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_ Corresponding author: Alban Agazzi, Universit de Nantes-Laboratoire de thermocintique de Nantes, La Chantrerie, rue Christian Pauc, BP 50609, 44306 Nantes cedex 3-France, phone : +332 40 68 31 71, fax :+332 40 68 31 41 email : alban.agazziuniv-nantes.fr A METHODOLOGY FOR THE DESIGN OF EFFECTIVE COOLING SYSTEM IN INJECTION MOULDING A.Agazzi1*, V.Sobotka1, R. Le Goff2, D.Garcia2, Y.Jarny1 1 Universit de Nantes, Nantes Atlantique Universits, Laboratoire de Thermocintique de Nantes, UMR CNRS 6607, rue Christian Pauc, BP 50609, F-44306 NANTES cedex 3, France 2 Ple Europen de Plasturgie, 2 rue Pierre et Marie Curie, F- 01100 BELLIGNAT, France ABSTRACT: In thermoplastic injection moulding, part quality and cycle time depend strongly on the cooling stage. Numerous strategies have been investigated in order to determine the cooling conditions which minimize undesired defects such as warpage and differential shrinkage. In this paper we propose a methodology for the optimal design of the cooling system. Based on geometrical analysis, the cooling line is defined by using conformal cooling concept. It defines the locations of the cooling channels. We only focus on the distribution and intensity of the fluid temperature along the cooling line which is here fixed. We formulate the determination of this temperature distribution, as the minimization of an objective function composed of two terms. It is shown how this two antagonist terms have to be weighted to make the best compromise. The expected result is an improvement of the part quality in terms of shrinkage and warpage. KEYWORDS: Inverse problem, heat transfer, injection moulding, cooling design 1 INTRODUCTION In the field of plastic industry, thermoplastic injection moulding is widely used. The process is composed of four essential stages: mould cavity filling, melt packing, solidification of the part and ejection. Around seventy per cent of the total time of the process is dedicated to the cooling of the part. Moreover this phase impacts directly on the quality of the part 12. As a consequence, the part must be cooled as uniformly as possible so that undesired defects such as sink marks, warpage, shrinkage, thermal residual stresses are minimized. The most influent parameters to achieve these objectives are the cooling time, the number, the location and the size of the channels, the temperature of the coolant fluid and the heat transfer coefficient between the fluid and the inner surface of the channels. The cooling system design was primarily based on the experience of the designer but the development of new rapid prototyping process makes possible to manufacture very complex channel shapes what makes this empirical former method inadequate. So the design of the cooling system must be formulated as an optimization problem. 1.1 HEAT TRANSFER ANALYSIS The study of heat transfer conduction in injection tools is a non linear problem due to the dependence of parameters to the temperature. However thermophysical parameters of the mould such as thermal conductivity and heat capacity remain constant in the considered temperature range. In addition the effect of polymer crystallisation is often neglected and thermal contact resistance between the mould and the part is considered more often as constant. The evolution of the temperature field is obtained by solving the Fouriers equation with periodic boundary conditions. This evolution can be split in two parts: a cyclic part and an average transitory part. The cyclic part is often ignored because the depth of thermal penetration does not affect significantly the temperature field 3. Many authors used an average cyclic analysis which simplifies the calculus, but the fluctuations around the average can be comprised between 15% and 40% 3. The closer of the part the channels are, the higher the fluctuations around the average are. Hence in that configuration it becomes very important to model the transient heat transfer even in stationary periodic state. In this study, the periodic transient analysis of temperature will be preferred to the average cycle time analysis. It should be noticed that in practice the design of the cooling system is the last step for the tool design. Nevertheless cooling being of primary importance for the quality of the part, the thermal design should be one of the first stages of the design of the tools. DOI10.1007/s12289-010-0695-2Springer-VerlagFrance2010Int J Mater Form (2010) Vol. 3 Suppl 1: 16 131.2 OPTIMIZATION TECHNIQUES IN MOULDING In the literature, various optimization procedures have been used but all focused on the same objectives. Tang et al. 4 used an optimization process to obtain a uniform temperature distribution in the part which gives the smallest gradient and the minimal cooling time. Huang 5 tried to obtain