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Evidence for a helical crystallographic texture in uranium bars of magnox clad fuel elements
JOURNAL
OF NUCLEAR
EVIDENCE
Central
20 (1966)
MATERIALS
307-313.
FOR A HELICAL
OF MAGNOX
F.
C. DUCKWORTH
Generating
Board,
effects
of irradiation
bars containing analysed.
The
of the resulting and
compared
on uranium
a helical relationship axial
an empirical
mined from experimental fuel
elements
Bradwell
reactors.
from
that
about
texture
the nature
and of this
3 June
3t l’appui hhlicoidale.
Der
evidence
Cette
Einflusz
enthielten Textur.
to be
geleitet de
l’irradiation
1’618ment combustible cristallographique relation axiale relation
entre la grandeur empirique
des
barreaux
contenant sont
obtenue par
et deform&
it partir
des
une texture
les barres
cristallographique
fournit
de plus
sur la nature
Reaktorbestrahlung
eine
schraubenfarmige
des
de cette
axialen
und
Diese
empirische
don&es
in
erhalten
stiitzt
UranstBbe
eine
besitzen
Riickschliisse
des reacteurs de Berkeley
ab-
Beziehung wurde
aus
und
Bradwell-
waren.
deutlich
Informationen
den resul-
die an bestimmten
die Hypothese,
schraubenf6rmige
phische Textur
com-
Beziehung
Berkeley-
worden
Brenn-
Die Proben
Spannungen
bestimmten
Daten entnommen,
Der Vergleich die
zwischen
torsionalen
und mit einer empirisch
Reaktoren
auf
kristallographische
die Beziehung
Brennstoffelementen
La A une
Blbments
einer
experimentellen
resultantes
et comparee
certains
dans
une texture
analysees.
des deformations
est ddduite
fournies
bustibles irradi&
1.
uranium helicoidale
et de torsion
experimentales
sur
des preuves
laquelle
aus Uran wurde untersucht.
verglichen. effets
UK
donne
suivant
comparaison
Es wurde
tierenden
deduced.
Les
comparaison
supplementaires
elementstiibe
a
further
texture
Glos.,
texture.
and
bars contain
La
contiennent
informations
irradiated
enables
IN URANIUM
Berkeley,
de l’hypothkse
d’uranium
deter-
Berkeley
CO., AMSTERDAM
1966
et de Bradwell.
are
is derived
provides
the uranium
Laboratories,
magnitudes
strains
the
SPEIGHT
Nuclear
element
relationship
The comparison
crystallographic
information
the
TEXTURE
and M. V.
Berkeley
texture
data from certain
discharged
for the hypothesis helical
between
and torsional
with
fuel
crystallographic
PUBLISHING
CLAD FUEL ELEMENTS
Received
The
NORTH-HOLLAND
CRYSTALLOGRAPHIC
BARS
Electricity
0
und ermiiglicht auf
die
dass
kristallogradurch weitere Natur
dieser
Textur.
Introduction
Post-irradiation investigations on uraniummagnox fuel elements irradiated in CEGB reactors have revealed the incidence of untwist in the helically finned cans. Speight, Hines and Greenwood 1) have attributed this to a corresponding torsion in the uranium bar caused by irradiation growth effects resulting from the presence of a preferred crystallographic orientation in the form of a helix. They demonstrated how this arrangement of grains could lead to bar torsion but did not consider how such a texture could also produce changes in length and diameter of the bar. In the particular case when the orthogonal helices containing respectively a preponderence of [OlO] (“growth”) and [ 1001 (“shrinkage”) crystallographic di307
rections are each inclined at 45” to the bar axis, the effect of irradiation is to produce bar torsion only. In the more general case, however, when the helices are not symmetrically inclined with respect to the bar axis, dimensional changes will accompany bar torsion. For a particular orientation of the helices the contribution to the overall bar dimensional changes from this source is therefore related to the concurrent bar torsion. This relationship is derived below and used in the analysis of fuel element monitoring data. 2.
Theoretical changes
prediction
of dimensional
The texture of a section of the bar is illustrated in fig. 1. AC represents the bar circumference,
I?. C.
308 so that
in effect
DUCKWORTH
A and C coincide,
AND
helices
of which
AB
and
On irradiation,
and the
contraction
portional
to their original
CB
solved
of CB
of AB
by amounts
lengths
with bar torsion
pro-
If the growth
and shrinkage
amount
dir, of AB has a, re-
representing
an increase
is to increase dr given
in
the bar radius
by an
by
Zndr = dir cos 8 - dla sin 8.
change.
directions
component
net effect
result in the
and length
in length,
has a resolved component corresponding to a decrease dla sin 19in the circumference and the
movement of the point B relative to points A and C. Components of this movement can be identified
YPEIGHT
circumference by an amount dli cos 8. Simultaneous contraction of BC by an amount dZz
are respective
the extension
V.
