annals of nuclear energy - wordpress.com · 2016. 4. 29. · regulatory commission, 2001) and trace...

Numerical implementation, verification and validation of two-phaseflow four-equation drift flux model with Jacobian-free Newton–Krylovmethod

Ling Zou ⇑, Haihua Zhao, Hongbin ZhangIdaho National Laboratory, P.O. Box 1625, Idaho Falls, ID 83415-3870, USA

a r t i c l e i n f o

Article history:Received 1 June 2015Received in revised form 27 July 2015Accepted 28 July 2015Available online 24 August 2015

Keywords:Drift flux modelJacobian-free Newton–Krylov methodVerificationValidation

a b s t r a c t

This paper presents a numerical investigation on using the Jacobian-free Newton–Krylov (JFNK) methodto solve the two-phase flow four-equation drift flux model with realistic constitutive correlations(‘closure models’). The drift flux model is based on Isshi and his collaborators’ work. Additional constitutivecorrelations for vertical channel flow, such as two-phase flow pressure drop, flow regime map, wall boil-ing and interfacial heat transfer models, were taken from the RELAP5-3D Code Manual and included tocomplete the model. The staggered grid finite volume method and fully implicit backward Euler methodwas used for the spatial discretization and time integration schemes, respectively. The Jacobian-freeNewton–Krylov method shows no difficulty in solving the two-phase flow drift flux model with a discreteflow regime map. In addition to the Jacobian-free approach, the preconditioning matrix is obtained byusing the default finite differencing method provided in the PETSc package, and consequently thelabor-intensive implementation of complex analytical Jacobian matrix is avoided. Extensive and success-ful numerical verification and validation have been performed to prove the correct implementation of themodels and methods. Code-to-code comparison with RELAP5-3D has further demonstrated the successfulimplementation of the drift flux model.

! 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Accurate modeling and simulation of the two-phase flow phe-nomena are critical to the safety analysis of nuclear power reac-tors. Two-phase flow problems can generally be formulated usingdrift flux models or two-fluid models. Several existing reactorsafety system analysis codes, such as RELAP5 (U.S. NuclearRegulatory Commission, 2001) and TRACE (U.S. NuclearRegulatory Commission, 2010), have achieved great successes byemploying the two-fluid six-equation two-phase flow model thattreats the two phases separately with the interfacial interactionsconsidered by constitutive correlations (‘‘closure models’’). Thedrift flux models (Zuber and Findlay, 1965; Ishii, 1977; Ishii andHibiki, 2011), on the other hand, treat the two phases as a mixture,and the models are formulated to consider the conservation lawsof the mixture. The relative motion between the two phases is trea-ted by constitutive correlations. Although the drift flux modelshave limitations in certain applications, they are widely used inmany applications due to their simplicity and applicability to a

wide range of two-phase flow problems. For example, theRETRAN-3D (Electric Power Research Institute, 1998) code usesthe drift flux models and has many applications in reactor tran-sient analyses including a small break loss-of-coolant accident.The TASS/SMR system analysis code is developed based on athree-equation drift flux model (with an additional mass equationfor the non-condensable gas) for the system-integrated modularadvanced reactor, SMART (Chung et al., 2012; Chung et al., 2013).Drift flux models have also been widely used in the subchannelanalysis of boiling water reactor (BWR) fuel bundles Khan and Yi,1985; Hashemi-Tilehnoee and Rahgoshay, 2013a,b, BWR core sim-ulators (Galloway, 2010), and two-phase flow instabilities analyses(Nayak, 2007; Wang et al., 2011).

Comparing to the more complex two-fluid six-equation model,the drift flux models are relatively easier to solve and implementinto a computer code. However, due to the nonlinear closure mod-els, iterative methods are normally used in solving such equationsystems to achieve convergence (Galloway, 2010; Talebi et al.,2012). The Jacobian-free Newton–Krylov (JFNK) method has gainedmany successes in solving nonlinear systems in different disci-plines (Knoll and Keyes, 2004). Mousseau has done severalpioneering works (Mousseau, 2004, 2005) to use such a method

http://dx.doi.org/10.1016/j.anucene.2015.07.0330306-4549/! 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (L. Zou).

Annals of Nuclear Energy 87 (2016) 707–719

Contents lists available at ScienceDirect

Annals of Nuclear Energy

journal homepage: www.elsevier .com/locate /anucene

http://crossmark.crossref.org/dialog/?doi=10.1016/j.anucene.2015.07.033&domain=pdfhttp://dx.doi.org/10.1016/j.anucene.2015.07.033mailto:[email protected]://dx.doi.org/10.1016/j.anucene.2015.07.033http://www.sciencedirect.com/science/journal/03064549http://www.elsevier.com/locate/anucene

to solve the two-fluid problems implicitly. In our previous works(Zou et al., 2015a,b), we have successfully demonstrated the appli-cations of the JFNK method in several simplified two-fluid prob-lems using a high-resolution spatial discretization scheme on thestaggered grid. It should be pointed out that by far only simplifiedand continuous closure models have been used in applying theJFNK method to solve two-phase flow problems with the two fluidmodels (Mousseau, 2004, 2005; Zou et al., 2015a,b). Recently, theJFNK method has also been used in newly developed reactor sys-tem analysis code (Idaho National Laboratory, 2012, 2014) andits application in multi-physics simulations of nuclear reactors(Gaston et al., 2015). However, in these works, only single-phaseflow model or simplified two-phase flow model was used. Thereare no published works showing JFNK applications with realisticclosure models such as those used in the RELAP5 code with the fullspectrum of flow regimes. The discontinuities in solution space,due to the discrete flow regimes, could potentially prevent theNewton’s method from converging. Additionally, an effective pre-conditioning scheme is required to help the Krylov method con-verge efficiently. The same challenges are also present whenapplying the JFNK method to solve the drift the drift flux models.Based on these discussions, we believe that it is still lack of fullunderstanding of the JFNK method in the applications of solvingrealistic two-phase flow problems. Resolving the potential issuesaforementioned are critical to prove the practicability of the JFNKmethod applied to solving the realistic two-phase flow problems.

In this work, our objective is to investigate application of theJFNK method to solve the four-equation drift flux model with real-istic and discrete closure models. Identifying and resolving numer-ical issues are also the purpose of this work. Section 2 providesmodel descriptions for the drift flux model along with the kine-matic closure correlations developed by Ishii and Hibiki (2011),as well as additional constitutive correlations required to fullyclose the system. Section 3 presents the numerical methods tosolve the drift flux model. Section 4 presents the numerical verifi-cation and validation of the code using experimental data, as wellas code comparison to the RELAP5-3D (RELAP5-3D, 2012a).Section 5 presents discussions and conclusions.

