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Diffusion theory of fission gas migration in irradiated nuclear fuel UO2
Journal of Nuclear Materials 135 (1985) 140-148 North-Holland. Amsterdam
140
DIFFUSION THEORY FUEL UOz
OF FISSION GAS MIGRATION
IN IRRADIATED
NUCLEAR
K. FORSBERG and A.R. MASSIH AB ASEA-ATOM,
Box 53, S-721 04 Vasteras, Sweden
Received 7 February 1985; accepted 4 May 1985
The migration of stable fission product-gases in irradiated nuclear fuel (UO,) is studied by considering the diffusing equation for the equivalent sphere model. The boundary condition (b.c.) was constructed partially following Turnbull. However, unlike Turnbull, our formulation allows the concentration of gas atoms at the grain boundary, N(r), to vary with time; thus making the b.c. time dependent. A general formalism for treating this type of differential equation with time dependent source term and diffusion constant, subject to a time varying b.c. is presented and a formal solution on grain boundary is derived. An integro-differential equation for the density of gas particles is derived rigorously. Analytic expressions for N(r) are derived for short and long times in the regime where the ratios, /3(r)/D(t) and b(t)/D(t) are constants in time (B, D and b are production rate, diffusion constant and resolution parameter respectively). Finally the derived formal solution is treated numerically to obtain the rate of the accumulation of gas atoms at the grain boundary. The saturation of grain boundary is used as the criterion for release.
1. Introduction Despite extensive developments in the calculational techniques of the diffusion theory of fission product gas migration in irradiated nuclear fuel such as UO*, quantitative success has been achieved chiefly in the quasistationary regime and in the domain where the physical parameters, governing the rate of diffusion, are non-time-varying. In the past few years a number of investigators have treated the situation where gas atoms diffuse through a spherical grain of irradiated UO, under time varying conditions. These have been the circumstances where both the production rate of gas atoms and the gas diffusion coefficient are time dependent [1,2]. However, these works were confined to the case of a perfect sink boundary condition (b.c.) - the standard Booth model- whereby the gas atoms are released upon arrival at the grain boundaries [3]. In this paper we wish to extend the formalism and calculations developed and presented earlier by us [2] (and referred to as paper I) to the situation where the grain boundary is no longer a perfect sink. The perfect sink assumption does not conform with
scanning electron microscopy observations; these show rather that the gas atoms accumulate continuously at grain boundaries, i.e., precipitate into intergranular bubbles. These bubbles grow, overlap, coalesce, and eventually release the gas they contain [4]. Also, it has long been realized that the as-fabribated porosity of UO, and the clusters of defects produced by irradiation act as static traps for the diffusing gas particles, giving rise to the formation of intragranular bubbles. Moreover, the recognition that the precipitated gas atoms- both in inter- and intragranular bubbles-can be dispersed by fission fragments throughout the grain matrix, irradiation induced resolution, makes the application of the standard Booth model to the real situation a strong simplification. The first steady state theory of fission gas migration, extending the work of Booth to include the precipitation and resolution effects was given by Speight [5]. This theory assumes that the gas diffusion coefficient, production rate and resolution rate are time independent parameters. For quasi-equilibrium between the flow of gas into intragranular trai>s and the resolution of gas from the traps, Speight’s model shows that the gas diffusion coefficient, Din, modifies
0022-3 115/85/$3.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
141
K. Forderg, A.R. MassihlDifusion theory of fision gas according D
_
to
b’& (b’ + g) ’
where b’ is the resolution probability from intragranular traps (s-l), and g is the capture rate (s-l) for the migrating gas particles. For spherical traps (Smoluchowski [6J) g = 4 ?rD,J?ii,
(2)
where fi and J? are the average concentration and radius of the intragranular bubbles, respectively. Further, Speight [5] proposed an expression for the rate of accumulation of gas particles at a grain boundary for a given temperature, i.e.
(3) where N is the number of gas particles per unit area of grain on the grain boundary (m-‘), b the resolution rate from intergranular bubbles (s-l), A the resolution layer depth from grain boundary (m), p the gas production rate (s-l) and f0 is the Booth flux [3]. For a grain of radius “a”,
(4) The gas arriving at grain boundaries, with the rate given by eq. (3), eventually saturates the grain boundaries through a network of interconnected bubbles. If the ideal gas equation of state is assumed, the density (per area) of gas particles over the whole grain boundary at saturation may be determined by [7]
(5) where y is the surface tension of the bubble, r its radius, k, the Boltzmann constant, T the temperature and f(0) takes into account the nonsphericality of the bubbles on grain boundaries. For a spherical bubble 0 = 90” and f(90) = 1. P,,, is any externally applied hydrostatic pressure and V, is the fractional coverage of grain boundary at saturation. The release is assumed to occur when N reaches the value N,, the saturation density. Eqs. (3) and (4) have been used as a basis for fission gas release prediction in several computer codes [g-lo].
