The design of the nuclear reactor core requires the application of complex methods for reliable reconstruction of the geometry and material composition of the actual reactor core. The greater the complexity of the core geometry is, the longer the computation time of neutron transport will be, especially when using Monte Carlo methods. Additionally, the execution of many input files with varied key parameters is necessary for a comprehensive parametric study. The problem is particularly significant in the modelling of modular high-temperature gas-cooled reactors (MHTRs) with graphite fuel compacts containing spherical TRi-structural ISOtropic (TRISO) particles, called double heterogeneity [1]. In the paper, an alternative method called volumetric homogenization is presented for reducing complexity of the numerical model and the resulting computation time. The method assumes the elimination of double heterogeneity of TRISO particles embedded in the fuel compact. In this method, materials of TRISO particles are admixed to the graphite matrix of the compact and by this way, the complicated geometrical structure of multilayer TRISO particles is eliminated and replaced by one homogenized material. This, in turn, simplifies the numerical model and speeds up the simulations. The method was applied for the modelling of the MHTR core with thorium-uranium fuel [2, 3]. This kind of fuel mix is envisaged as an option for the currently designed nuclear reactors, including MHTR [4, 5]. Example results comprise the time evolutions of the effective neutron multiplication factor
In this section, the volumetric homogenization method is described from the mathematical point of view. Table 1 shows the input parameters necessary to obtain the weight fractions of each isotope in the homogenized material mix and the density of the homogenized material for the Monte Carlo modelling of neutron transport. The presented mathematical set-up was transferred to the numerical script, which automatically calculates the aforementioned parameters using the input data from Table 1.
Input data for the volumetric homogenization method
A. Material data |
Uranium enrichment |
Natural abundances of isotopes Atomic weights of isotopes |
Weight fractions of each dioxide in the dioxide mix |
Densities of fuel and no-fuel materials |
B. Geometry |
TRISO kernel radius |
TRISO layers thickness |
TRISO packing fraction |
Fuel compact radius |
Fuel compact height |
The calculations begin with the assessment of the volumes of each TRISO material in the fuel compact. The Appendix 1 contains list of symbols used in the applied equations. Initially, the volume of the spherical TRISO fuel kernel
Next, the volumes
The volume
In addition, Eq. (1) is used for the calculation of the TRISO particle volume
The volumes calculated above may be used in Eq. (4) for calculations of the number of TRISO particles in the fuel compact
Then, the volume of all TRISO particles in the compact
Then, with the given densities of each material in the compact
For the input of the Monte Carlo simulations, the density of the homogenized compact ρHOMO is also needed. It can be calculated using given densities ρ
In the next step, the material isotopic composition in the fuel compact is calculated. Initially, the isotopic composition of (U,Th)O2 dioxide for given weight fractions of each dioxide
Further, the atom fractions
Then, the atom fraction
Afterwards, the density of mixed dioxides ρ(HM)O2 for given densities of each dioxide ρ
Additionally, natural abundances of no-fuel compact and TRISO materials
The last stage of calculations considers the preparation of the weight fractions of the homogenized material mix
The weight fractions of carbon isotopes from different materials are finally summarized to simplify the input for numerical simulations and by this way, the final isotopic composition of the homogenized mix of fuel compact material was defined.
The example calculations were performed for 180 MWth MHTR core, as depicted in Fig. 1. The core comprises of 310 hexagonal fuel blocks including 60 blocks with control rods (CR) holes. The numerical Monte Carlo simulations were performed for a two-year irradiation cycle. The MCB code was equipped with JEFF3.2 nuclear data libraries for neutron transport simulations. The reactor core was divided into four radial and 10 axial fuel zones, which gives a total of 40 fuel zones. The geometrical parameters of the core and the TRISO fuel are presented in Tables 2 and 3, respectively.
