Appendix C – Extension of the Mixing Model to particular situations
C.1 Introduction
In this appendix some additional results obtained at NRG (Petten) are presented. On the other hand, this work needs a deeper analysis because the results are just qualitative and an improvement of physic assumptions and explanations of the Mixing Model are necessary. It is not trough and trough clear why the Mixing Model needs the exponential and the related physic explanation. An other important question is the efficiency of Mixing Model to predict a mixing cell containing pebbles with different initial composition (like Uranium or Plutonium-Uranium Pebble). Also the results show the possibility to find some interesting “special” solution in the case of c-factor (that typically it is constant). C-factor probably is a function of the spectrum correction and this needs a deeper theoretical analysis. The last version of Mixing Model is based on a simple one-group theory. The application of a two-groups theory probably could give better chances to have a more efficient model. This implementation offers the possibility in future to find a fast full core k-effective predictive model.
C.2 Uranium pebble behaviour
The second step in order to implement a suitable Mixing Model is the study for different mixture, special in presence of different element (like Th or U).
This study shows that the Mixing Model could predict this k-infinitive behaviour for pebbles with a different initial composition but it supplies better if there is only one type of pebbles or in the case of a specific c-factor for a well-defined type of initial composition. The Table C.1 shows some results of the study. Unfortunately for time less the obtained results, even if they appear as very interesting, will have to be further cionfirmed by deeper analyses.
It is interesting to note the different behaviour between of U-based pebbles and Pu-based ones. That is the consequence of the c-factor values. There is an other interesting results: without adopting correction factor, the error for this type of mixture is not so high. The conclusion is the Mixing Model is, at least in principle, able to predict different type of initial compositions (not only U), but, to be really applicable a targeted study on the correction factor (c) is needed (see C.3).
Table C.1 : k-infinite mixing model results for Uranium only
Uranium Only
Mixture Correction factor K Mixing
Model MCNP Reference Value Error
1 1,243218299 1,25442 0,011201701 Uranium 4x0 GWd/t - 4x90 GWd/t
-5,054189127 1,255413617 1,25442 -0,000993617
1 1,168583817 1,17449 0,005906183 Uranium 6x0 GWd/t - 2x90 GWd/t
-5,054189127 1,176221397 1,17449 -0,001731397
1 1,33666443 1,34787 0,01120557 Uranium 2x0 GWd/t - 6x90 GWd/t
-5,054189127 1,347823076 1,34787 4,69239E-05
C.3 Uranium and plutonium pebble behaviour
After having considered a single pebble type of fixed initial composition, the next step is the implementation of pebbles mixtures with two initial compositions. The results seems to be really interesting, as shown in the Table C.2.
Table C.2: k-infinite mixing model results for Uranium an Plutonium pebbles Plutonium - Uranium
Mixture Correction factor K Mixing Model MCNP Reference Value Error 1 1,178226043 1,16392 -0,014306043 1,71 1,178113106 1,16392 -0,014193106 -5,054189127 1,179279702 1,16392 -0,015359702 Uranium 1x0 - 1x90 Plutonium
2x600 - 1x360 - 1x320 - 1x160 - 1x0 GWd/t
286,2946267 1,164080084 1,16392 -0,000160084 1 1,213751512 1,19926 -0,014491512 1,71 1,213307501 1,19926 -0,014047501 -5,054189127 1,217703571 1,19926 -0,018443571 Uranium 2x0 - 1x90 Plutonium
1x600 - 1x440 - 1x360 - 1x160 - 1x0 GWd/t
28,98972109 1,199133733 1,19926 0,000126267 1 1,03058411 1,01518 -0,01540411 1,71 1,033525414 1,01518 -0,018345414 -5,054189127 1,002559149 1,01518 0,012620851 Uranium 1x0 - 1x90 Plutonium
2x760 - 1x720 - 1x680 - 1x480 - 1x400 GWd/t
-2,508820599 1,014993031 1,01518 0,000186969 1 1,012788256 0,952 -0,060788256 1,71 1,016113487 0,952 -0,064113487 -5,054189127 0,982381264 0,952 -0,030381264 Uranium 2x0 - 1x90 Plutonium
2x760 - 2x720 - 1x680 GWd/t
-10,49383796 0,952118572 0,952 -0,000118572
The error introduced with different compositions without adopting a correction factor is, in some cases really high . But with an opportune c-factor is possible to reduce the error to a reasonable level.
