Math & Physics Proofs

The 400 m public-access line: bigger than the lab's land, and a state highway runs inside it

What the document says

Valar's accident analysis computes public doses at a site boundary 400 m from the reactor, which it calls 'the nearest point of public access', with the nearest residence at about 800 m and the nearest group at 1,200 m. It concludes the 400 m boundary 'provides adequate protection with substantial margin'.

Given

• r = 400 m, the distance to the nearest point of public access (NSDA pp. 10-11, 37-39)
• 1 acre = 4,046.856 m² (43,560 square feet; 1 ft = 0.3048 m exactly) (definition)
• A_site = 20.6 acres, the lab site DOE describes (DOE categorical exclusion DOE-ID-26-005)
• A_lab = 34.25 acres, the state lab parcel 04-0019-0029 (State of Utah DFCM) that holds the lab's address (Emery County parcel map via UGRC (outline); the state's government-owned-parcel layer records 34.2 acres (the county layer gives no acreage for this parcel))
• A_valar = 112.72 acres, parcel 04-0019-0030, which Emery County approved selling to Valar (Emery County parcel map via UGRC)
• SR 57 = Coal Haul Road beside the lab: State Route 57, a UDOT state highway (funding class A), paved, 65 mph, no gate or authorized-only flag (UGRC/UDOT road centerlines)

Working

1. A_circle = π r² = π × (400 m)² = 502,655 m²
   Everything within 400 m of the reactor is a circle of radius 400 m.
2. 502,655 m² ÷ 4,046.856 m² per acre = 124.21 acres
   The zone the analysis assumes the public stays out of.
3. 124.21 ÷ 20.6 = 6.0 124.21 ÷ 34.25 = 3.6
   The zone is 6 times the site DOE describes and 3.6 times the whole lab parcel, so the 400 m line must cross land outside the lab wherever the reactor is.
4. 34.25 + 112.72 = 146.97 acres > 124.21 acres
   Adding the parcel Emery County approved selling to Valar gives more area than the circle, so area alone does not settle it; the shapes do.
5. For each spot c on the lab parcel: A_out(c) = A_circle − A(circle ∩ lab) − A(circle ∩ Valar parcel)
   5,569 spots on a 5 m grid covering the parcel. Each circle was clipped against the official parcel outlines (each circle drawn with 180 sides; the best spot re-done with 1,440). A second method, counting 2 m grid points inside each circle, agrees within 0.05 acre.
6. min A_out = 21.8 acres (18%) median ≈ 56 acres max = 86.5 acres (70%)
   Even the best spot for Valar, on the lab parcel's north edge, leaves 21.8 acres inside the line on other land (the best 5 m grid spot gives 21.9; refining it to 0.2 m gives 21.8). None of that land is listed as government land in the state's government-owned-parcel layer; the county data we can read does not name the owners.
7. d(spot, SR 57) over 5,569 spots (5 m grid): min 17.2 m, max 496.2 m; d ≤ 400 m for 86.7% of the parcel
   State Route 57 runs along the lab parcel's west edge; the parcel's edge comes within about 16 m of its centerline. At the lab's address point it is 149 m away; in the largest patch of new 2026 change, about 365 to 390 m depending on the exact point, just inside the line.
8. Nearest residential address point: 987 m from the lab parcel's edge; none within 800 m
   Credit where due: the NSDA's '800m to nearest residence' (p. 10) is on the safe side. (Counted from the state's address points typed Residential; no address is shown. No address point of any type other than the lab's own lies within 800 m.)

Result

Wherever Ward 250 sits on the lab parcel, at least 21.8 acres inside its 400 m line lie outside both the state lab parcel and the land Emery County approved selling to Valar, and for about 87% of the possible spots part of SR 57, a public state highway, lies inside the line. The nearest address point the state lists as residential is 987 m from the lab parcel, farther than the 800 m Valar assumed.

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · pp. 10-11, 37-39
• NEPA Determination DOE-ID-26-005: Valar Ward250 Research Reactor (categorical exclusion B5.26) (U.S. Department of Energy, Office of Nuclear Energy (Idaho), 2026-04-22) · pp. 1-2 (20.6-acre site, fenced area, controlled access), p. 5 (DSA offsite doses)
• Emery County parcels (UGRC LIR): 04-0019-0029 and 04-0019-0030 and their neighbors (Utah Geospatial Resource Center / Emery County, 2024-07)
• Government-owned parcels (statewide) (Utah Geospatial Resource Center)
• Utah road centerlines: Coal Haul Rd = SR 57, UDOT, class A, 65 mph (Utah Geospatial Resource Center / UDOT)
• Utah address points (residential points counted by distance only) (Utah Geospatial Resource Center)
• Sentinel-2 L2A scene, 20 Sept 2026 (and 24 Sept 2021: S2A_MSIL2A_20210924T181101_R084_T12SVJ_20210925T143632) (European Union Copernicus programme / ESA (contains modified Copernicus Sentinel data 2021, 2026))
• USGS The National Map, imagery (NAIP orthoimagery; service refreshed June 2024; acquisition date not stated) (U.S. Geological Survey)

Checked: We checked the parcel, road, government-land and address layers again. We then recomputed every area with an exact circle-and-polygon method in a different map projection, and checked it against a fine raster count. We re-read the quoted pages of Valar's safety agreement and DOE's determination as printed. We also redid the satellite change detection. The findings hold; the true minimum is 21.8 acres rather than 21.9, and the captions now disclose how the figure center was chosen and carry the image attributions. Revised before publication (2026-10-01): wording made more exact against the cited records.

The paper's 400 m dispersion factor is 3 to 12 times below the standard rural method for its own stated weather

What the document says

For the worst-case accident, the paper assumes stable night air ('Stability Class F'), a 1 m/s wind, a ground-level release at 1.5 m and no credit for buildings stirring the air, and uses a dispersion factor of about 2.5 × 10⁻⁴ s/m³ at the 400 m boundary. It reports the public dose there as under 1 mSv (0.1 rem), and in an earlier, preliminary section under 0.5 rem, and compares it with a 25 rem guideline and a 1 rem figure it labels 'USNRC NPUF' (Table 4, p. 38).

Given

• x = 400 m = 0.4 km, the site boundary (NSDA p. 36)
• u = 1 m/s wind speed (NSDA p. 34)
• Stability = Pasquill class F ('moderately stable atmosphere') with a 1 m/s wind; ground-level release at 1.5 m; no plume rise or building-wake credit (from: NSDA pp. 11, 34, 36) (NSDA pp. 11, 36)
• χ/Q (paper) = 2.5 × 10⁻⁴ s/m³ (NSDA p. 36; Valar's draft value (NSDA Rev 02), not the DOE-approved analysis)
• σ_y, σ_z = the standard rural Pasquill-Gifford plume widths for class F (EPA's ISC3 fits) (EPA ISC3 User's Guide, Vol. II, Tables 1-1 and 1-2)
• M = 4, the low-wind meander allowance for class F at wind ≤ 2 m/s (NRC Regulatory Guide 1.145 Rev. 1, Reg. Position 1.3.1 and Figure 3 (p. 1.145-9))

Working

1. χ/Q = 1 ÷ (π · σ_y · σ_z · u)
   The textbook ground-level, plume-centerline form of the Gaussian plume model, the model the paper says it uses (p. 11).
2. θ = 0.017453293 × (4.1667 − 0.36191 × ln 0.4) = 0.07851 rad σ_y = 465.11628 × 0.4 × tan θ = 14.64 m
   Sideways spread at 400 m for class F (EPA ISC3, Table 1-1).
3. σ_z = 14.457 × 0.4^0.78407 = 7.05 m
   Vertical spread at 400 m for class F (EPA ISC3, Table 1-2).
4. χ/Q = 1 ÷ (π × 14.64 × 7.05 × 1) = 3.09 × 10⁻³ s/m³
   12.3 times the paper's 2.5 × 10⁻⁴.
5. With low-wind meander (σ_y × 4): χ/Q = 3.09 × 10⁻³ ÷ 4 = 7.71 × 10⁻⁴ s/m³
   Following RG 1.145's rule (take the higher of Eqs. 1 and 2, then the lower of that and Eq. 3), this is the most generous allowance NRC guidance gives for class F in light winds: still 3.1 times the paper's number. DOE-STD-3009-2014 allows meander only 'consistent with the accident release duration', and the paper assumes an instantaneous release (p. 34).
6. u needed for 2.5 × 10⁻⁴ = 3.09 × 10⁻³ ÷ 2.5 × 10⁻⁴ × 1 m/s = 12.3 m/s
   ... Longer averaging times do not close the gap: DOE-STD-3009-2014 sets a nominal 2-hour exposure (8 hours at most), and even a 24-hour correction (σ_y ∝ t^0.2 from a 10-minute base) leaves it 4.6 times low. Curves that do reach 2.5 × 10⁻⁴: the urban (city) family at about 451 m, or rural curves with the class F meander allowance at about 780 m (2.40 × 10⁻⁴ at 800 m, the paper's nearest-residence distance). The paper names neither.
7. Dose scales with χ/Q: 0.1 rem × 3.1 = 0.31 rem; 0.1 rem × 12.3 = 1.23 rem 0.5 rem × 3.1 = 1.5 rem; 0.5 rem × 12.3 = 6.2 rem Table 4 margin against 1 rem: '>10×' becomes about 3.2×, or below 1
   ... The 1 rem figure matches one of the NRC's two criteria for sizing an emergency planning zone for NRC applicants (1 rem over 96 hours, 10 CFR 50.33(g)(2)(i)(A)): a planning test, not a limit. The paper's own case for on-site-only emergency planning (p. 54) rests on its 0.5 rem boundary bound, which rescales to 1.5 to 6.2 rem.

