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The effect of irradiation-induced re-solution on fission gas release
37
Journal of Nuclear Materials 110 (1982) 37-46 North-Holland Publishing Company
THE EFFECT OF IRRADIATION-INDUCED
RE-SOLUTION
ON FISSION GAS RELEASE
D.M. DOWLING School of Mathematics,
R.J. WHITE
University of Bath, Claoerton Down, Bath BA2 7A Y, UK
and M.O. TUCKER
Central Electricity Generating Board, Berkeley Nuclear Laboratories, Received
20 January
1982; accepted
17 March
Berkeley, Glos. GLI3 9PB, UK
1982
A finite-difference technique is used to compute UO, nuclear fuel during steady reactor operation.
exact solutions to the diffusion equation describing fission gas release from The resolution of gas atoms from grain-boundary bubbles is treated in two
alternative ways, and the results of the parallel calculations compared. Predictions of gas release using simple analytical models are compared with the numerical results and are found in general to describe the process very accurately.
1. Introduction
An important factor determining the safe operation of UO, nuclear fuel elements is the pressure exerted on the cladding directly or indirectly by accumulated gaseous fission products. In an attempt to quantify the amount of gas being released from isothermally irradiated fuel, Booth [l] treated a single grain of UO, as a sphere in which the fission gas atoms were being produced uniformly and were individually diffusing to the grain boundary. This “equivalent sphere” approach was used to calculate the fractional releases of stable gas atoms both during irradiation and during a post-irradiation anneal. If it is assumed that at the moment a gas atoms diffuses to the boundary of the spherical grain it is effectively released from the fuel, for instance by way of thermally induced cracks, the model predictions are frequently found to be in poor agreement with experiment. The proposal of Barnes [2] and Whapham [3] that, in the absence of cracking, release is delayed by the accumulation of gas in bubbles on grain boundaries, was confirmed by the microstructural observations of Reynolds and Bannister [4]. Speight [5] proposed a modified diffusion model incorporating the influence both of a fine dispersion of intragranular gas bubbles and of a coarser distribution of lenticular intergranular bubbles decorating the grain faces. Assuming resolution
0022-3 115/82/0000-0000/$02.75
0 1982 North-Holland
of gas from these bubbles is irradiation-induced [6,7], Speight demonstrated that the simple gas-release model of diffusion from a sphere proposed by Booth [l] was valid only if a lower effective diffusion coefficient and a non-zero gas atom concentration at the surface of the sphere were assumed, to account for the effects of resolution from intragranular and grain-boundary gas bubbles, respectively. The effective surface gas atom concentration so defined depends in magnitude upon the instantaneous number of gas atoms held in grain-boundary bubbles, thus rendering the diffusion problem non-linear and impossible to solve analytically. Speight [5] suggested a simple approximate expression for the rate of accumulation of gas atoms in these bubbles, and his formula has been adopted in the gas-release model of Hargreaves and Collins [8] which forms the bases of the MINIPAT computer code. The simplification has never been rigorously tested for accuracy and in any case applies only up to the stage at which the grain faces become saturated with fission gas, when the lenticular bubbles start to interlink and gas release begins on a considerable scale. The purpose of the present paper is to describe the results of the “exact” numerical solution of the diffusion model of gas release when resolution from grainboundary bubbles is included, and to establish the effectiveness of simpler models to describe the process both before and after grain face saturation.
D. M. Dowling et al. / Effect oj w-solution on fission-gas release
38
2. The retention of fission gas atoms on grain faces
2.2. The effect of irradiation-induced re-solution on the incubation time for grain face saturation
2.1. The concentration of gas at saturation Scanning electron micrographs of the fracture surfaces of irradiated UO, fuel indicate typically that the lenticular fission gas bubbles occupy at most a fraction f, of about 50-608 of the grain face area, and constitute about 3% fractional swelling AV/V. Amongst such bubble populations there are usually signs of bubble coalescence and it is reasonable to assume that a boundary in this condition is very close to being saturated by gas atoms. To calculate the number of fission gas atoms per unit area of grain boundary which are present at saturation the bubbles are assumed to be identical, lenticular-shaped pores comprising spherical caps of radius r which are joined in the plane of the boundary with dihedral angles 28 = 100” [4]. The volume V, of each of these bubbles is given by the expression Vb=47rr3f(e)/3
The net rate of accumulation of gas atoms in grainboundary bubbles is reduced by irradiation-induced resolution. Speight [5] suggested an approximate treatment of this effect and his model was more fully developed by Olander [9], but no exact analytical treatment is at present available. Assuming that gas atoms residing in the grain boundary have a probability b of being knocked a distance X back into one of the adjacent grains, Speight argued that an equilibrium situation is reached when the diffusion flux of gas atoms from the local concentration gradients is exactly balanced by the flux due to re-solution. If instantaneously there are N atoms per unit area of face the resolution flux is Nb/2 into each grain, whilst if the gas atom concentration is zero at the boundary and C” a distance X from it, the diffusion flux is approximately DC&/X from Fick’s law. Thus at equilibrium, when the two fluxes are assumed to be equal,
(1) C” = NbX/2
where f(8)= 1 -$cost?+f cos30. Ignoring for simplicity the possibility that part of the boundary will be occupied by grain-edge porosity, the number of bubbles, n, associated with a single grain of radius a may be written n=2a2h/r2sin28,
(2)
and thus the fractional swelling (AV/V),,, due to grain-face bubbles at saturation is given by the equation (A?‘/‘)
SB,=2f
rf(e) ’ a sin2 e
The gas contained in the lenticular bubbles will be in mechanical equilibrium with both the surface tension associated with the bubble surface specific energy y and any externally applied hydrostatic stress P,.,, due either to fuel pin gas pressure or to mechanical interaction between the fuel and its containment. Thus the pressure PBBsof the fission gas in each grain-boundary bubble is given by the expression ps,, = 2 y/r + p,,, .
(4)
Assuming this gas behaves ideally, the number atoms N”’ per unit area of grain face when saturated may be written Nm
,4
f,f(e)r 3 ,,(?+p=J
of gas this is
D.
%=4/3(
F)“‘(
of grain size.
1-g).
Making the substitutions A = /laD=/8(
bh)3 N”‘, u = 4bh ( t,‘Da)“2,
cp= N/N”
eq. (6) becomes d+/du++=Au’
(7)
which, for the simplest solution
case when + = 0 at u = 0, has the
+=A[u2-2u+2-2exp(-u)].
