Transport Methods for Nuclear Reactor Analysis Marvin L. Adams Texas A&M University [email protected]...

Transport Methods for Nuclear Reactor Analysis

Marvin L. AdamsTexas A&M [email protected]

Computational Methods in TransportTahoe City, September 11-16, 2004

Acknowledgments

• Thanks to Frank for organizing this!

• Kord Smith taught me much of what I know about modern nuclear reactor analysis.

3 Texas A&M Nuclear Engineering

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Marvin L. Adams

Outline

Bottom Line

Problem characteristics and solution requirements

Modern methodology

Results: Amazing computational efficiency!

Summary


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Modern methods are dramatically successful for LWR transport problems.

Today’s 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 < 4 hours

This is phenomenal computational efficiency!


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Outline

Bottom Line

Problem characteristics and solution requirements Geometry is challenging Physics is challenging Requirements are challenging

Modern methodology


Summary


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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


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For physics, it helps to remember a simplified neutron life cycle.

fast neutron(few MeV)

leaksdoesn’t leak

absorbed fast

abs. in fuel

causes fission

slows to thermal(< 1 eV)leaks

abs. in junk

capturedabsorbed

abs. in fuel

causes fission

abs. in junk

captured

f fast neutronsth fast

neutrons


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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

1E+9

1E+11

1E+13

1E+15

1E+17

1E+19

1E+21

1E-4 1E-2 1E+0 1E+2 1E+4 1E+6 1E+8

E (eV)

f(E

) [n

/cm

^2-

s-M

eV]


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

Resonances: 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: ’s depend very sensitively on neutron energy and material temperature!

fU235

0.1

1000

10

0.1 100010neutron energy (eV)

= microscopic cross section (area/nucleus)

= macroscopic cross section = N (nuclei/vol) (area/nucleus)

= reactions / neutron_path_length


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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.

’s

T and

f

Reaction Rates

Heat Source

Coolant Flow

Heat Transfer


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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 n’s Thermal n’s 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 is huge At resonance energies, most n’s enter fuel from coolant captures

occur first in outer part of pellet U239 Np239 Pu239, and Pu239 is fissile “Rim effect”


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Transient calculations present further challenges. Delayed neutrons are important!

Small fraction of n’s from fission are released with significant time delays prompt neutrons (>99%) are released at fission time a released neutron takes < 0.001 s to either leak or be absorbed delayed neutrons (<1%) are released 0.01 – 100 s after fission they are emitted during decay of daughters of fission products (delayed-

neutron precursors)

Doesn’t affect steady state.

Delayed neutrons usually dominate transient behavior. slightly supercritical reactor would be subcritical without dn’s 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)


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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)


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Outline

Bottom Line


Modern methodology Divide & Conquer Sophisticated averaging Factorization / Superposition Coupling, searches, and iterations


Summary


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Divide-and-Conquer approach relies on multiple levels of calculation.

A-L Code (2D transport, high-res.)

2-grp ’s, etc.

2-grp ’s, etc.

2-grp ’s, etc.

2-grp ’s, etc.

Table for each assy type:

{’s, DF’s, power shapes}

as functions of

{burnup, boron, Tmod, Tfuel, rmod, power, Xe, histories, ...}

Different assembly types

C-L Code (2-grp Difn.) (with T/H feedback) SOLUTIONS

Pre-computed fine-group ’s.


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How can 2-group diffusion give good answers to such complicated transport problems?

1. 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. 2-group diffusion parameters come from fairly accurate reference solution: f(r,E) from single-assembly calculation

3. 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 ’s.

’s are averaged over “fast” and “thermal” energy ranges:

thermal: (0,1) eV fast (1,10000000) eV

Assemblies are “homogenized” by spatially averaging their ’s:

If averaging function has same “shape” as the real solution, then averaged ’s produce the correct reaction rates in low-order calculation. Assuming that net flow rates are correct ...

,{ }

{ }

fasti a i

i regions

f

fasta

ii regi

asti

fasti

ons

V

V

f

f

,{ }

,

{ }

ga i

g thermal gr

gi

gi

oupsslowa i

g thermal groups

f

f


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Marvin L. Adams

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

Even with perfectly averaged ’s, 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!

exact

homogenized

Ff

f


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Marvin L. Adams

We generate DF’s from the single-assembly problems.

Single Assembly: uses reflecting boundary fine-mesh fine-group transport generates “exact” f this generates homogenized 2-group ’s 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)

2 hom* continuousCM G DF f

single assy

FMFGhetT

CM2GhomDF

ff


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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.

,

,exact

homogenized

x y

x y

ff


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In the core, every assembly is different.

Core-level code needs ’s and F’s 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 d/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.


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Marvin L. Adams

Outline

Bottom Line


Modern methodology Divide & Conquer Sophisticated averaging Factorization / Superposition Coupling, searches, and iterations


Summary


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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 f’s, and surface leakages

Use surface leakages and region-avg f’s 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 < 1 hr (<2 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 < 4 s per 3D problem on cheap PC

k errors <0.1%. Pin-power errors <5% (RMS avg < 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.

Transport Methods for Nuclear Reactor Analysis Marvin L. Adams Texas A&M University [email protected]...

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Transcript of Transport Methods for Nuclear Reactor Analysis Marvin L. Adams Texas A&M University [email protected]...