An increase
and y re-
presents the length of the bar between successive intersections of the growth and shrinkage portions.
M.
The length
are
symmetrically aligned with respect to the bar axis, then the resultant irradiation growth strain has no component parallel to this axis and hence no change in radius occurs. The complete displacement of point B can then be represented in fig. 1 (with O-45”) with vectors A6 and 6C representing the displacement components in the growth and shrinkage directions respectively. The resultant AC then a purely torsional motion. When signifies 0 # 45”, however, an overall length change must occur and this will be accompanied by a change in radius. If this radius changes, dr, is positive, it can accommodate a fraction fr of the total length increase dir along AB, so that only the remaining fraction of l-fr manifests itself as a movement along the direction AB to produce bar torsion and length change. Similarly, an increase in bar radius causes movement of the point B in the direction BC, in addition to that engendered by the decrease in length, dls=dlr tan 8, of BC.
increase
increase
(I)
of AB absorbed
by this
in radius is 2n cos 0dr so that only the
remaining portion, dsr = dir - 2n cos Bdr, is accommodated by movement of the material in the direction AB. Prom eq. (l), dsr = dir - (dir cos = 2dlr sin2 0.
8-
d/asin 0)
cos e
(2)
The concurrent movement dss of the point B in the direction BC comprises the irradiation induced shrinkage d/s of the length BC plus a component 27~sin 8dr resulting from the increase in bar radius. Hence from eq. (l), dss = dla + (dli = 2dla co92
cos
8 -
d12 sin 0) sin 0
8.
(3)
The actual magnitude and direction of the movement of point B in the bar surface is compounded of the movements dsr along AB and dss along BC. Since dir =lclr and dls= kls, where k describes the net irradiation growth strain along a growth or shrinkage helix, and from fig. 1, II= y/sin 0 and 12= y/cos 0, then dsi
bar
t [OIOJ
2
OXil
1
(growth
(sh!it direct
Fig.
1.
Helical
direction)
faq/ ion)
texture
B
of uranium
bar.
Fig.
2.
Vector representation
of movement
of point B.
HELICAL
and dss are respectively
CRYSTALLOGRAPHIC
TEXTURE
2ky sin 0 and 2ky cos 0.
Hence the displacement of B with respect to A and C is 2ky and its direction is such that it makes an angle (ISO’-- 20) with the direction of the bar axis. The resultant represented
motion
and its components
by a geometrical
fig. 2 which is developed
construction
is in
from fig. 1. D’B’ and
B’E’ represent in both direction and magnitude the components of the movement of B along
dy/y=
2k cos (180” - 20).
(4)
Similarly, resolving perpendicularly to the bar axis gives the degree of rotation of B with respect to A and C. The shear strain 4 is the torsion incurred by the bar per unit length and is therefore given by 4 = 2k sin (180” - Se). The fractional the bar can be noting from fig. comes d+=k =
(5)
radius increase exhibited by calculated from eq. (l), and 1 that Ii= 2nr cos 0, this be(~0~20-sin28) -k cos (isoo- 28).
(6)
Eqs. (4) and (6) are consistent with conservation of volume. The constant k may be written as k = ko( T)It,,
(7)
where ko(T) is a function of temperature and represents the strain per unit irradiation dose for a uranium single crystal, I is the irradiation dose and tz describes the fractional departure
URANIUM
from randomness
309
BARS
in the arrangement
along the prescribed
of grains
helical directions.
Eqs. (4) and (5) may be used to predict the observed relationship between the axial and torsional
strains in an irradiated
fuel element.
The axial strain, E, consists of a contribution due to the helical
texture
given
by dy/y
in
eq. (4) in addition to the strain EOderived from other sources, e.g. swelling and irradiation creep. Hence,
the growth and shrinkage directions respectively, with D’E’ their resultant. It is seen that if the angle of the growth helix is greater than 45”, bar torsion is accompanied by axial expansion and when 0 is less than 45” axial contraction occurs. The situation 8= 45” represents the condition for maximum bar torsion for a given irradiation strain, with no overall change in length or radius. The increase in length, dy, of a section of bar of original length y is obtained by resolving the resultant motion, 2ky, along the direction of the bar axis. Hence,
IN
&=&g+2ko(T)Its
COS
(m”-28).
(8)
If h and r are the length and radius respectively of the fuel element bar, the measured angle of can untwist, a degrees, is given from eq. (5) by 7~. (a+oco)/(lsO h)=$ = 2ko (T) It, sin (180” - 20). where
(9)
0~0is the angle of bar twist produced
during manufacture. Combining eqs. (8) and (9) we obtain the relations &/I = &O/I+ (+/I) Cot (180” - 20) [(E- ~a)/112+ [d/l]” = 4k&?.
