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Research Papers: Multiphase Flows

Comparison of Moment-Based Methods for Representing Droplet Size Distributions in Supersonic Nucleating Flows of Steam

[+] Author and Article Information
Ali Afzalifar

School of Energy Systems,
Lappeenranta University of Technology,
Lappeenranta 53850, Finland
e-mail: ali.afzalifar@lut.fi

Teemu Turunen-Saaresti

School of Energy Systems,
Lappeenranta University of Technology,
Lappeenranta 53850, Finland
e-mail: teemu.turunen-saaresti@lut.fi

Aki Grönman

School of Energy Systems,
Lappeenranta University of Technology,
Lappeenranta 53850, Finland
e-mail: aki.gronman@lut.fi

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received July 30, 2016; final manuscript received May 6, 2017; published online October 19, 2017. Assoc. Editor: Olivier Coutier-Delgosha.

J. Fluids Eng 140(2), 021301 (Oct 19, 2017) (13 pages) Paper No: FE-16-1486; doi: 10.1115/1.4037979 History: Received July 30, 2016; Revised May 06, 2017

This paper investigates the performance of moment-based methods and a monodispersed model (Mono) in predicting the droplet size distribution and behavior of wet-steam flows. The studied moment-based methods are a conventional method of moments (MOM) along with its enhanced version using Gaussian quadrature, namely the quadrature method of moments (QMOM). The comparisons of models are based on the results of an Eulerian–Lagrangian (E–L) method, as the benchmark calculations, providing the full spectrum of droplet size. In contrast, for the MOM, QMOM, and Mono an Eulerian reference frame is chosen to cast all the equations governing the phase transition and fluid motion. This choice of reference frame is essential to draw a meaningful comparison regarding complex flows in wet-steam turbines as the most important advantage of the moment-based methods is that the moment-transport equations can be conveniently solved in an Eulerian frame. Thus, the moment-based method can avoid the burdensome challenges in working with a Lagrangian framework for complicated flows. The main focus is on the accuracy of the QMOM and MOM in representing the water droplet size distribution. The comparisons between models are made for two supersonic low-pressure nozzle experiments reported in the literature. Results show that the QMOM, particularly inside the nucleation zone, predicts moments closer to those of the E–L method. Therefore, for the test case in which the nucleation is significant over a large proportion of the domain, the QMOM provides results in clearly better agreements with the E–L method in comparison with the MOM.

