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Dynamic Modelling of Die Melt Temperature Profile inPolymer Extrusion: Effects of Process Settings, ScrewGeometry and MaterialDOI:10.1016/j.apm.2013.08.004

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Citation for published version (APA):Abeykoon, C., Martin, P. J., Li, K., & Kelley, A. L. (2014). Dynamic Modelling of Die Melt Temperature Profile inPolymer Extrusion: Effects of Process Settings, Screw Geometry and Material. Applied Mathematical Modelling,38(4), 1224-1236. https://doi.org/10.1016/j.apm.2013.08.004

Published in:Applied Mathematical Modelling

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Applied Mathematical Modelling 38 (2014) 1224–1236

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Dynamic modelling of die melt temperature profile in polymerextrusion: Effects of process settings, screw geometry andmaterial

0307-904X/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.08.004

⇑ Corresponding author at: School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast BT9 5AH, UK. Tel.: +44 28909E-mail addresses: yabeykoon01@qub.ac.uk, Y.Abeykoon@bradford.ac.uk (C. Abeykoon).

Chamil Abeykoon a,⇑, Peter J. Martin a, Kang Li b, Adrian L. Kelly c

a School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Belfast BT9 5AH, UKb School of Electronics, Electrical Engineering and Computer Science, Queen’s University Belfast, Belfast BT9 5AH, UKc IRC in Polymer Science and Technology, School of Engineering, Design and Technology, University of Bradford, Bradford BD7 1DP, UK

a r t i c l e i n f o a b s t r a c t

Article history:Received 29 November 2011Received in revised form 13 June 2013Accepted 5 August 2013Available online 20 August 2013

Keywords:Polymer extrusionProcess monitoringMelt temperature profileDynamic modellingThermal homogeneityProcess settings

Extrusion is one of the major methods for processing polymeric materials and the thermalhomogeneity of the process output is a major concern for manufacture of high qualityextruded products. Therefore, accurate process thermal monitoring and control are impor-tant for product quality control. However, most industrial extruders use single point ther-mocouples for the temperature monitoring/control although their measurements arehighly affected by the barrel metal wall temperature. Currently, no industrially establishedthermal profile measurement technique is available. Furthermore, it has been shown thatthe melt temperature changes considerably with the die radial position and hence point/bulk measurements are not sufficient for monitoring and control of the temperature acrossthe melt flow. The majority of process thermal control methods are based on linear modelswhich are not capable of dealing with process nonlinearities. In this work, the die melttemperature profile of a single screw extruder was monitored by a thermocouple meshtechnique. The data obtained was used to develop a novel approach of modelling the extru-der die melt temperature profile under dynamic conditions (i.e. for predicting the die melttemperature profile in real-time). These newly proposed models were in good agreementwith the measured unseen data. They were then used to explore the effects of process set-tings, material and screw geometry on the die melt temperature profile. The resultsshowed that the process thermal homogeneity was affected in a complex manner bychanging the process settings, screw geometry and material.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Extrusion is a method of processing polymeric materials and is used in the final production of many polymer-based prod-ucts such as pipes, films, sheets, tubes, rods, etc. Presently, it is one of the most important production methods due to theincreasing popularity of polymeric materials (i.e. due to their ability of saving energy over convectional raw materials such asmetal, glass, etc). As a result, a significant amount of research and development has been directed to improving the operationof polymer processes. An extruder is a machine which processes material by melting and conveying it along a screw andforcing it through a die. Currently, various types of polymer processing extruders are used in industry: single/multi screwextruders, disk/drum extruders, etc. Of these extruders, single screw continuous extruders are the most commonly used

74236.

Hopper

Screw

Heaters

Barrel Gear box

Breakerplate/Screen

AdapterSensors (Pressure/Temperature)

Die

Motor

12

3

1

2

3

Solids conveying

Melting

Melt conveying

Fig. 1. Basic components of a single screw extruder (barrel sectioned for clarity).

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1225

due to their low purchase and maintenance costs, simplicity of operation and the reliability [1,2]. Moreover, single screwextruders are commercially available in many different sizes [3]. Basic components of a single screw extruder are shownin Fig. 1. In addition to these components, various other equipment (e.g. gear pumps, screws with mixers units, sensorsfor process monitoring and control) are also used in modern extrusion lines. More details on the basic process mechanisms,process operation and process-related components of polymer extrusion can be found in the literature [4–6].

1.1. Process thermal monitoring

Despite significant developments in polymer extrusion over the last few decades, accurate process thermal monitoringand control with different materials and operating conditions still remains an issue. Process operators have to face chal-lenges in achieving the required thermal quality of the melt at high throughput. Achieving good thermal stability is a majorrequirement of the extrusion process in order to form high quality products. Even small variations in melt temperature cancause poor product quality [7]. Therefore, continuous monitoring of process thermal stability is an essential requirement foradvanced process control to achieve a high quality melt output.

As revealed in previous studies [6,8–14], the thermal homogeneity of the melt is considerably affected by the process set-tings and melt flow temperature is different at different radial locations of the die. Therefore, study of the entire melt tem-perature profile as a measure of process thermal stability is more appropriate than a single point or a bulk measurement toensure high quality products. Unfortunately, it is difficult to monitor a die melt temperature profile within a productionenvironment and most extruders are instrumented only with conventional wall mounted thermocouples. These are highlyaffected by the barrel wall temperature and they are not capable of measuring a melt temperature profile or detecting rapidvariations in the melt temperature [5,15]. As alternatives to the point/bulk measurements, some thermal profile measure-ment methods (e.g. a thermocouple mesh [13], a fluorescence technique [16]) have been proposed, but these are not yet ro-bust enough to use in a production environment due to constraints such as their complexity, limited durability, accessrequirements, disruptive effects to the melt flow and output, etc. However, some of these techniques have been used to gath-er valuable process thermal information in a research setting.

