Computer modeling of wire and cable extrusion.

Many extrusion process engineers never get the opportunity to design an extruder screw, given that prices for custom-made replacements can rival the cost of a new SUV. Improved screw designs for wire and cable extrusion have typically come from extruder equipment manufacturers or polymer compound suppliers, whose main focus has been on pelletized plastics. Over the past 35 years, a variety of screw designs for plastics extrusion have been developed and marketed, typically centered around the classic metering-type geometry with a deep feed section, tapered transition section and shallow metering section. Most of the design effort has focused on improving mixing by various means, ranging from simple mixing pins imbedded in the screw channel to a fluted mixing section attached to the discharge end of the screw. Other types, such as barrier flight screws, compression-relief and two-stage designs have been used, each with varying degrees of success.

Rubber extruder screw design, by comparison, has received relatively little attention. Generally speaking, rubber screws fall into two categories, including the traditional constant channel depth, varying pitch, double-flighted design and the constant pitch, varying channel depth, double-flighted metering type design. Intensive shear mixing sections are usually not employed on wire and cable rubber extruders, as the high discharge pressures and compound viscosities commonly seen would lead to excessive extrudate temperatures. And since the processing of strip-fed rubber normally requires a double-flighted screw for proper feeding, the potential design variables here are fairly constrained.

Many processors have put their efforts into developing pelletized versions of their rubber compounds. However, this usually involves adding plastics, mineral fillers or other materials which "stiffen" the formulation to facilitate pelletizing. These additives often have an undesirable effect on the physical characteristics of the final product. Also, the material may still not be suitable for bulk storage and handling and may not feed well at the extruder. Rubber-insulated wire and cable producers often find it necessary to just live with the inconveniences associated with strip fed materials.

Although there are a number of commercially-available computer programs for modeling pelletized plastics extrusion, which can be of great utility in screw design, the cost of the software and the effort and expense required to obtain the numerous compound flow and frictional variables necessary to run the programs tend to deter many processors, particularly the smaller ones. The author's own experience with these programs is that they can provide valuable information regarding the relative effects changes in screw geometry and/or compound characteristics will have on the process. However, absolute accuracy in terms of predicting extruder output, extrudate temperature, stability or degree of mixing is not possible, for a number of reasons which will be discussed later. Most extrusion process engineers find such programs interesting and fairly useful for understanding extrusion, but not as tools suitable for frequent use in day-to-day problem-solving. And pelletized plastic extrusion models do not work very well when applied to rubber (and strip-fed rubber in particular).

When dealing with strip-fed rubber materials, a different approach is required. As it turns out, many of the problems associated with modeling plastics extrusion are not encountered with rubber. However, rubber compounds tend to have some more complex rheological characteristics which must be taken into account. In order to understand the rubber extrusion process, one invariably has to make some comparisons to plastics extrusion.

Polymer rheology and melt conveying models

Plastics and rubbers both behave as non-Newtonian fluids when heated to a fluid state and forced to flow. Where Newtonian fluids such as water or mineral oil exhibit a linear relationship between shear stress [tau] and shear rate [gamma] (i.e., their viscosity does not change with shear rate), the types of polymers used in wire and cable insulations exhibit a non-linear relationship--their viscosity decreases as shear rate increases. However, when a plot of log [tau] versus log [gamma] remains linear over a substantial range, the polymer may be classified as a simple "power law" fluid whose behavior can be defined by the Ostwald and de Waele flow model (ref. 1):

(1) [tau] = m x [[gamma].sup.n]

where m is defined as the flow consistency (viscosity) and n is the flow index. Equation 1 reduces to the Newtonian case when n = 1.

While the plastics used in wire and cable can often be treated as simple power law fluids, many of the rubber compounds cannot. In addition to the viscosity varying with shear rate, the flow index of a typical synthetic rubber compound also varies significantly with shear rate, further complicating the mathematical model. However, all of the flow parameters needed for a rubber extrusion model can be readily obtained using a capillary rheometer, typically a simple die arrangement attached to a common Brabender lab extruder. By extruding and weighing samples of material at different pressures, the apparent shear rate [[gamma].sub.a] and apparent shear stress [[tau].sub.a] can be determined using the equations:

(2) [[tau].sub.a] = [DELTA]P R/2L

and (3) [[gamma].sub.a] = 4q/[pi][R.sup.3]

where R and L are the die radius and die length, [DELTA]P is the driving pressure, and q is the volumetric flow rate. The apparent viscosity [[mu].sub.a] is defined by the equation:

(4) [[mu].sub.a] = [pi][R.sub.4][DELTA]P/8qL

While it is normally assumed that the apparent shear stress is the true shear stress at the wall, die entrance and exit effects can introduce an error in determining the true pressure gradient. Several texts have shown a means of correcting for these effects (ref. 2).

