Computer-aided Engineering in Injection Moulding

14 Computer-aided Engineering in Injection Moulding GEORG MENGES and HANS RECKER Institut fiir Kunststoffverarbeitung, Aachen, FRG 14.1

451

INTRODUCTION

14.2 MOULD DESIGN AND THE USE OF COMPUTERS

452

14.3 ANALYSIS OF THE FILLING PROCESS OF THE MOULD

454 455 456 456

14.3.1 Fundamentals 14.3.2 The Flow Pattern Method 14.3.3 Segmentation - Rheological Calculation of Basic Geometries 14.3.4 Computer-aided Graphical Simulation of the Filling Process of the Mould in Thermoplastic Injection Moulding 14.3.5 Filling Process in Elastomer Injection Moulds

14.4 ANALYSIS OF THE HOLDING PRESSURE PHASE 14.4.1 Fundamentals 14.4.2 Calculation of the Holding Pressure 14.4.3 Evaluation of Shrinkage

14.5 THERMAL ANALYSIS 14.5.1 Fundamentals 14.5.2 Layout of Thermoplastic Moulds 14.5.3 Procedure for the Layout of the Heating/Cooling System 14.5.4 Layout of the Heating/Cooling System 14.5.4.1 Simulation calculation 14.5.4.2 Heating/cooling of corners 14.5.5 Temperature and Reaction Conversion in Elastomeric Injection Moulding

461 464 466 467 468 470 473 473 474 474 476 478 479 480

14.6 MECHANICAL LAYOUT OF MOULDS 14.6.1 Aids for the Calculation of Mould Dimensions 14.6.1.1 Superposition of loads 14.6.1.2 Deformation in the case of complex geometry

483 484 485 485

14.7 PROSPECTS FOR THE FUTURE

487

14.8 APPENDIX: ABBREVIATIONS AND SYMBOLS

487

14.9 REFERENCES

488

14.1

INTRODUCTION

An essential prerequisite for the rational and qualitatively high grade production of plastic parts is a mould layout tailored to the production process and to costs. To date, empirical values have generally been taken as a basis in both the conception and the design, thereby calling for highly qualified staff, on one hand, and necessitating frequent, time-consuming and costly touch-up work to the finished mould, on the other. Experience acquired so far with computer-aided mould design shows that even a designer with only little experience will reach his target more rapidly and more reliably, and can benefit from the results of demanding calculation and simulation methods which it was previously impossible to carry out. 451

Unit Operations of Polymer Processing

452

This paper outlines new developments in computer-aided mould design. It concentrates chiefly on the layout of thermoplastic and elastomer injection moulds. First, an overall analysis of the design process is presented. The second section looks into the simulation of filling methods. Programs for establishing threedimensional filling pictures, which will run on a minicomputer, are presented with sample applications. First of all the flow pattern is calculated and then, in subsequent steps, the pressure and velocity distribution in the filling phase is worked out from this. The system has been developed for mould layout in conjunction with the design of injection moulds for thermoplastics processing. A program system is also presented with which the same design steps can be followed for the design of elastomer moulds on a microcomputer. The simulation calculations are conducted in a twodimensional lay-flat of the moulding. The third section presents an analysis and simulation of the holding pressure phase in thermoplastics injection moulding. The temporal development of the pressure inside the mould is calculated at any desired point in the moulded part, thereby allowing for optimization of the holding pressure phase. This calculated pressure distribution is an important prerequisite for estimating shrinkage; the volume shrinkage can be estimated directly. Conclusions can then be drawn from this as to processing shrinkage. In the fourth section a thermal analysis of the processing methods is conducted. First of all, the temperature development in the moulding is investigated; from this, the reaction curve for crosslinking moulding compounds is derived. An investigation of the temperature distribution in the mould provides a guide to an optimized layout of the cooling or heating circuits, the shortest possible cycle time and the best quality. Finally the mechanical analysis of moulds is discussed. Here, programs are available for deformation calculations on basic cases and these will frequently be sufficient. For more complex cases of loading and deformation, finite element programs can be used.

14.2 MOULD DESIGN AND THE USE OF COMPUTERS The large number of tasks to be managed by a designer in the course of the process of mould design can be seen from the many functions a finished mould must be able to perform. The fundamental tasks of injection moulds comprise of the following (Figure 1): admission and distribution of the melt; moulding; cooling; and demoulding.

Main functions Sprue gate

Mould cavity

Secondary functions Force adsorpt ion

Motion transfer

Temperature control / GUiding and centring Demoulding

Figure 1 Functional complexes of a split follower

The most important functional complexes for performance of the main tasks are the following: runner and gate; mould cavity; heating/cooling system; and demoulding system. The specific demands on the single functional complexes can be derived in accordance with the demands to be placed on the finished product (for example keeping within tolerances in size, optical properties, etc.) and on the production process itself (for example secured filling of the cavity, short cycles, etc.). The following may serve as an ex~mple.

Mould cavity: correction of shrinkage effects; stiffnesses; position of the split level; positions of gates and knock-outs; joint seams; and deaeration.

Computer-aided Engineering in Injection Moulding

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Runner/gate: equal pressure demand; equal filling time; simultaneous start of the flow process; and adjusted sealing point. Temperature control: homogeneity; constant temperature level; and optimum cooling/heating time. Demoulding system: short demoulding times; short movements; and low forces. The variety of demands placed on the shaping mould illustrates the complexity of the tasks inherent in the design of injection moulds and hence the high demands placed on design and designer. If one intends to utilize computers for performing all these tasks, it becomes necessary to systematically analyze the design process and divide it into different stages with clearly defined tasks (Figure 2). 1,2

fr

Finding of mould principle

00

Rheological layout en en

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S

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Thermal layout

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on c= '2 .9 en c=

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Mechanical layout

("t")

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Manufacturing documents

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Figure 2

Main steps of mould design

The task of the first stage is the determination of the 'mould principle'. In this stage the designer decides, for example, which type of mould (standard mould, hot runner or cold runner, sucker pintype mould, etc.)3 serves the respective purposes in the best way, how many split levels are necessary for demoulding of the part and where he will place sucker pins and core pullers. The second stage of the design process includes dimensioning of the functional complexes and component groups selected in the course of the first stage. The third stage involves the drawing up of documentation for the manufacture (such as, for example, drawings, part lists etc.). The tasks of the first stage are predominantly intellectual and creative; it is not readily possible to describe the inherent processes with mathematical laws and data or to analyze them in the corresponding way; hence appropriate algorithms can hardly be established. Therefore targets of the first stage are not yet accomplished with the aid of computer programs. However, in the near future there will be programs leading the designer to the necessary decision along a logical path. This means that the designer advances step by step towards the solution by going through various possibilities both graphically and by means of calculation. As already stated, the goal to be achieved during the second stage of mould design is the dimensioning of the mould and its component parts. The corresponding tasks can be described and calculated, for example, by flow-related and thermal or mechanical equations; consequently they can be immediately solved by computers. Figure 3 depicts examples for single tasks, as they result from breaking down the dimensioning process into single steps; all of the listed tasks can be solved by means of computer programs. The rheological mould layout appears to be the appropriate initial design step, since there will be hardly any restrictions from results of the other steps. The following quantities are examples for results from a mathematical simulation of the flow process in the mould: information concerning position and number of welding lines; flow behaviour, as described by pressure requirements and temperature profiles; and stress in the melt. Corresponding to these results, the designer is given the immediate possibility of going through the flow process of the mould by repeatedly performing the appropriate mathematical simulation with varying boundary conditions until the results of the simulation calculation meet his

Unit Operations of Polymer Processing

454

Finding of mould principle

I

Rheological layout

Filling behaviour of the cavity (qualitative)

+

Filling behaviour of the' cavity (quantitative)

•

Gate system layout

•

Pressure drop in the nozzle

Energy balance of the total mould

+

Thetmallayout

First layout of the tempering system

• •

Exact segmented layout of the tempering system Check of homogeneity

Cinematic

•+

Rigidity rectangular to clamping direction Mechanical layout

I

Rigidity in clamping direction

+

Ejection forces, inertial forces, area pressure

Manufacturing documents Figure 3 Steps of mould design during the dimensioning phase

demands. 4 - 1o , 12 Variations the designer may apply are, for example, modification of the thermal boundary conditions, utilization of flow aids or flow restraints and modification of position and number of gates. In addition he can apply the so-called optimization programs. With these programs the computer automatically varies specific input quantities (for example, cross-sections of runners or parts of the moulding) until the required goals are achieved, such as equal volume flows or pressure gradients in certain sections of the mould. Accordingly, calculation programs assist in the thermal layout of moulds by determining position and number of heating/cooling channels, establishing the resulting homogeneity of the heating/ cooling and if necessary initiating modifications before manufacturing the mould. 5 - 8 ,lo-12 The last step of the design calculations offers the possibility to determine by means of calculating programs the deformations, for example, inside the mould cavity, as they result from the calculated forces and pressures and to find out exactly which measures (for example higher or lower,clamping forces, installation of reinforcing devices) are the consequent, appropriate counteractions to ensure effective use of the mould. 5

14.3 ANALYSIS OF THE FILLING PROCESS OF THE MOULD The polymer processing industry uses moulding processes for all applications involving the large volume manufacture of mouldings with complicated geometry. This processing technique is particularly advantageous here, because it allows the raw material to be moulded into the finished product in a single operation. The flowable raw material is introduced into the mould cavity at one or several locations; from these points the material spreads over the entire mould until the cavity is completely filled. Additional thermal treatment - a cooling or heating process with subsequent cross-linking - gives the final dimensional stability of the plastic and thus produces a copy of the mould cavity.

