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A generalized analysis of the cumulative diffusional release of fission product gases from an “Equivalent Sphere” of UO2
Joutnal of Nuclear Materials 88 (1980) 299-308 0 North-Holland Publishing Company
A GENERALIZED ANALYSIS OF THE CUMULATIVE DIFFUSIONAL RELEASE OF FISSION PRODUCT GASES FROM AN “EQUALED SPHERE” OF UO2 *
G.V. KIDSON * Department of physics, Brock University, St. Catherine&Ontario, Canada Received 6 July 1979; in revised form 22 October 1979
The “Equivalent Sphere Model” introduced by Booth in 195‘7 and used extensively in anaiytical studies related to the diffusional release of fission product gases from UOz reactor fuels, has been generalized to describe the behaviour of the m members of a radio-active chain duringk cycles of reactor operation. Expressions are obtained for the concentration of the species in the equivalent sphere and the cumulative fractional release from the sphere. These are shown to contain, as special cases, all results previously considered as well as some that have not been previously considered and provide a basis for a systematic comparison with experiment under more general conditions.
radioactive isotope during a single cycle of reactor operation. Since then the model has been extended in several ways; for example, Eichenberg et al. f3] obtained a time-dependent solution of the diffusion equation for the release of a single radioactive species during a single cycle of reactor operation, while Beck [4] reviewed the results of both Booth and Eichenberg et al. and presented alternative expressions for certain sums in the solutions that gave better convergence for short times. More recently Nobel [S] and Rim et al. [6] have discussed the equations for the cumulative release of a single stable isotope over several cycles of reactor operation while Friskney and Speight [7] considered the behaviour of the mth member of a radioactive chain during a single cycle of operation. ‘l%edevelopment of these purely phenomenological analyses has been complemented by parallel studies of the diffusion processes themselves; in particular the effects of radiation damage, trapping and resolution of the diffusing gas atoms have been considered. Capart [8] has shown that these latter effects may be incorporated into the phenomenological equations by introducing an “effective diffusion coefficient”, which is the product of the normal diffusion coefficient in the absence of trapping and resolution and the frac-
1. introduction During the burn-up of UO, reactor fuels, a fraction of the gas atoms produced directly from fission and/or by the decay of radio-active precursors, diffuse to the boundaries of the grains of the fuel pellets, where they are released to the ~ter~~~ar pores and eventua~y migrate to the region between the pellets and the encapsulating sheath wall. Considerable effort, both experimental and theoretical, has been directed toward an understan~ng of the behaviour of these gases. An idealized model of a UOZ pellet, originally introduced by Booth [ 1,2] and used extensively in analytical studies is based on the assump tion that the controlling process in the release of the gases is the diffusion from the interior of the grains to the grain surfaces. This “Equivalent Sphere Model” represents the average behaviour of the grains by a set of equivalent spheres whose total area is comparable to the total grain boundary surface area. Booth’s original studies concerned the release of a stable isotope and the steady-state beha~our of a
* Current address: Materials Science Branch, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE lL0, Canada. 299
300
G. V. Kidson / Diffusional release of fission product gases from UO2
tion of time the diffusing atom is in the untrapped, mobile state. While the several approaches used by the above are generally internally consistent, the development of the solutions, particularly for the fraction of gas released, has sometimes been sufficiently different from one another as to obscure their essential equivalence. Moreover, certain effects that may be of importance have sometimes been neglected. The primary purpose of the present paper is to extend the generalization of the Booth model to its logical limits by considering the cumulative release of the mth member (which may be radioactive or stable) of a radioactive chain over, say, k cycles of reactor operation. This seems not only desirable but also necessary if meaningful comparisons are to be made of the analytical predictions and measured release behaviour of the gases. In so doing, we obtain a result that can readily and systematically be reduced to any of the previous results obtained for less general treatments simply by putting appropriate parameters equal to one or zero. In section 2, below, we specify the model, and obtain the solutions to the diffusion equations. In section 3, the general expression for the cumulative fractional release F:(t) of the mth member of the chain up to a time t during the kth cycle of operation is developed; this is applied in section 4 to a variety of special cases and where appropriate, compared with results obtained in previous studies.
parameters governing the rates of trapping and resolution during the kth cycle. The annihilation of species 1 is assumed to occur both as a result of radioactive decay to species 2 at a rate X1,and by transmutation by neutron absorption at a rate 8, the latter being a function of the neutron capture cross section and neutron flux during the kth cycle. All other members of the chain are governed by corresponding parameters except that they will be produced as a result of the radioactive decay of their precursors as well as directly by fission. We obtain, then, a set of m coupled equations for the concentration distributions in the sphere:
acflat = pf + 0: v2 C: - g:cf ...
ac&lat = im_,c&_,
,
+p; +o&v2c; - &c&
Here C is the concentration in atoms per unit volume, V2 is the Laplacian operator in spherical coordinates, and l= (h t Y).We seek solutions to these equations in the form CL (r, t) = 5
n=1
ti(n, r) I!&(n. t)
,
We consider the production, diffusion, annihilation and release of the m members of a radioactive chain in and from an equivalent sphere of radius a, up to a time t, measured from the beginning of the kth cycle of reactor operation. The initial member of the chain is produced solely as a result of fission at a rate 0: which is determined by the total fission cross section, the probability of production of species 1, the concentration of fissionable uranium and the neutron flux associated with the Mh cycle of operation. The rate of diffusion of the species is characterized by the effective diffusion coefficient 0: determined by the temperature and the
(2)
subject to the boundary condition Ck(r=a,
t)=O,
(3)
and initial condition: C&(r, t = 0) = nql $4n, r) by&,
2. Specification of the model and solutions of the diffusion equations
.(I)
t = 0).
(4)
It is shown in Appendix A that the Jl(n, r), n = 1, 2 , .... 00,form the orthonormal set of spherical Bessel functions 9(n, r) =
ww41
Ww~Y~
9
(5)
while the time-dependent functions bk(n, t) are governed by the m coupled equations dbfldt
+p:b:
- K: = 0 ,
dbzldt +p;b’: - K’; - Al bf = 0, ... dbfS,ldt+pk,b~-K;-A,_,bk,_l=O.