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C.4 Power Fraction Mixing Model results
An other future development needed for the improvements of NRG PANTHERMIX is a proper power fraction model. The basis formula for this model is similar to that one implemented in the previously shown Mixing Model, but with some differences:
∑
=⋅
⋅
⋅
⋅
⋅ Σ
⋅
⋅
⋅ Σ
= 8
1 , ,
) ( tanh
) ( tanh
n
n x c n
n n n
a
n x c n
n n n
a M
f x e
x
f x e
x f
n n
(C.1)
Where:
f stands for power fraction of an octant of Pebble (M in the case of a Mixture and n for the single Case reference).
Sa is the absorption Cross Section.
x function is the Medium Cord length times 4 per Migration length.
c is an opportune correction factor.
In this case we compare directly the fraction of power produced by a single pebble in a Control Volume (see Chapter 4). It is necessary for PANTHERMIX efficiency predict the contribution of the power of all single pebble presents in the volume.
The results reported in Table C.3 shows that the proposed Power Fraction Model is efficient for simple configuration. Unfortunately for more complex combinations the model does not work properly (see Table C.4). So, as future development, it is necessary a further improvement of this model.
Table C.3: Power Fraction simple cases results First Case
Burn-up octant Power Fraction with Mixing Power Fraction with MCNP Error
760 0,014392152 0,01946468 -0,0051 760 0,014392152 0,019501152 -0,0051 760 0,014392152 0,019519198 -0,0051 760 0,014392152 0,019522731 -0,0051
0 0,235607848 0,23090182 0,0047
0 0,235607848 0,229829685 0,0058
0 0,235607848 0,231136991 0,0045
0 0,235607848 0,230123743 0,0055
Second Case
760 0,025812343 0,031853669 -0,0060 760 0,025812343 0,031713002 -0,0059 760 0,025812343 0,031721517 -0,0059 760 0,025812343 0,031907039 -0,0061 760 0,025812343 0,031885402 -0,0061 760 0,025812343 0,031958434 -0,0061
0 0,422562971 0,40440805 0,018
0 0,422562971 0,404552887 0,0180
Table C.4: Power Fraction Model complex cases results First Case
Burn-up octant Power Fraction with Mixing Power Fraction with MCNP Error
600 0,093309447 0,000172051 -0,093
0 0,165111905 0,492806466 0,33
400 0,130645408 0,000344487 -0,13 320 0,138929347 0,009620675 -0,13 160 0,150324007 0,486875274 0,34 360 0,135060991 0,009662525 -0,13 600 0,093309447 0,000172551 -0,093 600 0,093309447 0,00034597 -0,093
Second Case
760 0,022644831 0,000247472 -0,022 400 0,29332485 0,795390577 0,50 760 0,022644831 0,000232003 -0,022 760 0,022644831 0,000494985 -0,022 480 0,268336976 0,176376833 -0,092 680 0,140811081 0,013791903 -0,13 680 0,140811081 0,013001002 -0,123 720 0,088781517 0,000465226 -0,088
C.5 Conclusions
To conclude this preliminary analysis, the Mixing Model works efficiently for k-mixture and it is able to predict the behavior of several mixtures but an Exponential Correction Factor is necessary (and a physical explanation of that is needed). In the frame of the exponential formulation it would be necessary to implement a proper c-function (not only a c-factor), probably dependent ________________________________________________________________________________ 113
from the flux spectrum shape. Unfortunately the Mixing Model for Power Fractions works only for simple mixtures. An improvement is needed for more complex mixtures. An improvement can be expected from the use of diffusion area and Fermi age (i.e. Two Group Theory) for both the models. Further it will be necessary to implement the new Mixing model in PANTHERMIX using Two Group Theory in order to split the contribution of the fast neutrons (Fermi Age) from that of th thermal neutron that enter in the cell (Diffusion Area).
Eventually on the basis of the previous considerations it is possible to apply the k-infinitive mixing model also to other cases (such as the presence some moderator pebbles and/or Thorium-Plutonium pebbles) and to estimate Full Core k-effective.
