Fuel & Materials · C-FM-007

U-235 Consumption and Enrichment Change — Fissions, Capture (α) and Remaining Enrichment in One Calculation

From thermal power to fissions, from fissions to U-235 consumed with capture, and from there to the remaining enrichment, with worked examples.

Concept C-FM-007

Definition

U-235 consumption is the amount of U-235 lost by neutron absorption, the sum of fissions and radiative captures: consumption = fissioned amount × (1 + α), with α = σγ/σf ≈ 0.169; the enrichment after burnup is the remaining U-235 mass divided by the remaining U mass.

Principle and mechanism

Starting point — from thermal power to fission rate

The recoverable energy per fission is E_f ≈ 200 MeV = 3.204×10⁻¹¹ J.
Fission rate R_f = P / E_f → 1 W ≈ 3.12×10¹⁰ fissions/s. A rounded conversion factor of 3.1×10¹⁰ fissions/W·s corresponds to about 201 MeV per fission.
1 MWd = 8.64×10¹⁰ J ≈ 2.70×10²¹ fissions → mass of U-235 fissioned = 2.70×10²¹ × 235.04 / 6.022×10²³ ≈ 1.05 g/MWd.

This note carries the same constants (200 MeV, 1.05 g/MWd, 0.8815) throughout.

Absorption = fission + capture

Not every thermal neutron absorbed by U-235 causes fission. Some are radiative captures (n,γ) that produce U-236, which does not fission with thermal neutrons and acts only as a parasitic absorber.

Quantity	Definition	2200 m/s value (textbook)
α	σγ / σf	98.6 / 582.2 = 0.169
Fission probability per absorption	σf / σa = 1/(1+α)	0.855
η	ν / (1+α)	2.43 / 1.169 = 2.08

So per MWd, consumption = 1.052 × 1.169 ≈ 1.23 g, and the 0.178 g captured becomes 0.179 g of U-236 (heavier by the neutron mass). Radiative capture also releases gamma energy, but it is small compared with fission energy and already partly included in E_f (200 MeV). If σa and σf are given instead, 1+α = σa/σf (for example 678/577 = 1.175).

Flow diagram: 1.230 g of U-235 consumed by neutron absorption per MWd splits into a wide fission branch of 1.052 g (85.5 %), which supplies 2.70×10²¹ fissions × 200 MeV = 1 MWd of heat, and a narrow radiative-capture branch of 0.178 g (14.5 %) that becomes 0.179 g of U-236; a side box shows the U-238 to Pu-239 chain outside the simple model, and equation cards give α = σγ/σf = 98.6/582.2 ≈ 0.169 and σf/σa = 1/(1+α) ≈ 0.855 — Where absorbed U-235 goes in the simple model. Cross sections are 2200 m/s thermal values (Lamarsh); use a stated α when one is given.

How enrichment falls

The calculation has three steps.

N_f = P_th·t / E_f (number of fissions; P_th is thermal power, t = capacity factor × operating days)
m_f = N_f × 235/N_A, Δm₅ = (1+α)·m_f (without α, Δm₅ = m_f)
ε = (m₅,₀ − Δm₅) / (M_U,0 − m_f) × 100 % — the denominator is the remaining U, i.e. the initial U minus only the fissioned mass (captured U-235 stays as U-236, so it is not subtracted)

Worked example: 100 t of 3 % UO₂, 4,000 MWt, 90 % capacity factor, 300 days.

M_U = 100 × 0.8815 = 88.15 tU; initial U-235 = 2,644 kg
Energy released 1.08×10⁶ MWd → burnup 12,252 MWd/tU
U-235 fissioned = 1.08×10⁶ × 1.052 g = 1,137 kg
Remaining 1,508 kg ÷ (88,147 − 1,137) kg = 1.73 %
Including capture with α = 0.169: consumption 1,329 kg, enrichment 1.51 %

Two shorter worked results at 200 MeV per fission:

1,000 MWt for one day: 2.70×10²⁴ fissions, 1.05 kg of U-235 fissioned, 1.23 kg consumed
Fissioning all of 1 g of U-235 releases about 5.12×10²³ MeV; a 3,000 MWt core fissions about 3.16 kg of U-235 per day

In this simple model 3 % fuel runs out of U-235 at about 28.5 GWd/tU counting fission only, or about 24.4 GWd/tU including capture. Real PWR fuel, however, burns much longer, because other nuclides such as Pu-239 bred from U-238 capture also contribute to fission (the Pu share is around half at end of cycle); fuel with an initial 3.2 % discharged at about 33 GWd/tU is known to retain roughly 0.8–1 % U-235 (an approximate value common in the literature).

