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A compilation of the solutions to the important equations in low-temperature gas atom diffusion
Journal of Nuclear Materials 98 (1981) 2-10 North-Holland Publishing Company
A COMPILATLONOF THE SOLUTIONS TO THE IMPORTANT EQUATIONS IN LOW-TEMPERATURE GAS ATOM DIFFUSION N. BEATHAM UKAEA, Wndorrk Nuclear Pow
Development L&oratories, Sellajikld, Ssn~le,
Chmbrkr CA20 IPF, UK
Received 2 February 1981
This paper collects together the solutions for the diffusion of unstable gas atoms within a ~rni~~te solid. Allowance is made for one precursor. These solutions are appropriate to the diffusion of unstable gas atoms from inside the fuel into the pin voidage in situations where the radius of curvature of the surface can be neglected. The solutions are particularly relevant to low-temperature gasatom diffusion and are thus of considerable value to coolant analysis at commercial AGR stations.
1. Introduction
of reactor start-up and shutdown in addition to the approach to equilibrium at a new power level.
For low-rated UOa fuelled pins, the operating fuel temperatures are sufficiently low (T< 1000°C) to exclude everywhere the fo~ation of ~ter~~~ar porosity and to establish a temperature-~de~ndent but fmion-rate sensitive process as the dominant mode of fission product release from the fuel. The diffusion coefficient is small and this implies that diffusion depths’ are also small, compared to the reciprocal of the surface-to-volume ratio of the fuel. As a result, the radius of curvature of the surface can be neglected and the appropriate diffusion equations to use are those for a semi-infinite solid. This paper collects together the* timedependent solutions for the most relevant of the diffusion equations, and includes allowance for one precursor. It is assumed that the surface-to-volume ratio of the fuel remains constant during the course of the irradiation, which should be E good approximation for low temperature release. The most important application of this work is to the release from the fuel into the pm voidage of short-lived fission products such as 13rI. The solutions for these unstable isotopes are required in the analysis of radioactive release into the coolant from low-temperature pins. ‘Ihe solutions described here cover the situations 0 022-311 S/8 1~~~~0~$02.50
0 North-Holland
2. The diffusional release of unatabie gas atoms from within a semi-infmite solid 2.1. No precursor - the approach to equilibrium from initialstart-up
The diffusion equation within a semi-infiite solid for a single unstable isotope generated by fission is
ac _D a2c ,,+fl+,
arwhere
c is the concentration (me3), D is the diffusion
coefficient (m2 s-l), /3 is the generation rate per unit volume (mq3 s-r), X is the decay rate (s-r), t is the time (s), x is the distance into the fuel from the fuel/pm voidage interface (m). The equation lowing boundary c=Oatx=O
is required to be solved with the folconditions: and
c=Owhent=O.
3
N. Beatham /Solutions to equationsin low-temperature gasatomdiffusion
As is customary with these types of equation, it is expedient to apply the substitution C=ueAt. &A _&+pe”‘,
(2)
at
which is the corresponding solution for the diffusion of stable species. The most important property to be derived from eq. (4) is the flux J (m” s-l) of unstable atoms released from the fuel into the pin voidage. This is computed from the relationship J = tD@clax),,
with u=Oatx=O
and
u=Owhent=O.
Eq. (2) can be readily solved by application of the
Laplace tr~sfo~tion is
[If . The appropriate solution
.
The positive sign in eq. (7) arises from the fact that we have defined the flux J as that coming from the fuel into the pm voidage. The flux proves to be J = ~(D~~) erf(~~t))
-
(3)
which implies that
Although these solutions are appropriate to the case of the semi-infinite solid, they essentially apply to situations where the radius of curvature of the fuel surface can be neglected. The total release rate R (s-l) of atoms leaving the fuel is thus given by R = ST = S&I(D/X) erf(&r))
(4)
+ =$/@/A)
c=Oatx=O
since B = @V.
c=Owhent=O.
There are two very important limiting cases for eq. (4). The first is that as t + 00, the steady state concentration profile is obtained: e =;
[1 _ e-x&D’]
,
(5)
‘Ike second is that when X is set to zero, i.e. the isotope is stable, eq. (4) decomposes to yield
(6)
,
(9)
where S is the surface area of the fuel (m2). If Y is the volume of the fuel (m3), then the ratio of release rate to birth rate is given by
It may be readily shown that this solution does satisfy the respective boundary conditions and
@s)
The two important limiting cases of this equation are: (i) when t + 00, and the flux is then simply M(D/A) m-’ s-l, (ii) when X+ 0, and the flux becomes 2fl(Dt/n)1’2 mW2s-l.
erfc
+ exJ@fP) erfc
(7)
erf(&t))
(10)
For stable atoms, it is customary to express the release in terms of the fraction f of those generated: f= JR 0
dt’/VPt = ;+(Dt/n)
.
01)
2.2. Srirgle precutsor - the uppmach to equilib~um from initiai start-up The majority of unstable atoms have a precursor with an appreciable half-life and it is important that this should be included in the analysis. The appropriate pair of diffusion equations are as
4
N. Beatham /Solutions to equations in low-temperaturegas atom diffusion
follows:
acl -=
-a26 a2 +P-xlcl ,
(12)
a2c2 -t hlC1- xzcz, ax2
(13)
D
at
An expression for c2 can thus be constructed from eqs. (3), (14) and (16). The release rate R2 (s-l) of the daughter product from the fuel in to the pin voidage is given by
and k2
stzD
where cl is the concentration of the precursor (m-j), hr is the decay rate of the precursor (s-l), c2 is the concentration of the daughter product (mm3), X2is the decay rate of the daughter product (s-l). l’he corresponding boundary conditions are: cl =Oandc2 =O
atx=O,
cl =Oandc,
whent=O.
