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The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12) Log Number: 47

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

ADVANCES IN THE MODELING OF CLADDING HEAT

TRANSFER AND CRITICAL HEAT FLUX IN BOILING WATER

REACTOR FUEL ASSEMBLIES

Andrey Ioilev, Maskhud Samigulin, Vasily Ustinenko

Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

ioilev@socc.ru

Polina Kucherova

Sarov Laboratories, Sarov, Russia

Adrian Tentner

Argonne National laboratory, Argonne, IL, USA

tentner@anl.gov

Simon Lo, Andrew Splawski

CD-adapco, London, UK

simon.lo@uk.cd-adapco.com

ABSTRACT

This paper presents recent advances in the modeling of cladding-to-coolant heat

transfer and critical heat flux that have been implemented in the advanced

Computational Fluid Dynamics (CFD) computer code CFD-BWR. The CFD-BWR

code is being developed as a customized module built on the foundation of the

commercial CFD-code STAR-CD, which provides general two-phase flow modeling

capabilities, for the detailed analysis of the two-phase flow and heat transfer

phenomena in Boiling Water Reactor (BWR) fuel assemblies. These phenomena

include coolant phase changes and multiple flow regimes that directly influence the

coolant interaction with the fuel pins and, ultimately, the reactor performance.

The cladding-to-coolant heat transfer is described by a wall heat partitioning model

which, used in conjunction with various local flow topologies that range from bubbly

to droplet to film flow, allows the prediction of a wide range of cladding-to-coolant

heat transfer regimes, including the onset of Critical Heat Flux (CHF), without the use

of empirical correlations traditionally used in sub-channel codes. Results of recent

analyses of experiments that have measured the axial distribution of wall temperature

in two-phase upward flow in a vertical channel with a heated wall are presented,

illustrating the ability of the cladding-to-coolant heat transfer model to capture the

onset of CHF. The paper concludes with a discussion of results and plans for future

work.

KEYWORDS

Computational Fluid Dynamics, Boiling Water Reactor, Two-phase Flow, Critical

Heat Flux

The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12) Log Number: 47

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

1. INTRODUCTION

A new code, CFD-BWR [1, 2] is being developed for the fine-mesh, 3-dimensional

simulation of the two-phase flow phenomena that occur in a Boiling Ware Reactor

(BWR) fuel assembly. These phenomena include coolant phase changes and multiple

flow topologies that directly influence the reactor performance. The CFD-BWR code

is being developed as a specialized module built on the foundation of the commercial

CFD code STAR-CD [3] which provides general two-phase flow modeling

capabilities. A first generation of models describing the inter-phase mass,

momentum, and energy transfer phenomena specific for bubbly flow topologies have

been previously implemented in the CFD-BWR module and described in [1, 2]. A

second generation of boiling models referred to as the Extended Boiling Framework

(EBF) has been implemented recently and has been described in [4]. The EBF models

allow STAR-CD and CFD-BWR to simulate a wide spectrum of local flow topologies

expected in a BWR fuel assembly and an overview of these models is included below

in Section 2.

A model describing the heat transfer between the heated wall and the coolant has been

developed previously and described in [1, 2, 5]. The heat flux from the wall to the

liquid is divided into three parts according to a wall heat partitioning model which

includes convective heat for the liquid, evaporative heat for generation of steam and

quench heat for heating of liquid in the nucleation sites. The Extended Boiling

Framework has also extended the wall cladding-to-coolant heat transfer model,

including models for the heat flux transferred to the vapour phase and allowing the

modeling of dry-out situations. The extended cladding-to coolant heat transfer model

is described in Section 3. The EBF models have been used in analyses of experiments

that have measured the axial distribution of wall temperature in two-phase upward

flow in a heated vertical channel and the results presented in Section 4 illustrate the

ability of the cladding-to-coolant heat transfer model to capture the onset of CHF

2. THE EXTENDED BOILING FRAMEWORK MODELS

2.1 Methodology

The concept of a local inter-phase surface topology map [1, 2, 4] is employed to

determine the local flow configuration (bubbly, mist, annular film, etc.) as a function

of flow conditions and to prescribe which models and properties are relevant for each

computational cell. The continuous liquid bubbly topology, with vapour bubbles

flowing in a continuous liquid, already has an established base of CFD modelling

experience. The current inter-phase surface topology map includes, in addition to the

bubbly topology, a mist or droplet topology and a transition topology. The droplet

topology consists of liquid droplets flowing in a continuous vapour stream.

