1. Chapter 3
Isotope Separation Methods for Nuclear Fuel
Shuichi Hasegawa
Glossary
Isotope Nuclei of a chemical element which have the same number
of protons but different number of neutrons. Some isotopes
are stable; some are radioactive.
Separation factor A ratio of a mole fraction of an isotope of interest to that of
non-interest in an enriched flow divided by that in
a depleted flow from a separation unit. The factor should
be larger than unity for the unit to result in isotopic
enrichment.
Separation capability A measure of separative work by a cascade per unit time.
Mean free path An average distance of a moving gas molecule between its
collisions.
Molecular flow Low-pressure phenomenon when the mean free path of
a gas molecule is about the same as the channel diameter;
then a molecule migrates along the channel without inter-
ference from other molecules present.
This chapter was originally published as part of the Encyclopedia of Sustainability Science and
Technology edited by Robert A. Meyers. DOI:10.1007/978-1-4419-0851-3
S. Hasegawa (*)
Department of Systems Innovation, School of Engineering, The University of Tokyo,
7-3-1 Hongo Bunkyo-ku, Tokyo, Japan
e-mail: hasegawa@sys.t.u-tokyo.ac.jp
N. Tsoulfanidis (ed.), Nuclear Energy: Selected Entries from the Encyclopedia 59
of Sustainability Science and Technology, DOI 10.1007/978-1-4614-5716-9_3,
# Springer Science+Business Media New York 2013

2. 60 S. Hasegawa
Definition of the Subject
Isotope separation, in general, means enrichment of a chemical element to one of its
isotopes (e.g., 10B in B; 6Li in Li, 157Gd, etc). In the case of uranium, isotope
separation refers to the enrichment in the isotope 235U, which is only 0.711% of
natural uranium; today’s nuclear power plants require fuel enriched to 3–5%
in 235U. Uranium enrichment is the subject of this article.
Efficiencies of sorting out different isotopes of the element (separation factor)
are usually very low. For practical enrichment plants, a gaseous diffusion process
has been successfully employed to obtain enriched uranium. A gas centrifugation
process is the preferred method of enrichment today due to reduced energy con-
sumption. A new process using lasers, which can have a high efficiency of separa-
tion, is under development and has the potential to replace the current enrichment
methods.
Introduction
The fuel used today by commercial nuclear power plants is the fissile isotope 235U.
Unfortunately, 235U is only 0.711% of natural uranium, the rest of which is,
essentially, 238U. Light water reactors (LWR) operating dominantly all over the
world require isotope enrichment processes because the isotopic ratio of 235U for
their fuels should be 3–5%. The processes used to elevate the 235U content from
0.711% to 3–5% are called isotope separation or enrichment processes. Table 3.1
shows the current trends of isotope separation capabilities of the world. The main
countries performing the process are Russia, France, US, and URENCO (Germany,
Table 3.1 World Enrichment capacity (thousand SWU/year) [1]
Country 2010 2015 2020
France (Areva) 8,500* 7,000 7,500
Germany, Netherlands, UK (Urenco) 12,800 12,200 12,300
Japan (JNFL) 150 750 1,500
USA (USEC) 11,300* 3,800 3,800
USA (Urenco) 200 5,800 5,900
USA (Areva) 0 >1,000 3,300
USA (Global Laser Enrichment) 0 2,000 3,500
Russia (Tenex) 23,000 33,000 30–35,000
China (CNNC) 1,300 3,000 6,000–8,000
Pakistan, Brazil, Iran 100 300 300
Total approx. 57,350 69,000 74–81,000
Requirements (WNA reference scenario) 48,890 55,400 66,535
Source: WNA Market Report 2009; WNA Fuel Cycle: Enrichment plenary session WNFC April
2011
*Diffusion

