scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A novel heat transfer coefficient identification methodology for the profile extrusion calibration stage

TL;DR: In this paper, a new method to compute heat transfer coefficients of the profile extrusion process calibration stage, in conjunction with a prototype calibration system, is proposed, which involves two major ingredients: a numerical modeling code and a fitting procedure.
About: This article is published in Applied Thermal Engineering.The article was published on 2016-06-25 and is currently open access. It has received 8 citations till now. The article focuses on the topics: Heat transfer coefficient & Finite volume method.

Summary (4 min read)

1. Introduction

  • As in other industrial areas, the numerical simulation of the extrusion process is a fundamental tool to support the design and development of extruders, extrusion dies, and calibration/cooling systems, in order to optimize the production rate while providing high quality products [3, 4].
  • The finite volume method is a popular technique due to its built-in conservative property [9].
  • The cooling stage is critical since it generally determines the production rate, i.e. it is the limiting stage of the extrusion process, and may have a strong influence on the profile quality as it determines the degree of residual stresses present in the final product.
  • Therefore, in the simulations of the calibration/cooling stage, accurate values of hint, determined in well controlled conditions, should be considered.
  • The study considers an aluminium prototype calibrator cooling a polymer tape, where the authors seek for an accurate optimization of the unknown parameters, based on data collected in extrusion runs performed with the referred prototype [1], with a view to determine accurate values for the heat transfer coefficient at the polymer-calibrator interface.

2. The process modelling

  • As referred in the previous section and illustrated in Fig. 1(b), in a profile extrusion line the polymer profile passes through a calibrator after leaving the extrusion die.
  • It encompasses also several slots through which vacuum is applied to assure the contact between the hot polymer profile and the cold calibration surface.
  • The heat transfer process occurs essentially in the path through the calibrator, and also (at a minor extent) in the air path before and after the calibrator.

2.1. Heat transfer problem

  • The authors then prescribe the following boundary conditions.
  • The heat transfer between the polymer and the calibrator is defined as a thermal contact resistance, which originates a discontinuity in the temperature domain.
  • The temperature distribution on Γsup is a given polynomial function Tsup “ T px1q, experimentally determined with the data provided by thermocouples embedded in the calibrator block [1].
  • For the polymer inflow left boundary, the authors assume a constant prescribed temperature T “ Tin on Γin which corresponds to the extrusion temperature, whereas they assume an adiabatic condition for the polymer outflow on the right boundary ´kp BTp Bnp “ 0 on Γout.

3.3. Flux computation

  • Cell centred finite volume methods use the cell centre as collocation points for the unknowns.
  • The numerical fluxes are then evaluated based on the two vectors T and Θ. .
  • For each cell ci the authors define the affine function rTipx1, x2q “ Ti ` rCi,1 px1 ´mi,1q ` rCi,2 px2 ´mi,2q , where rCi,1 and rCi,2 are the coefficients that minimize a quadratic functional and correspond to the best approximation in the least squares sense of the temperature θk at the vertices of the cell.
  • In the case of the interface between the polymer and the calibrator, the authors have to distinguish two cases regarding to the side where the flux comes from (see [2] for the details).
  • The authors obtain a matrix-free scheme and the affine problem is solved by applying a preconditioned GMRES procedure.

4. Parameter identification procedure

  • The authors assume that the cooling process is mainly governed by five parameters: the heat transfer coefficients hint and hair, the temperatures Tair and Tin, and the velocity u.
  • In practice, some coefficients are measured (temperature, velocity) while the other parameters have to be deduced from the experimental tests.
  • More precisely, the authors seek the set of m parameters H that minimize the error between the measured temperatures and its the numerical approximation given by the following functional F pHq “ C ÿ̀ `“1 ” THp pq`q ´ T̂p` ı2 , (4) where C` is the number of sensors.
  • Indeed, one can consider a problem with two parameters as H “ phint, hairq if the others parameters are given or a problem with four parameters as H “ phint, hair, Tair, Tinq when the temperatures also are unknown.
  • Several optimization techniques will be considered to achieve this goal.

4.1. The Newton-Raphson method

  • The Newton-Raphson technique is a generic procedure to provide the zeros of a vector-valued function using successive linear approximations.
  • PHq, where ∇F is the gradient vector and ∇2F denotes the Hessian matrix.
  • To evaluate the derivatives, the finite differences were adopted.
  • Notice that the optimization procedure requires to evaluate 1`2m2 times the function F in each iteration, which means that the thermal problem has to be solved the same amount of times leading to an important and unnecessary computational effort.

4.2. The Gauss-Newton method

  • The Gauss-Newton method is a specific modification of the Newton-Raphson method for nonlinear least squares problems.
  • The main advantage is that much less computational effort is required since it avoids the calculation of second derivatives.
  • (8) Notice that calculation of ∆H only requires the Jacobian matrix and the Gauss-Newton method is less computational consuming when compared with Newton-Raphson technique, since for each iteration the thermal problem has to be solved 1`m times.