The direct simulation of transition from slug flow, through churn flow to annular flow

is not within the scope for this phase of the project. Instead, for transition-topology

cells a topology-based combination of the terms appropriate for the basic topologies,

The 12th International Topical Meeting on Nuclear Reactor Thermal Hydraulics (NURETH-12) Log Number: 47

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

bubbly and droplet is used. This can be interpreted as having a transition topology cell

where a fraction of the cell volume presents the bubbly topology while the remaining

volume presents the droplet topology. An alternative interpretation is that the map is

prescribing the probability of being in one topology or the other while solving

equations for the time-averaged flow.

The use of the local inter-phase surface topology map allows the typical sub-channel

annular flow regime to be resolved into a distinct core flow region in which the gas

phase is continuous and the local droplet topology is used, separated by transition

topology cells from a liquid film on the wall where the local bubbly topology is used.

The modelling of the sub-channel annular flow regime, which covers a significant

fraction of a typical BWR channel, has been divided into three stages.

The first stage has been to implement suitable wall boundary conditions for describing

stress and boiling heat transfer in the presence of a liquid film on the wall. While the

film is thick enough to be resolved on the grid, the local inter-phase surface topology

map will control whether core flow or film flow terms are relevant in each cell. When

the film is thinner than the grid cell thickness, then the film flow equations will be

replaced by wall functions. The first stage was completed and used in test cases

presented in this paper. Details of the mathematical models implemented are

described in the sub-sections below.

The second stage is to predict the thickness of the film by modelling droplet

entrainment and deposition. The third stage is to improve this model through

additional conservation equations for the film mass or for the mass fraction of liquid

phase that is held in the film. These stages will be considered in future work.

2.2 Eulerian Two-Phase Flow Model

The STAR-CD Eulerian two-phase solver tracks the mass, momentum, and energy of

the liquid and vapour phases in each cell. Full details of the Eulerian two-phase flow

models in STAR-CD can be found in [1] and [2]. The main equations solved are the

conservation of mass, momentum and energy for each phase.

Mass Conservation

The conservation of mass equation for phase is: k

()( )(

∑

=

−=∇+

∂

∂N

iikkikkkkk mmu

t1

.&&

ραρα

)

(1)

where k

α

is the volume fraction of phase , kk

ρ

is the phase density, is the phase

velocity, and are mass transfer rates to and from the phase, and is the total

number of phases.

k

u

ki

m

&ik

m

&N

Momentum Conservation

The conservation of momentum equation for phase is: k

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

()( )

(

)

(

)

Mgpuuu

tkkk

t

kkkkkkkkkk ++∇−=+∇−∇+

∂

∂

ραατταραρα

.. (2)

where k

τ

and are the laminar and turbulence shear stresses respectively,

t

k

τ

p

is

pressure,

g

is gravitational acceleration and

M

is the sum of the inter-phase forces.

Energy Conservation

The conservation of energy equation for phase k is:

()( )( )

QTeue

tkkkkkkkkkk =∇∇−∇+

∂

∂

λαραρα

.. (3)

where is the phase enthalpy,

k

ek

λ

is the thermal conductivity, is the phase

temperature and is the inter-phase heat transfer.

k

T

Q

2.3 Inter-Phase Surface Topology Map

The inter-phase surface topology map is used to control which phase is continuous

and which is dispersed so that bubbly flow and mist flow configurations can be

represented. The current implementation of the map described in Ref. [4] uses a single

variable 1

θ

that controls the transition from a continuous liquid bubbly flow topology

(0

1=

θ

) to a continuous vapor droplet flow topology ( 1

1

=

θ

).

It is important for convergence that this variable is a smooth continuous function of its

independent variables. In the current model, the topology variable 1

θ

is modeled as:

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎭

⎬

⎫

⎩

⎨

⎧

−

−

=INVINV

INV

12

1

1,1min,0max

αα

αα

θ

(4)

where the breakpoints have been set as 3.0

1

=

INV

α

and 7.0

2

=

INV

α

.

The topology variable 1

θ

is used in calculation of the local inter-phase surface area

and other topology-related geometrical features, affecting all the physical models as

illustrated in Ref. [4]. For example the inter-phase surface area in a cell is computed

as:

iDropletiBubblyi AAA 11 )1(

θ

θ

+

−

= (5)

2.4 Turbulence Model

A basic mixing length model is used to specify turbulent viscosity for modelling a

flow with phase topology inversion. First, the turbulent viscosity for each phase-k is

specified as if it were the continuous phase

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

(

)

2

1lDC kk

kc

Tk

ρµ

= (6)

The mixing length is based on two length scales. Near walls, the scale is distance

from the wall, , as used in the derivation of the law of the wall. Away from the wall

the length scale is limited to some fraction, , of the equivalent diameter, , for the

core flow.

y

2

Ce

d

),min( 2e

dCyl

κ

=

(7)

Here 419.0=

κ

, as for the law of the wall, and the model constant =0.08 is used.