3. 3 Isotope Separation Methods for Nuclear Fuel 61
Netherland, and UK). A number of separation processes have been studied so far,
but the principles of the current isotope separation processes mainly use gaseous
diffusion or gas centrifugation. The diffusion process was commercialized first but
the centrifugation is taking over because of less energy consumption. This article
following mainly [2, 3] describes the principles of the two processes and cascade
theory, which explains why it is required to repeat the process many times (using
successive stages/cascades) to obtain a certain desired enrichment fraction such as
3–5% because a single step provides only a small incremental enrichment. The new
enrichment technology using lasers will be described at the end.
Principles of the Separation Processes
Gaseous Diffusion
Figure 3.1 shows the schematic diagram of the gaseous diffusion process. Consider
a chamber divided into two compartments by a porous membrane. When dilute
gases are introduced into the bottom compartment of the chamber, the pores of the
membrane (membrane) make dependency of the transmission of the gases on their
molecular masses.
If we have a mixture of two molecules in a gas with the same kinetic energy
(kinetic energy is determined by kT, k = Boltzmann constant; T = temperature in K;
(1/2 mv2 $ kT)), the lighter molecule is faster than the heavier one. Therefore, their
frequencies of hitting the membrane is higher for the lighter than for the heavier
molecule. However, the mass preference phenomena occur only when
the mean free path of the gas molecule is longer than the diameter of the pores 2r
and the thickness of the membrane l. The mean free path, l of the molecule can be
written as [2]
kT
l ¼ pffiffiffi (3.1)
4 2ps2 p
where k is the Boltzmann constant, T is the absolute temperature, s is the radius of
the molecule, and p is the gas pressure in the chamber. In this condition, a molecule
Product
p′
Feed Waste
p′′
Fig. 3.1 A single gaseous
diffusion stage

4. 62 S. Hasegawa
cannot collide with others during the transmission through the membrane so that its
dynamics can be considered as a single molecule process. This process is called
molecular flow. The flux of the molecular flow through the flow path with circular
cross section is derived by Knudsen as [3]
8rDp
Gmol ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3.2)
3l 2pmRT
where Gmol is the molecular flow velocity, m is the molecular mass, R is the gas
constant, and Dp ¼ p00 À p0 is the pressure difference between the bottom and top
compartments of the chamber. Equation 3.2 shows that the flow velocity depends
on the mass of the gas molecules so that the ratio of the molecules in the mixture
transmitted to the upper compartment of the chamber is changed compared with
that of the feeding gas. The opposite condition where flows do not depend on the
molecular mass is called viscous flow.
We will derive the ideal separation factor in the case of 235UF6 and 238UF6 [3],
the gas molecules used for uranium enrichment. On the ideal condition where p00 is
very small and p0 can be neglected compared with p00 , when we have a binary
mixture of gases which consist of 235UF6 (molecular mass: m235 = 349, mole
fraction: x) and 238UF6 (molecular mass: m235 = 352, mole fraction: 1 À x), the
molecular flow velocities of 235UF6 and 238UF6 are
00 00
ð1ÀxÞ
G235 ¼ pffiffiffiffiffiffiffi ;
ap x
m235 G238 ¼ ap ffiffiffiffiffiffiffi
p
m238 (3.3)
where the constant a includes factors in Eq. 3.2. The ratio of the molecular flow of
235
UF6 to the whole can be written as
pffiffiffiffiffiffiffiffiffi
x x
G235 m235 1 À x ffiffiffiffiffiffiffiffiffi
s¼ ¼ ¼ r (3.4)
x 1Àx
G235 þ G238 pffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffi x þ m235
m235 m238 1 À x
m238
Therefore, the ideal separation factor a0 of the gaseous diffusion process can be
derived as the separation factor of the molecular flow of the porous media
s rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi
m238 352
a0 ¼ 1Às
x ¼ ¼ ¼ 1:00429 (3.5)
1Àx m235 349
The separation factor depends on the ratio of the molecular masses so that this
method is more effective for the isotope separation of lighter elements. For heavier
elements, a larger number of repeated processes is required to obtain sufficiently
enriched products.
However, in reality, the real value of the separation factor is smaller than that
given by Eq. 3.5 due to reverse molecular flow from the upper compartment to the