Result

For the weather Valar's paper says it assumes, the standard rural method gives an air concentration at 400 m 12.3 times the paper's figure, or 3.1 times with the low-wind allowance. The public-dose bounds scale up by the same factors: still below the 25 rem guideline, so the paper's no-Safety-Class conclusion holds, but the 1 rem figure the paper cites (the NRC's emergency-planning-zone test for research reactors it licenses) could shrink from more than 10 times to about 3 times, or be lost at the bound. These are bounds, not predicted doses.

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · pp. 11, 34, 36-38
• User's Guide for the Industrial Source Complex (ISC3) Dispersion Models, Vol. II (EPA-454/B-95-003b), Tables 1-1 to 1-4 (U.S. Environmental Protection Agency, 1995)
• Regulatory Guide 1.145, Atmospheric Dispersion Models for Potential Accident Consequence Assessments at Nuclear Power Plants (Rev. 1) (U.S. Nuclear Regulatory Commission)
• DOE-STD-3009-2014, Preparation of Nonreactor Nuclear Facility Documented Safety Analysis, s. 3.2.4.2 Radiological Dose Consequence (Atmospheric Dispersion), pp. 21-24 (U.S. Department of Energy)
• 10 CFR 50.33(g)(2): emergency planning zone for non-power reactors (1 rem over 96 hours) (Legal Information Institute (Cornell))

Checked: We recomputed every number, using EPA's ISC3 Vol. II coefficient tables, NRC Regulatory Guide 1.145 Rev. 1 (Figure 3 checked on the page) and DOE-STD-3009-2014 s.3.2.4.2. We re-read NSDA pp. 11, 34-38, 47 and 54 as printed. The 12.3x and 3.1x factors, the 12.3 m/s wind, the 451 m and roughly 780 m fits and the dose rescaling all hold. The corrected wording attributes the paper's own conservatism claims, narrows the emergency-planning statement and fixes the DOE standard citation. Revised before publication (2026-10-01): wording made more exact against the cited records.

The fuel: Valar's safety paper says 4.95%; the fuel DOE's shipping review approved for shipment to Ward 250 is up to 19.9%

What the document says

The joint Valar-Los Alamos NOVA release on Valar's site calls the NOVA core 'HALEU TRISO-fueled' and says NOVA uses 'the same fuel' as Ward250.

Given

• e (paper) = 4.95% U-235 (NSDA Table 6, p. 52)
• e (DOE) = up to 19.9% U-235 by weight ('less than 20 weight percent') (DOE Model 9979 review, p. 2 (HALEU, shipments to the Ward 250 site), p. 5 (19.9 wt.% maximum; the same page also says '20.0 wt.% maximum'), p. 9 (condition 4, 'less than 20 weight percent'))
• HALEU = uranium enriched from 5% to less than 20% (DOE Office of Nuclear Energy)

Working

1. 19.9% ÷ 4.95% = 4.02
   The fuel DOE cleared for shipment to Ward 250 can be about four times as enriched as the paper says.
2. 4.95% < 5% ⇒ not HALEU; 5% ≤ 19.9% < 20% ⇒ HALEU
   By DOE's own definition, the paper's figure is ordinary low-enriched fuel and the fuel cleared for shipment is high-assay (HALEU), the grade Valar's NOVA release names for the fuel it says Ward 250 shares.
3. Copies checked, 7 Nov 2025 to late September 2026: 8 (seven Internet Archive captures plus the live file), all identical; copies with a different enrichment: 0
   No revised public copy was found after DOE's shipping review (20 May 2026) or DOE's 18 June 2026 announcement of Ward 250's first criticality.

Result

The only public safety paper for Ward 250 lists 4.95% fuel, while DOE's shipping review covers HALEU fuel for Ward 250 at up to 19.9%, as much as about four times as enriched. The paper calls its fuel assumptions preliminary, to be checked against Ward 250's actual fuel specifications in a later analysis (the PDSA) and updated if the fuel differs from the AGR-program specification; no public copy of that analysis was found.

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · p. 52 (Table 6)
• Safety Evaluation Report, Letter Amendment of Certificate of Compliance No. 9979 for the Model 9979 Package, Docket 26-35-9979 (U.S. Department of Energy, Packaging Certification Program, 2026-05-20) · pp. 5, 9
• What is High-Assay Low-Enriched Uranium (HALEU)? (U.S. Department of Energy, Office of Nuclear Energy, 2024-12-03)
• Los Alamos National Laboratory and Valar Atomics Announce Project NOVA Criticality Milestone in Nevada (joint release on Valar's site) (Valar Atomics / Los Alamos National Laboratory, 2025-11-18)
• Ward250 NSDA Rev 02: latest of seven Internet Archive captures (7 Nov 2025 to 18 Sept 2026) (Valar Atomics (via Internet Archive), 2025-10)

Checked: We re-read NSDA pages 11, 26 and 52 and DOE's shipping review (pages 1, 2, 5 and 9) as printed, checked the current versions of DOE's HALEU definition and Valar's NOVA release, and compared every Internet Archive copy of the safety paper: all 8 public copies are identical and say 4.95%. The ratios and the energy-to-fission arithmetic were recomputed.

The temperature: a goal 100 °C above its own limit

What the document says

The paper sets 'Maximum outlet temperature: 650°C' as a safety limit (p. 25; Table 6, p. 52, '650°C nominal'). Its objectives say 'Demonstrate core outlet temperatures of 750°C under normal operations' (p. 8), and its creep check uses 750 °C as the 'maximum operating temperature' (p. 49).

Given

• T_limit = 650 °C (NSDA p. 25 (§5.4 safety limit); p. 52 (Table 6, '650°C nominal', basis 'Material limits'))
• T_goal = 750 °C 'under normal operations' (NSDA p. 8)

Working

1. T_goal − T_limit = 750 °C − 650 °C = 100 °C
   The stated goal exceeds the stated limit.
2. Creep check at 750 °C ≥ 650 °C limit
   The vessel creep check uses the higher figure. Creep worsens with temperature, so the check also covers operation at the 650 °C limit. The gap is an inconsistency in the paper, not evidence of unsafe operation.

Result

One paper gives two outlet temperatures for normal operation, 100 °C apart: a 650 °C safety limit (p. 25) and a 750 °C goal (p. 8). Within the paper, the safety limit outranks the goal. The Technical Safety Requirements that set actual operating limits were planned for submittal on 6 January 2026 (p. 20); no public record of them found.

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · pp. 8, 25, 49, 52
• Ward250 Nuclear Safety Design Agreement (Valar web summary) (Valar Atomics, 2025-10-13)

Checked: We compared the source PDF as of late September 2026 against the audited copy (the same file) and re-read NSDA pp. 8, 20, 25, 49 and 52 as printed. Every quote is exact, and 750 − 650 = 100 °C was recomputed. Valar's web summary ('up to 650°C') was confirmed and added as Valar's side, and the closing paragraph was reworded so it does not claim more than the paper shows about which figure binds.

How long it may run: the accident inventory assumes one tenth of the paper's own limit

What the document says

The paper's limit is 'Fuel burnup limit: 30 megawatt-days' (p. 25). Its worst-case accident inventory is built on '3 MWd total burnup' from 'Conservative 30 EFPD operation' (p. 34).

Given

• P = 100 kW = 0.1 MW continuous (NSDA p. 25)
• t = 30 effective full-power days (NSDA p. 34)
• B_limit = 30 MWd (NSDA p. 25)

Working

1. B_inventory = P × t = 0.1 MW × 30 d = 3.0 MWd
   The inventory basis is internally consistent.
2. B_limit ÷ B_inventory = 30 ÷ 3 = 10; 30 MWd ÷ 0.1 MW = 300 full-power days
   The limit allows ten times the running the accident inventory assumes.
3. At constant power, activity at shutdown ∝ 1 − e^(−ln2·t/T½). I-131 (T½ 8.0252 d): (1 − e^(−ln2·300/8.0252)) ÷ (1 − e^(−ln2·30/8.0252)) = 1.000 ÷ 0.925 = × 1.08; Cs-137 (30.08 y): × 9.92; Sr-90 (28.91 y): × 9.91
   Iodine-131 is already 92.5% of its steady level after 30 days, so the nuclide the paper names 'primary dose contributor' (p. 34) barely changes; long-lived nuclides grow almost in proportion to total running. Counting the other iodines and the noble gases, the combined inhalation-plus-cloud dose rises about 4 to 5% (estimate from standard fission yields and EPA dose coefficients).