2. Physical model descriptions

2.1. One-dimensional four-equation drift flux model

The one-dimensional four-equation drift flux model used in thiswork is directly adapted from the models developed by Isshi andhis coworkers (Ishii, 1977; Ishii and Hibiki, 2011; Hibiki andIshii, 2003, 2005). The original set of equations for theone-dimensional drift flux model includes two continuity equa-tions (mixture and the dispersed phase), one mixture momentumequation and one mixture energy equation. In this work, the orig-inal set of equations has been simplified and is given as Eqs. (1–4):

@qm@tþ @ðqmvmÞ

@x¼ 0 ð1Þ

@ðaqgÞ@t

þ@ðaqgvmÞ

@xþ @@x

aqgqfqm

Vgj! "

¼ Cg ð2Þ

@vm@tþ vm

@vm@x¼ % 1qm

@p@x% gx %

f m2D

vmjvmj

% 1qm@

@xaqgqfð1% aÞqm

Vgj2# $

ð3Þ

@ðqmemÞ@t

þ @ðqmhmvmÞ@x

þ @@x

aqgqfqm

Dhgf Vgj# $

¼ q00waw þ vm þaðqf % qgÞ

qmVgj

# $@p@x

ð4Þ

in which the subscripts m, g and f denote the mixture, gas phase,and liquid phase, respectively. Vgj is the mean drift velocity of thegas phase. Comparing to the original equations in Hibiki andIshii’s 2003 journal article (Hibiki and Ishii, 2003) and Ishii’s 2011book (Ishii and Hibiki, 2011), there are several noticeable changesthat need explanations: (1) the mixture momentum equation hasbeen rewritten in the primitive form, which was done in a similarway to obtain the primitive momentum equation for thesingle-phase Euler equations; (2) in the mixture energy equation,

Nomenclatures

aw heating surface density, [1/m]e specific internal energy, [J/kg]f friction factor, [non-dimensional]||E|| error normEd entrainment rateF nonlinear functionG mass flux, [kg/m2-s]h specific enthalpy, [J/kg]hji two-phase volumetric flux, [m/s]J Jacobian matrixp pressure, [Pa]v velocity, [m/s]t time, [s]x axial distance, [m]q00 heat flux, [W/m2]T temperature, [K]U unknown vectorVgj drift velocity of gas phase, [m/s]Vgj mean drift velocity of gas phase, [m/s]v Krylov vectorxe equilibrium quality

Greek symbolsa void fraction, [non-dimensional]e mean absolute errorDx finite volume cell size, [m]Dt time step size, [s]2 perturbation parameter in Jacobian-free Newton–Krylov

methodq density, [kg/m3]Cg volumetric vapor generation rate, [kg/m3-s]r surface tension, [N/m]

Subscriptsf liquid phaseg gas phasei finite volume cell indexinlet inlet conditionm mixturesat saturation conditionw wall

708 L. Zou et al. / Annals of Nuclear Energy 87 (2016) 707–719

the two transient terms of enthalpy and pressure were combined toobtain the transient term of the internal energy; (3) the stress ten-sor and covariance terms were ignored; and (4) the drift terms inthe gas phase mass equation and the mixture energy equation havebeen moved to the left-hand side of the equations, which will benumerically treated as advection terms in the numerical implemen-tations (discussed later).

The primary variables to be solved from this set of equations arep, a, vm, and T, which are pressure, void fraction (volume fraction ofthe gas phase), mixture velocity, and temperature, respectively. Itis noted that the phasic temperatures, Tl and Tg, have not beenproperly defined yet. There are two possible choices for definingthe phasic temperatures. For one choice, it is assumed that thephasic temperatures are the same, such that, Tf = Tg = T. For theother choice, the vapor phase is always at the saturation condition,such that Tg = Tsat(p) and the liquid phase temperature can deviatefrom the saturation temperature (e.g., subcooled liquid) and Tf = T.More discussion can be found in Section 2 of Ref. (Hirt et al., 1979).According to that report, there is only little difference betweenthese two choices. However, for flow boiling problems, it is obviousthat the later choice is more appropriate. For example, under sub-cooled flow boiling conditions, it is incorrect to assume that thevapor temperature is the same as the (subcooled) liquidtemperature.

Hibiki and Ishii’s works (Ishii and Hibiki, 2011; Hibiki and Ishii,2003) provide a framework for the drift flux model, along withkinematic closure models for different flow regimes. However,other key constitutive models are absent. These include the criteriafor the transition of the flow regimes, two-phase pressure dropcoefficients, wall boiling model, and interfacial heat transfer coef-ficients, etc. To obtain a complete system of equations for numer-ical implementation into a computer code, the water/steamproperties and additional constitutive correlations are needed.These are summarized in the following subsections.

2.2. Water/steam properties

The water/steam properties (such as phasic density and phasicspecific internal energy) are provided as functions of mixture pres-sure and phasic temperature. For example, the liquid phase densityis obtained as:

qf ¼ qf ðp; Tf Þ ð5Þ

The water/steam properties used in this work is calculated byfast and accurate bilinear interpolations of the water/steam prop-erties provided by the International Association for the Propertiesof Water and Steam (IAPWS) (The International Association forthe Properties of Water and Steam). Water/steam properties areprepared in tables from the highly accurate yet expensive IAPWSfunction calls. When water/steam properties are needed, they areinterpolated from the pre-prepared tables. Implementation detailsare documented in Zou et al.’s 2014 report (Zou et al., 2014).

2.3. Drift flux model constitutive correlations and flow regimes

In this work, only the vertical flow regimes are considered.Implementation of the vertical flow regime map is based on thatused in the RELAP5-3D code (RELAP5-3D, 2012a; RELAP5-3D,2012b) (see Section 3.2 of Ref. (RELAP5-3D, 2012b) for moredetails). It is noted that, in RELAP5-3D, the slug flow regime andchurn flow regime are combined as a single a single flow regimecalled slug flow. To maintain consistency with RELAP5-3D, thesame strategy is adopted in this work. Consequently, the verticalflow regimes considered include single-phase liquid flow, bubblyflow, slug flow, transition region between slug and annular mist

flow, and annular mist flow. The derivations of Vgj and correspond-ing closure correlations for the two-phase flow regimes are sum-marized below.

As suggested by Hibiki and Ishii (Hibiki and Ishii, 2003), the Vgjterm used in Eqs. (2–4) have to be expressed as a function of otherknown variables and constitutive correlations. For bubbly and slugflow regimes, this can be done by combing Eqs. (9) and (15) inHibiki and Ishii’s 2003 journal article (Hibiki and Ishii, 2003) toobtain the following useful correlation:

Vgj ¼qmVgj þ qmðC0 % 1Þvmqm % ðC0 % 1Þaðqf%qgÞ

ð6Þ

in which, Vgj and C0 are the weighted mean drift velocity of the gasphase and the distribution parameter, respectively. A complete setof constitutive correlations for the weighted mean drift velocity ofthe gas phase (Vgj) and distribution parameter (C0) are provided inHibiki and Ishii (2003) work for all flow regimes. Eq. (6) has alsobeen obtained in Talebi et al. (2012) work.

