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7/25/2019 Optimum Operating Conditions for a Laser
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O P T I M U M O P E R A T I N G C O N D I T I O N S F O R A L A S E R
U R A N I U M E N R I C H M E N T P L A N T
KIMIO YAMADA an d NORIHIKO OZAKI
Atomic Energy Research Laboratory, Hitachi Ltd, Ozenji, Tama-ku,Kawasaki, Kanagawa Japan)
a n d
MANABU YAMAMOTO an d KI ICHI UEYANAGI
Central Research Laboratory, Hitachi Ltd, Higashi-Koigakubo, Kokubunji, Tokyo Japan)
S UMMA R Y
Operating conditions o f the laser uranium enrichment plant to obtain cheaper enriched
uranium are optimised by using the standard optimisation procedure. A simple kinetic
model is given to obtain the ion production rate as a function of the laser energy
density, ultraviolet light energy density, atomic density and depth and height of the
reaction region. The unit cost of enriched uranium is chosen as a value imction instead
of the unit cost of the separative work. The construction cost is expressed by means of
an exponential fi~nction to take the scale merit into account.
Two numerical results are given. In case 1, the laser power and efficiency are subject
to the restraints determined by the present technical levels and in case
2,
they arej}'ee.
The unit cost of the enriched uranium is higher than those of the gaseous dijfitsion and
gas centrifuge methods by a factor of 2 ~ 11. Results indicate that laser uranium
enrichment is probably competitive with the other uranium enrichment methods,
provided that the laser efficiency is improved by up to 1 and the laser lijetime is
extended several times.
1 . I N T R O D U C T I O N
Th e t ech n i q u e fo r l a s e r i s o t o p e s ep a ra t i o n h as b een d ev e l o p ed an d r ecen t l y t h e
sepa ra t ion o f u ran ium i so top es wa s ach ieved by the se lect ive two-s tep
pho to ion i sa t ion p rocess , in the Uni ted S ta tes .1
A po in t c om m on to the laser iso top e separa t ion i s to use i so tope sh if t , and exc ite
select ively specif ic iso top es with the laser . La ser iso t op e sepa rat io n involves va rious
methods , depend ing on the d i f fe rences in separa t ion , such as two-s tep
287
Applied Energy ( 3 ) ( 1 9 7 7 ) - - A p p l i e d S c i e n c e P u b l i s h e r s L t d , E n g l a n d , 1 9 7 7
P r i n t e d i n G r e a t B r i t a i n
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288
KIMIO YAMADA, NORIH IKO OZAKI, MANABU YAMAMOTO, KI1CHI UEYANAGI
p h o t o i o n i s a t io n , p h o t o d e fl e c ti o n , p h o t o d i s s o c i a t i o n a n d p h o t o c h e m i s t r y . T h e
t w o - s te p p h o t o i o n i s a t i o n m e t h o d f o r t he u r a n i u m a t o m is t he m o s t p r o m i s i n g o f
t h e se m e t h o d s . T h e r e a s o n s a r e : ( 1) th i s m e t h o d h a s a l r e a d y a c h i e v e d su c c es s o n a
l a b o r a t o r y s c a l e; (2 ) t h e u r a n i u m a t o m h a s w el l k n o w n a n d r e s ol v e d is o t o p e s h i ft s i n
e l e c t ro n i c l ev e ls i n c o m p a r i s o n w i t h u r a n i u m m o l e c u l e s .
T h e p u r p o s e o f t h is p a p e r is t o d is c u s s th e o p t i m u m c o n d i t i o n s f o r th e l a s e r
u r a n i u m e n r i c h m e n t p l a n t. T h e t w o - s t e p p h o t o i o n i s a t i o n m e t h o d o f t he u r a n i u m
a t o m i s s e le c te d , a n d t h e C o m p l e x m e t h o d 2 is e m p l o y e d a s a n o p t i m i s a t i o n
p r o c e d u r e . T h e o p t i m u m c o n d i t i o n s f o r t w o t y p ic a l c a s e s a r e s u r ve y e d . I n ca s e 1 , t h e
l a s er p o w e r a n d e ~ c i e n c y a r e s u b j e c t t o r e s t r a in t s d e t e r m i n e d b y th e p r e s e n t
t e c h n i c a l l ev e ls , a n d i n c a s e 2 t h e y a r e f r e e f r o m t h e t e c h n i c a l r e q u i r e m e n t s ; i .e . n o
t e c h n i c a l r e s t r i c t i o n s w e r e p u t o n t h e m .
2.
LASER ISOTOPE SEPARATION PROCESS AS APPLIED TO URANIUM ENRICHMENT
2 1 Uranium energy levels
E n e r g y l ev els f o r t h e u r a n i u m a t o m e m p l o y e d i n th e s e l ec ti ve t w o - s t e p
p h o t o i o n i s a t i o n p r o c e s s a r e s h o w n i n F i g . 1 . T h e f a i r l y s t r o n g t r a n s i t i o n ( 7s ) 2 5 L 6
I
E
4 x 1 0 4
e--
2 x 1 0 4
X
I J . J
0
1 s t i o n i z a t i o n l e v e l
U V L i g h t
_ ~ ~ 3 1 0 0
E x c i t e d l e v e l
Is o to p e h i ft ~ ~ k
\ 0 . 0 8 A ) J
L a s e r l i g h t / / ~ 7 p 7 M 7
s 9 1 5 4 U G r o ,u n d l e v e l
i t s ) 2 5 L 6
6
In
4 a ~
-
_ t
O
o i
2 . ~ -
X
0
Fig. 1. U ra ni um energy levels for the selective two-step photoionisation proces s. The
235U
atom is
selectively excite d to the 7s7p7Mr level with dye las er light of wavelength 5915-4A. The excited atom is
successively ionised by the ultraviolet light o f wavelength between 2000 and 3100A.
