RPSGAe A Multiobjective Genetic Algorithm with Elitism: Application to Polymer Extrusion

Transcription

1 RPSGe Multiobjective Genetic lgorithm with Elitism: pplication to Polymer Extrusion. Gaspar-Cunha, J.. Covas Dept. of Polymer Engineering, University of Minho, Guimarães, PORTUGL. bstract. The application of a Multiobjective Optimisation Genetic lgorithm to polymer extrusion is presented. The aim is to implement an automatic optimisation scheme of the process capable to define the values of important parameters, such as operating conditions and screw geometry, yielding the best performance in terms of prescribed attributes. This problem is solved using a multiobjective optimisation genetic algorithm with elitism, Reduced Pareto Set Genetic lgorithm (RPSGe). The results obtained for specific case studies have physical meaning and correspond to a successful optimisation of the process. Keywords: Polymer extrusion, Screw design, Multiobjective optimisation, Genetic algorithms. 1- INTRODUCTION Extrusion is a major plastics production technology. It is used to produce widespread products such as tubing, pipes and profiles, film, sheet, filaments and fibres, electrical wires and cables. Plastics compounding, involving incorporation of additives in a polymer matrix in order to obtain materials with improved properties, is also carried out in extruders. The experimental and theoretical studies carried out during the last three decades allowed the understanding of the physical, thermal and mechanical phenomena occurring inside the extruder, and the development of mathematical models able to describe the entire process [1-4]. It is now possible to predict with good accuracy the values of important variables, such as mass output, power consumption, melt temperature, residence time distribution, pressure profiles and degree of mixing, for a given combination of polymer properties, operating conditions and screw configuration. However, process optimisation, i.e., the definition of the best screw configuration and/or operating conditions for a given application is still a trial and error procedure, where the above variables are changed, either experimentally or using the computer, until they meet the desirable performance. In this work, an automatic optimisation methodology of the polymer extrusion process, using a Multiobjective Optimisation Genetic lgorithms approach is proposed. For that purpose, the Reduced Pareto Set Genetic lgorithm (RPSG) proposed earlier [5,6] was modified in order to incorporate elitism, avoiding in such a way the deterioration of the fitness during the successive generations [7-11]. In the RPSG algorithm, the N individuals of the population, in each generation, are reduced to a pre-defined number of ranks (r=1,2,...,n Ranks ), then the value of the objective function is calculated using a ranking function. Elitism is introduced in the RPSG by maintaining an external population of size 2*N (Figure 1). The algorithm starts by the random definition of an internal population of size N and the formation of an empty external population. Then, in each generation the best 2*N/N ranks individuals, obtained by reducing the internal population with the clustering algorithm [12], are copied to the external population. This process is repeated until the number of individuals of the external population reaches 2*N. t this point, the RPSG is applied in order to sort the individuals of the external population. The best N/N ranks individuals of this population are incorporated in the internal population by replacing the individuals with lower fitness. Simultaneously, only the best N/N ranks are maintained in the external population. This algorithm was used to optimise the operating conditions and to design screws for a specific polymer extrusion problem.

2 a) Pre-define the number of required ranks, N ranks ; b) Pre-define the size of the Elitist population, N e ; c) Make N elite =; d) Make Rank[i]= for all the N individuals of the main population; e) First iteration, r =1; f) Do: 1. Calculate N R =r*(n/n ranks ); 2. Reduce the population to N R individuals using a clustering algorithm; 3. Make i=1; 4. Do: - If (Rank[i]=) Make Rank[i]=r; - Make i=i+1: 5. While (i<n R ); 6. Go to the next iteration, r=r+1; g) While (r<n ranks ); h) Make i=1; i) Do: 1. If (Rank[i]=) Make Rank[i]=N R ; 2. Make i=i+1: j) While (i<n); k) ssign a Fitness value for individual i using an linear or exponential ranking function, i.e., F i =f(rank[i]); l) Make Fi=F i /m i (where m i is the niche count); m) Copy the best 2*N/N ranks individuals to the elitist population; n) Make N elite =N elite + 2*N/N ranks ; o) If (N elite >=N e ) 1. Make Rank[i]= for all the N e individuals of the elitist population; 2. pply steps e) to l) to the Elitist population; 3. Copy the best N/N ranks to the main population; 4. Make N elite =N/N ranks ; p) Select the individuals for reproduction using for example roulette-well selection; Figure 1- Reduced Pareto Set Genetic lgorithm. 2- POLYMER EXTRUSION ND MODELING Figure 2 represents a longitudinal cut of a conventional plasticating extrusion unit. The solid polymer (pellets or powder) is fed in the hopper (i), and by gravity flows into the barrel, where an rchimedestype screw rotates at a given speed. Consequently, the solid polymer is dragged along the screw s helical channel (ii), where it starts melting (iii) the molten material subsequently accumulating in a pool, segregated from the surviving solids (iv). This melt is homogenized, pressurized (v) and forced to pass through the die, which gives the desired shape to the product (vi). For modelling purposes, the correct description of heat transfer and mass flow (through the mass, momentum and energy governing equations) in each functional zone must be guaranteed together with the adequate definition of the boundary conditions, as these ensure a physical coherent sequence along the screw. Polymers are strongly non-newtonian and highly viscous, hence it is of paramount importance to select an appropriate constitutive equation and to consider viscous dissipation effects. Consequently, the complexities of modelling plasticating screw extrusion will not be persued here, a detailed discussion being available elsewhere [6]. i) Hopper arrel Heaters Transversal cuts ii) iii) iv) v) Melt Solids Delay Melting Conveying Conveying Figure 2- Physical phenomena inside a typical polymer extruder. vi) Die