For bubbly and slug flow regimes, the weighted mean driftvelocity Vgj is the following:

Bubbly flow:

Vgj ¼ffiffiffi2p Dqgr

q2f

!1=4ð1% aÞ1:75 ð7Þ

Slug flow:

Vgj ¼ 0:35Dqgrqf

!1=2ð8Þ

The simplified version of the distribution parameter (C0) forround tube is used for both the bubbly and slug flow regimes:

C0 ¼ 1:2% 0:2ffiffiffiffiffiffiffiffiffiffiffiffiffiqg=qf

qð9Þ

For the annular mist two-phase flow regime, the mean driftvelocity of the gas phase, Vgj, is given as a function of thetwo-phase volumetric flux, hji,

Vgj &ð1% aÞð1% EdÞaþ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiqg=qf

q hjiþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDqgDð1% aÞð1% EdÞ

0:015qf

s !ð10Þ

Where hji is an unknown as well. In this case, the value of Vgj can beobtained by combining Eqs. (10) and (15) in Hibiki and Ishii’s 2003journal article (Hibiki and Ishii, 2003) to cancel hji:

Vgj ¼C1vm þ C1C21% C1

aðqf%qg Þqm

ð11Þ

where, the two constants C1 and C2 are terms in the Eq. (64) ofHibiki and Ishii’s 2003 journal article (Hibiki and Ishii, 2003),defined as:

C1 ¼ð1% aÞð1% EdÞaþ 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiqg=qf

q ð12Þ

C2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDqgDð1% aÞð1% EdÞ

0:015qf

s

ð13Þ

It is noted that, in the annular mist flow regime, the entrain-ment rate of liquid droplet in the vapor core region is needed.The model to predict the entrainment rate is based on the modelproposed by Wallis (U.S. Nuclear Regulatory Commission, 2010).

More details and discussions can be found in Section 1.2.1 ofChapter 14 in Ishii and Hibiki’s 2011 book (Ishii and Hibiki,

L. Zou et al. / Annals of Nuclear Energy 87 (2016) 707–719 709

2011). For the value of Vgj at the slug-annual mist transition region,linear interpolation based on void fraction is used.

2.4. Frictional pressure drop

Constitutive correlations for the pressure drop due to frictionare needed for both the single- and two-phase flows. The correla-tions used in this work are mainly based on those used inRELAP5-3D (RELAP5-3D, 2012a). The single-phase flow frictionalfactor is calculated in three flow regimes, laminar flow, turbulentflow and a transition region between these two. The two-phaseflow frictional pressure drop is based on a two-phase multiplierapproach:

dpdx

! "

two phase¼ /2f

dpdx

! "

f¼ /2g

dpdx

! "

gð14Þ

In the context of the drift flux model, the wall friction term inthe momentum equation becomes:

f m2D

vmjvmj ¼1qm

dpdx

! "

two phaseð15Þ

in which, /f and /g are the liquid-alone and vapor/gas-alonetwo-phase Darcy-Weisbach friction multipliers, respectively. Moredetails of the implementations of frictional pressure drop can befound in Section 3.3.8 of Ref. (RELAP5-3D, 2012a).

2.5. Vapor generation rate

Additional constitutive correlations are needed to determinethe vapor generation rate, Cg term, in the gas phase continuityEq. (2). In RELAP5-3D RELAP5-3D, 2012a, this term is modeled intwo regions: the near wall region and bulk region. A similar con-cept is used here,

Cg ¼ Cw þ Cig ð16Þ

in which, Cw and Cig are the volumetric vapor generation rate in thenear wall region and bulk region, respectively. The near wall regionvolumetric vapor generation rate is determined by the method pro-posed by Lahey, combining with the Saha-Zuber correlation todetermine the necessary conditions for net void generation (seeSection 4.7.1.1 of Ref. (RELAP5-3D, 2012b):

Cw ¼

0 hf < hcrq00wawðhf%hcrÞ

ðhf ;sat%hcrÞð1þepÞhfghcr 6 hf 6 hl;sat

q00wawhfg

hf > hl;sat

8>>><

>>>:ð17Þ

in which, ep is the pumping term, defined as:

ep ¼qf ½hf ;sat %minðhf ;hf ;satÞ(

qghfgð18Þ

The vapor generation rate in the bulk region is determined bythe heat transfer between the two phases and interface, which isassumed to be at the saturation condition. In the drift flux model,the gas phase is assumed to be always at the saturation conditionin the boiling region, and thus the volumetric vapor generation inthe bulk region is reduced to:

Cig ¼HilðTl % TsatÞ

hfgð19Þ

in which, Hil is the volumetric interfacial heat transfer coefficient forthe liquid phase. This parameter is also flow regime dependent, andthe RELAP5-3D constitutive correlations are used (see Section 4.1 ofRef. (RELAP5-3D, 2012b).

3. Numerical and solution methods

Numerical and solution methods include: (1) the finite volumediscretization scheme based on the staggered grid mesh arrange-ment; (2) first-order fully implicit backward Euler time integrationscheme; and (3) JFNK method to solve the nonlinear equationsystem.

3.1. Finite volume method using staggered grid mesh

In this work, the finite volume method based on the staggeredgrid mesh has been used for the spatial discretization of the equa-tion system. For the staggered grid mesh arrangement, scalar vari-ables (such as pressure, void fraction, and temperature) arearranged at the cell centers, while vector variables (such as mix-ture velocity) are arranged at the cell edges, which are schemati-cally shown in Fig. 1.

With pressure, void fraction, and temperatures being defined atthe cell centers, phasic thermodynamic quantities are also definedat the cell centers and calculated from the water/steam propertiesfunctions. For example, the liquid density at the ith volume is calcu-lated as:

qf ;i ¼ qf ðpi; Tf ;iÞ ð20Þ

Similarly, the mixture thermodynamic properties are alsodefined at the cell centers and calculated from their definitions,for example:

qm;i ¼ aiqg;i þ ð1% aiÞqf ;i ð21Þ

It is worth noting that the weighted mean drift velocity of thegas phase (Vgj) and the distribution parameter (C0) are generallycorrelated with local phasic thermodynamic properties and voidfraction, etc., and therefore both have been defined at the cellcenters:

Vgj;i ¼ Vgjðai;qf ;i;qg;i; . . .Þ ð22Þ

C0;i ¼ C0ðai;qf ;i;qg;i; . . .Þ ð23Þ

Consequently, the mean drift velocity of the gas phase, Vgj,is also defined at the cell centers. When the value of a vari-able is required at the cell centers (or cell edges), but thevariable is not defined there, an arithmetic average is used,for example,

qm;iþ1=2 ¼12ðqm;iþ1 þ qm;i%1Þ ð24Þ

vm;i ¼12ðvm;i%1=2 þ vm;iþ1=2Þ ð25Þ

In this work, all advection terms (including those due to drift) inthe equations are treated similarly using the classic donor cell con-cept. It is a first-order but very stable spatial discretization scheme.Using the gas phase mass equation as an example, the mixtureadvection term on the left-hand side of the equation is discretizedas:

@ðaqgvmÞ@x

&&&&i¼ 1

DxðvmaqgÞ

)iþ1=2 % ðvmaqgÞ

)i%1=2

h ið26Þ

and

ðvmaqgÞ)iþ1=2 ¼ vm;iþ1=2

aiqg;i if vm;iþ1=2 > 0aiþ1qg;iþ1 otherwise

(ð27Þ

The drift term is also numerically treated as an advection term,and thus:


@

@xaqgqfqm

Vgj! "

i¼ 1

Dxaqgqfqm

Vgj! ")

iþ1=2%

aqgqfqm

Vgj! ")

i%1=2

" #ð28Þ

and

aqgqfqm

Vgj! ")

iþ1=2¼ Vgj

&&iþ1=2

aiqg;iqf ;iqm;i

if Vgjjiþ1=2 > 0aiþ1qg;iþ1qf ;iþ1

qm;iþ1otherwise

8<

: ð29Þ

The advection term and drift term in the energy equation can betreated similarly. The advection term of the mixture momentumequation is treated slightly differently as it is expressed in theprimitive form, for example,

vm@vm@x

&&&&iþ1=2

¼ 1Dx

vm;iþ1=2vm;iþ1=2 % vm;i%1=2 ifvm;iþ1=2 > 0vm;iþ3=2 % vm;iþ1=2 otherwise

(

ð30Þ

For all other source terms appearing in the equation, the centraldifferencing scheme is used when a gradient appears.

