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    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  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)
    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
     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. From the formula (11), it can be seen that the cooling time only
    depends on the thermal properties of a plastic, part thickness, and
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