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Design of the Cooling Channels in Nonrectangular Plastic Flat Injection Mold Zone-Ching Lin and Ming-Ho Chou, Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C. Abstract The complex variables and parameters with respect to the design of cooling channels in nonrectangular plastic flat injection mold are investigated. Vector and simple mathe- matical calculations were used to solve problems related the deployment of cooling channels caused by different geomet- ric dimensions of molded products. Furthermore, the basic geometry characteristic surface symbols and database were established. Next, the basic geometry characteristic surface was used to compose and present the shapes of molded product on the basis of a rectangular plastic flat. The conver- sion concept of equivalent area was also introduced to sim- plify the selection of model and deployment of cooling chan- nels for nonrectangular plastic flat containing milder changes in shape. The optimization of cooling time in the first stage is based on the energy balance. A concise formula was used, with empirical algorithm as the constraint of optimization design, to locate the optimal cooling time and the required optimal geometric factor constraints. Then, the optimization of channel deployment was derived to achieve the require- ment of fast and uniform cooling of the mold. The method proposed in this paper is capable of handling a nonrectan- gular plastic flat product through the conversion to an equiv- alent rectangular area. This method simplifies the channel deployment problem of a molded product caused by nonuni- form distribution of heat source and reduces the instances of trial and error. Furthermore, the method proposed for the system framework is capable of completing the optimization faster than conventional finite difference method, which saves the time spent in designing the cooling channels and achieves fast and uniform cooling of finished products. Keywords: Injection Mold, Energy Balance, Optimal Design Introduction The design of an optimal cooling system and operation conditions of the cooling process are very crucial to the injection molding process (Rosato and Rosato 1985) because they both affect the quality of molded products and the production efficiency of this process. The design of an efficient cooling sys- tem reduces the cooling time needed. Uniform cool- ing improves such defects as the shrinkage of mold products, thermal residual stress, and warpage deformation. Thus, the design of an optimal cooling system must achieve the goals of the shortest cool- ing time possible and uniform cooling at the same time. There are a number of parameters during the injection molding process related to cooling. The most important factor affecting the uniform cooling of mold products is the deployment of the cooling system channels. A well-designed cooling channel deployment, plus appropriate operation condition, can achieve the goals of fast and uniform cooling so as to shorten the process cycle and improve mold product quality. By the early 198Os, research attention turned toward the thermal interactions between the cooling system and the mold cavity. During this time, most simulation programs dealt with only one-dimensional or two-dimensional average steady-state analysis or analysis of the quasi-steady state of the cycle (Austin 1985). The methods used include the finite difference method, finite element method, and cycle-averaged boundary element method (Singh 1987, Singh and Wang 1982, Barone and Caulk 198 1, 1982). The gen- eral analytical steps appearing in studies published after 1985 already adopted the so-called two-stage or three-stage analysis (Kwon, Shen, Wang 1986). The initial analysis mostly used the shape f actor method to evaluate the efficiency of the cooling system (Kwon Shen, Wang 1986; Chen, Hu. Davidot f 1990). The analytical results of this time were already controlled based on the operation conditions of the cooling sys- tem and the positions and size of the cooling chan- nels. The analytical approaches were all one-dimen- sional or two-dimensional analysis of the steady state or quasi-steady state (Kwon, Shen, Wang I 986; Chen, Hu, Davidoff 1990; Himasekhar, Wang, Lottey 1989; Himasekhar? Hiber, Wang 1989: Chen and Hu 199 1). A number of finished simulation analysis software packages became available after 1990 (Himasekar 1989; Himasekhar and Wang 199(I), including MCAP, MOLDCOOL, MOLDTEMP, POLYCOOL2, and C-COOL3D. The analytical methods include finite difference method, finite element method, and boundary element method (Turng and Wang 1990; Himasekhar, Lottey, Wang 1992). Sometimes the combination of the finite element method and shape 167 Journal of Manufacturing Systems Vol. 21Mo. 3 2002 factor method is used (Glavill and Denton 1977). These analytical programs were all developed with the reduction of memory space and CPU time in mind and tried to produce analytical results that matched the realistic conditions (Himasekhar, Lottey, Wang 1992). This paper combines the ener- gy balance and empirical algorithms to construct system modules. It also uses vector and concise mathematical calculations to handle such a complex system and gets rid of complex numerical computa- tions and analysis. Conventionally, mold manufacturing relies on experience and intuitions after numerous times of “trial and error.” Thus, it often results in perfor- mance and economic burdens. Nowadays, thanks to the rapid development of computers, molds can be designed through theoretical and numerical methods, and their results can be predicted or directly simulated through mold flow packaged software, thus improving the cost effectiveness and product quality. Among the numerical meth- ods, the finite element method has better stability and convergence than the finite difference method. But it also has more complicated theories and requires greater computer