3.2. Boundary conditions

This subsection is not intended to provide a complete set ofboundary conditions for the drift flux model. However, it is focusedon describing the mass flux inlet and pressure outlet boundaryconditions generally found in many applications. Given the liquidphase inlet mass flux Gf,inlet, and vapor phase inlet mass flux,Gg,inlet, the inlet boundary condition for the mixture mass equationis computed as:

ðqmvmÞinlet ¼ Gf ;inlet þ Gg;inlet ð31Þ

For the vapor mass equation, the inlet boundary condition is givenby:

aqgvm þaqgqfqm

Vgj! "

inlet¼ Gg;inlet ð32Þ

For the mixture energy equation, the inlet boundary condition isgiven by:

qmhmvm þaqgqfqm

Dhgf Vgj! "

inlet¼ Gf ;inlethf ;inlet þ Gg;inlethg;inlet ð33Þ

where, hf,inlet and hg,inlet are the specific enthalpy for the liquid andgas phase evaluated at the inlet, respectively. When evaluatingthese two parameters, the inlet temperatures are normally givenand the inlet pressure comes from the numerical iterations (i.e.,the inlet pressure is part of the unknowns of the system). For theoutlet, an outlet pressure boundary is normally used for themomentum equation.

3.3. Time integration schemes

In this work, a fully implicit backward Euler time integrationscheme is used. Using the mixture phase continuity equation asan example, the semi-discretized form is expressed as:

qnþ1m % qnmDt

þ @ðqmvmÞnþ1

@x¼ 0 ð34Þ

in which, Dt is the time step size, and superscripts n and n + 1denote the old time step and current time step, respectively.

3.4. Jacobian-free Newton–Krylov method

The resulted discretized nonlinear equation from the implicitbackward Euler time integration scheme requires an iterativemethod to solve. In this work, the JFNK method is used, whichhas proved its capability in many disciplines (Knoll and Keyes,2004) and has been used in our previous works in solvingtwo-phase flow problems (Zou et al., 2015a,b). This subsectionbriefly discusses the JFNK method to solve nonlinear system. Fora discretized nonlinear system, such as the discretized drift fluxmodel equations, one solves:

FðUÞ ¼ 0 ð35Þ

for the unknown vector, U = [p, a, vm, T]T. The solution to the nonlin-ear system is obtained by iteratively solving a series of Newton’slinear correction equations:

JkdUk ¼ %FðUkÞ ð36Þ

where, Jk is the Jacobian matrix; Uk is the kth nonlinear step solu-tion; and dUk is the correction vector. In the JFNK frame, the linearsystem, Eq. (36), could be effectively solved with a Krylov’s method.In the Krylov’s method, only a matrix–vector product is requiredand can be approximated as:

Jkv & FðUkþ 2 vÞ % FðUkÞ

2ð37Þ

in which, v is the Krylov vector. After the correction vector, dUk, issolved from the linear system, the (k + 1)th nonlinear step solutioncould be updated as:

Ukþ1 ¼ Uk þ dUk ð38Þ

One of the advantages of using the JFNK method is that theexplicit formation of the Jacobian matrix could be avoided. Thederivation and code implementations of the analytical Jacobianmatrix could be a cumbersome and error-prone task fortwo-phase flow problems, since many thermal–hydraulics correla-tions have quite complicated forms. In our implementation of theJFNK method, the scientific computational toolkit PETSc (Balayet al., 2013) is used to solve the discretized nonlinear fluid equa-tions. It is important to point out that, the JFNK method generallyneeds a good preconditioning scheme to be effective and efficient.There are different ways to precondition the JFNK method, whichwere reviewed in Knoll and Keyes’ 2004 journal article (Knolland Keyes, 2004). In our work, the preconditioning matrix isobtained by using the default finite differencing method providedin the PETSc package, which showed a satisfactory performance.Convergence difficulties are sometimes observed in the simula-tions when a discontinuous solution space (jumping from one flowregime to another one) first appears, which could normally beresolved by reducing the time step size temporarily for severaltime steps.

Fig. 1. Schematic drawing of the staggered grid mesh arrangement.


4. Results and discussions

This section presents the numerical verification on the numeri-cal method in Section 4.1, extensive code validation by comparingthe numerical results with the experimental data in Section 4.2,and additional code-to-code comparison with the RELAP5-3D codein Section 4.3.

4.1. Numerical verification

Numerical verification is an essential part of the modern verifi-cation and validation (V&V) process. In general, numerical verifica-tion can be performed by the investigation on the convergencerates of the time integration and spatial discretization schemes.Rigorous numerical verification is a very good process to verifythe correctness of code implementation. Ideally, convergence ratesare obtained by comparing the numerical results with known ana-lytical solutions. However, for the drift flux model and those verycomplicated constitutive correlations used in this work, it is diffi-cult to find such analytical solutions. In such situations, numerical

results obtained from fine meshes and small time steps are nor-mally used as the reference solutions.

This subsection focuses on the numerical verification of the spa-tial discretization scheme, which is a first-order upwind scheme asdescribed in the previous sections. For a simulation case (Case 6,described in Section 4.2), steady-state solutions were obtainedusing 320 finite volume cells, which serve as the reference solu-tions for convergence study. Additional simulations, using finitevolume cells from 5 to 80, were then obtained. Following the rec-ommendation given in LeVeque’s 2004 book (LeVeque, 2004), theL-1 numerical error norm is used, which is defined as:

kE1k ¼ DxXN

i¼1j~ui % uref ðxiÞj ð39Þ

The convergence rates of the mesh refinement results are sum-marized in Table 1, for two main parameters in two-phase flowsimulations, equilibrium quality (xe) and void fraction (a). For bothparameters, convergence rates of approximately 1 were obtained,which are expected for the first-order spatial scheme.

4.2. Code validation and error analysis

The code validation work is performed comparing the numeri-cal results to the experimental data. The Bartolomei experimentaldata (Bartolomei et al., 1982; Bartolomei and Chanturiya, 1967) ofsubcooled flow boiling in vertical pipes have been chosen for thepurpose of validation. These experiments covered a wide rangeof conditions: (1) pressure from 1.5 to 15 MPa, (2) mass flux from405 to 2123 kg/(m2s), (3) wall heat flux from 0.38 to 2.21 MW/m2,

Table 1Spatial convergence rate of the Case 6 problem.

Ncell Dx (=L/Ncell) L-1 norm (xe) Rate L-1 norm (a) Rate

80 0.01875 5.4228E%03 – 1.8873E%03 –40 0.0375 1.2670E%02 1.22 4.4086E%03 1.2220 0.075 2.7365E%02 1.11 9.4696E%03 1.1010 0.15 5.6538E%02 1.05 1.9661E%02 1.05

5 0.3 1.1394E%01 1.01 4.0188E%02 1.03

Table 2Simulation cases based on experimental conditions (Bartolomei et al., 1982; Bartolomei and Chanturiya, 1967).