5. 3 Isotope Separation Methods for Nuclear Fuel 63
bottom one and viscous flow not depending on the molecular mass; these two
phenomena work in a direction negating the enrichment process. Furthermore
operating conditions (porous media performance, working pressures, etc.) affect
the value of the separation factor. The energy consumption to run the process is
very high due to pressure controlling of the gases, small separation factors and so on
(see discussion about Separative Work Unit). Because of the relatively high energy
consumption, uranium enrichment by gaseous diffusion is on the way out and is
replaced by the gas centrifugation method.
Gas Centrifugation
The principle of gas centrifugation is based upon centrifugal forces that are created
inside a rotating cylinder containing two different gas molecules, forces that depend
on the molecular mass. Let’s see how it works in detail [2]. When we have a mixture
of two gas molecules in a rotating cylinder (centrifuge), pressure gradients develop
with respect to the radial direction. The pressures can be written as
dp
¼ o2 rr (3.6)
dr
where p is the pressure, r is the radial distance, o is the angular frequency of
rotation, and r is the density of the gases. By substituting the equation of state r
¼ pm=RT into the differential equation, we can derive the following equation,
dp mo2
¼ rdr (3.7)
p RT
When we integrate this differential equation from the radial distance r (pressure
pr) to the inner radius of the cylinder a (pressure pa), we can obtain this expression,
& r 2 '!
pr 1 mv2
¼ exp À a
1À (3.8)
pa 2 RT a
where the speed of the outer circumference of the cylinder na = oa This equation
shows that the ratio of the pressure at radius r to that of radius a depends on the
molecular mass of the gases.
If we have the gases which consist of 235UF6 (molecular mass: m235 = 349, mole
fraction: x) and 238UF6 (molecular mass: m235 = 352, mole fraction: 1 À x), their
ratios of the partial pressures at the radius r to the radius a can be derived as
r 2 '!
pr x r 1 m235 v2
¼ exp À a
1À (3.9)
pa x a 2 RT a

6. 64 S. Hasegawa
Table 3.2 The local separation factor of 235UF6 and 238UF6 at r/a with T = 300K, ua = 700m/s
r/a 0 0.5 0.8 0.9 0.95 0.98 0.99 1.0
a 1.343 1.247 1.112 1.058 1.029 1.012 1.006 1.0
r 2 '!
pr ð1 À xr Þ 1 m238 v2
¼ exp À a
1À (3.10)
pa ð1 À xa Þ 2 RT a
Therefore, the local separation factor at radial distance r of radius a is given by
xr r 2 '!
1 À xr ðm238 À m235 Þv2
a¼ xa ¼ exp a
1À (3.11)
2RT a
1 À xa
which depends on the difference of their molecular masses, Dm = m238Àm235 = 3.
Values of the local separation factor of 235UF6 and 238UF6 with T = 300 K and na =
700 m/s are given in Table 3.2. This feature is superior to the gaseous diffusion
method when the difference of the masses is large, (e.g., for heavier elements). The
separation factor increases as the speed of the outer circumference increases.
However, the maximum speed vmax is limited by stresses created to the cylinder
from the force of the centrifugation and can be written as [2]
rffiffiffi
s
vmax ¼ (3.12)
r
where r is the density of the material of the cylinder, and s is the tensile strength.
Although most molecular gases are localized at a % 1 because of the centrifugation,
r
the values of the separation factor could be higher than those obtained from the
gaseous diffusion method. These values can be enhanced if we make use of
a countercurrent flow in the vertical direction. Figure 3.2 shows the schematic
diagram of countercurrent centrifugation method. Gernot Zippe performed
pioneering work on the development of the centrifugation first in the Soviet
Union during 1946–1954, and from 1956 to 1960 at the University of Virginia.
The countercurrent flow can be induced by heating and cooling centrifuges, or pipes
drawing off flows in centrifuges. The temperature control can adjust the flow
deliberately but the equipment becomes more complicated than that of the flow
control by the pipes (Fig. 3.2). This countercurrent flow makes enrichment of the
lighter isotopes at inner radius as the flow descending along the axis direction, and
the heavier isotopes are being enriched at the circumference as the flow ascending.
These enriched gases are collected at different radial positions of the both ends (at
outer radius for heavier isotope and at inner radius by baffle for lighter isotope).
When the centrifuge has a length L, the maximum separative power dUmax can be
derived as [2, 4]
2
p Dmv2
dUmax ¼ LrD a
(3.13)
2 2RT

