Westinghouse AP1000 Pressurized Water Reactor Steam Generator

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Westinghouse AP1000 Pressurized Water Reactor
Steam Genera
tor
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





ii
























© Copyright
2013

by

Andrea J. Dalton

All Rights Reserved



iii

CONTENTS

CONTENTS

................................
................................
................................
.....................

iii

LIST OF TABLES

................................
................................
................................
.............

v

LIST OF FIGURE
S

................................
................................
................................
..........

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 throu
gh 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



iv

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

Summar
y of Results

................................
................................
...................

85

6.2.2

Model Limitations due to Hardware

................................
..........................

89

6.3

Results of Problem

................................
................................
............................

92

7.

Conclusions
................................
................................
................................
................

93

8.

References

................................
................................
................................
..................

95




v

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




vi

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



vii

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



viii

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,
an
d 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



ix

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: Sl
ice 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






x

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

Differenc
ing

Scheme

SIMPLE

Semi
-
Implicit Method for Pressure
-
Linked Equations





xi

LIST OF SYMBOLS

Fr

Froude Number

(dimensionless)

V

Velocity

(ft/s, m/s)

L

Characteristic Length

(ft, m)

g

Acceleration due to Gravity

(ft/s
2
, m/s
2
)

M

Mass

(lbm, kg)

t

Time

(s)

Φ

Extensive Property

(dimensionless)

Ω
cm

Volume of Control Mass

(ft
3
, m
3
)

ρ

Density

(lbm/ft
3
, kg/m
3
)

ϕ

Intensive Property

(dimensionless)

Ω
cv

Volume of Control Volume

(ft
3
, m
3
)

s
cv

Surface Enclosing Control Volume

(
ft, m
)




Velocity Vector

(ft/s, m/s)





Orthogonal Unit Vector

(dimensionless)



Nabla (Vector) Operator

(dimensionless)

u
i

Velocity Component

(ft/s, m/s)

x
i

Coordinate Direction Component

(ft, m)

mv

Momentum

(lbm
-
ft/s, kg
-
m/s)

f

Forces

(lbf, N)





Body Forces per Mass Unit

(lbf, N)

τ

Stress Tensor

(lbf/ft
2
, N/m
2
)

τ
ij

Viscous Component of the Stress Tensor

(lbf/ft
2
, N/m
2
)

p

Pressure

(psi, Pa)

g
i

Gravitational Acceleration Vector Component

(ft/s
2
, m/s
2
)

µ

Dynamic Viscosity

(lbm/ft
-
s, kg/m
-
s)

δ
ij

Kronecker Symbol

(dimensionless)

x

Cartesian Coordinate Direction

(ft, m)

y

Cartesian Coordinate Direction

(ft, m)

z

Cartesian Coordinate Direction

(ft, m)



xii

k

Turbulence Kinetic Energy

(lbf
-
ft
2
/lbm
-
s
2
, J/kg)

ε
=
噩獣潵猠⡋楮敭i瑩c⤠䑩獳a灡瑩潮⁒a瑥
=
⡦(
2
/s
3
, m
2
/s
3
)

D

Inertial Diffusive
Transport

(
ft/s
2
, m/s
2
)

P
k

Production by Shear Stress

(lbf, N)

H

Denotes Higher Order Terms

(dimensionless)

D

Tube Inner Diameter

(ft, m)





xiii

LIST OF KEYWORDS

Westinghouse

Westinghouse Electric Company LL
C

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
-
v
iscous (
k
inematic)
d
issipation
r
ate

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 D
rive

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 Nav
ier
-
Stokes equations in this study

Reynolds Number

Flow characteristic based on velocity, geometry, and fluid
properties

Froude Number

Flow characteristic quantifying gravitational inertia




xiv

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.



xv

ABSTRACT

This thesis reports on wor
k carried out to model the details of flow phenomena in a
Westinghouse AP1000 Pressurized Water Reactor
(PWR) Steam G
enerator
(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

resu
lts 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.

1


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 R
V 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]



2

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]
.


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.
OUTLET NOZZLE



3

This creates the opportunity for air to be introduced into the RCS, which must be water
-
solid during normal operation to avoid damage to the RCPs from ai
r 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 firs
t 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 p
ump
operation is dic
tated 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


4

outlet nozzle, and to a downstream relief v
alve 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, on
ly 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 wat
er 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 A
P1000 plants presents a
new
challenge
because
the RCPs
installed at CE plants have a single drive speed
, but t
he 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
conce
rns, 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
operation
al experience and developed plant
-
specific dynamic venting procedures.




5

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 tube
s. 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 l
egs
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 outle
t
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 ene
r
gy
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.

6


2.