Case** P (MPa) Mass flux (kg/m2-s) Wall heat flux (MW/m2) Tinlet (K) DTsub,inlet (K)! Notes

1 6.89 985 1.13 464 94.04 Fig. 1, row 1, Bartolomei et al. (1982)2 6.78 1071 1.13 465 91.96 Fig. 1, row 2, Bartolomei et al. (1982)3 6.84 961 1.13 465 92.55 Fig. 1, row 3, Bartolomei et al. (1982)4 6.84 995 1.15 466 91.55 Fig. 1, row 4, Bartolomei et al. (1982)5 6.81 998 0.44 521 36.25 Fig. 2(a), row 1, Bartolomei et al. (1982)6 6.89 965 0.78 493 65.04 Fig. 2(a), row 2, Bartolomei et al. (1982)7 6.84 961 1.13 466 91.55 Fig. 2(a), row 3, Bartolomei et al. (1982)8 6.74 988 1.7 416 140.58 Fig. 2(a), row 4, Bartolomei et al. (1982)9 7.01 996 1.98 434 125.24 Fig. 2(a), row 5, Bartolomei et al. (1982)

10 14.79 1878 0.42 603 11.26 Fig. 2(b), row 1, Bartolomei et al. (1982)11 14.74 1847 0.77 598 16.00 Fig. 2(b), row 2, Bartolomei et al. (1982)12 14.75 2123 1.13 583 31.06 Fig. 2(b), row 3, Bartolomei et al., 198213 14.7 2014 1.72 545 68.79 Fig. 2(b), row 4, Bartolomei et al., 198214 14.99 2012 2.21 563 52.37 Fig. 2(b), row 5, Bartolomei et al. (1982)15 6.89 405 0.79 421 137.01 Fig. 3(a), row 1, Bartolomei et al. (1982)16 6.89 1467 0.77 519 39.06 Fig. 3(a), row 3, Bartolomei et al. (1982)17 6.79 2024 0.78 520 37.10 Fig. 3(a), row 4, Bartolomei et al. (1982)18 11.02 503 0.99 494 97.43 Fig. 3(b), row 1, Bartolomei et al. (1982)19 10.81 966 1.13 502 88.01 Fig. 3(b), row 2, Bartolomei et al. (1982)20 10.81 1554 1.16 563 27.05 Fig. 3(b), row 3, Bartolomei et al. (1982)21 10.84 1959 1.13 563 27.26 Fig. 3(b), row 4, Bartolomei et al. (1982)22 3.01 990 0.98 445 62.49 Fig. 4(a), row 1, Bartolomei et al. (1982)23 4.41 994 0.9 463 66.55 Fig. 4(a), row 2, Bartolomei et al. (1982)24 14.68 1000 1.13 533 80.65 Fig. 4(a), row 5, Bartolomei et al. (1982)25 6.81 2037 1.13 504 53.30 Fig. 4(b) row 1, Bartolomei et al. (1982)26 1.5 900 0.38 448.47* 23.47 Fig. 5(a), row 1, Bartolomei and Chanturiya (1967)27 3.0 900 0.38 481.55* 25.75 Fig. 5(a), row 2, Bartolomei and Chanturiya (1967)28 4.5 900 0.38 506.14* 24.66 Fig. 5(a), row 3, Bartolomei and Chanturiya (1967)29 1.5 900 0.78 419.07* 52.89 Fig. 5(b), row 1, Bartolomei and Chanturiya (1967)30 3.0 900 0.78 452.98* 54.32 Fig. 5(b), row 2, Bartolomei and Chanturiya (1967)31 4.5 900 0.78 476.16* 54.64 Fig. 5(b), row 3, Bartolomei and Chanturiya (1967)

* Based on the inlet quality data from Larsen and Tong, 1969.** Case 1–25, vertical round pipe, pipe length = 1.5 m, pipe I.D. = 0.012 m Case 26–31, vertical round pipe, pipe length = 2.0 m, pipe I.D. = 0.024 m.

! The inlet subcooling levels are calculated from the inlet pressure, which is not available in the references and were obtained from numerical results. These values arepresented for purpose of references only, and they should not be used as ‘known’ experimental conditions.


(4) inlet subcooling level from 11 to 137 K!, and (5) maximum out-let void fraction up to 0.6. A summary of all experimental conditionsof the Bartolomei experimental data is presented in Table 2. It isnoted that the Bartolomei experimental data (Bartolomei et al.,1982; Bartolomei and Chanturiya, 1967) only provide experimentalmeasurements on void fraction distribution along the pipe length.All experimental data of void fraction were plotted against the localequilibrium quality. Measured void fraction data uncertainties werealso discussed in Bartolomei and et al. (1982), in which it was statedthat the maximum absolute errors of the void fraction measure-ments do not exceed ±0.04. For all the simulation cases, 20 finite vol-ume cells are used. A typical time step of 10%1 s, and 40 time stepsare used in transient simulations, at the end of which thesteady-state solutions could be obtained. Steady-state results areobtained in the order of 1 s of computer (wall) time (with 2.5 GHzIntel Core i7 CPU).

In this work, experimental data of void fraction are used forcode validation. Numerical results of void fraction are plottedagainst local equilibrium quality, and they are compared withexperimental data. In order to quantitatively estimate the errorbetween numerical results and experimental data, the mean abso-lute error (MAE) is used. The mean absolute error of the void frac-tion between the numerical results and experimentalmeasurements is defined as,

ea ¼1N

XN

i¼0janum;i % aexp;ij ð40Þ

in which, subscripts ‘num’ and ‘exp’ denote for numerical resultsand experimental data, respectively. Interpolation of numericalresults is used when necessary. N is the total data number in thetwo-phase region. It is noted that the single-phase region data areexcluded from the error analysis for two reasons. Firstly, only trivialerrors of void fraction exist in this region. If these trivial errors wereto be considered, the mean absolute error would be underestimatedwith bias. Secondly, many of the nonphysical measurement data,i.e., negative void fraction values should not be used for error anal-ysis. Mean absolute errors for all cases are summarized in Table 3.Overall, a good agreement was found between numerical resultsand experimental data. For most of cases, the MAE is within or closeto the maximum absolute measurement error, 0.04. The discrepan-cies of other cases will be discussed later in this subsection.

The comparisons are also visually presented in Fig. 2 throughFig. 10. Fig. 2 shows four cases with similar pressure, mass flux,wall heat flux, and subcooling. As explained in the original paper(Bartolomei et al., 1982), the scattering of the experimental datacould be resulted from the measurement errors of operationalparameters, as well as the inaccuracy of maintaining them. Thiscould also be due to the stochastic nature of the two-phase flowphenomenon. As it can be seen in Fig. 2, the spreading of the

experimental results for the four cases is much larger than thespreading of simulation results. Such a difference, scattering indata vs. non-scattering in simulation results, suggests the draw-back of the commonly used deterministic thermal–hydraulicscodes, which do not normally consider the uncertainties of thephenomenon. Detailed discussions on the remaining groups of dataare provided as following. Fig. 3 shows cases with similar pressure(around 7 MPa) and mass flux but with increasing wall heat flux

Table 3Mean absolute error (MAE) of void fraction, ea, between numerical results andexperimental data.