4.3. The Levenberg-Marquardt method

  • The Levenberg-Marquardt method is also a specific technique to solve nonlinear least squares problems where the parameters variation is an interpolation between the GaussNewton method and the gradient descent method.
  • This factor provides a better stability and increases the admissible convergence basin where one can choose the initial condition.
  • The critical point consists in choosing the damping factor since too large values will reduce the efficiency of the method and dramatically increase the computational cost.
  • There are several versions for the Levenberg-Marquardt method.

5. Synthetic tests

  • The authors propose several synthetic benchmarks to assess the efficiency and robustness of the minimization methods.
  • The authors then select 10 points on the upper and lower surfaces of the polymer they shall use in the identification algorithm.
  • The first benchmark aims to compare the three minimization methods in terms of convergence and computational cost where the authors show that the Gauss-Newton method is the best compromise.
  • The second benchmark deals with the basin of convergence.
  • Random initial parameters are prescribed and the authors assess the method ability to recover the reference parameters.

5.1. The manufactured reference solution

  • The thermal model was evaluated with the given parameters and the temperature data was extracted at the 10 points given in Fig. 6 and reported in Table 1 (coordinates and temperatures).
  • The authors define the dimensionless error estimator En “ ÿ i 2 ` Hn`1i ´Hni ˘ Hn`1i `Hni and they stop the iterative procedure when the error between two consecutive iterations is lower than 1ˆ 10´4.

5.2. Minimization methods comparison

  • Convergence tests for the Newton-Raphson method, the Gauss-Newton method, and the Levenberg-Marquardt method are carried out to select the most performavit algorithm.
  • For this benchmark, a set of four parameters is considered, namely: the heat transfer coefficient at the polymer/calibrator interface, the convection heat transfer coefficient of the air, the polymer inflow temperature, and the temperature of the air.
  • Computation is then carried out and the authors evaluate the number of iterations to reach the reference values.
  • The authors proceed in the same way with the Gauss-Newton technique and they report in Fig. 8 the error curves with respect to the number of iterations.

5.3. Test with a control parameter

  • Beyond the parameters studied in the previous tests, the extrusion velocity will also be included in the identification process.
  • The velocity has not the status of a unknown parameter but acts as a control parameter which assess the solution quality.
  • Therefore, the dimensionless value of each parameters for initial approximation can vary between 0.5 and 1.5 or 0.9 and 1.1.
  • The additional information shows whether the control parameter coincides with the reference one or not.

6. Experimental case study

  • The procedures proposed in this work, to find the optimal parameters that meet the experimental data, are now tested in a real situation, with data obtained with the prototype system, conceived to measure the heat transfer coefficient at the polymercalibrator interface [1].
  • For that purpose the prototype system was operated under specific process conditions and the average temperature at several points, identified in Fig. 6, was registered after achieving steady state conditions, during a measurement period of 20 min.
  • Regarding the fitting approach, in all tests performed in this section the Gauss-Newton method was considered, since it was the one that showed the best performance on the previous studies, described in the previous section.
  • The location of the measured points as well as the system properties, closely follows the data used in the synthetic tests (cf. Table 1).
  • Moreover, the temperature errors obtained for the 2+2 points fitting at points 9 and 10 is also lower than the ones obtained for the 10 points counterpart.

7. Conclusion

  • Namely the interface polymer-calibrator and the air convection, was proposed.
  • The fitting procedure employs experimental data collected with a previously developed experimental prototype system [1].
  • The novel methodology required the development of a modeling code for the heat transfer process, which involves discontinuous temperature and velocity fields and discontinuous materials properties.
  • For this purpose a new second-order finite volume.

Did you find this useful? Give us your feedback

Citations
More filters
Journal ArticleDOI
TL;DR: The results provide the optimal very high-order of convergence and prove the capability of the method to handle arbitrary curved interfaces and imperfect thermal contacts.

13 citations

Journal ArticleDOI
TL;DR: In this article, a thermal and strength analysis of the fluid catalytic cracking regenerator was conducted to identify the refractory lining state and the heat transfer coefficient between the FCC zeolite catalyst particles and the regenerator walls.

10 citations

Journal ArticleDOI
TL;DR: In this article , a modified hybrid conjugate gradient algorithm (MHCGA) was proposed to solve the optimization problem and then proved the global convergence of this MHCGA. But the results of the simulation experiments clearly show that this method can reduce running time and iteration number.

5 citations

Journal ArticleDOI
TL;DR: In this article, a modified hybrid conjugate gradient algorithm (MHCGA) was proposed to solve the optimization problem and then proved the global convergence of this MHCGA. But the results of the simulation experiments clearly show that this method can reduce running time and iteration number.

5 citations

Journal ArticleDOI
TL;DR: An extrusion-calendering process was developed to continuously manufacture unidirectional continuous fiber-reinforced polypropylene single-polymer composites (PP SPCs).
Abstract: An extrusion–calendering process was developed to continuously manufacture unidirectional continuous fiber-reinforced polypropylene single-polymer composites (PP SPCs). The process combined “underc...