(8)
It is interesting which is independent
(5)
By arguing that gas release will not occur until the intragranular gas atom concentration has exceeded the resolution barrier C” given by eq. (5) with N = Nm, Hargreaves and Collins [8] define the incubation time t, = C”/p = bX N “‘/2 Dp where /3 is the fission gas atom generation rate per unit volume. A rather more detailed treatment of gas atom accumulation on grain boundaries may be obtained using Speight’s [5] suggestion that the flux of atoms to the boundary from each grain is given by the flux in the absence of resolution [l] modified by the fraction (1 - Cx/Co) where Co is the concentration distant from the boundary. Since to a first approximation the Booth flux is 2/3( Dt/n)‘/= and Co = /3t the rate of accumulation of gas per unit area of boundary is
condition
to note that when us 1 the saturation N = N m or qb= 1 is reached when Au2 = 1,
D.M. Dowling et al. / Effect of re-solution on fission-gas release when the saturation time 1, is identical to that assumed by Hargreaves and Collins [8] and demonstrates that their assumptions are only valid when resolution is predominant. Olander (91 had attempted a more direct solution of the diffusion problem by making the simplifying assumption that in the narrow region of width X adjacent to the grain boundary the concentration profile is quasi-stationary, that is, that aC/at = 0 in the diffusion equation for that layer. In the notation used in eq. (8)
the solution
given by Olander
39
may be written
~=~{uz_~(l-~)+~[l-~+~(~)2] X[l-exp(g)
erfc(y)]}.
When resolution predominates ( I( z~ 1) this solution has the same limiting form as eq. (8). This is demonstrated in fig. 1 where +/A has been plotted as a function of u for various values of A’= 6X2/D, and from which it is also apparent that solutions (8) and (9) are essentially indistinguishable, provided b#l*/D e 0.01. 2.3. Gas release after grain-boundary saturation It was assumed by Hargreaves and Collins [8] that all gas arriving at grain boundaries once these had become saturated would be released, and a calculation by Turnbull [lo] tends to support this supposition. This calculation was of gas release after a period of irradiation at low temperature during which a uniform gas atom concentration Co was established throughout each grain. Turnbull further assumed that the effective grain-boundary concentration Cm = N m bX/2 D owing to resolution and showed that the fractional release F, that is the ratio of the number of gas atoms released to those generated in the period t since the start of the higher temperature irradiation, was given by the expression
F=l_$
5 d?-_ i Do
(CO- Cm)
n--l
X[l-exp( For t
MVS
and
(nr’)
-?)I. this equation
may
be written
more
X=0.0
(11) Fig. 1. The predicted variation of the density of gas atoms N held in bubbles on the grain face with (time)‘/‘, expressed through the quantities +/A and u defined in section 2.2, using eq. (9) due to Olander [9]. Curves are shown for different values of the parameter x = bA2/D and are compared with that marked MVS calculated from the simpler treatment due to Speight [S].
In the regime considered by Hargreaves and Collins [8] in which resolution is predominant, when grain-face saturation is reached the gas atom concentration profile will be more or less uniform across the grain. The Turnbull calculation is thus suitable to describe the subsequent gas release. In this instance however Co = Cm and, as shown by eq. (1 l), release will occur accord-
40
D.M. Dowling et al. / Effect of re-solution on fission-gas release
ing to the familiar formula due to Booth [I], but with I being measured from the moment that grain-boundary saturation occurs. When resolution is less predominant the profiles across the grain will be more arched at saturation and thus eqs. (10) and (11) will less accurately describe the gas-release process.
3. Tbe finite difference solution boundary
Although the model of Olander [9] attempts to describe the transient build-up of the concentration CA it is approximate in that it treats the concentration in the layer adjacent to the boundary as being quasi-static. Exact analytica treatment is not possible and so, if the accuracy of the approximate models is to be checked, numerical solution of the problem is desirable. In seeking such a solution it is possible either to assume that all atoms undergoing resolution are deposited a characteristic depth h from the grain boundary or that they are precipitated uniformly throughout a layer of thickness 2X. These approaches will be referred to respectiveiy as the ‘Olander model’ and the ‘Smeared model’,
JI *
‘II
b
JI = ‘ii
Resolution boundary
3.1. The Olander model
Chin boundary
The model is shown schematically in fig. 2(a). Solutions C, and C,, to the diffusion equations
ac, Da -r*c r* ar ( 1+P,=% act
(i = I, II)
(12)
Fig. 2. The two treatments of the resolution problem, (a) the Olander model, and (b) the smeared model.
entitled
are sought for regions I and II respectively, such that the concentration gradient is zero at the sphere centre, ac,,&=O
when 1=0,
the concentration C,,=O
at the grain boundary
(13)
3.2. The smeared model
(14)
In this case the boundary between the two regions is at r = a - 2X as shown in fig. 2(b). Solutions to eq. (12) are sought which satisfy conditions (13) and (14), but for which
is zero,
whenr-a,
and whilst the concentrations must be identical at depth X, the fluxes J, and J,, in fig. 2(a) differ by the resolution flux, that is C,=C,,whenr=a-X
CIIr=o--2X=CnIr=u-ah
(17)
and
(151
and 4aD(a--X)
“( ~Ir_~-~-~/,~a_-h)
=4ruZqN.
(16) In this model the production the same value 8.
rates in each region have
replace eqs. (15) and (16). However, the production rates are no longer the same in the two regions; they are given by the expressions &, = /? + bN/4X.
(19)
In each model, prior to saturation
of the grain faces,
P,=8
and
41
D.M. Dowling et al. / Effect of re-solution on fission-gas release
the value of N is calculated instantaneously by subtracting the integrated gas concentration in the grain from the total amount produced in it since the onset of irradiation. Once the saturation level N = N m has been reached, however, this value is assumed in eqs. (16) and (19), and all gas atoms arriving at the grain faces after this incubation time are deemed to have been released from the face for the purposes of the present calculation.
continuity conditions between the inner and outer regions, calculating the fluxes as first-order Taylor expansions about the adjacent points. In the Olander model, to speed up the calculation, the mesh was occasionally coarsened for one time step and the values of I,L at intermediate nodes estimated by interpolation. The generation of machine error necessitated the use of soluteconserving smoothing in the smeared model, thus removing noise and rounding errors from Ihe diffusion profiles.
