Core Neutronics Model
The Exitech Core Neutronics model, AMON™ (Advanced Model of Neutronics) is used at several simulators and faithfully reproduces core data, parameters,and transients. Example sections of the AMON handbook are provided here.
Table of Contents
1.0 Introduction
1.1 Core Neutronics Model Nodalization
2.0 Neutron Kinetics Equations
2.1 Amplitude-Function Equations Development
2.2 Solution of the Amplitude-Function
2.3 Shape-Function Equation Development
2.4 Solution of the Shape-Function (Eigenvalue and Eigenfunction)
3.0 Nodal Multiplication Constant with Feedback
4.0 Nodal Multiplication Constant Weighting Factor
5.0 Nodal Power Calculations
5.1 Prompt fission power
5.2 Decay heat
5.3 Energy distribution in the fuel and moderator
6.0 Nodal Flux
7.0 Fission Product Poisons
8.0 Nuclear Instrumentation
9.0 Assumptions and Simplifications
10.0 References
APPENDIX A Neutronics Data Formalism
A1.0 Introduction
A2.0 Nodal Property Dependent Data
A3.0 Core-Averaged Data
A3.1 Boron Reactivity Correction Data
A3.2 Doppler Reactivity Correction
A3.3 Xenon & Samarium Reactivity Corrections
A3.4 Equivalent One-Group Neutron Flux
A3.5 Delayed Neutron Precursor Data
A3.6 Decay Heat Group Data
A3.7 Nuclear Fission-Energy And Neutron Poison Data
APPENDIX B Parameterization And Optimization Data Request List
B1.0 Introduction
B2.0 AMON Parameterization Data
B3.0 AMON Optimization Data
1.0 Introduction
This document provides the engineering equation development for the EXITECH AMON^{TM} (Advanced Model Of Neutronics). This model calculates the neutron flux level and flux distribution within the reactor core as well as the thermal power deposited in the fuel and coolant. The thermal power values are passed to the Thermal-Hydraulics model. Core neutron flux level and distribution are passed to the NIS (Nuclear Instrumentation System) Model which simulates the source range (SRM), intermediate range (IRM) and local power range (LPRM) detectors and instrument channels. The AMON receives fuel temperature, coolant density, coolant void and boron concentration (of importance during ATWS events) as the major interface parameters from the Thermal-Hydraulics model. The AMON also receives control rod position information from the CRDH (Control Rod Drive Hydraulic System) Model.
1.1 Core Neutronics Model Nodalization
Figure 1 and 2 show the 137 (radial) by 8 (axial) nodalization utilized for the AMON^{TM} BWR Core Neutronics Model.
8.0 Nuclear Instrumentation
The core nuclear instrumentation SRM, IRM, and LPRM Detector responses, as well as the core power calibration TIP machine probe signals are calculated dynamically. For each of the nuclear instrumentation detectors, a total of nine (9) signals are calculated, seven (7) dynamically varying averaged-signals inside the active core region plus two (2) fixed-zero-signals at extrapolated distances from the top and bottom of the core. These nine (9) signals are curve fitted by the cubic spline method. The curve fit coefficients for each of the detector channels are used to reconstruct the detector signal at specific detector positions
The SRM/IRM detectors are movable detectors. Each of the seven (7) dynamically varying averaged-signals inside the core are calculated by averaging the eight (8) surrounding AMON^{TM} nodal neutron fluxes (four (4) neighboring AMON^{TM} nodes on each of the two (2) neighboring layers). If the detector resides within the reactor core, the reconstructed flux level signal of the detector is multiplied by a fixed detector efficiency constant to convert the flux signal into a counts per second (cps) signal (for SRM detectors) or a percentage power signal (for IRM detectors). If the detector is located below the reactor core bottom, the flux signal will be calculated by using the core bottom position neutron flux signal and attenuating this signal according to the detector’s distance below the core bottom. This attenuated neutron flux signal will then be multiplied by the same detector efficiency constants to convert the flux signal into cps (for SRM detectors) or percentage power (for IRM detectors).
The LPRM detectors are fixed in the reactor core. In each LPRM string, there are four (4) detectors located at four elevations in the core to measure the core fission power in units of watts per centimeter square (watts/cm^{2}). Seven (7) dynamically varying averaged-unit-area-fission-power values in the vicinity of the LPRM string are calculated by averaging the eight (8) surrounding AMON^{TM} nodal fission powers (four (4) neighboring AMON^{TM} nodes on each of the two (2) neighboring layers). Nine (9) cubic-spline-method curve fit coefficients for each LPRM string are used to reconstruct the four (4) LPRM detector signals at the specific detector elevations.