4 citations


Cites methods from "A novel heat transfer coefficient i..."

  • ...The air temperature of 20 C was used, and the heat transfer coefficient of the air over a surface at low speed flow was 50 W (m(2) K) (1).(42) The thermal properties of PP were from the material database of Moldex 3D (CoreTech System Co....

    [...]

References
More filters
Journal ArticleDOI

28,888 citations


"A novel heat transfer coefficient i..." refers methods in this paper

  • ...Convergence tests for the Newton-Raphson method, the Gauss-Newton method, and the Levenberg-Marquardt method are carried out to select the most performavit algorithm....

    [...]

  • ...At last, the Levenberg-Marquardt method is tested for the same situations....

    [...]

  • ...(10) There are several versions for the Levenberg-Marquardt method....

    [...]

  • ...Thus, the Levenberg-Marquardt method also need to solve the thermal problem 1`m times in each iteration....

    [...]

  • ...Heuristic approaches to evaluate the the factor have been given in [18] To avoid slow convergence in the direction of small gradients, Marquardt [19] proposed to scale the components of the gradient substituting the identity matrix by a diagonal matrix with the elements of the diagonal of JTJ, and Eq....

    [...]

Journal ArticleDOI
TL;DR: In this article, the problem of least square problems with non-linear normal equations is solved by an extension of the standard method which insures improvement of the initial solution, which can also be considered an extension to Newton's method.
Abstract: The standard method for solving least squares problems which lead to non-linear normal equations depends upon a reduction of the residuals to linear form by first order Taylor approximations taken about an initial or trial solution for the parameters.2 If the usual least squares procedure, performed with these linear approximations, yields new values for the parameters which are not sufficiently close to the initial values, the neglect of second and higher order terms may invalidate the process, and may actually give rise to a larger value of the sum of the squares of the residuals than that corresponding to the initial solution. This failure of the standard method to improve the initial solution has received some notice in statistical applications of least squares3 and has been encountered rather frequently in connection with certain engineering applications involving the approximate representation of one function by another. The purpose of this article is to show how the problem may be solved by an extension of the standard method which insures improvement of the initial solution.4 The process can also be used for solving non-linear simultaneous equations, in which case it may be considered an extension of Newton's method. Let the function to be approximated be h{x, y, z, • • • ), and let the approximating function be H{oc, y, z, • • ■ ; a, j3, y, ■ • ■ ), where a, /3, 7, • ■ ■ are the unknown parameters. Then the residuals at the points, yit zit • • • ), i = 1, 2, ■ • • , n, are

11,253 citations

Book
01 Jun 1995
TL;DR: This chapter discusses the development of the Finite Volume Method for Diffusion Problems, a method for solving pressure-Velocity Coupling in Steady Flows problems, and its applications.
Abstract: *Introduction. *Conservation Laws of Fluid Motion and Boundary Conditions. *Turbulence and its Modelling. *The Finite Volume Method for Diffusion Problems. *The Finite Volume Method for Convection-Diffusion Problems. *Solution Algorithms for Pressure-Velocity Coupling in Steady Flows. *Solution of Discretised Equations. *The Finite Volume Method for Unsteady Flows. *Implementation of Boundary Conditions. *Advanced topics and applications. Appendices. References. Index.

7,412 citations

01 Jan 1999
TL;DR: An Introduction to Computational Fluid Dynamics as discussed by the authors is a textbook for advanced undergraduate and first-year graduate students in mechanical, aerospace and chemical engineering that emphasizes understanding CFD through physical principles and examples.
Abstract: Introduction to Computational Fluid Dynamics-Anil W. Date 2005-08-08 Introduction to Computational Fluid Dynamics is a textbook for advanced undergraduate and first year graduate students in mechanical, aerospace and chemical engineering. The book emphasizes understanding CFD through physical principles and examples. The author follows a consistent philosophy of control volume formulation of the fundamental laws of fluid motion and energy transfer, and introduces a novel notion of 'smoothing pressure correction' for solution of flow equations on collocated grids within the framework of the well-known SIMPLE algorithm. The subject matter is developed by considering pure conduction/diffusion, convective transport in 2-dimensional boundary layers and in fully elliptic flow situations and phase-change problems in succession. The book includes chapters on discretization of equations for transport of mass, momentum and energy on Cartesian, structured curvilinear and unstructured meshes, solution of discretised equations, numerical grid generation and convergence enhancement. Practising engineers will find this particularly useful for reference and for continuing education.

395 citations


"A novel heat transfer coefficient i..." refers methods in this paper

  • ...The finite volume method is a popular technique due to its built-in conservative property [9]....

    [...]

Frequently Asked Questions (1)
Q1. What have the authors contributed in "A novel heat transfer coefficient identification methodology for the profile extrusion calibration stage" ?

A real case study demonstrates the advantages of using the new proposed methodology when compared with the previously applied [ 1 ].