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Modeling of the solids-conveying section of a starve-fed single-screw plasticating extruder.

A new model extends the classical Darnell-Mol analysis, including the effect of starve-feeding by removing the forces acting on the trailing flight in the extruder channel.

Starve-feeding of an extruder is a process option whereby solid feed is metered into the feed throat using a gravimetric or auger type feeder at a rate less than the solids-conveying capacity of the screw. The operation causes pressures and temperatures to be lower in the feed section of the screw as compared to the flood-fed case. As a consequence of the lower pressures and temperatures, resin will compact further downstream in the screw channel. Starve-fed extrusion has provided process enhancements for several polymers, including thermally sensitive resins (1).

The starve-fed mode of operation can offer several advantages (1-3) over flood-fed extrusion: reduced power requirements for the same feed rate; faster melting after an initial delay period; improved venting of volatiles and air through the hopper, reducing or eliminating bubbles in the extrudate; reduced die pressure fluctuations; and reduced levels of solid bed breakup. Starve-feeding also has some disadvantages. The operation is more complicated in that an external metering device is necessary to feed resin into the hopper. Extruder throughput is obviously reduced below its capacity, which may not be cost-effective. Also, the performance advantages cited above may not be maintained for degrees of starvation greater than about 10%.

Experimental observations (3) suggest that the extruder channel fills gradually during starve feeding. Frictional forces cause a void to develop on the trailing side of the channel. The void decreases in size along the channel because of compaction, and eventually disappears. The void allows flow of volatiles and air back through the hopper, maintains a low pressure in the feed channel, and minimizes frictional heating.

This article describes a numerical model for predicting pressure and temperature profiles in a single-screw plasticating extruder operated in the starve-fed mode. Predictions are then compared with experimental data, and models are presented as a first step in understanding the pressure growth mechanisms in starve-fed single-screw extruders.

Model Development

The starve-fed model developed here closely follows existing models of flood-feed extrusion. The classic work in the area of solids conveying was done by Darnell and Mol (4). They assumed the solids moved as a uniform plug, and predicted an exponential pressure rise along the extruder channel. Tadmor and Klein (5) extended the Darnell-Mol analysis to include effect of flight width and different coefficients of friction at the barrel and screw. The analyses used force and torque balances on an element of the polymer in the channel to determine pressure rise as a function of volumetric flow rate, rotation speed, and screw geometry. This simple approach uses pressure, rather than stress, as the primary variable. The Darnell-Mol approach requires specification of the initial pressure in the channel, which may not be well known. Models for predicting this initial pressure by calculating the pressure at the base of the feed hopper have been incorporated into some simulations (6), but these models are not appropriate for the starved-fed case, in which the hopper is empty. Lovegrove and Williams (7) noted that channel pressure calculation can be improved by introducing coefficients of friction accounting for anisotropy of the imposed strain. They also demonstrated how gravitational forces could be incorporated into a model for predicting pressure growth in an extruder with zero initial pressure.

The study presents a model for the solids-conveying section that accounts for a partially filled channel. Two modifications to existing flood-fed models are proposed. First, it is assumed that width of the polymer bed is reduced by an amount dependent on degree of starvation, defined as

S = (1 - |G.sub.s~/|G.sub.0~) (1)

where |G.sub.0~ and |G.sub.s~ are mass flowrates for flood-fed and starve-fed cases at the same screw rotation speed, respectively. For this analysis, a simple linear relationship between effective polymer bed width (w) and degree of starvation is proposed:

w = |w.sub.c~(1 - S) (2)

where |w.sub.c~ is the average channel width. A comparison of simulation results with experimental data helps determine the validity of this assumption. The second modification is the removal of the forces acting at the trailing flight surface, on the assumption that an air gap exists between the polymer plug and the trailing flight.

Force and torque balances on a differential polymer element have been used to develop a differential equation for the pressure in the extruder channel. The analysis is based on the analysis first presented by Darnell and Mol (4), as modified by Tadmor and Klein (5). The model assumes the polymer behaves as a continuum and moves in plug flow down the extruder channel. The model results in a one-dimensional differential equation for the pressure by ignoring pressure variations across the channel. The previous derivations have been modified here for the case where the trailing flight is not in contact with the polymer. By assuming that the forces acting on the trailing flight are zero (forces |F.sub.4~ and |F.sub.8~), expressions similar to those of Tadmor and Klein are obtained, but with different |A.sub.1~, |A.sub.2~, |B.sub.1~, and |B.sub.2~ coefficients. Equation 3 relates pressure (P) to helical distance (z) along the extruder channel for the starve-fed case.

|Mathematical Expression Omitted~

The constants are defined as follows:

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In these expressions, subscripts b and s refer to quantities at the barrel and screw, respectively, and the overbar indicates an average over the channel height. The formula depends on the geometry of the extruder through the helix angle (|theta~), channel depth (H), screw diameter (D), and effective channel width (w). The operating conditions influence the polymer coefficients of friction (|f.sub.b~ and |f.sub.s~), and the forwarding angle (|phi~), which is a measure of the velocity difference between the screw barrel and the polymer plug. A mass balance yields the following relation (5) between volumetric flow rate (|Q.sub.s~), screw rotation rate (N), flight width (e), and forwarding angle:

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Equation 3 can be integrated along the length of helical extruder channel given an initial pressure value, and the functional dependence of the equation parameters with distance. In starve-fed mode, the initial pressure is expected to be of the order of the hydrostatic pressure because of the height of one diameter of the polymer.