(IO) (11)
Assuming that EOis porportional to irradiation dose then so/I is constant. Hence a plot of E/I against +/I for elements in a particular position in a reactor channel should yield a straight line if the texture helix angle is constant and tz is allowed to vary [eq. (lo)], or a circle radius 2k& if the degree of texture is constant and 0 allowed to vary [eq. (ll)]. These two cases are illustrated in fig. 3. If, as has been suggested r), the proposed helical texture arises as a consequence of the bar manufacturing process, then because of the close control on this process 2) it is expected that 8 should be roughly constant. From eqs. (8) and (9) therefore it follows that the existence in practice of a positive linear correlation between E/I and 4/I for fuel elements from a particular channel position can be unambiguously attributed to the texture factor tz, this being the only factor which is contained in the expressions for E/I and 4/I and which can vary from one irradiated element to another. The existence of such a correlation
310
F.
C’.
DI-CKWORTH
AND
ill.
3.
Y.
SPEIGHT
Results The relationship
untwist data
+x
was
from
between
analysed elements
axial
strain
st,atisticnlly from
certain
and
using the channel
positions discharged from the Berkeley and Bradwell reactors 3). Axial strains were derived from the measured length changes and torsional strains + were derived from measured angles of twist or untwist having allowed for the most probable
twist initially
using the relation
Fig.
3.
Theoretically
derived
relationship
axial and torsional
between
strains.
is then indicative of the presence of the postulated helical texture. It is seen from fig. 1 that the growth helix must be left-handed for the bar torsion produced to promote untwist of the can (the fins of which are right-handed helices). Regarding this type of torsion as positive and defining 0 as the angle between the growth helix and a plane perpendicular to the bar axis, then 0~ 8 < 45” produces length decrease, whereas 45” < 8 < 90” gives rise to an increase, as illustrated in fig. 4. Using this diagram, information about the nature and extent of the texture can be derived from measurements of E/I and +/I for a particular set of fuel element bars. BAR GROWTH CAN TWIST.
9
EAR CAN
t
GROWTH UNTWIST
Fig.
4.
Relationship
BAR CAN
between
torsion and texture
length
SHRINKAGE UNTWIST.
change,
orientation.
$ =~nr(a i- a~)/( 180 h). Errors
exist in the measured values of both F/I and +/I, the principal error in the value of C$arising from an uncertainty in the value of 20 of & 3”. Regression analysis was used to test the linear correlation between F/I and +/I predicted by eq. (10). The analysis was conducted for sets of elements from six channel positions from the Berkeley and four from the Bradwell reactors, numbered [l]-[B] and [l]-[4] respectively from the bottom of the channels. The results are shown in table 1. For all Berkeley elements the correlation coefficient is greater than 0.5 for all positions. For the Bradwell elements the values are not so high. but this is due in part to the greater numbers of elements examined. The first notable feature is that for each of the ten sets of data analysed, a positive correlation coefficient is obtained. With zero correlation there would be an equal chance of any set giving a negative coefficient. The population correlation coefficients, Q, were obtained from table A-17 of “Experimental Statistics” by Natrella 4). The mean value of Q is seen to be
(riqht grort
BAR SHRINKAGE CAN TWIST.
present on production,
bar
positive for all positions, and the 95 y0 confidence limits for positive correlation do not contain zero for all Berkeley positions and for Bradwell positions [Z] and [4]. For Bradwell [l] and [3] a positive correlation coefficient does exist but the certainty of being able to interpret this as indicative of the existence of a linear relationship is less than for the other positions. It can be concluded that the statistical correlation between axial and torsional strains is sufficiently high and consistent over all sets of data to confirm the relationship predicted by eq. (10).
HELIC4L
CRYSTALLOGRAPHIC
TEXTURE
IN
URANIUM
311
BARS
TABLE 1 Results
Bradwell
4.
Population
correlation
coefficient
Sample
Numbers of elements
coefficient
examined
[ 1]
+ 0.56
18
+0.13
f0.80
PI
f0.75
18
+0.42
+0.90
$0.66
[31
+0.55
18
+0.13
+o.t30
+ 0.47
[41
1-0.71
19
$0.36
to.87
+0.62
[51
+0.52
17
+ 0.02
+0.79
+0.41
[61
+0.52
17
+ 0.02
+ 0.79
+0.41
in
channel Berkeley
analysis.
