Keisuke Okumura, Yoshiaki Oka, and Yuki Ishiwatari

Abstract The most fundamental evaluation quantity of the nuclear design calculation is the effective multiplication factor (kf and neutron flux distribution. The excess reactivity, control rod worth, reactivity coefficient, power distribution, etc. are undoubtedly inseparable from the nuclear design calculation. Some quan­tities among them can be derived by secondary calculations from the effective multiplication factor or neutron flux distribution. Section 2.1 treats the theory and mechanism to calculate the effective multiplication factor and neutron flux distri­bution in calculation programs (called codes). It is written by Keisuke Okumura.

The nuclear reactor calculation is classified broadly into the reactor core calcu­lation and the nuclear plant characteristics calculation. The former is done to clarify nuclear, thermal, or their composite properties. The latter is done to clarify dynamic and control properties, startup and stability, and safety by modeling pipes and valves of the coolant system, coolant pump, their control system, steam turbine and condenser, etc. connected with the reactor pressure vessel as well as the reactor core. The reactor core, plant dynamics, safety analysis and fuel rod analysis are described in Sect. 2.2. It is written by Yoshiaki Oka and Yuki Ishiwatari.

Yoshiaki Oka, Sadao Uchikawa, and Katsuo Suzuki

Abstract Summary of development and improvement of light water reactors is described in Sect. 3.1. It is written by Yoshiaki Oka.

Design and management of a boiling water reactor (BWR) core is described in Sect. 3.2. It includes design criteria, design of fuel lattice and assembly, reactivity change with burn-up, control of power distribution and history, future trends in core design, core and fuel management. The author of the section is Sadao Uchikawa.

The core nuclear design of PWR is written in Sect. 3.3. The features of PWR core and basic criteria of PWR core design are presented. The design setup of core, fuel lattice, and fuel assembly follows. Control rods and chemical shim are described in the reactivity characteristics. Power distribution control is explained. In addition, evolution and future trend, core management, and fuel management are shown briefly. This section is written by Katsuo Suzuki.

As mentioned, the reactor power of BWRs can be controlled by control rods and core flow rate. BWR operation management must accurately calculate power distribution in each part of the core and achieve efficient fuel burnup by control of control rods and core flow rate. A reactor operation scheme is prepared by using computational codes to simulate the reactor operation state; a time variation in Xe concentration as power is evaluated and 3D nuclear and thermal-hydraulic coupled core calculations are performed.

In the early BWR startup, core power was slowly increased while applying the PCIOMR, accumulating Xe. It took 3-4 days to reach the rated power from start of the electric generator. The startup period was reduced to about 1 day by improvement in fuel performance and power rise speed and by effective combination of control rod operation and core flow rate adjustment.

After startup, the long term reactivity is controlled by changing the control rod location and depth (control rod pattern) and the short term reactivity by core flow rate. Reactor operation management during operation involves preparing the period and scheme for control rod pattern adjustment. In the past the control rod pattern was adjusted mainly at low core power, but recently it is conducted at nearly the rated core power because of improved fuel performance.

(vi) Advanced PWR (APWR) (adoption of neutron reflector)

The implementation of the low leakage core and gadolinia-added fuel is discussed next, and the APWR (neutron reflector), high burnup, MOX-fueled core, and power up-rating issues are mentioned following that. The long cycle length and load following operation were touched in the list [1] of Sect. 3.3.2 and [4] of Sect. 3.3.5 respectively.