Computer-aided Engineering in Injection Moulding

455

Bearing this in mind, it becomes evident that the properties of the finished product depend significantly on the mould and are determined by the flow processes occurring during the filling of the mould (and during the holding pressure phase, see below). It is therefore sensible to optimize the mould-filling process and thus accomplish the required high quality mouldings. Unfavourable flow conditions and processes may lead to, for example, reduced strength ofjoint seams and air inclusions in the moulding, cause misorientation of fillers or fibres or result in deposit of residual material in the mould. 13 ,14 Due to the pressure of costs and time, the plastic processing industry is departing more and more from examining only the finished moulds in favour of an optimized flow process. The expensive and time-consuming short shots and more so the generally necessary touch-up work to the finished mould are reduced or become superfluous through computer simulation of the filling process of the mould..With this, both the manufacturing process as well as the costs and the product quality are optimized during the design of the mould. A number of research papers, programs and reports introduce aids and methods allowing for the performance of such computer simulations. 4 - 1o ,12,15-19,21 The techniques offered here can be classified into two major fields of application. One group serves to describe qualitatively the filling process of the mould for various filling stages by means of flow front prediction. The techniques of the second group are mostly based upon the results of the qualitative filling simulation; they yield quantitative information on the flow processes occurring during filling of the mould. This includes data on pressures, velocities, temperatures and other quantities depending thereof, such as the shear rate. Depending on the respective manufacturing process, the flow processes have to be taken into account in both the runner system and the actual mould cavity.

14.3.1

Fundamentals

The flow of plastics in a mould cavity can be described completely with a set of three differential equations and an appropriate material law for the description of the material dependent viscosities. They are the laws of conservation of mass, momentum and energy.20 The exact solution to the set of equations would require consideration of all equations simultaneously. This, however, involves immense expenses which are unjustified and even unnecessary in practical manufacturing. In general, the mouldings are thin-walled, three-dimensional shell-type bodies; therefore the exact solution method can be simplified significantly without considerable error. Locally, the melt flow can be described as a two dimensional problem of flow. The flow perpendicular to the wall of the mould cavity can be neglected here and thus the initial threedimensional set of differential equations reduces to a two-dimensional problem in a threedimensional space. The flow pattern method constitutes a further simplification; it is derived from the (onedimensional) Hagen-Poisseuille law. Deduced from this law, instructions can be specified with which it becomes possible to construct the flow front in a two-dimensional lay-flat of the moulding 4,5 or in a three-dimensional mesh. 5 In spite of the seemingly attractive three-dimensional representation, which is favoured by potential users, it is often preferable to represent the flow pattern in a lay-flat; it is less costly and often more informative. The calculation method for the simulation of the filling process is based upon the reduced laws of conservation of mass and momentum. 21 They are Continuity

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Momentum in direction of x

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Momentum in direction of y

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(2a)

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(2b)

The construction directions concerning the graphical flow pattern method are derived from a material law and a few geometrical relations (3)

456

Unit Operations of Polymer Processing

This means that the local flow front propagation is proportional to the local wall thickness. The set of equations can be solved by means of iteration; this requires knowledge of the boundary and initial conditions and determination of certain spatial boundaries within which the balancing assumptions are valid. However, it is a major difficulty here that the boundaries have to be constantly determined anew due to the instationary flow process necessitating repeated iteration calculation; the consequence is untenably long computing times.

14.3.2 The Flow Pattern Method The prerequisite for the application of this method is an areal representation of the moulding - the so-called lay-flat. Such a lay-flat is depicted in Figure 4; it represents a cup-shaped moulding.

Body

Pin - point gate

Bose

Figure 4

Lay-flat of a cup

The flow pattern resulting from application of the manual low pattern method (Figure 5) gives the designer important information about the nature of the flow process to be expected in the mould. As can be seen from Figure 7, the body of the cup is filled first. Filling starts from a pin-point gate located in the centre of the bottom. As soon as the flow front reaches the line A-B the material begins to flow into the handle and splits up into three single melt flows. The body of the cup is then filled shortly after the flow front reaches line 16. The further flow process exclusively fills the cup handle. At this point, three joint seams are formed as a result of the rejoining melt flows in the handle. Owing to the fact that the melt flows are diverted and mixed behind the 21st flow front, the resulting welding line is less critical than the one forming behind the 22nd flow front. Previous results of the graphical flow pattern method show that the exclusive consideration of the processes at the flow front is sufficient for a correct qualitative description of the filling process in the predominant number of cases. Departure from the theoretical results from actual flow fronts occurs only when there are great differences in wall thickness or very low injection speeds.

14.3.3 Segmentation - Rheological Calculation of Basic Geometries In this step (segmentation) the part is subdivided into basic geometrical bodies (circular segments, plates, cylinders) according to the flow pattern 15 (Figure 6). The structure of the single elements is acquired here and entered into the computer as input values together with the geometry data. 5,8 -10,12

Computer-aided Engineering in Injection Moulding

Figure 5

457

Filling pattern of a cup

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Figure 6

Segmentation of a cup

In order to determine, for example, temperatures, shear rates and pressures in the single elements, the calculating program requires input of additional information concerning the expected process parameters such as melt temperature, wall temperature of the mould and filling time. The following values were chosen for these quantities in the following example: 5 material = polypropylene, melt temperature = 230°C, wall temperature of the mould = 50°C and filling time = 0.5 s. In the calculation the single flow paths for the cup-shaped moulding are treated separately: one flow path to the rim of the cup body and three flow paths in the region of the cup handle. The simulation calculation yields the results in Table 1 for the flow path in the body of the cup.

458

Unit Operations of Polymer Processing Table 1 Results of Calculations for the Flow Path in the Body of the Cup

Geometry

1 disk 2 plate 3 disk 4 plate 5 plate

Length

Time

Pressure

(mm)

(s)

(bar)

Pressure gradient (barm- l )

24.5 28.5 73.5 78.5 84.5

0.0633 0.0849 0.3838 0.4240 0.4781

61.9 73.2 201.1 216.1 234.0

2782.4 2829.3 2909.5 3023.4 3024.6

Temperature

Dissipation

CC)

CC)

Shear rate (S-l)

226.0 223.5 197.7 193.3 188.3

4.7 0.9 10.2 1.2 1.5

858.4 858.6 583.6 576.3 513.6

107401 109211 112307 116704 116749

Shear rate

Stress

Stress (Pa)

Table 2 Results of Calculations for the Flow Path in the Cup Handle Geometry

1 disk 2 plate 3 disk 4 plate 5 plate 6 plate 7 plate 8 plate 9 plate 10 plate 11 plate 12 plate

Length

Time

Pressure

(mm)

(s)

(bar)

Pressure gradient (barm- l )

24.5 28.5 73.5 78.5 84.5 89.5 94.5 99.5 104.5 109.5 114.5 117.0

0.0633 0.0849 0.3838 0.4240 0.4781 0.4823 0.4856 0.4887 0.4919 0.4968 0.4989 0.5000

61.9 73.2 201.1 216.1 234.0 262.2 292.0 322.0 351.2 377.3 409.1 425.0

2782.4 2829.3 2909.5 3023.4 3024.6 5628.0 5922.1 5961.6 5823.1 5203.7 6320.0 6343.4

Temperature

Dissipation

CC)

CC)

(S-l)

226.0 223.5 197.7 193.3 188.3 190.0 191.8 193.6 195.4 196.6 198.7 199.7

4.7 0.9 10.2 1.2 1.5 2.3 2.4 2.4 2.4 2.1 2.6 1.3

585.4 858.6 583.6 576.3 513.6 5531.3 7018.2 7522.3 7166.3 4805.1 10571.2 11011.7

(Pa) 107401 109211 112307 116704 116749 217243 228593 230116 224771 200862 243950 244854

The filling of the cup body is concluded after 0.4781 s; the required pressure for this process is 234 bar(l bar = 105 N m -2). The melt temperature decreases to 188°C. The corresponding calculations yield the results in Table 2 for the flow path across line B-C in the cup handle. Up to the fourth segment, both flow paths are identical. After the conclusion of filling of the cup body the handle region is filled in the remaining 0.0219 s, owing to the fact that the entire volume flow is then forced into this area. Consequently, the resulting shear rates and shear stresses are very high; this causes the melt temperature in the handle region to rise to 200 °C and leads to a required pressure as high as 425 bar (1 bar = 10 5 N m - 2). As a result of these calculations it was possible to realize and eliminate weak points before manufacturing the mould. Subsequent experiments with the cup-shaped mould cavity confirmed all of the following problem areas established as a result of the simulation calculation. (i) At the rim of the cup the moulding is most likely to be subject to the formation of flashes, due to the extremely high pressure acting upon the rim after filling the cup body. (ii) The joint seam behind the 22nd flow front appears to be a critical area. If the filling time is too long the melt temperature drops and insufficient welding of the flow fronts produces a poor joint seam: The welding line becomes clearly visible and the cup is likely to break at this seam. (iii) High injection speeds are particularly critical for the region of the handle: the resulting increased temperatures (due to high dissipation) may entail damage of the material in the form of 'burned spots'. The first design features the welding line in an area which is subject to high reverse bending stress through use of the cup. One measure to eliminate these weak points would be, for example, to change the position of the joint seam by modifying the wall thickness of the cup handle. A new simulation calculation confirms the predicted effect. Increasing the wall thickness in the lower part of the handle from 1 to 2 mm gives higher flow rates in this region (Figure 7). As a result, the welding lines form in an uncritical part of the handle. The disadvantages of such a measure lie in the fact that the volume of the moulding increases by 0.3 cm 3 and that the required cooling time is 4 s longer than before. For the cup body the new simulation calculation with unaltered process parameters yields the results in Table 3.

Computer-aided Engineering in Injection Moulding

459

Welding lines 20 \

15

. 1J

Increase in wall thickness from I mm to 2 mm

!