The signals for the moveable TIP probe for each of the TIP channels (coexisting along with the LPRM strings in the core) are calculated by using the nine (9) cubic spline curve fit coefficients generated in the calculation of the corresponding LPRM string detector signals. The core unit area fission power along the channel direction is calculated by inserting the gradually changing probe position into the cubic spline curve fit reconstruction formula. If the probe has traveled below the bottom of the core, the core bottom position is inserted into the formula to generate a base value and the base value is attenuated according to the TIP detector’s distance below the bottom of the core.
A rod shadow effect is modeled when the tip of a control rod comes within two notches (of the axial distance) around the detector. Four rods that surround the detector is considered:
where D is the detector reading and F_{i} is the reading contribution from the i^{th} rod that surround the detector. Each F_{i} is weighted evenly by 0.25 since four neighboring rods contribute to the detector reading evenly.
X_{i} is a number ranged between -1 and 1 that represents the normalized axial distance between the i^{th} rod tip and the detector with respect to the distance of two notches of the control rod. If the rod tip is withdrawn but within two notches of the detector, X equals 1. If the rod tip is at the same axial position as the detector, X equal 0. If the rod tip is inserted but within two notches of the detector, X is -1.
The constant C is used to control the effectiveness of the rod shadow effect. It is implemented as a local constant in the nuclear instrument routine. To turn off the rod shadow effect, this constant is set to zero.
9.0 Assumptions and Simplifications
The following Assumptions and Simplifications are applicable to the AMON^{TM} core neutronics model:
1. The neutron flux distribution within the core will be calculated using the nodalization shown in Figure 1.
Purpose: Makes possible a solution of the neutron kinetics equations within the constraints of real-time simulation in the given simulation computer system.
Effect: Given correct Parameterization of the model to the reference plant core physics data, the neutron flux distribution and time response, as observable on the simulator control room instrumentation and process computer, will reflect that of the reference plant.
2. The spatial and time dependent neutron flux calculation is accomplished through use of the Improved-Quasi-Static Method^{3,4}. In this method, the space and time-dependent neutron production rate N_{R}(t) is factored into an "amplitude" function Φ(t) and a "shape" function ψ_{R}(t):
N_{R}(t) / Φ(t)ψ_{R}(t)
such that Φ(t) contains the primary "fast" time dependence of the total neutron production rate. Therefore, the time dependent term ψ_{R}(t) has to account only for the relatively "slow" time dependent spacial variations of neutron density.
Purpose: Makes possible a solution of the neutron kinetics equations within the constraints of real-time simulation in the given simulation computer system.
Effect: Given correct Parameterization of the model to the reference plant core physics data, the neutron flux distribution and time response, as observable on the simulator control room instrumentation and process computer, will reflect that of the reference plant.
3. Simulation of the core Thermal-Hydraulics model cell decay heat distribution will be accomplished using RETRAN model eleven (11) group data. Nodal decay heat distribution will be calculated based on the nodal power distributions.
Purpose: Permits solution of the nodal decay heat within the constraints of real-time simulation.
Effect: The decay heat levels as perceivable on the simulator control room instrumentation and process computer will match those of the reference plant.
4. In calculation of the fission product poison concentrations, the neutron absorption cross-sections of both Iodine and Promethium are ignored. In addition, both Iodine and Promethium are assumed to be produced directly from fission.
Purpose: Simplifies the calculations of fission product poison concentrations to save computer execution time.
Effect: The inaccuracy produced by this simplification will not be noticeable on the simulator control room instrumentation or process computer.
10.0 References
S. Glasstone, A. Sesonske, "Nuclear Reactor Engineering," §5.11, Van Nostrand Reinhold Compony, New York, 1980.
D. L. Delp, D. L. Fisher, J. M. Harriman and M. J. Stedwell,"FLARE, A Three-Dimensional Boiling Water Reactor Simulator,"GEAP-4598, July, 1964.
S. G. Wagner, A. T. Shesler, "A Three-Dimensional Improved Quasi-Static Core Model for Nuclear Reactor Control Room Simulators," Proceedings of Fifth Power Plant Dynamic Testing and Control Symposium, Knoxville, Tennessee, 1983.
K. O. Ott, D. A. Meneley, "Accuracy of the Quasi static Treatment of Spatial Reactor Kinetics," Nucl. Sci. Eng., 36:402-411, 1969.
