OpenMC–Morana SRE-derived comparison

This example evaluates Morana against an OpenMC continuous-energy (CE) reference for a small, bare, sodium-cooled graphite reactor model. Selected dimensions and material conditions are drawn from the Sodium Reactor Experiment (SRE), but the model includes only the features needed for this tutorial. It is not a historical reconstruction, benchmark, or validation case.

The calculation connects the full workflow: a heterogeneous OpenMC unit cell supplies homogenized multigroup cross sections (MGXS), Morana solves the corresponding 61-cell diffusion model, and an independently simulated heterogeneous OpenMC mini-core supplies the reference multiplication factor and fission-neutron production profiles.

Problem definition

Geometry

The transverse unit cell is an 11-in-pitch, point-up hexagon. It contains a graphite prism and zirconium can, one central and six peripheral uranium-metal fuel rods, a NaK bond and stainless-steel tube around each rod, a zirconium process channel, sodium coolant, and an axially invariant representation of the documented spiral spacer wire. The outer sodium sliver apportions half of the 0.170-in gap on each side of the can to the cell.

The controlling dimensions follow Starr and Dickinson (1958) unless the source column identifies a derived treatment:

Feature	Selected value	Source or treatment
Fuel-element pitch	11.000 in	Table 2-1
Zirconium can wall	0.035 in	§2-2.2
Inter-can sodium gap	0.170 in	§2-2.2
Graphite prism across flats	10.760 in	Pitch less two can walls and the sodium gap
Fuel diameter	0.750 in	Table 2-1
NaK bond / steel tube thickness	0.010 / 0.010 in	Table 2-1
Process-channel ID / wall	2.800 / 0.035 in	Table 2-1
Spacer-wire diameter	0.091 in	§5-4.1; transverse surrogate
Wire-to-channel-wall clearance	0.033 in	Derived from tangent outer wires and the selected process-channel ID
Active height	72.0 in	Six-foot principal core height

The three-dimensional mini-core is a complete five-ring cluster of 61 copies of this cell. Its 156 shared internal edges are transmissive, and the 54 exposed radial edges are vacuum boundaries. Vacuum conditions also apply at the top and bottom. There is no enclosing sodium region, reflector, control or safety element, shim element, or special peripheral cell. This intentionally simple layout isolates homogenization and diffusion effects.

Materials and physical state

Every material is at 658.15 K, the arithmetic mean of the documented 500 °F inlet and 950 °F outlet temperatures in Starr and Dickinson (1958), §2-2.1. The tutorial compositions and solid-material densities use Detwiler et al. (2021) where identified below.

Material	Composition	Density [g/cm³]	Source or treatment
Fuel	U metal, 2.8 wt% U-235	18.944	Enrichment rounded from SRE Table 2-1; density from PNNL
Graphite	Reactor-grade graphite with 1 ppm B	1.700	PNNL
Zirconium	Natural Zr	6.52	PNNL
Steel	SS-304	8.03	PNNL entry 331, p. 226
NaK	78.6 wt% K / 21.4 wt% Na	0.78143	Composition and 658.15-K correlation from Black and Wulff (1972)
Sodium	Natural Na	0.861170	658.15-K saturation-curve correlation from Fink and Leibowitz (1995)

The recorded OpenMC calculations used a locally corrected OpenMC 0.16.0 build for MGXS continuations (see the restart requirements) with ENDF/B-VIII.1 continuous-energy data. OpenMC applied the c_Graphite thermal-scattering treatment and interpolated the available nuclear data to the common material temperature.

Simulation approach

The comparison has three calculation stages. The energy-group and axial-mesh studies choose their resolutions without fitting them to the CE results.