2

C

The velocity gradient scale for phase-k, , (representing

k

D yu

∂

∂

/ in a fully developed

channel flow), is obtained from the second invariant of the phase rate of strain tensor

by

sk

Π

skk

DΠ= 2 (8)

The topology variable 1

θ

is used to compute the liquid-phase turbulent viscosity for

both the liquid-continuous (indicated by subscript lc) and the gas-continuous (gc)

regions.

(

)

(

)

gc

Tl

lc

TlTl

µ

θ

µ

θ

µ

11 )1(

+

−

=

(9)

A similar equation is used for the gas phase turbulent viscosity:

(

)

(

)

gc

Tg

lc

TgTg

µ

θ

µ

θ

µ

11)1(

+

−

=

(10)

3. CLADDING-TO-COOLANT HEAT TRANSFER MODELS

The calculation of wall heat flux and the partitioning of heat flux between the phases

are determined by the following three heat transfer coefficients: a) , for the heat

transfer from wall to the liquid phase; b) , for heat transfer from wall to the gas

phase; c) , for heat transfer from wall to the boiling interface. These coefficients are

specified per unit wall area and they become zero as the corresponding phase is lost or

as the phase-wall contact is lost. The wall heat flux is given by:

o

L

h

o

G

h

o

I

h

)()()( SATW

o

ILW

o

LGW

o

GW TThTThTThq −+−+−= (11)

If the wall temperature is unknown (e.g. for fixed flux boundary conditions), the heat

balance is solved iteratively to obtain the wall temperature.

3.1 Wall regime variables

The wall regime variable DRY

θ

which limits heat flux to liquid phase and boiling

interface when cells are nearly dry, is modelled as

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎭

⎬

⎫

⎩

⎨

⎧

−

−

=DRYDRY

DRY

DRY

12

1

,1min,0max

αα

αα

θ

(12)

where the breakpoints have been set as and .

9.0

1=

DRY

α

95.0

2=

DRY

α

3.2 Wall boiling

For the purpose of wall boiling, we distinguish resolved films and unresolved films by

the following definitions:

FILMfilmthin

θ

θ

θ

1_

=

(13)

filmthinfilmthick __ 1

θ

θ

−

=

(14)

The wall regime variable FILM

θ

is obtained by comparing film thickness with cell

thickness

WALL

CELL

FILML

FILM

A

V,

~

δ

α

= (15)

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎭

⎬

⎫

⎩

⎨

⎧

−

−

=THICKFILMTHICKFILM FILM

THICKFILM

FILM

12

2

~

,1min,0max

αα

αα

θ

(16)

In the current model the film thickness mm

FILML 1.0

,

=

δ

is prescribed. The

breakpoints are set at and .

5.0

1=

THICKFILM

α

0.1

2=

THICKFILM

α

The overall heat transfer coefficients to each phase, , and to the boiling

interface at the wall, , are assembled as follows.

o

L

o

Ghh ,

o

I

h

1

GDRY

o

Ghh

θ

= (17)

1

_)1()1( LnbfilmthickDRY

conv

Lhh

θθθ

−−= (18)

quenchfilmthickDRY

quench

Lhh _

)1(

θθ

−= (19)

quench

L

conv

L

o

Lhhh += (20)

nbDRY

nb

Ihh )1(

θ

−= (21)

quenchfilmthinDRY

quench

Ihh _

)1(

θθ

−= (22)

filmfilmthinDRY

film

Ihh _

)1(

θθ

−= (23)

film

I

quench

I

nb

I

o

Ihhhh ++= (24)

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

Definitions for the heat transfer coefficients for nucleate boiling, , quenching

and the nucleation site area factor

nb

h

quench

hnb

θ

can be found in Refs. [1] and [5] and

will not be repeated here.

3.3 Thin film boiling

The film heat transfer coefficient is based on Whalley’s application of the law of the

wall and of Reynolds' analogy to the film [6]:

FILML

L

filmfilm k

Nuh

,

δ

= (25)

+

+

=T

Nu Lfilm

δ

Pr (26)

where is the liquid Prandtl number, and

L

Pr +

δ

+

T

are dimensionless wall distance

and universal temperature profile.

