ADVANCED POWER PLANT MODELING WITHAPPLICATIONS TO THE ADVANCED BOILING
WATER REACTOR AND THE HEAT EXCHANGER
By
Prasanna Kumar Muralimanohar
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: ELECTRICAL POWER ENGINEERING
Approved:
Joe H. Chow, Thesis Adviser
Rensselaer Polytechnic InstituteTroy, New York
December 2009(For Graduation December 2009)
CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
ACKNOWLEDGMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. Advanced Boiling Water Reactor - General Description . . . . . . . . . . . 3
2.1 Modifications to the BWR [1] . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Description of Major Components . . . . . . . . . . . . . . . . . . . . 3
2.3 Functioning of an ABWR Plant . . . . . . . . . . . . . . . . . . . . . 5
3. Component Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Temperature Wave with Lateral Heat Transfer . . . . . . . . . . . . . 7
3.2 One-Dimensional Continuity Wave Equation for Boiling Mixtures . . 10
3.2.1 Derivation of Equation . . . . . . . . . . . . . . . . . . . . . . 10
3.2.2 Solution of Equations . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.3 Wave Propogation Time . . . . . . . . . . . . . . . . . . . . . 14
3.3 Pipe Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.1 Heat Exchanger - Code description . . . . . . . . . . . . . . . 18
3.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Boiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5.2 Analytical Model - Fundamental Equations . . . . . . . . . . . 24
3.5.3 Equations Used in BOIL . . . . . . . . . . . . . . . . . . . . . 29
3.5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 34
3.6 Nuclear Steam Turbine-Generator system . . . . . . . . . . . . . . . . 37
3.6.1 NSTGSYS - Code Description . . . . . . . . . . . . . . . . . . 37
3.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Control Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
ii
4. Advanced Boiling Water Reactor . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Model description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Description of variables . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Control Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Model Block Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
LITERATURE CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
APPENDICES
A. Wave Equation Solutions and Wave Movement Sequence . . . . . . . . . . 58
iii
LIST OF TABLES
3.1 Physical characteristic inputs - Heat Exhanger . . . . . . . . . . . . . . 20
3.2 Physical characteristic inputs - Boiler . . . . . . . . . . . . . . . . . . . 34
3.3 Sequencing switches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.4 Physical characteristic inputs - NSTGSYS . . . . . . . . . . . . . . . . 41
4.1 Base values of ABWR model . . . . . . . . . . . . . . . . . . . . . . . . 50
iv
LIST OF FIGURES
2.1 Advanced Boiling Water Reactor [2] . . . . . . . . . . . . . . . . . . . . 4
2.2 Plant Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Transient response of temperatures . . . . . . . . . . . . . . . . . . . . 16
3.2 Steady-state temperature profile as a function of distance . . . . . . . . 16
3.3 Transient response of fluid variables . . . . . . . . . . . . . . . . . . . . 17
3.4 Counter Flow Heat Exchanger . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Steady state temperature profiles . . . . . . . . . . . . . . . . . . . . . 21
3.6 Inlet and outlet temperatures vs time . . . . . . . . . . . . . . . . . . . 22
3.7 Temperature profiles at the end of simulation . . . . . . . . . . . . . . . 22
3.8 Total heat flow vs time . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.9 A Simple Boiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.10 Material Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.11 BOIL code steady-state run . . . . . . . . . . . . . . . . . . . . . . . . 35
3.12 Simulation result over time in sec . . . . . . . . . . . . . . . . . . . . . 36
3.13 BOIL code steady-state result after transients . . . . . . . . . . . . . . 36
3.14 Generator steady state phasor diagram . . . . . . . . . . . . . . . . . . 38
3.15 Machine in a Isolated lossless system . . . . . . . . . . . . . . . . . . . 39
3.16 Power Demand Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.17 Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.18 Voltage Regulator/Exciter . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.19 Angle Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.20 Power Deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.21 Voltage and Current deviations . . . . . . . . . . . . . . . . . . . . . . . 44
3.22 Control circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
v
4.1 Normal run: 100% to 90% power . . . . . . . . . . . . . . . . . . . . . . 52
4.2 Partial Load Rejection: 100% to 75% power . . . . . . . . . . . . . . . . 52
4.3 Partial Load Rejection: 100% to 75% power . . . . . . . . . . . . . . . . 53
4.4 Feedwater Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Downcomer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.6 Reactor Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 Heat flow into heating channel . . . . . . . . . . . . . . . . . . . . . . . 54
4.8 Water Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.9 Turbine/Generator bypass flow . . . . . . . . . . . . . . . . . . . . . . . 55
vi
ABSTRACT
The components of a modern Advanced Boiling Water Reactor (ABWR) nuclear
power plant are modeled in this thesis. The modeling involves the use of wave equa-
tions in the plant component flow path. The simulation procedure employed here
provides exact solutions for the differential equations permitting larger time interval
simulations for all components, including synchronous machines. A multivariable
control structure featuring dynamic switching with a constant gain matrix is devel-
oped here, i.e., recirculation flow is varied above 70% load demand, while control
rods are varied below 70% load demand.
For thermal performance of fluid systems a coordinate system that moves with
the fluid (Lagrangian) is used. This coupled with the use of the exact solutions to
the differential equations results in “Continuity Wave Equations”. The design of all
the component models relies on this approach.
A simplified model of the components of an ABWR plant is presented and
simulations have been performed. Results are shown for steady state and transient
conditions displaying the robustness of the design.
vii
ACKNOWLEDGMENT
I would like to express my gratitude and sincere thanks to my advisor, Prof. Joe
H. Chow, for the invaluable guidance, support and inspiration provided during the
course of this work. Working with him has been a true privilege and an enriching
experience. I am deeply thankful to Mr. T.D. Younkins for his contributions and
expert advice without which this project would not have been possible.
viii
1. Introduction
Modeling methods for producing equivalent models of large scale systems are
developed for performing cost-effective studies. This thesis presents one such model
of the components of a modern ABWR nuclear power plant. The modeling employed
here involves the use of wave equations which define the plant component flow paths.
The simulation focuses on the use of the exact solutions to the differential equations
which permits larger time interval simulation for all components thereby reducing
the need for computational power.
The ABWR is the evolutionary design of the last generation of Boiling Water Re-
actors (BWR). The existing BWR control systems and the load following capability
of the BWR have been discussed in [3]. The ABWR is an improved and enhanced
version of the conventional BWR. The multivariable control method implemented
here is a simple constant gain matrix with dynamic switching of the control struc-
ture.
The use of a coordinate system that moves with the fluid (Lagrangian frame of
reference) in the model rather than a coordinate system that is fixed in space and
time (Eulerian frame of reference) allows consistency in the use of coordinates and
this coupled with the use of the exact solutions to the differential equations results
in ‘Continuity wave equations’.
The remainder of the thesis is organized as follows. Chapter 2 presents a prelim-
inary description of the modifications made on the Boiling Water Reactor (BWR)
to achieve the Advanced Boiling Water Reactor (ABWR). Then the components
of an ABWR power plant are discussed and finally the functioning of an ABWR
plant is reviewed. Chapter 3 presents the details of the wave equations approach
used in the modeling of the component models. Here we also discuss about the
1
2
one-dimensional continuity wave equations for boiling mixtures. Then the details of
the design features for each of the component models (Pipe, Heat Exchanger, Boiler
and Nuclear Steam Turbine Generator (NSTGSYS)) are presented in the remainder
of this chapter. Also the simulation results for each of the component simulations
are reported. The control circuits used for modeling the ABWR and the NST-
GSYS are also presented in Chapter 3. Chapter 4 includes the design features and
control methodology of the ABWR model and finally the simulation results which
demonstrate the robustness of the control features implemented in the ABWR. Con-
clusions are drawn in Chapter 5. The solutions to the wave equations and the wave
movement sequence are included in the appendix A.
2. Advanced Boiling Water Reactor - General Description
The Boiling Water Reactor (BWR) is a single-cycle, forced circulation, light-water
nuclear reactor designed by the General Electric Company (GE). The Advanced
Boiling Water Reactor (ABWR) (Fig. 2.1) is an improved design of the BWR,
allowing better control of the nuclear reaction in the fuel core.