7. 3 Isotope Separation Methods for Nuclear Fuel 65
Fig. 3.2 Schematics of gas
centrifuge with Heads (Product)
countercurrent flow
Feed
Tails (Waste)
238UF 238UF
6 6
238
UF6 235
UF6 235 238
UF6
UF6
235UF 235UF
6 6
where D is diffusion coefficient. The maximum separative power is proportional to
the height of the centrifuge. It is preferable to have a taller centrifuge in the vertical
direction, but the length is imposed on the resonant vibration of the centrifuge. The
resonant conditions can be written as [3]
pffiffiffiffirffiffiffiffiffiffiffi
L 4 E
¼ li li ¼ 22:0; 61:7; 121:0; 200:0; 298:2;Á ÁÁ (3.14)
a i 2s0
where E is coefficient of elasticity. A taller centrifuge can give a larger separative
power although excellent mechanical properties are required to overcome the
resonant conditions.
Cascade Theory
The present isotope separation plants make use of these principles of enrichment
with small separation factors. In order to obtain high enrichment ratios, cascade
theory is necessary [3]. According to the theory, we can enhance the ratios by
iterating a single physical stage many times. Figure 3.3 shows a simple scheme

8. 66 S. Hasegawa
Heads P1 Feed Fi Heads Pi Feed Fn
xP
1 i
xF i
xP n
xF
Feed F Product P
stage1 stage i stage n
Feed xF Product xP
Waste W1 Waste Wi Waste Wn
xW1 xWi xWn
Fig. 3.3 Simple scheme of the cascade
of a cascade. An original material “feed” is provided to the system. The isotope of
interest is enriched as going through many separation stages and a final output
“product” is obtained. Another output which mainly contains unnecessary isotopes
is called “waste.” Each flow F, P and W should have the following equation
F¼PþW (3.15)
and with mole fractions of the isotope of interest in each flow xF , xP , xW , we can
obtain
FxF ¼ PxP þ WxW (3.16)
In this system, we have four independent parameters to define. In order to obtain
necessary flow of Product “P ” and mole fraction “xP” of the isotope of interest, we
need the design methodology to construct stages of separation units. The product of
a single stage (unit) is called heads and the waste of that is tails. The ratios of the
isotope of interest in the product are usually most important. If, for instance, we
have two isotopes “1” and “2,” and want to enrich the “1” isotope, we would focus
on the variation of the mole fraction ratio of the two isotopes, x1 , which can be
x2
x1
rewritten as 1Àx1 . The capability of each enrichment unit is described as separation
factor a. This factor is defined as the ratios of the isotopes of interest to that of
not-interest in the heads (product) divided by those in the tails (waste)
xP
1 À xP
a ¼ xW (3.17)
1 À xW
In a similar way, we can define the ratio of the heads (product) to the feed as
heads separation factor b, and that of the feed to the tails as tails separation factor g,
xP xF
1 À xP 1Àx
b¼ xF ; g ¼ xW F and a ¼ bg (3.18)
1 À xF 1 À xW

