0
Research Papers: Gas Turbines: Turbomachinery

# Nonrealizability Problem With Quadrature Method of Moments in Wet-Steam Flows and Solution Techniques

[+] 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 Turbomachinery Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received June 20, 2016; final manuscript received June 22, 2016; published online August 16, 2016. Editor: David Wisler.

J. Eng. Gas Turbines Power 139(1), 012602 (Aug 16, 2016) (9 pages) Paper No: GTP-16-1250; doi: 10.1115/1.4034196 History: Received June 20, 2016; Revised June 22, 2016

## Abstract

The quadrature method of moments (QMOM) has recently attracted much attention in representing the size distribution of liquid droplets in wet-steam flows using the $n$-point Gaussian quadrature. However, solving transport equations of moments using high-order advection schemes is bound to corrupt the moment set, which is then termed as a nonrealizable moment set. The problem is that the failure and success of the Gaussian quadrature are unconditionally dependent on the realizability of the moment set. First, this article explains the nonrealizability problem with the QMOM. Second, it compares two solutions to preserve realizability of the moment sets. The first solution applies a so-called “quasi-high-order” advection scheme specifically proposed for the QMOM to preserve realizability. However, owing to the fact that wet-steam models are usually built on existing numerical solvers, in many cases modifying the available advection schemes is either impossible or not desired. Therefore, the second solution considers correction techniques directly applied to the nonrealizable moment sets instead of the advection scheme. These solutions are compared in terms of accuracy in representing the droplet size distribution. It is observed that a quasi-high-order scheme can be reliably applied to guarantee realizability. However, as with all the numerical models in an Eulerian reference frame, in general, its results are also sensitive to the grid resolution. In contrast, the corrections applied to moments either fail in identifying and correcting the invalid moment sets or distort the shape of the droplet size distribution after the correction.

<>

## References

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.
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.
Young, J. B. , 1992, “ Two-Dimensional, Nonequilibrium, Wet Steam Calculations for Nozzles and Turbine Cascades,” ASME J. Turbomach., 114(3), pp. 569–579.
Bakhtar, F. , and Mohammadi Tochai, M. T. , 1980, “ An Investigation of Two-Dimensional Flows of Nucleating and Wet Steam by the Time-Marching Method,” Int. J. Heat Fluid Flow, 2(1), pp. 5–18.
McGraw, R. , 1997, “ Description of Aerosol Dynamics by the Quadrature Method of Moments,” Aerosol Sci. Technol., 27(2), pp. 255–265.
Gerber, A. G. , and Mousavi, A. , 2006, “ 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.
Gerber, A. G. , and Mousavi, A. , 2007, “ Representing Polydispersed Droplet Behavior in Nucleating Steam Flow,” ASME J. Fluids Eng., 129(11), pp. 1404–1414.
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.
McGraw, R. , 2006, “ Correcting Moment Sequences for Errors Associated With Advective Transport,” Brookhaven National Laboratory, Upton, NY.
Wright, D. L. , 2007, “ Numerical Advection of Moments of the Particle Size Distribution in Eulerian Models,” J. Aerosol Sci., 38(3), pp. 352–369.
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.
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.
Kah, D. , Laurent, F. , Massot, M. , and Jay, S. , 2012, “ A High Order Moment Method Simulating Evaporation and Advection of a Polydisperse Liquid Spray,” J. Comput. Phys., 231(2), pp. 394–422.
Becker, R. , and Döring, W. , 1935, “ Kinetische Behandlung der Keimbildung in Übersättigten Dämpfen,” Ann. Phys., 416(8), pp. 719–752.
Zeldovich, J. B. , 1943, “ On the Theory of New Phase Formation: Cavitation,” Acta Physicochim. URSS, 12, pp. 1–22.
Young, J. B. , 1982, “ The Spontaneous Condensation of Steam in Supersonic Nozzles,” PhysicoChem. Hydrodyn., 3(1), pp. 57–82.
Bakhtar, F. , Young, J. B. , White, A. J. , and Simpson, D. A. , 2005, “ Classical Nucleation Theory and Its Application to Condensing Steam Flow Calculations,” Proc. Inst. Mech. Eng. Part C, 219(12), pp. 1315–1333.
Kantrowitz, A. , 1951, “ Nucleation in Very Rapid Vapor Expansions,” J. Chem. Phys., 19(9), pp. 1097–1100.
Courtney, W. G. , 1961, “ Remarks on Homogeneous Nucleation,” J. Chem. Phys., 35(6), pp. 2249–2250.
Gyarmathy, G. , 1960, “ Grundlagen einer Theorie der Nassdampfturbine,” Ph.D. thesis, ETH Zürich, Zürich, Switzerland.
Vukalovich, M. P. , 1958, Thermodynamic Properties of Water and Steam, 6th ed., Mashgis, Moscow, Russia.
Bakhtar, F. , and Piran, M. , 1979, “ Thermodynamic Properties of Supercooled Steam,” Int. J. Heat Fluid Flow, 1(2), pp. 53–62.
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, NY.
White, A. J. , 2003, “ A Comparison of Modeling Methods for Polydispersed Wet-Steam Flow,” Int. J. Numer. Methods Eng., 57(6), pp. 819–834.
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.
Marchisio, D. L. , and Fox, R. O. , 2013, Computational Models for Polydisperse Particulate and Multiphase Systems, Cambridge University Press, Cambridge, UK.
Liou, M. S. , and Steffen, C. J. , 1993, “ A New Flux Splitting Scheme,” J. Comput. Phys., 107(1), pp. 23–39.
Roe, P. L. , 1986, “ Characteristic-Based Schemes for the Euler Equations,” Ann. Rev. Fluid Mech., 18(1), pp. 337–365.
Koren, B. , 1993, Numerical Methods for Advection-Diffusion Problems, Vieweg, Braunschweig, Germany, Chap. 5.
Shohat, J. A. , and Tamarkin, J. D. , 1943, The Problem of Moments, American Mathematical Society, New York, NY.
Moore, M. J. , Walters, P. T. , Crane, R. I. , and Davidson, B. J. , 1975, “ Predicting the Fog Drop Size in Wet Steam Turbines,” Institute of Mechanical Engineers (UK), Wet Steam 4 Conference, University of Warwick, Paper No. C37/73.

## Figures

Fig. 1

The nozzle B geometry

Fig. 2

Comparison of weight distributions along the nozzle centerline using different grid sizes; Ng  indicates the grid size

Fig. 8

Comparison of weight distributions, along the nozzle centerline, applying the realizable quasi-second-order advection scheme, denoted by QS, and Wright's moment correction, denoted by WM

Fig. 9

Comparison of pressures (top) and  d32  (bottom), along the nozzle centerline, applying the realizable quasi-second-order advection scheme, denoted by QS, and Wright's moment correction, denoted by WM

Fig. 3

Comparison of radii distributions along the nozzle centerline using different grid sizes; Ng  indicates the grid size

Fig. 4

Comparison of pressure distributions along the nozzle centerline using different grid sizes; Ng  indicates the grid size

Fig. 5

Comparison of Sauter mean diameter distributions along the nozzle centerline using different grid sizes; Ng  indicates the grid size

Fig. 6

Distributions of radii, critical radius (top), and weights (bottom) applying McGraw's moment correction

Fig. 7

Comparison of radius distributions in the nucleation zone, applying the realizable quasi-second-order advection scheme, denoted by QS, and Wright's moment correction, denoted by WM

## Errata

Some tools below are only available to our subscribers or users with an online account.

### Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related Proceedings Articles
Related eBook Content
Topic Collections