在不规则塑料平板进行注塑模具冷却水道的设计外文文献翻译、中英文翻译
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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) &=L pg) Better part in shorter time c%g I) Jj=j& Poor part in longer time Figure 2 Relation Between Proper and Effective Cooling, Product Quality, and Production Quantity ness direction. Figure 4 shows the situation of mold thermal conductivity in general. It is also assumed that the thermal energy is all directly taken away by the coolant, i.e., the transfer of thermal energy brought in by the plastic material only, excluding the 5% of energy transferred outside the mold due to radiation, convection, and heat conduction. Hence, the paths of thermal conductivity can be simplified to the following: thermal energy is transferred from the melted plastic material through thermal conductivity to the interface between the melt plastic material and mold. The thermal energy passes through this inter- face, then the mold body, and then to the interface between the mold and coolant through thermal con- ductivity. Next, it is transferred to the coolant from the interface through convection. Finally, the thermal energy is completely brought out of the mold body through coolant flow. Precautions for the Design of Cooling Channels in an Injection Mold Injection molding is almost always used in mass production. Thus, the most important concern is how to raise production to achieve good economic effectiveness. The most direct and effective way of raising production is by reducing the cooling time to achieve rapid product cooling. At the same time, to ensure uniform product temperature and maintain quality, the way to maintain uniform cooling is also an essential requirement. As far as mold cooling is concerned, mold design engineers need to determine the following design parameters: cooling channel position; 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 Figure 7 (Rohsenow and Hartnett 1973). S.f = 2?rLJ 2Psinh In ( 1 Yrd p 1 Dd (5) Cooling channel where d: the cooling diameter D: the distance between the center of cooling channel and melt plastic material P: pitch between two cooling channels Energy Balance Principle In this paper, the cooling derived from Eq. (3), supported by the energy balance principle (Chang 1985, Glavill and Denton 1977), is used as the mathematical theory of the cooling process. The solution of cooling time uses the time it takes for the average temperature in the product thickness direction to cool to T, as the basis. The tempera- tures and thermal energy at various parts of mold, as shown in Figure 4, is as the general heat transfer approach (Chang 1985). ql: The average thermal energy conducted from the plastic material to the cavity face in a time unit. wqml 41 =- t ( G-Q c (6) q2: The thermal energy conducted from the cav- ity face to the cooling channel wall surface. q2 = ks,qTw -T) (7) q3: The thermal energy conducted from the cool- ing channel wall surface to coolant. q4: The thermal energy taken away by coolant q4 = PqT,u, - cl) (9) 172 where T,: average temperature of mold cavity T,: initial temperature of the melt plastic material T, : temperature of molded product T;,: temperature at coolant entrance Tout: temperature at coolant exit T,.: T,: w: Y: PI L: d: s,: average temperature of cooling chan- nel wall average water temperature, r, =;ir., +? 1/ ,) plastic material weight coolant flow rate coolant density length of cooling channel diameter of cooling channel conductive shape factor Position and Size of Cooling Channels Position and size of coolant channels can become very complicated as a result of mold shape. In most cases, drills are used to drill holes, which are then refined by a radial boring machine. Determining the scale of a cooling system is no simple task. As for the issue of heat transfer, part of the problem can now be solved through computer use. However, dif- ficulty in finding a solution still exists as the bound- ary conditions vary. In light of this, the position and size of cooling channels are discussed in this study on the basis of concise thermal conductivity theories introduced in the abovementioned paragraphs and the empirical algorithms used in the industry as the basis of design so as to obtain better cooling effi- ciency. The empirical algorithms used in the study for the diameter and position of cooling channel are as follows: 1. Diameter of cooling channel (d): the diame- ter of cooling channels must allow a sufficient flow rate of coolant to produce turbulent flow. Generally, the diameter of cooling chan- nels depends on the average thickness of molded product. Given the average product thickness of H and the cooling channel diameter d, then the empirical algorithm is written as: where usually it is set dmaxI 14 mm to maintain the turbulent flow and get higher cooling efficiency. 