correlation
Element position
of statistical
(95 9‘36 confidence limits) to
from
:
mefm $0.47
[ 1]
+0.29
40
-0.04
+0.55
+ 0.26
[21
to.36
44
+0.06
+ 0.59
+0.33
r31
to.26
32
-0.13
+0.56
+0.22
[41
+ 0.47
34
j-0.12
$0.70
+0.41
L
Discussion
The linearity of the E/I vs +/I relationship is dependent upon EO being proportional to irradiation. This assumption is valid for Berkeley elements whose length changes exhibit a linear dependence on irradiation but is rather a crude approximation for the Bradwell elements 3). This would account for the lower correlation coefficients obtained for the sets of Bradwell elements. Fig. 5 shows a typical plot of E/I vs 4/I for irradiated elements discharged from a particular channel position with the abscissae and ordinates having the same scales. As 8 is liable to vary slightly from one element to another, regression analysis does not yield a single line representing the functional relationship predicted by eq. (10). Two regression lines are obtained as shown in fig. 5 and the line through the mean bisecting the angle between them gives a rough indication of the empirically predicted relationship between the two variables. Comparing this with figs. 3 and 4 gives information about the nature of the texture, and allows rough estimates of 0 and EOto be made. The line AA’ from fig. 5 is superimposed on fig. 4. Most of this appears in the quadrant 8=45”90” indicating that the growth direction of the texture, i.e. the [OlO] helix, is more axially aligned than the shrinkage di-
*y-4o, Fig.
5.
Typical
i,
, , :
axial strain (e/I) vs torsional strain
(+/I) plot for a set of elements irradiated under similar conditions to a burn-up of 1000 MWD/tonne.
rection. Thus the texture is generally so disposed as to produce bar growth and can untwist. A small proportion of the elements contribute points to the quadrant 8 = 135’-1 SO”, i.e. they undergo simultaneous bar shrinkage
312
/ i
Element position
Mean element
in
t,emperature
channel
Berkeley
Bra&elf
texture
rim (” C)
Est~imatc of
’
helix
(deg) [l]
210
121
250
131 [41 151 I61
290
68 67 65’ 61% 69 72
320 360 390
[l ]
255
121
330
[31 [41
455
676 68 70 66
410
L
’
angle, 0
I
Intercept
on
I
EIJII((lh) -0.15 -0.18 -0.11 - 0.08 0.00
0.07 -0.58 -0.72 -0.46 -0.13
Calculated percentage
E/I axis
j
texture
of
(So)
0.3 0.3 0.2 0.3 0.3 0.5 0.3 0.2 0.5 0.3
L
and can twist. This suggests that in some elements the growth and shrinkage directions are the reverse of the con~g~rations described above, so that the shrinkage direction is the more axially aligned. The angle of the texture, 8, can be estimated from the fact that AA’ is inclined at (180” - 20) to the E/I axis, as shown in fig. 3. Values of 8 measured from each set of data are shown in table 2. The constancy of these values is notable, their means being 67” -& 3” for the Berkeley elements and 68” -$: lo for the Bradwell ones, or 68” & 3” for all the elements, i.e. the [OlO] preferred direction is inchned at about 22” to the bar axis irrespective of the fuel element’s reactor history. It appears therefore that the helical texture arises as a consequence of the manufacturing process. The intercepts, corresponding to CO/I in eq. (10) and illustrated in fig. 3, have been measured for all sets of data and are given in table 2. They are seen to be generally negative, i.e. elements with no helical texture undergo a shrinkage. This is consistent with the existence of an axially aIigned erystallogra~hic texture in the core of the bars producing a shrinkage on irradiation. At the lowest irradiation temperatures this overrides the tendency to increase in length due to fission-gas swelling. Bradwell elements further undergo irradiation creep due
to the stack load they have to support. Thus the negative intercepts are greater than for the Berkeley elements. The circles of fig. 4 a,re of radii 2tEot,, so the average degree of texture, tz, may be calculated from the mean point of the plots like fig. 5 for each element position using experimentally derived values of k&“) for the appropriate temperatures 5). These are listed in table 2. As tz may in practice decrease with irradiation, then these values will be slight underestimates of the actual percentage textures. The amount of texture necessary to produce the average untwist is seen to be about 0.3 */* and the larger observed untwists correspond to a net growth texture of about 1 yo.
5. 1.
Conclusions Untwist in irradiated uranium-magnox fuel elements arises as a consequence of a helical crystallographic texture existing in the uranium bar.
2. The texture probably originates from the bar manufacturing process and is such that the grains are preferentially orientated with their [OlO] directions at about 22’ to the bar axis. 3. On average
the degree of texture
is about
HELICAL
CRYSTALLOGRAPHIC
TEXTURE
IN
URANIUM
313
BARS
0.3 y,, but it may be as high as 1 oh in some
References
elements.
l) M .V. Speight, G. F. Hines and G. W. Greenwood, J. Nucl.
Acknowledgements
and V. W. (1963)
The authors are grateful to Dr. G. W. Greenwood for reading the manuscript and making many helpful comments and criticisms. This paper is published by permission Electricity Generating Board.
Mat.
2, H. K. Hardy,
of the Central
20 (1966)
126
J. F. W.
Eldred,
Bishop,
J. Brit.
private
4, M. G. Natrella, Department
9
Pickman
Energy
Sot. 2
33
3, G. F. Hines, J. C. Wood, Uglow,
D. 0.
Nucl.
“Experimental
of Commerce,
S. N. Buckley,
J. Yelland
and A.