=O
Eq. (12) of course has already been solved in the previous section. As regards eq. (13), it is expedient to apply the substitutions
X
h/W&
) erf(d(b
0)
-
~4%~
1 erf(d2
O)l . (17)
If one wishes to calculate the amount of daughter product present in the pin voidage at any moment in time, it must be remembered that there will be two contributions to the generation rate of the daughter product in the pin voidage. The first is due to direct release of the isotope from the fuel and the second is from the decay of precursor released to the pin voidage. 2.3. No precursor - the approach to a new equilibri-
Cl=uedhlt ,
um level
and
The cases that have been studied in the last two sections represent the approach to steady state equilibrium from an initial zero concentration profde. A very important situation is when equilibrium has been reached at one fission rate and then the fuel rating is altered to a new level. In the first example, we shah take the case for no precursor and assume that the diffusion coefficient is independent of fission rating. The relevant equation is
c2
=&
[X,u eshlt + A2 u e-A2tl,
(14)
which results in a new form of eq. (13):
au _=D?k -i!!!&t at
ax2 x2
’
(15)
with u=Oatx=O
and
u=Owhent=O.
This equation is of the same type as eq. (2) and the solution can be immediately written down:
ac
z=D$tf12
-AC,
where p? is the new generation rate (me3 s-l) with the boundary conditions c=o
atx=O,
c =+ ~1_ e-xJGVo)l
whent=O.
or is the old generation rate (m-j s-l) and this initial concentration profile represents the equilibrium profile for the old generation rate.
5
N. Beatham /Solutions to equations in low-temperature gas atom diffusion
01
If we apply the substitution p=C-$
-x&W) I,
[l-e
-X&PI
c=T[l-e (20)
‘1
whent=O.
D, is the old diffusion coefficient (m2 s-l).
The first step in the analysis is to make the standard substitution c = uemA’.This results in the transformed equation
then eq. (19) is transformed into
(21)
au
z=D2$t&e"',
with p=Oatx=O
and
with the boundary conditions
p=Owhent=O.
This equation is also of the same type as eq. (2) and hence WWx),=c
= 0% - A ) L&W) erf(&W)
Therefore, the release rate R (s-l) becomes
(25)
.
u=o
atx=O,
u =F [1 _ e-d@/Dr)]
whenr=O.
(22)
from the fuel
‘Ihis equation may be solved by the standard Laplace transform technique but the solution may be readily obtained by inspecting the function
W=Y--
Ml u + t pz
~1 _ e-xJW’r)l
cur ,
(26)
where h is the ratio D2/D1, and y is the solution of = S d@/Q[PI + Go,-
PI) erf(dW)l .
(23)
au=D2 ~+&e”‘, a5 with
@r&/W
y=Oatx=O
,
(27)
at
‘Ihis equation shows how the release rate starts at the old steady-state equilibrium level of
and y=Owhent=O,
and approaches the new equilibrium level of
and u is the solution of
W&VU
2 -=av D2 st&e”‘, at
.
The equation automatically includes the situation of reactor shutdown for in this case & = 0. For the second example we shall take the more general case for no precursor, where the diffusion coefficient at the new ftion rating is different from the previous one. This is particularly appropriate for the athermal diffusion process where the diffusion coefficient is considered to be directly proportional to fission rating. The relevant equation is & -=
at
with
u=Oatx=O
and
u=Owhent=O.
It may easily be proved that
a2w D2 -ax2 +&$‘-*=o
at
w=o
atx=O,
’
with the boundary conditions atx=O
D2 $3+c,
where D2 is the new diffusion coefficient (m2 s-l) with the boundary conditions c=O
(28)
w =$
[l _ e--XJ@Pr)]
when t = 0 ,
which implies that u = w is the solution of eq. (25). The solution to eq. (24) is thus
N. Beatham / Solutions to equations in low-temperature gasatomdiffusion
6
c2
+t
[l _ e-&4)]
=W(D2/MP2
--e"1 pz
[l _
e-xJG2/D)]
+ &
(29)
eW-lP.
The release rate R (s-l)
=$
[e-XJ@IIQ
_
e-d@2/D)]
when t = 0 .
from the fuel is given by
Both of the initial concentration profiles represent the equilibrium profiles for the old generation rate, The first equation (3 1) has already been solved in the previous section. If we apply the substitutions
erf(d(W)
"(h-1)tSt/(D2/hA))2
erf(&ht)) p1
= cl
$ [l
_
_
e-d@l/D)] ,
(33)
1
+ eh(*-l)t
or&[1
- erf(&Irr))]}
.
(30)
:
P2 = c2 -
[1 _ e_XJ62/D)]
&
-
[e--d@l/D)_ e-xJ(h/D)] ,
(34)
2.4. Single precursor - the approach to a new equilibrium level
then eq. (32) is simplified to yield
a2p2 ap,_ --+bPl D
‘Ihe first case to be analysed in this section repre-
sents the situation where equilibrium has been reached at one fission rate and then the fuel rating is altered to a new level. The analysis includes allowance for a single precursor but assumes that the diffusion coefficient is independent of rating. ‘Ihe relevant pair of differential equations is as follows:
ac, a2cl - =Dtp2 -h1c* at ax2
,
(31)
and p2 = 0 when t = 0 ,
which is coupled to
8th
_D
at-
a%
--Alp, ax2
+pz
-01
9
when t = 0 ,
e--xJw~q
and p1 =Owhenr=O.
p1 =Oatx=O
and
ac2 _
at-
D
-a2c2+ h,c, ax2
- h2c2 ,
(32)
with c2 =o
(36)
‘Ihis pair of equations is exactly similar to eqs. (12) and (13) in section 2.2. As a result the release rate R2 (s-l) of the daughter product from the fuel is given by
atx=O,
Cl = 0 [1 _
p2 =Oatx=O
(35)
9
with
with
Cl =?
