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Boiling water reactors
CHAPTER
Boiling water reactors
13
13.1 Introduction Numerous Boiling Water Reactors (BWRs) are operated in the U.S. and other countries. About 30% of the commercial nuclear reactors in the U.S. are BWRs. Several different generations of BWRs have been built or planned. This chapter addresses the important common features and their influence on the dynamic characteristics of BWRs.
13.2 History of BWR design evolution Unlike PWR development, BWRs have undergone an evolution of designs with significant changes along the way. Two experimental boiling water reactors were built at Argonne National Laboratory to test the viability of this reactor type (Borax-1 in 1953 and EBWR, the Experimental Boiling Water Reactor in 1956). General Electric (GE) entered BWR development with construction of the Vallecitos prototype BWR in 1957. GE then embarked on design and construction of commercial BWR power plants. As of this writing, seven designs of commercial power plants by General Electric, some with significant changes from its predecessors, have been built. These are designated as BWR-1 through BWR-6 and ABWR (Advanced Boiling Water Reactor). A new BWR (the ESBWR or Economic Simplified BWR) has been designed. The main differences in the various designs are containment features, forced circulation vs. natural circulation, and in-vessel jet pumps with flow driven by external pumps vs. integral mechanical pumps. The evolution of GE power plants is described below:
13.2.1 BWR-1 These were early low-power BWRs (all less than 200 MWe). Three versions were built and had different design features, but all are designated as BWR-1. Differences in BWR-1 designs includes direct cycle (reactor steam goes directly to the turbine) or indirect cycle (reactor steam feeds a separate steam generator) and natural circulation of water flow into the core region or forced circulation. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00013-5 # 2019 Elsevier Inc. All rights reserved.
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13.2.2 BWR-2 BWR-2 produces greater power levels (greater than 500 MWe) than BWR-1. It uses mechanical recirculation pumps and Mark I containment (See below). BWR-2 and four subsequent designs (BWR-3 through BWR-6)) are all considered to be Generation II reactors.
13.2.3 BWR-3 BWR-3 produces greater power levels (800 MWe) than BWR-2 and was the first BWR to use jet pumps for recirculation flow. It uses Mark I containment.
13.2.4 BWR-4 BWR-4 is similar to BWR-3, but it operates at higher power (1100 MWe). BWR-4 reactors use either Mark I or Mark II containment.
13.2.5 BWR-5 BWR-5 is similar to BWR-4. It operates at the same power level as BWR-4 power (1100 MWe). BWR-5 reactors use either Mark I or Mark II containment.
13.2.6 BWR-6 BWR- 6 is available in different configurations having power levels of 600– 1400 MWe. BWR-6 uses Mark III containment.
13.2.7 ABWR ABWR is a Generation III reactor. The ABWR employs internal mechanical recirculation pumps. It uses Mark III containment and the power level is 1500 MWe.
13.3 Characteristics of BWRs 13.3.1 General features of a BWR Since there are five different Generation II BWRs, it is necessary to pick one for providing an overview of BWR characteristics. BWR-6 was chosen, but its dynamic behavior and control strategy is typical of all Generation II BWRs. The main difference is that it operates at a higher power level than earlier designs. Fig. 13.1 shows a typical BWR-6 system. Fig. 13.2 shows a BWR-6 reactor vessel and internals [1]. Subcooled water enters the bottom of the core. The flow rate and pressure are such that boiling begins near the entrance. Boiling continues along the rest of the passage through the core. A steam-water mixture exits the core region. This mixture
13.3 Characteristics of BWRs
Containment Cooling System
Steam Line Reactor Vessel
Turbine Generator Separators & Dryers
Heater Condenser
Feedwater 3
Condensate Pumps
Core 1&2
Feed Pumps
Control Rods
Demineralizer
Recirculation Pumps Emergency Water Supply Systems
FIG. 13.1 Schematic of a typical boiling water reactor system. U.S. Nuclear Regulatory Commission: www.nrc.gov/reactors/bwrs.html.
then passes through steam separators and steam driers located above the core. These systems remove water by centrifugal force and by sudden reversals in flow direction. The removed water flows downward into an annular region between the vessel and a core shroud. This annular region is called the downcomer. Fig. 13.3 shows a BWR-6 fuel bundle. The fuel is Uranium oxide contained in Zircaloy tubes. A typical 1220 MWe BWR-6 core consists of about 750 fuel assemblies. Each assembly is enclosed in a fuel box with an 8 8 or a 10 10 array of fuel rods (pins). The fuel pins are similar to those in a PWR, and have an active length of 12 ft. The fuel ‘box’ constrains the flow of coolant in the assembly. Upper and lower tie plates provide structural support for the assembly, along with a few tie rods. Some spaces in the fuel assembly are taken by water rods to provide additional neutron moderation. A typical BWR contains 60–70 thousand fuel rods and
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STEAM DRYER LIFTING LUG VENT AND HEAD SPRAY
STEAM DRYER ASSEMBLY STEAM OUTLET
STEAM SEPARATOR ASSEMBLY
CORE SPRAY INLET
FEEDWATER INLET FEEDWATER SPARGER
LOW PRESSURE COOLANT INJECTION INLET
CORE SPRAY LINE
CORE SPRAY SPARGER TOP GUIDE JET PUMP ASSEMBLY CORE SHROUD
FUEL ASSEMBLIES
CONTROL BLADE
CORE PLATE JET PUMP/RECIRCULATION WATER INLET
VESSEL SUPPORT SKIRT
RECIRCULATION WATER OUTLET
SHIELD WALL
CONTROL ROD DRIVES
CONTROL ROD DRIVE HYDRAULIC LINES IN-CORE FLUX MONITOR
FIG. 13.2 BWR vessel and reactor internals. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
160 metric tons (160,000 Kg) of UO2. Design parameters of a typical BWR-6 are given in Appendix A. Fig. 13.3 also shows a BWR-6 control assembly. It is a cruciform-shaped structure that enters the core from below and is inserted in the space between four fuel assemblies. The below-core location is necessary since the region above the core
13.3 Characteristics of BWRs
FOUR-BUNDLE FUEL MODULE FUEL ROD WATER RODS Care Lattice
TIE RODS
FIG. 13.3 Four fuel assemblies (boxed channels) showing the control rod cruciform shaped assembly in the center of fuel four fuel assemblies. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
contains steam separators and driers. The location of control rod assemblies means that gravity-based insertion is impossible. Insertion from below places the control rod assembles enter through the subcooled region where control rod worth is greatest and facilitates refueling from above. The control assemblies contain boron carbide and are used for reactivity control and power flattening. Burnable poison (oxide of gadolinium – gadolinia) is mixed with the fuel for shim control.
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13.3.2 Recirculation flow and jet pumps The core flow in a BWR-6 is controlled by two recirculation pumps that distribute the water to a set of jet pumps surrounding the core. Each recirculation pump distributes water to one of two manifolds. Each manifold supplies water via pipes to jet pumps. A pair of jet pumps receives water from a single pipe. There is a total of 20 jet pumps, with a typical jet pump overall length of 19 ft. Mechanical pumps (recirculation pumps) withdraw water from the downcomer and pump it at elevated pressure into the jet pumps. The jet pumps have no moving parts, making them maintenance-free reactor components. Fig. 13.4 illustrates the principle of operation of a jet pump [2]. The recirculation flow enters the jet pump nozzle at a high pressure and increases to a high velocity as it flows through the narrow throat, and results in a pressure drop. The suction flow in the downcomer region enters the inlet nozzle at a low pressure. The pressure decreases as the suction flow passes through the converging section of this nozzle. The driving flow and the suction flow mix in the throat region (mixing
FIG. 13.4 Operation of a jet pump. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
13.4 Reactivity feedbacks in BWRs
section with a constant diameter) resulting in an increase in the fluid pressure due to a change in the fluid velocity in this section. A long diffuser is connected at the end of the mixing section, causing an increase in the fluid pressure that drives the coolant into the lower plenum and then up through the reactor core.
13.3.3 Other features of BWRs The following are typical core parameters for a BWR-6: • • • • • • • • • • •
Total coolant flow rate: 105 106 lbm/h Plant efficiency: 34% Core diameter: 193 in. Number of control rods: 177 Coolant pressure: 1040 psia Core-exit (steam) temperature: 551°F Feed water temperature: 420°F Average coolant exit quality: 15% Vessel diameter: 19 ft. Wall thickness: 5.7 in./6.46 in. Vessel height: 71 ft. Core power density: 54 kW/l.
