advanced power plant modeling with …

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


3.5 Boiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5.2 Analytical Model - Fundamental Equations . . . . . . . . . . . 24

3.5.3 Equations Used in BOIL . . . . . . . . . . . . . . . . . . . . . 29


3.6 Nuclear Steam Turbine-Generator system . . . . . . . . . . . . . . . . 37

3.6.1 NSTGSYS - Code Description . . . . . . . . . . . . . . . . . . 37


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

17

Figure 3.3: Transient response of fluid variables

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.


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

23

Figure 3.8: Total heat flow vs time

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)


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


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

43

Figure 3.19: Angle Deviations

Figure 3.20: Power Deviations

44

Figure 3.21: Voltage and Current deviations

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.

52

Figure 4.1: Normal run: 100% to 90% power

Figure 4.2: Partial Load Rejection: 100% to 75% power

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

54

Figure 4.5: Downcomer

Figure 4.6: Reactor Pressure

Figure 4.7: Heat flow into heating channel

55

Figure 4.8: Water Level

Figure 4.9: Turbine/Generator bypass flow

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.

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