**2.1. Monte Carlo methods for nuclear plants**

The principle behind the MCNP is statistical sampling, which makes use of the randomness of numbers. The MCNP records the trajectory of neutrons emitted by a source and determines whether the neutron is able to penetrate a shield after interacting with radioisotopes on the shield [8, 9]. The set-up for the simulation using MCNP5 in this study is made up of three main interacting packages of software obtained from the Radiation Safety Information and Com‐ putational Center of the Oak Ridge National Laboratory, Tennessee, USA. These three packages are MCNP1.05/MCNPX, CINDER 1.05 and MONTEBURNS 2.0.

The whole simulation process starts with an input data file to the MCNP5 as shown in **Figure 1**. Some of the parameters in the input data file include initial material compositions, geometry specifications (cell and surface cards), material specifications and source definition cards. Based on the output data required, MCNP produces a wide variety of output data files. Keys among these output files will be the group cross-sections and the neutron fluxes of materials. MONTEBURNS, which links MCNP5 and CINDER, transfers one group crosssection and neutron fluxes generated by MCNP to CINDER. This serves as the burnup or decay code for the set-up. CINDER processes these input files and generates output results such as burnup values and final material compositions. The resulting material compositions are transferred again to MCNP in a repeated or cyclic process. The end of a cycle repents the end of a burnup step. The next burnup step continues with the next cycle.

#### **2.2. The MCNPX code**

MCNPX is a software package popularly used to simulate various kinds of reactions pertaining to neutron, photon, electron transport and radioactive particles. This computer code package also has the capability of simulating nuclear burnup reactions and calculating the radionuclide inventory due to fission of fissile or fissionable isotopes and transmutation of parent nuclides. The code eliminates the need for the combination of the MCNP code with the nuclear burnup code ORIGEN or CINDER, in which one group cross-sections and fluxes are transferred from MCNP to the decay code ORIGEN or CINDER using monteburns. The MCNPX code has the CINDER.dat file responsible for nuclear fuel burnup as part of its buildup which allows the direct transfer of one-group reaction rates (cross-sections) and 63 group neutron fluxes to the depletion code cinder for decay calculations. In this study, the MCNPX is used to simulate nuclear fuel burnup calculations for a pressurized water reactor system using three different fuel grades: UOX, CEU and MOX.

**Figure 1.** Simplified experimental model using MCNP and MONTEBURNS.

The inventory of important nuclides would then be analyzed with burnup time for each fuel. This would be done by burning the fuel in the reactor core for 220 days using a 30-day time step. A 10-day time step is allowed for core maintenance operations. The performance of each fuel grade on the effective multiplicative factor, reactivity and reactor core life would also be analyzed based on the inventory of radionuclides for each fuel grade. The three fuel grades would also be compared based on reactor neutronic parameters such as the reactivity and neutron fluxes. Finally, the total radioactivity of the reactor core would be analyzed using different fuel grades. This would be the key in estimating the radiological hazard at the end of the core life of the reactor.

#### **2.3. The lattice cell structure**

Nuclear reactor cores are constructed as either rectangular or hexagonal lattices of assemblies. The lattice structure could consist of several different shapes or identical shapes. The lattice assembly consists of the fuel, control and instrumentation pins surrounded by water or other material that moderates neutron energy and carries away the heat generated as a result of the fission process. The basic shapes which make up the lattice are known as unit cells or pin cells [10].

Various types of assemblies are then arranged in a lattice structure to form the reactor core. The two most common lattice structures used for nuclear reactor cores are the rectangular lattice, used in water cooled reactors, or a hexagonal lattice, more common in sodium-and-gascooled reactors. The lattice geometry is described by three main parameters: the unit cell shape; the distance between the centers of adjacent unit cells, referred to as the lattice pitch; and the lattice dimension, which is a measure of how many units form the lattice.

