On the starting and operational transients at the core of the RMB based on 2D multigroup diffusion

RMB is a multipurpose reactor to be built by CNEN. As a multipurpose facility, unlike a nuclear power plant, some operations may result in transients, whether during the movement of materials from remote irradiation, extraction of neutrons through beam holes, reactor starting up, control rod accident, etc. For that, a methodology to visualize some variables like neutron fluxes, temperatures, power, etc., was developed. In this one present some preliminary results based on multigroup diffusion theory, thermal hydraulics feedbacks, and numerical methods, systematized in the DINUCLE code. DINUCLE couples a numerical code of multigroup spatial kinetics, for calculations of the flux of neutrons in the core of reactors, with a numerical code for the analysis of thermo hydraulic transients in rods and plates, considering the refrigerant always single-phase. Use was made of a programming in VBA (Visual Basic for Applications) from Excel, for data visualization in the transient.


INTRODUCTION
RMB is a Brazilian multipurpose reactor designed for radioisotope production, material testing, operation of various neutron beam roles, etc. [1]. As a multipurpose facility, each activity operation can impact core criticality. In this article, we look at some of these operational transients in the core.
For that, we modeled it in 2D multigroup spatial kinetic equations, with feedback, through the thermo-hydraulic equation. The expected transients would be the typical start-up of the reactor and those resulting from the introduction of samples to be irradiated, causing disturbances in the criticality of the RMB. The distribution of the neutron flux in the steady state and the transients are calculated numerically. It is worth mentioning that in the case of power reactors, in their operation, the aim is to have a spatial distribution of the neutron flux as flat as possible, avoiding the existence of power density peaks. On the other hand, in the case of RMB, the versatility of its operation seeks a very heterogeneous distribution of the neutron flux, whether for high flux for use in materials testing, in a region of the nucleus, from fast neutrons, as well as for production of radioisotopes, with thermal neutrons, in other regions of the nucleus. Thus, while in a power reactor the result of its operation is the energy delivered in the form of enthalpy, in a reactor such as RMB, its spatial behavior gains relevance, whether operational, reinforced by its visualization of the core of various parameters such as neutron fluxes, temperature distribution, etc. Again, for comparison purposes, usual power reactors operate with minimal excess of reactivity, aiming to have an excess of neutrons that compensates for their operations at nominal temperatures, burning of the fuel throughout their operating cycles, etc. In the case of RMB, this excess of reactivity, due to its multipurpose nature, is much higher, even with much shorter burnup cycles than power reactors. This is mainly related to the fact that, whether for testing materials or for producing radioisotopes, a huge amount of neutrons are demanded in these irradiations. For this, monitoring the criticality state of the RMB, through its reactivity, an equivalent integral parameter, is of fundamental importance for several irradiation operations and sample movement in the RMB core.

MATHEMATICAL MODELS
The neutron kinetic equations are given by [2]: where, in a position r  and time t , ) , ( t r g g     now is the direct neutron flux of energy group g during transients and ) ,  are: velocity, diffusion coefficient, removal macroscopic cross-section, scattering macroscopic cross-section, prompt neutron spectra, neutrons per fission numbers, fission macroscopic cross-section, respectively, of the g-th energy group. For the precursor parameters, agk  , k  , k  are: delayed neutron spectra, decay constant, delayed neutron fraction, respectively, of the k-th delayed group. Besides, At any time, and since that g  is the energy liberated per fission at the g-th group, the power density is given by: The thermal hydraulic model, of an average channel, is given by [3]: With that, a generic integration along the radius is given by: having in mind that m means f (fuel) or c (cladding), r distance from center, for a rod ( 1)   or for a plate  .
Here the properties are associated with the coolant:   (density); c  (specific heat); T  (bulk temperature); U  (fluid velocity) and q  (heat flux). Since h  and S T  are the heat transfer coefficient between surface plate/rod to coolant, A global parameter that gives important information about the reactor deviation from criticality is the reactivity, ( ) t  . Considering the direct fluxes and the macroscopic cross-sections time dependent, it is calculate by [4]: Since that, in the steady state, the neutron flux distribution are calculated by using the eigenvalue criticality equation, given by: Since here , the weight functions are calculated from the following equation [4]: Since that is the neutron fluxes at nominal condition, normalized power ( ) p t is given by: Part of this work was to develop a computational application for the visualization of spatial results during transients. For this, a program for processing in spreadsheets was developed, using programming in Visual Basic for Applications (VBA) [5].
Based on the colors and captions of Figure 2, the configuration of the RMB core is sketched in Figure 3, below:

Core composition and geometry
For purpose of thermal-hydraulics transient analysis at RMB we consider the core composition in Table I: Table I: Core material composition.
The captions and colors associated with the material regions are given in Figure 2 below:

Control rod bank distribution
The banks of control rods are distributed as given by Figure 4, below: For our analysis we consider that during the driving of the control rods, the spaces left by them are filled out by water. With that, the following approach was used for calculation of the macroscopic constants: where, in each energy group, one has:

Feedback effect models
where,  depend on mass and surface of control rod.
The scattering macroscopic cross section in the moderator region is directly dependent on coolant density: where, for each node at the core spatial meshes. In this way, assuming the existence of these channels in these nodes, both the temperatures in the cooling channels, as well as the temperatures in the fuel region, will feed back the cross sections, capture in the fuel, as well as the scattering cross sections in the moderator (cooling channel). Thus, the reactor reactivity is a variable generated as a byproduct of these calculations. This procedure is repeated at each time interval, performing the RMB transient.
In the validation of the CINESP module, we compared its results with two other codes: MITKIN [8,9] and TWIGL [10]. For this comparison we generate a transient for a heterogeneous, twodimensional reactor, for two energy groups, caused by the perturbation in the macroscopic capture cross section, in the thermal group. This transient is described in references [8,9]. Table II  Energy [11]. In this transient, proposed for several laboratories in Europe, the external source of neutrons is interrupted for a few seconds and resumed again. SIRER has been modified to handle the external source. This version is named SIRER-ADS [12]. Figure 5 shows the result of SIRER-ADS to be compared with the Benchmark, Figure 6. The comparison leaves no doubt that the SIRER module is very consistent with the other results, qualitatively.   At nominal operation, that is, in the steady state, all six control banks are part inserted that in such way that f=0.5837. In this configuration, the core criticality is 0.905577. Considering a power of 25 Mw and the design parameters of RMB [1], the thermal (g=4) real flux distribution is given by Figure 7. In this graph it was used the Origin 8.5, a software to analysis and graphing from OriginLab Corporation, released in 2010 [6]. This software has tool to view the graph in perspectives that we can observe the neutron flux from another angle. The data of the macroscopic constants of the energy groups, in addition to the spatial discretization parameters, and the kinetic and channel parameters of the plates, are condensed in the Appendix placed after the references.

RMB in steady state operation
To validate the reactivity calculation, Eq. (6), and the normalized power, Eq. (9), we simulate a 1D problem, calculated by the code WIGLE [13], as well as another result from another code, UNICIN [14]. It is a problem using two groups of energies, fast and thermal. Table III shows the results of reactivity and normalized power, the latter not being available for UNICIN. These results show that the CINESP program presents excellent agreement with the other codes for these global     Figure 10 shows the precursor associated with the delayed neutrons concentration distribution. Note that precursors of delayed neutron do not diffuse through the core.    Figure 13 shows the initial thermal flux (g=4). Figure 12 shows the centerline temperature distribution in a view from top. Note that as tempera-tures are calculated on fuel region, irradiation elements, control rods, etc., do not are used at temperature calculations. In this view we can note that temperatures are calculated where there is fuel, consequently, heat.        Figure 20 shows the centerline temperature distribution at fuel plates at 71.15 seconds.

Power Reactivity
The behavior of global variables such power and reactivity is showed in Figure 21.
The behavior of the centerline plate fuel temperature (TCL) and outlet channel temperature (TOL), located at maximum flux peak factor, is showed in Figure 22. Figure 23 shows a Zoom of the tran-sient between 65 and 71.15 seconds, to separate the TCL and TOL variations.

APPENDIX A
In this appendix we have the RMB basic data used in this paper. Using Figure 2, the 2D axes are orientated as: Figure A1: Definition of mesh discretization.
Based on figure A1, the mesh discretization, in cm, are given at Table A1 and Table A2, as follow: Table A1: Spatial mesh at x direction (DX0 = 13.73) Table A2: Spatial mesh at y direction (DY0 = 14.00) The multigroup constants are given in the following tables, assuming four groups: Table A3: Velocity, spectrum and neutron per fission, at the energy groups.  Table A4: Macroscopic cross-sections (cm -1 ), diffusion coefficient (cm), at energy group g=1. Table A5: Macroscopic cross-sections (cm -1 ), diffusion coefficient (cm), at energy group g=2.