Simulation of the Power Profile in Gadolinium-Doped Fuel Rods With

Simulation of the Power Proﬁ le in Gadolinium-Doped Fuel Rods with the Fuel Performance Code RTOP

V. Likhanskii, I. Evdokimov, V. Zborovskii, A. Sorokin, S. Tokarev SRC RF TRINITI, Moscow, Troitsk, Russian Federation

The paper covers a four-group model to calculate Burn-out of gadolinium isotopes in a fuel rod the radial heat rate profi le and the concentration of depends on the ratio of the thermal and epithermal the isotopes of gadolinium, uranium and plutonium neutron fl uxes. The interaction of 155Gd and 157Gd in a Gd-doped fuel rod within a fuel performance nuclei with the thermal neutrons is characterized code. The results of modelling are compared to by suffi ciently large absorption cross-sections. The the calculations by a neutron code. The model absorption cross-sections are much lower for the describes with satisfactory accuracy the time evo- epithermal energies. Therefore the interaction of lution of concentration of gadolinium and fi ssile epithermal neutrons with gadolinium nuclei has isotopes as well as linear heat rate of the gado- a smaller effect on the distribution of neutron fl ux linium-doped fuel rod, given the average heat rate over fuel pellet. of a fuel assembly (FA). The model was verifi ed At present either the neutron codes or special- against the data on FA operation in various WWER ly developed simplifi ed models are used to calcu- power units and can be used for FAs with various late the thermomechanical behavior of Gd-doped fuel enrichments. fuel rods at low burnups. For example, neutron diffusion model is used in the TRANSURANUS code [1, 2]. The available fuel performance codes, for 1. Introduction example FRAPCON-3 [3,4], TRANSURANUS [5]

and RTOP [6] describe the UO2 fuel rods well, but The fuel with burnable absorber is used to achieve they still need additional modifi cations to describe the extension of fuel cycles. Gadolinium oxide the thermomechanical behavior of the Gd-doped fuel rods with more accuracy. Gd2O3 serves as the burnable absorber in UO2 fuel for WWER units. The gadolinium isotopes have a The purpose of the present work was to de- large cross-section of neutron absorption in the velop the physically-based engineering model to range of thermal energies. The gadolinium burns solve the problems related to the behavior of the out without changing its chemical composition. Gd-doped fuel rods at low burnups: 155 157 The odd isotopes Gd and Gd are converted The calculation of spatial power profi le and the 156 158 into the even isotopes Gd and Gd, respec- concentration of isotopes in the Gd-doped fuel rod tively. given the neutron fl uxes incident on that fuel rod. The thermomechanical behavior of the Gd- The calculation of the temporal evolution of the doped fuel rod at elevated burnups is similar to linear heat rate for the Gd-doped fuel rod from the that of the UO fuel rod in many aspects. It can 2 known values of average FA heat rate and the ini- be described using the available fuel performance 235 tial enrichment in U for the UO2 fuel rods and the codes with a change of some material properties Gd-doped fuel rods within the given FA. and parameters in the models (in particular, the fuel heat conductivity, the fi ssion product diffusivity and the fuel thermal expansion coeffi cient). 2. Description of the Four-Group At low burnups, less than 10 MWd/kgU, the Model thermomechanical behavior of a gadolinium- doped fuel rod considerably differs from that of UO2 fuel rod. The gadolinium burns out layer-by-layer, The transport and absorption of neutrons in which results in a high radial non-uniformity of heat the fuel pellet and isotope burn-out are described generation rate across fuel pellets. Heat rate is the in the four-group approximation. The burn-out of key parameter affecting the fuel temperature. Con- two gadolinium isotopes, 155Gd and 157Gd is con- sequently, the temperature fi elds, evolution of fuel sidered. Due to neutron absorption they are con- microstructure, buildup and release of fi ssion gas verted into 156Gd and 158Gd, respectively. The differ signifi cantly from respective parameters of a cross-section of the thermal neutron absorption