A. F. Henry, "Nuclear Reactor Analysis," §7.6, MIT Press, Cambridge, Mass., 1975.
P. F. Zweifel, "Reactor Physics," §3.1, Mcgraw-Hill Book Co., 1973.
S. Bφrresen, L. Moberg, J. Rasmussen, "Methods of PRESTO-B, A Three-Dimension BWR Core Simulation Code," §7.1, Non-Proprietary Version, Scandpower Inc., Jan., 1983.
"RETRAN-02 A program for transient thermal hydraulic analysis of complex fluid systems", Vol 1, May 1981.
APPENDIX A Neutronics Data Formalism
A1.0 Introduction
The format of the terms in Eq. (3.0-1) to (3.0-6) along with the nodal averaged fuel-assembly migration area and macroscopic fission cross sections are shown in this appendix. Referring to Figure 1 and Figure 2 (Core Nodalization Diagrams), the core cycle dependent (BOC, MOC, EOC as defined by the established simulator initial conditions) data noted in the following sections are required to parameterize and optimize the AMON Core Neutronics Model to a specific reference plant.
The fuel exposure, void history and control history corrected equivalent one-group three-dimensional AMON model "coarse-node" (137 x 8) data should be generated from fuel exposure, void history and control history corrected two-group fine-mesh (560 x 25) data by a flux weighting scheme based upon the reference plant flux distributions at HFP and ARI cold conditions respectively. The nuclear data which are a function of moderator density can either be provided in the form of polynomial functions (as proposed herein) or in tabular form or in a combination of the two.
The amount of coarse-node nuclear data collapsed from two group fine-mesh data is large, and it needs to be reduced by some scheme in order to be loaded in the AMON Model. And in reducing the amount of nuclear data the fidelity of AMON Model and the characteristics of the core cycle loaded have to be maintained. In the model, there are two (2) reduced data types are used, the nodal property dependent data type and core averaged property data type. The nodal nuclear data which will affect nodal power or flux heavily and whose properties are sensitive to the fuel lattice design, such as k-infinity k_{4R }, migration area M^{2}_{R} and fission cross section 3_{fR} are categorized as the nodal property dependent data. The number of grouped nodal property dependent data is determined by the fuel loading symmetry condition in the core, e.g. mirror symmetry, reflection symmetry, rotational symmetry, etc. This approach will usually reduce the data required by a factor of four (4) or more. Nodal nuclear data which will affect nodal power or flux slightly and whose properties are not that sensitive to the fuel lattice design will be treated and used as core averaged property data in the model. These data include boron reactivity coefficients, Doppler reactivity coefficients, Xe microscopic cross section and reactivity coefficients, and Sm microscopic cross section and reactivity coefficients.
A2.0 Nodal Property Dependent Data
The homogenized nodal neutronic data k-infinity k_{4R }, migration area M^{2}_{R} and fission cross section 3_{fR} are processed in the same way in the AMON Model. Those nodal neutronic data, given for each symmetric group in the core, exclude Doppler (average fuel temperature) effects and Iodine, Xenon, Promethium, Samarium and Boron effects in node R. They are functions of exposure rate, void history, control history, moderator density, and control rod effective volumetric control fraction..
In the model, these data are grouped into four sets according to exposure (core-life dependency). Within each set of data at a given exposure rate, there are two subsets of base data given at ARO (uncontrolled), and ARI (controlled) conditions. Each subset of base data are fitted into a polynomial as a function of moderator density:
Where:
N_DATA is the nodal neutronic data of k_{4 }, M^{2} and 3_{f} respectively.
The A's are curve fit polynomial coefficients, and χ_{R} is the moderator density (gm/Cm^{3}) of node R. Similarly, for the controlled case (ARI):
The final nodal property data N_DATA_{R}(f_{R}) will be linearly interpolated between the base N_DATA_{R}'s of Eqs. (A2.0-1) and (A2.0-2) according to the nodal control rod effective volumetric control fraction f_{R}.
A3.0 Core-Averaged Data
The data cited in the following subsections are to be provided on a core-averaged basis.
