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Fig. 1 A conformal cooling channels surround a free-form part molded part. Besides SFF method, milling by CNC milling machine is an alternative method for making the conformal cooling channels, even though it is not flexible than the SFF method. However, milled groove cooling channels method can be applied to any material which is used to make the mold. In addition, it is not necessary to machine the cooling channel precisely, so the machining time can be reduced by applying high-speed machining and high material removal rate technology. Milled groove cooling channels pattern can be designed freely to avoid interfering with other features in the mold such as ejector pins or other components. The milled groove cooling channels are modeled by extruding the cooling layouts including lines and curves with a thickness d in XY plane up to the offset surface of the cavity impression. There are two ways to make U-shape milled groove cooling channels. The first method is applied when milled groove is deeper than the permitted length of ball end milling tool. In this case, the big pocket is machined first then the U-shape pockets are milled with a constant depth into the offset surface of the mold core and mold cavity as shown in Fig. 2. An inserted block is also machined and assembled to the main mold cavity part to make complete U-shape mill groove cooling channels for a haft mold. Sealing by O-rings is required to prevent coolant leakage. The second method should be used when the deepest milled groove is shorter than the length of the milling tool. In this method, the pocket milling operation is performed for the cooling channel groove in the cavity part as shown by a cross-section in Fig. 3. A similar pattern core insert on which their bottom surfaces must be offset from the mold impression to keep the constant cross-section of cooling paths is also milled. A complete U-shape cooing is obtained when the mold cavity part and core inserts part are assembled and fixed by socket screws. The cross-section of a U-shape cooling channels is depicted in Fig. 3. In heat transfer analysis and finite element method, cooling channels are treated as beam elements; hence equivalent diameter de based on equivalent cross-sectional area is defined as follows: 14224eddWHdππ −+ = (1) The shape factor defined by the ratio of perimeter of the U-shape cross-section and the perimeter of the circle is calculated as follow: 2( )cHdSd π+= (2) Milled groove cooling channel burdens the mold manufacturing cost compared to straight-drilled channels due to extra expenditure of mold material and milling operation. In plastic injection molding, trade-offs are sometimes required. It can be seen that the volume of material removal of two previously mentioned milled groove methods are the same and equal to the bounding box volume of the whole core inserts in Fig. 2. However, the second method can reduce the volume of removal if the parallel and zigzag types of insert are formed by bending a sheet of soft metal material with an appropriate thickness corresponding to milled groove. Subsequently, these inserts are fixed on a plate with similar grooves patterns, and then the whole insert block is milled to obtain the final shape. It is noticed that there can be three types of milled groove conformal cooling channels layout: parallel type, zigzag type and spiral type. This paper mainly focus on the parallel and zigzag type because it is easier to form the parallel inserts using sheet metal as previously mentioned for reducing the extra machining cost of U-shape conformal cooling channels. In this case, the control of mold temperature can be done by adjusting the position of cooling channels closer to or farther from the surface of molded part in Z direction. This method is more convenient for automatic optimization as the number of variables is reduced. 3. Physical and mathematical modeling In the physical sense, cooling process in injection molding is a complex heat transfer problem. To simplify the mathematical model, some of the assumptions are applied.4,6 The objective of mold cooling analysis is to find the temperature distribution in the molded part and mold cavity surface during cooling stage. When the molding process reaches steady-state after several cycles, the average temperature of mold is constant even though the true temperature fluctuates periodically during the molding process because of the cyclic interaction between the hot plastic and the cold mold. For the convenience and efficiency in computation, Fig. 2 Milled groove method for deep cooling groove Fig. 3 Cross-section of U-shape milled groove cooling channels cycle-averaged temperature approach is used for mold region and transition analysis is applied to the molded part.4,6,17 The general heat conduction involving