9. 3 Isotope Separation Methods for Nuclear Fuel 67
Feed Fi Heads Pi Feed Fi Heads Pi
i
xF i
xP xF+ 1
i xP+ 1
i
stage
stage i
i+1
Waste Wi Waste Wi
xWi xW+ 1
i
Fig. 3.4 Simple cascade of the i and i + 1 th stages
The ratio of the product to the feed is called “cut” y and defined as
P xF À xw
y ¼ (3.19)
F xP À xW
The simplest design to accomplish enrichment is to accumulate separation stages
in a single line such as Fig. 3.4. This scheme is called simple cascade.
Simple Cascade
In this scheme, the heads and the mole fraction of the i th stage are equal to the feed
flow and the mole fraction of the i + 1 th stage (Fig. 3.4).
Fiþ1 ¼ Pi ; xiþ1 ¼ xiP
F (3.20)
This cascade disposes of the tails of all stages so that the total amount of the
isotope of interest in the waste should be given sufficient attention. This can be
evaluated by means of the recovery rate of the i th stage ri
xiW
1À
Pi xiP xi xiF À xiW xiP xiF a i À bi
ri ¼ ¼ yi P ¼ ¼ i ¼ a À1 (3.21)
Fi xiF xiF xiP À xiW xiF x i
1À W
xiP
When we have n stages in the cascade, the total recovery rate r can be expressed as
P xP Pn xn P1 x1 P2 x2 Pn xn
r¼ ¼ P
¼ P P
¼ P
¼ r1 r2 Á Á Á rn (3.22)
F xF F1 x1 F1 x1 F2 x2 Fn xn
F F F F
The over-all separation factor of the cascade o can be derived as

10. 68 S. Hasegawa
Fig. 3.5 Countercurrent
recycle cascade
Stage n
Heads Pi
xP+ 1
i Feed Fi
xF+ 1
i
Stage i Heads Pns +1
Tails Wi
xP s+ 1
n
xW+ 1
i
Feed F Stage
xF ns + 1
Heads Pns
n
xP s Tails Wns+1
Stage ns xW s + 1
n
Feed Fns
Tails Wns n
xF s
n
xW s
Stage 1
xn
P
1 À xn
o¼ P
¼ b1 b2 Á Á Á bn (3.23)
x1
F
1 À x1
F
Therefore, if a, b do not depend on each stage, the total recovery rate can be
rewritten as
!n
a À o f ng
1
aÀb n
r¼ ¼ (3.24)
aÀ1 aÀ1
When the feed itself is available without any special cost, the simple cascade is
effective. But in case the wastes from each stage should not be disposed because,
for instance, it is valuable or the recovery rate has to be increased, the waste flows
are recycled as feed flow, which is called countercurrent recycle cascade (Fig. 3.5).
Countercurrent Recycle Cascade
Since the simple cascade cannot improve the recovery rate, the tail flow is recycled
into either stage to use it efficiently, which is called recycle cascade (Fig. 3.5). If b
(heads separation factor) is equal to g (tails separation factor) in all stages, we can
obtain xiþ2 ¼ xiþ1 ð¼ xiP Þ. So the tails flow of the i + 2 th stage can be merged to the
W F

11. 3 Isotope Separation Methods for Nuclear Fuel 69
heads flow of the i th stage and fed into the i + 1 th stage without any mixing loss.
We will consider the case that the tails flow of the second upper stage is refluxed to
the i th stage.
The flows and the fractions of the isotope of interest in each stage of enriching
sections should have the following relationships.
Pi ¼ Wiþ1 þ P; Pi xiP ¼ Wiþ1 xiþ1 þ PxP
w (3.25)
In a similar way, those in stripping sections can be expressed as
Wjþ1 ¼ Pj þ W; Wjþ1 xjþ1 ¼ Pj xjP þ WxW
W
(3.26)
Let’s estimate the number of stages. From these equations, we can derive
xP À xiP
xiP À xiþ1 ¼
W Wiþ1 (3.27)
P
At total reflux, where the reflux ratio is infinity,
Wiþ1
!1 (3.28)
P
the mole fraction of the heads flow at the i th stage xiP becomes equal to that of the
tails flow at the i + 1 th stage xiþ1 and the number of the stages is minimal.
W
xiþ1 xiþ1 xi xiÀ1
P
¼ a W iþ1 ¼ a P i ¼ a2 P iÀ1 ¼ Á Á Á (3.29)
1 À xiþ1
P 1 À xW 1 À xP 1 À xP
gives the following equation,
xP xW
¼ an (3.30)
1 À xP 1 À xW
and the minimum number of the stages at total reflux can be derived as
1 xP 1 À xW
n¼ ln (3.31)
ln a 1 À xP xW
On the contrary, the reflux ratio becomes minimum when the mole fraction of
the heads at the i + 1 th stage is equal to that of the heads at the i th stage ðxP ¼ xPÞ.
iþ1 i

