Fundamental Cascade Stage Theory in
I t
Isotope
Separation
S
ti
for ENU4930/6937: Elements of Nuclear Safeguards, NonNon-Proliferation, and Security
Presented byy
Glenn E. Sjoden, Ph.D., P.E.
Associate Professor and
FP&L Endowed Term Professor -- 2007.2010
2007 2010
Florida Institute of Nuclear
Detection and Security
Nuclear & Radiological Engineering
University of Florida
Overview
– Introduction
– Discussion
i
i off Fissile
i il Materials
i l – French
h Pub
b
– Nuclear Fuel Cycle
• Front End / Back End
• Reactor Centric
–
–
–
–
Conversion
C
i
Enrichment
R
Reprocessing
i
Summary
Enrichment is keyy to the Nuclear Fuel Cycle
y
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
p
factors of various technologies
g
• Single Stage separation factor for stage i:
• Alphai = (yi/(1
/(1-yyi)) / (xi/(1
/(1-xxi))
• Gaseous Diffusion
• U235F6 and U238F6 Gas Molecules have kinetic energy
E=1/2 m v2
• Based on velocity ratios, U235 strikes barrier more
often leads to Alphai = 1.00429
often,
1 00429
• Becker Nozzle Process
• Based on centrifugal
g velocityy of the nozzle design,
g with 5% UF6
and 95% hydrogen gas, and a pressure ratio of 3.5 (which drives
cost)
• Alpha
p i = 1.015
From Benedict, et al, Nuc. Chem. Engineering
Separation
p
factors of various technologies
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
y Cascade
• Overall material balance, kg/s
•F=P+W
• Plant Material balance on desired component (U-235) is, kg/s
• F zf = P yp + W xw
• For a standard Countercurrent Recycling Cascade
– Feed is heads from adjacent lower stage + tails from
adjacent
dj
hi
higher
h stage
– Stage 1 is bottom tails, with W kg/s at xw weight frac.
– Stage n is top product
product, with P kg/s at yp weight frac.
frac
– Stage 1 : ns = Stripping section; ns+1: n Enriching section
– Intermediate stages:
g
Mi kg/s
g/ at yi weight
g frac.,,
Ni kg/s at xi weight frac.
From Benedict, et al, Nuc. Chem. Engineering
Countercurrent Recycle Cascade
Mi yi
Ni+1 xi+1
Mj yj
Nj+1 xj+1
From Benedict, et al, Nuc. Chem. Engineering
Countercurrent Recycle
y Cascade
• Refer to Diagram on previous slide
• In a ggiven enrichingg section from the product
p
end down to
just above stage i:
• Mi = Ni+1 + P
• Mi yi = Ni+1 xi+1 + P yp
• Solve 2 eqns, 2 unknowns for xi+1
• 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
From Benedict, et al, Nuc.
• xj+1 = (1 - W/Nj+1 )yi + W xw/Nj+1
Chem. Engineering
Countercurrent Recycle
y Cascade: Reflux Ratio
• Reconsider the heads strip weight fraction result:
• xi+1 = ((1 + P/Ni+1 )yi - P yp/Ni+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
p
f Ratio becomes infinite
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)
– 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 plant
From Benedict, et al, Nuc.
Chem. Engineering
Infinite Reflux Ratio for minimum # stages
g
• Reconsider the heads strip weight fraction result:
• Let ηi = yi /(
/(1- yi) and ξi+1 = xi+1 /(
/(1- xi+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
• η 3 = α η 2 = α2 η 1
so that
h
ηn = αn-11 η1
• But
η1 = α ξ1 = α xw /(1- xw)
• This yields the “Underwood
Underwood Fenske
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 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
p
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
Q
Questions?