Nuclear Engineering: Mathematical Modeling and Simulation

Title-page
Copyright
Dedication
Contents
About the author
Foreword
1 The atom and nuclear radiation
1.1 The atom
1.1.1 Nuclear stability
1.1.2 Binding energy
1.2 Radioactive decay
1.2.1 Alpha decay
1.2.2 Beta decay
1.2.3 Gamma decay
1.2.4 Radioactive nuclides in nuclear technologies
1.3 Interaction of radiation with matter
1.3.1 Interaction of alpha rays with matter
1.3.2 Interaction of beta radiation with matter
1.3.3 Interaction of gamma radiation with matter
1.4 Sources and effects of radiation
1.4.1 Radiation dose
1.4.2 Absorbed dose
1.4.3 Equivalent dose
1.4.4 Effective dose
1.4.5 Radiation safety limits
1.4.6 Radiation detection
1.5 Atomic densities of elements and mixtures
1.6 Mathematical modeling and simulation
1.6.1 Alpha particle transport simulation
1.6.2 Interaction of electrons with matter
1.6.3 Interaction of gamma radiation with matter
1.6.4 Radiation dose from Calfornium-252 gamma source in water
Capabilities developed
Nomenclature
Problems
References
2 Interactions of neutrons with matter
2.1 Kinetic theory
2.2 Types of neutron interactions
2.2.1 Neutron scattering in the lab and center of mass systems
2.3 The microscopic cross-section
2.4 The macroscopic cross-section
2.5 Flux measurement
2.6 Reaction rates
2.7 Neutron slowing down, diffusion and thermalization
2.8 Resonance cross-section
2.9 Nuclear fission
2.9.1 The fission process
2.9.2 Critical energy
2.9.3 Fission yield
2.9.4 Number of neutrons emitted in fission
2.9.5 Fissile and fertile materials
2.9.6 The fission spectrum
2.10 Criticality
2.10.1 Diffusion theory
2.10.2 Transport theory
2.10.3 Monte Carlo simulation
Problems
Nomenclature
References
3 Nuclear reactors and systems
3.1 Status of nuclear power
3.1.1 Generations of nuclear power
3.1.2 Reactors shut down
3.1.3 The future of the nuclear power industry
3.2 Nuclear reactor systems
3.2.1 Pressurized water reactor
3.2.2 Boiling water reactor
3.2.3 Pressurized heavy water reactor
3.2.4 Gas cooled reactor
3.2.5 Fast breeder reactor
3.3 Marine propulsion reactors
3.3.1 Introduction
3.3.2 US nuclear submarine program
3.3.3 Former Soviet/Russian nuclear submarine program
3.3.4 Submarine programs: UK, France, China, India and Pakistan
3.3.5 Modern-day submarines
3.3.6 Technical features
3.3.7 HEU/LEU submarine reactors
3.4 Plutonium production reactors
3.5 Small modular reactors
3.5.1 Design features of SMRs
3.5.2 Very small modular reactor
3.5.3 Generation-IV reactors
3.5.4 Radiation source term
3.6 Nuclear fusion
3.6.1 The fusion reaction
3.6.2 Magnetic confinement fusion
3.6.3 Inertial confinement fusion
3.7 Space propulsion
3.7.1 Conventional rocket designs
3.7.2 Space exploration
3.7.3 Nuclear rocket designs for deep space exploration
3.8 Nuclear power systems in space
3.8.1 Radioisotope thermal generators
3.8.2 Small nuclear auxiliary power systems
3.8.2.1 Nuclear reactors for space
3.9 Conclusions
Problems
Nomenclature
English (lower case)
English (upper case)
Greek lower case
Abbreviations and acronyms
References
ANNEX: the physics of nuclear fusion
4 Mathematical foundations
4.1 Ordinary differential equations
4.1.1 The Poisson equation: steady-state heat conduction in 1-D
4.1.2 Coupled first-order ODEs: the point kinetics equations
4.2 Partial differential equations
4.2.1 Equations of fluid dynamics
4.2.2 The 1-D time-dependent heat conduction
4.2.3 Laplace equation: 2-D steady-state heat conduction
4.2.4 Heat conduction in 2-D and 3-D
4.2.4.1 Spherical geometry
4.2.5 Flux formulation
4.3 Integral equations
4.3.1 An important integral equation for neutron transport
4.3.2 Integral equations in neutron transport
4.4 Integro-differential equations
4.5 Numerical methods
4.5.1 The Finite Difference Method
4.5.1.1 Matlab program: Finite Difference Method
4.5.2 The Finite Element Method
4.6 Approximate methods
4.6.1 The Ritz method
4.6.2 The Rayleigh–Ritz variational method
4.6.3 The weighted residual method
4.7 The adjoint function
