Pore pressure calculation of the UO2 high burnup structure

Nuclear Engineering and Design 260 (2013) 11–15

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Pore pressure calculation of the UO2 high burnup structure Lijun Gao a,b,∗ , Bingde Chen c , Zhong Xiao d , Shengyao Jiang a , Jiyang Yu a a

Tsinghua University, 100084 Beijing, China Science and Technology on Reactor System Design Technology Laboratory, P.O. Box 622-500, 610041 Chengdu, China c Nuclear Power Institute of China, 610041 Chengdu, China d National Energy R&D Center on Advanced Nuclear Fuel, 610041 Chengdu, China b

h i g h l i g h t s • • • •

Pore pressure is calculated based on local burnup, density and porosity. Ronchi’s equations of state are used instead of van der Waals’ equation. Pore pressure increases as HBS transformation begins and then stays constant. A best approximated parameter used for pore pressure calculation is recommended.

a r t i c l e

i n f o

Article history: Received 21 August 2012 Received in revised form 18 February 2013 Accepted 12 March 2013

a b s t r a c t UO2 high burnup structure has an important impact on fuel behavior, especially in case of reactivity initiated accident (RIA). Pore relaxation enhances local fuel swelling and puts additional load to the fuel cladding, which makes fuel more susceptible to pellet–cladding mechanical interaction induced failure. Therefore, pore pressure calculation becomes vital when evaluating the fuel failure. In this paper pore pressure is calculated as a function of pellet radial local burnup based on the basic characteristics of HBS using Ronchi’s correlation. The results indicate that pore pressure will approach a stable value as HBS is developing. A best approximated C value of 55 N/m is recommended for pore pressure calculation. © 2013 Elsevier B.V. All rights reserved.

1. Background The discharge burnup of fuel assembly has been increasing from 30 GWd/tU to around 50 GWd/tU throughout the past tens of years in order to improve the economics of the power plant. The discharge burnup of 60–70 GWd/tU may be expected if the fuel cladding integrity can be maintained in the future. Longer fuel cycle brought about by increased discharge burnup reduces both the fuel fabrication and the fuel deposition cost by consuming less fuel assemblies in the whole life of the reactor. Two main phenomena (Liu, 2007) arise from increasing the discharge burnup: oxidation and hydrogen pickup of the fuel cladding and the formation of UO2 high burnup structure. For the former phenomenon, the oxidation layer developed in the outer surface of the fuel cladding degrades the heat transfer capability of the fuel rod and the hydride in the cladding makes the cladding brittle and susceptible to cracking. For the latter one, HBS (Hereafter, HBS is

∗ Corresponding author at: Science and Technology on Reactor System Design Technology Laboratory, P.O. Box 622-500, 610041 Chengdu, China. Tel.: +86 1510 828 8295. E-mail address: [email protected] (L. Gao). 0029-5493/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nucengdes.2013.03.028

short for “UO2 high burnup structure” in the present paper) initiates at the fuel pellet periphery due to the resonance absorption of epithermal neutron, which induces the production of Pu239 and thus much higher local burnup in the fuel pellet outer rim than in the center and intermediate radial position of the fuel pellet. However, HBS development can penetrate into the intermediate radial position, occupying considerable fraction of the whole fuel pellet volume, when the average pellet burnup exceeds 90 GWd/tU. HBS with high porosity reduces the fuel conductivity and causes higher gaseous fuel swelling rate than before HBS transformation. Gaseous fuel swelling tends to grow in case of temperature rise, such as RIA, and exacerbates the PCMI, which usually occurs after two cycle’s operation. Therefore, additional gaseous fuel swelling induced by HBS transformation has the propensity to do harm to the integrity of the fuel cladding and lead to reactor accidents, constituting a possible obstacle to further increase the discharge burnup. Several international collaboration programs, such as high burnup effect program (Barner, 1990), have been established to explore the HBS phenomenon since early 1980s and more and more experimental data of HBS transformation becomes available. Technical advance has allowed detailed observation and description of HBS transformation, which includes XRD, SEM, EPMA, SIMS and so on.