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OPERATING CONDITIONS FOR URANIUM ENRICHMENT
289
7 s 7 p 7 M 7 , 2 = 5 9 1 5 . 4 A , i s c h o s e n f o r i s o t o p e s p e c if i c e x c i t a t i o n , b e c a u s e t h i s
le ve l h a s a r e m a r k a b l e i s o t o p e s h i ft , a p p r o x i m a t e l y 0 -0 8 A , a n d a f a i r l y s t r o n g
a b s o r p t i o n c r o s s - s e c t i o n . A l s o t h e e x c i t a t i o n e n e r g y o f t h i s l e v e l a g r e e s w i t h t h e
w a v e l en g t h o f th e m a x i m u m o u t p u t p o w e r o f t h e d y e l as er . T h e i o n i s a t io n l im i t f r o m
t h is e x c i t a ti o n l e v e l i s 3 1 0 0 A f o r a si ng le q u a n t u m p h o t o i o n i s a t i o n . T h e l o w e r l im i t o f
w a v e l e n g t h t o p r e v e n t d i r e ct i o n i s a t i o n f r o m t h e g r o u n d s t a te is 2 0 0 0 A . T h e
u l t r a v i o l e t li g h t in th i s w a v e l e n g t h r a n g e i s o b t a i n a b l e b y f r e q u e n c y d o u b l i n g o f
v i s ib l e l a se r l i gh t such a s the Ar ion l a se r l i ne s .
T h e 7 s 7 p 7 M 7 e x c i t e d le v el o f 2 3 5U a t o m i s s p l i t i n t o e i g h t h y p e r f i n e s t r u c t u r e s d u e
t o t h e i n t e r a c t i o n b e t w e e n s p i n s o f t h e e l e c t r o n s a n d n u c l e u s . I n t h e s e l e ct iv e t w o - s t e p
p h o t o i o n i s a t i o n p r o ce s s , o n l y o n e c o m p o n e n t o f t h e h y p e r fi n e s tr u c t u re is us e d .
2 .2 .
h o t o i o n p r o d u c t i o n r a t e
I n t h i s s e c t io n , a s i m p le k i n e t ic m o d e l t o c a l c u l a te t h e p h o t o i o n p r o d u c t i o n r a t e i s
g iv e n . T h e t h e r m a l i s a t i o n p r o c e s s e s to b e c o n s i d e r e d a r e i l l u s tr a t e d i n F i g. 2 . M a j o r
t r a n s i t i o n p r o c e s s e s a r e r e s o n a n t e x c it a t i o n , p h o t o i o n i s a t i o n , i n d u c e d e m i s s i o n a n d
t h e r m a l r e l a x a t io n . T h e r e s o n a n t e x c i t a t io n a n d p h o t o i o n i s a t i o n t a k e p l ac e w h e n
t h e a t o m a b s o r b s a p h o t o n w i t h e n e r g y h v= a n d h v r e sp e c ti v el y .
I f w e n e g l ec t c h a r g e t r a n s f e r a n d e n e r g y t r a n s f e r b e t w e e n 235U i n t h e g r o u n d s t a te ,
s t i o n i z a t io n C h a r g e t r a n s f e r
l a v e l / / ~ / / ~ , ~ ~ / ~ ~ / / ~ ~
Vi
E x c i te d l e v e l
W i
E n e r g y t r a n s f e r
l 1 V e
p o n t a n e o u s d e c a y
I n d u c e d e m i s s i o n
W e
G r o u n d l e v e l
35U 38u
Fig. 2. A three-levelsystem llustrating a selective wo-step photoionisationprocess. M ajor transition
processes are resonan t excitation, photoionisation, induced emission and thermal relaxation.
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O P E R A T I N G C O N D I T IO N S F O R U R A N I U M E N R I C H M E N T 291
3.
O P T I M I S A T I O N P R O C E D U R E
3 1 Op t imisat ion calculat ion
T h e c o m p l e x m e t h o d 2 w a s u s e d a s t he o p t i m i s a t i o n p r o c e d u r e . I n o r d e r t o
o p t i m i s e w i t h t h i s m e t h o d , t w o p r e r e q u i s i t e s a r e i n g e n e r a l n e c e s s a r y . F i r s t l y , o n e
h a s t o d e f in e t he i n d e p e n d e n t v a r i a b l e s o f t h e o p t i m i s a t i o n p r o b l e m . T h e n e c e s s a r y
c o n d i t i o n s f o r i n d e p e n d e n t v a r i a b l e s a r e : ( 1) t he e f f ec t u p o n t h e v a l u e f u n c t i o n is
g r e a t a n d ( 2) th e y h a v e f in i te o p t i m u m v a l u e s. S e c o n d l y , a s u i t a b l e v a l u e f u n c t i o n h a s
t o b e d ef in e d . O p t i m i s a t i o n o f t h e o p e r a t i n g c o n d i t i o n s o f th e l a s er u r a n i u m
e n r i c h m e n t p l a n t m e a n s t h e d e t e r m i n a t i o n o f t h e o p t i m u m c o m p o s i t i o n s a n d t he
s p e c i fi c a ti o n s w h i c h m i n i m i s e t h e u n i t c o s t o f e n r i c h e d u r a n i u m . I t is th e r e f o r e
d e s i r a b l e t o c h o o s e t h e u n i t c o s t o f e n r ic h e d u r a n i u m i n s t ea d o f t h e u n i t c o s t o f
s e p a r a t i v e w o r k a s a v a lu e f u n c t i o n . T h e v a l u e f u n c t i o n is , o f c o u rs e , e x p r e s s e d a s a
f u n c t i o n o f th e i n d e p e n d e n t v a r i a b le s .
