# Understanding Plastics Engineering Calculations

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1 Natti S. Rao Nick R. Schott Understanding Plastics Engineering Calculations Hands-on Examples and Case Studies Sample Pages from Chapters 4 and 6 ISBNs HANSER Hanser Publishers, Munich Hanser Publications, Cincinnati

2 4 Analytical Procedures for Troubleshooting Extrusion Screws Extrusion is one of the most widely used polymer converting operations for manufacturing blown film, pipes, sheets, and laminations, to list the most significant industrial applications. Fig. 4. shows a modern large scale machine for making blown film. The extruder, which constitutes the central unit of these machines, is shown in Fig The polymer is fed into the hopper in the form of granulate or powder. It is kept at the desired temperature and humidity by controlled air circulation. The solids are conveyed by the rotating screw and slowly melted, in part, by barrel heating but mainly by the frictional heat generated by the shear between the polymer and the barrel (Fig. 4.3). The melt at the desired temperature and pressure flows through the die, in which the shaping of the melt into the desired shape takes place. Figure 4. Large scale blown film line [2]

3 68 4 Analytical Procedures for Troubleshooting Extrusion Screws Figure 4.2 Extruder with auxialiary equipment [3] 4. Three-Zone Screw Basically extrusion consists of transporting the solid polymer in an extruder by means of a rotating screw, melting the solid, homogenizing the melt, and forcing the melt through a die (Fig. 4.3). The extruder screw of a conventional plasticating extruder has three geometrically different zones (Fig. 4.4), whose functions can be described as follows: Feed zone: Transport and preheating of the solid material Transition zone: Compression and plastication of the polymer Metering zone: Melt conveying, melt mixing and pumping of the melt to the die However, the functions of a zone are not limited to that particular zone alone. The processes mentioned can continue to occur in the adjoining zone as well. Although the following equations apply to the 3-zone screws, they can be used segmentwise for designing screws of other geometries as well. Figure 4.3 Plasticating extrusion [4]

4 4. Three-Zone Screw 69 Figure 4.4 Three-zone screw [0] 4.. extruder Output Depending on the type of extruder, the output is determined either by the geometry of the solids feeding zone alone, as in the case of a grooved extruder [7], or by the solids and melt zones to be found in a smooth barrel extruder. A too high or too low output results when the dimensions of the screw and die are not matched with each other Feed Zone A good estimate of the solids flow rate can be obtained from Eq. (4.) as a function of the conveying efficiency and the feed depth. The desired output can be found by simulating the effect of these factors on the flow rate by means of Eq. (4.). Calculated Example The solids transport is largely influenced by the frictional forces between the solid polymer and barrel and screw surfaces. A detailed analysis of the solids conveying mechanism was performed by Darnell and Mol [8]. The following example presents an empirical equation that provides good results in practice []. The geometry of the feed zone of a screw (Fig. 4.5) is given by the following data: Barrel diameter D b = 30 mm Screw lead s = 30 mm Number of flights = Flight width w FLT = 3 mm Channel width W = 28.6 mm Depth of the feed zone H = 5 mm Conveying efficiency F = Screw speed N = 250 rpm Bulk density of the polymer r o = 800 kg/m 3

5 70 4 Analytical Procedures for Troubleshooting Extrusion Screws Figure 4.5 Screw zone of a single screw extruder [5] The solids conveying rate in the feed zone of the extruder can be calculated according to [4] = 60 2 W G ro N hf π H Db ( Db H) + sin cos (4.) W w with the helix angle ( π ) = tan s/ D b (4.2) The conveying efficiency F in Eq. (4.) as defined here is the ratio between the actual extrusion rate and the theoretical maximum extrusion rate attainable under the assumption of no friction between the solid polymer and the screw. It depends on the type of polymer, bulk density, barrel temperature, and the friction between the polymer, barrel and the screw. Experimental values of F for some polymers are given in Table 4.. Table 4. Conveying efficiency F for some polymers Polymer Smooth barrel Grooved barrel LDPE HDPE PP PVC-P PA PET PC PS Using the values above with the dimensions in meters in Eq. (4.) and Eq. (4.2) we get G = π Hence G 50 kg/h FLT

6 6.3 Injection Mold Screw Dimensions Essential dimensions of injection molding screws for processing thermoplastics are given in Table 6.6 for several screw diameters []. Table 6.6 Significant Screw Dimensions for Processing Thermoplastics Diameter (mm) Flight depth (feed) h F (mm) Flight depth (metering) h M (mm) Flight depth ratio : : : : : : 0.25 > 20 max 4 max 5.6 max 3 : 0.25 Radial flight clearance (mm) Figure 6.8 Dimensions of an injection molding screw [] 6.3 Injection Mold The problems encountered in injection molding are often related to mold design whereas in extrusion, the screw design determines the quality of the melt as discussed in Chapter 5. The quantitative description of the important mold filling stage has been made possible by the well-known software MOLDFLOW [0]. The purpose of this section is to present practical calculation procedures that can be handled even by handheld calculators Runner Systems The pressure drop along the gate or runner of an injection mold [5] can be calculated from the same relationships used for dimensioning extrusion dies (Chapter 5).

