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Design and Simulation-Based Optimization of Cooling Channels for Plastic Injection Mold
Hong-Seok Park and Xuan-Phuong Dang
University of Ulsan
South Korea
1. Introduction
Injection molding has been the most popular method for making plastic products due to high efficiency and manufacturability. The injection molding process includes three significant stages: filling and packing stage, cooling stage, and ejection stage. Among these stages, cooling stage is very important one because it mainly affects the productivity and molding quality. Normally, 70%~80% of the molding cycle is taken up by cooling stage. An appropriate cooling channels design can considerably reduce the cooling time and increase the productivity of the injection molding process. On the other hand, an efficient cooling system which achieves a uniform temperature distribution can minimize the undesired defects that influence the quality of molded part such as hot spots, sink marks, differential shrinkage, thermal residual stress, and warpage (Chen et al., 2000; Wang & Young, 2005).
Traditionally, mold cooling design is still mainly based on practical knowledge and designers’ experience. This method is simple and may be efficient in practice; however, this approach becomes less feasible when the molded part becomes more complex and a high cooling efficiency is required. This method does not always ensure the optimum design or appropriate parameters value. Therefore, many researchers have proposed some optimization methods to tackle this problem. Choosing which optimization method was used mainly depends on the experience and subjective choice of each author. Therefore, finding appropriate optimization techniques for optimizing cooling channels for injection molding are necessary.
This book chapter aims to show the design optimization method for designing cooling channels for plastic injection molds. Both conventional straight-drilled cooling channels and novel conformal cooling channels are focused. The complication of the heat transfer process in the mold makes the analysis to be difficult when using the analytical method only. Therefore, using numerical simulation tools or combination of analytical and numerical simulation approach is one of the intelligent choices applied to modern mold cooling design.
The contents of this book chapter are organized as follows. Cooling channels layout and the foundation of heat transfer process happening in the plastic injection mold are presented systematically. Physical and mathematical modelings of the cooling channels are also introduced. This section supports the reader the basic governing equations related to the
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20 New Technologies – Trends, Innovations and Research
cooling process and how to build an appropriate simulation model. Subsequently, the simulation-based optimizations of cooling channels are presented. In this section, the state- of-art of cooling channels design optimization is reviewed, and then the systematic procedure of design optimization and optimization methods based on simulation are proposed. Two optimization approaches applied to cooling channels design optimization are suggested: metamodel-based optimization and direct simulation-based optimization. The characteristics, advantages, disadvantages, and the scope of application of each method will be analyzed. Finally, two case studies are demonstrated to show the feasibility of the proposed optimization methods.
2. Cooling channels layouts
2.1 Mold cooling system overview
Mold cooling process accounts for more than two-thirds of the total cycle time in the production of injection molded thermoplastic parts. An efficient cooling circuit design reduces the cooling time, and in turn, increases overall productivity of the molding process. Moreover, uniform cooling improves part’s quality by reducing residual stresses and maintaining dimensional accuracy and stability (see Fig. 1).
Poorer part in longer cooling time
Fig. 1. Proper cooling design versus poor cooling design (Shoemaker, 2006) A mold cooling system typically consists of the following items:
- Temperature controlling unit
- Pump
- Hoses
- Supply and collection manifolds
- Cooling channels in the mold
The mold itself can be considered as a heat exchanger, in which the heat from the hot polymer melt is taken away by the circulating coolant.
Figures 2 illustrates the components of a typical cooling system.
Design and Simulation-Based Optimization of Cooling Channels for Plastic Injection Mold 21
Collection
manifold
Baffles
Normal cooling channels
Supply manifold
Pump
Temperature control
Fig. 2. A typical cooling system in injection molding
2.2 Conventional straight-drilled cooling channels
The common types of straight-drilled cooling channels are parallel and series.
2.2.1 Parallel cooling channels
Parallel cooling channels are drilled straight channels that the coolant flows from a supply manifold to a collection manifold as shown in Fig. 3c. Due to the flow characteristics of the parallel cooling channels, the flow rate along various cooling channels may be different, depending on the flow resistance of each individual cooling channel. This varying of the flow rate, in turn, causes the heat transfer efficiency of the cooling channels to vary from one to another. As a result, cooling of the mold may not be uniform with a parallel cooling- channel configuration.
2.2.2 Serial cooling channels
Cooling channels that are connected in a single loop from the coolant inlet to its outlet are called serial cooling channels (see Fig. 3b). This type of cooling channel network is the most commonly used in practice. By design, if the cooling channels are uniform in size, the coolant can maintain its turbulent flow rate through its entire length. Turbulent flow enables the heat to be transferred more effectively. For large molds, more than one serial cooling channel may be required to assure a uniform coolant temperature and thus uniform mold cooling.