Result

The accident inventory the paper calls conservative covers one tenth of the burnup its own limit allows. For iodine-131, which the paper names as the primary dose contributor, the difference is about 8%; for long-lived cesium-137 and strontium-90, about ten times (constant power assumed).

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · pp. 8, 25, 34
• NuDat 3 decay data (ENSDF): I-131 T1/2 8.0252 d; Cs-137 30.08 y; Sr-90 28.91 y (National Nuclear Data Center, Brookhaven National Laboratory)
• Ward 250 By the Numbers (Valar Atomics, 2026-06-28)
• Valar Atomics + SRERC in Emery County (slides posted on the Utah Public Notice Website) (Valar Atomics, 2025-07-08)

Checked: Every number was recomputed. 0.1 MW x 30 d = 3.0 MWd, and 30 MWd at 0.1 MW = 300 full-power days. Growth factors from NuDat 3 half-lives: I-131 x1.081, Cs-137 x9.92 and Sr-90 x9.91, with a combined iodine plus noble-gas dose of x1.044. The quotes on NSDA pp. 8, 25 and 34 were checked against the pages of the current PDF. Valar's 30 full-power-day statements were confirmed on its June 2026 web page and its July 2025 county slides.

The on-site-only emergency request cites a 0.5 rem whole-body figure; the paper does not address EPA's thyroid (potassium iodide) guide

What the document says

Valar asks DOE for 'On-site emergency response only' (pp. 45, 54). Its dose justification: 'MHA analysis demonstrates site boundary doses <0.5 rem TEDE, well below Protective Action Guidelines' (p. 54); it also cites the 800 m distance to the nearest residence, a limited source term and a short operating period. The paper names 'I-131 (thyroid), noble gases (external)' as primary contributors to the 400 m dose (p. 37).

Given

• D = 0.5 rem whole-body (TEDE) at 400 m, the figure the emergency argument uses (NSDA pp. 11, 54 (p. 11 places the site boundary at 400 m); Valar's draft value (NSDA Rev 02), not the DOE-approved analysis)
• PAG = 1 to 5 rem projected dose for shelter or evacuation; 5 rem child thyroid dose for potassium iodide (KI) (EPA PAG Manual 2017, Table 1-1)
• Thyroid ÷ whole body = about 16 to 27 for the paper's noble-gas-plus-iodine release, with the gases escaping in fuel-inventory proportion (26.9 with FGR 11 weighting; about 16 to 18 with the 0.05 thyroid weighting of DOE's 10 CFR 835.2). Iodine-131 alone: 2.92 × 10⁻⁷ ÷ 8.89 × 10⁻⁹ Sv/Bq = 32.8 (FGR 11 Table 2.1, adult inhalation, p. 136) (EPA FGR 11 Table 2.1 (p. 136); 10 CFR 835.2 tissue weighting factors; inventory proportions: our own calculation, assuming gases escape in fuel-inventory proportion as the paper's release assumption suggests (p. 36); not a Valar or DOE value)

Working

1. Whole-body margin: 1 rem ÷ 0.5 rem = 2
   'Well below' the lowest EPA guide for sheltering or evacuating is a factor of two.
2. Iodine-131: 2.92 × 10⁻⁷ ÷ 8.89 × 10⁻⁹ = 32.8; noble-gas-plus-iodine mix in fuel-inventory proportion: 26.9 (FGR 11 weighting, thyroid 0.03) or about 16 to 18 (thyroid 0.05, as in 10 CFR 835.2)
   If iodine and noble gases escape in proportion, as the paper's failed-fuel assumption suggests ('100% of gaseous and volatile inventory', p. 36), iodine supplies about nine-tenths of the dose and the thyroid dose is many times the whole-body figure.
3. Adult thyroid ≈ 0.5 rem × 16 to 26.9 = 8 to 13 rem > 5 rem; child: roughly twice that (EPA PAG Manual s. 2.2.1, p. 15)
   If a release reached the paper's 0.5 rem and was driven by iodine, the child thyroid dose would be above the 5 rem level at which EPA says KI should be considered. The child figure stays above 5 rem as long as iodine supplies more than about a quarter of the 0.5 rem. The paper does not mention potassium iodide or give a thyroid dose.
4. With the paper's other figure, 0.1 rem: child thyroid ≈ 3 to 5 rem
   At the tighter figure (400 m, p. 37) the child thyroid dose is at or just under the guide; the emergency argument on p. 54 does not use it. If the dispersion factor is understated (proof pf-002), both figures would rise.

Result

The paper's dose argument for on-site-only emergency planning uses its looser 0.5 rem figure (p. 54; it also cites the 800 m distance to the nearest residence, a limited source term and a short operating period). If iodine and noble gases escape in fuel-inventory proportion, as the paper's failed-fuel assumption (p. 36) suggests, that figure implies roughly 8 to 13 rem to an adult thyroid at the 400 m boundary and about twice that for a young child, above EPA's 5 rem potassium-iodide guide. The child figure stays above 5 rem as long as iodine supplies more than about a quarter of the dose. The paper does not mention potassium iodide or give a thyroid dose. These are upper bounds derived from Valar's stated ceiling; Valar says its source term 'substantially exceeds any realistic release scenario' (p. 11). At the paper's other boundary figure, 0.1 rem (p. 37), the child thyroid dose is about 3 to 5 rem. Valar says DOE approved the NSDA (13 October 2025); no DOE record of what DOE accepted on emergency planning was found.

Sources

• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release) (Valar Atomics, 2025-10) · pp. 11, 20-21, 36, 37, 45, 54
• PAG Manual: Protective Action Guides and Planning Guidance for Radiological Incidents (EPA-400/R-17/001), Table 1-1 (U.S. Environmental Protection Agency, 2017)
• Federal Guidance Report No. 11 (EPA-520/1-88-020), Table 2.1: inhalation dose coefficients (U.S. Environmental Protection Agency, 1988)
• Ward250 Nuclear Safety Design Agreement (web summary, Valar library) (Valar Atomics, 2025-10-13)
• 10 CFR 835.2, Definitions (tissue weighting factors; thyroid 0.05) (Legal Information Institute (Code of Federal Regulations))

Checked: We re-read the pages cited from Valar's safety agreement (11, 36, 37, 45, 54) as printed, along with EPA's 2017 PAG Manual (Table 1-1, p. 6; s. 2.2.1, p. 15) and FGR 11 Table 2.1 (p. 136), all from EPA's own PDFs. The thyroid-to-effective ratios (32.8 for iodine-131, 26.9 for the release mix, about 16 to 18 under DOE's 0.05 thyroid weighting) and the resulting 8 to 13 rem adult and 3 to 5 rem child figures were recomputed in late September 2026. Revised before publication (2026-10-01): wording made more exact against the cited records.

Five minutes holding Ward One's used fuel: by our math, 8 to 33 times a CT scan's top dose a day after shutdown, not one; no supporting calculation was found on Valar's site

What the document says

Valar's April 2025 post describes Ward One as a 100 kWt high-temperature gas reactor using TRISO fuel, with a planned operational lifetime of less than a month, and says: "Our analysis indicates that holding the spent fuel from this system for five minutes" gives the same radiation exposure as a CAT (CT) scan. The post does not say how much fuel is held, how long the reactor ran, how long the fuel cooled, how far it is from the body, or which dose is meant, and the analysis it cites was not found on Valar's website as of late September 2026. The text is the same in the live page (late September 2026) and in Internet Archive captures of 23 March and 22 July 2026.