1. Continuous-energy reference. OpenMC models every material region in the heterogeneous 61-cell mini-core, which is 72.0 in (182.88 cm) high. A calculation with 100 million active histories directly tallies nu-fission in 100 uniform axial bins integrated over the core and in the 61 root cells integrated over height. It uses 20,000 particles per generation, 5,200 batches (200 inactive), and seed 31415. The result qualified for final comparison only if the \(k_\mathrm{eff}\) standard deviation was at most 10 pcm and every direct axial and planar tally had a relative standard deviation of at most 1%.
2. Homogenized MGXS. A separate two-dimensional unit cell has reflecting radial boundaries. OpenMC tallies the CASMO-70 structure and condenses the same reaction rates and flux data to CASMO-40, CASMO-25, and CASMO-8. The exported record contains TransportXS, absorption, consistent ordinary P0 scattering, scattering multiplicity, and a general incident-to-outgoing nu-fission transfer matrix. OpenMC’s transport correction is stored in the runtime total field, so Morana imports it with diffusion="total" and does not apply another P1 correction.
3. Diffusion calculation. Morana assigns the one homogenized record to every planar cell, uses 20 uniform axial layers for the selected result, and applies vacuum conditions on every exterior face.

The selected MGXS statistics use three independent seeds, each with 40 million active histories, 20,000 particles per generation, five generations per batch, and 100 inactive batches. Each 40-million-active-history MGXS sample continues its corresponding 20-million-active-history sample.

Comparison measures

Both codes report fission-neutron production rather than recoverable fission power. Morana forms the cell production from the volume-integrated action of the imported fission-transfer matrix on the multigroup flux. OpenMC tallies the corresponding nu-fission response directly. The planar and axial profiles are each normalized to unit total before comparison.

For a candidate profile \(x_i\) and reference profile \(r_i\), the reported normalized RMS difference is

\[ E_{\mathrm{RMS}}= 100\frac{\sqrt{N^{-1}\sum_i(x_i-r_i)^2}} {N^{-1}\sum_i r_i}. \]

Multiplication-factor differences are reported in pcm as \(10^5(k_{\mathrm{candidate}}-k_{\mathrm{reference}})\). The CE 100-bin axial profile is summed exactly into the candidate Morana layers; no interpolation is used.

Discretization studies

The convergence studies use the same 20-million-history MGXS sample with seed 31415. Consequently, the observed changes predominantly reflect discretization rather than differences between independent Monte Carlo samples.

The production RMS values in the plots below use finer Morana calculations as their references, not the CE reference: CASMO-70 for the energy-group study and the 100-layer CASMO-25 calculation for the axial-mesh study.

Energy groups

At 20 axial layers, the coarser structures are compared with CASMO-70:

Structure	\(\Delta k_\mathrm{eff}\) from CASMO-70 [pcm]	Axial-production RMS [%]	Planar-production RMS [%]
CASMO-8	964.460	0.035329	0.059310
CASMO-25	243.610	0.007129	0.008907
CASMO-40	203.118	0.006664	0.007964
CASMO-70	-	-	-

Even CASMO-40 differs from CASMO-70 by about 203 pcm, so CASMO-70 is retained. The production shapes are much less sensitive than the eigenvalue: spatially integrated reaction-rate errors can remain small while group condensation perturbs the neutron balance that determines \(k_\mathrm{eff}\).

The need for a relatively fine energy structure is consistent with the direct MGXS treatment. The reflected unit cell is strongly multiplying, with \(k_\infty\approx1.295\) in the three selected MGXS samples, whereas leakage reduces the bare mini-core to \(k_\mathrm{eff}\approx0.98\). Homogenization consists of direct reaction-rate and flux condensation with the reflected unit-cell spectrum; it includes no leakage-informed spectral iteration or equivalence correction. A finer energy mesh limits the amount of spectral information discarded by condensation. This interpretation is consistent with the spatial and energy weighting issues discussed by Boyd et al. (2019), but the present study does not isolate each possible source of bias.

Axial mesh

The axial study uses CASMO-25 and compares each mesh with 100 uniform layers:

Uniform layers	\(\Delta k_\mathrm{eff}\) from 100 layers [pcm]	Axial-production RMS [%]	Planar-production RMS [%]
5	336.870	0.192660	0.000894
10	86.311	0.059705	0.000228
20	21.205	0.013911	0.000058
50	2.659	0.001775	0.000007
100	-	-	-

Twenty layers leave only a 21-pcm eigenvalue difference and a 0.014% axial profile RMS relative to 100 layers. Further axial refinement is small beside the energy-condensation and final CE-to-diffusion differences, so 20 layers are used for the selected calculation.

