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## How to Calculate Neutrons

In order to design a nuclear reactor properly, the prediction how the **neutrons** will be **distributed **throughout the system is of the **highest importance**. This is a very difficult problem, because the neutrons interacts with differently with different environments (moderator, fuel, etc.) that are in a reactor core. Neutrons undergo various interactions, when they migrate through the multiplying system. To a **first approximation **the overall effect of these interactions is that the neutrons undergo a kind of **diffusion** in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the **diffusion approximation** and it is based on the **neutron diffusion theory**. This approximation allows solving such problem using **the diffusion equation**.

In this chapter we will introduce the **neutron diffusion theory **and we will examine the **spatial migration of neutrons** to understand the relationships between **reactor size**, **shape**, and **criticality**, and to determine the spatial flux distributions within power reactors. The diffusion theory provides theoretical basis for a **neutron-physical computing** of nuclear cores. It must be added there are many neutron-physical codes that are based on this theory.

First, we will analyze the spatial distributions of neutrons and we will consider a **one group diffusion theory** (**monoenergetic neutrons**) for a **uniform non-multiplying medium**. That means that the neutron flux and cross sections have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum.

See also: Neutron Flux Spectra

Moreover, mathematical methods used to analyze a **one group diffusion equation** are the same as those applied in more sophisticated and accurate methods such as **multi-group diffusion theory**. Subsequently, the one-group diffusion theory will be applied on an uniform multiplying medium (a homogeneous “nuclear reactor”) in simple geometries. Finally, the multi-group diffusion theory will be applied on an non-uniform multiplying medium (a heterogenous “nuclear reactor”) in simple geometries.

## Analytical Solution of Diffusion Equation

One of the most effective ways of calculating the **neutron diffusion** in thermal reactors is by the **multigroup diffusion method**. In this method, the entire range of neutron energies is divided into **N intervals**. All of neutrons within each interval are l**umped into a group** and in this group **all parameters** such as the diffusion coefficients or cross-sections are averaged.

As an illustrative example we will show a **two group diffusion equation** and we will briefly demonstrate its solution. In this example we consider a thermal energy group, and combine all neutrons of higher energy into a fast energy group.

In steady state, the diffusion equations for the fast and thermal energy groups are:

The equations are coupled through the **thermal fission term** the **fast removal term**. In this system of equations we assume that **neutrons appear in the fast group** as the result of fission induced by thermal neutrons (therefore Φ_{2}(x)). In the fission term, **k _{∞}** is to infinite multiplication factor and

**p**is the resonance escape probability. The fast absorption term expresses actually neutrons that are lost from the fast group

**by slowing down**.

**Σ**is equal to the

_{a1}Φ_{1}**thermal slowing down density**.

Consider the second equation (thermal energy group). Neutrons enter the thermal group as a result of **slowing down** out of the fast group, therefore the term ** pΣ_{a1}Φ_{1}** in this equation comes from fast group. It represents the source of

**neutrons that escaped to resonance absorption**.

To solve this system of equations we assume for a uniform reactor, that both groups of the fluxes in the core have a **geometrical buckling B _{g}** satisfying:

Since the geometrical buckling is the same for both the thermal and fast fluxes, the diffusion equations can be rewritten as:

For a uniform reactor, the vanishing of the neutron flux on the boundary requires that the neutron flux in **both groups** satisfies:

Since the **geometrical buckling** is the same for both the thermal and fast fluxes, the **thermal flux** and the **fast flux** are then **proportional** everywhere for the** bare reactor**. It can be derived that:

## Numerical Solution of Diffusion Equation

The **design** and** safe operation** of nuclear reactors is based on detailed and accurate knowledge of the **spatial** and **temporal** behavior of the **core power distribution** everywhere within the core. This knowledge is necessary to ensure that:

- the reactor can be
**safely operated**at certain power - the power density in localized regions does not exceed the
**limits of fuel integrity** - the fission chain reaction can be
**quickly shutdown** - the power plant is prepared to withstand all
**anticipated transients**, which are of importance in reactor safety and in must be proved and declared in the**Safety Analysis Report**(SAR).

Nowadays reactor core analyses and design are often performed using **nodal two-group diffusion methods**. These methods are based on **pre-computed assembly homogenized cross-sections**, **diffusion coefficients** and **assembly discontinuity factors** (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Highly absorbing control elements are represented by effective diffusion theory cross-sections which reproduce transport theory absorption rates. These **pre-computed data** (discontinuity factors, homogenized cross-sections, etc.) are calculated by **neutron transport codes** which are based on a more accurate **neutron transport theory**. In short, neutron transport theory is used to **make diffusion theory work**. The neutron transport equation is the most fundamental and exact description of the distribution of neutrons in space, energy, and direction (of motion) and is the starting point for approximate methods.

Two methods exist for calculation of the pre-computed assembly cross-sections and pin factors.

**Deterministic methods**that solve the Boltzmann transport equation.**Stochastic methods**that are known as Monte Carlo methods that model the problem almost exactly.