(1) Low Leakage Core

Originally the PWR fuel loading pattern was generally an out-in pattern (fresh fuel is loaded in the outermost core region and burned fuel in the core inside region). This pattern can flatten the core radial power distribution and reduce the power peaking factor. However, the loading of fresh fuel in the outermost core region leads to a high power in the core periphery and large neutron leakage to the core outside, and therefore it gives a disad­vantage in reactivity. Hence, since the late 1980s, a new fuel loading

Fig. 3.58 Arrangement of gadolinia-addded fuel rods in fuel assembly [36] (17 x 17 type and 24 gadolinia-added fuel rods) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

pattern, in which burned fuel assemblies are loaded in the core periphery within the design limit such as power peaking, has been implemented in refueling instead of the out-in pattern. The new fuel loading pattern is advantageous regarding reactivity because of small neutron leakage to the core outside. The higher burnup and larger number of burned fuel assem­blies loaded in the outermost core region can lead to higher reactivity and a smaller number of discharged fuel assemblies (i. e. smaller number of fresh fuel assemblies). For example, number of fresh fuel assemblies needed can be reduced by two applying such a low leakage loading pattern. Various fuel loading patterns are investigated for advantageous reactivity within the basic criteria of core design [35].

The functions of the low density PyC layer (first layer) are to protect the inner high density PyC (second layer) layer by adsorbing fission fragments coming from the fuel kernel, and to provide a space for containing FP gas coming from the fuel kernel. The inner high density PyC (second layer) is provided for retention of FP gas. Short half-life of FP gas nuclei and their small diffusion coefficients in the high density PyC layer, as well as retention capability of the fuel kernel, ensure that FP gas release from the fuel is actually no problem as long as the second layer has no defects.

The SiC layer (third layer) has high retention capability for FP gas and metallic FP fragments. Due to the short half-life of these gas nuclei and their small diffusion coefficients, the SiC layer, the third layer has good retention capability of them just as the second layer does. The outer high density PyC layer (fourth layer) compresses the SiC layer externally by shrinking during irradiation. That prevents the SiC layer from experiencing a tensile load due to the internal gas pressure and hence avoids failure of the coated particle fuels.

In PWRs, the reactivity control is accomplished in part by varying the concentration of boric acid (H3BO3) dissolved in the coolant. Such a chemical shim cannot be made to respond as quickly as control rods. Therefore, the chemical shim is used to control the long-term reactivity changes due to such factors as fuel burnup and Xe transients. It substantially reduces the number of control rods required in the reactor. Furthermore, because boric acid is more or less uniformly distributed through the reactor core, its concentration changes can be made without disturbing the neutron flux in the core.

Figure 1.9 shows an example of the changes of the boric acid concentration in a PWR. It is observed in the figure that the chemical shim compensates for the negative reactivity due to 135Xe and 149Sm and for the reactivity decrease with fuel burnup.

If a chemical shim is present, a decrease in coolant density due to temperature rise will also lead to a decrease in boric acid concentration. Thus it has a positive reactivity effect on moderator temperature coefficient as shown in Fig. 1.10. The requirement for a negative moderator temperature coefficient will limit the amount of boric acid concentration allowed.

The lattice burnup calculation prepares few-group homogenized cross sections with burnup on the infinite lattice system of fuel assembly. The series of lattice calculation procedures described in Fig. 2.8 are repeatedly done considering changes in fuel composition with burnup.

However, the lattice burnup calculation is not carried out in the design calculation of LMFRs which have a high homogeneity compared with LWRs and do not lead to a large change in neutron spectrum during burnup. Hence a macroscopic cross section at a position (r) in the reactor core can be formed by Eq. (2.30) using the homogenized microscopic cross section prepared in the lattice calculation. At this time, the homogenized atomic number densities are calculated according to burnup of each region during the reactor core calculation.

Ъ^(^)=^Ыи, и,то(г)о1х:вто (2.30)

і

By contrast, when considering 235U in a fuel assembly of a LWR as an example, the constituents of UO2 fuel, MOX fuel, or burnable poison fuel (UO2-Gd2O3) each lead to different effective cross sections and composition variations with burnup. It is difficult to provide common few-group

homogenized microscopic cross sections and homogenized atomic number densities suitable for all of them. Moreover, space-dependent lattice calcula­tions by a fuel assembly model during the core calculation require an enormous computing time. In the design of reactors with this issue of fuel assembly homogenization, therefore, the lattice burnup calculation is performed in advance for the fuel assembly type of the core and then few-group homoge­nized macroscopic cross sections are tabulated with respect to burnup, etc. to be used for the core calculation. In the core calculation, macroscopic cross sec­tions corresponding to burnup distribution in the core are prepared by interpo­lation of the table (the table interpolation method of macroscopic cross sections) [15].