2.1 Modifications to the BWR [1]
• The reactor internal pumps (RIPs) have been included inside the Reactor Pres-
sure Vessel (RPV), which allows significant volume reduction in containment
and reactor building.
• The Fine Motion Control Rod Drive (FMCRD), which is an electro-hydraulic
system, has enhanced the control rod adjustment.
• In order to improve plant efficiency, performance, and economy, the turbine
design incorporates a 52-inch last-stage bucket design which has resulted in
increased plant output.
• Improved core and fuel design has resulted in increasing operating efficiency,
operability and fuel economy.
2.2 Description of Major Components
• Reactor Pressure Vessel (RPV)
This is a thick-walled cylindrical steel vessel enclosing the reactor core which
contains the fuel rods and the coolant. This structure is designed to withstand
high pressures and temperatures ensuring the containment of the nuclear fis-
sion reaction taking place inside the vessel. Placement of the recirculation
pumps inside the RPV has significantly reduced the occupational radiation
exposure levels.
3
4
Figure 2.1: Advanced Boiling Water Reactor [2]
• Recirculation Pump/Reactor Internal Pumps (RIPs)
These are variable-speed pumps that aid in recirculating the feedwater back to
the reactor core. The recirculating pumps are also used to control the reactor
power and provide cooling to the reactor core in the off-normal modes.
• Reactor Core
The reactor core is that portion of the RPV which houses the reactor fuel
assembly. The nuclear fission reaction takes place inside the reactor core.
• Fuel Assembly
An ABWR fuel assembly consists of a square array of fuel rods, held together
by the upper and lower tie plates and interim spacers, and surrounded by a
fuel channel. The bottom of the assembly serves to regulate the flow through
the assembly.
• Control Rods and FMCRD
A control rod is a rod made of chemical elements capable of absorbing neu-
trons thereby controlling the rate of fission reaction. Because these elements
have different capture cross sections for neutrons of varying energies, the com-
5
positions of the control rods must be designed for the neutron spectrum of
the reactor it is supposed to control. The ABWR control rod is a four-bladed
assembly containing neutron absorber rods. This assembly is driven from the
bottom of the reactor vessel by the FMCRD.
The FMCRD is an electro-hydraulic system which is used to precisely posi-
tion the control rods to provide a wide-ranging control for the reactor thermal
output.
• Feedwater
The feedwater consists of varying proportions of recovered condensate water
and fresh water which has been purified to varying degrees. It is converted to
steam inside the RPV.
• Steam Separator Assembly
This is the device used for separating water droplets from steam. This ensures
high quality (moisture free) steam to be supplied to the turbine thereby causing
minimal erosion on the turbine blades. This steam separator assembly is
housed inside the RPV.
2.3 Functioning of an ABWR Plant
The ABWR nuclear power plant consists of the RPV with the internal recircu-
lation pumps, the nuclear steam turbine-generator system, and the condenser. Fig.
2.2 is a block diagram showing the major parts of an ABWR power plant.
The basic function of a nuclear reactor is the release of thermal energy from
each fission reaction that occurs in the reactor core. This large amount of energy is
used to convert the feedwater into steam when it passes through the reactor core.
The ABWR flow path begins with preheated feedwater entering the reactor
vessel above the top of the reactor core and near the bottom of the steam drum. The
feedwater mixes with the recirculating saturated water, and the sub-cooled water
flows down the downcomer, through internal pumps, and then turns upwards into the
bottom of the reactor core. The water flows up through the core where it is heated to
saturation and then partially boiled into saturated steam. The steam/water mixture
6
Figure 2.2: Plant Block Diagram
flows up out of the core, through the outlet plenum, and then through the steam
separators. The separated steam flows out of the upper part of the reactor vessel
and then through piping to the steam turbine control valves. The separated water
flows into the steam drum, where the feedwater control system maintains the water
level to a constant set point. For transient over-pressure conditions, steam can also
flow through the bypass valves, in parallel with the steam turbine, to the condenser.
The steady flow rating of the bypass system is 30%.
The reactor internal pumps are variable-speed centrifugal pumps, which are
powered by variable speed motors below the reactor vessel.The reactor pressure ves-
sel of an ABWR contains not only the core assembly but also devices for separating
and drying steam (Steam Separator Assembly). The steam generated is separated
from the liquid by a structure of steam separators which are positioned above the
core. Steam from the separator then passes through a dryer assembly which removes
moisture from the steam. The dry steam then proceeds outside the reactor vessel
to the steam turbine which drives the generator which in turn produces electrical
power output.
3. Component Models
The various components of an ABWR will be modeled using a wave equation, which
is decomposed into two ordinary differential equations. Given a time step in sim-
ulation, the solution of the differential equations can be computed explicitly. The
details of the derivation are given in Appendix A.
3.1 Temperature Wave with Lateral Heat Transfer
The basic one-dimensional wave equation [4] is
∂Ts∂t
+ Vf1∂Ts∂z
= Co (3.1)
where
Co (oF/sec) is the Heating/cooling rate
and
Co =Tw − Tsτs
(3.2)
where
τs = heat transfer time constant = (CpρAf )s/hsp
C ′s is the fluid heat capacity/length (Btu/ft-oF )
hs = (Btu/ft-oF -sec)
P is the flow area(cross-sectional area)periphery (ft). Also P is the fluid/wall
heat transfer area divided by the pipe length
Combining (3.1) and (3.2) we get
∂Ts∂t
+ Vf1∂Ts∂z
=Tw − Tsτs
(3.3)
transforming from partial to total derivatives we get
dt =dz
Vf1
=τsdTsTw − Ts
(3.4)
7
8
from which we getdz
dt= Vf1 (3.5)
τsdTsdt
+ Ts = Tw (3.6)
Solving (3.5) and (3.6) we have
z = zo + V dt (3.7)
Ts = Tw(1− e(−t/τs)) + Tsoe(−t/τs) (3.8)
where zo and Tso are the previous wave’s final value before dt starts. Note that the
next Tw is used if z >∑dL and (3.8) is for the new wave front and the new time t.
Ts is the fluid temperature (oF) at x
x is the distance (ft) along the pipe
V is the fluid and wave velocity at t
t is the time (sec)
Tw is the average temperature of wall section dL long (oF)
Ta is the average temperature of fluid along dL (oF)
τ (sec) is C ′/hP , where C ′ (Btu/ft−oF) = CpρAF and
h (Btu/sec-ft2-oF) is the heat transfer coefficient (fluid-wall)
P (ft) is the perimeter
Cp (Btu/lb-oF) is specific the heat capacity
ρ (lb/ft3) is the density
AF (ft2) is the flow area (fluid), cross section area (pipe).
Note that h = 1/(1/hf + 1/hw), where hf is the fluid film coefficient and hw is
the pipe wall surface layer coefficient. Also hw = kw/tw, where kw is the pipe wall
conductivity (Btu/sec-ft-oF) and tw is a fraction of the pipe wall thickness.
For a section of length “dL” the average fluid temperature is given by
Tsav =1
dl
∫ dL
0
Tsdz (3.9)
9
which is
Tsav =1
dl
∫ dL
0
Tw(1− e(−t/τs)) + Tsoe(−t/τs) dz
which on integration gives
Tsav =Ts1 − Ts0e−z1/ls − (Ts1 − Ts0)τs/τt(1− e−z1/ls)
(1− e−z1/ls)(3.10)
Here Ts is now behind the wave front for the dz corresponding to dt, and Tso is now
the new up stream Ts at z = 0 in (3.9). Here again note that the next Ts is used if
z <∑dL. Also ls is the attenuation length (ft) = Vf1 τs.
The differential equation for the wall is
τw∂Tw∂t
= Ta − Tw (3.11)
whose solution is given by
Tw = Ta(1− e(−t/τw)) + Twoe(−t/τw) (3.12)
where τw is the wall heat transfer time constant (sec) and Two is the previous wave’s
final value before dt starts.
Similarly we can get the average temperatures for the wall section. Thus we
have the average temperatures of the wall and the fluid which can be used to model
a temperature wave with lateral heat transfer. In the computer codes, t is replaced
with ∆t, where ∆t is the time interval.