(6)
Here, pk z & + n2n2(Dk,/a2) ,
(7)
301
G. V. Kidson / Diffisional release of fission product gases from UO2
= 1 - Q/$7ra”Cik(t),
and K;
= (-l”+‘/r~n~)(2~a)~‘~ /3& .
(8)
Solutions to the set of equations (6) are obtained in Appendix B by use of the Laplace Transform method and may be put in the form:
where
(i
q=l
(13)
Q = bdt/(2na)l n$lbk(t) jr sin(nnr/a> dr , 0
= [4n/d(2na)]
x
02)
n$l (-ln+1a2/nn)
b:(t)
(14)
.
The CGk are, of course, also governed by a set of m coupled equations:
bp”o ii &&‘$> r=q
dCTkldt t .$:ck
- 0: = 0,
dClk/dt t ,$C;’ - 0’: - hr CTk = 0 , ... dCGk/dt t &C&k -0; -A,_rc$!_r =o.
(15)
These are identical in form to those found for the b;(t), given in eqs. (6). Thus by substituting C’Gk for bk, C;“,for pk and & for Ki in eq. (9), we can
where Ap+p;-pjk.
(10)
We note that the dependence of b;(t) on all of its precursors during the preceding (k - 1) cycles is contained in the b& terms. See section 4.3 below for a full discussion.
immediately write:
Cik(t).= (X,)-l [%
Aj exp(-t/Q)
j=l
3. Fractional release To ensure consistency with the several definitions proposed in the past [l-8], we adopt the following for the cumulative fractional release F&(t) of the mth member of the radioactive chain: “F&(t) is the ratio of the total number of atoms of the mth species released from the volume of the equivalent sphere during all k cycles of operation, and still surviving at time t in the kth cycle to the total number of atoms of that species produced in the sphere during all k cycles and still surviving at the same time t. ” To form F:(t), let Cik(t) be the total number of atoms per unit volume of the mth species at time t in the kth cycle. We then define F:(t) as:
4 7ra3Cik(t) - jt47rr2 CL (r, t) dr F:(t)
=
0 4 7ra3CGk(t)
(11)
The final form for FL(t) can now be written as [6/(2na)3/2] 2 F;(t)
= 1-
n=1
(-1 ‘+r/n) bk, (t)
GkU)
9 (17)
where b% and C’G”are given by eqs. (9) and (16) respectively. In section 4 this general expression is applied to a variety of special cases many of which have been previously discussed in the literature.
302
G. V. Kidson / Diffusional release of fission product gases from UO2
comparison with their result, we incorporate these conditions into eq. (20) by setting vj = 0, so that [j = Ai, & = 0 for Q # 1 and letting
4. Application to special cases
4.1. Release of the mth member of a radioactive chain during a single cycle of operation
X=hi+n21r2(Dja2).
We set the number of cycles k = 1, drop the superscript and put bqo, C$ = 0. Then the eq. (9) reduces to
Then with minor rearrangement, eq. (20) reduces to:
rk=l
1
-
F,(t)
b
m
m
(18)
m
m
X l-l @r/X,) ,Fr 11 - exP(-Xjr)l r=1
!Jr XtJu*j +i
where we have used equation (8) for K, . Similarly, eq. (16) becomes m
CA(t)= (~m)-l~l(X//tj)[l
-
expt-[jr)1
With the use of the result obtain@ in Appendix C (Srivastava [9]), (19)
E(l?i
j=l
so that the fraction of the mth species released can be written as:
i-=1 +j
E j=l
( fi r=l +j
X,/AX,)=
13
(22) &/Ax,) = 1 ,
eq. (22) becomes identical to the expression reported by Friskney and Speight [7].
X [l - exP(-pi01 $rP,
4.2. Release of a single radioactive species during a
1 b/&rj
‘=Q +j
single cycle of operation This case readily follows from eq. (20) by setting
X 5 (Aj/tj) i l - exP(-tjt)l Cj=1
m =
1, whence
F(t)
(20) 2
In the model discussed by Friskney and Speight [7], the loss of atoms through transmutation by neutron absorption was ignored, and only the first member of the radioactive chain was assumed to be produced by fission while all other members were assumed to be produced solely by the decay of their precursors. For
=
1
(6p/n2
R2)(l /n’ n2 t p)( 1 - exp(-pt))
_n=1 1 - exp(-[t) (23)
where p = [a’/0
.
(24)
303
C. V. K&on / Diffusional release of fission product gases from UO2
It is shown in Appendix D that the conventional expression for this case, as discussed by Beck [4], for example, may be obtained by rearranging eq. (23) to give: k,m=l
1 =
F(t)
where fore convenience, we put Rk = (Kk/py [ 1 - exp - (pt)k] .
(33)
Clearly, we can also write b k-1 = bkV2 exp(-pt)k-l
+Rk--l
,
3 KU&) coth dp - (l/r-l)1
be,,, C;e = 0 [
6~ 5 1 - exp(-4) -exp(ht) - 1 n=r n2rr2(n2rr2 +Ir) ’
and so on, for each preceding cycle, so as finally to obtain (25)
k
bk = bb
where Q = n2 rr2(Dt/a’) ,
(34)
(26)
exp(-S2k) + zIR”
exp(-AS2k”) ,
(35)
where
and we have made use of the relation from ref. [lo] L&pt)l,
5
(n2n2 t /.+-’ = +[(l/&)
coth&
- l/p] .
r=l
(27)
n=1
(36)
A@ES2k-S2S.
4.3. Release of a single, radioactive species during k cycles of operation
Since the initial concentration of the species in the first cycle of operation is zero, we put b: = 0, to get k
bk(t) = Sq RS exp(-AfinkS) ,
We set m = 1 in eq. (9) to obtain: bk(t) = bff exp(-pt)k
+ (Kk/p’?[l
-
exp(-pOkl ,(28) = s$l (KS/~%1
and from eq. (16): Pk(t)
= Clk exp(-&)k
+ @“/[k>[l - exp(-&?)R] .
- exp - (pt)“j
exp(-AakS).
(37)
Applying the same arguments to eq. (29) for C’k(t),
(29)
gives
Then, Fk(t)[m = l] = 1 - (6/(2naj3p) X {bt expG-pt)k + (Kk/pk)[l
$I
(-l”“/n)
-
exp(-Nkl)
X {C,‘” exp(-&)k + Qk&k)[l - exp(-t;t)k]}-l
C*k(t) = s$l @%W
- ex&Wl
,
(38)
so that the fractional release of a single radioactive species up to time tin the Mh cycle is .