The steam produced in the core and separated from liquid water passes through a control valve into the turbine. The steam pressure is maintained at a constant value by throttling the steam valve. Exhaust steam passes into a condenser and the condensate passes through a series of feedwater heaters before returning to the reactor vessel. Note that a BWR system shares general features with U-tube steam generators used in most PWRs (a heated riser, steam separators and driers, and a downcomer). A BWR containment consists of a concrete “drywell” that encloses the reactor. If steam escapes from the reactor vessel or related piping it flows into the drywell. The drywell has piping that connects it to a large pool of water called the suppression pool. The suppression pool water condenses the steam and reduces pressure in the drywell. Three different types of suppression pool are used, Mark I, II and III [3]. See Fig. 13.5.
13.4 Reactivity feedbacks in BWRs The BWR fuel temperature coefficient of reactivity is due to the Doppler effect (fuel temperature feedback reactivity) and is always negative. Typically, the Doppler coefficient in BWRs is around 2 105 Δρ/°C. The magnitude of the negative Doppler coefficient increases as fuel temperature increases. BWRs are under-moderated. Increases in moderator/coolant temperature cause increased boiling and reduced in-core liquid water density. Thus, an increase in
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moderator/coolant temperature decreases neutron moderation (a negative reactivity mechanism) and decreases neutron absorptions (a positive reactivity mechanism). The moderation effect is larger, causing a net negative moderator/coolant temperature coefficient of reactivity. Typically, the moderator/coolant temperature coefficient is around 3 104 Δρ/°C. The void coefficient depends strongly on reactor conditions, but it also is always negative. Typically, the void coefficient is around 1.4 103 Δρ/% voids. Pressure also affects reactivity because pressure affects boiling. For example, a pressure increase causes a reduction in core voids. Because the void coefficient is negative, a pressure increase causes a reactivity increase. That is, the pressure coefficient of reactivity is positive. The control strategy in BWRs is to maintain constant pressure by modulating the steam valve.
DRYWELL HEAD
DRYWELL FLANGE
DRYWELL SHEAR LUG SUPPORT
REACTOR PRESSURE VESSEL
DRYWELL SHIELD WALL
CORE
RADIAL BEAM
MANWAY RADIAL BEAM VACUUM BREAKER
JET DEFLECTOR VENT
VENT HEADER
DOWNCOMERS WATER LEVEL
(A)
SUPPRESSION CHAMBER (TORUS)
FIG. 13.5 (A) BWR Mark I containment with toroidal pressure suppression chamber. (continued)
13.4 Reactivity feedbacks in BWRs
The moderator temperature also affects the thermal neutron spectrum. Increases in moderator temperature cause hardening of the thermal neutron spectrum. This causes a negative component of the reactivity change due to changes in U-235 absorptions and a positive component due to changes in Pu-239 absorptions. See Section 7.3. As in all thermal spectrum power reactors, burnup and production of Xe-135 causes a reactivity feedback effect. See Section 6.2. The power coefficient for BWRs is negative, thereby ensuring that reactor power achieves a new steady state level following a change in external reactivity.
DRYWELL HEAD
DRYWELL
REACTOR VESSEL
SACRIFICIAL SHIELD WALL STEEL LINER REACTOR PEDESTAL
DRYWELL DECK
S/R VALVE TAILPIPE (18) EQUIPMENT HANDLING PLATFORM DOWNCOMER (VENT)
VACUUM BREAKERS (5) SUPPORT COLUMN (12)
PRESSURE SUPPRESSION CHAMBER
WATER LEVEL
QUENCHER (18) REINFORCED CONCRETE
(B) FIG. 13.5, cont’d (B) BWR Mark II containment chamber. (continued)
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CONTAINMENT SPRAY SHIELD BUILDING 125 TON CRANE W/15 TON AUX HOOK
CONTAINMENT UPPER POOL
DRYWELL HEAD FUEL TRANSFER POOL
REACTOR VESSEL REACTOR SHIELD DRYWELL BOUNDRY
WEIR WALL S/R VALVE LINE
DRYWELL FUEL TRANSFER TUBE
SUPRESSION POOL HORIZONTAL VENT
(C) FIG. 13.5, cont’d (C) BWR Mark III containment. Courtesy of GE Hitachi Nuclear Energy Americas LLC (General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ ML1414/ML14140A181.pdf).
13.5 Reactivity and recirculation flow BWRs can use control rods to change reactivity, but an alternate way is to change recirculation flow rate. Increasing the recirculation flow rate increases the amount of liquid water in the core relative to steam. Since the BWR is under-moderated, this increases reactivity, thereby increasing reactor power and steam production. Thus, BWRs have two ways to change reactivity by external means, whereas PWRs have one (control rod motion). In forced circulation BWRs recirculation pumps are used to draw water from the lower downcomer region and distribute the water to a set of jet pumps at an elevation above the pump suction location. Thus, a BWR is a variable flow system, with the flow modulation facilitating start-up and load-following operations. Two recirculating pumps distribute water to jet pumps, through a sparger ring. Changing the pumping power (hence, the coolant flow rate) causes a change in reactivity through a change in core voids. See Section 13.3.2 for a description of jet pump operation.
13.7 BWR dynamic models
13.6 Total reactivity balance For steady state, zero reactivity is required. The total reactivity balance is as follows: ρ ¼ Control rod reactivity + Recirculation flow reactivity + Feedback reactivity ¼ 0
The externally-controlled reactivity may be achieved by a combination of control rod reactivity and recirculation flow reactivity. Thus, a desired reactivity setting for either externally-controlled reactivity can be achieved by adjusting the other externally-controlled reactivity.
13.7 BWR dynamic models Detailed BWR dynamic models include treatment of all of the complex neutronic and thermal-hydraulic effects that contribute to the dynamics of the system. Both linear and nonlinear models exist. Detailed models are too complex for inclusion here. Interested readers can find information in the literature [4, 5]. Linear models provide estimates of the small-perturbation time response and frequency response. An approximate, low-order model provides simple simulation capability. It accounts for the neutronic and thermal-hydraulic processes that determine feedback reactivity. A low-order model [4] was developed by fitting a loworder transfer function to match the closed-loop frequency response calculated with a detailed model [5]. The results obtained with the low-order model are essentially identical with results from the detailed model. Note that the author of Ref. [4] chose to express the frequency in Hz rather than in rad/s as used elsewhere in this book. Figures in this chapter use rad/s for frequency units. The low-order closed loop transfer function used in the fit [4] is Gc ðsÞ ¼
K ðs2 + as + bÞ ðs + cÞ + ds + eÞ ðs + f Þ ðs + gÞ
ðs 2
(13.1)
The following values of the low-order model parameters are typical for BWR simulation [4]: K ¼ a gain that varies with power level a ¼ 0.36 b ¼ 0.1055 c ¼ 0.03 d ¼ 0.09 e ¼ 0.1044 f ¼ 0.25 g ¼ 21.0 The low-order model from Ref. [4] provides the capability to further investigate BWR dynamics and the cause for potential stability issues. The low-order model
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of Ref. [4] was modified to provide the amplitude in units of % power/cent of reactivity as used throughout this book. Fig. 13.6 shows the power-to-reactivity frequency response obtained with the low-order model. Note the resonance at around 0.3 rad/s. The resonance grows if changes in conditions cause the system to move toward instability. Recall that the closed loop transfer function for a system with feedback is given by Gc ¼
Go 1 + Go H
(A)
(B) FIG. 13.6 (A) BWR frequency response amplitude, (B) BWR frequency response phase.
(13.2)
13.8 BWR stability problem and impact on control
where Gc ¼ closed-loop transfer function (feedback effects included). Go ¼ open-loop transfer function (the zero-power transfer function). H ¼ feedback (cents/% power). At low frequencies, the magnitude of Go is large. Consequently, Gc ¼ 1/H at low frequencies. The feedback frequency response can be calculated using H¼
1 1 Gc Go
(13.3)
The resulting feedback frequency response appears in Fig. 13.7. Note that the phase lag is more than 90 deg. at frequencies above around 0.1 rad/s. As shown in Section 3.8 such phase shifts in system feedback can cause instability if the feedback gain is large enough. The frequency response for various feedback conditions can be deduced by applying a multiplicative factor, K to the feedback term (GoH in Eq. (13.2)). Gc ¼
Go 1 + K Go H
(13.4)
Note that K ¼ 1 for the original low-order model. Fig. 13.8 shows the closed-loop gain for various values of K. Clearly the resonance at around 0.3 rad/s grows as K increases and shifts to higher frequencies. Ref. [4] shows that the system becomes unstable at a value of K 2.25. The above discussion reveals that useful insights can be deduced if basic principles of dynamic analysis of feedback systems are understood and employed.
