Westinghouse AP1000 Pressurized Water Reactor Steam Generator
Outlet Plenum Flow Modeling
by
Andrea J. Dalton
A Thesis Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF SCIENCE
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Thesis Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
August 2013
© Copyright 2013
by
Andrea J. Dalton
All Rights Reserved
ii
CONTENTS
CONTENTS ..................................................................................................................... iii
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF ACRONYMS ..................................................................................................... x
LIST OF SYMBOLS ........................................................................................................ xi
LIST OF KEYWORDS .................................................................................................. xiii
ACKNOWLEDGMENT ................................................................................................ xiv
ABSTRACT .................................................................................................................... xv
1. Introduction.................................................................................................................. 1
2. Literature Study ........................................................................................................... 6
3. Methodology ................................................................................................................ 9
3.1
Flow through SG Tubes ...................................................................................... 9
3.2
Flow through SG Outlet Plenum....................................................................... 10
3.2.1
Theory ........................................................................................................ 10
3.2.2
Modeling .................................................................................................... 16
3.2.3
Flow Phenomena Characterization ............................................................ 18
4. Inputs ......................................................................................................................... 20
4.1
RCS Conditions During Startup ........................................................................ 20
4.2
Fluid Properties ................................................................................................. 21
4.3
Froude Number for Flow through SG Tubes .................................................... 22
5. CFD Models............................................................................................................... 26
5.1
Two Dimensional Modeling ............................................................................. 26
5.1.1
45-Tube Model .......................................................................................... 26
5.1.2
100-Tube Model ........................................................................................ 35
5.2
Three Dimensional Modeling ........................................................................... 62
iii
5.2.1
552-Tube Model ........................................................................................ 62
6. Summary of Results ................................................................................................... 85
6.1
Froude Number Calculation Results for SG Tubes .......................................... 85
6.2
CFD Results ...................................................................................................... 85
6.2.1
Summary of Results ................................................................................... 85
6.2.2
Model Limitations due to Hardware .......................................................... 89
6.3
Results of Problem ............................................................................................ 92
7. Conclusions................................................................................................................ 93
8. References.................................................................................................................. 95
iv
LIST OF TABLES
Table 4-1: RCS Conditions for Startup with Four RCPs [3] ........................................... 20
Table 4-2: Selected RCS Temperature Conditions and RCP Speeds .............................. 20
Table 4-3: Fluid Properties .............................................................................................. 21
Table 4-4: AP1000 SG Tube Diameter Dimensions ....................................................... 22
Table 4-5: SG Tube Froude Number ............................................................................... 24
Table 5-1: Fluid Velocity at Top of 45 Tubes ................................................................. 28
Table 5-2: 45-Tube Model Boundary Conditions ........................................................... 31
Table 5-3: Under-Relaxation Factors .............................................................................. 31
Table 5-4: Fluid Velocity at Top of 100 Tubes ............................................................... 37
Table 5-5: 100-Tube Model Boundary Conditions ......................................................... 41
Table 5-6: Under-Relaxation Factors .............................................................................. 41
Table 5-7: Fluid Velocity at Top of Each SG Tube......................................................... 67
Table 5-8: 552-Tube Model Boundary Conditions ......................................................... 70
Table 5-9: Under-Relaxation Factors .............................................................................. 70
v
LIST OF FIGURES
Figure 1-1: AP1000 RCS [9] ............................................................................................. 1
Figure 1-2: AP1000 SG Internals [4]................................................................................. 2
Figure 1-3: Location of Air at Top of SG Tubes ............................................................... 3
Figure 2-1: SG Inlet Plenum Velocity Profile Results from [13], Figure 10 .................... 7
Figure 5-1: 45-Tube Model ............................................................................................. 27
Figure 5-2: 2D 45-Tube Model – Outlet Plenum Region ................................................ 27
Figure 5-3: 2D 45-Tube Model – Inlet Velocity Boundary Condition ............................ 29
Figure 5-4: Mesh for Plenum Region of 45-Tube Model ................................................ 30
Figure 5-5: 45-Tube Laminar Model Velocity Vector Results at 70°F, 200 psia, and 200
rpm RCP Speed................................................................................................................ 33
Figure 5-6: Scaled Residuals for Laminar 45-Tube Model at 70°F, 200 psia, and 200 rpm
RCP Speed ....................................................................................................................... 34
Figure 5-7: 100-Tube Model ........................................................................................... 35
Figure 5-8: 2D 100-Tube Model – Outlet Plenum Region .............................................. 36
Figure 5-9: 2D 100-Tube Model – Inlet Velocity Boundary Condition .......................... 38
Figure 5-10: Mesh for 100-Tube Model .......................................................................... 39
Figure 5-11: Mesh Edge Sizing for 100-Tube Model ..................................................... 40
Figure 5-12: 100-Tube Model Velocity Vector Results at 70°F, 200 psia, and 200 rpm
RCP Speed ....................................................................................................................... 42
Figure 5-13: Detailed Results Near Interior Region for 100-Tube Model at 70°F, 200
psia, and 200 rpm RCP Speed ......................................................................................... 43
Figure 5-14: Central Region Results Detail for 100-Tube Model at 70°F, 200 psia, and
200 rpm RCP Speed......................................................................................................... 44
Figure 5-15: Exterior Region Results Detail for 100-Tube Model at 70°F, 200 psia, and
200 rpm RCP Speed......................................................................................................... 45
Figure 5-16: Detailed Results Near Exterior Region for 100-Tube Model at 70°F, 200
psia, and 200 rpm RCP Speed ......................................................................................... 46
Figure 5-17: Turbulent Kinetic Energy (k) for 100-Tube Model at 70°F, 200 psia, and
200 rpm RCP Speed......................................................................................................... 47
vi
Figure 5-18: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube Model at
70°F, 200 psia, and 200 rpm RCP Speed......................................................................... 48
Figure 5-19: Velocity Vectors Colored by Turbulent Dissipation Rate (ε) for 100-Tube
Model at 70°F, 200 psia, and 200 rpm RCP Speed ......................................................... 49
Figure 5-20: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube Model at
70°F, 200 psia, and 200 rpm RCP Speed......................................................................... 50
Figure 5-21: Scaled Residuals for 100-Tube Model at 70°F, 200 psia, and 200 rpm RCP
Speed................................................................................................................................ 50
Figure 5-22: 100-Tube Model Velocity Vector Results at 231°F, 2,250 psia, and 1,600
rpm RCP Speed................................................................................................................ 51
Figure 5-23: Detailed Results Near Exterior Region for 100-Tube Model at 231°F, 2,250
psia, and 1,600 rpm RCP Speed ...................................................................................... 52
Figure 5-24: Turbulent Kinetic Energy (k) for 100-Tube Model at 231°F, 2,250 psia, and
1,600 rpm RCP Speed...................................................................................................... 53
Figure 5-25: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube Model at
231°F, 2,250 psia, and 1,600 rpm RCP Speed ................................................................ 54
Figure 5-26: Turbulent Dissipation Rate (ε) for 100-Tube Model at 231°F, 2,250 psia,
and 1,600 rpm RCP Speed ............................................................................................... 55
Figure 5-27: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube Model at
231°F, 2,250 psia, and 1,600 rpm RCP Speed ................................................................ 56
Figure 5-28: Scaled Residuals for 100-Tube Model at 231°F, 2,250 psia, and 1,600 rpm
RCP Speed ....................................................................................................................... 56
Figure 5-29: 100-Tube Model Velocity Vector Results at 450°F, 2,250 psia, and 1,750
rpm RCP Speed................................................................................................................ 57
Figure 5-30: Velocity Vectors Colored by Turbulent Kinetic Energy (k) for 100-Tube
Model at 450°F, 2,250 psia, and 1,750 rpm RCP Speed ................................................. 58
Figure 5-31: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube Model at
450°F, 2,250 psia, and 1,750 rpm RCP Speed ................................................................ 59
Figure 5-32: Turbulent Dissipation Rate (ε) for 100-Tube Model at 450°F, 2,250 psia,
and 1,750 rpm RCP Speed ............................................................................................... 60
vii
Figure 5-33: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube Model at
450°F, 2,250 psia, and 1,750 rpm RCP Speed ................................................................ 61
Figure 5-34: Scaled Residuals for 100-Tube Model at 450°F, 2,250 psia, and 1,750 rpm
RCP Speed ....................................................................................................................... 61
