Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2020 / Module NE4 (Fuel cycle, waste and decommissioning)

Original solutions to both printed questions. The University of Cambridge release occupies physical PDF pages 37–40, with the question body on pages 38–39. ^[ref]

The fuel-cycle account uses the stated assays and explicitly identifies the assumed full-power year. The chemical-separation answer has a visible conceptual scope limitation.

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.

A parent initially at rest emits an alpha particle and a recoiling heavy daughter. momentum conservation gives equal and opposite momenta, so their nonrelativistic kinetic energies satisfyHere $Q$ is the kinetic energy available to this branch; excitation of the daughter reduces it and may be followed by photon emission. For an illustrative $Q=5$ MeV, $m_{\alpha }=4\mathrm{u}$ and $M_d=236\mathrm{u}$, the energies are $4.917$ MeV in the alpha particle and $83.3$ keV in the daughter. The heavy daughter receives much less energy, but that does not imply much less atomic damage.

The fast alpha particle loses most of its energy through electronic stopping: ionization and electronic excitation along a relatively long path. It also undergoes nuclear collisions, particularly towards the end of its range, and can produce displacement damage. The slower, heavy, highly charged recoil couples strongly to surrounding nuclei and generates a compact displacement cascade. Successive knock-on atoms create vacancies, interstitial defects and clusters. Some Frenkel pairs recombine as the cascade relaxes; the initial collision count is therefore not the number of defects that remain.

A quantitative illustration comes from brannerite rather than an assumed universal ceramic. Simulations in the primary natural-mineral study used a $4.5$ MeV alpha particle and an $80$ keV daughter recoil. The corresponding ranges were about $13$ micrometres and $23$ nanometres. At the study's assumed $50$ eV threshold displacement energy, its tabulated results give roughly $560$–$571$ recoil atomic displacements and $91$–$100$ alpha particle atomic displacements per radioactive decay. The numbers depend strongly on the displacement threshold and material, but the shorter recoil path contains the dominant and much denser ballistic radiation damage.^[ref]

In a susceptible ceramic, cascade cores can accumulate into an amorphous fraction; defects and associated relaxation can also change volume, thermal conductivity and mechanical response. molecular dynamics simulations of zirconolite found a strongly disordered core surrounded by more isolated point defects, supporting this distinction between local cascade damage and an entirely transformed specimen.^[ref]

The alpha particle eventually becomes a helium atom after slowing and charge exchange. helium accumulation can have later microstructural effects distinct from the initial knock-on cascade. Material-dependent recombination and annealing limit surviving disorder; a change in diffraction or radiation-induced swelling is not automatically radiation-induced amorphization, and radiation-induced amorphization is not automatically dissolution or release of the contained radionuclide.

Consider a representative bulk volume $\mathcal{V}$ and let $f_a(t)$ be its amorphous fraction. With alpha-decay event rate density $R$, during $dt$ the expected number of independent alpha decay events is $R\mathcal{V}dt$. Each produces effective amorphization volume per decay $V_{\alpha }$, but only the fraction $1-f_a$ of a randomly placed damage region was previously crystalline. The expected new amorphous volume is thereforeDividing by $\mathcal{V}dt$ givesSeparate variables and integrate:Hence, with cumulative alpha-decay event density $D=Rt$,The product $Rt{V}_{\alpha }$ is dimensionless: events per cubic metre accumulated over time multiply cubic metres damaged per event. Here “dose” is an event density, not an absorbed energy per kilogram.

An independent probability interpretation reaches the same result. A particular point is covered by a number of random damage events with a Poisson distribution of mean $D{V}_{\alpha }$. Its probability of remaining crystalline is the zero-hit probability $e^{-D{V}_{\alpha }}$; the complementary fraction has been hit at least once. Overlapping events do not create a second copy of already amorphous material.

At small dose, $f_a\simeq D{V}_{\alpha }$; at larger dose, overlap bends the curve towards one. For example, $D{V}_{\alpha }=1$ gives $f_a=0.6321$, not full radiation-induced amorphization, and $f_a=0.99$ requires $D{V}_{\alpha }=\ln 100=4.6052$.

The derivation assumes a homogeneous, initially crystalline material; independent random event locations; one-event radiation-induced amorphization of a fixed effective volume; and no recovery of that volume. It is the direct-impact model, not a model requiring several prior hits to cross a damage threshold. If $R$ varies while $V_{\alpha }$ stays fixed, replace $Rt$ by ${\int }_0^{t}R(s)ds$. If annealing, material evolution or spatially correlated actinide distributions matter, this simple law need not hold. The subscript on $V_{\alpha }$ indexes an alpha decay event: it does not require the emitted alpha particle itself to produce the damaged volume.

The assumption is a leading approximation for concentrated ballistic radiation damage, not an exact statement that the alpha particle displaces no atoms. The daughter has only about $m_{\alpha }/M_d$ times the alpha particle's kinetic energy, but it is much slower and loses a much larger fraction of that energy in collisions with target nuclei. Those collisions initiate closely packed recoil cascades. The alpha particle deposits most of its much larger energy into the electronic system over a path hundreds of times longer, with a less concentrated displacement contribution.

Thus the dense region susceptible to one-event loss of crystalline order can reasonably be associated chiefly with the daughter recoil. This explains why the effective damaged volume in the direct-impact amorphization model can be estimated or fitted from recoil-dominated cascades. It is the spatial density and stopping channel of the damage, not simply which particle carries more energy, that matters.

The approximation has a measurable limit. Turcotte's comparison of actinide-dioxide self-damage and external alpha irradiation found a nonzero alpha contribution, reporting about $19\%$ of total damage in the studied case. Comparison of diffraction-derived lattice changes with internal-reference measurements helps separate real radiation-induced swelling from instrument shifts, while the distinction between self-decay and external-alpha irradiation probes the omitted recoil channel.^[ref]

Consequently “all ballistic radiation damage” is useful as the stated model assumption but should not be promoted to an exact material-independent law. electronic excitation can also couple to structural change, and recovery alters what survives. The derivation in part (b) remains valid for an effective one-event volume fitted consistently to the relevant regime; it does not establish that all ceramics or spent fuels share the same $V_{\alpha }$ or the same eventual amorphous fraction.