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L. Gao et al. / Nuclear Engineering and Design 260 (2013) 11–15

Accompanying the experiments theoretical analysis and modeling work have already unfolded in order to reveal the mechanism of HBS transformation and implement it into the fuel performance code. Such theoretical and modeling work would hopefully provides valuable instrument to evaluate the potential risk resulting from increasing the discharge burnup and benefits the future design process of more robust fuel rods. 2. State of the art HBS has been under extensive investigation since its discovery, especially on its effects on fuel swelling (Koo et al., 2001; Spino et al., 2005), fission gas release (FGR) (Jernkvist and Massih, 2002; Uffelen, 2002) and PCMI in case of RIA (Lemoine, 1997; Liu, 2007; Une et al., 1997). General consensus has been reached about the basic characteristics and formation conditions of HBS, despite its debatable formation mechanism, for which several hypotheses (Lee and Jung, 2000; Matzke and Kinoshita, 1997; Nogita and Une, 1995; Rest and Hofman, 2000; Spino et al., 1996) have been put forward partly explaining the underlying physical process of HBS formation but all suffering more or less from self-consistency problems. Currently detailed data of HBS transformation created by different individuals are available for the average pellet burnup range from 40 GWd/tU to more than 100 GWd/tU, which show good agreement with each other and thus provide experimental support for the fuel performance analysis. It is generally accepted that HBS is characterized by the three following processes: - A subdivision of the original fuel grain from 10 ␮m to submicron range (∼0.3 ␮m). - Creation of overpressurized pores with micron-sized diameters and resulting high porosity. - Depletion of Xe in the fuel matrix and simultaneous retention of Xe in the overpressurized pores. It is revealed both experimentally and theoretically that the majority of Xe depleted from the fuel matrix is retained in the newly created pores in HBS. Specially designed Knudsen cell experiments (Kinoshita et al., 2004) quantitatively confirmed that depleted Xe from the fuel matrix is relocated and captured in the HBS pores. However, excessive porosity may induce pore-interconnection forming channels for Xe in the pores to be released to the external free space in the fuel rod. According to the rim porosity and pore size distribution Koo’s analysis (Koo et al., 2003) using the Monte Carlo method indicated that the threshold porosity for channel formation for FGR in HBS is around 24%. But subsequent three-dimension reconstruction of the pore space by Spino et al. (2004) reveals that 24% is too low and 25% is possible for a most conservative estimation. For fuel irradiated to average burnup of 66.6 GWd/tU, therefore exhibiting maximum porosity of about 15% (Spino et al., 1996) at the outermost periphery of the pellet, almost full retention of occluded Xe would be ensured. The maximum porosity would not exceed 25% until the average pellet burnup goes beyond 100 GWd/tU (Spino et al., 2006). Xe depleted from the fuel matrix and retained in the pores in HBS causes overpressurized pores, which would have a propensity for high local swelling at increased temperature and leading to PCMI failure of fuel rods in case of RIA. As a possible explanation, pore growth (or pore relaxation) is assumed to occur with high temperature induced overpressure through dislocation-punching mechanism (Liu, 2007), just as the intragranular bubbles (Losonen, 2000). Therefore the initial pore pressure is of great importance for the pore growth modeling. Nogita and Une (1995) observed the presence of an extremely high density of dislocations on the surface of coarsened pores

in HBS, and estimated the excessive pore pressure using the dislocation-punching condition for bubbles, which is thereafter adopted as an important reference for pore pressure despite his mistakenly calculating the equilibrium pressure and overestimating the pore pressure with radius of 0.5–2.0 ␮m in his example. Koo calculated the pore pressure using van der Waals’ equation by equating the depleted Xe to that in the pores, which showed the pore pressure at the pellet average burnup of 67 GWd/tU is 477 MPa at the typical rim temperature 800 K in a pore with radius 0.5 ␮m, seeming unrealistically high. Liu (2007) reevaluated the pore pressure using almost the same method but different burnup and porosity distribution and got more reasonable results, which showed the pore pressure with radius 0.5 ␮m at the pellet average burnup of 72 GWd/tU is roughly 94 MPa. Using averaged pore size and Ronchi’s (Ronchi, 1981) equations of state for rare gases, Une et al. (1997) estimated that the pore pressure, for the pore radius 0.5 ␮m, pore density 6 × 107 /mm3 , fractional gas retention of 30–50% in the pores and a pellet average burnup of 80 GWd/tU, is 90–210 MPa. This work would devote efforts to calculate the pore pressure of the UO2 HBS based on some corrections of the published results, which, however, includes corrections on fuel density, Xe concentration and the choice of the equation of state (EOS) and emphasizes the significance of local burnup. 3. Procedure for the pore pressure calculation Pore pressure depends on the gas inventory stored in pores, pore size and porosity. Temperature plays an important role on gas pressure but is fixed at the typical rim temperature 800 K since steady operation is considered here. The gas inventory is determined by Xe production and depletion. Xe concentration as a function of local burnup is given by Lassmann et al. (1995) based on fitting of a pool of PWR and BWR data, as is shown below. Xe(Bu) = c · Bu − c

