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Figure 7 depicts the DOE results of the mold temperature variations with respect to different locations of cooling channels. It can be seen that if the pitch x is large and the depth y is small, the mold wall temperature variation increases. As a result, the mold temperature distribution and molded part temperature distribution are uneven. Response surface methodology (RSM) was used to find approximated mathematical equations that express the response of mold temperature to the cooling channels’ configuration. The mold temperature is represented by the quadratic equation. Tw = – 0.268x2 + 0.257y2 + 0.157xy +3.430x + 4.131y + 13.6 (15) The R-squared coefficient that indicates the goodness of fit of this model is 0.983. The fidelity of RSM model was verified by comparing this model with the analytical model obtained from equation (13). The shape of analytic and RSM surfaces were drawn on the same graph for the comparison purpose as shown in Fig. 8. The maximum error between analytic surface and RSM surface is about 7.4%; however, most of the points in design space have the error below 3.6%. Therefore, the analytic model can be applicable for estimating the behavior of mold temperature towards cooling channels’ configuration. DOE method and FEM simulation also gave the results of temperature variation in mold surface. For a specific input data given in Table 1, the variation of mold temperature is represented by the quadratic response surface equation: ∆TW =0.149x2 + 0.448y2 - 0.540xy + 1.162x – 1.604y +0.505 (16) Figure 9 shows the shape of mold temperature variation of the equation (16). Fortunately, this surface has an extremum point in the middle region of design space. For example, if the designer wants the mold temperature variation to be lower than 0.5°C, the feasible region to select the pitch and the depth of cooling channels Fig. 6 FEM model to identify mold temperature distribution (1) (9) (5) (13) (14) (10) (2) (6) (3) (11) (15) (7) (12) (16) (8) (4) Fig. 7 Mold surface temperature variations for different coolingchannel depth y and pitch x Fig. 8 Comparison between analytic surface and RSM surface Fig. 9 Shape of response surface of mold temperature variation will be the one in Fig. 10. It is clear that the feasible region for the solution of equation (13) is narrowed down when the uniformity of mold temperature is considered strictly. The more the temperature uniformity is required, the less the feasible region is. The optimal cooling channels’ configuration should be in the feasible region. There exists a line in the feasible region that there is smallest temperature variation (see Fig. 10). This line shows the good combinations between y and x and it can be estimated by an approximate linear equation: 0.7 1.6 yxd =+ (17) Although approximate equations (15), (16) and (17) are estimated based on a particular polymer and mold material, the thermal behavior of the mold towards cooling channels’ configuration for other polymer material and molding condition also has a similar response as shown in Fig. 5 and Fig. 10. Consequently, equation (17) can be used as a guideline to support the selection of x and y. In some cases, to satisfy other constraints, the linear equation (17) that represents a line should be widened into its left and right side to make a ribbon area (see Fig. 10) described by inequality equations: 0.7 (1.6 )0.7 (1.6 )y xdy xdδδ≥+− ≤++ (18) Even though the feasible region is getting smaller when stricter constraints are considered, there are still many combinations of cooling channels location and diameter as above mentioned. The raising question is that what are the best values of pitch x, depth y, and diameter d for a particular mold cooling design. This issue is discussed in the optimization method section. 4.2 Optimization method for designing conformal cooling channel 4.2.1 Objective function The aim of mold cooling design optimization is obtaining uniform temperature distribution of the part surface, achieving target mold temperature, and minimizing the cooling time. Even though the required cooling time is calculated by formula (11), improper cooling channels design will result in longer actual cooling time due to the uneven cooling and high temperature at some location in the part surface. Satisfying uniform cooling, a strong point of conformal cooling, somewhat satisfies the requirement of reducing cooling time. Optimization conformal cooling channels focuses on how to make the mold cool uniformly and to meet the target average mold temperature. Cooling process of a curved surface is different from those of a flat part due to 3D effect in heat transfer. With the same cooling channel deployment, the cooling effect of the inner and outer surface on the molded part is different. The difference of temperature distribution in both sides will cause residual stress and bend the product after cooling. For above reasons, the design goals include: - Obtaining the target average mold temperature W T represented in equation (13) - Minimizing the difference of average temperature between the inner faces and outer faces of molded part: 11() ( ) 0NMii o jijTTNM ==−= ∑∑ (19) where N and M, Ti and To are the number of elements, the temperature at corresponding elements in the inner and outer surfaces, respectively. The temperature at any element is obtained by querying the simulation results of 3D CAE analysis. 