G.
communication
private
NBS
Statistics”, (1963)
communication
U.S.
OF NUCLEAR
EVIDENCE
Central
20 (1966)
MATERIALS
307-313.
FOR A HELICAL
OF MAGNOX
F.
C. DUCKWORTH
Generating
Board,
effects
of irradiation
bars containing analysed.
The
of the resulting and
compared
on uranium
a helical relationship axial
an empirical
mined from experimental fuel
elements
Bradwell
reactors.
from
that
about
texture
the nature
and of this
3 June
3t l’appui hhlicoidale.
Der
evidence
Cette
Einflusz
enthielten Textur.
to be
geleitet de
l’irradiation
1’618ment combustible cristallographique relation axiale relation
entre la grandeur empirique
des
barreaux
contenant sont
obtenue par
et deform&
it partir
des
une texture
les barres
cristallographique
fournit
de plus
sur la nature
Reaktorbestrahlung
eine
schraubenfarmige
des
de cette
axialen
und
Diese
empirische
don&es
in
erhalten
stiitzt
UranstBbe
eine
besitzen
Riickschliisse
des reacteurs de Berkeley
ab-
Beziehung wurde
aus
und
Bradwell-
waren.
deutlich
Informationen
den resul-
die an bestimmten
die Hypothese,
schraubenf6rmige
phische Textur
com-
Beziehung
Berkeley-
worden
Brenn-
Die Proben
Spannungen
bestimmten
Daten entnommen,
Der Vergleich die
zwischen
torsionalen
und mit einer empirisch
Reaktoren
auf
kristallographische
die Beziehung
Brennstoffelementen
La A une
Blbments
einer
experimentellen
resultantes
et comparee
certains
dans
une texture
analysees.
des deformations
est ddduite
fournies
bustibles irradi&
1.
uranium helicoidale
et de torsion
experimentales
sur
des preuves
laquelle
aus Uran wurde untersucht.
verglichen. effets
UK
donne
suivant
comparaison
Es wurde
tierenden
deduced.
Les
comparaison
supplementaires
elementstiibe
a
further
texture
Glos.,
texture.
and
bars contain
La
contiennent
informations
irradiated
enables
IN URANIUM
Berkeley,
de l’hypothkse
d’uranium
deter-
Berkeley
CO., AMSTERDAM
1966
et de Bradwell.
are
is derived
provides
the uranium
Laboratories,
magnitudes
strains
the
SPEIGHT
Nuclear
element
relationship
The comparison
crystallographic
information
the
TEXTURE
and M. V.
Berkeley
texture
data from certain
discharged
for the hypothesis helical
between
and torsional
with
fuel
crystallographic
PUBLISHING
CLAD FUEL ELEMENTS
Received
The
NORTH-HOLLAND
CRYSTALLOGRAPHIC
BARS
Electricity
0
und ermiiglicht auf
die
dass
kristallogradurch weitere Natur
dieser
Textur.
Introduction
Post-irradiation investigations on uraniummagnox fuel elements irradiated in CEGB reactors have revealed the incidence of untwist in the helically finned cans. Speight, Hines and Greenwood 1) have attributed this to a corresponding torsion in the uranium bar caused by irradiation growth effects resulting from the presence of a preferred crystallographic orientation in the form of a helix. They demonstrated how this arrangement of grains could lead to bar torsion but did not consider how such a texture could also produce changes in length and diameter of the bar. In the particular case when the orthogonal helices containing respectively a preponderence of [OlO] (“growth”) and [ 1001 (“shrinkage”) crystallographic di307
rections are each inclined at 45” to the bar axis, the effect of irradiation is to produce bar torsion only. In the more general case, however, when the helices are not symmetrically inclined with respect to the bar axis, dimensional changes will accompany bar torsion. For a particular orientation of the helices the contribution to the overall bar dimensional changes from this source is therefore related to the concurrent bar torsion. This relationship is derived below and used in the analysis of fuel element monitoring data. 2.
Theoretical changes
prediction
of dimensional
The texture of a section of the bar is illustrated in fig. 1. AC represents the bar circumference,
I?. C.
308 so that
in effect
DUCKWORTH
A and C coincide,
AND
helices
of which
AB
and
On irradiation,
and the
contraction
portional
to their original
CB
solved
of CB
of AB
by amounts
lengths
with bar torsion
pro-
If the growth
and shrinkage
amount
dir, of AB has a, re-
representing
an increase
is to increase dr given
in
the bar radius
by an
by
Zndr = dir cos 8 - dla sin 8.
change.
directions
component
net effect
result in the
and length
in length,
has a resolved component corresponding to a decrease dla sin 19in the circumference and the
movement of the point B relative to points A and C. Components of this movement can be identified
YPEIGHT
circumference by an amount dli cos 8. Simultaneous contraction of BC by an amount dZz
are respective
the extension
V.
An increase
and y re-
presents the length of the bar between successive intersections of the growth and shrinkage portions.
