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Effect of porosity on the thermal conductivity of UO2
JOURNAL
OF NUCLEAR
EFFECT
MATERIALS
24 (1967) 109-l 12. 0
OF POROSITY
ON THE
NORTH-HOLLAND
THERMAL
PUBLISHING
CONDUCTIVITY
CO., AMSTERDAM
OF UOz
J. R. Iv$acEWAN, R. L. STOUTE rmndM. J. P. NOTLEY Atomic Energy of Canada Ltd., Chalk River Nuclear Laboratory, Ontario, Canada Received 19 June 1967
Maxwell’s
expression
1) for
electrical
about 2 for UOs sintered in the conventional manner. Ross found that most of the porosity
con-
duction in a heterogeneous material has been used by Eucken 2) to show that the thermal conductivity of a crystal, Ac, is reduced by a dispersion of internal pores with zero conductivity according to &=GI
-P)l[lI
+p,fB-
111,
in the lower density UOs was irregularly shaped and located at grain boundaries with only a small volume fraction present as isometric intragranular pores of 0.01 to 0.05 ,um diameter. However, the conductivities of two specimens, prepared by a special route that yielded only large spherical pores (0.5-1.0 ,um), obeyed the Loeb relationship. Thus, we expected the Loeb equation to apply to UOa sinters with pore volume fractions below 0.05, since the pores in our highly sintered fuel have isometric shapes.
(1)
where & is the conductivity of a porous crystal with a pore volume fraction up. /? is equal to 1.5 for spherical pores and is larger for irregular pores a). More commonly, an expression deduced by Loeb 4) &=&(l
-p),
(2)
has been employed to calculate the thermal conductivity of the matrix from measurements made on porous sinters 5). More complex expressions are provided by Eucken 2) and Loeb 4) for instance where the pores have finite conduetivities, Measurements by Franc1 and Kingery 6) on specially fabricated alumina samples having large isometric pores, 310 ,um diameter, confirmed the validity of Loeb’s relationship for pore volume fractions from 0.123 to 0.487. However, the relationship was not substantiated by measurements made on sintered samples of Be0 7) and UOz 8) prepared by conventional processes. Rather the results fitted an empirical relationship &==&(1-~2)),
(3)
where a was a numerical constant with a value greater than unity [for p Q 0.1, eq. (1) approximates to eq. (3) with oc= @ = 1.51. Powell’s measurements 7) on Be0 are consistent with an LYvalue of about 1.9, while Ross s) reported
The purpose of this note is to draw attention to more recent results by Vogt et al. 9) and of our own that indicate high values of a can be present at low pore volume fractions, contrary to our expectations. Vogt et al. 9) presented laboratory evidence that indicated a was about 3 over the temperature range 200 to 1200” C for pore volume fractions from 0.021 to 0.112. Two of us 10) compared the performance of UOa fuel elements containing pore volume fractions from 0.02 to 0.05 (fuel batch A*). The behaviour, as evidenced by the extent of microstructural change during irradiation, led us to conclude that 01 had a value of 2.6 rt 0.8 averaged over the temperature range 500 to 1500” C. When a similar high value was deduced from a second irradiation experiment (fuel batch B*) ii), we decided to confirm the results by a direct * The fuel for each irradiation was prepared from a single batch of powder. Differences in porosity were achieved by varying the compaction pressure and the sintering temperature. 109
110
J
R.
MacEWAN
ET
AL.
measurement of the thermal conductivity of unirradiated archive pellets retained from the two experiments, since the interpretation irradiation results required assumptions the kinetics
I
of the about
A
l
BATCH
A
BATCH
6
I I /
of grain growth 10).
