Journal of NUCLEAR SCIENCE and TECHNOLOGY, 3[5], p.184~188 (May 1966). A Method For the Burnup Analysis of Power Reactors in Equilibrium Operation Cycles Shoichiro NAKAMURA* Received February 7, 1966 This paper describes a method based on the consistent thermal neutron flux concept for burnup analysis of power reactors in equilibrium operation cycle. Radial flux distributions are discussed, as well as the variation of isotopic densities in the fuel, and considerations are given on control rod distribution for maintaining a consistent thermal flux distribution. The present method can be applied to most power reactors like BWR and PWR in which a multiple-batch refueling scheme is adopted. I. INTRODUCTION The main objective of burnup calculation is to determine the characteristics of the core that vary with operation time. Burnup calculation consists of the main parts: Flux distribution calculation with a diffusion code and isotopic density calculation with burnup equation. From the burnup calculations we obtain the variations of flux distribution, the isotopic densities, the excess reactivity, the available operation time and the total energy output produced by a unit volume of fuel. Since the eigenvalue (keff) of the reactor should be maintained at unity, excess reactivity is invariably cancelled by the control rods during full power operation. The control rods having as they do a strong influence on flux distribution, flux calculations in burnup analysis must be performed with consideration given to the effect of the control rods. When the operation cycles of the core are stabilized, refueling is performed periodically. We define gone operation cycle h as the period of operation between one refueling and the next. If an identical process of physical property variation is repeated in each cycle, calculations for only one cycle should suffice. Excess reactivity being large at the beginning of the cycle, a large number of control rods are inserted, and it is easy to modify the flux distribution. The flux distribution decreases its flexibility with operation time, and finally, it becomes quite rigid at the end of the cycle when all the control rods are fully withdrawn. The flux distribution at the end of the cycle is uniquely determined by the history of the flux distribution during the cycle. It has been shown since our previous study(1) that this fact is compatible with burnup calculations based on consistent thermal flux distribution. By gconsistent flux his meant in this instance the regid shape of the thermal flux distribution maintained throughout the cycle. Using the two-group theory, this paper discusses the method based on the consistent flux distribution of radial coordinates. The control rod patterns required for maintaining the consistent thermal flux are also discussed. The present concept not only simplifies the burnup calculations as a whole, but also has another merit in that the consistent thermal flux is the flattest among all the possible flux distributions during the cycle, as it can easily be proved. II. THEORY In the one dimensional two-group diffusion approximation, the flux distributions on radial coordinates satisfy the equation * Nuclear Reactor Design Section, Hitachi Works, Hitachi Ltd., Hitachi-shi, Ibaraki-ken. 14
Vol. 3, No. 5 (May 1966) 185 (1) into a number of zones bounded by concentric circles, each numbered (represented by the symbol k) sequentially from center to periphery. The spent fuel is discharged and replaced by fresh fuel in some of these where (2) In the above equations, Sa is the absorption cross section, SIS is the absorption cross section of Xe and Sm. Assuming that the control rods can be approximated by a homogeneous poison, the absorption is represented by Sc. We consider that the variations of the absorption and fission cross sections are functions of exposure, so that zones. Shuffling of partly irradiated fuel is also undertaken. It is assumed that the shuffled fuel is completely mixed and therefore uniform in in each zone. The refueling schemes are represented by the matrix designated as gshuffling matrix h, the element Gkk Œbeing unity when the fuel in the k-th zone has been previously irradiated in the k Œ-th zone, and otherwise zero. If we assume that, the thermal flux where t is the thermal neutron flux time of exposure of the fuel, N the isotopic density, and SMa the absorption cross section of materials other than fuel. The dependence of the fast neutron cross sections on exposure is neglected. The relation of the isotopic densities to is obtained(2) from the burnup equations (3) (4) (5) (6) t distribution is maintained constant throughout, in conformity with the consistent flux concept, the exposure of the fuel at the end of a cycle is given by (12) where g is the position in the k-th zone, T the duration of one cycle, and <P2>k Œthe average thermal flux in the k Œ-th zone. With usual power reactors, the reactivity of the core at the end of a cycle is a decreasing function of the operation time, so that it may be assumed that the reactor is just critical at the full power with Xe and Sm at equilibrium, and with all the control rods fully withdrawn. Therefore, we can obtain the consistent thermal flux P02 by solving (13) (7) (8) (9) (10) (11) where P1 is the non-leakage factor for the neutrons slowing down from fission energy to resonance. All the refueling schemes considered here are multiple-batch. The core is divided which can be done by letting keff=1.0 and Sc =0 in Eq.(1). Here we consider that the neutron flux is normalized so as to satisfy where Pt, is the thermal output in MW. (14) If the neutron flux distribution is constant, the total power varies with time depending upon the variation of the fission cross section. Constant power can nevertheless be maintained if desired by changing the neutron flux level and without changing the shape of the flux distribution. 15