T h e s t a n d a r d d e v i a t i o n o f i n d e p e n d e n t v a r i a b l e s is a p p l i e d t o a c o n v e r g e n c e
j u d g e m e n t in th e o p t i m i s a t i o n c a l c u l a t io n . T h e c o n v e n t i o n a l j u d g e m e n t f o r
c o n v e r g e n c e i s t o c o m p a r e t h e s t a n d a r d d e v i a t i o n o f th e v a l u e f u n c t i o n w i th a s m a l l
p o s i ti v e n u m b e r . H o w e v e r , it is d e s ir a b l e t o u s e t h e s t a n d a r d d e v i a t io n o f t h e
i n d e p e n d e n t v a r i a b l e s a s a j u d g e m e n t c o n d i t i o n f o r c o n v e r g e n c e in c a se o n e i s
i n t e r e st e d in t h e c o n v e r g e n c e v a lu e s o f th e i n d e p e n d e n t v a r ia b l e s.
l
S ~ l o n c o lle c to r
R e a c t i o ne g i o n
S l i t
ruc ib le
Fig. 3. The schem aticview of the isotope separation region. The uranium atom evaporating from the
high temp erature crucible is irradiated by laser light and ultraviolet light in the reaction region , and
ionised 235U atoms are collected by the ion collector electrodes. Distance a = 50mm .
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2 9 2 KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIICHI UEYANAGI
3.2.
ndependent variables
Figure 3 shows the schematic view of the isotope separation region. The
arrangement of the laser, ultraviolet light and atomic beam and the related
independent variables are shown in Fig. 4. The isotope separation region is
composed of a crucible for making the uranium vapour, a slit for the atomic beam
collimation and ion collector. The uranium metal in the crucible is heated using the
electron gun. The uranium atoms evaporating from it enter the reaction region with
a cross-section ofSuv cm in width and 1 cm in depth, and are irradiated by laser and
ultraviolet beams. The laser light, being perpendicular to the atomic beam, is Suv cm
in width and Sa cm in height. The ultraviolet light with cross-section of 1 cm in
depth and S a cm in height meets orthogonally both the atomic beam and the laser
light. The
235U
ions are selectively produced in the reaction region and are collected
by the ion collector located downstream of the reaction region.
Five independent variables are selected according to the reasons described in
section 3.1 ; these are laser energy density, ultraviolet light energy density, atomic
density, depth 1 and height S, of the reaction region. The important variables which
might have influence on the unit cost of enriched uranium besides the independent
variables are distance between the crucible and the reaction region, gap o f the ion
collector electrodes and width Su~ of the reaction region. It is, of course, desirable for
React ion region
W i d t h
t y = P i
Laser l i gh t
E n , r . d . , . , = p , )
tomic beam
( n e n s i t y = N o )
F i g . 4 . T h e i n d e p e n d e n t v a r i a b le s f o r o p t i m i s a t i o n o f l as er u r a n i u m e n r i c h m e n t p l a n t . T h e u r a n i u m
atom is i r rad ia ted by laser l i gh t and u l t rav io le t ] i gh t w i th energy dens i t i es Pc and P~ , respec t i ve ly .
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OPERATING CONDITIONS FOR URANIUM ENRICHMENT
293
a t o m i c d e n s i t y t o r e d u c e t h e d i s t a n c e b e t w e e n t h e c r u c i b le a n d t h e r e a c t i o n r e g i o n .
H o w e v e r , i t i s n e c e s s a r y fo r t h e t h e r m a l s h i el d a n d c o l l i m a t i n g sl it to s e p a r a t e t h e m
b y a t l e a st 5 c m . T h e d e n s i t y o f 2 3 5U i o n s d e c r e a s e s b y c h a r g e t r a n s f e r a s t h e y g o
a c r o s s th e a t o m i c b e a m t o t h e i o n c o l le c t o r . T h e r e f o r e , th e s m a l l e r g a p o f t h e i o n
co l l ec to r e l ec t r o d es i s d e s i r a b l e . T h e w id th S uv o f t h e r e a c t i o n r eg io n i s 1 cm . Th e
u l t ra v i o l e t l ig h t e n e r g y d e n s i t y d if fe r s o n b o t h s id e s o f t h e r e a c t i o n r e g i o n b e c a u s e o f
t h e a b s o r p t io n o f th e e xc it e d u r a n i u m a t o m s . H o w e v e r, th e a m o u n t o f a t t e n u a t i o n
lo s s w i ll s c a r ce ly b e n o t i c ed s i n ce t h e p h o to io n i s a t i o n c r o s s - s ec t i o n o f u l t r a v io l e t
l ig h t is e x t r e m e l y s m a l l. A s l o n g a s t h e a m o u n t o f a t t e n u a t i o n l os s c a n b e n e g l e c te d ,
u l t r a v io l e t l i gh t m a y b e u s ed . A s a r e s u l t , i t is p o s s ib l e t o u s e u l t r a v io l e t li g h t a t l e a s t
500 t imes.
3.3.
Value un ction
(1) Prod uction rate o f enricheduranium.
T h e c o n c e n t r a t i o n o f 2 35 U i o n s d e c r e a se s
b y c h a r g e t r a n s f e r c o l l is i on s b e t w e e n z 3 5U i o n s a n d 2 3 8 U a t o m s w h e n i o n s a r e
s e p a r a t e d f r o m t h e n e u t r a l b e a m b y t h e s t a t ic e l e ct ri c fi el d. T h e n u m b e r o f z 35 U i o n s
co l l e c t ed b y t h e i o n co l l e c to r p e r u n i t t im e i s g iv en b y t h e f o l l o w in g eq u a t io n ( s ee
A p p e n d i x ) :
n
1 1 0 )
n
~ S, . S..V(N~5 + N~) - A , (11)
A s = n
1
w h e r e n i s t h e n u m b e r o f d i vi s io n s o f d e p t h S Z n t h e r e a c t i o n r e g i o n , N d a n d d a r e t h e
n u m b e r a n d t h e g a p o f t h e i o n co l l e c to r e l ec t r o d es , r e sp ec t iv e ly , v i s t h e a v e r a g e
t h e r m a l v e l o c i ty o f t h e a t o m i c b e a m , a n d a r~ is t h e c h a r g e t r a n s f e r c ro s s -s e c t io n .