Fk(t)[m = l] = 1 - $l
(6/n2n2)
(30) This result can be put into a more compact form by noting that the initial value of b at the onset of the kth cycle is just equal to the final value at the end of the (k - 1)th cycle; i.e., we can write b;(tk
=
0) = bk-l(tk-')
.
(31)
- exp(-pt)s1 exp(-AS2’s)ln2 tr2 t Cp
k X {s~lWtfW
- wGWW~
where we have used eq. (8) for K and put
Then eq. (28) becomes bk(t) = bk-’ exp(-pt)k
k
X gl @/P)[l
+ Rk ,
(32)
p = (D/a2)(n2n2 + p),
p = ta2/D .
304
G.V. Kidson/ Diffusionalreleaseof fissionproductgasesfrom UOz
4.4. Release of a single, stable species during k cycles of operation This may be treated in exactly the same way as section 4.3, by putting A = 0 so that t = u and
=1 -
Fk(t) X
C (6az/nzn2)
(45)
4.5. Release of a single, stable species during a single
n=l
cycle of operation
s$l GcrD%l- exp(-WI
It would appear that all previous treatments of this case have ignored the loss of atoms by transmutation; if such losses are taken into account we can use the results obtained in section 4.2 by putting f = Yto get
X exp(-ASZ’ks)/n2 n* + r X (6 @IyF)[l S=l
Aq,ks=@k-@
exp(-@I}-’ ,
(40) flt)r;::;,Azo]
where k
Y~v+(D/a2)n2n2,
s2’kzZ(Yt)r,
r=1
(41)
2 (6{/n*n*)[l - exp(-Yt)]/(n*n* = 1 _n=l [ 1 - exp(-vt)]
and
+ 5)
(46)
5 = (w*/D) .
(42)
Nobel [5] and Rim et al. [6] have discussed a slightly more restricted case in which the loss of atoms by transmutation was ignored. Their result follows from eq. (40) by setting v = 0, noting that lim [ 1 - exp(-vt)] /v = t ,
This can be rearranged as in eq. (25) to give: Wr;;;;O,h=O]
=3ry-i]
6t - [exp(vt) - l] (43)
V-+0
X ncl [l - exp(-9)1/[n*s*(n*n’
while
Y(u=0)=(D/a2)n2n2,
{=O,
(47)
where, again 0 E (Dt/a2) n* n* .
and (Yt) = 9,
+ 01 ,
as defined in eq. (26).
Then, 1 - C (6u2/n4n4) n=l
In the conventional treatment of this case it has been assumed that v = 0. The fraction released then follows from eq. (45) for k = 1; i.e. F$z;::;,WOj = 1 - (6a*/Dt) ne1(l/n4n4)[l
- exp(-$)]
= 1 - (a*/lSDt) t nel exp(-@)/n4n4 .
(48)
305
G. V. Kidxw / Diffisional release of fission product gases from UO2
The final form in eq. (48) makes use of the Riemann zeta function [l l] Z(4) = $
(6 log 2/(2vra>3/2),$ dj e&-P+) =I__
m
,
(53)
(l/?r4) = 7r4/90 )
and is identical to that discussed by Beck [4].
where we have used the result that
4.6. Effect of a reactor shut-down cycle If the reactor is shut-down during, say the ith cycle of operation, both the production and the diffusional release of the gases will cease, although, of course, radioactive decay will continue. These effects can be examined by setting K6, Pb, Db, IJ: = 0, for 4= 1,2,..+?. Then ,$, pf + hi and eq. (9) reduces to b’,(t) = (l&JJgrdi
exp(-$t) ,
(49)
where
(50) #j
while eq. (16) becomes
(51) We note from eq. (49) that b&(t) is independent of the summation index n; thus on insertion into eq. (2) we get m
CL(rpt)’ (~/LI)[~~~ $h r>l ,g
dj exp(-Ajt)
, (52)
which simply says that there is an overall decrease in concentration, with no diffusional redistribution. From eqs. (5 l), (49) and (17), we ontain for the fractional release:
c
(-l”+‘/?z) = log 2 .
n=1
5. Discussion It is recognized that the Booth “Equivalent Sphere” model is an over-simplification of the complex phenomena associated with the release of fission product gases in U02 fuels. For example, while the behaviour of all those spheres located in an annular shell between R and R t dR from the center of a cylindrical pellet may be assumed to be governed by a common set of conditions, separate conditions must be assigned to each contributing shell to take into account the temperature and neutron flux distributions as a function ofR. Moreover, only the diffusional release to the surfaces of the equivalent spheres is considered, leaving untouched the qeustion of the subsequent fate of the gases, which is clearly influenced by effects such as grain growth, the nucleation and growth of bubbles, cracking of the oxide and so on. Nevertheless the Booth model can play a role in establishing meaningful comparisons between calculated and measured gas release behaviour; the generalized expressions contained in the eqs. (9), (16) and (17) provide a common basis for a systematic discussion within the limitations of the model. Work in progress is directed toward the preparation of a computer program for the generalized release equation and for the summation of the release for a complete pellet.
Appendix A
(i) We seek solutions for
(A-1)
G. V. Kidson / Diffusional release of fission product gases from UO2
306
where we have used the result
in the form OD
(119) V2$ = -@r/a>“‘,
Cl(C t) = nqI $(n, r) bk,(% 0 *
(A.2)
Taking the time derivative of eq. (A.l), to eliminate p”, and using eq. (A.2) gives
and put p& = & + (Dh/a2) n2n2 . Multiplying eq. (A.lO) by J/(m, r) and again integrating over the volume of the sphere leads to h; +p;bk,
Am_&;_, =0 1
-
(A.3)
)
where we have used the * to indicate differentiation with respect to time. Since the $(n, r) form a linearly independent set, we require that the term in the bracket be zero; this then leads to (l/Jl) V2$ = (l/D:)[b:/h& -
&,I
- Xm_lb&_I
=
(A.4)
integral on the right-had side can readily be evaluated to give The
s
la 47rr2& $(n, r) dr = K;
0
Since the J/ are functions of r only, while the b; are functions oft only, we put (I/$) V2J, = constant = -oi
.
