Skip to main content
SpringerLink
Log in

Random Field Sampling for a Simplified Model of Melt-Blowing Considering Turbulent Velocity Fluctuations

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

In melt-blowing very thin liquid fiber jets are spun due to high-velocity air streams. In literature there is a clear, unsolved discrepancy between the measured and computed jet attenuation (thinning). In this paper we will verify numerically that the turbulent velocity fluctuations causing a random aerodynamic drag on the fiber jets—that has been neglected so far—are the crucial effect to close this gap. For this purpose, we model the velocity fluctuations as vector Gaussian random fields on top of a k–ϵ turbulence description and develop an efficient sampling procedure. Taking advantage of the special covariance structure the effort of the sampling is linear in the discretization and makes the realization possible. Numerical results are discussed for a simplified melt-blowing model consisting of a system of random ordinary differential equations.

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

Similar content being viewed by others

References

  1. Antman, S.S.: Nonlinear Problems of Elasticity. Springer, New York (2006)

    Google Scholar 

  2. Arne, W., Marheineke, N., Meister, A., Wegener, R.: Numerical analysis of Cosserat rod and string models for viscous jets in rotational spinning processes. Math. Models Methods Appl. Sci. 20, 1941–1965 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arne, W., Marheineke, N., Schnebele, J., Wegener, R.: Fluid-fiber-interactions in rotational spinning process of glass wool manufacturing. J. Math. Ind. 1, 2 (2011)

    Article  MathSciNet  Google Scholar 

  4. Arne, W., Marheineke, N., Wegener, R.: Asymptotic transition of Cosserat rod to string models for curved viscous inertial jets. Math. Models Methods Appl. Sci. 21, 1987–2018 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Arne, W., Marheineke, N., Meister, A., Wegener, R.: Finite volume approach for the instationary Cosserat rod model describing the spinning of viscous jets. arXiv:1207.0731 (2012)

  6. Asmussen, S., Glynn, P.W.: Stochastic Simulation: Algorithms and Analysis. Springer, Berlin (2007)

    MATH  Google Scholar 

  7. Audoly, B., Clauvelin, N., Brun, P.T., Bergou, M., Grinspun, E., Wardetzky, M.: A discrete geometric approach for simulating the dynamics of thin viscous threads. arXiv:1202.4971 (2012)

  8. Bansal, V., Shambaugh, R.L.: On-line determination of diameter and temperature during melt blowing of polypropylene. Ind. Eng. Chem. Res. 37, 1799–1806 (1998)

    Article  Google Scholar 

  9. Bresee, R.R., Ko, W.-C.: Fiber formation during melt blowing. Int. Nonwovens J. 12, 21–28 (2003)

    Google Scholar 

  10. Cameron, C.: Relative efficiency of Gaussian stochastic process sampling procedures. J. Comput. Phys. 192, 546–569 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Chhabra, R., Shambaugh, R.L.: Experimental measurements of fiber threadline vibration in the melt-blowing process. Ind. Eng. Chem. Res. 35, 4366–4374 (1996)

    Article  Google Scholar 

  12. Elliott, F., Majda, A.J.: A new algorithm with plane waves and wavelets for random velocity fields with many spatial scales. J. Comput. Phys. 117, 146–162 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Elliott, F., Horntrop, D.J., Majda, A.J.: A Fourier-wavelet Monte Carlo method for fractal random fields. J. Comput. Phys. 132, 384–408 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Entov, V.M., Yarin, A.L.: The dynamics of thin liquid jets in air. J. Fluid Mech. 140, 91–111 (1984)

    Article  ADS  MATH  Google Scholar 

  15. Ferziger, J.H., Perić, M.: Computational Methods for Fluid Dynamics, 3rd edn. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  16. Frisch, U.: Turbulence. The Legacy of A.N. Kolmogorov. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  17. Horntrop, D.J., Majda, A.J.: An overview of Monte Carlo simulation techniques for the generation of random fields. In: Muller, P., Henderson, D. (eds.) Monte Carlo Simulations in Oceanography, Proceedings of the Ninth ‘Aha Huliko’ a Hawaiian Winter Workshop, pp. 67–79 (1997)

    Google Scholar 

  18. Kaneda, Y.: Lagrangian and Eulerian time correlations in turbulence. Phys. Fluids A 5, 2835–2845 (1993)

    Article  ADS  MATH  Google Scholar 

  19. Kase, S., Matsuo, T.: Studies on melt spinning. 1. Fundamental equations on the dynamics of melt spinning. J. Polym. Sci., Part A 3, 2541–2554 (1965)

