Nuclear Fuel Cycle

The Separative Work Unit (SWU) is the standard measure of the physical effort (work) required to separate isotopes during enrichment. SWU has units of kg (or sometimes written as kg-SWU).

The cost of enrichment services is quoted in $/SWU (or pounds/SWU).

Mass Balance Equations

For any enrichment process, conservation of total mass and conservation of $^{235}$U mass give us two fundamental equations:

Total mass balance:

$F = P + T$

$^{235}$U mass balance:

$F \cdot x_f = P \cdot x_p + T \cdot x_t$

Where:

• F = mass of feed (natural uranium), kg
• P = mass of product (enriched uranium), kg
• T = mass of tails (depleted uranium), kg
• x(f) = weight fraction of $^{235}$U in feed (generally 0.00711 for natural uranium)
• xₚ = weight fraction of $^{235}$U in product (desired enrichment)
• xₜ = weight fraction of $^{235}$U in tails (set by enricher or customer)

From these, the feed factor (feed per unit product) and tails factor (tails per unit product) can be derived:

$\frac{F}{P} = \frac{x_p - x_t}{x_f - x_t} \quad \text{(Feed factor)}$

$\frac{T}{P} = \frac{x_p - x_f}{x_f - x_t} \quad \text{(Tails factor)}$

Economic Note: The tails assay xₜ is the key adjustable variable. If the price of natural uranium (feed) is high, a low tails assay is chosen to extract more $^{235}$U from each kg of feed — but this requires more separative work (more SWU, higher electricity cost). There is therefore an economic optimum tails assay that balances feed cost against enrichment cost.

The SWU Formula

The number of Separative Work Units required is calculated using the value function V(x):

$V(x) = (2x - 1) \cdot \ln\left(\frac{x}{1 - x}\right)$

The total separative work is:

$\boxed{SW = P \cdot V(x_p) + T \cdot V(x_t) - F \cdot V(x_f)}$

Where:

• SW = Separative Work (in kg-SWU)
• V(xₚ) = value function evaluated at the product assay
• V(xₜ) = value function evaluated at the tails assay
• V(x(f)) = value function evaluated at the feed assay