-15

Figure 7

Change of welding line position by changing the wall thickness

Table 3 Geometry

Length (mm)

1 disk 2 plate 3 disk 4 plate 5 plate

24.5 28.5 73.5 78.5 84.5

Time

Pressure

(s)

0.0621 0.0833 0.3765 0.3922 0.4741

Pressure gradient

(bar)

(bar m- 1 )

62.3 73.6 201.8 207.7 234.2

2795.6 2841.5 2912.0 2985.6 3005.0

Temperature

Shear rate (S-l)

Stress

eC) 4.7 0.9 10.2 0.5 2.2

875.2 875.4 595.1 587.6 509.1

107909 109680 112402 115245 115991

Dissipation

(OC)

226.2 223.5 198.5 196.4 188.8

(Pa)

Modification of the volume of the moulding and the accompanying modified volume flow have little effect on the calculated values in the main body of the cup. The flow path in the handle region, however, shows a number of considerable changes (Figure 8). The required pressure decreases by 75 bar (1 bar = 10 5 N m - 2). Consequently, the risk of flash formation reduces at the rim of the cup. The material temperature at the joint between cup and

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20

40

60 80 100 Flow length I (mm)

120

Reduction of filling pressure through wall thickness optimization: - ~ constant wall thickness, -0- = changed wall thickness. For polypropylene: OM = 230°C, Ow = 50°C, t F = 0.5 s

Unit Operations of Polymer Processing

460

handle (segment 4) increases by 3 °C on account of the higher volume flow. The temperature at reaching the welding line is increased as well (203.7°C); this leads to improved welding at the joint seam. Thus the reduction of strength at the welding line becomes less critical. Moving the welding line into a less critical area has positive effects on the moulding quality and the process control, but it will have to be examined by means of profitability calculations whether these advantages actually outweigh the disadvantages of increased moulding volume and longer cycles. The possibility for applying optimization programs is a further advantage resulting from the performance of simulation calculations. The pressure drop during the injection phase is an example to illustrate the need for optimization; the drop depends very strongly on the filling time. (i) Slow injection leads to extreme cooling of the material during the flow process of the mould, this in turn leading to the increase in viscosity and required pressure. (ii) Rapid injection causes high losses due to friction, resulting in increased pressure demand. The effect is more significant than the decrease of required pressure occurring as a result of the viscosity reduction caused by the high temperature. The required pressure increases if the filling time becomes very short. The actual pressure requirement as a function of the filling time is determined by these two counteracting effects. An optimization calculation offers the possibility to determine the filling time which entails the minimum pressure drop (Figure 9).

o

I

2

Filling time IF (s)

Figure 9 Pressure drop vs. filling time of a cup: for PS454H, OM = 200 °c and Ow = 50°C; for P5200, OM = 240 °c and Ow = 50°C

Within the range of processing conditions, the viscosity of amorphous materials is more temperature-dependent than the viscosity of semicrystalline materials. Hence amorphous materials show a distinct pressure minimum; this is particularly true for mouldings with thin walls. When optimizing the filling time, temperature and shear stress have to be taken into account in addition to the pressure drop. For the cup-shaped moulding, increasing the filling time to 0.7 s would reduce the required pressure to a little extent, but the accompanying considerable reduction of the temperature would also produce a low quality joint seam. Multi-cavity moulds, mouldings with multiple gates and balanced runner systems for composite moulds cause additional problems. Calculating programs for the performance of balancing help to solve these problems (Figure 10). The aim of these programs is to have the melt cover the different flow distances within the same time (and with the same pressure drop) by adapting the runner geometry. Figure 11 depicts an example for the balancing of the system runner moulding. Before balancing, the smaller moulding was filled first. By reducing the runner diameter from 7 mm to 4 mm it was possible to have the flow process - and therefore the filling of both mouldsconcluded at the same time. This reduces the risk of flash formation. Cold runner manifolds are balanced for simultaneous, identical pressure drop. The goal of balancing hot runner systems for multi-cavity moulds with identical cavities and multiple gating is to obtain equal pressure drops for equal volume flows at all gates. This balancing therefore yields simultaneous filling of the cavities of multi-cavity moulds and uniform filling of the cavity in the case of multiple gating.

Computer-aided Engineering in Injection Moulding

Gates

JL

Different mouldings

461

Side - by - side runners

Multiple gates

Hot runner systems

Figure 10 Balancing

Before balancing

15,

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20.... /20 25.... /

25

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After balancing

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20_ . . 25 25-. 29 29....

Figure 11

Balancing a runner system for two different mouldings (4); = diameter ; mm)

Balancing is an obviously important criterion of rheological optimization for this type of processing.

14.3.4 Computer-aided Graphical Simulation of the Filling Process of the Mould in Thermoplastic Injection Moulding When applying the simplifying assumptions used in the manual construction of the flow pattern to computer-aided determination, one finds that a simple forward' calculation is all that has to be

462

Unit Operations of Polymer Processing

performed. The continuity equation and the theorem of conservation of momentum have to be satisfied locally at the flow front by means of a reduced iteration loop. Therefore the computer simulation performed, together with the graphical method of determination of flow pattern, is subject to the same restrictions in terms of accuracy as this manual method. The results can not be significantly more accurate. The decisive advantage of the calculation method lies, in addition to time saving accomplished by it, in being able to make the qualitative process of flow pattern determination accessible to a subsequent quantitative analysis, without having to perform time-consuming tasks such as segmentation of the moulding into single, basic geometries. 5,9 Thus the pressure distribution behind the flow fronts can be determined from the potential equation by means of the calculated flow fronts and from the boundary conditions prevailing in injection moulding. (4)

The distribution of the velocities can be obtained from the pressure distribution. 2,3 A number of three-dimensional representation programs have been developed for analysis of the filling process of thermoplastic moulds. A program developed at the IKV analyzes the filling process in a three-dimensional shell-type model of the moulding. In order to do so, the moulding geometry has to be described by a mesh structure consisting of nodes and elements as known from finite element calculation. The program uses linear shell-type triangles for the simulation calculation; the local wall thickness of the moulding can be assigned to each triangle as a boundary condition. Figure 12 shows such a triangle structure of a housing with a rib and two cut-outs. Structures of this kind can be generated with standard mesh generators. .

Figure 12 Triangle structure

The programs for simulation of the flow processes use the information from the FEM structure to determine the flow pattern and the pressure distribution at arbitrary stages of the filling phase; this is accomplished on the basis of the equations quoted above. Local analysis of the flow conditions in the various triangles yields the flow fronts to be determined; the results are then transcribed into a three-dimensional coordinate system which includes the entire structure. Figure 13 represents an exemplary application of the flow pattern analysis; the filling process of the mould for production of a cup-shaped moulding was analyzed here. 5 The cup is gated centrally in the cup bottom. The melt propagates in the form of a source flow and· spreads evenly over the main body of the cup until the flow front reaches the cup handle. At this point the melt divides into two separate flows which fill the upper and the lower part of the handle; the flows rejoin at the point marked by an arrow and form a welding line. The position of this joint seam determined by computer simulation was compared to the results of short shots conducted with the same material; the experimental joint seam was found to be only two millimeters away from the theoretical seam.

Computer-aided Engineering in Injection Moulding

463

Figure 13 Flow pattern of a cup

The determined flow pattern, however, actually yields no information concerning flow processes behind the flow front; these processes largely determine the orientations, in particular the orientations in fibre-filled thermoplastics. The main directions of fibre and molecule layers areespecially with these reinforced (as well as with self-reinforcing) polymers-of decisive significance to the strength of the produced components and hence of great interest in the prediction of the orientation of the glass fibres. The high susceptibility to orientation inherent in these materials and the resulting reinforcing effect provides the possibility to selectively apply such materials to lightweight constructions; it must however be ensured here, by means of flow-related measures, that the direction of the reinforcing effect coincides with the direction of stress in all parts of the moulding. 23,1 Consequently, the simplified calculation of the filling process as well as the representation of the flow pattern have to be extended. The calculations also have to determine the pressures and velocities prevailing in the cavity sections which are already filled. A subsequently added FEM program solves the differential equation. 4 These calculations yield the pressure field in the parts of the mould cavity which are already filled; the pressures are represented in the form of isobaric. lines. Figure 14 shows the result of such a calculation; it represents the pressures immediately before the moulding is completely filled. The isobaric lines show clearly that from the gate the melt flows almost directly to the cup handle at this stage of the injection phase. The densities of the isobaric lines in the handle provide further evidence of the fact that the melt (due to the extremely high local pressure gradient) flows with a much higher velocity in the handle than it does in the residual parts of the moulding.

Figure 14 Isobaric lines of a cup

Unit Operations of Polymer Processing

464

The example used here points out the difference between the 'flow pattern' and the 'representation of isobaric lines'. The flow pattern method shows the progression of the filling process; the representation of isobaric lines yields information about an instantaneous flow state. 5,22 Conclusions can be drawn from this information concerning the orientation behaviour of, for instance, glass fibres as it results in the various significant sections of the component. When considering the flow over a cross-section of the wall, three main layers of different directions of orientations can be distinguished. 24 The two layers next to the walls are oriented mainly in the direction of flow. The centre area of the core is oriented radially in a convergent flow - hence in the direction offlow-when the velocity vectors concur. Divergent source flow, however, for example in the vicinity of the gate, yields tangential orientation of the fibres in the centre layer. The flow in the body of the cup is mainly a source-type flow, while strong shear flow prevails in the cup handle. Therefore the fibres in the core area of the cup will be oriented tangentially; the shear flow in the handle is a condition that favours orientation of the fibres in the direction of flow. Compared to short shots and the manual flow pattern method, these programs offer the designer additional means of predicting the flow processes occurring during filling of the mould and the effect these processes have on the properties of the finished moulding. Besides enhancing the product quality, the programs allow for substantial time saving in the performance of a flow analysis. It is true that the time necessary for description of the moulding geometry and generation of the mesh is the same as the time required for conducting the manual flow pattern analysis by means of lay-flat, but once the geometry of the moulding is generated in the computer, modifications of the wall thickness as well as changing.of gate positions can be performed quickly. Calculation of the filling process with altered boundary conditions then only takes a few minutes computing time.

14.3.5 Filling Process in Elastomer Injection Moulds An appropriate program system 4 has been developed for the design of moulds for elastomer injection moulding as well; it also allows the filling process to be simulated. The simulation is performed in the two-dimensional lay-flat of the moulding by a personal computer; this requires the making out of a lay-flat of the moulding. Screening of the lay-flat into discrete elements, as is necessary for the calculation, is carried out automatically by a preprocessor (Figure 15).

