R. L. BALLMAN and TEVlS SHUSMAN Monsanto Chemical Co., Springfield, Mass. H. L. TOOR Carnegie Institute of Technology, Pittsburgh 13, Pa.
Injection Molding Flow of a Molten Polymer info a Cold Cavity This study of the flow of a molten polymer into a cold cavity may lead to an understanding of the relationship between rheological properties and iniection molding
INJECTION
J
molding is a major polymer fabrication operation. Many of the broad relationships among the varied phenomena that occur in the process have been reported (Z), but to predict the effect of changing variables such as temperature or pressure, or whether a polymer with particular rheological and thermal properties can be successfully molded in a particular die and press, demands a much deeper understanding than is now in the literature. Previous studies of separate parts of the process have been reviewed (7). This research program, the initial stages of which are reported here, is aimed at understanding the basic mechanisms in terms of the fundamental rheological and thermal properties of the polymers. Possibly the most complex and important phase of the process is the flow into the die. The viewpoint of this study is that the die in an injection molding machine is a nonisothermal rheometer, and that the flow in a particular die will be completely defined once the inlet die pressure and temperature are specified.
du/dr = f( T , r )
(1 1
does this equation hold a t every point in unsteady, nonisothermal flow in a die where the mean and local velocity vary strongly with time and the local temperature varies with time and position? This would be true only if relaxation effects were negligible in the sense that the time necessary for the polymer to readjust to time-varying conditions is much less than the time scale of the experiment. (It is assumed that the flow is always laminar, as the highest Reynolds numbers obtained with these very viscous fluids are well below the critical value for turbulence. Actually, if the flow regimes in the rheometer and molding machine differ, the following experiment should indicate it.) To answer the original question, two commercial polystyrene molding powders, compounds A and B, were run in a n isothermal capillary rheometer a t a series of different temperatures and shear rates. Over the range of measurement (400 to 500’ F., shear rate 50 to 3000 sec.-l) the increase in temperature required to give A the same
apparent viscosity as B varied only from to 34’ F. Thus, if rheological measurements are of significance to injection molding, the same shift of the temperature scale in molding should cause these materials to behave in a like fashion. I n molding, if ram pressure and cylinder temperature are low enough, the polymer cannot fill the die, and it will freeze at some point inside the cavity. This length of flow, called “fill-out,” will depend strongly upon all the parameters controlling flow and heat transfer and, consequently, is a convenient measure of the gross behavior of the polymer in unsteady, nonisothermal flow. Because the temperature varies from the hot cylinder temperature to the cold die temperature, the fill-out measures the integrated effect of the rheological properties over wide ranges of shear stress and temperature. The die used was a bar 0.075 inch thick and 1 inch wide, connected by a narrow channel to the nozzle of a 3ounce press (Figure 1). Figure 2 shows the fill-out for compound B, measured as length of bar, as a
28’
Preliminary Experiments A preliminary experiment was carried out to determine whether it is possible, a t least in principle, to describe the flow in terms of the steady-state rheological properties of the polymer-Le., if the relationship between local shear stress and rate of strain measured in an isothermal steady-state capillary rheometer is
The complete article, here condensed, will be published in Modern Plastics.
COOLED DIE SOLID POLYMER
HEATERS
I
R
MOLTEN
P~LYMER
u
> L
DIE CAVITY
In the injection molding process aolid polymer particles are forced into the heating cylinder and the molten polymer which is displaced from the cylinder flows into the die, where it is frozen into the finished product VOL. 51, NO. 7
JULY 1959
847
"
RUNNERS 5/16" OIAM. HALF- ROUND
RAM PRESSURE= 18,300 PSI Tc To a 300Y
-
R A M PRESSURE. 11,700 psi Tc - To =4OO'F
301
WIDE FAN GATE 0.120"THICK
28
-
E
0 ,-26 3
1
-
24-
B
i-dA I
a t 430'F
1
I
U
28
-
I
I
FATj 3 3 ~
20 I
380
POSITION
i
I
1 ,
400
420
CYLINDER
1
I
,
440
460
480
500
520
540
560
T E M P E R A T U R E , "F
Figure 2. The increase in temperature required to give material A the same fill-out as material B was essentially the same as predicted by rheological measurements
i WIDE FAN GATE 0,120" THICK
Figure 1.