1 a



+ Bu0 −

1 a





e−a(Bu−Bu0) ,

Bu ≥ Bu0

(1)

where Xe(Bu) is matrix Xe concentration in wt.%, c is the Xe creation rate (1.46 × 10−2 wt.% per GWd/tU), a is a fitting parameter of 0.0584, Bu is the local burnup and Bu0 is the threshold burnup for HBS transformation. The burnup factor in the radial direction of a typical fuel pellet is given by the TRANSURANUS burnup model (Schubert et al., 2008) as follows. f (r) = 1 + p1 exp(−p2 (R − r)p3 )

(2)

where r is the radial position in the fuel pellet, R is the fuel outer radius, and p1 , p2 and p3 are constants derived after comprehensive comparisons with measured data of irradiated fuel slice (p1 = 3.45, p2 = 3.0 and p3 = 0.45), which indicates that p1 , p2 and p3 show little dependence on burnup history and uranium enrichment below 7% (Lassmann et al., 1994). Considering normalization of the dimensionless burnup factor, for a fuel pellet with average burnup Buav the burnup profile in the radial direction is given by R2 f (r) Bu(r) = Buav  R 2rf (r)dr 0

(3)

The pore radius distribution data of different burnups have been fitted to the lognormal distribution, which shows excellent agreement (Spino et al., 1996). The probability density function of the pore radius is shown below. f (rp ) = √

1 2rp



exp

−

(ln rp − )2 2 2



(5)

L. Gao et al. / Nuclear Engineering and Design 260 (2013) 11–15

where rp is the pore radius,  and  2 are the statistical mean and variance for lnrp respectively. Quantitative metallography shows only slight increase of the pore radius with burnup and radial position at pellet average burnup from 40.3 GWd/tU to 66.6 GWd/tU, and therefore a fixed pore radius distribution is assumed in this work, implying varied pore density with fuel pellet radius making up for the increased porosity.  and  2 are determined in the following way ln rpl =  + 3

(6)

ln rpu =  +  2

(7)

where rpl and rpu are respectively the lower bound and upper bound of the pore radius. Experimental results show that rpl = 0.2 ␮m and rpu = 2.0 ␮m would be reasonable. The porosity as a function of radial position and pellet average burnup is fitted to exponential function by Liu (2007) shown below. P

r R



= a1 + exp a2 + a3

 r  R

a1 = 0.001144Buav − 0.02287

(8) (9)

a2 = 1.05Buav − 100.6

(10)

a3 = −1.057Buav − 99.01

(11)



rpu

V= rpl

4 3 r f (rp )drp 3 p

(12)

The mechanical equilibrium of a bubble in solids (Olander, 1976) is p=

2 + ph rp

(13)

where p is the gas pressure in the bubble,  is the surface energy of solids (1 N/m is adopted for UO2 ), ph is the uniform hydrostatic stress. Greenwood et al. (1959) showed that the pore overpressure should be larger than Gb rp for dislocation punching to be energetically favorable, where G is the shear module and b is Burgers vector (0.39 nm for UO2 (Olander, 1976)). It is generally accepted that Greenwood’s analysis is valid for pores with intermediate radius. Nogita and Une’s experimental observation revealed that the dislocation punching around the coarsened pores in HBS is the very evidence of overpressurized pores since the irradiation temperature at the fuel pellet rim is too low to activate vacancy diffusion to contribute to pore coarsening process. Dislocation punching around the pores also suggests pore pressure would have been equal or slightly larger than Gb rp . So pores in HBS would relieve their overpressure by punching dislocation loops in the fuel matrix around them, which is also believed to be the pore relaxation mechanism in case of RIA by Liu (2007). According to Nogita and Une and ignoring the hydrostatic pressure prior to PCMI, the pore pressure in HBS can be roughly estimated by p=

2 Gb + rp rp

(14)

Therefore, considering the effects of temperature and burnup on dislocation punching the pore pressure in HBS is assumed to take the following form in the present paper p=

C rp

Table 1 Calculation results. Buav (GWd/tU)

Calculated max burnup (GWd/tU)

Upper bound of C (N/m)

Best approximated value of C (N/m)

56 66 76

148.8 159.4 162.3

60.5 62.8 60.0

51.0 56.0 56.5

where C (N/m) is the function of local burnup and temperature. By equating the depleted Xe to that in the pores in HBS at each radial position of the fuel pellet, we attain the following expression. Xe(Bu)NA

(15)