10
3.2 One-Dimensional Continuity Wave Equation for Boiling
Mixtures
3.2.1 Derivation of Equation
The continuity equations for saturated steam and water with change of phase [4,
5] are∂α
∂t+∂Jg∂z
=Γgρg
(3.13)
−∂α∂t
+∂Jf∂z
=Γgρf
(3.14)
Jg = Qg/At = AgVg/At = αVg (3.15)
Jf = Qf/At = AfVf/At = (1− α)Vf (3.16)
where
J is the volumetric flux (ft/sec)
Q is the volumetric flow (ft3/sec)
α is the steam void fraction which is “steam flow area (Ag)/Total flow area (At)”
Vg is the steam velocity (ft/sec)
ρg is the steam density (lb/ft3)
Vf is the water velocity (ft/sec)
ρf is the water density (lb/ft3)
Γg is the steam generation rate (lb/ft3sec)
The Zuber-Findlay (Z-F) drift flux model [6] is given by
Vg = Jm/KB + Vd (3.17)
where
Jm is the mixture volumetric flux, i.e., Jm = Jg + Jf
Vd is the vertical steam drift flux (ft/sec)
KB is the Bankoff void model constant [7] and KB = 1/Co from Z-F
11
Adding (3.13) and (3.14) we get
∂Jm∂z
= (Γg/ρg)p = (Γg/ρg)(1− q) (3.18)
where
p = 1− ρg/ρf = 1− q and q = ρg/ρf = 1− p (3.19)
Integrating (3.19) we get
Jm = (Γg/ρg)p(z − zb − Vtt) + Vf0 (3.20)
where
Vf0 is the water velocity at boiling boundary (ft/sec)
zb is the height at which boiling begins and where α is 0
Vt is the velocity of zb (ft/sec).
Combining (3.17) and (3.20) gives the equation for the steam velocity
Vg = (Γgp/(ρgKB))(z − zb − Vtt) + Vf0/KB + Vd (3.21)
Here note that Vg is directly proportional to Γg and z. Finally combining (3.21) and
(3.13) we get∂α
∂t+ Vg
∂α
∂z= (Γg/ρg)(1− pα/KB) (3.22)
In order to make equation solving easier, the following transformations are used
αp/KB = a (3.23)
(Γgp/(ρgKB)) = g (1/sec) (3.24)
Vf0/KB = Vk (ft/sec) (3.25)
12
Then (3.21) and (3.22) become
Vg = g(z − zb − Vtt) + Vk + Vd (ft/sec) (3.26)
∂a
∂t+ Vg
∂a
∂z= g(1− a) (1/sec) (3.27)
Equations (3.21) or (3.26), with (3.22) or (3.27), is the one-dimensional continuity
wave equation for boiling mixtures. Note from (3.26) that, at z = zb, where a =
α = 0, the initial steam velocity
Vg0 = Vk + Vd − g0Vtt (3.28)
where g0 is the value of g at z = zb.
3.2.2 Solution of Equations
In accordance with [8], (3.22) implies
dt = dz/Vg(z, t) = da/(g(1− a)) (3.29)
because dt, dz, and da are arbitrary. Equation (3.29) constitutes three ordinary
differential equations, but only two of them are independent since “t” is the inde-
pendent variable, while z = wave position, and a = a at z are dependent variables.
Thus from (3.29) we get
dz
dt− Vg(z, t) = 0 or
dz
dt− gz = Vk + Vd − gzb − g0Vtt (3.30)
and
da/dt+ ga = g (3.31)
are the two differential equations to be solved. The solutions to (3.30) and (3.31),
including boundary conditions are
zv = z1 + (z1 − zb + (Vk + Vd − (g0/g)Vt)/g)(egt − 1) + (g0/g)Vtt (3.32)
13
av = 1− (1− a1)e−gt (3.33)
where zv is the wave front and av is a at zv.
Also at t = 0, zv = z1 and av = a1. These are the equations used for boiling
flow in the computer code. For uniform power distribution up the flow channel
g0/g = 1 (3.34)
The third dependent differential equation from (3.29) is
da/dz = g(1− a)/Vg(z) (3.35)
Since (3.35) applies at any fixed time, the Vtt term in Vg drops out. The solution to
(3.35) can be used to determine a for any z between two waves, zv1, the upstream
wave and zv2, the downstream wave (zv2 > zv1)
az = 1−(zv2−zv1)(1−av2)(1−av1)/((av2−av1)(z−zv1)+(zv2−zv1)(1−av2)) (3.36)
This equation is used to calculate the void fraction ae, αe at the exit of the heated
channel, He (ft). The equation for Vt, involves a derivative, which is to be avoided,
if possible, as fundamentally destabilizing
Vt = (zb(new)− zb(old))/dt (3.37)
Here Vt comes from the preheat region. Vt is the velocity of the saturation tem-
perature, Tsat. A more sophisticated equation for Vt comes from the basic wave
equation [9] for the preheat region
dT/dt = q(z, t)/(Cp ∗ ρ) =
[∂T
∂t
]z
+
[Vf∂T
∂z
]t
(3.38)
where
T is the temperature (oF)
14
q(z, t) is the heating rate (Btu/ft3sec)
Cp is the specific heat (Btu/lb-oF).
Now
dz/dt = Vf =
[∂z
∂T
]t
dT
dt+
[∂z
∂t
]T
(3.39)
Solving (3.39) for Vt = [∂z/∂t]T
Vt = Vf −[∂z
∂t
]t
dT
dt= Vf −
[dT
dt
∂T
∂z
]t
(3.40)
Combining (3.38) and (3.40)
Vt = Vf − (q(z, t)/(Cp ∗ ρ))/
[∂T
∂z
]t
= −[∂T
∂t
]z
[∂T
∂z
]t
(3.41)
However (3.41) still includes derivatives. In the ABWR computer model |Vt| = 0,
unless it is above a limit. This avoids minor oscillatory effects on void reactivity
and reactor power.
3.2.3 Wave Propogation Time
The wave time, τv for traversing the entire boiling region is
τv =
∫ He
Vg
dz/vg = (He − zb)(1− pαav/KB)/(Vf0/KB + Vd) (3.42)
This is essentially the bubble rise time through the equivalent height of solid water
in the boiling region. It is exactly for this p→ 1, Vf0 → 0, and KB → 1.
15
3.3 Pipe Model
The PIPE code (computer code for simulating a temperature wave with lateral
heat transfer) calculates the propagation of variable temperature waves through a
pipe with a single phase fluid flowing at variable velocity and includes the effect of
pipe wall heat capacity. This PIPE code models temperature waves in a section of a
pipe. We will introduce a transient change in temperature and velocity and simulate
the temperature wave behavior till final time tf . There are two plots generated by
the code: the first plot shows fluid and wall temperatures along the length of the
pipe at a specific time and the second plot shows the inlet fluid, outlet fluid, and
wall temperatures as a function of time.
3.3.1 Simulation Results
The physical characteristic inputs given to the pipe code are
Length of pipe L = 20 ft
Fluid velocity = 5 ft/sec
Fluid heat transfer time constant τs = 2 sec
Pipe wall heat capacity/fluid heat capacity, cws = 0.2
New inlet temperature Tin = 30 oF
New velocity v= 8 ft/sec
Transient time tr= 2 sec
Time increment dt = 0.25 sec
Final time tf = 10 sec
Figs. 3.2 and 3.3 are the plots generated by the pipe code. It can be seen that
Fig. 3.3 shows the wave structure of the fluid and wall temperatures along the
pipe length. Fig. 3.4 shows the fluid variable changes that occur throughout the
simulation.
16
Figure 3.1: Transient response of temperatures
Figure 3.2: Steady-state temperature profile as a function of distance
18
3.4 Heat Exchanger
3.4.1 Heat Exchanger - Code description
The heat exchanger code simulates a heat exchanger that combines two PIPE
models with a common wall (each of length L and heat transfer area AHT ) to make
a counterflow heat exchanger, with flow areas AF and velocities V that can vary.
The code divides the length of the heat exchanger L, such that dL = L/5.
