Title: Transport Methods for Nuclear Reactor Analysis
1Transport Methods for Nuclear Reactor Analysis
- Marvin L. AdamsTexas AM University
- mladams_at_tamu.edu
- Computational Methods in Transport
- Tahoe City, September 11-16, 2004
2Acknowledgments
- Thanks to Frank for organizing this!
- Kord Smith taught me much of what I know about
modern nuclear reactor analysis.
3Outline
- Bottom Line
- Problem characteristics and solution requirements
- Modern methodology
- Results Amazing computational efficiency!
- Summary
4Modern 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!
5Outline
- Bottom Line
- Problem characteristics and solution requirements
- Geometry is challenging
- Physics is challenging
- Requirements are challenging
- Modern methodology
- Results Amazing computational efficiency!
- Summary
6Reactor 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
7For 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
8It 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
9Neutron-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
10A 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
11Temperature dependence makes this a
coupled-physics problem.
T and r
Coolant Flow
Ss
Heat Transfer
f
Heat Source
Reaction Rates
12Depletion 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
13Transient 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)
14Solution 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)
15Outline
- Bottom Line
- Problem characteristics and solution requirements
- Modern methodology
- Divide Conquer
- Sophisticated averaging
- Factorization / Superposition
- Coupling, searches, and iterations
- Results Amazing computational efficiency!
- Summary
16Divide-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
17How 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.
18Assembly-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!
19Fine-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 ...
20Even 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!
21We 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)
22Global 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.
23In 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
24Still 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.
25Outline
- Bottom Line
- Problem characteristics and solution requirements
- Modern methodology
- Divide Conquer
- Sophisticated averaging
- Factorization / Superposition
- Coupling, searches, and iterations
- Results Amazing computational efficiency!
- Summary
26Coupling 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!
27Results 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)
28Summary
- 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.