P(r/R) f (Bu) = MXe V



rpu

rpl

4 3 r f (rp )(T, p)drp 3 p

(16)

where f (Bu) is the fuel density at local burnup Bu, NA is Avogadro’s number, MXe is the Xe atomic mass (134 g/mol) and (T,p) is the molecular density of Xe, which is the function of temperature and Xe pressure. Local burnup is the function of radial position as shown in Eq. (3), and consequently Xe(Bu) is also the function of radial position. f (Bu) is reduced from theoretical density of UO2 for the factors: (1) thermal expansion of the fuel, (2) increased porosity, (3) matrix swelling due to fission product, which has been evaluated by Spino et al. (2005) in detail. So the following expression is obtained. f (Bu) =

The average pore volume V is

13

t,UO2 (1 − P(r/R)) 3

(1 + ˛T ) (1 + rs1 Bu0 + rs2 (Bu − Bu0))

(17)

where t,UO2 is the theoretical density of UO2 (10.96 g/cm3 ), ˛ is the linear thermal expansion rate (10−5 /K), T is the temperature difference referred to zero expansion temperature (300 K), rs1 is the matrix swelling rate prior to HBS transformation (1.0%) and rs2 is the matrix swelling rate after HBS transformation is completed (0.32%). Van der Waals equation has been commonly used without checking its applicability as the EOS of Xe under the conditions of nuclear reactor operation for its simple expression, however, considerable deviation from the experimental data can occur for high temperature and high density conditions. Ronchi developed an EOS for rare gases based on perturbated hard sphere model using Lennard–Jones potential, which showed good agreement with available experimental data of Xe. New data become available due to molecular dynamics simulation (MD) (Oh et al., 2008), which keep consistent with the experiment data even at the pressure up to 3.02 GPa at room temperature and therefore are believed to serve as a standard for evaluating the applicability of other EOS. MD results show that the applicable range of Ronchi’s EOS goes up to 280 MPa at 900 K with deviation from the MD data less than 15%. So Ronchi’s EOS is adopted in this work. 4. Results and discussion Three cases with different average fuel pellet burnups are studied, for which the pellet radius is set as 4.65 mm. The results are shown in Table 1. The range 56–76 GWd/tU of average fuel pellet burnups is chosen considering both the potential discharge burnup in the near future and data availability for HBS characteristics. Besides, three different threshold burnups of HBS formation account for scattering of Xe concentration data in the fuel matrix with increase of burnup (Lassmann et al., 1995) and 68 GWd/tU would be treated as the best estimated threshold burnup since the fitting curve with it goes through the center of the data area. The best approximated value of C is calculated as the average of the results when the best estimated threshold burnup is used and data for the local burnup below 90 GWd/tU are discarded. Table 1 shows that nearly identical upper bound and best approximated

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L. Gao et al. / Nuclear Engineering and Design 260 (2013) 11–15

Fig. 1. C-curves of Buav = 56 GWd/tU.

Fig. 3. C-curves of Buav = 76 GWd/tU.

value of C are obtained for three different average fuel pellet burnups. So the maximum pore pressure is around 280 MPa, which occurs when the minimum pore radius 0.2 ␮m is used. According to p–V–T correlation by MD (Oh et al., 2008), this value will reduce to around 240 MPa, however, 153 MPa is the upper bound if the dislocation punching mechanism is applied. C is plotted as the function of the local burnup in Figs. 1–3, which is named C-curve in this paper. In Figs. 1 and 2 the C-curves rise sharply below 90 GWd/tU, for which the enhanced Xe depletion accompanying the HBS formation accounts. After that the C-curves gradually become flat indicating that the HBS formation approaches completion and the pore pressure for pores with the same radius is fixed. The C-curves for different threshold burnups tend to converge after a local burnup of 120 GWd/tU because the saturated Xe concentration of the fuel matrix 0.25 wt.% is reached, equating the Xe concentration in pores despite different threshold burnups. C-curves in Fig. 3 differ slightly from those in Figs. 1 and 2 in that C curves rise continuously with the increase of local burnup without implying the presence of a steady stage, which can be explained by the fact that the porosity profile and the pore size distribution used are fitted with HBS data below 66 GWd/tU and at the burnup of 76 GWd/tU deviation from these are considerable. However, the gradient of C-curves becomes much smaller after a local burnup of 110 GWd/tU is reached so that for the local burnup range