4
%NARROW FAN GATE 0.030" T H I C K
BRASS PLUG
This die was used for flow studies
function of cylinder temperature for two ram pressures. Each point is the average of 30 shots, reproducible to about 5%. The die temperature was varied with cylinder temperature to keep a constant 300' F. difference between the two temperatures at the high pressure and 400" F. a t the low pressure. Material -4 was run with the same differences between the cylinder and die temperatures; as both materials have essentially the same thermal properties, heat transfer should be identical in both cases if the flow is identical, for the reduced temperature parameters are similar. Points for material A at the low and high pressures are shown; at the low pressure, A gives the same fill-out as B when its temperature is 33' F. higher than B's and a t the high pressure 30" F. increase is necessary. These differences are substantially the same as those in the rheometer, indicating that the entire injection molding process depends largely on the rheological relationship obtained in the rheometer, and that relaxation and other effects, if not negligible, at least vary in the same manner for both materials. Two materials \vith similar thermal and rheological properties which differ as above, ("rheologically similar") ill "mold" the same way, whatever the measure of molding may be, if the temperature scale is shifted to make the rheological properties the same.
b Figure 3. Typical distance-time curves Points are reproducible to about 5%
RFT,
steady-state rheological relationship for the polymers under consideration can be correlated by a simple power law. For unidirectional flow, duldr = - A ( T )
T*
(2)
where A ( T)is the temperature-dependent term. Consider the flow into a nonisothermal cavity, at an instant of time in which the length of cavity that has been filled is X. The cavity map be a tube or two wide, parallel flat plates. If Equation 2 is combined with a force balance and a number of reasonable assumptions, integration across and along the filled section yields the mean velocity as a function of time and X, (3)
where
and V is the time derivative of X. T h e evaluation of R(e) in nonisothermal flow is a complex heat transfer problem, as the conduction equations are strongly coupled with the velocity profiles through parameter A and A normally varies over several orders of magnitude during the flow. Because for a finite cavity inlet pressure Equation 3 makes Vinfinite, the filling of an empty cavity cannot be considered in isolation, but must be considered in terms of its interaction with the delivery system. By applying the above equations to the channels preceding the cavity, the cavity inlet pressure can be removed. There results
where the total resistance of the system preceding the cavity is given by
R(sj =
Theory
The above results allow a preliminary analysis of the flow problem. The
848
sec.
INDUSTRIAL AND ENGINEERING CHEMISTRY
(4)
There are two possible types of flow described by Equation 5. I n one the applied pressure, P,', is constant as the
N O N - N E W T O N I A N FLUIDS
10
6
005
I
0075
2
350 70 400 BO
%’ 0
2
Figure 4 . The velocity fell exponentially with time as the cavity filled
>
cavity is being filled-“die-controlled flow.” In the other the system resistance is so low that the delivery system cannot deliver material fast enough to satisfy Equation 5 for any set value of P,’. I n this case P,’rises as the wave front moves down the cavity; this is called “machine-controlled’’ flow because the variation of P,’depends upon the delivery system. The ram pressure differs from P,’ because of a pressure loss through the solid particles in the cylinder ( 3 ) .
Equipment and Experimental Procedure The distance-time relationship in the die was obtained by measuring the length of flow into a cavity as a function of the ram forward time, RFT, measured from the instant the ram started forward. The pressure on the ram is released while the flow is taking place and the length of the molding is taken as the distance the wave front had penetrated the cavity when the ram was stopped. The method assumes that negligible forward flow takes place after the ram pressure is released. This was found to be true. Two presses were used, one rated at a 3-ounce shot capacity and one at an 8-ounce capacity. Both presses had booster pumps which increased the rate of build-up to the set pressure. The die was designed with interchangeable parts, so that various cavity thicknesses, channels, and flow paths could be used (Figure 1). The polymer entered the sprue perpendicular to the plane of the drawing, flowed into the semicircular cross-section runner, and then entered the bar cavity on the left. The cavities were all bars, 1 inch wide, 12 inches long, and 0.150, 0.075,
b Figure 5 . The velocity varied with time in a manner different from what would have been expected for isothermal flow
e, and 0.05 inch thick. These width-thickness ratios should be large enough for the velocity profile in the wide plane to be negligible. Most of the runs were made with polystyrene, compound B in Figure 2. Polyethylene was also used.