13.8 BWR stability problem and impact on control BWRs exhibit instability at conditions of high power and low recirculation flow. This instability is caused by a complex coupling of neutronics and thermalhydraulics. The basic cause of BWR instability is time lagged flow and reactivity feedbacks. Recall that positive feedbacks usually cause stability problems, but negative feedbacks also can cause instability if their effect is delayed (and the feedback gain is large enough). See Section 3.8. Subcooled water enters a BWR channel at the bottom. As it flows upward, boiling occurs and the void fraction increases. A disturbance (typically inlet flow change, inlet subcooling change, or power change) causes a localized change in steam bubble concentration in the lower portion of the channel. The propagation of this bubble packet as it travels up the channel is called a density wave [2]. This density wave causes changes in the local pressure drops as it propagates. Consider a disturbance that causes an increase in steam bubble formation in the lower portion of the channel. The resulting density wave travels upward, and it travels faster than the flow prior to the disturbance. As a result, the total channel pressure drop
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(A)
(B) FIG. 13.7 (A) Feedback frequency response amplitude. (B) Feedback frequency response phase shift.
increases, but it is delayed relative to the input disturbance. If the pressure drop is large and it lags the input disturbance by 180 degrees, it destabilizes the channel. The density wave also affects reactor power by the void feedback effect on reactivity. The local reactivity and reactor power change follow the change in void density, but the heat transfer from the fuel to the fluid lags because of the fuel’s heat transfer lag. The fuel heat transfer time constant in a BWR is typically 6–10 s. So, the neutronic response is also lagged, thereby introducing its additional component of the lagged response. The coupled thermal-hydraulic and neutronic response can cause instability, especially at low flow and high power conditions. At low flow and high power, resulting lagged thermal-hydraulic and neutronic feedbacks have values that induce instability.
13.9 The power flow map and startup
FIG. 13.8 Closed-loop frequency response amplitude for various values of K.
Fig. 13.9 shows the feedback paths that determine stability. Instabilities of different nature can occur. These include single channel instability, in-phase core-wide instability (the whole core responds in unison) and out-ofphase core instability (different regions respond out-of-phase with one another). The processes involved in coupled thermal-hydraulic and neutronics are very complex and not amenable to a simple analysis. Very detailed computer codes are necessary and several analysis codes have been developed. Many publications have addressed the BWR stability problem, its analysis and its mitigation. The magnitude of the effort to deal with BWR instability illustrates the importance of the problem. Details may be found in the literature (see Refs. [2, 4, 6–10]). The typical conditions for instability are power levels of 35–60% and core flow rates of 30–45%. The strategy for avoiding instability is to avoid operation in the range of reactor power levels and core flow rates where instability occurs. Specifically, the reactor power is kept below the instability threshold at low core flows by using control rods to adjust power.
13.9 The power flow map and startup A reactor with a negative power coefficient (such as a BWR) experiences a specific new steady state power following a reactivity change (see Chapter 7). Since BWR externally controlled reactivity depends on control rod positions and recirculation flow there are many different paths to change reactor power. Control rods are used for reactivity adjustment to keep power low at low flows to avoid the instability
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Reactivity
NEUTRONICS
Doppler
Power
Fuel temp
FUEL
Void reactivity CORE T-H
Direct heat
Inlet enthalpy Outlet pressure
RECIRCULATION LOOP
Inlet flow
FIG. 13.9 BWR dynamics block diagram showing the various feedback paths.
region. When flow increases beyond around 30% of full flow, control rods are used to increase power. Subsequently, recirculation flow changes are used to induce reactivity changes and, consequently, power changes. Explanation of the power flow map for flows above around 30% conveniently begins with consideration of the required control rod reactivity needed to achieve some desired power at 100% flow. For example, achieving 100% power at 100% flow requires a specific amount of control rod reactivity. If the control rod reactivity remains constant while flow decreases, the power follows a fixed trajectory down to the flow control minimum (around 30%). This trajectory is called the 100% line. The same logic applies for other lines. For example, there is a specific required control rod reactivity needed to achieve 50% power at 100% flow. This trajectory is called the 50% line, and the power again follows a fixed trajectory down to the flow control minimum. Fig. 13.10 shows a typical BWR power-flow map. This map is for the Advanced Boiling Water Reactor. Similar maps apply for other BWR designs. The map shows the strategy for using control rods and core flow to achieve specific reactor power levels. The shaded area is the region to be avoided because of instability problems for the indicated range of core flows and power levels. The startup trajectory provides a useful means to explain the power-flow map. First consider a hypothetical reactor startup for a BWR with a power flow map as shown in Fig. 13.10. Startup involves achieving criticality by withdrawal of control rods while natural circulation provides core flow (slightly above 30% of full flow). Then control rods are withdrawn and core flow is increased to around 40% using recirculation pumps until the reactor power reaches around 65% of full power. Subsequently, core flow is used to induce positive reactivity and power increases.
13.10 On-line stability monitoring
130 NINE OF TEN INTERNAL PUMPS OPERATING
120
PERCENT PUMP SPEED 0 0 NATURAL CIRCULATION 1 30 2 35 3 40 100% POWER = 3926 MWt 4 50 5 60 100% FLOW = 52.2 X 106 kg/h 6 70
110 100
PERCENT POWER
90
7 80 100% SPEED = 157 rad/s 8 90 9 95 10 100 3 2 PERCENT ROD LINE 1 A 102 B 100 C 80 0 D 60 E 40 F 20 REGION III
80 70 60 50 40
8
9
10
7 6 5 4 A B C
REGION IV
D
E
30
REGION II
REGION I
20
F
TYPICAL STARTUP PATH
10
STEAM SEPARATOR LIMIT
0 0
10
20
30
40
50 60 70 PERCENT CORE FLOW
80
90
100
110
120
FIG. 13.10 A BWR power-flow map. Courtesy of GE Hitachi Nuclear Energy Americas LLC (ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997).
This scenario involves the use of control rods to reach the 100% line before switching to recirculation flow changes to increase power. Now consider an alternate scenario. In this case use control rods to reach the 50% line. Then increase core flow to some power level at or below 50%. Then withdraw control rods until the reactivity increase corresponds to the 100% line. The power flow map for BWRs is roughly analogous to the steady state program for PWRs. The BWR situation is more complicated because of the need to avoid instability and because two reactivity control measures are available in BWRs. The BWR power flow map indicates a range of acceptable conditions while a PWR steady state program indicates desired conditions.
13.10 On-line stability monitoring Stability in an operating BWR may be monitored by analyzing the natural fluctuations in measured signals. This process is usually called reactor noise analysis. Two analysis methods are available: spectrum analysis and time-series analysis. Spectrum analysis uses Fourier transforms of the measured fluctuations to provide the power spectrum (signal energy vs. frequency). Time series analysis involves the estimation using the following model, called an auto-regression (AR) model, from the measured data.
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xðtÞ ¼
n X
ai xðt iΔtÞ + vðtÞ
(13.5)
i¼1
where x(t) is a stationary random signal (measured by neutron detectors). v(t) is a model prediction error. {ai, i ¼ 1, 2, …, n} is a set of model parameters. Δt ¼ data sampling interval (sec). n ¼ model order. The model parameters {ai, i ¼ 1, 2, …, n}and the AR model order n are estimated using the measurements such that the model prediction error is minimized. The least-squares approach uses the given sampled measurements {x(1), x(2), …, x(N)}, where N is the total data points. More general forms of the AR model, called an auto-regression moving average (ARMA) model, are used in some applications. See Ref. [11] for details. An example illustrates the evaluation of a time series model. Consider the form of Eq. (13.5) for a second-order fit. The analysis begins with the third measured value of x(t). xð3Þ ¼ a1 xð2Þ + a2 xð1Þ + vð3Þ xð4Þ ¼ a1 xð3Þ + a2 xð2Þ + vð4Þ xð5Þ ¼ a1 xð4Þ + a2 xð3Þ + vð5Þ ⋮ xðN Þ ¼ a1 xðN 1Þ + a2 xðN 2Þ + vðN Þ
(13.6)
Note that a1 and a2 are unknown parameters whose values are sought by using all the available measurements {x(1), x(2), …, x(N)}. An efficient approach for estimating the parameters is to minimize the error between the left hand side and the right hand side (also called the model prediction error by minimizing a squared error function shown below, with respect to (a1, a2): Min J ¼
N X
ðxðkÞ a1 xðk 1Þ a2 xðk 2ÞÞ2 ða1, a2Þ
(13.7)
k¼3
The two parameters are estimated by solving the two equations obtained from ∂J ∂J ¼ 0 and ¼0 ∂a1 ∂a2
(13.8)
The two equations are then simplified by collecting the terms multiplying a1 and a2 and solving for the 2-dimensional vector (a1, a2) to give the following solution: 31 2 N 3 N X X xðk 1Þxðk 2Þ 7 6 xðkÞxðk 1Þ 7 6 7 6 k¼3 7 6 a1 k¼3 k¼3 7 6 7 ¼6 7 7 6 6 N N N X a2 5 4X 5 4X 2 xðk 1Þxðk 2Þ xðk 2Þ xðkÞxðk 2Þ 2
N X xðk 1Þ2
k¼3
k¼3
k¼3
(13.9)
13.11 Power maneuvering
As the number of data points, N, increases, the estimates of (a1, a2) converge to the actual values with least error. Once the AR coefficients are determined, the resulting model can be used to compute the impulse response of the system. The above discussion shows the basic idea of time series modeling from observed data. Recursive parameter estimation techniques are available to compute the (n + 1)-th order model from the n-th order model; these do not require the inversion of large matrices. These are computationally fast and more accurate than methods using direct matrix inversion [12]. An AR analysis of neutron power fluctuations using average power range monitor (APRM) detector signals from two operating BWRs provides the powerto-reactivity impulse response. The developed modeled may be used directly to compute this impulse response in a time recursive fashion. The impulse response may then be used to estimate a decay ratio of the neutron power response to a change in the reactivity. The decay ratio is defined as the ratio between successive positive peaks or successive negative peaks calculated from the impulse response function. For stable reactor operation the decay ratio must be less than 1, and must be less than a value specified by the regulatory agency. An increased power-to-flow ratio indicates a system with a smaller stability margin. A case study [10] provides a stochastic time series model of a measured neutron signal. The developed model was then used to generate the response to an impulse change in the reactivity as its input. Fig. 13.11 shows impulse response results for the case study. Data from two BWRs operating at different power-to-flow ratios were processed using the AR model. The upper plot in Fig. 13.11 shows the power-to-reactivity impulse response of a BWR-4 plant operating at 100% power and 100% recirculation flow rate. The calculated decay ratio is, DR ¼ 0.024. The lower plot in Fig. 13.11 shows the impulse response of a BWR-4 plant operating at 100% power and 65% recirculation flow rate (this is a test case). The calculated decay ratio is, DR ¼ 0.37. The decay ratio of the impulse response function is less than one, and thus both the systems are stable. As the power-to-flow ratio increases this stability margin decreases, indicating a change in the reactor operating characteristic. The flow-topower ratio for the two operating cases are 100% and 65%, respectively. Both BWRs are rated around 1100 MWe. This method of stability monitoring is recommended as a criterion applied to operating reactors by the U.S. Nuclear Regulatory Commission [13].
