ADVANCED POWER PLANT MODELING WITH APPLICATIONS 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 Institute Troy, New York December 2009 (For Graduation December 2009)
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ADVANCED POWER PLANT MODELING WITHAPPLICATIONS TO THE ADVANCED BOILING
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
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
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
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
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