below 160 GWd/tU the value of C is still in reasonable agreement with those in Figs. 1 and 2. Actually the pore size goes larger and larger with the increase of local burnup (Romano et al., 2007; Spino et al., 2006), especially after the local burnup around 110 GWd/tU is exceeded, which necessitates some modifications to the porosity profile and pore size distribution assumed in this paper. When C value of 55 N/m is used, the calculated pore pressure with radius 0.5 ␮m is 110 MPa, which is slightly larger than Liu’s calculation but lies in the pressure range of Une’s calculation. It should be noted that average fuel parameters rather than local ones, such average fuel pellet burnup and average porosity in the rim, are used in Liu’s and Une’s calculation, leaving some difficulty to comparison between their calculation and that of this work. However, it is estimated that Liu’s calculation would have produced higher pore pressure than that of this paper if local burnup had been used in his calculation, which can be explained by the following facts: (1) usage of Van der Waals equation. Among several EOS of Xe, Van der Waals equation gives the highest pressure at the same condition. (2) Ignorance of fuel matrix swelling. Ignorance of fuel matrix swelling in the calculation tends to overestimate the fuel density and thus the produced Xe inventory. (3) Using 52 GWd/tU as the threshold burnup for HBS transformation and thus leading to excessive Xe depletion fraction. In spite of this, Liu’s calculation results together with that of this work consistently reveal that pore pressure grows with burnup and greatly exceeds the pressure that can be held by the surface energy of UO2 . It is predicted in the paragraph above that the pore overpressure is roughly limited by the upper bound set by dislocation punching mechanism, which estimates C value of around 30 N/m well below the best approximated ones in this paper. Besides, irradiatedinduced creep of UO2 fuel would start at typical rim temperature at around 100 MPa, which may serve as another evidence that C value lager than 30 N/m would probably overestimate the pore pressure. Several factors may be responsible for this discrepancy: (1) porosity data are subjected to considerable uncertainty due to sample preparation, the operator and the magnification selection in quantitative metallography and the error cannot be set below ±1–2 vol.% (Spino et al., 2005); (2) the EOS of Xe in the range of high density and high pressure is not yet well validated due to lack of experimental data though much theoretical advance has been achieved and proved applicable to other rare gases; (3) moreover, the pore size distribution and thus the porosity profile are measured at room temperature and pore shrinkage would happen during the decrease of fuel temperature from the reactor operation temperature leading to smaller pore volume and overestimated pore pressure.

Fig. 2. C-curves of Buav = 66 GWd/tU.

L. Gao et al. / Nuclear Engineering and Design 260 (2013) 11–15

Local burnup instead of rim average burnup is used to calculate the pore pressure because the structure changes dramatically toward the fuel pellet periphery, especially the Xe distribution in the fuel. Distinctive stages of HBS development in the radial direction of the fuel pellet have been observed and pore pressure should also be among the indicators of this progression which is supported by the calculation of this paper. Temperature variation across the pellet radius in the rim region is not taken into consideration of this paper and the identical temperature 800 K is adopted. Actually, the calculated average pellet burnup does not exceed 76 GWd/tU and at the same time the best threshold burnup of HBS transformation is set at 68 GWd/tU, which altogether results in a very thin layer of HBS and small temperature variation in the layer, which is estimated to be around 100 K. So using a single temperature value does not incur too much error to the calculation results. However, when the average pellet burnup exceeds 90 GWd/tU, HBS region penetrates into the intermediate radial position and covers temperature drop up to 200 K, which makes usage of more realistic radial temperature profile a must when calculating the pore pressure. The mechanism of HBS transformation, which may hopefully provide key clues to several mysterious phenomena, is still under debate, and future experiments may give new evidence to help build a persuasive explanation to the mechanism and at the same time clarify the pore overpressure problem. 5. Conclusions Pore pressure of HBS has been calculated based on the experimental data of HBS characteristics and conclusions listed below can be drawn: The pore pressure increase sharply first due to Xe depletion and then stays at a relatively stable value independent of local burnup and average fuel pellet burnup, which indicates that pore pressure is one of the signals of the HBS transformation. On the base of experimental data, a best approximated C value of around 55 N/m can be used to calculate the pore pressure using the formula p = rCp after the local burnup exceeds 90 GWd/tU. However, it should be noted that considerable discrepancy still exists in the pore pressure calculation results, depending firstly on the chosen method and secondly on a lack of reliable experimental data. C value obtained in this paper is valid in pore pressure calculation only when the identical hypotheses with this work are assumed. A few corrections are made to the available method of pore pressure calculation, which makes the pore pressure calculation more consistent with underlying physical process, and as a result the calculated pore pressure seems more realistic and closer to that obtained by dislocation punching mechanism. Usage of Ronchi’s EOS is one of the important corrections, which produces lower pressure value than Van der Waals equation at the same condition.

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