Temperature profile within a plane channel of an experimental extrusion die under polymer processing conditions

Abstract

The flow of melted polyethylene (Dowlex 2042^E) through an extrusion die is investigated both experimentally and numerically. This article deals with the reconstruction of thermal history of the melted polymer in the channel die and the influence of the extrusion parameters on the inlet temperature profile. The inverse heat transfer problem is formulated as an optimization problem by considering both the heat transfer and Navier-Stokes equations. It is solved by using the classical conjugate gradient algorithm. Experimental results are presented and the article focuses on the importance of checking the adequacy of the modelling equations in experimental data processing in order to avoid convergence of the inverse algorithm toward solutions which have no physical significance. The effects of the pressure drop combined with the viscous dissipation effect along the channel die are then discussed.

Keywords:

• extrusion die
• inlet temperature profile
• flow rate
• inverse method

Nomenclature

Cp	=	heat capacity (J kg−1°C−1)
F	=	source term of energy (Pa s−3)
J	=	functional to be minimized (°C2)
lc	=	length of the die connector (m)
lb	=	width of the zone (m)
lp	=	perimeter (m)
Lhp	=	length of the heating-plate (m)
Ns	=	number of sensors
P	=	pressure (Pa)
Q	=	flow rate (g s−1)
T	=	temperature (°C)
	=	velocity (m s−1)
u, v	=	axial and radial velocities (m s−1)
	=	measure temperatures (°C)

c	=	connector
m, p	=	sensor locations
0	=	inlet extrusion die
hp	=	heating plate
melt	=	melted polymer
n	=	iteration number
n	=	vector normal
p	=	polymer
s	=	outlet extrusion die
w	=	die wall

	=	shear rate (s−2)
η	=	dynamic viscosity (Pa s)
λ	=	conductivity (W m−1°C−1)
ρ	=	density (kg m−3)
ψ	=	adjoint variable
θ	=	sensitivity variable
δ	=	dirac delta function
Ω	=	spatial domain

1. Introduction

During polymer extrusion, even after the melting zone, heat transfer takes an important place at the outlet of the extruder 1,2. Due to high viscous dissipation and very low heat conductivity, the temperature profile within the channel die can be quite sharp. Overheated areas can be generated and degradation of the polymer could occur. For such creeping flow the temperature field is affected far downstream from the entrance of the die, so to predict accurately and to control the temperature rise, the inlet temperature profile has to be taken into account. However, due to the history in the extruder, the material temperature at the die entrance is rarely uniform. Hence, the determination of this temperature profile is referred to as a boundary inverse heat convection–conduction problem. There are numerous works on the initial temperature profile restoration, for example, in 3 the inlet temperature profile is estimated in the laminar duct flow and subsequent investigations 4–6 examine various aspects of this problem. Recently, Hsu et al. 7 presented a two-dimensional inverse least squares method to estimate both inlet temperature and wall heat flux in a steady laminar flow in a circular duct. Huang and Chen 8 have solved a non-stationary Navier-Stokes equation, to provide coefficients for energy equation, but the velocity field does not depend on temperature. Gejadze and Jarny 9 have presented a detailed analysis of IHCP coupled problem, when the velocity field depends on temperature field through viscosity and Navier-Stokes equations for a non-Newtonian fluid. Recently, Nguyen and Prystay 10 estimated the initial temperature profile and its evolution in polymer processing using the surface temperature measurement and the conjugate gradient method was employed to search for the minimum of the functional. In a previous work 11 the authors studied the numerical resolution of such an inverse heat convection–conduction problem to estimate the temperature field within the polymer flow from temperature measurements taken inside the die wall. The solution was computed according to the classical conjugate gradient method [CGM]. An iterative numerical procedure was used to solve the direct, the adjoint and the sensitivity equations. The modelling equations for the fluid flow in the channel as well as for the heat flow within the melted polymer and the wall regions were solved simultaneously 12,13. The melted polymer was considered as an incompressible pseudo-plastic fluid, and the flow was assumed steady and laminar. The sensitivity analysis led to important results to design the experimental die and to decide the thermocouple locations within the die 13. The method was validated using numerically generated data for different thermal and flow conditions 14. It was shown that the influence of the initial guess on the estimated inlet profile was very weak. In this article, the same inverse problem is considered and the same algorithm is used but the goal is to reconstruct the inlet polymer temperature profile within an experimental die channel. The article aims to show the importance of checking the model adequacy when experimental data have to be processed. In this problem, it is observed that the inverse algorithm cannot converge towards physical realistic solutions without modifying the heat conduction model equations. This modification consists of taking into account the lateral heat losses of the die. With these new model equations, the reconstructed inlet profile is satisfactory and consistent with a priori knowledge of the polymer flow.