4.2.2 Constraints Proper constraints have to be enforced upon all design variables in form of equality or inequality constraints in order to keep the design optimization realistic from the design and manufacturing point of view. Constraints on cooling channels design includes the lower and upper limit of the pitch distance x of channels, the distance from the cavity surface to the channels y, and cooling channel diameter d. Diameter of cooling channels should be properly selected to ensure heat removal and to allow sufficient flow rate and turbulent flow. In reality, the cooling channel diameter depends on average part thickness s, and it can be determined by empirical formulation as follows5: 28 10410 12610 14s mm mm d mms mm mm d mms mm mm d mm≤⇒≤≤ ≤⇒ ≤≤ ≤⇒ ≤≤ (20) Furthermore, according to the manufacturers’ view point, the diameter of milled groove cooling channels also depends on how deep the milled grooves are because the standard length of milling tools depends on the tool diameter. The range of validity for the pitch x and depth y of cooling channels vary within the range described in formula (14). The distance of pitch x and depth y are also confined by the requirement of avoiding the interference of cooling channels with other components such as ejector pins or sliders. Fig. 10 An example of feasible region to select the parameters ofcooling lines 4.2.3 Systematic procedure for optimization With the advancement of both computer’s hardware and CAE software, three-dimensional computation of heat transfer and flow simulation in injection molding is widely used in mold design engineering. Instead of performing the simulation for each cross-section, the entire mold with all cooling channels is simulated. The advantage of 3D simulation over analytical computation is the more accurate simulation of cooling conditions, especially for complex part.16 In fact, the thermal properties of polymers are time-dependent, and this characteristic significantly influences the analysis results.24,25 Three-dimensional CAE simulation software tackles this problem very well, so the precise result can be obtained in comparison with applying the analytical method with constant thermal properties. Moreover, the ability of coupling with filling and packing computation, warpage and residual stress analysis as well as convenient graphical visualization are also the strong points of 3D CAE simulation. However, the computation cost of 3D analysis for each run is still very high while the optimization process always requires a loop of analysis, modifying input, and reanalysis for searching the optimum design point. In addition, when the number of design variable increases, the number of iterations and computation cost also increase correspondingly. To improve the accuracy of analysis and reduce the computation time, combination approach in which analytical method and CAE simulation-based method is proposed in this paper. The cooperation between analytic approach and CAE simulation-based approach can be carried out by two steps. In the first step, analytical method is used to determine the initial configuration of cooling channels including pitch x, depth y, and diameter d. Even though this step is called initial design, its result tends to come up to the optimal design since the analytical method has been reported to be applicable for simple molded part.5 This statement was also confirmed in Section 4.1.2 by DOE method. In the second step, the number of design variables is reduced to only one (the distance from the cavity surface to the channels y). The target mold temperature and the uniformity of temperature between the top and bottom faces of molded part can be reached after a few iterations by adjusting the variables y using linear interpolation method (Regula Falsi method). Consequently, the number of simulations is reduced and the computational efficiency increases. The systematic procedure for optimizing conformal cooling channels design is shown in Fig. 11. 4.2.4 Implementation Returning to how to determine the cooling channels’ configuration, the possible roots of equation (13) is the intersection of the surface W T =f (x,y) and the plane W T = const when a given cooling channel diameter d and a target mold temperature W T are predefined (see Fig. 12). Applying the constraint condition that the Fig. 11 Strategy for optimizing conformal cooling channels Fig. 12 Shape of solution curve of equation (13) for a given d andW T Fig. 13 Example of the possible solutions of equation (13) whenconsidering mold temperature variation variation of mold wall temperature does not exceed a given allowance as illustrated in Fig. 9 and Fig. 10, the possible solution curve must be in the feasible region as shown in Fig. 13. It can be seen that the solution space is narrowed down. For instance, the initial design space of the pitch x and depth y of cooling channels are 2d to 5d and 1d to 5d down to approximately 2d to 4.2d and 2.9d to 3.7d respectively for above example. If W T increases, for example, W T =45°C, the possible solution curve moves upward and the range of feasible x and y also changes as shown in Fig. 13. It can be seen that the range of choice of pitch x is wider than those of depth y. If the target mold temperature W T increases, the upper limit of pitch x and depth y can also increases and vice versa. For milled groove cooling channels, the upper range of pitch x should be selected in order to reduce the number of cooling paths or reduce the machined cost of the cooling system. When proper values of d and x are selected as experienced before, the equation (13) becomes an equation with one variable y. If the solution y violates the constraint of y, x must be re-selected and this equation is solved again. An alternative method is that it is no need to specify a certain value of x; the solver will search the solution of equation (13) with two variables x and y to satisfy all constraints and optimality conditions (18) and (20). Besides solving explicit equation for finding the good initial cooling channels configuration,