-h2P2
ax2
at-
$ atx=O,
(1 _
eexJR21D))
N. Beatham /Solutions to equations in low-temperature gapatom diffision lA=O
xJwD) _ e--xJW4
t-el,!A(-
u
=*{
&Vhi
-;(oh2N31
)[Pr + dB2
+ 032
-
Pd
-
011
PI 1 erf(\/(h
eWCX2t))ll
(37)
.
For the final example, we shall take the general case for a single precursor, where the diffusion coefficient at the new fusion rating is different from the previous one. l’he relevant coupled equations are
acl
a%
_D
at-
atx=O,
)11,=o
2
2--~1c1 ax2
to2 ,
(38)
atx=O,
C, =$ ~1_ e_XJ(“~/D1)]
whent=O,
[1
_
e-xJ@~IDI)l
whent=O,
and av,D2
at
&eAZ',
E$_”
(41)
x2
with U=O
atx=O,
~=d$!~~-e-~JC’W4)]~
whent=O.
Both eq. (41) and eq. (40) are similar to eq. (25) which was solved in the previous section. ‘Ihe release rate R2 (s-r) of the daughter product from the fuel is thus
with cr = 0
=t
7
and
a2c2 2-+ax2 hct
ac2
_D
Z-
-
A2c2
(39)
9
= &[A1
(g)x;,
eShlt + A2(:),=,
eeA2’]
with c2
=o
c2 =$
atx=O, t eAl(h-l)t g,fi[
(1 _ e-XJ(A1ID1))
-
4(D2 /X2 ) h
1 - erf(d(hX1 erf(&
t eA2@--l)' fl,fi[1
when r=O
= ueeA1
2))
- erf(&As
t))] 11 .
(42)
lhis solution covers the most general case of the examples analysed in this report. All the other solutions may be derived from eq. (42) by the appropriate simplification.
.
If we apply the standard substitutions Cl
t))l 1
f ,
and 3. Illustrativeexample ,UeShrt t X2rje-Ast] [A to these equations, then
au at
-_=D
with
a2u
-t
2 ax2
&e”l’,
(40)
‘Ihe following example is included to show how the solutions may be applied to a typical reactor situation. The case in question concerns the release rate from the fuel of ls31 and ‘j3Xe during: (i) reactor startup to a fuel rating of 10 W/gU (fuel t moderator); and
N. Beatham /Solutions to equationsin low-temperature gasatomdiffusion
8
(ii) a subsequent uprating to 12 W/gU (f t m) after equilibrium has been reached at 10 W/gU.
Table 1 Valuesfor the release rates from the fuel after reactor start-up to a rating of 10 W/gU (f + m)
3.1. Reactor start-up
Time
If one assumes a value of 204 MeV for the average energy generated per fission and a fuel density of 10.67 t me’, then a rating of x W/gU (f t m) is equivalent to a fusion rate f of 2.88 X 10”~ mm3s-l. ‘Ihe approximation is adopted that all ‘j31 is generated directly from fission with a fractional yield of 0.066 and that ‘33Xe is produced only from the decay of 1331. An appropriate choice for the athermal diffusion coefficient associated with low-temperature release is 2 X 104’fm2 s-l. It is assumed that both isotopes possess the same diffusion coefficient. lhe parameter set for the approach to equilibrium at a rating of 10 W/gU (f t m) is as follows: fll = 1.90 X 10” mm3 s-l , Dl = 5.76 X lo-” Xl = 9.21 X 10T6 s-l ,
m2 s-l
A2 =1.52x10-6s-‘.
‘Ihe appropriate equation for the release rate, RI (me2 s-l), of 13jI from the fuel is eq. (9): RI =&&DA)
erf(&t))
RI
RXe
Ge
0 0.21 0.41 0.71 0.94 1.09 1.19 1.32 1.39 1.49 1.50 1.50 1.50 1.50 1.50 1.50
0 0.001 0.01 0.05 0.12 0.21 0.31 0.50 0.69 1.30 1.86 2.17 2.53 2.62 2.63 2.63
0 0.21 0.48 0.75 1.05 1.27 1.45 1 .I4 1.97 2.57 3.05 3.31 3.62 3.69 3.70 3.10
(days) 0 0.02 0.10 0.25 0.50 0.75 1 .oo 1.50 2.00 4.00 7.00 10 20 35 50 . 100
Release rates in units of lo9 mB2 s-l.
RI refers to the release of ’ 331 from the fuel. Rre refers to the release of ’ 33Xe from the fuel. RXe is the calculated release of ’ 33Xe from the fuel if the presence of the iodine precursor is neglected.
.