(A.1 1)
0
+ t&
.
=4nr2& J/(?r, r) dr .
s
=
(-l”+‘/nn2)(2na)3’2pk nl.
(A.12)
(A.3
Eq. (A.5) can readily be put in the form of Bessel’s equation of order (0 t i), whose solutions are
Appendix B
Use of the boundary condition (3), i.e.
Let L{b,(t)}=fq(s) be the Laplace transform of Then from eqs. (6), we obtain the relations in transform space:
NW)
(s +~l)fi
= (ho +Kds)
(s + ~2)fz ...
-
$((y,r) = A sin(o&/r .
64.6)
=0 ,
b,(t).
3
requires that on = @m/u) , n = 1,2, ...) .
(A.7)
To evaluate A, we normalize the $ over the grain volume :
@+Pm)fm
Ad-1
= (b2o
+ K2/s)
(B-1)
>
-Arn-lfrn-1=(brno+KmIS).
These equations can be expressed in matrix form as
(I s a
47rr2Jl(n; r) $(??Z,r) dr = 6mn 3
(A.@
AF=D,
(B.2)
where
where 6,, is the Kronecker delta. This the gives A = 1/d(2rru) so that Jl(n, r) = [ 1/d(27ra)] sin(nnr/u)/r .
a1
(A.9
-A1
(ii) Returning to eq. (A.l) and again using eq. (A.2)
we get
A=
~ 0
0
0
...
0
0
a2
0
...
0
0
-X2
a3
.. .
0
0
0
0
... -L-r
... c
n=l
[Jlhk +p&$bL
-A,_~~b~_J
=P”, , (A.lO)
0
1
am
,
307
G. V. Kidson / Diffisional release of fission product gases from UO2
and and uj = (s + pi) , di E (bio + KifS) e
(B.3)
We find F = A-‘D by construction of the inverse matrix A-‘, where
X{bn
exp(-PGq
+ &I1 - ex&PQql/Pq).
(B.7)
It is convenient to reorder the sums for comparison of our results with others. It is straightforward to show that eq. (B.7) can be put in the form
A-’
Al
-1
ala2
a2
- x1x2 ala2a3
0
0
- 12
-1
o
a2a3
a3
...
:
h~2~3
---?‘2h3
A3
ala2a3a4
a2a3a4
a3a4
1
X (bqo exp(--Wi + Kq 11-
. ..
...
exp(-pt)il/piI - (B-8)
Appendix C
a4
Consider the set of m distinct, real numbers {x,}, r = 1, 2, ... . m. We require to show that
Thus, we find
E
=1
pi(O)
(B-4)
j=l
(C-1)
3
where To obtain the inverse transform of eq. (B.4), we expand the product terms as partial fractions, in the form m
m
q:
x - x, pi(X)=
Clearly Pi(X) =
gq = ii U/44,) r=j fq
Then,
(C-2)
X,
0 ifXj=Xr
where
45,
-
#i
(B-5)
lKs+PSI=
ii -. r=l Xi
= Pr - &
>
(B.6)
( 1 ifXj=Xi,forj=
(C-3)
1, 2, ....m .
Now consider the function m
(C-4)
. BY eq. (C.31, P(Xl)=P(X*)=
.a.=P(Xm)= 1 *
(C.5)
G. V. Kidson/ Diffusi~nalreleaseof fissionproductgasesfrom UC4
308
We let f(x) = P(x) - 1. Then, by eq. (CS), we see that the set of numbers fx,} are aIf roots of f(x). Consider the first (m - 1) roots; we know that (x - xi), (x - xa), .... (x - x, _r) are factors off(x) and since f(x) is of degree (m - l), we can write m-1 j(x) = P(x) - 1 = c Pvr (x - x,) .
X
nGlCl -
exp(-$)l/n2
n2(n2 n2 t lfl) .
@W
the relation from ref. [lo],
Using
.T;, Wa2
+4 =$I(l/v/Et)cothu/v
b’/4 >
then gives eq. (25). To evaluate C, we make use of the remaining mth root of Rx), to write m-l nxm)=P(xm_l~=cr~l
(x-x,)=0.
(C.6)
Acknowledgements
Since the {x,.}, are distinct, eq. (C.6) requires that C = 0. Thus P(x) = 1 for all x and hence for x = 0 in particular.
The author is grateful to M.J. Motley and I. Hastings of Chalk River Nuclear Laboratories (CRNL) of Atomic Energy of Canada Limited for their interest and support in this work. The content of Appendix C was kindly supplied by Professor K. Srivastava, of the Department of Mathematics, Brock University. The study was done with support by CRNL and Ontario
Appendix D
Hydro
.
Let References
Pt = (& f @It/C?) n%r2) = e + 14,. Then eq. (23) can be written as
[I] A.H. Booth, Chalk River Report CRDC-721 (1957). [ 2] A.H. Booth, Atomic Energy of Canada Limited Report,
F(t) = 1 - 2
[ 31 J.D. Eichenberg, P.W. Frank, T.J. Kisiel, B. Lustman and
n=l
AECL DCI-27 (1957).
{(6/n27r2) - [l/(n2n2 +@)I}
x [l - exp(4)
exp(-4)) [l - exp(-@)I-’
. (D.l)
Expanding eq. (D.l), making use of the Riemann zeta function in ref. [l I], Z(2) = n$r (l,n2) = n2/6 , and rearranging, we can put (D.1) into the form ~0) = 6 m$i Ill(n2n2
+A1 - CQ&vW - 111
K.H. Vogel, Westin~ouse Report WAPD-183 (1957). [4] S.D. Beck, Batelle Memorial Institute Report, BMi-1433 (1960). [S] L.D. Nobel, Status Report ANS 5.4 (1977) p. III A-l. [6] C.S. Rim and W. Preble, Status Report ANS 5.4 (1977) p. III B-l. [7] C.A. Friskney and M.V. Spelght, J. Nucl. Mater. 62 (1976) 89. [ 81 G. Capart, Belge Nucleaire Report BN7311-02. 19 J K. Srivastava, private communication. [IO] See, for example, E.T. Whittaker and G.N. Watson, A Course in Modern Analysis (Cambridge Univ. Press, Cambridge, 1927) p. 136. [ 1l] See, for example, G. A&en, Mathematical Methods for Physicists (Academic Press, 1970) p. 285.
