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

References
1.
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
.
2.
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
.
3.
Young
,
J. B.
,
1992
, “
Two-Dimensional, Nonequilibrium, Wet Steam Calculations for Nozzles and Turbine Cascades
,”
ASME J. Turbomach.
,
114
(
3
), pp.
569
579
.
4.
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
.
5.
McGraw
,
R.
,
1997
, “
Description of Aerosol Dynamics by the Quadrature Method of Moments
,”
Aerosol Sci. Technol.
,
27
(
2
), pp.
255
265
.
6.
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
.
7.
Gerber
,
A. G.
, and
Mousavi
,
A.
,
2007
, “
Representing Polydispersed Droplet Behavior in Nucleating Steam Flow
,”
ASME J. Fluids Eng.
,
129
(
11
), pp.
1404
1414
.
8.
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
.
9.
McGraw
,
R.
,
2006
, “
Correcting Moment Sequences for Errors Associated With Advective Transport
,”
Brookhaven National Laboratory
, Upton, NY.
10.
Wright
,
D. L.
,
2007
, “
Numerical Advection of Moments of the Particle Size Distribution in Eulerian Models
,”
J. Aerosol Sci.
,
38
(
3
), pp.
352
369
.
11.
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.
12.
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
.
13.
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
.
14.
Becker
,
R.
, and
Döring
,
W.
,
1935
, “
Kinetische Behandlung der Keimbildung in Übersättigten Dämpfen
,”
Ann. Phys.
,
416
(
8
), pp.
719
752
.
15.
Zeldovich
,
J. B.
,
1943
, “
On the Theory of New Phase Formation: Cavitation
,”
Acta Physicochim. URSS
,
12
, pp.
1
22
.
16.
Young
,
J. B.
,
1982
, “
The Spontaneous Condensation of Steam in Supersonic Nozzles
,”
PhysicoChem. Hydrodyn.
,
3
(1), pp.
57
82
.
17.
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
.
18.
Kantrowitz
,
A.
,
1951
, “
Nucleation in Very Rapid Vapor Expansions
,”
J. Chem. Phys.
,
19
(
9
), pp.
1097
1100
.
19.
Courtney
,
W. G.
,
1961
, “
Remarks on Homogeneous Nucleation
,”
J. Chem. Phys.
,
35
(
6
), pp.
2249
2250
.
20.
Gyarmathy
,
G.
,
1960
, “
Grundlagen einer Theorie der Nassdampfturbine
,”
Ph.D. thesis
, ETH Zürich,
Zürich, Switzerland
.
21.
Vukalovich
,
M. P.
,
1958
,
Thermodynamic Properties of Water and Steam
,
6th ed.
,
Mashgis
,
Moscow, Russia
.
22.
Bakhtar
,
F.
, and
Piran
,
M.
,
1979
, “
Thermodynamic Properties of Supercooled Steam
,”
Int. J. Heat Fluid Flow
,
1
(
2
), pp.
53
62
.
23.
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
.
24.
White
,
A. J.
,
2003
, “
A Comparison of Modeling Methods for Polydispersed Wet-Steam Flow
,”
Int. J. Numer. Methods Eng.
,
57
(
6
), pp.
819
834
.
25.
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
.
26.
Marchisio
,
D. L.
, and
Fox
,
R. O.
,
2013
,
Computational Models for Polydisperse Particulate and Multiphase Systems
,
Cambridge University Press
,
Cambridge, UK
.
27.
Liou
,
M. S.
, and
Steffen
,
C. J.
,
1993
, “
A New Flux Splitting Scheme
,”
J. Comput. Phys.
,
107
(
1
), pp.
23
39
.
28.
Roe
,
P. L.
,
1986
, “
Characteristic-Based Schemes for the Euler Equations
,”
Ann. Rev. Fluid Mech.
,
18
(
1
), pp.
337
365
.
29.
Koren
,
B.
,
1993
,
Numerical Methods for Advection-Diffusion Problems
, Vieweg, Braunschweig, Germany, Chap. 5.
30.
Shohat
,
J. A.
, and
Tamarkin
,
J. D.
,
1943
,
The Problem of Moments
,
American Mathematical Society
,
New York, NY
.
31.
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.
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