1.2. Melt temperature modelling

As reported in the literature, only a minor amount of work has been carried out so far on the modelling of an extruder’smelt temperature profile under static or dynamic conditions. Instead, the development of static or dynamic models has beenreported [17–27] to predict a single point or bulk melt temperature. Essentially, some of these modelling attempts are basedon first principles while others are data driven models based on point or bulk melt temperature measurements taken fromthermocouples fixed at the end of the extruder barrel or the die. Accuracy of the first principle models is constrained due tonumerous simplifying assumptions and also they are computationally too expensive to implement in real-time which limitstheir practical applications [28]. While some of the other point/bulk thermal models may be used in practice, these provideno detailed information on the actual process thermal stability. Furthermore, most of these existing thermal models are lin-ear models and hence these may not be sufficient to represent the nonlinear processing behaviour.

Additionally, several computer simulation software packages are presently available to model the polymer extrusion pro-cess. Usually, finite element methods are used by most of these simulation packages to obtain solutions of relevant differentialequations. Kelly et al. [29] carried out a 2-D computational fluid dynamics (CFD) simulation (i.e. based on the Tadmor meltingmodel [30]) to predict the die melt temperature profile with relevant boundary conditions by using the CompuplastTM Flow2000 software. An experimentally measured die melt temperature profile obtained by using a thermocouple mesh and thepredicted 2-D CFD profile gave good agreement at lower screw speeds. At higher screw speeds, the software was able to pre-dict the melt temperature at the centre of the die with good accuracy, but the differences between the predicted and measured

1226 C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236

melt temperatures closer to the die wall increased as the screw speed increased. These differences are thought to be due to thedifficulty in determining the correct boundary conditions to apply to the CFD model. Conversely, Vlachopoulos [31] arguedthat these simulation software packages were poor at representing real processing behaviour due to their inherent shortcom-ings such as the inability of representing shear thinning behaviour of polymer flows, the difficulty of representing contact be-tween the polymer melt and metal wall, inabilities of predicting phenomena such as sharkskin, die lip build-up, melt fractureand die resonance. Most of these shortcomings remain unchanged with the currently available simulation software packagesdue to the difficulties/challenges of modelling such processing phenomena accurately.

Previous studies by the author [10,32] developed a static model to predict the die melt temperature profile in single screwextrusion. Melt temperatures at different radial locations of the die were predicted from readily measurable process vari-ables (i.e. screw speed and barrel set temperatures) and good agreement was achieved between the experimental and modelpredicted die melt temperature profiles. According to the authors’ knowledge, there has not been any other melt tempera-ture profile modelling work reported in the literature. Therefore, further studies on die melt temperature profile may assistin the development of polymer extrusion process monitoring and control.

In this work, a thermocouple mesh technique [13] was used to measure melt temperature profiles across the die meltflow. The data obtained were used to develop nonlinear dynamic models to predict the die melt temperature profile overdifferent process operating conditions, screw geometries and materials. The study reported in this paper was focused onthe melt temperature of a single screw extruder as it is the most common type used in the current polymer processing indus-try. Two polymers and two different screw geometries were used in the experiments.

2. Equipment & procedure

All measurements were carried out on a 63.5 mm diameter (D) single screw extruder (Davis Standard BC-60). A barrierflighted (BF) screw with a spiral Maddock mixer (a general purpose screw with a 2.5:1 compression ratio) and a single flight-ed tapered gradual compression (GC) screw (with 3:1 compression ratio) were used to process the materials. More details ofthe screws are shown in Fig. 2 with the screw channel depths (SCD) of the solids conveying and metering zones.

The extruder was fitted with a 38 mm diameter adaptor (with a heater of 1.4 kW) by using a clamp ring (with a heater of0.9 kW) prior to a short rod die (with a heater of 0.2 kW) with a 6 mm diameter bore. A schematic showing the of thearrangement of these components is shown in Fig. 3 together with their dimensions.

4D 10D

Solids conveying

(SCD = 10.53mm)

Melting Metering

10D

(SCD = 3.46mm)

5D 6D 13D

(SCD = 12.19mm) Melting

Metering (SCD = 4.90mm)

Maddock mixer

(a)

(b)

Solids conveying

Extruder barrel

Extruder barrel centreline

Extruder barrel

Extruder barrel centreline

(Barrier flighted)

Fig. 2. Details of the screws (a) The gradual compression screw (b) The barrier flighted screw with a Maddock mixer.

638

174 57

Extruder barrel Adapter

Die

All dimensions are in millimetres Thermocouple mesh

63.5

Clamp ring 20

Fig. 3. Extruder barrel, die, adapter and thermocouple mesh.

1460 All dimensions are in millimetres

Z 1 Z 2 Z 3 Z 4

8030 30 30

.5

90145

Zone 1 Zone 2 Zone 3 Zone 4

635Metering Zone

254Feeding Zone

635Melting Zone

63

4 x heaters, 300 mm length, 3000W/240VWater cooled feed throatMetering ZoneFeeding Zone Melting Zone

Fig. 4. BC-60 extruder barrel and heater arrangement.

Table 1Dimensions of the TCM junctions.

TCM Distance to the each mesh junction (J) from die centre (mm)

1 0 (J1), 3 (J2), �4.5 (J3), 8.8 (J4), �11 (J5), 14.7 (J6), �16.5 (J7)2 0 (J1), 2.4 (J2), �5.4 (J3), 7.6 (J4), �11.2 (J5), �17 (J6)

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1227

The extruder barrel has four separate temperature zones (each with a heater of 4 kW) equipped with dual-therm temper-ature controllers. Each of the clamp ring, adapter and die is also equipped with a separate temperature controller allowingtheir set temperature to be individually controlled. The extruder barrel dimensions and the arrangement of heaters areshown in Fig. 4.