The rectangular lattice for water-cooled reactors is used in this research. The lattice geometry is set up using the lattice fill matrix of the MCNPX visual editor. The total number of unit cells is hence NxNy. The lattice cell is shown below:

> 3 0 8 7 10 9 U 1 lat 1 fill 24 : 24 24 : 24 0 : 0 - - == =- - -

The first number is the cell number, which is 3. The second number 0 is the material number indicating a void material. The next four numbers are the surface numbers of the planes which are the sides of the rectangular shape. The U card specifies the universe to which the cell belongs, which in this case is 1. A cell is filled with a universe which is either a lattice or an arbitrary collection of cells. There are 24 unit cells on either side of the center lattice element in the x and y direction as indicated by the fill card with no unit cells in the z direction. **Figure 2** shows the lattice geometry together with the cell and surface numbers as can be seen when using the MCNPX visual editor. This shows the position of cells and surfaces in the reactor core lattice and how they are related in the system geometry.

**Figure 2.** MCNPX simplified geometry showing both cell and surface numbers.

**Figure 1.** Simplified experimental model using MCNP and MONTEBURNS.

of the core life of the reactor.

40 Nuclear Material Performance

**2.3. The lattice cell structure**

[10].

The inventory of important nuclides would then be analyzed with burnup time for each fuel. This would be done by burning the fuel in the reactor core for 220 days using a 30-day time step. A 10-day time step is allowed for core maintenance operations. The performance of each fuel grade on the effective multiplicative factor, reactivity and reactor core life would also be analyzed based on the inventory of radionuclides for each fuel grade. The three fuel grades would also be compared based on reactor neutronic parameters such as the reactivity and neutron fluxes. Finally, the total radioactivity of the reactor core would be analyzed using different fuel grades. This would be the key in estimating the radiological hazard at the end

Nuclear reactor cores are constructed as either rectangular or hexagonal lattices of assemblies. The lattice structure could consist of several different shapes or identical shapes. The lattice assembly consists of the fuel, control and instrumentation pins surrounded by water or other material that moderates neutron energy and carries away the heat generated as a result of the fission process. The basic shapes which make up the lattice are known as unit cells or pin cells

Various types of assemblies are then arranged in a lattice structure to form the reactor core. The two most common lattice structures used for nuclear reactor cores are the rectangular lattice, used in water cooled reactors, or a hexagonal lattice, more common in sodium-and-gascooled reactors. The lattice geometry is described by three main parameters: the unit cell shape;

## **2.4. The critical reactor core system**

A critical reactor system usually has an effective multiplication factor of 1. This usually means that the neutron production in one generation is equal to the neutron lost through either absorption or leakage in the preceding generation. While the sub-critical reactor might need an external neutron supply through accelerator-driven systems, the critical reactor system is self-sustaining. Usually, the size of the fuel cylinder and the density are much smaller for critical systems. The geometry of the critical pressurized water reactor system is modeled using the MCNPX visual editor. The material composition of the reactor core showing nuclide atom fractions for the three fuel grades and also for the clad material and moderator used is shown in **Table 1**.


**Table 1.** Material composition of reactor core.

## **2.5. Normalization of tally plots**

The tally plots are used to specify what needs to be investigated from the MCNP calculations. There are seven standard tallies available for use in MCNP. These tally cards are specified by Fn cards, where n specifies the tally number. SD cards are also used together with tally cards where necessary. The SD card is used to input a constant (a new area or volume) to divide the tally in cases where MCNP cannot calculate the area or volume for the regions. In all, three tally cards are used in this simulation. These are the f2, f4 and f7 tally cards. These were used to find out the flux across a surface, the track length in a cell and the track length estimate of fission energy deposition, respectively. The flux actually gives an idea of the flow of a physical property with time variation. The unit is given as quantity/(area × time). Some very important deductions were made from the flux calculations.

The tally bin width is usually normalized by dividing by the energy bin width. This gives a much broader representation of the particle distribution with energy. When a logarithmic scale is used for both axes, the visual representation is further obscured, with the tallies seen to decrease gradually with increasing energy or spread wide across the entire plot area. The two methods for normalizing energy-dependent tally are divided by the width of each energy bin or dividing by the logarithmic width of each energy bin [11].