The true shear rate at the wall is defined by the Mooney-Rabinowitsch

Rabinowitsch equation:

(5) [[gamma].sub.t] = [[gamma].sub.a] [3n' + 1 / 4n']

where, for true power law fluids, n' = n. Since most polymers, and robbers in particular, have non-constant power indices, it is necessary to confine laboratory measurements to narrow windows of shear rates at several points over the expected range of shear rates one expects to encounter in the extrusion process. When using the data for extruder process modeling, however, one has to account for the large differences between screw channel and flight clearance shear rates, and it becomes necessary to develop a mathematical model of the change in shear index with shear rate from the rheological data.

A capillary die rheometer is also useful in determining a polymer's thermal sensitivity. At any constant shear rate, the viscosity-temperature relationship is defined by the equation

(6) [mu] = [[mu].sub.o] exp[[a.sup.*](T - [T.sub.o])]

or alternatively

(7) m =[m.sub.o] exp[[a.sup.*](T - [T.sub.o])]

where [m.sub.o] is the apparent viscosity at T = [T.sub.o] and [gamma] = 1.

The temperature sensitivity factor [a.sup.*] defines how a material's viscosity changes with temperature. Plastics, especially the more crystalline types, tend to have much larger values for [a.sup.*] and are, therefore, more sensitive to spatial temperature gradients within a flow channel. Rubber compounds, on the other hand, tend to exhibit smaller values for [a.sup.*], making their viscosity far less sensitive to temperature variations.

In the classical Newtonian superposition theory, isothermal conditions are assumed, and the reduction in output due to back pressure is accounted for by the concept of a negative pressure flow up the extruder channel which opposes the down-channel drag flow imparted by the relative barrel to screw velocities. This concept is presented as:

Q (net flow rate) = [Q.sub.d] (drag flow) minus [Q.sub.p] (pressure flow)

Making the appropriate accommodations for screw geometry, the output equation becomes (ref. 3):

(8) Q = [F.sub.d][[pi].sup.2][D.sup.2]Nh(1- b/t)sin[phi]cos[phi]/2 - [F.sub.p][pi]D[h.sup.3] (1-pb/t)[sin.sup.2][phi][DELTA]P/12[mu][L.sub.m]

where [F.sub.d] and [F.sub.p] are drag and pressure flow shape factors to correct for the fact that more polymer melt will flow through a channel which is almost full than one which is completely full and D = barrel diameter; N = screw speed in RPMs; h = screw channel depth; p = number of flights; b = flight width; t = flight lead length; [phi] = helix angle of screw flights; [L.sub.m] = effective length of melt conveying section; and [mu] = melt viscosity.

Applying the above Newtonian analysis to almost any plastic or rubber extruder will yield results which grossly over-estimate the machine's output. Since the output of an extruder is the integral of the fluid velocity profile over the depth of the channel times the width of the channel, the equations for down-channel and cross-channel flow must be solved simultaneously by numerical iteration on a computer. Using the Runge-Kutta method as employed by Griffith (ref. 4), a graph of the reduced flow rate due to non-Newtonian behavior versus a dimensionless pressure gradient may be constructed. Figure 1 shows the ratio of the actual output rate, Q, to the calculated value of the Newtonian drag flow rate, [Q.sub.d], plotted against a dimensionless pressure gradient [G.sub.2]/cos[phi], where [G.sub.z] is defined by the equation:

[G.sub.z] = [h.sup.n+1]/m[([pi]DN/60).sup.n] [differential]P/[differential]z

where

(9) [differential]P/[differential]z [equivalent to] [DELTA]P sin[phi]/[L.sub.m]

The curves in figure 1 were plotted for a relatively narrow range of helix angle values, and isothermal conditions were assumed. While application of the isothermal approximation to a wide variety of polymers may not be appropriate, it is generally applicable to the rubber insulating and jacketing materials used in wire and cable products. The typically smaller values for [a.sup.*] common to most rubber compounds (as compared to plastics), along with the smaller barrel-screw temperature differences employed in rubber extrusion justify assuming isothermal conditions for calculating flow rates in the melt conveying zone (ref. 5).

[FIGURE 1 OMITTED]

Although this type of numerical analysis is employed in various plastics extrusion models, it can only simulate what takes place in an extruder after the polymer is completely melted. The pelletized plastic extrusion process is, however, much more complicated. As melting of pellets begins to take place in the middle, transition zone of a plastics extruder, a pool of melt forms in the rear of the screw channel as a result of screw geometry. Ideally, and in a properly designed screw, this pool remains segregated from the unmelted pellets, gradually growing in size as material progresses down the channel. By the time it reaches the discharge end of the screw, the melt pool has completely replaced the solid plug of plastic pellets and is thermally homogenized.