M.
The length
are
symmetrically aligned with respect to the bar axis, then the resultant irradiation growth strain has no component parallel to this axis and hence no change in radius occurs. The complete displacement of point B can then be represented in fig. 1 (with O-45”) with vectors A6 and 6C representing the displacement components in the growth and shrinkage directions respectively. The resultant AC then a purely torsional motion. When signifies 0 # 45”, however, an overall length change must occur and this will be accompanied by a change in radius. If this radius changes, dr, is positive, it can accommodate a fraction fr of the total length increase dir along AB, so that only the remaining fraction of l-fr manifests itself as a movement along the direction AB to produce bar torsion and length change. Similarly, an increase in bar radius causes movement of the point B in the direction BC, in addition to that engendered by the decrease in length, dls=dlr tan 8, of BC.
increase
increase
(I)
of AB absorbed
by this
in radius is 2n cos 0dr so that only the
remaining portion, dsr = dir - 2n cos Bdr, is accommodated by movement of the material in the direction AB. Prom eq. (l), dsr = dir - (dir cos = 2dlr sin2 0.
8-
d/asin 0)
cos e
(2)
The concurrent movement dss of the point B in the direction BC comprises the irradiation induced shrinkage d/s of the length BC plus a component 27~sin 8dr resulting from the increase in bar radius. Hence from eq. (l), dss = dla + (dli = 2dla co92
cos
8 -
d12 sin 0) sin 0
8.
(3)
The actual magnitude and direction of the movement of point B in the bar surface is compounded of the movements dsr along AB and dss along BC. Since dir =lclr and dls= kls, where k describes the net irradiation growth strain along a growth or shrinkage helix, and from fig. 1, II= y/sin 0 and 12= y/cos 0, then dsi
bar
t [OIOJ
2
OXil
1
(growth
(sh!it direct
Fig.
1.
Helical
direction)
faq/ ion)
texture
B
of uranium
bar.
Fig.
2.
Vector representation
of movement
of point B.
HELICAL
and dss are respectively
CRYSTALLOGRAPHIC
TEXTURE
2ky sin 0 and 2ky cos 0.
Hence the displacement of B with respect to A and C is 2ky and its direction is such that it makes an angle (ISO’-- 20) with the direction of the bar axis. The resultant represented
motion
and its components
by a geometrical
fig. 2 which is developed
construction
is in
from fig. 1. D’B’ and
B’E’ represent in both direction and magnitude the components of the movement of B along
dy/y=
2k cos (180” - 20).
(4)
Similarly, resolving perpendicularly to the bar axis gives the degree of rotation of B with respect to A and C. The shear strain 4 is the torsion incurred by the bar per unit length and is therefore given by 4 = 2k sin (180” - Se). The fractional the bar can be noting from fig. comes d+=k =
(5)
radius increase exhibited by calculated from eq. (l), and 1 that Ii= 2nr cos 0, this be(~0~20-sin28) -k cos (isoo- 28).
(6)
Eqs. (4) and (6) are consistent with conservation of volume. The constant k may be written as k = ko( T)It,,
(7)
where ko(T) is a function of temperature and represents the strain per unit irradiation dose for a uranium single crystal, I is the irradiation dose and tz describes the fractional departure
URANIUM
from randomness
309
BARS
in the arrangement
along the prescribed
of grains
helical directions.
Eqs. (4) and (5) may be used to predict the observed relationship between the axial and torsional
strains in an irradiated
fuel element.
The axial strain, E, consists of a contribution due to the helical
texture
given
by dy/y
in
eq. (4) in addition to the strain EOderived from other sources, e.g. swelling and irradiation creep. Hence,
the growth and shrinkage directions respectively, with D’E’ their resultant. It is seen that if the angle of the growth helix is greater than 45”, bar torsion is accompanied by axial expansion and when 0 is less than 45” axial contraction occurs. The situation 8= 45” represents the condition for maximum bar torsion for a given irradiation strain, with no overall change in length or radius. The increase in length, dy, of a section of bar of original length y is obtained by resolving the resultant motion, 2ky, along the direction of the bar axis. Hence,
IN
&=&g+2ko(T)Its
COS
(m”-28).
(8)
If h and r are the length and radius respectively of the fuel element bar, the measured angle of can untwist, a degrees, is given from eq. (5) by 7~. (a+oco)/(lsO h)=$ = 2ko (T) It, sin (180” - 20). where
(9)
0~0is the angle of bar twist produced
during manufacture. Combining eqs. (8) and (9) we obtain the relations &/I = &O/I+ (+/I) Cot (180” - 20) [(E- ~a)/112+ [d/l]” = 4k&?.