Atomic number densities of heavy metals and fission products (FPs) are calculated along a burnup pathway as shown in Fig. 2.12, called the burnup chain.

The time variation in atomic number density (Ni) of a target nuclide (i) can be expressed as the following differential equation.

where the first term on the right-hand side expresses the production rate of nuclide i due to radioactive decay of another nuclide j on the burnup chain. Aj is the decay constant of nuclide j and /j! i is the probability (branching ratio) of decay to nuclide i. The second term is the production rate of nuclide i due to the nuclear reaction x of another nuclide k. The major nuclear reaction is the neutron capture reaction and other reactions such as the (n,2n) reaction can be considered by necessity. {okxф), which is the microscopic reaction rate of nuclide k integrated over all neutron energies, is calculated from the fine-group effective cross section and neutron flux in the lattice calculation. gk! is the probability of transmutation into nuclide i for nuclear reaction x of nuclide k. The third term is for the production of FPs. Fl is the fission rate of heavy nuclide l and yl! 1 is the production probability (yield fraction) of nuclide i for the fission reaction. The last term is the loss rate of nuclide i due to radioactive decay and absorption reactions.

Applying Eq. (2.31) to all nuclides including FPs on the burnup chain gives simultaneous differential equations (burnup equations) corresponding to the number of nuclides. Ways to solve the burnup equations include the Bateman method [16] and the matrix exponential method [17]. This burnup calculation, called the depletion calculation, is performed for each burnup region in case of multiple burnup regions like a fuel assembly and it gives the variation in material composition with burnup in each region. The lattice calculation is carried out with the material composition repeatedly until reaching maximum burnup expected in the core.

The infinite multiplication factor calculated during the lattice calculation is described as the ratio between neutron production and absorption reactions in an infinite lattice system without neutron leakage.

/too = Production Rate / Loss Rate

= Production Rate / (Absorption Rate + Leakage Rate from System) = Production Rate/ (Absorption Rate + 0.0)

(2.32)

Figure 2.13 shows the infinite multiplication factor obtained from the lattice burnup calculation of a BWR fuel assembly. Since thermal reactors rapidly produce highly neutron absorbing FPs such as 135Xe and 149Sm in the beginning of burnup until their concentrations reach the equilibrium values, the infinite

Fig. 2.14 Tabulation of few-group homogenized macroscopic cross sections in lattice burnup calculations

multiplication factor sharply drops during a short time. Then the infinite multiplication factor monotonously decreases in the case of no burnable poison fuel (UO2 + Gd2O3). Meanwhile, it increases once in burnable poison fuel because burnable poisons burn out gradually with burnup, and then it decreases when the burnable poisons become ineffective.

The few-group homogenized macroscopic cross sections are tabulated at representative burnup steps (E1, E2, E3, …) by the lattice burnup calculation and stored as a reactor constant library for the core calculation, as shown in Fig. 2.14. Three different moderator densities (p 1, p2, p3) for one fuel type are assumed in the lattice burnup calculation of BWRs. Moderator density of BWRs

Fig. 2.15 Branch-off calculation of moderator density

is provided as a function of void fraction a for two densities pl and pg in liquid and vapor phases, respectively, in the steam table. That is, it is given by

р = рг(1-а) + ра« (2.33)

where p 1, p2, and p3 are the moderator densities corresponding to each void fraction at the core outlet (a = 0.7), as an average (a = 0.4), and at the inlet (a = 0.0) of typical BWRs. Since this moderator density is assumed to be constant as an average value during each lattice burnup step, it is called a

historical moderator density.

The few-group homogenized macroscopic cross sections are tabulated according to two historical parameters (burnup E and historical moderator density p) by energy group (G) or cross section type (x). The cross section tables are prepared individually according to fuel type (F), control rod insertion or withdrawal, etc. Furthermore, homogenized microscopic cross sections and atomic number densities of 135Xe, 149Sm, 10B, etc. can be tabulated by necessity.

