Design and Optimization Study of 10,000MWe Very Large Fast Reactor Core

Design and Optimization Study of 10,000MWe Very Large Fast Reactor Core Yasuhiro KOBAYASHI, Shunsuke KONDO and Yasumasa TOGO Department of Nuclear Engineering, Faculty of Engineering, University of Tokyo* Received February 17, 1973 Revised June 25, 1973 To answer the increasing demand for electric power in Japan, Very Large Fast Reactors of 10,000 MWe unit capacity are expected to make their appearance in due course. The paper describes the method and results of a design study on a 10,000MWe Liquid Metal Fast Breeder Reactor. First, a reference design was obtained for this unit of unprecedented capacity by extrapolating the various characteristics of a 1,000MWe LMFBR and the nuclear characteristics thereof were studied. It was in the central part of the core, and that the increase of reactivity and the decrease of breeding ratio with time were rather large for the initial loading core. Secondly, a design optimization procedure was developed based on complex method of nonlinear programming, and the method was applied to the Very Large Fast Reactor. The process resulted in a relatively large core height and fuel pin diameter, while the power cost was improved due to enhanced breeding gain. The fuel center temperature and the coolant velocity were found close to the upper boundaries of their prescribed ranges. These results concurred qualitatively with calculations using more straightforward optimization techniques. KEYWORDS: fast reactors, LMFBR type reactors, design, nuclear characteristics, burnup analysis, optimization, reactor cores, reactor reactivity, breeding ratio, fuel assemblies I. INTRODUCTION In the field of fast reactor research and development, a notable amount of work has been carried around the world. Various conceptual designs for commercial fast breeder reactors have been studied, including that for the liquid-metal cooled fast breeder reactor (LMFBR). These design studies have greatly contributed to the development of the LMFBR, by indicating the state of the art and by establishing clear targets of research and development in many associated fields, as well as by suggesting various inspiring concepts that could lead to innovations in FBR system design. The present study is one such attempt, and deals with the design of an LMFBR of the unprecendented unit capacity of 10,000 MWe. So far, interest has generally been focused on the first generation LMFBR's expected to be realized in the early 1980's, and consequently, the studies have rarely extended to ranges much beyond 1,000MWe. To the authors' knowledge, the largest unit ever treated is a 10,000MWt LMFBR for a steam supply system(1). The LMFBR with a unit capacity of 10,000MWe, which we will term hereafter the "Very Large Fast Reactor" (VLFR), can well be expected to make its appearance in due course to answer the increasing demand for electric power in Japan. According to one estimation(2), they would be introduced in the early part of the next century, with the first practical unit becoming a reality in the 2030's. In this paper the method and results of a design study on a 10,000MWe VLFR are presented and discussed. The second chapter is devoted to a discussion of the characteristics of the reference design, obtained by extrapolating the various characteristics of a 1,000MWe LMFBR. In the third chapter, a design optimization procedure is developed and applied to the VLFR in order to obtain a consistent evaluation of the optimal design and the effects * Hongo, Bunkyo-ku, Tokyo.

J. Nucl. Sci. Technol., of various design limits. Several studies(3)(4) have been published in the past regarding design optimization by means of linear and nonlinear programming techniques. These procedures have effectively and consistently contributed to improving reactor design with the view to minimizing (or maximizing) certain performance index such as power cost or fuel cycle cost. Optimization of reactor design is particularly important for commercial reactor development because even a slight improvement in the design may bring about significant saving of cost. The procedure described here is based on the complex method, and is applied to obtain well balanced values for the design variables of this reactor of unprecedented size. Table 1 Reference design of 10,000MWe Very Large Fast Reactor II. REFERENCE DESIGN 1. Parametric Survey for Reference Design As little information is available for determining the reference design of the VLFR with consistent parameters, a preliminary parametric study was necessary concerning the nuclear and thermohydraulic characteristics of a reactor core of such size. Calculations were performed with use made of the one-dimensional diffusion code AIM-6(5) (modified version) and the twodimensional diffusion and burnup code 2DB(6), the one-dimensional perturbation code PERT(7) (modified version), a thermohydraulic code and a fuel cycle cost code. The Russian cross section set(8) and fewer-group sets collapsed thesefrom were used in the neutronic calculations. The reference design is summarized in Table 1 and the core configuration depicted in Fig. 1. Fig. 1 Configuration of reference core (1) Core Geometry Several core shapes have been proposed in the past for large LMFBR-including (a) cylindri- In the present study, a flat cylindrical shape has been taken up, for simplicity in form and compactness, and this configuration is further believed to possess significant advantages in nuclear performance. In respect of average power density, currently adopted values are around 450 kw/l, examples being found in the French prototype fast reactor Phenix of 430 kw/l and in the UK prototype PFR of 480kW/l. It is considered difficult to raise the power density of the VLFR to a level significantly higher than these values, and the core size of the VLFR must