(IO) (11)
Assuming that EOis porportional to irradiation dose then so/I is constant. Hence a plot of E/I against +/I for elements in a particular position in a reactor channel should yield a straight line if the texture helix angle is constant and tz is allowed to vary [eq. (lo)], or a circle radius 2k& if the degree of texture is constant and 0 allowed to vary [eq. (ll)]. These two cases are illustrated in fig. 3. If, as has been suggested r), the proposed helical texture arises as a consequence of the bar manufacturing process, then because of the close control on this process 2) it is expected that 8 should be roughly constant. From eqs. (8) and (9) therefore it follows that the existence in practice of a positive linear correlation between E/I and 4/I for fuel elements from a particular channel position can be unambiguously attributed to the texture factor tz, this being the only factor which is contained in the expressions for E/I and 4/I and which can vary from one irradiated element to another. The existence of such a correlation
310
F.
C’.
DI-CKWORTH
AND
ill.
3.
Y.
SPEIGHT
Results The relationship
untwist data
+x
was
from
between
analysed elements
axial
strain
st,atisticnlly from
certain
and
using the channel
positions discharged from the Berkeley and Bradwell reactors 3). Axial strains were derived from the measured length changes and torsional strains + were derived from measured angles of twist or untwist having allowed for the most probable
twist initially
using the relation
Fig.
3.
Theoretically
derived
relationship
axial and torsional
between
strains.
is then indicative of the presence of the postulated helical texture. It is seen from fig. 1 that the growth helix must be left-handed for the bar torsion produced to promote untwist of the can (the fins of which are right-handed helices). Regarding this type of torsion as positive and defining 0 as the angle between the growth helix and a plane perpendicular to the bar axis, then 0~ 8 < 45” produces length decrease, whereas 45” < 8 < 90” gives rise to an increase, as illustrated in fig. 4. Using this diagram, information about the nature and extent of the texture can be derived from measurements of E/I and +/I for a particular set of fuel element bars. BAR GROWTH CAN TWIST.
9
EAR CAN
t
GROWTH UNTWIST
Fig.
4.
Relationship
BAR CAN
between
torsion and texture
length
SHRINKAGE UNTWIST.
change,
orientation.
$ =~nr(a i- a~)/( 180 h). Errors
exist in the measured values of both F/I and +/I, the principal error in the value of C$arising from an uncertainty in the value of 20 of & 3”. Regression analysis was used to test the linear correlation between F/I and +/I predicted by eq. (10). The analysis was conducted for sets of elements from six channel positions from the Berkeley and four from the Bradwell reactors, numbered [l]-[B] and [l]-[4] respectively from the bottom of the channels. The results are shown in table 1. For all Berkeley elements the correlation coefficient is greater than 0.5 for all positions. For the Bradwell elements the values are not so high. but this is due in part to the greater numbers of elements examined. The first notable feature is that for each of the ten sets of data analysed, a positive correlation coefficient is obtained. With zero correlation there would be an equal chance of any set giving a negative coefficient. The population correlation coefficients, Q, were obtained from table A-17 of “Experimental Statistics” by Natrella 4). The mean value of Q is seen to be
(riqht grort
BAR SHRINKAGE CAN TWIST.
present on production,
bar
positive for all positions, and the 95 y0 confidence limits for positive correlation do not contain zero for all Berkeley positions and for Bradwell positions [Z] and [4]. For Bradwell [l] and [3] a positive correlation coefficient does exist but the certainty of being able to interpret this as indicative of the existence of a linear relationship is less than for the other positions. It can be concluded that the statistical correlation between axial and torsional strains is sufficiently high and consistent over all sets of data to confirm the relationship predicted by eq. (10).
HELIC4L
CRYSTALLOGRAPHIC
TEXTURE
IN
URANIUM
311
BARS
TABLE 1 Results
Bradwell
4.
Population
correlation
coefficient
Sample
Numbers of elements
coefficient
examined
[ 1]
+ 0.56
18
+0.13
f0.80
PI
f0.75
18
+0.42
+0.90
$0.66
[31
+0.55
18
+0.13
+o.t30
+ 0.47
[41
1-0.71
19
$0.36
to.87
+0.62
[51
+0.52
17
+ 0.02
+0.79
+0.41
[61
+0.52
17
+ 0.02
+ 0.79
+0.41
in
channel Berkeley
analysis.