Figure 3.4: Counter Flow Heat Exchanger
The physical characteristic inputs given to the heat exchanger code are given
in Table 3.1. The total heat transfer Q is given by
Q = U ∗ AHT ∗ LMTD (3.43)
where
U = Overall heat transfer coefficient
AHT = Heat transfer area
LMTD = Log Mean temperature difference
The expression for LMTD for a counter-flow heat exchanger is
LMTD = ((Tpi − Tso)− (Tpo − Tsi))/ ln((Tpi − Tso)/(Tpo − Tsi)) (3.44)
19
where
Tpi - Primary inlet temperature
Tpo - Primary outlet temperature
Tsi - Secondary inlet temperature
Tso - Secondary outlet temperature.
The basic fluid wave equation is given by
∂T
∂t+ V
∂T
∂x=
(Twf − T )
τf(3.45)
where we have
T is the fluid temperature (oF)
x is the distance (ft) along flowpath
V is the fluid velocity at t (ft/sec)
Twf is the average temperature of wall section dL
τf is the fluid heat transfer time constant.
Equation (3.45) applies to both the primary and secondary flowpaths.
The basic equation for the wall [9] is
τw∂Tw∂t
+ Tw = τw(Tpτwp
+Tsτws
) (3.46)
where
1τw
=[
1τws
+ 1τwp
]τwp = C ′w/hp ∗ Pτws = C ′w/hs ∗ PTp and Ts are average fluid temperatures
The basic equation for fluid temperature (T ) [9] is
T = Twf (1− e−dt/τf ) + Toe−dt/τf (3.47)
20
Input Parameters Symbols and Units ValuesLength of the heat exchanger L (ft) 20Fluid heat capacity/length C ′
s (Btu/ft−o F) 500Flow area Afs (ft2) 10Fluid heat transfer time constant τs(sec) 2Pipe wall heat capacity/fluid heat capacity(Secondary side) cws 0.2Pipe wall heat capacity/fluid heat capacity(Primary side) cps 0.4Fluid secondary side inlet temperature tsi(oF ) −30Fluid secondary side outlet temperature tso(oF ) −10Secondary side fluid velocity Vs (ft/sec) 5Fluid primary side Inlet temperature tpi (oF) 40Primary side fluid velocity Vp(oF ) 5
Table 3.1: Physical characteristic inputs - Heat Exhanger
where To is the final value at the previous time interval. To is also the final value of
the upstream wave. Putting dt = dx/V in T the solution yields the wavefront T vs
x. Then by putting new T for the upstream wave into To yields spatial T vs x at
time t. The change to a different wave within dL is accounted for when calculating
average T . For output display, only wavefront T ’s are recorded in lists ‘tpl’ and ‘tsl’
and these are connected by straight lines in the plots of Tp and Ts vs x.
The basic solution for Tw is
Tw = τw(Tpτwp
+Tsτws
)(1− e(−dt/τw)) + Twoe(−dt/τw) (3.48)
where Two is the final value at the previous time interval. There are two average
wall section temperatures which are
Tws = Twso + (Tw − Two) (3.49)
Twp = Twpo + (Tw − Two) (3.50)
The wall thermal center is τw/τwstw from the secondary side wall surface, where
tw is the wall thickness. If hp ≈ hs then τwp = τws = 2τw and the thermal center is
at tw/2.
The heat exchanger code initiates in a steady state. The operating data input
21
into this code for the initial steady state are Tpi, Tsi, Tso, Vp, and Vs. (Tpo is calculated
by the code.) For subsequent transients, new values of Tpi, Vp, Tsi and Vs, at the
end of ramp time Tr, can be input.
3.4.2 Simulation Results
For the given physical input characteristics in Table 3.1, the steady-state tem-
perature profiles are shown in Fig. 3.5.
Figure 3.5: Steady state temperature profiles
The transient inputs given to the heat exchanger are
Primary inlet temperature changes to 50 oF
Secondary inlet temperature changes to −20 oF
Primary and secondary velocities change to 8 ft/sec
Ramp time tr = 2 sec
Time increment dt = 0.25 sec
Final time tf = 10 sec
The results of the simulation due to the transient changes are shown in Figs.
3.6-3.8.
22
Figure 3.6: Inlet and outlet temperatures vs time
Figure 3.7: Temperature profiles at the end of simulation
24
3.5 Boiler
3.5.1 Introduction
A boiler generates saturated steam from cooler feedwater by the application
of heat. A boiler is a pressure vessel with internal parts. Throughout the vessel
the pressure is essentially the same which is the saturation pressure Psat (psia).
The temperature ranges from saturation temperature Tsat (oF) down to a somewhat
cooler feedwater temperature Tfw (oF) at the feedwater inlet nozzle.
The BOIL computer code calculates the subcooling, the steam quality, void
fraction, and flow. It also calculates other important variables for the heating section
of the boiler with axially uniform or non-uniform heat input. The BOIL code uses
finite-element solutions of the energy flow equation [9] for the boiler preheat region
and two-phase continuity equations [5] and Zuber-Findlay void model [6] for the
boiling region.
The basic result is a series of forward wave equations that calculate sequences
of wave positions and corresponding values of subcooling, steam void fraction, and
velocity up the flow channel. The BOIL code is applicable for steady state or
transient solutions of the performance of the heating section of the steam generators,
boiling water reactors, and hot channels of pressurized water reactors.
3.5.2 Analytical Model - Fundamental Equations
Subcooled Region (Preheat Region)
The temperature variation (incompressible flow) is given by
∂T
∂t+ Vfo
∂T
∂z= Co =
Qf
Cpρf(3.51)
where
z = vertical distance (ft); inlet is z = 0
t = time (sec)
T = water temperature (oF); T = To at z = 0
Vfo = inlet water velocity (ft/sec), constant throughout region
Co = water heating rate (oF/sec); Co can vary with z
25
Qf = water heat input per unit volume (Btu/ft3-sec)
Cp = water specific heat (Btu/lb-oF)
ρf = water density (lb/ft3)
Equation (3.51) is a continuity wave equation with waves moving only in the
positive z direction with constant velocity Vfo. For solving (3.51) we need one
initial condition and a boundary condition. Consider a pipe of length zb, that is,
the pipe varies from z = 0 to z = zb. The initial condition we require is that
we need the temperature profile of this entire pipe at time t = 0. The boundary
condition requires knowing the temperature of water at z = 0 at all times. Using
these conditions we can solve (3.51). (In the BOIL code, T is actually the subcooling
enthalpy ratio, T = ∆hsc/hfg, a negative variable, where ∆hsc = hw−hf ; hfg = hg−hf ; hw is the water enthalpy (Btu/lb); hg is the saturated steam enthalpy (Btu/lb);
hf is the saturated water enthalpy (Btu/lb). Here hfg is the latent heat and is
contant for a particular pressure. Thus Co is the time rate change of the subcooling
ratio due to Qf . Note that T is a negative extension of the steam quality into the
subcooled region.)
The velocity of boiling boundary is
VT =∂zb∂t
(3.52)
where
VT is the velocity (ft/sec)
zb is the height at which boiling begins (ft), at the exit of the preheat region
(where T = 0)
For these equations Q, Vfo, and To are input forcing functions that can vary
with time. Inside the boiler we force Qf to get T = 0 at the exit (z = zb).
Boiling Region
This is the region starting from the water level in the boiler which is at z = zb till
the height of the boiller which is at z = He.
26
Equations in the boiling region
Steam continuity
The steam continuity equation is defined for two-dimensional flows. With this equa-
tion we can get the volumetric flux as a function of time and distance. Consider a
boiler to be a stack of circular sections as shown in Fig. 3.11. Here the region of
consideration is the boiling region which is between z = zb and z = He. Inside the
boiling region let us take a particular section at a height z = zx and of thickness δz
and write the Material Balance equation.
