Fundamental Cascade Stage Theory in Isotope Separation for ENU4930/6937: Elements of Nuclear Safeguards, Non-Proliferation, and Security Presented by Glenn E. Sjoden, Ph.D., P.E. Associate Professor and FP&L Endowed Term Professor -- 2007.2010 2010 Florida Institute of Nuclear Detection and Security Nuclear & Radiological Engineering University of Florida
Introduction Overview Discussion i of Fissile il Materials French Pub Nuclear Fuel Cycle Front End / Back End Reactor Centric Conversion Enrichment Reprocessing Summary
Enrichment is key to the Nuclear Fuel Cycle From Reilly, et al, Passive NDA of Nuclear Materials, NRC Press, March 1991
The 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) 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 technologies Single Stage separation factor for stage i: Alpha i = (y i /(1-y i )) / (x i /(1-x i )) Gaseous Diffusion U 235 F 6 and U 238 F 6 Gas Molecules have kinetic energy E=1/2 m v 2 Based on velocity ratios, U 235 strikes barrier more often, leads to Alpha i = 1.00429 Becker Nozzle Process Based on centrifugal velocity of the nozzle design, with 5% UF 6 and 95% hydrogen gas, and a pressure ratio of 3.5 (which drives cost) Alpha i = 1.015 From Benedict, et al, Nuc. Chem. Engineering
Separation factors of various technologies 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 Cascade Overall material balance, kg/s F = P+W Plant Material balance on desired component (U-235) is, kg/s F z f = P y p + W x w For a standard Countercurrent Recycling Cascade Feed is heads from adjacent lower stage + tails from adjacent higher h stage Stage 1 is bottom tails, with W kg/s at x w weight frac. Stage n is top product, with P kg/s at y p weight frac. Stage 1 : n s = Stripping section; n s +1: n Enriching section Intermediate stages: M i kg/s at y i weight frac., N i kg/s at x i weight frac. From Benedict, et al, Nuc. Chem. Engineering
Countercurrent Recycle Cascade M i y i N i+1 x i+1 M j y j N j+1 x j+1 From Benedict, et al, Nuc. Chem. Engineering
Countercurrent Recycle Cascade Refer to Diagram on previous slide In a given enriching section from the product end down to just above stage i: M i = N i+1 + P M i y i = N i+1 x i+1 + P y p Solve 2 eqns, 2 unknowns for x i+1 x i+1 = (1 + P/N i+1 )y i - P y p /N i+1 In stripping section where flow is reversed, stage balancing at a strip stage j yields: M j = N j+1 -W M j y j = N j+1 x j+1 + W x w Solve 2 eqns, 2 unknowns for x i+1 x j+1 = (1 - W/N j+1 )y i + W x w /N j+1 From Benedict, et al, Nuc. Chem. Engineering
Countercurrent Recycle Cascade: Reflux Ratio Reconsider the heads strip weight fraction result: x i+1 = (1 + P/N i+1 )y i -P y p /N i+1 Solve this for y i -x i+1 : y i -x i+1 = (y p -y i )/(N i+1 /P) (N i+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 infinite When this occurs, heads at i = tails at i+1, or y i = x i+1 We can use this to derive the minimum number of stages needed for a given enrichment scenario and separation technology (alpha) Optimization of the Reflux Ratio (N i+1 /P) relative to the desired amount of top product P is essential for designing feed and throughput into an enrichment plant From Benedict, et al, Nuc. Chem. Engineering
Infinite Reflux Ratio for minimum # stages Reconsider the heads strip weight fraction result: Let η i = y i /(1- y i ) and ξ i+1 = x i+1 /(1- x i+1) ) With (N i+1 /P) -> Infinity for a maximum Reflux Ratio y i = x i+1 and η i = ξ i+1 But then η i+1 = α η i, and η 2 = α η 1 η 3 = α η 2 = α 2 η 1 so that η n = α n-1 η 1 But η 1 = α ξ 1 = α x w /(1- x w ) This yields the Underwood Fenske equation: η p = y p /(1- y p ) = α n x w /(1- x w ) Solving for α n min = yp (1- x w ) / ((1- y p ) x w ) (where n is a minimum) From Benedict, et al, Nuc. Chem. Engineering
Product mass withdrawal limited with yp Mass Feed through the plant is based on the value function for separative work kg U separative work/kg U fed or (SWU kg/kg) Proportional to Value Function: From Benedict, et al, Nuc. Chem. Engineering
Summary Simple Stage Isotope Separation theory considered A complex process when optimizing for each technology Optimum heads, tails flow, etc 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?