I n d . E n g . C h e m . Res. 1989, 28, 174-178
174
MATERIALS AND INTERFACES New Process for Ultradrawn Polyethylene Structures A. Rudin* University of Waterloo, Department of Chemistry, Waterloo, Ontario N 2 L 3G1, Canada
W. J. Tchir and R. Gagnon Dow Chemical Canada, Western Canada Division, Fort Saskatchewan, Alberta T 8 L 2P4, Canada
H. P. Schreiber Ecole Polytechnique, Chemical Engineering Department, Montreal, Quebec H3L 3A7, Canada
R. Collacott Dow Chemical Canada, Sarnia Division, P.O. Box 3030, Sarnia, Ontario N 7 T 7M1, Canada
Polyethylene filaments with moduli greater than 100 GPa can be made by a process in which the polyolefin is blended with paraffin wax to produce mixtures that contain 30 wt 7% or more polyethylene. These blends are shear modified and then extruded and stretched. Wax is removed from the resulting filaments by extraction with solvents like hexane. The extracted products are given a multiple stretch a t controlled temperatures. The wax apparently functions as a diluent that facilitates disentanglement of polymer chains during the shear modification step. These experiments indicate that mechanical shearing of polymer melts under appropriate conditions is a partial alternative to dilution as a means for obtaining relatively disentangled species that can be used to make articles with high orientations and superior mechanical properties. Most articles made from plastics have relatively low moduli, of the order of lo9 Pa. By contrast, steel has a modulus near 10” Pa, which is somewhat higher than the modulus of ordinary glass fibers. The moduli of polymer-based products can be enhanced by orientation, as in conventional fiber spinning and production of biaxially oriented films. Another expedient involves reinforcement of thermosets or thermoplastic3 with rigid fibers (like glass) that have a high aspect ratio. Both procedures are limited in their effectiveness, and moduli of the order of l o l l Pa are not accessible. Another way to produce very high modulus polymers is to make so-called rigid rod species, with very stiff molecules. These generally incorporate aromatic structures into the polymer backbone. Poly(p-phenylene terephthalamide) is a well-known example of such a material, which has been commercialized under the tradename “Kevlar”. The moduli of Kevlar fibers are about 1.3 X 10’l Pa. Various other aromatic structures have also been synthesized, based on linking groups other than the amide moiety. These materials, like aromatic polyesters, polyamide-hydrazides, and others, also provide rodlike species and can be spun into ultrahigh strength fibers. More recently, low-cost fibers with very high strength have been produced from flexible, semicrystalline polymers. This report describes a new method for making such products from polyethylene (PE). Processes to produce ultrahigh strength polyethylene structures can be classified broadly as gel/solution spin-
* Address communications to this author. 0888-5885/89/2628-0174$01.50/0
ning, melt spinning, and solid-state extrusion techniques. Gel spinning is a technique that yields a moderately oriented fiber from a composition that contains a low concentration of polymer. This is followed by a drawing procedure for the final high strength product. There are two methods available for the production of polyethylene gel fibers. Fibers may be obtained by passing the polymer-solvent solution through a spinnerette or by seedinitiating crystal growth in the solution and subsequently pulling fiber directly out. The cooled fibers are high in solvent content. The solvent is removed by evaporation or liquid extraction prior to the drawing process. Gel spinning techniques for PE were first reported in the 1960’s (Zwick, 1967). The polyethylene solvents used were naphthalene, paraffin wax, and stearic acid. Polymer concentrations were of the order 1-12% PE lo5) and 3-4% PE (Mwlo6). Fiber drawing was not performed, and physical properties were not listed. Recent accounts of solution (gel) spinning are given in reports by Smith and Lemstra (1979, 1980a). Here PE is dissolved in decalin at 150 “C at concentrations less than 2% w/w. The solution is extruded through a capillary at 130 “C into a water quenching bath. This cooled gel filament is stretched in a hot air oven (120 “ C ) to yield a solvent-free fiber. Fiber cannot be drawn successfully from solutions with concentrations >lo% w/w PE in decalin at a draw ratio (DR) of greater than 5. D R s greater than 20 are required to obtain high strength properties, however. Later patent disclosures outline critical operating conditions in more detail (Smith et al., 1982a,b). The polymer must be highly linear and have Mw of lo6 for the best
