Past exam of the MPhil in nuclear energy course of the University of Cambridge / 2021 / Module NE1 / 4

The first balance treats the fertile parent as being removed by neutron absorption. The second has a production contribution associated with capture in the parent and a removal contribution from absorption in the fissile population. Capture and total absorption are not interchangeable: in general absorption includes capture and fission, and a capture channel is the one that feeds the specified transmutation chain.

The lumped description assumes spatially averaged populations and flux, fixed effective reaction coefficients, no transport of material into or out of the system, and a shortened chain in which intermediate decay delays are neglected. It omits explicit precursor accumulation, competing reaction chains and the changing neutron spectrum. It also prescribes flux rather than solving a coupled power-and-composition feedback problem. Neglecting intermediate lifetimes requires them to be short relative to the variation being modelled; that assumption can fail during a rapid change in irradiation.

The first equation is a prescribed first-order removal law. For a constant absorption coefficient,Integrating from the initial condition givesand henceThe neutron fluence $\Phi$ is the time integral of scalar neutron flux; it has inverse-area units, making $\sigma \Phi$ dimensionless. Constant flux is not needed for this integration, but a constant effective coefficient is.

Use neutron fluence $\Phi$ as the independent variable and abbreviate $a={\sigma }_{a\mathrm{Th}}$, $b={\sigma }_{a\mathrm{U}}$ and $c={\sigma }_{c\mathrm{Th}}$. Substituting the parent solution into the second linear differential equation givesThe integrating factor is $e^{b\Phi }$. Integrating from the initial state yieldsThe first term is survival of the initial population; the second accumulates production at earlier exposures, weighted by subsequent survival. For $b=a$, the continuous limit is $(N_{U0}+c{N}_{Th0}\Phi )e^{-a\Phi }$. The same formula applies to a time-varying prescribed flux through $\Phi (t)$; intervals of zero flux leave both populations unchanged.

The production rate is ${\sigma }_{c\mathrm{Th}}\varphi N_{\mathrm{Th}}$, while depletion includes both capture and fission of the fissile population, ${\sigma }_{a\mathrm{U}}\varphi N_{\mathrm{U}}$. Thus the conversion ratio in this stipulated model isAt zero instantaneous flux this expression describes the limiting ratio of the two reaction rates. The total absorption coefficients are $a=3.08+0.02=3.10\mathrm{b}$ and $b=7.10+57.10=64.20\mathrm{b}$, soIt is below one: fissile nuclei are initially removed slightly faster than this simplified chain replaces them. Consequently this model is not initially self-replacing in fissile inventory. Inventory replacement is distinct from the instantaneous neutron balance that determines criticality.

At this point $N_{\mathrm{Th}}/N_{Th0}=0.95$, so $a\Phi =-\ln 0.95$ and $e^{-b\Phi }=0.95^{b/a}=0.345671$. The solution from part (c) givesThereforeThe fissile population has declined by about $4.51\%$ from its initial value, while the conversion ratio has decreased slightly from $0.9595$. The replacement shortfall persists over this interval. Maintaining a specified long-term fuel inventory would require replenishment or a different balance of reactions; the prescribed-flux equations alone do not establish a constant-power operating state. Intermediate decay delays and the changing neutron spectrum were excluded by the original lumped model.