T h o u g h h i g h e n r i c h m e n t is o b t a i n e d a t t h e i o n c o l l e c to r , t h e e n r i c h m e n t r e q u i r e d
f o r t h e f u e l u s ed i n a n u c l ea r p o w e r p l a n t i s a b o u t 3 % . Th e r e f o r e , i t i s n ece s s a r y t h a t
h i g h l y e n r i c h e d u r a n i u m i s b l e n d e d w i t h n a t u r a l u r a n i u m t o f o r m a n u c l e a r f u e l. T h e
q u a n t i t i e s o f 3 % p r o d u c t a n d n a t u r a l u r a n i u m f o r b l e n d i n g c a n b e w r i t t e n a s :
M s M s A s ( I - ~ e) - ~ e ( 3 M s A s +
M~A8)
w . = 1 2 )
N a[M sM s(~ e - ce .) - 9a . ae]
M s A 5 + M s A s
W e = W n + (13)
N ,
w h e r e W . a n d /,Iz a r e t h e q u a n t i t i e s o f n a t u r a l u r a n i u m a n d 3 % p r o d u c t
r e sp e c ti v el y , M s a n d M s t h e m a s s n u m b e r s o f 2 3 sU a n d z 3 s u r e sp e c ti v el y , a e t h e
e n r i c h m e n t o f t h e u r a n i u m o b t a i n e d , ~ , th e n a t u r a l a b u n d a n c e r a ti o o f u r a n i u m a n d
N a A v o g a d r o ' s n u m b e r .
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2 9 4 K I M I O Y A M A D A , N O R I H I K O O Z A K I , M A N A B U Y A M A M O T O , K I l C H I U EY A N A GI
(2 ) O p e r a t i n g c o s t . T h i s i n c l u d e s c o s t s f o r m a i n t e n a n c e , e m p l o y m e n t , m a t e r i a l s
a n d e l e ct r ic p o w e r . T h e e q u i p m e n t t o b e m a i n t a i n e d i n c lu d e s t h e d y e l a se r ,
u l t r a v i o le t la s e r a n d e v a c u a t i o n s y s t e m s . T h e m a i n t e n a n c e c o s t c a n b e w r i t te n a s
m r = s t C l / t l + ~;,vCuv/tuv + ~vCv/tv (14 )
w h e r e C , 5, t a r e r e s p e c t i v e l y u n i t c o s t , t h e r a t i o o f m a i n t e n a n c e c o s t t o i n i ti a l c o s t
a n d t h e l i f e t im e o f t h e e q u i p m e n t : t h e s u b s c r i p t s 1, u v , v r e p r e s e n t t h e d y e l a s e r,
u l t r a v i o l e t l a s e r a n d e v a c u a t i o n s y s t e m s , r e s p e c t i v e l y .
I t is a s su m e d t h a t t h e e m p l o y m e n t c o s t is p r o p o r t i o n a l t o t h e n u m b e r o f l as er s. A s
a l a s e r i r r a d i a t e s t h e a r e a o f S a x S a a t t h e r e a c t i o n r e g i o n , t h e n u m b e r o f l a s er s is
g iven a s
N , = [ S , / ( . a p ) ]
+ [Suv/Sa] (15)
I[ 1]: ga uss sy m bo l
w h e r e AD is t h e n u m b e r o f r e - u t il i s a ti o n s o f th e u l t r a v i o l e t l a se r , W i t h t h e n u m b e r o f
l a se r s , th e e m p l o y m e n t c o s t c a n b e w r i t t e n a s :
G e m : A e m N l + B e rn (16)
w h e r e
Aem
a n d
Bern are
c o n s t a n t .
A m a t e r i a l c o s t is g i ve n b y n a t u r a l u r a n i u m q u a n t i t i e s p a s s in g t h r o u g h t h e
r e a c t i o n r e g i o n a n d t h a t f o r b l e n d i n g a s :
C m a = U m ( W n -'~ M s N o V S r / N ~ ) (17)
w h e r e IV , is t h e n a t u r a l u r a n i u m q u a n t i t y f o r b l e n d i n g c a l c u l a te d f r o m e q n . ( 1 2) , U m
t h e u n i t c o s t o f n a t u r a l u r a n i u m a n d S r t h e c r o s s - s e ct i o n o f t h e a t o m i c b e a m .
T h e e l e c t r i c p o w e r c o s t i s
E p : U p P e S e / ~ e + P i S i / t ] i A p ) + E g + O r ) (18)
w h e r e P a n d S a r e e n e r g y d e n s i t y a n d i r r a d i a t i o n a r e a r e s p e c ti v e ly , q t h e l a s e r
e f f ic i e n c y , s u b s c r i p t s e a n d i r e p r e s e n t t h e d y e l a s e r a n d u l t r a v i o l e t l a s e r r e s p e c ti v e l y ,
U p t h e u n i t c o s t o f e l ec t ri c it y , E g t h e p o w e r c o n s u m e d b y t h e e l e c t r o n g u n a n d O t t h e
e l e c tr i c p o w e r c o s t in c l u d i n g t h a t f o r e v a c u a t i o n a i r c o n d i t i o n e r a n d c o o l i n g w a t e r
s y s t e m s .
