)/co. c, = heat capacity at constant pressure. E = reaction activation energy. H = mold thickness. H , = heat of reaction.

t.

K = thermal conductivity. k =kinetic rate constant evaluated at

ko = pre-exponential factor in kinetic con-

kTo = kinetic rate constant evaluated at To. k* = dimensionless reaction rate: k/kTo. n = reaction order. R = gas constant. s = reciprocal of the usual power-law model

parameter shear stress a (shear rate)’/s. T = temperature. To = initial material temperature. T* =dimensionless temperature: (T - To)/

t = current filling time. tf = total filling time. t,, =residence time of material particle in

t* = dimensionless residence time: t,/t. u =velocity. uave = average velocity. u* = dimensionless velocity: u/uaue. x = axial direction. x, = position of the front line. y = transverse direction.

Greek Letters (Y = thermal diffusivity: klpc,. ATad = total adiabatic temperature rise: H K ~ /

.$ = dimensionless transverse position:

7 = dimensionless axial position: x/xP p =density.

Subscripts above = above the envelope. below = below the envelope. cr e = envelope.

Superscripts ’ = particle position after experiencing the front

Dimensionless Numbers D a = Damkoehler number, Da = H2kT,co”Hr/

P e = Peclet number, P e = ~ a u e H / 2 ( ~ .

T k,e-E/RT.

stant.

AT,,.

mold.

PC,.

Y /W/2).

= position of u* = 1 .

flow.

4 kATad.

REFERENCES 1. R. S. Spencer, and G. D. Gilmore, J. Colloid Sci., 6. 1 18

(1951). 2. R. L. Ballman, T. Shusman, and H. L. Toor, Mod. Plast.,

37, 105 (1959). 3. R. L. Ballman, T. Shusman, and H. L. Toor, Ind. Eng.

Chem., 51, 847 (1959). 4. W. Rose, Nature, 191, 242 (1961). 5. L. R. Schmidt, Polym. Eng. Sci., 14,797 (1974). 6. 2. Tadmor, J. Appl. Polym. Sci., 18, 1753 (1974). 7. S. Bhattacharji, and P. Savic, “Real and Apparent Non-

Newtonian Behavior in Viscous Pipe Flow of Suspen- sions Driven by a Fluid Piston,” Proc. of the 1965 Heat Transfer and Fluid Mech. Inst., 248 (1965).

1234 POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, Vol. 27, NO. 16

Space-Time Distribution in Filling a Mold

8. J. M. Castro, “Mold Filling and Curing Studies for the Polyurethane RIM Process,” PhD Thesis, University of Minnesota (1 980).

9. B. Silliman, “Viscous Film Flows with Contact Lines: Finite Element Simulation, a Basis for Stability Assess- ment and Design Optimization,” PhD Thesis, University of Minnesota (1 979).

10. C. F. Huang, “Simulation of the Cavity Filling Process with the Marker-and-Cell Method in Injection Molding,” PhD Thesis, Stevens Institute of Technology (1978).

11. L. T. Manzione, Polym. Eng. Sci., 21 , 1234 (1981). 12. M. R. Kamal, E. Chu, P. G. Lafleur, and M. E. Ryan,

SOC. Plast. Eng. Annual Technical Conf. 31, 818 (1 985).

13. J. D. Domine, “Computer Simulation of the Injection Molding of a Liquid Undergoing a Linear Step Polym- erization,” PhD Thesis, Stevens Institute of Technology (1 976).

NOTE IN PROOF Recent work on the kinematics of fountain

flow by Coyle et al. (21) and Blake (22) provide an additional test of the simplified fountain flow model presented in this paper. Finite ele- ment solutions of the Navier Stokes equations give the detailed flow field, from which material line deformation can be calculated. A compari- son similar to Fig. 3 is shown in Fig. 11. To capture all of the flow, the tracer line was se- lected to span the mold thickness. As in Fig. 3 there is poor agreement in the front flow detail, but good agreement in terms of the cumulative effect of the front flow.

14. Z . Tadmor, and C. G. Gogos, “Principles of Polymer Processing,” p. 568, John Wiley, New York (1979).

15. W. H. Beyer, ed., “CRC Standard Mathematical Tables,” p. 10, CRC Press, Boca Raton (1981).

16. R. L. Burden, J. D. Faires, and A. C. Reynolds. “Numer- ical Analysis,” p. 34, Prindle, Weber, and Schmidt, Boston (1981).

17. C. W. Gear, and L. R. Petzold, SIAM J. Numer. Anal., 21, 716 (1984).

18. J. R. A. Pearson, “Mechanics of Polymer Processing,” p. 605, Elsevier Applied Science, (1985).

19. J. M. Castro, and C. W. Macosko, SOC. Plast. Eng. Annual Technical Conf. 26. 434 (1980).

20. J. M. Castro, and C. W. Macosko, AlChE J., 28, 250 (1982).

21. D. J. Coyle, J. W. Blake, and C. W. Macosko, AlChE J. 33. 1168.

22. J. W. Blake, “Studies in Reaction Injection Mold Filling,” PhD Thesis, University of Minnesota (1987).

The finite element solutions of Coyle et al. (21) were also used to generate the space-time distribution shown in Fig. 12 for comparison to Fig. 5. The average front line, 7 = 1 in Fig. 12, corresponds to the product of the average fluid velocity in the main flow and the current filling time, and is fixed by requiring that the fountain flow area between the wetting line and the tip of the advancing front be halved by this line. The same average front line is used in Fig. 11. Discrepancies between Figs. 5 and 12 arise because the residence time in the front is ig- nored in Fig. 5. These results show further that the simplified fountain flow model adequately describes the cumulative effects of the fountain flow.

-3 k 2 H N = 1=3HN

_ - _ _ _ _ --_ t=8H/Y

_ - - _ _ _ _ - - ~ _ _ _ Fig. 1 1 . Schematic representation of the motion for a material line spanning the gap thickness. Solid curves are results obtained using the fountain flow model of Coyle et al. ( 1 987) and dashed curves are results obtained using the present model (Eqs 5 and 7).

up I- I (3 W I - cn cn W

0 1

DIMENSIONLESS LENGTH,q Fig. 12. Space-time distribution for mold filling with a Newtonianfluid using the fountainflow model of Coyle et al. (1987).

1235 POLYMER ENGINEERING AND SCIENCE, MID-SEPTEMBER, 1987, Vol. 27, NO. 16