13.11 Power maneuvering The scenario following opening of the main steam valve in an uncontrolled BWR is as follows: • •
Main steam valve opening " Steam flow to turbine "
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CHAPTER 13 Boiling water reactors
2 DECAY RATIO,DR = 0.024 DAMPING COEFFICIENT,
IMPULSE RESPONSE
x = 0.51
1
0
–1 0
2
4
6
4
6
TIME (S)
4 DECAY RATIO,DR = 0.37 DAMPING COEFFICIENT,
IMPULSE RESPONSE
186
x = 0.16
2
0
–2
0
2 TIME (S)
FIG. 13.11 Impulse response to reactivity perturbation of two BWR-4 systems at different power-to-flow operations. The estimated decay ratios are: 0.024 (top), 0.37 (bottom).
• • • • • •
Turbine power " Steam pressure # Moderator/coolant boiling " In-core voids " Reactivity # Reactor power #
This scenario shows that the inherent initial response in a BWR is a reduction in reactor power in response to an increase in steam flow. The control engineer’s job is to overcome this behavior through appropriate control action.
13.13 BWR safety
13.12 BWR control strategy We have seen that a BWR with no control action responds initially in the wrong direction following an increase in steam flow. So, the basic idea in BWR control is to increase reactor power before releasing more steam to the turbine following an increase in demand. A reactor operated in this way is called a “turbine following boiler”. That is, following a power demand maneuver, the reactor power is adjusted first. The turbine waits until the reactor changes power level before experiencing a change in steam flow. The main control systems in a BWR are the reactor power controller, the feedwater controller, and the pressure controller. The reactor controller uses control rod motion and core flow adjustment to control reactivity. As shown above, the choice of control action (control rods or core flow) depends on reactor condition. The power-flow map provides information on allowable flows at all power levels. The feedwater controller is a so-called three element controller. Measurements provide the downcomer level, the feedwater flow rate and the steam flow rate. Feedwater flow rate is adjusted to eliminate a deviation in level from its set point and to eliminate a mismatch between feedwater flow rate and steam flow rate. This type of control is necessary because shrink and swell occur in BWRs (just like in U-tube steam generators). The pressure controller adjusts the steam valve to maintain constant steam pressure. Called the electro-hydraulic controller, it modulates the steam valve to achieve constant pressure. Consider the response to an increase in power demand. The first action is an increase in core flow. This increases reactivity, power level and steam production. The resulting pressure increase causes the pressure controller to open the steam valve and consequently provide the steam flow needed to satisfy the increase in power demand. The feedwater flow is regulated to maintain the downcomer level at a set point. This scenario illustrates the “turbine following boiler” approach used in BWRs.
13.13 BWR safety Like PWRs, generation II plants require emergency power supplies to provide coolant water pumping in the event of an accident. As shown in Section 11.5.3 emergency power was lost due to a tsunami at the Fukushima Dai-ichi power plant in Japan. This led to catastrophic failures, but it was a result of placing emergency power facilities in a vulnerable location rather than a failure of BWR safety philosophy. Changes at other generation II BWRs provides increased security of emergency power supplies. New BWR designs eliminate the need for electrically driven emergency coolant pumps by using gravity feed and flow from pressurized tanks.
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13.14 Advantages and disadvantages BWRs are sometimes touted as being simpler than PWRs (fewer components, including no steam generators or pressurizers), operation at lower pressure, and being well-suited for power maneuvers. The main disadvantage is the need to operate in a way that avoids instability. Procedures are in place to deal with this problem, but they certainly represent a departure from simplicity. Also, because steam entering the turbine contains radioisotopes produced in the core (mainly nitrogen 16), the turbine becomes radioactive and maintenance and repairs are affected, but the radiation drops rapidly after reactor shutdown. Since N-16 has a half-life of 7.1 s, it decays quickly, permitting safe access to the turbine for maintenance. BWRs and PWRs are competitors around the world. Since both are in operation and being built, the relative advantages and disadvantages of these two types must be viewed as comparable.
Exercises 13.1. Explain why Fig. 13.7 for the feedback phase shift indicates that large feedback gains can cause instability. 13.2. Compare the frequency response plots for PWRs and BWRs and discuss the implications of any major differences.
References [1] GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor. [2] R.T. Lahey Jr., F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, American Nuclear Society, LaGrange Park, 1977. [3] NRC, General electric advanced technology manual, Chapter 4.3, Power Oscillations, U.S. NRC, n.d. https://www.nrc.gov/docs/ML1414/ML14140A074.pdf. [4] J.A. March-Leuba, Dynamic Behavior of Boiling Water Reactors, Doctoral Dissertation, The University of Tennessee, Knoxville, 1984. available at: http://trace.tennessee.edu/ utk_graddiss/1655. [5] P.J. Otaduy, Modeling of the Dynamic Behavior of Large Boiling Water Reactor Cores, PhD Dissertation, University of Florida, 1979. [6] J.A. March-Leuba, Density-wave instabilities in boiling water reactors, Published as Oak Ridge National Laboratory Report ORNL/TM-12130 and as U.S. Nuclear Regulatory Commission report NUREG/CR-6003, October, 1992. [7] C. Kao, A Boiling Water Reactor Simulator for Stability Analysis, PhD dissertation, The Massachusetts Institute of Technology, 1996. February. [8] R. Hu, Stability Analysis of the Boiling Water Reactor: Methods and Advanced Designs, Doctoral dissertation. MIT, 2010. June. [9] J. March-Leuba, A reduced-order model of boiling water reactor linear dynamics, Nucl. Technol. 75 (1986) 15–22.
Further reading
[10] B.R. Upadhyaya, M. Kitamura, Stability monitoring of boiling water reactors by time series analysis of neutron noise, Nucl. Sci. Eng. 77 (1981) 480–492. [11] B.R. Upadhyaya, T.W. Kerlin, Estimation of response time characteristics of platinum resistance thermometers by the noise analysis technique, ISA Trans. 17 (1978) 21–38. [12] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1970. [13] U.S. Nuclear Regulatory Commission Standard Review Plan, Boiling Water Reactor Stability, NUREG-0800, March, 2007.
Further reading [14] International Atomic Energy Agency, Boiling Water Reactor Simulator Training Course Series No. 23, available at, www.pub.iaea.org/MTCD/publications/PDF/TCS-23_web. pdf. [15] General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ML1414/ ML14140A181.pdf. [16] ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997.
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13
13.1 Introduction Numerous Boiling Water Reactors (BWRs) are operated in the U.S. and other countries. About 30% of the commercial nuclear reactors in the U.S. are BWRs. Several different generations of BWRs have been built or planned. This chapter addresses the important common features and their influence on the dynamic characteristics of BWRs.