Figure 3.9: A Simple Boiler
Consider three sections of the boiler shown in Fig. 3.12. For the middle section
i there is steam input from a similar section i − 1 below it and steam output from
this section i to a section i + 1 above it, and there is also steam generated in this
section i itself. We can write the material balance asSTEAMINPUT
+
STEAM
GENERATED
=
STEAM
OUTPUT
+
STEAM
ACCUMULATION
where
Steam Input = jgρ at z = zx
Steam Output = jgρ at z = zx + δz
27
Steam Generation = Γg/ρg
Figure 3.10: Material Balance
From the material balance equation we can derive the steam continuity equa-
tion
∂α
∂t+∂jg∂z
=Γgρg
= G (3.53)
where
α = steam void fraction; α = 0 at z = zb
jg = α Vg = steam volumetric flux (ft/sec)
ρg = steam density (lb/ft3)
Γg = steam generation rate per unit mixture volume (lb/ft3sec); Γg = Qb/hfg,
where Qb (Btu/ft3) is the heat input per unit mixture volume in the boiling region
G = per unit volumetric steam generation rate (ft3/sec)/(ft3)
Water Continuity
Similarly we can derive the water continuity equation for the evaporation of
water as we proceed through the boiling region
−∂α∂t
+∂jf∂z
= −Γgρf
= G (3.54)
where
ρf = water density
28
jf = (1− α)Vf = water volumetric flux (ft/sec)
Vf = water velocity (ft/sec); Vf = Vfo at z = zb
Zuber-Findlay steam void model
When the two phases are considered to have different velocities (e.g., liquid and
gas), the relation between the void fraction and steam quality is not analytically
computable, thus requiring some empirical data which links void and quality. A
large number of empirical and semi-empirical methods have been suggested over the
last fifty years. The semi-empirical model which seems to have the most physical
basis is the drift flux model. It relates the gas-liquid velocity difference to the drift
flux (or “drift velocity”) of the vapor relative to the liquid, e.g., due to buoyancy
effects. The equation representing this model is
Vg =1
Kb
jm + Vd (3.55)
where we have
Vg = steam velocity (ft/sec)
jm = jg + jf = mixture volumetric flux (ft/sec) = mixture volume flow/flow area
Vd = vertical steam drift velocity (ft/sec)
1Kb
= Vg and α transverse distribution parameter
Kb = Bankoff parameter, Kb ≤ 1
In all equations, all values are average across the flow area, Af . Also ρg, ρf , Vd, VT ,
Vfo, Q, and To are assumed constant over the time interval ∆t, although Vfo, Q,
and To can vary with time.
Equations (3.63)-(3.65) can be combined and transformed to be
∂a
∂t+ Vg
∂a
∂z+ aQ = Q (3.56)
Vg = Q(z − VT t− zb) + V′
fo + Vd (3.57)
29
Also we have
x = (q
p)[
aVg(1− a)Vg − Vd
] (3.58)
where
a = α pKb
= α pk
q = ρg
ρf; p = (1 - q)
Q = Gpk (sec−1); note that Q can vary with z
V′
fo =Vfo
Kb(ft/sec)
x = steam quality = lb steam/ lb mixture = (hmix − hf )/hfg
3.5.3 Equations Used in BOIL
Heat Addition
The heat addition up the flow path is divided into 10 segments of equal length,
∆H, from z = 0 at the entrance of the preheat region to z = He at the exit of the
boiling region. The user inputs the power factor, fp, for each section, such that
1
10
10∑n=1
fpn = 1.0
The code calculates the average value of G, Q, and Co from other user supplied
input; then, using fpn (input given to the code), a listQL of 10 values ofQn is created
for use in the total heating region. In the preheat region, each Qn is multiplied by
cq = Co
Q, when used. After the initial steady state performance is calculated, the
user can input time varying G, and QL is accordingly modified at each subsequent
step.
Preheat Region
There are two solutions for (3.51): one for wave position zi and one for Ti at
zi, as follows
zi = zoi−1 + Vfo∆t (3.59)
30
Ti = Toi−1 + cq ·Q(n) ·∆t−Kp(δp
Pb) (3.60)
where zoi−1 and Toi−1 are the values of z and T at the beginning of each time step,
∆t. A new wave, zi, is started from zoi−1, and the first new wave, z1, is always
started at z = 0. Note that zoi−1 is the position of the wave zi−1 at the end of the
last time step. Kpδp/Pb is the correction of subcooling for change in pressure δp/Pb
in dt
Kp =Pbhfg
∂hf∂p
At Pb = 1000 psia, Kp = 0.24.
To account for the different values of Q(n) up the flow channel, two things
are done. First, a list KL is set up that contains K values of n, one for each
wave position (from lowest to highest) and this list is updated during each time
step, ∆t. Second, (3.60) is inverted to solve for the incremental time, and then the
cumulative time to go from zoi−1 to each successively higher value of n∆H. When
the cumulative time exceeds time step ∆T , zi is found from (3.59) directly, using
∆t = ∆t - (next to last cumulative time) and (n− 1) ∆H for zoi−1.
For each successive value of n, (3.60) is used to calculate a successively higher
value of T , using appropriate value of Q(n), starting with n from the list KL, and
using the incremental time from Eqn (3.60), inverted. The last value calculated is
Ti, corresponding to zi.
The total number of waves K, increases by one at each time step, but at the
end of each time step, K may be reduced so that there is no more than one wave
with Tk > 0. Then, a new value of zb called z1 is found as follows. If Tk = 0, z1 =
zk, but then if Tk−1 = 0, z1 = zk−1 and K is reduced by one. If neither Tk nor Tk−1
= 0, then z1 is found by interpolation
Tz = (Tk−1zk − Tkzk−1
zk − zk−1
) + (Tk − Tk−1
zk − zk−1
) · z (3.61)
where we have zk−1 ≤ z ≤ zk; Tk > 0; Tk−1 < 0.
31
Solving for z with Tz = 0 yields
z1 =
[Tkzk−1 − Tk−1zk
Tk − Tk−1
](3.62)
This linear interpolation is exact unless Q(n) varies from z = 0 to zb; it is
approximate then. However, Eqn (3.62) is exact for non-uniform Q(n) if ∆H is an
integer multiple of Vfo ·∆t. Also, Eqn (3.62) is exact if just Q at zb and Q(n) below
zb differ. The code puts wave zk−1 at the lower boundary of Q for zb. After z1
is found it replaces zk, and 0 replaces Tk. The new value of To is substituted into
TL(1). The list KL is also updated each time step.
Finally, the velocity of the boiling boundary is found to be
VT =z1 − zb
∆t(3.63)
and the wave time is given by
τs =zbVfo
(3.64)
Boiling Region
There are two solutions [8] for (3.56), one for wave position wi = z and one
for ai at wi, as follows
wi = woi−1 +Q(no)
Q(n)VT∆t+
[Vg(no)
Q(n)− Q(no)VT
Q2(n)
](eQ(n)∆t − 1) (3.65)
ai = 1− [1− aoi−1]e−Q(n)∆t −(λoi−1
xoi−1
)[1− λoi−1
(Vgi
Vgi − Vd
)]Kp
δp
Pb(3.66)
where woi−1 and aoi−1 are the values of w and a at the beginning of each time step
∆t. A new wave wi is started from woi−1, which is the position of the wave (woi−1),
that is, the position of the wave (wi−1) at the end of the last step. The first wave,
w1, always starts at zb, where α = 0. In Eqn (3.65), Vg(no) is Vg at z = zb, where
32
Q(n) = Q(no)
Vg(no) = Vd + V′
fo (3.67)
and, in general, from (3.67)
Vg(z,∆t) = Q(n)[z − zb]−Q(no)VT ·∆t+ Vg(no) (3.68)
Equations (3.65) and (3.68) cannot be used because of the possible changes
in n as the wave w advances during ∆t. Therefore, (3.65) is inverted to solve for
the incremental and cumulative time to go from woi−1 to the top of each successive
Q(n), at n∆H and finally to wi. Thus we have
δt =1
Q(n)ln[(x− VT1δt)/V4 + 1] (3.69)
where
x = wn - wo
VT1 = Q(no) VT / Q(n)
V4 = [Vg(no) - Q(no) VT / Q(n)]/Q(n)
wo = wave position at beginning of δt
wn = n∆H = wave position at end of δt.