2. Experimental device

2.1. Extrusion die and data acquisition system

The main parts of the experimental extrusion die are described in Figures 1 and 2. A rectangular channel (0.002 × 0.03 m²) is connected to the extruder outlet through a cone of length l_c = 0.051 m. The total channel length is L_ch = 0.24 m. The channel walls are made of four stainless steel blocks, instrumented by thermocouples (K), length, thickness and width of each block are L = 0.2 m, H = 0.019 m and l_b = 0.05 m. Two electrical heater plates ensure the heating of the die on its upper and lower faces. To reduce the lateral heat side losses of the die, a composite material (polystyrene, thickness: 0.002 m) and a porous material (foam-glass, thickness: 0.03 m) surround the non-heated side surfaces, except around the measurement connectors. The pressure P_melt and the temperature T_melt of the polymer are measured near the channel entrance (extruder outlet), both parameters give first information on the thermal state of the polymer. According to the previous numerical study 13,14, heat transfer within the die and the channel was first assumed to be bi-dimensional (lateral heat losses were neglected) and only the median plane (B-B) of the extrusion die (Figure 3) is instrumented. The temperature is measured in several places of the upper and lower die walls by using 56 thermocouples K (Chromel–Alumel φ = 50 µm). The wires pass in grooves of 4 × 10⁻⁴ m of diameter. They are welded to metal in holes of 0.001 m of diameter. Figure 4 shows the accurate location of each welding. The locations of these thermocouples are specified as follows: 24 thermocouples are close to the upper and lower faces of the channel, 20 thermocouples are close to the upper and lower heating plates and 8 thermocouples are located at the outer face of the die.

Figure 1. Extrusion die (model 3D), (a) thermocouple connectors, (b) inlet of the die, (c) insulator, (d) heating plate, (e) composite material: polystyrene.

Figure 2. B–B plane of the experimental die–sensor locations.

Figure 3. Lower face of extrusion die.

Figure 4. Welding location of thermocouple inside the die wall.

The measurements of inlet and outlet pressure in the channel are essential since they are used as boundary conditions of the mathematical model. The Dynisco melt pressure transducer for space-restricted areas is used to make pressure measurements of molten polymers up to 400°C. They incorporate a 350-ohm, bonded foil strain gage Wheatstone bridge. This proven technology provides an output of 0–3.33 mV/V (nominal), proportional to melt pressure (within the specified error band). All the measurements were numerically recorded using a high-speed data acquisition system (scanner HP 34970A and Nanovolt/Micro-ohm-meter type HP 34420A), controlled by a VEE-Pro software. The integration time is specified in a number of power line cycles (NPLC). For a steady flow and a better resolution, the longest time is used, i.e. NPLC = 10. The thermophysical properties of the metallic die are assumed to be constant in the temperature range of working: the thermal conductivity is λ_w = 16 W m⁻¹ K⁻¹.