Given

• P₀ = 100 kW thermal (Valar post, Ward One paragraph)
• Fuel = TRISO fuel in a high-temperature gas reactor; fuel mass, enrichment and element type not stated (Valar post, Ward One paragraph)
• T = run of 1 to 30 full-power days (the post gives only a planned operating lifetime of less than a month); break-even run length also solved (Valar post; the range is an assumption)
• t_hold = 5 minutes = 300 s (Valar post)
• f = share of the core's fission products held: the whole core (the literal reading), 1/76 of the core (one of Ward 250's 76 fuel elements, as a stand-in), one 6 cm pebble holding 7 g of uranium (1/35,700 of a 250 kg core), and the break-even share (assumption; Ward 250 NSDA Table 6, p. 52; INL/EXT-20-60236 Table 2)
• t = cooling time after shutdown: 1 day to 1 year shown, break-even solved; less than 1 day not modeled (it would give more) (assumption (the post states none))
• M = 250 kg of uranium in the core (stand-in; a larger core spreads the fission products more thinly per gram) (Ward 250 NSDA Table 6, p. 52 (a different Valar reactor of the same 100 kW))
• E_f = 200 MeV per fission, so 3.12 × 10¹⁰ fissions per watt-second (Lamarsh & Baratta, Introduction to Nuclear Engineering, ch. 3)
• y, T½, gamma lines = U-235 thermal cumulative fission yields (e.g. Zr-95 6.50%, Ba-140 6.21%, Te-132 4.30%, I-133 6.70%); half-lives and principal gamma lines of 43 fission products (ENDF/B-VII.1 and JEFF-3.1 yields; ENSDF decay data via NuDat 3)
• μ/ρ, μ_en/ρ = graphite and uranium attenuation; air and soft-tissue energy absorption (tissue = 0.990 × water, ICRU-44) (NIST, Hubbell & Seltzer (NISTIR 5632))
• E/K_a = 1.00 to 1.43 Sv of effective dose per Gy of air kerma, front-on (AP) exposure (ICRP Publication 74, Table A.17)
• Pebble = 6 cm ball: 2.5 cm fueled zone inside a 0.5 cm fuel-free shell; matrix graphite 1.70 g/cm³ (1.75 used); 7 g of uranium (the HTR-PM value; INL's benchmark pebble holds 9 g) (INL/EXT-20-60236 Rev. 1, Tables 2-3)
• d = body reference point 50 cm from the near surface of what is held (30 cm as a sensitivity) (assumption)
• CT = 1 to 10 mSv effective dose for typical diagnostic CT; FDA table values from 2 mSv (head) to 16 mSv (coronary CT angiogram) (FDA, What are the Radiation Risks from CT?, text and Table 1)

Working

1. F = 1.0 × 10⁵ W ÷ (200 MeV × 1.602 × 10⁻¹³ J/MeV) = 3.12 × 10¹⁵ fissions/s 30 full-power days: 0.1 MW × 30 d = 3.00 MWd = 8.09 × 10²¹ fissions; 1 full-power day: 0.10 MWd
   The radioactive fragments in spent fuel depend on how many uranium atoms were split, that is on power × running time, not on how much uranium was loaded. 200 MeV per fission is the textbook value; 193 MeV would raise every dose below by about 4%. The post's 'less than a month' is taken as a run of 1 to 30 full-power days.
2. A_i(T, t) = y_i × F × (1 − e^(−λ_i T)) × e^(−λ_i t) daughters (La-140 from Ba-140, Nb-95 from Zr-95, I-132 from Te-132 and others) by the two-member Bateman equations
   Activity of each gamma-emitting fission product after running for T and cooling for t, from U-235 fission yields and half-lives. Gases and iodine are taken to stay inside the intact fuel particles.
3. S_γ = Σ_i A_i × (gamma energy per decay)_i whole core, 30-day run: 164 W at 1 day of cooling, 25 W at 30 days, 0.49 W at 1 year
   Gamma-ray power given off by the fission products. Cross-check: half of the Way-Wigner decay-heat formula, 0.0622 × [t^−0.2 − (t + T)^−0.2] × 100 kW (t and T in seconds), gives 159 W at 1 day and 21 W at 30 days; gamma rays carry roughly half of fission-product decay heat. At 1 day lanthanum-140 (34%) and iodine-132 (21%) give most of it; at 30 days, lanthanum-140, zirconium-95 and niobium-95.
4. One CT in five minutes: (1 to 10 mSv) ÷ (5/60 h) = 12 to 120 mSv/h
   FDA gives typical effective doses of 1 to 10 mSv for diagnostic CT. The claim is counted as met at any five-minute dose up to 10 mSv, the top of that range.
5. Piece held = a solid ball of fuel: R = [3 f M ÷ (4π ρ_HM)]^(1/3), ρ_HM = 7 g ÷ 113.1 cm³ = 0.0619 g/cm³, M = 250 kg whole core R = 98.8 cm; 1/76 of the core R = 23.3 cm; one pebble: 2.5 cm fueled zone inside a 0.5 cm graphite shell μ = 1.75 g/cm³ × (μ/ρ)_graphite + ρ_HM × (μ/ρ)_uranium = 0.139 per cm at 0.7 MeV
   f is the share of the core's fission products held. Ward One's fuel mass and form were not found in public records, so Valar's Ward 250 figures (250 kg of uranium, 76 fuel elements; a different reactor) and a standard 6 cm pebble stand in. A ball of fuel with no gaps, at a graphite density above INL's 1.70 g/cm³, absorbs more of its own radiation than a real core would, which lowers every dose here.
6. φ(d) = (S_v ÷ 2μ) ∫₀^(π/2) sin α × (R/d) × (cos β ÷ cos α) × e^(−μs) × (1 − e^(−2μR cos β)) dβ, sin α = (R/d) sin β, s = path through the pebble's fuel-free shell (zero for a uniform ball)
   The exact rate at which gamma energy leaves the ball without scattering and reaches a point at distance d from its center (S_v is the gamma power per cm³). Checks: it reproduces the closed-form contact result, the point-source result far away, and an independent shell integral to six digits.
7. Whole body: E = 300 s × Σ φ(R + 50 cm) × (μ_en/ρ)_air × (E/K_a)_AP Hand: H = 300 s × Σ φ(R) × (μ_en/ρ)_tissue, (μ_en/ρ)_tissue = 0.990 × (μ_en/ρ)_water
   Effective dose, the quantity FDA uses for CT, is estimated from the air dose at a point 50 cm from the near surface of the fuel, converted with ICRP Publication 74's front-on factors (1.0 to 1.4 Sv per Gy). The hand dose is the dose to the skin where it touches the fuel. It is not an effective dose and is not compared with the CT figure.
8. 30-day run, whole-body dose in five minutes: whole core: 334 mSv at 1 day of cooling, 54 mSv at 30 days, 0.99 mSv at 1 year 1/76 of the core: 64, 10.3 and 0.19 mSv one 7 g pebble: 0.77, 0.12 and 0.0024 mSv
   Against a 10 mSv CT, the whole core gives 33 times as much at 1 day and 5.4 times at 30 days; one pebble is below FDA's 1 to 10 mSv range. Hand on the fuel at 1 day: 1,379 mSv (core surface), 1,182 mSv (1/76) and 316 mSv (pebble). The whole core and the 1/76 piece (about 100 kg in this model) cannot be lifted; they are shown because the post says 'the spent fuel'.
9. 1-day run, whole-body dose in five minutes: whole core: 75 mSv at 1 day of cooling, 3.0 mSv at 30 days; 1/76 of the core: 14.6 and 0.56 mSv; one pebble: 0.18 and 0.006 mSv
   The low end of 'less than a month'. A shorter or lower-power run gives less (see the break-even run length below).
10. Solve E(f) = 10 mSv for the share f held: 30-day run, 1 day of cooling: f = 1/1,589, about 157 g of uranium or 22 pebbles' worth in a 250 kg core (a ball 17 cm across) 30-day run, 30 days: f = 1/81 (3.1 kg); 1-day run, 1 day: f = 1/156 (1.6 kg) for a 1 mSv CT: 8.7 g, 85 g and 50 g
    How small a piece makes five minutes equal one CT, as whole-body dose. Held 30 cm from the body instead of 50 cm, the 157 g falls to 48 g; against FDA's highest table value (16 mSv, coronary CT angiogram) it rises to 310 g.
11. One pebble-sized piece: E = 10 mSv when it holds 1/2,759 of the core's fission products (30-day run, 1 day of cooling)
    So the single-pebble result needs a core of at least about 2,760 pebbles' worth of fuel (about 19 kg of uranium at 7 g each). A 250 kg core holds about 35,700. Ward One's core size was not found in public records.
12. Solve E(t) = 10 mSv for the cooling time t, and E(T) = 10 mSv for the run length T
    Whole core: 128 days of cooling after a 30-day run, 9.6 days after a 1-day run (365 and 62 days for a 1 mSv CT). 1/76 of the core: 31 days and 1.4 days. One day after shutdown, the whole core matches a 10 mSv CT only if the reactor ran about 2.1 full-power hours (8.7 kWd).
13. Hand dose, one 7 g pebble, 30-day run, 1 day of cooling: 316 mSv in five minutes; for the 157 g break-even piece: 833 mSv
    Skin dose at the point of contact from gamma rays alone; beta particles would add more if bare fuel were touched. It is a different quantity from a CT's whole-body effective dose. For scale, the NRC's annual limit for the skin of a worker's hands is 500 mSv (10 CFR 20.1201). The pebble's hand dose falls to 10 mSv after about 129 days of cooling (30-day run) or 9 days (1-day run).
14. Sensitivities, 30-day run, 1 day of cooling (factor on the whole-body dose): body 30 cm away instead of 50 cm: × 1.4 (core) to × 2.6 (pebble); scattered photons (buildup): × 1.35 to × 2.4 neptunium-239 from U-238 capture: × 1.04 to × 1.12; Ward 250's core volume instead of a solid ball: × 1.1 to × 1.2 9 g pebble (INL benchmark) instead of 7 g: × 1.27; only the 18 strongest emitters counted: × 0.89 to × 0.90
    Each choice the post leaves open was set to lower the dose; these factors show how far each could move the results. None of them moves a case across the 10 mSv line at 1 day of cooling.