transition heat transfer problem is governed by the partial differential equation. The cycle-averaged temperature distribution can be represented by the steady-state Laplace heat conduction equation. When the heat balance is established, the heat flux supplied to the mold and the heat flux removed from the mold must be in equilibrium. Figure 4 shows the sketch of configuration of cooling system and heat flows in an injection mold. The heat balance is expressed by equation: 0 mce QQQ ++= (3) where ,m Q c Q and e Q are the heat flux from the melt, the heat flux exchange with coolant and environment, respectively. The heat from the molten polymer is taken away by the coolant moving through the cooling channels and by the environment around the mold’s exterior surfaces. The heat exchange with the coolant is taken place by force convection, and the heat exchange with environment is transported by convection and radiation at side faces of the mold and heat conduction into machine platens. In application, the mold exterior faces can be treated as adiabatic because the heat lost through these faces is less than 5%.6,18 Therefore, the heat exchange can be considered as solely the heat exchange between the hot polymer and the coolant. The equation of energy balance is simplified by neglecting the heat lost to the surrounded environment. 0 mc QQ += (4) Heat flux from the molten plastic into the coolant can be calculated as5 310 [ ( ) ]2mpMEmsQcTTix ρ −=−+ (5) Heat flux from the mold exchanges with coolant in the time tc amounts to6: ()1 133111010cc WCst eQt TTdkS απ− −−−=− (6) In fact, the total time that the heat flux transfers to coolant should be cycle time including filling time tf, cooling time tc and mold opening time to. By comparing the analysis results obtained by the analytical method using the formula (6) and the analysis result obtained by commercial flow simulation software, the formula (6) under-estimates the heat flux value. On the contrary, if tc in (6) is replaced by the sum of tf , tc and to, the formula (6) over-estimates the heat flux from the mold exchanges with coolant. The reason is that the mold temperature at the beginning of filling stage and mold opening is lower than other stages within a molding cycle. The under-estimation or over-estimation is considerable when the filing time and mold opening time is not a small portion compared to the cooling time, especially for the large part with small thickness. For this reason, the formula (6) is adjusted approximately based on the investigation of the mold wall temperature of rectangular flat parts by using both practical analytical model and numerical simulation. ()1331111102310cf co W Cst eQttt TTdkS απ−−− =++ + − (7) The influence of the cooling channels position on heat conduction can be taken into account by applying shape factor19 Se 22 sinh(2 / )lne Sx yxdπππ= (8) Heat transfer coefficient of water is calculated by20: 0.8 31.395e Rdα = (9) where the Reynolds number edR uν= (10) The cooling time of a molded part in the form of plate is calculated as16,20: 4ln MWcEWsTTtaTT ππ −= − (11) where mpkac ρ= (12) From the formula (11), it can be seen that the cooling time only depends on the thermal properties of a plastic, part thickness, and process conditions. It does not directly depend on cooling channels configuration. However, cooling channels’ configuration influences the mold wall temperature ,W T so it indirectly influences the cooling time. By combining equations from (4) to (12), one can derive the following equation: 0.82 sinh(2 ) [( ) ]11 2 ln2 0.03139pM E mWC st ey sx cT T i xxTT k d Rπ ρππ π −+ + − 411ln23MWf oEWsTTttaTT ππ −=+ + − (13) Fig. 4 Physical modeling of heat flow and the sketch of coolingSystem Mathematically, with preset TM, TE, ,W T predefined tf and to, and others thermal properties of material, equation (13) presents the relation between cooling time tc and the variables related to cooling channels configuration including pitch x, depth y and diameter d. In reality, the mold wall temperature W T is established by the cooling channels configuration and predefined parameters TM, TE, tf , to, and thermal properties of material in equation (13). The value of ,W T in turn, results in the cooling time calculated by the formula (11). This issue will be analyzed more details in the next sections. The purpose of design optimization of cooling channels is how to obtain the target mold temperature, how to reduce the cooling time and minimize the non-uniformity of the part surface temperature distribution. Before proceeding with optimization method, it is necessary to understand thoroughly the reaction of the thermal behavior of the mold to cooling channels’ configuration physically and mathematically. 