In the core calculations with a huge amount of space-dependent data (cross section and neutron flux), the effective cross sections are processed, with a little degradation in accuracy as possible, by using the results from the multi-group lattice calculation. There are two processing methods. One is homogenization to lessen the space-dependent information and the other is group-collapsing to reduce the energy-dependent information as shown in Fig. 2.11. The funda­mental idea of both methods is to conserve neutron reaction rate. In the homogenization, a homogenized neutron flux фк°то is first defined as averaged flux weighted by volume Vk of region (k). Next, a homogenized cross section ^°то is determined as satisfying Eq. (2.23) to represent the reaction rate in the homogenized whole region of volume vhomo. The fine-group neutron flux фё к is used to calculate the homogenized cross section in Eq. (2.24).

2.22)

k / k k 47

homo Л homo T T homo — > V A T 7

x, g <Pg У — Zj^x, g,k(Pg, kVk

Heterogeneous Cross Sections

Homogenized Cross Sections

(for Core Calculation)

Few-Group Cross Sections

Collapsing

G = 3

C = 2

Ns —- g ——- 321 N„ ^——— 321

Similarly, the homogenized microscopic cross section 0-і^ото which repre­sents the homogenized region is defined. First, the homogenized atomic number density Nl, homo of nuclide i is defined from the following conservation of number of atoms

Ni, homo=^NlVk/V homo (2.25)

k v • /

The homogenized microscopic cross section is then given from the conser­vation of microscopic reaction rate of nuclide i as

(2.26)

Such homogenized neutron flux ^h°m° and cross section £ hgmo (or ^lJg>mo) terms are used in performing the energy-group collapse. The fine-groups (g) are first apportioned to few-groups (G) for the core calculation as shown in Fig. 2.11. Since the multi-group neutron flux was obtained by the integration over its energy group, the few-group homogenized neutron flux can be represented by

2.27)

g^G v ’

The next step is to consider the conservation of reaction rate in energy group G in the same manner as that in the homogenization.

The few-group homogenized macroscopic and microscopic cross sections are then given by

V homo — л v homo a. homo / ж homo — > v homo м homo / л A homo bx, G уд 1YG ~Zu^x, g <Pg / Zj Yg

g^G g^G / g(EG

i, hom, o— Xі i, homo a. homo / a. homo — Xі i, homo a. homo / Xі a. homo

vx, G — Li°x, g Yg / Yg ~ZaGx, g Yg ZjYg

g^G g(EG / g(EG

The number of few-groups depends on reactor type and computation code. Two or three groups are adopted for the nuclear and thermal-hydraulic coupled core calculation of LWRs and about 18 groups are used for the core calculation of LMFRs. It should be noted that the conservation of reaction rate has been considered under the assumption that the lattice system can be represented as an infinite array of identical lattice cells. If the neutron spectrum in the actual core system is very different from the multi-group neutron spectrum calculated in the infinite lattice system, the applicability of such few-group homogenized cross sections to the core calculation is deteriorated. In this case, it is effective to increase the number of energy groups in the core calculation, but that gives a more costly computation.