The wall-vapour heat transfer coefficient is evaluated by

gggdryg UCpSth

ρθ

=

0 (27)

where is the Stanton number. In calculations (see below) the Stanton number was

varied to match the experimental data for the wall temperature.

St

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

4. CRITICAL HEAT FLUX EXPERIMENTS ANALYZED

To validate the wall heat transfer models included in the CFD-BWR code we

analyzed the static dryout experiments conducted by Becker, et al., [7, 8]. The

experimental setup and results used in this paper are based on the description given by

Hoyer [9]. These experiments were designed to study CHF and post-dryout heat

transfer in vertical circular pipes. The loop consisted of a 7 m long test section, a

condenser, feed water and main recirculation pumps, flow measuring devices and a

preheater. Subcooled water was fed at the bottom of the test section. The wall was

heated uniformly and all typical to BWR flow regimes were produced in the upward

water/steam flow. In the experiments the outer wall temperature was measured, and

the inner wall temperature has been calculated assuming an adiabatic boundary

condition. The experimental data are presented as axial distributions of the inner wall

temperature.

Figure 1 Schematic of

experiments

Schematic of experiments is presented in Fig.1. Two

experiments in a vertical channel 0.01 m in diameter

and 7 m in length with a uniformly heated wall were

used as verification test-cases. Pressure in both

experiments was 7.01 MPa. The experiments differed

in inlet mass flux G (497 kg/(s⋅m2) and 1495 kg/(s⋅m2))

and wall heat flux (350000 W/m2 and 797000 W/m2).

The following thermal-physical properties were used

for water and vapour (at pressure 7 МPa):

• water: density is 739 kg/m3, viscosity is

0.9⋅10-3 (Pa⋅s), specific heat is 5490 J/(kg⋅K),

thermal conductivity is 0.562 W/(m⋅K),

• vapour: density is 36.6 kg/m3, viscosity is

0.19⋅10-4 (Pa⋅s), specific heat is 5580 J/(kg⋅K),

thermal conductivity is 0.066 W/(m⋅K).

Saturation temperature at pressure 7 МPa is 559 K,

vaporization heat is 1504000 J/kg, surface tension

coefficient is 0.018 N/m.

The CFD analyses presented were performed using a 2-dimensional representation of

the upward flow in the test section, assumed to be axisymmetric.

5. EXPERIMENT ANALYSIS RESULTS AND DISCUSSION

The calculated void fraction distributions are shown for the two experiments analyzed

in Figs.2 and 6, respectively, with the test section shown horizontally. The

characteristic flow regimes in a pipe with heated walls are simulated. Since the

experiments were focused on the dryout and post dryout heat-transfer, the bubbly

flow regime was not practically produced, but the slug, annular-mist and mist flow

regimes were clearly observed, as well as liquid film on the wall. The calculated wall

temperature is also shown in these figures, illustrating the fact that the sharp rise in

the wall temperature coincides with the disappearance of the calculated liquid film in

both cases.

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

In the simulation of the first experiment two wavy film regions are clearly noticeable

(see Fig.2). These waves are unlikely to be a numerical effect, since:

• the peak-to-peak distance is about 7-10 times the longitudinal cell size,

• the waves do not disappear with mesh refinement, and

• the waves are steady, i.e. they do not change noticeably with the number of

iterations.

There are no film waves in the second calculation (see Figs.6), which was performed

in the same manner but at higher heat flux and higher inlet mass flux.

The calculated wall temperature for the two experiments analyzed is compared with

the corresponding measured temperature in Figs.3 and 7 respectively. The dryout

location is close to experimental data for both experiments. As for the post-dryout

heat-transfer, there is rather good agreement between computed and measured wall

temperatures in the first experiment, as illustrated in Fig.3. In the second experiment,

the calculated wall temperature peak is near the outlet, while the experimental

temperature peak is near the dryout location, after which wall temperature decreases

(Fig.7).

According to Hoyer [9], the wall temperature non-monotony is caused by evaporation

of water droplets in superheated steam after the dryout elevation. This evaporation

makes steam mass and, therefore, its velocity, rapidly increase. In its turn, this

increases wall-steam heat transfer coefficient and decreases wall temperature. Another

potential reason for the post-dryout wall temperature decrease in the presence of many

droplets is the direct heat transfer from the wall due to the impinging droplets. This

mechanism agrees with the dryout quality we obtain in our computations.

In post-processing of results of calculations, cross-section-averaged flow quality was

calculated as:

g

g

G

xGG

=+f

, (28)

where

gggg

dA

A

1

GW

A

=αρ

∫∫ ,

fgff

A

1

G(1)WdA

A

=−αρ

∫∫ , (29)

and Wf and Wg are the vertical (axial) components of fluid and vapor velocity.