3.3. The finite-difference calculation Making substitutions $=
DC/Da=;
each of the equations the simple form
4. Results and discussion
of the form
w = Dt/a’;
v = r/a
for C, and C,, may be written
in
(20) in both the Olander and smeared models, though in the latter the right-hand side of eq. (20) has the additional term bN/4/3A in the outer layer. In order to model accurately the steep concentration gradients in the vicinity of the grain boundary it is necessary that mesh points be bunched close together in that region. This was achieved in the present calculation by assuming the radial position v, of the n th mesh point is given by the expression ?&=v*(l
_&“‘I)/(1
_E),
where E < 1 and v is below the resolution layer. For the outermost region the mesh points were evenly spaced at the distance equal to the ultimate mesh spacing in the inner fuel region. The method of solution adopted was to represent the normalised concentration $ at the n th node by a quadratic expression in v passing through the (n - l)th, n th and (n + 1)th concentration levels. With the aid of this equation the differential coefficient a#Jdo may be evaluated by means of eq. (20), and the profile # updated by means of the first-order Taylor expansion ~~(o+Aw)=~~(w)+(a~~/aw)Aw.
(21)
The timestep Aw was chosen such that the fractional change in 4 did not exceed a value (Y, that is that Aa = +5/(a+~/a~)limln. In practice, when 100 mesh points were used, 01 was chosen not to exceed -0.3. In both the Olander and smeared models the concentration at the resolution boundary was not updated by means of eq. (21). Instead the value of $J was evaluated directly from the flux
The values of physical parameters used in the following illustrative calculations are listed in table 1, and were chosen more for convenience than for their absolute physical validity. The assumed temperature dependence of the gas atom diffusion coefficient is shown in fig. 3. 4.1. Gas release during the incubation period The variation with normalized time w = Dt/a’ of the fraction-face saturation + = N/N,,, is shown in figs. 4-6 for values of b = 10-6, lo-’ and lop4 s-’ respectively, for the single resolution depth X = lop8 m. Results obtained for the finite-difference calculations using the smeared model are shown as continuous curves and coincide with those obtained for the Olander model for all but the lowest temperature 800°C, for b = 10e6 s-’ and 10e4 s-‘. Also shown, as discrete points, are results calculated by means of eq. (8). Where the two finite-difference models disagree, at low temperatures where Table 1 Values of physical parameters assumed in example calculations Grain radius, a Fraction coverage at grain faces, f, Surface energy, y Semi-dehedral angle, B Gas atom diffusion coefficient, D
4 4 D2
E; Resolution Resolution
parameter, depth, X
b
5X10-6m 0.50 0.1 J m-’ 50” Do+& exp(-Q,/kT) + D2 edQ2/W 2.04X lo-” m2 s-’ 2.1 X lo-’ m2 SC’ 2.31 X lo-‘s m2 s-’ 3.85 eV 1.37 eV 10-6-10-4
s-’
1O-8-1O-5
m
D.M. Dowling et al. / Effeci of re-soiution on fission-gas release
/ IO-’
Fig. 5. The development of fractional for b = 10e4 s-’ and h = IO-’ m.
I
I.0
Invrrs*
x10-‘
*0
¶.O
mnp.rotw*
IO'
10-'
grain-face
saturation
+
release is strongly controlled by resolution, it may be seen from the diagrams that the Olander approach more nearly agrees with the predictions of eq. (S), which was derived on very similar physical assumptions. Comparing fig. 6 with the others, the removal of the
Fig. 3. The assumed temperature dependence of the gas atom diffusion coefficient used in the example calculations.
800 “C (Smeared
!
)
0
I
0.a Fig. 4. The fractional saturation cp = N/W” of gram faces by fission gas shown as a function of time through the parameter o = Dt/a’. The resolution parameter and depth are assumed to be 10m6 SC’ and 10-s m respectively.
Fig. 6. The time-dependence of fractional cp for b = 10m4 SC’ and h = 10-s m.
grain-face
saturation
AM, Dowiing et ai. / Effect of re-solution on fission-gas release temperature dependence because of the predominance of resolution is evident. In these circumstances the quantity of gas on the boundaries increases linearly with time, corresponding to large values of u in eq. (8). On the other hand the curves in fig.4 representing situations in which resolution is unimportant, exhibit the t312 dependence suggested by the diffusion model of Booth [l] for short irradiation times. From fig. 1 it appears that discrepancies between the finite-difference calculations and the approximate model described by eq. (8) will occur only for values of bA2/D > 0.1, and thus are likely to be worse the larger the resolution depth A. In fig. 7 the variation of + with w is drawn for T= 1OOO’C and bh= 10-‘3m s-’ for various values of X for which eq. (8) is a rather poor approximation. Included on this graph are discrete points calculated from the better analytic approximation given by eq. (9). Clearly, although the equation slightly overestimates the rate of gas accumulation, it provides a satisfactory description in this regime. The adequacy of the simple approximation given by eq. (8) is clearly demonstrated in fig. 8, in which most of the data from figs. 4-7 are presented. In this case the finite-difference results are shown as discrete points whilst the continuous curve has been calculated from eq. (8). The group of nine points represented by filled circles which lie to the left of the main body of data are for the extreme case when X = IO-’ m-‘. The dotted line associated with these points has been calculated from eq. (9) for the case when bA2/D = 0.92, and dem-
43
onstrates the validity of this expression at such high values of A. The temperature dependence of the incubation time necessary to reach grain-face saturation (+ = 1) is shown in fig.9 for X = 10e8 m and b = lo-*, 10e5 and 10m6 The slight variation between the finite difference S -‘. results for the Olander and smeared models has been omitted for b = 10e6 s-t for the sake of clarity. The predictions obtained solving eq. (8) when cp=: 1 are shown as discrete points in fig. 9 and demonstrate once again that, at least for h = 10d8 m, the overall agreement is excellent. 4.2. Gas release after grain-boundary saturation If the finite-difference calculations are extended beyond the point when the grain faces become saturated, fractional release curves are obtained as shown in fig. 10. By “fractional release” is meant the fraction of the total fission gas atoms generated in a grain which are either resident on the grain boundary or which, since saturation, have escaped from the body of the fuel. The calculations refer to fuel for which b= 10e5 s”“’ and X = 10P8 m. The point of incubation, shown by arrows in the diagram, is characterised by a fairly rapid change of gradient which is particularly striking at low temperatures. Included for the purpose of comparison is the equivalent prediction of the Booth model given by eq. (IO) with Co = Cm. The finite-difference results tend asymptotically to this curve at longer irradiation times.
Fig. 7. The time dependence of Q,for various values of resolution depth h calculated for bX = 1O-‘3 m s-’ and for a temperature of IOOO~C.