All release rates quoted in this section are per unit surface area. For 133Xe, the release rate, Rxe, from
the fuelis given by eq. (17):
diffusion theory because the decay rate of the precursor enters the equations in the form of the square root. 3.2. The approach to a new equilibrium level
RJk = &
[d(Di /Ai ) erf(&
- d(D 1/A2 ) erf(d&
0)
t))l -
However, if one had neglected the presence of the iodine precursor, then the release rate, R&, of rJ3Xe would have been calculated from the equation: %
= Prd(Dr/&)
erf(&st))
For this case it is assumed that the fuel has been operated at 10 W/gU (f t m) for sufficient time for the 1331and “‘Xe release rates to have attained their equilibrium values. The fuel is then uprated to 12 W/gU (f t m). ‘Ihe parameter set for the approach to the new equilibrium release rates is as follows:
.
fir = 1.90 X 10” Table 1 gives the three different release rates as a function of time. The release rate for 1331reaches its equilibrium value after only a few days whereas it takes of the order of 40 days for “‘Xe to attain its steady state release rate. The major aspect of the table is that there is a considerable discrepancy between Rxe and R&. ‘Ihis highlights the importance of including the effects of the iodine procursor in the analysis. Precursor effects are more pronounced in
me3 s-l ,
D, = 5.76 X 1O-22 m2
s-l ,
Al = 9.21 X lo+
s-l ,
p1 = 2.28 X 10”
mm3 s-l ,
D2 = 6.91 X 1O-22 m2 A2 = 1.52 X lO-‘j s-l , h=Dz/D,
= 1.20.
s-l ,
9
N. Beatham /Solutions to equations in low-temperature gas atom diffusion
The appropriate equation for the release rate, RI, of l 33I from the fuel is eq. (30): RI = 0s
/Xi ) U% erf(&r
0)
+ ehi(h-l)t f.I,&[ 1 - erf(&
hf))] ) .
For ’ 33Xe, the release rate, R Xe, from the fuel is given by eq. (42): Rxe = &
Ws
/Ai ) (Pz erf(&
t ehi(h-l)t /I,*[1
0)
- erf(\l(hX, t))]}
- T/(& /As) Co, erf(A t eAs@-l)’ &fi[l
0)
- erf(&rh2t))]}]
.
&in if one had neglected the presence of the iodine precursor then the release rate, Rie, of 13’Xe would have been calculated from the equation:
Rk, = &D2 /X2) cP2 eNO2 t eh2@-1)t
& &[
tzqacjax),=,
Time (days)
RI
Rxe
RZGe
0 0.02 0.10 0.25 0.50 0.75 1.00 1.50 2.00 4.00 7.00 10 20 3s so 100
1.80 1.81 1.83 1.85 1.88 1.90 1.92 1.94 1.95 1.97 1.98 1.98 1.98 1.98 1.98 1.98
3.16 3.15 3.14 3.13 3.12 3.12 3.12 3.13 3.15 3.23 3.32 3.31 3.44 3.46 3.46 3.46
4.44 4.44 4.45 4.41 4.49 4.51 4.52 4.56 4.58 4.61 4.15 4.19 4.85 4.86 4.86 4.86
0)
1 - erf(~(/rX2t))]}] .
‘Ihe variations with time of the three release rates are presented in table 2. The first point to mention is that as soon as the fuel rating is increased, the release rates immediately become a factor of 1.2 higher. This is because although the concentration profile is still the same at the point of the uprating, the flux from the fuel is given by J =
Table 2 Values for the release rates from the fuel after an uprating from 10 to lZ/W/gU (f + m)
,
and the diffusion coefficient is immediately increased by a factor of 1.2. Again it is discernible that there is a considerable discrepancy in both the absolute values and in the time dependence of the release rates Rxe and R&, which underlines the importance of allowing for the effects of precursors with a significant half-life. The most interesting feature of the table is however the decrease in Rxe during the first day at the new rating level. ‘Ihe reason for this can be traced to the exponential terms eA1(r’-l)t and eh2(h-1)t in eq. (42). For very small times it may readily be shown that the change in release rate is proportional to (h - 1) (6-A). As h is greater than 1 and x1 > A2, then there is a decrease in the release rate for very small times. The physical basis behind this decrease is as follows. Immediately after the uprating,
Release rates in units of lo9 mm2 s-l. RI refers to the release of ’ 331from the fuel. Rxe refers to the release of ’ 33Xe from the fuel. R k, is the calculated release of ’ 3 ’ Xe from the fuel if the presence of the iodine precursor is neglected.
the fuel has an increased diffusion coefficient but still retains the concentration profile from the previous rating level. The increased diffusion coefficient will tend to reduce the concentration gradient at the surface, until the effects of the increased generation rate of 13’Xe at the new rating level can be transmitted through the iodine precursor.
4. Conclusions The analytical solutions of the more important diffusion equations for the release of unstable atoms from a semi-infinite solid have been presented in the previous section. Although the expressions for the concentration profiles of the unstable atoms within the bulk may be slightly complex, those for the total release rate’ from the fuel into the pin voidage are relatively simple. These may be readily employed in codes modelling the subsequent leakage of unstable volatile isotopes from a failed fuel pm into the reactor coolant.
10
N. Beatham / Solutions to equations in low-tempemture gas atom diffusion
The solutions cover the important cases of reactor start-up and shutdown in addition to the approach to equilibrium at a new reactor power level. Allowance has been made in the analysis for the fact that the diffusion coefficient will not necessarily remain the same at the new power level although it is assumed that the surface to volume ratio of the fuel remains constant. ‘Ihis is particularly the case for the release of unstable atoms from low rated UOz fuel pins where the operating temperatures are sufficiently low (T < 1000°C) to establish a temperature-independent but ftion rate sensitive process as the dominant mode of fission product release from the fuel. The diffusion coefficient in this case is directly proportional to rating.
Acknowledgement ‘Ihis paper is published by permission of the United Kingdom Atomic Energy Authority.