Some years ago Turnbull [ 1 l] suggested that if one replaces the perfect sink b.c. by an imperfect one, the Booth model may be extended to treat the resolution effect. Specifically, he set the concentration of gas particles at the grain surface equal to: bhN/2D which he assumed to be a value constant in time - and solved the diffusion equation for gas concentration in a spherical grain of radius “a”, obtaining the release fraction. In this paper we attempt to extend Turnbull’s formulation to a time varying situation. More precisely we consider,
ack 4 -
at
= D(t)A,C(r,
t) + P(t)
with A =??,?? ’ ar2-rar subject to C( r, 0) = 0, (7)
C(a, t) = (b(t)AN(t)/2D(r)),
where C(r, t) is the concentration of gas atoms and it is assumed that gas generated uniformly throughout a grain of radius “a”. In. section 2, we give a general formalism for treating the differential equation (6) with a time dependent boundary condition, eq. (7). This formalism allows us to readily use and extend the numerical scheme, presented in our paper I, to the present situation. Further, from this formalism we are able to find rigorously the corresponding generalization to eq. (3) in a form of integro-differential equation. In section 3 we present the short and long time limit of N(r) for a semi-steady-state condition. In section 4 we shall apply the results obtained in section 2 to calculate the accumulation of gas on grain boundaries and release to downward and upward cascading power histories.
2. General equations of dihsion The diffusion equation more convenient form
and resolution
(6) can be resealed
to a
142 am ~
K. For&erg, A. R. ~sih/~i~ion 7) at
= A,C(r, 7) + P,(T)
theory of jission gas
(8)
(13
with
with
C( t, 0) = 0,
C&a, 7) = 0.
(367)
(9)
= h,(dN(W,
Now eq. (11) can be invoked (16) as
(16) to write eqs. (15) and
where h, = b(~)A/~(A), PC= P(r)lD(r) and Combining T=
‘D(&,)d$.
eqs. (13), (14) and (17) we have
(IO)
Jf,
It was shown earlier that for the case of a perfect sink boundary condition, i.e., C(a, 7) = 0, eq.’ (8) may be transformed to an integral equation of the form [2]
(18) where
‘47rr’C(r,
J0
r)dr=
J0TK(T--7o)Pe(dds
where the kernel K(r-
(11)
71,)is given by
(12) To treat eqs. (8) and (9) we assume that a certain quantity of gas N(r) has been accumuiated on the grain boundary at some time I, and N(T) < N, where N, is the amount of gas required to saturate the grain boundary or N(r),,,,, = N,. Thus we have the identity: 4rra2Pl(r) = 2
(
47ra' 3
r
J Pe(rtJd~,--4P
J
r’C(r,
7)dr
(19) The integral equation (18) is the basic equation used in this note to determine the time dependence of the gas particle density on the grain boundary prior to the release. Further examination of eq. (18) yields to an interesting result. Specifically, under the realistic assumption that /3/b is constant in time, the time derivative of eq. (18) gives
1,
0
(13) where the factor of 2 on the right hand side accounts for release into the boundary from two adjacent grains. Setting C,,(r, 7) = ctr, 7) - eta,
7)
eq. (8) for C,,(r, T) becomes
(14)
The first term on the right hand side of eq. (20) corresponds to the Speight equation (3) in r space, i.e. for the case where D and J3 are time dependent. For example it can be seen than when p and D are constants Kz=
folP,
(21)
K. Forsberg, A.R. Massih/DifJusion theory of fission gas
where f,) is the flux given by eq. (4). Applying consideration to eq. (20) yields
this
where (see the Appendix)
R,=a’
coth (u’s)“’
1
--
((1*3)“*
Substituting
The first term in eq. (22) is identical to eq. (3), and the second term vanishes if N(t) is assumed to be a linear function of f. This result is noteworthy, as Speight’s derivation of eq. (3) was based purely on plausibility arguments [5] whereas eq. (20) is an exact mathematical result based on solving the diffusion equation with the appropriate boundary condition. The difference between ours and Speight’s is that Speight assumed a constant /3 and a linearly time dependent N(t). This latter assumption, however, is approximately true when the resolution effect becomes predominant. Then N(t) evolves, approximately, linearly with time.
143
(a’s)’ >.
(25)
eq. (25) into eq. (24) yields
For short times, large s, eqs. (A8) and (26) can be utilized to write [12] N(T)=?(T+& - h2 exp ( h: 7) erfc( h, T”‘) + h, exp (ht 7) erfc ( - h2T”2) h, h,(h, + h,) + o(T”), (27)
where l/2 h,=_++
!!j+;
,
(
3. Analyticsolutions
l/2 h3++(;+2)
In this section we shall apply the integral equation derived in the last section, eq. (18), to the case where PC= p( t)/D( t) and h, = ‘Ab(t)/D( t) are constants of time. This assumption is not unrealistic, because when fuel experiences higher powers, it leads to higher production rate p, larger fission rate and higher temperatures, whereby larger D emerges, e.g., if Arrhenius law is envisaged. We shall present here a detailed computation of this steady state case, where we obtain an analytical expression for N(t) in short and long times limits. We start by taking the Laplace transform in time of eq. (18)
where in deriving eq. (23) we recalled: h,(~)N(7)/2 and used the notation:
)
C(a, T) =
,
and _ erfc (i) = ?G, I
e-l2 dt.
The reader may check (by series expansion of the exponential and error functions) that in the absence of resolution, h, = 0, eq. (27) yields the Booth solution, i.e.
Eq. (27) provides a working equation for determining the density of gas atoms on a spherical grain boundary of the fuel material. The integration of eq. (26) can be carried out by residue calculation to give the long time limit for N(T)
e-“‘f( 7) dT. N(T)=-
If
/$(T)
and
h1(7)
ti= 2/3,Ez,Is(l+
Wea
are constants we have d&h,),
m
(24)
+c
(
T-
a*
S(3 + h, a)
4pca3
e-wa)*r
??I=, u;[ u’, + ah,(3 + ah,)]
(28)
K. For&erg, A.R. MassihlDifusion
144
theory of fission gas
Adding eqs. (31) and (34)
with u/arctan(*)+rnn.