Results: five minutes holding Ward One's spent fuel (mSv; a typical CT scan is 1 to 10 mSv whole-body)

Whole-body = effective dose, the quantity FDA gives for CT, at a point 50 cm from the near surface of the fuel. The last column is the skin dose where the hand touches the pebble: a different quantity, not comparable with a CT's whole-body figure. 'All the spent fuel' and 'one element' (about 100 kg in this model) could not be lifted by hand; they show what the post's words 'the spent fuel' mean taken literally. Scattered photons, beta particles and neptunium-239 are left out, so every figure is a lower estimate. Run length is in full-power days at 100 kW.

Result

With the post's 100 kWt and a run of 1 to 30 full-power days, five minutes with all of the reactor's spent fuel, 50 cm from the body, gives a whole-body effective dose of about 75 to 334 mSv one day after shutdown and 3 to 54 mSv after 30 days, against FDA's typical 1 to 10 mSv for a CT. It comes down to one 10 mSv CT only for a piece holding no more than about 1/1,600 of the core's fission products one day after a 30-day run (1/156 after a one-day run; 1/81 after 30 days of cooling), about 157 g of uranium or 22 pebbles' worth if the core held 250 kg; or, for the whole core, after about 10 to 128 days of cooling. One 7 g pebble gives about 0.8 mSv whole-body, below FDA's typical CT range, while the skin of the hand touching it receives about 316 mSv in the same five minutes, a different dose quantity. Scattered photons, beta particles and neptunium-239 are left out, so these are lower estimates.

Sources

• Valar Atomics is Suing the NRC (Valar Atomics, 2025-04-07) · 'Our Vision' section, Ward One paragraph
• Valar Atomics is Suing the NRC: Internet Archive capture of 23 March 2026 (Valar Atomics (via Internet Archive), 2025-04-07) · 'Our Vision' section, Ward One paragraph
• Valar Atomics is Suing the NRC: Internet Archive capture of 22 July 2026 (Valar Atomics (via Internet Archive), 2025-04-07) · 'Our Vision' section, Ward One paragraph
• What are the Radiation Risks from CT? (U.S. Food and Drug Administration, 2017-12-05) · Radiation Dose from CT Examinations; Table 1
• Radiation Dose from X-Ray and CT Exams (RadiologyInfo.org (Radiological Society of North America and American College of Radiology), 2025-04-15) · Effective radiation dose in adults (table, and its note citing ICRP Publication 103 on partial-body exposure)
• The nuclear fuel cycle (Nuclear explained) (U.S. Energy Information Administration, 2023-10-26) · The back end of the nuclear fuel cycle
• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft; issued for public release); used only for stand-in core figures (Valar Atomics, 2025-10) · p. 34 (3 MWd, 30 EFPD inventory basis); p. 52, Table 6 (100 kWth; core 1.5 m by 1.8 m; 76 fuel elements; 250 kg heavy metal)
• Benchmark for Fuel Shuffling and Depletion for Pebble-Bed Reactors, INL/EXT-20-60236 Rev. 1 (R. Sonat Sen) (Idaho National Laboratory (U.S. Department of Energy), 2020-09) · Tables 1-3 (6 cm pebble, 2.5 cm fuel zone, 0.5 cm fuel-free shell, matrix graphite 1.70 g/cm3, 9 g heavy metal per pebble)
• NuDat 3: half-lives, gamma-ray energies and intensities (ENSDF) (National Nuclear Data Center, Brookhaven National Laboratory) · ENSDF half-lives and gamma lines, entered from the evaluated data; no copy of NNDC's data pages is kept for this check
• X-Ray Mass Attenuation Coefficients (Hubbell & Seltzer, NISTIR 5632): graphite, uranium, air, water (National Institute of Standards and Technology) · values from the published tables for graphite, uranium, air and water; No copy of NIST's page is archived
• ICRP Publication 74: Conversion Coefficients for use in Radiological Protection against External Radiation (International Commission on Radiological Protection, 1996) · Table A.17 (effective dose per air kerma, front-on). The publication is not openly published, so no copy is archived; using a flat 1.0 Sv per Gy instead changes the results by about 2%
• 10 CFR 20.1201: occupational dose limits for adults (Legal Information Institute (Cornell)) · (a)(2)(ii): 50 rem (500 mSv) shallow-dose equivalent to the skin of any extremity per year
• A startup wants to build a massive nuclear-powered data center on public land in Utah (NPR, 2026-09-30) · quotes checked against the article; no copy is archived here

Checked: Two independent calculations worked out the doses, each entering the nuclear data separately; a third check re-derived every case and traced each difference between them to a stated modeling choice (distance measured from the fuel's center or surface, density and size of the fuel ball, how many nuclides were counted, a point-source shortcut). With matching choices the third check reproduces both within 5%, and the two calculations' gamma-ray data agree within about 4%. The total gamma output agrees with the standard decay-heat formula within 3% one day after shutdown. Valar's post was compared across the live page (late September 2026) and Internet Archive captures of 23 March and 22 July 2026: the text is identical. FDA's CT figures, EIA, Valar's Ward 250 safety paper, the INL pebble benchmark and the NRC's 10 CFR 20.1201 were read from archived copies. The fission-yield, decay and photon-attenuation tables come from the standard evaluated data (ENDF/ENSDF, NIST); no copies of the IAEA, NIST and NNDC pages are kept here, so those tables were not re-checked against an archived copy. An error of a few percent in them would move the results by about the same few percent; none of the sensitivity checks moves any case across the CT line. The analysis its post cites was not found on Valar's website, so which reading it used cannot be checked.

This proof has its own page, with its pictures first: Radiation Exposure →

Project Beehive's 456 reactors, as NPR reported them: by our math, the radioactive leftovers of 6 to 10 large reactors, in less uranium but much more bulk

What the document says

NPR reported on 30 September 2026, citing a proposal from Valar to federal regulators that NPR reviewed, that Valar's 'Project Beehive' near Price would include data centers and "some 456 small nuclear reactors", with a fuel-production facility and waste-storage facilities, on over 9,000 acres of Bureau of Land Management land. NPR describes "Valar's 25-megawatt electric units" and says that in aggregate the reactors would produce "around 9.6 gigawatts of power for the data centers" and more heat for industry, with the first reactors in 2028 and the full site possibly by 2032. The proposal itself was not found in public records. BLM's public case layer lists 'Project Beehive Small Modular Reactor/Data Center' as a pending Valar application. Valar describes its Ward 250 test reactor as a helium-cooled reactor fueled by TRISO particles, but on the pages and papers we read it has published no commercial unit's heat output, fuel form, enrichment, burnup, refueling schedule or spent-fuel plan. At a 29 September 2026 town hall in Price, Valar officials said used fuel from its Carbon County project would first be stored on site in dry, sealed casks under federal licensing, and that no specific long-term disposal site outside the area had been identified (ETV News, 30 September 2026). This note connects NPR's numbers to the used fuel and radiation they imply, with every unknown shown as a range.

