## Boundary Conditions – Diffusion Equation

To solve the diffusion equation, which is a second-order partial differential equation throughout the reactor volume, it is necessary to specify certain **boundary conditions**. It is very dependent on the complexity of certain problem. **One-dimensional problems** solutions of diffusion equation contain **two arbitrary constants**. Therefore, in order to solve one-dimensional one-group diffusion equation, we need two boundary conditions to determine these coefficients. The most convenient boundary conditions are summarized in following few points:

The diffusion equation is mostly solved in media with high densities such as neutron moderators (H_{2}O, D_{2}O or graphite). The problem is usually bounded by air. The **mean free path** of neutron in air is **much larger** than in the moderator, so that it is possible to treat it** as a vacuum** in neutron flux distribution calculations. The vacuum boundary condition supposes that** no neutrons are entering a surface**.

If we consider that no neutrons are reflected from the vacuum back to the volume, the following condition can be derived from the **Fick’s law**:

Where **d ≈ ⅔ λ _{tr}** is known as the

**extrapolated length**. For homogeneous, weakly absorbing media, an exact solution of the mono-energetic transport equation in this case yields

**d ≈ 0.7104 λ**. The geometric interpretation of the previous equation is that the relative neutron flux near the boundary has a slope of

_{tr}**-1/d**, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary. This

**zero flux boundary**condition is more straightforward and is can be written mathematically as:

If d is not negligible, physical dimensions of the reactor are increased by d and extrapolated boundary is formulated with dimension **R _{e} = R + d** and this condition can be written as

**Φ(R + d) = Φ(R**

_{e}) = 0.It may seem the flux goes to 0 at an **extrapolated length** beyond the boundary. **This interpretation is not correct.** The flux **cannot go to zero** in a vacuum, because there are no absorbers to absorb the neutrons. The flux only appears to be heading to the zero value at the extrapolation point.

Note that, the equation **d ≈ 0.7104 λ _{tr}** is applicable to plane boundaries only. The formulas for curved boundaries can differ slightly, however, the difference is small unless the radius of curvature of the boundary is of the same order of magnitude as the extrapolated length.

**Typical values of the extrapolated length:**

The most common moderators have following diffusion coefficients (for thermal neutrons):

D(H_{2}O) = 0.142 cm

D(D_{2}O) = 0.84 cm

D(Be) = 0.416 cm

D(C) = 0.916 cm

The thermal neutron **extrapolated lengths** are given by:

d ≈ 0.7104 λ_{tr} = 0.7104 x 3 x D

**therefore:**

**H _{2}O: d ≈ 0.30 cm**

**D _{2}O: d ≈ 1.79 cm**

**Be: d ≈ 0.88 cm**

**C: d ≈ 1.95 cm**

As can be seen, this approximation is valid when the dimension L of the diffusing medium is much larger than the extrapolated length, **L >> d**.

**reasonable values**i.e. must be real, non-negative, and single valued. Also the solution must be finite in those regions where the equation is valid, except perhaps at artificial singular points of a source distribution. This boundary condition can be written mathematically as:

**This conditions are often used to eliminate **unnecessary functions **from solutions.**

**interface**between

**two different media**. At interfaces between two different diffusion media (such as between the reactor core and the neutron reflector), on physical grounds the neutron flux and the normal component of the neutron current must be

**continuous**. In other words,

**φ and J**are not allowed to show a jump.

It must be added, as **J** must be continuous, the flux gradient will show a jump if the diffusion coefficients in both media differ from each other.

**neutron source**. But the presence of the neutron source can be used as a

**boundary condition**, because it is necessary that all neutrons flowing through

**bounding area**of the source must come from the neutron source. This boundary condition depends on the source geometry and can be written mathematically as:

**neutron reflector**. The reflector

**reduces the non-uniformity**of the power distribution in the peripheral fuel assemblies,

**reduces neutron leakage**and reduces a coolant flow bypass of the core. The neutron reflector is a

**non-multiplying medium**, whereas the reactor core is a multiplying medium.

On this special interface we shall apply an **albedo boundary condition** to represent the neutron reflector. Albedo, the latin word for “whiteness”, was defined by Lambert as the fraction of the incident light reflected diffusely by a surface.

In reactor engineering, **albedo**, or the **reflection coefficient**, is defined as the **ratio** of **exiting to entering neutrons** and we can express it in terms of **neutron currents** as:

For sufficiently thick reflectors, it can be derived, that albedo becomes

where **D _{refl}** is the

**diffusion coefficient**in the reflector and the

**L**is the

_{refl}**diffusion length**in the reflector.

If we are not interested in the neutron flux distribution in the reflector (let say in the slab B) but only in the effect of the reflector on the neutron flux distribution in the medium (let say in the slab A), the albedo of the reflector can be used as a boundary condition for the diffusion equation solution. This boundary condition is similar to the **vacuum boundary condition**, i.e. **Φ(R _{albedo}) = 0**, where

**R**and

_{albedo}= R + d_{e}**Nuclear and Reactor Physics:**

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