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.