13.2 History of BWR design evolution Unlike PWR development, BWRs have undergone an evolution of designs with significant changes along the way. Two experimental boiling water reactors were built at Argonne National Laboratory to test the viability of this reactor type (Borax-1 in 1953 and EBWR, the Experimental Boiling Water Reactor in 1956). General Electric (GE) entered BWR development with construction of the Vallecitos prototype BWR in 1957. GE then embarked on design and construction of commercial BWR power plants. As of this writing, seven designs of commercial power plants by General Electric, some with significant changes from its predecessors, have been built. These are designated as BWR-1 through BWR-6 and ABWR (Advanced Boiling Water Reactor). A new BWR (the ESBWR or Economic Simplified BWR) has been designed. The main differences in the various designs are containment features, forced circulation vs. natural circulation, and in-vessel jet pumps with flow driven by external pumps vs. integral mechanical pumps. The evolution of GE power plants is described below:
13.2.1 BWR-1 These were early low-power BWRs (all less than 200 MWe). Three versions were built and had different design features, but all are designated as BWR-1. Differences in BWR-1 designs includes direct cycle (reactor steam goes directly to the turbine) or indirect cycle (reactor steam feeds a separate steam generator) and natural circulation of water flow into the core region or forced circulation. Dynamics and Control of Nuclear Reactors. https://doi.org/10.1016/B978-0-12-815261-4.00013-5 # 2019 Elsevier Inc. All rights reserved.
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13.2.2 BWR-2 BWR-2 produces greater power levels (greater than 500 MWe) than BWR-1. It uses mechanical recirculation pumps and Mark I containment (See below). BWR-2 and four subsequent designs (BWR-3 through BWR-6)) are all considered to be Generation II reactors.
13.2.3 BWR-3 BWR-3 produces greater power levels (800 MWe) than BWR-2 and was the first BWR to use jet pumps for recirculation flow. It uses Mark I containment.
13.2.4 BWR-4 BWR-4 is similar to BWR-3, but it operates at higher power (1100 MWe). BWR-4 reactors use either Mark I or Mark II containment.
13.2.5 BWR-5 BWR-5 is similar to BWR-4. It operates at the same power level as BWR-4 power (1100 MWe). BWR-5 reactors use either Mark I or Mark II containment.
13.2.6 BWR-6 BWR- 6 is available in different configurations having power levels of 600– 1400 MWe. BWR-6 uses Mark III containment.
13.2.7 ABWR ABWR is a Generation III reactor. The ABWR employs internal mechanical recirculation pumps. It uses Mark III containment and the power level is 1500 MWe.
13.3 Characteristics of BWRs 13.3.1 General features of a BWR Since there are five different Generation II BWRs, it is necessary to pick one for providing an overview of BWR characteristics. BWR-6 was chosen, but its dynamic behavior and control strategy is typical of all Generation II BWRs. The main difference is that it operates at a higher power level than earlier designs. Fig. 13.1 shows a typical BWR-6 system. Fig. 13.2 shows a BWR-6 reactor vessel and internals [1]. Subcooled water enters the bottom of the core. The flow rate and pressure are such that boiling begins near the entrance. Boiling continues along the rest of the passage through the core. A steam-water mixture exits the core region. This mixture
13.3 Characteristics of BWRs
Containment Cooling System
Steam Line Reactor Vessel
Turbine Generator Separators & Dryers
Heater Condenser
Feedwater 3
Condensate Pumps
Core 1&2
Feed Pumps
Control Rods
Demineralizer
Recirculation Pumps Emergency Water Supply Systems
FIG. 13.1 Schematic of a typical boiling water reactor system. U.S. Nuclear Regulatory Commission: www.nrc.gov/reactors/bwrs.html.
then passes through steam separators and steam driers located above the core. These systems remove water by centrifugal force and by sudden reversals in flow direction. The removed water flows downward into an annular region between the vessel and a core shroud. This annular region is called the downcomer. Fig. 13.3 shows a BWR-6 fuel bundle. The fuel is Uranium oxide contained in Zircaloy tubes. A typical 1220 MWe BWR-6 core consists of about 750 fuel assemblies. Each assembly is enclosed in a fuel box with an 8 8 or a 10 10 array of fuel rods (pins). The fuel pins are similar to those in a PWR, and have an active length of 12 ft. The fuel ‘box’ constrains the flow of coolant in the assembly. Upper and lower tie plates provide structural support for the assembly, along with a few tie rods. Some spaces in the fuel assembly are taken by water rods to provide additional neutron moderation. A typical BWR contains 60–70 thousand fuel rods and
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STEAM DRYER LIFTING LUG VENT AND HEAD SPRAY
STEAM DRYER ASSEMBLY STEAM OUTLET
STEAM SEPARATOR ASSEMBLY
CORE SPRAY INLET
FEEDWATER INLET FEEDWATER SPARGER
LOW PRESSURE COOLANT INJECTION INLET
CORE SPRAY LINE
CORE SPRAY SPARGER TOP GUIDE JET PUMP ASSEMBLY CORE SHROUD
FUEL ASSEMBLIES
CONTROL BLADE
CORE PLATE JET PUMP/RECIRCULATION WATER INLET
VESSEL SUPPORT SKIRT
RECIRCULATION WATER OUTLET
SHIELD WALL
CONTROL ROD DRIVES
CONTROL ROD DRIVE HYDRAULIC LINES IN-CORE FLUX MONITOR
FIG. 13.2 BWR vessel and reactor internals. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
160 metric tons (160,000 Kg) of UO2. Design parameters of a typical BWR-6 are given in Appendix A. Fig. 13.3 also shows a BWR-6 control assembly. It is a cruciform-shaped structure that enters the core from below and is inserted in the space between four fuel assemblies. The below-core location is necessary since the region above the core
13.3 Characteristics of BWRs
FOUR-BUNDLE FUEL MODULE FUEL ROD WATER RODS Care Lattice
TIE RODS
FIG. 13.3 Four fuel assemblies (boxed channels) showing the control rod cruciform shaped assembly in the center of fuel four fuel assemblies. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
contains steam separators and driers. The location of control rod assemblies means that gravity-based insertion is impossible. Insertion from below places the control rod assembles enter through the subcooled region where control rod worth is greatest and facilitates refueling from above. The control assemblies contain boron carbide and are used for reactivity control and power flattening. Burnable poison (oxide of gadolinium – gadolinia) is mixed with the fuel for shim control.
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13.3.2 Recirculation flow and jet pumps The core flow in a BWR-6 is controlled by two recirculation pumps that distribute the water to a set of jet pumps surrounding the core. Each recirculation pump distributes water to one of two manifolds. Each manifold supplies water via pipes to jet pumps. A pair of jet pumps receives water from a single pipe. There is a total of 20 jet pumps, with a typical jet pump overall length of 19 ft. Mechanical pumps (recirculation pumps) withdraw water from the downcomer and pump it at elevated pressure into the jet pumps. The jet pumps have no moving parts, making them maintenance-free reactor components. Fig. 13.4 illustrates the principle of operation of a jet pump [2]. The recirculation flow enters the jet pump nozzle at a high pressure and increases to a high velocity as it flows through the narrow throat, and results in a pressure drop. The suction flow in the downcomer region enters the inlet nozzle at a low pressure. The pressure decreases as the suction flow passes through the converging section of this nozzle. The driving flow and the suction flow mix in the throat region (mixing
FIG. 13.4 Operation of a jet pump. Courtesy of GE Hitachi Nuclear Energy Americas LLC (GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor).
13.4 Reactivity feedbacks in BWRs
section with a constant diameter) resulting in an increase in the fluid pressure due to a change in the fluid velocity in this section. A long diffuser is connected at the end of the mixing section, causing an increase in the fluid pressure that drives the coolant into the lower plenum and then up through the reactor core.
13.3.3 Other features of BWRs The following are typical core parameters for a BWR-6: • • • • • • • • • • •
Total coolant flow rate: 105 106 lbm/h Plant efficiency: 34% Core diameter: 193 in. Number of control rods: 177 Coolant pressure: 1040 psia Core-exit (steam) temperature: 551°F Feed water temperature: 420°F Average coolant exit quality: 15% Vessel diameter: 19 ft. Wall thickness: 5.7 in./6.46 in. Vessel height: 71 ft. Core power density: 54 kW/l.
The steam produced in the core and separated from liquid water passes through a control valve into the turbine. The steam pressure is maintained at a constant value by throttling the steam valve. Exhaust steam passes into a condenser and the condensate passes through a series of feedwater heaters before returning to the reactor vessel. Note that a BWR system shares general features with U-tube steam generators used in most PWRs (a heated riser, steam separators and driers, and a downcomer). A BWR containment consists of a concrete “drywell” that encloses the reactor. If steam escapes from the reactor vessel or related piping it flows into the drywell. The drywell has piping that connects it to a large pool of water called the suppression pool. The suppression pool water condenses the steam and reduces pressure in the drywell. Three different types of suppression pool are used, Mark I, II and III [3]. See Fig. 13.5.