Cross-section-averaged flow quality and void fraction are presented in Figs.5 and 9.

In the first calculation, where the experimental wall temperature is monotonous,

dryout quality is about 0.95, i.e. the mass of droplets is small and they cannot

noticeably change wall temperature. In the second calculation dryout quality is about

0.57, i.e. there are many droplets present in the vapor stream. The fact that the

calculated wall temperature for the second experiment (Fig.7) does not show the

measured post-dryout decrease may be explained by the lack of the wall-to-droplet

heat transfer component which will be added to the wall heat transfer model in the

future.

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

The variation of the calculated wall heat flux components along the length of the test

section for the two experiments analyzed is shown in Figs.4 and 8 respectively. In

both cases the vaporization heat flux component dominates prior to dryout, with the

heat transferred to the liquid phase decreasing up the dryout location and the heat

transferred to the vapor phase increasing rapidly at dryout.

The correctness of our modeling of dryout quality is confirmed by the data of

Sugawara [10]. Sugawara [10] has correlated the results of experiments conducted by

Wurtz [11] and Tompson, et al. [12] in pipes of diameter 10 mm at pressure 7 MPa as

dryout quality versus boiling length-to-diameter ratio for various inlet mass fluxes. In

our cases (G=497 kg/(s⋅m2) and 1495 kg/(s⋅m2), these correlations yield dryout quality

of around 0.95 and 0.67, which agree reasonably well with the computed values

(Figs.5 and 9).

Note that the presence of oscillations of cross-section-averaged flow quality and void

fraction in the first calculation (Fig. 5) is associated with the “wavy film regions” that

can be observed in the void fraction distribution (Fig. 2). These wavy film regions

could be the effect of physical instability corresponding to dryout process (“vapor

chimney” mentioned by Sugawara [10]) but could also be partially caused by a local

numerical effect. Their cause will be investigated in future work. No wavy film

regions are observed in the void fraction distribution obtained in the second

calculation (Fig. 6).

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

Figure 2 Calculation 1: Void fraction distribution and wall temperature

500

550

600

650

700

750

800

850

01234567

z, m

Tw, K

run 1

exp

Figure 3 Wall temperature: experiment and calculation 1

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

0

50000

100000

150000

200000

250000

300000

350000

400000

01234567

z, m

Heat flux, Wt/m2

Qwi

Qwl

Qwg

Figure 4 Calculation 1: Heat flux from wall: Qwi for vaporizing, Qwl to water,

and Qwg to vapor

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0123456

z, m

7

quality

void fraction

Figure 5 Calculation 1: Thermal hydraulic void fraction and quality

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

Figure 6 Calculation 2: Void fraction distribution and wall temperature

500

550

600

650

700

750

800

850

01234567

z, m

Tw, K

run 2

exp

Figure 7 Wall temperature: experiment and calculation 2

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

0

100000

200000

300000

400000

500000

600000

700000

800000

900000

01234567

z, m

Qw

Qi

Ql

Qg

Figure 8 Calculation 2: Heat flux from wall: Qi for vaporizing, Ql to water, and

to Qg vapor

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0123456

z, m 7

quality

void fraction

Figure 9 Calculation 2: Thermal hydraulic void fraction and quality

6. CONCLUSIONS

We presented recent advances in the modeling of cladding-to-coolant heat transfer

and critical heat flux that have been implemented in the advanced Computational

Fluid Dynamics (CFD) computer code CFD-BWR. The cladding-to-coolant heat

transfer is described by a wall heat partitioning model which is used in conjunction

with various local flow topologies that range from bubbly to droplet to film flow. Two

experiments involving upward boiling water flow and dryout in a heated circular

channel were analyzed. Comparisons of the calculated wall temperatures with the

corresponding measured values show that the cladding-to-coolant heat transfer model

implemented allows the prediction of a wide range of cladding-to-coolant heat

transfer regimes, including the onset of Critical Heat Flux (CHF), without the use of

empirical correlations traditionally used in sub-channel codes. Future work will

enhance the cladding-to-coolant heat transfer model to include models of droplet

dynamics and droplet interactions with the liquid film and the walls.

Sheraton Station Square, Pittsburgh, Pennsylvania, U.S.A. September 30-October 4, 2007.

ACKNOWLEDGMENTS

This work was performed under partial financial support of the ISTC Project 2601p,

sponsored by the U.S. Department of Energy IPP Program.

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