The melt temperature at different radial locations of the melt flow at the end of the adapter (denoted as die melt tem-perature throughout this paper) was measured using a thermocouple mesh placed in-between the adapter and the die asshown in Figs. 3. As was previously confirmed by Kelly et al. [12], the die melt temperature measurements are symmetricalacross the thermocouple mesh centreline when averaged over a significantly long period of time. Therefore, seven (i.e. forTCM 1) and six (i.e. for TCM 2) thermocouple wires were placed asymmetrically across the die melt flow to generate junc-tions along the diameter of the mesh and this asymmetric placement of wires gave the opportunity to increase the number ofeffective temperature measurements across the melt flow by mirroring them over the centreline to obtain the complete diemelt temperature profile. Details of the thermocouple meshes used are shown in Table 1.

The die wall set temperature was used as the melt temperatures at the ±19 mm radial positions. Then, the final temper-ature profile was obtained by 15 and 13 radial positions across the melt flow for tests with TCM 1 and TCM 2, respectively.The arrangement of the TCM 1 is illustrated in Fig. 5.

A data acquisition program developed in LabVIEW was used to communicate between the experimental instruments anda personal computer (PC). Screw speed and all temperature signals were acquired at 10 Hz using a 16-bit DAQ card (NationalInstruments (NI) PCI-6035E) through a thermocouple connector box (NI TC-2095) and a low-noise signal conditioning box(NI SCXI-1000).

Tm3-Tm16.5

-Tm11 Tm0

Twall:Tm19

-Tm4.5

38m

m

Negative wire

Positivewires

Solid lines – Real mesh junctions Dashed lines – Mirror images of real mesh junctions

Mesh junctions

Tm8.8

Tm14.7

Fig. 5. The thermocouple mesh arrangement.

1228 C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236

2.1. Materials and experimental conditions

Experimental trials were carried out on a virgin high density polyethylene (HDPE), (ExxonMobil HYA 800), (density:0.961 g/cm3, melt flow index (MFI): 0.7 g/10 min @ (190 �C,2.16 kg)) and a regrind polypropylene (PP), (density: 0.850 g/cm3, MFI: 5.71 g/10 min @ (230 �C,2.16 kg)). The extruder barrel temperature settings were fixed as described in Table 2 un-der three different set conditions for each material and denoted as A, D (high temperature); B, E (medium temperature); andC, F (low temperature). These settings were selected in order to generate realistic processing conditions (i.e. avoiding con-veying and melting problems) whilst covering the full operating range of the extruder (i.e. 0–100 rpm). This therefore al-lowed investigation of melting performance at low throughputs where melting is dominated by conduction from thebarrel and screw, and intermediate and high throughputs where melting is primarily achieved by viscous shearing.

In this experimental programme, tests 1 and 2 were carried out using the same BF screw (i.e. by using TCM 1), but withdifferent barrel set temperatures and materials. These tests were performed to allow comparison of the effects of differentmaterials and process settings on the process melting conditions. It was not possible to perform these tests at the same settemperature due to the differing melting behaviours of HDPE and PP materials. Moreover, a separate test (test 3) was carriedout using the same HDPE material and process settings as test 1 (i.e. with TCM 2), but with a different screw geometry. Thisallowed exploration of the effects of different screw geometries on process melting behaviour.

Each experiment was started with the temperature setting A or D (i.e. condition A for tests 1 and 3, condition D for test 2)and the data was recorded with the screw stationary for 1 minute. Then, the screw speed was increased from 0 to 90 rpmwith steps of between ±5 and 40 rpm and in different barrel set temperatures with the extruder running for 193 minutesin tests 1 and 2, and 151 minute in test 3 continuously. The extruder was allowed to stabilise for 15 minute after each settemperature change while it was held for about 7 minute at each other different condition. These time periods were chosenby considering the time requirement for the process to become steady after applying the screw speed and barrel set tem-perature step changes. The magnitudes of the applied process changes were also determined by ensuring that the processcan run under normal conditions after applying these changes. Additionally, the rate of material consumption of the extruderat each speed was also considered in deciding the experimental time periods to minimise the waste of material. The processsetting matrices for tests 1 and 3 are shown in Figs. 6 and 7, respectively.

Table 2Extruder barrel temperature settings.

Test No-Material-Screw TCM Temperature settings Set temperatures (�C)

Barrel zones Clamp ring Adapter Die

1 2 3 4

1-HDPE-BF 1 A 110 130 180 230 230 230 230B 105 125 175 215 215 215 215C 100 120 170 200 200 200 200

2-PP-BF (Regrind PP) 1 D 150 190 215 240 240 240 240E 145 180 200 220 220 220 220F 140 170 185 200 200 200 200

3-HDPE-GC 2 A 110 130 180 230 230 230 230B 105 125 175 215 215 215 215C 100 120 170 200 200 200 200

Time (min)

Rp (

mm

), SS

(rpm

), Se

t tem

pera

ture

s (o C

)

-190

19

50

100

150

200

240T4

T3

T1

T2

A B C

Screw speed (SS)

Radial position (Rp)

Training Validation

65 129 193 0

Fig. 6. Model input matrices of test 1.

0 44 65 87 108 130 151-19

019

50

100

150

200

240

Time (min)

Rp (

mm

), SS

(rpm

), Se

t tem

pera

ture

s (o C

)

T4

T3

T1

T2

Screw speed (SS)

Radial position (Rp)

Training Validation

Fig. 7. Model input matrices of test 3.