13.4 Reactivity feedbacks in BWRs The BWR fuel temperature coefficient of reactivity is due to the Doppler effect (fuel temperature feedback reactivity) and is always negative. Typically, the Doppler coefficient in BWRs is around 2 105 Δρ/°C. The magnitude of the negative Doppler coefficient increases as fuel temperature increases. BWRs are under-moderated. Increases in moderator/coolant temperature cause increased boiling and reduced in-core liquid water density. Thus, an increase in
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moderator/coolant temperature decreases neutron moderation (a negative reactivity mechanism) and decreases neutron absorptions (a positive reactivity mechanism). The moderation effect is larger, causing a net negative moderator/coolant temperature coefficient of reactivity. Typically, the moderator/coolant temperature coefficient is around 3 104 Δρ/°C. The void coefficient depends strongly on reactor conditions, but it also is always negative. Typically, the void coefficient is around 1.4 103 Δρ/% voids. Pressure also affects reactivity because pressure affects boiling. For example, a pressure increase causes a reduction in core voids. Because the void coefficient is negative, a pressure increase causes a reactivity increase. That is, the pressure coefficient of reactivity is positive. The control strategy in BWRs is to maintain constant pressure by modulating the steam valve.
DRYWELL HEAD
DRYWELL FLANGE
DRYWELL SHEAR LUG SUPPORT
REACTOR PRESSURE VESSEL
DRYWELL SHIELD WALL
CORE
RADIAL BEAM
MANWAY RADIAL BEAM VACUUM BREAKER
JET DEFLECTOR VENT
VENT HEADER
DOWNCOMERS WATER LEVEL
(A)
SUPPRESSION CHAMBER (TORUS)
FIG. 13.5 (A) BWR Mark I containment with toroidal pressure suppression chamber. (continued)
13.4 Reactivity feedbacks in BWRs
The moderator temperature also affects the thermal neutron spectrum. Increases in moderator temperature cause hardening of the thermal neutron spectrum. This causes a negative component of the reactivity change due to changes in U-235 absorptions and a positive component due to changes in Pu-239 absorptions. See Section 7.3. As in all thermal spectrum power reactors, burnup and production of Xe-135 causes a reactivity feedback effect. See Section 6.2. The power coefficient for BWRs is negative, thereby ensuring that reactor power achieves a new steady state level following a change in external reactivity.
DRYWELL HEAD
DRYWELL
REACTOR VESSEL
SACRIFICIAL SHIELD WALL STEEL LINER REACTOR PEDESTAL
DRYWELL DECK
S/R VALVE TAILPIPE (18) EQUIPMENT HANDLING PLATFORM DOWNCOMER (VENT)
VACUUM BREAKERS (5) SUPPORT COLUMN (12)
PRESSURE SUPPRESSION CHAMBER
WATER LEVEL
QUENCHER (18) REINFORCED CONCRETE
(B) FIG. 13.5, cont’d (B) BWR Mark II containment chamber. (continued)
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CONTAINMENT SPRAY SHIELD BUILDING 125 TON CRANE W/15 TON AUX HOOK
CONTAINMENT UPPER POOL
DRYWELL HEAD FUEL TRANSFER POOL
REACTOR VESSEL REACTOR SHIELD DRYWELL BOUNDRY
WEIR WALL S/R VALVE LINE
DRYWELL FUEL TRANSFER TUBE
SUPRESSION POOL HORIZONTAL VENT
(C) FIG. 13.5, cont’d (C) BWR Mark III containment. Courtesy of GE Hitachi Nuclear Energy Americas LLC (General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ ML1414/ML14140A181.pdf).
13.5 Reactivity and recirculation flow BWRs can use control rods to change reactivity, but an alternate way is to change recirculation flow rate. Increasing the recirculation flow rate increases the amount of liquid water in the core relative to steam. Since the BWR is under-moderated, this increases reactivity, thereby increasing reactor power and steam production. Thus, BWRs have two ways to change reactivity by external means, whereas PWRs have one (control rod motion). In forced circulation BWRs recirculation pumps are used to draw water from the lower downcomer region and distribute the water to a set of jet pumps at an elevation above the pump suction location. Thus, a BWR is a variable flow system, with the flow modulation facilitating start-up and load-following operations. Two recirculating pumps distribute water to jet pumps, through a sparger ring. Changing the pumping power (hence, the coolant flow rate) causes a change in reactivity through a change in core voids. See Section 13.3.2 for a description of jet pump operation.
13.7 BWR dynamic models
13.6 Total reactivity balance For steady state, zero reactivity is required. The total reactivity balance is as follows: ρ ¼ Control rod reactivity + Recirculation flow reactivity + Feedback reactivity ¼ 0
The externally-controlled reactivity may be achieved by a combination of control rod reactivity and recirculation flow reactivity. Thus, a desired reactivity setting for either externally-controlled reactivity can be achieved by adjusting the other externally-controlled reactivity.
13.7 BWR dynamic models Detailed BWR dynamic models include treatment of all of the complex neutronic and thermal-hydraulic effects that contribute to the dynamics of the system. Both linear and nonlinear models exist. Detailed models are too complex for inclusion here. Interested readers can find information in the literature [4, 5]. Linear models provide estimates of the small-perturbation time response and frequency response. An approximate, low-order model provides simple simulation capability. It accounts for the neutronic and thermal-hydraulic processes that determine feedback reactivity. A low-order model [4] was developed by fitting a loworder transfer function to match the closed-loop frequency response calculated with a detailed model [5]. The results obtained with the low-order model are essentially identical with results from the detailed model. Note that the author of Ref. [4] chose to express the frequency in Hz rather than in rad/s as used elsewhere in this book. Figures in this chapter use rad/s for frequency units. The low-order closed loop transfer function used in the fit [4] is Gc ðsÞ ¼
K ðs2 + as + bÞ ðs + cÞ + ds + eÞ ðs + f Þ ðs + gÞ
ðs 2
(13.1)
The following values of the low-order model parameters are typical for BWR simulation [4]: K ¼ a gain that varies with power level a ¼ 0.36 b ¼ 0.1055 c ¼ 0.03 d ¼ 0.09 e ¼ 0.1044 f ¼ 0.25 g ¼ 21.0 The low-order model from Ref. [4] provides the capability to further investigate BWR dynamics and the cause for potential stability issues. The low-order model
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of Ref. [4] was modified to provide the amplitude in units of % power/cent of reactivity as used throughout this book. Fig. 13.6 shows the power-to-reactivity frequency response obtained with the low-order model. Note the resonance at around 0.3 rad/s. The resonance grows if changes in conditions cause the system to move toward instability. Recall that the closed loop transfer function for a system with feedback is given by Gc ¼
Go 1 + Go H
(A)
(B) FIG. 13.6 (A) BWR frequency response amplitude, (B) BWR frequency response phase.
(13.2)
13.8 BWR stability problem and impact on control
where Gc ¼ closed-loop transfer function (feedback effects included). Go ¼ open-loop transfer function (the zero-power transfer function). H ¼ feedback (cents/% power). At low frequencies, the magnitude of Go is large. Consequently, Gc ¼ 1/H at low frequencies. The feedback frequency response can be calculated using H¼
1 1 Gc Go
(13.3)
The resulting feedback frequency response appears in Fig. 13.7. Note that the phase lag is more than 90 deg. at frequencies above around 0.1 rad/s. As shown in Section 3.8 such phase shifts in system feedback can cause instability if the feedback gain is large enough. The frequency response for various feedback conditions can be deduced by applying a multiplicative factor, K to the feedback term (GoH in Eq. (13.2)). Gc ¼
Go 1 + K Go H
(13.4)
Note that K ¼ 1 for the original low-order model. Fig. 13.8 shows the closed-loop gain for various values of K. Clearly the resonance at around 0.3 rad/s grows as K increases and shifts to higher frequencies. Ref. [4] shows that the system becomes unstable at a value of K 2.25. The above discussion reveals that useful insights can be deduced if basic principles of dynamic analysis of feedback systems are understood and employed.