22 New Technologies – Trends, Innovations and Research
(a) Straight-drilled cooling channels
Fig. 3. Conventional straight cooling channels
(c) Straight parallel cooling channels
(b) Straight series cooling channels
2.3 Conformal cooling channels
To obtain a uniform cooling, the cooling channels should conform to the surface of the mold cavity that is called conformal cooling channels. The implementation of this new kind of cooling channels for the plastic parts with curved surfaces or free-form surfaces is based on the development of solid free-form fabrication (SFF) technology. On the other hand, conformal cooing channels can also be made by U-shape milled groove using CNC milling machine (Sun et al., 2004).
Fig. 4. A layout of conformal cooling channels
The conformal cooling channels are different from straight-drilled conventional cooling channels. In conventional cooling channels, the free-form surface of mold cavity is surrounded by straight cooling lines machined by drilling method. It is clear that the distance from the cooling lines and mold cavity surface varies and results in uneven cooling in molded part. On the contrary, for the conformal cooling channels, the cooling paths match the mold cavity surface well by keeping a nearly constant distance between cooling paths and mold cavity surface (see Fig. 4). It was reported that this kind of cooling channels gives better even temperature distribution in the molded part than the conventional one.
Design and Simulation-Based Optimization of Cooling Channels for Plastic Injection Mold 23
Figure 5 shows an example of molds with conformal cooling channels made by direct metal laser sintering method. It was said that this cooling channels not only ensure the high quality of the product but also increase the productivity by 20 %.
Fig. 5. Molds with conformal cooling channels made by laser sintering (Mayer, 2009)
3. Physical and mathematical modeling of cooling channels
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 (Park & Kwon, 1998; Lin, 2002). 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 the steady-state after several cycles, the average temperature of the 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, cycle-averaged temperature approach is used for mold region and transition analysis is applied to the molded part (Park & Kwon, 1998; Lin, 2002; R?nnar, 2008).
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. The coupling of cycle-averaged and one- dimensional transient approach was applied since it is computationally efficient and sufficiently accurate for mold design purpose (Qiao, 2006; Kennedy, 2008). Heat transfer in the mold is treated as cycle-averaged steady state, and 3D FEM simulation was used for analyzing the temperature distribution. The cycle-averaged approach is applied because after a certain transient period from the beginning of the molding operation, the steady-state cyclic heat transfer within the mold is achieved. The fluctuating component of the mold temperature is small compared to the cycle-averaged component so that cycle-averaged temperature approach is computationally more efficient than periodic transition analysis (Zhou & Li, 2005). Heat transfer in polymer (molding) is considered as transient process. The temperature distribution in the molding is modeled by following equation:
?T = a
?t
?2T
?z2
(1)
The partial difference equation (1) can be solved conveniently by finite difference method. Due to the nature of thermal contact resistance between polymer and mold, a convective boundary condition (Kazmer, 2007) was applied instead of isothermal boundary condition.
24 New Technologies – Trends, Innovations and Research
This boundary condition expresses the nature of the heat transfer in mold-polymer interface better than isothermal boundary condition.
h éT
- T ù = -k ?T
(2)
c ? ps m ?
p ?z
where Tps and Tm are molded part surface temperature and mold temperature, respectively;
kp is the thermal conductivity of polymer.
The inversion of the heat transfer coefficient hc is called thermal contact resistance (TCR). It is reported that the TCR between the polymer and the mold is not negligible. TCR is the function of a gap, roughness of contact surface, time, and process parameters. The values of TCR are very different (Yu et al., 1990; C-MOLD, 1997; Delaunay et al., 2000; Sridhar & Narh, 2000; Le Goff et al., 2005; Dawson et al., 2008; Hioe et al., 2008; Smith et al., 2008), and they are often obtained by experiment.
The heat flux across the mold-polymer interface is expressed as follows.
q = -k
where n is the normal vector of the surface.
?T
p ?n
(3)
The cycle-averaged heat flux is calculated by the equation:
t
1 c
t
q = ò qdt
c 0
(4)
The required cooling time tc is calculated as follows (Menges et al., 2001; Rao & Schumacher, 2004).
s2 é 4 ? T - T ?ù
tc =
ln ê
? i m ÷ú
(5)
pa ê?p è Te - Tm ?ú?
where a = km
rcp
is the thermal diffusivity of polymer
An example solution of the system of Eq. (1) to (5) for a specific polymer and a given process parameters is depicted in Fig. 6.
Fig. 6. Typical temperature profile and heat flux of a given molding obtained by finite difference method
Design and Simulation-Based Optimization of Cooling Channels for Plastic Injection Mold 25
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 7 shows the sketch of configuration of cooling system and heat flows in an injection mold. The heat balance is expressed by equation.
Q& m + Q& c + Q& e = 0
(6)
where Q& m , Q& c and Q& e are the heat flux from the melt, the heat flux exchange with coolant
and environment respectively.
Fig. 7. Physical modeling of the heat flow and the sketch of cooling system
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 exchanges with the coolant is taken place by force convection, and the heat exchanges 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% (Park & Kwon, 1998; Zhou & Li, 2005). 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 loss to the surrounding environment.
