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Modeling of Flow Structure, Bubble Distribution, and Heat Transfer in Polydispersed Turbulent Bubbly Flow Using the Method of Delta Function Approximation

Abstract

The results of modeling of flow structure, air bubble distribution over the pipe cross section, and heat transfer in a vertical polydispersed gas-liquid flow are presented. Themathematical model is based on the Euler description with allowance for the back effect of bubbles on the averaged characteristics and turbulence of the carrier phase. The polydispersity of two-phase flow is described by the delta approximation method with consideration of bubble break-up and coalescence. The turbulence of the carrier phase is predicted using the Reynolds stress transport equations. The results of the modeling showed good agreement with experimental and numerical data of other works.

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References

  1. 1.

    Nigmatulin, R.I., Dynamics of Multiphase Media, vol. 1, CRC Press, 1990.

  2. 2.

    Ishii, M. and Hibiki, T., Thermo-Fluid Dynamics of Two-Phase Flow, Berlin: Springer, 2011.

    MATH  Book  Google Scholar 

  3. 3.

    Ramkrishna, D., Population Balances. Theory and Applications to Particulate Systems in Engineering, N.Y.: Acad. Press, 2000.

    Google Scholar 

  4. 4.

    Carrica, P.M., Drew, D.A., Bonetto, F., and Lahey, R.T., Jr., A Polydisperse Model for Bubbly Two-Phase Flow around a Surface Ship, Int. J. Multiphase Flow, 1999, vol. 25, pp. 257–305.

    MATH  Article  Google Scholar 

  5. 5.

    Politano, M., Carrica, P., and Converti, J., A Model for Turbulent Polydisperse Two-Phase Flow in Vertical Channel, Int. J. Multiphase Flow, 2003, vol. 29, pp. 1153–1182.

    MATH  Article  Google Scholar 

  6. 6.

    Yeoh, G.H. and Tu, J.Y., Population Balance Modelling for Bubbly Flows with Heat and Mass Transfer, Chem. Eng. Sci., 2004, vol. 59, pp. 3125–3139.

    Article  Google Scholar 

  7. 7.

    Yeoh, G.H. and Tu, J.Y., Numerical Modelling of Bubbly Flows with and without Heat and Mass Transfer, Appl. Math. Model., 2006, vol. 30, pp. 1067–1095.

    MATH  Article  Google Scholar 

  8. 8.

    Das, A.K., Das, P.K., and Thome, J.R., Transition of Bubbly Flow in Vertical Tubes: New Criteria through CFD Simulation, ASME J. Fluids Eng., 2009, vol. 131, no. 9, p. 091303.

    Article  Google Scholar 

  9. 9.

    Deju, L., Cheung, S.C.P., Yeoh, G.H., and Tu, J.Y., Capturing Coalescence and Break-Up Processes in Vertical Gas–Liquid Flows: Assessment of Population BalanceMethods, Appl.Math. Model., 2013, vol. 37, pp. 8557–8577.

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Gafiyatov, R.N., Gubaidullin, D.A., and Gubaidullina, D.D., AcousticWaves of Various Geometry in Multi- Fraction Bubbly Liquids, Fluid Dyn., 2018, vol. 53, pp. 119–126.

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Gubaidullin, D.A. and Snigerev, B.A., Numerical Simulation of the Turbulent Upward Flow of a Gas- Liquid Bubble Mixture in a Vertical Pipe: Comparison with Experimental Data, High Temp., 2018, vol. 56, pp. 61–69.

    Article  Google Scholar 

  12. 12.

    Krepper, E., Lucas, D., Frank, T., Prasser, H.-M., and Zwart, P.J., The InhomogeneousMUSIGModel for the Simulation of Polydispersed Flows, Nucl. Eng. Des., 2008, vol. 238, pp. 1690–1702.

    Article  Google Scholar 

  13. 13.

    McGraw, R., Description of Aerosol Dynamics by the Quadrature Method of Moments, Aerosol Sci. Technol., 1997, vol. 27, pp. 255–265.

    ADS  Article  Google Scholar 

  14. 14.

    Selma, B., Bannaru, R., and Proulx, P., Simulation of Bubbly Flows: Comparison between Direct Quadrature Method of Moments (DQMOM) and Method of Classes (CM), Chem. Eng. Sci., 2010, vol. 65, pp. 1925–1941.

    Article  Google Scholar 

  15. 15.

    Piskunov, V. and Golubev, A., The Technique for Specifying Dynamic Parameters of Coagulating Systems, Dokl. Phys., 1999, vol. 366, pp. 341–344.

    Google Scholar 

  16. 16.

    Zaichik, L.I., Mukin, R.V., Mukina, L.S., and Strizhov, V.F., Development of a Diffusion-Inertia Model for Calculating Bubble Turbulent Flows: Isothermal Polydispersed Flow in a Vertical Pipe, High Temp., 2012, vol. 50, pp. 621–630.

