Estimation of Injection Pressure During Mold Filling*+

C . TITOMANLIO

Istituto di lngegneria Chimica Unioersita di Palerrno, Italy

Dimensionless diagrams for estimating the bulk tempera- ture of the Row front aiid injectioir pressure i n the linii t of s111a11 viscous generation are ol)tained. Also, a criterion for ne- glecting viscous generation is identified. The cli~igranis, based on the Lord and Williatns model, refer to rectangular geometry i ind amorphous materials. A satisfactory coniparison is oh- tained with literature data taken on polystyrene. A reasonable estimate of polyethylene injection pressure was obtained by roughly accounting for latent heat of crystallization through i n ocli fi e tl thermal d i ffu s i vi ty .

INTRODUCTION njection molding is the first polymer processing op- I eration as Ear as mass production is concerned. This

priority motivated the recent effort toward the under- standing of phenomena occurring during the process cycle whose optimization is still far from being achieved.

Several analyses of mold filling have been recently developed (1-21) obtaining a good agreement with ex- periments. However, these analyses do not give results that can be currently utilized, but rather suggest com- plicated numerical procedures that have to be adopted case by case. As a conscquence, the processing condi- tions are often empirically estimated by operators.

Only recently (22, 23), some effort has been made in the direction of providing, on a scientific basis, normal- ized diagrams and computational methods that can be used for direct evaluation of important processing variables.

This work is a further step in this direction; in particu- lar, on the basis of the model suggested by Lord and Williams (14), a normalized injection pressure curve is obtained in the limit of small viscous generation.

Governing Equations

During mold filling, hot molten polymer is forced to flow into the mold cavity whose walls are at a tempera- ture. T,,., lower than the polymer solidification temper- ature. The simulation of this process obviously requires the equations of motion, continuity, and energy; fur- thermore, the rheological constitutive equation also has to be specified.

The model considered here refers to a cavity having rectangular geometry, with thickness, 2 Y, much sinaller than both the width, w , and the length, L; a schematic drawing of the cavity is shown in Fig . 1 . La- tent heat of crystallization is not accounted for in the

analysis that therefore refers to amorphous materials. The fluid is considered non-Newtonian but viscous.

The balance equations, available in Ref. 14, were de- veloped under the main hypothesis that the flow is es- sentially undirectional along the x direction. This hy- pothesis is not correct very close to the gate but soon becomes a good approximation to the flow field. The equations will be summarized here directly in dimen- sionless form; the case of a constant flow rate will be considered. Bold face characters will be adopted for di- mensionless quantities.

Continuity and momentum equations are

l ' u d y = 1 (continuity) (1)

(momentum) (2) dP dx !> f dy

u = --

where

u = u / u (3)

U being the cross-section average viscosity, 7,. a refer- ence viscosity, which will be specified below, and (Y

I I

POLYMER ENGINEERING AND SCIENCE, JULY. 1982, Vol. 22, No. 10

G. Titomanlio

the thermal diffusivity. The dimensionless axial posi- tion x is the inverse of Greatz number.

Equation 2 has been obtained under the hypothesis that inertial, body forces, and pressure variations along the thickness direction are negligible.

Neglecting longitudinal conduction and transverse convection, the energy equation becomes

(8)

(9)

t = at/y2 (10)

(1 1)

Ti being the inlet polymer temperature, which is con- sidered constant. The dimensionless quantity B is Bridgman number evaluated on the basis of the refer- ence viscosity 77,.

dT dT dP d u a t ax ay2 dx dy

T = (T - T,.)/(Ti - TIC)

B = qrU2/K(Ti - Tic)

- + u - = a2T + B y - -

where

The thermal boundary conditions are:

dT - (x, 0, t) = 0 (Axial symmetry) (12)

T(x, 1, t) = 0 (wall temperature) (13)

T(0, y, t) = 1 (Inlet temperature) (14)

dY

T(Xf, Y> tf) =% x7) (15)

Equations 12 to 14 are self-explanatory. Equation 15 defines the temperature of the volume element filled at time, tf, as equal to the bulk temperature of the fluid element slightly upstream, which in Eq 15 is indicated by its axial position x i .

Equations 1 , 2 , 8 , and 12 to 15 were solved numeri- cally following the procedure reported in (14). The computer program, as in (14), was checked for the case of Greatz problem for fluid flowing in a rectangular slit: B = 0 and q = const.

VISCOSITY EQUATION

For most of the calculations performed here, and ex- cept when explicitly specified, the effect of tempera- ture and shear rate on viscosity was described as fol- lows.