A3.1 Boron Reactivity Correction Data
Since boron is transported and resides within the liquid phase of the coolant or moderator only, even if the concentrations remain constant for a given amount of boron in the system, the differences in void fraction will yield different boron number densities. The boron reactivity worth is calculated as the product of boron number density and the boron worth coefficient which is a function of the moderator density and boron number density, f:
where
in which:
where
and F_{B}, the boron worth coefficient, is interpolated between two (2) families of fitted polynomial curves corresponding to two (2) different boron number density (for example at 660 ppm and 1000 ppm). The two (2) polynomials are curve fitted as a function of moderator density and are formatted as follows:
where N_{B}^{R} is the boron number density of node R, N_{B}^{600}, N_{B}^{1000} are the corresponding boron number density at 600, 1000 ppm concentration.
A3.2 Doppler Reactivity Correction
The Doppler correction is a function of the core-averaged moderator temperature, density as well as the nodal average fuel temperature. The format of the reactivity calculation is as follows:
where
A3.3 Xenon & Samarium Reactivity Corrections
The Xenon and Samarium reactivity worth are proportional to Xenon and Samarium number density as follows:
where
Xenon and Samarium reactivity coefficients are treated as a function of nodal moderator density only and have the following formats:
Xenon and Samarium microscopic cross sections used in fission product poisons calculation Eq. (7.0-2) and (7.0-4) are treated in a similar way using same formats:
A3.4 Equivalent One-Group Neutron Flux
The flux is an equivalent one-group value converted from two-group data. Since this flux is an initial value used to establish the reference case simulator IC only (e.g. for model initialization during testing), the core-averaged value at hot full power will be sufficient.
A3.5 Delayed Neutron Precursor Data
Core-averaged six group delayed neutron precursor data are required in the form of a table of constants for use in Parameterization of the amplitude function.
where the λ_{i}, β_{i} and Λ are the decay, yield and neutron lifetime constants, respectively.
A3.6 Decay Heat Group Data
The core-averaged eleven group decay heat precursor data are required in the form of a table of constants. If no reference plant specific data is provided, EXITECH will utilize the eleven group decay heat data presented in revision 2 of the RETRAN user's manual.
where the λ_{i}^{D} and Y_{i}^{D} are the decay and yield constants, respectively.
A3.7 Nuclear Fission-Energy And Neutron Poison Data
The following data are required as core-averaged constant values.
where κ is the core-averaged nuclear fission-energy and the λ's and Y's are the decay and yield constants for the fission product poisons, respectively.
APPENDIX B Parameterization And Optimization Data Request List
B1.0 Introduction
The performance and accuracy of the EXITECH Advanced MOdel of Neutronics (AMON) code is mainly dependent on the level of detail and accuracy of the reference plant reactor core design and performance data provided by the licensee. Data required for implementation of the AMON code may be divided into two (2) types:
Type 1: Data required to parameterize the AMON code for a specific reference plant core configuration;
Type 2: Data required to optimize the AMON model to match reference plant operating conditions.
B2.0 AMON Parameterization Data
The data required to parameterize the AMON code for a specific reference plant reactor core fuel load must reflect the status of the specific fuel load cycle. As a precursor to the Licensee data generation process, the Licensee simulator training organization typically will define the simulator core life states (e.g. BOC, MOC, EOC, ...) required to support the Licensee’s operator training and examination program. In addition, for each core life state, the following global NSSS thermal hydraulic data at Hot Full Power (HFP) must be provided:
- reactor power level* (MW Thermal);
- 00EF, and 500EF moderator temperature;
- Hot Full Power cases with 0%, 40%, 60% and 80% instantaneous void.
To provide a high fidelity simulation in the real-time environment, the AMON code requires core-life dependent equivalent one (1) group nuclear data. This data may be divided into the following two (2) categories:
Category 1. Nodal dependent nuclear data. This data includes:
a. 3D nodal base K_{4} data for both controlled and uncontrolled cases. The base K_{4} is defined as the nodal K_{4} excluding Xe, Sm, Doppler and Boron worth effects.
b. 3D nodal migration area M ^{2} and nodal macroscopic fission cross section 3_{f} for both controlled and uncontrolled cases. These data must also exclude Xe, Sm, Doppler and Boron effects.