4.8 Random processes, probability, and statistics
4.8.1 Random processes
4.8.2 Markovian processes
4.8.3 Population and sample
4.8.4 Random variables, PDF, and CDF
4.8.5 Random numbers
4.8.5.1 Matlab program
4.8.6 Sampling from PDFs
4.8.6.1 Sampling from analytic PDFs
4.8.6.2 Sampling from nonanalytic PDFs
4.8.7 Kullback–Leibler divergence for uniform random numbers
4.8.8 The law of large numbers
4.8.8.1 Application of the law of large numbers
4.8.9 The central limit theorem
4.8.9.1 Matlab program: mean and variance
4.9 Evaluation of integrals
4.9.1 The Monte Carlo method for numerical integration
4.9.1.1 Matlab program
Problems
Nomenclature
References
5 The neutron diffusion equation
5.1 The conservation equation
5.2 The one-group diffusion equation
5.2.1 Nonmultiplying systems
5.2.1.1 Finite slab
5.2.1.2 Finite cylinder
5.2.1.3 Point source in an infinite medium
5.2.2 Multiplying systems
5.2.2.1 Slab reactor
5.2.2.2 Cylindrical reactor
5.2.2.3 Spherical reactor
5.2.3 One-group criticality
5.3 The two-group diffusion equation
5.3.1 Nonmultiplying systems
5.3.1.1 Computer programming example
5.3.1.2 Temperature effects on neutron flux
5.3.2 Multiplying systems
5.3.3 Two-group criticality
5.4 The multigroup diffusion equation
5.4.1 Numerical solution of the multigroup diffusion equations
5.5 Effect of fuel concentration on critical mass
5.5.1 Goertzel’s theorem
5.5.2 Nonuniform fuel distribution: a slab model
5.5.3 Nonuniform fuel distribution: a spherical model
5.5.4 Critical core with flat thermal flux loading
5.6 The two-group adjoint diffusion equations
5.7 Core neutronics with diffusion equations
Problems
Nomenclature
References
6 The neutron transport equation
6.1 Structure of the neutron transport equation
6.1.1 An integro-differential form of the neutron transport equation
6.1.2 The two-group transport equation
6.1.3 The integral form of the transport equation
6.1.4 Multigroup form of the integral transport equation
6.2 Exact solutions of the transport equation
6.2.1 The classic albedo problem
6.2.2 Infinite medium with a plane isotropic source
6.2.3 Finite sphere with a point isotropic source
6.3 Numerical methods for solving the transport equation
6.3.1 The discrete ordinates method
6.3.2 The Spherical harmonics method
6.3.3 The DPN method
6.3.4 The BN method
6.3.5 The finite element method
6.3.6 The nodal method with transport theory
6.3.7 Hybrid methods
6.3.8 Criticality estimates
6.4 Transport theory for reactor calculations
6.4.1 Collision probability method
6.4.2 Method of characteristics
Problems
Nomenclature
References
7 The Monte Carlo method
7.1 Stochastic simulation
7.1.1 Markov processes
7.1.2 Events in a random walk
7.1.3 The physics of interactions
7.1.4 Nuclear interaction data
7.1.5 How do we know an answer is good?
7.2 Simulation of a random walk
7.2.1 Monte Carlo simulation
7.2.2 Estimators and tallies
7.2.3 Sampling a source
7.2.4 Sampling the “distance to collision”
7.2.5 Determining the type of event
7.2.6 Determining the nuclide of interaction
7.2.7 Processing a scattering event
7.2.8 Processing a fission event
7.2.9 Processing a capture event
7.2.10 Processing an escape-from-system event
7.2.11 Mean and variance
7.2.12 Batch, history, random walk and events
7.3 Modeling the geometry
7.3.1 Geometries for illustration of Monte Carlo simulation
7.4 Demonstration
7.5 Variance reduction methods
7.6 Estimating perturbations with Monte Carlo simulation
7.7 Conclusions
Problems
Nomenclature
References
8 Computer codes
8.1 Neutron and radiation transport codes
8.1.1 ANISN
8.1.2 DOT
8.1.3 TORT
8.1.4 PARTISN
8.1.5 MCNP
8.1.6 TART
8.1.7 MORSE
8.1.8 KENO
8.1.9 Other Monte Carlo codes
8.2 Time-dependent reactor kinetics codes
8.3 Thermal hydraulics codes
8.4 Radiological protection codes
8.5 Performance and safety analyses
8.6 Nuclear data
8.6.1. MCNP
8.7 Conclusion
Problems
Nomenclature
References
9 Optimization and variational methods
9.1 Introduction
9.2 Deterministic optimization
9.2.1 Deterministic optimization without constraints