The logarithmic width of the energy bin is referred to as the lethargy width. The governing equations are as follows (Eq. 1) [11]:

$$\mathbf{T}\_i = \prod\_{\text{fit}}^{\text{E}\_{\text{at}}} \mathbf{f}\left(\mathbf{E}\right) \mathbf{dE} \tag{1}$$

where *E*utand *E*lt are the upper and lower energy bin width, and *f*(*E*) is the flux or reaction rate. The tallies (*T*<sup>i</sup> ) tend to be small for small energy bins and large for large energy bins as can be seen from the above equation.

The normalized energy-dependent tally is calculated by dividing by the width of the energy bin and is shown below (Eq. 2):

$$\mathrm{I}\left(\mathrm{E}\right) = \int\_{\mathrm{E\_{\mathrm{h}}}}^{\mathrm{E\_{\mathrm{f}}}} \frac{\mathrm{f}\left(\mathrm{E}\right)\mathrm{dE}}{\int\_{\mathrm{E\_{\mathrm{h}}}}^{\mathrm{E\_{\mathrm{u}}}} \mathrm{dt}} = \frac{\mathrm{T\_{\mathrm{i}}}}{\mathrm{E\_{\mathrm{u}}} - \mathrm{E\_{\mathrm{h}}}} \text{ (tally units)/unit energy} \tag{2}$$

For lethargy normed tallies, a definition is required for lethargy. Neutron lethargy is defined in the analysis of nuclear reactors as the logarithmic energy loss of neutrons scattered elasti‐ cally. Mathematically, this is defined as in Eq. 3:

$$\mathbf{U} = \text{In} \, \frac{\mathbf{E}\_0}{\mathbf{E}} = \text{In} \left(\mathbf{E}\_0 - \mathbf{E}\right) \tag{3}$$

where U is the neutron lethargy.

**2.4. The critical reactor core system**

42 Nuclear Material Performance

**Material name Nuclide atom fraction**

UO2 16O 5.85402 × 10-3; 235U 3.862438 × 10-2 238U 0.9555216

deductions were made from the flux calculations.

or dividing by the logarithmic width of each energy bin [11].

234U 3 × 10-4; 235U 2.96 × 10-2; 238U9.701 × 10-1

Cladding (zirconium alloy) 50Cr 9.98 × 10-4; Fe 1.499 × 10-3; Zr 0.982499; 112Sn 0.014999

Mixed oxide fuel 16O 5.88402 × 10-3; 235U 2.5 × 10-3; 238U 0.9386; 238Pu 1.1147 × 10-4; 239Pu 4.150 × 10-2; 240Pu 7.9657 × 10-4; 241Pu 1.001 × 10-2; 242Pu 5.6388 × 10-4

H 4.7716 × 10-2; 16O 2.3858 × 10-2; 10B 3.6346 × 10-6; 11B 1.6226 × 10-5

The tally plots are used to specify what needs to be investigated from the MCNP calculations. There are seven standard tallies available for use in MCNP. These tally cards are specified by Fn cards, where n specifies the tally number. SD cards are also used together with tally cards where necessary. The SD card is used to input a constant (a new area or volume) to divide the tally in cases where MCNP cannot calculate the area or volume for the regions. In all, three tally cards are used in this simulation. These are the f2, f4 and f7 tally cards. These were used to find out the flux across a surface, the track length in a cell and the track length estimate of fission energy deposition, respectively. The flux actually gives an idea of the flow of a physical property with time variation. The unit is given as quantity/(area × time). Some very important

The tally bin width is usually normalized by dividing by the energy bin width. This gives a much broader representation of the particle distribution with energy. When a logarithmic scale is used for both axes, the visual representation is further obscured, with the tallies seen to decrease gradually with increasing energy or spread wide across the entire plot area. The two methods for normalizing energy-dependent tally are divided by the width of each energy bin

in **Table 1**.