2.2. Polymer properties

The melted polymer used in this work is a low-density polyethylene (LDPE Dowlex 2042^E extrusion grade) suitable for the production of blow film. To measure the viscosity of PE material, we used two kinds of measuring apparatus: a capillary rheometer (Ceast Co., Italy) and Rubber process analyser (RPA 2000). We consider the shearing viscosity because flow pattern in a die extrusion of polymer melts is mainly shear flow. It is defined as the ratio of the local shear stress to the shear rate at that point. The viscosity function , is temperature and shear-rate dependent. The rheological properties were measured over the 177–237°C temperature range with shear rates varying from 0.1 s⁻¹ to 1000 s⁻¹. Figure 5 presents the melt flow curves at various temperatures. It can be seen that the logarithmic plots are quite similar in shape, showing the higher shear rate, the lower the shear viscosity. The shear viscosity irregularly decreases with increasing apparent shear rate, indicating a shear thinning (i.e. pseudoplastic) nature due to the random orientation and highly entangled state of the molecules which under high shear rate get disentangled and become orientated, leading to a reduction of viscosity 15. With a rise of temperature, the value of shear viscosity decreases, especially at relatively lower apparent shear rate, demonstrating that increasing temperature improves the flow behaviour of the polymer melts. However, the effect of temperature on shear viscosity changes with the shear rate. The data indicate that the temperature sensitivity of shear viscosity is higher in the lower shear rate region, and drops at higher shear rates. Determination of the viscosity curve, Figure 5, over a wide range of shear rate values is essential for solving the flow Equations (3–9). The data measurements were fitted (inverse method) to the Cross WLF model (Equation 1). Thermal and rheological parameters of the PE are given in Table 1. (1) (2) The thermal conductivity of the melted PE was measured under moulding conditions (high pressure and high temperature) using a specific experimental apparatus and a numerical method 14. The variation of thermal conductivity is shown in Figure 6. Phase change occurs in the range [115°C, 130°C] and the maximum value of thermal conductivity is in the same temperature range. The heat capacity C_pp(T) (Figure 7) was measured with a differential scanning calorimeter (Pyris 6 DSC). Data measurements were fitted in the temperature range [150°C, 250°C]. The temperature-dependent heat capacity is represented by Equation 2.

Figure 5. Shear viscosity vs. shear rates for PE melt at various temperatures.

Figure 6. Thermal conductivity of PE Dowlex vs. temperature.

Figure 7. Heat capacity of PE Dowlex vs. temperature.

Table 1. Thermal and rheological properties of PE Dowlex 2042^E.

Thermal properties at T = 220°C		Rheological properties
λp(W m^− 1 K^− 1)	0.3	a (Pa s)	54329
Cpp(J kg^− 1 K^− 1)	2674	b (K^−1)	0.0095
ρp(g cm^− 3)	0.745	λ (s^−1)	0.1657
		n	0.36

2.3. Experimental conditions and procedure

All the experiments were carried out with a single-screw extruder (screw diameter = 3 cm and length = 78 cm). The minimum and the maximum flow rates are 0.5 and 15 kg h⁻¹ respectively, and the maximum rotation speed is 100 rpm. The polymer in the form of small pellets about 4–7 mm in size is fed from a hopper by gravity into the extruder. Flow rate is measured by weighing the mass of polymer exiting at the channel die outlet. The pressure drop (ΔP) along the channel die is measured by using two pressure sensors located at the inlet and outlet of the channel die. The transducers are rated at up to 80 MPa and are interfaced with the control system, which displays the polymer pressures. The heating units controlled the temperature of the upper and lower faces of the die. Several experiments on the PE Dowlex 2042^E have been carried out at different pressures from 10 to 25 MPa. Some experimental conditions are listed in the Table 2.

Table 2. Polymer processing conditions of PE Dowlex 2042^E.

Tests	Screw rotation (rpm)	Tmelt (°C)	Pmelt (MPa)	ΔP (MPa)	Q (g s^−1)
1	10	201 ± 1	9.9	8.8	0.15
2	19	201 ± 1	15.6	13.9	0.29
3	26	202 ± 1	20	17.7	0.43
4	37	201 ± 1	25	21.3	0.53

Figures 8 and 9 show the temperature profiles measured within the die wall in the flow direction, both near the channel and heating plates. The temperature is maximal at the die inlet and then decreases along the flow axis. This decrease results in non-controlled heat losses at the outer face of the die. But it can be observed that increasing the pressure drop in the channel raises the flow axis temperature profiles at the die outlet, near the channel. This temperature rise results in a shear heating effect within the polymer flow, which increases with the pressure drop. Figures 10 and 11 show the thermal boundary conditions on the upper and lower faces, used in the modelling equations.