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1229

Separate tests were carried out for model training and validation (i.e. two experimental trials under each test).

3. Effects of process settings on die melt temperature profile

As mentioned above, melt temperature profiles of the extruder output melt flow at the end of the 38 mm diameter adap-ter were observed and profiles over some processing conditions (i.e. average melt temperature at each mesh junction of thelast 2 min at each screw speed) of tests 1 and 2 are shown in Fig. 8. The data collected over the last two minutes of the dif-

-19 -15 -10 -5 0 5 10 15 19210

220

230

240

-19 -15 -10 -5 0 5 10 15 19195

200

210

220

230

-19 -15 -10 -5 0 5 10 15 19220

230

240

250

255

-20 -15 -10 -5 0 5 10 15 20230

235

240

245

250

255

260

-20 -15 -10 -5 0 5 10 15 20215

220

225

230

235

240

245

-20 -15 -10 -5 0 5 10 15 20195

200

205

210

215

220

10rpm 30rpm 50rpm 70rpm 90rpm

Radial position (mm)

Ave

rage

die

mel

t tem

pera

ture

(o C)

Ave

rage

die

mel

t tem

pera

ture

(o C)

Ave

rage

die

mel

t tem

pera

ture

(o C)

Radial position (mm)

1-HDPE-BF-A

1-HDPE-BF-B

1-HDPE-BF-C

2-PP-BF-D

2-PP-BF-E

2-PP-BF-F

Fig. 8. Average die melt temperature profiles over the last 2 min at different processing conditions from 10–90 rpm.

1230 C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236

ferent experimental conditions were used to create these plots as the process signals included transients during the first fewminutes, followed by the applied step changes to the process variables. Melt temperature profiles only at 10, 30, 50, 70 and90 rpm screw speeds are shown in Fig. 8. The figure legends are in the format of: test number-material-screw-set temper-ature condition.

These temperature profiles show the effects of process settings on the temperature profile of the extruder melt outputand it is evident that the process thermal homogeneity significantly varied with process settings. At some processing con-ditions (e.g. 1-HDPE-BF-A-70,1-HDPE-BF-A-90,2-PP-BF-D-50,2-PP-BF-D-70 and 2-PP-BF-D-90), the melt temperature atsome of the radial positions decreased below the adapter wall set temperature which caused poor melt temperature homo-geneity across the melt flow. Ideally, these should be flat profiles under all process conditions to ensure optimum melthomogeneity. However, as can be seen from Fig. 8, melt temperature varied across the melt flow, with highest temperaturemeasured in the centre of the flow. Temperature close the die wall reflected the set temperature of the metal whereas melttemperature in the intermediate positions between 5 and 15 mm from the centre of flow reflected the state of the bulk of thepolymer melt exiting the extruder screw, which was seen to vary significantly with set temperature and screw rotationspeed. Conventionally, increasing extruder screw rotation speed is assumed to increase bulk melt temperature of the poly-mer. However, for the melt temperature profiles shown in Fig. 8, only the profiles with the lowest barrel set temperatureconditions for both materials (i.e. condition C for HDPE and condition F for PP) show agreement with this fact. At higherset temperature conditions, the highest average temperatures across the melt flow occurred at 30 rpm for HDPE and at10 rpm for PP, i.e. at low screw rotation speeds. These temperature profiles reflect the complex nature of polymer heatingand melting in extrusion, being influenced by both conduction from the extruder barrel and screw and from viscous shearheating from polymer conveyed along the screw channel. At low screw rotations speeds, the polymer experiences a long res-idence time within the extruder, allowing into be heated predominantly by conduction. At higher screw rotation speeds, thepolymer experiences a much shorter residence time and hence heating/melting relies more on viscous shear, which in turn ishighly dependent upon rheological and thermal properties of the polymer itself. Here, these temperature profiles can be con-sidered as a good illustration of the unpredictable nature of the polymer extrusion process due to the effects of complexlycoupled process variables. Clear differences between the melting behaviour of the two polymer types are evident due to thedifferences in material properties as described above. Obviously, control of this type of dynamic processes is challenging inpractice. More information on the typical variability of the temperature profile across the die melt flow over different pro-cessing conditions was previously discussed in detail by the author [6,8–11,32,33].

4. Modelling

In general, the melt temperature (Tm;j) at a particular die radial position (Rp;j) which is jmm away from the melt flow cen-tre can be represented as a function of xsc , Rp;j and Tb:

Tm;j ¼ f ðxsc;Rp;j; TbÞ ð1Þ

where xsc is the screw speed and Tb represents the barrel set temperatures (subscript b represents different barrel zones T1-T4). Six model inputs (xsc;Rp;j; T1; T2; T3; T4) and one output (Tm;j), �19 mm 6 j 6 19 mm, were considered as illustrated inFig. 9.

In this study, the set temperature of zone 4 of the barrel, clamp ring, the adapter, and the die were always equal to T4.Therefore, all of these four different temperature zones were considered as a single input to the model.

4.1. Model development and operation

As a large amount of data was generated, both training and validation data were down sampled into 1 Hz (original datawas collected at 10 Hz sampling speed) for the ease of data processing. In fact, this would not adversely affect the modellingwork since the frequency of process thermal fluctuations were found to be slow in nature (lower than 0.5 Hz despite thechanges of screw speed (from 0–90 rpm) and barrel set temperatures). The model should predict melt temperature valuesat each radial position assigned by the radial position input. Fifteen or thirteen radial positions make a complete melt tem-perature profile across the die melt flow depending on the mesh dimension used in each test. The model should estimate themelt temperature values of these fifteen/thirteen points individually by only changing the radial position input while screw

Thermocouple mesh

T1 T2 T3

Motor

Tm,j

scω

T4

Zone

-1

Rp,j Rp,j

Zone

-2

Zone

-3

Zone

-4

Cla

mp

Rin

g

Ada

pte r

Die

Fig. 9. Extruder model with the selected inputs and output.