UO2 fuel rod. by the gadolinium isotopes with even numbers is

562 equal to 6.1·104 barn for 155Gd and 2.54·105 barn for 157Gd. As the neutron temperature increases, the cross-sections decrease. In the absence of de- tailed calculations of neutron spectra the effective cross-sections can be treated as free parameters of the model. We assume the absorption cross- sections of thermal neutron to be equal to 2,18·104 barn for 155Gd and 8,5·104 barn for 157Gd. The absorption cross-sections of epithermal neutrons are assumed to be equal to resonance integrals [10]. The integrated contribution of thermal neutrons to ﬁ ssion of uranium and plutonium is considered in the approximation of Maxwellian spectrum with

effective temperature Tn taking into account West- Figure 1. Spatial distribution of 239Pu in a fuel cott g-factor [7]: pellet of Gd-doped fuel rod at the 1/2  burnup of 10 MWd/kgU [7,8] TT293 , (1) inTgT  ni 293  4Tn T small and therefore they do not make any signifi - here  293 is the cross-section at neu- cant effect on the neutron distribution. To calculate i tron speed of 2200 m/s, neutron temperature is the heat rate the fi ssion of 235U and 239Pu nuclei is 235 239 taken into account. Tn = 700 K, g( U) = 0.918, g( Pu) = 1.777 [7]. The signifi cant non-uniformity of radial heat profi le in a Gd-doped fuel rod is observed only for 2.2. Main Equations that Govern Burn-Out of early stage of fuel rod operation. In the present Isotopes and Radial Profi le of Neutron version of the model the concentration of plutoni- Flux um 239Pu produced from 238U is assumed for these conditions to increase linearly with burnup accord- The kinetics of the isotope burn-out in a fuel pellet ing to model [7,8]. The relative radial distribution is described by the equation of 239Pu is described by a fi xed profi le [7]. Figure 1 235 155 157 239 gives the typical spatial distribution of Pu isotope CCCkkkTTkkFkFk,,,   ,{U,Gd,Gd}  produced in the fuel when the burnup of Gd-doped (2) fuel rod is about 10 MWd/kgU. Here С is the concentration of the isotope k, Two groups of neutrons, in the thermal and in k are the cross-sections of thermal neutron ab- the epithermal regions, are considered for each  kT, isotope. The thermal neutron fl ux is the common sorption,  kF, are the cross-sections of epither- value for all the considered isotopes. As far as the mal neutron absorption, T is the thermal neutron epithermal neutrons are concerned, an assumption fl ux,  are the epithermal neutron fl uxes. is made that the peaks of resonance absorption of kF, The thermal neutron fl ux distribution in a fuel epithermal neutrons by 235U and gadolinium nuclei pellet is found from the transport equation overlap weakly. Under the given assumption it is possible to consider a separate group of epither- fT fCT Gd155 Gd 155, T  C Gd 157 Gd 157, T  C U 235 U 235, T  C Pu 239 Pu 239 , mal neutrons for each isotope. Thus, four groups of x (3a) neutrons are regarded in the model: thermal neutrons due to which the fi ssion of 235U, 239Pu takes with the boundary condition place and 155Gd and 157Gd isotopes burn out, and f   1. (3b) also three groups of epithermal neutrons: one for T rRx,0 uranium and one for each odd gadolinium isotope. here fT is the relative fl ux distribution in the pellet, r is the radial position of the point the distri- 2.1. The Choice of Cross-Sections bution is considered in, ψ is the angle between the direction of the neutron fl ux and the radius of the The cross-sections of thermal neutron absorption by the gadolinium isotopes are provided in [9] for point and xr cos . the neutrons with energy 0.0253 eV and they are The thermal fl ux is obtained from the relative