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0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 175 Table I. Physical Property Comparison of Fibers Made by Different Techniques tensile strength, techniaue MJMn DR GPa 1.5 x 106/2 x 105 gel spinning (Smith and Lemstra, 1980a) 32 3.1 5 X 105/6 X lo4 gel spinning (Smith et al., 1982a) 37 1.9 1.5 x 106/2 x 105 solution cast (Lemstra and Smith, 1980) 72 3.2 5 x 105 gel spinning (Kavesh and Prevorsek, 1983) 32 1.7 1 x 106 gel spinning (Kavesh and Prevorsek, 1983) 28 3.0 100 3.8 1.5 x 106/2 x 105 suspension spin (Smook and Pennings, 1983) 4 x 106 70 4.1 gel spinning (Smook et al., 1980a) 4 x 106 5 4.7 surface growth (Smook et al., 1980b) 2 x 106 150 7.0 gel spinning (Bershtein et al., 1984) 2.0 x i 0 6 p x 105 247 0.9 solid state (Kanamoto et al., 1983) 1.5 X 105/3.3 X lo4 52 0.6 solid state (Southern and Porter, 1970) 1.1 X 106/2.8 X lo4 melt spinning (Wu and Black, 1979) 32 1.6 0.7 1.6 X 105/0.8 X lo4 melt spinning (Wu and Black, 1979) 16 1.0 X 105/0.6 X lo5 30 hot drawing (Capaccio and Ward, 1974) 6.9 X 105/1.4 X lo5 1.4 30 melt spinning (Cansfield et al., 1983)
physical properties. The gels should contain at least 80% solvent and be stretched at temperatures higher than 75 "C. Highly oriented PE fibers are obtained of the order of 20-pm diameter. A patent of Kavesh and Prevorsek (1983) outlines a gel spinning and drawing line of pilot plant scale. This process is claimed to have advantages over the work of Smith et al. (1982b) in greater control and reliability in the drying and stretching steps, plus the ability to produce gel fibers with more uniform polymer concentration. Polyethylene fibers made by conventional monofilament extrusion techniques have tensile strengths and moduli of the order of 1.0 and 10 GPa, respectively, because the draw ratios that can be achieved are limited. The production of high tensile strength and high modulus PE fibers by melt processing has commanded attention only recently. Wu and Black (1980,1981) reported a technique used to obtain high-strength PE fibers (Wu and Black). The better physical properties obtained by this method are due to the higher DR achieved. In the examples given, the PE has a A?,, of at least 20 000 and a Mwof less than 125000. The temperature of the drawing section is kept within 115-132 "C. Ward and co-workers have also used melt spinning to produce high-strength fibers. Ward's extensive work covers a wide range of molecular weights. In two separate patent specifications, the ranges are broken down into Mw/M,,in the range (3-10) X 105/(l.&10) X lo4 (Ward and Capaccio, 1980) and (5-15) X 104/(5-15) X lo3 (Capaccioand Ward, 1978), respectively. A spinning temperature of 210 "C is used to produce monofilaments 0.7-1.0 mm in diameter. Wards concern for polymer crystallization prior to drawing has led him to investigate different thermal treatments of the isotropic PE. The monofilament is crystallized at different rates in a glycerol bath at 120 "C before being drawn to a DR as high as 30. Solid-state extrusion is usually carried out on a small scale and produces very high modulus fibers. Processing equipment can be no more than a rheometer barrel with a specially designed die. A preformed PE billet placed in the barrel is extruded under great pressure at temperatures just below or near the melting point of the polymer. The DR is determined by the barrel and die dimensions. Production rates of fiber are very small and depend on the polymer molecular weight (0.01-0.1 m/min). The highpressure crystallization of the polymer near its melting point and extrusion in a semicrystalline form produces highly oriented, optically clear fibers of PE. The majority of the high-pressure crystallization/extrusion technique research has been conducted in the laboratories of R. S. Porter (Southern and Porter, 1970;
tensile modulus, GPa 90 60 120 42.5 170 124 106 119 144 222 65 71 18 65 45
melt temm O C 145.5
drawing temD. O C 120 120
147 149
120 120-50 100-40 100-48 100-48
142 134 138
115 134 127 127 75 120
Porter et al., 1975). Related work has also been performed by Keller (Ode11 et al., 1979). Porter places the preformed billet in the rheometer barrel and heats to temperatures of 134 "C. Extrusion pressure is of the order 0.2-0.3 GPa. Draw ratios are of the order 50-60. The molecular weight of the PE used in this technique is fairly low (AXw (0.6-1.4) x 105). Very high moduli and tensile strengths have been reported recently by starting with single-crystal mats of ultrahigh molecular weight polyethylene. The process involves first solid-state coextrusion at low temperatures, followed by tensile extensions at higher temperatures (Kanamoto et al., 1988). Table I compares physical properties of fibers made by the different techniques. Several methods that are mentioned in the table have not been described in the text because they do not appear to have commercial potential, despite their