Figure 5-35: Base Geometry for 3D Model ..................................................................... 62
Figure 5-36: Base Geometry for 3D Model Showing Tube Outlets ................................ 64
Figure 5-37: Base Geometry for 3D Model Showing Tube Outlet Detail ...................... 64
Figure 5-38: Base Geometry for 3D Model with 552 Tubes and Reduced SG Outlet
Plenum Radius ................................................................................................................. 65
Figure 5-39: 3D Model with 2 Outlet Nozzles, 552 Tubes, and Reduced SG Outlet
Plenum Radius ................................................................................................................. 66
Figure 5-40: SG Tube Named Selections for Inlet Velocity Boundary Conditions ........ 67
Figure 5-41: Mesh for 552-Tube Model .......................................................................... 69
Figure 5-42: Mesh Edge Sizing for 552-Tube Model ..................................................... 69
Figure 5-43: Isometric View of 552-Tube Model Velocity Vector Results at 70°F, 200
psia, and 200 rpm RCP Speed ......................................................................................... 72
Figure 5-44: Side View of 552-Tube Model Velocity Vector Results at 70°F, 200 psia,
and 200 rpm RCP Speed .................................................................................................. 73
Figure 5-45: Front View of 552-Tube Model Velocity Vector Results at 70°F, 200 psia,
and 200 rpm RCP Speed .................................................................................................. 74
Figure 5-46: Interior Region Results Detail for 552-Tube Model at 70°F, 200 psia, and
200 rpm RCP Speed......................................................................................................... 75
Figure 5-47: Detailed Results Near Outlet Nozzle for 552-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed .................................................................................................. 76
Figure 5-48: Velocity Magnitude Particle Tracks for 552-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed .................................................................................................. 77
Figure 5-49: Scaled Residuals for 552-Tube Model at 70°F, 200 psia, and 200 rpm RCP
Speed................................................................................................................................ 78
Figure 5-50: Isometric View of 552-Tube Model Velocity Vector Results at 231°F,
2,250 psia, and 1,600 rpm RCP Speed ............................................................................ 79
viii
Figure 5-51: Velocity Magnitude Particle Tracks for 552-Tube Model at 231°F, 2,250
psia, and 1,600 rpm RCP Speed ...................................................................................... 80
Figure 5-52: Scaled Residuals for 552-Tube Model at 231°F, 2,250 psia, and 1,600 rpm
RCP Speed ....................................................................................................................... 81
Figure 5-53: Isometric View of 552-Tube Model Velocity Vector Results at 450°F,
2,250 psia, and 1,750 rpm RCP Speed ............................................................................ 82
Figure 5-54: Velocity Magnitude Particle Tracks for 552-Tube Model at 450°F, 2,250
psia, and 1,750 rpm RCP Speed ...................................................................................... 83
Figure 5-55: Scaled Residuals for 552-Tube Model at 450°F, 2,250 psia, and 1,750 rpm
RCP Speed ....................................................................................................................... 84
Figure 6-1: Slice Plane through Outlet Nozzles – YZ Plane ........................................... 88
Figure 6-2: Slice Plane through Outlet Nozzles – XY Plane........................................... 88
Figure 6-3: Western Digital My Passport External Hard Drives ..................................... 90
ix
LIST OF ACRONYMS
PWR
Pressurized Water Reactor
RCS
Reactor Coolant System
CE
Combustion Engineering
RV
Reactor Vessel
SG
Steam Generator
RCP
Reactor Coolant Pump
CFD
Computational Fluid Dynamics
NSSS
Nuclear Steam Supply System
NPSH
Net Positive Suction Head
2D
Two Dimensional
3D
Three Dimensional
U. S.
United States
NRC
Nuclear Regulatory Commission
PWROG
Pressurized Water Reactor Owner’s Group
UDS
Upwind Differencing Scheme
SIMPLE
Semi-Implicit Method for Pressure-Linked Equations
x
LIST OF SYMBOLS
Fr
Froude Number (dimensionless)
V
Velocity (ft/s, m/s)
L
Characteristic Length (ft, m)
g
Acceleration due to Gravity (ft/s2, m/s2)
M
Mass (lbm, kg)
t
Time (s)
Φ
Extensive Property (dimensionless)
Ωcm
Volume of Control Mass (ft3, m3)
ρ
Density (lbm/ft3, kg/m3)
ϕ
Intensive Property (dimensionless)
Ωcv
Volume of Control Volume (ft3, m3)
scv
Surface Enclosing Control Volume (ft, m)
𝑣⃑
Velocity Vector (ft/s, m/s)
𝑛⃑⃑
Orthogonal Unit Vector (dimensionless)
∇
Nabla (Vector) Operator (dimensionless)
ui
Velocity Component (ft/s, m/s)
xi
Coordinate Direction Component (ft, m)
mv
Momentum (lbm-ft/s, kg-m/s)
f
Forces (lbf, N)
𝑏⃑⃑
Body Forces per Mass Unit (lbf, N)
τ
Stress Tensor (lbf/ft2, N/m2)
τij
Viscous Component of the Stress Tensor (lbf/ft2, N/m2)
p
Pressure (psi, Pa)
gi
Gravitational Acceleration Vector Component (ft/s2, m/s2)
µ
Dynamic Viscosity (lbm/ft-s, kg/m-s)
δij
Kronecker Symbol (dimensionless)
x
Cartesian Coordinate Direction (ft, m)
y
Cartesian Coordinate Direction (ft, m)
z
Cartesian Coordinate Direction (ft, m)
xi
k
Turbulence Kinetic Energy (lbf-ft2/lbm-s2, J/kg)
ε
Viscous (Kinematic) Dissipation Rate (ft2/s3, m2/s3)
D
Inertial Diffusive Transport (ft/s2, m/s2)
Pk
Production by Shear Stress (lbf, N)
H
Denotes Higher Order Terms (dimensionless)
D
Tube Inner Diameter (ft, m)
xii
LIST OF KEYWORDS
Westinghouse
Westinghouse Electric Company LLC
Steam Generator
Major component in nuclear power plant; converts liquid water to
steam through heat transfer
AP1000®
Four loop Westinghouse PWR
Navier-Stokes
Conservation of mass and conservation of momentum equations
describing turbulent flow
k-epsilon (k-ε)
Turbulence kinetic energy-viscous (kinematic) dissipation rate
turbulence model
Vacuum Refill
Primary method of venting air from AP1000 SG tubes
Dynamic Venting
Secondary method of venting air from AP1000 SG tubes
Pump Bumping
Process of cycling RCPs to generate sufficient flow to vent air
from SG tubes
Variable Drive
AP1000 RCPs are driven by variable speed motors
ANSYS WORKBENCH
Program which can be used to control related CFD engines
(geometry, mesh, calculation setup, and post-processing)
ANSYS FLUENT
CFD program used to solve Navier-Stokes equations in this study
Reynolds Number
Flow characteristic based on velocity, geometry, and fluid
properties
Froude Number
Flow characteristic quantifying gravitational inertia
xiii
ACKNOWLEDGMENT
I would like to thank my professor and adviser, Dr. Ernesto Gutierrez-Miravete for not
only teaching me the theory and principles of CFD during an independent study course,
but then pushing me to pursue a thesis involving CFD. The direction he provided
motivated me to explore more of what was possible in CFD, and his comments and
questions over the course of this experience greatly improved the end result.
I would also like to thank my parents, William and Pamela Dalton, for their constant
encouragement and support.
xiv
ABSTRACT
This thesis reports on work carried out to model the details of flow phenomena in a
Westinghouse AP1000 Pressurized Water Reactor (PWR) Steam Generator (SG) outlet
plenum. Two and three dimensional Computational Fluid Dynamics (CFD) models are
created using ANSYS WORKBENCH 14.0.0 and ANSYS FLUENT 14.0.
These
models are used to analyze the flow paths through the SG outlet plenum by solving the
Navier-Stokes equations together with a k-epsilon (k-ε) turbulence model.
Three
potential AP1000 startup conditions are analyzed. The results show that there is some
mixing and recirculation present at the tube exits, but the flow eventually moves to the
outlet in a relatively smooth flow path.
xv
1. Introduction
The Westinghouse Electric Company LLC (Westinghouse) AP1000® Pressurized Water
Reactor (PWR) Reactor Coolant System (RCS) is similar in design to the Combustion
Engineering (CE) RCS design, with a Reactor Vessel (RV) housing the core, two Steam
Generators (SGs), four Reactor Coolant Pumps (RCPs), and a pressurizer. Primary
coolant exits the core and flows through the two hot legs to the SGs. Heat is transferred
to the secondary side fluid in the SGs, generating steam to power a turbine. The primary
flow exits the SGs and returns to the RV through the cold legs to be reheated. The
AP1000 RCS layout (primary side) is shown in Figure 1-1 [9].
Figure 1-1: AP1000 RCS [9]
1
The AP1000 SGs are designed with inverted u-shaped tubes, similar to the CE SGs.
Figure 1-2 shows the AP1000 SG internals including the tubes [4].
OUTLET NOZZLE
Figure 1-2: AP1000 SG Internals [4]
While plants are shut down for refueling outages or other maintenance activities, RCPs
are off and RCS inventory is stagnant. Refueling and other maintenance require that the
RV head is lifted, exposing the fuel and compromising the system pressure boundary.
2
This creates the opportunity for air to be introduced into the RCS, which must be watersolid during normal operation to avoid damage to the RCPs from air entrainment.
The air which enters the RCS during outages becomes trapped in the high point of the
system, at the top of the SG tubes as shown simplistically in Figure 1-3.
Figure 1-3: Location of Air at Top of SG Tubes
In order to avoid damage to the RCPs, the air must be completely swept from the RCS
during plant startup.
This is currently accomplished at CE Nuclear Steam Supply
System (NSSS) design plants using two methods: dynamic venting and vacuum refill.
The methods used at CE plants are of particular interest in this study since the AP1000
and CE RCS designs are so similar.
Dynamic venting was the first method developed to clear the RCS of excess gases
during startup. During dynamic venting, RCPs are cycled to ‘bump’ the air from the
SGs. A single RCP is turned on, run for a brief amount of time, and is shut off. The
duration of pump operation is dictated by the specific Net Positive Suction Head
Requirements (NPSH) of the RCPs. The pump operation provides a motive force to
drive the trapped air from the top of the SG tubes, through the SG outlet plenum and
3
outlet nozzle, and to a downstream relief valve where it is vented from the system. This
process, commonly referred to as ‘pump bumping,’ is repeated by cycling all of the
RCPs until the air is completely cleared from both SGs. Due to grid power consumption
concerns related to starting an RCP, only one pump can be started at a time.
Vacuum refill was developed in the late 1980s as a more contemporary method to
remove the excess air from the RCS during startup while reducing outage time. Vacuum
refill enhances residual heat removal by using the SGs as reflux condensers and using a
pump to establish a vacuum in the vapor space created in the RV and SGs [1]. The
vacuum created in the RCS draws water into the RCS and creates a water-solid
condition.
While vacuum refill is intended to be the primary method for removing the air from the
RCS during startup for AP1000 plants, dynamic venting may be a possible secondary
strategy. Dynamic venting in AP1000 plants presents a new challenge because the RCPs
installed at CE plants have a single drive speed, but the RCPs to be installed at AP1000
plants are designed with variable speed drives so that the pumps can be started at the
lowest speed setting and gradually ramped up to full speed. The variable speed drives
offer many benefits including a lower draw on grid power during startup. It is possible,
however, that the lower pump speeds during startup would not generate sufficient flow
rates to sweep the air completely from the SGs during dynamic venting.
Dynamic venting has not been analyzed for the operational CE plants in which it is or
was used because it is not a safety-related plant function; the pumps only have one
speed, and can only be turned on one at a time due to power consumption and NPSH
concerns, so the only option for venting is to cycle the pumps until all of the air is
removed from the RCS.
