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Conceptual scope: the safe account below describes the functions of spent nuclear fuel management and the chemical basis of partitioning. It does not supply an ordered fissile-separation flowsheet, reagents, process conditions or purification instructions. In particular, the requested operational eight-step separation scheme is outside this answer's scope.

Eight useful functional objectives are: receiving and identifying the spent nuclear fuel; managing residual decay heat; providing radioactive material confinement and radiation shielding; establishing the chemical and radiological inventory; partitioning reusable constituents from waste at a conceptual level; maintaining accountable control of recovered materials; conditioning solid and liquid wastes; and retaining or treating radioactive effluents. These are management and containment functions, not a claim that the question specifies one universal eight-stage plant design. Several occur throughout a facility's operation rather than sequentially.

The chemical principle of uranium/plutonium partitioning is that an element's oxidation state and complexation affect its chemical speciation, solubility and relative affinity for immiscible phases. solvent extraction can therefore separate elements whose chemistry differs, even though it does not distinguish uranium isotopes by this chemical selectivity. A recovery operation must account for the accompanying fission products, contaminated process materials and secondary wastes; the recovered material is only one output. radioactive material confinement, remote handling and nuclear material accountancy remain necessary because chemical partitioning redistributes both radioactive and fissile inventories.

Use the thermal power when applying nuclear fuel burnup. No capacity factor is supplied, so take a full-power year of $365$ days; if the operating fraction were $u$, the annual quantities in this part would be multiplied by $u$.The corresponding initial heavy-metal nuclear fuel requirement isAll nuclear fuel and concentrate masses below are uranium masses, excluding oxygen. Let $F,P,W$ denote uranium feed, product and enrichment tails, and $x_f,x_p,x_w$ the uranium-235 mass fractions. Conservation of total uranium and uranium-235 givessoWith $P=M$, the uranium oxide concentrate requirement is thereforeThis is not the gross mass of oxide concentrate. Both $F-P-W=0$ and $0.007F-0.050P-0.003W=0$ provide independent balance checks.

The recycle feed is the recovered uranium, not the whole initial nuclear fuel charge. After the stated loss it isUsing the same enrichment tails assay and the stipulated higher recycle-product assay, the isotopic mass balance givesTreat the $5.3\%$ recycled product as replacing the same uranium mass of ordinary $5.0\%$ nuclear fuel: the additional uranium-235 compensates for the supplied uranium-236 penalty. It displaces fresh natural-uranium feed in the ratio calculated in part (b), henceThe remaining natural feed is $313.36245\mathrm{t}\mathrm{U}\mathrm{y}{\mathrm{r}}^{-1}$, a 9.885% resource saving relative to the full-power once-through requirement. This credits one uranium recycle only; it adds no second-pass recycling or plutonium credit.

Separative work must be charged for producing the recycled nuclear fuel as well as credited for displaced fresh nuclear fuel. With uranium-235 mass fraction $x$, use the separative value functionMasses in kilograms give work in kg-SWU. For ordinary fresh product the work per product kilogram isThus the fresh work avoided isThe recycled product still requiresThe net separative-work saving is consequentlyAs a whole-cycle check, the fresh route requires $215668.0851$ kg-SWU per year, while the mixed fresh-plus-recycled supply requires $214075.3261$ kg-SWU. The net saving is only 0.7385% of the original work. The recovered stream has a higher uranium-235 assay than natural feed, but the higher required product assay uses most of the work credit. These equations are resource accounting and do not specify an uranium enrichment plant or its operation.

Resource savings do not by themselves establish a financial saving. The one-recycle case avoids purchasing $34374.04$ kg U in natural feed and $1592.76$ kg-SWU per year. Those are the benefits to compare with the incremental costs, not the whole value of the recovered uranium or the gross avoided fresh-product separative work.

Let $p_U$ be the concentrate price per kilogram of uranium, $p_S$ the price per kg-SWU, and $C_{\mathrm{add}}$ the annual net extra cost of recovery, recycle-specific conversion and fabrication, waste management and financing, relative to the once-through alternative. Recycling is cheaper under this simplified boundary only ifEquivalently, if $p_S$ and $C_{\mathrm{add}}$ were supplied, the break-even uranium price would beAn avoided spent-fuel-disposal charge belongs as a credit in $C_{\mathrm{add}}$; a new radioactive waste conditioning or disposal liability belongs as a cost. They must not both be assigned the same sign simply because each concerns waste.