Vol. 10, No. 10 (Oct. 1973) consequently be very large. Extension of the cylindrical core can be accomplished by two means. One is to increase the active height, and the other to extend the radius. The core height cannot in general be increased indiscriminately on account of the pressure drop through the core and the coolant void coefficient that would become excessive. In the reference design, the core height was set at 150cm to keep within reasonable limits the coolant temperature rise and coolant velocity through the core, determined by parametric studies on the thermohydraulic chracteristics, and in consideration of the reported design experience obtained on the 1,000MWe LMFBR. On the other hand, radial extension of the core affects such values as the radial peaking factor, and the radius of the reactor vessel. Having already chosen the average power density and the core height, the outer core radius is determined and a value of 344cm was obtained, which is considered to be reasonable for reactor vessel design(1). To attain the given average power density, radial power flattening is necessary and a flat neutron flux in the core region should be favorable for reducing the critical mass. Radial power flattening is ensured in the present study by the adoption of a two-region core to minimize complication of the core composition, though needless to say, three or more radial zones should further reduce radial peaking(9). In Fig. 2 is presented the power peaking factor as function of the inner core radius, based on the result of the parametric survey. It indicates that to realize Fig. 2 Result of parametric survey on radial power flattening the required power flattening, the volumetric ratio Vin./V between inner and total core should be between 0.75 and 0.85, when the fissile enrichment of the inner core is 0.09 and that of outer core 0.12. This combination of parameters has been determined from considerations mainly of power flattening and criticality in the initial cycle. The value of Vin./V does not change significantly in the course of burnup, because in a core with flat power distribution, the radial peaking factor does not change very sharply with burnup. (2) Blanket To determine the proper thickness of the radial blanket in the VLFR, the relation between the breeding ratio and the radial blanket thickness was calculated, with the result shown in Fig. 3. Fig. 3 Result of parametric survey on radial blanket thickness From this figure, it can be seen that beyond 40 cm, it becomes increasingly ineffective to extend the radial blanket. A similar result was obtained for the effect on breeding gain provided by changing the thickness of the axial blanket. Hence a thickness of 40cm was adopted in the reference design for both axial and radial blankets. (3) Fuel Design Both mixed oxide and mixed carbide have been proposed as fuel for the LMFBR. The former is considered to be the more orthodox, and the latter the more advanced. In the present reference design, the mixed oxide formula is adopted, with mixed carbide considered in parallel for the purpose of comparing the nuclear characteristics. Vented fuel is adopted, since it not only permits lowering of the internal pressure of the fuel pin but is also conducive to a decrease in fuel pin length, and further proved(9)(10) also

J. Nucl. Sci Technol.,. to contribute to the safety and economics of the LMFBR. An outer diameter of 6.4mm has been chosen for the fuel pin, this value being typical in previous reports of 1,000MWe LMFBR design studies(9)(11) (4) Thermohydraulics In Fig. 4 is shown the results of a parametric survey on the thermohydraulics of a sodium cooled core with mixed oxide fuel. It represents the relation between the spacing of the fuel pins and several core parameters, such as fuel volume fraction and coolant velocity. (5) Components The components of a nuclear power plant Fig. 4 Typical result of parametric survey of thermohydraulic characteristics include such items as reactor vessel, intermediate heat exchanger, steam generator, sodium pump and turbine generator. These components call for detailed studies including, the possibilities of extension in unit capacity, in keeping with the size of the nuclear power plant as a whole. Scanty information available today in this connection renders such detailed study difficult. 2. Nuclear Characteristics To obtain an outline of the nuclear characteristics of the reference core, a typical mixed oxide fueled LMFBR design (1,000MWe) and a mixed carbide fueled LMFBR design (10,000MWe) are taken up for purposes of comparison. The result is summarized in Table 2. Since the core enrichment is inversely proportional to the volume of the core, the infinite multiplication factor also decreases in keeping, and approaches 1.0: it is Table 2 Comparison of several nuclear parameters. 1.10 in the inner core and 1.27 in the outer core in the case of the mixed oxide fueled VLFR. The neutron energy spectrum is softened in this very large core, as clearly shown by the

