Transport Methods for Nuclear Reactor Analysis

1
Transport Methods for Nuclear Reactor Analysis

• Marvin L. AdamsTexas AM University
• mladams_at_tamu.edu
• Computational Methods in Transport
• Tahoe City, September 11-16, 2004

2
Acknowledgments

• Thanks to Frank for organizing this!
• Kord Smith taught me much of what I know about
  modern nuclear reactor analysis.

3
Outline

• Bottom Line
• Problem characteristics and solution requirements
• Modern methodology
• Results Amazing computational efficiency!
• Summary

4
Modern methods are dramatically successful for
LWR transport problems.

• Todays codes calculate
• power density (W/cm3) in each of the 30,000,000
  fuel pellets
• critical rod configuration or boron concentration
• nuclide production and depletion
• as a function of time for a full one- to two-year
  cycle
• including off-normal conditions
• including coupled heat transfer and coolant flow
• with accuracy of a few
• using a 1000 PC
• in lt 4 hours
• This is phenomenal computational efficiency!

5
Outline

• Bottom Line
• Problem characteristics and solution requirements
• Geometry is challenging
• Physics is challenging
• Requirements are challenging
• Modern methodology
• Results Amazing computational efficiency!
• Summary

6
Reactor geometry presents challenges.

• Fuel pins are simple (cylindrical tubes
  containing a stack of pellets)
• but there are 50,000 of them and we must compute
  power distribution in each one!
• Structural materials are complicated
• grid spacers, core barrel, bundle cans
• Instrumentation occupies small volumes

7
For physics, it helps to remember a simplified
neutron life cycle.
fast neutron(few MeV)
leaks
doesnt leak
nf fast neutrons
nth fast neutrons
absorbed fast
slows to thermal(lt 1 eV)
abs. in junk
leaks
abs. in fuel
absorbed
captured
abs. in junk
causes fission
abs. in fuel
captured
causes fission
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It also helps to know something about the answer.

• Cartoon (not actual result) of basic dependence
  on energy in thermal reactor
• Fission-spectrum-ish at high energies
• 1/E-ish in intermediate energies
• Maxwellian-ish at low energies
• 10 orders of magnitude in domain and range

9
Neutron-nucleus interaction physics presents
challenges cross sections are wild!

• Resonances
• s changes by 2-4 orders of magnitude with
  miniscule changes in neutron energy (really total
  kinetic energy in COM frame)
• arise from discrete energy levels in compound
  nucleus
• effectively, become shorter and broader with
  increasing material temperature (because of
  averaging over range of COM kinetic energies)
• Bottom line ss depend very sensitively on
  neutron energy and material temperature!

1000
10
sfU235
0.1
0.1
1000
10
neutron energy (eV)
s microscopic cross section (area/nucleus)S
macroscopic cross section N (nuclei/vol) s
(area/nucleus) reactions /
neutron_path_length
10
A milder challenge scattering is anisotropic.

• Scattering is isotropic in the center-of-mass
  frame for
• light nuclides
• neutron energies below 10s of keV
• Not for higher energies.
• Not for heavier nuclides.
• Almost never isotropic in lab frame!

O-16 elastic
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Temperature dependence makes this a
coupled-physics problem.
T and r
Coolant Flow
Ss
Heat Transfer
f
Heat Source
Reaction Rates
12
Depletion and creation of nuclides adds to the
challenge.

• Example depletion of burnable absorber (such
  as Gd)
• Some fuel pellets start with Gd uniformly
  distributed
• Very strong absorber of thermal ns
• Thermal ns enter from coolant ? n-Gd absorptions
  occur first in outer part of pellet
• Gd depletion eats its way inward over time
• Example U238 depletion and Pu239 buildup
• Similar story
• Most n capture in U238 is at resonance energies,
  where S is huge
• At resonance energies, most ns enter fuel from
  coolant ? captures occur first in outer part of
  pellet
• U239 ? Np239 ? Pu239, and Pu239 is fissile
• Rim effect

13
Transient calculations present further
challenges. Delayed neutrons are important!

• Small fraction of ns from fission are released
  with significant time delays
• prompt neutrons (gt99) are released at fission
  time
• a released neutron takes lt 0.001 s to either leak
  or be absorbed
• delayed neutrons (lt1) are released 0.01 100 s
  after fission
• they are emitted during decay of daughters of
  fission products (delayed-neutron precursors)
• Doesnt affect steady state.
• Delayed neutrons usually dominate transient
  behavior.
• slightly supercritical reactor would be
  subcritical without dns
• subcritical reactor behavior limited by decay of
  slowest precursor
• must calculate precursor concentrations and decay
  rates as well as neutron flux (and heat transfer
  and fluid flow)

14
Solution requirements are challenging.

• To license a core for a cycle (1-2 years), must
  perform thousands of full-core calculations
• dozens of depletion steps
• hundreds of configurations per step
• Each calculation must provide enormous detail
• axial distribution of power for each of 50,000
  pins
• depletion and production in hundreds of regions
  per pin
• includes heat transfer and coolant flow
• includes search for critical (boron concentration
  or rod position)
• Transient calculations are also required
• Simulators require incredible computational
  efficiency (real-time simulation of entire plant)

15
Outline

• Bottom Line
• Problem characteristics and solution requirements
• Modern methodology
• Divide Conquer
• Sophisticated averaging
• Factorization / Superposition
• Coupling, searches, and iterations
• Results Amazing computational efficiency!
• Summary

16
Divide-and-Conquer approach relies on multiple
levels of calculation.
Different assembly types
A-L Code (2D transport, high-res.)
Table for each assy type Ss, DFs, power
shapes as functions of burnup, boron, Tmod,
Tfuel, rmod, power, Xe, histories, ...
SOLUTIONS
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How can 2-group diffusion give good answers to
such complicated transport problems?

