Transport Methods for Nuclear Reactor Analysis Marvin L. Adams Texas A&M University Computational Methods in Transport Tahoe 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.

Texas A&M Nuclear Engineering 1876 Marvin L. Adams3 Outline  Bottom Line  Problem characteristics and solution requirements  Modern methodology  Results: Amazing computational efficiency!  Summary

Texas A&M Nuclear Engineering 1876 Marvin L. Adams4 Modern methods are dramatically successful for LWR transport problems.  Today’s codes calculate  power density (W/cm 3 ) 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!

Texas A&M Nuclear Engineering 1876 Marvin L. Adams5 Outline  Bottom Line  Problem characteristics and solution requirements  Geometry is challenging  Physics is challenging  Requirements are challenging  Modern methodology  Results: Amazing computational efficiency!  Summary

Texas A&M Nuclear Engineering 1876 Marvin L. Adams6 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams7 For physics, it helps to remember a simplified neutron life cycle. fast neutron (few MeV) leaks doesn’t leak absorbed fast abs. in fuel causes fission slows to thermal (< 1 eV) leaks abs. in junk captured absorbed abs. in fuel causes fission abs. in junk captured f fast neutrons th fast neutrons

Texas A&M Nuclear Engineering 1876 Marvin L. Adams8 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams9 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!  f U neutron energy (eV)  = microscopic cross section (area/nucleus)  = macroscopic cross section = N (nuclei/vol)  (area/nucleus) = reactions / neutron_path_length

Texas A&M Nuclear Engineering 1876 Marvin L. Adams10 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams11 Temperature dependence makes this a coupled-physics problem.  ’s T and   Reaction Rates Heat Source Coolant Flow Heat Transfer

Texas A&M Nuclear Engineering 1876 Marvin L. Adams12 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: U 238 depletion and Pu 239 buildup  Similar story  Most n capture in U 238 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  U 239  Np 239  Pu 239, and Pu 239 is fissile  “Rim effect”

Texas A&M Nuclear Engineering 1876 Marvin L. Adams13 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 < 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)

Texas A&M Nuclear Engineering 1876 Marvin L. Adams14 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)

Texas A&M Nuclear Engineering 1876 Marvin L. Adams15 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams16 Divide-and-Conquer approach relies on multiple levels of calculation. A-L Code (2D transport, high-res.) 2-grp  ’s, etc. Table for each assy type: {  ’s, DF’s, power shapes} as functions of {burnup, boron, T mod, T fuel, r mod, power, Xe, histories,...} Different assembly types C-L Code (2-grp Difn.) (with T/H feedback) SOLUTIONS Pre-computed fine- group  ’s.

Texas A&M Nuclear Engineering 1876 Marvin L. Adams17 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:  (r,E) from single-assembly calculation 3.Diffusion is reasonably accurate given large homogeneous regions.

Texas A&M Nuclear Engineering 1876 Marvin L. Adams18 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!

Texas A&M Nuclear Engineering 1876 Marvin L. Adams19 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, ) 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...

Texas A&M Nuclear Engineering 1876 Marvin L. Adams20 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!

Texas A&M Nuclear Engineering 1876 Marvin L. Adams21 We generate DF’s from the single-assembly problems.  Single Assembly:  uses reflecting boundary  fine-mesh fine-group transport generates “exact”   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)

Texas A&M Nuclear Engineering 1876 Marvin L. Adams22 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.

Texas A&M Nuclear Engineering 1876 Marvin L. Adams23 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams24 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 (4 th -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.

Texas A&M Nuclear Engineering 1876 Marvin L. Adams25 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

Texas A&M Nuclear Engineering 1876 Marvin L. Adams26 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  ’s, and surface leakages  Use surface leakages and region-avg  ’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!

Texas A&M Nuclear Engineering 1876 Marvin L. Adams27 Results demonstrate truly amazing computational efficiency.  Assembly-level code:  D 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%)

Texas A&M Nuclear Engineering 1876 Marvin L. Adams28 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.