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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.

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## 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. 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

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

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