Given

• N = 456 reactor units (NPR: 'some 456 small nuclear reactors') (RECORD: NPR, 30 Sept 2026, citing a Valar proposal NPR reviewed; the proposal was not found in public records)
• P_unit = 25 MW of electricity per unit (RECORD: NPR ('Valar's 25-megawatt electric units'))
• P_e = both carried: 9.6 GWe (NPR's aggregate 'for the data centers') and 11.4 GWe (456 x 25 MWe); the 1.8 GW difference is not explained in the article (RECORD: NPR; arithmetic ours)
• Reactor type = helium-cooled, TRISO-fueled high-temperature gas reactor (said of Ward 250); Valar's mission page speaks of 'hundreds of reactors on one Gigasite'; no commercial-unit design figures found in Valar's public documents (RECORD: Valar, 'Ward 250 By the Numbers' and mission page)
• Valar fuel element = Ward 250 block as DOE cleared it for shipment: 49.052 kg, 13.13 in across flats by 11.81 in tall, 170 compacts about 0.5 in by 2 in, at most 234.94 g of U-235 at up to 19.9-20%, so, if both are at their maximums, about 1.18 kg of uranium per block and 6.7 to 7.0 g per compact (the review also prints 1.33 g U-235 per compact, which does not multiply to 234.94 g) (RECORD: DOE Model 9979 safety evaluation report, 20 May 2026 (derived))
• η = heat-to-electricity efficiency 0.33 to 0.45 (central 0.40) (ASSUMPTION, bounded by REFERENCE values: AVR 15/46 MW, Peach Bottom 40/115, Fort St. Vrain 330/842, THTR 300/750 (INL/EXT-10-19329); design targets 45% (Sandia); EIA's 33% example)
• CF = capacity factor 0.80 to 0.95 (central 0.90) (ASSUMPTION; U.S. fleet 90.8% to 93.4% in 2016-2025 (EIA, REFERENCE); past gas-cooled reactors ran far less)
• B = discharge burnup 80 to 170 GWd per metric ton of heavy metal (central 120) (ASSUMPTION within REFERENCE values: pebble-bed benchmark cases 80 to 204 (INL/EXT-20-60236), Xe-100 168 (X-energy report to NRC), prismatic MHTGR about 120 (INL catalog via Sandia; the only archived prismatic value))
• p = core specific power 50 to 130 MW of heat per metric ton (central 100) (ASSUMPTION; pebble-bed references 99 to 128 (derived from INL, ANL, Sandia tables); no archived value for a low-enriched prismatic power plant)
• Pebble = 6 cm pebble: 2.5 cm fuel zone inside a 0.5 cm graphite shell; 5 to 9 g of uranium (central 7); about 202 g whole; loose packing 0.60 to 0.74 (central 0.61) (REFERENCE: INL/EXT-20-60236 Tables 1-3, 8-10; ANL/NSE-22/98 Table 3-1; Sandia Table 3-1 (pebble mass derived))
• Prismatic volume = 8.4 to 24.5 m³ per metric ton of uranium: Fort St. Vrain's thorium-bearing elements to the Ward 250 block as cleared (central: the cleared block) (REFERENCE: NWTRB Fort St. Vrain fact sheet; DOE Model 9979 review (derived))
• y = chain yields per fission, U-235 / Pu-239 (thermal): Cs-137 6.189% / 6.614%; Sr-90 5.782% / 2.104%; also Zr-95, Ru-103, Ru-106, Cs-133 and Ce-144 (REFERENCE: England & Rider, Evaluation and Compilation of Fission Product Yields 1993 (Los Alamos, October 1994), Table 7 (a scanned table))
• Pu share = share of fissions in plutonium, averaged over the fuel's life: 0 to 0.4 (central 0.3) (ASSUMPTION; the one archived inventory implies about 0.3 to 0.4)
• Inventory = at about 120 GWd/t, g per metric ton: Cs-137 4,410; Sr-90 2,100; Cs-134 421; Ru-106 355; Eu-154 73; Sb-125 21.6 (REFERENCE: Sandia SAND2023-08602R Table A-1 (INL fuel cycle catalog, one prismatic design))
• Capture products = Cs-134 matched to that inventory x 0.6 to 1.8; Np-239 0.35 to 0.55 per fission (0.47 derived); Eu-154 x 0.5 to 1.5; Sb-125 x 0.7 to 1.3; Cs-134 gamma 1.50 to 1.60 MeV per decay (ASSUMPTION (one archived inventory; enrichment and table time point unknown))
• T½ = Cs-137 30.08 y; Sr-90 28.9 y; Cs-134 2.0652 y (REFERENCE: Sandia Table A-1 (a secondary DOE-laboratory compilation))
• E_f = 200 MeV per fission (±3% assumed), so 2.70 x 10^21 fissions per MWd (REFERENCE: DOE-HDBK-1019/2-93; ±3% ASSUMPTION)
• Dose engine = pf-007's published calculation, unchanged: gamma lines of 43 fission products, photon attenuation, ICRP 74 front-on conversion; whole body at 50 cm from the near surface; skin at contact (RECORD: pf-007 (its data limits carry over))
• CT = 1 to 10 mSv effective dose for a typical CT scan (REFERENCE: FDA, What are the Radiation Risks from CT?)
• Comparators = large U.S. reactors: median 3,411 MW of heat (middle half 2,797 to 3,630; 94 units), efficiency 0.300 to 0.343, burnup 50 to 60 GWd/t; U.S. spent fuel 2,000 to 2,200 t a year; U.S. nuclear capacity 98,436 MWe (2025) (REFERENCE: NRC NUREG-1350 App. A; ANL; NWTRB; DOE; EIA Table 6.07.B)
• Decay heat = 10 years after discharge: Xe-100 32.2 and reference PWR 40.6 kW per GWe-year, so 12.9 to 13.8 kW per GW-year of heat (REFERENCE: ANL/NSE-22/98 Table E-2 (prepared for DOE))
• Life = 20 and 40 years per unit; units start evenly from 2028 to 2032 (ASSUMPTION (no Valar operating life was found in public records); dates RECORD (NPR))
• Pool = Olympic pool taken as 2,500 m³ (50 m x 25 m x 2 m) (ASSUMPTION (a nominal size))