A GENERALIZED ANALYSIS OF THE CUMULATIVE DIFFUSIONAL RELEASE OF FISSION PRODUCT GASES FROM AN “EQUALED SPHERE” OF UO2 *
G.V. KIDSON * Department of physics, Brock University, St. Catherine&Ontario, Canada Received 6 July 1979; in revised form 22 October 1979
The “Equivalent Sphere Model” introduced by Booth in 195‘7 and used extensively in anaiytical studies related to the diffusional release of fission product gases from UOz reactor fuels, has been generalized to describe the behaviour of the m members of a radio-active chain duringk cycles of reactor operation. Expressions are obtained for the concentration of the species in the equivalent sphere and the cumulative fractional release from the sphere. These are shown to contain, as special cases, all results previously considered as well as some that have not been previously considered and provide a basis for a systematic comparison with experiment under more general conditions.
radioactive isotope during a single cycle of reactor operation. Since then the model has been extended in several ways; for example, Eichenberg et al. f3] obtained a time-dependent solution of the diffusion equation for the release of a single radioactive species during a single cycle of reactor operation, while Beck [4] reviewed the results of both Booth and Eichenberg et al. and presented alternative expressions for certain sums in the solutions that gave better convergence for short times. More recently Nobel [S] and Rim et al. [6] have discussed the equations for the cumulative release of a single stable isotope over several cycles of reactor operation while Friskney and Speight [7] considered the behaviour of the mth member of a radioactive chain during a single cycle of operation. ‘l%edevelopment of these purely phenomenological analyses has been complemented by parallel studies of the diffusion processes themselves; in particular the effects of radiation damage, trapping and resolution of the diffusing gas atoms have been considered. Capart [8] has shown that these latter effects may be incorporated into the phenomenological equations by introducing an “effective diffusion coefficient”, which is the product of the normal diffusion coefficient in the absence of trapping and resolution and the frac-
1. introduction During the burn-up of UO, reactor fuels, a fraction of the gas atoms produced directly from fission and/or by the decay of radio-active precursors, diffuse to the boundaries of the grains of the fuel pellets, where they are released to the ~ter~~~ar pores and eventua~y migrate to the region between the pellets and the encapsulating sheath wall. Considerable effort, both experimental and theoretical, has been directed toward an understan~ng of the behaviour of these gases. An idealized model of a UOZ pellet, originally introduced by Booth [ 1,2] and used extensively in analytical studies is based on the assump tion that the controlling process in the release of the gases is the diffusion from the interior of the grains to the grain surfaces. This “Equivalent Sphere Model” represents the average behaviour of the grains by a set of equivalent spheres whose total area is comparable to the total grain boundary surface area. Booth’s original studies concerned the release of a stable isotope and the steady-state beha~our of a
* Current address: Materials Science Branch, Whiteshell Nuclear Research Establishment, Pinawa, Manitoba, ROE lL0, Canada. 299
300
G. V. Kidson / Diffusional release of fission product gases from UO2
tion of time the diffusing atom is in the untrapped, mobile state. While the several approaches used by the above are generally internally consistent, the development of the solutions, particularly for the fraction of gas released, has sometimes been sufficiently different from one another as to obscure their essential equivalence. Moreover, certain effects that may be of importance have sometimes been neglected. The primary purpose of the present paper is to extend the generalization of the Booth model to its logical limits by considering the cumulative release of the mth member (which may be radioactive or stable) of a radioactive chain over, say, k cycles of reactor operation. This seems not only desirable but also necessary if meaningful comparisons are to be made of the analytical predictions and measured release behaviour of the gases. In so doing, we obtain a result that can readily and systematically be reduced to any of the previous results obtained for less general treatments simply by putting appropriate parameters equal to one or zero. In section 2, below, we specify the model, and obtain the solutions to the diffusion equations. In section 3, the general expression for the cumulative fractional release F:(t) of the mth member of the chain up to a time t during the kth cycle of operation is developed; this is applied in section 4 to a variety of special cases and where appropriate, compared with results obtained in previous studies.
parameters governing the rates of trapping and resolution during the kth cycle. The annihilation of species 1 is assumed to occur both as a result of radioactive decay to species 2 at a rate X1,and by transmutation by neutron absorption at a rate 8, the latter being a function of the neutron capture cross section and neutron flux during the kth cycle. All other members of the chain are governed by corresponding parameters except that they will be produced as a result of the radioactive decay of their precursors as well as directly by fission. We obtain, then, a set of m coupled equations for the concentration distributions in the sphere:
acflat = pf + 0: v2 C: - g:cf ...
ac&lat = im_,c&_,
,
+p; +o&v2c; - &c&
Here C is the concentration in atoms per unit volume, V2 is the Laplacian operator in spherical coordinates, and l= (h t Y).We seek solutions to these equations in the form CL (r, t) = 5
n=1
ti(n, r) I!&(n. t)
,
We consider the production, diffusion, annihilation and release of the m members of a radioactive chain in and from an equivalent sphere of radius a, up to a time t, measured from the beginning of the kth cycle of reactor operation. The initial member of the chain is produced solely as a result of fission at a rate 0: which is determined by the total fission cross section, the probability of production of species 1, the concentration of fissionable uranium and the neutron flux associated with the Mh cycle of operation. The rate of diffusion of the species is characterized by the effective diffusion coefficient 0: determined by the temperature and the
(2)
subject to the boundary condition Ck(r=a,
t)=O,
(3)
and initial condition: C&(r, t = 0) = nql $4n, r) by&,
2. Specification of the model and solutions of the diffusion equations
.(I)
t = 0).
(4)
It is shown in Appendix A that the Jl(n, r), n = 1, 2 , .... 00,form the orthonormal set of spherical Bessel functions 9(n, r) =
ww41
Ww~Y~
9
(5)
while the time-dependent functions bk(n, t) are governed by the m coupled equations dbfldt
+p:b:
- K: = 0 ,
dbzldt +p;b’: - K’; - Al bf = 0, ... dbfS,ldt+pk,b~-K;-A,_,bk,_l=O.
(6)
Here, pk z & + n2n2(Dk,/a2) ,
(7)
301
G. V. Kidson / Diffisional release of fission product gases from UO2
= 1 - Q/$7ra”Cik(t),
and K;
= (-l”+‘/r~n~)(2~a)~‘~ /3& .