2. Position of cooling channel: in general, to obtain higher cooling efficiency, cooling holes should be as close to the mold cavity surface of product as possible and the distance between cooling holes should be kept at a minimum, as shown in Figure 7. The order of empirical algorithms for the posi- tion of cooling channel are as following: (1) Pitch between cooling channels (P): it is better that the triangular shape forming by the centers of cooling channels and the mold cavity surface covers the mold cavity surface as shown in Figure 7. A value between 3d and 5d is com- monly recommended for the value of pitch P. (2) Distance between hose and product (D): in general, the closer the cooling channel is to the product surface, the better the cooling effect. However, if ns 16 tnm. significant temperature variation easily occurs. Hence, a value between 1.5d and 3d is commonly rec- ommended for the value of distance D. (3) Length (L) and number (N) of cooling channel: In general, the arrangement of these variables is based on having almost the same total sur- face area of effective cooling channel length (.4,.) and total heat transfer area ofproduct (A,), i.e., A&4, (Ioannis and Qin 1990). Deployment of Cooling Channels for Nonrectangular Plastic Flat An ideal cooling system not only has to satisfy economic concern (short cooling time) but also achieve uniform and thorough product cooling. Cooling quality depends on the design of cooling channels. Generally speaking, to obtain better results, design is based on the basis that the coolant should be able to take away all heat generated by the product. Let the energy taken away by coolant be q4 and the energy brought to the mold by melt plastic material be ql, then q,Tq, (Ioannis and Qin 1990). If the surface of cooling channel represents a “cool- ing surface,” and the surface of product or mold cavi- ty represents the “warming surface,” then the thermal flow between two planes occurs on the surface. Without considering the factor of pressure drop, the empirical algorithm based on Eq. (IO) (Ioannis and 173 Journal of Manufacturing Systems Vol. 21mo. 3 2002 Qin 1990) is used as one of the basic conditions for design so as to maintain a better uniformity of the tem- perature at any given location of the product. A6, + de, 5 20C (10) where A,: A0,: Temperature variation produced by the heat transfer in the distance between mold and cooling channel Temperature variation produced by coolant flowing Ae,+AO,: denotes the total temperature variation According to experiences in general, when the total area of cooling channel A, equals the total heat diffusion area of the product A, a concise and opti- mal arrangement can be obtained. Thus, A, had bet- ter equal or greater A, as follows (Rohsenow and Hart&t 1973): A, 2 A, (11) where A,: Total heat diffusion area of product A,: Total surface area of cooling channels A,=xdL (12) where d : diameter of cooling channel L : total length of cooling channel Hence, total cooling channel length Ld$ (13) In general, plastic product s thickness is very small in relation to its size. Besides, it is assumed that heat is conducted only in the thickness direc- tion. Thus, to solve the total heat diffusion area A, of product, we first established the basic shape charac- teristics (surface characteristics). Designer can then proceed with the conversion of the equivalent rec- tangle and development of the length according to the product shape. Next, basic mathematical compu- tations are used, under the energy balance principle and through the combination of surface characteris- tics, to derive the value of total heat diffusion area A,. Finally, it can obtain the corresponding total length of cooling channels, L, from Eq. (13). Basic Shape Characteristic Surfaces In this paper, cross product of vectors and simple mathematical equations are used to derive the area of all basic characteristic surfaces. These basic char- acteristic surfaces are then combined to obtain the heat diffusion area of the molded product. The fol- lowing sections describe the calculation for the area of basic characteristic surfaces. 