scientific value. The values chosen for this table are the maxima claimed for a particular set of experimental variables. The highest tensile modulus reported is 222 GPa, from a solid-state coextrusion technique (Bershtein et al., 1984). This value approaches the theoretically calculated figures of 240-300 GPa (Sakurada et al., 1966; Shimanouchi et al., 1962; Boudreaux, 1973). Tensile strength values as high as 7.0 GPa were reported, but this does not approach the tensile strength of polyethylene single-crystal mats, at 13-19 GPa (Boudreaux, 1973; Peterlin, 1969). A high draw ratio is necessary for outstanding physical properties. The relation between draw ratio and tensile modulus or tensile strength is well documented; increases in draw ratio, up to a limit, always provide improved mechanical properties, but not necessarily in a linear fashion (Smith and Lemstra, 1980a; Lemstra and Smith, 1980; Smook et al., 1980b; Wu and Black, 1979). The data in Table I do not all fall on the same relation between draw ratio and modulus because the process itself plays a part in determining the physical properties of the final product and because draw ratio is not always defined on the same basis by different workers. The maximum DR achievable will be in accordance with the degree of entanglement density present before drawing and that which is removed during the drawing process. The molecular mobility in the melt state is more restricted compared to that of a polymer in solution. The high density of trapped entanglementsin the melt state impedes the large deformations required for chain extension and alignment (Smith et al., 1982a-c). Crystallization from dilute solutions allows for a greater disentanglement of the molecular chains and higher obtainable DR than that achievable from the melt state (Smith and Lemstra, 1979,
176 Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 Table 11. Polymer Molecular Weights HDPE M" M-. A 173000 302 000 B 21 000 171000
M. 685 000 332 000
1980b; Kanaomoto et al., 1988; Smith et al., 1981). A minimum amount of entanglements is necessary, however, in order that a network may be drawn out from solution (Smook et al., 1981; Smook and Pennings, 1982). Entanglements, which limit the draw ratio that is attainable, can be reduced by shear modification. This is a process whereby polymer melts or solutions are deliberately sheared to reduce melt elasticity and, by inference, entanglement density, without significant changes in molecular weight (Rudin and Schreiber, 1983). Similar effects can be achieved by recovering polymers from dilute solution (Schreiber et al., 1987). Polyethylenes respond to such shear or solution treatments, and changes that are observed in polymer properties are consistent with a reduction in entanglement frequency (Schreiber et al., 1965,1987; Teh et al., 1984). Shear modification by conventional process equipment may become impractical with higher molecular weight polymers because the shear field intensity is not great enough to cause disentanglements or because the total shear strain requirements exceed the dwell times of the polymer melt in the apparatus. In such cases, shear refinability can be enhanced if the coupling density or energy is reduced by some additional means (Rudin and Schreiber, 1983). Among such decoupling modifiers is included the use of solvating additives. In the present case, linear polyethylenes were effectively decoupled enough to permit the production of high-strength fibers by shear modification of blends of these polymers with paraffii wax. Wax was chosen as the diluent in this case because it is inexpensive, easily removed and recovered, compatible with polyethylene, and nonhazardous.
Experimental Section Polyethylene blends were made by using a corotating twin screw extruder in the fully intermeshing configuration. For ease of handling, such blends contained no more than 60 wt 740 wax. These blends thus contained significantly more polyethylene than in gel spinning processes. Shear modification was in a laboratory scale single screw extruder at 150 "C. This extruder contained a 27/1 lengthldiameter screw with 1.27-cm diameter and 3:l compression ratio. An 80-mesh screen pack was used to filter the melt. The die used had an LID ratio of 25 ( L = 1.3 cm, D = 0.05 cm) and an entrance angle tapered at 120 "C. Smaller dies were used to make finer denier fibers. A 1-in.-long barrel spacer (i.d. = 1.27 cm) was inserted between the screen pack and the smaller dies to reduce elastic memory effects. Fluoroelastomer processing aids (Rudin et al., 1985) were used to reduce melt processing problems such as melt fracture and sharkskin. Figure 1 shows the experimental
0
A I R FLOW
"
P
EXTRUDE%
OdiV
T A K E - UP SPOOL
2 nd GODET
1st
L-L
'
R RING
GDDET
Figure 1. Schematic diagram of fiber spinning apparatus.