12. 70 S. Hasegawa
Ideal Cascade
Ideal cascade satisfies the condition that the values of b (heads separation factor) at
all stages are constant and the mole fraction of the heads flow at the i + 1 th stage is
equal to those of the tails flow at the i À 1 th stage and of the feed flow at the i th
stage ðxiþ1 ¼ xiÀ1 ¼ xiF Þ . In this instance, each separation factor satisfies the
p W
following relationship.
pffiffiffi
b¼ a¼g (3.32)
In a similar way to the previous section, we can obtain the total number of the
stages for an ideal cascade
1 xP 1 À xW
n¼ ln À1
ln b 1 À xP xW
(3.33)
2 xP 1 À xW
¼ ln À1
ln a 1 À xP xW
The number of stages in stripping nS and enriching nE = n À nS sections can be
derived as
1 xF 1 À xW
nS ¼ ln À1 (3.34)
ln b 1 À xF xW
1 xP 1 À xF
nE ¼ n À nS ¼ ln (3.35)
ln b 1 À xP xF
The reflux ratio Eq. 3.27 can be rewritten using xiP ¼ xiþ1 and b as
F
'
Wiþ1 xP À xiP 1 xP bð1 À xP Þ
¼ i ¼ À (3.36)
P xP À xiþ1 b À 1 xiþ1
W W 1 À xiþ1
W
Mccabe–Thiele Diagram
It is useful to draw McCabe–Thiele diagram to investigate the design of the
cascade, the mole fractions of the stages and so on. Figure 3.6 shows a typical
McCabe–Thiele diagram. In this graph, the horizontal and vertical axes correspond
to the mole fractions of the heads flow xiP and of the tails flow xiW , respectively.

13. 3 Isotope Separation Methods for Nuclear Fuel 71
1
xP
Equilibrium line
x P+1
i
x iP x iW
= α
1 x iP 1 x iW
x iP
Operating line
Heads mole fraction
i
xP
1 x iP
x P−1
i
i +1
xW
= α
i +1
1 xW
i +1 i −1
xW = x P
xW
0
0 1
xW xW−1
i
x iW xW+1
i
xW+2
i xP
Tails mole fraction
Fig. 3.6 McCabe-Thiele diagram
First, the following equation is satisfied at the enrichment process of the i th
stage because of the definition of the separation factor
xiP xiW
i ¼ a ðEquilibrium lineÞ (3.37)
1 À xP 1 À xiW
Second, the condition that the tail (waste) flow at the i + 1 th stage is the feed of
the i th stage ðxiF ¼ xiþ1 Þ defines the relationship between the mole fractions of the
W
tail (waste) and head (product) flows at different stages as follows
xiP xiF pffiffiffi xiþ1
i ¼ b i ¼ a W iþ1 ðOperating lineÞ (3.38)
1 À xP 1 À xF 1 À xW