The WLF equation was used to express the depen- dence of viscosity upon temperature. In particular, with reference to the equation

r) = r), exp{-C,(T - T,)/(C, + T - T,)} (16)

the values of 20.4 and 101.6"C were given to the con- stants C1 and C,; with these values, the reference tem- perature T,s is T,y -- T , + 45°C (24).

As for the effect of shear rate 7 , the relation derived by Bueche (25) was adopted. This relation is of the type

where r),, is the value of r) at the same temperature and zero shear rate. I' is a dimensionless quantity having expression

where M is molecular weight, p is density, and G,, a modulus defined by Eq 18.

In conclusion, the following equation was adopted

r) = q r exp{-Cl(T - TJ/(Cz + T - TJ) * f(r) whose dimensionless form is

T.I = exp{-C,(T - T,)/

(19)

(T,C + T - T,J} . f?BI"Zq),, (20)

where

Z = {Kq, (Ti - Tlr)1'2}/YCo

C = 1O1.6"C/(Ts - Tll.)

T,v = ( T,s - Tf,.)/(Tj - TI,.)

(21)

(22)

(23)

i , = 7 Y / U (24

RESULTS AND DISCUSSION

The temperature has been non-dimensionalized so as to make the boundary conditions to Eq 8 independent of the inlet and the wall temperature Ti and Tll. . How- ever, C and T, are determined by T,s - Ti,. and Ti - TI, . . Furthermore, since Cis proportional to the inverse of T , - T,,., T , - TIC is easily related to the value of C, consid- ering that T , = T , + 45°C (24); moreover, T i - T, can be obtained from C and T, by the relation

Obviously, C and T,7 determine both the value of q at the cavity entrance, where T = 1, and the rate of viscos- ity changes with T; both may vary by orders of magni- tude for reasonable changes of T, and C.

Apart from C and T,, the mathematical problem con- tains two additional parameters: B, which is a scale for the viscous generation, and Z (or ZB"'), which deter- mines the non-Newtonian behavior.

All the equations are tightly coupled; in particular, the energy equation, Eq 8, is linked to the others through both the axial convection and the viscous gen- eration terms. The latter is proportional to the pressure gradient, which decreases with the viscosity (or some average viscosity on the cavity cross-section), and, when it is sufficiently small, the viscous generation can be neglected in Eq 8.

On a qualitative basis, one can say that the pressure gradient increases as the melt temperatrire decreases; furthermore. if the viscous generation is negligible, the cross section average temperature T is smaller for larger values of the axial position x. The pressure gradient, then, assumes its maximum value close to the melt

-front, on which a criterion for viscous generation omis- sion has to be based.

Several calculations were performed for very differ- ent values of the parameters, both neglecting and re- taining the viscous dissipation in Eq 8. It was observed that the effect of viscous dissipation on the pressure Pi at the cavity entrance is negligible for small values of B . q(Tf, ? = 3) where q(Tf, ? = 3) has the meaning of the value of r) evaluated at both the bulk average tempera-

-

590 POLYMER ENGINEERING AND SCIENCE, JULY, 1982, Vol. 22, No. 10

E s t i m u t i o n of Injection Pressure During Mold Fil l ing

ture'T, of the flow front and the dimensionless shear rate i. = 3. The effect of viscous dissipation starts to be relevant for

V = B . q(Tf, = 3) 2- 0 .1 (26) In this limit, three parameters are left and the cou-

pling of the energy equation with the others is related only to the axial convection term, u dT/dx. I t was, how- ever, observed that, for V smaller than 0.1, the bulk temperature Tf at the flow front is essentially inde- pendent from the three parameters left (i. e . , C, T,, and ZB"'), at least for reasonable values of Ti - T, and Ti - TI+ order to show the deviations between the values of Tf, some curves, covering the ranges 7O"C-17O0C, and 15"C-2OO0C for T , - T,,, and Ti - T,, respectively, are reported in Fig . 2 vs the flow-front position, xf. Larger values of xf have not been considered in F i g . 2 , as xf = 0.5 is already a very large value that, except for cases when Ti - T,7 is exceptionally higher than T, - T K , corresponds already to enormous injection pressures.

The fact that the curves of Fig . 2 are essentially coin- cident may appear somewhat surprising; the explana- tion is related to the following stabilizing mechanism. If q changes more rapidly with y, a parallel increase of the rate of change of u from the center line to the wall is produced; as aconsequence, far from the x axis, the con- vection term is reduced and the local temperature can decrease more rapidly with x. On the other hand, be- cause of the more rapid decrease of u with y z h e weight of the center line temperature in evaluating Tfbecomes larger, and the previous effect (smaller temperature close to the walls) is counter-balanced as far as the final value ofTf is concerned.