References [l] H.S. Car&w and J.C. Jaeger, Conduction Solids, 2nd ed. (OUP, 1959).
of Heat in
A COMPILATLONOF THE SOLUTIONS TO THE IMPORTANT EQUATIONS IN LOW-TEMPERATURE GAS ATOM DIFFUSION N. BEATHAM UKAEA, Wndorrk Nuclear Pow
Development L&oratories, Sellajikld, Ssn~le,
Chmbrkr CA20 IPF, UK
Received 2 February 1981
This paper collects together the solutions for the diffusion of unstable gas atoms within a ~rni~~te solid. Allowance is made for one precursor. These solutions are appropriate to the diffusion of unstable gas atoms from inside the fuel into the pin voidage in situations where the radius of curvature of the surface can be neglected. The solutions are particularly relevant to low-temperature gasatom diffusion and are thus of considerable value to coolant analysis at commercial AGR stations.
1. Introduction
of reactor start-up and shutdown in addition to the approach to equilibrium at a new power level.
For low-rated UOa fuelled pins, the operating fuel temperatures are sufficiently low (T< 1000°C) to exclude everywhere the fo~ation of ~ter~~~ar porosity and to establish a temperature-~de~ndent but fmion-rate sensitive process as the dominant mode of fission product release from the fuel. The diffusion coefficient is small and this implies that diffusion depths’ are also small, compared to the reciprocal of the surface-to-volume ratio of the fuel. As a result, the radius of curvature of the surface can be neglected and the appropriate diffusion equations to use are those for a semi-infinite solid. This paper collects together the* timedependent solutions for the most relevant of the diffusion equations, and includes allowance for one precursor. It is assumed that the surface-to-volume ratio of the fuel remains constant during the course of the irradiation, which should be E good approximation for low temperature release. The most important application of this work is to the release from the fuel into the pm voidage of short-lived fission products such as 13rI. The solutions for these unstable isotopes are required in the analysis of radioactive release into the coolant from low-temperature pins. ‘Ihe solutions described here cover the situations 0 022-311 S/8 1~~~~0~$02.50
0 North-Holland
2. The diffusional release of unatabie gas atoms from within a semi-infmite solid 2.1. No precursor - the approach to equilibrium from initialstart-up
The diffusion equation within a semi-infiite solid for a single unstable isotope generated by fission is
ac _D a2c ,,+fl+,
arwhere
c is the concentration (me3), D is the diffusion
coefficient (m2 s-l), /3 is the generation rate per unit volume (mq3 s-r), X is the decay rate (s-r), t is the time (s), x is the distance into the fuel from the fuel/pm voidage interface (m). The equation lowing boundary c=Oatx=O
is required to be solved with the folconditions: and
c=Owhent=O.
3
N. Beatham /Solutions to equationsin low-temperature gasatomdiffusion
As is customary with these types of equation, it is expedient to apply the substitution C=ueAt. &A _&+pe”‘,
(2)
at
which is the corresponding solution for the diffusion of stable species. The most important property to be derived from eq. (4) is the flux J (m” s-l) of unstable atoms released from the fuel into the pin voidage. This is computed from the relationship J = tD@clax),,
with u=Oatx=O
and
u=Owhent=O.
Eq. (2) can be readily solved by application of the
Laplace tr~sfo~tion is
[If . The appropriate solution
.
The positive sign in eq. (7) arises from the fact that we have defined the flux J as that coming from the fuel into the pm voidage. The flux proves to be J = ~(D~~) erf(~~t))
-
(3)
which implies that
Although these solutions are appropriate to the case of the semi-infinite solid, they essentially apply to situations where the radius of curvature of the fuel surface can be neglected. The total release rate R (s-l) of atoms leaving the fuel is thus given by R = ST = S&I(D/X) erf(&r))
(4)
+ =$/@/A)
c=Oatx=O
since B = @V.
c=Owhent=O.
There are two very important limiting cases for eq. (4). The first is that as t + 00, the steady state concentration profile is obtained: e =;
[1 _ e-x&D’]
,
(5)
‘Ike second is that when X is set to zero, i.e. the isotope is stable, eq. (4) decomposes to yield
(6)
,
(9)
where S is the surface area of the fuel (m2). If Y is the volume of the fuel (m3), then the ratio of release rate to birth rate is given by
It may be readily shown that this solution does satisfy the respective boundary conditions and
@s)
The two important limiting cases of this equation are: (i) when t + 00, and the flux is then simply M(D/A) m-’ s-l, (ii) when X+ 0, and the flux becomes 2fl(Dt/n)1’2 mW2s-l.
erfc
+ exJ@fP) erfc
(7)
erf(&t))
(10)
For stable atoms, it is customary to express the release in terms of the fraction f of those generated: f= JR 0
dt’/VPt = ;+(Dt/n)
.
01)
2.2. Srirgle precutsor - the uppmach to equilib~um from initiai start-up The majority of unstable atoms have a precursor with an appreciable half-life and it is important that this should be included in the analysis. The appropriate pair of diffusion equations are as
4
N. Beatham /Solutions to equations in low-temperaturegas atom diffusion
follows:
acl -=
-a26 a2 +P-xlcl ,
(12)
a2c2 -t hlC1- xzcz, ax2
(13)
D
at
An expression for c2 can thus be constructed from eqs. (3), (14) and (16). The release rate R2 (s-l) of the daughter product from the fuel in to the pin voidage is given by
and k2
stzD
where cl is the concentration of the precursor (m-j), hr is the decay rate of the precursor (s-l), c2 is the concentration of the daughter product (mm3), X2is the decay rate of the daughter product (s-l). l’he corresponding boundary conditions are: cl =Oandc2 =O
atx=O,
cl =Oandc,
whent=O.