(8)
Solutions to the set of equations (6) are obtained in Appendix B by use of the Laplace Transform method and may be put in the form:
where
(i
q=l
(13)
Q = bdt/(2na)l n$lbk(t) jr sin(nnr/a> dr , 0
= [4n/d(2na)]
x
02)
n$l (-ln+1a2/nn)
b:(t)
(14)
.
The CGk are, of course, also governed by a set of m coupled equations:
bp”o ii &&‘$> r=q
dCTkldt t .$:ck
- 0: = 0,
dClk/dt t ,$C;’ - 0’: - hr CTk = 0 , ... dCGk/dt t &C&k -0; -A,_rc$!_r =o.
(15)
These are identical in form to those found for the b;(t), given in eqs. (6). Thus by substituting C’Gk for bk, C;“,for pk and & for Ki in eq. (9), we can
where Ap+p;-pjk.
(10)
We note that the dependence of b;(t) on all of its precursors during the preceding (k - 1) cycles is contained in the b& terms. See section 4.3 below for a full discussion.
immediately write:
Cik(t).= (X,)-l [%
Aj exp(-t/Q)
j=l
3. Fractional release To ensure consistency with the several definitions proposed in the past [l-8], we adopt the following for the cumulative fractional release F&(t) of the mth member of the radioactive chain: “F&(t) is the ratio of the total number of atoms of the mth species released from the volume of the equivalent sphere during all k cycles of operation, and still surviving at time t in the kth cycle to the total number of atoms of that species produced in the sphere during all k cycles and still surviving at the same time t. ” To form F:(t), let Cik(t) be the total number of atoms per unit volume of the mth species at time t in the kth cycle. We then define F:(t) as:
4 7ra3Cik(t) - jt47rr2 CL (r, t) dr F:(t)
=
0 4 7ra3CGk(t)
(11)
The final form for FL(t) can now be written as [6/(2na)3/2] 2 F;(t)
= 1-
n=1
(-1 ‘+r/n) bk, (t)
GkU)
9 (17)
where b% and C’G”are given by eqs. (9) and (16) respectively. In section 4 this general expression is applied to a variety of special cases many of which have been previously discussed in the literature.
302
G. V. Kidson / Diffusional release of fission product gases from UO2
comparison with their result, we incorporate these conditions into eq. (20) by setting vj = 0, so that [j = Ai, & = 0 for Q # 1 and letting
4. Application to special cases
4.1. Release of the mth member of a radioactive chain during a single cycle of operation
X=hi+n21r2(Dja2).
We set the number of cycles k = 1, drop the superscript and put bqo, C$ = 0. Then the eq. (9) reduces to
Then with minor rearrangement, eq. (20) reduces to:
rk=l
1
-
F,(t)
b
m
m
(18)
m
m
X l-l @r/X,) ,Fr 11 - exP(-Xjr)l r=1
!Jr XtJu*j +i
where we have used equation (8) for K, . Similarly, eq. (16) becomes m
CA(t)= (~m)-l~l(X//tj)[l
-
expt-[jr)1
With the use of the result obtain@ in Appendix C (Srivastava [9]), (19)
E(l?i
j=l
so that the fraction of the mth species released can be written as:
i-=1 +j
E j=l
( fi r=l +j
X,/AX,)=
13
(22) &/Ax,) = 1 ,
eq. (22) becomes identical to the expression reported by Friskney and Speight [7].
X [l - exP(-pi01 $rP,
4.2. Release of a single radioactive species during a
1 b/&rj
‘=Q +j
single cycle of operation This case readily follows from eq. (20) by setting
X 5 (Aj/tj) i l - exP(-tjt)l Cj=1
m =
1, whence
F(t)
(20) 2
In the model discussed by Friskney and Speight [7], the loss of atoms through transmutation by neutron absorption was ignored, and only the first member of the radioactive chain was assumed to be produced by fission while all other members were assumed to be produced solely by the decay of their precursors. For
=
1
(6p/n2
R2)(l /n’ n2 t p)( 1 - exp(-pt))
_n=1 1 - exp(-[t) (23)
where p = [a’/0
.
(24)
303
C. V. K&on / Diffusional release of fission product gases from UO2
It is shown in Appendix D that the conventional expression for this case, as discussed by Beck [4], for example, may be obtained by rearranging eq. (23) to give: k,m=l
1 =
F(t)
where fore convenience, we put Rk = (Kk/py [ 1 - exp - (pt)k] .
(33)
Clearly, we can also write b k-1 = bkV2 exp(-pt)k-l
+Rk--l
,
3 KU&) coth dp - (l/r-l)1
be,,, C;e = 0 [
6~ 5 1 - exp(-4) -exp(ht) - 1 n=r n2rr2(n2rr2 +Ir) ’
and so on, for each preceding cycle, so as finally to obtain (25)
k
bk = bb
where Q = n2 rr2(Dt/a’) ,
(34)
(26)
exp(-S2k) + zIR”
exp(-AS2k”) ,
(35)
where
and we have made use of the relation from ref. [lo] L&pt)l,
5
(n2n2 t /.+-’ = +[(l/&)
coth&
- l/p] .
r=l
(27)
n=1
(36)
A@ES2k-S2S.
4.3. Release of a single, radioactive species during k cycles of operation
Since the initial concentration of the species in the first cycle of operation is zero, we put b: = 0, to get k
bk(t) = Sq RS exp(-AfinkS) ,
We set m = 1 in eq. (9) to obtain: bk(t) = bff exp(-pt)k
+ (Kk/p’?[l
-
exp(-pOkl ,(28) = s$l (KS/~%1
and from eq. (16): Pk(t)
= Clk exp(-&)k
+ @“/[k>[l - exp(-&?)R] .
- exp - (pt)“j
exp(-AakS).
(37)
Applying the same arguments to eq. (29) for C’k(t),
(29)
gives
Then, Fk(t)[m = l] = 1 - (6/(2naj3p) X {bt expG-pt)k + (Kk/pk)[l
$I
(-l”“/n)
-
exp(-Nkl)
X {C,‘” exp(-&)k + Qk&k)[l - exp(-t;t)k]}-l
C*k(t) = s$l @%W
- ex&Wl
,
(38)
so that the fractional release of a single radioactive species up to time tin the Mh cycle is .