Also we have
Vgn = Q(n)(wn − wo − VT1δt) + Vgo (3.70)
1− an = (1− ao)e−Q(n)δt (3.71)
where
Vgo and ao are Vg and a at beginning of δt
Vgn and an are Vg and a at end of δt.
33
If VT = 0, Eqn (3.69) is solved directly; if VT 6= 0, Eqn (3.69) is solved by a
simple iteration scheme that converges to δt with an error less than 0.0001 sec in
less than 3 cycles.
When the cumulative value of δt’s exceeds ∆t, wi is found from Eqn (3.65)
using ∆t = ∆t− (next to last cumulative time), and wo for woi−1. For each successive
value of n, Eqn (3.70) and Eqn (3.71) are used to calculate higher values of Vg and
a. The last values calculated are Vgi and ai, corresponding with wi. Then the steam
quality xi is calculated as
xi =
(q
p
)ai
Vgi(1− ai)Vgi − Vd
(3.72)
Solving for ai we obtain
ai =(Vgi − Vd)xiVgi(xi + q
p)
The values of w, a, x, and n are contained in the list WL and in lists AL,
XL, and JL, respectively, which correspond with WL. All lists are updated at
each ∆t. The list V L contains steam velocity Vg(n), which corresponds with the
inlet (bottom) of heating section n, from zb to He.
The total number of waves, J , increases by one each time step, but at the end
of each time step, J may be reduced so that there is no more than one wave with wJ
> He. Note that, for wJ > He, Q(n = 10) is assumed to extend above He. Then,
αe at z = He. Then, αe at z = He can be expressed in terms of the two waves, wJ
at z ≥ He and wJ−1 at z < He, as
ae = 1− (Vgj − Vgj−1)(1− aj−1)(1− aj)(aj − aj−1)(Vge − Vgj−1) + (Vgj − Vgj−1)(1− aj)
; αe =aePk
(3.73)
The exit steam velocity, Vge, is calculated using Eqn (3.70) with n = 10; the
exit quality, xe, is calculated with Eqn (3.72), using ae and xe; and then
jge = αeVge (3.74)
Finally, He is placed in WL(J); ae into AL(J); zb into WL(1) and ZL(K);
and no corresponding to z1 is placed into JL(1) and KL(K). For output, the list
34
Input Parameters Symbols ValuesBoiler exit height He(ft) 20Steam quality fraction at exit Xe 0.2Preheat region inlet velocity Vfo(ft/sec) 4Preheat region inlet subcooling ratio To −0.05Steam/water density q 0.05Drift velocity Vd (ft/sec) 1Bankoff constant KB 0.8Pressure correction factor Kp 0
Table 3.2: Physical characteristic inputs - Boiler
AL is converted from a to α = a/Pk.
The average a, aav, is found using a cubic regression of AL from zb to He and
the boiling region wave time, τv, is
τv = (He − zb)(1− av)/(Vd + V′
fo) = (He − zb)(1− αavP/Kb)/(Vd + Vfo/KB)
(3.75)
3.5.4 Simulation Results
The physical characteristic inputs given to the BOIL code are given in Table
3.2. With the inputs given, the BOIL code calculates temperature waves in the boiler
preheat region, steam void fraction, and steam quality waves in the preheat region
with variable axial power distribution and moving boiling boundary in response to
changes in the heat input, inlet flow, and temperature.
Fig. 3.11 is the plot showing the variation of steam quality, steam void fraction,
and subcooling ratio as a function of wave heights in steady state for the values in
Table 3.2.
The transient conditions given are
Boiling Function, g = 1.25 (sec−1)
Preheat region inlet velocity, vf = 5 (ft/sec)
Preheat region inlet subcooling ratio, To = −0.06
Transient time, tr = 2 (sec)
Time step = 0.25 (sec)
35
Figure 3.11: BOIL code steady-state run
Final time = 10 (sec).
The variation of the steam quality, steam void fraction, and boiling function
over time is shown in Fig. 3.12 for the transient conditions. Fig. 3.13 shows the
variables shown by Fig. 3.11 at the end of the simulation.
36
Figure 3.12: Simulation result over time in sec
Figure 3.13: BOIL code steady-state result after transients
37
3.6 Nuclear Steam Turbine-Generator system
The Nuclear Steam Turbine-Generator system (NSTGSYS) code simulates the
functions of a steam turbine which converts thermal energy generated at the steam
generators to kinetic (rotation) energy and that of a generator which eventually
converts the kinetic energy to electrical energy.
3.6.1 NSTGSYS - Code Description
The generator can be connected to an infinite bus or an isolated load, and
with this latter connection, partial load rejections (PLR) can be simulated. With
either connection, power maneuvers can be simulated, and balanced faults applied
to the load bus and cleared, with selected fault impedance and clearing time, and
with doubled post-fault system impedance from generator to load bus.
All model constants are built in, except the voltage regulator/exciter gain,
which must be entered. These constants are typical for a nuclear steam turbine
with a 4-pole generator. The model has 7 states which are the machine angle, speed,
turbine reheater/moisture separator, exciter, power system stabilizer, generator field
and the q-axis amortissuer. The turbine has a simplified control valve that responds
to load demand ramps and speed governing. The generator has the automatic
voltage regulator/exciter control system. Note that with only a q-axis amortissuer,
x′q = x′d.
The model initiates with the following inputs:
- Load connection (isolated system or an infinite bus)
- Voltage regulator (VR) gain (KA)
- Load power
- Load power factor.
To enhance accuracy and thus solution stability, some initial calculations are
repeated. The model then runs in steady state for one second to demonstrate
solution stability. Using dt = 0.1, variables repeat within < 10−14. After this initial
run transient inputs may commence. The steady state phasor diagram of a generator
is shown in Fig. 3.14.
38
Figure 3.14: Generator steady state phasor diagram
Even with the built-in integral solutions, the high VR gain influences numerical
stability and accuracy. So the following limits on incremental time, dt are imposed
- for a normal run dt = 0.1 sec
- for KA > 50, dt = 0.05 sec
- for KA > 100, dt = 0.02 sec
- also, after a fault removal or during a PLR, max dt = 0.05 sec.
The following labels are used to define the sequence in which the code runs.
39
- Label r → Display output
- Label ra → Display plots (at end of simulation)
- Label rc1 → Scheduler (this selects next label to go to)
- Label rc2 → Transient selection
- Label rc3 → Fault application
- Label rc4 → Fault removal
- Label rc41 → Calculate post switch VARS at switch time, after rc3 and rc4
- Label rc5 → Breakpoint - go to rc4 if fault is cleared
- Label rc51 → Main transient time - set system impedance seen by generator at et
- Label rc52 → Calculate transient performance.
Also the code has four built-in switches used in sequencing. They are shown in the
Table 3.3.
All time constants and gain values are initialized and after the inputs parame-
ters are given, the NSTGSYS code calculates the folowing values for an inital steady
state run (the values of the switches s1 = 1 and s2 = 1). Fig. 3.15 shows the gener-
ator connected to an isolated lossless system which represents the electrical system
model. After the initial steady state run the code displays the machine and load
Figure 3.15: Machine in a Isolated lossless system
parameters as output. When a no-fault transient input is given the code goes to the
Main Transient time sequence rc51 and sets the system impedances as seen from the
generator at et. Then the code continuously calculates the transient performance
rc52. Here a time increment is made and the new system parameter is calculated
using two control circuits which are explained in Section 3.7.
The model of the turbine used in the code is shown below.
40
SWITCH S1Value Sequence1 Following initial conditions2 (and if s2 = 1) Display output after 1 sec steady state
transient and set s1 = 1 and s2 = 23 Fault applied4 Remove Fault>4 Goto Label rc51 (Main transient time)6 Post fault set dt as min(dt, .05) and then set s1 = 6161 End the run if change in machine angle is too large
SWITCH S2Value Sequence1 (and s1 = 2), Display output after 1 sec steady state
transient and set s1 = 1 and s2 = 22 Goto rc2 (Transient selection)
SWITCH S3Value Sequence1 Infinite bus2 Isolated system
SWITCH S4Value Sequence1 No PLR2 PLR (When a PLR occurs, δm remains constant)
SWITCH pswValue Sequence1 Display output after t = 0, 1 and then every dt
with plots at end of run2 Display output after t = 0, 1 with plots at end of run
Table 3.3: Sequencing switches
3.6.2 Simulation Results
The characteristic inputs and system type given to the NSTGSYS code are
shown in Table 3.5. The code runs for 1 sec in steady state with a time step
dt = 0.1 (sec) and results of the steady state run are shown below.