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1231

speed and barrel set temperatures remain constant. There are 25 different processing situations for tests 1 and 2 and eachinput signal contained 173,700 (i.e. 193� 60� 15) data points. Only 19 situations and 117,780 (i.e. 151� 60� 13) datapoints are available for test 3. The model input matrices for training and validation data of tests 1 and 3 are shown in Figs. 6and 7, respectively. Each model output contains the measured melt temperatures at the different radial positions corre-sponding to the relevant inputs and the signal length is the same as the input signals’ length.

Selection of an efficient modelling technique is highly important in this type of work. One of the possibilities was the useof a modelling approach based on first principles. Such models based on first principles are possible but have limitations inpractical applications for real-time process monitoring and control due to several constraints such as computational com-plexity, difficulty of obtaining closed form solutions, etc. [28]. Alternatively, use of a data driven modelling approach waspossible and a comprehensive review on model selection approaches for non-linear system identification is available inthe literature [34]. Moreover, a number of different data driven modelling techniques (e.g. time series, transfer function,state-space, grey box) which have been used in the polymer extrusion field could be found in the previous research and someof them were discussed previously [9]. However, most of these attempts encountered some problems and therefore a newmodelling approach was sought. After considering several alternatives, a technique which can be used to develop linear/non-linear polynomial models with a linear-in-the-parameters (LITP) model structure was selected to use in this study [35,36].Here, a two stage algorithm is used in the model selection and refinement. In the first stage, a fast recursive algorithm (FRA)was used for the selection of the model structure and for estimation of the model parameters. This solved the problem recur-sively and did not require matrix decomposition as was the case for OLS techniques [37]. However, the model developedincluded a constraint that the terms added later were based on previously selected ones. As a result, the correlations amongthese selected terms make the model less compact. Then, in the second stage a backward model refinement procedure wascarried out to eliminate non-significant terms to build up a compact model. The significance of each selected model term wasreviewed and compared with those remaining in the candidate term pool and all insignificant terms were replaced, leadingto improved performance without increasing the model size. It has been shown that this method is computationally moreefficient and numerically stable than other techniques such as orthogonal least squares (OLS). For these reasons and its pop-ularity in the modelling of nonlinear systems, it was anticipated that this polynomial modelling technique would be suitablefor this particular work. A brief description on the modelling technique is given in Section 4.2.

4.2. Fast recursive algorithm (FRA)

Suppose a general nonlinear discrete-time dynamic multi-input–single-output (MISO) system can be expressed as:

yðtÞ ¼ f ðyðt�1Þ;yðt�2Þ; . . .yðt�n1Þ; . . .yðt�naÞ;uiðt�nikÞ;uiðt�nik�1Þ; . . .uiðt�nik�n1Þ; . . .uiðt�nik�nibÞÞ ð2Þ

where yðtÞ is the system output at time t;uiðtÞ ui; i ¼ 1; . . . ;m are the system input variables at time t (m is the total numberof inputs to the system), na is the number of poles, nib is the number of zeros plus 1 and nik is the corresponding delays (i.e.number of input samples that occur before each input affects the output) of each input. By using a polynomial function, thisequation can be approximated using a Linear-In-The-Parameters (LITP) model:

yðtÞ ¼XM

i¼1

hiuiðxðtÞÞ þ eðtÞ ð3Þ

where uið�Þ; i ¼ 1; . . . ;M are all candidate model terms, xðtÞ ¼ ½u1ðtÞ; . . . ;umðtÞ�T is the model input vector, and eðtÞ is themodeling residue.

Suppose N data samples are used for model training, then (3) can be re-written as:

y ¼ Uhþ e ð4Þ

where U ¼ ½/1; . . . ;/M� 2 RN�M is the regression matrix with column vectors /i ¼ ½uiðxð1ÞÞ; . . . ;uiðxðNÞÞ�T ,

y ¼ ½yð1Þ; . . . ; yðNÞ�T 2 RN is the desired output, h ¼ ½h1; . . . ; hM�T 2 RM is the model coefficients and

e ¼ ½eð1Þ; . . . ; eðNÞ�T 2 RN is the residual vector.The well known least-square method solves the problem by minimising the cost function

JðhÞ ¼ eT e ð5Þ

and the corresponding solution is given by:

h ¼ ðUTUÞ�1UTy ð6Þ

However, due to the noise and correlations between a large number of regressors, the information matrix UTU is always ill-conditioned in practice, which may lead to inaccurate calculation of the model coefficients h. Ridge regression can preventthis problem, but gives a biased solution. Therefore, use of a subset selection algorithm eliminates this problem by selectingthe most relevant and significant terms.

1232 C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236

The fast recursive algorithm utilised in this study can be presented by defining a recursive matrix, Mk, and a residual ma-trix, Rk:

Mk ,UTkUk k ¼ 1; . . . ;M ð7Þ

Rk , I �UkM�1k UT

k R0 , I ð8Þ

where Uk 2 RN�k contains the first k columns of the full regression matrix U, and the cost function in (5) can be rewritten as:

JðPkÞ ¼ yTRky ð9Þ

where Pk ¼ ½p1; . . . ;pk�; k ¼ 1; . . . ; n;n represents the total number of terms included in the final model, and pi represents theselected model terms from all the candidates of those remaining in the pool denoted as, /i; i ¼ 1; . . . ;M.