Lay - flat Moulding

Scanning pattern image

Identification of cutting edges

Figure 15 Preprocessing for the simulation of the filling process

Figure 16 shows one result of the computer simulation of the filling process, the graphical representation of flow fronts in an experimental moulding. When comparing the theoretical flow fronts to actual fronts determined through short shots, potential errors in the calculation become evident. In the present example, the area of the film gate is subject to such an error; it is caused by the fact that the continuity equation is no longer satisfied after the flow front has propagated past the thin gate. The error in the calculation depends on the difference in wall thickness between the single sections of the moulding; the greater this difference is, the more the theoretical flow fronts will differ from the actual fronts here.

Computer-aided Engineering in Injection Moulding

465

Short shots

Calculated filling pattern

Figure 16 Comparison between calculated flow pattern and short shots

The influence of the theoretical error in the vicinity of the gate becomes less significant with increasing distance between the flow fronts and the point of discontinuity. Thus it is possible to predict the problem spots of a moulding with reasonable accuracy, even if the preconditions of correct application of the calculation method are not satisfied due to the geometry of the moulding. Figure 17 depicts the velocity field prevailing in the experimental moulding as it is derived from the flow pattern generated previously. The calculation was based on the assumption of constant injection speed. Consequently, the flow velocity is bound to increase significantly at the end of the filling process because the mould cavity is almost completely filled then, leaving only very small areas of the moulding while the entire volume flow is still applied by the machine. Furthermore the program system computes the pressure distribution over the entire lay-flat of the moulding that prevails at the end of the filling phase (Figure 18). This representation of the pressures is particularly significant to the mechanical layout of a mould because it indicates the local stress acting upon the mould cavity as a result of the, mostly very high,

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Figure 17 Distribution of velocity PS 7-P

466

Unit Operations of Polymer Processing

Figure 18 Distribution of pressure at the end of the filling phase

injection pressure. The pressure distribution therefore also allows the stresses acting upon forming parts of the mould (such as sucker pins, jaws, cores, etc.) to be determined directly and thus makes it possible to size the stress-absorbing components of the mould appropriately. Criteria for dimensioning of such elements may be, for example~ the admissible gap in the split level for flashless injection moulding. Finally the programs offer detailed calculation of the velocities, the temperatures and - with the fundamentals discussed below - the degree of cross-linking as they result over the cross-section (thickness) of the moulding; the calculation is performed along flow lines here. The flow lines have to be determined from the velocity fields first. The recalculation serves to determine the material state if the flow pattern, the representation of velocities or the calculation of the required pressure show evidence of any extreme values. With this, it becomes possible to draw conclusions as to where potential manufacturing problems may arise, for example, due to high temperatures (peaks) resulting from high dissipation or extended retention times in the vicinity of flow obstacles.

14.4 ANALYSIS OF THE HOLDING PRESSURE PHASE The process conditions prevailing in the holding pressure phase have a particular, decisive influence on the final dimensions of mouldings made of thermoplastic material as well as on shrinkage, warping and sink marks. Consequently, programs computing this phase of the moulding process are an important prerequisite for evaluation of the moulding quality in terms of truth to dimensions and form, and for determination of the corresponding manufacturing parameters. Thus the programs for simulation of the holding pressure phase improve the quality of dimensioning of the mould. The holding pressure simulation discussed in this paper links the quantities pressure, specific volume and temperature (p, v, T characteristic) with the flowability of the material. It must be clearly understood, however, that while this purely thermodynamic approach is necessary here, it can not be a sufficient means to directly calculate the target quantities quoted above, for example, truth to dimensions and form, in other words shrinkage and warping. Nevertheless the computer simulation of the holding pressure phase is very apt to point out tendencies, indicate problem areas and determine potential framing conditions of the production if the results are interpreted and evaluated on the basis of knowledge acquired through practical experience. The projected target is to be able to predict the process conditions and their effects with such high accuracy that t~e required dimensions of the moulding can be realized without touch-up work to the mould, simply by modification of the machine setting (adjustment parameters).

Computer-aided Engineering in Injection Moulding

14.4.1

467

Fundamentals

The moulding materials cool off, subsequent to the actual filling process of the mould; the areas closer to the cavity walls cool faster than the centre of the cross-section. Amorphous thermoplastics contract by some 5-10% by volume; partially crystalline materials contract by some 10-20%. The volume decrease can be compensated for - to a certain degree - by the holding pressure applied by the machine. Thus additional material is being pushed into the mould cavity through the still liquid centre of the moulding cross-section. The amount of material to be injected additionally (volume flow) is determined here by the thermodynamical characteristics of the moulding compound. The pressure loss over the flow path depends on the flow behaviour. Basically, the mathematical description of these processes on a computer must therefore include: (i) a cooling calculation for determination of the remaining 'free' cross-section and the volume to be injected thermodynamically; and (ii) a flow calculation for determination of the pressure drop over the flow path. The description of the flow behaviour-together with the calculation of the filling phase-is based upon the viscosity equation according to Carreau and on the temperature shift according to WLF. 2s ,26 The thermal diffusivity is used in the form of an effective value 27 and the p, v, T characteristic is described by an expression with seven coefficients according to equation (5).28 v(T, p)

=

+

(5)

The holding pressure process is concluded either when the material temperature in the centre of the cross-section falls below the solidification point (no more liquid melt in the cavity) or when the total pressure drop in the cavity exceeds the applied holding pressure, thus stopping the material transport into the mould. These are the criteria to end the·flow calculation. Figure 19 shows the significant steps which were used to translate the conception discussed above into a calculation program. 28 This was largely based upon experiences reported on in refs. 29 and 31. Input of geometry, material and processing data Calculation of the filling phase

Calculation of the packing phase

Calculation of the cooling next time step

Calculation of the free cross-section

Calculation of the thermodynamic change of state

Calculation of the volume flow

Calculation of the pressure drop

Iteration loop for local pressure equalizing Figure 19 Calculation scheme for the holding pressure phase

468

Unit Operations of Polymer Processing

The effective holding pressure time is subdivided into discrete time intervals. Within these intervals the thermal and thermodynamical changes (relations) are linearized. The pressure adjust~ ment as a function of time and location has to be performed by means of iteration, since both the temperature of solidification as well as the resulting volume contraction depend on the pressure determined by the flow conditions prevailing along the flow path. The time - as a function of the location - after which a certain point in the mould cavity reaches ambient pressure is of great importance to the predetermination of the potential shrinkage. Once ambient pressure is reached, the moulding or the corresponding section of the moulding contractsfrom a thermodynamic point of view - purely in terms of volume, until the conditions of demoulding or ambient conditions are reached. The shrinkage by volume can then be determined by relating the corresponding quantities appropriately, From this volume decrease it now becomes possible to deduce important information on the shrinkage characteristics in terms of dimensions; the quantitative description of this dimensional shrinkage, however, requires a thermomechanic analysis (as opposed to the thermodynamic analysis discussed here). The program referred to here is linked with the program system. 5 Subsequent to calculation of the filling phase and following the transfer of the additional material data and boundary conditions, the program system continues with the calculation of the behaviour under holding pressure and the volume shrinkage. 14.4.2 Calculation of the Holding Pressure In the following section the application and the options of the calculation of the holding pressure phase shall be illustrated by means of a straightforward model moulding. Figure 20 shows a plate such as the one used for investigations into shrinkage in the reports. 31,32 The special design of the manifold element brings about the advantage of parallel flow over the entire width of the plate in both the filling phase as well as the holding pressure phase. This is a very desirable feature 'here since the simulation calculation of the holding pressure phase determines the variables of state at the boundary of a segment. It is thus possible, without further theoretical considerations, to compute a segmentation (see below) that is particularly matched to the calculation of the holding pressure phase. Figure 20 also shows the necessary preprocessing operations -lay-flat, generation of the flow pattern, segmentation, coding - which are already known from calculation of the filling phase. 26 , 17,33 Figures 21 and 22 depict exemplarily a comparison between calculated and measured internal mould pressures vs. time at two different locations; and for two different types of material (amorphous and partially crystalline, see Figure 20 for position of the pressure transducers).28 It

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III

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1111

. 3·A

Loy-flat including filling pattern and segmentation

Figure 20 Preprocessing for analysis of the holding pressure phase

Computer-aided Engineering in I njection Moulding

469

Stabilization of the calculation

300

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300

Stabilization of the calculation Close to the gate

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4

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Figure 22

Measured and calculated internal mould pressure vs. time in a moulded plate: material = HDPE, T M = 230°C, Tw = 58°C, Po = 270 bar; - measurement, --Q---(}- calculation, -0- -0- calculation isochore

is not at all surprising that - from a technical point of view - there is good agreement between calculated and actual pressures; in spite of the considerable reductions in the theoretical description, the significant quantities were taken into consideration in the program. Disagreement, however, is more distinct at the beginning and at the end of the calculation. The theoretical error at the beginning of the calculation must be attributed to the absence of a correct initial value of the pressure. The program therefore sets the initial holding pressure of all segments at the same value (750/0 of the pressure in the prechamber of the screw derived from the hydraulic pressure; empirical values according to refs. 28 and 30. Consequently, the pressure calculation does not correlate to the actual conditions before the second time interval (calculation step). The criteria to end the calculation quoted above constitute a further source of uncertainty. Changeover to isochoric calculation within the running program leads to high theoretical pressure drops due to the discreteness in respect of time and location. In actual processing the changes are more flowing; moreover the precondition of an almost perfectly stiff mould were not satisfied in the case of this model mould either (in practical processing most moulds can not be considered as perfectly stiff). This makes it clear that it can not be the object of this calculation to aim directly at quantitative calculation of certain values (this applies to simulation calculations in general). It became apparent through practical experience that there are a number of influencing parameters which on the one hand, can not be quantified accurately, for example, temperature of solidification, holding pressure, flow characteristics at very low shear rates, etc., but, on the other hand, may easily cause disagreement between measured and calculated conditions of up to 500/0. In practical applications it is therefore always necessary to work up to a conclusive, logical result by means of systematic variation of the boundary conditions (the operator convenience and the short computing time of the programS are a great advantage in this aspect).