arrangement used to extrude, stretch, and collect polyethylenelwax fibers. Extrusion was a t 190 "C. The extrudate passed through an air coil, to freeze the fiber. The quenched extrudate was taken up on the first godet, passed through a forced air oven with countercurrent air a t 114 "C, and then was taken up on the second godet. The fiber was then collected on hollow metal frame spools. Godet speeds were adjusted to achieve the maximum possible stretching of the fibers in the oven. Draw ratio was measured as the ratio of the second godet speed to the speed of the first godet. After sample collection, the spools were placed in agitated hexane at 60 "C, for 10 h. The spools were then dried overnight in a vacuum oven at 45 "C. Fibers produced to this point required further stretching to achieve good physical properties. This could not usually be accomplished by means of a second or third pass through the forced air oven because the fibers that were made were very thin and fluttered in the turbulent air stream. As a laboratory expedient, these fibers were stretched further in a tensile tester equipped with an oven. The extracted fibers were cut into sections, and bundles were drawn at constant temperature in a small tensile tester. The oven temperature and rate of extension were varied to produce maximum draw ratios. A systematic study of the effects of extension rates was not made since these data cannot be extrapolated to continuous drawing processes. Extension was measured as the ratio of the final to initial jaw separations of the tensile apparatus. The same tensile tester was used, at room temperature, to measure tensile strength and modulus. Bundles of five fibers were tested at one time. The initial slope of the stressstrain curve was used to obtain the tensile modulus. Fiber diameter was estimated by weighing a bundle of fibers of known length and assuming a circular cross section and a polymer density of 970 kg/m3.
Results and Discussion Table I1 lists molecular weight parameters and fiber properties for two linear polyethylenes that were studied in this work, while Table I11 gives the properties of filaments produced under various conditions. The equipment used here was too crude to allow useful attempts to optimize the process. Nevertheless, these exploratory experiments indicate that shear modification is a partial alternative to dilution as a means for obtaining relatively disentangled polyethylenes that can be used to make fibers with high draw ratios and superior tensile strengths and moduli. The effects of polymer molecular weight have not been explored in this work. It is interesting, however, that the molecular weights of the poly-
Table 111. Filament Production Conditions and Filament Properties extrusion draw temp, "C draw ratio material temp, "C 1st 2nd 3rd 1st 2nd 3rd HDPE A/wax 30/70 182 116 147 16.1 2.7 195 116 147 145 12.5 2.4 2.0 HDPE A/wax 40/60 190 114 130 130 HDPE B/wax 40/60 190 114 100 6.7 3.1 6.7 3.0 114 90 6.7 3.9 114 74 6.7 4.3 114 70
total 43 58 61 21 20 26 29
tensile strength, GPa 1.9 2.1 1.4 1.0
tensile modulus, GPa 53 106 114
1.1
100
1.5
111
2.5
137
85
Ind. Eng. Chem. Res., Vol. 28, No. 2, 1989 177 ethylenes that were used were well below those that are reported to be required for optimum results in gel spinning processes. Two patents to Mitsui Petrochemical Industries Ltd. (Motooka et al., 1985, 1986) came t o our attention after the work reported here was largely completed. The Mitsui process bears some resemblance to that described here, but there are also some very significant differences. The cited patents are confined to the use of ultrahigh molecular weight polyethylene (UHMWPE). Although this qualification is not defined in molecular weight terms, it refers presumably to M , values of lo6 or more, which are much greater than the molecular weights of the polyethylenes used in our research. In the Mitsui process, UHMWPE mixtures with paraffii wax are blended in an extruder a t temperatures higher than those that we find to be optimum for shear modification of the polyethylene/wax mixtures of our own work. The UHMPE/wax blends are then extruded and solidified, and the solid extrudate is stretched at 60-140 "C. Some of the wax may be removed in subsequent stretching steps. The moduli and tensile strengths reported in the examples of these patents are generally lower than those found in the process described in this report. Although the two methods both use wax as a diluent, its purpose if different in the two cases. The paraffin wax appears to be used as a flow improver in the Mitsui process, in order to make the UHMWPE extrudable. In our method, which uses lower molecular weight, processable polyethylenes, the wax is employed to facilitate shear modification and the consequent reduction of entanglement density and enhancement of attainable draw ratios. Shear modification is a necessary step in this process and is preferably carried out at as low a temperature as possible (Rudin and Schreiber, 1983). It is also necessary to remove substantially all the wax after the stretching process that accompanies extrusion, unlike the Mitsui process where the inherent strength of UHMPE permits the inclusion of some residual diluent without drastic deterioration of the mechanical properties of the final articles. Registry No. PE, 9002-88-4.