memory space. Though the packaged software used by the indus- try can rapidly solve the problem of mold design, it cannot effectively help engineers simplify the complexity and fuzziness of the system. Besides, the packaged software for simulation is, in fact, only capable of solving 60%70% of the prob- lems encountered in design. The “trial and error” part is unavoidable, and the lack of experience among engineers in design still becomes an influ- ential variable. Using the rectangular plastic flat as the research subject, this paper simplified the design variables and influential parameters encountered by the design of cooling channel deployment in a mold to help design engineers quickly complete the writing of the optimal flowchart and program with respect to the cooling channels and obtain the optimal cooling time and cooling channel deployment. The concept was adopted to simplify the deployment of cooling channels in the injection mold. This study combined the concise energy balance algorithm, shape factors, and empirical values to construct the modular relations that serve as the reference mod- ule for the design of optimal deployment of cooling channels. Mold close and injection time Mold opening and ejection time Mold cooling time Figure I Relation Between Mold Cooling Time and Molding Cycle Theory of the Cooling System in an Injection Mold A mold cooling system usually contains a tem- perature-controlling unit, pump, coolant supply manifold, hoses, cooling channels, and collection manifold. Among the deployment in the mold the temperature-controlling system affects the uniform distribution of mold temperature and degree of sta- bility, An increase in the cooling efficiency in the temperature-controlling system increases its produc- tion efficiency. Cooling of the Injection Mold In injection molding, the mold cooling time usual- ly takes up about 70%80% of the time of the entire cycle. Figure I shows the relationship between mold cooling time and a molding cycle (Chow 1999). An effective cooling loop design reduces the cooling time and effectively increases the production rate and reduces cost. In addition, uniform cooling prevents the product from suffering such defects as shrinkage, warpage, and deformation due to thermal stress, which then increases the dimension precision and reliability of molded products and also improves product quality. Figure 2 shows the relationship between the presence of effective cooling and the quality of molded products and mass production (Chow 1999). In general, a mold consists of the three parts of the mold body, cooling channels and plastic material. Figure 3 illustrates a simplified mold cool- ing mechanism. In this paper, cooling is analyzed under the assumption that no thermal energy is lost through the edges of molded products. That is, cool- ing (thermal conductivity) occurs only at the thick- 168 TKl I) cooling channel size; cooling channel type; cooling channel deployment and connection; length of cooling channel loop; and flow rate of coolant. It is necessary to note here that standard sizes of cooling channels must be used to allow the use of Plastic material / Figure 3 Illustration of Mold Cooling Mechanism Figure 4 Thermal Conductivity of Ordinary Molds working tools and connecting parts of standard spec- ifications and rapid mold change. The rapid and uni- form cooling are the major guidelines for the mold cooling design. Because the cooling process takes up 70%80% of the time of the entire molding cycle, if the cooling system can rapidly cool off the product, i.e., a small improvement in cooling time, can greatly shorten the time of the entire molding cycle and increase production, Thus, the way to shorten the cooling time is crucial to the designer and also the subject of discussion in this paper. If unbalanced cooling occurs during the cooling process of molded products, they produce a thermal stress, causing shrinkage and warpage. Thus, it is necessary to maintain uniform cooling of molded products so as to reduce the thermal stress sustained by the products and the ensuing shrinkage and warpage. In other words, the temperature difference between two sides of the molded product should be small to achieve uniform mold temperature. Empirically, the temperature difference must not exceed 10C. The easiest and most effective method is to match the thermal conductivity surface area of the cooling channel (A,.) and that of the molded 169 Journal of Manufacturing Systems Vol. 2uNo. 3 2002 product (AJ, which is the basis of maintaining product uniform cooling in this paper (Ioannis and Qin 1990). Theory of the Concise Computation of Injection Mold Cooling Channels Every stage of the molding injection process contains a cooling process. Thus, the cooling time is generally explained as follows: “The melt plastic material starts to cool as soon as it is injected into the cavity, and the cooling continues during the stages of filling, post-fill, and cooling throughout the entire molding cycle till the molded product is hard enough to push out of the cavity, which is considered the end of cooling time.” As shown in Figure I, cooling time t, takes up about 70%80% of the entire molding cycle. Hence, the shortening of cooling time t, by a few percentage points can have a tremendous impact on the entire mold- ing effect. The shortening of cooling time is the most direct and significant factor affecting the cost of molded products. In this paper, the optimal cooling time is used as the basis of the design of cooling channels in an injec- tion molding cooling system. Basic Assumptions of the Design of Injection Mold Cooling Channels Correlation factors that affect J c;g Thermal H Heat con1 _ properties properties viscosity Molded product Geometric shape factors Cooling channel Pitch _ distance _ Diameter length Operation _ conditions Mold material Mold temperature Entrance temperature/ Exit temperature I I The goal of injection mold design is to minimize the cooling time. There are a number of factors affecting cooling time. Here, factors related to cooling time are listed in brief in Figure 5 to serve as the basis of design considerations. These factors are described below (Chang 1985): 1. Thickness H of molded product. The thicker the molded product, the longer the cooling time needed. 