470

Unit Operations of Polymer Processing

Once such appropriate working points have been established--this is obviously the case in the present example - a wealth of information can be acquired concerning dimensioning of the mould, optimization of the process and finally, the evaluation of the shrinkage to be expected. The possibility of computing even the most complex components is the particular advantage of the presented simulation of the holding pressure phase based upon segmentation of the moulding. It is, for example, possible with this calculation program to determine, for different gates and gate positions, the behaviour under the holding pressure and consequently the different shrinkage of the moulding in terms of volume. However, the segmentation of the moulding (or of the flow patterns in the lay-flat) is not very straightforward; it has a stronger influence on the computed, final result of the holding pressure phase than it has on that of the filling phase. In actual processing, the flow pattern in the holding pressure phase is quite different from the calculation because the pressure gradients and pattern of short shots do not agree any more already at the end of the filling phase; the three-dimensional filling simulation (isobaric lines, see Section 14.3.4) proves this. This means that it is necessary, in principle, to base the calculation of the holding pressure phase on modified flow successions from one time interval (calculation step) to the next. But at the moment this would still involve disproportionately high expense. Today, parts of appropriate programs are still in the experimental stage. The present calculations were therefore carried out with set segmentation of the small box. Figure 23 depicts the segmentation of the box-shaped moulding with pin-point gating at the long side.

Figure 23

Flow pattern and segmentation of a box gated at the long side: hi

= 3 mm, h2 = 2 mm; -.- welding line

14.4.3 Evaluation of Shrinkage It has already been discussed in the previous section that values concerning the shrinkage of the moulding in terms of volume are derived from the results of the calculation of the pressure profile in an additional step. Figures 24 and 25 show the dependence of the shrinkage on mass and wall temperature or the materials ABS and HDPE. The reasons for these shrinkage dependencies are discussed at length in ref. 32. In this case good correlation was found between the calculated shrinkage by volume and the measured linear shrinkage 23 (at the moment this only applies to the plate moulding; the boundary conditions were ideal here). From these results a correlation model can be derived describing the interrelation between volume shrinkage and processing shrinkage. This correlation then serves to determine the processing shrinkage from the calculated volume shrinkage and the processing shrinkage measured at a certain working point. 32 Figure 26 represents the result of this approach. The disagreement between theoretical and experimental values is between 10 and 150/0 here. The pressure calculation and the shrinkage calculation of the box-type moulding were carried out with the restrictions concerning segmentation that were quoted previously. Figure 27 gives an example of a descriptive form of representation of the differences in volume shrinkage for gating at the long side. In accordance with what was to be expected, the volume shrinkage - the main force behind linear shrinkage-increases with the flow distance (difference in height and duration of holding pressure).

Computer-aided Engineering in Injection Moulding

471

4

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260

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Level of back pressure (bar)

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Influence of material temperature on the shrinkage: Sy = volume shrinkage, SL = processing shrinkage, material = ABS, T w = 71°C; o. = calculated, 0 = measured

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Figure 25

Influence of wall temperature on the shrinkage: Sy = volume shrinkage, SL = processing shrinkage, material = HDPE, TM = 230°C; 0 = calculated, D = measured

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Figure 26 Correlation of measured and calculated processing shrinkage of HDPE: SLO(PO = 220 bar, TMO = 230°C, Two = 57°C) = 2.25%, k = 2.2, ltyp = - 0.00470/0 bar -1, ltYTM = - 0.01 % °C - 1, ltYTW = + 0.031 % °C - 1

Furthermore the shrinkage is obviously enhanced in the region of higher wall thickness (3 mm) in the bottom area. This can be put down to the fact that, in this area, more material reaches ambient pressure at a higher temperature level than in areas which are 2 mm thick. This, however, does not necessarily mean that the linear shrinkage is enhanced as well; as long as the moulding is inside the mould, the core entirely impedes shrinkage of the bottom area of the box. Furthermore, the increased shrinkage potential favours the occurrence of stresses; but the higher material temperature in these areas, on the other hand, allows the stresses to relax better. 32 There is

472

Unit Operations of Polymer Processing

~

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Volume shrinkage of the box-type moulding: material = PP, Po = 450 bar, TM = 225°C, Tw = 30°C, t F = 1.5 s, hi = 3 mm, h2 = 2 mm

also the risk of sink marks and voids (in sensitive materials such as standard polystyrene) and of stress cracking. With the simulation calculation it not only becomes possible to realize such problems at an early stage, but the computation also yields information on the effectiveness of potential remedial measures. Even so, it is not possible, at this stage, to make any quantitative statements. Figure 28 shows that gating the box centrally at the bottom by means of a hot runner reduces the total degree of shrinkage and optimizes the shrinkage homogeneity. This coincides well with practical experience. Obviously the simulation program presented here enables the user to gain information about, and understanding of, potential problem areas and weak points of the production during the design of mould and moulding; the necessary modifications can thus be carried out early.

14

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Influence of the gate position on the shrinkage: material = PP, Po = 450 bar, TM = 225°C, Tw = 30°C, t F = 1.5 s, hi = 3mm, h2 = 2mm

Computer-aided Engineering in Injection Moulding

473

14.5 THERMAL ANALYSIS It is the aim of the thermal analysis to determine temperatures and heat flows as functions of time and location in both the mould as well as in the moulding. In the case of reactive moulding materials the course of the reaction is closely linked with these temperatures; it is therefore subject to discussion in the following section.

14.5.1

Fundamentals

The energy equation represents a general description of the temperature field as a function of time and location. d

- (p C p 8)

dt

=

(8)

[b ( Abx bx

(8)

+ -b ( Aby

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.

+
(6)

The source term must be taken into account if mouldings made of reactive moulding materials are concerned; it is proportional to the reaction speed. The reaction conversion, C, is a function of time and temperature; it is described by equation (7).21, 34,35 bC

-

bt

=

k(T)(l

-

C)"

(7)

According to an equation by Arrhenius the coefficient of the reaction speed k(T) depends on the temperature (8)

In the mould (as opposed to the cavity) and in non-reactive moulding materials (thermoplastics) the source term equals zero; even with elastomeric materials the reaction energy is trifling, so that the source term can be neglected in comparison with the other terms. With the characteristic material values (which are generally temperature dependent) and the boundary conditions, the resulting set of differential equations can basically be solved. Boundary conditions have to be defined between moulding and mould and between the mould and the environment, as well as - after demouldingbetween the moulding and the environment. The heat flows per unit of area have to be equal at the surfaces of the mould cavity and the moulding; this yields

(9)

The heat transfer coefficients to the environment and the heating/cooling elements constitute further boundary conditions (10)

The boundary conditions may be subject to changes in the course of the moulding process; this has to be taken into account by the program. The given equations describe the three-dimensional, instationary temperature fields in both mould and moulding; after separation the set of equations can be solved by means of numerical calculation methods. However, the computing times and the memory requirements necessary for such a numerical solution are extremely high and are actually not justified in the design of moulds. But a number of simplifying reductions can be made. The most common solutions are: (i) temperature at a certain, fixed point at different times (referring to a certain location, instationary); (ii) temperature profile along a line through mould and moulding at different times (one-dimensional, instationary); and (iii) temperature field in a crosssection at a certain time (two-dimensional, referring to a certain time). With this the computing expenses can be drastically reduced without losing much accuracy and quality of the information given by the results. However, various reductions can be made according to the respective processing and the moulding compound. PS 7-P·

474

Unit Operations of Polymer Processing

14.5.2 Layout of Thermoplastic Moulds The efficiency and hence the design of the heating/cooling system of the mould has a significant influence on the economy of the manufacturing process and the quality of the moulding. Short cycles are important for economically efficient production; but a carefully designed and calculated heating/cooling system is also a guarantee for the production of high quality mouldings: (i) the wall temperature of the mould influences the surface quality of the moulding; and (ii) temperature differences on the surface of the moulding lead to inhomogeneous properties and warping, and increase the risk of stress cracking as a result of internal stresses. In many cases a selective layout of the heating/cooling system is not carried out when designing a mould; time pressure often prohibits this measure, but the time necessary for this layout can be significantly reduced by the application of programs designed for pocket calculators. 26 The costs saved by abandoning thermal layout involves high consequential costs on account of loss of moulding quality and extended cycles. The efficiency of the heating/cooling system design in terms of moulding quality can not easily be described with mathematical terms; in terms of cooling time however, the characteristic value in equation (11) can be used for quantification of the quality of the heating/cooling system (11)

For a given material the lower limit of this characteristic value can be determined by means of calculation of the cooling time. If, for example, the calculated value of CT is 1.5 s mm - 2 aQ-d in operation C T is 3 s mm - 2, the output can almost be doubled by improving the heating/cooling system. In the case of unfilled amorphous thermoplastics, a good design of the heating/cooling system is characterized by values of CT below 3 s mm - 2

14.5.3 Procedure for the Layout of the Heating/Cooling System The layout of the heating/cooling system can be carried out in a number of single steps. The calculations used are similar for different materials, but the sequence in which single steps are applied may vary. The thermallayout S - S ,10-12 is based upon the geometry of the moulding and the properties of the material to be used. This information serves to calculate the minimum cooling time necessary (criterion of economic efficiency) for cooling the moulding to the point of dimensional stability (moulding quality) at a specific material-dependent wall temperature of the mould (surface quality). The positions of heating/cooling channels are then to be calculated which are necessary to realize this wall temperature of the mould; the 'homogeneity' of the heating/cooling system is an additional criterion. The goal is to obtain a uniform temperature distribution along the heating/cooling channels as well as in th~ region between the channels. In addition to position and dimensions of the heating/cooling channels the programs allow for calculation of the characteristic demands on the heating/cooling system such as the temperature of the heat transfer agent, the pressure loss and required power. Furthermore, the thermal layout is able to provide information concerning the temperatures at external mould surfaces and the heat flows to the environment. The exact calculation of these quantities, however, can only be conducted after the final determination of the external mould dimensions. Since the latter only result from the mechanical layout and the selection of standards, the heat exchange with the environment can only be estimated at this point in the layout. An additional step pro~ides the possibility of analyzing the thermal behaviour of the mould during start-up and operational stops. S The single steps for layout of the heating/cooling system are depicted in Figure 29. S (1) Calculation of the cooling time. The minimum, and therefore economically most efficient cooling time required, can be determined from the material data of the used plastic, the melt temperature and the highest wall thickness of the moulding by means of calculation. Shorter cooling times of the moulding entail losses in surface quality on account of the necessary low wall temperature. (2) Balance ofheat flows. During the cooling time, energy has to be extracted from the plastic; this is accomplished by conduction of heat from the moulding to the mould. Heat will also be exchanged with the environment; at the moment the corresponding heat flow has to be estimated for the reasons quoted above. The heat flow conducted to the heating/cooling agent results from the fact that the sum of all heat flows in the moulding equals zero.