Vol. 10, No. 10 (Oct. 1973) value of median neutron flux energy. Figure 5 shows the neutron flux energy spectrum in the center of the reference core, together with those of the other cores. The breeding ratio is high owing to the large fraction of fertile material 238U in the initial loading core. This spectrum softening also causes enhancement of the Doppler coefficient and prolongation of the prompt neutron lifetime. The other point to be noted is that a high neutron flux is required to obtain the specified power density, since the core has a relatively Fig. 5 Comparison of neutron flux energy spectra small macroscopic fission cross section. Also a somewhat high neutron fluence is necessary to attain a specified burnup of fuel, and the maximum allowable fuel burnup may thus become smaller in the VLFR than in orthodox sized reactors. (1) Void Coefficient In the early days of fast reactor development, it was considered that the sodium void coefficient should have to be negative both locally and in overall value. More recent advances in safety studies and the development of incore instrumentation would now allow it to be positive, to a limited extent. In Fig. 6 are shown the values of the void coefficients of the reference design calculated by one-dimensional perturbation theory, together with the relation between change of reactivity and void radius in the core. The void coefficients have been evaluated by two-dimensional diffusion calculations to cover two particular cases of sodium voiding, and the results are given in Table 3. It has also been found that voiding from seven subassemblies in the central part of the core, corresponding to a void radius of about 20 cm, induces a reactivity increase of about 6. (2) Burnup Analysis Burnup calculations concerning the reference core have been performed for the initial cycle and for equilibrium state. Before describing these results, the procedures adopted for the burnup calculation will be described briefly. The equilibrium cycle was examined by considering the effect of fuel shuffling and by calculating the

J. Nucl. Sci. Technol., reaching about 4% in the initial cycle. This is because the density of the fissile material increases, while that of the fertile material-which mainly consists of 238U-decreases during burnup, owing to the high breeding ratio in the initial stage. The changes in the breeding ratio are shown in Fig. 8 for the initial and equilibrium cycles. The breeding ratio decreases with burnup, averaging 1.25 in the equilibrium fuel cycle, though it is higher in the initial cycle. Fig. 6 Reactivity change caused by voiding of central radial region Table 3 Sodium void coefficient Fig. 8 Change of breeding ratio with time core equilibrium using a simplified burnup program. For the sake of simplicity, three fuel batches, a cycle time of 300 days for each region of the core, and uniform scatter loading were assumed. It was found that equilibrium condition would be approximately established after five cycles. In Fig. 7 are shown the changes of reactivity with time. The increase of reactivity is large, Figures 9 (a), (b), (c) and (d) illustrate the changes seen in the fuel composition, as represented in terms of the density changes with time of 239Pu, 240Pu, 241Pu and 238U in the inner core, in both initial and equilibrium cycles. According to these results obtained on the burnup performance, the change of nuclear characteristics is large during the initial cycle as compared with the equilibrium cycle. The high excess reactivity possible in this reference core should mean so much more room for improvement of the burnup program. III. OPTIMIZATION STUDY 1. Optimization Procedure Fig. 7 Change of effective multiplication factor with time (1) Complex Method The complex method(12)(13) of nonlinear programming is a search technique used extensively in constrained optimization problems. It has been adopted for the present optimization problem, on account of its effective and simple algorithm for calculation. The following sequence is used where n is

Vol. 10, No. 10 (Oct. 1973) Fig. 9 Change of atomic density with time the number of independent variables in the problem. (a) First, potential points that satisfy all the constraints must be provided to derive (b) Then the function is evaluated at each of these points, and the vertex of worst value Xh is determined. (c) The worst point is replaced by its overreflection in the centroid of the remaining (d) If this trial point Xnew satisfies all the constraints, the performance index is evaluated there, and the whole process repeated, while if (e) The trial point happens to be the worst point or violates some constraints in the new configuration, the trial point is moved halfway further on towards the centroid. (2) Description of Problem There are many design parameters involved in the core design; we here choose several of them as independent variables to describe the core design, with a number of. assumptions adopted for the sake of simplicity. These selected parameters are thought to influence the performance index strongly and are required for design calculations, In this optimization problem, the following parameters are used as independent variables. (a) Core height (cm) (b) Fuel pin diameter (mm) (c) Fuel pin pitch to diameter ratio (d) Average linear power (W/cm) (e) Ratio between outer and inner core volumes (f) Difference of enrichment between outer and inner cores (g) Excess multiplication factor at initial loading Other design variables such as breeding ratio and maximum fuel center temperature are dependent variables to be determined in the course of calculation. Seven among the dependent pa-