Literature Study

Liquid
-
only s
team 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 mode
led 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 non
-
uniform.
A
ir 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]

i
s a
United States (
U. S.
) Nuclear Regulatory Commission
(N
RC
)

funded study
, published in 2003,

which developed a CFD model of an SG inlet
plenum
based on a Westinghouse 1/7
th

scale tes
t 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 res
ults of
the CFD model were compared to the 1/7
th

model test data to validate the model. The
CFD model results were generally within 5% of the 1/7
th

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.




7


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
s
hould be

sufficient to clear any air which might be trapped in the SG tubes before the pumps are


8

fully
enga
ged

and operating normally
.

The results of this study
are

compared to the
results presented in previous studies to determine if there are any simila
rities or
differences between the modeled flow paths.

9


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

need
s

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 proce
ss to be successful.
For the
tube and outlet nozzle portions of this system, F
roude 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]
:







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.







10

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 i
s 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:






Equation
2

Where:

M = mass

t = time









Equation
3

Where:

Φ = extensive property

Ω
cm

= volume of control mass

ρ = density

ϕ = intensive property (= 1 for conservation of mass; = v for conservation of momentum)


The conservation of mass becomes:



11






















Equation
4

Where:

Ω
cv

= volume of control volume

s
cv

= surface enclosing control volume




= fluid velocity

vector





= orthogonal unit vector


The conservation of mass written in the form of Gauss’
Divergence Theorem is:







(



)





(



)






Equation
5

Where:



= nabla (vector) operator =
(








)

u
i

= velocity component

x
i

= 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:


(



)





(



)



(



)



(



)


Equation
6





12

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:


(

)





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 i
s:





























Equation
8



















































Equation
9

Where:





= body forces per unit mass

τ = stress tensor


The conservation of momentum can also be written as:








(





)



















Equation
10





13

Where:

τ
ij

= viscous component of the stress tensor (defined in

Equation
11
)

p = pressure

g
i

= gravitational acceleration vector component


The viscous component of the stress tensor is:





(













)












Equation
11

Where:

µ = dynamic viscosity

δ
ij

= 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:





















































Equation
12





















































Equation
13





















































Equation
14

For incompressible fluids, the components of the viscous component of the stress tensor
are:











Equation
15



14











Equation
16











Equation
17









(









)

Equation
18









(









)

Equation
19









(









)

Equation
20


The conservation of momentum equations in rectangular Cartesian coordinates
(
Equation
12
,
Equation
13
, and
Equation
14
) become:


































(























)








Equation
21

































(























)








Equation
22



15
































(























)








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]
:




















(

)











(


)



(

)




(

)

E
quation
24




16






̅










[



(











)
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅








̅
̅
̅
̅
̅
̅
̅
]







̅
̅
̅
̅
̅
̅
̅



̅





(





























̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
̅
)

Equation
25

Where:






(















)

Equation
26

The k
-
ε model simplifies to:









̅













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 occur
ring
in the SG outlet

plenum, 2D and 3D
CFD hydraulic model
s

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 i
terations are dictated by the computing power.
The models are only run as steady state scenarios due to computer processor limitations.



17

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 ϕ
e

is approximated as:




{





(






)








(






)




Equation
28


A Taylor series expansion about P for Cartesian coordinates with
(






)


> 0 and where
H denotes higher order terms is:







(





)
(


)


(





)


(






)




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 Se
mi
-
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:









Equation
30

5.

Calculate u, v, and w from their starred values using:



18










(







)

Equation
31










(







)

Equation
32










(







)

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 conv
erged 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

fl
ow
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 jum
p region are developed in
[7]
.
Correlations between fluxes of kinetic ener
gy before and after a hydraulic jump are
described in

[6]
.




19

It is appropriate to model the flow of water to determine if air can be successfully
push
ed

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 t
he water (shown as velocity vector results in Section
5

of t
his report) would
force the air
to follow
a path

of similar direction

as the water
.

20


4.

Input
s

4.1

RCS Conditions During Startup

AP1000
RCS temperature conditions vary from 70°F to 557°F during RCP
st
artup

[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]


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 p
re
-
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 Temperat
ure Conditions and RCP Speed
s

RCS
Temperature
(°F)

RCP Speed
(rpm)

SG Flow
(gpm)

70

200

18,000

231

1,600

148,000

450

1,750

157,500

Time (min)

RCS
Temperature
(°F)

Speed (rpm)

Head (ft)

Flow/Pump
(gpm)

Total

Flow
(gpm)

0

70

200

5

9,000

36,000

3

70

1,600

322

74,000

288,000

360

231

1,600

322

74,000

288,000

361

231

1,680

335

75,500

302,000

840

450

1,680

335

75,500

302,000

841

450

1,750

365

78,750

315,000

1,080

557

1,750

365

78,750

315,000


21


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 a
re 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 flu
id properties based on
these conditions are taken from
[14]

and are shown in
Table
4
-
3
.
Density and viscosity
are provided in b
oth English and metric units in
Table
4
-
3
.