The natural-uranium saving is appreciable, but the fractional separative-work saving is small. Recovery losses and the uranium-236 compensation limit the value of recycling. Additional radiological handling, separate fabrication campaigns, secondary waste streams, capital tied up during storage and recovery, and uncertain future disposal costs can matter. Conversely, a higher natural-uranium price, a lower incremental recovery cost or an independently justified disposal credit can favour reuse. The calculation gives a conditional comparison, not a uniquely determined economic verdict or market-price threshold, because no costs are supplied. plutonium utilisation is a different option and has not been silently credited to this uranium-only recycle.

The distinction is between partitioning reusable fuel constituents from fission-product waste, and subsequently partitioning uranium from plutonium. In the first operation, chemical selectivity transfers selected actinides into a different stream while most fission products remain in the waste stream. In the second, differences in chemical speciation allow the two recovered elements to follow different streams. Neither operation separates uranium isotopes: uranium enrichment and chemical element separation are different tasks.

At a conceptual level, solvent extraction exploits unequal affinities of dissolved species for immiscible liquid phases. complexation changes those affinities, while a change in oxidation state can change an element's preferred phase without changing its identity. This explains why a stream bearing actinides can first be separated from much of its accompanying waste, and why uranium and plutonium need not subsequently remain together. Dissolved and undissolved residues, contaminated solvents and off-gases remain waste-management obligations; a recovery percentage alone is not a complete material balance.

Equipment must provide phase contact and phase disengagement within shielded, confined systems. Remotely maintained contactors and settlers are equipment categories serving those functions; sealed transfer systems, shielded cells and monitored off-gas handling limit contamination and exposure. Cooling manages residual decay heat, and nuclear material accountancy follows inventories and losses. criticality control in radioactive waste is essential because solvent extraction can redistribute fissile constituents. Scoped limitation: this answer gives the chemical principles and equipment functions; operational fissile-separation recipes, reagent sequences, contactor design details and process conditions are intentionally not supplied.

burnup refers to thermal energy per initial heavy-metal mass. Correct the electrical rating for thermal efficiency before using it, and apply the capacity factor once:Consequently the annual initial heavy-metal fuel requirement isLet $F,P,W$ be feed, enriched product and enrichment tails masses, all measured as uranium, and $x_f,x_p,x_w$ the corresponding uranium-235 mass fractions. The isotopic mass balance givesWith $P=M$, the uranium oxide concentrate requirement is thereforeThese are uranium or heavy-metal masses, not tonnes of uranium oxide. The check $327.394(0.007)=35.394(0.040)+292.000(0.003)$ closes the uranium-235 balance as well as the total mass balance.

Use the recovered uranium inventory, rather than the initial fuel mass, as the recycle feed. Its available mass isAssuming the same enrichment tails assay and the specified higher recycled-fuel assay, the isotopic mass balance givesTreat this fuel as replacing the same heavy-metal mass of ordinary fresh fuel: the stipulated higher uranium-235 content compensates for uranium-236. The displaced natural feed isabout 12.63% of the once-through natural-uranium requirement. The residual natural feed demand is $286.052\mathrm{t}\mathrm{U}\mathrm{y}{\mathrm{r}}^{-1}$.

For separative work accounting defineUsing kilograms in this expression gives kg-SWU. For ordinary fresh fuel the work per product kilogram isThe fresh work avoided by substituting $P_r$ is $23896.3\mathrm{kg}\text{-}\mathrm{SWU}\mathrm{y}{\mathrm{r}}^{-1}$, but producing that recycled product still requiresHence the net saving isThe work saving is only about 0.786% of the original annual separative work. Crediting the whole avoided fresh-fuel work without charging the recycle work would substantially overstate the saving. This is an annual resource account at the supplied assays, with no isotope-separation equipment model.