Unfortunately, this does not always take place in many plastics extruders. Extruders with poorly-designed screws can exhibit random and premature break-up of the solid plug, allowing the unmelted pellets to become dispersed in the melt pool, rather than remaining segregated from it. This is a primary cause of poor thermal mixing and output instability (surging) in plastics extruders. In common practice, the problem is addressed by screw cooling, which tends to stabilize the solid plug and prevent premature breakup, and by the use of mixing devices, which add energy to help complete melting and thermally homogenize the melt prior to discharge.

The melting mechanism is very difficult to model, particularly as it applies to predicting heat transfer within the rotating melt pool, spatial segregation of the melt pool from the solid pellet plug, and breakup and mixing of the solid pellet plug. The strip-fed rubber extrusion process is much simpler to model, in this respect, since this melt pool/unmelted pellet segregation mechanism is not a factor. The rubber essentially goes from being a solid plug to a sheared fluid at some point in the process. Although there are ways to mathematically predict this, our own experience shows that including this analysis in a computer simulation is usually unnecessary, as the transition point can be estimated by empirical means.

Friction and the solids conveying mechanism

In both plastics and rubber extrusion, the transport mechanism of the solids conveying zone is treated as plug flow. In plug flow, the basic assumption is made that the polymer behaves as a solid, elastic mass, undergoing no internal shear, but slipping along the screw surface, dragged down-channel by friction with the extruder barrel. Figure 2a shows an idealized "unwrapped" model for the solids conveying mechanism. The upper plate, representing the barrel surface, moves with a velocity Vb across the screw channel at an angle equal to the helix angle [phi]. The plug slides down-channel with a velocity of Vp/sin [phi], where Vp is the axial plug velocity. The angle [theta] is the angle of movement of the plug relative to the barrel. The volumetric throughput [Q.sub.s] is defined by the equation (ref. 6):

(10) [Q.sub.s] = [[pi].sup.2]NhD(D - h)[tan[phi] tan[theta]/tan[phi] + tan[theta]](1 - pb / [pi] (D - h) sin[phi])

The last term of the equation is a correction factor for flight width. The angle [theta] is a complex function of the screw geometry and friction that is too involved to discuss in this article. However, the term [tan[phi] tan[theta] / tan[phi] + tan[theta]] is a determining factor in the solids conveying rate, and is frequently shown plotted against the helix angle to illustrate the latter's effect on extruder feeding, as shown in figure 2b. The curves in 2b were plotted for a particular screw diameter and feed channel depth, and where [f.sub.s] (polymer coefficient of friction with screw) equals [f.sub.b] (coefficient of friction with barrel). In practice, the values for [f.sub.s] and [f.sub.b] are almost never equal, given the differences in barrel and screw temperatures and relative surface velocities, and tend to change anyway as the material moves down-channel and internal extruder pressure develops.

[FIGURE 2 OMITTED]

Finding the right values for [f.sub.s] and [f.sub.b] can be problematic, even though laboratory instruments designed for testing solid polymer frictional properties are commercially available. Duplicating the exact roughness of extruder screw and barrel surfaces, pressure conditions and other factors is not possible, and the cost of such equipment can be prohibitive to management. An alternative is to measure the open discharge output of an extruder with a constant channel depth or low-compression screw and use these data to estimate the effective coefficients of friction. Although the accuracy may be limited, my experience has shown the data obtained in this manner to be useful for approximating the relative magnitude of change in solids conveying capacity that a change in screw geometry would cause.

In the days before computers facilitated feed rate and melt conveying capacity calculations, screw designers relied on simple rules of thumb to determine compression ratios, helix angles and channel depths. They recognized early-on that an extruder operating in a feed-restricted, or starved condition (where the melt pumping capacity exceeds the feeding capacity) tends to surge badly. They also recognized that screws with excessive compression ratios had their own problems, such as bailing-up at the feed throat of a strip-fed rubber extruder, or pushing unmelted pellets through a pelletized plastic extruder. Since pelletized materials have about half the bulk density of strip-fed rubbers, plastic screws had to have higher compression ratios to compensate. The old 3:1 or 4:1 compression ratio for plastics and 2:1 or 2.5:1 ratio for strip-fed rubber still dominate the thinking of many screw designers, as does the practice of making the flight lead equal to the screw diameter for plastics, or one and a half times the diameter on a rubber screw.

Unfortunately, such designs may not be suitable for a particular material. Double-flighted rubber screws, in particular, may have output stability problems associated with their characteristically long flight leads. The helix angle for such screws is often outside the optimum range for solids conveying capacity, and a feed-restricted operating condition results.