1. Shape: Triangle Characteristic symbol: A A, = $2, X j,l hYJ & = (x, - x,)i + (y, - y()J 8, = (x* -x,)i +(y* - y,)J .e, =(x2-x$+(yz-y,)J whereifj,*j,*O, 2&+0,and j,j,fO,thenit is a regular triangle. If&.j,=O, j2-e3=0,and Q,*j,=O,thenitisa “-$ ?T+$ I&, then it is an isosceles tri- angle. If lj,l=I&i=l- 1 h !, , t en it is an equiangular triangle. 2. Shape: parallelogram Characteristic symbol: B 4 = (2, x j2/ j, =(x, -x,)i+(Y, -YdJ j2 qx2-xl)i+(Y2-YJ 5, =(x2-x3)r+(Y2-Y3) g4 =(x3 -x,)i-+(y, -YJI 174 whereif,=,or j2=j,and ?,*?z+O or j3*?4#0, then it_ is a parallelogram. If I, = !, or 7, =i, and ?, j2 = 0 or jj .P, = 0, and li/=li:l*i,l=7,1 then it is a square. And if 7/=li:l*7;=/i., then it is a rectangle. 7, =(x, -x,)F+(y, -_I(,).; c2 =(x2 -X,)7+() , -y,)J 7; =(x2 -X$f(_v2 -v$ r, = (x3 -x,)i +(_I$ -v,)J where if iI #nI and i2 f ITT, then it is a reg- ular tetragon. If 7, # 7, 7_ = ni, then it is a trapezoid. If 7, = ,I?. . 2 + 7 I then it is a trapezoid. If f, = jli:, li( = Ii,/, then it is an If ,=l/, . lk/=/7;!, then it is an isosceles trapezoid. isosceles trapezoid. 4. Shape: n multilateral (n25 Characteristic symbol: D and an integer) IZ multilateral A eTotd = Ap, + A(, + A?, A”eTrml = A, + A,2 + A?, + A, A”eTotul = A?, + A(,? + AP3+.+Ac.,_2 where A?, = $i, XT?/ A, = ;I& x?,l A c,i-? 5. Shape: Circle Characteristic symbol: E A,=nah L_-_-/Jb1“ 1,: where if “=“= z, then it is a circle and LI is the diameter. If (I # h , then it is an ellipse. Combination of Characteristic Surfkces The product shape discussed in this paper primarily contains milder changes in shape without drastic shape changes. Products containing more drastic changes are beyond the scope of this study. Here, a simple flat com- posed of a number of basic geometric shapes is used as an example to explain the combination of characteris- tic surfaces. The symbols are listed in Table I. The product shape is shown in Figure 8. During the com- position, the area of all basic characteristic surfaces listed in Table I are derived to establish a database. Next, the correlation among the characteristic surfaces is clearly described based on the characteristic surface database established and the treelike structure as shown in Figure 9. Then the heat diffusion areas of the mold- ed products can be derived through simple combina- tions of mathematical calculations. Equivalent Rectangle Conversion of Nonrectangular Plastic Flat If the molded product is a flat on the same plane and its shape belongs to a geometric shape with mild changes and without drastic changes, it must be con- sidered the shape to be composed of some simple basic characteristic shapes-triangles, tetragons, and circles. Among the shape factors of a flat rela- tive to the product shape, a symmetrical rectangle achieves the most uniform thermal conductivity behavior within the same domain. In this paper, a rectangle is used as the basic characteristic surface to implement the deployment of channels. As for nonrectangular product shapes, an equivalent rectan- gle is used in replacement. Figure IO shows a non- rectangular plastic flat, while Figuw f I shows the equivalent rectangular shapes of various domains of the nonrectangular plastic flat shown in Figure JO. 175 _IL_ upwop puoaag u!eluop JsJ!j zooz E w/lZ PA stuawQ 8ugnpvjnuvpq jo punof c 1 C-l rigure Y Relationship Between Tree-Like Combinations of Characteristic Surfaces Step 7: Calculate the cooling channel length Lj required for various domains. If=3 Total cooling channel length L = c L, = L, + Lz + L, where, *=I A L,=K,L,K,=-“ + A f/rrvo/ I A A L,rzK2L,KKz=*= A e/lo/o/ AC A L, = K,L, K3 = _?!? = % A (14) (15) (16) (6) Constraint: 9, = Y2 = q, q, -Y, 2 0 T, - 7;, - 5 5 0 n I T, O, output the optimal result oft, T, S, Re and proceed with the optimization of positions of cooling channels, as shown in F &re f3. Optimization of Positions of Cooling Channels After the optimal cooling time t, is obtained the cor- responding conductive shape factor S,. is derived. Next, the random search method is used in the optimization search of the position and size of cooling channels. At this time, the design parameters are: e, d, and S, and the design variables are P and D. The objective function is: (17) 179 Journal of Manufacturing Systems Vol. 21/No. 3 2002 Heat transfer properties of plastic Injection temperature of melt Heat transfer properties of mold Heat transfer properties C, p. of coolant a, C, p, I, Temperature of coolant entrance set initial mold temperature: T, ie YES ( t Calculate initial target value , T, = T, + 1 L I 4 Calculate initial