The diffusion equation of the eigenvalue problem by the one-group theory [Eq. (2.67)] can be extended to an equation based on the multi-group theory and its numerical solutions are discussed here.

In the eigenvalue problem of the multi-group theory, it is necessary to describe the fission source term (и£ф in the multi-group form as well as the loss and production terms due to the neutron slowing-down. Since the fission reaction occurs in all energy groups, the total production rate of fission neutrons in the unit volume is given by Eq. (2.70).

P=v’LfA<l>1+v’Lf,2<l>2+v’Lf,3<l)3—=%v’Zf. g<l>!i’ (2.70)

The probability that a fission neutron will be born with an energy in group g is given by

(2.71)

where x(E) is the fission neutron spectrum normalized to unity and therefore the sum ofxg is the same unity.

Xz»=l-0

Hence, the fission source of group g can be expressed as Eq. (2.73).

ХдЕ Хд^У^д’Фд’

g

For the three-group problem (fixed-source problem) of Fig. 2.20, replace­ment of the external source by the fission source is considered. If Sg is substituted by xgP/keff in Eqs. (2.57), (2.58), and (2.59), then the following diffusion equations are given for three-group eigenvalue problem:

9 X2P

Group 2: — D2V202 + (Га2ф2 + Г2->3Фг) = т— + ^і^2Фі

Keff

Group 3: — D3V203 + (£а, зФз)

7Г—- ^ і^зФі + ^г^гФі)

K-eff

where

P — vZf гфг + 202 + [4] [5]^/,303 (2.77)

In the three-group fixed-source problem, it was seen that the diffusion equations are solved in consecutive order from group 1. However, Eqs. (2.74), (2.75), and (2.76) have the fission source P which includes 01, 02, and 03, and moreover kef is unknown. Hence, iterative calculations are performed as the next sequence.

(i) Guess kf for kef, and 010), 020), and 03°for determining P. It is common

to begin with an initial guess of k^f = 1.0 and a flat distribution of the neutron fluxes (constant values).

(ii) Since neutron fluxes are relative and arbitrary values in eigenvalue problems, normalize the initial neutron fluxes to satisfy Eq. (2.78).

k{$ =fcore vZf. і фТ+vlLf, 2 (pf+vZf, з фТ<Ж (2.78)

Since fission source terms are divided by keff in eigenvalue diffusion equations, Eq. (2.78) indicates that the total neutron source in the system is normalized to a value of 1 to fit into the next diffusion equation.

(iii) Provide P(0) of Eq. (2.79) with the RHS of Eq. (2.80) and then solve this

for 011).

P(0) = v£/дф1(0) + v2>,20<o) + v2>>3^0) (2.79)

У p(°)

(2.80)

keff

(2.81)

Substitute again the solutions 0^ and 021) from Eqs. (2.80) and (2.81) for the RHS of Eq. (2.82) and then solve this for 031).

Y P

(2.82)

ke/f

(vi) Recalculate kejf by ф^, , and ф^ obtained in (iii) to (v) like

Eq. (2.83)

=fcore VLf, і фГ+ vXf, 2 (jW+v-Lf, з tffdV (2.83)

Here, keff is interpreted as the number of neutrons that will be born in the next generation finally, considering moderation and transport in the whole core, when assuming that one neutron is given as a fission source to the whole core. This is the definition of the effective multiplication factor.

Those solutions are not the correct ones yet because they are based on the initial guesses. The calculations from (iii) to (vi) are iteratively performed until keff and all of the group fluxes converge. This iterative calculation is known as the outer iteration calculation. The outer iteration test for convergence is done by comparing values at an iteration (n) with those at its previous iteration (n — 1):

where ek and Єф are the convergence criteria of the effective multiplication factor and neutron flux, respectively.

The relative neutron flux distribution ф^ (r) has absolute values due to the thermal power Q of the reactor. The normalization factor A of ф^ (r) to an absolute neutron flux distribution Og(r) can be determined by

Фд(г)=Афд(г) (2.86)

Q=Kfcore Zf-e ^ ФЄ ^dV=KAfcore ъг-и (r)dV (287)

where к is the energy released per fission (about 200 MeV). Finally, the absolute neutron flux distribution for the thermal power can be given as the following.

(2:88)

Fig. 2.21 Balance of neutron flux in group g

The distribution of the thermal power within the reactor core is

(2.89)

9

which is used as a heat source for the thermal-hydraulic calculation.

A general form of the multi-group diffusion equation in the case of the eigenvalue problem is given by Eq. (2.90), which can be solved in the same way considering the balance between neutron production and loss in group g as shown in Fig. 2.21. It is seen that Eqs. (2.74), (2.75), and (2.76) are also represented by the general form.

(2.90)