Figure 8. Measured temperature vs. pressure drop.

Figure 9. Measured temperatures near the lower heating-plate: .

Figure 10. Measured temperature at the outer die face .

Figure 11. Measured temperature at the outer die face .

3. Mathematical modelling equations

A steady laminar flow model of an incompressible polymer is considered. Figure 12 shows the 2D spatial domain (Ω = Ω₁ ∪ Ω₀) of the extrusion die in a median plane parallel to the flow direction.

Figure 12. Extrusion die/2D spatial domain.

The polymer melt enters the extrusion die at x = 0 with a temperature profile T₀(z). The velocity and the temperature fields are governed by the coupled equations of mass, momentum and energy, i.e. (3) (4) (5) (6) The term F is due to viscous dissipation. According to the rheological behaviour, this term can take various forms. In our study, it is given by: (7) The term S_p,w was not considered in the previous study, because heat losses through the connectors of the non-heated side face of the die were neglected. This new term is introduced here for taking into account the lateral heat side losses of the die. The lateral face is divided in four zones (straight rectangular fins) as shown in Figures 13–15, l_p, l_b and S^zone are the perimeter, thickness and surface area of each fin (Table 3), respectively. (8) (9) is a heat transfer coefficient for each zone. Their values are determined by the thermal theory of the rectangular fin with uniform cross section 16.

Figure 13. Cross face of the extrusion die: locations of heat losses.

Figure 14. (a) Insulator (foam glass), (b) insulator (polystyrene), (c) die wall, (d) heat flux from melted polymer to the connectors and insulators, (e) aluminium to fix the connectors, (f) thermocouples, (g) connectors.

Figure 15. Zone heat loses: (a) polymer, (b) die wall, (c) composite: polystyrene (2mm) (d) aluminium and connectors, (e) foam-glass (30 mm).

Table 3. Heat losses coefficients.

zones		S^zone(m²) × 10^− 6	lp(m)	ξ^zone(W m^−3 K^−1)
1	6.51	640	0.112	130
2	87	6600	0.410	932
3	32	760	0.118	657
4	4.98	8000	0.48	99.65

The following boundary conditions are then considered. For the velocity field:

•	At the channel inlet: (10)
•	At the channel outlet: (11)
•	At the internal surface, (usual no-slip boundary condition): (12)

For the temperature field, the boundary conditions are taken as follows:

•	At the channel entrance:

(13) The heat flux and the surface temperature of the die are fixed: (14) (15) (16) (17)

4. Inverse heat transfer problem

The inlet temperature profile T₀(z) has to be determined from the temperature given by Ns sensors located inside the extrusion die wall at (x_m, z_p), m = 1, …, Ns/2, p = 1, 2. The inverse problem is formulated by considering the following least square criterion: (18) where T(x_m,z_p;T₀) are the temperatures computed from the direct problem equations, at the measurement locations, with the fixed inlet temperature profile T₀(z). The conjugate gradient method is used to search for the minimum of the functional. This method has been well documented elsewhere 14,17,18 and will not be repeated here. The method is based on the computation of the gradient of the functional, which is obtained by solving the sensitivity problem and the adjoint problem defined as follows.

4.1. The sensitivity problem

In order to develop the sensitivity problem equations, a variation ϵδT₀ of the inlet temperature profile is considered, and the resulting temperature is denoted T(x, z; T₀ + ϵδT₀).

The sensitivity is then defined by 14,17: (19) Developing the direct problem Equations (3–9) for T(x, z; T₀) and T(x, z; T₀ + ϵδT₀) and then subtracting the resulting expressions leads to: (20) (21) (22) (23) (24) (25) where is the velocity profile in the channel die.