These arguments hold also if a form different from E q 19 is adopted for viscosity. Furthermore, depending on the values given to the parameters and, for a given set of parameters, depending on the flow-front posi- tion, different zones of the Bueche viscosity curve are involved in determining the main features of the veloc- ity distribution in each section. As the Bueche curve goes from a Newtonian behavior for small values of r to a negative slope of about 0.6 (on a log versus log r plot) for larger r values, the correlation of F i g . 2 does not have to be considered as related to the particular viscosity cquation adopted; for instance, with reference to the case C = 1.5, T,,, = 0.25, and ZB"' = 0.1, only the Newtonian zone of the Bueche viscosity curve deter- mined the main characgr of the velocity distribution, and thus the values of Tf reported in F i g . 2 .

0 1 , X I

0 1 2 3 4 6

Fig . 2 . B u l k uceruge teniperutrire Tiofflow f ron t L'S i t s p(isition Coniputeti d u e s (ire irtdicuted li!/ synibols.

However, in the case of a viscosity constant over the whole mold, a small but significant increase of the rate of change of Tf with xf was found.

Obviously, above the limit defined by E q 14, values - higher than those reported in Fig . 2 were obtained for Tr.

Contrary to what was observed with reference t o y f , the pressure Pi at the cavity entrance is enormously influenced by C, T,, and ZB"', especially in the limit ofsmall viscous generation. This is because the pressure gradient follows q, which, for a given T, may induce change of orders of magnitude depending on the values of C, T,, and ZB"'.

I t was verified that, insofar as xfis very small a n d y f i s still very close to 1, a suitable normalizing factor for Pi is q(T = 1, = 3) .

This, together with the expression of V, may suggest the use of q = q(T, i. = 3) as good measure for the pressure gradient. In order to obtain a normalization of the entrance pressure also for large values of xf, onc should thus consider the ratio between Pi and the in- stantaneous valEe of the integral of the cross-section a j - erage viscpity 4 over the rangc 0 - xp Unfortunately, Tf and thus qf, are functions of time; a t y p g l behavior is shown in F i g . 3 , wkere the temperature Tf, the steady- state temperature T,, and a series of curves giving for each time the average temperature distribution over thc plastic are reported. Obviously, the latter curves e n d u p over that of Tf.

Performing exactly the integration of 6 would be of little practical use, as the integral, for each time t, would not be an analytical function of C, T,, and Z . B"*. O n the other hand,as shown in F i g . 3 , near the flow front, the curves T(x) show a nearly horizontal zone; furthermore, just before the horizontal zone of these curves, when T, is approached, an extended solidified layer is already formcd, and the higher tem- perature is somehow compensated for as far as the pres- sure gradient is concerned. Most of the contribution on the pressure drop is concentrated in both these zones, which become more extended as xf grows.

These observations suggested normalizing Pi with qf. Xf.

The quantity -

N = Pi/qf. xf (27)

0 1 a 3 4 6

Fig. 3 . B u l k uvei-uge ter~i~)eruture us f~mct io i r oft i i i ic tund p s i - tiori x; t!/picu/ cuse. T,, teniperutrrre ofJo1L: f r o n t ; T,, t e ~ i r p e r ~ ticre distribirtiori a f t e r infinite time; ---, teriiperutctre d i s t r i h - tioii ut time t.

POLYMER ENGINEERING AND SCIENCE, JULY, 1982, Vol. 22, No. 70 591

G. Titomanlio

with q,evaluated, for each value of xf, on the basis ofy , given by the fill1 line reported in Fig . 2 , was calculated for several sets of parameters covering the range of C, D, and ZB”’, already considered in Fig. 2. The values of N resulted closely within 20 percent to the curve re- ported in Fig . 4 .

The shape of the curve N(xf) is roughly determined by the variation with xfof both the fraction of the mold sec- tion really available for the flow and t h e extension-of the zone having average temperature T close to T,. Should the former be constant and the latter propor- tional to xf, N should be essentially independent from xf. Actually, a solidified layer forms on the mold walls, whose thickness grows with xf; this reduces the area available for the molten polymer, thus raising the value of N. On the other hand, for large values of xf, the ex- teEion of the x zone having avcrage temperature close to Tf grows very weakly with xf; this tends to progres- sively decrease the slope of the N(x,) curve. The combi- nation of these two effects generates the maximum in Fig. 4 .

