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Transport Methods for Nuclear Reactor Analysis

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Title: 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
8
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
11
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
17
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.

18
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!
19
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 ...

20
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.

23
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

24
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!

27
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)

28
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.
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