563 fl ux distribution by the following: water [7], Q is the thermal neutron source, P is   the coeffi cient of thermal neutron absorption. We  0T fr(, ) d, (4) consider a semi-infi nite problem, with the bound- TT  0 ary condition at infi nity (9) and on the outer surface of the fuel rod (10). here 0T is the thermal neutron fl ux incident on the surface of Gd-doped fuel rod. The boundary condition (10) on the outer boundary of the fuel column in a Gd-doped fuel rod includes The epithermal neutron fl ux distributions in the the κ factor which is the ratio of absorbed to incident pellet are described by similar equations: number of neutrons. The value of the given factor is calculated by solving the problem of neutron trans- f port and absorption in a pellet (see Section 2.2). kF, fC , (5a) x kF,, k kF The source of thermal neutrons Q is deter- f   1, (5b) mined by thermalization of fast neutrons that are kF, rRx,0 born due to fi ssion. The thermal neutron absorption depends on the presence of boric acid in cool-  0,kF , ant, accumulation of poisons in the fuel during re-  frd(, ) . (6) kF,,  kF actor operation and it is also affected by the control 0 rod position in the reactor. 235 155 157 The solution to diffusion problem (8)-(10) at k {U,Gd,Gd}, 0,kF , are the epithermal neutrons fl uxes, incident on the surface of each moment of time can be found analytically as Gd-doped fuel rod. follows: The radial profi le of heat rate is given by (11) 00 CK r , 238 235 239 qCCqkvkkTTkkkFkFkv,,,    ,{U,Pu} k (7) P CKrKr01000/     ,   llD here is the energy generated per one fi s-  k (12) sion of the isotope k. The concentration of urani- here K are the modifi ed Bessel functions of um is calculated from Eq.(2), the concentration of ν the second kind, and the factor κ is defi ned above. 239Pu is found according to Section 2. 238 is the qv The diffusion coeffi cient in the model is as- correction to heat rate due to fi ssion of 238U nuclei. sumed to be equal to 5 m2/s [11]. An assumption is made that the value of the 2.3. Model of Thermal Neutron Distribution in absorption function P varies little in the equilib- the Space between Fuel Rods rium fuel cycles of the reactor. Therefore, having defi ned the absorption function for one Gd-doped fuel rod during the cycle, we can apply it to calcula- The distribution of the thermal neutrons in the tions for other FAs and Gd-doped fuel rods in simi- space between the Gd-doped fuel rod and adja- lar cycles of other WWER units. The value of func- cent UO fuel rods is considered in the diffusion 2 tion Q is assumed to be proportional to FA power. approximation. The equations and the boundary conditions are as follows: 1  3. Determination of Linear Heat Rate rD Q P 0 , (8) rr r for Gadolinium-Doped Fuel Rod

QP/ , (9) The further stage of solution is the determination r 0 of the linear heat rate of Gd-doped fuel rod either

from the available linear heat rate of adjacent UO2   fuel rods, or from the average FA heat rate. This  . (10) rr problem arises because the basic parameters cal- rl0 rr 0 culated for a fuel cycle are the average linear heat

here r0 is the radius of the fuel pellet of a fuel rate of FA and the power peaking factor. However, rod, Φ is the thermal neutron ﬂ ux, D is the diffu- to use the fuel performance code for calculations sion coefﬁ cient of thermal neutron in water, l = 0.29 of fuel rod thermomechanical behavior one needs cm is the scattering length of thermal neutrons in the rod-by-rod data.