target value 1 T, = T, I2 I I S,=Sf+E, f Re = Re + &Re YES NO Figure I2 Flowchart of the Optimization Program of Cooling Time The constraint conditions are: g, = G( P, 0) - 1.5e I 0 g, = G( P, 0) - 0.5e 2 0 alPlb,nlDlm where e is the program s convergence adjusted error. The correlation among d, P, and D are shown in (18) Table 2. The message box is used to allow the program and user to obtain real-time communication. Given 180 Set conductive shape factor Input information r = 0, (PI. D,)” i r=r+l Calculate initial objecbve function value G(P,D,) I 10 I I - I I I Opbmization search I I , Yes Flowchart of the Optimization of Cooling Channel Position different molding conditions, the derived optimal cooling time and corresponding conductive shape factor 5” the corresponding design parameters e, d, and ,$values and the range of design variables P and D can be entered instantaneously under the window operation environment. One can also learn about information of the optimization search via the mes- sage box. The optimal P and D values can then be learned through the designated output file. The opti- mization flowchart of positions of cooling channels is shown in Figure 8 with the following steps: Step 1: Enter the values of design parameters e, d, S, Step 2: Enter the maximum and minimum LI, h, n, and m values of design variables P and D. Step 3: Connect to the optimization program and pro- ceed with the iteration computation of the optimiza- tion search. Step 4: Output the optimal P and D values to the out- put file and end the program. Case Study of the Optimization of Cooling Channels in Nonrectangular Plastic Flat Injection Mold The research subject in this paper is a nonrectangular plastic flat. The execution steps and results from the method proposed in this paper are as follows. Basic Information of Case Study of Nonrectangular Plastic Flat (Rohsenow and Hartnett 1973) Given the dimension and shape of the molded product as shown in Figure 8: 1. 2. 3. 4. Basic heat transfer properties of plastic PS material: Density pm: 1080 kg/m” Heat conductive coefficient / 10000 Transient flow: 23OORe 10000 Laminar flow: lOORe2300 Static flow: RelOO Execution Steps and Results In this case study, the empirical algorithm and fundamental heat transfer behavior are combined through optimization to derive the results, as shown in Tables 3, 4, and 5. The correlation is shown is Figure 15. The operating requirements are as follows: Injection temperature of the melt plastic material: T,=22O”C Molded product temperature: T,=80C Coolant entrance temperature: Ti,=2O”C Execution steps: Step 1: Step 2: Calculate the size of basic characteristic sur- faces as listed in Table 6. Determine the maximum width Cmaxi and the equivalent width Z, of various domains: First domain: Maximum width Cmaxl = 30 mm, Equivalent width Z, = 30 mm Second domain: Maximum width CmavZ = 44 mm, Equivalent width Z, = 44 mm Step 3: Derive the heat transfer area & and equiva- lent heat transfer area A, of the domains of the molded product as listed in Table 7. n=2 AP=xAi=A,I+AeZ Aei = Aefl where A, =l,_, +2(1,_,)+2(1,)+4(E,-,) Aex = lC,_, + 14-1 + Ae = lB,-, + 242 + 24 + lC,_, + Q-, + Step 4: Convert the equivalent length zig of various domains where the area of the equivalent rectangle is shown in Figure 14. 182 Table 3 Selection ofAverage Thickness. Cooline Channel Diameter d, Pitch P, and the Distance D Between Cooling Channel and Mold Cavitv Molded Product Thickness H (mm) HC2 X5(/ IO Cooling Channel Cooling Channel Diameter d (mm) Pitch P (mm) 8 24 2 P I40 9 27 I P 545 10 30 $ P 50 Distance Between Cooling Channel and Mold Cavity, D (mm) lZii)24 1.3.5 5; II 5 2: I 5 i 0 c 30 10 30 5 P 5 50 Ii 33 P 5 55 12 366 14 42 P I 70 31 c: I42 Cooling Channel Diameter ti (mm) T,( C, Table 4 Results of Cooling Time Optimization AT,.( “C) To, (“C) A T,u,( “c ) Flow rate LI( m”/sec.) .s, /%.