    Google Scholar 

  20. Kolonko, M.: Stochastische Simulation: Grundlagen, Algorithmen und Anwendungen, 1st edn. Vieweg & Teubner, Wiesbaden (2008)

    Book  MATH  Google Scholar 

  21. Kramer, P.R.: A review of some Monte Carlo simulation methods for turbulent systems. Monte Carlo Methods Appl. 7, 229–244 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kruse, S.: A generalized parametrix method, smoothness of random fields and applications to parabolic stochastic partial differential equations. PhD thesis, Universität Mannheim (2001)

  23. Kurbanmuradov, O., Sabelfeld, K.: Stochastic spectral and Fourier-wavelet methods for vector Gaussian random fields. Monte Carlo Methods Appl. 12, 395–445 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Launder, B.E., Sharma, B.I.: Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transf. 1, 131–138 (1974)

    Article  ADS  Google Scholar 

  25. Lee, Y., Wadsworth, L.C.: Effects of melt-blowing process conditions on morphological and mechanical properties of polypropylene webs. Polymer 33, 1200–1209 (1992)

    Article  Google Scholar 

  26. Lu, Q.Q.: An approach to modeling particle motion in turbulent flows—I. Homogeneous isotropic turbulence. Atmos. Environ. 29, 423–436 (1995)

    Article  Google Scholar 

  27. Majda, A.J.: Random shearing direction models for isotropic turbulent diffusion. J. Stat. Phys. 75, 1153–1165 (1994)

    Article  ADS  MATH  Google Scholar 

  28. Majda, A.J., Kramer, P.R.: Simplified models for turbulent diffusion: theory, numerical modelling and physical phenomena. Phys. Rep. 314, 237–574 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  29. Malkan, S.R.: An overview of spunbonding and meltblowing technologies. Tappi J. 78, 185–190 (1995)

    Google Scholar 

  30. Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: general modeling framework. SIAM J. Appl. Math. 66, 1703–1726 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Marheineke, N., Wegener, R.: Asymptotic model for the dynamics of curved viscous fibers with surface tension. J. Fluid Mech. 622, 345–369 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Marheineke, N., Wegener, R.: Modeling and application of a stochastic drag for fiber dynamics in turbulent flows. Int. J. Multiph. Flow 37, 136–148 (2011)

    Article  Google Scholar 

  33. Marla, V.T., Shambaugh, R.L.: Three-dimensional model of the melt-blowing process. Ind. Eng. Chem. Res. 42, 6993–7005 (2003)

    Article  Google Scholar 

  34. Matovich, M.A., Pearson, J.R.A.: Spinning a molten threadline. Steady-state isothermal viscous flows. Ind. Eng. Chem. Fundam. 8, 512–520 (1969)

    Article  Google Scholar 

  35. Müller-Gronbach, T., Novak, E., Ritter, K.: Monte Carlo-Algorithmen. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  36. von Neumann, J.: Various techniques used in connection with random digits. In: Monte Carlo methods. Appl. Math. Ser. vol. 12, pp. 36–38. U.S. Dep. of Commerce, N.B.S, Washington (1951)

    Google Scholar 

  37. Pinchuk, L.S., Goldade, V.A., Makarevich, A.V., Kestelman, V.N.: Melt Blowing: Equipment, Technology and Polymer Fibrous Materials. Springer Series in Materials Processing. Springer, Berlin (2002)

    Google Scholar 

  38. Pismen, L.M., Nir, A.: On the motion of suspended particles in stationary homogeneous turbulence. J. Fluid Mech. 84, 193–206 (1978)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Rao, R.S., Shambaugh, R.L.: Vibration and stability in the melt blowing process. Ind. Eng. Chem. Res. 32, 3100 (1993)

    Article  Google Scholar 

  40. Ribe, N.M.: Coiling of viscous jets. Proc. R. Soc. Lond. A 2051, 3223–3239 (2004)

    MathSciNet  ADS  Google Scholar 

  41. Ribe, N.M., Huppert, H.E., Hallworth, M.A., Habibi, M., Bonn, D.: Multiple coexisting states of liquid rope coiling. J. Fluid Mech. 555, 275–297 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Royston, J.P.: An extension of Shapiro and Wilk’s W test for normality to large samples. Appl. Stat. 31, 115–124 (1982)

