Fundamental Cascade Stage Theory in Fundamental Cascade Stage Theory in I t S tiI t S tiIsotope SeparationIsotope Separation

for ENU4930/6937: Elements of Nuclear Safeguards, Nonfor ENU4930/6937: Elements of Nuclear Safeguards, Non--Proliferation, and SecurityProliferation, and Security

Presented byPresented byyyGlenn E. Sjoden, Ph.D., P.E.Glenn E. Sjoden, Ph.D., P.E.

Associate Professor andAssociate Professor andFP&L Endowed Term ProfessorFP&L Endowed Term Professor ---- 2007 20102007 2010FP&L Endowed Term Professor FP&L Endowed Term Professor 2007.20102007.2010

Florida Institute of Nuclear Florida Institute of Nuclear Detection and SecurityDetection and SecurityDetection and SecurityDetection and Security

Nuclear & Radiological EngineeringNuclear & Radiological EngineeringUniversity of FloridaUniversity of Florida

OverviewOverview

– Introductioni i f i il i l h b– Discussion of Fissile Materials – French Pub

– Nuclear Fuel Cycle/• Front End / Back End

• Reactor Centric

C i– Conversion– Enrichment

R i– Reprocessing– Summary

Enrichment is key to the Nuclear Fuel CycleEnrichment is key to the Nuclear Fuel Cycley yy y

From Reilly, et al, Passive NDA of Nuclear Materials, NRC Press, March 1991

The Nuclear Fuel Cycle: Uranium EnrichmentThe Nuclear Fuel Cycle: Uranium Enrichment

• Most nuclear reactors need higher concentrations of U235 than found in natural uranium • U235 is "fissionable," meaning that it starts a nuclear reaction and keeps it going.

– Normally, the amount of the U235 isotope is enriched from 0.7% of the uranium mass to about 5%, as illustrated in this diagram of the enrichment process.

• The three processes often used to enrich uranium are – Gaseous diffusion (the only process currently in the United States for commercially

enrichment)enrichment)– Gas centrifuges (as often reported in Iran) and Becker Nozzle (South Africa)– AVLIS (Atomic Vapor Laser Isotope Separation)

From USNRC, April 2010

Separation factors of various technologiesSeparation factors of various technologiesp gp g

• Single Stage separation factor for stage i:• Alphai = (yi/(1-yi)) / (xi/(1-xi))Alphai (yi/(1 yi)) / (xi/(1 xi))

• Gaseous Diffusion• U235F6 and U238F6 Gas Molecules have kinetic energy 6 6

E=1/2 m v2

• Based on velocity ratios, U235 strikes barrier more often leads to Alpha = 1 00429often, leads to Alphai = 1.00429

• Becker Nozzle Process• Based on centrifugal velocity of the nozzle design, with 5% UF6g y g 6

and 95% hydrogen gas, and a pressure ratio of 3.5 (which drives cost)

• Alphai = 1.015p i

From Benedict, et al, Nuc. Chem. Engineering

Separation factors of various technologiesSeparation factors of various technologiesp gp g• Gas Centrifuge

• Analysis shows that

– Δm=3– The separation constant (alpha) is based on a mass difference and va, the tangential speed of

rotation at the rotating drum surface, so that alpha is a function of the radius r where the product is scooped, up to a radius of the centrifuge r = a. (R is gas constant and T is absolute temp)

– Resonant frequency speeds (depending upon length and diameter) must be avoided; motor drives of sufficient power to accelerate/decelerate centrifuges quickly through resonant speeds are needed

From Benedict, et al, Nuc. Chem. Engineering

Countercurrent Recycle CascadeCountercurrent Recycle Cascadeyy

• Overall material balance, kg/s• F = P + WF P + W

• Plant Material balance on desired component (U-235) is, kg/s• F zf = P yp + W xwf p w