Field Voltage = 2.35 per unit (pu)
Terminal bus voltage = 1.07 pu
Load Bus Voltage = 1.00 pu
Electrical Power Output = 0.9 pu
41
Figure 3.16: Power Demand Model
Figure 3.17: Turbine Model
Input Parameters ValuesSystem type Isolated systemVoltage Regulator Gain 75Initial Load 0.9Load Power factor 0.95
Table 3.4: Physical characteristic inputs - NSTGSYS
Mechanical Power Output = 1.06 pu
Terminal current = 0.95 pu
After the steady state run we perform a transient run for a system with no
fault, no PLR and with a new load power with a ramp time and final time. The
transient input conditions are given and they are listed below. The plots show the
system parameter’s deviations at the end of the transient run.
42
Figure 3.18: Voltage Regulator/Exciter
No fault system
No PLR
New load power = 0.92 pu
Ramp Time = 2 sec
Final Time = 20 sec
psw = 1
45
3.7 Control Circuits
The two control circuits used in the NSTGSYS and ABWR computer code
are
- Lead-lag
- Lag-rate
Figure 3.22: Control circuits
The two control system transfer functions of the lead-lag circuit and the lag-rate
circuits are shown in Fig. 3.21
Lead-lag Circuit : y =1 + τ2s
1 + τ1sx (3.76)
Lag-rate Circuit : y =τ2s
1 + τ1sx (3.77)
The use of a ramp as a function input type preserves function universality, but
introduces some small error when the output of the upstream control element is not
a ramp but this error is small. This enables the use of large time intervals for the
simulations.
4. Advanced Boiling Water Reactor
The basic description of the functioning of a ABWR plant was discussed in Chapter
2. In this chapter we shall discuss the detailed design features of the model used in
the simulation of the ABWR computer program.
4.1 Model description
There are 2 types of transients that can be run with ABWR which are listed
below
1. Normal power maneuvering, between 50% and 100% power
2. Partial load rejections (PLR’s) between 50% and 100% power.
For normal power maneuvering, Pe = Pt (which is the turbine power output)
and the speed error ds = 0. While actual power set-point rates are limited to
±10%/minute (±1%/6 sec), double of these rates can be used with the ABWR
model. Note that the control will automatically switch between the flow and rods
at 70% power. In the model, ∆k values are set to 0 for the initial power level.
For any generator connected to a utility power system, a sudden sustained
load decrease on the generator can result in
1. Complete load rejection caused by the generator high-voltage side breaker
opening
2. Partial load rejection caused by power systems breaking up into islands with
the generator unit remaining synchronized to a generation rich island.
For PLR in the ABWR model a new Pe, less than the initial value by no more than
30%, is entered and kept constant. Because Pt > Pe, ds increases. After Pt settles
out at Pe (usually < 30 sec), another transient segment can be run, restoring the
power set point Pst to Pe (and ds, bsf to 0, where bsf is the bypass steam flow).
46
47
The reactor kinetics equations determine the transient behavior of reactor
power, φ, in response to excess reactivity, ∆k. In the ABWR model, there are 3, in-
cluding 2 equivalent “delayed neutron group” first-order differential equations. The
third equation is algebraic and includes the “prompt jump effect”, which accurately
calculates transient φ for cumulative ∆k < 0.4, a value much higher than for normal
transients.
The “Excess reactivity” ∆k directly controls the fission process with the fol-
lowing implications:
• Increase in ∆k denotes increase in fission rate, φ
• Decrease in ∆k denotes decrease in fission rate,
• ∆k = 0 means steady state
The excess reactivity is given by
∆k = ∆kv + ∆kr + ∆kd
where ∆kv is the void reactivity, ∆kr is the rod reactivity, and ∆kd is the doppler
reactivity. ∆kv comes from the change in average steam void fraction, ∆av, which
is the fraction of the core boiling region fluid volume occupied by steam. ∆kv
is changed by the reactor power and reactor flow. ∆kd is a “negative feedback”
reactivity directly proportional to the fuel rod temperature, and thus reactor power.
This is an important safety feature of the reactor, as reactor power increases, so does
∆kd, which tends to shut the reactor down. All 3 ∆k values are relative, and the
ABWR model sets them = 0 at rated 100% reactor power.
4.2 Description of variables
All variables in the ABWR model are given in per unit, except some associated
with the wave models for the reactor heating flow channel which are
g is boiling region steam generation rate (lb/ft3sec)/ρg(lb/ft3)
z and w are wave heights (ft)
48
zb is the height at which boiling begins (ft)
vf is the preheat region water velocity (ft/sec)
vg is the boiling region steam velocity (ft/sec)
vt is the velocity of zb (ft/sec)
(To avoid minor disorder in the void reactivity, if |vt| < 0.01 then vt = 0)
The dimensional constants are core height, h = 12.17 ft, and steam vertical drift
velocity, vd = 1.3 ft/sec. The wave model in ABWR is the same as described in
the BOIL write-up except for the simpler uniform power distribution up the heating
channel in ABWR. The wave model includes the integral solutions of 4 differential
equations, two of which is for the preheat region
z (wave height) and t (pu subcooling), wave time, τs = 0.579 → 0.352 sec
two in the boiling region
w (wave height ) and α (void fraction), wave time, τv = 0.876→0.518 sec
Outside the heating channel, there are 15 states in the ABWR model, including
7 within the reactor. These 7 states, with their associated time constants, are
- downcomer subcooling (to, τdc = 12.44/cf sec, where cf is the per unit core flow)
- water level ( dl, τw = 0.5 sec )
- reactor pressure (pr, τpr = 12.71 sec)
- control rod drive reactivity (dkr, τkr = 1 sec )
- two delayed neutron group decay rates (ly1 and ly2 where f1=0.224, τ1 = 43.4 sec;
f2=0.776, τ2 = 4.15 sec) and
- fuel rod heat flow (qf , τq = 7 sec )
Outside the reactor are two states with the steam turbine output power, Pt ( fhp =
0.3, flp = 0.7) which are
- moisture separator/reheater, τp = 2.8 sec and
- speed ds, ht = 3.83 sec.
Three states with the feedwater control system which are
- level controller ( lc, τlc = 20 sec)
49
- flow error (fe, τfe = 2 sec) and
- flow controller (fc, τfc = 10 sec)
Three states with the current ABWR multivariable controller v2, v4, and v6 will be
described later. The ABWR model denoting these states are shown in the section
“Model Block Diagrams”.
Other non-state variables of interest are
Pst the plant power setpoint
Pe the plant power output
sf the reactor steam flow
sfe the core exit steam flow
ff the reactor feedwater flow
tsf the turbine steam flow
bsf the bypass system steam flow
Ph(φ) the reactor power
xe the core exit steam quality
αe the core exit steam void fraction
dw the net water flow rate setting the water level
The rated values of important variables are
xer = 0.1435
αavr = 0.4098
tor = −0.0347
The base value of core flow, cfb = 0.6085
The dimensionless constants are kb = 0.8 which is the Bankoff constant for Zuber-
Findlay steam void fraction model, q = .05 = ρg/ρf which is steam/water density
ratio and kp = .24 = (Pb/hfg) where (∂hf/∂P ) is the correction factor for the effect
of dpr on xe, to and αe, where pb = base pressure. The base values for the ABWR
model are given in Table 4.1
50
Parameters ValuesReactor power 3800 MWReactor flow 31970 lb/secReactor steam/feedwater flow 4,587 lb/secTurbine/generator power 1300 MWReactor water level range ±15 in
Table 4.1: Base values of ABWR model
4.3 Control Structure
The multivariable control structure modeled using output feedback is ex-
pressed as
ur = Kmvr (4.1)
where vr, Km and ur are all matrices whose values are
vr = [v1 v2 v3 v4 v5 v6]T
Km is the gain matrix =
1 0.4 1
0.3 0 0.3
0 0.2 0.1
0 0.1 0.5
−1 4 20
−0.3 0.3 1
ur = [u1 u2 u3]T
Also we have
v1 = pst - Pe - 20 ds
v2 =∫v1 dt
v3 = (0.5) bsf + Pe - Ph
v4 =∫v3 dt
v5 = 1 - pr
v6 =∫v5 dt
Pst + ur(1, 1) = ld
51
cf0 + ur(2, 1) = cf
-0.002 + ur(3, 1) = dkr0
where cf is the core flow, ld is the steam turbine load demand and dkr is the control
rod reactivity.
