SIMULATION OF THE HEATING STEP WITHIN THE THERMOFORMING PROCESS USING THE FINITE DIFFERENCE METHOD

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1 SIMULATION OF THE HEATING STEP WITHIN THE THERMOFORMING PROCESS USING THE FINITE DIFFERENCE METHOD A. Fertschej 1 *, G.R. Langecker 1 University of Leoben artur.fertschej@u-leoben.at; Franz-Josef Strasse 18, 87 Leoben, Austria Langecker@a1.net The heating of the sheet up to the ideal teperature is the first step within the theroforing process. It has a significant influence on the therofored part. The teperature profile of the heating station affects the reuested wall thickness distribution of the sheet generated while theroforing. The reuired teperatures of the heating units can be predeterined. Using the ethod of finite differences (FDM) a siulation progra was developed to calculate the heating step within the theroforing process. The progra can be upgraded or odified and iplies aong others the possibilities to calculate the heating of one-layer or ulti-layer sheets in a single- or ulti-station achine and to optiize the teperature distribution of the heating units. Using the progra the influence of various paraeters (aterial, heating unit and calculation paraeters) on the teperature distribution of the sheet was analysed. The results were copared with a finite eleent ethod calculation. Introduction Alongside extrusion and injection olding theroforing is one of the ost iportant ethods of processing polyers. Heating the sei-finished product to the optiu foring teperature is the first step of the theroforing process and has a deciding influence on the final result. After heating the sei-finished product should be in an elastoeric state which, depending on the aterial properties, lies within a ore or less narrow teperature window. The teperature profile of the radiative heaters can be used to influence the desired wall thickness distribution after theroforing and the necessary teperature for the individual heating eleents can be deterined through calculations in advance. Infrared radiation is preferred for the heating process. The heating eleents used are ceraic, uartz and halogen radiators which differ in respect of their service teperature, control and the spectru of the eitted radiation (long, ediu and short wave lengths). Heating with infrared radiation is universally applicable for all polyers. The spectral absorption coefficient and the eission coefficient are aterial specific paraeters. Using infrared heating the theral conductivity plays a subordinate role in the heating process in coparison to other heating ethods (convection and conduction) since the radiative energy penetrates into the aterial. Shorter heating ties are possible due to infrared radiation s higher heat flux density. Daage to the sei-finished product is kept within liits when it is heated uickly with infrared radiation. In order to prevent aterial daage a axiu teperature ust not be exceeded at any point during the heating of the sei-finished product. With double sided heating of the sheet using infrared radiation the lowest teperature occurs in the core. This should however be high enough to avoid crack foration through a cold stretching process [1,]. Experiental Basics of the Calculation of the Teperature Distribution of the Sheet The cross section of the sheet is divided into differential layers parallel to the surface having the width dx. At the interfaces the following heat fluxes occur (Figure 1): radiation of the radiative heater radiation exchange between the layers abient radiation heat conduction between the layers abient convection at the surface of the sheet. Because of the large difference of the teperature between the radiative heater and the sheet the radiation exchange between the layers and the abient radiation of the sheet can be neglected. The Polyer Processing Society 3rd Annual Meeting

2 To calculate the teperature field the euation of the one-diensional transient heat conduction including a position dependent heat source has to be used: T T ρ c p = λ + Φ(x, t) ( 1 ). t x x The heat source Φ(x,t) considers the radiation flux that is absorbed in a voluetric eleent da dx in the differential layers. It results fro the derivation of the Labert-Bouguer-law: Φ(x, t) = ( λ) k( λ) x [ 1 ρ( λ) ] k( λ) e dλ ( ). The decrease of the intensity of the radiation can be described with the help of the spectral absorption coefficient k(λ). The reciprocal value of the spectral absorption coefficient characterizes the distance at that the intensity of the radiation declines to 36.8% of the original incoing radiation. It is called penetration depth. The entire density of the radiation is calculated fro the Stefan-Boltzann-law: T heater = c1 ( 3 ). Tsheet 1 For alone-standing surfaces (as in theroforing) the radiation exchange nuber c 1 can be obtained fro the following euation: c 1 ε1 ε cs ϕ1 = 1 (1 ε ) (1 ε ) ϕ 1 1 A A 1 ( 4 ). To calculate the view factor ϕ 1 for two parallel rectangular surfaces the following euation can be used [5]: ϕ 1 = π x + y ln y (1 + x (1 + x ) (1 + y 1+ x + y ) arctan y 1+ x ) + x (1 + y ) arctan 1+ y x arctan(x) y arctan(y) x + ( 5 ), with x = a/h and y = b/h, whereas a and b represent the length and the width of the surfaces and h is the distance of the surfaces. To deterine the various view factors for each sheet eleent the following procedure is ipleented: Both the surfaces of the heating unit and the sheet are divided into two parts which are uadratic and of the sae size (Figure, left). Then the view factor fro heating eleent 1 for sheet eleent can be calculated. The solution of the view factor for the entire surfaces and for the single surfaces fro heating eleent 1 to sheet eleent 1 is known. It is deeed that the incoing energy fro the entire surface is eual to the su of the incoing energy fro the sub surfaces: A heater = A heater1ϕ1 1 + A heater1ϕ1 + A heaterϕ + A heaterϕ 1 ϕ ( 6 ). With respect to the syetry ϕ 1 1 = ϕ and ϕ 1 = ϕ 1 as well as the correlation for the surfaces A heater1+ = A heater1 + A heater and A heater1 = A heater the resulting view factor is obtained: ϕ = ϕ ϕ ( 7 ) The Polyer Processing Society 3rd Annual Meeting