Fig. 9. The variation of incubation time with temperature for X = IO-* m and for various values of the resolution parameter h.
10
%
ia
provide the value of +, from which the fractional F is given by the expression F= 3cgNm/2aflt.
1c
(ii)
Following
release (221
incubation
at
time
t = tint the frac-
tional release is given by the sum of the quantities F in eq. (11) and (22) where, in the former, t is remeasured from rilx: and the concentrations Co and Cm are given by the expressions
rti
Co = Ptinc - 3Nm/2a Cm = bXNm/2D.
16
1C
16
.5
101
I
-1 10
10°
3
10'
I
s.9
,
ld
-v-
Fig. 8. The inclusion of most of the finite-difference calculation results shown in figs. 4-7 as discrete points on a single plot of $/A against U. The continuous and broken curves have been calculated from eqs. (8) and (9) respectively.
Eqs. (8) and (11) have been used to provide the discrete points in fig. 10 in the following manner; (i) For times before incubation, eq. (8) is used to
(23)
The agreement between eqs. (8) and (11) and the finite-difference calculations is very good at low temperatures but shows considerable discrepancy immediately after incubation at 12OO’C. The reason for this is because, at higher temperatures, the release of gas is more dependent upon diffusion, and by the time incubation has been achieved there will exist a marked variation in gas atom concentration across the grain. E?.q.(23) however effectively smears the profile into a uniform level at all points. At lower temperatures, when resolution predominates, the profile is always near-uniform and this explains the accuracy of the approximation. The predictions of the finite-difference and approximate calculations over times equivalent to threeyear irradiations at XKPC and 1200°C are compared in fig. Il. Again a value of X = lo-’ m has been assumed, and solid lines indicate the results of the finite-difference approach. The releases determined using eqs. (8)
D, M. Dowling et al. / Effect of re-solution an fission-gas release
0.c
10-J
10-L
10-z
10-t
-wFig. 10. Fractional release of fission gas atoms before and after grain-face saturation (indicated by arrows) for b = low5 s-’ and X = IO-* m. Continuous curves are from finite-element calculations whilst discrete points are derived from eqs. (8) and (11) as described in section 4.2.
and (11) are shown as broken lines, and also show the overestimate immediately after incubation has been achieved. It is encouraging that, for the 1200°C calculation with b = 10e6 s-‘, the finite-difference calculation
o-
., -
c /
u*y
I! -
._._._.
800
b
SO0 10-b
e
1200
d
1200
104 10-b
b I
II -
$0‘ 4
10-6
0
lo-’
-w-
lo-=
10-1
Fig. 1 I. The time-dependence of fractional release of fission gas atoms predicted numerically (&id lines) and analyticatly (broken lines) at 800°C and 1200°C for different values of the resolution parameter.
and eq. (11) differ by only 0.1% when o = 0.2, when the fractional release is 71%. This‘is surprising since eq. (11) is strictly only valid for ~~0.1 because of the mathematical simplifications made on its formulation. For curves labelled b, at 8OO’C when b = lop4 SK’, incubation was still incomplete after three years, and as with curves a, c and d the two approaches are in very acceptable agreement. Finally, in fig. 12, fractional release curves are shown for computations at lOOO”C, for h = 10e8 m except in the one case indicated. The discrete points calculated from eq. (8) and (i 1) are again in excellent agreement with the numerical results shown in solid lines except immediately following incubation. Included in fig. 12 are also the results for b = low6 and X = lo-‘, comparing the finite-difference predictions (broken line) with those of eq. (9). It is interesting to note that this approximate equation correctly predicts the presence of a ~nimum in the fractional release prior to incubation as suggested by the more accurate calculation. In general then the simple analytic models discussed in section 2 very adequately describe gas release from UO, being irradiated under steady reactor conditions. The application of these same ideas to variable conditions, of the kind experienced in practice even when the irradiation is nominally steady, needs more complex treatment and is considered elsewhere [ 111.
D. M. Dowling et al. / Effect of re-solurion on fission-gas
46
(2) Under such low-temperature conditions, the grain-boundary concentration increases linearly with time, whereas at higher temperatures the t3i2 dependence characteristic of diffusional release is exhibited. (3) Prior to saturation of the grain boundaries by fission gas there is excellent agreement between the numerical and analytical models, though differences begin to emerge both as resolution begins to dominate diffusional release, that is at low temperatures, and as the resolution depth is increased. (4) The simple model used to describe fission gas behaviour following grain-boundary saturation tends at higher temperatures to overestimate the release rate immediately following this event, but accurately represents the situation at later times.
0.2!
0.2,
f
O.,!
L
z
s .;
O.IO
release
I -
I;
O.OI
Acknowledgement
10‘
10-J
lo-‘
1
-W-
Fig. 12. Fractional
releases of fission gas at 1000°C.
5. Conclusion
Computer calculations at diffusional fission gas release to grain boundaries in UO, fuel irradiated under steady reactor conditions are reported. In the analysis two types of irradiation-induced resolution models were considered, one in which the affected atoms residing in grain-boundary bubbles are deposited at a single resolution depth h and the other in which these atoms are deposited uniformly over a depth 2h. The results of each approach have been compared with each other and with the predictions of simple analytical models, and the following conclusions drawn. (1) The numerical calculations were insensitive to the resolution model chosen except at low temperatures where, since gas atom diffusion is very slow, release is strongly controlled by resolution.
This paper is published by permission Electricity Generating Board.
of the Central
References
111A.H. Booth, AECL Rep. 496 (1957). [21 R.S. Barnes, J. Nucl. Mater. 11(1964) 135. [31 A.D. Whapham, Nucl. Appl. 2 (1966) 125. [41 G.R. Reynolds and G.H. Bannister, J. Mater. Sci. 5 (1970) 84. [51 M.V. Speight, Nucl. Sci. Engrg. 3 1 (1969) 180. [61 A.D. Whapman and B.E. Sheldon, UKAEA Rep. AERE R4970 (1965). Turnbull and R.M. Cornell, CEGB Rep. [71 J.A. RD/B/N1716 (1970). [81 R. Hargreaves and D.A. Collins, J. Brit. Nucl. Energy Sci. 15 (1976) 311. Aspects of Nuclear Reactor [91 D.R. Olander, Fundamental Fuel Elements (ERDA, USA, 1976) p. 314. [lOI J.A. Turnbull, J. Nucl. Mater. 50 (1974) 63. [ill R.J. White, D.J. Dowling and M.O. Tucker, to be published (1982).