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Frictional heat is generated at the screw and barrel surfaces, causing resin temperature to increase until a melt film develops. The melting location can be predicted by solving an energy conservation equation for the polymer plug. The temperature field in the extruder depends on the convection of heat along the channel due to the motion of the polymer, and conduction of heat due to temperature gradients. We assume that cross-channel and curvature effects can be neglected, and that conduction in the down-channel (z) direction can be neglected compared with that in the radial direction (y). Assuming constant thermal conductivity, the following partial-differential equation describes the temperature field in the unwound channel:

|Mathematical Expression Omitted~

Here T is the temperature, | is the down-channel velocity of the polymer bed, and |alpha~ is the thermal diffusivity of the polymer.

The heat generation rate due to friction at the barrel and the screw is assumed to be proportional to the pressure and the relative velocity between the polymer and the barrel. Several boundary conditions could be used to solve Equation 5, depending on the type of information available. For this article, a solution was obtained by assuming that temperatures were known at the inlet, screw root, and barrel probe locations. The temperature probes were not located at the polymer/barrel interface, but an expression relating the temperature gradient in the polymer to the interface temperature can be obtained as follows. The rate of frictional heat generation per unit area at the interface is given by:

|Mathematical Expression Omitted~

where |V.sub.bx~ and | are cross-channel and down-channel barrel velocities, P is the pressure, and |f.sub.b~ is the coefficient of friction at the barrel. By equating frictional heat generation to the difference between heat fluxes in the barrel and the polymer, we obtain:

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where |k.sub.p~ and |k.sub.b~ are thermal conductivities of the polymer and metal, respectively. Conductive heat flux in the barrel is estimated from the temperature probe value (|T.sub.b~), assuming a linear temperature profile:

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where |T.sub.i~ is the unknown polymer temperature at the interface, and |delta.sub.p~ is the radial distance from the probe to the inner barrel surface. The polymer temperature gradient at the interface is estimated from |T.sub.i~ and interior polymer temperature values using the discretized first derivative of the temperature. The resulting difference equation can be used to express the interface temperature in terms of other nodal values.

Equations 3 and 5 were solved numerically, taking into account variations in the screw geometry, the effect of temperature and pressure on the polymer bulk density (8), and using appropriate friction coefficient data (9). The pressure gradient Equation 3 was integrated using a fifth-order Runge-Kutta scheme with adaptive step size based on a local error estimate. The partial-differential equation for the channel temperature was solved using an implicit Crank-Nicolson finite-difference approach. Frictional heat generation at the barrel couples the temperature and pressure profiles. The solution procedure was repeated with an updated temperature profile until convergence was obtained.


The experiments were performed on a 2.5-in-diameter, 21:1 length-to-diameter ratio extruder. The extruder was highly instrumented, had three barrel temperature control zones, and was controlled and monitored using a Dow CAMILE data acquisition and control system. These zones were operated at set-points of 150|degrees~C, 165|degrees~C, and 180|degrees~C for zones 1, 2, and 3, respectively. The positions of the sensors for this extruder were given previously (10). An auxiliary auger feeder was used to meter the resin to the hopper for the starve-fed experiments. A single-flighted, square-pitch screw was used. This screw had six constant depth feed flights at 8.89 mm (0.350 in), eight flights of taper, and seven flights of constant depth meter at 3.18 mm (0.125 in).

A low-density polyethylene (LDPE) resin was used for the study. The resin had a solid density of 0.922 g/|cm.sup.3~, and a melt index of 2.

The flood-fed base case was established by adjusting the screw rotation rate to match the feed rate to the hopper. Steady-state operation was confirmed by monitoring the height of the polymer in the hopper. Various degrees of starve feeding were introduced by increasing the rotation rate while maintaining the flood-fed mass flowrate. The experimental results are reported in terms of a degree of starvation based on the ratio of screw rotation rate for the flood-fed case (|N.sub.0~) to the starve-fed case (|N.sub.s~):

|Mathematical Expression Omitted~

This quantity is not the true degree of starvation based on mass flowrates as defined by Equation 1, but is expected to be a reasonable estimate.


This section describes the experimental extrusion data for both the flood-fed and starve-fed cases. The experimental data are used to show how well the model describes the starve-fed process.

Figure 2 shows experimental pressure as a function of time at selected distances along the extruder barrel for the flood-fed case (30 rpm, 47 lb/hr). Linear regressions based on a 12-min window are plotted to illustrate that steady-state was obtained. The observed oscillations were caused by passage of the screw flights across the pressure transducers. Figure 3 shows the measured pressure in the extruder operating at 37 rpm (20% starvation) for the same axial locations as the flood-fed case in Fig. 2. The die exit pressure for the two cases was approximately the same, but pressure at the upstream locations for the starve-fed case was much lower. The same die exit pressures for the experiments are expected because mass flow rates were the same for both runs.