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References

Moore, M. J. , and Sieverding, C. H. , 1976, Two-Phase Steam Flow in Turbines and Separators, Hemisphere Publishing Corporation, Washington, DC.
White, A. J. , Young, J. B. , and Walters, P. T. , 1996, “ Experimental Validation of Condensing Flow Theory for a Stationary Cascade of Steam Turbine Blade,” Philos. Trans. R. Soc. London, Ser. A, 354(1704), pp. 59–88. [CrossRef]
Bakhtar, F. , Henson, R. J. K. , and Mashmoushy, H. , 2006, “ On the Performance of a Cascade of Turbine Rotor Tip Section Blading in Wet Steam—Part 5: Theoretical Treatment,” Proc. Inst. Mech. Eng. Part C, 220(4), pp. 457–472. [CrossRef]
Young, J. B. , 1992, “ Two-Dimensional, Nonequilibrium, Wet Steam Calculations for Nozzles and Turbine Cascades,” ASME J. Turbomach., 114(3), pp. 569–579. [CrossRef]
White, A. J. , and Young, J. B. , 1993, “ Time-Marching Method for the Prediction of Two-Dimensional, Unsteady Flows of Condensing Steam,” J Propul. Power, 9(4), pp. 579–587. [CrossRef]
Mccallum, M. , and Hunt, R. , 1999, “ The Flow of Wet Steam in a One‐Dimensional Nozzle,” Int. J. Numer. Methods Eng., 44(12), pp. 1807–1821. [CrossRef]
Gerber, A. G. , 2008, “ Inhomogeneous Multifluid Model for Prediction of Nonequilibrium Phase Transition and Droplet Dynamics,” ASME J. Fluids Eng., 130(3), pp. 1–11. [CrossRef]
Dykas, S. , and Wróblewski, W. , 2011, “ Single- and Two-Fluid Models for Steam Condensing Flow Modeling,” Int. J. Multiphase Flow, 37(9), pp. 1245–1253. [CrossRef]
White, A. J. , and Hounslow, M. J. , 2000, “ Modelling Droplet Size Distributions in Polydispersed Wet-Steam Flows,” Int. J. Heat Mass Transfer, 43(11), pp. 1873–1884. [CrossRef]
Simpson, D. A. , and White, A. J. , 2005, “ Viscous and Unsteady Flow Calculations of Condensing Steam in Nozzles,” Int. J. Heat Fluid Flow, 26(1), pp. 71–79. [CrossRef]
McGraw, R. , 1997, “ Description of Aerosol Dynamics by the Quadrature Method of Moments,” Aerosol Sci. Technol., 27(2), pp. 255–265. [CrossRef]
Gerber, A. G. , and Mousavi, A. , 2007, “ Application of Quadrature Method of Moments to the Polydispersed Droplet Spectrum in Transonic Steam Flows With Primary and Secondary Nucleation,” Appl. Math. Model., 31(8), pp. 1518–1533. [CrossRef]
Gerber, A. G. , and Mousavi, A. , 2007, “ Representing Polydispersed Droplet Behavior in Nucleating Steam Flow,” ASME J. Fluids Eng., 129(11), pp. 1404–1414. [CrossRef]
White, A. J. , 2003, “ A Comparison of Modeling Methods for Polydispersed Wet-Steam Flow,” Int. J. Numer. Methods Eng., 57(6), pp. 819–834. [CrossRef]
Becker, R. , and Döring, W. , 1935, “ Kinetische Behandlung der Keimbildung in Übersättigten Dämpfen,” Ann. Phys., 416(8), pp. 719–752. [CrossRef]
Zeldovich, J. B. , 1943, “ On the Theory of New Phase Formation: Cavitation,” Acta Physicochim. URSS, 12(1), pp. 1–22. https://www.degruyter.com/view/books/9781400862979/9781400862979.120/9781400862979.120.xml
Kantrowitz, A. , 1951, “ Nucleation in Very Rapid Vapor Expansions,” J. Chem. Phys., 19(9), pp. 1097–1100. [CrossRef]
Courtney, W. G. , 1961, “ Remarks on Homogeneous Nucleation,” J. Chem. Phys., 35(6), pp. 2249–2250. [CrossRef]
Gyarmathy, G. , 1960, “ Grundlagen einer Theorie der Nassdampfturbine,” Ph.D. thesis, ETH Zürich, Zürich, Switzerland.
Young, J. B. , 1982, “ The Spontaneous Condensation of Steam in Supersonic Nozzles,” PhysicoChem. Hydrodyn., 3(1), pp. 57–82.
Vukalovich, M. P. , 1958, Thermodynamic Properties of Water and Steam, 6th ed., Mashgis, Moscow, Russia.
Keenan, J. H. , Keyes, F. G. , Hill, P. G. , and Moore, J. G. , 1978, Steam Tables: Thermodynamics Properties of Water Including Vapor, Liquid and Solid Phases, Wiley, New York.
Young, J. B. , 1988, “ An Equation of State for Steam for Turbomachinery and Other Flow Calculations,” ASME J. Eng. Gas Turbines Power, 110(1), pp. 1–7. [CrossRef]
Liou, M. S. , and Steffen, C. J. , 1993, “ A New Flux Splitting Scheme,” J. Comput. Phys., 107(1), pp. 23–39. [CrossRef]
Koren, B. , 1993, Numerical Methods for Advection-Diffusion Problems, Vieweg, Braunschweig, Germany, Chap. 5.
Gerber, A. G. , 2002, “ Two-Phase Eulerian/Lagrangian Model for Nucleating Steam Flow,” ASME J. Fluids Eng., 124(2), pp. 465–475. [CrossRef]
Hill, P. G. , 1966, “ Condensation of Water Vapour During Supersonic Expansion in Nozzles,” J. Fluid Mech., 25(3), pp. 593–620. [CrossRef]
Hulburt, H. M. , and Katz, S. , 1964, “ Some Problems in Particle Technology: A Statistical Mechanical Formulation,” Chem. Eng. Sci., 19(8), pp. 555–574. [CrossRef]
Marchisio, D. L. , Pikturna, J. T. , Fox, R. O. , Vigil, R. D. , and Barresi, A. A. , 2003, “ Quadrature Method of Moments for Population-Balance Equations,” AIChE J., 49(5), pp. 1266–1276. [CrossRef]
Marchisio, D. L. , and Fox, R. O. , 2013, Computational Models for Polydisperse Particulate and Multiphase Systems, Cambridge University Press, Cambridge, UK. [CrossRef]
Moore, M. J. , Walters, P. T. , Crane, R. I. , and Davidson, B. J. , 1975, “ Predicting the Fog Drop Size in Wet Steam Turbines,” Wet Steam 4 Conference, Coventry, UK, Apr. 3–5, Paper No. C37/73.
Moses, C. , and Stein, G. , 1978, “ On the Growth of Steam Droplets Formed in a Laval Nozzle Using Both Static Pressure and Light Scattering Measurements,” ASME J. Fluids Eng., 100(3), pp. 311–322. [CrossRef]
Kermani, M. J. , and Gerber, A. G. , 2003, “ A General Formula for the Evaluation of Thermodynamic and Aerodynamic Losses in Nucleating Steam Flow,” Int. J. Heat Mass Transfer, 46(17), pp. 3265–3278. [CrossRef]
Desjardins, O. , Fox, R. O. , and Villedieu, P. , 2008, “ A Quadrature-Based Moment Method for Dilute Fluid-Particle Flows,” J. Comput. Phys., 227(4), pp. 2514–2539. [CrossRef]
Vikas, V. , Wang, Z. J. , Passalacqua, A. , and Fox, R. O. , 2010, “ Development of High-Order Realizable Finite Volume Schemes for Quadrature-Based Moment Method,” AIAA Paper No. 2010-1080. http://lib.dr.iastate.edu/cbe_conf/10/
Vikas, V. , Wang, Z. J. , Passalacqua, A. , and Fox, R. O. , 2011, “ Realizable High-Order Finite Volume Schemes for Quadrature-Based Moment Methods,” J. Comput. Phys., 230(13), pp. 5328–5352. [CrossRef]
Kah, D. , Laurent, F. , Massot, M. , and Jay, S. , 2012, “ A High Order Moment Method Simulating Evaporation and Advection of a Polydisperse Liquid 522 Spray,” J. Comput. Phys., 231(2), pp. 394–422. [CrossRef]
Afzalifar, A. , Turunen-Saaresti, T. , and Grönman, A. , 2016, “ Non-Realizability Problem With Quadrature Method of Moments in Wet-Steam Flows and Solution Techniques,” ASME J. Eng. Gas Turbines Power, 139(1), p. 012602. [CrossRef]
Afzalifar, A. , Turunen-Saaresti, T. , and Grönman, A. , 2016, “ Origin of Droplet Size Underprediction in Modeling of Low Pressure Nucleating Flows of Steam,” Int. J. Multiphase Flow, 86, pp. 86–98 [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Normalized dimensions of nozzles (top) and distributions of supersaturation across the nozzles (bottom) obtained by the E–L method

