For a reference LWR: assume a thermal efficiency is $33\%,$ power $1150$ MWe, uranium is $$70/$lb$.$ U$3$O$8,$SWU is $$120/$kgSWU$,$ conversion is $$40/$kgU$,$ losses in conversion $0\mathrm{.}15\%,$ product enrichment of $4\mathrm{.}95$wt$\%,$ tails assay of $0\mathrm{.}2$wt$\%,$ reference core contains $90$ tonne fuel, capacity factor is $0\mathrm{.}92.$ An average economical grade of uranium ore is $0\mathrm{.}2\%.$ The average fuel discharge burnup is$55,000$ MWD$/$MTU$.$ For centrifuge enrichment, assume $50$ kWh$/$kgSWU$.$

a$.$ What is the requirement of natural uranium, SWU, and ore to provide the enriched uranium for the reactor fuel for a typical reload of one$-$third of the core? Be sure to calculate showing all work intermediate quantities such as feed, waste, and SWU factors.

b$.$ What is the cost of the enriched uranium on a per kg basis and total requirement for this reload?

c$.$ What is the optimal tails assay to minimize the cost of the enriched uranium $($not necessarily the value given above and used in your analysis here$)?$ Plot $($using Excel or similar$)$ the cost of enriched fuel vs$.$ tails assay and include in your exam. Clearly outline the methodology $($formulae$)$ used in your calculation to produce the plot.

d$.$ Calculate the energy input for the enrichment of the fuel for this reload.

e$.$ Compare the natural uranium requirement to the available world reported $$Identified Resources$$ from the current OECD Redbook. Cite referenced$/$reported value including page number.

f$.$ Suppose you are given $($for free$)$ high assay tails $0\mathrm{.}5\%.$ What would be the cost per kg to reenrich as a substitute for the natural uranium required for this reload? Compare to the costs for natural uranium. Assume your waste stream has the same tails assay as evaluated earlier.