564 Figure 2. Absorption function Calculation results by the four-group model plotted in dashed line, calculations by the neutron code plotted in open cyrcles. Figure 3. Linear heat rate of the gadolinium-doped fuel rod from FA 1

Calculation results by the four-group model plotted in Calculation results by the four-group model plotted in dashed line, calculations by the neutron code plotted in dashed line, calculations by the neutron code plotted in open cyrcles. open cyrcles. Figure 4. Linear heat rate of the gadolinium-doped Figure 5. Linear heat rate of the gadolinium-doped fuel rod from FA 2 fuel rod from FA 3

The absorption function P presented in Figure The ﬁ gures below present the linear heat rates 2 remains constant in all calculations. The value of the Gd-doped fuel rods calculated by the four- of function Q is chosen so that the calculated heat group model. The results of the calculation by the rate of the UO2 fuel rod agrees with the results of PERMAK neutron code [9] are given for compari- the calculation by the PERMAK neutron code [9]. son. A good agreement between the model and Next, the parameters Q and P are used to solve the neutron code can be seen. the problem for the Gd-doped fuel rod. Figure 3 shows the calculated linear heat rate The average value of the linear heat rate of of the Gd-doped fuel rod from FA 1. FA 1 consisted several UO2 fuel rods that surround the Gd-doped of 240 UO2 fuel rods with 4.95 % enrichment in 235 fuel rod is used to determine the value of Q for the U, 66 UO2 fuel rods with 4.4% enrichment and 6 average UO2 fuel rod. The use of the mean heat Gd-doped fuel rods with 3.6 % enrichment and 5% rate value is caused by the power proﬁ le across weight percent of Gd2O3. The geometry of the UO2 the FA. fuel rod and of the Gd-doped fuel rod pellets was

565 identical, the outer pellet radius was 3.8 mm, the for the gadolinium isotopes concentration and the radius of the central hole in pellets was 0.6 mm. heat rate in a Gd-doped fuel rod. The linear heat rate of the adjacent UO fuel rods was assumed to Figures 4 and 5 show the calculated linear 2 heat rates of the Gd-doped fuel rods from FAs be constant and equal to 18 kW/m. The absorp- 2 and 3. FAs 2 and 3 were fabricated under the tion function P used in calculations is presented in Figure 2. same design, they consisted of 306 UO2 fuel rods with 4.4% enrichment and 6 Gd-doped fuel rods The results of the calculations are provided be- with 3.6 % enrichment and 5% weight percent of low. These results were compared to the calculations [2] using the neutron code HELIOS and also Gd2O3. The geometry of the pellets in the UO2 fuel rods and in the Gd-doped fuel rods was the same. using the diffusion model incorporated into the fuel The pellet outer diameter was 7.57 mm, the cen- performance code TRANSURANUS. Figures 6 and 7 provide the spatial distributions of the gado- tral hole diameter was 1.4 mm. linium isotopes at different values of burnup. Fig- ures 6-8 present the calculations by the described model, by the neutron code HELIOS [2], and also 4. Burn-Out of Gadolinium and by the TRANSURANUS code [2]. The calculations Evolution of Heat Rate Profi le for by the model are plotted with the solid blue line. Gd-Doped Fuel Rod Figure 8 provides the calculated radial profi le of heat rate in the Gd-doped fuel rod. The calcu- The model described in Sections 2, 3 was used lated values are presented for different values of to calculate time evolution of the spatial profi les burnup up to 10 MWd/kgU.

Burnup: a) 0 MWd/kgU, b) 0.4 MWd/kgU, c) 0.8 MWd/kgU, d) 3.1 MWd/kgU Figure 6. Spatial distribution of the isotope 155Gd

566 Burnup: a) 0 MWd/kgU, b) 0.4 MWd/kgU, c) 0.8 MWd/kgU, d) 3.1 MWd/kgU Figure 7. Spatial distribution of the 157Gd isotope