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Fig. 2

Comparison of weights from QMOM using grid sizes of 1000, 2000, and 3000 elements for nozzle A. Ng denotes the grid size.

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Fig. 3

Comparison of radii from QMOM using grid sizes of 1000, 2000, and 3000 elements for nozzle A. Ng denotes the grid size.

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Fig. 4

Comparison of pressure distributions in nozzle A

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Fig. 5

Comparison of droplet mean diameter distributions in nozzle A

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Fig. 6

Distributions of normalized moments, relative to the E–L, of MOM (top) and QMOM (bottom) over the nucleation zone in nozzle A

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Fig. 7

Comparison of distributions for weights (top) and radii (bottom) between QMOM and MOM in nozzle A

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Fig. 8

Comparison of discrete size distributions using Gaussian quadrature for QMOM and MOM with E–L at X/s = 0.3(top), X/s = 1.5 (center) and X/s = 7.0 (bottom)

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Fig. 9

Distributions of nucleation rates and growth rates of droplet sizes present in calculations of QMOM and MOM in Nozzle A

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Fig. 10

Distributions of nucleation rates and growth rates of droplet sizes present in calculations of QMOM and MOM in Exp. 252

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Fig. 11

Distributions of normalized moments, relative to the E–L, of MOM (top) and QMOM (bottom) over the nucleation zone in Exp. 252

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Fig. 12

Comparison of distributions for weights (top) and radii (bottom) between QMOM and MOM in Exp. 252

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Fig. 13

Comparison of pressure distributions in Exp. 252

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Fig. 14

Comparison of droplet mean diameter distributions in Exp. 252

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