(3 ) P l a n t c o n s t r u c t i o n c o s t a n d c a p i t a l c o s t. T h e c o n s t r u c t i o n c o s t h a s t o b e d e f in e d
a s a f u n c t i o n o f t h e i n d e p e n d e n t v a r ia b l e s . F o r t h i s p u r p o s e , w e e x p r e s s t h e
c o n s t r u c t i o n c o s t b y m e a n s o f a n e x p o n e n t i a l f u n c t i o n
Cj = A j (DPj)kJ (19)
I n t h i s e x p r e s s i o n , t h e c o s t e x p o n e n t k j is a d i m e n s i o n l e s s q u a n t i t y b e t w e e n z e r o a n d
u n i t y w h i c h d e t e r m i n e s t h e v a r i a t i o n s i n t h e c o s t o f c o n s t r u c t i o n w i th t h e s i ze o f t h e
c o m p o n e n t . B o t h A j a n d k j m u s t b e d e t e r m i n e d e m p i r ic a l ly b y m e a n s o f c o s t
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OPERATING CONDITIONS FOR URANIUM ENRICHMENT 2 9 5
e v a l u a t i o n s t u d ie s . T h e d e s i g n p a r a m e t e r s , DPj a r e q u a n t i t ie s c h a r a c t e r i s i n g t h e c o s t
o f t h e i n d iv i d u a l c o m p o n e n t s a n d m u s t t h em s e l ve s b e th e i n d e p e n d e n t v a ri a b le s o r
f u n c t i o n s o f t h e m .
F i g u r e 5 s h o w s t h e e m p i r i c a l c o s t f u n c t i o n s w h i c h a r e u s e d t o d e s c r i b e t h e
c o n s t r u c t i o n c o s t s o f th e v a r i o u s p l a n t c o m p o n e n t s . T h e a b s c is s a e in d i c a t e t h e
d e si g n p a r a m e t e r o f t h e c o n s t r u c t i o n c o s t f u n c t i o h ; t h e o r d i n a t e s s h o w t h e c o st o f
c o m p o n e n t s . T e n d i f f e re n t c o s t f u n c t i o n s a r e c o n t a i n e d i n F ig . 5 , e a c h o f t h e m
d e r i v e d b y t h e u s e o f g o o d s o n t h e m a r k e t . H o w e v e r , t h e e x p o n e n t k j o f th e e le c t ri c
p o w e r s o u r c e a n d c o o l i n g w a t e r s y s t em w a s q u o t e d f r o m a p a p e r b y M .
M ~t r t ensson . 3
A s s u m i n g t h e p l a n t l i f e ti m e a n d a n n u a l c a p i t a l c h a rg e , w e c a n d e d u c e t h e c a p i ta l
c o s t f r o m t h e c o n s t r u c t i o n c o s t.
( 4 ) The unit cost of enriched uranium. T h e u n i t c o s t o f e n r i c h e d u r a n i u m is
o b t a i n e d f r o m t h e c o s t d e s c r i b e d a b o v e a s f o l l o w s :
U = (Cop + Cp /We (20)
w h e r e C op i s t h e o p e r a t i n g c o s t a n d C v t h e c a p i t a l c o s t .
4 .
CASESTUDY
T a b l e 1 s h o w s t h e p a r a m e t e r s u s e d t o o p t i m i s e t h e o p e r a t i n g c o n d i t i o n s o f t h e la s e r
u r a n i u m e n r i c h m e n t p l a n t . T h e m o s t f a v o u r a b l e o n e s a t t h e p r e s e n t t i m e a r e
e m p l o y e d i n t h is c a l c u l a t io n , b u t s o m e o f t h e m p o s s e ss t h e p o s s i bi l it y o f c h a n g i n g t o
TABLE
PARAMETERS USED IN THE COMPUTATION
No tatio n xplanation Value
Ap
Up
Um
i
T
T I
T u v
? v
The number o f re-utilisations of the ultraviolet laser
Unit cost of electricity, S/kWh
Unit cost of natural uranium, /kg
Fixed charge rate,
Lifetime, years
Dye laser
Ultraviolet laser
Evacuation systems
500
0.017
33
10
0.5
0-5
2
m o r e f a v o u r a b l e v a l u e s in t h e f u t u r e . F o r e x a m p l e , i t is p o s s i b le t o e x t e n d t h e l if e ti m e
o f a l a s e r b y a f a c t o r o f 5 if t h e l a s e r t u b e i s i m p r o v e d . H o w e v e r , th e u n i t c o s t s o f
e l e c t ri c i ty a n d n a t u r a l u r a n i u m r is e g r a d u a l l y , a n d t h e r is e s i n th e s e p ri c es h a v e
m i n o r i n fl u en c e s u p o n t h e u n i t c o s t o f en r i c h e d u r a n i u m .
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2 9 6 K I M I O Y A M A D A , N O R I H I K O O Z A K 1 , M A N A B U Y A M A M O T O , KII CH 1 U E Y A N A G I
106
10s
Dye laser
k= 0.67
J
I
J
J
/
101 102 103
Dye lase r po w er ( w )
105
1 4
J
i
U ltrav io le t I ~ ' ~
k -0 .67
f
1 2 1 3 1 4
Ultrav io le t l as e r pow er ( w )
F i g . 5 . T h e e m p i r i c a l c o n s t r u c t i o n c o s t f u n c t i o n s . I n t h e fi g u r e s, t h e a b s c i s s a i n d i c a t e s t h e d e s i g n
p a r a m e t e r o f t h e c o n s t r u c ti o n c o s t f u n c t io n . T h e o r d i n a t e s h o w s t he c o s t o f c o m p o n e n t s i n d ol la r s.
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OPERATING CONDITIONS FOR URANIUM ENRICHMENT
297
10 5
10 4
E l e c t r i c p o w e r s o u r c e
k = 0 . 6
/
/
/
/
/
I
w
10 6 10 7 10 8
T o ta l e l e c tr ic p o w e r ( w )
B u i l d i n g
k = 0 . 8
10 3
10 2
/
/
J
10 0
/
J
f
101
F loo r space ( m 2 )
10 2
Fig.
5 contd.
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7
z
~
e
o
2
,
o
i
~
0
~
I
-
7
o
e
I
i
A
v
I
-
7
.