14. 72 S. Hasegawa
And third, the feed flow at the i th stage consists of the tails flow of the i + 1 th
stage and the heads flow of the i À 1 th stage and their mole fractions are the same.
xiþ1 ¼ xiÀ1
W P (3.39)
These three formulae can be shown in the McCabe–Thiele diagram as shown in
Fig. 3.6. We can estimate the number of necessary stages, mole fractions of the
stages, and overview the total processes through the graphical construction.
Separative Work Unit
The total flow in the cascade can be derived as
X
bþ1 xW
ðPi þ Wi Þ ¼ Wð2xW À 1Þ ln
i
ðb À 1Þ ln b 1 À xW
!
xP xF
þPð2xP À 1Þ ln ÀFð2xF À 1Þ ln (3.40)
1 À xP 1 À xF
The first term of Eq. 3.40 including b indicates the difficulty of the separation
and increases as the value of b approaches to unity. The second term corresponds to
the amount of work for separation, and it has the same dimension as flow rates and
is called separative capacity or separative power. This value is important because it
is considered to be proportional to the initial cost of the plant. When we use the unit
of the amounts of material (mole, kg, etc.) instead of flow rates, this is called
separative work. The sum of the annual investment and operation costs can be
expressed by the product of the separative work SW (kg SWU/year) and unit price
of separative work cs ($/kg SWU). SWU is the abbreviation of Separative Work
Unit. The separative work is defined as
SW ¼ WfðxW Þ þ PfðxP Þ À FfðxF Þ (3.41)
where f(xi) is called separation potential and written as
xi
fðxi Þ ¼ ð2xi À 1Þ ln (3.42)
1 À xi
When we use kg SWU/year for the separative work, the unit of W, P, and F
should be kg/year.
For operating the plant, we need the raw materials, the amount of which is F (kg/
year) and unit price of the raw materials cF ($/kg). The total cost per year c ($) can
be written as
c ¼ SWcS þ FcF (3.43)

15. 3 Isotope Separation Methods for Nuclear Fuel 73
When the amount of the product per year is P (kg), the unit cost of the product
cP ¼ P : could be derived as
c
'
SWcs FcF fðxF Þ À fðxW Þ
cP ¼ þ ¼ ðfðxP Þ À fðxF ÞÞ À ðxP À xF Þ cs
P P xF À xW
xP À xW
þ cF (3.44)
xF À xW
Example
With the ideal cascade of the gaseous diffusion method (a = 1.00429), the mole
fraction of the feed flow 0.711% (xF = 0.00711) would be enriched to 3% (xP = 0.03)
and the mole fraction of the waste is planned to be 0.3% (xW = 0.003). In this case, the
necessary moles of the feed and the waste to obtain the product of 1 [mol] are
PðxP À xW Þ 1 Â ð0:03 À 0:003Þ
F¼ ¼ ¼ 6:569½molŠ
xF À xW 0:00711 À 0:003
PðxP À xF Þ 1 Â ð0:03 À 0:00711Þ
W¼ ¼
xF À xW 0:00711 À 0:003
¼ 5:569ð¼ 6:569 À 1Þ½molŠ
The total number of stages n and the number of stages in stripping section nS and
in enriching section nE are calculated as
Stripping Section
2 xF 1 À xW
nS ¼ ln À1
lna 1 À xF xW
2 0:00711 1 À 0:003
¼ ln À 1 ¼ 404
ln 1:00429 1 À 0:00711 0:003

16. 74 S. Hasegawa
Enriching Section
2 xp 1 À xF
nE ¼ ln
ln a 1 À xp xF
2 0:03 1 À 0:00711
¼ ln ¼ 683:5
ln 1:00429 1 À 0:03 0:00711
The total number of stages
2 xp 1 À xW
n¼ ln À1
ln a 1 À xp xW
2 0:03 1 À 0:003
¼ ln À 1 ¼ 1087:5
ln 1:00429 1 À 0:03 0:003
The heads flow rate in the enriching section can be written as
Pi ¼ P þ Wiþ1
P
¼Pþ fxp ð1 À biÀn Þ þ ð1 À xp ÞbðbnÀi À 1Þg
bÀ1
and that in the stripping section
W È É
Pi ¼ xW bðbi À 1Þ þ ð1 À xW Þð1 À bÀi Þ
bÀ1
These flows as a function of the number of the stages can be shown as Fig. 3.7 in
this example.
When we need higher concentration, such as 5%, F = 11.436[mol], W = 10.436
[mol], n = 1336 and nE = 932.
Laser Isotope Separation (LIS)
The photon absorbing frequencies of isotopes show small differences caused by
shifts of atomic electron energies due to the differences in the number of neutrons
among isotopes. This is called isotope shift. The invention and development of
lasers enable to resolve the isotope shift sufficiently and make isotope-selective
photo-chemical reaction possible. Laser Isotope Separation may lead to almost
100% isotope separation in a single stage. Mainly, two methods such as Atomic
Vapor Laser Isotope Separation (AVLIS) and Molecular Laser Isotope Separation
(MLIS) were intensively studied. AVLIS uses uranium atomic vapor that is struck
by lasers of such wavelength that only 235U atoms are excited and then ionized;