Experimental : Cylindrical
conductivity
speci-
mens, 7.4 mm dia. by 7.4 mm long, were prepared from archive fuel pellets. Conventional methods were employed structure parative,
and
to determine macroscopic
longitudinal
grain size, pore density.
heat-flow
A
apparatus
IRRADIATED
com-
*-I
was
used to measure the thermal conductivity at 60” C! 8). While the reproducibility on the same specimen was within rlr:lo/,, the absolute accuracy is uncertain: all values are relative to the value selected by Ross 8) for Zircaloy-2 (0.14 W7/cm0” C). Following these measurements, two type A specimens, pore volume fractions of 0.0241 and 0.0501, were irradiated at a temperat~e below 150” C to a burnup of 4.5 x 1017 fissions/cm3 and the conductivities at 60” C redetermined. Details of the irradiation proeedure are given
I
0.01
Fig.
1.
I
I
0.03 PORE
VOLUME
I 0.05
I
J
0.07
FRACTION
Variation of the thermal conductivity of UOz at 60” C with pore volume fraction.
Recuts and ~~sc~~~o~- The measured values of thermal conductivity of the unirradiated samples are plotted against pore volume fraction in fig. 1. Both type A and B fuels exhibited the same dependence on porosity. Fitting all the points to eq. (3) by a least-squares analysis gave an 01value of 2.98 with a standard deviation
analysis 3) of the thermal resistance of irregular pores would require all pores to be ellipsoids, with a major to minor axis ratio of at least 5 : 1 and all major axes perpendicular to the direction of heat flow to explain even an a value of 2.0. An axialratio of 8 : 1 would be required for an LX value of 3.0. Thus, the magnitude of the results is inexplicable on the basis of pore shape alone. The thermal conductivities of two type A specimens with pore volume fractions of 0.024
of 5 0.1. Individual fits gave respective values of 2.78 and 3.02 for type A and B fuels. ~etallographic examination did not reveal evidence of cracking or any other macroscopic defect that would explain high LX values. Electron micro~aphs of fractured gross-sections, figs. Za, b, indicate a predominance of spherical pores in material with a pore volume fraction of 0.02 compared with a mixture of isometric and elongated pores in material with a port volume fraction of 0.05. However, there was no suggestion of a preferred orientation for the elongated pores and the ratio of major to minor axes was less than 5 : 1. Biancheria’s theoretical
and 0.050 were reduced 28 and 27% respectively (fig. 1) by an irradiation exposure to 4.5 x 1017 fissions~cm3 at a temperat~e below 150’ C. Reductions of similar magnitude were obtained previously for other stoichiometric UOs specimens irradiated under similar con~tionslz}. While the value of LX is more sensitive to experimental scatter when the number of samples is small, the LYof 2.3s calculated from the two irradiated samples is in reasonable agreement with the value of 2.78 obtained for the four ~lnirr~iated samples. Both values agree with the LX value of 2.6 i 0.8 derived from the bohaviour of fuel elements employing
in r2).
EFFECT
OF
POROSITY
ON
THE
THERMAL
CONDUCTIVITY
OF
U02
111
b Fig. 2.
Electron
micrographs
showing
the
pore
morphology
of
Batch
A
material
with
a pore volume
fraction of 0.024 (a) and 0.050 (b).
the same batch of UOs la). Thus, the irradiation induced reductions in thermal conductivity appear to be independent of pore volume fraction, so the same large corrections for porosity must be applied to those regions of sintered UOs fuel elements that retain their porosity during irradiation, i.e. for fuel temperatures below 1500 “C. The discrepancy between experimental results and the theoretical expressions is probably due to deficiencies in the theory rather than to the effect of some other imperfection whose concentration varies in the same manner as fuel porosity. Although the grain boundary surface area of a sinter usually decreases with a reduction in porosity, only a boundary with an appreciable segregation of impurities could affect the conductivity, since a clean boundary has a negligible thermal resistance. Even so, the boundary region would have to have both the low conductivity of an amorphous material and an appreciable thickness (- 0.3 ,um) to explain the discrepancy. Further, the evidence
that a given irradiation exposure causes equal percentage decreases in conductivity for specimens with different pore volume fractions would be expected if the imperfection (e.g. conductivity. The porosity) has negligible result would be unlikely if the imperfection has an appreciable thermal conductivity, since both the matrix and the imperfection would have to be equally affected by irradiation damage. Since the present results are not quantitatively fitted by existing theories, the work has two important practical implications. The first is that extrapolation of the line shown on fig. 1 to zero porosity will not necessarily give the correct thermal conductivity for the matrix. The second is that comparisons of the conductivity values for samples with different porosities is not straightforward unless the individual porosity corrections are known. We do not believe that the value of OL = 3 found here need be applicable to all UOs sinters ; different fabrication routes might yield different values.