Category 2. Core-wide-averaged nuclear data. This data includes:
a. Flux-weighted core averaged σ_{Xe}, σ_{Sm} microscopic cross section data for both controlled and uncontrolled cases. These data must exclude any boron effects.
b. Flux-weighted core averaged Xe and Sm worths for both controlled and uncontrolled cases. Each worth must exclude the effects of the other worth as well as exclude the effects of Doppler and boron.
c. Flux-weighted core averaged Doppler worth for both controlled and uncontrolled cases. The worths must exclude the effects of Xe, Sm, and boron.
d. Flux-weighted core averaged boron worths for a minimum of two (2) different boron concentrations (for example 600ppm and 1000 ppm) for both controlled and uncontrolled cases. This data must also exclude Xe, Sm, Doppler effects.
e. Flux-weighted core averaged six (6) group delayed neutron precursor effective fission yields β_{i}, decay constants λ_{i}, and neutron generation time Λ.
f. Flux-weighted core averaged I^{135}, Xe^{135}, Pm^{149}, Sm^{145} isotopic fission yields and decay constants.
g. Core average fission energy κ .
h. If the "flux-weighted" data previously specified above is based on thermal neutron flux instead of an equivalent one-group neutron flux, then the fast to thermal neutron flux ratio as a function of moderator density for each of the different core-life cases must also be provided.
The AMON Parameterization data specified in this section can be provided by one of three (3) methods dependent upon the licensee’s data generation capabilities:
Method A. The controlled and uncontrolled AMON Parameterization data sets as a function of moderator density can be provided collapsed to the AMON nodalization (4 axial by 137 radial nodes for Brunswick).
Method B. The controlled and uncontrolled AMON Parameterization data sets as a function of moderator density can be provided as quarter (¼) core, 3D fine-mesh (N/4 fuel assembly x 25 axial mesh) data sets.
Method C. The controlled and uncontrolled AMON Parameterization data sets can be provided as fuel lattice data. Each set of the fuel lattice data will be a function of fuel exposure, density history and control history. The fuel lattice data must be a function of moderator density, including the range of moderator temperatures and instantaneous voids specified previously.
In order for EXITECH to process the fuel exposure, density history and control history corrections, the following 3D information must also be provided on a quarter (¼) core, 3D fine mesh (N/4 fuel assembly x 25 axial mesh) basis:
3D fine mesh fuel exposure distributions for each of the required fuel-cycle states (BOC, MOC, EOC, etc.).
3D fine mesh moderator density (void) history distributions for each of the required fuel-cycle states (BOC, MOC, EOC, etc.).
3D fine mesh control (rod) history distributions for each of the required fuel-cycle states (BOC, MOC, EOC, etc.).
3D fuel lattice core loading map, including the number of fuel rods within each type of fuel assembly for the fuel cycle to be simulated.
B3.0 AMON Optimization Data
In order to optimize the AMON model to match reference plant operating conditions, the following multi-dimensional distribution nuclear data are required. These data can be provided either collapsed to the AMON nodal configuration or as quarter (¼) core, 3D fine mesh (N/4 fuel assembly x 25 axial mesh) data. These data must be corrected for the corresponding fuel exposure, void (density) history and control history.
3D power or neutron flux (fast and thermal) distribution at HFP rated conditions and COLD ARI conditions for each of the required core-life states (BOC, MOC, EOC, etc.).
3D fuel temperature distributions at HFP rated conditions for each of the required fuel-cycle states (BOC, MOC, EOC, etc.). Alternatively, a fuel temperature correlation formula together with all correlation coefficients for each fuel lattice as well as a 3D fuel lattice core loading map may be provided.
3D fuel element inner and outer radius, i.e. fuel pin and cladding dimensions.
Two dimensional core inlet coolant flow distributions in radial direction for each of the required fuel-cycle states (BOC, MOC, EOC, etc.).
HFP rated condition core average neutron flux for each of the core-life states (BOC, MOC, EOC, etc.). Note that this neutron flux, whether thermal or total, must be consistent with the neutron flux used to generate the previously (Section II) noted core averaged cross sections and reactivity worths.
3D core instrumentation layout for the source range, intermediate range, power range, and movable in-core probe detectors.
Controlled and uncontrolled core bypass-channel and in-channel flow areas for each of the fuel lattice loaded in the core.
Predicted HFP LPRM values or reference plant actual LPRM readings corresponding to the specific fuel cycle for each of the core-life states (BOC, MOC, EOC, etc.).
Control rod withdrawal sequence.
Cold critical rod notch position and corresponding moderator temperature for each of the required core-life states (BOC, MOC, EOC, etc.).
Core Eigenvalue at Cold ARI and HFP rated conditions for each of the required core-life states (BOC, MOC, EOC, etc.).
Neutron source strengths and locations (if applicable).
Core pressure drop, Dome pressure, Inlet subcooling or Fedwatter enthalpy for each of the required core-life states (BOC, MOC, EOC, etc.).
Total core flow and bypass core flow fraction.