3 However, although the authors are able to solve the direct problem, i.e., to predict the process performance for a given combination of polymer properties, screw geometry and operating conditions, the pertinent question here is to solve the inverse problem, i.e., to define the operating conditions (screw speed and barrel temperature profile) and/or the screw geometry that yield the best extruder performance. Due to the complexity of the process, where various physical phenomena develop sequentially inside the extruder, this cannot be made explicitly, i.e., it is not possible to solve the governing equations in order to the operating conditions and/or screw geometry parameters. In fact, this is multimodal and multiobjective optimisation problem with conflicting objectives (e.g., maximizing the output while minimising the mechanical power consumption). 3- OPTIMIZTION OF OPERTING CONDITIONS s an example, the case study illustrated in Figure 3 will be used. The aim is to optimise the screw speed (N) and the barrel temperature in three zones (T 1, T 2 and T 3 ) in order to maximize the mass output (Q) and the mixing degree (WTS) and to minimize the power consumption (Power). The range of variation allowed for the parameters to optimise are indicated between square brackets, screw speed can range between 1 and 5 rpm and the barrel temperatures between 15 and 21 ºC. The problem considers a situation where a given polymer is processed in an extruder with a fixed geometry. The G parameters used are the following: number of generations, 5, share,.4, crossover rate,.7, mutation rate,.5, population size, 2, chromosome length for each variable, 1 and N ranks, 5. Two optimisation runs will be performed taking into account two criterions at time. Mass output will be used in both runs since it can be considered the most important criterion. In the first run the objective is to maximize Q and to minimize Power, in the second run the latter criterion was substituted by the maximization of the degree of mixing (WTS). However, due to physical and process reasons some constraints need to be considered, i.e., only the solutions that have Power lower than 92W and where the polymer melts completely before the end of screw (i.e., L melting <.9m) will be take into account in the optimisation procedure. s in real problems the optimal frontiers are not known a priori, 6 solutions are randomly generated in the search space allowing the plot of an approximate feasible criteria space in order to be possible the comparison with the optimal set obtained. N= [1,5] rpm D = 36 mm Heater band 1.3D 7D 6D 2.7D 235 mm T1=[15,21] ºC T2=[15,21] ºC T3=[15,21] ºC 2 mm L = 936 mm Figure 3- Optimisation of the operating conditions (L is the total screw length and D is the internal barrel diameter). The results obtained for the first optimisation run are presented in Figure 4. s expected, these two criteria are contradictory, since the aim is to maximize Q and minimize Power but when the mass output increases the mechanical power necessary to rotates the screw increases as well. The comparison between the feasible search space (Figure 4-) and the optimal Pareto frontier (Figure 4- ) shows that the optimisation algorithm is able to make a good approximation of the Pareto front. lso, the Pareto-optimal set obtained goes beyond the front defined by the feasible search space, with a clear improvement of the solutions found. Figure 5- presents the optimal Pareto frontier for the maximization of mass output and mixing degree. In this case the Pareto-optimal frontier is discontinuous, however the comparison with the feasible search space (Figure 5-) allows one to conclude that the RPSGe is able to find the Pareto set. gain, the optimal set presents some improvements when compared with the feasible search space defined