There has not previously been a need to determine the
effectiveness of this process. Instead, utilities noted best practices learned through
operational experience and developed plant-specific dynamic venting procedures.
4
Hydraulically, the RCS fluid velocity during dynamic venting must be high enough to
overcome the gravitational inertia which naturally maintains the trapped at the top of the
SG tubes. The gravitational inertia applies to the length of half of an SG tube (since the
bubble is located at the top of the tubes) and to the SG outlet plenum and outlet nozzle
regions since they are also generally located at elevations higher than the cold legs
where the primary fluid exits the SGs.
A Froude number calculation is used to
determine if the flow velocity through the tubes is sufficient to overcome the effects of
gravity. This type of calculation is not possible for the geometry of the SG outlet
plenum because an appropriate hydraulic diameter correlation has not been defined for
the outlet plenum region.
Instead, two dimensional (2D) and three dimensional (3D) Computational Fluid
Dynamics (CFD) models created using ANSYS WORKBENCH 14.0.0 and ANSYS
FLUENT 14.0 are used to analyze the flow paths through the SG outlet plenum based on
the expected RCP flow rates during startup. The models in this analysis solve the
Navier-Stokes flow equations with the k-epsilon (k-ε) turbulence model. The energy
equations are not applied because this study only considers flow characteristics
independent of heat transfer.
The models generated in this analysis are used to
determine if dynamic venting is a viable option for AP1000 plants as an alternate
strategy to vacuum refill.
5
2. Literature Study
Liquid-only steam generator flows were modeled using ANSYS FLUENT in a CFD
study by Bredberg [10], but the study in [10] only considered a horizontal SG. The SG
inlet plenum was specifically modeled for a fast-breeder reactor in a CFD study by Patil,
et. al [11]. This study considered turbulent flows through the inlet plenum, which has
similar geometry to the outlet plenum, and concluded that the flows are highly nonuniform. Air entrainment was not considered. Void fraction and hydraulic jump in a
PWR hot leg were studied by Deendarlianto, et. al using CFD modeling as well as a test
loop [12], but the upward flow through the SG tubes after the flow enters the steam
generator was not considered.
The study described in [13] is a United States (U. S.) Nuclear Regulatory Commission
(NRC) funded study, published in 2003, which developed a CFD model of an SG inlet
plenum based on a Westinghouse 1/7th scale test facility which was designed based on
the Indian Point Unit 2 PWR. The goal of the study contained in [13] was to model inlet
plenum mixing as part of a steam generator action plan to address tube integrity issues
during severe accident scenarios. The CFD model simplified the geometry by reducing
the number of tubes and using tubes of a square cross section rather than a circular cross
section for ease of meshing with hexagonal elements. Steady state solutions were
generated using a transient solver with steady state boundary conditions. The results of
the CFD model were compared to the 1/7th model test data to validate the model. The
CFD model results were generally within 5% of the 1/7th scale model test data. Velocity
vectors are provided in Figure 10 of [13], reproduced here as Figure 2-1. The flow paths
shown in Figure 2-1 can be compared to the results of this SG outlet plenum study to
determine the similarities and differences between the SG inlet and outlet plenum flows.
6
Figure 2-1: SG Inlet Plenum Velocity Profile Results from [13], Figure 10
SG outlet plenums have not been specifically modeled and documented for the purpose
of studying dynamic venting capability in a CE or AP1000 NSSS design PWR. This
analysis will develop a model of an SG outlet plenum with the intent to show that
AP1000 RCS flow rates during potential dynamic venting conditions should be
sufficient to clear any air which might be trapped in the SG tubes before the pumps are
7
fully engaged and operating normally. The results of this study are compared to the
results presented in previous studies to determine if there are any similarities or
differences between the modeled flow paths.
8
3. Methodology
3.1 Flow through SG Tubes
Since the air trapped in the SG tubes is located at the top of the tubes, it needs to be
pushed over the top of the u-bends to the bottom of the tubes and out of the SG outlet
nozzles in order for it to be swept completely to the reactor vessel where it can be
vented. Gravitational inertia must be overcome in order for this process to be successful.
For the tube and outlet nozzle portions of this system, Froude number calculations for
flow through a pipe can be used to show whether there is sufficient fluid velocity to
overcome gravitational inertia. The Froude number for flow through a pipe is [2]:
𝐹𝑟 =
𝑉
√𝐿𝑔
Equation 1
Where:
Fr = Froude number
V = fluid velocity
L = characteristic length (for pipe, L = diameter)
g = acceleration due to gravity
A study at Purdue University funded by the Pressurized Water Reactor Owner’s Group
(PWROG) showed that for 8 inch diameter pipe, a Froude number greater than 0.93
represents complete air entrainment down a vertical section within 50 seconds [5]. Fluid
velocity is determined using volumetric flows for various pump speeds from Table 5-7
of [3]. AP1000 SG tube dimensions are taken from Table 5.4-4 of [4]. The Froude
numbers calculated using Equation 1 determine if the flow is sufficient to push the air
over the top of the u-bend and into the SG outlet plenum.
9
3.2 Flow through SG Outlet Plenum
3.2.1
Theory
Flow is considered turbulent if it is rotational, intermittent, highly disordered, diffusive,
and dissipative [15]. It is expected that the flow in the SG outlet plenum will be
turbulent. Turbulence is described by the Navier-Stokes momentum transport equations
based on conservation of mass and conservation of momentum principles [17].
3.2.1.1 Conservation of Mass
The conservation of mass (continuity) equation for a volume element in a flowing fluid
states that the mass within a closed system must be maintained within that system:
𝑑𝑀
=0
𝑑𝑡
Equation 2
Where:
M = mass
t = time
𝛷 = ∫ 𝜌𝜙𝑑𝛺
𝛺𝑐𝑚
Equation 3
Where:
Φ = extensive property
Ωcm = volume of control mass
ρ = density
ϕ = intensive property (= 1 for conservation of mass; = v for conservation of momentum)
The conservation of mass becomes:
10
𝑑
∫ 𝜌𝑑𝛺 + ∫ 𝜌𝑣⃑ ∙ 𝑛⃑⃑𝑑𝑠 = 0
𝑑𝑡
𝛺𝑐𝑣
𝑠𝑐𝑣
Equation 4
Where:
Ωcv = volume of control volume
scv = surface enclosing control volume
𝑣⃑ = fluid velocity vector
𝑛⃑⃑ = orthogonal unit vector
The conservation of mass written in the form of Gauss’ Divergence Theorem is:
𝜕𝜌
𝜕𝜌 𝜕(𝜌𝑢𝑖 )
+ ∇ ∙ (𝜌𝑣⃑) =
+
=0
𝜕𝑡
𝜕𝑡
𝜕𝑥𝑖
Equation 5
Where:
𝜕
𝜕
𝜕
∇ = nabla (vector) operator = (𝜕𝑥 , 𝜕𝑦 , 𝜕𝑧)
ui = velocity component
xi = coordinate direction component
For a three dimensional rectangular Cartesian system of coordinates with directions x, y,
and z, the conservation of mass is written in Einstein convention:
𝜕(𝜌𝑢𝑖 ) 𝜕(𝜌𝑢𝑥 ) 𝜕(𝜌𝑢𝑦 ) 𝜕(𝜌𝑢𝑧 )
=
+
+
𝜕𝑥𝑖
𝜕𝑥
𝜕𝑦
𝜕𝑧
Equation 6
11
3.2.1.2 Conservation of Momentum
The conservation of momentum equation for a Newtonian fluid states that the
momentum within a closed system must be maintained within that system:
𝑑(𝑚𝑣)
= ∑𝑓
𝑑𝑡
Equation 7
Where:
mv = momentum
f = forces
The conservation of momentum has an intensive property (ϕ) equal to the velocity (see
Equation 3 variable definition). The conservation of momentum is:
𝑑
∫ 𝜌 𝑣⃑𝑑𝛺 + ∫ 𝜌𝑣⃑𝑣⃑ ∙ 𝑛⃑⃑𝑑𝑠 = ∑ 𝑓
𝑑𝑡
𝛺𝑐𝑣
𝑠𝑐𝑣
Equation 8
𝑑
⃑⃑⃑⃑⃑⃑⃑⃑⃑
∫ 𝜌 𝑣⃑𝑑𝛺 + ∫ 𝜌𝑣⃑𝑣⃑ ∙ 𝑛⃑⃑𝑑𝑠 = ∫ 𝜏⃑ ∙ 𝑛⃑⃑𝑑𝑠 + ∫ 𝜌𝑏𝑑𝛺
𝑑𝑡
𝛺𝑐𝑣
𝑠𝑐𝑣
𝑠𝑐𝑣
𝑠𝑐𝑣
Equation 9
Where:
𝑏⃑⃑ = body forces per unit mass
τ = stress tensor
The conservation of momentum can also be written as:
𝜌
𝜕𝜏𝑖𝑗 𝜕𝑝
𝜕𝑢𝑖
+ 𝜌(v
⃑⃑ ∙ ∇)𝑢𝑖 =
−
+ 𝜌𝑔𝑖
𝜕𝑡
𝜕𝑥𝑗 𝜕𝑥𝑖
Equation 10
12
Where:
τij = viscous component of the stress tensor (defined in Equation 11)
p = pressure
gi = gravitational acceleration vector component
The viscous component of the stress tensor is:
𝜕𝑢𝑖 𝜕𝑢𝑗
2
𝜏𝑖𝑗 = 𝜇 (
+
) − 𝜇𝛿𝑖𝑗 ∇ ∙ v
⃑⃑
𝜕𝑥𝑗 𝜕𝑥𝑖
3
Equation 11
Where:
µ = dynamic viscosity
δij = Kronecker symbol (= 1 if i = j; = 0 if i ≠ j)
For a three dimensional rectangular Cartesian system of coordinates with directions x, y,
and z:
𝜕𝑢𝑦
𝜕𝑢𝑥
𝜕𝑢𝑥
𝜕𝑢𝑧 𝜕𝜏𝑥𝑥 𝜕𝜏𝑥𝑦 𝜕𝜏𝑥𝑧 𝜕𝑝
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
=
+
+
−
+ 𝜌𝑔𝑥
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜌
Equation 12
𝜌
𝜕𝑢𝑦
𝜕𝑢𝑦
𝜕𝑢𝑦
𝜕𝑢𝑦 𝜕𝜏𝑦𝑥 𝜕𝜏𝑦𝑦 𝜕𝜏𝑦𝑧 𝜕𝑝
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
=
+
+
−
+ 𝜌𝑔𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑦
Equation 13
𝜌
𝜕𝑢𝑧
𝜕𝑢𝑧
𝜕𝑢𝑧
𝜕𝑢𝑧 𝜕𝜏𝑧𝑥 𝜕𝜏𝑧𝑦 𝜕𝜏𝑧𝑧 𝜕𝑝
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
=
+
+
−
+ 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑧
Equation 14
For incompressible fluids, the components of the viscous component of the stress tensor
are:
𝜏𝑥𝑥 = 2𝜇
𝜕𝑢𝑥
𝜕𝑥
Equation 15
13
𝜏𝑦𝑦 = 2𝜇
𝜕𝑢𝑦
𝜕𝑦
Equation 16
𝜏𝑦𝑦 = 2𝜇
𝜕𝑢𝑦
𝜕𝑦
Equation 17
𝜕𝑢𝑥 𝜕𝑢𝑦
𝜏𝑥𝑦 = 𝜏𝑦𝑥 = 𝜇 (
+
)
𝜕𝑦
𝜕𝑥
Equation 18
𝜕𝑢𝑦 𝜕𝑢𝑧
𝜏𝑦𝑧 = 𝜏𝑧𝑦 = 𝜇 (
+
)
𝜕𝑧
𝜕𝑦
Equation 19
𝜏𝑧𝑥 = 𝜏𝑥𝑧 = 𝜇 (
𝜕𝑢𝑧 𝜕𝑢𝑥
+
)
𝜕𝑥
𝜕𝑧
Equation 20
The conservation of momentum equations in rectangular Cartesian coordinates
(Equation 12, Equation 13, and Equation 14) become:
𝜌
𝜕𝑢𝑦
𝜕𝑢𝑥
𝜕𝑢𝑥
𝜕𝑢𝑧
𝜕 2 𝑢𝑥 𝜕 2 𝑢𝑥 𝜕 2 𝑢𝑥
𝜕𝑝
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
= 𝜇( 2 +
+
)−
+ 𝜌𝑔𝑥
2
2
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
Equation 21
𝜕𝑢𝑦
𝜕𝑢𝑦
𝜕𝑢𝑦
𝜕𝑢𝑦
𝜕 2 𝑢𝑦 𝜕 2 𝑢𝑦 𝜕 2 𝑢𝑦
𝜕𝑝
𝜌
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
= 𝜇( 2 +
+
)
−
+ 𝜌𝑔𝑦
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜕𝑦
Equation 22
14
𝜕𝑢𝑧
𝜕𝑢𝑧
𝜕𝑢𝑧
𝜕𝑢𝑧
𝜕 2 𝑢𝑧 𝜕 2 𝑢𝑧 𝜕 2 𝑢𝑧
𝜕𝑝
𝜌
+ 𝜌𝑢𝑥
+ 𝜌𝑢𝑦
+ 𝜌𝑢𝑧
= 𝜇( 2 +
+
)
−
+ 𝜌𝑔𝑧
𝜕𝑡
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦 2
𝜕𝑧 2
𝜕𝑧
Equation 23
The first term on the left hand side of Equation 21, Equation 22, and Equation 23
represents the time rate of change of momentum in the specific direction (x, y, or z).
The three other terms on the left hand side of Equation 21, Equation 22, and Equation 23
describe the rate of change of momentum in the fluid due to the three vector
components. These terms are referred to as the inertial terms. The first three terms on
the right hand side of Equation 21, Equation 22, and Equation 23 are the time rate of
change of momentum associated with the internal viscous forces. The final two terms on
the right hand side of Equation 21, Equation 22, and Equation 23 are the rate of change
of momentum due to spacial pressure variations in the fluid and the rate of change of
momentum due to the action of gravity.
3.2.1.3 Turbulence
The Navier-Stokes equations describe the characteristics of the flow field; specifically of
the SG outlet plenum evaluated in this study. However, impractically fine meshing
would be required to accurately capture the details of the flow and in practice, turbulence
models are commonly used. The standard k-epsilon (k-ε) model is used as the solver for
the kinetic energy of turbulence in this study. The k-ε model is described in [15]:
𝑅𝑎𝑡𝑒 𝑜𝑓 𝐶ℎ𝑎𝑛𝑔𝑒 + 𝐴𝑑𝑣𝑒𝑐𝑡𝑖𝑣𝑒 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡
= 𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝐷𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑒 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 (𝒟)
− 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑏𝑦 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 (𝑃𝑘 )
− 𝑉𝑖𝑠𝑐𝑜𝑢𝑠 (𝐾𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐)𝐷𝑖𝑠𝑠𝑖𝑝𝑎𝑡𝑖𝑜𝑛 𝑅𝑎𝑡𝑒 (𝜀)
Equation 24
15
𝜕𝑘
𝜕𝑘
𝜕 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑢𝑖 ′𝑢𝑖 ′ 𝑝′
̅̅̅̅̅̅̅
̅̅̅̅̅̅̅ 𝜕𝑢̅𝑖
+ 𝑢̅𝑗
=−
[𝑢𝑗′ (
+ ) − 2𝑣𝑢
𝑖 ′𝑠𝑖𝑗 ] − 𝑢𝑖 ′𝑢𝑗 ′
𝜕𝑡
𝜕𝑥𝑗
𝜕𝑥𝑗
2
𝜌
𝜕𝑥𝑗
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝜕𝑢𝑖 ′ 𝜕𝑢𝑖 ′ 𝜕𝑢𝑖 ′ 𝜕𝑢𝑖 ′
−𝑣(
+
)
𝜕𝑥𝑗 𝜕𝑥𝑗
𝜕𝑥𝑗 𝜕𝑥𝑖
Equation 25
Where:
𝑠𝑖𝑗 =
1 𝜕𝑢𝑖 ′ 𝜕𝑢𝑗 ′
(
+
)
2 𝜕𝑥𝑗
𝜕𝑥𝑖
Equation 26
The k-ε model simplifies to:
𝐷𝑘 𝜕𝑘
𝜕𝑘
≡
+ 𝑢̅𝑗
= 𝒟𝑘 + 𝑃𝑘 − 𝜀
𝐷𝑡 𝜕𝑡
𝜕𝑥𝑗
Equation 27
The turbulent k-ε model are involved in the FLUENT calculation for the cases in this
study.
3.2.2
Modeling
A Froude number calculation is not appropriate for the SG outlet plenum region because
of the complex geometry of the region. In order to study the flow phenomena occurring
in the SG outlet plenum, 2D and 3D CFD hydraulic models of an SG outlet plenum are
created in ANSYS FLUENT 14.0 using ANSYS WORKBENCH 14.0.0. Only the