J.
112
1%. MacEWAN
ET
8)
References
A.
AL.
M.
Ross,
(Canada) 1) 2)
J. C. Maxwell, A treatise on Magnetism, 3rd ed. 1 (1891) p. A.
Eucken,
Forsch.
Geb.
Forschungsheft
no.
3)
A.
Biancheria,
Trans.
4)
A.
L. Loeb,
5)
For example: T. G. Kollie, J. Am.
Cer.
6)
J. Franc1
and
37 (1954)
99
7)
R. W. Powell,
353
J. Am.
Electricity 440
Ingenieurw.
ANS Cer.
Sot.
48 (1965) D. Kingery,
Vogt,
L.
B 3,
Tech.
15
37 (1954)
10)
96
W. Fulkerson, D. L. McElroy,
11)
Cer. Sot.
Brit,. Cer. Sot. 53 (1954)
389
Report
Nuclear
Grandell
and
Laboratories
(1960) U.
Runfors,
in Uranium Series
Dioxide,
no.
AU
Vienna.
1966,
59
M. J. F. Not,ley and J. R. MacEwan, 2 (1966)
297 J. Am.
River
AECL-1096
Atomenergi (Sweden) Int. Report RMB-527 cited by IAEA Panel on Thermal (1964),
12) Trans.
J.
Conductivity
9 (1966)
G. Godfrey, J. P. Moore and W.
9)
(1932)
T.
Sot.
and
Chalk
Report
Nucl.
Appl.
117
M. J. F. Notley,
R. Des Haies and J. R. MacEwan,
Chalk
River
Nuclear
Report
AECL-2662
J. R. MacEwan (1967)
70
Laboratories
(Canada)
(1967)
and R. L. Stoute,
J. Nucl.
Mat.
OF NUCLEAR
EFFECT
MATERIALS
24 (1967) 109-l 12. 0
OF POROSITY
ON THE
NORTH-HOLLAND
THERMAL
PUBLISHING
CONDUCTIVITY
CO., AMSTERDAM
OF UOz
J. R. Iv$acEWAN, R. L. STOUTE rmndM. J. P. NOTLEY Atomic Energy of Canada Ltd., Chalk River Nuclear Laboratory, Ontario, Canada Received 19 June 1967
Maxwell’s
expression
1) for
electrical
about 2 for UOs sintered in the conventional manner. Ross found that most of the porosity
con-
duction in a heterogeneous material has been used by Eucken 2) to show that the thermal conductivity of a crystal, Ac, is reduced by a dispersion of internal pores with zero conductivity according to &=GI
-P)l[lI
+p,fB-
111,
in the lower density UOs was irregularly shaped and located at grain boundaries with only a small volume fraction present as isometric intragranular pores of 0.01 to 0.05 ,um diameter. However, the conductivities of two specimens, prepared by a special route that yielded only large spherical pores (0.5-1.0 ,um), obeyed the Loeb relationship. Thus, we expected the Loeb equation to apply to UOa sinters with pore volume fractions below 0.05, since the pores in our highly sintered fuel have isometric shapes.