Computer-aided Engineering in Injection Moulding

475

-,-- ..

Cooling time

Cooling throughput

Temperature of cool ing channels

Diameter of cool ing channels

Position of cool I ng channels

Pressure drop

2: 0 =0

Heat balance

Heat exchange with the environment

Figure 29 Thermal layout

(3) Throughput of heating/cooling agent. The heat exchange between mould and heating/cooling agent causes an increase of the temperature between inlet and outlet of the agent. This temperature rise should not exceed 5°C; otherwise inhomogeneity will occur as a result of the increased cooling at the inlet and the reduced cooling efect of the agent at the outlet. The required throughput of heating/cooling agent can be calculated if the admissible temperature difference of the agent is given. (4) Diameter of heating/cooling channels. Usually each company uses their own standard diameters of heating/cooling channels (6,8, 10, 12 mm). The minimum diameter can be calculated if an admissible pressure drop is given. The demand for turbulent flow determines the maximum admissible diameter. (5) Temperature of the heating/cooling channels. The heat transfer coefficient of the heating/ cooling agent can be calculated with the diameter of the heating/cooling channels and the temperature and material data of the heating/cooling agent. With this information the temperature at the wall of the heating/cooling channels can be determined, if the temperature of the heating/ cooling agent is given. (6) Posit.ion of heating/cooling channels. Three criteria serve to determine the distance between heating/cooling channels and the distance between channels and the cavity wall: (i) the heat flow from the moulding has to be conducted at the calculated temperature difference between the cavity wall (surface temperature of moulding) and the wall of the heating/cooling channels; (ii) the geometry has to be calculated in such a manner that the admissible inhomogeneity is not exceeded between the heating/cooling channels (cooling error); and (iii) the number of heating/cooling channels should be kept as small as possible in order to minimize the production costs of the mould and the pressure drop in the heating/cooling agent. (7) Pressure drop. The total pressure loss in the heating/cooling system including all pipes and fittings is calculated in this step. The result is to be compared to the pressure rating of the supplying system. (8) Heat exchange with the environment. After determination of the external dimensions of the mould the accuracy of the previously estimated heat flow to the environment can be checked. The result of the layout of the heating/cooling system can be checked and optimized with simulation programs determining the temperature profile in the mould. Such a simulation program also allows the starting process to be analyzed. The corresponding steps are realized by two different types of programs. (1) Programs with alphanumerical output are used for the steps between the calculation of the cooling time and the calculation of the heat flow to the environment (step 1 to step 8) because the amount of data obtained is quite limited here.

476

Unit Operations of Polymer Processing

(2) The simulation calculation yields the temperatures in both mould and moulding as a function of time and location. It is not possible to alphanumerically represent such quantities of data in a clear, readily comprehensible way; therefore it is appropriate to issue the results of simulation programs graphically (see Figure 32). Alphanumerical programs operate on the basis of simplifying assumptions. Thus the moulding can be reduced to a plate in order to perform the initial, quick layout; the calculation steps are carried out for this plate first. 36 In the case of complex mouldings with corners, ribs and the like, this approach can effectively be used as an approximation, but it is not sufficient. The approximation may serve as a basis on which an exact layout of the heating/cooling system can be worked out. The accuracy can be increased by subdividing the moulding into areas (segments) with little mutual influence. Each of these segments includes one heating/cooling element (heating/cooling channel, cooling finger, etc.); the single segments are characterized by the form of the moulding area to be cooled. The following are taken into consideration here. (i) Different moulding thicknesses which require different heat flows. (ii) The change in the temperature of the heating/cooling agent in the heating/cooling system. In terms of homogeneity this change is minor but it has quite some influence; this will be discussed in the following section. (iii) Different kinds of segments such as corners, core areas, circular sections of the moulding. (iv) Different heating/cooling elements. The first segment may comprise a twisted sheet metal partition, the second could have a tube-type cooling finger and the temperature of a third may be controlled by means of a simple single, or double, helix cooling finger. (v) Different heat flows. For example, the core sections conduct less heat to the environment than the cavity sections. Thus subdivision into segments allows for greatly increased accuracy of the layout. However, the assumption that the elements do not influence one another results in error. The simulation calculation makes it possible to quantify the errors because it considers the mould as a whole. It is therefore perfectly suitable for checking and optimization.

14.5.4 Layout of the Heating/Cooling System The results of thermal layout by means of a plain-shaped moulding (Figure 30) have been discussed. 5 I

- .ED· I

~

~

1~1~r-!

!

I

i

I

--
i

~so=Ji4-.---7~ J

I

I

I

Figure 30

T

r.LsoJJ ~ 2~

- .(±)- --(j)- _.$- .Ef)i i i

.-$- .--E!)I

27

I

4- .%- --'-I

.

Iii

Layout of a cooling system (all measurements are in mm)

The results of a layout with a simplified moulding are: in the case of polystyrene the calculation of the cooling time yields a minimum time of 26.3 s for a melt temperature of 230°C, a wall temperature of the mould of 40°C and a mean demoulding temperature of 80°C. The cooling time is determined by the area of maximum wall thickness of the moulding. The cycles last 31.3 s including additional time. Hence a heat flow of 1243 W has to be conducted from the moulding. Assuming that 5% of the total heat is conducted to the environment, the heating/cooling agent absorbs a heat flow of 1181 W. A volume flow of 60 cm 3 s -1 entails 5 °C difference in temperature of the heating/cooling agent between inlet and outlet. All heating/cooling channels with diameters less than 33 mm have turbulent flow conditions. A channel diameter of 12 mm was given. The layout of the simplified moulding also yields a distance between channels and cavity wall of 22 mm, if the heating/cooling agent enters the mould with 20°C sufficient homogeneity is ensured here. The cooling error is an indication for the homogeneity; for amorphous materials it should not

Computer-aided Engineering in Injection Moulding

477

exceed 5 to 10%. In our example it was 2.3%. The temperature at the wall of the heating/cooling channel is 29 ac. The flow rate of 53 cm s- 1 causes a pressure drop of 0.4 bar. The required power output of the supplying system amounts to 1245 W; 3 Ware necessary 'to make up for the pressure drop, in the heating/cooling agent the residual power is necessary for cooling of the agent. For these calculations the moulding was assumed to be a plate of uniform thickness for the purpose of simplification. Figure 31 shows the result of a segmented layout which takes the different thicknesses into account. 6

5

I

I

4

2

3

I I --$--Cf)l -{}} --$- f -ffi·-- t --C9I I I I 39

+4 t

Figure 31

+4 ~

+3

I

t

i

-4- .$- --~ --4l. --$ ~. i

i

I

7

8

31

.

I

I

9

10

30

t i2

_.1.

I

II

12

Segmented layout of a cooling system (all measurements are in mm)

As opposed to the calculation with the simplified moulding the positions of the heating/cooling channels resulting from segmented layout vary for each section. The following values were calculated with series connection of the heating/cooling channels (inlet of the heating/cooling agent at the first segment). The segmented layout takes the increase of the temperature of the heating/cooling agent into account. In order to realize equal heat flows, the channels have to be closer to the moulding, as the temperature of the heating/cooling agent rises. Despite the fact that the temperature difference is only 5 ac between inlet and outlet of the cooling agent, the distances between cooling channels and cavity wall differ as much as 9 mm (segment 1 compared to segment 12). (See Table 4.) Table 4 Segment

1 2 3

4 5 6 7 8 9 10 11 12

Distance between cavit y wall and heating/cooling channel (mm)

Temperature of heating/ cooling agent at end of segment

Temperature at the wall of heating/ cooling channel

CC)

CC)

31 30.5 38.5 37 28 27 26 25 31.5 30.5 22.5 21.5

20.4 20.8 21.2 21.6 22.0 22.4 22.8 23.2 23.6 24.0 24.4 24.8

25.8 26.2 27.0 27.3 27.4 27.8 28.2 28.6 29.3 29.7 29.9 30.3

In areas with different wall thicknesses a uniform distribution of surface temperatures can be realized by adapting the distances between the channels and the cavity wall. The distance of the third segment has to be increased relative to the distance of the first segment, because the heat flow must decrease for identical temperature differences, due to the reduced wall thickness. Hence alphanumerical layout can be performed at two levels of accuracy: (i) layout with simplified moulding; and (ii) segmented layout.

Unit Operations of Polymer Processing

478

The example indicates that, even for plain-shaped mouldings, the increased accuracy of segmented layout entails improved cooling systems. This advantage is even more significant for complex mouldings.

14.5.4.1

Simulation calculation

The temperature in the mould as a function of time and location can be calculated and represented with simulation programs. The geometries of mould, moulding and heating/cooling system as well as the boundary conditions on the part of the environment and the initial conditions (for example the melt temperature) are entered into the computer. With these data the temperatures, as a function of time and location, are calculated by means of difference equations. The result is issued in the form of isothermal lines prevailing in a desired cross-section of mould and moulding. The isothermal lines can be represented for different stages (times) of the process. The simulation calculation clearly points out areas of heating/cooling inhomogeneity. The effectiveness of single modifications of the heating/cooling system can be checked by conducting a simulation calculation with the improved mould. Figure 32 exemplarily depicts the isothermal lines before and after optimization. Clearly the temperature differences at the cavity wall are very high in the case of the heating/cooling system depicted in the left part of the figure. The modified cooling channels - the positions are changedyield reduced temperature differences; this means enhanced homogeneity.