E
~
x
O
0
,
z 0
0 0
> z> >
> 0
o
m
>
z> 0
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OPERATING CONDITIONS FOR URANIUM ENRICHMENT
2 9 9
1
10 3
Vacuum vessel
k = 0 . 4 J
~P
f
10 0 101 10 2
Vo lume in vacuum v ess e l (m 3 )
E vacuatio n s y s t ~
10 4
J
10 3
10 0 101 10 2
Volume in vacu um vess e l ( m 3 )
Fig.
5 contd.
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300
K I M I O Y A M A D A , N O R I H I K O O Z A K I , M A N A B U Y A M A M O T O , K II C HI U E Y A N A G I
1 4
103
; C o o l i n g w a t e j
system
k=0.78 J
J
i f
1 5 1 6 1 7
Power of cooling wa ter pum p(w )
105
1 4
f
Electron g u n j
k=0.68
. /
/
f
105 10 6
Electric gun power ( w
J
10 7
Fig .
5 contd.
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O P E R A T IN G C O N D I T I O N S F O R U R A N I U M E N R I C H M E N T
301
Table 2 shows the physical quantities used in the optimisation programme. The
excitation cross-section of uranium is from the paper by Tvccio
e t a l 1
The
photoionisat ion cross-section is the one approximated to theoretically. The charge
transfer cross-section is estimated from the experimental values of the other
elements. 4 The energy transfer cross-section is the calculated value f rom those of
inert gases. These values are possibly overestimated, and this has a minor influence
on the unit cost of enriched uranium.
T A B L E
P H Y S I C A L Q U A N T I T I E S
npu t da ta N um e r i c a l v a lue
C r o s s - s e c t i o n , c m z
Ex c i t a t i on 1 .3 x 10 ~4
l o n i s a t i o n 1 . 5
x 1 0 - 1 7
C h a r g e t r a n s f e r 1 .3 x 1 0 13
E n e r g y t r a n s f e r 1 . 0 x 1 0 13
W a v e l e n g t h , A
D y e l a s e r 5 9 1 5 . 4
U l t r a v i o l e t l a s e r 2 6 0 0
E
. ,m
m
e-
,,-=
e-
e-
F.
2 5 0 0
2 3 0 0
2 1 0 0 ,
T
1 0 - 2 1 0 -1 1 0 0
E f f ic i e n c y o f d y e l a s e r ~ x
F i g . 6 . T h e u n i t c o s t o f e n r i c h e d u r a n i u m p l o t t e d a s a f u n c t i o n o f d y e l a s e r e t ~c i en c y . T h e u n i t co s t o f
e n r i c h e d u r a n i u m i s o p t i m i s e d v i a t h e f iv e i n d e p e n d e n t v a r i a b l e s . T h e u l t ra v i o l e t l a se r et ~ c ie n c y is t a k e n a s
5 x 1 0 - 2 % .
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3 0 2 KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIICHI UEYANAGI
F i g u r e 6 s h o w s th e u n i t c o s t o f e n r ic h e d u r a n i u m a s a f u n c t i o n o f t h e d y e la s e r
e f fi c ie n c y , a n d F i g . 7 s h o w s t h e u n i t c o s t o f e n r i c h e d u r a n i u m v e r s u s u l t r a v i o l e t l a s e r
e f f i c i e n c y . O n t h i s c a l c u l a t i o n , a n y t e c h n i c a l r e s t r i c t i o n s a r e n o t i m p o s e d o n t h e
i n d e p e n d e n t v a r i a bl e s . T h e u n i t c o s t o f e n r i c h e d u r a n i u m d e c r e a s e s e x p o n e n t i a l l y as
l a s e r e f fi c ie n c y i n c r e a s e s t o 1 ~o w h e r e i t t a k e s t h e a s y m p t o t i c v a l u e . I t m i g h t b e s a i d
t h a t t h e e f f o r t s t o i m p r o v e t h e l a s e r e t fi c ie n c y a b o v e 1 ~o a r e r a t h e r f r u i tl e s s f o r
i s o t o p e s e p a r a t i o n .
H o w e v e r , t e c h n i c a l r e s t r i c t i o n s e x i s t w i t h r e s p e c t t o l a s e r e n e r g y d e n s i t y a n d
e f fi c ie n c y . F o r e x a m p l e , t h e u l t i m a t e v a l u e o f th e d y e l a s e r e ff ic i en c y i s a b o u t
2 x 10-3~o , a n d t h e u l t r a v i o l e t l a s e r e f fi c ie n c y i s a b o u t 3 x 1 0 - 2 ~ o f o r t h e
p r e s e n t t e c h n i c a l l e v e l s . I t i s h i g h l y f a v o u r a b l e f o r e c o n o m i c l a s e r u r a n i u m
e n r i c h m e n t t o i m p r o v e t h e l a s e r e f fi c ie n c y u p t o o n e p e r c e n t .
= 6 0 0 0
4 0 0 0
C
2 0 0 0
0
E
1 0 2 1 0 1 1 0 0
E f f i c i e n c y o f u l t r a v i o l e t l i g h t s o u r c e I ~
Fig. 7. The un it cost of enriched uranium plotted as a func tion of ultraviolet laser efficiency.A un it cost
of enriched uranium is optimised by the five ndepen dent variables. The dye las er efficiency s taken a s 2-5
10 2~, .