Main geometrical parameters of the core
Reactor core |
|
Baffle radius (cm) | 200 |
Active height (cm) | 800 |
Reflector thickness (top/bottom) (cm) | 120/160 |
Number of blocks (without CR/with CR) | 250/60 |
Number of CR (core/reflector) | 12/18 |
Fuel block | |
Apothem (cm) | 18 |
Height (cm) | 80 |
Number of cooling channels per block: | |
– without CR: small/large | 6/102 |
– with CR: small/large | 7/88 |
Cooling channel radius (small/large) (cm) | 0.635/0.8 |
Pitch (cm) | 1.88 |
Radius of CR channel (cm) | 6.5 |
Radius of fuel channel (cm) | 0.635 |
Number of fuel channels (without CR/with CR) | 216/182 |
Fuel compact | |
Radius (cm) | 0.6225 |
Height (cm) | 5 |
Geometry of TRISO particles
TRISO | Thickness (mm) | Density (g/cm3) |
---|---|---|
Fuel | 600/700 (diameter) | 10.42 (UO2)/9.5 (ThO2) |
BPC | 95 | 1.05 |
IPyC | 40 | 1.90 |
SiC | 35 | 3.18 |
OPyC | 40 | 1.90 |
Table 4 shows four versions of the homogenized thorium-uranium fuel composition. The main criterion is a similar mass of fissionable 235U in the fresh reactor core – about 205 kg. This may be achieved by a variation of the input parameters for the volumetric homogenization method, i.e. kernel radius, weight fraction of each dioxide in the dioxide mix, enrichment and packing fraction.
Fuel parameters for the volumetric homogenization method
Case | Mass 235U (kg) | Mass 238U (kg) | Mass 232Th (kg) | |||||
---|---|---|---|---|---|---|---|---|
1 | 0.35 | 50 | 50 | 20 | 15 | 204.76 | 816.46 | 1018.70 |
2 | 0.30 | 40 | 60 | 20 | 15 | 205.05 | 820.22 | 681.43 |
3 | 0.35 | 55 | 45 | 20 | 17 | 207.68 | 832.01 | 1267.00 |
4 | 0.35 | 60 | 40 | 25 | 15 | 203.35 | 610.04 | 1216.20 |
Figure 2 shows the evolution of
The evolution of 233U bred from 232Th is presented in Fig. 3. The mass of the produced 233U depends strictly on the initial mass of 232Th. The higher the initial mass of 232Th, the faster the production of 233U because of the higher absorption macroscopic cross section of 232Th. Therefore, the fastest production of 233U occurs in the case C3 and the lowest is observed in the case C2. The decrease of 235U due to the fuel burnup is shown in Fig. 4. The characteristics of all cases C1–C4 are similar due to the presence of almost the same amount of fissionable 235U in the reactor core. The initial mass of 235U decreases by about 60% during the irradiation period of two years. The production of 239Pu depends on the initial mass of 238U, as shown in Fig. 5. The case C4 with the lowest initial mass of 238U shows the lowest production of 239Pu and 241Pu (Fig. 6). The production of 239Pu and 241Pu is similar in other cases, where the initial mass of 238U is similar. However, during the ongoing fuel burnup, a difference in the shapes of the plutonium curves appears, which is related to the depletion of fissionable 235U and its replacement by 233U and plutonium isotopes as secondary fuel. This process is unambiguously observed in the case C2, where the mass of 239Pu starts to decrease after reaching its peak at about 500 days of irradiation. In this case, the production of 233U is the lowest; therefore, 235U is replaced by fissionable plutonium isotopes, mainly 239Pu.
The volumetric homogenization method for the modelling of the MHTR core was presented in this study. This method was applied for the numerical Monte Carlo modelling of neutron transport in the MHTR core with thorium-uranium fuel. The example results of the modelling were presented. The results prove the reliability of the method for the initial screening of the reactor core performance. The universal character of this method makes it suitable for the numerical modelling of any type of material fuel composition and geometry. Thus, the method can be used not only for the MHTR modelling but also for the modelling of any fissionable system with a complicated fuel geometry, especially using linear chain method [8]. Further study on this method will focus on benchmarking with a full 3D MHTR core model with double heterogeneity of the TRISO fuel. This will allow the verification of the simplified models developed and their use as a replacement of detailed models. In the benchmarking process, the condition under which a defined replacement may happen will be also determined. The benchmarking will define the areas of the reactor core physics that can be reliably modelled using the volumetric homogenization method, e.g. radiotoxicity of the spent nuclear fuel [9], which may facilitate the whole MHTR design methodology.