The diffusion equation can be derived by adding an additional assumption that the angular flux has a linearly anisotropic directional dependence in problems with isotropic sources and scattering. This allows the removal of the directional variables from the neutron density and simplifies the governing equation and associated numerical methods.

It is common practice, these methods can be divided into **two classes** based on the relative spatial** mesh size** over which the numerical approximations are valid.

**Finite difference methods**are generally restricted to very small spatial meshes (fine mesh method) in reactor applications — typically about 1cm.**Nodal methods**are based on a higher-order (or even analytical) expansion of the solution in the spatial variable and are applied to meshes much larger (coarse-mesh method) than a mean free path.

## Nodal Method in Neutron Diffusion

The steady-state core analysis of both new reactor designs and the reload cores of operating reactors involves a **large number** of whole-core calculations in order to optimize loading patterns and determine reload safety parameters. It must be noted that:

- There are about
**60 000 fuel pins**in a reactor core. In order to evaluate thermal margins these fuel pins have to be divided to about**50 axial levels**(axial nodalization). At one state point coupling of neutronics/fuel heat conduction/coolant hydraulics/structural mechanics is needed to evaluate thermal margins. - Each fuel cycle have to be analyzed for about tens of states (fuel burnup) and about 300 isotopes must be tracked.
- Millions of cycle depletion simulation have to be performed for designing loading design and optimization.
- Hundreds of startup and maneuver simulations have to be performed.
- Hundreds of transient accident simulations have to be performed for safety analysis.

In result, at the current speed of computational machinery, it is absolutely impractical to perform all of these calculations by applying **fine-mesh transport methods** to a model containing detail at the level of individual fuel rods, control elements, and coolant regions in an entire reactor core.

Reference: Scott W. Mosher, A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations. Georgia Institute of Technology, 2004.

For this reason, **nodal methods** are currently widely used **to predict neutronic behavior of a reactor core**. In general** nodal methods** are based on a multi-phase approach:

**Lattice Cell Homogenization.**In the first phase the reactor core is**decomposed**into relatively small sub-regions of the core, called**lattice cells**. A lattice cell typically contains a**single fuel assembly**plus half of the surrounding coolant gap and is precisely modeled in two-dimensional geometry with materials characterized by fine-group cross sections (100s of energy group). The**reflective boundary condition**(infinite lattice) is used, it is equivalent to a problem involving an infinitely large core composed of a single type of assembly. These**calculations**are performed by two-dimensional**neutron transport codes**which are based on a more accurate**neutron transport theory**. The neutron flux distribution from these fine-mesh calculations is used to spatially homogenize and condense (with respect to energy) cross sections and to calculate**pin power factors**. In this phase, self-shielding corrections are applied on the flux distribution. The homogenized lattice cell data are then used in a simplified core model to which less expensive diffusion theory is applied in the second phase, the nodal calculations.**Nodal Calculations.**A homogenized lattice cell (one single node) is typically represented by a 20 cm high section of a single fuel assembly. The nodal approach involves a high-order or analytical expansion of the intra-nodal flux shape in order to achieve a higher degree of accuracy, for a given node size, than the conventional finite difference approach to discretizing the spatial variable. The**nodal balance equation**, which is derived from 3D steady-state multigroup neutron diffusion equation and the**nodal balance equation**is solved within each node. There is a set of equations for the surface average currents instead of solving 3D finite difference equations directly. In approaching surface averaged current, the average of flux derivative on a surface is approximated as the derivative of average flux at the surface. It is more efficient to work with the diffusion equation for average flux rather than diffusion equation for 3D point wise fluxes. A transverse integration procedure is often employed to reduce the multi-dimensional equations to a set of coupled one-dimensional equations. The resulting system is then solved on a three-dimensional core model consisting of homogeneous nodes characterized by the generalized equivalence theory (GET) constants generated in the first phase.**Flux Reconstruction.**In this phase it is necessary to predict the neutron flux in individual fuel rods (**pin-by-pin**). Therefore we need the**heterogeneous solution**. The heterogeneous solution is reconstructed from given homogenized solution. The detailed, or heterogeneous, flux distribution can be approximated by modulating the smooth nodal solution with the fine-mesh transport solutions using a variety of techniques. The figure shows superposition of lattice flux (form functions) determined by fine-mesh transport codes with nodal fluxes determined by nodal calculations.

The efficiency of nodal methods is very high and comparable with the finite difference methods. It is very fast and allows us to compute such **huge number of reactor states**. It must be added, using this method one can get only approximate information about the neutron flux in a single node (or coarse-mesh area, usually a single fuel assembly of a 20 cm height). This analysis requires many subsequent calculations of the flux and power distributions for the fuel assemblies while there is no need for detailed distribution within the assembly. For obtaining detailed distribution within the assembly the heterogeneous **flux reconstruction** must be applied. However homogenization of fuel assembly properties, required for the nodal method, may cause difficulties when applied to fuel assemblies with many burnable absorber rods, due to very high absorption cross section (especially with Gd – burnable absorbers) and very strong heterogeneity within the assembly.

These methods are very efficient and accurate when applied to the current Pressurized Water Reactors (PWRs) or Boiling Water Reactors (BWRs).