uniform temperature distribution in the part and high production efficiency i.e a minimal cooling time. Lin 6 summarized the objectives of the mould designer in 3 facts. Cool the part the most uniformly, achieve a desired mould temperature so that the next part can be injected and minimize the cycle time. The optimal cooling system configuration is a compromise between uniformity and cycle time. Indeed the longer the distance between the mould surface cavity and the cooling channels is, the higher the uniformity of the temperature distribution will be 6. Inversely, the shorter the distance is, the faster the heat is removed from the polymer. However non uniform temperatures at the mould surface can lead to defects in the part. The control parameters to get these objectives are then the location and the size of the channels, the coolant fluid flow rate and the fluid temperature. Two kinds of methodology are employed. The first one consists in finding the optimal location of the channels in order to minimize an objective function 47. The second approach is based on a conformal cooling line. Lin 6 defines a cooling line representing the envelop of the part where the cooling channels are located. Optimal conditions (location on the cooling and size of the channels) are searched on this cooling line. Xu et al. 8 go further and cut the part in cooling cells and perform the optimization on each cooling cell. 1.3 COMPUTATIONAL ALGORITHMS To compute the solution, numerical methods are needed. The heat transfer analysis is performed either by boundary elements 7 or finite elements method 4. The main advantage of the first one is that the number of unknowns to be computed is lower than with finite elements. Only the boundaries of the problem are meshed hence the time spent to compute the solution is shorter than with finite elements. However this method only provides results on the boundaries of the problem. In this study a finite element method is preferred because temperatures history inside the part is needed to formulate the optimal problem. To compute optimal parameters which minimize the objective function Tang et al. 4 use the Powells conjugate direction search method. Mathey et al. 7 use the Sequential Quadratic Programming which is a method based on gradients. It can be found not only deterministic methods but also evolutionary methods. Huang et al. 5 use a genetic algorithm to reach the solution. This last kind of algorithm is very time consuming because it tries a lot of range of solution. In practice time spent for mould design must be minimized hence a deterministic method (conjugate gradient) which reaches an acceptable local solution more rapidly is preferred. 2METHODOLOGY 2.1 GOALS The methodology described in this paper is applied to optimize the cooling system design of a T-shaped part (Figure 1). This shape is encountered in many papers so comparison can easily be done in particularly with Tang et al. 4. Figure 1 : Half T-shaped geometryBased on a morphological analysis of the part, two surfaces 1 and 3 are introduced respectively as the erosion and the dilation (cooling line) of the part (Figure 1). The boundary condition of the heat conduction problem along the cooling line 3 is a third kind condition with infinite temperatures fixed as fluid temperatures. The optimization consists in finding these fluid temperatures. Using a cooling line prevents to choose the number and size of cooling channels before optimization is carried out. This represents an important advantage in case of complex parts where the location of channels is not intuitive. The location of the erosion line in the part corresponds to the minimum solidified thickness of polymer at the end of cooling stage so that ejectors can remove the part from the mould without damages. 2.2 OBJECTIVE FUNCTION In cooling system optimization, the part quality should be of primarily importance. Because the minimum cooling time of the process is imposed by the thickness and the material properties of the part, it is important to reach the optimal quality in the given time. The fluid temperature impacts directly the temperature of the mould and the part, and for turbulent fluid flow the only control parameter is the cooling fluid temperature. In the following, the parameter to be optimized is the fluid temperature and the determination of the optimal distribution around the part is formulated as the minimization of an objective function S composed of two terms computed at the end of the cooling period (Equation (1). The goal of the first term S1 is to reach a temperature level along the erosion of the part. The second term S2 used in many works 47 aims to homogenize the temperature distribution at the surface of the part and therefore to reduce the components of 14thermal gradient both along the surface 2 and through the thickness of the part. These two terms are weighted by introducing the variablerefT. It must be noted that when refT the criterion is reduced to the first term. On the contrary the weight of the second term is increased when 0refT. ()?