• For a standard Countercurrent Recycling Cascade– Feed is heads from adjacent lower stage + tails from

dj hi hadjacent higher stage – Stage 1 is bottom tails, with W kg/s at xw weight frac.– Stage n is top product with P kg/s at y weight frac– Stage n is top product, with P kg/s at yp weight frac.– Stage 1 : ns = Stripping section; ns+1: n Enriching section– Intermediate stages: Mi kg/s at yi weight frac., g i g/ yi g ,

Ni kg/s at xi weight frac. From Benedict, et al, Nuc. Chem. Engineering

Countercurrent Recycle CascadeCountercurrent Recycle Cascade

Mi yii i

Ni+1 xi+1

Mj yj

Nj+1 xj+1

From Benedict, et al, Nuc. Chem. Engineering

Countercurrent Recycle CascadeCountercurrent Recycle Cascadeyy• Refer to Diagram on previous slide• In a given enriching section from the product end down to g g p

just above stage i:• Mi = Ni+1 + P• Mi yi = Ni+1 xi+1 + P yp

• Solve 2 eqns, 2 unknowns for xi+1

• x = (1 + P/N )y P y /N• xi+1 = (1 + P/Ni+1 )yi - P yp/Ni+1

• In stripping section where flow is reversed, stage balancing at a strip stage j yields:

• Mj = Nj+1 - W• Mj yj = Nj+1 xj+1 + W xw

Solve 2 eqns, 2 unknowns for xi+1

• xj+1 = (1 - W/Nj+1 )yi + W xw/Nj+1From Benedict, et al, Nuc. Chem. Engineering

Countercurrent Recycle Cascade: Reflux RatioCountercurrent Recycle Cascade: Reflux Ratioyy• Reconsider the heads strip weight fraction result:

• xi+1 = (1 + P/Ni+1 )yi - P yp/Ni+1i+1 ( i+1 )yi yp i+1

• Solve this for yi - xi+1 :• yi - xi+1 = (yp - yi )/(Ni+1/P)

• (Ni+1/P) is the “Reflux Ratio”– As P -> 0 (minimal product mass flow, at maximum interstage to

product flow ratio) then the Reflux Ratio becomes infinitep ) f f– When this occurs, heads at i = tails at i+1, or yi = xi+1

– We can use this to derive the minimum number of stages needed for a given enrichment scenario and separation technology (alpha)a given enrichment scenario and separation technology (alpha)

– Optimization of the Reflux Ratio (Ni+1/P) relative to the desired amount of top product P is essential for designing feed and throughput into an enrichment plantthroughput into an enrichment plant

From Benedict, et al, Nuc. Chem. Engineering

Infinite Reflux Ratio for minimum # stagesInfinite Reflux Ratio for minimum # stagesgg• Reconsider the heads strip weight fraction result:

• Let ηi = yi /(1- yi) and ξi+1 = xi+1 /(1- xi+1)ηi yi /( yi) ξi+1 i+1 /( i+1)• With (Ni+1/P) -> Infinity for a maximum “Reflux Ratio”

• yi = xi+1 and ηi = ξi+1

• But then ηi+1 = α ηi , and η2 = α η1

• η3 = α η2 = α2 η1

h 1so that ηn = αn-1 η1

• But η1 = α ξ1 = α xw /(1- xw)• This yields the “Underwood Fenske” equation:• This yields the Underwood Fenske equation:

ηp = yp /(1- yp) = αn xw /(1- xw) (where n is a minimum)

• Solving for αnmin = yp (1- xw) / ((1- yp) xw )

From Benedict, et al, Nuc. Chem. Engineering

Product mass withdrawal limited with Product mass withdrawal limited with yypp

• Mass Feed through the plant is based on the value function for separative work—

• kg U separative work/kg• kg U separative work/kg U fed or (SWU kg/kg)

• Proportional to Value Function:

From Benedict, et al, Nuc. Chem. Engineering

SummarySummary

Simple Stage Isotope Separation theory considered• A complex process when optimizing for each• A complex process when optimizing for each

technology– Optimum heads, tails flow, etcp , ,– Unit failure rates, complexities of maintenance

• Can be analyzed for minimum #stages in a straight forward manner

• Separative work measured in SWU-kg/kg

Questions?Questions?QQ