Graph of U-235 enrichment from 0 to 3.5 wt% against burnup from 0 to 35 GWd/tU with two calculated straight-ish lines from 3 %: fission only at 1.05 g/MWd reaching zero at 28.5 GWd/tU and with capture at 1.23 g/MWd reaching zero at 24.4 GWd/tU; a worked-example point at 12.25 GWd/tU giving 1.73 % (fission only), a marker for real discharged fuel of initial 3.2 % at about 33 GWd/tU retaining 0.8 to 1 %, and a card with the three calculation steps and the worked example of 1,137 kg fissioned and about 1.73 % — Simple model of U-235 depletion for 3 wt% fuel (fission in U-235 only, 200 MeV). The worked-example point counts fission only; the real-fuel range is an approximate literature value and the model zero is not an operating limit.

What it means in a plant

Core design: cycle length is set by the balance between the fissile inventory and the build-up of poisons. When choosing the initial enrichment and loading, U-235 depletion and Pu production are tracked with depletion codes.
Reactivity management: U-235 depletion and the build-up of U-236 and fission products reduce the excess reactivity, which PWRs compensate by lowering the boron concentration. The parasitic absorption of U-236 is a reactivity penalty when reprocessed uranium is reused.
Kinetics: as the Pu-239 share grows toward end of life, βeff decreases.
Nuclear material accountancy: the U-235 and Pu inventories of spent fuel are calculated with depletion codes and used for accountancy and safeguards declarations.
Reactor type examples: fresh OPR1000 and APR1400 fuel is low-enriched below 5 wt% (example), while CANDU uses natural U (0.711 %) and has a low discharge burnup of about 7 GWd/tU (example).

Regulatory and design criteria

The items below are examples: US NRC rules and guidance, and the general definition of low-enriched uranium.

Basis	Key point
10 CFR 50.68(b)(7)	Nominal enrichment of fresh fuel assemblies 5.0 wt% U-235 or less (condition for the 50.68 criticality alternative). Higher enrichment needs an exemption — 2023 exemption for 6 wt% lead test assemblies at Vogtle Unit 2
RG 1.183 Rev. 1 (2023)	Extends the applicability of accident source-term release fractions to 8 wt% enrichment and a rod-average 68 GWd/MTU
Low-enriched uranium (LEU)	Less than 20 % U-235

Fuel enrichment is moving beyond the 5 wt% limit toward LEU+ (5–10 %).

Easily confused pairs

	Fissioned amount	Consumption (absorption)	Depletion	Enrichment
Definition	U-235 that fissioned	Fission + capture	Process of enrichment falling with burnup	Raising the U-235 fraction by isotope separation
Per MWd	1.05 g	1.23 g (α 0.169)	—	—
Basis of calculation	P·t / E_f	× (1+α)	ε = remaining U-235 / remaining U	Mass balance and SWU
Common mistake	Confusing it with consumption	Omitting α (about 15 % too low)	Not converting UO₂ to U	Confusing it with depletion

Not converting UO₂ to U changes the result considerably: in one worked case the remaining enrichment is 0.74 % with the conversion and 1.01 % without it.

Key points at a glance

Definition: U-235 consumption = neutron absorptions = fissions + radiative captures (→ U-236).
Fission rate: R_f = P/E_f; 1 W ≈ 3.1×10¹⁰ fissions/s; 1 MWd ≈ 2.7×10²¹ fissions.
Fissioned amount: 1 MWd ≈ 1.05 g of U-235 (fissioning all of 1 g ≈ 5.12×10²³ MeV ≈ 0.95 MWd).
α: σγ/σf ≈ 0.169 → consumption = fissions × 1.169 ≈ 1.23 g/MWd.
U-236: fissioned amount × α ≈ 0.18 g/MWd, a parasitic absorber.
Enrichment: ε = (m₅,₀ − Δm₅)/(M_U,0 − m_f) × 100 %, with M_U,0 = M_UO₂ × 0.8815 (check whether a mass is given as U or as UO₂).
Example: 3 % UO₂ 100 t, 1.08×10⁶ MWd → 1,137 kg fissioned, 1.73 %.
Limits: ignoring Pu-239 and U-238 fast fission → the real decrease is slower; precise values need depletion calculations.
Regulation: 5 wt% limit for fresh fuel (10 CFR 50.68), inventory calculation for accountancy.

Sketch: a box for U-235 absorbed (1+α) branching to fission 1 → 200 MeV and capture α → U-236, followed by the equations Nf = P·t/Ef, mf = Nf·235/NA, Δm5 = (1+α)·mf and ε = (m5,0 − Δm5)/(MU,0 − mf) × 100 %, with α ≈ 0.169 and 1.052 g fissioned / 1.230 g consumed per MWd — Absorption branch and the three calculation equations.