correlation
Element position
of statistical
(95 9‘36 confidence limits) to
from
:
mefm $0.47
[ 1]
+0.29
40
-0.04
+0.55
+ 0.26
[21
to.36
44
+0.06
+ 0.59
+0.33
r31
to.26
32
-0.13
+0.56
+0.22
[41
+ 0.47
34
j-0.12
$0.70
+0.41
L
Discussion
The linearity of the E/I vs +/I relationship is dependent upon EO being proportional to irradiation. This assumption is valid for Berkeley elements whose length changes exhibit a linear dependence on irradiation but is rather a crude approximation for the Bradwell elements 3). This would account for the lower correlation coefficients obtained for the sets of Bradwell elements. Fig. 5 shows a typical plot of E/I vs 4/I for irradiated elements discharged from a particular channel position with the abscissae and ordinates having the same scales. As 8 is liable to vary slightly from one element to another, regression analysis does not yield a single line representing the functional relationship predicted by eq. (10). Two regression lines are obtained as shown in fig. 5 and the line through the mean bisecting the angle between them gives a rough indication of the empirically predicted relationship between the two variables. Comparing this with figs. 3 and 4 gives information about the nature of the texture, and allows rough estimates of 0 and EOto be made. The line AA’ from fig. 5 is superimposed on fig. 4. Most of this appears in the quadrant 8=45”90” indicating that the growth direction of the texture, i.e. the [OlO] helix, is more axially aligned than the shrinkage di-
*y-4o, Fig.
5.
Typical
i,
, , :
axial strain (e/I) vs torsional strain
(+/I) plot for a set of elements irradiated under similar conditions to a burn-up of 1000 MWD/tonne.
rection. Thus the texture is generally so disposed as to produce bar growth and can untwist. A small proportion of the elements contribute points to the quadrant 8 = 135’-1 SO”, i.e. they undergo simultaneous bar shrinkage
312
/ i
Element position
Mean element
in
t,emperature
channel
Berkeley
Bra&elf
texture
rim (” C)
Est~imatc of
’
helix
(deg) [l]
210
121
250
131 [41 151 I61
290
68 67 65’ 61% 69 72
320 360 390
[l ]
255
121
330
[31 [41
455
676 68 70 66
410
L
’
angle, 0
I
Intercept
on
I
EIJII((lh) -0.15 -0.18 -0.11 - 0.08 0.00
0.07 -0.58 -0.72 -0.46 -0.13
Calculated percentage
E/I axis
j
texture
of
(So)
0.3 0.3 0.2 0.3 0.3 0.5 0.3 0.2 0.5 0.3
L
and can twist. This suggests that in some elements the growth and shrinkage directions are the reverse of the con~g~rations described above, so that the shrinkage direction is the more axially aligned. The angle of the texture, 8, can be estimated from the fact that AA’ is inclined at (180” - 20) to the E/I axis, as shown in fig. 3. Values of 8 measured from each set of data are shown in table 2. The constancy of these values is notable, their means being 67” -& 3” for the Berkeley elements and 68” -$: lo for the Bradwell ones, or 68” & 3” for all the elements, i.e. the [OlO] preferred direction is inchned at about 22” to the bar axis irrespective of the fuel element’s reactor history. It appears therefore that the helical texture arises as a consequence of the manufacturing process. The intercepts, corresponding to CO/I in eq. (10) and illustrated in fig. 3, have been measured for all sets of data and are given in table 2. They are seen to be generally negative, i.e. elements with no helical texture undergo a shrinkage. This is consistent with the existence of an axially aIigned erystallogra~hic texture in the core of the bars producing a shrinkage on irradiation. At the lowest irradiation temperatures this overrides the tendency to increase in length due to fission-gas swelling. Bradwell elements further undergo irradiation creep due
to the stack load they have to support. Thus the negative intercepts are greater than for the Berkeley elements. The circles of fig. 4 a,re of radii 2tEot,, so the average degree of texture, tz, may be calculated from the mean point of the plots like fig. 5 for each element position using experimentally derived values of k&“) for the appropriate temperatures 5). These are listed in table 2. As tz may in practice decrease with irradiation, then these values will be slight underestimates of the actual percentage textures. The amount of texture necessary to produce the average untwist is seen to be about 0.3 */* and the larger observed untwists correspond to a net growth texture of about 1 yo.
5. 1.
Conclusions Untwist in irradiated uranium-magnox fuel elements arises as a consequence of a helical crystallographic texture existing in the uranium bar.
2. The texture probably originates from the bar manufacturing process and is such that the grains are preferentially orientated with their [OlO] directions at about 22’ to the bar axis. 3. On average
the degree of texture
is about
HELICAL
CRYSTALLOGRAPHIC
TEXTURE
IN
URANIUM
313
BARS
0.3 y,, but it may be as high as 1 oh in some
References
elements.
l) M .V. Speight, G. F. Hines and G. W. Greenwood, J. Nucl.
Acknowledgements
and V. W. (1963)
The authors are grateful to Dr. G. W. Greenwood for reading the manuscript and making many helpful comments and criticisms. This paper is published by permission Electricity Generating Board.
Mat.
2, H. K. Hardy,
of the Central
20 (1966)
126
J. F. W.
Eldred,
Bishop,
J. Brit.
private
4, M. G. Natrella, Department
9
Pickman
Energy
Sot. 2
33
3, G. F. Hines, J. C. Wood, Uglow,
D. 0.
Nucl.
“Experimental
of Commerce,
S. N. Buckley,
J. Yelland
and A.
G.
communication
private
NBS
Statistics”, (1963)
communication
U.S.