Journal of Nuclear Materials 110 (1982) 37-46 North-Holland Publishing Company
THE EFFECT OF IRRADIATION-INDUCED
RE-SOLUTION
ON FISSION GAS RELEASE
D.M. DOWLING School of Mathematics,
R.J. WHITE
University of Bath, Claoerton Down, Bath BA2 7A Y, UK
and M.O. TUCKER
Central Electricity Generating Board, Berkeley Nuclear Laboratories, Received
20 January
1982; accepted
17 March
Berkeley, Glos. GLI3 9PB, UK
1982
A finite-difference technique is used to compute UO, nuclear fuel during steady reactor operation.
exact solutions to the diffusion equation describing fission gas release from The resolution of gas atoms from grain-boundary bubbles is treated in two
alternative ways, and the results of the parallel calculations compared. Predictions of gas release using simple analytical models are compared with the numerical results and are found in general to describe the process very accurately.
1. Introduction
An important factor determining the safe operation of UO, nuclear fuel elements is the pressure exerted on the cladding directly or indirectly by accumulated gaseous fission products. In an attempt to quantify the amount of gas being released from isothermally irradiated fuel, Booth [l] treated a single grain of UO, as a sphere in which the fission gas atoms were being produced uniformly and were individually diffusing to the grain boundary. This “equivalent sphere” approach was used to calculate the fractional releases of stable gas atoms both during irradiation and during a post-irradiation anneal. If it is assumed that at the moment a gas atoms diffuses to the boundary of the spherical grain it is effectively released from the fuel, for instance by way of thermally induced cracks, the model predictions are frequently found to be in poor agreement with experiment. The proposal of Barnes [2] and Whapham [3] that, in the absence of cracking, release is delayed by the accumulation of gas in bubbles on grain boundaries, was confirmed by the microstructural observations of Reynolds and Bannister [4]. Speight [5] proposed a modified diffusion model incorporating the influence both of a fine dispersion of intragranular gas bubbles and of a coarser distribution of lenticular intergranular bubbles decorating the grain faces. Assuming resolution
0022-3 115/82/0000-0000/$02.75
0 1982 North-Holland
of gas from these bubbles is irradiation-induced [6,7], Speight demonstrated that the simple gas-release model of diffusion from a sphere proposed by Booth [l] was valid only if a lower effective diffusion coefficient and a non-zero gas atom concentration at the surface of the sphere were assumed, to account for the effects of resolution from intragranular and grain-boundary gas bubbles, respectively. The effective surface gas atom concentration so defined depends in magnitude upon the instantaneous number of gas atoms held in grain-boundary bubbles, thus rendering the diffusion problem non-linear and impossible to solve analytically. Speight [5] suggested a simple approximate expression for the rate of accumulation of gas atoms in these bubbles, and his formula has been adopted in the gas-release model of Hargreaves and Collins [8] which forms the bases of the MINIPAT computer code. The simplification has never been rigorously tested for accuracy and in any case applies only up to the stage at which the grain faces become saturated with fission gas, when the lenticular bubbles start to interlink and gas release begins on a considerable scale. The purpose of the present paper is to describe the results of the “exact” numerical solution of the diffusion model of gas release when resolution from grainboundary bubbles is included, and to establish the effectiveness of simpler models to describe the process both before and after grain face saturation.
D. M. Dowling et al. / Effect oj w-solution on fission-gas release
38
2. The retention of fission gas atoms on grain faces
2.2. The effect of irradiation-induced re-solution on the incubation time for grain face saturation
2.1. The concentration of gas at saturation Scanning electron micrographs of the fracture surfaces of irradiated UO, fuel indicate typically that the lenticular fission gas bubbles occupy at most a fraction f, of about 50-608 of the grain face area, and constitute about 3% fractional swelling AV/V. Amongst such bubble populations there are usually signs of bubble coalescence and it is reasonable to assume that a boundary in this condition is very close to being saturated by gas atoms. To calculate the number of fission gas atoms per unit area of grain boundary which are present at saturation the bubbles are assumed to be identical, lenticular-shaped pores comprising spherical caps of radius r which are joined in the plane of the boundary with dihedral angles 28 = 100” [4]. The volume V, of each of these bubbles is given by the expression Vb=47rr3f(e)/3
The net rate of accumulation of gas atoms in grainboundary bubbles is reduced by irradiation-induced resolution. Speight [5] suggested an approximate treatment of this effect and his model was more fully developed by Olander [9], but no exact analytical treatment is at present available. Assuming that gas atoms residing in the grain boundary have a probability b of being knocked a distance X back into one of the adjacent grains, Speight argued that an equilibrium situation is reached when the diffusion flux of gas atoms from the local concentration gradients is exactly balanced by the flux due to re-solution. If instantaneously there are N atoms per unit area of face the resolution flux is Nb/2 into each grain, whilst if the gas atom concentration is zero at the boundary and C” a distance X from it, the diffusion flux is approximately DC&/X from Fick’s law. Thus at equilibrium, when the two fluxes are assumed to be equal,
(1) C” = NbX/2
where f(8)= 1 -$cost?+f cos30. Ignoring for simplicity the possibility that part of the boundary will be occupied by grain-edge porosity, the number of bubbles, n, associated with a single grain of radius a may be written n=2a2h/r2sin28,
(2)
and thus the fractional swelling (AV/V),,, due to grain-face bubbles at saturation is given by the equation (A?‘/‘)
SB,=2f
rf(e) ’ a sin2 e
The gas contained in the lenticular bubbles will be in mechanical equilibrium with both the surface tension associated with the bubble surface specific energy y and any externally applied hydrostatic stress P,.,, due either to fuel pin gas pressure or to mechanical interaction between the fuel and its containment. Thus the pressure PBBsof the fission gas in each grain-boundary bubble is given by the expression ps,, = 2 y/r + p,,, .
(4)
Assuming this gas behaves ideally, the number atoms N”’ per unit area of grain face when saturated may be written Nm
,4
f,f(e)r 3 ,,(?+p=J
of gas this is
D.
%=4/3(
F)“‘(
of grain size.
1-g).
Making the substitutions A = /laD=/8(
bh)3 N”‘, u = 4bh ( t,‘Da)“2,
cp= N/N”
eq. (6) becomes d+/du++=Au’
(7)
which, for the simplest solution
case when + = 0 at u = 0, has the
+=A[u2-2u+2-2exp(-u)].