Working

1. 456 x 25 MWe = 11.4 GWe; NPR's aggregate: about 9.6 GW; 9.6 GW ÷ 456 = 21.05 MWe per unit; gap 1.8 GW = 15.8% of 11.4 GWe
   NPR does not reconcile its two numbers. The 9.6 GW may be the data centers' share (NPR adds 'even more thermal heat' for industry), or the whole fleet at about 21 MWe a unit. Read as heat, it does not fit: 9.6 GW would be 21 MW of heat per unit, less than each unit's rated 25 MW of electricity. Every result carries both electric figures; neither is picked. The proposal was not found in public records.
2. Heat P_th = P_e ÷ η, η = 0.33 to 0.45 9.6 GWe: 21.3 to 29.1 GW of heat (central 24.0); 11.4 GWe: 25.3 to 34.5 (central 28.5); per unit 46.8 to 75.8 MW = 6.25 to 10.1 median large U.S. reactors (3,411 MW of heat); 5.88 to 12.4 across the middle half of U.S. units
   No commercial unit's heat output was found in Valar's public documents, so it follows from the electric figure and an efficiency. Gas-cooled plants have turned 33% (AVR), 35% (Peach Bottom), 39% (Fort St. Vrain) and 40% (THTR) of their heat into electricity; design targets reach 45%. Lower efficiency means more heat, and so more radioactive leftovers, for the same electricity. The fleet would be 9.8% (9.6 GWe) to 11.6% (11.4 GWe) of U.S. nuclear capacity.
3. Heat made per year E = P_th x CF x 365.25 days, CF = 0.80 to 0.95 = 7,890 GWd a year (9.6 GWe basis) and 9,370 GWd a year (11.4 GWe basis); 6,230 to 12,000 GWd a year across both; fissions = E x 2.70 x 10^21 per MWd
   CF is the share of the year at full power. U.S. reactors ran 90.8% to 93.4% of the year in 2016-2025 (EIA); past gas-cooled reactors ran far less, and a lower CF lowers every yearly total in proportion. Every split uranium or plutonium atom leaves two radioactive fragments, so the leftovers follow the heat made, whoever buys the power.
4. Used heavy metal per year = E ÷ B, B = 80 to 170 GWd per metric ton = 66 t (9.6 GWe basis) and 78 t (11.4 GWe basis); 37 to 150 t across both 20 years: 730 to 3,000 t; 40 years: 1,500 to 6,000 t (central 2,600 and 3,100); plus about half a core-load still in the reactors at the end: 82 to 350 t
   Burnup is the heat each metric ton of uranium gives before it is removed. This is 1.7% to 7.5% of the 2,000 to 2,200 t of spent fuel U.S. reactors discharge each year, and 0.29 to 0.75 of what a fleet of large water-cooled reactors making the same heat would discharge at 50 to 60 GWd/t. That lower uranium mass favors Valar's design type.
5. Pebbles = heavy metal ÷ (5 to 9 g); mass = number x 202 g; bulk volume = number x 113.1 cm³ ÷ packing (0.60 to 0.74) per year: 4.1 to 30 million pebbles, 820 to 6,100 t, 620 to 5,600 m³ = 0.25 to 2.3 Olympic pools (central 0.70 and 0.83) 40 years: 10.0 to 90 pools (central 28 and 33)
   A 6 cm pebble holds a few grams of uranium in about 200 g of graphite, which is why this kind of used fuel is light in uranium but bulky. Volumes are loose pebbles only; canisters, casks and spacing would add more.
6. Ward 250 blocks as DOE cleared them: about 1.18 kg uranium (at the maximum U-235 and enrichment), 49.052 kg and 0.0289 m³ each = 24.5 m³ and 41.6 t of block per metric ton of uranium per year: 31 to 130 thousand blocks, 1,500 to 6,200 t, 890 to 3,700 m³ = 0.36 to 1.5 pools; 40 years: 14 to 59 pools (central 26 and 31) denser prismatic fuel (8.4 m³ per metric ton): down to 310 m³ a year
   The block DOE cleared for shipment to Ward 250 is the only Valar fuel element on record; whether a commercial unit would use it, or pebbles, is not known. Valar's draft safety paper lists 250 kg of heavy metal in 76 elements for Ward 250 (3.29 kg each), which DOE's figures do not match. Standard power-plant blocks are larger, but no archived source gives their uranium loading for low-enriched fuel.
7. Cs-137 made = fissions x yield; yield 6.19% (U-235) to 6.61% (Pu-239); Sr-90 5.78% to 2.10% per year: Cs-137 230 to 482 kg = 19.9 to 41.8 million curies (central 305 and 363 kg); Sr-90 105 to 288 kg = 14.5 to 39.6 million curies
   Cesium-137 and strontium-90, with half-lives of about 30 and 29 years, are the main long-lasting radioactive leftovers of the first few centuries. As fuel burns, plutonium made from uranium-238 also splits; it makes slightly more cesium-137 and much less strontium-90. Each year's cesium-137 would be about 5.57% to 12.6% of what all U.S. reactors make, because the fleet's heat is that share of theirs.
8. Each unit runs L years; units start evenly over W = 4 years (2028 to 2032) Cs-137 left when the last unit stops = made x (1 − e^(−λL)) x (1 − e^(−λW)) ÷ (λ² W L), λ = ln 2 ÷ 30.08 y 40 years each: Cs-137 made 9,200 to 19,000 kg; left at the end 5,700 to 12,000 kg (498 to 1,040 million curies); 100 years later about 0.10 of that (570 to 1,200 kg); 300 years later 5.7 to 12 kg
   Cesium-137 decays while more is made, so not all of it is present at the end. Starting every unit at once would leave about 5% more. No Valar operating life was found in public records; per-year figures scale to any other life.
9. Decay heat of one year's used fuel 10 years later = heat made (GW-years) x 12.9 to 13.8 kW per GW-year = 220 to 450 kW (central 280 and 330); 100 years later 43 to 110 kW; per GWe-year 28.6 to 41.8 kW (reference PWR 40.6)
   The ANL/INL study gives 32.2 kW per GWe-year for Xe-100 at 40% efficiency and 40.6 for a reference PWR at 34%; per unit of heat the two differ by about 7%. Per unit of electricity this fleet's used fuel gives off less heat than a PWR's unless its efficiency is at the low end. Decay heat sets how closely used fuel can be packed in storage. At one year it is higher still, but no archived reference value anchors it, so no figure is given.
10. Activity of each fission product in one metric ton after running at specific power p to burnup B (T = B ÷ p) and cooling t: A = y x F x (1 − e^(−λT)) x e^(−λt) (pf-007) added: Pu-239 yields for five chains; Cs-134 from neutron capture on Cs-133, matched to the archived inventory; Eu-154 and Sb-125 tied to the same inventory; Np-239 from U-238 capture
    pf-007's calculation was built for a reactor that ran less than a month; commercial fuel runs for years. Over years, plutonium fissions and neutron capture add cesium-134, which carries about half the dose from a year-old fuel pebble. Without these additions the one-year dose would be 2.3 times too low (4.1 mSv instead of 9.2 at central inputs). Check: at 120 GWd/t and a plutonium share of 0.3, the model gives the archived inventory's Cs-137, Sr-90, Ru-106 and Cs-134 within 1.4% to 3.6%.
11. Five minutes: whole body = effective dose at a point 50 cm from the near surface (ICRP 74 front-on); skin = dose where the piece touches the skin one 6 cm pebble (2.5 cm fuel zone in a 0.5 cm shell); one Ward 250 compact (1.27 x 5.08 cm, 6.7 to 7.0 g uranium) and one whole block (49 kg) as spheres of the same volume
    Same geometry and dose convention as pf-007. The central value counts only photons that leave the piece unscattered, so it is lower than a full count; counting scattered photons raises the whole-body dose about 1.36 times, the top of each range. Beta particles are not counted: a pebble's graphite shell stops most, but for a bare compact they would probably add more than the gamma dose. On the compact's real cylinder the contact dose is about 0.97 of the sphere's.
12. One pebble, whole body, five minutes: 1 day 302 mSv (100 to 760); 30 days 86 mSv (29 to 220); 1 year 9.2 mSv (2.4 to 41); 10 years 1.6 mSv (0.63 to 5.1) skin touching it: 1 day 120,000 mSv (42,000 to 300,000); 1 year 3,800 mSv (980 to 17,000); 10 years 648 mSv (260 to 2,100) main sources: 1 day La-140 28%, I-132 14%, Nb-95 9%; 1 year Cs-134 49%, Cs-137+Ba-137m 17%, Nb-95 13%; 10 years Cs-137+Ba-137m 81%, Cs-134 14%
    Each range covers every combination of the uncertain inputs at their low and high ends (2,048 combinations). Specific power, burnup, uranium per pebble, the cesium-134 factor and scattered photons move it most. Held at 30 cm instead of 50 cm the whole-body dose is 2.6 times higher. For scale, FDA puts a typical CT at 1 to 10 mSv, and U.S. rules allow a radiation worker 500 mSv a year to the skin of the hands. Checked against published Oak Ridge National Laboratory (ORNL) calculations for spent pebbles: a year after removal our source gives 0.9 to 1.4 times ORNL's per-pebble gamma output, with cesium-137 within 4%; about four days after removal ORNL estimated 33.5 R/h at 1 m from one pebble, about half what this calculation gives for the same pebble (ORNL/SPR-2024/3615, ORNL/SPR-2023/2988).
13. Cooling time until five minutes = one 10 mSv CT (whole body): one pebble 0.93 years (0.35 to 5.9); one compact 1.1 years (0.52 to 5.1); 50 cm from a whole Ward 250-type block 81 years (60 to 140) until 1 mSv: pebble 22 years (3.8 to 63); until the skin touching a pebble gets under 500 mSv in five minutes: 15 years (2.6 to 54)
    This favors Valar: about a year after removal, one used fuel pebble held for five minutes gives a whole-body dose inside FDA's CT range. The longest times need every dose-raising choice at once. A whole block, or many pebbles together, takes far longer, and the skin of a hand touching one pebble receives more than a worker's yearly hand limit in those five minutes for years longer.
14. Independent checks at central inputs, one pebble: calculation A 9.0 mSv at 1 year, 0.91 years to one CT; calculation B 10.9 mSv, 1.10 years; this calculation 9.2 mSv, 0.93 years
    Two calculations were made independently from the same input table. With A's choices this calculation reproduces A exactly; with B's choices (half the scattered photons counted, neptunium-239, Eu-154 and Sb-125) it reproduces B within 2%. The differences are stated choices, mainly whether scattered photons count in the central value, not errors.
15. Used-fuel volume vs a same-heat fleet of large water-cooled reactors (0.44 m³ of assemblies per metric ton): pebbles 11 to 64 times (central 25); Ward 250 blocks 16 to 42 times (central 23); denser prismatic fuel from 5.6 times per GWe-year: 4.8 to 14 t of uranium (PWR 21.7); pebbles 81 to 520 m³ (PWR 9.58); the same arithmetic gives ANL's Xe-100 figures: 5.40 t and 118 m³ (ANL: 5.41 and 118)
    Published studies differ mainly by method: ANL/INL found 12.3 times for Xe-100 pebbles packed as tightly as possible; the National Academies cite about 23.5 times (a vendor figure); Wainwright et al. (peer-reviewed) found 25 to 30 times for prismatic blocks, with 40 to 50% less mass and a repository about 30% smaller; Krall et al. (peer-reviewed) found 2 to 30 times for water-, salt- and sodium-cooled designs, analyzed no gas-cooled design, and are disputed by the other two. Graphite reflector blocks could add up to 24.5 m³ per GWe-year (ANL), 190 to 270 m³ a year here, if they stay in the reactor for its whole life; Valar's reflector design was not found in public records.

Results (central value, with the low-to-high range over every uncertain input). Part 1: one used fuel piece, five minutes. Part 2: the 456-unit fleet's used fuel

Whole body = effective dose, the quantity FDA gives for CT, at a point 50 cm from the near surface of the piece. Skin = dose where the piece touches the skin, a different quantity, not comparable with a CT. Central values leave out scattered photons and beta particles, so from about a year after removal they are lower estimates; the one-day and one-month values also assume full power until removal, and could be somewhat high for a pebble that ran its last pass at lower power; the top of each dose range counts scattered photons. Fleet ranges run from the 9.6 GWe basis at its lowest to the 11.4 GWe basis at its highest. Volumes are loose pebbles or whole blocks, before any canisters or casks. No commercial Valar design was found in public records: every design quantity is a stated range.

