Skip to main content
SpringerLink
Log in

The Efficacy of a Three-Dimensional Static Liquid–Vapor Interface Model to Calculate the Static Liquid Holdup of a Packed Bed

  • Original Research Article
  • Published:
Metallurgical and Materials Transactions B Aims and scope Submit manuscript

Abstract

When liquid flows in an operating packed bed, some volume of the liquid remains in the bed at any instant of time. This is known as total liquid holdup. When the liquid flow is stopped, some portion remains static in the bed and the rest drains out. The first portion of the total liquid holdup is known as the static liquid holdup. The extent of the static liquid holdup affects mass transfer, heat transfer, permeability, as well as productivity of a packed bed reactor. Experimental and fundamental research has been performed to calculate the extent of static liquid holdup for last several decades. Many empirical correlations have been developed and mathematical models have been made in pursuit of quantifying this. However, significant difference in the experimental observations and modeling results can still be seen in the literature. In this work, a fundamental building block of a packed bed, i.e., a two-sphere contact, has been taken as the investigation domain. Experiments have been performed to understand how liquid films are formed, their shapes and variation of maximum static film volume before dripping, with respect to inclination for specific contact angles. Dripping involves formation and draining of droplets, and a volume remaining at the contact after dripping; these are measured for various inclinations of the sphere contact axis. Notionally, static liquid holdup is the volume of the liquid that can be held at the two-sphere contacts due to the interplay between static forces namely the surface tension and gravitational force. Hence, the mathematical model incorporating only static forces to capture the three-dimensional liquid shape at the two-sphere contacts, developed in a previous work of our group, is used in this work, and it is shown that the maximum volume of liquid before the liquid dripping can be very well captured for various inclinations. On the other hand, the remaining volume calculated from the mathematical model, with the assumption of quasistatic droplet detachment, results in much underprediction. When the two-sphere experimental values are extended to packed bed situations, it can be shown that the static liquid holdup calculated from the remaining volume after dripping matches are closer to the packed bed experimental values, whereas static liquid holdup calculated from maximum volume leads to overprediction.

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
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Notes

  1. Turner and Hewitt’s correlation is modified in terms of inclination (φ). The present author calculate φ from the vertical axis (φ = 0, when vertical) whereas Turner and Hewitt have calculated from horizontal axis.

Abbreviations

A :

Area (m2)

Bo:

Bond number (Dimensionless)

\( Bo^{\prime} \) :

Dimensionless Bond number (dimensionless)

C p :

Capillary number (dimensionless)

C pm :

Modified capillary number (dimensionless)

\( d_{p}^{{}} \) :

Sphere diameter (m)

Eo:

Eotvos number (dimensionless)

\( E_{\text{total}} \) :

Total energy of an element (joules)

\( g \) :

Gravitational acceleration (m/s2)

\( h_{s} \) :

Static liquid holdup (dimensionless)

\( H \) :

Mean curvature (1/m)

K :

Coefficient (dimensionless)

N :

Co-ordination number of the sphere in a packed bed (dimensionless)

\( N_{c} \) :

Dimensionless contact angle (dimensionless)

P :

Pressure (Pascal)

q :

Index (dimensionless)

\( R_{1} ,R_{2} \) :

Principal radii of curvature (m)

\( V \) :

Volume of the liquid (m3 or microliter)

\( V_{\dim } \) :

Volume of the film/volume of the sphere (dimensionless)

\( V_{\dim }^{\text{avg}} \) :

Dimensionless volume averaged over all the inclinations (dimensionless)

z :

Height of the element from the reference plane (m)

\( \theta \) :

Equilibrium contact angle (dimensionless/degrees)

\( \varphi \) :

Inclination (vertical to be 0) (dimensionless/degrees)

\( \rho \) :

Density of the liquid (kg/m3)

\( \varepsilon \) :

Void fraction (dimensionless)

\( \sigma \) :

Surface tension (N/m)

\( \delta \) :

Signifies the elemental quantity

*:

Dimensionless

o :

Reference

max:

Quantity corresponding to maximum volume

rem:

Quantity corresponding to remaining volume

l :

Liquid

s :

Solid

g :

Gas

References

  1. 1 S. Ghosh, N.N. Viswanathan, and N.B. Ballal: Steel Res. Int., 2017, vol. 88, pp. 1–25.

    Google Scholar 

  2. 2 H.L. Shulman, C.F. Ullrich, A.Z. Proulx, and J. Zimmerman: AIChE J., 1955, vol. 1, pp. 253–8.

    CAS  Google Scholar 

  3. 3 H.L. Shulman, C.F. Ullrich, and N. Wells: AIChE J., 1955, vol. 1, pp. 247–53.

    CAS  Google Scholar 

  4. 4 V.S. Patwardhan: AIChE J., 1963, vol. 9, pp. 331–7.

    Google Scholar 

  5. 5 V.S. Patwardhan and V.R. Shrotri: Chem. Eng. Commun., 1981, vol. 10, pp. 349–55.

    CAS  Google Scholar 

  6. 6 P.J. Mackey and N.A. Warner: Chem. Eng. Sci., 1973, vol. 28, pp. 2141–54.