(29)
GO(T)+[I + ~&(T)]Gs(T) The first term in eq. (29) suffices to determine the long time behavior of N(7). Again in the absence of resolution eqs. (28) and (29) yield the Booth solution for long times, i.e.
It is interesting to compare eqs. (27) and (28) with the analytical solution of the phenomenological equation (3) obtained by Dowling, White and Tucker [7]. 4. Numerical
treatment
A numerical method for applying eq. (18) to obtain the time evolution of gas atoms within grain boundaries is presented in this section. Sinoe the grain boundary area per unit volume of fuel is 3/2 a, where a is the radius of a spherical grain, the number of gas atoms per unit volume of fuel within the grain boundaries, GB, is G,, = 3N/2a.
(30)
Assuming that the ratio: b( T)/@(7) is time independent the integral equation (18) can be written in terms of G, as
Go
%)q(%) d%,
h, = abh/3/?.
K?(T) = ; &(T)
(36)
Eq. (35) may be interpreted as conservation of number of gas particles, i.e., the total number gas particles generated, the left hand side of eq. (35), is equal to the number of gas particles in the grain and on the grain boundary, the right hand side of eq. (35). For numerical computation we choose a sufficiently short time interval AT = TV - T, (> 0) during which the gas production rate,
I
r:’ 8,(T) d7
is estimated by:
(32)
@(‘I) + p(T2)
?2
pe(T)
dT
&
z
.
2
71
(37)
At is the difference in “real” time t corresponding to the interval AT. Exercising this assumption on eq. (35) results in
AC% + AG, + h4G~(&V% + h&(Tz)AGe
I
+ b(%)
At
(38)
.
2
(31)
where
(35)
with
= = 1,’ &(T--
= I’ &(%) d% 0
The reason for not approximating the integral in eq. (37) with ; &(T,) + &(T~)]AT is because this may give rise to numerical inaccuracies during start-up intervals, under which the diffusion coefficient, and subsequently &, vary within several orders of magnitude. As in paper I we will employ the approximation,
and l+&(T)=
9(T)= MT) -g
& [&(T)G~T)I.
(33)
i
n=,
&exp(
-$T),
(39)
where Similarily the number of gas atoms per unit volume of fuel within the grain, G,, can be obtained from eq. (17) G,,(T) = j ’ Cl - &(T - %)]q(%) d% 0
(34)
A, = 0.63003
B, = 9.9904,
A, = 0.2065 1
B2 = 64.488,
A, = 0.14776
B, = 511.61.
(40)
K. Forsbcrg, A. R. Massihl Diffusion theory of fissiongas
Applying form
eqs. (39) and (34) may be written
in the
determined
from eq. (35) i.e. through
GO(T)= i G”(T) n-l where
Finding q from eq. (45) it can be used in eqs. (43) and (44) to obtain G,,(T~) and GB(~2), respectively.
Similarly eq. (31) becomes:
Gs(T) = j’ q(%) d%- i G,,(T)* 0 “=I
(42)
By applying the numerical method presented in section 4 we compute here the rate of accumulation of gas atoms in the grain boundary bubbles. The density of gas contained in the fuel -grain, Ggrai,, can bk written as
Defining:
f.=exp(-$!h)-1. AG,, and AG, can be written, respectively
5. Saturation sod releaw
as
Gg,,i, = Go + hdB=GB,
(46)
where G,, h4 and GB were defined earlier in section 4. The density of gas within the grain boundaries at saturation is
AG = fnGn(d
-$ (72- d) q(Q)dTc,
(43) G. = & N,,
(47)
and
AG = -i
~,,G"(TJ “-1
where we employed the short time approximation for &(T~T,J,see the Appendix and eq. (35). Now the catch is that q in eqs. (43) and (44) depends on GB which is a complicated function of time, see eqs. (34) and (33). Even for the case under consideration where /3 is a linear function of f, q is a nonlinear function. To overcome this impediment we take q to be a constant within the time interval AT. Then q may be
where N, can be determined from the equation of state (5). The onset of gas saturation on the grain boundary may be explained by a percolation mechanism, where at a certain concentration of gas particles on the grain a network of interconnected bubbles sets in [13,14]. When G8(7J 2 G, the amount gas released per unit volume of fuel, &, may be written FR =
fG,
where 0 < f 5 1, and its upper bound corresponds to the condition that all the accumulated gas at grain boundaries would be released upon saturation. We have used eqs. (43) to (45) to obtain GB and eqs. (5) and (47) to calculate G,. The production rate used in this calculation [lo] is
146
K. Forsberg, A. R. MassihlDifision
06386 1+375E/W, where taken: atoms
>
W,, atoms/m”
s
W, is the rating (W/g) and E is the enrichment E = 0.03. The diffusion coefficients for gas are obtained from the following correlations
[lOI D=l.O9~10~“exp(-6614/T) D = 2.14 X lo-l3 exp (-22884/T)
D=l.SlXlO~“exp(-9508/T) Table 1 provides the numerical perature dependence of ratings
T > 1650 K, 1381< T < 1650 K, T < 1381 K. values for the temand the generation
theory of fission
gas
Table 1 Numerical values for ratings and generation rates Temperature (“C)
Linear rating Rating (kW/m) (W/g)
Generation rates (mol/m3 s)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
26.8
29.8
4.4x 10 (’
30.6 34.4 38.2 41.9 45.7 49.4 53.2 57.0 61.0 64.6
34.1 38.3 42.4 46.6 50.8 55.0 59.2 63.4 67.6 71.8
5.1 x 5.7 x 6.4 x 1.2 x 7.7 x 8.3 x 8.9 x 9.6 x 1.0x 1.1 x
lo-” lo- h lo-” 10-h 10-h lo-” lO-h 10-h 1o-5 1om5
In Eq.(5) a' = 0.6 J m-* cl=500 r = 0.5pm
ReleareMolalm3 -----
Pext =0 v, = 0.25 .1600
I ,160O 00 e a i -1400 g c
1.0 Exposure,Mwd/kgU
Fig. 1. Fraction of gas atoms on grain boundary, GS/GB, as a function of exposure for downward fuel cascading temperature history. y is the bubble surface tension, 28 is the angle where two free surfaces meet at a grain boundary, r is average bubble radius, V, is the fractional coverage of the grain boundaries at saturation and the grain radius is taken to be 5 pm.