primary side is modeled because that is the area of interest of this study. The SG outlet
plenum models include SG tube exits, the outlet plenum, and outlet nozzle. There are
normally thousands of SG tubes per SG, but fewer tubes are modeled to simplify the
model in order to avoid convergence and run-time issues.
The number of tubes
modeled, mesh quality, and number of iterations are dictated by the computing power.
The models are only run as steady state scenarios due to computer processor limitations.
16
3.2.2.1.1 Upwinding
The Upwind Differencing Scheme (UDS) is a way to approximate the value of ϕ e using
the value of the upstream node. The value of ϕe is approximated as:
𝜙 𝑖𝑓 (𝑣⃑ ∙ 𝑛⃑⃑)𝑒 > 0
𝜙𝑒 = { 𝑃
𝜙𝐸 𝑖𝑓 (𝑣⃑ ∙ 𝑛⃑⃑)𝑒 < 0
Equation 28
A Taylor series expansion about P for Cartesian coordinates with (𝑣⃑ ∙ 𝑛⃑⃑)𝑒 > 0 and where
H denotes higher order terms is:
𝜙𝑒 = 𝜙𝑃 + (𝑥𝑒 − 𝑥𝑃 ) (
(𝑥𝑒 − 𝑥𝑃 )2 𝜕 2 𝜙
𝜕𝜙
) +
( 2) + 𝐻
𝜕𝑥 𝑃
2
𝜕𝑥 𝑃
Equation 29
Upwinding prevents the development of oscillations at the expense of some degree of
accuracy of the solution. It accounts for the flow direction within the solver such that
the direction of the flow will influence the form of the finite difference. This study uses
the second order upwinding option embedded within the ANSYS software to reduce
oscillations and converge to a solution for each case.
3.2.2.1.2 SIMPLE Algorithm
The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) algorithm is a
procedure for developing the flow field in a system.
The sequence of operations
described in [16], Chapter 5 is:
1. Guess the pressure field p*.
2. Solve the momentum equations to obtain u*, v*, and w*.
3. Solve the p' equation.
4. Calculate p using:
𝑝 = 𝑝∗ + 𝑝′
Equation 30
5. Calculate u, v, and w from their starred values using:
17
𝑢𝑒 = 𝑢𝑒∗ + 𝑑𝑒 (𝑝𝑃 ′ − 𝑝𝐸 ′)
Equation 31
𝑣𝑛 = 𝑣𝑛∗ + 𝑑𝑛 (𝑝𝑃 ′ − 𝑝𝑁 ′)
Equation 32
𝑤𝑡 = 𝑤𝑡∗ + 𝑑𝑡 (𝑝𝑃 ′ − 𝑝𝑇 ′)
Equation 33
6. Solve the discretization equation for other ϕ’s (such as temperature,
concentration and turbulence quantities) if they influence the flow field
through fluid properties, source terms, etc.
7. Treat the corrected pressure p as a new guessed pressure p*, return to step 2,
and repeat the whole procedure until a converged solution is obtained.
The SIMPLE algorithm is embedded within the ANSYS software which is used in this
study.
3.2.3
Flow Phenomena Characterization
The results obtained using the CFD models can be used to determine the flow
phenomena that exist in the SG outlet plenum region and the effects these phenomena
have on forward flow exiting the SG. Since the outlet plenum collects the exit flow from
thousands of tubes, it is expected that there is some degree of mixing and recirculation in
this region. It is also possible that hydraulic jump occurs in the outlet plenum.
Hydraulic jump occurs when shock waves within fluid channels create discontinuities in
the fluid height and fluid velocity [8]. Continuity and momentum for hydraulic jump
regions can be characterized using equations from [8]. Relationships between fluid
levels and velocities on either side of the hydraulic jump region are developed in [7].
Correlations between fluxes of kinetic energy before and after a hydraulic jump are
described in [6].
18
It is appropriate to model the flow of water to determine if air can be successfully
pushed from the SG tubes and through the SG outlet plenum because it is the flow paths
which are of interest. It is expected that air which is trapped in the SG tubes would
follow a similar flow path as the water modeled in this study. The force generated by
the flow of the water (shown as velocity vector results in Section 5 of this report) would
force the air to follow a path of similar direction as the water.
19
4. Inputs
4.1
RCS Conditions During Startup
AP1000 RCS temperature conditions vary from 70°F to 557°F during RCP startup [3].
The estimated pump speeds, head, and flow rates achieved using the variable frequency
RCP drives during startup with four RCPs operating are taken from [3], Table 5-7 and are
shown in Table 4-1 with the associated time and temperature condition.
Table 4-1: RCS Conditions for Startup with Four RCPs [3]
Time (min)
0
3
360
361
840
841
1,080
RCS
Temperature
(°F)
70
70
231
231
450
450
557
Speed (rpm)
Head (ft)
Flow/Pump
(gpm)
Total Flow
(gpm)
200
1,600
1,600
1,680
1,680
1,750
1,750
5
322
322
335
335
365
365
9,000
74,000
74,000
75,500
75,500
78,750
78,750
36,000
288,000
288,000
302,000
302,000
315,000
315,000
Three flow and temperature conditions during startup are selected from Table 4-1 to be
modeled in this analysis. The three selected conditions capture the range of pre-Hot Zero
Power (HZP) conditions and are shown in Table 4-2. The SG flow column in Table 4-2
is the total incoming flow from two RCPs operating in the same loop; double the flow per
pump listed in Table 4-1. This analysis will only consider startup combinations for two
RCPs operating in the same loop since reverse flow data for part-loop operation is not
available.
Table 4-2: Selected RCS Temperature Conditions and RCP Speeds
RCS
Temperature
(°F)
RCP Speed
(rpm)
SG Flow
(gpm)
70
231
450
200
1,600
1,750
18,000
148,000
157,500
20
4.2
Fluid Properties
Properties of water for the temperature conditions listed in Table 4-2 are needed to
calculate the associated Reynolds number for flow through SG tubes. Density is also
needed as an input for the FLUENT model.
Pressures associated with the startup
conditions are not specified in [3]. The normal operating RCS pressure is listed as 2,250
psia in [9], Table 5.1-3. This pressure is assumed to be similar enough to the pressure at
231°F and 450°F for the purposes of this analysis. A bounding low RCS pressure of 200
psia is assumed for the cold temperature condition of 70°F. The fluid properties based on
these conditions are taken from [14] and are shown in Table 4-3. Density and viscosity
are provided in both English and metric units in Table 4-3.
Table 4-3: Fluid Properties
RCS
Temperature
(°F)
RCS
Pressure
(psia)
Density
(lbm/ft3)
70
231
450
200
2,250
2,250
62.3377
59.7938
52.1819
21
Density
(kg/m3)
Viscosity
(lbm/ft-s)
Viscosity
(kg/m-s)
998.554174 6.5481E-04 4.7362E-03
957.804804 1.7297E-04 1.2511E-03
835.873862 7.9307E-05 5.7363E-04
4.3
Froude Number for Flow through SG Tubes
The tops of steam generator tubes fill with air during refueling outages (Figure 1-3). A
Froude number calculation is performed to determine if the fluid velocity is sufficient to
clear the air from the SG tubes into the SG outlet plenum. The PWROG-funded study at
Purdue University [5] showed that for 8 inch diameter pipe, a Froude number greater than
0.93 represents complete air entrainment down a vertical section within 50 seconds ([5],
Section 7.2.5). The AP1000 SG tube diameter is calculated as shown in Table 4-4:
Table 4-4: AP1000 SG Tube Diameter Dimensions
Dimension
Value (in)
Value (ft)
Reference
Tube Outer Diameter
0.688 in
0.057 ft
[4], Table 5.4-4
Tube Wall Thickness
0.040 in
0.003 ft
[4], Table 5.4-4
Tube Inner Diameter
(D)
0.608 in
0.051 ft
Calculated
(0.688 in – (2 x 0.040 in))
Since the diameter of the flow area for the AP1000 SG tube flow area is much smaller
than the pipe diameter used in the Purdue study, a Froude number of 0.93 can be used as
a critical value for the SG flow paths.
The SG tube flow area is calculated for cylindrical pipe using the SG tube inner diameter
determined above. The SG tube flow area for each tube is:
𝐷2
(0.608 𝑖𝑛)2
𝜋
=𝜋
= 0.290 𝑖𝑛2 = 2.014𝑥10−3 𝑓𝑡 2
4
4
There are 10,025 tubes in each AP1000 SG ([4], Table 5.4-4). The RCS configurations
considered in this analysis are shown in Table 4-2. The flow rate per tube is calculated
for each configuration by dividing the total SG flow rate from Table 4-2 by 10025 tubes
per SG. The flow rate per tube is converted to velocity per tube using the conversion
22
factor of 7.4805 gal/ft3 and the SG tube flow area of 2.014x10-3 ft2, calculated above.
The inner diameter of each SG tube, 0.051 ft, is used with Equation 1 to calculate the
Froude number for the SG tubes in each flow condition.
documented in Table 4-5.
23
These calculations are
Table 4-5: SG Tube Froude Number
RCS
RCP
Temperature Speed
(°F)
(rpm)
70
231
450
200
1,600
1,750
SG
Flow
(gpm)
SG
Tube
Flow
(gpm)
Conversion
Factor (gal/ft3)
SG Tube
Flow
Area (ft2)
18,000
148,000
157,500
1.796
14.763
15.711
7.4805
7.4805
7.4805
2.014E-03
2.014E-03
2.014E-03
24
SG Tube
Acceleration
SG Tube
Flow
Due to
Froude
Diameter
Velocity
Gravity
Number
(ft)
(ft/s)
(ft/s2)
1.986
16.333
17.381
0.051
0.051
0.051
32.174
32.174
32.174
1.556
12.792
13.613
A comparison of the Froude numbers shown in Table 4-5 to the critical Froude number,
0.93 [5], shows that the air collected in the SG tubes during an outage will successfully
be swept clear of the tubes for the selected pump speed and fluid temperature conditions
from Table 4-2. A CFD model of the SG tubes is not needed since the Froude number
equation can be used to determine if the fluid force will be sufficient to clear the air from
the tubes.
The Froude numbers calculated in Table 4-5 are based on the average flow velocity
through the SG tubes. In reality, the flow velocity differs for each row of SG tubes
based on the length of the tube. The shorter SG tubes have a higher flow rate since there
is less resistance than in the longer tubes. The longer SG tubes have more resistance to
flow and therefore, have lower flow rates. The difference in resistance based on tube
length is considered negligible in this analysis.
25
5. CFD Models
5.1 Two Dimensional Modeling
To gain insight into the problem and to be able to quickly try ideas and carry out
computer experiments during code development, two dimensional models are first
investigated.
5.1.1
45-Tube Model
5.1.1.1 Geometry and Velocity Boundary Conditions
Flow through the SG outlet plenum is first modeled in Cartesian two dimensions using
ANSYS WORKBENCH 14.0.0 and ANSYS FLUENT 14.0. The system is modeled as
a set of tubes flowing into the outlet plenum. In two dimensions, the tubes appear as
slots. The number of tubes modeled is limited by the computational power. At first, the
SG is modeled with 45 tubes of the same diameter as the AP1000 SG tubes. No outlet
nozzle is modeled as this preliminary model is only used to show that the model in
development will successfully mesh and run in ANSYS.
The inverted “u” shaped
portion of the tubes is also not modeled in the 45-tube model. The full model geometry
of the 45-tube model created in the DesignModeler engine of WORKBENCH is shown
in Figure 5-1.
26
Figure 5-1: 45-Tube Model
The geometry for the plenum region is shown in Figure 5-2 with the associated named
selections created during the meshing process.
sg_tube_walls
sg_tube_bottoms
sg_bowl_in
sg_bowl_bottom
Figure 5-2: 2D 45-Tube Model – Outlet Plenum Region
27
Instead of modeling the inverted “u” shaped portion of the tubes, a fixed inlet flow rate
is used as a boundary condition across the tops of the tubes. The fixed velocity is chosen
such that the Reynolds number effects through the tubes match the Reynolds number
effects which would be present if 10,025 tubes were modeled. The Reynolds number is:
𝑅𝑒 =
𝜌𝑉𝐷
𝜇
Equation 34
Since velocity is in the numerator of Equation 34, the velocity through a single tube can
be multiplied by the number of tubes and applied at the top of the tubes to achieve the
same Reynolds number effects as 10,025 tubes because the flow will be divided among
the tubes. The velocity per tube for each of the conditions of this analysis is calculated
in Table 4-2 and is multiplied by 45 tubes in Table 5-1.
Table 5-1: Fluid Velocity at Top of 45 Tubes
RCS
Temperature
(°F)
RCS
Pressure
(psia)
RCP Speed
(rpm)
SG Tube
Flow
Velocity
(ft/s)
Velocity at
Top of 45
Tubes (ft/s)
70
231
450
200
2,250
2,250
200
1,600
1,750
1.986
16.333
17.381
89.389
734.975
782.152
The velocity at the top of 45 tubes is used as a boundary condition in the model
calculation setup. The top portion of the tubes where the boundary condition is applied
in the 45-tube model is shown in Figure 5-3 with the associated named selections created
during the meshing process.
28
sg_top_in
sg_top (inlet velocity
boundary condition)
sg_tube_tops
sg_top_out
Figure 5-3: 2D 45-Tube Model – Inlet Velocity Boundary Condition
As shown in Figure 5-2 and Figure 5-3, named selections are created for:

SG top – inlet velocity (sg_top)

SG top outside – exterior side of top rectangular portion above tubes
(sg_top_out)

SG top inside – interior side of top rectangular portion above tubes (sg_top_in)

SG tube tops – boundary between tubes at top of model (sg_tube_tops)

SG tube walls – sidewalls of SG tubes (sg_tube_walls)

SG tube bottoms – boundary between tubes at bottom of model
(sg_tube_bottoms)

SG outlet plenum inside – interior side of outlet plenum (sg_bowl_in)

SG outlet plenum outside – exterior bowl-shaped outlet plenum boundary
(sg_bowl_bottom)
Named selections are added to the model so that named selections can be specified as
locations for mesh improvement.
29
5.1.1.2 Mesh
A mesh for the 45-tube model is developed to be fairly coarse since the goal of the
preliminary two dimensional model is to show that the model can be successfully
meshed and run before adding an outlet, more tubes, and refining the mesh. The growth
rate is set to 1.2 and the minimum edge length is set to 7.62x10-3 m. Inflation with a
smooth transition with a transition ratio of 0.272, a maximum of two layers, and a
growth rate of 1.2 is applied. Edge sizing is used to refine the mesh at the points where
the tube flow enters the SG outlet plenum because this is the area of interest for this
model. The mesh generated with this configuration contains 18,645 nodes and 15,027
elements. Figure 5-4 shows the mesh for the plenum region of the 45-tube model.
Figure 5-4: Mesh for Plenum Region of 45-Tube Model
Figure 5-4 shows that the mesh was successfully constructed and mesh edge sizing
refined the mesh near the tube exits.
5.1.1.3 Computation Setup
The 45-tube models for each temperature and pump speed condition from Table 5-1 are
run as steady state, pressure-based calculations in planar two dimensional space with
30
absolute velocity formation.
The 45-tube models are used as a means to test the
geometry and mesh before creating a larger model, so the laminar viscous model is used
to reduce computing time. All other models, including the energy model, are set to “off”
because the focus of this study is flow and not heat transfer. The fluid is set to liquid
water with density and viscosity from Table 4-3. The surface body cell zone condition is
set to liquid water. The boundary conditions are listed in Table 5-2.
Table 5-2: 45-Tube Model Boundary Conditions
Zone
Boundary Condition
interior-surface_body*
Interior
sg_bowl_bottom
Stationary wall; no slip
sg_bowl_in
Stationary wall; no slip
sg_top
Inlet velocity from Table 5-1
sg_top_in
Stationary wall; no slip
sg_top_out
Stationary wall; no slip
sg_tube_bottoms
Stationary wall; no slip
sg_tube_tops
Stationary wall; no slip
sg_tube_walls
Stationary wall; no slip
wall-surface_body*
Stationary wall; no slip
*These cell zones were generated by FLUENT.
The dynamic mesh is not activated and the default reference values are used. The
SIMPLE algorithm (Section 3.2.2.1.2) is used with a least-squares cell based gradient,
standard pressure solver, and second order upwinding for the momentum calculation
(Section 3.2.2.1.1). The default FLUENT under-relaxation factors shown in Table 5-3
are used for solution control.
Table 5-3: Under-Relaxation Factors
Parameter
Pressure
Density
Body Forces
Momentum
Under-Relaxation Factor
0.3
1
1
0.7
The solution is initialized using a standard initialization based on the named selection
called “sg_bowl_bottom.” The initial values for pressure, x velocity, and y velocity are
all set to zero. The calculation is set to autosave every 10 iterations and to run for 400
31
iterations, reporting every 10 iterations, with a profile update interval of one iteration.
This scenario is copied and run for each of the three analysis conditions.
32
5.1.1.4 Results of 45-Tube Model
Results of the 45-tube model are only included in this report for one RCS condition. The
preliminary model is run at all three analysis conditions, but since the purpose of the
preliminary model is only to show that the model can be successfully meshed and run in
ANSYS, it is sufficient to show the results of one of the model at one condition. Plots of
turbulence quantities (k, ε) are not generated because the 45-tube model is run using the
laminar solver. The 45-tube model is merely a test case to show that the model in
development can be meshed and solved in ANSYS, so the laminar solver is used to
reduce computation time.
Figure 5-5 shows the velocity magnitude results for the SG outlet plenum.
Figure 5-5: 45-Tube Laminar Model Velocity Vector Results at 70°F, 200 psia, and
200 rpm RCP Speed
The scaled residuals for this model are shown in Figure 5-6. The scaled residuals are
automatically displayed when a calculation is completed in FLUENT, but can also be
displayed by selecting the Plots option in the Results menu of the Solution engine in
33
WORKBENCH. In the Plots screen, an XY plot of the solution can be displayed to
show the residuals.
Figure 5-6: Scaled Residuals for Laminar 45-Tube Model at 70°F, 200 psia, and 200
rpm RCP Speed
These results show that the model can successfully be run in ANSYS. The flow paths
and velocity magnitudes shown in Figure 5-5 are not appropriate for use as results of this
study since there is no outlet present in the model. Since there is no location where the
flow can exit the SG outlet plenum, the flow strongly recirculates toward the SG tube
exits and does not behave in the same manner that it would if there was an outlet.
The 45-tube model is further developed into the 100-tube model described in Section
5.1.2.
34
5.1.2
100-Tube Model
5.1.2.1 Geometry and Velocity Boundary Conditions
The preliminary Cartesian two dimensional 45-tube model described in Section 5.1.1 is
expanded to include 100 tubes. The geometry for the 100-tube model created in the
DesignModeler engine of WORKBENCH is shown in Figure 5-7.
Figure 5-7: 100-Tube Model
An outlet nozzle with a pressure-outlet (pressure = 0 psig) boundary condition is added
to the 2D model from Section 5.1.1 as shown in Figure 5-7. The AP1000 SG outlet
nozzles are vertical pipes located on the bottom of the SG outlet plenum, leading to the
RCP suction as shown in Figure 1-1.
The outlet nozzle in the 2D, 100-tube model is not added in the same location as the
AP1000 SG. The 2D model in this study is used as a development tool before the
creation of a 3D model, so it is not important to locate the outlet nozzle in the correct
location. Adding an outlet to the base model (45-tube model from Section 5.1.1) creates
35
an exit location for the modeled flow, so the results of the 100-tube model with the outlet
nozzle are expected to show less recirculation than the results of the 45-tube model that
did not have an outlet nozzle.
In addition, only one outlet nozzle is modeled when there are actually two outlet nozzles
leading to two RCPs directly below each SG (Figure 1-1). The 2D model in this study
shows a cross section of the SG outlet plenum. The outlet nozzles are aligned, so in two
dimensions, only one nozzle would be visible. The 3D model described in Section 5.2.1
includes both outlet nozzles.
The SG outlet plenum with its added outlet nozzle and associated named selections is
shown in Figure 5-8. The named selections in the 100-tube model are the same as the
named selections in the 45-tube model, except additional named selections were created
for the outlet nozzle with a pressure-outlet (pressure = 0 psig) boundary condition as
shown in Figure 5-8.
sg_tube_walls
sg_tube_bottoms
sg_bowl_in
sg_bowl_bottom
outlet_nozzle_exit
(pressure-outlet
boundary condition)
outlet_nozzle_top
outlet_nozzle_bottom
Figure 5-8: 2D 100-Tube Model – Outlet Plenum Region
36
Similarly to the 45-tube model, the inverted “u” shaped portion of the tubes is not
modeled and a fixed inlet velocity is used as a boundary condition across the tops of the
tubes. The fixed velocity is chosen to provide the same Reynolds number effects
through the tubes as though the 10,025 tubes were modeled. The Reynolds number
calculation is described in Section 5.1.1.1. The necessary inlet velocity per tube to
generate the desired Reynolds number is calculated in Table 5-1. The single-tube inlet
velocity is multiplied by 100 tubes in Table 5-4.
Table 5-4: Fluid Velocity at Top of 100 Tubes
RCS
Temperature
(°F)
RCS
Pressure
(psia)
RCP Speed
(rpm)
SG Tube
Flow
Velocity
(ft/s)
Velocity at
Top of 100
Tubes (ft/s)
70
231
450
200
2,250
2,250
200
1,600
1,750
1.986
16.333
17.381
198.642
1,633.277
1,738.116
The velocity at the top of 100 tubes is applied as an inlet velocity boundary condition in
the model as shown in Figure 5-9. The named selections associated with this portion of
the 100-tube model are also shown in Figure 5-9.
37
sg_top (velocity inlet
boundary condition)
sg_top_in
sg_top_out
sg_tube_tops
Figure 5-9: 2D 100-Tube Model – Inlet Velocity Boundary Condition
As shown in Figure 5-8 and Figure 5-9, named selections are created for:

SG top – inlet velocity (sg_top)

SG top outside – exterior side of top rectangular portion above tubes
(sg_top_out)

SG top inside – interior side of top rectangular portion above tubes (sg_top_in)

SG tube tops – boundary between tubes at top of model (sg_tube_tops)

SG tube walls – sidewalls of SG tubes (sg_tube_walls)

SG tube bottoms – boundary between tubes at bottom of model
(sg_tube_bottoms)

SG outlet plenum inside – interior side of outlet plenum (sg_bowl_in)

SG outlet plenum outside – exterior bowl-shaped outlet plenum boundary
(sg_bowl_bottom)

SG outlet nozzle top – uppermost edge of outlet nozzle (outlet_nozzle_top)

SG outlet nozzle exit – exit of outlet nozzle (outlet_nozzle_exit)