13.8 BWR stability problem and impact on control BWRs exhibit instability at conditions of high power and low recirculation flow. This instability is caused by a complex coupling of neutronics and thermalhydraulics. The basic cause of BWR instability is time lagged flow and reactivity feedbacks. Recall that positive feedbacks usually cause stability problems, but negative feedbacks also can cause instability if their effect is delayed (and the feedback gain is large enough). See Section 3.8. Subcooled water enters a BWR channel at the bottom. As it flows upward, boiling occurs and the void fraction increases. A disturbance (typically inlet flow change, inlet subcooling change, or power change) causes a localized change in steam bubble concentration in the lower portion of the channel. The propagation of this bubble packet as it travels up the channel is called a density wave [2]. This density wave causes changes in the local pressure drops as it propagates. Consider a disturbance that causes an increase in steam bubble formation in the lower portion of the channel. The resulting density wave travels upward, and it travels faster than the flow prior to the disturbance. As a result, the total channel pressure drop
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(A)
(B) FIG. 13.7 (A) Feedback frequency response amplitude. (B) Feedback frequency response phase shift.
increases, but it is delayed relative to the input disturbance. If the pressure drop is large and it lags the input disturbance by 180 degrees, it destabilizes the channel. The density wave also affects reactor power by the void feedback effect on reactivity. The local reactivity and reactor power change follow the change in void density, but the heat transfer from the fuel to the fluid lags because of the fuel’s heat transfer lag. The fuel heat transfer time constant in a BWR is typically 6–10 s. So, the neutronic response is also lagged, thereby introducing its additional component of the lagged response. The coupled thermal-hydraulic and neutronic response can cause instability, especially at low flow and high power conditions. At low flow and high power, resulting lagged thermal-hydraulic and neutronic feedbacks have values that induce instability.
13.9 The power flow map and startup
FIG. 13.8 Closed-loop frequency response amplitude for various values of K.
Fig. 13.9 shows the feedback paths that determine stability. Instabilities of different nature can occur. These include single channel instability, in-phase core-wide instability (the whole core responds in unison) and out-ofphase core instability (different regions respond out-of-phase with one another). The processes involved in coupled thermal-hydraulic and neutronics are very complex and not amenable to a simple analysis. Very detailed computer codes are necessary and several analysis codes have been developed. Many publications have addressed the BWR stability problem, its analysis and its mitigation. The magnitude of the effort to deal with BWR instability illustrates the importance of the problem. Details may be found in the literature (see Refs. [2, 4, 6–10]). The typical conditions for instability are power levels of 35–60% and core flow rates of 30–45%. The strategy for avoiding instability is to avoid operation in the range of reactor power levels and core flow rates where instability occurs. Specifically, the reactor power is kept below the instability threshold at low core flows by using control rods to adjust power.
13.9 The power flow map and startup A reactor with a negative power coefficient (such as a BWR) experiences a specific new steady state power following a reactivity change (see Chapter 7). Since BWR externally controlled reactivity depends on control rod positions and recirculation flow there are many different paths to change reactor power. Control rods are used for reactivity adjustment to keep power low at low flows to avoid the instability
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Reactivity
NEUTRONICS
Doppler
Power
Fuel temp
FUEL
Void reactivity CORE T-H
Direct heat
Inlet enthalpy Outlet pressure
RECIRCULATION LOOP
Inlet flow
FIG. 13.9 BWR dynamics block diagram showing the various feedback paths.
region. When flow increases beyond around 30% of full flow, control rods are used to increase power. Subsequently, recirculation flow changes are used to induce reactivity changes and, consequently, power changes. Explanation of the power flow map for flows above around 30% conveniently begins with consideration of the required control rod reactivity needed to achieve some desired power at 100% flow. For example, achieving 100% power at 100% flow requires a specific amount of control rod reactivity. If the control rod reactivity remains constant while flow decreases, the power follows a fixed trajectory down to the flow control minimum (around 30%). This trajectory is called the 100% line. The same logic applies for other lines. For example, there is a specific required control rod reactivity needed to achieve 50% power at 100% flow. This trajectory is called the 50% line, and the power again follows a fixed trajectory down to the flow control minimum. Fig. 13.10 shows a typical BWR power-flow map. This map is for the Advanced Boiling Water Reactor. Similar maps apply for other BWR designs. The map shows the strategy for using control rods and core flow to achieve specific reactor power levels. The shaded area is the region to be avoided because of instability problems for the indicated range of core flows and power levels. The startup trajectory provides a useful means to explain the power-flow map. First consider a hypothetical reactor startup for a BWR with a power flow map as shown in Fig. 13.10. Startup involves achieving criticality by withdrawal of control rods while natural circulation provides core flow (slightly above 30% of full flow). Then control rods are withdrawn and core flow is increased to around 40% using recirculation pumps until the reactor power reaches around 65% of full power. Subsequently, core flow is used to induce positive reactivity and power increases.
13.10 On-line stability monitoring
130 NINE OF TEN INTERNAL PUMPS OPERATING
120
PERCENT PUMP SPEED 0 0 NATURAL CIRCULATION 1 30 2 35 3 40 100% POWER = 3926 MWt 4 50 5 60 100% FLOW = 52.2 X 106 kg/h 6 70
110 100
PERCENT POWER
90
7 80 100% SPEED = 157 rad/s 8 90 9 95 10 100 3 2 PERCENT ROD LINE 1 A 102 B 100 C 80 0 D 60 E 40 F 20 REGION III
80 70 60 50 40
8
9
10
7 6 5 4 A B C
REGION IV
D
E
30
REGION II
REGION I
20
F
TYPICAL STARTUP PATH
10
STEAM SEPARATOR LIMIT
0 0
10
20
30
40
50 60 70 PERCENT CORE FLOW
80
90
100
110
120
FIG. 13.10 A BWR power-flow map. Courtesy of GE Hitachi Nuclear Energy Americas LLC (ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997).
This scenario involves the use of control rods to reach the 100% line before switching to recirculation flow changes to increase power. Now consider an alternate scenario. In this case use control rods to reach the 50% line. Then increase core flow to some power level at or below 50%. Then withdraw control rods until the reactivity increase corresponds to the 100% line. The power flow map for BWRs is roughly analogous to the steady state program for PWRs. The BWR situation is more complicated because of the need to avoid instability and because two reactivity control measures are available in BWRs. The BWR power flow map indicates a range of acceptable conditions while a PWR steady state program indicates desired conditions.
13.10 On-line stability monitoring Stability in an operating BWR may be monitored by analyzing the natural fluctuations in measured signals. This process is usually called reactor noise analysis. Two analysis methods are available: spectrum analysis and time-series analysis. Spectrum analysis uses Fourier transforms of the measured fluctuations to provide the power spectrum (signal energy vs. frequency). Time series analysis involves the estimation using the following model, called an auto-regression (AR) model, from the measured data.
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xðtÞ ¼
n X
ai xðt iΔtÞ + vðtÞ
(13.5)
i¼1
where x(t) is a stationary random signal (measured by neutron detectors). v(t) is a model prediction error. {ai, i ¼ 1, 2, …, n} is a set of model parameters. Δt ¼ data sampling interval (sec). n ¼ model order. The model parameters {ai, i ¼ 1, 2, …, n}and the AR model order n are estimated using the measurements such that the model prediction error is minimized. The least-squares approach uses the given sampled measurements {x(1), x(2), …, x(N)}, where N is the total data points. More general forms of the AR model, called an auto-regression moving average (ARMA) model, are used in some applications. See Ref. [11] for details. An example illustrates the evaluation of a time series model. Consider the form of Eq. (13.5) for a second-order fit. The analysis begins with the third measured value of x(t). xð3Þ ¼ a1 xð2Þ + a2 xð1Þ + vð3Þ xð4Þ ¼ a1 xð3Þ + a2 xð2Þ + vð4Þ xð5Þ ¼ a1 xð4Þ + a2 xð3Þ + vð5Þ ⋮ xðN Þ ¼ a1 xðN 1Þ + a2 xðN 2Þ + vðN Þ
(13.6)
Note that a1 and a2 are unknown parameters whose values are sought by using all the available measurements {x(1), x(2), …, x(N)}. An efficient approach for estimating the parameters is to minimize the error between the left hand side and the right hand side (also called the model prediction error by minimizing a squared error function shown below, with respect to (a1, a2): Min J ¼
N X
ðxðkÞ a1 xðk 1Þ a2 xðk 2ÞÞ2 ða1, a2Þ
(13.7)
k¼3
The two parameters are estimated by solving the two equations obtained from ∂J ∂J ¼ 0 and ¼0 ∂a1 ∂a2
(13.8)
The two equations are then simplified by collecting the terms multiplying a1 and a2 and solving for the 2-dimensional vector (a1, a2) to give the following solution: 31 2 N 3 N X X xðk 1Þxðk 2Þ 7 6 xðkÞxðk 1Þ 7 6 7 6 k¼3 7 6 a1 k¼3 k¼3 7 6 7 ¼6 7 7 6 6 N N N X a2 5 4X 5 4X 2 xðk 1Þxðk 2Þ xðk 2Þ xðkÞxðk 2Þ 2
N X xðk 1Þ2
k¼3
k¼3
k¼3
(13.9)
13.11 Power maneuvering
As the number of data points, N, increases, the estimates of (a1, a2) converge to the actual values with least error. Once the AR coefficients are determined, the resulting model can be used to compute the impulse response of the system. The above discussion shows the basic idea of time series modeling from observed data. Recursive parameter estimation techniques are available to compute the (n + 1)-th order model from the n-th order model; these do not require the inversion of large matrices. These are computationally fast and more accurate than methods using direct matrix inversion [12]. An AR analysis of neutron power fluctuations using average power range monitor (APRM) detector signals from two operating BWRs provides the powerto-reactivity impulse response. The developed modeled may be used directly to compute this impulse response in a time recursive fashion. The impulse response may then be used to estimate a decay ratio of the neutron power response to a change in the reactivity. The decay ratio is defined as the ratio between successive positive peaks or successive negative peaks calculated from the impulse response function. For stable reactor operation the decay ratio must be less than 1, and must be less than a value specified by the regulatory agency. An increased power-to-flow ratio indicates a system with a smaller stability margin. A case study [10] provides a stochastic time series model of a measured neutron signal. The developed model was then used to generate the response to an impulse change in the reactivity as its input. Fig. 13.11 shows impulse response results for the case study. Data from two BWRs operating at different power-to-flow ratios were processed using the AR model. The upper plot in Fig. 13.11 shows the power-to-reactivity impulse response of a BWR-4 plant operating at 100% power and 100% recirculation flow rate. The calculated decay ratio is, DR ¼ 0.024. The lower plot in Fig. 13.11 shows the impulse response of a BWR-4 plant operating at 100% power and 65% recirculation flow rate (this is a test case). The calculated decay ratio is, DR ¼ 0.37. The decay ratio of the impulse response function is less than one, and thus both the systems are stable. As the power-to-flow ratio increases this stability margin decreases, indicating a change in the reactor operating characteristic. The flow-topower ratio for the two operating cases are 100% and 65%, respectively. Both BWRs are rated around 1100 MWe. This method of stability monitoring is recommended as a criterion applied to operating reactors by the U.S. Nuclear Regulatory Commission [13].