O p t i m i s a t i o n o f t h e o p e r a t i n g c o n d i t i o n s is p e r f o r m e d f o r tw o c a s es : c a s e 1, t h e
l a s e r e n e r g y d e n s i t y a n d e f fi c ie n c y a r e s u b j e c t t o r e s t r a i n t s d e p e n d i n g o n t h e p r e s e n t
t e c h n i c a l l e v e ls ; w h e r e i n c a s e 2 t h e r e i s n o r e s t r i c t i o n . I n c a s e 2 , t h e e f fi c ie n c i e s o f t h e
d y e l a s e r a n d t h e u l t r a v i o l e t l a s e r a r e c h o s e n a s 1 ~,,, a c c o r d i n g t o t h e r e s u l ts
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OPERATING CON DITIONS FOR URAN IUM ENRICHMENT
303
TABLE 3
TECHNICALRESTRICTIONS
A s s u m p t i o n C a s e 1 C a s e 2
Power density, W /cm 2
Dye laser 200 free
Ultrav iolet laser 4000 free
Efficiency, %
Dye laser 2 10 -3 1
Ultraviolet laser 3 x 10 -2 1
TABLE 4
OPTIMUM OPERATINGCONDITIONSAND CONSEQUENTCOSTS
R e s u l t s C a s e 1 C a s e 2
Optimum values
Dye laser power density, W /cm 2 200 314
Ultraviolet laser power density, kW /cm 2 4.00 16.7
A tom ic density, cm -3 7-83 x 1013 8.23 x 1013
De pth Sj, cm 511 510
Height S a cm 1.02 0-998
3 % uraniu m prod uction , kg/year 998 2470
To tal electricity, M W 26.0 3.06
Co nst ruct ion cost, $/kg 197 132
Electric power cost, $/kg 3830 182
Ma intenance cost, $/kg 462 359
M ate rial cost, $/kg 888 395
U nit cost of enriched uranium, $/kg 5380 1070
m e n t i o n e d a b o v e . T h e t e c h n i c a l r e s t r ic t i o n s i m p o s e d o n t h e la s e r e n e r g y d e n s i t y a n d
e ff ic ie n cy a re s u m m a r i s e d i n T a b l e 3 .
T a b l e 4 s h o w s t h e o p t i m i s e d o p e r a t i n g c o n d i t i o n s o f t h e la s e r u r a n i u m
e n r i c h m e n t p l a n t . T h e a t o m i c d e n s i t y a n d d e p t h S l a n d h e i g h t Sa o f t h e r e a c t i o n
r e g i o n a r e h a r d l y a f f e c te d b y c h a n g i n g t h e l a s e r e n e r g y d e n s i t y a n d e ff ic ie n c y, a n d t h e
e l e c t r ic p o w e r c o s t d e c r e a s e s a s a r e s u l t o f t h e i m p r o v e m e n t o f l a s e r e f fi c ie n c y . T h e
o t h e r r e m a r k a b l e r e s u l t is t h a t t h e e n e r g y d e n s i t y r e q u i r e d b y t h e d y e la s e r a n d
u l t r a v i o l e t la s e r w i t h a n d w i t h o u t t h e r e s t r i c t i o n s a r e n o t a p p r e c i a b l y d if f er e n t. T h e
l a s e r e n e r g y d e n s i t y s h o w n i n T a b l e 4 i s p r o b a b l y r e a l is e d ev e n a t t h e p r e s e n t
t e c h n i c a l l e v e l s, w h e r e a s a t t a i n m e n t o f a l a s e r e f fi c ie n c y o f 1 ~o i s v e r y d i f fi c ul t. A h i g h
m a i n t e n a n c e c o s t e n s u e s i n c a s e 2 b e c a u s e t h e l i f e t i m e o f t h e l a s e r i s r e l a t i v e l y s h o r t .
H e n c e , i t is i m p o r t a n t t o e x t e n d t h e l i f e t i m e o f t h e l a s e r f o r t h e r e d u c t i o n o f th e u n i t
c o s t o f e n r i c h e d u r a n i u m .
T h e o p t i m i s e d u n i t c o s ts o f e n r i c h e d u r a n i u m a r e 5 3 8 0 5 / k g i n c a s e 1 a n d
1 07 0 $ / k g i n c a s e 2. T h e u n i t c o s t o f s e p a r a t i v e w o r k d o n e b y t h e g a s e o u s d i f f u s i o n
a n d g a s c e n t r i f u g a l m e t h o d s i s e s t i m a t e d a t a b o u t 1 00 $ / k g S W U w i t h a p o s s i b l e c o s t
e s c a l a t i o n . S u p p o s i n g t h a t t h e a s s a y i s 0 .3 ~o, w e n e e d s e p a r a t i v e w o r k 3 .5 k g S W U
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304
K I M I O Y A M A D A , N O R I H I K O O Z A K I , M A N A B U Y A M A M O T O , K I 1C HI U E Y A N A G I
and 10 kg of feed material for the production of 1.0 kg of 3.0 ~o product. Then the
unit cost of enriched uranium including the material cost is approximately 500 /kg.
Compared with the costs of the other uranium enrichment methods, the obtained
opt imum values are greater by a factor of 11 in case 1 and by a factor of 2 in case 2.
However, the opt imum value of the unit cost of enriched uranium depends largely on
the unit cost of electricity Up, the lifetime z of the laser and the efficiencies of the
lasers. The electricity cost of 0.017 S/kWh is probably high and the improvement of
the lifetime and efficiency of the laser can reasonably be expected in the near future.
It can be concluded from the results of case 2 that laser uranium enrichment is able to
compete with the other methods.
5 . C O N C L U S I O N
Optimum operating conditions of the laser uranium enrichment plant, in which the
selective two-step photoionisation process of atomic uranium is employed, has been
studied. The unit cost of enriched uranium at present is fairly expensive in
comparison with those by the gaseous diffusion and gas centrifuge methods.
However, we can reasonably say that laser uranium enrichment is competitive with
the other uranium enrichment methods provided that the following requirements
are satisfied:
(1) the laser efficiency is improved to about one percent.
(2) the laser lifetime is extended to about one year.
A C K N O W L E D G E M E N TS
The authors would like to thank Drs K. Taniguchi, S. Yamada and A. Doi of the
Atomic Energy Research Lab orat ory for their constant encouragement throughout
the study.
R E F E R E N C E S
1 . S . A . T v c c I o et al . T w o - s t e p s el ec ti v e p h o t o i o n i z a t i o n o f 235U n u r a n i u m v a p o r , 8th In terna t iona l
ConJerence on Quantum Electronics S a n F r a n c i s c o , U S A ( 1 9 7 4 ) .