Temperature and pressure data from the experiments were time-averaged and plotted vs. the axial distance to obtain the plots in Fig. 4. Several trends are worth noting in the data. The pressure growth in the channel is delayed for the starve-fed cases, with the higher degree of starvation cases delayed the most. The pressure profiles for degrees of starvation larger than 25% are essentially indistinguishable. A maximum appears in the pressure profile for the flood-fed case; it diminishes with the degree of starvation. The temperature of the polymer for the starve-fed cases is generally lower than the flood-fed case. The effect of starvation on the temperature profiles is complicated: Below turn 12 the temperature increases with degrees of starvation, while above it the effect is reversed.

A commercial extrusion simulation package called PASS-1 (11) was used to simulate the behavior of the extruder operating under the flood-fed conditions using the data from the experimental run. Experimental pressure and temperature profiles are plotted with the PASS-1 predictions in Fig. 5. The agreement between the simulation and the experimental data is quite good, although the PASS-1 simulation does not predict the observed maximum in the pressure profile. The delay zone from turn two to five is clearly visible in the simulation. The initial section of the pressure and temperature profiles corresponds to the solids-conveying zone, and the abrupt discontinuity at turn number two is the predicted melt location.

The starve-fed model described in this article was used to predict the pressure, temperature, and solid bed density profiles in the extruder under the same conditions as in the starve-feeding experiments. The initial pressure for the simulation, |P.sub.0~, at the start of the screw channel was estimated at about 50 Pa using the following:

|P.sub.0~ = pg|H.sup.*~

Where |H.sup.*~ is the characteristic polymer height at the beginning of the channel, p is the polymer density, and g is the gravitational acceleration. The extremely low pressure of 50 Pa results from using the channel depth (7 mm) as the characteristic height. The simulation results appear in Fig. 6. In this Figure the melting location is indicated by the circle at the terminus of each profile. The starve-fed predictions are plotted along with results from the flood-fed model (4, 5). The results show several trends: the melting position moves further downstream in the extruder as the degree of starvation is increased; the pressure at the melting point decreases with the degree of starvation; the polymer density increases in a sigmoidal fashion, with the compaction occurring further downstream in the extruder as the degree of starvation is increased. The starve-fed model for 0% starvation differs markedly from the flood-fed model, illustrating the large effect of removing forces at the trailing flight.

Figure 7 shows an expanded view of the pressure profiles in the first turn of the extruder. The starve-fed model results show a characteristic exponential behavior, with the dependence on the initial pressure clearly shown.

The experimental data were analyzed to obtain estimates of the melting location at the barrel wall. The melt location for the experiments was estimated as follows. Complete extruder models suggest that pressure in the extruder rises exponentially in the solids conveying zone, drops nearly to zero in a "delay zone," and rises again in the melting zone. The length of the delay zone for the flood-fed case was estimated to be 2.9 screw terms, based on a PASS-1 extruder simulation. The melt location for all experimental data was estimated by extrapolating the pressure profiles to zero pressure and subtracting 2.9 turns for the delay zone. The melt location estimates allow rough comparisons of the experimental data with the model predictions. The starve-fed model simulation results were also analyzed to obtain predictions of the melting location as a function of degree of starvation. Figure 8 shows the simulation predictions of the variation of the melting location with the degree of starvation plotted with the experimental estimates. A linear regression is plotted through the experimental data, indicated by the circles, while smooth curves are drawn through the simulation results.


1. S.R. Jenkins, K.S. Hyun, J.R. Powers, and J.A. Naumovitz, Polymers, Laminations and Coatings Conference, TAPPI, 501 (1989).

2. D.P. Isherwood, R.N. Pieris, and J. Kassatly, Trans. ASME, 106, 132 (1984).

3. J.M. McKelvey and S. Steingiser, SPE ANTEC Tech. Papers, 24, 507 (1978).

4. W.H. Darnell and E.A.J. Mol, SPE J., April 1956, p. 20.

5. Z. Tadmor and I. Klein, Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold Co., New York (1970).

6. E.E. Agur and J. Vlachopoulos, Polym. Eng. Sci., 22, 1084 (1982).

7. J.G.A. Lovegrove and J.G. Williams, J. Mech. Eng. Sci., 15, 114 (1973).

8. K.S. Hyun and M.A. Spalding, Polym. Eng. Sci., 30, 571 (1990).

9. M.A. Spalding, D.E. Kirkpatrick, and K.S. Hyun, accepted in Polym. Eng. Sci.

10. K.S. Hyun, W.A. Trumbull, and S.R. Jenkins, SPE ANTEC Tech. Papers, 31, 38 (1985).

11. PASS 1 is an extrusion software product of the Polymer Processing Institute at Stevens Institute of Technology, Hoboken, N.J.
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Author:Strand, Steven R.; Spalding, Mark A.; Hyun, Kun S.
Publication:Plastics Engineering
Date:Jul 1, 1992
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