=O
Eq. (12) of course has already been solved in the previous section. As regards eq. (13), it is expedient to apply the substitutions
X
h/W&
) erf(d(b
0)
-
~4%~
1 erf(d2
O)l . (17)
If one wishes to calculate the amount of daughter product present in the pin voidage at any moment in time, it must be remembered that there will be two contributions to the generation rate of the daughter product in the pin voidage. The first is due to direct release of the isotope from the fuel and the second is from the decay of precursor released to the pin voidage. 2.3. No precursor - the approach to a new equilibri-
Cl=uedhlt ,
um level
and
The cases that have been studied in the last two sections represent the approach to steady state equilibrium from an initial zero concentration profde. A very important situation is when equilibrium has been reached at one fission rate and then the fuel rating is altered to a new level. In the first example, we shah take the case for no precursor and assume that the diffusion coefficient is independent of fission rating. The relevant equation is
c2
=&
[X,u eshlt + A2 u e-A2tl,
(14)
which results in a new form of eq. (13):
au _=D?k -i!!!&t at
ax2 x2
’
(15)
with u=Oatx=O
and
u=Owhent=O.
This equation is of the same type as eq. (2) and the solution can be immediately written down:
ac
z=D$tf12
-AC,
where p? is the new generation rate (me3 s-l) with the boundary conditions c=o
atx=O,
c =+ ~1_ e-xJGVo)l
whent=O.
or is the old generation rate (m-j s-l) and this initial concentration profile represents the equilibrium profile for the old generation rate.
5
N. Beatham /Solutions to equations in low-temperature gas atom diffusion
01
If we apply the substitution p=C-$
-x&W) I,
[l-e
-X&PI
c=T[l-e (20)
‘1
whent=O.
D, is the old diffusion coefficient (m2 s-l).
The first step in the analysis is to make the standard substitution c = uemA’.This results in the transformed equation
then eq. (19) is transformed into
(21)
au
z=D2$t&e"',
with p=Oatx=O
and
with the boundary conditions
p=Owhent=O.
This equation is also of the same type as eq. (2) and hence WWx),=c
= 0% - A ) L&W) erf(&W)
Therefore, the release rate R (s-l) becomes
(25)
.
u=o
atx=O,
u =F [1 _ e-d@/Dr)]
whenr=O.
(22)
from the fuel
‘Ihis equation may be solved by the standard Laplace transform technique but the solution may be readily obtained by inspecting the function
W=Y--
Ml u + t pz
~1 _ e-xJW’r)l
cur ,
(26)
where h is the ratio D2/D1, and y is the solution of = S d@/Q[PI + Go,-
PI) erf(dW)l .
(23)
au=D2 ~+&e”‘, a5 with
@r&/W
y=Oatx=O
,
(27)
at
‘Ihis equation shows how the release rate starts at the old steady-state equilibrium level of
and y=Owhent=O,
and approaches the new equilibrium level of
and u is the solution of
W&VU
2 -=av D2 st&e”‘, at
.
The equation automatically includes the situation of reactor shutdown for in this case & = 0. For the second example we shall take the more general case for no precursor, where the diffusion coefficient at the new ftion rating is different from the previous one. This is particularly appropriate for the athermal diffusion process where the diffusion coefficient is considered to be directly proportional to fission rating. The relevant equation is & -=
at
with
u=Oatx=O
and
u=Owhent=O.
It may easily be proved that
a2w D2 -ax2 +&$‘-*=o
at
w=o
atx=O,
’
with the boundary conditions atx=O
D2 $3+c,
where D2 is the new diffusion coefficient (m2 s-l) with the boundary conditions c=O
(28)
w =$
[l _ e--XJ@Pr)]
when t = 0 ,
which implies that u = w is the solution of eq. (25). The solution to eq. (24) is thus
N. Beatham / Solutions to equations in low-temperature gasatomdiffusion
6
c2
+t
[l _ e-&4)]
=W(D2/MP2
--e"1 pz
[l _
e-xJG2/D)]
+ &
(29)
eW-lP.
The release rate R (s-l)
=$
[e-XJ@IIQ
_
e-d@2/D)]
when t = 0 .
from the fuel is given by
Both of the initial concentration profiles represent the equilibrium profiles for the old generation rate, The first equation (3 1) has already been solved in the previous section. If we apply the substitutions
erf(d(W)
"(h-1)tSt/(D2/hA))2
erf(&ht)) p1
= cl
$ [l
_
_
e-d@l/D)] ,
(33)
1
+ eh(*-l)t
or&[1
- erf(&Irr))]}
.
(30)
:
P2 = c2 -
[1 _ e_XJ62/D)]
&
-
[e--d@l/D)_ e-xJ(h/D)] ,
(34)
2.4. Single precursor - the approach to a new equilibrium level
then eq. (32) is simplified to yield
a2p2 ap,_ --+bPl D
‘Ihe first case to be analysed in this section repre-
sents the situation where equilibrium has been reached at one fission rate and then the fuel rating is altered to a new level. The analysis includes allowance for a single precursor but assumes that the diffusion coefficient is independent of rating. ‘Ihe relevant pair of differential equations is as follows:
ac, a2cl - =Dtp2 -h1c* at ax2
,
(31)
and p2 = 0 when t = 0 ,
which is coupled to
8th
_D
at-
a%
--Alp, ax2
+pz
-01
9
when t = 0 ,
e--xJw~q
and p1 =Owhenr=O.
p1 =Oatx=O
and
ac2 _
at-
D
-a2c2+ h,c, ax2
- h2c2 ,
(32)
with c2 =o
(36)
‘Ihis pair of equations is exactly similar to eqs. (12) and (13) in section 2.2. As a result the release rate R2 (s-l) of the daughter product from the fuel is given by
atx=O,
Cl = 0 [1 _
p2 =Oatx=O
(35)
9
with
with
Cl =?