4.2. The adjoint problem

The following adjoint problem equations are obtained 14: (26) (27) (28) (29) (30) (31)

with (32) (33)

5. Results and discussion

The computational technique used to solve the direct problem is based on a finite volume method (FVM) discretization of the governing mass, momentum and energy equations. More details can be found in previous studies 13,14. Comparisons between numerical and analytical solutions confirm the adequacy of the mesh grid size chosen for solving the modelling equations. The same numerical method and grids are used to solve the adjoint and the sensitivity problem. The feasibility and accuracy of the inverse analysis for estimating the unknown inlet distribution temperature, by using the CGM, was illustrated in several numerical experiments, including the case of non-Newtonian fluid 14.

5.1. Effect of heat losses in the model equations

First we will show the influence of the heat losses through the non-heated sides of the die, on the estimated inlet temperature profiles. Two sets of modelling equations are considered successively for solving the inverse problem: one by setting the lateral heat losses S_p,w(x, z; T) equal to zero, the other one by adding the heat losses S_p,w(x, z; T) according to Equation 8. In both cases, the iterative process is initialized with the uniform initial guess . In order to compare and to conclude about the validity of the estimated profiles that result from the minimization of the same criterion J(T₀), Equation 18, it is necessary to take into account the magnitude of the error measurements in the stopping rule of the iterative process. The errors on the temperature data measured in the die wall are supposed to be zero mean, and Gaussian. The standard deviation is not exactly known, but it is less than σ_max = 0.1 K and more than σ_min = 0.05 K. Then, without errors in the modelling equations, the asymptotic value J_s of the least square criterion J(T₀), computed with Ns = 40 sensors, is expected to be in the following range: J_s = Nsσ²⊂[0.1 K²,0.4 K²]. The stopping rule of the conjugate gradient algorithm used to minimize J(T₀) must be consistent with this expected asymptotic range. It is clear that if the asymptotic value J_s remains greater than the upper bound 0.4 K², the assumption of the model adequacy has to be rejected. But on the contrary, when the value J_s (almost) reaches this expected range, the adequacy of the modelling equations may be (almost) accepted. In practice, even in this favourable situation, the adequacy of the modelling equations remains imperfect, and the final value of J_s cannot be predicted exactly. Therefore the iterative process is stopped when no significant decrease of J(T₀) is observed within this expected range.

Case I: Model equations without heat losses, S_p,w(x, z, T) = 0

In Figure 16, a comparison is made between the initial guess of the temperature profile and the reconstructed temperature profiles for different pressure drops. For three different operating conditions: ΔP = 9.9, 15.6 and 20 MPa. Figure 17 compares the measured and the calculated temperatures and Figure 18 shows the least square criterion J(T₀) versus the iteration number n. It can be observed, for the three cases considered, that the asymptotic values J_s are different but all of them are reached after n = 3 iterations and these values are greater than the above expected range. From these observations, we are forced to conclude that the modelling equations without heat losses are not able to accurately predict heat transfer in the die wall. The model is biased, and the estimated inlet profiles computed with this model are not acceptable. In fact, this conclusion is confirmed by the shape of the estimated profiles (Figure 16), the temperature is very high near the walls (up to 260°C) and very low in the middle of the channel (130–187°C). These profiles have no physical significance. The hypothesis of the model adequacy is violated, and the resulting estimated inlet profiles strongly biased.

Figure 16. Estimated inlet temperatures profile, S_p,w = 0.

Figure 17. Measured and calculated temperatures, P_melt = 20 Mpa, S_p,w = 0.

Figure 18. Least square criterion J(T₀(z)), S_p,w = 0.

Case II: Model equations with lateral heat losses, S_p,w(x, z, T) ≠ 0

In Figure 19, comparison is made between the initial guess of the temperature profile and the reconstructed temperature profiles for a pressure drop P_melt = 20 MPa. It is observed in Figure 20 that the asymptotic value of the least square criterion in this case, is obtained after five iterations, and this value is very close to the best expected value J_s = 0.1 K². This result shows better adequacy of the new model equations (Figure 21), because now this value is consistent with the standard deviation of the measurement errors. Moreover, the estimated temperature profiles of the polymer now have a physical meaning: it is observed that the mean temperature of the outlet profile is inferior to that of the inlet profile because of the heat losses through the non-heated faces of the die, but contrary to the inlet profile, the distribution of the outlet profile shows a maximum at the channel centre. These observations are consistent with the heat transfer phenomena which occur within the die wall and the polymer flow under viscous dissipation. These results are confirmed with the other experiments.