• Homogenization Theory
• Low-order model can reproduce (limited features
  of) any reference high-order solution.
• Consider a reference solution generated by
  many-group fine-mesh transport for heterogeneous
  region.
• 2-group coarse-mesh diffusion on a homogenized
  region can reproduce
• reaction rates in coarse cell
• net flow across each surface of coarse cell
• Discontinuity Factors make this possible!
• 2-group diffusion parameters come from fairly
  accurate reference solution
• f(r,E) from single-assembly calculation
• Diffusion is reasonably accurate given large
  homogeneous regions.

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Assembly-level calculation has very high
fidelity.

• 2D long-characteristics transport
• Scattering and fission sources assumed constant
  (flat) in each mesh region
• Essentially exact geometry
• Dozens of energy groups
• Thousands of flat-source mesh regions

Biggest approximation reflecting boundaries!
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Fine-mesh fine-group assembly-level solution is
used to average the Ss.

• Ss are averaged over fast and thermal energy
  ranges
• thermal (0,1) eV
• fast (1,10000000) eV
• Assemblies are homogenized by spatially
  averaging their Ss
• If averaging function has same shape as the
  real solution, then averaged Ss produce the
  correct reaction rates in low-order calculation.
• Assuming that net flow rates are correct ...

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Even perfectly averaged Ss are not enough!
Also need correct net leakages.

• Even with perfectly averaged Ss, the homogenized
  problem cannot produce correct reaction rates and
  correct leakages.
• The solution is to specify a discontinuity in the
  scalar flux at assembly surfaces, using a
  discontinuity factor
• This is what makes homogenization work!

21
We generate DFs from the single-assembly
problems.

• Single Assembly
• uses reflecting boundary
• fine-mesh fine-group transport generates exact
  f
• this generates homogenized 2-group Ss
• then solve homogenized single-assembly problem
  with low-order operator (coarse-mesh 2-group
  diffusion)
• DF is ratio of exact to low-order solution on
  each surface
• Core Level
• we know that exact heterogeneous solution is
  continuous
• in each coarse mesh, this is approximated as the
  low-order solution times the DF for that assembly
  and surface
• continuity of this quantity means discontinuity
  of low-order solution (unless neighboring
  assemblies have the same DF)

22
Global calculation must produce pin-by-pin powers
as well as coarse-mesh reaction rates.

• Pin power reconstruction is done using form
  functions.
• Basic idea assume that
• depends weakly on assembly boundary conditions.
• We tabulate this form function for each fuel
  pin in the single-assembly calculation, then use
  it to generate pin powers after each full-core
  calculation.

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In the core, every assembly is different.

• Core-level code needs Ss and Fs as functions
  of
• fuel temperature
• coolant temperature
• boron concentration
• void fraction
• burnup
• various history effects
• etc.
• Assembly-level code produces tables using branch
  cases. Basic idea
• define base-case parameter values run base case
  and tabulate
• change one parameter re-run. Generates dS/dp
  for this p.
• repeat for all parameters

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Still must discretize 2-group diffusion
accurately on coarse homogenized regions.

• Lots of ways to do this well enough.
• Typical modern method
• high-order polynomials for fast flux (4th-order,
  e.g.)
• continuity conditions and spatial-moment
  equations determine the unknowns
• thermal equation is solved semi-analytically
• transverse integration produces coupled 1D
  equations
• each is solved analytically (giving sinh and cosh
  functions)
• transverse-leakage terms are approximated with
  quadratic polynomials
• Result is quite accurate for the large
  homogenized regions used in practice.

25
Outline

• Bottom Line
• Problem characteristics and solution requirements
• Modern methodology
• Divide Conquer
• Sophisticated averaging
• Factorization / Superposition
• Coupling, searches, and iterations
• Results Amazing computational efficiency!
• Summary

26
Coupling and search is rolled into eigenvalue
iteration in practice.

• Guess k, fission source, temperatures, and boron
  concentration.
• Solve 2-group fixed-source problem
• new k, fission source, region-avg fs, and
  surface leakages
• Use surface leakages and region-avg fs to define
  CMFD equations.
• Use CMFD equations to iterate on
• k
• fission source
• temperatures (coupled to heat transfer and fluid
  flow)
• boron concentration
• Update high-order solution repeat.
• This is incredibly fast!

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Results demonstrate truly amazing computational
efficiency.

• Assembly-level code
• 1600 2D transport calculations per PWR assembly
• hundreds of flat-source regions
• dozens of energy groups
• dozens to hundreds of directions per group
  0.2-mm ray spacing
• total run time lt 1 hr (lt2 s per 2D transport
  calculation) on cheap PC
• Core typically has 3-5 different kinds of
  assemblies.
• Core-level code
• thousands of 3D diffusion calculations per cycle
• 200 x 25 coarse cells
• high-order polynomial / analytic function
• coupled to heat transfer and fluid flow critical
  search done
• pin-power reconstruction
• lt 4 s per 3D problem on cheap PC
• k errors lt0.1. Pin-power errors lt5 (RMS avg lt
  1)

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Summary

• Reactor analysis methods are quite mature for
  commercial LWRs.
• They are really, really fast!
• They work very well for all-uranium cores.
• Still some challenges for MOX cores.