-h2P2
ax2
at-
$ atx=O,
(1 _
eexJR21D))
N. Beatham /Solutions to equations in low-temperature gapatom diffision lA=O
xJwD) _ e--xJW4
t-el,!A(-
u
=*{
&Vhi
-;(oh2N31
)[Pr + dB2
+ 032
-
Pd
-
011
PI 1 erf(\/(h
eWCX2t))ll
(37)
.
For the final example, we shall take the general case for a single precursor, where the diffusion coefficient at the new fusion rating is different from the previous one. l’he relevant coupled equations are
acl
a%
_D
at-
atx=O,
)11,=o
2
2--~1c1 ax2
to2 ,
(38)
atx=O,
C, =$ ~1_ e_XJ(“~/D1)]
whent=O,
[1
_
e-xJ@~IDI)l
whent=O,
and av,D2
at
&eAZ',
E$_”
(41)
x2
with U=O
atx=O,
~=d$!~~-e-~JC’W4)]~
whent=O.
Both eq. (41) and eq. (40) are similar to eq. (25) which was solved in the previous section. ‘Ihe release rate R2 (s-r) of the daughter product from the fuel is thus
with cr = 0
=t
7
and
a2c2 2-+ax2 hct
ac2
_D
Z-
-
A2c2
(39)
9
= &[A1
(g)x;,
eShlt + A2(:),=,
eeA2’]
with c2
=o
c2 =$
atx=O, t eAl(h-l)t g,fi[
(1 _ e-XJ(A1ID1))
-
4(D2 /X2 ) h
1 - erf(d(hX1 erf(&
t eA2@--l)' fl,fi[1
when r=O
= ueeA1
2))
- erf(&As
t))] 11 .
(42)
lhis solution covers the most general case of the examples analysed in this report. All the other solutions may be derived from eq. (42) by the appropriate simplification.
.
If we apply the standard substitutions Cl
t))l 1
f ,
and 3. Illustrativeexample ,UeShrt t X2rje-Ast] [A to these equations, then
au at
-_=D
with
a2u
-t
2 ax2
&e”l’,
(40)
‘Ihe following example is included to show how the solutions may be applied to a typical reactor situation. The case in question concerns the release rate from the fuel of ls31 and ‘j3Xe during: (i) reactor startup to a fuel rating of 10 W/gU (fuel t moderator); and
N. Beatham /Solutions to equationsin low-temperature gasatomdiffusion
8
(ii) a subsequent uprating to 12 W/gU (f t m) after equilibrium has been reached at 10 W/gU.
Table 1 Valuesfor the release rates from the fuel after reactor start-up to a rating of 10 W/gU (f + m)
3.1. Reactor start-up
Time
If one assumes a value of 204 MeV for the average energy generated per fission and a fuel density of 10.67 t me’, then a rating of x W/gU (f t m) is equivalent to a fusion rate f of 2.88 X 10”~ mm3s-l. ‘Ihe approximation is adopted that all ‘j31 is generated directly from fission with a fractional yield of 0.066 and that ‘33Xe is produced only from the decay of 1331. An appropriate choice for the athermal diffusion coefficient associated with low-temperature release is 2 X 104’fm2 s-l. It is assumed that both isotopes possess the same diffusion coefficient. lhe parameter set for the approach to equilibrium at a rating of 10 W/gU (f t m) is as follows: fll = 1.90 X 10” mm3 s-l , Dl = 5.76 X lo-” Xl = 9.21 X 10T6 s-l ,
m2 s-l
A2 =1.52x10-6s-‘.
‘Ihe appropriate equation for the release rate, RI (me2 s-l), of 13jI from the fuel is eq. (9): RI =&&DA)
erf(&t))
RI
RXe
Ge
0 0.21 0.41 0.71 0.94 1.09 1.19 1.32 1.39 1.49 1.50 1.50 1.50 1.50 1.50 1.50
0 0.001 0.01 0.05 0.12 0.21 0.31 0.50 0.69 1.30 1.86 2.17 2.53 2.62 2.63 2.63
0 0.21 0.48 0.75 1.05 1.27 1.45 1 .I4 1.97 2.57 3.05 3.31 3.62 3.69 3.70 3.10
(days) 0 0.02 0.10 0.25 0.50 0.75 1 .oo 1.50 2.00 4.00 7.00 10 20 35 50 . 100
Release rates in units of lo9 mB2 s-l.
RI refers to the release of ’ 331 from the fuel. Rre refers to the release of ’ 33Xe from the fuel. RXe is the calculated release of ’ 33Xe from the fuel if the presence of the iodine precursor is neglected.
.