J. Nucl. Sci. Technol., rameters are treated as constrained variables. The constraints to be considered explicitly in this study are and a step for evaluating the performance index, which, in this study, is the power cost. In each calculational cycle, the former step determines a perfomance index of the given design point after a series of calculations. The procedure is as follows. From the set (3) Calculation Procedure The procedure of calculation is shown in Fig. 10. First, the initial design points must be given to start the optimization calculation. In this study, 14 design points that satisfy all the constraints are used as vertices of the initial complex. These initial design points are established by choosing the ranges of the indepedent variables and using normalized random numbers. The optimization procedure comprises a step for setting a search point-i. e. a set of values of independent variables-by the complex method, design point and the latter the value of the of independent variables chosen, the outline of the core configuration and composition is determined. To determine the core enrichments, that for the inner core is adjusted by iteration to satisfy the criticality condition, keeping the difference of enrichment between the inner and outer cores constant. Adopting a fuel cycling method similar to that previously described, the core composition in the equilibrium cycle is obtained by one point burnup calculation, with the burnup and power density given in each region. Thermohydraulic calculations are performed to determine such variables as fuel center temperature and pressure drop through the core, while nuclear calculations provide information on radial and axial power profiles, reactivity, breeding ratio, fuel inventory etc. One-dimensional two-group diffusion codes are used in this procedure. Finally, power cost-which is mainly fuel cycle cost-is calculated from the results obtained from these calculations. Upon evaluation of the performance index of a given search point, the optimizer determines the next search point. If the design point under consideration violates more than one of the constraints imposed in this problem, the optimizer modifies the design point so as to satisfy all the constraints 2. Results in the course of the calculation. Fig. 10 Flow diagram of optimization procedure ( 1) Numerical Result 1 Optimization of the VLFR core in equilibrium cycle is performed by means of the procedure described in the previous section. In Table 4 is shown the result, representing the optimized core. The optimized design in this case is obtained by using a systematic procedure, in contrast to the case of the reference core. (2) Numerical Result 2. So far, the design optimization has been

Vol. 10, No. 10 (Oct. 1973) Table 4 Optimum design point (Numerical result 1) -undertaken for the equilibrium cycle. But under certain circumstances the economical evalution is performed in a simpler manner, for example, based on the initial core. For purposes of comparison, optimization without accounting for burnup has been carried out, with the result presented in Table 5. A large difference is revealed between the two cases of initial and equilibrium states, the optimized core based on the initial stage having a markedly greater core height, which is conducive to high breeding ratio. 3. Discussion Table 5 Optimum design point (Numerical result 2) (1) Optimization Method While the complex method is considered an effective algorithm for constrained optimization problems, the relevant procedure is not very efficacious, particularly in the vicinity of the optimum point. The convergence criterion of the optimizer is required to be small enough to attain sufficient accuracy and large enough to make convergence possible. In the present case, the criterion used was to see that the difference between the maximum and minimum costs in the complex must be less than 0.1% of the minimum value. The fuel center temperature and coolant velocity are given in Table 4, together with other parameters. They appear close to the upper boundaries of the constraints, but do not reach the limits by a very small margin on account of numerical errors. To study the convergence characteristics of the optimizer, the behavior of the design parameters in the course of the calculation (Numerical result 1) are given at intervals of about ten calculational cycles. The behavior of core height, fuel pin diameter, fuel center temperature and power cost is depicted in Fig. 11. This result indicates that the selected value of the convergence criterion is appropriate for our present purpose. (2) Optimized Core The following comments are offered on the optimized core for equilibrium cycle (Numerical result 1). (a) Compared with conventional LMFBR, the core height is rather great. This is due to the adoption of vent type fuel which has the particular advantage of permitting the realization of reasonable fuel pin length. (b) Constraints on fuel center temperature and coolant velocity are the most significant, and are often violated by these parameters in this calculation. (c) The optimized core has nearly the same fuel and coolant volume fractions as those of the reference core, by pure coincidence. (d) The optimized fuel pin diameter is rather large, fuel fabrication cost having been assumed to decrease in inversely proportion thereto. (e) Judging from the design parameters thus determined, such as breeding ratio and average power density, the power cost can be improved by increasing breeding gain rather than by decreasing fuel inventory. It is important to prove the uniqueness of the