    Article  ADS  MATH  Google Scholar 

  43. Royston, J.P.: Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Appl. Stat. 32, 121–133 (1983)

    Article  MATH  Google Scholar 

  44. Sinha-Ray, S., Yarin, A.L., Pourdeyhimi, B.: Meltblowing: I-basic physical mechanism and threadline model. J. Appl. Phys. 108, 034912 (2010)

    Article  ADS  Google Scholar 

  45. Sun, Y.F., Zeng, Y.C., Wang, X.H.: Three-dimensional model of whipping motion in the processing of microfibers. Ind. Eng. Chem. Res. 50, 1099–1109 (2011)

    Article  Google Scholar 

  46. Taylor, G.I.: The spectrum of turbulence. Proc. R. Soc. Lond. A 164, 476–490 (1938)

    Article  ADS  Google Scholar 

  47. Trujillo-Ortiz, A., Hernandez-Walls, R., Barba-Rojo, K., Cupul-Magana, L.: Roystest: Royston’s multivariate normality test. A MATLAB file. Website. http://www.mathworks.com/matlabcentral/fileexchange/loadFile.do?objectId=17811 (2007)

  48. Uyttendaele, M.A.J., Shambaugh, R.L.: Melt blowing: general equation development and experimental verification. AIChE J. 36, 175–186 (1990)

    Article  Google Scholar 

  49. Wente, V.A.: Manufacture of superfine organic fibers. Report PB111437, NRL-4364, US Department of Commerce, Office of Technical Services, Washington, DC (1954)

  50. Wilcox, D.C.: Turbulence Modeling for CFD, 2nd edn. DCW Industries, La Canada (1998)

    Google Scholar 

  51. Wu, T.T., Shambaugh, R.L.: Characterization of the melt blowing process with laser Doppler velocimetry. Ind. Eng. Chem. Res. 31, 379–389 (1992)

    Article  Google Scholar 

  52. Xie, S., Zeng, Y.: Turbulent air flow field and fiber whipping motion in the melt blowing process: experimental study. Ind. Eng. Chem. Res. 51, 5346–5352 (2012)

    Article  Google Scholar 

  53. Yarin, A.L.: Free Liquid Jets and Films: Hydrodynamics and Rheology. Longman, New York (1993)

    MATH  Google Scholar 

  54. Yarin, A.L., Sinha-Ray, S., Pourdeyhimi, B.: Meltblowing: II-linear and nonlinear waves on viscoelastic polymer jets. J. Appl. Phys. 108, 034913 (2010)

    Article  ADS  Google Scholar 

  55. Yin, H., Yan, Z., Bresee, R.R.: Experimental study of the melt blowing process. Int. Nonwovens J. 8, 60–65 (1999)

    Google Scholar 

  56. Zeng, Y.C., Sun, Y.F., Wang, X.H.: Numerical approach to modeling fiber motion during melt blowing. J. Appl. Polym. Sci. 119, 2112–2123 (2011)

    Article  Google Scholar 

  57. Ziabicki, A., Kawai, H.: High Speed Melt Spinning. Wiley, New York (1985)

    Google Scholar 

Download references

Acknowledgements

Special thanks go to the Department of Transport Processes, Fraunhofer ITWM for the air flow simulations of the melt-blowing process. This work has been supported by German Bundesministerium für Bildung und Forschung, Schwerpunkt “Mathematik für Innovationen in Industrie und Dienstleistungen”, Projekt 03MS606, and the Fraunhofer Innovationszentrum “Applied System Modeling”, Kaiserslautern.

Author information

Authors and Affiliations

  1. Fraunhofer ITWM, Fraunhofer Platz 1, 67663, Kaiserslautern, Germany

    Florian Hübsch & Raimund Wegener

  2. FB Mathematik, TU Kaiserslautern, Computational Stochastics, Paul-Ehrlich-Str. 31, 67663, Kaiserslautern, Germany

    Florian Hübsch & Klaus Ritter

  3. Lehrstuhl Angewandte Mathematik I, FAU Erlangen-Nürnberg, Cauerstr. 11, 91058, Erlangen, Germany

    Nicole Marheineke

Authors
  1. Florian Hübsch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nicole Marheineke
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Ritter
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Raimund Wegener
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nicole Marheineke.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hübsch, F., Marheineke, N., Ritter, K. et al. Random Field Sampling for a Simplified Model of Melt-Blowing Considering Turbulent Velocity Fluctuations. J Stat Phys 150, 1115–1137 (2013). https://doi.org/10.1007/s10955-013-0715-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-013-0715-y

Keywords

Search

Navigation