147
K. For&erg, A.R. MassihlDiffusion theory of fission gas
In Eq. 5 $
= 0.6
J m-’
e = 500 r = 0.5pm P
ext
v,
=o
= 0.25
I
00
1400
5 [
+
1200
1000
800
I
I
2.0
I
4.0
3.0 Exposure,
MwdlkgU
Fig. 2. Fraction of gas atoms on grain boundary and amount of gas released per unit volume of grain-as a function of exposure
for upward cascading fuel temperature history. The input data is as in fig. 1.
rate. In fig 1 we have displayed the result of our calculation for accumulation of gas on grain boundary, the ratio: GJG,, for the case of downward cascading fuel temperature history. The other input parameters used in the calculation are indicated in the figure caption and legend. Fig. 2 depicts our calculation for an upward cascading fuel temperature history. The release is indicated by a broken line. We assumed 100% release upon saturation. An oscillatory nature of gas accumulation on the grain boundary after the release is noted. This, however, is not expected to have an overall effect on fuel behavior, since it is believed to take place locally; nevertheless, experimental effort to observe this phenomenon can be worthwhile.
denoting the eigenvalues eigenvalue equation:
A,J/,= -W,
A formal formed solution of eq. (8) for the case of a perfect sink b.c., i.e. ~(a, T) = 0 may be obtained by
(Al)
and C(r, T) can be expanded in terms of the complete set of the eigenfunction
c
ar, 4 = i=, k,(~M(r) with
I
R +i*j =
Appendix: Derivation of the kernel K2
of A, by (hi) which gives the
&j
where the integral is carried out over the volume of the grain a. In paper I, (ref. [2D we showed
K. Forsberg,A. R. iUassih/Di@ion
148
I
C(r, 7) =
R
Ir
NT - s)Be(dds,
W)
theory of fissiongas
Taking the inverse Laplace transforms of eq. (A8),
,I
K2( T) = 2(i)
where for a sphere of radius “a” the kernel K
“’ - a
short times
(AIO)
and for long times we simply set n = 1 in eq. (A6) (A3) Kz(T)+$e-““‘“’ In I, we also showed that approximation
(A4) We wish to thank Dr. 6. Bernander for his helpful comments on the manuscript. The work was supported in parts by the Swedish Nuclear Inspectorate under contract No. B24 184.
with A; = 2.6391
B ; = 9.9905,
A; = 0.865
B; = 64.488,
A; = 0.6189
B; = 511.61,
(As)
accurately describes K(T) in all the regions of interest. Now substitution of eq. (A3) in eq. (19) gives
646) which can easily be Laplace transformed
&(s) = a3 -(a*s)-*+
to s space
coth(a’s)“’ (a2s)3’2 > ’
(A7)
For large s (short times) & =
,q-3/2
_
a-l
(AI I)
Acknowledgements
A,/ e-%(‘)
K( 7) = i
long times.
s-2
and for small s (long times);
(A@
References [l] L. Vlth, J. Nucl. Mater. 99 (1981) 324 and refs. therein. [2] K. Forsbexg and A.R. Massih, J. Nucl. Mater. 127 (1985) 137. [3] A.H. Booth, Atomic Energy of Canada Limited Report No. 496 (1957). [4] M.O. Tucker, Radiat. Effects 53 (1980) 251. [5] M.V. Speight, Nucl. Sci. and Eng. 37 (1969) 180. [6] M. von Smoluchowski, Phys. Z. 17 (1916) 557; 585. [7] D.M. Dowling, R.J. White and M.O. Tucker, J. Nucl. Mater. 110 (1981) 37. [8] R. Hargreaves and D.A. Collins, J. Br. Nucl. Energy Sot. 15 (1976) 311. [9] H. Nerman, in: Water Reactor Fuel Element Performance Computer Modelling, ed., J. Gittus (Applied Science Publishers Ltd, London, 1983). [lo] R.J. White and M.O. Tucker, J. Nucl. Mater. 1 (1983) 1 118. [l l] J.A. Turnbull? J. Nucl. Mater. 50 (1974) $2. [12] A. Abramowitz and I. Stegun, A Handbook of Mathematical Functions (Dover, New York, 1954). [13] C. Ronchi, J. Nucl. Mater. 84 (1979) 55. [14] A.R. Massih, J. Nucl. Mater. 119 (1983) 116.