Fk(t)[m = l] = 1 - $l
(6/n2n2)
(30) This result can be put into a more compact form by noting that the initial value of b at the onset of the kth cycle is just equal to the final value at the end of the (k - 1)th cycle; i.e., we can write b;(tk
=
0) = bk-l(tk-')
.
(31)
- exp(-pt)s1 exp(-AS2’s)ln2 tr2 t Cp
k X {s~lWtfW
- wGWW~
where we have used eq. (8) for K and put
Then eq. (28) becomes bk(t) = bk-’ exp(-pt)k
k
X gl @/P)[l
+ Rk ,
(32)
p = (D/a2)(n2n2 + p),
p = ta2/D .
304
G.V. Kidson/ Diffusionalreleaseof fissionproductgasesfrom UOz
4.4. Release of a single, stable species during k cycles of operation This may be treated in exactly the same way as section 4.3, by putting A = 0 so that t = u and
=1 -
Fk(t) X
C (6az/nzn2)
(45)
4.5. Release of a single, stable species during a single
n=l
cycle of operation
s$l GcrD%l- exp(-WI
It would appear that all previous treatments of this case have ignored the loss of atoms by transmutation; if such losses are taken into account we can use the results obtained in section 4.2 by putting f = Yto get
X exp(-ASZ’ks)/n2 n* + r X (6 @IyF)[l S=l
Aq,ks=@k-@
exp(-@I}-’ ,
(40) flt)r;::;,Azo]
where k
Y~v+(D/a2)n2n2,
s2’kzZ(Yt)r,
r=1
(41)
2 (6{/n*n*)[l - exp(-Yt)]/(n*n* = 1 _n=l [ 1 - exp(-vt)]
and
+ 5)
(46)
5 = (w*/D) .
(42)
Nobel [5] and Rim et al. [6] have discussed a slightly more restricted case in which the loss of atoms by transmutation was ignored. Their result follows from eq. (40) by setting v = 0, noting that lim [ 1 - exp(-vt)] /v = t ,
This can be rearranged as in eq. (25) to give: Wr;;;;O,h=O]
=3ry-i]
6t - [exp(vt) - l] (43)
V-+0
X ncl [l - exp(-9)1/[n*s*(n*n’
while
Y(u=0)=(D/a2)n2n2,
{=O,
(47)
where, again 0 E (Dt/a2) n* n* .
and (Yt) = 9,
+ 01 ,
as defined in eq. (26).
Then, 1 - C (6u2/n4n4) n=l
In the conventional treatment of this case it has been assumed that v = 0. The fraction released then follows from eq. (45) for k = 1; i.e. F$z;::;,WOj = 1 - (6a*/Dt) ne1(l/n4n4)[l
- exp(-$)]
= 1 - (a*/lSDt) t nel exp(-@)/n4n4 .
(48)
305
G. V. Kidxw / Diffisional release of fission product gases from UO2
The final form in eq. (48) makes use of the Riemann zeta function [l l] Z(4) = $
(6 log 2/(2vra>3/2),$ dj e&-P+) =I__
m
,
(53)
(l/?r4) = 7r4/90 )
and is identical to that discussed by Beck [4].
where we have used the result that
4.6. Effect of a reactor shut-down cycle If the reactor is shut-down during, say the ith cycle of operation, both the production and the diffusional release of the gases will cease, although, of course, radioactive decay will continue. These effects can be examined by setting K6, Pb, Db, IJ: = 0, for 4= 1,2,..+?. Then ,$, pf + hi and eq. (9) reduces to b’,(t) = (l&JJgrdi
exp(-$t) ,
(49)
where
(50) #j
while eq. (16) becomes
(51) We note from eq. (49) that b&(t) is independent of the summation index n; thus on insertion into eq. (2) we get m
CL(rpt)’ (~/LI)[~~~ $h r>l ,g
dj exp(-Ajt)
, (52)
which simply says that there is an overall decrease in concentration, with no diffusional redistribution. From eqs. (5 l), (49) and (17), we ontain for the fractional release:
c
(-l”+‘/?z) = log 2 .
n=1
5. Discussion It is recognized that the Booth “Equivalent Sphere” model is an over-simplification of the complex phenomena associated with the release of fission product gases in U02 fuels. For example, while the behaviour of all those spheres located in an annular shell between R and R t dR from the center of a cylindrical pellet may be assumed to be governed by a common set of conditions, separate conditions must be assigned to each contributing shell to take into account the temperature and neutron flux distributions as a function ofR. Moreover, only the diffusional release to the surfaces of the equivalent spheres is considered, leaving untouched the qeustion of the subsequent fate of the gases, which is clearly influenced by effects such as grain growth, the nucleation and growth of bubbles, cracking of the oxide and so on. Nevertheless the Booth model can play a role in establishing meaningful comparisons between calculated and measured gas release behaviour; the generalized expressions contained in the eqs. (9), (16) and (17) provide a common basis for a systematic discussion within the limitations of the model. Work in progress is directed toward the preparation of a computer program for the generalized release equation and for the summation of the release for a complete pellet.
Appendix A
(i) We seek solutions for
(A-1)
G. V. Kidson / Diffusional release of fission product gases from UO2
306
where we have used the result
in the form OD
(119) V2$ = -@r/a>“‘,
Cl(C t) = nqI $(n, r) bk,(% 0 *
(A.2)
Taking the time derivative of eq. (A.l), to eliminate p”, and using eq. (A.2) gives
and put p& = & + (Dh/a2) n2n2 . Multiplying eq. (A.lO) by J/(m, r) and again integrating over the volume of the sphere leads to h; +p;bk,
Am_&;_, =0 1
-
(A.3)
)
where we have used the * to indicate differentiation with respect to time. Since the $(n, r) form a linearly independent set, we require that the term in the bracket be zero; this then leads to (l/Jl) V2$ = (l/D:)[b:/h& -
&,I
- Xm_lb&_I
=
(A.4)
integral on the right-had side can readily be evaluated to give The
s
la 47rr2& $(n, r) dr = K;
0
Since the J/ are functions of r only, while the b; are functions oft only, we put (I/$) V2J, = constant = -oi
.
(A.1 1)
0
+ t&
.
