Skip to main content
SpringerLink
Log in

The heat balance integral method for cylindrical extruders

  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

In the hot end of a 3-D printer, polymer feedstock flows through a heated cylinder in order to become pliable. This setup determines a natural upper limit to the speed at which the polymer may be extruded. The case of polymers which undergo the crystalline-melt transition is considered; the resulting mathematical model is a Stefan-like moving boundary-value problem for the polymer temperature. Using the heat balance integral method provides an analytical approximation for the temperature. Several different conditions which use this temperature to establish the maximum velocity are considered; using a pointwise polymer exit temperature in the hot end matches well with experimental data.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Gibson I, Rosen DW, Stucker B (2009) Additive manufacturing technologies: rapid prototyping to direct digital manufacturing, 1st edn. Springer Publishing Company, Incorporated, New York, NY

    Google Scholar 

  2. Edwards DA, Mackay ME, Swain ZR, Banbury CR, Phan DD (2019) Maximal 3D printing extrusion rates. IMA J Appl Math 84(5):1022–1043

    Article  MathSciNet  Google Scholar 

  3. Mackay ME, Swain ZR, Banbury CR, Phan DD, Edwards DA (2017) The performance of the hot end in a plasticating 3D printer. J Rheol 61(2):229–236

    Article  Google Scholar 

  4. Wood AS (2001) A new look at the heat balance integral method. Appl Math Model 25(10):815–824

    Article  Google Scholar 

  5. Lotero F, Couenne F, Maschke B, Sbarbaro D (2017) Distributed parameter bi-zone model with moving interface of an extrusion process and experimental validation. Math Comput Modell Dyn Syst 23(5):504–522

    Article  MathSciNet  Google Scholar 

  6. Mu Y, Zhao G, Wu X, Hang L, Chu H (2015) Continuous modeling and simulation of flow-swell-crystallization behaviors of viscoelastic polymer melts in the hollow profile extrusion process. Appl Math Model 39(3–4):1352–1368

    Article  MathSciNet  Google Scholar 

  7. Sandoval Murillo JL, Ganzenmueller GC (2017) A convergence analysis of the affine particle-in-cell method and its application in the simulation of extrusion processes. In: Wriggers P, Bischoff M, Onate E, Owen D, Zohdi T (eds) V International conference on particle-based methods-fundamentals and applications (Particles 2017). European Community on Computatational Methods in Applied Sciences, International Association for Computational Mechanics, Barcelona, pp 397–408

    Google Scholar 

  8. Schoinochoritis B, Chantzis D, Salonitis K (2017) Simulation of metallic powder bed additive manufacturing processes with the finite element method: a critical review. Proc Inst Mech Eng B 231(1):96–117

    Article  Google Scholar 

  9. Alexiades V, Solomon AD (1992) Mathematical modeling of melting and freezing processes. Taylor & Francis, Washington

    Google Scholar 

  10. Goodman TR (1961) The heat-balance integral-further considerations and refinements. J Heat Transf 83(1):83–85

    Article  Google Scholar 

  11. Mitchell SL, Myers TG (2010) Application of standard and refined heat balance integral methods to one-dimensional stefan problems. SIAM Rev 52(1):57–86

    Article  MathSciNet  Google Scholar 

  12. Ren HS (2007) Application of the heat-balance integral to an inverse Stefan problem. Int J Thermal Sci 46(2):118–127

    Article  Google Scholar 

  13. Caldwell J, Chin C (1998) Solution of two-phase Stefan problems by the heat balance integral method. In: Tupholme GE, Wood AS (eds) Institute of mathematics and its applications conference series. Oxford University Press, Oxford, pp 131–137

    Google Scholar 

  14. Mosally F, Wood A, Al-Fhaid A (2002) An exponential heat balance integral method. Appl Math Comput 130(1):87–100

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the Office of Undergraduate Research and Experiential Learning at the University of Delaware for their support of this project. We also thank the reviewers for their insightful comments which helped improve this manuscript. Many of the calculations herein were performed with the assistance of Maple and Matlab.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Jacob W. Sitison & David A. Edwards

Authors
  1. Jacob W. Sitison
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Edwards
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David A. Edwards.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sitison, J.W., Edwards, D.A. The heat balance integral method for cylindrical extruders. J Eng Math 122, 1–16 (2020). https://doi.org/10.1007/s10665-020-10041-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-020-10041-y

Keywords

Search

Navigation