Table
4
-
3
: Fluid Properties

RCS
Temperature
(°F)

RCS
Pressure
(psia)

Density
(lbm/ft
3
)

Density
(kg/m
3
)

Viscosity
(lbm/ft
-
s)

Viscosity
(kg/m
-
s)

70

200

62.3377

998.554174

6.5481E
-
04

4.7362E
-
03

231

2
,
250

59.7938

957.804804

1.7297E
-
04

1.2511E
-
03

450

2
,
250

52.1819

835.873862

7.9307E
-
05

5.7363E
-
04







22


4.3

Froude Number for Flow through SG Tubes

The tops of s
team generator tubes f
ill 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
S
G 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 e
ntrainment 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 th
e 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
:








(





)






















There are 10
,
025 tubes in each AP1000 SG (
[4]
, Table 5.4
-
4
).

The RCS configu
rations
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

23


factor of 7.4805 gal/ft
3

and the SG tube flow area of 2.014x10
-
3

ft
2
,
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. These calculations are
documented in
Table
4
-
5
.

24


Table
4
-
5
: SG Tube Froude Number

RCS
Temperature
(°F)

RCP
Speed
(rpm)

SG
Flow

(gpm)

SG
Tube
Flow
(gpm)

Conversion
Factor (gal/ft
3
)

SG Tube
Flow
Area (ft
2
)

SG Tube
Flow
Velocity
(ft/s)

SG Tube
Diameter
(ft)

Acceleration
Due to
Gravity
(ft/s
2
)

Froude
Number

70

200

18,000

1.796

7.4805

2.014E
-
03

1.986

0.051

32.174

1.556

231

1
,
600

148,000

14.763

7.4805

2.014E
-
03

16.333

0.051

32.174

12.792

450

1
,
750

157,500

15.711

7.4805

2.014E
-
03

17.381

0.051

32.174

13.613



25


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 out
age 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 need
ed 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 high
er 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.





26


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 Bounda
ry 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 a
s
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
.



27



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
.



Figure
5
-
2
:

2D 45
-
Tube Model


Outlet Plenum Region


sg_tube_bottoms

sg_bowl_bottom

sg_bowl_in

sg_tube_walls


28


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:







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 a
chieve 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

200

200

1.986

89.389

231

2
,
250

1
,
600

16.333

734.975

450

2
,
250

1
,
750

17.381

782.152



The velocity at the top of 45 tubes
is used as a boundary condition in the model
calculation setup. T
he 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
.




29



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.


sg_top (inlet velocity
boundary condition)

sg_top_out

sg_top_in

sg_tube_tops


30


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
. T
he mesh generated with this configuration contains 18
,
645 nodes and 15
,
027
elements.

Figure
5
-
4

shows the mes
h 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 wi
th

31


absolute velocit
y 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 s
urface body c
ell 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
*

I
nterior

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 c
ontrol.

Table
5
-
3
: Under
-
Relaxation Factors

Parameter

Under
-
Relaxation Factor

Pressure

0.3

Density

1

Body Forces

1

Momentum

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 4
00

32


iterations, reporting every 10 iterations, with a profile update interval of one i
teration.
This scenario is copied and run for each of the three analysis conditions.





33


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 an
d run in
ANSYS, it is sufficient to show the results of one of the model at one condition.
Plots of
t
urbulence 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 re
sults 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 Pl
ots option in the Results menu of the Solution engine in

34


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
.




35


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
expan
ded 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 no
zz
le 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.
Addin
g an outlet to the base model (45
-
tube model from
Section
5.1.1
) creates

36


an exit location for the modeled flow, so the results of the 100
-
tube mode
l 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 dire
ctly 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, onl
y one nozzle would be visible. T
he 3D model described in Section
5.2.1

includes both outlet nozzles.



The SG outlet plenum with its added outlet nozzle and asso
ciated 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
-
o
utlet
(pressure = 0 psig)
boundary condition as
shown in
Figure
5
-
8
.



Figure
5
-
8
: 2D 100
-
Tube Model


Outlet Plenum Region


sg_tube_walls

sg_tube_bottoms

sg_bowl_in

sg_bowl_bottom

outlet_no
zzle_top

outlet_nozzle_exit
(pressure
-
outlet
boundary condition)

outlet_nozzle_bottom


37


Similarly to the 45
-
tube model, t
he 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

200

200

1.986

198.642

231

2
,
250

1,600

16.333

1,633.277

450

2
,
250

1,750

17.381

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
.



38



Figure
5
-
9
: 2D
100
-
Tube Model


Inlet Velocity Boundary
Condition


As shown in
Figure
5
-
8

and
Figure
5
-
9
, n
amed 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


exteri
or 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 (outl
et_nozzle_bottom)


sg_top (velocity inlet
boundary condition)