and 186 J. Nucl. Sci. Technol. In order to obtain the solution of Eq. (13), it is convenient to let Sc=0 in Eq. (1). Two iteration schemes are used in order to obtain the solution of Eq. (13): the first being an ginner iteration hto determine eff for a given value of T, and the k second an gouter iteration hto obtain the value of T that gives keff=1 for Sc=0 in Eq. (1). In the inner iteration, a normalized flux 2 is assumed in obtaining tt from Eq. (12), P upon which the exposure-dependent constants can be calculated. Solving Eq. (1) for this tt, a new q2 and eigenvalue are obtained. Then the new P2 is used for the next value of tt. This procedure is repeated until tt2 and keff converges for the given T. For some inputs, however, oscillation of the eigenvalue is experienced during the iterations performed by the above procedure. In order to remove this oscillation, we replace P2(g-1) (g being the iteration number) by the formula (15) for calculating tt in Eq. (12). It has been found that this modification not only removes the oscillation but also reduces the computing time for all cases to approximately one third of that required when P(g-1)2 is directly used. In the first and second steps of the outer iterations, certain values are assumed for T1 and T2 for each of which the corresponding and k2, - are calculated keff-i. with e k1 Eq. (1). After the second iteration, letting e be the iteration number, Te is extrapolated by where (17) (18) (19) In the above equations, t is the time measured from the beginning of the cycle. Assuming that koo is a decreasing function of exposure, it can be proved that the consistent flux has the lowest HSF (Hot Spot Factor). If the core is operated with a flux with HSF lower than that of the consistent flux, koo at flux peaking becomes higher than the case of consistent flux at the end of the cycle, so that the peaking level becomes higher than the consistent flux. Conversely, in order to obtain a lower peaking at the end of the cycle, a higher flux peaking must be maintained at the beginning of the cycle. III. NUMERICAL ILLUSTRATIONS For the purpose of illustrating the application of the present method, calculations were performed for a large power reactor with the characteristics shown in Table 1. Table 1 Characteristics of the Core (16) and the corresponding ke is calculated. This procedure is repeated until ke converges to unity. Once the solution of Eq. (13) is obtained, it can be maintained throughout the cycle by varying the control rod distribution. The time dependence of the control rod during the cycle can be obtained by solving of Eq. (1), letting P2=P02 and keff=1.0: Sc Two types of refueling schemes were considered: (1) 5-batch out-in The core is divided into five concentric circular or annular zones of equal area, During each refueling, the fuel in the first zone is discharged, the fuel in the second to fifth zones are shifted to the next inner zones, and fresh fuel is 16
Vol. 3 No. 5 (May 1966) 187 introduced in the fifth zone. The shuffling matrix for this scheme is (2) 5-batch scatter The core is divided into ten concentric zones of equal area for instance. Each of the zones are further divided into five subzones of equal area. During each refueling, fuel in the highest exposure subzone in each zone is discharged, the fuel in the remaining subzones shifted inwards, and the outermost subzone loaded with new fuel, as shown in Fig. 1. Fig. 2 The Consistent Thermal Flux and Control Rod Distributions for the 5-batch Out-in Scheme Fig. 1 Approximation for the 5-batch Scatter Scheme This refueling scheme is none other than an out-in refueling scheme performed in each zone, and can be expressed in terms of a diagonal matrix where This approximation for the scatter scheme is valid when the widths of the subzones are so small that the variation of thermal flux within the five adjacent subzones is negligible. Figures 2 and 3 show, for the two types of refueling schemes being considered, the consistent thermal flux distribution obtained from Eq. (13), as well as the control rod distribution required to maintain these flux distributions. Despite the appearance of Fig. 3 The Consistent Thermal Flux and Control Rod Distributions for the 5-batch Scatter Scheme (The shape of the small flux peakings will be changed if other calculation model is used.) negative values of Sc in Fig. 3, the calculated can still be used for control Sc rod programming, based on the following considerations: (1) Local variations in Sc hardly influence the results of the burnup calculation, so that Sc can be averaged in each of the small regions. When Sc is averaged for each zone, only positive values are obtained. (2) The actual control rods used in practice are heterogeneous, whereas in this paper they have been treated as homogeneous. The control rod pattern is determined in such manner as to satisfy the condition that the locally averaged values of the 17
188 J. Nucl. Sci Technol effective absorption cross section of the actual control rods are equal to the locally averaged values of Sc. ACKNOWLEDGEMENT The author wishes to thank his colleagues Mr. T. Kawai, Mr. M. Yokomi and Mr. T. Morita for their stimulating and valuable discussions. The numerical calculations were performed by Mr. H. Kobayashi. REFERENCES (1) NAKAMURA, S.: Effect of Fuel; Shuffling on Burnup of BWR, Preprint of 1961 National Nuclear Congress, Tokyo, Japan, A25, (1961). (2) BENEDICT, M., PIGFORD, T.H.: gnuclear Chemical Engineering h, 91 (1957), McGraw-Hill Book Co. 18