17. 3 Isotope Separation Methods for Nuclear Fuel 75
Feed
xF = 0.00711
1600
1400
1200
Heads flow rate
1000
800
600
Product
400 Waste xP = 0.03
xW = 0.003
200
0
0 100 200 300 400 500 600 700 800 900 1000 1100 1200
number of stages
Fig. 3.7 Heads flow rate
once ionized, the 235U ions are collected by an electromagnetic field. MLIS uses
UF6, and vibrationally excites and multiphoton-dissociates only 235UF6 into
235
UF5 by infrared lasers. The research to commercialize them has faded on
a global scale.
A new process called Separation of Isotopes by Laser Excitation (SILEX) is
under development. All details are not out in the open yet; but SILEX is considered
to be a kind of molecular LIS using UF6. The method only isotope-selectively
excites but not dissociates 235UF6. The separation factor announced by the company
has been 2–20 [5]. Silex Systems Ltd was originally established as a subsidiary of
Sonic Healthcare Limited of Australia in 1988. In 2007, the SILEX Uranium
Enrichment project was transferred to GE’s nuclear fuel plant in the United States.
Global Laser Enrichment (GLE) was formed as a subsidiary of GE-Hitachi in 2008
[5]. In June 2009, GE-Hitachi submitted a license application to construct a
commercial laser enrichment plant in Wilmington, NC. The NRC staff is currently
reviewing that application. They announced that they succeeded the initial mea-
surement program at Test Loop in 2010 and proceeded to evaluate the program to
decide the commercialization of the process [6].

18. 76 S. Hasegawa
Future Directions
As of today, the gaseous diffusion and centrifuge processes have been used on a
commercial scale. For the future, it seems that laser enrichment (the SILEX
process) may be the successor to current enrichment methods. Preliminary results,
based on enrichment by lasers, are encouraging. However, considerable
improvements are needed before this method achieves commercial competitive
status. Every uranium enrichment process is linked to nuclear proliferation issues.
It would be very beneficial for the world if a method of enrichment is devised
which inherently offers non- proliferation safeguards for nuclear materials.
Bibliography
1. World Nuclear Association, Uranium Enrichment, World Enrichment capacity - operational and
planned. http://www.world-nuclear.org/info/inf28.html
2. Villani S (1976) Isotope separation. American Nuclear Society, Hillsdale
3. Benedict M, Pigford TH (1957) Nuclear chemical engineering. Mcgraw-Hill, New York;
Benedict M, Pigford TH, Levi HW (1981) Nuclear chemical engineering (second edn.),
(trans: by Kiyose R into Japanese)
4. Kemp RS (2009) Gas centrifuge theory and development: a review of U.S. programs. Science
and Global Security 17, 1; Wood HG, Glaser A, Kemp RS (2008) The gas centrifuge and
nuclear weapons proliferation. Physics Today 40
5. Silex Systems Limited home page. http://www.silex.com.au/
6. World Nuclear News (2010) Initial Success from SILEX test loop, 12 April 2010. http://www.
world-nuclear-news.org/NN-Initial_success_from_SILEX_test_loop-1204104.html