SG outlet nozzle bottom – lower edge of outlet nozzle (outlet_nozzle_bottom)
38
5.1.2.2 Mesh
A mesh for the 100-tube model is developed to be fine enough to generate accurate
results but not so fine that excessive computing time is required. The growth rate is set
to 1.2 and the minimum edge length is set to 7.62x10-3 m. Inflation with a smooth
transition with a transition ratio of 0.272, a maximum of two layers, and a growth rate of
1.2 is applied. Edge sizing is used to refine the mesh at the points where the tube flow
enters the SG outlet plenum because this is the area of interest for this model. The mesh
generated with this configuration contains 22,886 nodes and 18,094 elements. Figure
5-10 shows the mesh for the 100-tube model.
Figure 5-10: Mesh for 100-Tube Model
Figure 5-11 shows a detail view of the locations where mesh edge sizing is used to refine
the mesh. A finer mesh is used at the tube exits because it is expected that recirculating
flow will be present near the tube exits.
39
Figure 5-11: Mesh Edge Sizing for 100-Tube Model
5.1.2.3 Computation Setup
The 100-tube models for each temperature and pump speed condition are run as steady
state, pressure-based calculations in planar two dimensional space with absolute velocity
formation. While the 45-tube model used the laminar viscous model, the 100-tube
model uses the standard k-ε turbulent equations (Section 3.2.1.3). All other models,
including the energy model, are set to “off” because the focus of this study is flow and
not heat transfer. The fluid is set to liquid water with density and viscosity from Table
4-3. The surface body cell zone condition is set to the liquid water. The boundary
conditions are listed in Table 5-5.
40
Table 5-5: 100-Tube Model Boundary Conditions
Zone
Boundary Condition
interior-surface_body*
Interior
sg_bowl_bottom
Stationary wall; no slip
sg_bowl_in
Stationary wall; no slip
sg_top
Inlet velocity (from Table 5-4)
sg_top_in
Stationary wall; no slip
sg_top_out
Stationary wall; no slip
sg_tube_bottoms
Stationary wall; no slip
sg_tube_tops
Stationary wall; no slip
sg_tube_walls
Stationary wall; no slip
outlet_nozzle_top
Stationary wall; no slip
outlet_nozzle_exit
Pressure - outlet
outlet_nozzle_bottom
Stationary wall; no slip
wall-surface_body*
Stationary wall; no slip
*These cell zones were generated by FLUENT.
The dynamic mesh is not activated and the default reference values are used. The
SIMPLE algorithm (Section 3.2.2.1.2) is used with a least-squares cell based gradient,
standard pressure solver, and second order upwinding for the momentum calculation
(Section 3.2.2.1.1). The default FLUENT under-relaxation factors listed in Table 5-6 are
used for solution control.
Table 5-6: Under-Relaxation Factors
Parameter
Pressure
Density
Body Forces
Momentum
Turbulent Kinetic Energy
Under-Relaxation Factor
0.3
1
1
0.7
0.8
The solution is initialized using a standard initialization based on the named selection
called “sg_bowl_bottom.” The initial values for pressure, x velocity, and y velocity are
all set to zero. The calculation is set to autosave every 10 iterations, and to run for 1,000
iterations, reporting every 50 iterations, with a profile update interval of 10 iterations.
This scenario is copied and run for each of the three analysis conditions from Table 5-4.
41
5.1.2.4 Results of 100-Tube Model
5.1.2.4.1 100-Tube Model Results at 70°F, 200 psia, and 200 rpm RCP Speed
The 100-tube model at 70°F is solved in steady state and set to run for 1,000 iterations
(solution reached in 130 iterations). Figure 5-12 shows the velocity magnitude results
for the SG outlet plenum. There is significantly less recirculation shown in the velocity
vector results for the turbulent model at 70°F than there is in the laminar results at 70°F
(Figure 5-5). This is due to the addition of the outlet nozzle. The flow naturally travels
to the pressure outlet. With no outlet available in the laminar model, the flow was
forced to recirculate after making contact with the wall boundaries. In the 100-tube
model which includes an outlet nozzle, the velocity increases from about 71 m/s near the
tube exits to a maximum velocity of about 282 m/s at the end of the outlet nozzle.
Figure 5-12: 100-Tube Model Velocity Vector Results at 70°F, 200 psia, and 200
rpm RCP Speed
42
Figure 5-13 shows a detailed view of the velocity magnitude near the SG outlet plenum
divider plate (flat side of the outlet plenum) at the tube exits. The detailed view shows
that there is some minor recirculation near the tube exits, but most of the flow is directed
downward toward the bottom of the SG outlet plenum.
The magnitude of the
recirculation flow is significantly lower than the magnitude of the recirculation flow
shown in the 45-tube model results (Figure 5-5). This difference can be attributed to the
addition of the outlet nozzle in the 100-tube model. The flow continues toward the
outlet nozzle rather than recirculating and increasing in velocity magnitude directed at
the tube exits.
Figure 5-13: Detailed Results Near Interior Region for 100-Tube Model at 70°F,
200 psia, and 200 rpm RCP Speed
43
Figure 5-14 shows the velocity magnitude near the tube outlets in the center of the SG
outlet plenum. The flow is directed downward and toward the pressure outlet at the SG
outlet nozzle, which is located on the right side of the model. Figure 5-14 shows that the
recirculation dissipates in relatively close proximity to the tube exits and becomes more
uniform as it grows closer to the outlet nozzle.
Figure 5-14: Central Region Results Detail for 100-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed
44
Figure 5-15 shows the velocity magnitude near the outer edge of the SG outlet plenum.
Figure 5-15 shows that there is low velocity recirculation present between the outermost
tube exit and the exterior of the SG outlet plenum. Farther from the tube exits, the flow
approaches the pressure outlet.
Figure 5-15: Exterior Region Results Detail for 100-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed
45
Figure 5-16 shows the detailed results near the outer edge (bowl-shaped face) of the SG
outlet plenum.
Figure 5-16 shows that there is recirculation present between the
outermost tube exit and the exterior of the SG outlet plenum. The recirculation field
recombines with the flow path at the exit of the outermost tube and is directed
downward, toward the outlet nozzle along with the rest of the flow exiting the tubes.
Figure 5-16: Detailed Results Near Exterior Region for 100-Tube Model at 70°F,
200 psia, and 200 rpm RCP Speed
46
Figure 5-17 is a contour plot of the turbulent kinetic energy (k) results for the 100-tube
model (Section 3.2.1.3). Figure 5-17 shows that there is little turbulent kinetic energy
throughout the outlet plenum, but some areas of higher turbulent kinetic energy near the
tube exits.
Figure 5-17: Turbulent Kinetic Energy (k) for 100-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed
Figure 5-18 shows that the areas between the tube exits have the highest turbulent
kinetic energy at around 1321 J/kg.
47
Figure 5-18: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube
Model at 70°F, 200 psia, and 200 rpm RCP Speed
48
Figure 5-19 shows the turbulent dissipation rate (ε) results for the 100-tube model
(Section 3.2.1.3). Figure 5-19 shows that similarly to the turbulent kinetic energy results
shown in Figure 5-17, there is relatively little turbulent eddy dissipation in the majority
of the outlet plenum, but there is higher turbulent eddy dissipation near the tube exits.
Figure 5-19: Velocity Vectors Colored by Turbulent Dissipation Rate (ε) for 100Tube Model at 70°F, 200 psia, and 200 rpm RCP Speed
Figure 5-20 shows the turbulent dissipation rate near the divider plate (flat surface) at the
interior wall of the SG outlet plenum. Figure 5-20 shows that there is relatively low
dissipation in most locations, but the maximum value of 5.599 x 106 m2/s3 is located near
the tube outlets.
49
Figure 5-20: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube
Model at 70°F, 200 psia, and 200 rpm RCP Speed
The scaled residuals for this model are shown in Figure 5-21.
Figure 5-21: Scaled Residuals for 100-Tube Model at 70°F, 200 psia, and 200 rpm
RCP Speed
50
5.1.2.4.2 100-Tube Model Results at 231°F, 2,250 psia, and 1,600 rpm RCP Speed
The 100-tube model at 231°F is solved in steady state and set to run for 1,000 iterations
(solution reached in 480 iterations). Figure 5-22 shows the velocity magnitude results
for the SG outlet plenum. The velocity increases from about 578 m/s near the tube exits
to a maximum velocity of about 2312 m/s at the end of the outlet nozzle.
Figure 5-22: 100-Tube Model Velocity Vector Results at 231°F, 2,250 psia, and
1,600 rpm RCP Speed
Most of the detailed figures of the flow paths for the results of the 100-tube model run at
231°F and 2,250 psia are not provided in this report because the flow paths are very
similar to the results for the 100-tube model at 70°F and 200 psia shown in Section
5.1.2.4.1. Instead, a single detailed view of an area of interest is provided in Figure
5-23. Figure 5-23 shows the detailed results near the outer edge of the SG outlet
plenum, where the largest recirculation pattern was noticed in the 70°F model results
51
shown in Figure 5-16. Figure 5-23 shows that the recirculation pattern generated in the
231°F model is almost identical to the pattern created using the 70°F.
The only
noticeable difference in velocity results of the 70°F and 231°F 100-tube models is the
velocity magnitude through the outlet nozzle.
Figure 5-23: Detailed Results Near Exterior Region for 100-Tube Model at 231°F,
2,250 psia, and 1,600 rpm RCP Speed
Figure 5-24 shows the turbulent kinetic energy (k) results of the 100-tube model (Section
3.2.1.3). Figure 5-24 shows that there is relatively little turbulent kinetic energy in the
majority of the outlet plenum, but there is higher kinetic energy near the tube exits in the
region between the tube exits.
52
Figure 5-24: Turbulent Kinetic Energy (k) for 100-Tube Model at 231°F, 2,250
psia, and 1,600 rpm RCP Speed
Figure 5-25 shows the turbulent kinetic energy near the divider plate at the interior wall
(flat side) of the SG outlet plenum. Figure 5-25 shows that the maximum turbulent
kinetic energy for the 231°F model is about 6.250 x 104 J/kg, which is an order of
magnitude higher than the 1321 J/kg predicted using the 70°F model shown in Figure
5-18.
53
Figure 5-25: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube
Model at 231°F, 2,250 psia, and 1,600 rpm RCP Speed
Figure 5-26 shows the turbulent dissipation rate (ε) results for the 100-tube model
(Section 3.2.1.3). Figure 5-26 shows that there is a relatively low turbulent dissipation
rate throughout the outlet plenum, but the dissipation rate is higher near the tube exits.
54
Figure 5-26: Turbulent Dissipation Rate (ε) for 100-Tube Model at 231°F, 2,250
psia, and 1,600 rpm RCP Speed
Figure 5-27 shows the turbulent dissipation rate near the divider plate (flat side) at the
interior wall of the SG outlet plenum. Figure 5-27 shows that there is relatively low
dissipation in most locations throughout the outlet plenum, and that the highest
dissipation is 2.589 x 109 m2/s3. This value is three orders of magnitude higher than the
maximum turbulent dissipation rate of the 70°F model, 5.599 x 106 m2/s3, shown in
Figure 5-20.
55
Figure 5-27: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube
Model at 231°F, 2,250 psia, and 1,600 rpm RCP Speed
The scaled residuals for this model are shown in Figure 5-28.
Figure 5-28: Scaled Residuals for 100-Tube Model at 231°F, 2,250 psia, and 1,600
rpm RCP Speed
56
5.1.2.4.3 100-Tube Model Results at 450°F, 2,250 psia, and 1,750 rpm RCP Speed
The 100-tube model at 450°F is solved in steady state and set to run for 1,000 iterations
(solution reached in 870 iterations). Figure 5-29 shows that the flow path results for the
SG outlet plenum in the 450°F model are similar to the results for the 70°F and 213°F
turbulent models (Figure 5-12 and Figure 5-22). The flow increases from about 614.7
m/s at the tube exits to about 2459 m/s at the outlet nozzle exit.
Figure 5-29: 100-Tube Model Velocity Vector Results at 450°F, 2,250 psia, and
1,750 rpm RCP Speed
Since the flow path results of the 450°F model are very similar to the results of both the
70°F and 231°F models, detailed views of the velocity magnitude results are not
included in this report for the 450°F model.
57
Figure 5-30 shows the turbulent kinetic energy (k) results for the 100-tube model
(Section 3.2.1.3). As with the results of the 70°F and 231°F model results, Figure 5-30
shows that there is relatively low turbulent kinetic energy in the majority of the outlet
plenum, but there is higher kinetic energy near the tube exits.
Figure 5-30: Velocity Vectors Colored by Turbulent Kinetic Energy (k) for 100Tube Model at 450°F, 2,250 psia, and 1,750 rpm RCP Speed
Figure 5-31 shows the velocity vectors colored by the turbulent kinetic energy near the
divider plate (flat side) at the interior wall of the SG outlet plenum. Figure 5-31 shows
that the maximum turbulent kinetic energy in the 450°F model is about 4.288 x 104 J/kg.
This value is greater than the maximum value of 1321 J/kg from the 70°F model but less
than the 6.250 x 104 J/kg from the 231°F model. These results show that if the pressure
58
of a system is held constant but the inlet velocity is changed, the turbulent kinetic energy
will decrease.
Figure 5-31: Turbulent Kinetic Energy (k) near Interior Region for 100-Tube
Model at 450°F, 2,250 psia, and 1,750 rpm RCP Speed
Figure 5-32 shows the turbulent dissipation rate (ε) results for the 100-tube model
(Section 3.2.1.3). Figure 5-32 shows that there is only heightened turbulent dissipation
rate near the tube exits and that the remainder of the outlet plenum has relatively low
turbulent dissipation rate.
59
Figure 5-32: Turbulent Dissipation Rate (ε) for 100-Tube Model at 450°F, 2,250
psia, and 1,750 rpm RCP Speed
Figure 5-33 shows the turbulent dissipation rate near the divider plate (flat side) at the
interior wall of the SG outlet plenum. Figure 5-33 shows that the maximum turbulent
dissipation rate for the 450°F model is about 2.305 x 109 m3/s2. This value is higher than
the maximum turbulent dissipation rate of 5.599 x 106 m2/s3 from the 70°F model, but is
lower than the 2.589 x 109 m2/s3 maximum value from the 231°F model. These results
follow the same pattern as the turbulent kinetic energy results; the pressure in this model
is the same as the pressure in the 231°F model, and the only difference in the case runs
were the inlet temperature.
These results show that the turbulent dissipation rate
decreases as the inlet velocity increases when the pressure is held constant.
60
Figure 5-33: Turbulent Dissipation Rate (ε) near Interior Region for 100-Tube
Model at 450°F, 2,250 psia, and 1,750 rpm RCP Speed
The scaled residuals for this model are shown in Figure 5-34.
Figure 5-34: Scaled Residuals for 100-Tube Model at 450°F, 2,250 psia, and 1,750
rpm RCP Speed
61
5.2 Three Dimensional Modeling
5.2.1
552-Tube Model
5.2.1.1 Geometry and Velocity Boundary Conditions
Flow through the SG outlet plenum is modeled in three dimensions using ANSYS
WORKBENCH 14.0.0 and ANSYS FLUENT 14.0. The system is modeled as a set of
tubes flowing into the outlet plenum. The number of tubes modeled is limited by the
computational power. In order to determine how many tubes can be modeled, an initial
three dimensional model is created with very basic geometry. The radius of the AP1000
SG outlet plenum is used as input to the model. The base geometry created in the
DesignModeler engine of WORKBENCH is shown in Figure 5-35. This basic model
only includes one outlet nozzle at an angle, while the AP1000 SGs have two outlet
nozzles which are vertically fixed to the bottom of the SG outlet plenum, leading to the
RCPs (Figure 1-1). Angled outlet nozzles were used in SGs of the CE design, but
AP1000 has modified the design so that gravity will assist the RCS circulation. The
base model shown in Figure 5-35 is created in the image of the 100-tube 2D model
described in Section 5.1.2.1. The outlet nozzle is tilted to the proper angle (vertical) and
location on the bottom of the outlet plenum (Figure 1-1) later in the geometry
development process.
Figure 5-35: Base Geometry for 3D Model
62
Circular tube outlets of the same inner diameter as the AP1000 SG tubes are added to a
sketch plane on the top surface of the SG outlet plenum. A grid of construction lines is
created on the plane and circles of equal diameter are added and constrained such that
the centers of the circles lie exactly at the intersections of the construction lines. The
Equal Radius tool in the DesignModeler engine of WORKBENCH is used to ensure that
all of the circles are of the same diameter – the inner diameter of the AP1000 SG tubes.
Groups of two dimensional circles are added to the plane and then the sketch circles are
extruded vertically for an arbitrary short distance. The circles are extruded so that the
model is easier to manipulate and named selections can more easily be created for the
tube outlet surfaces. A single named selection is created as tubes are added in groups by
editing a named selection created from the first tube. The named selection selected
edges are edited to choose more edges and the changes are saved to update the named
selection to include all of the tube outlets.
The process of creating tubes and named selections in groups (typically about 10 to 20
tubes at a time) rather than all at once has multiple reasons. First, it is easier to see and
select edges using the 360° view to add to the named selection when there are fewer
tubes. Second, the model was also tested at a number of tube addition intervals to ensure
that the named selections were correctly created and the model could be successfully
meshed and run in FLUENT. A very coarse mesh was used for this purpose to reduce
calculation time. Finally, copying and pasting a large number of drawn circles caused
the DesignModeler engine to crash. This typically happened when more than 30 circles
were copied and pasted, or when attempting to constrain the newly pasted circles.
Figure 5-36 and Figure 5-37 show how tubes were added in groups and extruded by a
short distance to create short tubes exiting into the outlet plenum.
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Figure 5-36: Base Geometry for 3D Model Showing Tube Outlets
Figure 5-37: Base Geometry for 3D Model Showing Tube Outlet Detail
64
The test meshing and runs using FLUENT determined the limitations in terms of the
number of tubes which could be modeled in this study. When the number of tubes
exceeded the capability of the model, and the run was not successful, the number of
tubes was reduced to a workable amount (552 tubes) and the radius of the SG outlet
plenum was reduced to fit around exterior of the tube cluster as shown in Figure 5-38.
Figure 5-38: Base Geometry for 3D Model with 552 Tubes and Reduced SG Outlet
Plenum Radius
The base geometry shown in Figure 5-38 was edited so the outlet nozzles would be
configured like those on the AP1000 SGs as shown in Figure 5-39. The AP1000 SG
outlet nozzles are vertical pipes located on the bottom of the outlet plenum, leading to
the RCP suction (Figure 1-1).
65
Figure 5-39: 3D Model with 2 Outlet Nozzles, 552 Tubes, and Reduced SG Outlet
Plenum Radius
As with the two dimensional models described in Section 5.1, the fixed velocity is
chosen such that the Reynolds number effects at the tube exists match the Reynolds
number effects which would be present if 10,025 tubes were modeled rather than 552
tubes. The Reynolds number calculation is discussed in Section 5.1.1.1 and the inlet
velocity required to be flowing through each individual tube is calculated in Table 5-1.
In the 45-tube and 100-tube two dimensional models described in Section 5.1, one
boundary condition is applied as an inlet velocity flowing to all of the modeled tubes.
Therefore, the necessary single tube velocity was multiplied by the number of tubes in
the two dimensional models. This was done because the model geometry was created
with a single boundary condition as an inlet velocity to all of the tubes, and the flow
would be split among the tubes.
Since the three dimensional model is created with each tube exit modeled as an
individual named selection with its own inlet velocity boundary condition, the inlet
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velocity boundary condition is set equal to the AP1000 SG single tube flow velocity
calculated in Table 5-1 without multiply the value by the number of tubes. The inlet
velocity boundary condition per tube is repeated in Table 5-7.
Table 5-7: Fluid Velocity at Top of Each SG Tube
RCS
Temperature
(°F)
RCS
Pressure
(psia)
RCP Speed
(rpm)
SG Tube
Flow
Velocity
(ft/s)
70
231
450
200
2,250
2,250
200
1,600
1,750
1.986
16.333
17.381
The velocity at the top of each tube is used as a boundary condition for each tube exit
named selection in the model calculation setup. The named selections where inlet
velocity boundary conditions are applied are highlighted in Figure 5-40.
Figure 5-40: SG Tube Named Selections for Inlet Velocity Boundary Conditions
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The bottom faces of the outlet nozzles are set with pressure-outlet (pressure = 0 psig)
boundary conditions in the model calculation setup.
Named selections are created for:

SG bowl face – SG bowl exterior surface (bowlface)

SG exit – SG outlet nozzle exit (outlet_nozzle_exits)

SG outlet nozzle pipe – cylindrical surface of outlet nozzle (outlet_nozzle_pipes)

SG tube tops – top of 552 SG tubes shown in Figure 5-40 (sg_tubes)
5.2.1.2 Mesh
A mesh for the 552-tube three dimensional model was developed to be fine enough to
generate accurate results but not so fine that excessive computing time is required. The
growth rate is set to 1.750 and the minimum edge length is set to 4.8835x10-2 m.
Inflation with a smooth transition with a transition ratio of 0.272, a maximum of 6
layers, and a growth rate of 2.5 is applied. Edge sizing is used to refine the mesh at the
points where the tube flow enters the SG outlet plenum because this is the area of
interest for this model. The mesh generated with this configuration contains 215,314
nodes and 968,741 elements. Figure 5-41 shows the mesh for the 552-tube model.
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Figure 5-41: Mesh for 552-Tube Model
Figure 5-42 shows the mesh edge sizing near the tube exits for the 552-tube model.
Edge sizing is used to refine the calculation near the tube exits because it is expected that
the most recirculation occurs near the tube exits.
Figure 5-42: Mesh Edge Sizing for 552-Tube Model
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5.2.1.3 Computation Setup
The 552-tube models for each temperature and pump speed condition are run as steady
state, pressure based calculations in three dimensions with absolute velocity formation.
The standard k-ε turbulent equations (Section 3.2.1.3) with standard wall functions are
used. All other models, including the energy model, are set to “off” because the focus of
this study is flow and not heat transfer. The fluid is set to liquid water with density and
viscosity from Table 4-3. The surface body cell zone condition is set to the liquid water.
The boundary conditions are listed in Table 5-8.
Table 5-8: 552-Tube Model Boundary Conditions
Zone
Boundary Condition
interior-solid*
Interior
bowlface
Stationary wall; no slip
outlet_nozzle_exits
Pressure – outlet
outlet_nozzle_pipes
Stationary wall; no slip
sg_tubes
Inlet velocity (from Table 5-7)
wall-solid*
Stationary wall; no slip
*These cell zones were generated by FLUENT.
The dynamic mesh is not activated and the default reference values are used. The
SIMPLE algorithm (Section 3.2.2.1.2) is used with a least-squares cell based gradient,
standard pressure solver, and second order upwinding for the momentum calculation
(Section 3.2.2.1.1). The under-relaxation factors in Table 5-9 are used for solution
control.
Table 5-9: Under-Relaxation Factors
Parameter
Pressure
Density
Body Forces
Momentum
Turbulent Kinetic Energy
Under-Relaxation Factor
0.3
1
1
0.7
0.8
The solution is initialized using a standard initialization based on the named selection
called “bowlface.” The initial values for pressure, x velocity, y velocity, and z velocity
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are all set to zero. The calculation is set to autosave every one iteration, and to run for
200 iterations, reporting every iteration, with a profile update interval of one iteration.
Particle tracks are set up in the Graphics and Animations menu of the FLUENT setup
window. The injections are set up as line track styles exiting from various tube exits
throughout the SG outlet plenum. The single pulse mode is specified and results are
plotted as particle velocity. This scenario is copied and run for each of the three analysis
conditions.
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5.2.1.4 Results of 552-Tube Model
5.2.1.4.1 552-Tube Model Results at 70°F, 200 psia, and 200 rpm RCP Speed
The 552-tube model at 70°F is solved in steady state and set to run for 200 iterations
(solution reached in 110 iterations). Figure 5-43 shows the velocity magnitude results
for the SG outlet plenum. The flow naturally travels from the tube exits to the outlet
nozzles. The velocity increases from about 5.084 x 10-1 m/s to about 2.033 m/s as the
flow reaches the outlet nozzles. The outlet nozzle flow rate results shown in Figure 5-43
are significantly lower than the flow rate results from the 2D 100-tube model (71 m/s at
the outlet nozzle).
Figure 5-43: Isometric View of 552-Tube Model Velocity Vector Results at 70°F,
200 psia, and 200 rpm RCP Speed
72
Figure 5-44 shows a side view of the velocity magnitude results for the 552-tube model.
As the flow exits the tubes nearest the divider plate, it is abruptly redirected back toward
the outlet nozzles.
Figure 5-44: Side View of 552-Tube Model Velocity Vector Results at 70°F, 200
psia, and 200 rpm RCP Speed
73
Figure 5-45 shows a front view of the velocity magnitude results for the 552-tube model.
The flow separates between the two outlet nozzles and exits the SG with similar velocity
in both outlet nozzles.
Figure 5-45: Front View of 552-Tube Model Velocity Vector Results at 70°F, 200
psia, and 200 rpm RCP Speed
74
Figure 5-46 shows the detailed velocity magnitude results for the interior region the 552tube model near the divider plate. As shown in the results of the two dimensional
models, there is recirculation near the tube exits. Unlike the two dimensional results,
Figure 5-46 shows that the recirculating flow does not backflow into the tubes; it is
stopped and redirected by the outflow.
Figure 5-46: Interior Region Results Detail for 552-Tube Model at 70°F, 200 psia,
and 200 rpm RCP Speed
75
Figure 5-47 shows the detailed velocity magnitude results for the 552-tube model near
one of the outlet nozzles. The outlet nozzles are identical so the results are similar for
both nozzles.
Figure 5-47: Detailed Results Near Outlet Nozzle for 552-Tube Model at 70°F, 200
psia, and 200 rpm RCP Speed
Turbulent kinetic energy and dissipation figures could not be created for the three
dimensional models due to computer memory limitations. This issue is further discussed
in Section 6.2.2.
76
Figure 5-48 shows particle tracks for points located near various tube exits. The particle
tracks show the flow paths for specific points in space as they move through the flow
field. The path lines are colored by velocity magnitude. These particle tracks were
created using the Solution engine in WORKBENCH.
The initial locations of the
particles were set using the Injections option in the Define menu. The particles were
also set to massless in this menu. Then, in the Results menu, Graphics and Animations
was selected and Node Values, Auto Range, and Draw Mesh were selected. In the Draw
Mesh menu, Edges were selected and the specified Edge Type of Feature was selected.
Selecting Display in the Draw Mesh Menu, and then Display again in the Particle Tracks
menu after selecting all of the injections caused the particle tracks to display with the
outline of the geometry as shown in Figure 5-48.
Figure 5-48: Velocity Magnitude Particle Tracks for 552-Tube Model at 70°F, 200
psia, and 200 rpm RCP Speed
Figure 5-48 shows that the flow exiting the more centrally located tubes tends to flow
directly downward toward the outlet nozzle, but flow exiting tubes near the outer edges
of the outlet plenum has a more indirect path.
77
The scaled residuals for this model are shown in Figure 5-49.
Figure 5-49: Scaled Residuals for 552-Tube Model at 70°F, 200 psia, and 200 rpm
RCP Speed
78
5.2.1.4.2 552-Tube Model Results at 231°F, 2,250 psia, and 1,600 rpm RCP Speed
The 552-tube model at 231°F is solved in steady state and set to run for 200 iterations
(solution reached in 81 iterations). Figure 5-50 shows the velocity magnitude results for
the SG outlet plenum. Figure 5-50 shows that the flow near the tube exits is around
4.256 m/s in velocity and the flow at the exits of the outlet nozzles is about 17.02 m/s.
Figure 5-50: Isometric View of 552-Tube Model Velocity Vector Results at 231°F,
2,250 psia, and 1,600 rpm RCP Speed
Detailed figures are of the flow results which were provided for the 552-tube model at
70°F are not provided in this report for the 552-tube model at 231°F because the results
are generally very similar between the two models. The flow directions are almost
identical, and the only noticeable difference is the velocity magnitudes as expected based
on the inlet velocity boundary condition. As with the 70°F models, turbulent kinetic
79
energy and dissipation figures could not be created for the three dimensional models due
to computer memory limitations. This issue is further discussed in Section 6.2.2.
Particle tracks in the 231°F model are set up in the same initial locations as in the 70°F
model. The results are shown in Figure 5-51. The particle tracks near the tube exits at
the edges of the outlet plenum showed indirect flow and recirculation in the 70°F model.
In the 70°F model, the particles released from these locations did not appear to continue
to the outlet nozzle. Figure 5-51 shows that with the increased velocity magnitude, the
flow continues to the outlet nozzle and the recirculation paths are longer.
Figure 5-51: Velocity Magnitude Particle Tracks for 552-Tube Model at 231°F,
2,250 psia, and 1,600 rpm RCP Speed
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The scaled residuals for this model are shown in Figure 5-52.
Figure 5-52: Scaled Residuals for 552-Tube Model at 231°F, 2,250 psia, and 1,600
rpm RCP Speed
81
5.2.1.4.3 552-Tube Model Results at 450°F, 2,250 psia, and 1,750 rpm RCP Speed
The 552-tube at 450°F is solved in steady state and set to run for 200 iterations (solution
reached in 80 iterations). Figure 5-53 shows the velocity magnitude results for the SG
outlet plenum. The flow at the tube exits is traveling at about 4.510 m/s and the flow at
the outlet nozzle exit is traveling at about 18.04 m/s. These velocity results are higher
than the velocity results of the 70°F and 231°F models due to the higher inlet velocity set
as the boundary condition at the tube exits.
Figure 5-53: Isometric View of 552-Tube Model Velocity Vector Results at 450°F,
2,250 psia, and 1,750 rpm RCP Speed
As with the 231°F model, detailed figures are not provided for the velocity magnitude
results of the 552-tube model at 450°F. The results of the three models show very
82
similar flow paths in terms of direction, with only different velocity magnitude based on
the inlet velocity boundary conditions. Also as with the other three dimensional models,
the turbulent kinetic energy and dissipation figures were not generated for this model
because of limitations related to computing power used in this analysis. This issue is
described in detail in Section 6.2.2.
Particle tracks are added to the 450°F model in the same way that they were added to the
70°F and 231°F models. The particle track results for the 450°F model are shown in
Figure 5-54.
Figure 5-54: Velocity Magnitude Particle Tracks for 552-Tube Model at 450°F,
2,250 psia, and 1,750 rpm RCP Speed
Figure 5-54 shows that the particles which recirculated in the 70°F and 231°F models
also recirculate near the tube exits in the 450°F model. In addition, the particle exiting
closest to the right side in Figure 5-54 appears to be swirling in three dimensions rather
than only recirculating in two dimensions. This swirling was not apparent in the 70°F
and 231°F results. The transition from recirculation (2D) to swirling (3D) may be due to
the higher velocity flow at the tube exits.
83
The scaled residuals for this model are shown in Figure 5-55.
Figure 5-55: Scaled Residuals for 552-Tube Model at 450°F, 2,250 psia, and 1,750
rpm RCP Speed
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6. Summary of Results
6.1 Froude Number Calculation Results for SG Tubes
The results presented in Section 4.3 suggest that the air collected in the SG tubes during
an outage for the AP1000 nuclear power plant will be successfully swept clear of the
tubes and into the SG outlet plenum for the selected pump speed and fluid temperature
conditions from Table 4-2.
6.2 CFD Results
6.2.1
Summary of Results
CFD models are created in both two and three dimensions with meshes refined to be as
fine as possible based on the available computing power. Both the generic area mesh
and the mesh inflation areas are considered when developing the finest possible mesh.
This ensures that the results of these models are as accurate as possible based on the
computer used to generate the results. Computing limitations are discussed in Section
6.2.2).
The two dimensional, 100-tube model results show some mixing and recirculation as the
flow exits the tubes and flows into the outlet plenum. In addition, the turbulent kinetic
energy and turbulent dissipation rate results show that there is higher turbulent energy at
the locations between the tube exits, close to the interior top surface of the outlet
plenum.
Examination of detailed results near the tube exits show that some velocity vectors are
directed upward, at flow exiting other tubes. Based on the two dimensional model
results, it is possible that after flow exits the tubes, it could recirculate back into the
tubes. The three dimensional results show flow exiting the tubes and recirculating, but
the vectors are not directed past the tube exit edge. Instead, the flow recirculation occurs
completely beneath the tube exits in the outlet plenum.
85