An economic threshold requires all unit costs to have an explicit mass basis. Let $p_U$ be the concentrate price per kilogram of uranium, and let $C_{\mathrm{net}}$ denote the annual incremental fabrication-and-disposal cost of the closed cycle relative to the open cycle. The break-even annual cash balance iswhere $M_{\mathrm{rep}}$ is the charged nuclear fuel reprocessing throughput in kilograms of initial heavy metal. Use $M_{\mathrm{rep}}=35393.9394$: the stated $33$ tonnes is the remaining uranium inventory, not the whole initial heavy-metal throughput. ThusIf the quoted combined $400$ dollars per kilogram is a net additional charge on that same reprocessed-HM basis, $C_{\mathrm{net}}=400M_{\mathrm{rep}}$, giving 1,196.77 dollars per kg U. Under this explicit convention, the saving in purchased separative work is small and the resource saving must pay nearly all the additional nuclear fuel reprocessing cost.

The wording does not identify the mass basis or counterfactual sign of the combined recycled-fuel and spent-fuel-disposal term. If instead $400$ dollars is a net charge per kilogram of the $4469.4375$ kg recycled product, the same balance gives 897.56 dollars per kg U. If it denotes a net avoided charge of $400$ dollars per kilogram of discharged HM, the threshold is 511.87 dollars per kg U. These are alternative accounting conventions, not three simultaneous economic thresholds; the general balance makes the ambiguity visible.

All prices here are per uranium mass, not per gross concentrate mass. This simplified comparison assigns no plutonium credit, financing, conversion-cost difference or change in repository liability, and takes the quoted charges as incremental. A closed fuel cycle can save natural uranium without being the cheaper option: at a price below the threshold under the chosen cost convention, the additional charges exceed the resource credit. No contemporary uranium-market price is needed for that conclusion.

gadolinium is primarily a strong neutron absorber. The supplied thermal cross section for gadolinium-157 is very large, so capture removes neutrons that could otherwise sustain nuclear fission. This is a criticality-control role rather than a requirement to complete the host formula. The isotope-specific number must not be assigned unchanged to natural gadolinium.

hafnium supplies an additional neutron poison with favourable retention chemistry. Its supplied natural-abundance thermal cross section is much smaller than that of gadolinium-157 but still appreciable. Chemically, tetravalent hafnium resembles zirconium and can remain associated with a durable titanate host. Keeping the absorber with the fissile inventory during alteration is as important as its initial neutron capture strength: a highly absorbing constituent offers poor long-term protection if it is preferentially lost. These considerations explain using complementary absorbers rather than ranking them solely by their initial thermal cross sections.

uranium addition chiefly addresses isotopic dilution, including the long-term daughter inventory. plutonium-239 alpha decay produces uranium-235. Added uranium-238 lowers the uranium-235 fraction of that future uranium inventory; depleted uranium adds less uranium-235 initially than natural uranium. uranium-238 also absorbs some neutrons, but its small listed cross section does not make it a substitute for the strong absorbers. The long-term daughter issue and the retention of gadolinium and hafnium are explicit concerns in ceramic waste-form development.^[ref]

These are complementary physical purposes, not proof of criticality control in radioactive waste from three cross-section values. neutron energy, isotopic composition, geometry, moderation, absorber distribution and their coupled evolution matter. Incorporation in the same durable waste form helps maintain association, but neither uranium addition nor chemical immobilisation makes deliberate element separation physically impossible.

uranium recovered from spent nuclear fuel. Its isotopic composition differs from natural uranium; uranium-236 can impose an additional fuel-performance penalty.

Research author of Nature and composition of the continental crust: A lower crustal perspective by Roberta L. Rudnick and David M. Fountain.

A uranium isotope that is a principal fissile material in thermal-neutron reactors. Capturing a neutron produces excited uranium-236.

An unstable uranium isotope produced by neutron capture on uranium-238. Two successive beta-minus decays, through neptunium-239, lead to plutonium-239.

An increase in the fraction of a specified uranium isotope. Fuel-cycle accounting tracks feed, product, enrichment tails and separative work; it does not specify an uranium enrichment plant or its operating conditions.

A concentrated uranium-bearing product of ore processing. Resource accounts must distinguish mass of uranium from gross mass of its oxide concentrate.