    Article  Google Scholar 

  17. 17.

    Mukin, R.V., Modeling of Bubble Coalescence and Break-Up in Turbulent Bubbly Flow, Int. J. Multiphase Flow, 2014, vol. 62, pp. 52–66.

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zaichik, L.I., Skibin, A.P., and Solov’ev, S.L., Simulation of the Distribution of Bubbles in a Turbulent Liquid Using a Diffusion-InertiaModel, High Temp., 2004, vol. 42, pp. 111–118.

    Article  Google Scholar 

  19. 19.

    Zaichik, L.I., Simonin, O., and Alipchenkov, V.M., Turbulent Collision Rates of Arbitrary-Density Particles, Int. J. HeatMass Transfer, 2010, vol. 53, pp. 1613–1620.

    MATH  Article  Google Scholar 

  20. 20.

    Kocamustafaogullari, G. and Ishii, M., Foundation of the Interfacial Area Transport Equation and Its Closure Relations, Int. J. Heat Mass Transfer, 1995, vol. 38, pp. 481–493.

    MATH  Article  Google Scholar 

  21. 21.

    Wu, Q., Kim, S., Ishii, M., and Beus, S.G., One-Group Interfacial Area Transport in Vertical Bubbly Flow, Int. J. Heat Mass Transfer, 1998, vol. 41, pp. 1103–1112.

    MATH  Article  Google Scholar 

  22. 22.

    Lehr, F. and Mewes, D., A Transport Equation for the Interfacial Area Density Applied to Bubble Columns, Chem. Eng. Sci., 2001, vol. 56, no. 3, pp. 1159–1166.

    Article  Google Scholar 

  23. 23.

    Lahey, R.T., Jr. and Drew, D.A., The Analysis of Two-Phase Flow andHeat TransferUsingMultidimensional, Four Field, Two-FluidModel, Nucl. Eng. Des., 2001, vol. 204, pp. 29–44.

    Article  Google Scholar 

  24. 24.

    Pakhomov, M.A. and Terekhov, V.I., Numerical Simulation of the Flow and Heat Exchange in a Downward Turbulent Gas-Fluid Flow in a Pipe, High Temp., 2011, vol. 49, no. 5, pp. 715–721.

    Google Scholar 

  25. 25.

    Lobanov, P.D. and Pakhomov, M.A., Experimental and Numerical Study of Heat Transfer Enhancement in a Turbulent Bubbly Flowin a Pipe Sudden Expansion, J. Eng. Thermophys., 2017, vol. 26, pp. 277–290.

    Article  Google Scholar 

  26. 26.

    Hibiki, T., Ishii, M., and Xiao, Z., Axial Interfacial Area Transport ofVertical Bubbly Flows, Int. J. HeatMass Transfer, 2001, vol. 44, pp. 1869–1888.

    Article  Google Scholar 

  27. 27.

    Lucas, D., Krepper, E., and Prasser, H.M., Development of Co-Current Air-Water Flow in a Vertical Pipe, Int. J. Multiphase Flow, 2005, vol. 31, pp. 1304–1328.

    MATH  Article  Google Scholar 

  28. 28.

    Pakhomov, M.A. and Terekhov, V.I., Modeling of Turbulent Structure of an Upward PolydisperseGas-Liquid Flow, Fluid Dyn., 2015, vol. 50, pp. 229–239.

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Zaichik, L.I. and Alipchenkov, V.M., Modeling of the Motion of Light-Weight Particles and Bubbles in Turbulent Flows, Fluid Dyn., 2010, vol. 45, no. 4, pp. 574–590.

    ADS  MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Zaichik, L.I. and Alipchenkov, V.M., A StatisticalModel for Predicting the Fluid Displaced/Added Mass and Displaced Heat Capacity Effects on Transport and Heat Transfer of Arbitrary-Density Particles in Turbulent Flows, Int. J. HeatMass Transfer, 2011, vol. 54, pp. 4247–4265.

    MATH  Article  Google Scholar 

  31. 31.

    Zaichik, L.I., A Statistical Model of Particle Transport and Heat Transfer in Turbulent Shear Flows, Phys. Fluids, 1999, vol. 11, no. 6, pp. 1521–1534.

    ADS  MATH  Article  Google Scholar 

  32. 32.

    Derevich, I.V., Statistical Modelling of Mass Transfer in Turbulent Two-Phase Dispersed Flows. 1. Model Development, Int. J. Heat Mass Transfer, 2000, vol. 43, no. 19, pp. 3709–3723.

    MATH  Article  Google Scholar 

  33. 33.

    Lopez de Bertodano, M., Lee, S.J., Lahey, R.T., and Drew, D.A., The Prediction of Two-Phase Turbulence and Phase Distribution Using a Reynolds Stress Model, ASME J. Fluids Eng., 1990, vol. 112, no. 1, pp. 107–113.