Results Forty-eight distance-time curves were obtained. Typical operating conditions are given in Figures 4 and 5 and the table contains measured values of X,. In most runs the ram pressure had reached its maximum value before the flow entered the cavity under study. The shape of the wave front remained fairly constant during a run and from one run to another. Figure 3 shows typical curves where the distance the wave front has moved into the cavity is plotted against RFT. Each point is the average of 30 trials and the points are reproducible to about 5%. All the data shown are for polystyrene except run 36, which is for polye thylene, The velocity time curves shown in Figures 4 and 5 were obtained directly from the raw distance time data by dividing the change in X between two subsequent points by the change in RFT. This average velocity over the
sec.
interval is plotted against the arithmetic average of the initial and final RFT for that interval.
Discussion The sample velocity-time curves for polystyrene shown in Figures 4 and 5 are extremely simple, considering the complex operation taking place. The data are correlated very well by the equation
where constant B is defined by 1/B =
- 1/V
X dV/de
(8)
This same functional form was obtained in all die-controlled experiments. Variations of cylinder and die temperature, ram pressure, gate size, cavity thickness, and press type merely caused variations in parameters VOand B. Polyethylene yielded the same equation as polystyrene. Because the runners preceding the cavity in which the flow is being measured are nonisothermal, Equation 5 indicates that the velocity should depend upon the geometry and heat transfer in the runners as well as in the cavity. However, Equation 7 was found to hold in the runners alone as well as in a cavity preceded by an earlier VOL. 51, NO. 7
JULY 1959
849
cavity (position 2, Figure 1). Shifting position did not alter the functional form but merely changed the constants. Thus, for a constant applied pressure, Equation 7 describes the filling of a cold cavity with hot polymer under very general conditions. I t indicates that the time required for the velocity to fall to zero is infinite (if the Row is not stopped by a wall). Whether the velocity does follow Equation 7 as time approaches infinity is immaterial; all that is required is that the equation describes the velocity until it falls to a very small value. This can be checked by noting that the mean velocity is the rate of change of wave front position, so that Equation 7 may be integrated to give wave front position as a function of time,
which at infinite time reduces to
x, =
IQ 'B
(10)
where X, is the fill-out, the maximum length of flow that can be obtained for any operating conditions. It can he shown from the above equations that in almost all runs the time was long enough so that the maximum length of flow was at least 99% of X,. Consequently for practical purposes Equation 7 can be considered to give a complete description of the flow. Combination of Equations 7, 9, and 10 yields the surprisingly simple velocitydistance relationship, VIVO = 1
- X/Xf
(11)
which can be used in place of Equation 7 to describe the velocity in the cavity. All the data have been plotted against RFT rather than time measured from the instant the wave front entered the cavity, because of the difficulty of determining the zero time point, However, Equation 10 allowed calculation of VO from measured values of X, and B, and Bo could then be determined from the velocity-time graphs. Under certain conditions it was possible to obtain machine-controlled flow by decreasing the pumping capacity of the 8-ounce press; here, as expected, the velocity did not fall exponentially with time, but at a rate controlled by the press characteristics. Although the nature of the flow into the cavities has been determined, the formidable problem of explaining its simplicity and relating constants VO and B to fundamental properties remains. B must depend at least upon the rheological and thermal properties of the polymer, the die and inlet polymer trmperatures, and the cavity geometry and temperature. The cavity inlet velocity, VO.cannot be affected by cavity temperature or geometry (except for the area factor. s) although it may
850
depend upon all the other variables which affect B. V O must also depend upon ram pressure. A useful starting point is a comparison of the experimental results with those expected for isothermal flow. If both the cavity being filled and the preceding channels are isothermal, integration of Equation 5 yields
lationship between the experimental results and material properties and system parameters must depend strongly upon the heat transfer in the die This coupling of the equation of motion and conduction will be considered later. Acknowledgmenf The authors thank R. I. Dunlap, R. L. Heider, and G. W. Ingle, Monsanto Chemical Co., for help in carrying out this work.