22

..

22 ••••• '

24

•.~

'

,

....

, '"

....

...

.

.•...'" '.

".....

.•....... '0.

.....

'.

..'

, .'

........

,

,,

:'

Figure 32 Optimization of the heating/cooling system: step size = 2 mm, grid size = 80 x 80, number of cycles = 5, Aw = 50 Wm- 1 K -1, ('J.w = 15mm 2 S-1, AF = 0.3 W m- 1 K -1, ('J.F = 0.08 mm 2 S-1, ()w,o = 20°C, ()F,O = 200°C, t K = 9s, t N = 4s, ()u = 20°C

Thus the simulation program can be used to compare local temperature fields resulting from different heating/cooling systems. Similarly, the temperature field can be issued at different times for one and the same heating/cooling system. It becomes clear how the mould heats up cycle by cycle; thus the time can be determined after which quasi-stationary operation prevails, in other words the time after which the starting process has ceased to be of any influence. Simulation programs are suitable for checking and optimization of the heating/cooling system design. In addition they allow for representation of the temperature profiles as functions of time (Figure 33).

Computer-aided Engineering in Injection Moulding

479

Positions of the calculated courses of mould temperatures

Middle of the mould, adiabatic

"'"

Edges of the mould, convective

3 50

10

~

~ rAtW~;::::;=;:::::;:::::;=:;:::::;;::h.~-J Q)

a.

E ~

Time (s)

72

Time (s)

72

Time (s)

72

Figure 33 Mould temperature as a function of time

14.5.4.2 Heating/cooling of corners The equations used in the segmented layout can also be utilized in the determination of dimensioning criteria for special problem areas. The simulation calculation serves as a checking device here. The layout of the heating/cooling of corners is one example of such an application. At corners, the temperatures in the core differ from those in the cavity if the distance between the heating/cooling channels and the moulding is identical in the two areas of the mould. The simulation calculation shows these temperature differences (Figure 34). The temperature increase leads to slower cooling down of the material in the range of the core. The moulding warps in such a way as to decrease the angle. In many cases this warping can be eliminated by lowering the core temperature. The moulding is then cooled faster in the straight sections at the side of the core. The moulding now warps in these regions and arches toward the side of the cavity or is subject to increased residual stresses. This may lead to stress cracking during use of the part.

Adiab~

Figure 34 Inhomogeneous temperature profile in corners

480

Unit Operations of Polymer Processing

The homogeneity of the heating/cooling can be improved through segmented layout. To that end the moulding is subdivided into segments (Figure 35). The heat flows conducted from the moulding during cooling are proportional to the volume to be cooled. b1

Q1

Q2

2(b

~)

+

2

(12a)

b

0.5-': if s b2

~

(12b)

b2

Temperature fields and electric potential fields can be considered as analogous. A 'heat resistance' can be defined in temperature fields analogously to the ohmic resistance in potential fields: =

Potential field U Temperature field

-4·-

Q x

=

~()

IxR

WW

--$-I

I

._.ED- I 2

Figure 35 Segmentation of a corner

Identical temperature difference between moulding wall and heating/cooling channel wall in both segments yields (13)

The heat resistances of the segments can be described as functions of the geometry and the thermal conductivity of the mould materia1. 26 ,37 WW 1 WW2

1

1 [2b --In -sinh(2na 1 /bd] 2nLl

1 [2b 0] = --In -slnh(n) 2

2nLl

(14)

nd

nd

x

2

[°0

077 2

+

+

In(b 2 /d)] In(b 2 /d)

(15)

If the distances a 1 and b1 are given and if the diameter d is known, the required distance b2 can be calculated with computing programs for segmented layout. 26 Increasing the heat flow in the range of the corner reduces distortion of the angle and yields equal temperatures of core and cavity heating/cooling. The simulation calculation clearly points out the improvement (Figure 36). The temperature in the corner decreases from 46°C to 34 °C (see Figure 34).

14.5.5 Temperature and Reaction Conversion in Elastomeric Injection Moulding4 When analyzing the temperature distribution in injection moulding one finds that the separate consideration of mould and moulding has advantages. Initially, the temperature at the surface

Computer-aided Engineering in Injection Moulding

481

Figure 36 Improvement of the homogeneity in corners: step size = 2 mm, grid size = 50 x 50, number of cycles = 5, Aw = 50 Wm- 1 K -1, (J.w = 15 mm 2 S-1, AF = 0.3 Wm -1 K -1, (J.F = 0.08 mm 2 S-1, ()w,o = 20°C, ()F,O = 200°C, ()TK = 25°C, ()u = 20°C, t K = 9 S, t N = 4 s

between mould and moulding can be assumed as approximately constant since, after the first heating process, the temperature changes in the mould are minor compared to the temperature gradient in the moulding. Figure 37 shows the temperature profile over the cross-section of the moulding at different times, t, to, t 6 , as it is obtained in elastomer injection moulding. 21 Heating up of the material begins as the side of the hot mould wall. The reaction conversion can be derived from the course of the temperature (right of Figure 37). The conversion also increases with the time. 180~i

10

~J60~!A80~ f .2 E CD

60

140

CD

>

c: o u c:

~

E

t!

.~

120

40

~ o

CD

0::

100

20

o

80 Wall thickness of the moulding

Figure 37

Distribution of temperature and degree of cross-linking

Figure 38 shows the temperature at discrete points, s, of the moulding cross-section as a function of time: 13 S = 0 refers to the centre of the moulding; S = 2.5 mm refers to the mould wall. Initially the temperature in the outer layers of the moulding rises very quickly while the internal areas of the moulding heat up later. After demoulding, the outer layers conduct heat to the environment; this leads to enhanced cooling of the corresponding region. The development of the temperature has an effect on the cross-linking (Figure 39). In the outer moulding layers cross-linking takes place at a very high rate. After demoulding, the resulting temperature drop ensures that conversion in these layers remains at the previous level. The internal areas of the moulding, on the other hand, cross-link later, continuing to do so after demoulding. The conversion in the moulding is therefore balanced. Compared to the previous one-dimensional analysis of a plate-type moulding the twodimensional analysis yields additional information.

Unit Operations of Polymer Processing

482

Demoulding 180

+

0

~ 155

!

:3

5

"0 i

130

0-

E

{!

= 2.3 mm

~S=1.9mm 5 = 1.5 mm

105

5

= I mm

5

= 0.6 mm =0.2 mm

5

2

3

4

Time (min)

Figure 38 Temperature before and after demoulding Demoulding 100

t

~

t..

5

75

.~ Q)

>

'5

(,)

50

c

g 0

25

Q)

0::::

2

Time (min)

Figure 39 Conversion before and after demoulding

189°C 188 187 186 185 184

Figure 40 Isothermal lines in a cross-section of a moulding

Figure 40 shows the two-dimensional representation of the temperature at a certain time. The central area of the T-shape constitutes a concentration of melt, which heats up less. Figure 41 shows that this leads to a lower degree of cross-linking. Thermal analysis of the moulding (one- or two-dimensional) yields valuable information on the appropriate process control, particularly on the duration of the heating process required by the moulding. The length of the heating time has an influence on the formation of voids, for example. The wall temperature of the mould is an input variable of the previously discussed calculation. It has to be reached during the initial heating phase. Figure 42 shows the local temperature profile at different times of the initial heating phase as it occurs in a typical mould for processing of elastomer

Computer-aided Engineering in Injection Moulding

483

----96% 92

88 84 80 76

Figure 41

Lines of equal conversion in a cross-section of a moulding

60

100

Temperature (Oe)

Figure 42

'::;:::;:;:::::;:::::::::::::::::::::::::::;:;:::::;:;::1

Heating up of an indirectly heated mould: heating up time = 48 min I;:;:;:;:::::::::::::::::::::::::::::::;:;:::::::::::::::1

Mould closed

Mould open

_.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:-

Temperature (Oe)

Temperature (Oe) ( a) After inject ion

After end of delay time

( b) end of heating phase

Figure 43

Operating behaviour

plastics (flat, heated by heating plates). The required heating time is determined with this calculation. The temperature of the mould wall is subject to cyclic fluctuations even in quasi-stationary operation. Figure 43 shows the local temperature profile in the closed and the open mould. Immediately after the injection the temperature drops sharply du~ to contact with the colder material. The mould wall temperature increases with rising temperature of the rubber mixture until the end of the heating phase. After demoulding of the part, the mould wall temperature drops due to heat exchange with the ambient air.

14.6 MECHANICAL LAYOUT OF MOULDS Reliable and secure functioning of moulds is also a question of appropriate design and dimensioning of its component parts. This includes first the design of the motion mechanism; in moulds with sucker pins and core pullers, for example, the sequence of motions as well as the possibly necessary gear units have to be determined. However, the designer's major concern is to ensure that the mould does not deform more than the admissible values during operation.

Unit Operations of Polymer Processing

484

Two design criteria must be taken into account here. On the one hand, it has to be ensured that the elastic deformations, occurring at right angles to the direction of clamping as a result of the internal mould pressure, are reversed completely in order to compensate for thermal contraction of the plastic, otherwise demoulding may cause problems. On the other hand, bending of the mould plates due to forces in direction of clamping must be kept sufficiently low in order to avoid undue gaps in the split level which produce flashes. Finally, the designer has to take into account the forces of friction and inertia acting on the moulding and the elements of the mould such as sucker pins, guide bushing, etc. Many of the quoted design complexes are already covered by calculating programs. 5 The possibilities offered by these programs are discussed in the following sections. 14.6.1

Aids for the Calculation of Mould Dimensions

The various design criteria show, that dimensioning can be determined in a quick and straightforward way by means of analytical expressions. Examples for such values are the maximum deflection of the mould plates, the expansion of the cavity and the compression of the core at right angles to the direction of clamping. Some of the corresponding equations are given in Figure 44.