140
DIFFUSION THEORY FUEL UOz
OF FISSION GAS MIGRATION
IN IRRADIATED
NUCLEAR
K. FORSBERG and A.R. MASSIH AB ASEA-ATOM,
Box 53, S-721 04 Vasteras, Sweden
Received 7 February 1985; accepted 4 May 1985
The migration of stable fission product-gases in irradiated nuclear fuel (UO,) is studied by considering the diffusing equation for the equivalent sphere model. The boundary condition (b.c.) was constructed partially following Turnbull. However, unlike Turnbull, our formulation allows the concentration of gas atoms at the grain boundary, N(r), to vary with time; thus making the b.c. time dependent. A general formalism for treating this type of differential equation with time dependent source term and diffusion constant, subject to a time varying b.c. is presented and a formal solution on grain boundary is derived. An integro-differential equation for the density of gas particles is derived rigorously. Analytic expressions for N(r) are derived for short and long times in the regime where the ratios, /3(r)/D(t) and b(t)/D(t) are constants in time (B, D and b are production rate, diffusion constant and resolution parameter respectively). Finally the derived formal solution is treated numerically to obtain the rate of the accumulation of gas atoms at the grain boundary. The saturation of grain boundary is used as the criterion for release.
1. Introduction Despite extensive developments in the calculational techniques of the diffusion theory of fission product gas migration in irradiated nuclear fuel such as UO*, quantitative success has been achieved chiefly in the quasistationary regime and in the domain where the physical parameters, governing the rate of diffusion, are non-time-varying. In the past few years a number of investigators have treated the situation where gas atoms diffuse through a spherical grain of irradiated UO, under time varying conditions. These have been the circumstances where both the production rate of gas atoms and the gas diffusion coefficient are time dependent [1,2]. However, these works were confined to the case of a perfect sink boundary condition (b.c.) - the standard Booth model- whereby the gas atoms are released upon arrival at the grain boundaries [3]. In this paper we wish to extend the formalism and calculations developed and presented earlier by us [2] (and referred to as paper I) to the situation where the grain boundary is no longer a perfect sink. The perfect sink assumption does not conform with
scanning electron microscopy observations; these show rather that the gas atoms accumulate continuously at grain boundaries, i.e., precipitate into intergranular bubbles. These bubbles grow, overlap, coalesce, and eventually release the gas they contain [4]. Also, it has long been realized that the as-fabribated porosity of UO, and the clusters of defects produced by irradiation act as static traps for the diffusing gas particles, giving rise to the formation of intragranular bubbles. Moreover, the recognition that the precipitated gas atoms- both in inter- and intragranular bubbles-can be dispersed by fission fragments throughout the grain matrix, irradiation induced resolution, makes the application of the standard Booth model to the real situation a strong simplification. The first steady state theory of fission gas migration, extending the work of Booth to include the precipitation and resolution effects was given by Speight [5]. This theory assumes that the gas diffusion coefficient, production rate and resolution rate are time independent parameters. For quasi-equilibrium between the flow of gas into intragranular trai>s and the resolution of gas from the traps, Speight’s model shows that the gas diffusion coefficient, Din, modifies
0022-3 115/85/$3.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
141
K. Forderg, A.R. MassihlDifusion theory of fision gas according D
_
to
b’& (b’ + g) ’
where b’ is the resolution probability from intragranular traps (s-l), and g is the capture rate (s-l) for the migrating gas particles. For spherical traps (Smoluchowski [6J) g = 4 ?rD,J?ii,
(2)
where fi and J? are the average concentration and radius of the intragranular bubbles, respectively. Further, Speight [5] proposed an expression for the rate of accumulation of gas particles at a grain boundary for a given temperature, i.e.
(3) where N is the number of gas particles per unit area of grain on the grain boundary (m-‘), b the resolution rate from intergranular bubbles (s-l), A the resolution layer depth from grain boundary (m), p the gas production rate (s-l) and f0 is the Booth flux [3]. For a grain of radius “a”,
(4) The gas arriving at grain boundaries, with the rate given by eq. (3), eventually saturates the grain boundaries through a network of interconnected bubbles. If the ideal gas equation of state is assumed, the density (per area) of gas particles over the whole grain boundary at saturation may be determined by [7]
(5) where y is the surface tension of the bubble, r its radius, k, the Boltzmann constant, T the temperature and f(0) takes into account the nonsphericality of the bubbles on grain boundaries. For a spherical bubble 0 = 90” and f(90) = 1. P,,, is any externally applied hydrostatic pressure and V, is the fractional coverage of grain boundary at saturation. The release is assumed to occur when N reaches the value N,, the saturation density. Eqs. (3) and (4) have been used as a basis for fission gas release prediction in several computer codes [g-lo].