4.1 The relation between thermal behaviors of the mold and cooling channels configuration There are some factors that affect the cooling system performance such as the layout of channels, coolant parameters, and mold material into which cooling channels are cut. This paper mainly focuses on the physical layout of cooling channels because of its importance. Once the cooling channels were cut, they could not be reconfigured or adjusted as other factors. For mold cooling design, the important things are how to achieve a desired target mold temperature, and how to minimize the cycle time, and uniformly cool the part. To investigate the relation of thermal behaviors of the mold and cooling channels’ configuration thoroughly and check the feasibility of the analytical model, both analytical method and design of experiment (DOE) method in conjunction with CAE simulation were used. 4.1.1 Analytical method It is known that minimum cooling time depends on mold wall temperature ; W T lower mold temperature can significantly reduce the cooling time21 according to formula (11). ,W T in turn, depends on the configurations of cooling channels.22 Therefore, the problem is optimizing the cooling channels layout and their configurations to satisfy a pre-determined .W T To have a general look at this problem, the relation between thermal behaviors of the mold and cooling channels’ configuration is investigated first. Considering equation (13), this equation has three unknowns including x, y, and d, so countless roots exist. It means that there are many combinations of x, y, and d to satisfy the preset .W T In other words, different configurations of cooling channels result in different average mold surface temperature. However, a good combinations of x, y, and d is the one that makes a uniform temperature distribution in the mold cavity surfaces. In mold design practice, the pitch x, the depth y, and diameter d of cooling channels are chosen as: 8mm 14mmand 2 515dxdydββββ≤≤ = ≤≤ = ≤≤ (14) The figure 5 illustrates the effect of cooling channels’ configuration on W T for a given molded part thickness. Thermal properties of materials and specific processing conditions are shown in Table 1. This graph is drawn by using the system of equations (13) and (14). The graph shows that when the pitch x and depth y increase, the mold temperature W T increases. In other words, when the cooling channels locate near the mold cavity surface and/or the pitch of cooling channels are small, the mold temperature decreases and the cooling time also decreases because the time required for solidifying the product decreases.23 The effect of the depth y on the mold temperature is greater than those of the pitch x because the slope of the temperature surfaces in Fig. 5 in y/d axis is greater than those of x/d axis. It can be seen that analytical method based on explicit function (13) only gives the average mold temperature. In cooling design, as previously mentioned, not only the target average mold temperature but also so the uniformity of temperature of mold cavity surfaces should be obtained. To investigate the influence of cooling Table 1 Example values of parameters that are used in calculating cooling channels configuration Parameters Value Molded part thickness s (mm) 3.0 Melt temperature TM (°C) 230.0 Demolding temperature TE (°C) 97.0 Specific heat of the melt cp (KJ/(kg°K)) 1.79 Melt density ρ (g/cm3) 0.929 Thermal conductivity of the melt km (W/m.°K) 0.189 Thermal conductivity of the steel kst (W/m.°K) 45.0 Kinetic viscosity of water ν (m2/s) 1.2×10-6 Velocity of cooling water u (m/s) 1.0 Temperature of cooling water TC (°C) 20 Fig. 5 Effect of cooling channels configuration on TW for a specific processing conditions channels’ configuration to mold temperature variation and temperature distribution, simulation and design of experiment were used as a supplementary method. 4.1.2 Simulation and design of experiment method The analytic equations (13) is derived from the two-dimensional heat transfer problem of mold cooling for a plastic plate, so a FEM model for identifying mold temperature distribution on the cavity surface was built for a flat molded part. The mold cross-section is considered, and two-dimensional heat transfer computation is considered as a suitable model because it is unnecessary to perform the simulation for the whole mold. Figure 6 depicts the FEM model for calculating the mold surface temperature distribution. Instead of spending a lot of time to investigate all the value of d in its range, d = 10 mm was chosen to analyze the affect of cooling channels location to the temperature distribution of the mold wall. This choice retains the generality since the pitch x and depth y are multiples of diameter d. When d is fixed, there are only two factors left including x and y. Full factorial design of experiment for two factors and four levels was selected so that there were sixteen experiments. The necessary input data is shown in Table 1.