7 64 6 Analytical Procedures for Troubleshooting Injection Molding Calculated Example For the following conditions, the isothermal pressure drop 0 and the adiabatic pressure drop are to be determined: For polystyrene with the following viscosity constants according to [29] A 0 = A = A 2 = A 3 = A 4 = c = c 2 = 33.2 T 0 = 90 C flow rate m = kg/h melt density r m =.2 g/cm 3 specific heat c pm =.6 kj/(kg K) melt temperature T = 230 C length of the runner L = 0.6 mm radius of the runner R = 5.08 mm Solution: a) Isothermal flow a from 4 Q a = = = s 3 3 π R 3.6 π ( Q = volume flow rate cm 3 /s) a T from c( T T0) ( ) c2+ ( T T0) ( ) at = 0 = 0 = 0 = 0.02 Power law exponent [29] n = h a, viscosity [29] h = 32 Pa s a

8 6.3 Injection Mold 65 shear stress = Pa K: K = Die constant G circle from Table.4 G circle ( ) π = = with Q = m /s from Equation 5.: ( ) ( ) = = 42 bar b) Adiabatic flow The relationship for the ratio D p 0 ln L = 0 L is [7] where L 0 = + r c m pm Temperature rise from Equation (5.20): 42 D T = = = 2.34 K 0 r c For polystyrene m = K L Finally, pm = =.06 ln L 42 ln.06 = 0 = = bar 0.06 L In the adiabatic case, the pressure drop is smaller because the dissipated heat is retained in the melt.

9 66 6 Analytical Procedures for Troubleshooting Injection Molding Mold Filling As already mentioned, the mold filling process is treated extensively in commercial simulation programs and by Bangert [3]. In the following sections the more transparent method of Stevenson is given with an example. To determine the size of an injection molding machine in order to produce a given part, knowledge of the clamping force exerted by the mold is important, as this force should not exceed the clamping force of the machine. Injection Pressure The isothermal pressure drop for a disc-shaped cavity is given as [4] ( n ) nr K 360 Q 2 r R r2 = ( nr) N Θ 4 π n b R r2 b + (6.0) The fill time is defined as [4] V a = 2 Q b (6.) The Brinkman number is given by [4] Br = b Kr Q ( TM TW) N Θ π b r2 + nr (6.2) Calculated Example with Symbols and Units Given data: The part has the shape of a round disc. The material is ABS with n R = , which is the reciprocal of the power law exponent n. The constant K r, which corresponds to the viscosity h p, is K r = Constant injection rate Q = 60 cm 3 /s Part volume V = 60 cm 3 Half thickness of the disc b = 2. mm Radius of the disc r 2 = 20 mm Number of gates N =

10 6.3 Injection Mold 67 Inlet melt temperature Mold temperature T M = 58 K T W = 323 K Thermal conductivity of the melt = 0.74 W/(m K) Thermal diffusivity of the polymer a = cm 2 /s Melt flow angle [4] Θ = 360 The isothermal pressure drop in the mold is to be determined. Solution: Applying Eq. (6.0) for ( ) = = 254 bar 0 ( ) π Dimensionless fill time from Eq. (6.): = = Brinkman number from Eq. (6.2): Br = = π From the experimental results of Stevenson [4] the following empirical relation was developed to calculate the actual pressure drop in the mold ln = Br 2.6 Br (6.3) The actual pressure drop is therefore from Eq. (6.3) =.574 = = 400 bar Clamping Force The calculation of clamping force is similar to that of the injection pressure. First, the isothermal clamping force is determined from [4] 2 n R F ( r ) = 0 π r nr (6.4) where F (r 2 ) = isothermal clamping force (N).

11 68 6 Analytical Procedures for Troubleshooting Injection Molding F (r 2 ) for the example above is with Eq. (6.4) 2 ( ) π F r = = kn The actual clamping force can be obtained from the following empirical relationship, which was developed from the results published in [4]. F ln = Br Br F (6.5) Hence the actual clamping force F from Eq. (6.5) F = = kn The above relationships are valid for a disc-shaped cavity. Other geometries of the mold cavity can be taken into account on this basis in the manner described by Stevenson [4]. 6.4 Flow Characteristics of Injection Molding Resins One of the criteria for resin selection to make a given part is whether the melt is an easy flowing type or whether it exhibits significantly viscous behavior. To determine the flowability of the polymer melt, the spiral test, which consists of injecting the melt into a spiral shaped mold shown in Fig. 6.9, is used. The length of the spiral serves as a measure of the ease of flow of the melt in the mold, and enables mold and part design suited to material flow. The parameters involved in the flow process are resin viscosity, melt temperature, mold wall temperature, axial screw speed, injection pressure, and geometry of the mold. To minimize the number of experiments required to determine the flow length, a semi-empirical model based on dimensional analysis is given in this section. The modified dimensionless numbers used in this model taking non-newtonian melt flow into account are the Graetz number, Reynolds number, Prandtl number, Brinkman number, and Euler number. Comparison between experimental data obtained with different thermoplastic resins and the model predictions showed good agreement, confirming the applicability of the approach to any injection molding resin [28]. The experimental flow curves obtained at constant injection pressure under given melt temperature, mold temperature, and axial screw speed are given schematically in Fig. 6.0 for a resin type at various spiral depths with melt flow rate of the polymer brand as a parameter. By comparing the flow lengths with one another at any spiral depth also called wall thickness, the flowability of the resin brand in question with reference to another brand can be inferred [8, 8].

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