Literature Cited Bershtein, V. A.; Savitsky, A. V.; Egorov, V. M.; Gorshkova, I. A.; Demicheva, V. P. Polymer Melting at an Equilibrium Temperature in a Zero Melting Range as well as Inter- and Intramolecular Stages of the Process. Polym. Bull. 1984,12,165-172. Boudreaux, D. S. Calculations of the Strength of the Polyethylene Molecule. J. Polym. Sci. Phys. Ed. 1973,11, 1285-1292. Cansfield, D. L. M.; Ward, I. M.; Woods, D. W.; Buckley, A,; Pierce, J. M.; Wesley, J. L., Tensile Strength of Ultra High Modulus Linear Polyethylene Filaments. Polym. Commun. 1983, 24, 130-131. Capaccio, G.; Ward, I. M. Preparation of Ultra-High Modulus Linear Polyethylenes, Effect of Molecular Weight and Molecular Weight Distribution on Drawing Behaviour and Mechanical Properties. Polym. (London) 1974,15,233-238. Capaccio, G.; Ward, I. M. Polymer Materials. Br. Patent 1,498,628, January 25, 1978, to National Res. Dev. Corp. Kanamoto, T.; Tsuruta, A.; Tanaka, K.; Takeda, M.; Porter, R. S. On Ultra-High Tensile Modulus by Drawing Single Crystal Mats of High Molecular Weight Polyethylene. Polym. J. 1983, 25, 327-329. Kanamoto, T.; Tsuruta, A,; Tanaka, K.; Takeda, M.; Porter, R. S. Superdrawing of Ultrahigh Molecular Weight Polyethylene. I. Effect of Techniques on Drawing of Single Crystal Mats. Macromolecules 1988,21,470-477. Kavesh, S.; Prevorsek, D. C. High Tenacity, High Modulus Polyethylene and Polypropylene Fibers and Intermediates Thereof. US Patent 4,413,110, November 1, 1983, to Allied Corporation. Lemstra, P. J.; Smith, P. Ultra-drawing of High Molecular Weight Polyethylene. BF.Polym. J . 1980,12,212-214.
Motooka, M.; Mantoku, H.; Ohno, T. Process for Producing Stretched Articles of Ultra-high Molecular Weight Polyethylene. US Patent 4,545,950, October 8, 1985, to Mitsui Petrochemical Industries Ltd. Motooka, M.; Mantoku, H.; Ohno, T. Process for Producing Stretched Articles of Ultrahigh-Molecular Weight Polyethylene. US Patent 4,612,148, September 16,1986, to Mitsui Petrochemical Industries, Ltd. Odell, J. A.; Grubb, D. T.; Keller, A. High Modulus Through Lamellar Structures Nucleated by Flow Induced Fibrous Substrates. Polym. Eng. Sci. 1979,6,433-435. Peterlin, A. Folded Chain Model of Highly Drawn Polyethylene. Polym. Eng. Sci. 1969,9, 172-181. Porter, R. S.; Southern, J. H.; Weeks, N. Polymer Modulus and Morphology: The Tensile Properties of Polyethylene. Polym. Eng. S C ~1975, . 3,213-218. Rudin, A.; Schreiber, H. P. Shear Modification of Polymers. Polym. Eng. Sci. 1983,23, 422-430. Rudin, A.; Worm, A. T.; Blacklock, J. E. Fluorocarbon Elastomer Processing Aid in Film Extrusion of Linear Low Density Polyethylenes. J . Plast. Film Sheet. 1985,I , 189-204. Sakurada, I.; Ho, T.; Nakamae, K. Elastic Moduli of the Crystal Lattices of Polymers. J. Polym. Sci., C 1966,15,75-91. Schreiber, H. P.; Ajji, A.; Li, Y.; Rudin, A. Property Modifications in Polystyrene Recovered from Solution. In Current Topics in Polymer Science; Ottenbrite, R. M., Utracki, L. A,, Inoue, S., Eds.; Hanser: Munich, 1987; Vol. 11. Schreiber, H. P.; Rudin, A.; Bagley, E. B. Separation of Elastic and Viscous Effects in Polymer Melt Extrusion. J . Appl. Polym. Sci. 