3 The view factor calculation for a diagonally opposite eleent is analogue as described before (Figure, right). Using the already entioned conditions the view factor becoes in this case: ϕ = ϕ ϕ ϕ ( 8 ) Applying this way of solution conseuently each abitrary view factor for two parallel rectangular surfaces can be achieved [3]. To solve euation (1) and () one initial condition and two boundary conditions are reuired. The teperatures in each layer are predefined as initial condition: T (x, t ) = = ( 9 ). T The boundary conditions result fro a theral balance at the surfaces. The convective heat transfer becoes [1]: x T α o (Tx= TU ) = λ ( 1 ), x x= T α u (Tx= s TU ) = λ ( 11 ). x The differential euations for the calculation of the teperature gradient across the thickness of the sheet were solved using the finite difference ethod. The differentials of the differential euations are replaced within this discretisation by differences. The transfored euation (1) thus becoes T n,i 1 = T n,i t λ T + ρ c p n+ 1,i T x n,i + T x= s n 1,i t + ρ c p Φ + ( 1 ), whereas n is the index for the layer thickness and i is the index for the tie. Siplifying constant aterial paraeters (that eans independent of the wave length) are assued. Also the lateral heat conduction is neglected, because plastics poorly conduct heat and the lateral diensions are large in relation to the thickness. The Coputation Progra The core of the heating calculation progra was written in C++ in order to achieve higher coputational speeds. The input interface for both data entry and analysis of the results was written in the Microsoft Excel executable prograing language Visual Basic. The coputation progra can be extended and odified without significant effort. It coprises: the calculation of a static or continuous heating process (single- or ultiple-station achine) of the sheet, the iterative calculation of the reuired heating eleent teperatures for a desired aterial teperature distribution, the autoatic analysis of the results (Figure 3). Paraetric Studies In order to deterine both the ualitative and uantitative influence of the individual paraeters a nuber of paraetric studies were conducted. The aterial properties (ABS) and relevant coputation paraeters are suarized in Table 1. The teperatures of the heating eleents were selected in such a way as to achieve a constant teperature of 17 C across the surface of the sheet (Figure 4). As the sheet is syetric only the teperature profiles fro the surface to the iddle of the sheet are shown in the presentation of the results. n The Polyer Processing Society 3rd Annual Meeting

4 Results and Discussion The size of the tie intervals has a negligible influence on the end results, however an unduly sall tie interval increases the coputation tie considerably. The nuber of layers, that is the size of the layer thickness step, has a very considerable influence on the end result and the teperatures can vary by several degrees. The nuber of layers should therefore be chosen to be neither too sall nor too large. The nuber of layers is linked to the tie interval by a convergence criterion. As the nuber of layers increases the tie interval ust fall. The coputational tie therefore increases exponentially with the nuber of layers. A higher specific heat capacity reduces the teperature level of the sheet, since ore energy is reuired to heat up the polyer. The teperature profile however changes only arginally across the thickness of the sheet (Figure 5). Analogue results to those for specific heat capacity were calculated for variations in density. The consideration of the penetration depth covered the range fro.1 to (Figure 6). With very sall penetration depths the radiation heats only the outerost layers, with larger penetration depths the layers in the iddle are also reached. When the penetration depth is nearly zero the heating of the sheet is de facto only through conduction. The absorbed energy and rate of heating increase as the eission coefficient rises. Where the eission coefficient is zero the teperature seen at the start reains unchanged, whilst the eission coefficient is one the axiu energy transfer occurs. All the previous values raise or lower the teperature of the sheet but barely change the gradient of the teperature profile across the sheet thickness. In contrast to this theral conductivity influences the unifority of the teperature across the sheet thickness. With higher theral conductivity the teperature differential between the centre and the surface of the sheet is reduced (Figure 7). With increasing wall thickness the teperature differential between the centre and the surface of the sheet also increases (Figure 8). The distance between the radiative heaters and the polyer sheet can be altered in order to control the heating process. Whilst a larger distance results in a lower teperature level the heat energy can distribute itself ore uniforly into the interior of the sheet. Figure 9 shows this relationship for two different wall thicknesses of the sheets. Larger distances lead to a gentler heating cycle. The heating tie however increases due to the reduced energy transfer. The energy losses for the heating of the sheet increase. The heat transfer coefficient characterizes the effects of cooling air flows over the sheet surface. A range of industrially relevant values were calculated, starting with a heat transfer coefficient of 1 W/(²K) for still air going up to 1 W/(²K) for blown air flow with a correspondingly higher cooling effect. Whilst at low values for the heat transfer coefficient the polyer can heat up with a negligible interference, at higher flow rates the air reoves substantial aounts of energy before this can spread into the sheet core (Figure 1). The heating eleent teperatures are the easiest process paraeters to alter. Current technology offers individual teperature control of the heating plates. In contrast to previous paraetric studies an optiized teperature distribution was not assued but rather a constant teperature for all of the heating eleents. Only the teperature of the heating eleents was altered in order to deterine the effect on the teperature of the sheet and the teperature profile across the thickness of the sheet. Figure 11 shows for two different wall thicknesses of the sheet that the teperature of the sheet rises sharply with increasing heating eleent teperature but that the hoogeneity of the teperature profile decreases. Whilst in principle the coputationally deterined results of the evaluations conducted are only valid for the ABS aterial used, their agnitude and the ualitative insights are generally applicable [4]. Coparison to FEM To ensure the validity of the finite difference ethod another calculation using FEM software ABAQUS, version 6.4-1, was perfored and the results were copared. For this an analogue odel for the FEM siulation was created. Both results for FEM and FDM are very siilar. Figure 1 displays the teperature distribution of the surface of the sheet calculated with FDM respectively FEM. If the tie effort is considered the advantages of the FDM progra is clearly recognizable. The creation of the FEM odel and the calculation can last hours whilst the FDM progra provides the odel and results within inutes. The Polyer Processing Society 3rd Annual Meeting