All release rates quoted in this section are per unit surface area. For 133Xe, the release rate, Rxe, from
the fuelis given by eq. (17):
diffusion theory because the decay rate of the precursor enters the equations in the form of the square root. 3.2. The approach to a new equilibrium level
RJk = &
[d(Di /Ai ) erf(&
- d(D 1/A2 ) erf(d&
0)
t))l -
However, if one had neglected the presence of the iodine precursor, then the release rate, R&, of rJ3Xe would have been calculated from the equation: %
= Prd(Dr/&)
erf(&st))
For this case it is assumed that the fuel has been operated at 10 W/gU (f t m) for sufficient time for the 1331and “‘Xe release rates to have attained their equilibrium values. The fuel is then uprated to 12 W/gU (f t m). ‘Ihe parameter set for the approach to the new equilibrium release rates is as follows:
.
fir = 1.90 X 10” Table 1 gives the three different release rates as a function of time. The release rate for 1331reaches its equilibrium value after only a few days whereas it takes of the order of 40 days for “‘Xe to attain its steady state release rate. The major aspect of the table is that there is a considerable discrepancy between Rxe and R&. ‘Ihis highlights the importance of including the effects of the iodine procursor in the analysis. Precursor effects are more pronounced in
me3 s-l ,
D, = 5.76 X 1O-22 m2
s-l ,
Al = 9.21 X lo+
s-l ,
p1 = 2.28 X 10”
mm3 s-l ,
D2 = 6.91 X 1O-22 m2 A2 = 1.52 X lO-‘j s-l , h=Dz/D,
= 1.20.
s-l ,
9
N. Beatham /Solutions to equations in low-temperature gas atom diffusion
The appropriate equation for the release rate, RI, of l 33I from the fuel is eq. (30): RI = 0s
/Xi ) U% erf(&r
0)
+ ehi(h-l)t f.I,&[ 1 - erf(&
hf))] ) .
For ’ 33Xe, the release rate, R Xe, from the fuel is given by eq. (42): Rxe = &
Ws
/Ai ) (Pz erf(&
t ehi(h-l)t /I,*[1
0)
- erf(\l(hX, t))]}
- T/(& /As) Co, erf(A t eAs@-l)’ &fi[l
0)
- erf(&rh2t))]}]
.
&in if one had neglected the presence of the iodine precursor then the release rate, Rie, of 13’Xe would have been calculated from the equation:
Rk, = &D2 /X2) cP2 eNO2 t eh2@-1)t
& &[
tzqacjax),=,
Time (days)
RI
Rxe
RZGe
0 0.02 0.10 0.25 0.50 0.75 1.00 1.50 2.00 4.00 7.00 10 20 3s so 100
1.80 1.81 1.83 1.85 1.88 1.90 1.92 1.94 1.95 1.97 1.98 1.98 1.98 1.98 1.98 1.98
3.16 3.15 3.14 3.13 3.12 3.12 3.12 3.13 3.15 3.23 3.32 3.31 3.44 3.46 3.46 3.46
4.44 4.44 4.45 4.41 4.49 4.51 4.52 4.56 4.58 4.61 4.15 4.19 4.85 4.86 4.86 4.86
0)
1 - erf(~(/rX2t))]}] .
‘Ihe variations with time of the three release rates are presented in table 2. The first point to mention is that as soon as the fuel rating is increased, the release rates immediately become a factor of 1.2 higher. This is because although the concentration profile is still the same at the point of the uprating, the flux from the fuel is given by J =
Table 2 Values for the release rates from the fuel after an uprating from 10 to lZ/W/gU (f + m)
,
and the diffusion coefficient is immediately increased by a factor of 1.2. Again it is discernible that there is a considerable discrepancy in both the absolute values and in the time dependence of the release rates Rxe and R&, which underlines the importance of allowing for the effects of precursors with a significant half-life. The most interesting feature of the table is however the decrease in Rxe during the first day at the new rating level. ‘Ihe reason for this can be traced to the exponential terms eA1(r’-l)t and eh2(h-1)t in eq. (42). For very small times it may readily be shown that the change in release rate is proportional to (h - 1) (6-A). As h is greater than 1 and x1 > A2, then there is a decrease in the release rate for very small times. The physical basis behind this decrease is as follows. Immediately after the uprating,
Release rates in units of lo9 mm2 s-l. RI refers to the release of ’ 331from the fuel. Rxe refers to the release of ’ 33Xe from the fuel. R k, is the calculated release of ’ 3 ’ Xe from the fuel if the presence of the iodine precursor is neglected.
the fuel has an increased diffusion coefficient but still retains the concentration profile from the previous rating level. The increased diffusion coefficient will tend to reduce the concentration gradient at the surface, until the effects of the increased generation rate of 13’Xe at the new rating level can be transmitted through the iodine precursor.
4. Conclusions The analytical solutions of the more important diffusion equations for the release of unstable atoms from a semi-infinite solid have been presented in the previous section. Although the expressions for the concentration profiles of the unstable atoms within the bulk may be slightly complex, those for the total release rate’ from the fuel into the pin voidage are relatively simple. These may be readily employed in codes modelling the subsequent leakage of unstable volatile isotopes from a failed fuel pm into the reactor coolant.
10
N. Beatham / Solutions to equations in low-tempemture gas atom diffusion
The solutions cover the important cases of reactor start-up and shutdown in addition to the approach to equilibrium at a new reactor power level. Allowance has been made in the analysis for the fact that the diffusion coefficient will not necessarily remain the same at the new power level although it is assumed that the surface to volume ratio of the fuel remains constant. ‘Ihis is particularly the case for the release of unstable atoms from low rated UOz fuel pins where the operating temperatures are sufficiently low (T < 1000°C) to establish a temperature-independent but ftion rate sensitive process as the dominant mode of fission product release from the fuel. The diffusion coefficient in this case is directly proportional to rating.
Acknowledgement ‘Ihis paper is published by permission of the United Kingdom Atomic Energy Authority.
References [l] H.S. Car&w and J.C. Jaeger, Conduction Solids, 2nd ed. (OUP, 1959).
of Heat in