In this forward stepwise selection, polynomial terms are selected one by one based on their contributions to the finalmodel. As shown in Li et al. [35,36], if one more regressor /j from the candidate term pool is to be selected, the net contri-bution of /j to the cost function can be calculated as:

DJkþ1ð/jÞ ¼ yTðRkþ1 � RkÞy ¼ðyT/

ðkÞj Þ

2

/Tj /ðkÞj

ð10Þ

where /ðkÞj , Rk/j; kþ 1 6 j 6 M. The above net contribution can be further simplified by defining an auxiliary matrix

A 2 Rk�M and a vector b 2 RM�1 with their elements given by:

ai;j ,ðpði�1Þ

i ÞTpj; 1 6 j 6 k

ðpði�1Þi Þ

T/j; k < j 6 M

8<: ð11Þ

bi ,ðpði�1Þ

i ÞTy; 1 6 i 6 k

ð/ðkÞi ÞTy; k < i 6 M

8<: ð12Þ

According to the properties of Rk [35,36], ak;j and bk can be updated recursively:

ak;j ¼ pTk/j �

Xk�1

l¼1

al;kal;j=al;l k ¼ 1; . . . ;n; j ¼ 1; . . . ;M: ð13Þ

bk ¼ pTky �

Xk�1

l¼1

al;kbl=al;l k ¼ 1; . . . ;n: ð14Þ

Now, substituting (13) and (14) into (10), the net contribution of /j; j ¼ kþ 1; � � � ;M to the cost function can be expressed as:

DJkþ1ð/jÞ ¼ �b2

j

aj;jð15Þ

The model term that gives the largest contribution is then selected, and this procedure is continued until some criterion ismet (e.g., Akaike’s information criterion (AIC) [38]) or a pre-set maximum number of terms are selected. To further reducethe calculation complexity in term selection, at the ðkþ 1Þth step, aðkþ1Þ

j;j and bðkþ1Þj (j ¼ kþ 1; . . . ;M) can be updated recur-

sively instead of being computed from (13) and (14):

aðkþ1Þj;j ¼ aðkÞj;j � a2

k;j=ak;k ð16Þ

bðkþ1Þj ¼ bðkÞj � ak;jbk=ak;k ð17Þ

At the end of each selection, these terms are updated and stored for the next comparison/selection. After a satisfactory modelhas been constructed, the model coefficients can now be computed recursively.

hj ¼ bj �Xk

i¼jþ1

hiaj;i

!,aj;j; j ¼ k; k� 1; . . . ;1: ð18Þ

where the terms aj;j and bj in (18) are similar notations for the aðkÞj;j and bðkÞj terms used above. Finally, a more compact model(i.e. terms and coefficients) which includes only the most significant terms is selected based on the defined process inputsand outputs by the sub model selection algorithm.

Initially, both linear and nonlinear polynomial models were adopted to approximate the function f. However, linear mod-els did not show satisfactory performance. Nonlinear polynomial models were then developed and 2nd order models with 20terms were selected for further study. The author has used the same modelling technique for the modelling of the melt tem-perature [32,10], melt pressure [39,40] and motor power consumption [41] in polymer extrusion under static conditions andgood results have been achieved.

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1233

5. Results and discussion

For the dynamic model selection, a number of different model combinations (i.e. with different orders and number ofterms) were studied. Two past output terms and one past input term from each input were used to predict the current output(i.e. na=2 and nb for each input is equal to 1) and these two variables can be changed as required. Then the maximum delays(nk) attributed to each model input had to be determined. Melt temperature variations at each radial position followed byscrew speed and barrel set temperature changes were observed from the experimentally measured data. Melt temperaturechanged soon after change in screw speed. Also, melt temperature was affected by the barrel set temperatures but it took alonger period of time to change the barrel zone temperatures once any change was made. Therefore, the selection of delayswere quite complex and hence values were selected to reflect the information collected from the measured signals and theother details observed during the process. The values which were selected for delays attributed to each input are:d�xsc=10 s, d� Rp=0 s, d� T1=150 s, d� T2=120 s, d� T3=90 s and d� T4=60 s. These delays can be adjusted as requireddepending on the screw geometry, material, processing condition, etc. Ideally, the modelling algorithm would determinethe delays automatically depending on the processing conditions and this will be considered under the future work.

To test the model accuracy, the the root mean squared errors (RMSE) of the models were determined by Eq. (19).

Table 32nd ord

Term

RMSE ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1

½ðyiðtÞ � yiðtÞÞ�2

vuut ð19Þ

where yiðtÞ and yiðtÞ are the measured and predicted melt temperatures at time t respectively, and N is the number of datapoints. Then, three models of the same size (i.e. a 2nd order model with 20 terms for each test) were selected for the dis-cussion and are shown in Table 3. The RMSE and the percentage fit of each model to the training and validation data aregiven in Table 3 and also the model terms are arranged in descending order of term coefficients. Moreover, the measuredand the model estimated temperature profiles over 200 data points with test 1 under the set temperature condition C areshown in Fig. 10 together with the modelling error (ME).