Q& m + Q& c = 0
Heat flux from the molten plastic into the coolant can be calculated as (Rao et al., 2002)
(7)
s
2
Q& m = 10-3[cp(TM - TE ) + im ]r x
(8)
Heat flux from the mold that changes with coolant in the time tc amounts to (Park & Kwon, 1998):
? -1
?-1
Q& = 10-3t
? 1 1 ÷
(T - T )
(9)
c c ? 10-3ap d
kstSe ÷ W C
è ?
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 t0. By comparing the analysis results
26 New Technologies – Trends, Innovations and Research
obtained by the analytical method using the formula (9) and the analysis result obtained by commercial flow simulation software, the formula (9) under-estimates the heat flux value. On the contrary, if, tc in (9) is replaced by the sum of tf, tc and to, the formula (9) 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 stage is lower than others 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 (Park & Dang, 2010). For this reason, the formula (9) 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.
? 1 1
?? 1 -1 1 ?-1
Q& c = 10-3 ?
t f + tc +
to ÷? -3
÷ (TW - TC )
(10)
è 2 3
?? 10
ap d
kstSe ÷
è ?
The influence of the cooling channels position on the heat conduction can be taken into account by applying shape factor Se (Holman, 2002)
Se = 2p
lné 2xsinh(2p y /x)ù
(11)
ê p d ú
? ?
Heat transfer coefficient of water is calculated by (Rao & Schumacher, 2004):
e
a = 31.395 R 0.8
d
(12)
where the Reynolds number
d Re = un
(13)
The cooling time of a molded part in the form of plate is calculated as (Menges et al., 2001; Rao & Schumacher, 2004):
s2 é 4 ? TM - TW ?ù
tc = p 2a lnêp ?? T - T
÷÷ú
(14)
?ê è E W ?ú?
From the formula (14), 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 TW , so it indirectly influences the cooling time.
By combining equations from (7) to (14), one can derive the following equation:
s ì é y ù ü
[cp(TM - TE ) + im ]r 2 x ? 1
ê 2x sinh(2p ) ú
1 ? s2
é 4 ? T - T ?ù 1 1
(15)
x M W
T - T
í p k
ln ê ú +
p d
. y =
ln ê ?
p T - T
÷ú +
t f + to
W C ? 2 st
ê ú 0.03139p Re0 8 ?
p 2a
?ê è
E W ??ú 2 3
?? ê? ú? ?t
Design and Simulation-Based Optimization of Cooling Channels for Plastic Injection Mold 27
Mathematically, with preset TM, TE, TW , predefined tf and to, and others thermal properties of material, equation (15) 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 TW
is established by the cooling channels configuration
and predefined parameters TM, TE, tf, to, and thermal properties of material in equation (15). The value of TW , in turn, results in the cooling time calculated by the formula (14).
4. Simulation-based optimization of cooling channels
4.1 Cooling system design and optimization: The state-of-the-art
For many years, the importance of cooling stage in injection molding has drawn a great attention from researchers and mold designers. They have been struggling for the improvement of the cooling system in the plastic injection mold. This field of study can be divided into two groups:
· Optimizing conventional cooling channels (straight-drilled cooling lines).
· Finding new architecture for injection mold cooling channels (conformal cooling channels).
The first group focuses on how to optimize the configuration of the cooling system in terms of shape, size, and location of cooling lines (Tang et al., 1997; Park & Kwon, 1998; Lin, 2002; Rao et al., 2002; Lam et al., 2004; Qiao, 2005; Li et al., 2009; Zhou et al., 2009; Hassan et al., 2010). These studies used some of methods from semi-analytical method to finite difference, boundary element method (BEM), and finite element method (FEM). Rao N. (Rao et al., 2002) proposed the optimization of cooling systems in injection mold by using an applicable analytical model based on 2D heat transfer equations. Most studies mainly focus on the numerical methods. Park and Kwon (Park & Kwon, 1998) proposed the optimization method for cooling system design in injection molding process by applying design sensitive method. The heat transfer was treated as 2D problem. Boundary element method is preferred to solve the heat transfer problem in mold cooling design (Qiao, 2005; Zhou et al., 2009). BEM is effective for calculating heat transfer in the mold because: (a) the discretization associated with BEM does not extend to the interior region of the mold that there is no need for mesh generation when the cooling channels are rearranged, (b) BEM method reduces the input data due to the reduction of total nodes so that the computation cost is reduced in comparison to finite element method. Although the BEM can extend to 3D application as the new feature of most of commercial injection molding software, these works are mainly based on 2D case studies that are not always practical. Moreover, most of case studies are simple.
For 3D analysis in heat transfer in injection mold, 3D simulation based on professional or commercial software is the common approach. Nowadays, commercial simulation software can help the designer to calculate the temperature distribution and cooling time. Nevertheless, it is only the simulation tools, and these tools themselves are often confined in a single simulation. The optimization task needs a scientific strategy and methodology to obtain a believable result. Lam Y. C. et al. (Lam et al., 2004) proposed an evolutionar
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