=4nr2& J/(?r, r) dr .
s
=
(-l”+‘/nn2)(2na)3’2pk nl.
(A.12)
(A.3
Eq. (A.5) can readily be put in the form of Bessel’s equation of order (0 t i), whose solutions are
Appendix B
Use of the boundary condition (3), i.e.
Let L{b,(t)}=fq(s) be the Laplace transform of Then from eqs. (6), we obtain the relations in transform space:
NW)
(s +~l)fi
= (ho +Kds)
(s + ~2)fz ...
-
$((y,r) = A sin(o&/r .
64.6)
=0 ,
b,(t).
3
requires that on = @m/u) , n = 1,2, ...) .
(A.7)
To evaluate A, we normalize the $ over the grain volume :
@+Pm)fm
Ad-1
= (b2o
+ K2/s)
(B-1)
>
-Arn-lfrn-1=(brno+KmIS).
These equations can be expressed in matrix form as
(I s a
47rr2Jl(n; r) $(??Z,r) dr = 6mn 3
(A.@
AF=D,
(B.2)
where
where 6,, is the Kronecker delta. This the gives A = 1/d(2rru) so that Jl(n, r) = [ 1/d(27ra)] sin(nnr/u)/r .
a1
(A.9
-A1
(ii) Returning to eq. (A.l) and again using eq. (A.2)
we get
A=
~ 0
0
0
...
0
0
a2
0
...
0
0
-X2
a3
.. .
0
0
0
0
... -L-r
... c
n=l
[Jlhk +p&$bL
-A,_~~b~_J
=P”, , (A.lO)
0
1
am
,
307
G. V. Kidson / Diffisional release of fission product gases from UO2
and and uj = (s + pi) , di E (bio + KifS) e
(B.3)
We find F = A-‘D by construction of the inverse matrix A-‘, where
X{bn
exp(-PGq
+ &I1 - ex&PQql/Pq).
(B.7)
It is convenient to reorder the sums for comparison of our results with others. It is straightforward to show that eq. (B.7) can be put in the form
A-’
Al
-1
ala2
a2
- x1x2 ala2a3
0
0
- 12
-1
o
a2a3
a3
...
:
h~2~3
---?‘2h3
A3
ala2a3a4
a2a3a4
a3a4
1
X (bqo exp(--Wi + Kq 11-
. ..
...
exp(-pt)il/piI - (B-8)
Appendix C
a4
Consider the set of m distinct, real numbers {x,}, r = 1, 2, ... . m. We require to show that
Thus, we find
E
=1
pi(O)
(B-4)
j=l
(C-1)
3
where To obtain the inverse transform of eq. (B.4), we expand the product terms as partial fractions, in the form m
m
q:
x - x, pi(X)=
Clearly Pi(X) =
gq = ii U/44,) r=j fq
Then,
(C-2)
X,
0 ifXj=Xr
where
45,
-
#i
(B-5)
lKs+PSI=
ii -. r=l Xi
= Pr - &
>
(B.6)
( 1 ifXj=Xi,forj=
(C-3)
1, 2, ....m .
Now consider the function m
(C-4)
. BY eq. (C.31, P(Xl)=P(X*)=
.a.=P(Xm)= 1 *
(C.5)
G. V. Kidson/ Diffusi~nalreleaseof fissionproductgasesfrom UC4
308
We let f(x) = P(x) - 1. Then, by eq. (CS), we see that the set of numbers fx,} are aIf roots of f(x). Consider the first (m - 1) roots; we know that (x - xi), (x - xa), .... (x - x, _r) are factors off(x) and since f(x) is of degree (m - l), we can write m-1 j(x) = P(x) - 1 = c Pvr (x - x,) .
X
nGlCl -
exp(-$)l/n2
n2(n2 n2 t lfl) .
@W
the relation from ref. [lo],
Using
.T;, Wa2
+4 =$I(l/v/Et)cothu/v
b’/4 >
then gives eq. (25). To evaluate C, we make use of the remaining mth root of Rx), to write m-l nxm)=P(xm_l~=cr~l
(x-x,)=0.
(C.6)
Acknowledgements
Since the {x,.}, are distinct, eq. (C.6) requires that C = 0. Thus P(x) = 1 for all x and hence for x = 0 in particular.
The author is grateful to M.J. Motley and I. Hastings of Chalk River Nuclear Laboratories (CRNL) of Atomic Energy of Canada Limited for their interest and support in this work. The content of Appendix C was kindly supplied by Professor K. Srivastava, of the Department of Mathematics, Brock University. The study was done with support by CRNL and Ontario
Appendix D
Hydro
.
Let References
Pt = (& f @It/C?) n%r2) = e + 14,. Then eq. (23) can be written as
[I] A.H. Booth, Chalk River Report CRDC-721 (1957). [ 2] A.H. Booth, Atomic Energy of Canada Limited Report,
F(t) = 1 - 2
[ 31 J.D. Eichenberg, P.W. Frank, T.J. Kisiel, B. Lustman and
n=l
AECL DCI-27 (1957).
{(6/n27r2) - [l/(n2n2 +@)I}
x [l - exp(4)
exp(-4)) [l - exp(-@)I-’
. (D.l)
Expanding eq. (D.l), making use of the Riemann zeta function in ref. [l I], Z(2) = n$r (l,n2) = n2/6 , and rearranging, we can put (D.1) into the form ~0) = 6 m$i Ill(n2n2
+A1 - CQ&vW - 111
K.H. Vogel, Westin~ouse Report WAPD-183 (1957). [4] S.D. Beck, Batelle Memorial Institute Report, BMi-1433 (1960). [S] L.D. Nobel, Status Report ANS 5.4 (1977) p. III A-l. [6] C.S. Rim and W. Preble, Status Report ANS 5.4 (1977) p. III B-l. [7] C.A. Friskney and M.V. Spelght, J. Nucl. Mater. 62 (1976) 89. [ 81 G. Capart, Belge Nucleaire Report BN7311-02. 19 J K. Srivastava, private communication. [IO] See, for example, E.T. Whittaker and G.N. Watson, A Course in Modern Analysis (Cambridge Univ. Press, Cambridge, 1927) p. 136. [ 1l] See, for example, G. A&en, Mathematical Methods for Physicists (Academic Press, 1970) p. 285.