4 Figure 4- Output vs. power consumption: -feasible search space; -Pareto-optimal frontier Mixing degree Mixing Degree Figure 5- Output vs. mixing degree: -feasible search space; -Pareto-optimal frontier. 4- SCREW DESIGN Figure 6 presents the case study used for screw design, i.e., the optimisation of the screw geometrical parameters. The aim is to optimise the screw length of zone 1 and 2 (L 1 and L 2 ), the internal screw diameter of section 1 and 3 (D 1 and D 3 ), the screw pitch (P) and the flight thickness (e). The operating conditions are fixed at N=5rpm and T i =17ºC. The range of variation allowed for the geometrical parameters are indicated between square brackets. In this case only the maximization of the output and the minimization of the power consumption will be used as criteria. The constraints referred above for the remaining criteria are, also, considered. D = 36 mm D1 = [2,26] mm D3 = [26,32] mm Heater band 1.3D 7D 6D 2.7D L1= [15,4] mm L2 = [25,4] mm 235 mm L = 936 mm Pitch: Flight thickness: P = [3,42] mm e = [3,4] mm Figure 6- Conditions used for screw design. Figure 7 presents the same type of results of Figure 4, but now for the design of screws. In this case the optimal values obtained for output are limited, ranging between 7.5 and 1 kg/hr, because in this case the operating conditions are fixed and the variation of mass output is principally due to variations on screw speed (fixed at 5 rpm). However, for identical values of power consumption the variations on geometry produce high changes in mass output. 1 2 mm

5 It is important to note that these results have physical meaning. Since the maximization of output is obtained for geometries with small internal diameter for screw section 3 (D 3 ), the channel where the polymer is forced to cross offers lower resistance to the flow, which decreases the mechanical power consumption needed to rotate the screw Figure 6- Output vs. power consumption for screw design: -feasible search space; -Pareto-optimal frontier. 5- CONCLUSIONS In this work an elitist multiobjective genetic algorithm, called Reduce Pareto Set Genetic lgorithm with Elitism (RPSGe), was applied with some degree of success in the optimisation of the operating conditions and in screw design of a polymer extrusion process. The optimisation of the operating conditions and the screw design of polymer extrusion can be considered a multiobjective optimisation problem. The optimisation methodology proposed is able to find solutions with physical meaning. 6- REFERENCES 1. mellal, K., Lafleur, P.G., rpin,.: Computer ided Design of Single-Screw Extruders, in.. Collyer, L.. Utracki (eds): Polymer Rheology and Processing, Elsevier (1989) Rauwendaal, C.: Polymer Extrusion, Hanser Publishers, Munich (1986) 3. O rian, K.: Computer Modeling for Extrusion and Other Continuous Polymer Processes, Carl Hanser Verlag, Munich (1992) 4. gassant, J.F., venas, P., Sergent, J.: La Mise en Forme des Matiéres Plastiques, 3 rd edn, Lavoisier, Paris (1996) 5. Gaspar-Cunha,., Oliveira, P., Covas, J..: Use of Genetic lgorithms in Multicriteria Optimization to Solve Industrial Problems, Seventh Int. Conf. on Genetic lgorithms, Michigan, US (1997) 6. Gaspar-Cunha,.: Modeling and Optimization of Single Screw Extrusion, Ph. D. Thesis, University of Minho, Guimarães, Portugal (2) 7. Deb, K., grawal, S., Pratap,., Meyarivan, T.: Fast and Elitist Multi-Objective Genetic lgorithm: NSGII, Technical Report 21, Indian Institute of Technology, Kampur, India (2) 8. Deb, K., grawal, S., Pratap,., Meyarivan, T.: Fast Elitist Non-dominated Sorting Genetic lgorithm for Multi- Objective Optimization: NSGII, Proceedings of the Parallel Problem Solving from Nature VI (PPSNVI) (2) Zitzler, E.: Evolutionary lgorithms for Multiobjective Optimization: Methods and pplications, Ph. D. Thesis, Zürich, Switzerland: Swiss Federal Institute of Technology (1999) 1. Zitzler, E., Laumanns, M., Thiele, L.: SPE2: Improving the Strength Pareto Evolutionary lgorithm, TIK report no. 13, Swiss Federal Institute of Technology, Zürich, Switzerland (21) 11. Knowles, J.D., Corne, D.W.: pproximating the Non-dominated Front using the Pareto rchived Evolutionary Strategy, Evolutionary Computation Journal, 8 (2) Roseman, M.., Gero, J.S.: Reducing the Pareto Optimal Set in Multicriteria Optimization, Eng. Optim., 8 (1985)