DD transients

Similarly to what is done for the cross section there is the perturbation of the diffusion lengths.

(a) DD transients (b) DD_C transients

Figure 5.4: DD and DD_C transients

The increase of diffusion length means that neutrons can continue for its path without making any collision: the consequence is the lower probability to make fission or absorption. This is the reason why the effect on amplitude is not so

similar to a real case. This is the reason of the sizes chosen and also the kind of transients performed. They are two: one where there is the augmentation of absorption cross section, the other where there is the rise of fission cross section.

The figures 5.7 and 5.8, at the end of the chapter, and the table 5.2 show the data important for the evaluation of the amplitude.

Two groups data Dg [cm] Σa,g [cm-1] Σg→g+1 [cm-1] νΣf [cm-1] χ [-]

Core 1 1.01 0.01 0.008 0.003946 1.0

2 1.2 0.1 - 0.165 0

Reflector 1 0.9 0.01 0.0095 0.0 1.0

2 0.8 0.1 - 0.0 0

Table 5.2: two groups data.

The first transient studied wants reproduce the first one of two dimensional cases, so it is extended in x direction and at the edge of this. In this one it is studied just the increase of absorption cross section. The other study, wants to observe the effect on fission cross section. This tries to simulate the removal of a control rod from the center of the reactor and the amplitude response is seen.

The distribution of flux is not reported for space region but it is found again the symmetry of the quantities due to the configuration.

5.2.1 3D1

Here it is reported the study on an increase of the removal cross section.

Figure 5.5: 3D_1 transient

The region increase is small and it is similar to the one of HE1. The results are the one expected: the amplitude tends to decrease. Again the space dis-cretization does not correspond to the other types, giving lower results. The reason of this is due to the bad factorization. Using only the space to study the amplitude trend considers that in the system there is only the increase of the removal of neutrons. On the other side with the energy discretization, the decrease is slower, since a neutron does not only disappear but it can just pass to the other group.

5.2.2 3D2

In this section it is presented the increase of fission cross section. It is considered like the expulsion of a rod.

Figure 5.6: 3D_1 transient

The graph show what is obtained before. The space discretization overestimates the results given by the energy and space energy factorization. However it is noted that the errors committed are sligtly lower than the ones realized in the transients FA1 and FA2. At instant t = 0.002 s they are 3.87 % and 30.5 %.

Maybe is due to the perturbation size, that is extended along y direction so

(a) Section B-B of the system (b) Section B-B of the system Figure 5.7: 3D systems

In this chapter some final conclusions are done, considering what is simulated and the results extracted in the previous two chapters. Finally also a list of the needed updates is done.

Most of the studies are the ones that consider two groups in a 2D system.

Overall it is noted that the EQS method can offer a good prediction of the results. Not strange effect are noted like decrease of power when it is expected to rise. The space discretization seems to offer overestimation or underestimation of the results respect to the two other types of factorization. However the errors are not so big, if it is considered that only a perturbation is activated and the consequent trend of amplitude is expected. These tend to increase with time and with the magnitude of the perturbation. In case of reactivity compensation it is noted that the method is not so good to predict the expected trend that wants that the amplitude sets up to one. This is achieved well by the space and space-energy discretization but not always. It is noted that in case of an horizontal wide and not centred perturbation it works well, even in case of mono-energetic approximation despite some errors. The energy discretization seems to fail. However if the perturbation is small and distributed it seems to work. Conversely if the modification is localized while the fission cross section is increased in a large space the compensation fails. Even space energy and energy discretization differ since this one is not able to report the trend of quantities on y direction. Indeed it was noted that bigger differences appear when the perturbation is limited on vertical direction. The failure is bigger depending also on the location: if it is more displaced, towards the edge of the system, the errors are bigger. In case of fission and scattering cross section rise the space discretization seems to differ a lot respect to the one of energy and space-energy.

These offer always lower results respect to the first one: maybe this is due to the capability to distinguish for all the period of study the division in groups. A reversed behaviour is expected in case of decrease of scattering and fission cross section. Finally it seems that the diffusion length can be studied equally by

the three types of discretization. Moreover it is noted that space discretization give more different results when the properties changed are the ones that have a direct effect on coupling. The insertion of delayed generally seem to agree with the prompt evaluations. Also the one group case seem to offer results that do not differ from the expectations. The three dimensional problem is studied well and the behaviours similar to previously cited are again found. The errors between the space and the others two discretizations are lower respect to the 2D case in case of fission cross section augmentation. Maybe this is due to the further averaging that integration on another direction causes.

Concerning the needed updates to the code, many things should be inserted.

Starting from the begin of the solver, the possibility to consider more complex system should be investigated. A big lack of the code is the impossibility to consider a source of neutron. Even if this is rare inside a reactor, it could be important for didactic purposes. This should be considered from the reference flux evaluation, modifying the code. Another interesting developing could be the increase of number of groups studied. Actually they are just two but more of them could offer simulations that are closer to the real physics. To this, also the possibility to the cross sections to change on time enhances a lot prediction of a true reactor system. Finally as it can be seen by the hypothesis 2.9 and the successive of this type, the flux is factorized by the amplitude and the reference flux that is the one of the solution at instant equal to zero. This approximation is good for limited time period studied, that in case of EQS method could be also larger respect to the QS study, but of course for longer ones there is the need of updates. This is quite difficult in OpenFOAM® because it requests a continuous change of mesh, internal to the code.

[1] Greenshields C. J., 2015, OpenFOAM The Open Source CFD Toolbox -Programmer’s guide, OpenFOAM

[2] Wikipedia,2021, GNU General Public License, fromhttps://en.wikipedia.o rg/wiki/GNU_General_Public_License.

[3] Wikipedia, 2021,OpenFOAM, from https://en.wikipedia.org/wiki/OpenFOAM.

[4] Wikipedia, 2021, Fluidodinamica computazionale, fromhttps://it.wikipedi a.org/wiki/Fluidodinamica_computazionale.

[5] OpenFOAM Wiki, Nuclear Technical Committee, fromhttps://wiki.openfo am.com/Nuclear_Technical_Committee

[6] Fiorina C., Clifford I., Aufiero M., Mikityuk K., 2015, GeN-Foam: a novel OpenFOAM® based multi-physics solver for 2D/3D transient analysis of nuclear reactors, Nuclear Engineering and Desgin, 294, pp. 24-37.

[7] Aufiero M., Brovchenko M., Cammi A. Clifford I., Geoffroy O., Heuer D., Laureau A., Losa M., Luzzi L., Merle-Lucotte E., Ricotti M. E., Rouch H., 2013, Calculating the effective delayed neutron fraction in the Molten Salt Fast Reactor: Analytical , deterministic and Monte Carlo approaches, Politecnico di Milano, from https://re.public.polimi.it/retrieve/handle/1 1311/761453/386932/Calculating%20the%20effective%20delayed%20neut ron%20fraction%20in%20the%20Molten%20Salt%20Fast%20Reactor_113 11-761453_Luzzi.pdf

[8] Ma Y., Wang Y., Yang J., 2021, ntkFoam: An OpenFOAM based neutron transport kinetics, Computers and Mathematics with Applications, 81, pp.

512-531

[9] Cosgrove P., Shwageraus E.,2017, Development of MoCha-Foam: a new Method of Characteristics Neutron Transport Solver for OpenFOAM, Inter-national Conference on Mathematics & Computational Methods Applied to Nuclear Science & Engineering,Jeju, Korea.