Each group of 15 data points in the figure shows a complete temperature profile across the die melt flow and only 200data points are shown for clarity. It is evident that the model can predict the real-time die melt temperature profile withgood accuracy. Therefore, these can be used to further study the processing behaviour under dynamic conditions.

er 20 terms model details for three different tests.

number Coefficient and variable/s in each term1-HDPE-BF 2-PP-BF 3-HDPE-GC

[fit - 81.53%, RMSE - 2.43 (with trainingdata)]

[fit - 84.65%-, RMSE - 2.42 (with trainingdata)]

[fit - 67.59%, RMSE - 5.15 (with trainingdata)]

[fit - 79.39%, RMSE - 2.70 (with unseendata)]

[fit - 73.20%-, RMSE - 2.98 (with unseendata)]

[fit - 65.57%, RMSE - 5.77 (with unseendata)]

1 þ1:53521�xscðt � 10Þ þ0:78324� T4ðt � 60Þ þ4:53978� T4ðt � 60Þ2 þ1:18080� T4ðt � 60Þ þ0:31054� Tm;jðt � 1Þ �2:72557� T2ðt � 120Þ3 �0:45767� Rp;jðtÞ �0:28595� Rp;jðtÞ �1:24484� Tm;jðt � 1Þ4 �0:18916� Tm;jðt � 2Þ �0:03320� T3ðt � 90Þ � T4ðt � 60Þ �0:64746�xscðt � 10Þ5 �0:03612� Tm;jðt � 1Þ � T4ðt � 60Þ þ0:02662� T3ðt � 90Þ2 �0:04566� Tm;jðt � 1Þ � T4ðt � 60Þ6 �0:02385� Rp;jðtÞ2 þ0:02157� T4ðt � 60Þ2 þ0:02726� Tm;jðt � 2Þ � T4ðt � 60Þ7 þ0:01811� T4ðt � 60Þ2 �0:01566� Tm;jðt � 1Þ � T3ðt � 90Þ þ0:02066� Tm;jðt � 1Þ2

8 þ0:01803� Tm;jðt � 1Þ2 �0:01562� Tm;jðt � 1Þ � T4ðt � 60Þ �0:01884� Rp;jðtÞ2

9 �0:01322� Tm;jðt � 1Þ � Rp;jðtÞ �0:01556� Rp;jðtÞ2 �0:01719� Tm;jðt � 2Þ �xscðt � 10Þ10 �0:01154�xscðt � 10ÞT3ðt � 90Þ þ0:01520� Tm;jðt � 1Þ2 þ0:01643� Tm;jðt � 1Þ � T2ðt � 120Þ11 þ0:01042� Tm;jðt � 2Þ � Rp;jðtÞ �0:01477� Tm;jðt � 1Þ � Rp;jðtÞ þ0:01530� Tm;jðt � 1Þ �xscðt � 10Þ12 þ0:00737� Tm;jðt � 1Þ þ0:00936� Tm;jðt � 2Þ � Rp;jðtÞ �0:01160� Tm;jðt � 2Þ2

13 þ0:00503�xscðt � 10Þ � T4ðt � 60Þ �0:00681� Tm;jðt � 2Þ �xscðt � 10Þ �0:00976� Tm;jðt � 2Þ � Rp;jðtÞ14 �0:00483� T1ðt � 150Þ � T4ðt � 60Þ þ0:00598� Rp;jðtÞ � T4ðt � 60Þ þ0:00924� Tm;jðt � 1Þ � Rp;jðtÞ15 þ0:00438� Tm;jðt � 1Þ � T1ðt � 150Þ þ0:00504� Tm;jðt � 1Þ �xscðt � 10Þ þ0:00783�xscðt � 10Þ � Rp;jðtÞ16 þ0:00436� Rp;jðtÞ � T4ðt � 60Þ þ0:00413�xscðt � 10Þ � T1ðt � 150Þ �0:00657� Tm;jðt � 2Þ � T3ðt � 90Þ17 �0:00411� Tm;jðt � 2Þ �xscðt � 10Þ þ0:00229�xscðt � 10Þ � Rp;jðtÞ þ0:00552�xscðt � 10Þ � T4ðt � 60Þ18 þ0:00166�xscðt � 10Þ � Rp;jðtÞ þ0:00165� Tm;jðt � 2Þ � T4ðt � 60Þ �0:00387�xscðt � 10Þ2

19 þ0:00150� Tm;jðt � 1Þ �xscðt � 10Þ �0:00125�xscðt � 10Þ � T2ðt � 120Þ þ0:00199� Tm;jðt � 1Þ � Tm;jðt � 2Þ20 þ0:00058� T3ðt � 90Þ2 �0:00069� Tm;jðt � 2Þ2 �0:00041� Rp;jðtÞ � T4ðt � 60Þ

118,300 118,360 118,420 118,500-2

2

5118,300 118,360 118,420 118,500190

200

210

220

235

Mel

t tem

pera

ture

(o C)

ME

(o C)

Data points

A temperature profile (-19mm to 19mm)

Each 15 data points shows a profile Temperature set

condition C

Fig. 10. Estimated and measured melt temperature profiles with modelling error for test condition C over 200 data points.

1234 C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236

These models show the significance of each parameter on the shape and the level of the melt temperature profile acrossthe melt flow over the different processing conditions. The effects of screw speed and barrel set temperatures on the level ofthe melt temperature are significant for all tests and these affects the shape of the melt temperature profile as well (seeFig. 8). Moreover, the presence of radial position input (Rp) in a large number of model terms with relatively large coefficientshighlights the significance of the position specific thermal variability across the melt flow and this is evident from Fig. 8 aswell. Of the other 5 model input variables (i.e. except Rp), variable/s which are included in the most significant 5 terms (i.e.based on the coefficients of model terms) of each model are shown in Table 4.