(1)
where & is the conductivity of a porous crystal with a pore volume fraction up. /? is equal to 1.5 for spherical pores and is larger for irregular pores a). More commonly, an expression deduced by Loeb 4) &=&(l
-p),
(2)
has been employed to calculate the thermal conductivity of the matrix from measurements made on porous sinters 5). More complex expressions are provided by Eucken 2) and Loeb 4) for instance where the pores have finite conduetivities, Measurements by Franc1 and Kingery 6) on specially fabricated alumina samples having large isometric pores, 310 ,um diameter, confirmed the validity of Loeb’s relationship for pore volume fractions from 0.123 to 0.487. However, the relationship was not substantiated by measurements made on sintered samples of Be0 7) and UOz 8) prepared by conventional processes. Rather the results fitted an empirical relationship &==&(1-~2)),
(3)
where a was a numerical constant with a value greater than unity [for p Q 0.1, eq. (1) approximates to eq. (3) with oc= @ = 1.51. Powell’s measurements 7) on Be0 are consistent with an LYvalue of about 1.9, while Ross s) reported
The purpose of this note is to draw attention to more recent results by Vogt et al. 9) and of our own that indicate high values of a can be present at low pore volume fractions, contrary to our expectations. Vogt et al. 9) presented laboratory evidence that indicated a was about 3 over the temperature range 200 to 1200” C for pore volume fractions from 0.021 to 0.112. Two of us 10) compared the performance of UOa fuel elements containing pore volume fractions from 0.02 to 0.05 (fuel batch A*). The behaviour, as evidenced by the extent of microstructural change during irradiation, led us to conclude that 01 had a value of 2.6 rt 0.8 averaged over the temperature range 500 to 1500” C. When a similar high value was deduced from a second irradiation experiment (fuel batch B*) ii), we decided to confirm the results by a direct * The fuel for each irradiation was prepared from a single batch of powder. Differences in porosity were achieved by varying the compaction pressure and the sintering temperature. 109
110
J
R.
MacEWAN
ET
AL.
measurement of the thermal conductivity of unirradiated archive pellets retained from the two experiments, since the interpretation irradiation results required assumptions the kinetics
I
of the about
A
l
BATCH
A
BATCH
6
I I /
of grain growth 10).
Experimental : Cylindrical
conductivity
speci-
mens, 7.4 mm dia. by 7.4 mm long, were prepared from archive fuel pellets. Conventional methods were employed structure parative,
and
to determine macroscopic
longitudinal
grain size, pore density.
heat-flow
A
apparatus
IRRADIATED
com-
*-I
was
used to measure the thermal conductivity at 60” C! 8). While the reproducibility on the same specimen was within rlr:lo/,, the absolute accuracy is uncertain: all values are relative to the value selected by Ross 8) for Zircaloy-2 (0.14 W7/cm0” C). Following these measurements, two type A specimens, pore volume fractions of 0.0241 and 0.0501, were irradiated at a temperat~e below 150” C to a burnup of 4.5 x 1017 fissions/cm3 and the conductivities at 60” C redetermined. Details of the irradiation proeedure are given
I
0.01
Fig.
1.
I
I
0.03 PORE
VOLUME
I 0.05
I
J
0.07
FRACTION
Variation of the thermal conductivity of UOz at 60” C with pore volume fraction.
Recuts and ~~sc~~~o~- The measured values of thermal conductivity of the unirradiated samples are plotted against pore volume fraction in fig. 1. Both type A and B fuels exhibited the same dependence on porosity. Fitting all the points to eq. (3) by a least-squares analysis gave an 01value of 2.98 with a standard deviation
analysis 3) of the thermal resistance of irregular pores would require all pores to be ellipsoids, with a major to minor axis ratio of at least 5 : 1 and all major axes perpendicular to the direction of heat flow to explain even an a value of 2.0. An axialratio of 8 : 1 would be required for an LX value of 3.0. Thus, the magnitude of the results is inexplicable on the basis of pore shape alone. The thermal conductivities of two type A specimens with pore volume fractions of 0.024
of 5 0.1. Individual fits gave respective values of 2.78 and 3.02 for type A and B fuels. ~etallographic examination did not reveal evidence of cracking or any other macroscopic defect that would explain high LX values. Electron micro~aphs of fractured gross-sections, figs. Za, b, indicate a predominance of spherical pores in material with a pore volume fraction of 0.02 compared with a mixture of isometric and elongated pores in material with a port volume fraction of 0.05. However, there was no suggestion of a preferred orientation for the elongated pores and the ratio of major to minor axes was less than 5 : 1. Biancheria’s theoretical
and 0.050 were reduced 28 and 27% respectively (fig. 1) by an irradiation exposure to 4.5 x 1017 fissions~cm3 at a temperat~e below 150’ C. Reductions of similar magnitude were obtained previously for other stoichiometric UOs specimens irradiated under similar con~tionslz}. While the value of LX is more sensitive to experimental scatter when the number of samples is small, the LYof 2.3s calculated from the two irradiated samples is in reasonable agreement with the value of 2.78 obtained for the four ~lnirr~iated samples. Both values agree with the LX value of 2.6 i 0.8 derived from the bohaviour of fuel elements employing
in r2).