+?=222112.dTTTdTTTTTSrfejecinjejecfluid(1)ejecT: Ejection temperature, injT: Injection temperature, refT: Reference temperature, infT: Fluid temperature, T: Temperature field computed with the periodic conditions(),0(, 0XtTXTf+=21X, and ft, 0 is the cooling period, =?dTT22.1: Average surface temperature of the part at the ejection time ft. 3NUMERICAL RESULTS Numerical results are compared with those of Tang et al 4 who consider the optimal cooling of the T-shaped part by determining the optimal location of 7 cooling channels and the optimal fluid flow rate of the coolant. The first step was to reproduce their results (left part of Figure 2) obtained with the following conditions (case w=0.75 in 4): KTfluid303= , fluid flow rate scmQ/3643= in each cooling channels,s 5 .23=ft. Figure 2: Geometry Tang (left) and cooling line (right) Case 1: Cooling line versus finite number of channels for a constant fluid temperature (fluidT). The average distance (cmd5 . 1=) between the 7 channels and the part surface in the cooling system determined by Tang is adopted in our system for locating the cooling line 3. Moreover, the fluid temperature and the heat transfer coefficient values issued from Tang are imposed on the dilation of the part 3. In Figure 3 the temperature profiles along the part surface 2 are compared at the ejection time ft. All the temperature profiles along the surfaces 3 , 2 , 1 =ii are plotted counter-clockwise only on the half part from iAto iB(Figure 1) and at the ejection time. We observe that the magnitude of the temperature is less uniform with the cooling line than with the 7 channels (15K instead of 5K). Hence the optimal cooling configuration computed with a finite number of channels is better than this with the cooling line and it will be then considered as a reference. Figure 3: Temperature profiles along the part surface2Case 2: Cooling line with a variable fluid temperature ()(sTfluid) and the weighting factorrefT. The fluid temperatures )(sTfluid are computed by minimizing the objective function of Equation 1 where the second term is ignored. The results are plotted in Figures 4 and 5. Figure 4: Temperature profiles along the erosionFigure 5: Temperature profiles along the part surface In Figure 4 the temperature profile on the erosion is much uniform and close to the ejection temperature with our method (-511.79.10S=) than with Tangs method (-512.32.10=S). However in both cases a peak remains between 0.12m and 0.14m which corresponds to the top of the rib (B1in Figure 1). This hotspot is due to the geometry of the part and is very difficult to cool. Nevertheless in Figure 5 we notice that the profile of temperature at the part surface is less uniform than in 15case 1 (20K instead of 15K). In conclusion, the first term is not sufficient to improve the homogeneity at the part surface but it is adequate for achieving a desired level of temperature in the part. Case 3: Cooling line with ()(sTfluid) and the weighting factorsKTref10= and KTref100=. The fluid temperatures )(sTfluidare now computed by minimizing the objective function of Equation 1 with KTref10= and KTref100=. Results are plotted in Figures 6 and 7. Figure 6: Temperature profiles along the part surface Figure 7: Temperature profiles along the erosion The influence of the term S2 is shown in Figure 6. This term makes the surface temperature of the part uniform. Indeed in case KTref10= temperature is quasi-constant all over the surface 2 except for the hotspot as explained previously. However for this value of refT, the temperature on the erosion is not acceptable, the mean temperature being too high (340K for a desired level of 336 K). Then the second term improves the homogeneity at the interface but hedges the solution. Making uniform the temperature at the interface meanwhile extracting the total heat flux needed to obtain a desired level of temperature in the part, become antagonistic problems if this level is too low. The best solution will be a compromise between quality and efficiency. For example, by setting KTref100=the level of temperature (ejecT) in the part is reached whereas the solution becomes less uniform than with the value of KTref10=. Nonetheless this solution remains more uniform than the solution given by Tang. The optimal fluid temperature profile along the dilation of the half part is plotted (Figure 8). Figure 8: Optimal fluid temperature profile 4 CONCLUSIONS In this paper, an optimization method was developed to determine the temperature distribution on a cooling line to obtain a uniform temperature field in the pa
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