where
Nomenclature '4
= rheological
parameter,
1 'sec.
(s*)n
This isothermal equation needs three constants for its description, so that adding the complexitirs of heat transfer has removed a constant and simplified the form of equation. Equation 12 is compared with experiment in Figure 5. The three runs differed only in the die temperature. The average V3 was used in the isothermal calculations and R was calculated at 350' F. from rheological data. The actual pressure applied to the molten polymer is estimated to be in the range of 10,000 to 18,000 p.s.i. and isothermal curves are presented for two values of Pr' to show that the actual presssure is relatively unimportant-in all cases the experimental velocity falls off much more rapidly than the isothermal velocity. (It is easy to show that the isothermal fill-out is infinite.) The difference between the calculated and experimental curves is caused mainly by the heat transfer in the die, and as time increases the cooling of the polymer increases and the experimental values diverge increasingly from the isothermal ones. As the die temperature is increased, the experimental curves shift toward the isothermal and in the limit where the die temperature equals the cylinder temperature, the flow would be isothermal (except for heat generation). Thus somewhere between 180' and 350' F. the velocity time curve should shift from the exponential toward the hyperbo1 c type which is characteristic of isothermal flow. The qualitative as well as quantitative differences between isothermal and non isothermal flow indicate that the re-
Typical Measured Values Expt
€3,
hTo
Sec.
2 32 34 36
1.69
48a
INDUSTRIAL AND ENGINEERING CHEMISTRY
a
0.38 0.67 4.12
Polyethylene.
x/,
T'O,
eo
Cm. C'm./Sec. Sec. 20.6 12.2 1.3 14.5 Machine Control 23.3 61.5 5.4 6.0 9.1 2.4 23.2 5.6 0.9
B
reciprocal fractional rate of decrease of velocity, seconds BO = value of E at time zero for isothermal flow d = channel diameter or thickness of wide flat plates f = functional symbol K = dimensionless constant m = n 1 for Rat plates, n 2 for a tube n = rheological parameter = number of channels preceding P cavity P = pressure, p.s.i. Pr = ram pressure = pressure applied to molten polyp: mer, p.s.1. r = radial distance coordinate, inches R = flow resistance per unit length, p .s.i . (sec./ in.7z + l ) 1' = total flow resistance preceding RT cavity p.s.i. (sec./in."+l)l'n RF7' = ram forward time, seconds = ratio of cavity cross section to Si cross section of channel i. = temperature, ' E'. 1 T, = cylinder temperature, ' F. TD = die temperature, ' F. = point velocity, inches/sec. U = mean velocity, inches/sec., V cm./sec. = axial distance coordinate measured from cavity inlet, inches = reduced axial distance, x / X X' X = length of cavity filled at any time, inches or cm. = fill-out, inches or cm. = reduced radial distance, 2 r / d = time, seconds 0 = RFT when wave front enters $0 cavity = shear stress, p.s.i. 7 =
+
+
'2
SUBSCKIPTS = entrance to cavity or at instant 0 polymer enters cavity i = channel index Literature Cited
(1) Eveland, J., Karma, H. J., Beyer, C. E,, SPE \/OUrnQl 12, NO. 5, 30 (1956). (2) Spencer, R. S., Gilmore, G. D., J . Collozd Sci. 6 , 118 (1951). (3) Spencer, R. S., Gilmore, G. D., Wiley, R. W., J . A)$. Phys. 21, 527 (1950). RECEIVED for review January 2 , 1959 h C C E P T E D March 2, 1959