(8)
It is interesting which is independent
(5)
By arguing that gas release will not occur until the intragranular gas atom concentration has exceeded the resolution barrier C” given by eq. (5) with N = Nm, Hargreaves and Collins [8] define the incubation time t, = C”/p = bX N “‘/2 Dp where /3 is the fission gas atom generation rate per unit volume. A rather more detailed treatment of gas atom accumulation on grain boundaries may be obtained using Speight’s [5] suggestion that the flux of atoms to the boundary from each grain is given by the flux in the absence of resolution [l] modified by the fraction (1 - Cx/Co) where Co is the concentration distant from the boundary. Since to a first approximation the Booth flux is 2/3( Dt/n)‘/= and Co = /3t the rate of accumulation of gas per unit area of boundary is
condition
to note that when us 1 the saturation N = N m or qb= 1 is reached when Au2 = 1,
D.M. Dowling et al. / Effect of re-solution on fission-gas release when the saturation time 1, is identical to that assumed by Hargreaves and Collins [8] and demonstrates that their assumptions are only valid when resolution is predominant. Olander (91 had attempted a more direct solution of the diffusion problem by making the simplifying assumption that in the narrow region of width X adjacent to the grain boundary the concentration profile is quasi-stationary, that is, that aC/at = 0 in the diffusion equation for that layer. In the notation used in eq. (8)
the solution
given by Olander
39
may be written
~=~{uz_~(l-~)+~[l-~+~(~)2] X[l-exp(g)
erfc(y)]}.
When resolution predominates ( I( z~ 1) this solution has the same limiting form as eq. (8). This is demonstrated in fig. 1 where +/A has been plotted as a function of u for various values of A’= 6X2/D, and from which it is also apparent that solutions (8) and (9) are essentially indistinguishable, provided b#l*/D e 0.01. 2.3. Gas release after grain-boundary saturation It was assumed by Hargreaves and Collins [8] that all gas arriving at grain boundaries once these had become saturated would be released, and a calculation by Turnbull [lo] tends to support this supposition. This calculation was of gas release after a period of irradiation at low temperature during which a uniform gas atom concentration Co was established throughout each grain. Turnbull further assumed that the effective grain-boundary concentration Cm = N m bX/2 D owing to resolution and showed that the fractional release F, that is the ratio of the number of gas atoms released to those generated in the period t since the start of the higher temperature irradiation, was given by the expression
F=l_$
5 d?-_ i Do
(CO- Cm)
n--l
X[l-exp( For t
MVS
and
(nr’)
-?)I. this equation
may
be written
more
X=0.0
(11) Fig. 1. The predicted variation of the density of gas atoms N held in bubbles on the grain face with (time)‘/‘, expressed through the quantities +/A and u defined in section 2.2, using eq. (9) due to Olander [9]. Curves are shown for different values of the parameter x = bA2/D and are compared with that marked MVS calculated from the simpler treatment due to Speight [S].
In the regime considered by Hargreaves and Collins [8] in which resolution is predominant, when grain-face saturation is reached the gas atom concentration profile will be more or less uniform across the grain. The Turnbull calculation is thus suitable to describe the subsequent gas release. In this instance however Co = Cm and, as shown by eq. (1 l), release will occur accord-
40
D.M. Dowling et al. / Effect of re-solution on fission-gas release
ing to the familiar formula due to Booth [I], but with I being measured from the moment that grain-boundary saturation occurs. When resolution is less predominant the profiles across the grain will be more arched at saturation and thus eqs. (10) and (11) will less accurately describe the gas-release process.
3. Tbe finite difference solution boundary
Although the model of Olander [9] attempts to describe the transient build-up of the concentration CA it is approximate in that it treats the concentration in the layer adjacent to the boundary as being quasi-static. Exact analytica treatment is not possible and so, if the accuracy of the approximate models is to be checked, numerical solution of the problem is desirable. In seeking such a solution it is possible either to assume that all atoms undergoing resolution are deposited a characteristic depth h from the grain boundary or that they are precipitated uniformly throughout a layer of thickness 2X. These approaches will be referred to respectiveiy as the ‘Olander model’ and the ‘Smeared model’,
JI *
‘II
b
JI = ‘ii
Resolution boundary
3.1. The Olander model
Chin boundary
The model is shown schematically in fig. 2(a). Solutions C, and C,, to the diffusion equations
ac, Da -r*c r* ar ( 1+P,=% act
(i = I, II)
(12)
Fig. 2. The two treatments of the resolution problem, (a) the Olander model, and (b) the smeared model.
entitled
are sought for regions I and II respectively, such that the concentration gradient is zero at the sphere centre, ac,,&=O
when 1=0,
the concentration C,,=O
at the grain boundary
(13)
3.2. The smeared model
(14)
In this case the boundary between the two regions is at r = a - 2X as shown in fig. 2(b). Solutions to eq. (12) are sought which satisfy conditions (13) and (14), but for which
is zero,
whenr-a,
and whilst the concentrations must be identical at depth X, the fluxes J, and J,, in fig. 2(a) differ by the resolution flux, that is C,=C,,whenr=a-X
CIIr=o--2X=CnIr=u-ah
(17)
and
(151
and 4aD(a--X)
“( ~Ir_~-~-~/,~a_-h)
=4ruZqN.
(16) In this model the production the same value 8.
rates in each region have
replace eqs. (15) and (16). However, the production rates are no longer the same in the two regions; they are given by the expressions &, = /? + bN/4X.
(19)
In each model, prior to saturation
of the grain faces,
P,=8
and
41
D.M. Dowling et al. / Effect of re-solution on fission-gas release
the value of N is calculated instantaneously by subtracting the integrated gas concentration in the grain from the total amount produced in it since the onset of irradiation. Once the saturation level N = N m has been reached, however, this value is assumed in eqs. (16) and (19), and all gas atoms arriving at the grain faces after this incubation time are deemed to have been released from the face for the purposes of the present calculation.
continuity conditions between the inner and outer regions, calculating the fluxes as first-order Taylor expansions about the adjacent points. In the Olander model, to speed up the calculation, the mesh was occasionally coarsened for one time step and the values of I,L at intermediate nodes estimated by interpolation. The generation of machine error necessitated the use of soluteconserving smoothing in the smeared model, thus removing noise and rounding errors from Ihe diffusion profiles.