Figure 19. Estimated inlet and outlet temperatures, P_melt = 20 MPa, S_p,w ≠ 0.

Figure 20. Least square criterion J(T₀(z)), S_p,w ≠ 0.

Figure 21. Measured and calculated temperatures, P_melt = 20 MPa, S_p,w ≠ 0.

5.2. Effect of the flow rate on the inlet temperature (S_p,w(x, z, T) ≠ 0)

In these experiments, we varied the pressure drop ΔP = P₀ − P_s (flow rate) along the channel die by increasing the screw rotation of the extruder. In all these cases, the temperature of the polymer in the inlet die-connector is strongly affected by both the screw rotation and the temperature of the screw heating-zones. When the screw rotation is 37 rpm, to keep a polymer temperature close to 200°C at the extruder outlet, the heating-zones temperature must be controlled to a low value equal to 193°C. Figure 10 shows the influence of pressure drops on the temperature profiles along the flow axis, near the heating plates. The temperature is maximal at the die inlet and then decreases slightly because of lateral heat losses.

Figures 22 and 23 show the different estimated inlet temperature profiles according to the different pressure drops considered: ΔP = 9.9, 15.6 and 20 MPa. When the pressure drop increases, the flow rate increases and the internal heating within the polymer flow changes both the viscosity and the temperature profile. It is thought that the observed differences in the estimated inlet temperature profile could directly result in the flow rate changes and the shear heating effect. Figures 24–26 show the comparison between the measured and calculated temperatures in the die wall and the least square criterion versus the iteration number.

Figure 22. Estimated inlet temperatures for PE Dowlex 2042^E.

Figure 23. Calculated outlet temperatures for PE Dowlex 2042^E.

Figure 24. Least square criterion vs. iterations.

Figure 25. Measured and calculated temperatures, P_melt = 15.6 MPa, S_p,w ≠ 0.

Figure 26. Measured and calculated temperatures for PE Dowlex 2042^E P_melt = 9.9 MPa, S_p,w ≠ 0.

The evolutions of the least square criterion are similar: they converge after five iterations. The asymptotic values are different, but they remain in the range of the expected value, and are compatible with the evaluated standard deviation of the error measurements. Nevertheless, the adequacy of the model equations is not perfect and some improvements should be considered.

6. Conclusion

An inverse heat convection–conduction algorithm and a specific experimental apparatus have been described in order to reconstruct the inlet temperature profile of a melted polymer in the channel of an extrusion die. Temperature data are measured in the metallic wall of the die. The estimated inlet profiles have no physical significance when the adequacy of the model equations is not sufficient. This can be evaluated by observing the asymptotic value of the least squares criterion, which must be close to the expected value, defined by the number of sensors and the variance of the error measurements.

The reconstructed profiles by considering heat losses in the heat transfer model equations are consistent with a priori knowledge of polymer melt flow in such experimental conditions. The effects of viscous dissipation and flow rate on inlet temperature profiles can be analysed, thanks to this inverse approach. When the pressure drop increases in the channel, the flow rate and the outlet mean temperature increase as well. The estimated inlet profile T₀(z) is also changed because of a different melted thermal history in the extruder.

To our knowledge, there is no experimental device for directly measuring the temperature inlet profiles in such polymer melt flow conditions. However, several improvements could be added to this experiment, for particularly better control of the heat losses of the die. Additional measurements would be helpful to improve our confidence in the estimated results, for example, the surface temperature of the melted polymer at the outlet of the die could be compared to the predicted value by the model equations.

Finally this experimental study illustrates the interest in developing such an inverse approach based on the numerical resolution of the heat convection–conduction problem by the well known conjugate gradient algorithm. It shows how to investigate and to analyse complex experimental phenomena during thermoplastic polymer processing, which combined flow mechanisms, viscous dissipation and heat transfer.