The following labels are used to define the sequence in which the code runs.
-Label rc1 → Sets constants and continues to additional transients
-Label rc2 → Sets type of transient and new power set-point
-Label rc21 → Sets new timing for transients
-Label rc3 → New time begins and continues ramp
-Label rc4 → Calculates transient time performance
4.4 Simulation Results
Two inputs, initial power φ (%) and ∆t (sec) are needed to initiate an ABWR
model run. The input values given are
- Initial power, φ = 100 %
- ∆t = 0.2 sec
The transient and new power set point inputs given to the model are
- Normal run
- New power set point = 95
- Time to reach new set point tr = 15 sec
- Final time tf = 40 sec
- Time step dt = 1/3 sec
For each transient segment two plots are made, one for the segment and one
for the entire transient. After each segment the segment plot is shown and after
the last segment, the plot for the entire transient is shown. The plots show the
deviations of the ABWR variables with respect to time. Here only the first segment
of the transient is shown.
53
Figure 4.3: Partial Load Rejection: 100% to 75% power
4.5 Model Block Diagrams
This section shows the block diagrams of the components of an ABWR.
Figure 4.4: Feedwater Control
5. Conclusions
The wave solution approach was used to model the components of an ABWR power
plant and all model simulations were presented. In this thesis, the main design
features for the model used are as follows:
1. The ABWR multivariable control was able to perform smooth load following
maneuvers in response to power setpoint variations.
2. A significant achievement of this multivariable control scheme is the variable
control structure. With a constant gain matrix, the control structure switches
dynamically
(a) above 70% load demand core flow is varied, but is held constant below
70% load demand,
(b) control rod position is held constant above 70% load demand, but is
varied below 70% load demand.
3. The use of exact solutions to the differential equations to the wave model has
permitted the use of larger time intervals without the loss of accuracy and
requires lesser computational time.
4. The multivariable control was tested with an isolated load model (a demanding
power system model) which requires the ABWR to perform all the frequency
regulation.
56
LITERATURE CITED
[1] S. A. Hucik, “Advanced boiling water-reactor, the next generation - status andfuture,” pp. 1377–1382, 1991, IEEE Nuclear Science Symposium and MedicalImaging Conference, Santa Fe, NM, Nov, 1991.
[2] http://www.nytimes.com/2007/09/25/washington/25nuke.html. Last accessedon November 27, 2009.
[3] D. G. Carroll, R. G. Serenka, and H. R. Propst, “BWR ManeuveringCapability,” Proceedings of the American Power Conference, vol. 41, pp. 73–78,1979.
[4] G. B. Wallis, One-Dimensional Two-Phase Flow. McGraw-Hill, 1969.
[5] N. Zuber and F. W. Staub, “An analytical investigation of transient responseof volumetric concentration in a boiling forced-flow system,” Nuclear Scienceand Engineering, vol. 30, no. 2, pp. 268–278, 1967.
[6] N. Zuber and J. A. Findlay, “Average volumetric concentration in 2-phase flowsystems,” Journal of Heat Transfer, vol. 87, pp. 453–468, 1965.
[7] S. G. Bankoff, “A variable density single fluid model for two-phase flow withparticular reference to Steam-Water flow,” Journal of Heat Transfer, vol. 82, p.265, 1960.
[8] F. B. Hildebrand, Advanced Calculus for Engineers. Prentice-Hall, 1949.
[9] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena. J.Wiley, 1962.
57
APPENDIX A
Wave Equation Solutions and Wave Movement Sequence
The derivation shown in this appendix is provided by Mr. T. D. Younkins, with
notes dated November 29, 2008. The “continuity” wave equation in one spatial di-
mension has one unknown, or dependant variable, Y, and two independent variables,
t (sec), x (ft) which is given as
∂Y
∂t+ V ∗ ∂Y
∂x= Q (A.1)
where, in general we have V = V (x, t). (In the simulation code, Q is constant over
∆t, but can be changed for the next ∆t. Also in most code, Q can be changed over
discrete increments in x. In single-phase flow, V is also constant over ∆t, but can
be changed for the next ∆t.)
There are two solutions to this wave equation, which can be broken down into
two ordinary differential equations of which the first one being
dt = dx/V = dY/Q = constant, (A.2)
because dt, dx, dY are arbitrary. Then from dt and dx we have
dx/dt = V (A.3)
which on integration becomes ∫V dt = ∆Xvi (A.4)
where ∆Xvi is the change in wavefront or wave position and ∆Xvi = Xvi − Xvi0,
where Xvi0 is the initial value of Xvi at t−∆t and i is the wave number.
From (A.2) we have
dY/dt = Q (A.5)
58
59
which on integration we get ∫Qdt = ∆Yvi (A.6)
where ∆Yvi is the change in Y corresponding to ∆Xvi and ∆Yvi = Yvi − Yvi0. (The
solution for dt and dx, the homogeneous part of the wave equation, can also be
written as the classic constant wavefront parameter u = (x−∫V dt).
In addition, a third dependent ordinary differential equation can be written
from dx and dY which is
dY/dx = Q/V (A.7)
The solution to this equation is∫(Q/V )dx = Y (X1)− Y (X0) (A.8)
This equation is used to obtain the variation of Y with x at a constant t, where X1
= Xvi at time t and X0 = Xvi−1 at time t.
All the computer code that has wave equation solutions uses the same basic
method to sequence the movement of the waves from one set of positions to the
next set of positions over time interval ∆t = dt. Two primary lists (xvl,yvl) are used
for Xv and corresponding Yv. Additional lists may be used for other corresponding
variables.
These lists are initialized with at least 2 values each, one at the inlet and the
other at the exit of the flowpath. Other values within the flowpath maybe added at
discrete intervals, or the two waves maybe advanced in “false time” until the wave
at the inlet of the flowpath passes the exit of the flow path. Then this last wave
is repositioned at the exit of the flowpath. The total number of waves is placed in
index label “K”, which is a coded variable.
At every value of time (t), a new wave starts at the inlet of the flowpath. The
inlet value Yi comes from elsewhere in the model, and is put into yvl[1] at the end
of the wave movement sequence, which is described in the following paragraph.
At new time t, K is increased by 1, with nothing in xvl[newK] and yvl[newK].
Then the following “for” loop is executed, starting at the exit of the flowpath and
indexing down to wave position 2.
60
For i = K:-1:2
j = i-1
xvl[i] = xvl[j] + ∆Xv
yvl[i] = yvl[j] + ∆Yv
End For
Note that no essential information is lost or overwritten. Also, the number
of each wave increases by one for each ∆t, as the wave progresses through the flow
channel. xvl[1] remains the same at the inlet of flowpath. A new value of inlet Yi is
put into yvl[1]. Another for loop then sets K such that there is one value of Xv at
or beyond the exit, Xe. The exit value of Y is found by using the foregoing solution
to the third differential equation, where Xvi = Xe.
The value of ∆Q will usually change over n sections of ∆x, between inlet
and exit, requiring at least two more lists for ∆Q and ∆Q position number. ∆Q
is calculated for each wave and part wave in a separate “for” loop that accounts
for the proper ∆x′s and part ∆x′s between two waves. This separate “for” loop is
inside the foregoing wave movement loop.
If the heat transfer between the wall and fluid is involved, ∆Q requires two
lists for average section temperature of the wall and fluid. An additional “for” loop
calculates these average temperatures for the next time (t), starting with the initial
values.