3.3. The finite-difference calculation Making substitutions $=
DC/Da=;
each of the equations the simple form
4. Results and discussion
of the form
w = Dt/a’;
v = r/a
for C, and C,, may be written
in
(20) in both the Olander and smeared models, though in the latter the right-hand side of eq. (20) has the additional term bN/4/3A in the outer layer. In order to model accurately the steep concentration gradients in the vicinity of the grain boundary it is necessary that mesh points be bunched close together in that region. This was achieved in the present calculation by assuming the radial position v, of the n th mesh point is given by the expression ?&=v*(l
_&“‘I)/(1
_E),
where E < 1 and v is below the resolution layer. For the outermost region the mesh points were evenly spaced at the distance equal to the ultimate mesh spacing in the inner fuel region. The method of solution adopted was to represent the normalised concentration $ at the n th node by a quadratic expression in v passing through the (n - l)th, n th and (n + 1)th concentration levels. With the aid of this equation the differential coefficient a#Jdo may be evaluated by means of eq. (20), and the profile # updated by means of the first-order Taylor expansion ~~(o+Aw)=~~(w)+(a~~/aw)Aw.
(21)
The timestep Aw was chosen such that the fractional change in 4 did not exceed a value (Y, that is that Aa = +5/(a+~/a~)limln. In practice, when 100 mesh points were used, 01 was chosen not to exceed -0.3. In both the Olander and smeared models the concentration at the resolution boundary was not updated by means of eq. (21). Instead the value of $J was evaluated directly from the flux
The values of physical parameters used in the following illustrative calculations are listed in table 1, and were chosen more for convenience than for their absolute physical validity. The assumed temperature dependence of the gas atom diffusion coefficient is shown in fig. 3. 4.1. Gas release during the incubation period The variation with normalized time w = Dt/a’ of the fraction-face saturation + = N/N,,, is shown in figs. 4-6 for values of b = 10-6, lo-’ and lop4 s-’ respectively, for the single resolution depth X = lop8 m. Results obtained for the finite-difference calculations using the smeared model are shown as continuous curves and coincide with those obtained for the Olander model for all but the lowest temperature 800°C, for b = 10e6 s-’ and 10e4 s-‘. Also shown, as discrete points, are results calculated by means of eq. (8). Where the two finite-difference models disagree, at low temperatures where Table 1 Values of physical parameters assumed in example calculations Grain radius, a Fraction coverage at grain faces, f, Surface energy, y Semi-dehedral angle, B Gas atom diffusion coefficient, D
4 4 D2
E; Resolution Resolution
parameter, depth, X
b
5X10-6m 0.50 0.1 J m-’ 50” Do+& exp(-Q,/kT) + D2 edQ2/W 2.04X lo-” m2 s-’ 2.1 X lo-’ m2 SC’ 2.31 X lo-‘s m2 s-’ 3.85 eV 1.37 eV 10-6-10-4
s-’
1O-8-1O-5
m
D.M. Dowling et al. / Effeci of re-soiution on fission-gas release
/ IO-’
Fig. 5. The development of fractional for b = 10e4 s-’ and h = IO-’ m.
I
I.0
Invrrs*
x10-‘
*0
¶.O
mnp.rotw*
IO'
10-'
grain-face
saturation
+
release is strongly controlled by resolution, it may be seen from the diagrams that the Olander approach more nearly agrees with the predictions of eq. (S), which was derived on very similar physical assumptions. Comparing fig. 6 with the others, the removal of the
Fig. 3. The assumed temperature dependence of the gas atom diffusion coefficient used in the example calculations.
800 “C (Smeared
!
)
0
I
0.a Fig. 4. The fractional saturation cp = N/W” of gram faces by fission gas shown as a function of time through the parameter o = Dt/a’. The resolution parameter and depth are assumed to be 10m6 SC’ and 10-s m respectively.
Fig. 6. The time-dependence of fractional cp for b = 10m4 SC’ and h = 10-s m.
grain-face
saturation
AM, Dowiing et ai. / Effect of re-solution on fission-gas release temperature dependence because of the predominance of resolution is evident. In these circumstances the quantity of gas on the boundaries increases linearly with time, corresponding to large values of u in eq. (8). On the other hand the curves in fig.4 representing situations in which resolution is unimportant, exhibit the t312 dependence suggested by the diffusion model of Booth [l] for short irradiation times. From fig. 1 it appears that discrepancies between the finite-difference calculations and the approximate model described by eq. (8) will occur only for values of bA2/D > 0.1, and thus are likely to be worse the larger the resolution depth A. In fig. 7 the variation of + with w is drawn for T= 1OOO’C and bh= 10-‘3m s-’ for various values of X for which eq. (8) is a rather poor approximation. Included on this graph are discrete points calculated from the better analytic approximation given by eq. (9). Clearly, although the equation slightly overestimates the rate of gas accumulation, it provides a satisfactory description in this regime. The adequacy of the simple approximation given by eq. (8) is clearly demonstrated in fig. 8, in which most of the data from figs. 4-7 are presented. In this case the finite-difference results are shown as discrete points whilst the continuous curve has been calculated from eq. (8). The group of nine points represented by filled circles which lie to the left of the main body of data are for the extreme case when X = IO-’ m-‘. The dotted line associated with these points has been calculated from eq. (9) for the case when bA2/D = 0.92, and dem-
43
onstrates the validity of this expression at such high values of A. The temperature dependence of the incubation time necessary to reach grain-face saturation (+ = 1) is shown in fig.9 for X = 10e8 m and b = lo-*, 10e5 and 10m6 The slight variation between the finite difference S -‘. results for the Olander and smeared models has been omitted for b = 10e6 s-t for the sake of clarity. The predictions obtained solving eq. (8) when cp=: 1 are shown as discrete points in fig. 9 and demonstrate once again that, at least for h = 10d8 m, the overall agreement is excellent. 4.2. Gas release after grain-boundary saturation If the finite-difference calculations are extended beyond the point when the grain faces become saturated, fractional release curves are obtained as shown in fig. 10. By “fractional release” is meant the fraction of the total fission gas atoms generated in a grain which are either resident on the grain boundary or which, since saturation, have escaped from the body of the fuel. The calculations refer to fuel for which b= 10e5 s”“’ and X = 10P8 m. The point of incubation, shown by arrows in the diagram, is characterised by a fairly rapid change of gradient which is particularly striking at low temperatures. Included for the purpose of comparison is the equivalent prediction of the Booth model given by eq. (IO) with Co = Cm. The finite-difference results tend asymptotically to this curve at longer irradiation times.
Fig. 7. The time dependence of Q,for various values of resolution depth h calculated for bX = 1O-‘3 m s-’ and for a temperature of IOOO~C.