It was already mentioned above that 0.5 is a large value for xf: except for cases when very large values of both the inlet temperature Ti and the mold wall tem- perature T,r are adopted, the region beyond the maxi- mum has no interest. Furthermore, before any use of Fig. 4 , one should verify that the viscous generation may be neglected by comparing the value of V with 0.1 (see E q 27) .

COMPARISON WITH DATA The use of Figs . 2 and 4 is trivial if the operating con-

ditions correspond exactly to the hypothesis under which they have been obtained.

Unfortunately, data taken under constant flow rate were not found in the literature together with all the needed information.

A comparison is, however, made with experimental results reported in (19), for which the flow-front posi- tion q w a s measured as a function of time and is plotted in the bottom ofFigs. 5 and 6. At low times, the original curves of X I versus t (dotted lines) display an inflection due to the transition from radial to parallel flow (26); the extrapolation (as shown in Figs. 5 and 6 ) of the large- time behavior was considered more accurate in eval- uating the flow rate at these times than the use of the experimental curve.

Making use of the definitions of Pi, xf, and vf, Ey 27 may be written as

10

6

20

10

0

iI t ,se t

0 1 2 3 4

Fig . 5 . Pressurc. u t cucity entruirce P , us injection time t ;Jou;- frnnt position c s time is plotted ilr the Imttoni; u t snrull times, dattr (tfottcd / ine) huce been smoothed.

^^ I U I

15

- exp, kuo and kamal(19)

- - -computed , 1

0 0

-4 , a.6.5.10

0 -4 0 . _ = . *

1 -

0 1 2 3

F i g . 6. Presszm ut cul;ity etitrunce P, c.s ii!jection time t . Flow- f r o n t position v.9 time is plotted in the bottom; cit s l i d 1 times, datu (dotted lirre) huce been snioothecl.

which is similar to the expression of the pressure drop undergone by a Newtonian fluid in a rectangular slit.

E q 28 was held for the comparison. As for the factor N , it depends essentially upon the energy transport process, and conceivably has to be related to time rather than position.

xf was then evaluated as

xr = p at (29)

POLYMER ENGINEERING AND SCIENCE, JULY, 1982, Vol. 22, No. 10

Estimation of Injection Pressure During Mold Filling

which, for the case of constant flow rate, is equivalent to the definition given above, E 9 7 .

Polystyrene and low-density polyethylene are con- sidered in Figs. 5 and 6, respectively. The former is amorphous and has a non-Newtonian behavior very close to that described by the Bueche relation used for the calculations. The latter crystallizes upon cooling and has a considerably less pronounced non-Newtonian behavior. Steady shear viscosities as a function of tem- perature and shear rate are available in (19) for both res- ins; these have been used directly for evaluation of r)f, whose values were needed for both verifying the value of V and calculating Pi according to E 9 28. The experi- mental results and the values ofPi obtained by means of E9s 28 and 29 and ofFigs. 2 and 4 are shown in the top of Figs. 5 and 6.

For the polystyrene resin, the comparison is satis- factory.

As for polyethylene (semicrystalline), together with the data, three calculated curves are reported, each of them being evaluated for a different value of the thermal diffusivity a. The higher one corresponds to the thermal diffusivity of the material in the molten state; a was considered a fitting parameter for the lower one. As for the third, the thermal diffusivity was evaluated by replacing the melt-specific heat by

where T , indicates the melting temperature, A the la- tent heat of fusion, and ( a degree of crystallinity to which the value 0.5 was given. The resultant Pj curve is not very close to the experimental one but is nearly within the approximation declared for Fig. 4 .

CONCLUSIONS The Lord and Williams model (14), which refers to an

amorphous material filling a rectangular geometry mold cavity, has been considered. In the limit of small viscous generation, identified by E9 26, the mathemat- ical dimensionless problem contains three parameters. It was found that, within this limit, the bulk dimension- less temperature of the flow front does not depend sig- nificantly upon them. Also, a proper dimensionless in- let pressure accounting of the main effects of the parameters was identified.

The results, summarized in Figs. 2 and 4, allow the immediate evaluation of the inlet pressure; a compari-

son with literature (19) data taken on polystyrene (amorphous) is shown in Fig. 5 .

A tentative effort is made to extend the use of Figs. 2 and 4 also to semicrystalline polymers; in particular, crystallinity is roughly accounted for through a modi- fied thermal diffusivity.

ACKNOWLEDGMENT Thanks are due to ing. M . Gibbardo for performing

most of the computer work.

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