uranium

Commercially enriched

Moderator (light water) <sup>1</sup>

**Table 1.** Material composition of reactor core.

**2.5. Normalization of tally plots**

A critical reactor system usually has an effective multiplication factor of 1. This usually means that the neutron production in one generation is equal to the neutron lost through either absorption or leakage in the preceding generation. While the sub-critical reactor might need an external neutron supply through accelerator-driven systems, the critical reactor system is self-sustaining. Usually, the size of the fuel cylinder and the density are much smaller for critical systems. The geometry of the critical pressurized water reactor system is modeled using the MCNPX visual editor. The material composition of the reactor core showing nuclide atom fractions for the three fuel grades and also for the clad material and moderator used is shown

The difference in the log of the energy bins is related to the neutron lethargy as follows (Eq. 4):

$$\ln(\mathcal{E}\_{\rm ut}) - \ln(\mathcal{E}\_{\rm it}) = \mathcal{U}\_{\rm it} - \mathcal{U}\_{\rm ut} \tag{4}$$

where *U*lt is the lethargy at *E*lt and *U*ut is the lethargy at *E*ut. The lethargy normed value (FI) is then given by the relation (Eq. 5):

$$\text{E}\_{\text{i}}\left(\text{U}\right) = \frac{\text{T}\_{\text{i}}}{\text{In}\left(\text{E}\_{\text{ut}} - \text{E}\_{\text{it}}\right)} = \frac{\text{T}\_{\text{i}}}{\text{U}\_{\text{h}} - \text{U}\_{\text{ut}}} \text{(tally units)}\\\text{(unit length)}\tag{5}$$

#### **2.6. Setting up a criticality problem**

The criticality calculation generally gives an idea about the ability of the reactor core to maintain a self-sustaining nuclear chain reaction. This is represented by the value of the effective multiplication factor, *K*eff. For reactors, which are infinitely large, an infinite multi‐ plication factor represents the criticality because it assumes that no neutrons leak out of the reactor. For a complete description of the life cycle of a real finite reactor, it is necessary to account for the neutrons that leak out. The effective multiplication factor takes this into account. Mathematically, *K*eff is defined as follows (Eq. 6):

$$\mathbf{K}\_{\text{eff}} = \frac{\text{neutron production from fission in one generation}}{\text{neutron absorbed} + \text{neutron leakage in preceding generation}} \tag{6}$$

The calculation of *K*eff consists of estimating the mean number of fission neutrons produced in one generation per fission neutron started. The *K*eff cycle is thus the computational equivalent of a fission generation, where a cycle is used to denote the computed estimate of an actual fission generation. MCNPX uses three different estimates to set up *K*eff (absorption, collision and track length) estimate. The final result is the statistically combined result for the three estimates.

The KCODE card is used together with a number of cards to set up a criticality problem. These cards specify the initial spatial distribution of fission points and include the KSRC card, SDEF card and SRCTP card. The KSRC card sets the initial x, y, z locations of fission points. The SDEF card is used to define points uniformly in volume whilst the SRCTP card is defined from a previous MCNPX criticality calculation. A typical KCODE card used together with a KSRC card has the following format:

## KCODE: 10000 1.000000 70 150 KSRC: 0.0000 0.0000 0.0000

The card above indicates that the nominal number of source histories is 10000 with the initial *K*eff guess kept as 1.00. The number of inactive cycles skipped before active *K*eff accumulation is 70 and the total number of cycles that run in the problem is 150. The KSRC card indicates that the x, y, z locations for initial fission source points were taken from the origin. The criticality calculations were performed with help from Bunde Kermit at the U.S Department of Energy.

In this research, different fuel grades and different cladding materials were used for the same reactor core configuration to investigate the effect of these on the criticality. The different fuel grades used were MOX, UOX and CEU. The materials used for cladding include zirconium, zircaloy and stainless steel. Successive fission cycles were run for determination of the criticality. **Table 2** shows the neutron absorption cross-sections and thermal conductivities for different clad materials at 25°C [12].


**Table 2.** Neutron absorption cross-sections and thermal conductivities for common clad materials at 25°C.