Case ea Case ea Case ea Case ea

1 0.020 9 0.015 17 0.032 25 0.0322 0.069 10 0.023 18 0.062 26 0.0533 0.048 11 0.014 19 0.041 27 0.0304 0.040 12 0.022 20 0.056 28 0.0205 0.011 13 0.021 21 0.027 29 0.0936 0.050 14 0.032 22 0.012 30 0.0427 0.045 15 0.096 23 0.036 31 0.0418 0.030 16 0.037 24 0.072 – –

Fig. 2. Comparisons between numerical results and experimental measurements,Group 1. EXP denotes for experiments; NUM denotes for numerical results. (Thesame for all the following figures.)

Fig. 3. Comparisons between numerical results and experimental measurements,Group 2.

! See the footnote of Table 2.


and inlet subcooling. The model simulation correctly captured theeffects of varying wall heat flux and subcooling. Fig. 4 shows caseswith similar pressure (around 15 MPa) and mass flux but withincreasing wall heat flux and inlet subcooling. Fig. 5 shows caseswith similar pressure (around 7 MPa) but with different mass fluxand inlet subcooling. Fig. 6 shows cases with similar pressure(around 11 MPa) but with different mass flux and inlet subcooling.Fig. 7 shows cases with similar mass flux (around 1000 kg/m2-s)but with different pressure. Fig. 8 shows cases with the same wall

heat flux (1.13 MW/m2) and mass flux (around 2000 kg/m2-s) butwith different pressure and inlet subcooling. Figs. 9 and 10 showtwo groups of low pressure (1.5–4.5 MPa) cases with the samemass flux (900 kg/m2-s). Cases in Fig. 9 have lower wall heat flux(0.38 MW/m2) and inlet subcooling than cases in Fig. 10(0.78 MW/m2). It can be observed that the numerical results agreevery well with the experimental data in general, with a few excep-tions (Cases 15, 18, 24, 26, and 29 for example). The cases showinglarger discrepancy between the experimental data and the






simulation results can be categorized into two groups. Cases 15, 18,and 24 have large sub cooling and low mass flux. Cases 26 and 29have very low pressure. For Cases 15, 18, and 24, although themodel predicts that the inception of subcooled boiling is laggedcompared to experimental data, the slope of the void fraction stillagrees well with experimental data. This suggests that the modelpredictions could be further improved by using a better model tocapture the onset of nucleate boiling (e.g., a modified Saha-Zubermodel as used in Talebi’s work (Talebi et al., 2012). However, forCases 26 and 29, the possible sources of discrepancies betweenmodel predictions and experimental results are not clear.

4.3. Comparison to RELAP5-3D

The RELAP5-3D (RELAP5-3D, 2012a) code has been developedas a best-estimate reactor system analysis code. It has gainedworldwide success in supporting reactor safety analyses, as wellas design and licensing of new reactors. It employs a two-fluidtwo-phase flow six-equation system, including one continuity,one momentum and one energy equation for each phase (liquidand gas). In terms of spatial discretization, the RELAP5-3D codeuses the similar staggered grid upwind method as that used in thiswork. In terms of time integration scheme, the RELAP5-3D codeemploys a semi-implicit method. As discussed in Section 2, thedrift flux model itself is not a complete system, and many constitu-tive correlations have been adapted from the existing system


Fig. 9. Comparisons between numerical results and experimental measurements, Group 8.

Fig. 10. Comparisons between numerical results and experimental measurements, Group 9.


analysis code (such as RELAP5-3D) to close the system. In this sub-section, the numerical results from this work and from RELAP5-3Dare compared for several cases, as shown in Figs. 11–14. It can beobserved that the RELAP5-3D code gives slightly better results inthe low void fraction region; however, it tends to over-predict atthe larger void region. Nevertheless, the numerical results obtainedin this work are fairly close to those obtained from the RELAP5-3Dcode. This is somewhat expected as several key constitutive corre-lations used in this work, such as the Saha-Zuber model and flow

regime maps, are identical to those used in RELAP5-3D. The goodagreements between this work and the RELAP5-3D results furthervalidate the correct implementation of the physical models andnumerical methods used in this work.

It is also worth pointing out that, in RELAP5-3D, for the purposeof numerical stability, a width of 0.05 in void fraction is used forthe transition between the slug and annular mist flow regimes.In this work, this transition region was reduced to 0.01, and theJFNK method still showed very good numerical stability. Aspointed out by Levy (Levy, 1999), p. 131], ‘‘to avoid sudden changesin flow regime that may be deleterious to numerical stability, tran-sition regions covering a span Da of at least 0.1 are included incomputer system codes and that required computational flexibilityovershadows any accuracy that might be derived from more exactflow regime maps.’’ The excellent numerical stability demon-strated in this work, even with a much smaller transition region(0.01 compared to 0.05), shows that the JFNK method is a promis-ing numerical method to reduce numerical errors related to theflow regime transitions. It is expected that similar treatment couldbe used to solve the two fluid six-equation two-phase flow equa-tions, which will be investigated in our future work.

Additional numerical results comparisons between this workand the RELAP5-3D code are also provided in terms of phasic veloc-ities and flow regimes, for two representative cases, Case 9 and 19.Figs. 15 and 17 show the flow regimes along the pipe length pre-dicted in this work and from RELAP5-3D. In both cases, predictedflow regimes agree with each other pretty well with only one mis-match. For case 9, onset of nucleate boiling lag one cell predicted inthis work compared to that predicted using RELAP5-3D. For case19, flow regime in the last cell is predicted as transition betweenslug and annular mist flow, while annular mist flow is predictedin RELAP5-3D. Figs. 16 and 18 show the phasic velocities betweennumerical results obtained in this work and those from RELAP5-3Dpredictions. The overall trends are similar, however a major notice-able difference can be found in the region where boiling starts,between numerical results obtained using drift flux model andfrom RELAP5-3D. In this region, local vapor phase velocities

Fig. 11. Comparisons between numerical results and RELAP5 results, Cases 5, 7, and9. RELAP5 denotes for the numerical results obtained from RELAP5-3D. (The samefor the following figures.)

Fig. 12. Comparisons between numerical results and RELAP5 results, Cases 6 and 8.

Fig. 13. Comparisons between numerical results and RELAP5 results, Cases 18 and20.


predicted using Ishii’s model (this work) are larger than local liquidphase velocities. On the contrary, RELAP5-3D code predicts that thelocal vapor phase velocities are smaller than the local liquid phasevelocities. Such a difference is mainly due to the different drift fluxmodels used in RELAP5-3D and in this work, i.e., Ishii’s model. InRELAP5-3D, EPRI drift flux model (RELAP5-3D, 2012a; Chexal andLellouche, 1986) is used to estimate the interfacial drag in the ver-tical bubbly and slug flow regimes. EPRI drift flux model considersconditions when void concentrates near the wall as nucleate boil-ing takes place, and such that C0 < 1 (see Discussion on page 2–2 of

Chexal and Lellouche (1986)). Thus the vapor velocity is greatlyaffected by the liquid velocity near the wall region, which is muchsmaller than the liquid phase average velocity. This agrees wellwith Zuber’s original analysis (Zuber and Findlay, 1965), and themodel gives reasonable prediction on vapor phase velocity (smallerthan the liquid phase average velocity) in this region. However, forIshii’s drift flux model used in this work, the value of C0 is *1.2,which is larger than 1. This seems to violate Zuber’s original con-cept. As a result, RELAP5-3D using EPRI drift flux model gives bet-ter results in almost all cases in this region. Another difference inthe phase velocities profiles can be observed in regions where flowregime transits from one to another, which could be due to several

Fig. 14. Comparisons between numerical results and RELAP5 results, Cases 19 and21.