Some years ago Turnbull [ 1 l] suggested that if one replaces the perfect sink b.c. by an imperfect one, the Booth model may be extended to treat the resolution effect. Specifically, he set the concentration of gas particles at the grain surface equal to: bhN/2D which he assumed to be a value constant in time - and solved the diffusion equation for gas concentration in a spherical grain of radius “a”, obtaining the release fraction. In this paper we attempt to extend Turnbull’s formulation to a time varying situation. More precisely we consider,
ack 4 -
at
= D(t)A,C(r,
t) + P(t)
with A =??,?? ’ ar2-rar subject to C( r, 0) = 0, (7)
C(a, t) = (b(t)AN(t)/2D(r)),
where C(r, t) is the concentration of gas atoms and it is assumed that gas generated uniformly throughout a grain of radius “a”. In. section 2, we give a general formalism for treating the differential equation (6) with a time dependent boundary condition, eq. (7). This formalism allows us to readily use and extend the numerical scheme, presented in our paper I, to the present situation. Further, from this formalism we are able to find rigorously the corresponding generalization to eq. (3) in a form of integro-differential equation. In section 3 we present the short and long time limit of N(r) for a semi-steady-state condition. In section 4 we shall apply the results obtained in section 2 to calculate the accumulation of gas on grain boundaries and release to downward and upward cascading power histories.
2. General equations of dihsion The diffusion equation more convenient form
and resolution
(6) can be resealed
to a
142 am ~
K. For&erg, A. R. ~sih/~i~ion 7) at
= A,C(r, 7) + P,(T)
theory of jission gas
(8)
(13
with
with
C( t, 0) = 0,
C&a, 7) = 0.
(367)
(9)
= h,(dN(W,
Now eq. (11) can be invoked (16) as
(16) to write eqs. (15) and
where h, = b(~)A/~(A), PC= P(r)lD(r) and Combining T=
‘D(&,)d$.
eqs. (13), (14) and (17) we have
(IO)
Jf,
It was shown earlier that for the case of a perfect sink boundary condition, i.e., C(a, 7) = 0, eq.’ (8) may be transformed to an integral equation of the form [2]
(18) where
‘47rr’C(r,
J0
r)dr=
J0TK(T--7o)Pe(dds
where the kernel K(r-
(11)
71,)is given by
(12) To treat eqs. (8) and (9) we assume that a certain quantity of gas N(r) has been accumuiated on the grain boundary at some time I, and N(T) < N, where N, is the amount of gas required to saturate the grain boundary or N(r),,,,, = N,. Thus we have the identity: 4rra2Pl(r) = 2
(
47ra' 3
r
J Pe(rtJd~,--4P
J
r’C(r,
7)dr
(19) The integral equation (18) is the basic equation used in this note to determine the time dependence of the gas particle density on the grain boundary prior to the release. Further examination of eq. (18) yields to an interesting result. Specifically, under the realistic assumption that /3/b is constant in time, the time derivative of eq. (18) gives
1,
0
(13) where the factor of 2 on the right hand side accounts for release into the boundary from two adjacent grains. Setting C,,(r, 7) = ctr, 7) - eta,
7)
eq. (8) for C,,(r, T) becomes
(14)
The first term on the right hand side of eq. (20) corresponds to the Speight equation (3) in r space, i.e. for the case where D and J3 are time dependent. For example it can be seen than when p and D are constants Kz=
folP,
(21)
K. Forsberg, A.R. Massih/DifJusion theory of fission gas
where f,) is the flux given by eq. (4). Applying consideration to eq. (20) yields
this
where (see the Appendix)
R,=a’
coth (u’s)“’
1
--
((1*3)“*
Substituting
The first term in eq. (22) is identical to eq. (3), and the second term vanishes if N(t) is assumed to be a linear function of f. This result is noteworthy, as Speight’s derivation of eq. (3) was based purely on plausibility arguments [5] whereas eq. (20) is an exact mathematical result based on solving the diffusion equation with the appropriate boundary condition. The difference between ours and Speight’s is that Speight assumed a constant /3 and a linearly time dependent N(t). This latter assumption, however, is approximately true when the resolution effect becomes predominant. Then N(t) evolves, approximately, linearly with time.
143
(a’s)’ >.
(25)
eq. (25) into eq. (24) yields
For short times, large s, eqs. (A8) and (26) can be utilized to write [12] N(T)=?(T+& - h2 exp ( h: 7) erfc( h, T”‘) + h, exp (ht 7) erfc ( - h2T”2) h, h,(h, + h,) + o(T”), (27)
where l/2 h,=_++
!!j+;
,
(
3. Analyticsolutions
l/2 h3++(;+2)
In this section we shall apply the integral equation derived in the last section, eq. (18), to the case where PC= p( t)/D( t) and h, = ‘Ab(t)/D( t) are constants of time. This assumption is not unrealistic, because when fuel experiences higher powers, it leads to higher production rate p, larger fission rate and higher temperatures, whereby larger D emerges, e.g., if Arrhenius law is envisaged. We shall present here a detailed computation of this steady state case, where we obtain an analytical expression for N(t) in short and long times limits. We start by taking the Laplace transform in time of eq. (18)
where in deriving eq. (23) we recalled: h,(~)N(7)/2 and used the notation:
)
C(a, T) =
,
and _ erfc (i) = ?G, I
e-l2 dt.
The reader may check (by series expansion of the exponential and error functions) that in the absence of resolution, h, = 0, eq. (27) yields the Booth solution, i.e.
Eq. (27) provides a working equation for determining the density of gas atoms on a spherical grain boundary of the fuel material. The integration of eq. (26) can be carried out by residue calculation to give the long time limit for N(T)
e-“‘f( 7) dT. N(T)=-
If
/$(T)
and
h1(7)
ti= 2/3,Ez,Is(l+
Wea
are constants we have d&h,),
m
(24)
+c
(
T-
a*
S(3 + h, a)
4pca3
e-wa)*r
??I=, u;[ u’, + ah,(3 + ah,)]
(28)
K. For&erg, A.R. MassihlDifusion
144
theory of fission gas
Adding eqs. (31) and (34)
with u/arctan(*)+rnn.