EFFECT
OF
POROSITY
ON
THE
THERMAL
CONDUCTIVITY
OF
U02
111
b Fig. 2.
Electron
micrographs
showing
the
pore
morphology
of
Batch
A
material
with
a pore volume
fraction of 0.024 (a) and 0.050 (b).
the same batch of UOs la). Thus, the irradiation induced reductions in thermal conductivity appear to be independent of pore volume fraction, so the same large corrections for porosity must be applied to those regions of sintered UOs fuel elements that retain their porosity during irradiation, i.e. for fuel temperatures below 1500 “C. The discrepancy between experimental results and the theoretical expressions is probably due to deficiencies in the theory rather than to the effect of some other imperfection whose concentration varies in the same manner as fuel porosity. Although the grain boundary surface area of a sinter usually decreases with a reduction in porosity, only a boundary with an appreciable segregation of impurities could affect the conductivity, since a clean boundary has a negligible thermal resistance. Even so, the boundary region would have to have both the low conductivity of an amorphous material and an appreciable thickness (- 0.3 ,um) to explain the discrepancy. Further, the evidence
that a given irradiation exposure causes equal percentage decreases in conductivity for specimens with different pore volume fractions would be expected if the imperfection (e.g. conductivity. The porosity) has negligible result would be unlikely if the imperfection has an appreciable thermal conductivity, since both the matrix and the imperfection would have to be equally affected by irradiation damage. Since the present results are not quantitatively fitted by existing theories, the work has two important practical implications. The first is that extrapolation of the line shown on fig. 1 to zero porosity will not necessarily give the correct thermal conductivity for the matrix. The second is that comparisons of the conductivity values for samples with different porosities is not straightforward unless the individual porosity corrections are known. We do not believe that the value of OL = 3 found here need be applicable to all UOs sinters ; different fabrication routes might yield different values.
J.
112
1%. MacEWAN
ET
8)
References
A.
AL.
M.
Ross,
(Canada) 1) 2)
J. C. Maxwell, A treatise on Magnetism, 3rd ed. 1 (1891) p. A.
Eucken,
Forsch.
Geb.
Forschungsheft
no.
3)
A.
Biancheria,
Trans.
4)
A.
L. Loeb,
5)
For example: T. G. Kollie, J. Am.
Cer.
6)
J. Franc1
and
37 (1954)
99
7)
R. W. Powell,
353
J. Am.
Electricity 440
Ingenieurw.
ANS Cer.
Sot.
48 (1965) D. Kingery,
Vogt,
L.
B 3,
Tech.
15
37 (1954)
10)
96
W. Fulkerson, D. L. McElroy,
11)
Cer. Sot.
Brit,. Cer. Sot. 53 (1954)
389
Report
Nuclear
Grandell
and
Laboratories
(1960) U.
Runfors,
in Uranium Series
Dioxide,
no.
AU
Vienna.
1966,
59
M. J. F. Not,ley and J. R. MacEwan, 2 (1966)
297 J. Am.
River
AECL-1096
Atomenergi (Sweden) Int. Report RMB-527 cited by IAEA Panel on Thermal (1964),
12) Trans.
J.
Conductivity
9 (1966)
G. Godfrey, J. P. Moore and W.
9)
(1932)
T.
Sot.
and
Chalk
Report
Nucl.
Appl.
117
M. J. F. Notley,
R. Des Haies and J. R. MacEwan,
Chalk
River
Nuclear
Report
AECL-2662
J. R. MacEwan (1967)
70
Laboratories
(Canada)
(1967)
and R. L. Stoute,
J. Nucl.
Mat.