2. Shape of molded product. If the molded prod- uct has a complicated shape, then the cooling effect at some parts may appear less distinc- Figure 5 Factors Affecting Cooling Time tive, which may in turn affect the cooling time of the entire molded product. 3. Quality of plastic material melt. Because dif- ferent kinds of plastic materials have different thermal diffusivity, their thermal conductivity effects also differ. Plastic materials having a greater thermal diffusivity have greater ther- mal conductivity rates and require a shorter cooling time. 4. Injection temperature and ejection tempera- ture. The higher the injection temperature, the longer the cooling time required. In contrast, the lower the ejection temperature, the longer the cooling time required. 170 5. Mold material. Because different metal materi- als of the mold have different thermal conduc- tivity, their thermal conductivity effects also differ. Metals with a greater thermal conduc- tivity conduct heat faster and require a shorter cooling time. 6. Number, position, and size of cooling chan- nels. The design of cooling channels has a decisive affect on the overall cooling time. Generally speaking, the larger the number of cooling channels, the closer the cooling chan- nels are to the molded product or the larger the channel diameter, the better the cooling effect and the shorter the cooling time. 7. Quality of coolant. Different coolants have dif- ferent heat transfer coefficient, specific heat, density and viscosity, and thus, different heat transfer results. 8. Coolant flow rate and temperature. The coolant flow rate must reach the turbulent flow to increase the heat transfer effect. Besides, the lower the coolant temperature, the shorter the cooling time. The cooling stage involves very complicated issues. To simplify the process, the following assumptions are made in this study: 1. Because changes of the physical properties of mold materials as a result of temperature and pressure are not significant, they are consid- ered constants. 2. The energy released by the plastic materials is assumed to be completey absorbed by the coolant and mold material. 3. The mold surface temperature is assumed to be constant and so is the temperature of the cool- ing channel wall. 4. It is assumed that during the initial stage, both the mold and the plastic material have their own uniform temperature, and that the plastic material does not contain any solid part. 5. The inner pressure of mold cavity is assumed constant. Thus, the effect of pressure reduction at the boundary layer is ignored and the volume of plastic materials remains constant during the solidification process. 6. The solidification latent heat is calculated as part of the specific heat, without considering the dis- placement of boundary layer. 7. The plastic material is assumed to be in a static state throughout the entire cooling process. Hence, the thermal effect derived from flow is ignored. 8. The thermal effect derived from the crystallization process is ignored in this study. Computation of Cooling Time The computation of cooling time for liquid plastic inside a mold was once considered in the research by Ballman and Shusman (1959). Further studies on the subject have been published by a number of researchers (Kening and Kamal 1970). In this paper, a concise computation is made on cooling time. Here, the prob- lem of cooling and solidification of the plastic materi- al in a mold is written in a 3-D transient thermal con- ductivity equation as below: where C, = C,(T): denotes the heat capacity of the plastic material k: thermal conductivity of the plastic material p: plastic density of the plastic material X, X, and X3: the heat transfer conductivity direction of the molded product, respectively, where X, denotes the direction of thickness. Here, the crystallization rate of the plastic mate- L/x dX, a x, -L,A _ rial, cif dt rlt * and the latent heat of the plas- tic material, iw, are not taken into consideration. Besides, under most circumstances, cooling mainly occurs along the thickness direction (the X, direc- tion). During the injection molding process, the pri- mary thermal energy is removed during the cooling process. Therefore, the cooling during the short peri- od of time of filling can be ignored, and it is assumed that the plastic material cools off from a uniform tem- perature during the cooling process. Hence, the fol- lowing concise equation can be derived: ilT k d T x - PC, ax; =(x iE ” ax (2) k where a, =- pC, denotes the thermal diffusivity (cm*/sec.) of plastic material 171 Journal of Manufacturing Systems Vol. 21mo. 3 2002 1 X2 Mold surface I Mold bo 1 *x, or c:T =C IV Figure 6 Figure 7 Boundary Conditions of Molded Product Illustration of Cooling Channel Position The boundary conditions are shown in Figure 6 (Chang 1985). The cooling time relation Eq. (3) and (4) can be obtained through the first item of the Fourier development formula (Chang 1985). 1. The cooling time t, that it takes for the uniform temperature to cool to T, in the thickness direc- tion of the molded product is: (3) 2. If it takes for the center point in the product thickness direction to cool to T, as the basis, then the cooling time t, is: (4) where tc: cooling time (sec.) T,: initial temperature of plastic material (“C) T,: temperature of mold wall (“C) T,: ejection temperature, i.e., molded product temperature (“C) (average temperature in the X, thickness direction) Hz Average thickness of molded product (mm) 3. Conductive shape factor: the dimension of S is the length or nondimension that shows in the