( sec. ) 10 40 2.643 21.82 1.82 2.8x I o- I.XYl 75.46 II 41 2.37 21.63 I.63 3.0x10- 2.289 35.85 12 43 3.43 21.45 1.45 3.3x lo-s I .670 26.66 13 44 3.25 21.32 1.32 3.6 I (I- I .8X0 27.08 I4 45 3.10 21.22 1.22 3.9x lo-5 2.083 77.53 Cooling Channel Diameter d (mm) 10 11 I2 I3 Table 5 Results of Optimal Cooling Channel Deployment Cooling Channel Distance Between Cooling Cooling Channel Pitch P (mm) Channel and Mold Cavity, D (mm) Length L (mm) I, (sec.) - 44.01 20.93 206 25.46 47.09 18.31 I87 25.X5 56.94 30.37 172 x.24 41.94 22.14 159 27.09 Table 6 Size of Basic Characteristic Surfaces Symbol B,., B,_l B1.3 Cz.1 D7.I Et.1 Ez-1 EI-2 Ez-2 b-4 I?., :-_ E, _ Size (mm ) 3960 110 720 45 480 16x I57t Orr l87t 36x 45x I xon: First domain: 178.96 mm Equivalent length L, = Second domain: Equivalent length L, = 24.75 mm Step 5: Determine the cooling channel length alloca- tion percentage coefficient Kj of various domains. jy, - :y _ 2, ( /f where first domain: equivalent area A, = 5368.58 mm* Cooling channel length allocation percentage coefficient Ki = 0.83 137 Second domain: equivalent area A, = 1088.92 mm* Cooling channel length allocation percentage coef- ficient Ki = 0.16863 Step 6: Obtain the following optimal design variables through the optimization of cooling time. The results are listed in Table 4. The correlation is shown in Figure 15. As listed in Tuble 4, the cooling time at u = 10 mm is the shortest at 25.46 sec. Step 7: Connect the conductive shape factor 3, obtained in Step 6 under the shortest cooling time with the optimization program of conductive shape factor through the dialogue window on the panel. When the derived conductive shape factor equals 1.891, results of the cooling channel positions P and D can be obtained as listed in Tublr 3. 183 Journal of Manufacturing Systems Vol. 21/No. 3 2002 First domain Second domain ! ._ _ ._ 178.96 mm 24.75 mm I . Figure 14 Illustration of an Equivalent Rectangle Area Step 8: Calculate the cooling channel Li length required for the cooling channels of various domains. where first domain: cooling channel length L1 = KIL = 0.83137 X 0.206 = 0.17126 m Second domain: cooling channel length L2 = K,L = 0.16863 X 0.206 = 0.03474 m Step 9: Determine the number of cooling chan- nels for various domains Mi and the number of cooling channels in actual deployment A4worki. The number of cooling channels for various domains: M. = L, Z 28” 2 27. ? G 2 $ 2 26. E P 2 = ; 25. 2 10 11 12 13 14 15 16 Cooling channel diameter d(mm) (b) Relationship between cooling channel 1 diameter and cooling temperature The number of cooling channels actually deployed for various domains: iklworki If P 2 ti, then M, = M, If P tj and Mi + Mi, then Mworki = Mu + 1 If P 17 5 mm &Ill $4 9 which is capable of taking away all energy on the mold cavity surface of the second domain. The cooling channel deploy- ment is shown in Figure 16. Conclusion This study used vector and simple mathematical calculations to handle the geometric shape factors of molded products. Under the principle of energy balance, it uses combinations of the basic geomet- ric characteristic surfaces and its database con- structed in this paper to describe the shape and heat diffusion area of a nonrectangular flat with milder changes in shape. The nonrectangular flat products were converted into rectangular flats through the conversion of equivalent rectangle area. This method simplified the channel deployment problem caused by nonuniform distribution of heat source in the molded product. In this study, to simplify a complex cooling system, the system structure is divided into two stages: the first involves the optimization of cool- ing time and the second handles the optimization Table 10 Correction of Cooling Channel Length & and Adjustment of Flow Velocity Vi Design Cooling Actual Cooling Correction Cooling Design Flow Domain Name Channel Length Channel Length Required Channel Length Velocity () Li mm wrki mm AL; (mm) Voi (mkec. _ First domain 171.26 150 -2 1.26 0.3565 Second domain 34.74 44 9.26 0.3565 Work flow Velocity I ;.,rki (m/set.) 0.4064 0.2823 185 Journal of Manufacturing Systems Vol. 21iNo. 3 2002 of geometric shape factor and cooling channel position deployment. The optimization of cooling time in the first stage is based on energy balance. It then uses a concise formula, with empirical algorithm as the constraint of optimization design, to derive the optimal cooling time and the optimal geometric shape constraints required. The position deployment of cooling is then derived in the stage of the optimization of geometric shape factors and position deployment of cooling chan- nel to satisfy the requirement of fast and uniform cooling. This approach can rapidly complete opti- mization to save the time spent in designing cool- ing channels and allow the molded product to cool quickly and uniformly. References Austin, C. (1985). “Mold cooling.” Society of Plastics Engineers Technique Papers (31, 1985) 764-766. Ballman, B.L. and Shusman, T. (1959). “Easy way to calculate injection molding set up times.” Modern Plastics (37, n3), 126. Baron
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