The performed calculations show that the four- rods given the heat rate of the adjacent UO2 fuel group model allows calculation of the heat rate rods. distribution across the pellet of Gd-doped fuel rod This work makes it possible to determine the with satisfactory accuracy, as well as the time evo- histories of linear heat rate for the Gd-doped fuel lution of gadolinium isotopes concentration. The rod from the average heat rate of FA for various signifi cant relative error in calculated concentra- equilibrium WWER fuel cycles, power units and tions of Gd isotopes is observed for small absolute fuel enrichments. The presented model is imple- values of the concentrations. mented in the RTOP fuel performance code to calculate the thermomechanical behavior of the gadolinium-doped fuel rods. 5. Conclusions References The developed four-group model allows the calculation of the radial heat rate profi le and gadolinium burn-out in a Gd-doped fuel rod. The comparison [1] A. Schubert, Cs. Gyori, J. Van De Laar, P. Van of the present calculations with the results of cal- Uffelen, S. Bznuni, T. Safaryan, Extension of the TRANSURANUS burn-up model for Gd-doped UO2 culations by neutron codes shows that the model fuel in WWER reactors. 8th International Confer- describes the heat rate distribution and the gado- ence on WWER Fuel Performance, Modelling and linium burn-out with suffi cient accuracy and also Experimental Support. 28.09-03.10.2009, Albena, calculates the linear heat rate of Gd-doped fuel Bulgaria.

567 Burnup a) 0 MW·days/kgU, b) 0,4 MW·days/kgU, c) 0,8 MW·days/kgU, d) 3.1 MW·days/kgU, e) 10 MW·days/kgU Figure 8. Spatial distribution of heat rate in a pellet

568 [2] M. Ieremenko, I. Ovdiienko, Cross-checking of the [7] Galanin A.D. Introduction into the theory of thermal TRANSURANUS burn-up model for Gd-dopped neutron nuclear reactor. Rev. 2, M.: Energoatomiz- UO2 WWER-1000 fuel based on results of HELIOS dat, 1989 (in Russian). code. 10th International Conference on WWER Fuel [8] S.Yu.Kurchatov, V.V.Lihanskii, A.A.Sorokin, Performance, Modelling and Experimental Support. O.V.Horuzhii. “RTOP-Code simulation of the radial 09-13.09.2013, Sandanski, Bulgaria. distribution of heat release and plutonium isotope [3] Lanning D.D., Beyer C.E., Painter C.L., “FRAP- accumulation in high burnup oxide fuel”. Atomic En- CON-3: Modifi cations to Fuel Rod Material Proper- ergy, v. 92, issue 4, pp. 349-356. ties and Performance Models for High-Burnup Ap- plication”, NUREG/CR-6534, PNNL-11513, Volume [9] Handbook of physical quantities. Edited by 1, October, 1997. I.S.Grigoriev, E.Z.Meilikhov, M.: Energoatomizdat, [4] Berna G.A., Beyer C.E., Davis K.L., Lanning D.D., 1991 (in Russian). “FRAPCON-3: A Computer Code for the Calculation [10] Neutron data encyclopedia, ROSFOND, IPPE, of Steady-State, Thermal-Mechanical Behavior of 2006. Oxide Fuel Rods for High-Burnup”, NUREG/CR- [11] Merzlikin G.Ya. Fundamentals of the theory of nu- 6534, PNNL-11513, Volume 2, December, 1997. clear reactors. The course for the operational per- [5] Lassmann K., “TRANSURANUS: a fuel rod analy- sonnel of nuclear power plants. – Sebastopol: SI- sis code ready for use”, J. Nucl. Mat., v.188 (1992), YaiP, 2001 (in Russian). pp.295-302. [12] PERMAK-A code (version 1.3). Software certifi cate. [6] V.V. Likhanskii, T.N. Aliev, I.A. Evdokimov, V.D. Ka- nukova, A.A. Sorokin, V.G. Zborovskii, Simulation of TsEP registration number ОР No.518 of 21.02.2002. Fuel Rod Behavior by the RTOP Code for Transient Certifi cation registration number ОР No. 136 of Experiments Proc. LWR Fuel Performance Meeting/ 21.02.2002. Federal Regulatory Authority of Russia Top Fuel/WRFPM, Orlando, USA, 26-29 Sept., 2010 on nuclear and radiation safety. М., 2002.

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