Fig. 15. Flow regimes, comparisons between numerical results and RELAP5 results, Case 9.

Fig. 16. Phasic velocities, comparisons between numerical results and RELAP5results, Case 9.


possible reasons, e.g., a different drift flux model is used inRELAP5-3D code, different numerical treatments when flow regimetransits, and different width of void fraction used for the slug toannular mist transition as discussed in the previous paragraph.

5. Conclusions

We have successfully demonstrated the capability of applyingthe JFNK method to solve the two -phase four-equation drift fluxmodel with the full coverage of all the flow regimes for verticalupward channels. To our best knowledge, this work represents

the first successful application of the JFNK method to solvetwo-phase flow models with realistic closure models. The stag-gered grid finite volume method and fully implicit backwardEuler method was used for the spatial discretization and time inte-gration schemes, respectively. Numerical verification was per-formed in the form of convergence study on mesh refinement,from which the expected order of convergence was obtained.Extensive code validation was performed comparing the numericalresults to subcooled flow boiling experimental data in verticalpipes. For most simulation cases, the numerical results agree wellwith experimental data. Code-to-code comparison was also per-formed to verify the code implementation. Numerical results werecompared to those obtained using RELAP5-3D, and good agree-ment between them was obtained. Differences of the numericalresults using drift flux model and RELAP5-3D were also identifiedand discussed.

The JFNK method is robust and efficient even without imple-menting the analytical Jacobian matrix, which is typically requiredfor traditional Newton’s methods (Frepoli et al., 2003). TheJacobian-free feature yields significant savings in developing a fullyimplicit system analysis code. The work also demonstrates thatflow regime transition is not a critical issue for convergence,although it was observed that time step size reduces for better con-vergence when flow regime transits. The required transition spanfor the new method can be much narrower than that required intraditional system codes. This can potentially result in improvedsimulation accuracy by using more exact flow regime maps.

This work only represents the first attempt to apply the JFNKmethod in realistic two-phase flow simulations. Additional workon downward flow regimes and horizontal flow regimes is neededto complete the capability of the four-equation drift flux model tocover a wide range of applications. Extending this method to thesix-equation two-fluid model will be performed in the near future.

Acknowledgements

This work is supported by the United States (U.S.) Departmentof Energy, under Department of Energy Idaho Operations Office

Fig. 17. Flow regimes, comparisons between numerical results and RELAP5 results, Cases 19.

Fig. 18. Phasic velocities, comparisons between numerical results and RELAP5results, Case 19.


Contract DE-AC07-05ID14517. Accordingly, the U.S. Governmentretains a nonexclusive, royalty-free license to publish or reproducethe published form of this contribution, or allow others to do so, forU.S. Government purposes.

References

Balay, S., et al., ‘‘PETSc Users Manual, Reversion 3.4’’, ANL-95/11 (2013).Bartolomei, G.G., Chanturiya, V.M., 1967. Experimental study of true void fraction

when boiling subcooled water in vertical tubes. Therm. Eng. 14, 123–128.Bartolomei, G.G. et al., 1982. An experimental investigation of true volumetric vapor

content with sub-cooled boiling in tubes. Therm. Eng. 29 (3), 132–135.Chexal, B. and Lellouche, G. A Full-Range Drift-Flux Correlation for Vertical Flows

(Revision 1), EPRI NP-3989-SR, Electric Power Research Institute, September1986.

Chung, Y.J. et al., 2012. Development and assessment of system analysis code, TASS/SMR for integral reactor, SMART. Nucl. Eng. Des. 244, 52–60.

Chung, Y.J. et al., 2013. Applicability of TASS/SMR using drift flux model for SMARTLOCA analysis. Nucl. Eng. Des. 262, 228–234.

Frepoli, C., Mahaffy, J.H., Ohkawa, K., 2003. Notes on the implementation of a fully-implicit numerical scheme for a two-phase three-field flow model. Nucl. Eng.Des. 225, 191–217.

Galloway, J. D. 2010. Boiling Water Reactor Core Simulation with GeneralizedIsotopic Inventory Tracking for Actinide Management, PhD Dissertation,University of Tennessee, Knoxville.

Gaston, D.R. et al., 2015. Physics-based multiscale coupling for full core nuclearreactor simulation. Ann. Nucl. Energy 84, 45–54.

Hashemi-Tilehnoee, M., Rahgoshay, M., 2013a. Benchmarking a sub-channelprogram based on a drift-flux model with 8+8 NUPEC BWR rod bundle. Ann.Nucl. Energy 58, 202–212.

Hashemi-Tilehnoee, M., Rahgoshay, M., 2013b. Sub-channel analysis of 8+8 and9+9 BWR fuel assemblies with different two-phase flow model. Ann. Nucl.Energy 62, 264–268.

Hibiki, T., Ishii, M., 2003. One-dimensional drift-flux model and constitutiveequations for relative motion between phases in various two-phase flowregimes. Int. J. Heat Mass Transf. 46, 4935–4948.

Hibiki, T., Ishii, M., 2005. Erratum to: One-dimensional drift-flux model andconstitutive equations for relative motion between phases in various two-phaseflow regimes. Int. J. Heat Mass Transf. 48, 1222–1223.

Hirt C. W. et al., SOLA-LOOP: A Nonequillibrium, Drift-Flux Code for Two-PhaseFlow in Networks, LA-7659, June (1979).

Ishii, M., 1977. One-dimensional drift-flux model and constitutive equations forrelative motion between phases in various two-phase flow regimes, ANL-77-47,USA.

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

Khan, H.J., Yi, H.C., 1985. Subchannel analysis in BWR fuel bundles. Ann. Nucl.Energy 12 (10), 559–572.

Knoll, D.A., Keyes, D.E., 2004. Jacobian-free Newton–Krylov methods: a survey ofapproaches and applications. J. Comp. Phys. 193, 357–397.

Larsen, P.S., Tong, L.S., 1969. Void fraction in subcooled flow boiling. J. Heat Transfer,471–476.

LeVeque, R.J., 2004. Finite-Volume Methods for Hyperbolic Problems. CambridgeUniversity Press.

Levy, S., 1999. Two-Phase Flow in Complex Systems. John Wiley & Sons, ISBN 0-471-32967-3.

Mousseau, V.A., 2004. Implicitly balanced solution of the two-phase flow equationscoupled to nonlinear heat conduction. J. Comp. Phys. 200, 104–132.

Mousseau, V.A., 2005. A fully implicit hybrid solution method for a two-phasethermal–hydraulic model. J. Heat Transfer 127, 531–539.

Nayak, A.K., 2007. Study on the stability behaviour of two-phase natural circulationsystems using a four-equation drift flux model. Nucl. Eng. Des. 237, 386–398.

‘‘RELAP5/MOD3.3 Code Manual Volume I’’, NUREG/CR-5535 ed., U.S. NuclearRegulatory Commission, December, (2001).

‘‘RELAP5-3D Code Manual Volume I: Code Structure, System Models and SolutionMethods’’, INEEL-EXT-98-00834, Revision 4.0, June (2012a).