    CAS  Google Scholar 

  7. 7 P.J. Mackey and N.A. Warner: Metall. Trans. B, 1972, vol. 3, pp. 1807–16.

    Google Scholar 

  8. 8 G.C. Gardner: Chem. Eng. Sci., 1956, vol. 5, pp. 101–13.

    CAS  Google Scholar 

  9. 9 J.C. Charpentier and M. Favier: AIChE J., 1975, vol. 21, pp. 1213–8.

    CAS  Google Scholar 

  10. 10 T. Fukutake and V. Rajakumar: Trans. ISIJ, 1982, vol. 22, pp. 355–65.

    Google Scholar 

  11. 11 W. Xiong, X.-G. Bi, G.-Q. Wang, and F. Yang: Metall. Mater. Trans. B, 2012, vol. 43, pp. 562–70.

    Google Scholar 

  12. 12 H. Kawabata, K. Shinmyou, T. Harada, and T. Usui: Tetsu-to-Hagane, 2017, vol. 92, pp. 893–900.

    Google Scholar 

  13. 13 C.N. Schubert, J.R. Lindner, and R.M. Kelly: AIChE J., 1986, vol. 32, pp. 1920–3.

    CAS  Google Scholar 

  14. 14 S.W.P.M. Van, J.C. Charpentier, and J. Villermaux: Chem. Eng. Sci., 1969, vol. 24, pp. 1083–95.

    Google Scholar 

  15. 15 A. Janzen, M. Schubert, F. Barthel, U. Hampel, and E.Y. Kenig: Chem. Eng. Process. Process Intensif., 2013, vol. 66, pp. 20–6.

    CAS  Google Scholar 

  16. 16 A. Janzen, J. Steube, S. Aferka, E.Y. Kenig, M. Crine, P. Marchot, and D. Toye: Chem. Eng. Sci., 2013, vol. 102, pp. 451–60.

    CAS  Google Scholar 

  17. W.M. Husslage, M. A. Reuter, R.H. Heerema, T. Bakker, and G.S. Steeghs: Metall. Mater. Trans. B 2005, vol. 36, pp. 765–76.

    CAS  Google Scholar 

  18. 18 N. Standish: Chem. Eng. Sci., 1968, vol. 23, pp. 51–6.

    CAS  Google Scholar 

  19. 19 N. Standish: Chem. Eng. Sci., 1968, vol. 23, pp. 945–7.

    CAS  Google Scholar 

  20. 20 A.E. Sáez, M.M. Yépez, C. Cabrera, and E.M. Soria: AIChE J., 1991, vol. 37, pp. 1733–6.

    Google Scholar 

  21. 21 H. Kawabata, Z. Liu, F. Fujita, and T. Usui: ISIJ Int., 2005, vol. 45, pp. 1466–73.

    CAS  Google Scholar 

  22. 22 Y. Sato, T. Hirose, F. Takahashi, M. Toda, T. Hirose, and Y. Hashiguchi: J. Chem. Eng. JAPAN, 1973, vol. 6, pp. 315–9.

    CAS  Google Scholar 

  23. 23 Y. Eto, K. Takeda, and S. Taguch: ISIJ Int., 1993, vol. 33, pp. 681–6.

    CAS  Google Scholar 

  24. 24 G.X. Wang, D.Y. Liu, J.D. Lister, A.B. Yu, S.J. Chew, and P. Zulli: Chem. Eng. Sci., 1997, vol. 52, pp. 4013–9.

    CAS  Google Scholar 

  25. 25 S. Natsui, K.I. Ohno, S. Sukenaga, T. Kikuchi, and R.O. Suzuki: ISIJ Int., 2018, vol. 58, pp. 282–91.

    CAS  Google Scholar 

  26. 26 S. Natsui, H. Nogami, S. Ueda, J. Kano, R. Inoue, and T. Ariyama: ISIJ Int., 2011, vol. 51, pp. 41–50.

    CAS  Google Scholar 

  27. 27 T. Ariyama, S. Natsui, T. Kon, S. Ueda, S. Kikuchi, and H. Nogami: ISIJ Int., 2014, vol. 54, pp. 1457–71.

    CAS  Google Scholar 

  28. 28 D.D. Geleta, I.H. Siddiqui, and J. Lee: Metall. Mater. Trans. B, 2020, vol. 51, pp. 102–13.

    Google Scholar 

  29. Dong XF, Jayasekara A, Sert D, Ferreira R, Gardin P, Monaghan BJ, Chew SJ, Pinson D, Zulli P (2020) Metall. Mater. Trans. B 1:11 https://doi.org/10.1007/s11663-020-02009-1

    Article  Google Scholar 

  30. 30 L.H. George, B.J. Monaghan, R. Longbottom, and P.R. Austin: ISIJ Int., 2011, vol. 53, pp. 1172–9.

    Google Scholar 

  31. 31 H.L. George, R.J. Longbottom, S.J. Chew, and B.J. Monaghan: ISIJ Int., 2014, vol. 54, pp. 820–6.