5 Table 1 Coputational paraeters for the paraetric study (aterial data: ABS). Paraeter Value Sheet thickness 1. Distance heater-sheet 1 Heating tie 1 s Diensions (heater and sheet) 6 x 6 ² Nuber of eleents 6 x 6 Density 1.5 g/c³ Specific heat capacity 15 J/(kg K) Theral conductivity.18 W/( K) Eission coefficient.95 Penetration depth 1 Heat transfer coefficient 1 W/(² K) Abient teperature 3 C Figure 1 Occuring heat fluxes at the interfaces Figure Partition of the surfaces to calculate the view factor The Polyer Processing Society 3rd Annual Meeting

6 Figure 3 Autoatically created sheet with the results and diagras Figure 4 Teperatures of the heating plates in C to obtain a unifor sheet teperature of 17 C c p [J/kgK] Surface Thickness position [] Middle Figure 5 Teperature profiles across the sheet thickness for various specific heat capacities The Polyer Processing Society 3rd Annual Meeting

7 Penetration Depth [] Surface Thickness position [] Middle Figure 6 Teperature profiles across the sheet thickness for various penetration depths Sheet surface Middle of the sheet Heat conductivity [W/ K] Figure 7 Influence of the theral conductivity on the teperature of the surface and the iddle of the sheet The Polyer Processing Society 3rd Annual Meeting

8 15 1 Sheet surface Middle of the sheet Wall thickness [] Figure 8 Influence of the wall thickness on the teperature of the surface and the iddle of the sheet s=1., sheet surface s=1., iddle of the sheet s=4, sheet surface s=4, iddle of the sheet Distance heater - sheet [] Figure 9 Influence of the heating eleent distance on the teperature of the surface and the iddle of the sheet The Polyer Processing Society 3rd Annual Meeting

9 α [W/²K] Surface Thickness position [] Middle Figure 1 Teperature profiles across the sheet thickness for various heat transfer coefficients 7 s=1., sheet surface s=1., iddle of the sheet s=4, sheet surface s=4, iddle of the sheet Teperature of the heating eleents [ C] Figure 11 Influence of the heating eleent teperatures on the teperature of the surface and the iddle of the sheet The Polyer Processing Society 3rd Annual Meeting

10 Teperature [ C] Teperature [ C] Figure 1 Teperature distribution of the sheet surface calculated with FEM (left) and FDM (right) Conclusion Using the finite difference ethod a siulation progra for the heating step of theroforing was developed with which the user can predictively deterine the heating process and the ideal teperature distribution of the heating eleents in order to achieve the desired teperature distribution in the sheet. A paraetric study showed the various influences of different aterial and process variables. References 1. M. Küppers, PhD Thesis, Technische Hochschule Aachen, D. Weinand, PhD Thesis, Technische Hochschule Aachen, F.M. Duarte; J.A. Covas, Plastics, Rubber and Coposites Processing and Applications, IOM, London, 1997; Vol. 6 (5), J. Kertz, Diploa Thesis, Montanuniversität Leoben, VDI-Wäreatlas, Abschnitt Kb, Springer Verlag, Düsseldorf, The Polyer Processing Society 3rd Annual Meeting