Comparison results

The CE reference achieved \(k_\mathrm{eff}=0.9803155\pm0.0000974\) (9.742 pcm, one standard deviation). The maximum direct-tally relative standard deviations were 0.4244% in the 100 axial bins and 0.2792% in the 61 planar cells.

The selected CASMO-70, 20-layer results are:

MGXS seed	Morana \(k_\mathrm{eff}\)	Morana minus CE [pcm]	Axial-production RMS [%]	Planar-production RMS [%]
16180	0.9853717	+505.620	0.137999	1.013620
27182	0.9855220	+520.642	0.138037	1.013543
31415	0.9854358	+512.029	0.137953	1.013732
Mean	0.9854432	+512.763	0.137996	1.013632

Across the three MGXS samples, Morana \(k_\mathrm{eff}\) ranges from 0.9853717 to 0.9855220, and the Morana-minus-CE difference ranges from +505.620 to +520.642 pcm. This 15.022-pcm span is much smaller than the roughly 513-pcm CE-to-Morana difference. The residual difference is systematic for this model and may combine whole-cell homogenization, discretization, spectrum mismatch, the diffusion approximation, and boundary treatment; this comparison does not assign it to one mechanism.

Planar production

The maps below integrate production over the full active height and show the mean of the three selected Morana samples. The first two panels divide each 61-cell profile by its own cell mean. The error panel shows Morana minus OpenMC as a percentage of the OpenMC cell mean, matching the normalization used by the planar RMS measure.

Both calculations reproduce the strong center-to-edge falloff. Morana is slightly low in the inner core and high around much of the perimeter. The signed cell differences range from about -2.20% to +1.38% of the OpenMC cell mean, producing the 1.014% normalized planar RMS. The predominant smooth radial trend is consistent with a systematic leakage or homogenization effect; small symmetry-breaking features are consistent with statistical noise in the CE reference.

Axial production

The axial figure compares the direct 100-bin CE profile with the 20-layer Morana profile. Each is shown relative to its own axial-bin mean. The lower panel first rebins OpenMC exactly to 20 bins, then reports Morana minus OpenMC as a percentage of the OpenMC bin mean.

The broad vacuum-boundary shape agrees closely. Differences remain within about 0.30% of the OpenMC bin mean, and the normalized axial RMS is 0.138%. The small antisymmetric component is unphysical for this axially symmetric model and is consistent with statistical noise in the shared CE direct-tally reference. Because each comparison uses that same CE record, the component is present across the three MGXS samples and does not represent seed-to-seed Morana variation.

Reproduce the example

The reproduction workflow covers environment setup, geometry checks, CE and MGXS execution, restart rules, artifact locations, Morana execution, comparison commands, and documentation-figure regeneration for examples/openmc_comparison/. Raw statepoints, evaluated nuclear data, runtime-MGXS libraries, result archives, and NumPy arrays are deliberately not distributed.

References

Black and Wulff (1972). W. Z. Black and W. Wulff, Space Radiator Simulation System Analysis, NASA Contractor Report NASA-CR-128595, April 1972. NASA bibliographic record.

Boyd et al. (2019). W. Boyd, A. Nelson, P. K. Romano, S. Shaner, B. Forget, and K. Smith, “Multigroup Cross-Section Generation with the OpenMC Monte Carlo Particle Transport Code,” Nuclear Technology, 205(7), 928–944, 2019, DOI 10.1080/00295450.2019.1571828. Open full text.

Detwiler et al. (2021). R. S. Detwiler, R. J. McConn Jr., T. F. Grimes, S. A. Upton, and E. J. Engel, Compendium of Material Composition Data for Radiation Transport Modeling, 200-DMAMC-128170, PNNL-15870, Rev. 2, Pacific Northwest National Laboratory, April 2021. Open full text.

Fink and Leibowitz (1995). J. K. Fink and L. Leibowitz, Thermodynamic and Transport Properties of Sodium Liquid and Vapor, ANL/RE-95/2, Argonne National Laboratory, January 1995. Open full text.

Starr and Dickinson (1958). Ch. Starr and R. W. Dickinson, Sodium Graphite Reactors. Addison-Wesley Publishing Company, Inc., 1958. Open full text.