    CAS  Google Scholar 

  32. 32 I. Jeong, H. Kim, and Y. Sasaki: ISIJ Int., 2013, vol. 53, pp. 2090–8.

    CAS  Google Scholar 

  33. 33 I. Jeong and S. Jung: ISIJ Int., 2016, vol. 56, pp. 537–45.

    CAS  Google Scholar 

  34. Harris CK, Kotteda VMK, Shantha-kumar S, Kumar V, Bronson A (2021) Metall. Mater. Trans. B 1:11 https://doi.org/10.1007/s11663-021-02065-1

    Article  Google Scholar 

  35. 35 G.J. Kramer: Chem. Eng. Sci., 1998, vol. 53, pp. 2985–92.

    CAS  Google Scholar 

  36. 36 A.E. Sáez and R.G. Carbonell: J. Colloid Interface Sci., 1990, vol. 140, pp. 408–18.

    Google Scholar 

  37. 37 E. Bayramli, A. Abou-Obeid, and T.G.M. Van De Ven: J. Colloid Interface Sci., 1987, vol. 116, pp. 490–502.

    CAS  Google Scholar 

  38. P.S. Laplace: Méchanique Céleste, 1806.

  39. 39 Z.S. Mao, T.Y. Xiong, and J. Chen: Chem. Eng. Sci., 1993, vol. 48, pp. 2697–703.

    CAS  Google Scholar 

  40. 40 S. Ghosh, N.N. Viswanathan, and N.B. Ballal: Chem. Eng. Sci., 2020, vol. 212, pp. 115332–40.

    CAS  Google Scholar 

  41. 41 M.A. Erle, D.C. Dyson, and N.R. Morrow: AIChE J., 1971, vol. 17, pp. 115–21.

    Google Scholar 

  42. S. Ghosh, N.B. Ballal, and N.N. Viswanathan: Metall. Mater. Trans. B 2020, 46:1–27.

    Google Scholar 

  43. S. Ghosh, N.N. Viswanathan, and N.B. Ballal: in Proceedings for the 8th International Conference on Modeling and Simulation of Metallurgical Processes in Steelmaking, STEELSIM 2019, 2019, pp. 160–67.

  44. 44 S. Iliev and N. Pesheva: J. Colloid Interface Sci., 2006, vol. 301, pp. 677–84.

    CAS  Google Scholar 

  45. 45 S.D. Iliev: J. Colloid Interface Sci., 1997, vol. 194, pp. 287–300.

    CAS  Google Scholar 

  46. 46 G.A. Turner and G.F. Hewitt: Trans. Instn. Chem. Engrs, 1959, vol. 37, pp. 329–324.

    Google Scholar 

  47. 47 C.W. Extrand and A.N. Gent: J. Colloid Interface Sci., 1990, vol. 138, pp. 431–42.

    CAS  Google Scholar 

  48. A. Rohatgi: (http://arohatgi.info/WebPlotDigitizer/app/Accessed June 2 2014.), 2020, pp. 1–17.

  49. 49 R.G. Picknett and R. Bexon: J. Colloid Interface Sci., 1977, vol. 61, pp. 336–50.

    CAS  Google Scholar 

  50. M. Namadi: M.Tech Thesis, Indian Institute of Tecnology Bombay, 2020.

  51. 51 R.F. Benenati and C.B. Brosilow: AIChE J., 1962, vol. 8, pp. 359–61.

    CAS  Google Scholar 

  52. 52 A. de Klerk: AIChE J., 2003, vol. 49, pp. 2022–9.

    Google Scholar 

  53. 53 M. Suzuki, H. Kada, and M. Hirota: Adv. Powder Technol., 1999, vol. 10, pp. 353–65.

    Google Scholar 

  54. 54 G. Mason and W. Clark: Nature, 1965, vol. 207, p. 512.

    Google Scholar 

Download references

Author information

Author notes

  1. Snigdha Ghosh

    Present address: Department of Fuel, Minerals and Materials Engineering, Indian Institute of Technology (ISM) Dhanbad, Dhanbad, Jharkhand, 826004, India

  2. Akhilandeshwari Erram

    Present address: Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, 400076, India

  3. Manoja Namadi

    Present address: Department of Materials Engineering, Indian institute of Science, Bangalore, 560012, India

Authors and Affiliations

  1. Department of Metallurgical Engineering & Materials Science, Indian Institute of Technology Bombay, Mumbai, 400076, India

    Snigdha Ghosh, Akhilandeshwari Erram, Manoja Namadi, N. B. Ballal & N. N. Viswanathan

Authors
  1. Snigdha Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akhilandeshwari Erram
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Manoja Namadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. N. B. Ballal
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. N. N. Viswanathan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Snigdha Ghosh.

Additional information

Publisher's Note

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

Manuscript submitted December 13, 2020, accepted April 1, 2021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghosh, S., Erram, A., Namadi, M. et al. The Efficacy of a Three-Dimensional Static Liquid–Vapor Interface Model to Calculate the Static Liquid Holdup of a Packed Bed. Metall Mater Trans B 52, 2072–2086 (2021). https://doi.org/10.1007/s11663-021-02167-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11663-021-02167-w

Search

Navigation