1965,9,887-892. Shimanouchi, T.; Asahina, M.; Enamoto, S. Elastic Moduli of Oriented Polymers. I. The Simple Helix, Polyethylene, Polytetrafluoroethylene, and a General Formula. J. Polym. Sci. 1962,59, 93-100. Smith, P.; Lemstra, P. J. Ultra-High Strength Polyethylene Filaments by Solution Spinning/Drawing, 2, Influence of Solvent on Drawability. Makromol. Chem. 1979,180,2983-2986. Smith, P.; Lemstra, P. J. Ultra-high-strength Polyethylene Filaments by Solution Spinning/Drawing. J. Mater. Sci. 1980a,15,505-514. Smith, P.; Lemstra, P. J. Ultra-drawing of High Molecular Weight Polyethylene Cast from Solution. Colloid Polym. Sci. 1980b,258, 891-894. Smith, P.; Lemstra, P. J.; Booij, H. C. Ultradrawing of High-Molecular-Weight Polyethylene Cast from Solution. 11. Influence of Initial Polymer Concentration. J. Polym. Sci. Phys. Ed. 1981,19, 877-888. Smith, P.; Lemstra, P. J.; Kirschbaum, R.; Pijpers, J. P. L. Process for the Production of Polymer Filaments Having High Tensile Strength and Modulus. Europ. Patent Appl. 0,077,590, Filed October 10, 1982a, to Stamicarbon B.V. Smith, P.; Lemstra, P. J.; Pennings, A. J. Filaments of High Tensile Strength and Modulus. UK Patent GB 2,042,414 B, December 22, 1982b, to Stamicarbon B.V. Smith, P.; Lemstra, P. J.; Pjipers, J. P. L. Tensile Strength of Highly Oriented Polyethylene. 11. Effect of Molecular Weight Distribution. J . Polym. Sci., Phys. Ed. 1982c,20,2229-2241. Smook, J.; Pennings, A. J. The Effect of Temperature and Deformation Rate on the Hot-Drawing Behavior of Porous High-Molecular-Weight Polyethylene Fibers. J. Appl. Polym. Sci. 1982, 27, 2209-2228. Smook, J.; Pennings, A. J. Suspension Spinning of Ultra-High Molecular Weight Polyethylene. Polym. Bull. 1983, 10,291-297. Smook, J.; Flinterman, M.; Pennings, A. J. Influence of Spinning/ Hot Drawing Conditions on Tensile Strength of Porous High Molecular Weight Polyethylene. Polym. Bull. 1980a,2,775-783. Smook, J.;Torts, J. C.; van Hutten, P. F.; Pennings, A. J. Ultra-High Strength Polyethylene by Hot Drawing of Surface Growth Fibers. Polym. Bull. 1980b,2, 293-300. Smook, J.; Torts, J. C. M.; Pennings, A. J. Hot Drawing of Surface Growth Polyethylene Fibers, 2, Effect of Drawing Temperature and Elongational Viscosity. Makromol. Chem. 1981, 182. 3351-3359 Southern, J. H.; Porter, R. S. The Properties of Polyethylene Crystallized Under the Orientation and Pressure Effects of a Pressure Capillary Viscometer. J . Appl. Polym. Sci. 1970,14, 2305-2317. Teh, J. W.; Rudin, A.; Schreiber, H. P. Shear modification of a linear low density polyethylene. Plast. Rubb. Process. Appl. 1984,4, 157-163. Ward, I. M.; Capaccio, G. Oriented Polymer Materials. Br. Patent 1,568,964, June 11, 1980, to National Res. Dev. Corp.
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Wu, W.-L.; Black, W. B. High Strength Polyethylene. Polym. Eng. Sci. 1979,19, 1163-1169. Wu, W.-L.; Black, W. B. Process for Producing High Tenacity Polyethylene Fibers. US Patent 4,228,118, October 14, 1980, to Monsanto Co. Wu, W.-L.; Black, W. B. High Tenacity Polyethylene Fibers and Process for Producing Same. US Patent 4,276,348, June 30, 1981, to Monsanto Co.