As shown in Table 4, the significance of the barrel zone temperatures on melt temperature depends on the material andscrew geometry used. A HDPE with a BF screw was used for test 1. Test 3 was carried out with a GC screw while all otherconditions were the same as the test 1. Therefore, the differences in behaviour observed between tests 1 and 3 are due togeometrical differences between the screws. It has been found that the temperature of the screw also has a considerable im-pact on the melting process [42]. Although the same material and process settings were used, the temperature of the screwsmay differ as the mechanical heat generation may change with the differences in the arrangement of the solid beds andmaterial conveying over the different screw geometries. Moreover, the rate of mass throughput and the level of materielmixing also depend on the screw geometry [43,44]. For tests 1 and 2, the same screw was used but the materials and settemperatures were different. Considerable differences in the level of the melt temperature and temperature homogeneityacross the melt flow can be observed between tests 1 and 2 as well (see Fig. 8), and these highlight the effects of materialand barrel set temperature on the process melt thermal homogeneity. For all models, T4 was the most significant barrel zonetemperature while the second most significant barrel zone temperatures are T4; T3 and T2 for tests 1, 2 and 3 respectively.The extruder used for this study has four barrel temperature zones and a significant proportion of zone 3 (T3) and all of zone4 (T4) pertain to the metering zone as shown in Fig. 4. Therefore, it can be assumed that the metering zone temperature is themost significant barrel zone temperature which affects melt temperature level and homogeneity for all tests under dynamicprocessing conditions despite the differences in material, machine geometry and process settings. As was discussed in a pre-vious study [10], the melting zone temperature was recognised as the most significant barrel zone temperature on the melttemperature level and temperature homogeneity across the melt flow with a GC screw under the static processing condi-tions. However, the results observed under dynamic processing conditions showed that the metering zone temperature isthe most significant on the level and the homogeneity of process melt temperature with the same GC screw geometry. There-fore, these differences may be attributed to changes in processing behaviour at dynamic and static conditions and also to theeffects of different screw geometry and material.

Table 42nd order 20 terms model details for three different tests.

Test number-material-screw Order of significance

First Second Third Fourth Fifth

Coeffi. Vari. Coeffi. Vari. Coeffi. Vari. Coeffi. Vari. Coeffi. Vari.

1-HDPE-BF 1.5535 xsc 1.1808 T4 �0.0361 T4 0.0181 T4 �0.0115 xsc ; T3

2-PP-BF 0.7832 T4 �0.0332 T3; T4 0.0266 T3 0.0216 T4 �0.0157 T3

3-HDPE-GC 4.5398 T4 �2.7256 T2 �0.6474 xsc �0.0457 T4 0.0273 T4

C. Abeykoon et al. / Applied Mathematical Modelling 38 (2014) 1224–1236 1235

The findings of this study are in agreement with the previous findings of Rasid and Wood [14] (using a square pitchedscrew of 50 mm in diameter) and Crabtree et al. [45] (using a conventional single-flighted screw with a square-pitch leadlength and 63.5 mm in diameter), who found that screw speed was the most influential process variable on the level ofthe extruder melt temperature while the metering zone temperature was the most significant barrel zone temperature. Theyperformed experiments on industrial extruders by changing the process settings individually (i.e. screw speed and each bar-rel zone temperature) to investigate their effects on the level of the extruder melt output temperature under dynamic con-ditions. Although the metering zone temperature was recognised as the most significant barrel zone temperature on thelevel of melt temperature from this study as well, it seems that the melting zone temperature also has a significant impacton the level and homogeneity of the extruder melt temperature more so with the GC screw than the BF screw used in thisstudy. Moreover, higher melt temperature fluctuations were observed with the GC screw than the BF screw, particularly athigher screw speeds. The BF screw used in tests 1 and 2 had a spiral Maddock mixer at the end of the screw and specificallythese screws are designed to achieve better melting performance than conventional screws. Also, the better performance ofthe BF screw over the GC screw was confirmed by having a lower magnitude of fluctuations of melt temperature during theexperiments with the BF screw. As was reported in previous work [46,12], the BF screws perform favorably (e.g. efficientmelting and mixing) compared to conventional GC screws. Such differences may also be responsible for the variable natureof melt temperature profiles across the extruder die melt flow with different screw geometries. Overall, these results high-light the significance of differing thermal behaviour with the change of screws, materials and processing conditions, andhence the complexity of the extrusion process. Therefore, the selection of a proper screw and the process settings (i.e. shouldbe compatible with both machine and material) is a major requirement for achieving a good quality melt output for a par-ticular material.

6. Conclusions

A novel method was proposed to model the die melt temperature profile in polymer extrusion as a function of readilymeasurable process parameters under dynamic processing conditions. The experimental data collected over a wide operat-ing window on an industrial scale extruder was used in the modelling work. As shown from the experimental results andmodels, the effects of screw speed and barrel set temperature on the level of melt temperature and temperature homoge-neity across the melt flow are highly significant. Of the barrel zone temperatures, the metering zone temperature was rec-ognized as the most significant on the level of melt temperature and the temperature homogeneity of the extruder meltoutput. Moreover, the predictions of the proposed models are in agreement with previously reported experimental findingswhich confirms the accuracy of the proposed modelling technique. In fact, the proposed dynamic models seem to representthe actual processing conditions with good accuracy over a wide operating window. Also, these models are simple in struc-ture and could be used in real-time applications. Therefore, this newly proposed technique could help to demonstrate a po-tential method for determining the melt flow thermal homogeneity in real-time and to build-up a control strategy to obtainthe required melt flow homogeneity in polymer extrusion by manipulating the process settings while maintaining the re-quired average temperature across the die melt flow. In this study, the die melt temperature profile was modelled as a func-tion of major process variables (i.e. screw speed and barrel set temperatures) together with a machine geometricalparameter (i.e. die radial position). However, these models should be further generalised by adding other machine and mate-rial related parameters which would help to improve the model performance further and this will be considered in futurework.

Acknowledgements

This work was supported by the Engineering and Physical Sciences Research Council (EPSRC-UK) under Grant Nos. EP/F021070/1 and EP/G059330/1. The partial support provided by the Shanghai Science and Technology Commission underGrant No. 11ZR1413100 is also appreciated. Furthermore, the authors would like to thank all who supported the researchfrom the Queen’s University Belfast and the University of Bradford in various ways.

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