Application to design and problem solving 

Many traditional rubber screw designs can allow the extruder to operate in a feed-restricted mode under certain operating conditions. As mentioned earlier, the larger helix angle common to double-flighted rubber screws is often the cause. This, coupled with the low compression ratios of most rubber screws, can make the output rate highly sensitive to ever-present feed rate fluctuations. The ramifications of feed-restricted operation are often observed in the extrusion of larger polymer cross sections where head pressures are low. Minor variations in compounding or mixing, or the amount of lubricant used, can have a significant effect on extruder output rate or stability that cannot be explained by traditional laboratory tests, such as Mooney viscosity.

The dramatic effect of screw design on performance can best be illustrated by the cable diameter recording charts shown in figures 3a, 3b and 3c. In figure 3a, the recording instrument was located at a CV line where medium voltage, 500 MCM copper power cable was being insulated with a strip-fed EPDM rubber. The severe surging in diameter being experienced could not be associated with machinery speed or temperature control. This line had a history of chronic output variability, and a recent run had shown a 20% drop in output when changing to a batch of rubber mixed with a slightly outside of normal lot of raw polymer. However, the Mooney viscosity of the rubber was well within specified tolerances, and other lab results were good.

[FIGURE 3 OMITTED]

In the particular run charted in figure 3a, lowering the extruder output rate had no effect on the surging. The line's 4-1/2" primary extruder had a metering type, constant pitch, double-flight screw design which worked well in other plant CV lines insulating smaller wire and cable constructions with higher head pressures. This suggested that the machine might be operating in a feed-restricted condition. Computer simulations of the melt conveying capacity also indicated this to be the case, yielding calculated theoretical output values that were significantly higher than what was actually being observed at the extruder.

From previous screw design efforts, it was known that simply increasing the screw's compression ratio by deepening the feed section of this basic design created feeding problems, while decreasing the metering section channel depth would cause unacceptable increases in melt temperature. As an alternative, a new screw was designed, retaining the existing channel depths, section lengths and compression ratio, but with a slightly shorter flight lead to give a helix angle that simulations said would increase the feed rate. The new design resulted in a significant improvement in output stability, as shown in figure 3b.

The output rate with the new screw was also 18-20% higher than with the old design, even though decreasing the flight lead reduced theoretical melt conveying capacity. The magnitude of the operating performance improvements was particularly surprising, given the slight (two or three degree) change in helix angle. In continuation of this initial design effort, additional work was conducted to improve extruder feeding characteristics and to optimize strip material frictional proper: ties.

Figure 3c illustrates the improved output stability achieved through these optimization efforts. In addition to the obvious savings gained through reduced material usage, the improved diameter control was critical to the quality of power cable products manufactured on the line, particularly those which required the application of an extruded insulation shield at a separate operation.

In the years since this development work, the author has encountered numerous instances of feed-related output variability in both strip-fed rubber and pelletized plastics. Computer models have been invaluable tools for predicting the effects of screw design, polymer compound rheological and frictional properties, and other process variables on rubber extruder performance. They have also been useful for developing a better understanding of the extrusion process for both rubber and plastics.

In addition to calculating extruder throughput, computer modeling can also predict extrudate temperatures and horsepower requirements for different screw designs under various operating conditions using the non-Newtonian flow analysis described above in combination with numerical heat transfer and power dissipation analyses. Such programs calculate the energy added or removed though barrel/screw heating or cooling and frictional shear energy generated in the screw channel and the barrel/flight clearances. The absence of the complex plastics melting mechanism in rubber extrusion greatly simplifies the analysis and facilitates accuracy. The program used in the design of the improved rubber screw discussed here produced numbers which correlated exceptionally well with operating data from the extruder. Although the development of the software and associated laboratory work to quantify material rheological and frictional variables was very time-consuming and expensive, the improvements in output stability and material usage on just one CV line justified the investment.

Unfortunately, the competitive environment under which the wire and cable industry must operate tends to limit the resources available for such efforts today. Most domestic manufacturers find it necessary to focus their process engineering on supporting cost-cutting initiatives like Lean Manufacturing and Six-Sigma, often to the neglect of basic process design and development work.

References

(1.) Williams, D.J., Polymer Science and Engineering, pp. 353-355, Prentice-Hall, Inc., New Jersey, 1971.

(2.) ibid., p. 363.

(3.) Bernhardt, Earnest C., Ed., et. al., Processing of Thermoplastic Materials, p. 215, Van Nostrand Reinhold Publishing, New York, 1959.

(4.) Tadmor, Z. and Klein, I., Engineering Principles of Plasticating Extrusion, pp. 267-277, Van Nostrand Reinhold Publishing, New York, 1970.

(5.) Pearson, J.R.A., Mechanical Principles of Polymer Melt Processing, pp. 84-89, Pergamon Press, Oxford, 1965.

(6.) Tadmor, Z and Klein, I., Engineering Principles of Plasticating Extrusion, p. 57, Van Nostrand Reinhold Publishing, New York, 1970.

E. Alan McCaslin, consultant