‘‘RELAP5-3D Code Manual Volume IV: Models and Correlations’’, INEEL-EXT-98-00834, Revision 4.0, June (2012b).

RELAP-7 Level 2 Milestone Report: Demonstration of a Steady State Single PhasePWR Simulation with RELAP-7, INL/EXT-12-25924, May 2012, Idaho NationalLaboratory.

RELAP-7 Theory Manual, INL/EXT-14-31366, February 2014, Idaho NationalLaboratory.

‘‘RETRAN-3D – A Program for Transient Thermal-Hydraulic Analysis of ComplexFluid Flow Systems: Volume 1: Theory and Numerics (Revision 3)’’, NP-7450-V1R3, Electric Power Research Institute, October, (1998).

Talebi, S., Kazeminejad, H., Davilu, H., 2012. A numerical technique for analysis oftransient two-phase flow in a vertical tube using the drift flux model. Nucl. Eng.Des. 242, 316–322.

The International Association for the Properties of Water and Steam, http://www.iapws.org.

‘‘TRACE V5.0 Theory Manual’’, ML120060218, U.S. Nuclear Regulatory Commission,June 4, (2010).

Wang, J. et al., 2011. Numerical study of nuclear coupled two-phase flow instabilityin natural circulation system under low pressure and low quality. Nucl. Eng.Des. 241, 4972–4977.

Zou L., et al., ‘‘C++ Implementation of IAPWS Water/Steam Properties’’, INL/EXT-14-31502, February (2014).

Zou, L., Zhao, H., Zhang, H., 2015a. Applications of high-resolution spatialdiscretization scheme and Jacobian-free Newton–Krylov method in two-phaseflow problems. Ann. Nucl. Energy 83, 101–107.

Zou, L., Zhao, H., Zhang, H., 2015b. On the analytical solutions and numericalverifications of the two-phase water faucet problem. In: 16th InternationalTopical Meeting on Nuclear Reactor Thermalhydraulics, Augst 30 – September4, Chicago, Illinois, USA.

Zuber, N., Findlay, J.A., 1965. Average volumetric concentration in two-phase flowsystems. J. Heat Transfer, 453–468.


http://refhub.elsevier.com/S0306-4549(15)00397-7/h0010http://refhub.elsevier.com/S0306-4549(15)00397-7/h0010http://refhub.elsevier.com/S0306-4549(15)00397-7/h0015http://refhub.elsevier.com/S0306-4549(15)00397-7/h0015http://refhub.elsevier.com/S0306-4549(15)00397-7/h0025http://refhub.elsevier.com/S0306-4549(15)00397-7/h0025http://refhub.elsevier.com/S0306-4549(15)00397-7/h0030http://refhub.elsevier.com/S0306-4549(15)00397-7/h0030http://refhub.elsevier.com/S0306-4549(15)00397-7/h0035http://refhub.elsevier.com/S0306-4549(15)00397-7/h0035http://refhub.elsevier.com/S0306-4549(15)00397-7/h0035http://refhub.elsevier.com/S0306-4549(15)00397-7/h0045http://refhub.elsevier.com/S0306-4549(15)00397-7/h0045http://refhub.elsevier.com/S0306-4549(15)00397-7/h0050http://refhub.elsevier.com/S0306-4549(15)00397-7/h0050http://refhub.elsevier.com/S0306-4549(15)00397-7/h0050http://refhub.elsevier.com/S0306-4549(15)00397-7/h0050http://refhub.elsevier.com/S0306-4549(15)00397-7/h0055http://refhub.elsevier.com/S0306-4549(15)00397-7/h0055http://refhub.elsevier.com/S0306-4549(15)00397-7/h0055http://refhub.elsevier.com/S0306-4549(15)00397-7/h0055http://refhub.elsevier.com/S0306-4549(15)00397-7/h0055http://refhub.elsevier.com/S0306-4549(15)00397-7/h0060http://refhub.elsevier.com/S0306-4549(15)00397-7/h0060http://refhub.elsevier.com/S0306-4549(15)00397-7/h0060http://refhub.elsevier.com/S0306-4549(15)00397-7/h0065http://refhub.elsevier.com/S0306-4549(15)00397-7/h0065http://refhub.elsevier.com/S0306-4549(15)00397-7/h0065http://refhub.elsevier.com/S0306-4549(15)00397-7/h0080http://refhub.elsevier.com/S0306-4549(15)00397-7/h0080http://refhub.elsevier.com/S0306-4549(15)00397-7/h0085http://refhub.elsevier.com/S0306-4549(15)00397-7/h0085http://refhub.elsevier.com/S0306-4549(15)00397-7/h0090http://refhub.elsevier.com/S0306-4549(15)00397-7/h0090http://refhub.elsevier.com/S0306-4549(15)00397-7/h0095http://refhub.elsevier.com/S0306-4549(15)00397-7/h0095http://refhub.elsevier.com/S0306-4549(15)00397-7/h0100http://refhub.elsevier.com/S0306-4549(15)00397-7/h0100http://refhub.elsevier.com/S0306-4549(15)00397-7/h0105http://refhub.elsevier.com/S0306-4549(15)00397-7/h0105http://refhub.elsevier.com/S0306-4549(15)00397-7/h0110http://refhub.elsevier.com/S0306-4549(15)00397-7/h0110http://refhub.elsevier.com/S0306-4549(15)00397-7/h0115http://refhub.elsevier.com/S0306-4549(15)00397-7/h0115http://refhub.elsevier.com/S0306-4549(15)00397-7/h0120http://refhub.elsevier.com/S0306-4549(15)00397-7/h0120http://refhub.elsevier.com/S0306-4549(15)00397-7/h0155http://refhub.elsevier.com/S0306-4549(15)00397-7/h0155http://refhub.elsevier.com/S0306-4549(15)00397-7/h0155http://www.iapws.orghttp://www.iapws.orghttp://refhub.elsevier.com/S0306-4549(15)00397-7/h0170http://refhub.elsevier.com/S0306-4549(15)00397-7/h0170http://refhub.elsevier.com/S0306-4549(15)00397-7/h0170http://refhub.elsevier.com/S0306-4549(15)00397-7/h0180http://refhub.elsevier.com/S0306-4549(15)00397-7/h0180http://refhub.elsevier.com/S0306-4549(15)00397-7/h0180http://refhub.elsevier.com/S0306-4549(15)00397-7/h0185http://refhub.elsevier.com/S0306-4549(15)00397-7/h0185

Numerical implementation, verification and validation of two-phase flow four-equation drift flux model with Jacobian-free Newton–Krylov method1 Introduction2 Physical model descriptions2.1 One-dimensional four-equation drift flux model2.2 Water/steam properties2.3 Drift flux model constitutive correlations and flow regimes2.4 Frictional pressure drop2.5 Vapor generation rate

3 Numerical and solution methods3.1 Finite volume method using staggered grid mesh3.2 Boundary conditions3.3 Time integration schemes3.4 Jacobian-free Newton–Krylov method

4 Results and discussions4.1 Numerical verification4.2 Code validation and error analysis4.3 Comparison to RELAP5-3D

5 ConclusionsAcknowledgementsReferences

annals of nuclear energy - wordpress.com · 2016. 4. 29. · regulatory commission, 2001) and trace...

Documents