Advances in the modeling of cladding heat transfer and critical heat flux in boiling water reactor fuel assemblies

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.

Full text

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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

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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,

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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

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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

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(
)
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

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⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎭
⎬
⎫
⎩
⎨
⎧
−
−
=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)

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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

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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.

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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.

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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).

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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

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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

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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

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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.

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.
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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