Basic load conditions

Equations

S

3p[4

p[22.66

f

- -3 +

f

- -3 +

f

- - -3 +

f

pr{

32Es

x

1.2

8Es

P

f

S

12p[4

pP 2.66

384Es

x

1.2

8Es

·1

Ip

f

S

.....

I-

----d

12pd4

pd 2 2.66

1138 Es

16Es

-I

E

1

r.I 2

+ r2 a r/ ra 2

x

+

p

Figure 44 Simple calculation methods for mechanicallayout

~)

1.2

Computer-aided Engineering in Injection Moulding

485

The appropriate programs 5 can be selected from menus offered by the computer. They are designed for interactive communication in order to ensure simple, straightforward use. Use of the computer is particularly advantageous, if the programs are utilized to calculate the minimum dimension of a component (for example minimum thickness of mould plates or minimum outer radius of the mould cavity) with given admissible deformation and given load. This can only be accomplished by means of successive approximation; however, without the use of a computer this would involve too much time. These straightforward calculation procedures are based upon assuming loads for each single component of the mould; as long as the deformations of these single components can be considered separately, the results of the calculations are therefore sufficiently accurate.

14.6.1.1

Superposition of loads

The superposition method allows the single and total deformation to be calculated in structures composed of a number of 'basic blocks'. The method is based upon the programs for description of the previously discussed basic cases of loading. 26 With the superposition procedure, the designer is also able to obtain information about the maximum deformation occurring in the split level of an injection mould within a short time. Figure 45 represents this in a diagrammatic form.

Figure 45

i

Spring model of the c1ampside of the mould (4) = diameter)

The single elements of the mould can be considered as springs; by connecting the elements parallel or in series a spring model of the overall mould is obtained. This model, the characteristic value of the single elements (form, dimensions, type of loading) and the total load are all the computer requires for determination of the single deformations and the superposition of the latter which results in the required total deformation in the split level. If the calculated total deformation in the split level is too high, the maximum total deformation of the system in the calculating program can be limited and the computer can increase, for example, the thickness of the mould plates by means of successive approximation calculations (iteration method) until the stiffness of the entire system meets the requirements.

14.6.1.2 Deformation in the case of complex geometry For complicated structures, the superposition method yields a good approximation of the occurring deformation through reduced mathematical equations and assumptions. With such a

Unit Operations of Polymer Processing

486

computing program 26 the deformations can be calculated at the different points. The finite elen method makes higher accuracy possible. When processing with complex moulds for the production of car panel boards, bumper lining, it should be borne in mind that the deformation of the fringe area of the mould frequently cal flash formation. In addition, enhancing the stiffness of such moulds, even in the case of foan materials, often involves such extensive measures that the mould as a whole can then onl) described approximately by means of the quoted basic cases of loading. The immense expense terms of time and calculation required for the deformation analysis by means of the finite elen method is very well justified if the demands on the accuracy of the moulding are very high. Today, a great number of readily applicable FEM program packages are offered 1, 38 for deformation calculation, which constitutes an important step in the mould design. Figure 46 sh the results of an FEM deformation analysis performed with such a program;39 the analysis exam the behaviour of a mould cavity under the influence of clamping force and internal mould press In the present figure the deformations are enlarged to 50 times their size for the purpOSI representation.

--

-roo-

Unloaded

I I

r--r--_

-lli~~

~~~

~ ~-

y~~~ 1'1~

__

0-.

_.

I I

_._

--

~

I ~I,;i

4

Loaded by clamping force

I I

---

ill

-N~-"-

-~ >.~

- ._._._.

--

11"I

-

R

II

......

~

3.995

Loaded by clamping force and cavity pressure

Figure 46

Deformation of the cavity under the load of clamping force and pressure according to FEM

FE programs are a very valuable aid for the designer who has very little knowledge of diff( mould designs and constantly requires various mould dimensions and moulding shapes fOI moulds to be designed; finite element methods allow for the description of even highly com structures and make it possible to check the latter mathematically. In principle, the time requirel a FE analysis depends on the user comfort and on both the hardware and software used for processing. In practical application of the finite element method, however, it became apparent the computed results are influenced by the part description (the FE mesh) and the type oj elements used. The use of finite element methods requires some experience in respect of the] favourable structure of the mesh; but in addition it is absolutely essential to check all resul1 means of comparative values. For example, the designer may use the deformation values determ with the straightforward methods described above for assessment of the FEM calculation.

Computer-aided Engineering in Injection Moulding

487

14.7 PROSPECTS FOR THE FUTURE The options presented in this report of computer application for simulation and dimensioning purposes refer initially only to the field of design. However, the rationalizing effect of this system is not necessarily restricted to the design process. The effects will extend to the processes following the design, such as production planning, manufacturing of the mould and even the moulding production; these processes will benefit from the effects of improved design, enhanced quality of the processing data and from the increased use of standard components. The fields of moulding design, mould design and moulding production are linked through a data processing system; this constitutes a line of connection between already available solutions to entire single tasks. In the future, the computer will be used as a central data bank for the management of all in-plant data; it will thus correlate and inter-link all areas of a plant.

14.8 APPENDIX: ABBREVIATIONS AND SYMBOLS distance between centre of heating/cooling channel and moulding -width C - conversion CT - characteristical value of the quality of the heating/cooling system Cp - thermal capacity d - diameter of the heating/cooling channel E - modulus of elasticity Ea - activation energy f - deformation fges total deformation H - height of segment h -height i-current K -constant k ----c-constant L - segment length I - flow distance p -pressure Po - pressure, initial value ~p - pressure loss Q -heat flow q" - heat flow per unit of area R - gas constant, ohmic resistance S - shrinkage SL - processing shrinkage Sv - volume shrinkage s -local coordinate, wall thickness T - temperature TM - melt temperature T w - wall temperature t -time U - potential difference u -velocity v - velocity, specific volume WW - heat resistance x -local coordinates y -local coordinates -local coordinates z a

-

a

- heat transfer coefficient - heat transfer coefficient - difference - difference - temperature

b

P

~

b ()

488 8M 8w 80 8 00 A AF AKS Aw p

l'

Unit Operations of Polymer Processing melt temperature wall temperature temperature of medium ambient temperature - thermal conductivity thermal conductivity of the moulding thermal conductivity of the plastic - thermal conductivity of the mould -density shear stress

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14.9 REFERENCES 1. u. Lichius, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1983. 2. H. Bangert, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1981. 3. G. Menges and P. Mohren, 'Anleitung fiir den Bau von Spritzgiesswerkzeugen', Carl Hanser Verlag, Munich, 1974. 4. 'Cadgum', software developed by Institut fiir Kunststoffverarbeitung (IKV), Pontstrasse 49, 5100 Aachen, West Germany. 5. 'Cadmould-Mesfisto', software developed by Institut fiir Kunststoffverarbeitung (IKV), Pontstrasse 49, 5100 Aachen, West Germany. 6. 'Computer Aided Injection Molding System', software developed by Cornell University, Ithaca, NY, USA. 7. 'Microplast', software developed by Compagnie Internationale des Service Informatique (CISI), Paris, France. 8. 'Moldfill Mold-cool', software developed by Application Engineering Corp. (AEC), Elk Grove Village, IL, USA. 9. 'Moldflow', software developed by Moldflow Australia, PTY. LDT., Boronia, Australia. 10. 'Optimold', software developed by Graftek Inc., Boulder, CO, USA. 11. 'Policool', software developed by the General Electric Company (GEC), Schenectady, NY, USA. 12. 'TMC', software developed by Plastics and Computer Inc., Montclair, NJ, USA. 13. J. Rothe, Ph. D. Thesis, University of Stuttgart, West Germany, 1972. 14. P. Thienel, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1977. 15. L. Schmidt, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1981. 16. M. H. Naitove, Plast. Technol., 1984,30 (April), 74-79. 17. T. Schacht, U. Maier, K. Esser, O. Kretschmar and T. Schmidt, Adv. Polym. Technol., 1985,5,99. 18. G. Menges, U. Lichius and H. Bangert, Plastverarbeiter, 1980, 31, 671. 19. H. Bangert, Kunststoffe, 1985, 75, 325. 20. R. B. Bird, W. E. Steward and E. N. Lightfoot, 'Transport Phenomena', Wiley, New York, 1960. 21. W. Benfer, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1985. 22. G. Menges, T. Schacht and A. Storzer, Plastverarbeiter, 1985, 36, 14. 23. P. Geisbiisch, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1980. 24. T. Schacht, Ph. D. Thesis of the Techical University of Aachen, West Germany, 1986. 25. W. Kemper, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1982. 26. O. Kretzschmar, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1985. 27. G. Wiibken, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1976. 28. T. Schmidt, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1986. 29. P. Geisbiisch, R. Sarholz, E. Winkel, F. Buschhatis and W. Bongardt, 'Requirements on injection molding machines', preprint of the 10th Technical Conference of Plastics Processing of the Institut fiir Kunststoffverarbeitung (IKV), Pontstrasse 49, 5100 Aachen, West Germany, 1980. 30. S. Stitz, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1973. 31. J. Backhaus, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1985. 32. W.-B. Hoven-Nievelstein, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1984. 33. G. Menges, W.-B. Hoven-Nievelstein, T. Schmidt and O. Kretzschmar, 'Handbook of Injection Mold Calculations', Kunststoff Information, Bad Homburg, 1985. 34. C. W. Macosko, Int. J. Heat Mass Transfer, 1980, 23, 1479. 35. H. Schwesig, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1979. 36. E. Schiirmann, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1979! 37. S. S. Kutaletadze and V. U. Borishanskii, 'A Concise Encyclopedia of Heat Transfer', Pergamon Press, Oxford, 1966. 38. N. N. ISIS software report, Infratest Information Services, Munich, 1980. 39~ P. Steinke, Ph. D. Thesis of the Technical University of Aachen, West Germany, 1982.