2 . M . J . B o x , A n e w m e t h o d o f c o n s t r a i n e d o p t i m i z a t io n a n d a c o m p a r i s o n w i th o t h e r m e t h o d s ,
Co m p u t e r J . , 8 , 42 ( 1965) .
3. M . MARTEr~SSON,S w e d i sh s t u d i e s o n t h e e c o n o m i c s o f u r a n i u m e n r i c h m e n t ,
J . o f the Br i t ish Nuc lear
E n er g y S o c i e t y
10, 191 (1971) ,
4 . B . M. SMIRNOV an d M . 1 . CHIBISOV,R e s o n a n c e c h a r g e t r a n s f e r i n i n e r t g a s e s , S o v i e t P h ys . - Tech .
Phys . 10 , No . 1 , 88 ( 1965) .
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O P E R A T I N G C O N D I T I O N S F O R U R A N I U M E N R I C H M E N T 305
A P P E N D I X
Th e n u m b e r o f 23 5 U io n s co l l e c t ed a t t h e i o n co l l e c to r s i s g iven b y co n s id e r in g t h e
ch a r g e t r a n s f e r co l l i si o n s w h ich t a k e p l a ce w h en th e i o n is ex t r a c t ed f r o m th e n eu t r a l
b ea m . A s t h e r e i s n o ex c i t ed a to m in th e co l l e c t io n r eg io n , th e eq u a t i o n s d e s c r ib in g
t h e p o p u l a t i o n s o f t h e i o n i s a t io n a n d g r o u n d s t a t e s o f u r a n i u m i s o to p e s a s a
f u n c t i o n o f th e d i s t a n c e b e t w e e n t h e i o n a n d i o n c o l le c t o r c a n b e w r i t t e n a s :
dA~ _
d x a r ~(A s A g 5 - A g s A i s ) ( A . I )
d A ~ g i
d x - - a r ~ ( A s A 5 - A i s A ~ )
(A .2 )
d A ~
d x = a r ( A ~ A ~ - A g sA is )
(A .3 )
dAg8
d x - a v ~ (A ~ A ~ - A i s A ~ )
(A .4 )
w h e r e A i s t h e p o p u l a t i o n ( s u b sc r ip t s 5 a n d 8 in d i c a t e t h a t t h e q u a n t i t y b e l o n g s to
2 3 5 U a n d 2 3 S u r e s p ec t iv e ly , s u p e r s c r ip t s g a n d i i n d i ca t e g r o u n d a n d i o n i s a t i o n
s t a t e s r e s p ec t iv e ly ) , a n d x i s t h e d i s t a n ce b e tw een th e i o n a n d i o n co l l e c to r .
F r o m eq n s . ( A . I ) t o ( A . 4 ) , w e h a v e
dA~ dA~ dA~ dA~ dA~ dA~ dA~ dA~
+ - - - - + . . . . 0 (A .5 )
d x d x d x d x d x d x d x d x
Th e s o lu t i o n s o f eq n . ( A . 5 ) is giv en a s :
~ 1~ = A ~ + f l A ~ = 6 - A ~ A . 6 )
w h e r e 7 , fl, 7 a n d 6 a re c o n s t a n t . T h e n , w e o b t a i n t h e f o l lo w i n g e q u a t i o n :
dA~
+ ar~(cc + fl + 27) A~ - crrY(~ + 7) = 0 (A .7)
d x
E l i m i n a t i n g t h e c o n s t a n t s a , f l a n d 7 f r o m e q n . ( A . 7) b y u s i n g t h e s o l u t i o n s (A . 6) a n d
th e i n i t i a l co n d i t i o n t h a t a t x = 0 , t h e p o p u l a t i o n s A ~ = N ~ , A ~ = N ~a, A ~ = N ~ a n d
A ~ = N ~ , w e o b t a in t h e s o lu t i o n o f th e l i n ea r d i f fe r en t i a l eq u a t io n a s f o l lo w s :
A ~ 5 (x ) = N~ol [ ( N ~ N ~ - N ~ N ~ ) e x p ( - a r ~ N o x ) + (U g 5 +
N s ) ( N~ i + N i s ) ] ( A . 8 )
w h e r e N ~, N ~ , N~ a n d N ~ a r e t h e i o n a n d a t o m d e n s i ti e s a f t e r p a s s in g t h r o u g h t h e
r ea c t i o n r eg io n .
A s s u m i n g t h a t t h e d e n s i t y d i s t r i b u t i o n o f t h e i n c i d e n t i o n b e a m t o t h e i o n
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KIM10 YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KI1CH1 UEYANAGI
co l lec to r e lec t rode s is un i fo rm an d the ion ve loc i ty occurr ing f r om the e lec tr i c f ie ld o f
t h e i o n co l le c t o r e l ec t ro d es is m u ch f a s t e r t h an t h e t h e rm a l v e l o c it y o f a t o m i c b eam ,
we h av e t h e t o t a l n u m b er o f 2 3 s u i o n s :
n
2 s f
, t 5 = - ~ V N d A ~ ( x ) d x
0
1
n
= Y / ~ SvNdFNiN~-~NgsNinO L N o a r ~ ( 1 - e x p ( - o r ~ N o d ) ) + N g s + N i s ) N i + N i s ) d ]
(A.9)
wh e re v is th e av e rag e t h e rm a l v e l o c i t y o f t h e a t o m i c b eam , Nd t h e n u m b er o f th e i o n
co l l ec to r e l ec tro d es , d t h e e l ec t ro d e g ap , t h e s u m m a t i o n is b a s ed o n co n s i d e ri n g t h e
d ec reas e o f la s e r en e rg y d en s i t y i n t h e r e ac t i o n r eg i o n an d n is t h e n u m b er o f
d iv i s ions o f the dep th S , o f the reac t ion reg ion .