Fig. 9. The variation of incubation time with temperature for X = IO-* m and for various values of the resolution parameter h.
10
%
ia
provide the value of +, from which the fractional F is given by the expression F= 3cgNm/2aflt.
1c
(ii)
Following
release (221
incubation
at
time
t = tint the frac-
tional release is given by the sum of the quantities F in eq. (11) and (22) where, in the former, t is remeasured from rilx: and the concentrations Co and Cm are given by the expressions
rti
Co = Ptinc - 3Nm/2a Cm = bXNm/2D.
16
1C
16
.5
101
I
-1 10
10°
3
10'
I
s.9
,
ld
-v-
Fig. 8. The inclusion of most of the finite-difference calculation results shown in figs. 4-7 as discrete points on a single plot of $/A against U. The continuous and broken curves have been calculated from eqs. (8) and (9) respectively.
Eqs. (8) and (11) have been used to provide the discrete points in fig. 10 in the following manner; (i) For times before incubation, eq. (8) is used to
(23)
The agreement between eqs. (8) and (11) and the finite-difference calculations is very good at low temperatures but shows considerable discrepancy immediately after incubation at 12OO’C. The reason for this is because, at higher temperatures, the release of gas is more dependent upon diffusion, and by the time incubation has been achieved there will exist a marked variation in gas atom concentration across the grain. E?.q.(23) however effectively smears the profile into a uniform level at all points. At lower temperatures, when resolution predominates, the profile is always near-uniform and this explains the accuracy of the approximation. The predictions of the finite-difference and approximate calculations over times equivalent to threeyear irradiations at XKPC and 1200°C are compared in fig. Il. Again a value of X = lo-’ m has been assumed, and solid lines indicate the results of the finite-difference approach. The releases determined using eqs. (8)
D, M. Dowling et al. / Effect of re-solution an fission-gas release
0.c
10-J
10-L
10-z
10-t
-wFig. 10. Fractional release of fission gas atoms before and after grain-face saturation (indicated by arrows) for b = low5 s-’ and X = IO-* m. Continuous curves are from finite-element calculations whilst discrete points are derived from eqs. (8) and (11) as described in section 4.2.
and (11) are shown as broken lines, and also show the overestimate immediately after incubation has been achieved. It is encouraging that, for the 1200°C calculation with b = 10e6 s-‘, the finite-difference calculation
o-
., -
c /
u*y
I! -
._._._.
800
b
SO0 10-b
e
1200
d
1200
104 10-b
b I
II -
$0‘ 4
10-6
0
lo-’
-w-
lo-=
10-1
Fig. 1 I. The time-dependence of fractional release of fission gas atoms predicted numerically (&id lines) and analyticatly (broken lines) at 800°C and 1200°C for different values of the resolution parameter.
and eq. (11) differ by only 0.1% when o = 0.2, when the fractional release is 71%. This‘is surprising since eq. (11) is strictly only valid for ~~0.1 because of the mathematical simplifications made on its formulation. For curves labelled b, at 8OO’C when b = lop4 SK’, incubation was still incomplete after three years, and as with curves a, c and d the two approaches are in very acceptable agreement. Finally, in fig. 12, fractional release curves are shown for computations at lOOO”C, for h = 10e8 m except in the one case indicated. The discrete points calculated from eq. (8) and (i 1) are again in excellent agreement with the numerical results shown in solid lines except immediately following incubation. Included in fig. 12 are also the results for b = low6 and X = lo-‘, comparing the finite-difference predictions (broken line) with those of eq. (9). It is interesting to note that this approximate equation correctly predicts the presence of a ~nimum in the fractional release prior to incubation as suggested by the more accurate calculation. In general then the simple analytic models discussed in section 2 very adequately describe gas release from UO, being irradiated under steady reactor conditions. The application of these same ideas to variable conditions, of the kind experienced in practice even when the irradiation is nominally steady, needs more complex treatment and is considered elsewhere [ 111.
D. M. Dowling et al. / Effect of re-solurion on fission-gas
46
(2) Under such low-temperature conditions, the grain-boundary concentration increases linearly with time, whereas at higher temperatures the t3i2 dependence characteristic of diffusional release is exhibited. (3) Prior to saturation of the grain boundaries by fission gas there is excellent agreement between the numerical and analytical models, though differences begin to emerge both as resolution begins to dominate diffusional release, that is at low temperatures, and as the resolution depth is increased. (4) The simple model used to describe fission gas behaviour following grain-boundary saturation tends at higher temperatures to overestimate the release rate immediately following this event, but accurately represents the situation at later times.
0.2!
0.2,
f
O.,!
L
z
s .;
O.IO
release
I -
I;
O.OI
Acknowledgement
10‘
10-J
lo-‘
1
-W-
Fig. 12. Fractional
releases of fission gas at 1000°C.
5. Conclusion
Computer calculations at diffusional fission gas release to grain boundaries in UO, fuel irradiated under steady reactor conditions are reported. In the analysis two types of irradiation-induced resolution models were considered, one in which the affected atoms residing in grain-boundary bubbles are deposited at a single resolution depth h and the other in which these atoms are deposited uniformly over a depth 2h. The results of each approach have been compared with each other and with the predictions of simple analytical models, and the following conclusions drawn. (1) The numerical calculations were insensitive to the resolution model chosen except at low temperatures where, since gas atom diffusion is very slow, release is strongly controlled by resolution.
This paper is published by permission Electricity Generating Board.
of the Central
References
111A.H. Booth, AECL Rep. 496 (1957). [21 R.S. Barnes, J. Nucl. Mater. 11(1964) 135. [31 A.D. Whapham, Nucl. Appl. 2 (1966) 125. [41 G.R. Reynolds and G.H. Bannister, J. Mater. Sci. 5 (1970) 84. [51 M.V. Speight, Nucl. Sci. Engrg. 3 1 (1969) 180. [61 A.D. Whapman and B.E. Sheldon, UKAEA Rep. AERE R4970 (1965). Turnbull and R.M. Cornell, CEGB Rep. [71 J.A. RD/B/N1716 (1970). [81 R. Hargreaves and D.A. Collins, J. Brit. Nucl. Energy Sci. 15 (1976) 311. Aspects of Nuclear Reactor [91 D.R. Olander, Fundamental Fuel Elements (ERDA, USA, 1976) p. 314. [lOI J.A. Turnbull, J. Nucl. Mater. 50 (1974) 63. [ill R.J. White, D.J. Dowling and M.O. Tucker, to be published (1982).