(29)
GO(T)+[I + ~&(T)]Gs(T) The first term in eq. (29) suffices to determine the long time behavior of N(7). Again in the absence of resolution eqs. (28) and (29) yield the Booth solution for long times, i.e.
It is interesting to compare eqs. (27) and (28) with the analytical solution of the phenomenological equation (3) obtained by Dowling, White and Tucker [7]. 4. Numerical
treatment
A numerical method for applying eq. (18) to obtain the time evolution of gas atoms within grain boundaries is presented in this section. Sinoe the grain boundary area per unit volume of fuel is 3/2 a, where a is the radius of a spherical grain, the number of gas atoms per unit volume of fuel within the grain boundaries, GB, is G,, = 3N/2a.
(30)
Assuming that the ratio: b( T)/@(7) is time independent the integral equation (18) can be written in terms of G, as
Go
%)q(%) d%,
h, = abh/3/?.
K?(T) = ; &(T)
(36)
Eq. (35) may be interpreted as conservation of number of gas particles, i.e., the total number gas particles generated, the left hand side of eq. (35), is equal to the number of gas particles in the grain and on the grain boundary, the right hand side of eq. (35). For numerical computation we choose a sufficiently short time interval AT = TV - T, (> 0) during which the gas production rate,
I
r:’ 8,(T) d7
is estimated by:
(32)
@(‘I) + p(T2)
?2
pe(T)
dT
&
z
.
2
71
(37)
At is the difference in “real” time t corresponding to the interval AT. Exercising this assumption on eq. (35) results in
AC% + AG, + h4G~(&V% + h&(Tz)AGe
I
+ b(%)
At
(38)
.
2
(31)
where
(35)
with
= = 1,’ &(T--
= I’ &(%) d% 0
The reason for not approximating the integral in eq. (37) with ; &(T,) + &(T~)]AT is because this may give rise to numerical inaccuracies during start-up intervals, under which the diffusion coefficient, and subsequently &, vary within several orders of magnitude. As in paper I we will employ the approximation,
and l+&(T)=
9(T)= MT) -g
& [&(T)G~T)I.
(33)
i
n=,
&exp(
-$T),
(39)
where Similarily the number of gas atoms per unit volume of fuel within the grain, G,, can be obtained from eq. (17) G,,(T) = j ’ Cl - &(T - %)]q(%) d% 0
(34)
A, = 0.63003
B, = 9.9904,
A, = 0.2065 1
B2 = 64.488,
A, = 0.14776
B, = 511.61.
(40)
K. Forsbcrg, A. R. Massihl Diffusion theory of fissiongas
Applying form
eqs. (39) and (34) may be written
in the
determined
from eq. (35) i.e. through
GO(T)= i G”(T) n-l where
Finding q from eq. (45) it can be used in eqs. (43) and (44) to obtain G,,(T~) and GB(~2), respectively.
Similarly eq. (31) becomes:
Gs(T) = j’ q(%) d%- i G,,(T)* 0 “=I
(42)
By applying the numerical method presented in section 4 we compute here the rate of accumulation of gas atoms in the grain boundary bubbles. The density of gas contained in the fuel -grain, Ggrai,, can bk written as
Defining:
f.=exp(-$!h)-1. AG,, and AG, can be written, respectively
5. Saturation sod releaw
as
Gg,,i, = Go + hdB=GB,
(46)
where G,, h4 and GB were defined earlier in section 4. The density of gas within the grain boundaries at saturation is
AG = fnGn(d
-$ (72- d) q(Q)dTc,
(43) G. = & N,,
(47)
and
AG = -i
~,,G"(TJ “-1
where we employed the short time approximation for &(T~T,J,see the Appendix and eq. (35). Now the catch is that q in eqs. (43) and (44) depends on GB which is a complicated function of time, see eqs. (34) and (33). Even for the case under consideration where /3 is a linear function of f, q is a nonlinear function. To overcome this impediment we take q to be a constant within the time interval AT. Then q may be
where N, can be determined from the equation of state (5). The onset of gas saturation on the grain boundary may be explained by a percolation mechanism, where at a certain concentration of gas particles on the grain a network of interconnected bubbles sets in [13,14]. When G8(7J 2 G, the amount gas released per unit volume of fuel, &, may be written FR =
fG,
where 0 < f 5 1, and its upper bound corresponds to the condition that all the accumulated gas at grain boundaries would be released upon saturation. We have used eqs. (43) to (45) to obtain GB and eqs. (5) and (47) to calculate G,. The production rate used in this calculation [lo] is
146
K. Forsberg, A. R. MassihlDifision
06386 1+375E/W, where taken: atoms
>
W,, atoms/m”
s
W, is the rating (W/g) and E is the enrichment E = 0.03. The diffusion coefficients for gas are obtained from the following correlations
[lOI D=l.O9~10~“exp(-6614/T) D = 2.14 X lo-l3 exp (-22884/T)
D=l.SlXlO~“exp(-9508/T) Table 1 provides the numerical perature dependence of ratings
T > 1650 K, 1381< T < 1650 K, T < 1381 K. values for the temand the generation
theory of fission
gas
Table 1 Numerical values for ratings and generation rates Temperature (“C)
Linear rating Rating (kW/m) (W/g)
Generation rates (mol/m3 s)
1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000
26.8
29.8
4.4x 10 (’
30.6 34.4 38.2 41.9 45.7 49.4 53.2 57.0 61.0 64.6
34.1 38.3 42.4 46.6 50.8 55.0 59.2 63.4 67.6 71.8
5.1 x 5.7 x 6.4 x 1.2 x 7.7 x 8.3 x 8.9 x 9.6 x 1.0x 1.1 x
lo-” lo- h lo-” 10-h 10-h lo-” lO-h 10-h 1o-5 1om5
In Eq.(5) a' = 0.6 J m-* cl=500 r = 0.5pm
ReleareMolalm3 -----
Pext =0 v, = 0.25 .1600
I ,160O 00 e a i -1400 g c
1.0 Exposure,Mwd/kgU
Fig. 1. Fraction of gas atoms on grain boundary, GS/GB, as a function of exposure for downward fuel cascading temperature history. y is the bubble surface tension, 28 is the angle where two free surfaces meet at a grain boundary, r is average bubble radius, V, is the fractional coverage of the grain boundaries at saturation and the grain radius is taken to be 5 pm.