J. Nucl. Sci. Technol., Fig. 11 Behavior of parameters in early phase of calculation convergence point, which is difficult to show analytically by this procedure. Two techniques -grid search and trial and error method-can be applied to examine this aspect. In the former, some typical values are set for each independent variable, and a search point is determined from combinations of these values, the performance index is evaluated for each combination. In the Table 6 (a) Result of grid search Table 6 (b) Result of random search

Vol. 10, No. 10 (Oct. 1973) latter, the search point is established from random numbers and treated similarly. As applied to the numerical result 1, the two methods produce the data presented in Tables 6 (a) and (b), including the case of smallest power cost. Qualitative agreement is seen with the previously obtained design parameters, such as core height and fuel pin diameter, in spite of the rather simplified calculation. Optimization by these techniques is time consuming, and further comparison of the optimization methods is not studied in detail here. IV. CONCLUSION A description has been given of the method and results of a design study on a 10,000MWe Very Large Fast Reactor, which can be expected to make its appearance in due course to answer the increasing demand for electric power in Japan. First, a reference design of mixed oxide fueled Liquid Metal Fast Breeder Reactor whose unit capacity was 10,000MWe was obtained by extrapolating the various characteristics of a conventional 1,000MWe LMFBR. The nuclear characteristics of the reference core was examined in comparison with a mixed oxide fueled 1,000MWe LMFBR and a mixed carbide fueled 10,000MWe LMFBR cores. It was found that the neutron energy spectrum was softer in the reference-as compared with conventional-core and the prompt neutron lifetime was longer. In respect of the sodium void coefficient, the increase of reactivity was about 6 when seven subassemblies in the central part of the core were voided, which would become very much larger for large scale voiding. A steep increase of reactivity and decrease of breeding ratio with burnup were found for the period of initial loading, because of the large fraction of fertile material in the fuel composition. Such increase of reactivity should provide room for improvement of the burnup program. Compared with a highly enriched core, the low enrichment adopted here would require a higher neutron flux to obtain a specified power density, meaning a somewhat higher neutron fluence to attain a specified burnup of fuel. Thus the maximum allowable fuel burnup may become smaller in the VLFR than in a conventional LMFBR. Secondly, the method and the results of an optimization study have been presented. The optimization procedure adopted was complex method of nonlinear programming, which was then applied to the 10,000MWe VLFR core. The procedure comprised a step for seeking the design point of minimum power cost and a step for evaluating the power cost based on nuclear and thermohydraulic calculations. Seven independent variables and seven constraints were considered in this procedure. The former included such parameters as core height, average linear power and fuel pin diameter, and the latter represented limiting factors on the parameters, such as fuel center temperature, pressure drop through the core and coolant velocity. The process resulted in a fuel center temperature and coolant velocity that were close to the upper boundaries of their prescribed ranges. The dimensions of the core thus optimized represented a relatively large core height and fuel pin diameter, while the power cost was improved due to enhanced breeding gain. These results concurred qualitatively with calculations using simpler and straightforward but time consuming methods. It can be said that the optimization procedure developed by the present study should serve as a practical design tool that would contribute to establishing a systematic approach to the core design of the LMFBR. Finally, it should be pointed out that the present study has revealed a number of noteworthy problems, involved in the enlargement of the core size, the enhancement of neutron flux and the scaling-up of components. These problems remain to be solved through active and creative research and development efforts. ACKNOWLEDGMENT The authors wish to express their thanks to Prof. S. An, University of Tokyo, for his valuable suggestions and discussions throughout the performance of the present work. Thanks are also due to Mr. H. Nakagawa, Tokyo Electric Power Co. Ltd., for his valuable suggestions relative to the enlargement of the LMFBR. Numerical calculations were carried out using a HITAC 5020 and a HITAC 5020E in the Computer Center of the University of Tokyo.

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