13.11 Power maneuvering The scenario following opening of the main steam valve in an uncontrolled BWR is as follows: • •
Main steam valve opening " Steam flow to turbine "
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2 DECAY RATIO,DR = 0.024 DAMPING COEFFICIENT,
IMPULSE RESPONSE
x = 0.51
1
0
–1 0
2
4
6
4
6
TIME (S)
4 DECAY RATIO,DR = 0.37 DAMPING COEFFICIENT,
IMPULSE RESPONSE
186
x = 0.16
2
0
–2
0
2 TIME (S)
FIG. 13.11 Impulse response to reactivity perturbation of two BWR-4 systems at different power-to-flow operations. The estimated decay ratios are: 0.024 (top), 0.37 (bottom).
• • • • • •
Turbine power " Steam pressure # Moderator/coolant boiling " In-core voids " Reactivity # Reactor power #
This scenario shows that the inherent initial response in a BWR is a reduction in reactor power in response to an increase in steam flow. The control engineer’s job is to overcome this behavior through appropriate control action.
13.13 BWR safety
13.12 BWR control strategy We have seen that a BWR with no control action responds initially in the wrong direction following an increase in steam flow. So, the basic idea in BWR control is to increase reactor power before releasing more steam to the turbine following an increase in demand. A reactor operated in this way is called a “turbine following boiler”. That is, following a power demand maneuver, the reactor power is adjusted first. The turbine waits until the reactor changes power level before experiencing a change in steam flow. The main control systems in a BWR are the reactor power controller, the feedwater controller, and the pressure controller. The reactor controller uses control rod motion and core flow adjustment to control reactivity. As shown above, the choice of control action (control rods or core flow) depends on reactor condition. The power-flow map provides information on allowable flows at all power levels. The feedwater controller is a so-called three element controller. Measurements provide the downcomer level, the feedwater flow rate and the steam flow rate. Feedwater flow rate is adjusted to eliminate a deviation in level from its set point and to eliminate a mismatch between feedwater flow rate and steam flow rate. This type of control is necessary because shrink and swell occur in BWRs (just like in U-tube steam generators). The pressure controller adjusts the steam valve to maintain constant steam pressure. Called the electro-hydraulic controller, it modulates the steam valve to achieve constant pressure. Consider the response to an increase in power demand. The first action is an increase in core flow. This increases reactivity, power level and steam production. The resulting pressure increase causes the pressure controller to open the steam valve and consequently provide the steam flow needed to satisfy the increase in power demand. The feedwater flow is regulated to maintain the downcomer level at a set point. This scenario illustrates the “turbine following boiler” approach used in BWRs.
13.13 BWR safety Like PWRs, generation II plants require emergency power supplies to provide coolant water pumping in the event of an accident. As shown in Section 11.5.3 emergency power was lost due to a tsunami at the Fukushima Dai-ichi power plant in Japan. This led to catastrophic failures, but it was a result of placing emergency power facilities in a vulnerable location rather than a failure of BWR safety philosophy. Changes at other generation II BWRs provides increased security of emergency power supplies. New BWR designs eliminate the need for electrically driven emergency coolant pumps by using gravity feed and flow from pressurized tanks.
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13.14 Advantages and disadvantages BWRs are sometimes touted as being simpler than PWRs (fewer components, including no steam generators or pressurizers), operation at lower pressure, and being well-suited for power maneuvers. The main disadvantage is the need to operate in a way that avoids instability. Procedures are in place to deal with this problem, but they certainly represent a departure from simplicity. Also, because steam entering the turbine contains radioisotopes produced in the core (mainly nitrogen 16), the turbine becomes radioactive and maintenance and repairs are affected, but the radiation drops rapidly after reactor shutdown. Since N-16 has a half-life of 7.1 s, it decays quickly, permitting safe access to the turbine for maintenance. BWRs and PWRs are competitors around the world. Since both are in operation and being built, the relative advantages and disadvantages of these two types must be viewed as comparable.
Exercises 13.1. Explain why Fig. 13.7 for the feedback phase shift indicates that large feedback gains can cause instability. 13.2. Compare the frequency response plots for PWRs and BWRs and discuss the implications of any major differences.
References [1] GE Nuclear Energy, BWR-6: General Description of a Boiling Water Reactor. [2] R.T. Lahey Jr., F.J. Moody, The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, American Nuclear Society, LaGrange Park, 1977. [3] NRC, General electric advanced technology manual, Chapter 4.3, Power Oscillations, U.S. NRC, n.d. https://www.nrc.gov/docs/ML1414/ML14140A074.pdf. [4] J.A. March-Leuba, Dynamic Behavior of Boiling Water Reactors, Doctoral Dissertation, The University of Tennessee, Knoxville, 1984. available at: http://trace.tennessee.edu/ utk_graddiss/1655. [5] P.J. Otaduy, Modeling of the Dynamic Behavior of Large Boiling Water Reactor Cores, PhD Dissertation, University of Florida, 1979. [6] J.A. March-Leuba, Density-wave instabilities in boiling water reactors, Published as Oak Ridge National Laboratory Report ORNL/TM-12130 and as U.S. Nuclear Regulatory Commission report NUREG/CR-6003, October, 1992. [7] C. Kao, A Boiling Water Reactor Simulator for Stability Analysis, PhD dissertation, The Massachusetts Institute of Technology, 1996. February. [8] R. Hu, Stability Analysis of the Boiling Water Reactor: Methods and Advanced Designs, Doctoral dissertation. MIT, 2010. June. [9] J. March-Leuba, A reduced-order model of boiling water reactor linear dynamics, Nucl. Technol. 75 (1986) 15–22.
Further reading
[10] B.R. Upadhyaya, M. Kitamura, Stability monitoring of boiling water reactors by time series analysis of neutron noise, Nucl. Sci. Eng. 77 (1981) 480–492. [11] B.R. Upadhyaya, T.W. Kerlin, Estimation of response time characteristics of platinum resistance thermometers by the noise analysis technique, ISA Trans. 17 (1978) 21–38. [12] G.E.P. Box, G.M. Jenkins, Time Series Analysis: Forecasting and Control, Holden-Day, San Francisco, 1970. [13] U.S. Nuclear Regulatory Commission Standard Review Plan, Boiling Water Reactor Stability, NUREG-0800, March, 2007.
Further reading [14] International Atomic Energy Agency, Boiling Water Reactor Simulator Training Course Series No. 23, available at, www.pub.iaea.org/MTCD/publications/PDF/TCS-23_web. pdf. [15] General electric advanced technology manual, Chapter 6.2, BWR Primary Containments, U.S. Nuclear Regulatory Commission, https://www.nrc.gov/docs/ML1414/ ML14140A181.pdf. [16] ABWR Design Control Document, prepared by GE Nuclear Energy for the U.S. Nuclear Regulatory Commission, 1997.
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