Result

On NPR's figures (456 units; 9.6 or 11.4 GWe), the fleet would make 21.3 to 34.5 GW of heat. That equals 6.25 to 10.1 median large U.S. reactors, so it would make about the same radioactive leftovers: 230 to 482 kg of cesium-137 and 105 to 288 kg of strontium-90 a year, 5.57% to 12.6% of all U.S. reactors' output. Its used fuel would hold 37 to 150 t of uranium a year, 0.29 to 0.75 times what large reactors making the same heat would leave. Because that uranium sits in graphite, it would fill 0.25 to 2.3 Olympic pools a year as loose fuel pebbles (10.0 to 90 pools over 40 years), or 0.36 to 1.5 pools as Ward 250-type blocks. That is 11 to 64 times (pebbles) or 16 to 42 times (blocks) the volume of same-heat large-reactor fuel, before packaging. One used fuel pebble held for five minutes gives the whole body 302 mSv a day after removal (100 to 760) and 9.2 mSv after a year (2.4 to 41). It falls to one 10 mSv CT after 0.93 years (0.35 to 5.9), while the skin touching it receives 120,000 mSv at one day (42,000 to 300,000) and 3,800 mSv at one year (980 to 17,000). These numbers rest on NPR's account of a proposal not found in public records, and no commercial Valar design was found in public records, so every design quantity is a range.

In pictures

Sources

• A startup wants to build a massive nuclear-powered data center on public land in Utah (NPR, 2026-09-30) · quotes checked against the article; no copy is archived here
• Valar Atomics hosts town hall on proposed Carbon County site (ETV News, 2026-09-30) · quoted from the article
• BLM National MLRS Land Use Authorizations: Rights-of-Way (query by customer name) (U.S. Bureau of Land Management) · case UTUT106801248, Price Field Office, Pending
• Mission (Valar Atomics) · 'A New Model for Nuclear Power'
• Ward 250 By the Numbers (Valar Atomics, 2026-06-28) · 'What is Ward250?'
• Safety Evaluation Report: Letter Amendment of Certificate of Compliance No. 9979 for the Model 9979 Package (Ward 250 TRISO fuel block) (U.S. Department of Energy, 2026-05-20) · pp. 2-3 and 5 (block mass, 170 compacts, 234.94 g U-235, compact size)
• Ward250 Nuclear Safety Design Agreement, No. 100403, Rev 02 (status: Draft) (Valar Atomics, 2025-10) · p. 52, Table 6 (76 fuel elements, 250 kg heavy metal)
• NEPA Determination DOE-ID-26-005: Valar Ward250 Research Reactor (U.S. Department of Energy, 2026-04-22) · pp. 4-5 (Ward 250 only: one core, dry storage casks to a DOE-approved facility)
• Benchmark for Fuel Shuffling and Depletion for Pebble-Bed Reactors, INL/EXT-20-60236 Rev. 1 (Idaho National Laboratory (U.S. Department of Energy), 2020-09) · Tables 1-3, 6, 8-10, 16 (pebble geometry, loading, packing, burnup, power per particle)
• Nuclear Waste Attributes of SMRs Scheduled for Near-Term Deployment, ANL/NSE-22/98 Rev. 1 (Kim, Boing, Halsey, Dixon) (Argonne and Idaho National Laboratories, for DOE (copy hosted by INL), 2022-11-18) · Tables E-1, E-2, 3-1, 4-2 (a laboratory report, not a journal paper)
• Advanced Reactors Spent Fuel and Waste Streams Disposition Strategies, SAND2023-08602R (Sandia National Laboratories (U.S. Department of Energy), 2023-06-28) · Table 3-1; Appendix A.1 and Table A-1 (inventory and half-lives)
• Evaluation and Compilation of Fission Product Yields 1993 (T. R. England and B. F. Rider) (Los Alamos National Laboratory (scanned copy via OSTI, marked LA-SUB--94-170), 1994-10) · Table 7, printed pp. 19-20, rows 90, 95, 103, 106, 133, 137, 144 (scanned)
• DOE Fundamentals Handbook: Nuclear Physics and Reactor Theory, Vol. 2, DOE-HDBK-1019/2-93 (U.S. Department of Energy, 1993-01) · Module 4, printed p. 33
• High Temperature Gas-Cooled Reactors Lessons Learned Applicable to the Next Generation Nuclear Plant, INL/EXT-10-19329 Rev. 1 (Idaho National Laboratory, 2011-04) · Table 1 and sec. 1.4 (AVR, Peach Bottom, Fort St. Vrain, THTR power and output)
• Xe-100 Topical Report: TRISO-X Pebble Fuel Qualification Methodology, Revision 2 (ML21246A289) (X Energy, LLC (submitted to the NRC), 2021-09-02) · p. 33 table (burnup 168 GWd/MTU; enrichment 15.5%; loading redacted)
• TRISO Particles: The Most Robust Nuclear Fuel on Earth (U.S. Department of Energy, 2019-07-09) · AGR program section
• Electric Power Monthly, Table 6.07.B: Capacity Factors for Utility Scale Generators Primarily Using Non-Fossil Fuels (U.S. Energy Information Administration, 2026-09-24) · annual rows 2016-2025, nuclear column
• What is the efficiency of different types of power plants? (U.S. Energy Information Administration)
• NUREG-1350 Vol. 35, Appendix A: Commercial Nuclear Power Reactors, Operating Reactors (U.S. Nuclear Regulatory Commission, 2025) · Appendix A, licensed thermal power of 94 operating units
• Commercial Spent Nuclear Fuel (fact sheet, Rev. 2) (U.S. Nuclear Waste Technical Review Board) · pp. 2-3
• 5 Fast Facts about Spent Nuclear Fuel (U.S. Department of Energy, 2022-10-03) · fact 2
• Department of Energy-Managed Spent Nuclear Fuel at Fort St. Vrain (fact sheet, Rev. 1) (U.S. Nuclear Waste Technical Review Board) · pp. 2-4 (element size, 1,464 elements, about 15 MTHM, canister count)
• A multidisciplinary framework from reactors to repositories for evaluating spent nuclear fuel from advanced reactors (Wainwright, Christiaen, Atz and others), Scientific Reports 14:26904 (Scientific Reports (Springer Nature); peer-reviewed, 2024-11-06) · Results: SNF mass and volume; decay heat; repository footprint
• Nuclear waste from small modular reactors (Krall, Macfarlane and Ewing) (Proceedings of the National Academy of Sciences (copy in PubMed Central); peer-reviewed, 2022-05-31) · Abstract (no gas-cooled design analyzed; disputed by the ANL/INL study and Wainwright et al.)
• Merits and Viability of Different Nuclear Fuel Cycles and Technology Options and the Waste Aspects of Advanced Nuclear Reactors (National Academies of Sciences, Engineering, and Medicine (copy hosted by Sandia), 2022) · ch. 5, printed p. 168
• Advanced Reactor Spent Fuel Geologic Disposal Technical Issues (Pensado, Parmenter, Stothoff and Livingston) (Center for Nuclear Waste Regulatory Analyses, for the U.S. Nuclear Regulatory Commission, 2025-09) · sec. 2.2 (citing a PNNL study that is not archived here)
• What are the Radiation Risks from CT? (U.S. Food and Drug Administration, 2017-12-05) · Radiation Dose from CT Examinations
• 10 CFR 20.1201: occupational dose limits for adults (Legal Information Institute (Cornell)) · (a)(2)(ii): 500 mSv a year to the skin of any extremity
• NuDat 3: half-lives, gamma-ray energies and intensities (ENSDF) (National Nuclear Data Center, Brookhaven National Laboratory) · gamma lines in pf-007's calculation and for Cs-134, Eu-154 and Sb-125 here; no copy of NNDC's pages is kept here
• ICRP Publication 74: Conversion Coefficients for use in Radiological Protection against External Radiation (International Commission on Radiological Protection, 1996) · Table A.17 (as in pf-007); not openly published, so no copy is archived
• Identify and Assess Technical Challenges in Safeguards Measurements of Spent Advanced Reactor Fuels (ORNL/SPR-2024/3615) (Oak Ridge National Laboratory (via OSTI.gov), 2024) · Table 2, p.8
• Nuclear Material Control and Accounting for Pebble Bed Reactors (ORNL/SPR-2023/2988) (Oak Ridge National Laboratory (via OSTI.gov), 2023-12) · Pebble 6 measurement section

Checked: Two calculations were made independently from the same input table: one built on pf-007's published dose calculation, one written separately with its own nuclear data and shielding code. A third check re-ran both (each gave identical output on a second run), reproduced the first exactly and the second within 2% when given the same choices, and traced every difference to a stated choice, mainly whether scattered photons count in the central value. Every input was re-read in its archived copy; the fission yields come from a scanned table and were read from the scanned pages. The same arithmetic reproduces the ANL/INL study's Xe-100 figures (5.40 t and 118 m3 per GWe-year, against 5.41 and 118). NPR's quotes were checked against the article. Afterward we added pictures, built from the model's numbers and checked against them; reworded some plain-language sentences; now use the engineering word 'pebble' for the fuel spheres throughout; and added one table row (big reactors' used fuel in Olympic pools, worked out with the model's own formula and inputs). Separate reviews checked the pictures' numbers, wording and drawing, and the fixes were checked again. A later check against published Oak Ridge National Laboratory calculations for spent pebbles found the per-pebble figures within a factor of 2 of every usable one, and led to a note that the one-day and one-month values may be somewhat high.