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2-group axial reduction
General form
The reduction is formally obtained by summing the balance equations
(1,2) in each plane "z" of the core, after prior weighing
(16) by a
stationary (time independent) function
ψz(x,y,V) attached to each plane and, up to now, arbitrary.
This function must, however, satisfy the same continuity (of
ψ and
d grad ψ) and core lateral boundary conditions as those imposed to the direct flux.
In order to facilitate the representations, we introduce
(16) a weighted summation operator
< > to form the weighted summation of any reaction rate
R(x,V) in plane
z and over the macrogoup
g.
To insure consistency with static nuclear design we adopt the following partition in 2 macro-groups:
g=1: fast group (0.625 ev < V)
g=2: thermal group (V < 0.625 ev).
Planar summation must extend over the whole core transverse area, including radial reflector in case that the radial boundary conditions are applied to the outer boundary of reflector.
Reminding that prompt and delayed neutron are born in the fast group only, (
χ(V)=0 for V <.625 ev)
(10d) and operating
(9) the neutronic balances successively with
χ1 and
χ2, leads to
(10a:10d)).
It is not convenient to use this form of relations because of the presence of the individual spectra.
A first simplification is obtained by substituting (
14a) into the first term of
(20) so
F may be written as
(14d)). Index "2" can be dropped in (c) because of (d)
Introducing the "importance" function
In practice, planar summation apply
(26) to the various fuel assemblies "
a" or "nodes" of the core. Here the summation must not extend over the reflector because
f = 0 there.
Following simplification aims at expressing delayed fission source in (16) by means of the average stationary spectra χi
For this purpose, let us firstly define
(28) the relative contribution
αai of fissile isotope
i to assembly fission source, where the weighing by 1 simply represents the volumic summation (29) over
a.
The approximation
(30) amounts to assuming that
fi φ is uniform over
a, which is actually the case in the assembly homogenization program.
The
neutron importance Iaj of family
j in
a is defined from
(31).
It accounts for the characteristics of each line spectrum
χj as compared with the average assembly composite spectrum.
With these notations, we get (32).
With (33) giving the relative weighted contribution πa of "a" to the plane fission source
With
(34) defining the "mean effective"
β for family "j",.
(35) defining the corresponding "total effective" β~ corrected by importance, and (36) the total effective beta.
Thanks to these definitions, the reduced equations simplify to (37).
These somewhat tedious developments aim at detailing how the effective β~ may be calculated by means of the combination of the assembly and the nuclear diffusion programs.
(37) represent the general form of 2-group, axial reduction of the space kinetics equations.
Introduction of planar averages
The flux volume averages
(38) are obtained by planar summation with unit weighing function, where
<1,1> is simply the core cross-section area, including the reflector.
Operating
(37)) with
(39) and by defining (40), (41) leads to (42), or by making use of the fast group radial profile
(φ/Φ)1 leads to
(42a).
Remind that
ψz(x,y,V) is stationary, whilst the dynamic profile is time-dependant.
Introducing here the first approximation of the spatial reduction model as
(43) leads to (44).
Owing to this approximations, the delayed emitters balance simplifies to (45)
Let us now operate similarly for the removal terms
(46) to define the total macroscopic removal cross-section from group 1 by absorption and scattering to group 2.
With (47), (48) or (49) we get the macroscopic cross-sections Σ2 and Σr1 which depend only on individual fast and thermal profiles.
Planar average of diffusion coefficient for each group is defined from (50).
Accepting the approximation
(51) provides (52).
Define finally (53). Likewise (54) for the thermal group.
Further, define fast and thermal fission sources from
(55,56,57).
Defining (58) or, in terms of profiles (59) gives (60).
At the outcome of all these transformations, we finally arrive to a formal reduction of 2-group, spatial kinetics equation as
(61,62,63).
Besides the two minor approximations introduced above, this reduced system is just as exact as the original equations.
As such, there are not of practical use because the planar averages do depend on the radial profiles φ/Φ which are still undefined.
The model ψ=1 would correspond to strict conservation of individual reaction rates along the process of space and energy reduction. The 2-group assembly cross-sections generated by the assembly code correspond to this model.
Those cross-sections could also be used to feed a 3-D kinetics code on the condition to replace
(64a)by (64b).
Σ1 and
Σ2 are then the classical assembly macroscopic cross-sections which do not include radial cross-sections.
The other parameters
β ...are directly taken from the assembly program.