9.2.2 Deterministic optimization with algebraic constraints
9.2.3 Optimal solution with a system of first-order ordinary differential equation constraints
9.2.4 Optimal solution with a system of first-order ordinary differential equation constraints
9.2.5 Optimal discrete control (Pontryagin maximum principle)
9.3 Controller design and optimization
9.4 Dynamic programming
9.5 Stochastic optimization
9.5.1 Genetic algorithms
9.5.2 Particle swarm optimization
9.6 Applications of optimization in reactors
9.6.1 Multi-objective core optimization
9.6.2 Pressurized water reactor core pattern optimization
9.6.3 Controller proportional integral derivative
9.6.4 Radiation shielding
9.6.5 Some other applications of optimization
Problems
Nomenclature
References
10 Monte Carlo simulation in nuclear systems
10.1 Introduction
10.2 Bare critical assemblies
10.2.1 Godiva
10.2.2 Jezebel
10.3 Criticality safety
10.3.1 Storage of interacting units
10.3.2 Storage of uranium hexafluoride cylinders
10.4 Radiation moderation and shielding
10.4.1 Radiation moderation for a neutron generator
10.4.2 Radiation shielding
10.5 Nuclear fission applications
10.5.1 Unit lattice cell and fuel assembly of the AP1000 reactor
10.5.2 The Toshiba 4S reactor
10.5.3 Micronuclear reactor
10.6 Nuclear fusion applications
Problems
Nomenclature
Greek letters
Abbreviations and acronyms
References
Annex A MCNP listing for Godiva (Section 10.2.1)
Annex B MCNP input listing (Jezebel, Section 10.2.2)
Annex C MCNP input listing (BK10Shld, Section 10.5.1)
Annex D MCNP input listing (BK10AP10, Section 10.5.1)
11 Comparisons: Monte Carlo, diffusion, and transport
11.1 Introduction
11.2 Criticality in a bare sphere
11.2.1 One-group diffusion theory criticality
11.2.2 Two-group diffusion theory criticality
11.2.3 One-speed transport theory criticality
11.3 The classic albedo calculation
11.4 Flux in a slab
11.4.1 Diffusion theory
11.4.2 Transport theory
11.4.3 Monte Carlo simulation
11.4.4 Comparison
11.5 Flux in a finite sphere with a point isotropic source
11.5.1 Diffusion theory
11.5.2 Transport theory exact solution
11.5.3 Monte Carlo simulation
Problems
Nomenclature
References
Annex A MATLAB Program AlbedoSlabDiffTh.m (Section 11.3)
Annex B MCNP Input File BK11Albd (Section 11.2)
Annex C MATLAB Program CH11ExactSolSlabJan03.m (Section 11.4.4)
12 Exercises in Monte Carlo simulation
12.1 Sampling from a distribution function
12.1.1 Sampling from a normal distribution
12.1.2 Sampling from a Watt fission spectrum
12.2 Estimating the neutron flux in a non-multiplying sphere
12.2.1 The simulation process
12.2.2 MATLAB program for point source in a finite non-multiplying sphere
12.2.3 Results
12.3 Reflected assemblies
12.4 Reactor core modeling
12.4.1 Input file
12.4.1.1 Defining the bounding window
12.4.1.2 Defining the universe(s)
12.4.1.3 Defining the lattice
12.4.2 Surrounding cells
12.4.2.1 Surface cards
12.4.3 Source description
12.4.4 Plotting the geometry
12.4.5 Tally cards
12.4.6 Reaction rates
12.4.7 Plotting tallies
12.5 Radiation safety and shielding
12.6 Perturbation calculations
12.7 MCNP geometry plotting in core neutronics
Problems
Conclusions
Nomenclature
References
Annex A: MATLAB Program CH12_NormalSampling.m
Annex B MATLAB Program CH12_Watt Sampling.m
13 Optimization in nuclear systems
13.1 Introduction
13.2 Reactor core design optimization
13.3 Fusion neutronics design optimization
13.4 Radiation shielding design optimization
13.5 Fuel loading pattern optimization
13.5.1 Optimal distribution: Pontryagin’s maximum principle
13.6 Radiation detection or optimization
13.7 Controller design optimization
Problems
Nomenclature
References
14 Monte Carlo simulation in medical physics
14.1 Introduction
14.1.1 The production of radio-isotopes
14.1.2 Alpha radiation therapy
14.2 Brachytherapy
14.2.1 Monte Carlo simulation in brachytherapy
14.2.2 Monte Carlo simulation to calculate energy deposition and dose distribution for brachytherapy
Nomenclature
References
Index