147
K. For&erg, A.R. MassihlDiffusion theory of fission gas
In Eq. 5 $
= 0.6
J m-’
e = 500 r = 0.5pm P
ext
v,
=o
= 0.25
I
00
1400
5 [
+
1200
1000
800
I
I
2.0
I
4.0
3.0 Exposure,
MwdlkgU
Fig. 2. Fraction of gas atoms on grain boundary and amount of gas released per unit volume of grain-as a function of exposure
for upward cascading fuel temperature history. The input data is as in fig. 1.
rate. In fig 1 we have displayed the result of our calculation for accumulation of gas on grain boundary, the ratio: GJG,, for the case of downward cascading fuel temperature history. The other input parameters used in the calculation are indicated in the figure caption and legend. Fig. 2 depicts our calculation for an upward cascading fuel temperature history. The release is indicated by a broken line. We assumed 100% release upon saturation. An oscillatory nature of gas accumulation on the grain boundary after the release is noted. This, however, is not expected to have an overall effect on fuel behavior, since it is believed to take place locally; nevertheless, experimental effort to observe this phenomenon can be worthwhile.
denoting the eigenvalues eigenvalue equation:
A,J/,= -W,
A formal formed solution of eq. (8) for the case of a perfect sink b.c., i.e. ~(a, T) = 0 may be obtained by
(Al)
and C(r, T) can be expanded in terms of the complete set of the eigenfunction
c
ar, 4 = i=, k,(~M(r) with
I
R +i*j =
Appendix: Derivation of the kernel K2
of A, by (hi) which gives the
&j
where the integral is carried out over the volume of the grain a. In paper I, (ref. [2D we showed
K. Forsberg,A. R. iUassih/Di@ion
148
I
C(r, 7) =
R
Ir
NT - s)Be(dds,
W)
theory of fissiongas
Taking the inverse Laplace transforms of eq. (A8),
,I
K2( T) = 2(i)
where for a sphere of radius “a” the kernel K
“’ - a
short times
(AIO)
and for long times we simply set n = 1 in eq. (A6) (A3) Kz(T)+$e-““‘“’ In I, we also showed that approximation
(A4) We wish to thank Dr. 6. Bernander for his helpful comments on the manuscript. The work was supported in parts by the Swedish Nuclear Inspectorate under contract No. B24 184.
with A; = 2.6391
B ; = 9.9905,
A; = 0.865
B; = 64.488,
A; = 0.6189
B; = 511.61,
(As)
accurately describes K(T) in all the regions of interest. Now substitution of eq. (A3) in eq. (19) gives
646) which can easily be Laplace transformed
&(s) = a3 -(a*s)-*+
to s space
coth(a’s)“’ (a2s)3’2 > ’
(A7)
For large s (short times) & =
,q-3/2
_
a-l
(AI I)
Acknowledgements
A,/ e-%(‘)
K( 7) = i
long times.
s-2
and for small s (long times);
(A@
References [l] L. Vlth, J. Nucl. Mater. 99 (1981) 324 and refs. therein. [2] K. Forsbexg and A.R. Massih, J. Nucl. Mater. 127 (1985) 137. [3] A.H. Booth, Atomic Energy of Canada Limited Report No. 496 (1957). [4] M.O. Tucker, Radiat. Effects 53 (1980) 251. [5] M.V. Speight, Nucl. Sci. and Eng. 37 (1969) 180. [6] M. von Smoluchowski, Phys. Z. 17 (1916) 557; 585. [7] D.M. Dowling, R.J. White and M.O. Tucker, J. Nucl. Mater. 110 (1981) 37. [8] R. Hargreaves and D.A. Collins, J. Br. Nucl. Energy Sot. 15 (1976) 311. [9] H. Nerman, in: Water Reactor Fuel Element Performance Computer Modelling, ed., J. Gittus (Applied Science Publishers Ltd, London, 1983). [lo] R.J. White and M.O. Tucker, J. Nucl. Mater. 1 (1983) 1 118. [l l] J.A. Turnbull? J. Nucl. Mater. 50 (1974) $2. [12] A. Abramowitz and I. Stegun, A Handbook of Mathematical Functions (Dover, New York, 1954). [13] C. Ronchi, J. Nucl. Mater. 84 (1979) 55. [14] A.R. Massih, J. Nucl. Mater. 119 (1983) 116.