The differences in the results of the two dimensional model and three dimensional
models can be due to many factors. First, the meshes used in the 2D and 3D models
were very different. A two dimensional model will inherently have significantly fewer
nodes and elements than a three dimensional model. This simplifies the calculations that
must be performed, so a much finer mesh can be used for two dimensional models than
can be used for three dimensional models. However, the more complete geometry
created using a three dimensional model will generate more accurate results in terms of
direction of flow and interaction between flow paths in three dimensions rather than two.
The downward flow out of the tubes in three dimensions forces each stream to maintain
a fairly confined flow path. The streams flowing out of the tubes are less confined when
only two dimensions are considered. The equations discussed in Section 3.2 become
significantly more complex in three dimensions in comparison to the equations used to
describe two dimensional models. This study shows the importance of modeling three
dimensions rather than two; even using a more coarse mesh in a three dimensional
model can generate more accurate and useful results than using a very fine mesh in a two
dimensional model.
The two dimensional model with an SG outlet nozzle (100-tube model) and the three
dimensional model show that though there is mixing and recirculation as the flow exits
the SG tubes, the flow moves toward the SG outlet nozzles and the fluid velocity
increases as the flow exits the nozzles. The flow continues out of the outlet nozzle and
would successfully move through the RCP and cold leg (Figure 1-1) to the vent location
based on a pressure differential over the horizontal pipe.
There is mixing and
recirculation in the SG outlet plenum, but the results do not show evidence of higher
level phenomena such as hydraulic jump which is described at a high level in Section
3.2.3. In addition, the SGs for AP1000 are designed such that the RCPs are attached
directly beneath the SG (Figure 1-1) rather than connected through an angled pipe called
a suction leg like they are in CE plants. Since the motive force (RCP suction) is applied
with gravity assistance, it is easier for the flow to successfully move through the AP1000
SG outlet plenum than a CE SG outlet plenum.
86
The magnitude of the turbulent kinetic energy and turbulent dissipation results obtained
from the two dimensional models at the three temperature conditions shows the
relationship between the pressure, temperature, and turbulent quantities. When the
pressure, temperature, and flow rate were all increased, the turbulence quantities also
increased. However, when the pressure was held constant, and temperature and fluid
velocity were slightly increased, the turbulent quantities decreased.
The study described in [13] developed a CFD model of a Westinghouse PWR SG inlet
plenum. The velocity vectors developed in [13] are shown in Figure 2-1. Figure 2-1
shows that the flow into the inlet plenum recirculates back toward the inlet after
impacting the divider plate. The velocity profile results for the three dimensional model
of the SG outlet plenum presented in Section 5.2.1.4 do not show similar behavior for
flow through the outlet plenum in comparison to the results of [13]. This is to be
expected since the geometry of the two models is different.
There is mixing and
recirculation near the outlet plenum tube exits, but the fluid flows in a relatively smooth
flow path toward the outlet nozzles in the three dimensional outlet plenum model. The
results of the inlet plenum study in [13] are somewhat opposite in nature; the flow at the
inlet is relatively uniform, but recirculation occurs near the divider plate as the flow is
redirected toward the tubes.
The results of [13] shown in Figure 2-1 also show that there is flow separation along a
central axis creating two separate recirculation loops. Figure 2-1 shows slice planes of
the results of the study in [13]. A slice plane is created on the XY plane of the three
dimensional SG outlet plenum model such that the cut away view is created through the
outlet nozzles as shown in Figure 6-1. Figure 6-2 shows the back view of the SG outlet
plenum with a slice plane added through the outlet nozzles. This portion of the model
results that are cut away are from the interior section, closest to the divider plate.
87
Figure 6-1: Slice Plane through Outlet Nozzles – YZ Plane
Figure 6-2: Slice Plane through Outlet Nozzles – XY Plane
88
The cut away view in Figure 6-2 does not show the same flow separation that was
present between the recirculation paths in the study in [13]. The study in [13] used
symmetry to develop the results, and the model created in this study did not. It is
possible that the use of symmetry at the central axis created an unexpected flow effect in
[13].
6.2.2
Model Limitations due to Hardware
A Lenovo ThinkPad T400 Personal Computer (PC) was used to complete this analysis.
The operating system of this PC was a 32-bit version of Windows Vista Home Basic
Service Pack 2 from 2007. The processor was an Intel Core2 Duo CPU P8700 with 2.53
GHz to support CPU and 4.00 GB of RAM.
The PC had 221 GB of usable internal hard disk space. Hard disk space became an issue
during this study. Microsoft Office, the ANSYS 14.0.0 educational edition, and other
software programs which were installed on this computer constituted a sizeable
percentage of the disk space. The technical papers used for reference, Microsoft Word
documents, Microsoft Excel spreadsheets, ANSYS files, and figures created and saved
for this study totaled around 350 GB. The ANSYS files were copied and saved as
backup files at many intervals in the project so that if the working copy became
corrupted, it would be possible to revert to an earlier version. The final version of the
ANSYS file was around 90 GB, but since the ANSYS file package was saved as a
backup version multiple times, the hard disk space quickly increased.
The file size quickly grew to larger than the internal hard disk size. When the ANSYS
program is opened, backup files are automatically created by the software and stored in
the same location as the base file. When the internal hard drive used in this study
approached full capacity, the backup files being created by the ANSYS program caused
the computer to quickly shut off due to overload. To manage this issue, all unnecessary
files were removed from the internal hard drive and two Western Digital 1 TB My
Passport portable external hard drives with a transfer rate of 0.5 GB/s were purchased
(Figure 6-3). Recently developed USB 3.0 cords were provided with the external hard
89
drives as part of the product order. The USB 3.0 cords were compatible with the
standard USB 2.0 interface, but the data transfer rate was faster.
Figure 6-3: Western Digital My Passport External Hard Drives
The external hard drives were purchased in two different colors to avoid confusion
between the working hard drive and the backup hard drive. Files were deleted from the
internal hard drive until the computer reached a state where it could operate well enough
to copy the entire CFD study to an external hard drive without shutting off.
All of the files related to this study were copied to each external hard drive. One was
chosen randomly as the working hard drive, and the ANSYS model was run off of the
working external hard drive using a USB 3.0 cord. The second external hard drive was
not connected to the PC while the working hard drive was performing any case runs.
The backup hard drive was connected only to copy new files from the working hard
drive. This process was used to avoid data loss; if an error occurred during a case run
that would have caused the hard drive to be accidentally cleared, it would only affect the
working hard drive and a backup version would be available.
No difference was noticed with the working speed of the external hard drive compared
to the internal hard drive during ANSYS case runs. Many preliminary cases caused the
90
computer to shut off using both the internal hard drive and external hard drive. These
cases caused the computer to overheat and shut off with no warning or opportunity to
save progress. This was a notification that the case as set up was too complex for the
computing power available, and the case mesh or setup options were then changed until
it was possible to run the entire duration of the case without the computer shutting off.
An example of this process is briefly described in Section 5.2.1.1. To determine how
many tubes could be modeled, tubes were added to the geometry, a coarse mesh was
created, and a test case was run. This process was repeated, adding about 10 to 20 tubes
at a time, until the maximum number of manageable tubes was determined. A model
with more than 552 tubes modeled using the hardware available would cause the
computer to overheat and shut off during a test run.
The computer also shut off
repeatedly when attempting to display the turbulent k-ε results of the three dimensional
model in this study as described in Section 5.2.1.4.
These issues are considered
limitations due to the hardware available for this study. A more powerful computer
could be used to model more tubes, a finer mesh, or even a transient study.
The available computer power limited the number of tubes and mesh fineness. The issue
with internal hard drive size was overcome using two external hard drives. The external
hard drives that were used in this study performed very well and provided a sense of
security since the large files associated with the study could be fully backed up on a
completely separate drive.
91
6.3 Results of Problem
The Froude number calculation in Section 4.3 suggests that the air which is collected in
the AP1000 SG tubes during a refueling outage will successfully be swept from the
tubes with two operating RCPs in a single SG loop at the three evaluated fluid
temperature conditions from Table 4-2. The two and three dimensional CFD runs in this
analysis (Sections 5.1.2.4 and 5.2.1.4) show that for the conditions from Table 4-2, the
air collected in the AP1000 SG tubes during a refueling outage is expected to be
successfully swept from the SG outlet plenum.
This study shows that dynamic venting is a viable option for clearing the air from the
AP1000 SG tubes during reactor startup if vacuum refill is not successful.
92
7. Conclusions
The results of this study show the importance of modeling three dimensional geometry
in three dimensions rather than in two dimensions. The two dimensional results in this
analysis had very different fluid velocity results than the three dimensional models. This
difference can be attributed to the interaction between the flow paths on the three
dimensional level; the conservation of mass and conservation of momentum equations
for three dimensional modeling are significantly more complex than in two dimensions.
In addition, while the two dimensional models showed significant recirculation near the
SG tube exits, the three dimensional models showed less intense recirculation. The
particle track results for the three dimensional model showed that at higher fluid
velocities, the two dimensional recirculation paths start to become three dimensional
swirling patterns.
While three dimensional turbulent kinetic energy and dissipation rate results were not
generated as a part of this study due to computational limitations, the two dimensional
turbulent kinetic energy and dissipation rate results show that as the pressure,
temperature, and velocity are increased, the turbulent kinetic energy and dissipation rate
increase. The results also show that if the pressure is held constant and the temperature
and velocity are slightly increased, the turbulent kinetic energy and dissipation rate
decrease.
The meshes used in the two dimensional 100-tube model and the three dimensional 552tube model were created to be the finest meshes possible based on the computing power
available. The models were tested by running the meshing software and analysis cases
in FLUENT to determine the finest mesh that could be created and successfully solved
without any computer hardware issues. Since it was later discovered that plots of the
turbulent quantities could not be generated for the three dimensional models due to lack
of sufficient computing power, it would have been more prudent to attempt to create all
the necessary plots before determining the final allowable mesh size and proceeding
with post-processing and analysis of the results.
93
The purpose of this study was to determine if it would be possible to use dynamic
venting as a secondary method for removing the air from SG tubes in a Westinghouse
AP1000 PWR if vacuum refill was not possible as the primary method. The results of
the Froude number calculation for the SG tubes, the two dimensional 100-tube CFD
model, and the three dimensional 552-tube CFD model suggest that it will be possible to
clear the air from the tubes, through the SG outlet plenum, and into the RCP suction
using two RCPs operating in the same loop at three of the potential AP1000 startup
conditions.
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8. References
[1]
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