The fuel inventory (W) can be given by

W=pp xNBx Nrod x Lrod x ИДУ2)2 (3.3)

where NB is the number of fuel assemblies, Nrod is the number of fuel rods per fuel assembly, Lrod is the active height of fuel rod (= active core height), and Dp and pp are diameter and density of the fuel pellet, respectively. The fuel rod diameter Drod is determined from the pellet diameter DP, cladding thickness, and gap between cladding and pellet. The core volume Vcore and the equivalent core diameter Dcore can be calculated by

V^ore = Nb x LBx LBx Lrod (3:4)

Dm = 2x(NBxLBxLB /n)1/2 (3.5)

where LB is the fuel assembly pitch.

For a constant fuel inventory in the core, a long active core height Lrod reduces the number of fuel assemblies NB. This is desirable from the viewpoint of fuel management, but not from the viewpoints of nuclear and thermal — hydraulic design and mechanical design of fuel. A low neutron leakage leads to an effective usage of neutrons and enhances the efficiency from the

viewpoint of nuclear design. It is desirable that the surface area of the core is small, namely, the ratio between the active height and equivalent diameter of the core Lrod/Dcore is close to 1.0. From the viewpoint of thermal-hydraulic design, a long fuel rod causes a high pressure drop in the fuel assembly and has an effect on the primary coolant pump design. Since it also increases the buoyancy of the fuel assembly, a design consideration is needed so that it does not rise. From the viewpoint of mechanical design, a long fuel rod is easily bent and therefore that influences the fuel loading characteristics. The typical design has about 4 m core height and about 3.6-3.7 m active height of the fuel assembly including fissionable materials.

A large-sized fuel assembly reduces the number of fuel assemblies to be replaced in refueling and that leads to an improvement in reactor capacity factor. It is desirable to make the factor large to the extent possible. From the safety viewpoint for handling fuel assemblies outside the core, however, the fuel assembly cannot be sized too large. The fuel assembly is required to maintain subcriticality, even if it is flooded in fresh water, for which the fuel enrichment is an important parameter. The fuel enrichment of 5 % is considered as an upper limit and its corresponding fuel assembly size is limited to about 220 mm when not considering the burnable poison effect on reactivity control. The practical size of fuel assembly is about 135 mm.

The rod cluster control assembly (RCCA) is designed with appropriate limits on the maximum worth so that the core internal structures can provide core cooling without damage to the integrity of the coolant pressure boundary at an ejection of a RCCA. Secondly, it is designed with appro­priate limits on the maximum reactivity insertion rate so that the fuel integrity is secured at a simultaneous withdrawal of two banks of RCCAs at the maximum speed. Thirdly, it is also designed with appropriate limits on the maximum worth so that the fuel integrity is secured at a drop of a fully-withdrawn RCCA at the hot full power condition.

[1] Characteristics of fast reactor core

In the primary cooling system of the LMFBR, the high-temperature and low-pressure sodium heated in the core is cooled by the intermediate heat exchangers and then sent back to the core. In the secondary cooling system, the sodium receives heat from the primary cooling system through the inter­mediate heat exchangers and gives heat to water and steam through the steam generators. The energy of the generated steam is converted to electricity through the steam turbines and generators. The secondary cooling system in the LMFBR is necessary for two reasons: the high radioactivity of the primary coolant; and preventing the core from being directly affected by the accidental sodium-water reaction in the steam generator. Sodium as the primary coolant provides forced cooling of the core by single-phase flow under low pressure. It does not evaporate in the core due to its high boiling point.

Figure 4.3 illustrates the cross section of a fast reactor core. In this example, the core consists of 198 core fuel assemblies, 19 control rod assemblies, surrounding blanket fuel assemblies and neutron shieldings. The shape of the whole core is almost hexagonal. Each core fuel assembly contains 169 fuel elements in a hexagonal wrapper tube.