    Article  Google Scholar 

  34. 34.

    Ranz, W.E. and Marshall, W.R., Jr., Evaporation from Drops. Parts I and II, Chem. Eng. Progress, 1952, vol. 48, pp. 141–146, 173–180.

    Google Scholar 

  35. 35.

    Manceau, R. and Hanjalic, K., Elliptic Blending Model: A New Near-Wall Reynolds-Stress Turbulence Closure, Phys. Fluids, 2002, vol. 14, pp. 744–754.

    ADS  MATH  Article  Google Scholar 

  36. 36.

    Loth, E., Quasi-Steady Shape and Drag of Deformable Bubbles and Drops, Int. J. Multiphase Flow, 2008, vol. 34, no. 6, pp. 523–546.

    Article  Google Scholar 

  37. 37.

    Wallis, G.B., The Terminal Speed of Single Drops in an Infinite Medium, Int. J. Multiphase Flow, 1974, vol. 1, no. 4, pp. 491–511.

    Article  Google Scholar 

  38. 38.

    Kashinskii, O.N., Gorelik, R.S., and Randin, V.V., Phase Velocity in aGas-Liquid Bubbly Flow, J. Eng. Phys. Thermophys., 1989, vol. 57, pp. 732–734.

    Article  Google Scholar 

  39. 39.

    Drew, D.A. and Lahey R.T., Jr., The Virtual Mass and Lift Force on a Sphere in Rotating and Straining Inviscid Flow, Int. J. Multiphase Flow, 1987, vol. 13, no. 1, pp. 113–121.

    MATH  Article  Google Scholar 

  40. 40.

    Tomiyama, A., Tamai, H., Zun, I., and Hosokawa, S., Transverse Migration of Single Bubbles in Simple Shear Flows, Chem. Eng. Sci., 2002, vol. 57, no. 11, pp. 1849–1958.

    Article  Google Scholar 

  41. 41.

    Antal, S.P., Lahey, R.T., Jr., and Flaherty, J.E., Analysis of Phase Distribution in Fully Developed Laminar Bubbly Two-Phase Flow, Int. J. Multiphase Flow, 1991, vol. 17, no. 5, pp. 635–652.

    MATH  Article  Google Scholar 

  42. 42.

    Tomiyama, A., Struggle with Computational Bubble Dynamics, Procs. of the 3rd Int. Conf. onMultiphase Flow ICMF’98, Lyon, France, June 8–12, 1998.

    Google Scholar 

  43. 43.

    Nguyen, V.T., Song, C.-H., Bae, B.U., and Euh, D.J., Modeling of Bubble Coalescence and Break-Up Considering Turbulent Suppression Phenomena in Bubbly Two-Phase Flow, Int. J. Multiphase Flow, 2013, vol. 54, no. 1, pp. 31–42.

    Article  Google Scholar 

  44. 44.

    Yao, W. and Morel, C., Volumetric Interfacial Area Prediction in Upward Bubbly Two-Phase Flow, Int. J. Heat Mass Transfer, 2004, vol. 47, no. 2, pp. 307–328.

    MATH  Article  Google Scholar 

  45. 45.

    Hanjalic, K. and Jakirlic, S., Contribution Towards the Second-Moment Closure Modelling of Separating Turbulent Flows, Comp. Fluids, 1998, vol. 27, no. 2, pp. 137–156.

    MATH  Article  Google Scholar 

  46. 46.

    Anderson, D.A., Tannehill, J.C., and Pletcher, R.H., Computational Fluid Mechanics and Heat Transfer, 2nd ed., New York: Taylor and Francis, 1997.

    MATH  Google Scholar 

  47. 47.

    Ganchev, B.G. and Peresadko, V.G., Hydrodynamic and Heat Transfer Processes in Descending Bubbly Flows, J. Eng. Phys. Thermophys., 1985, vol. 49, pp. 181–189.

    Google Scholar 

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Acknowledgments

The numerical results were obtained with partial financial support from the grant of the Russian Foundation for Basic Research (RFBR project No. 18-08-00477), and the mathematical model was developed as part of the IT SB RAS state assignment (program AAAA-A17-117030310010-9). The authors thank Dr. R. Mukin (Paul Scherrer Institute, Villingen, Switzerland) for providing data from MTLOOP experiments and his own numerical calculations in electronic form.

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Correspondence to M. A. Pakhomov or V. I. Terekhov.

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Pakhomov, M.A., Terekhov, V.I. Modeling of Flow Structure, Bubble Distribution, and Heat Transfer in Polydispersed Turbulent Bubbly Flow Using the Method of Delta Function Approximation. J. Engin. Thermophys. 28, 453–471 (2019). https://doi.org/10.1134/S1810232819040015

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