Zwick, M. M. Spinning of Fibers from Polymer Solutions Undergoing Phase Separation. I. Practical Considerations and Experimental Study. In Fiber Spinning and Drawing; Coplan, M. J., Ed.; Applied Polymer Symposia 6; Interscience; New York, 1967.
Received for review May 10, 1988 Revised manuscript received September 8, 1988 Accepted October 27, 1988
PROCESS ENGINEERING AND DESIGN Linear Quadratic-Model Algorithmic Control Method: A Controller Design Method Combining the Linear Quadratic Method and the Model Algorithmic Control Algorithm Chun-Min Cheng* Department of Chemical Engineering, University of Washington, Seattle, Washington 98195
The linear quadratic (LQ) feedback control method and model algorithmic control (MAC) algorithm are combined to obtain the complementary properties of these two methods. This formulation allows for a simple method of on-line controller tuning and can affect the robustness property in a direct way, while keeping the structure of the LQ optimal control method. A theoretical analysis covering closed-loop stability, convergence to the set point, and robustness properties in the case of an inexact process model is included. I t is demonstrated, for example, that zero offset is achieved even if the model is inexact. The method is used to design a controller for a simulated multieffect evaporation process. In the past decade, a few internal model-based control methods have been developed: Model Algorithmic Control (MAC) (Richalet et al., 1978; Mehra et al., 1980, 1981; Mehra and Rouhani, 1980; Rouhani and Mehra, 1982); Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980; McDonald and McAvoy, 1987); Quadratic Matrix Control (QDMC) (Garcia et al., 1984; Cutler et al., 1983); Internal Model Control (IMC) (Garcia and Morari, 1982, 1985a,b);Quadratic Programming Internal Model Control (QPIMC) (Ricker, 1985). These methods have been successful in industrial applications (e.g., Lecrique et al. (1978); Mehra et al. (1978); Mehra and Eterno (1980); Garcia et al. (1984),and Ricker et al., 1986)-because they provide a high degree of flexibility and allow for on-line tuning. For example, MAC includes an adjustable “reference trajectory” through which one can affect the robustness of the feedback system in a direct way. Garcia and Morari (1982) gave an unified review of DMC, MAC, and IMC, pointing out common features and noting analogies to classical forms of optimal control. Stephanopoulos and Huang (1986) introduced a two-port control system and demonstrated that it can be equivalent with IMC. A shortcoming of the IMC-type methods is that an open-loop stable system is required; the nonminimumphase characteristics need to be “factored out” in advance in the IMC design procedures (Garcia and Morari, 1985a,b; Holt and Morari, 1985a,b). The well-studied linear
* Current address: Union Chemical Laboratories, ITRI, 321 Kuang F u Road, Section 2, Shinchu, Taiwan, R.O.C. 08S8-5885/S9/2628-0178$01.50/0
quadratic (LQ) feedback control method (e.g., Anderson and Moore, 1971; Edgar, 1976; Astrom and Wittenmark, 1984) can stabilize the open-loop unstable process and is applicable directly for processes with nonminimum-phase characteristics. Cheng and Ricker (1986) developed a combination of the LQ method and MAC method in their controller design for a multieffect evaporator to take advantage of the potential complementary properties of the LQ method and IMC-type methods. The LQ optimization method is used to formulate a feedback control law, based on a nominal “internal model” of the process. A reference trajectory similar to that used in MAC is included to allow for on-line tuning. In this paper, we discuss the structure, formulation, properties, and applications of the LQ-MAC method. A multieffect evaporator is used as an example to demonstrate the properties and applications of this method. A comparison of the LQ-MAC method with MAC and IMC methods is also discussed. Structure of LQ-MAC The structure of the LQ-MAC controller is similar to the MAC method that was proposed by Richalet et al. (1978) and was analyzed by Rouhani and Mehra (1982). It includes (1)an internal model for system representation and prediction, (2) a reference trajectory leading to the desired set point, (3) an optimality criterion for control law formulation, and (4) a resetting of the reference trajectory at regular intervals using the measurement of the process outputs. These components are described as follows. Internal Model. The internal model is a linear discrete-time state-space formulation: 0 1989 American Chemical Society
