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Diffusion Equation – Finite Cylindrical Reactor

Solutions of the Diffusion Equation – Multiplying Systems

In previous section it has been considered that the environment is non-multiplying. In a non-multiplying environment, neutrons are emitted by a neutron source situated in the center of a coordinate system and then freely diffuse through media. We are now prepared to consider neutron diffusion in the multiplying system containing fissionable nuclei (i.e., Σf ≠ 0). In this multiplying system, we will also study the spatial distribution of neutrons, but in contrast to the non-multiplying environment, these neutrons can trigger nuclear fission reactions.
Multiplying systems from criticality point of view
The required condition for a stable, self-sustained fission chain reaction in a multiplying system (in a nuclear reactor) is that exactly every fission initiate another fission. The minimum condition is for each nucleus undergoing fission to produce, on average, at least one neutron that causes fission of another nucleus. Also, the number of fissions occurring per unit time (the reaction rate) within the system must be constant.

This condition can be expressed conveniently in terms of the multiplication factor. The infinite multiplication factor is the ratio of the neutrons produced by fission in one neutron generation to the number of neutrons lost through absorption in the preceding neutron generation. This can be mathematically expressed as shown below.

Multiplication Factor

The infinite multiplication factor in a multiplying system measures the change in the fission neutron population from one neutron generation to the subsequent generation.

  • k < 1. Suppose the multiplication factor for a multiplying system is less than 1.0. In that case, the number of neutrons decreases in time (with the mean generation time), and the chain reaction will never be self-sustaining. This condition is known as the subcritical state.
  • k = 1. If the multiplication factor for a multiplying system is equal to 1.0, then there is no change in neutron population in time, and the chain reaction will be self-sustaining. This condition is known as the critical state.
  • k > 1. If the multiplication factor for a multiplying system is greater than 1.0, then the multiplying system produces more neutrons than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as the supercritical state.
Reactor criticality
Reactor criticality. A – a supercritical state; B – a critical state; C – a subcritical state
Diffusion Theory - Multiplying Systems
In this section, we will solve the following diffusion equation

diffusion equation - multiplying system

in various geometries that satisfy the boundary conditions. In this equation ν is number of neutrons emitted in fission and  Σf is macroscopic cross-section of fission reaction. Ф denotes a reaction rate. For example a fission of 235U by thermal neutron yields 2.43 neutrons.

It must be noted that we will solve the diffusion equation without any external source. This is very important because such an equation is a linear homogeneous equation in the flux. Therefore if we find one solution of the equation, then any multiple is also a solution. This means that the absolute value of the neutron flux cannot possibly be deduced from the diffusion equation. This is totally different from problems with external sources, which determine the absolute value of the neutron flux.

We will start with simple systems (planar) and increase complexity gradually. The most important assumption is that all neutrons are lumped into a single energy group, they are emitted and diffuse at thermal energy (0.025 eV). Solutions of diffusion equations in this case provides an illustrative insights, how can be the neutron flux distributed in a reactor core.

Solution for the Finite Cylindrical Reactor

Diffusion Theory - Cylindrical Reactor-minLet assume a uniform reactor (multiplying system) in the shape of a cylinder of physical radius R and height H. This finite cylindrical reactor is situated in cylindrical geometry at the origin of coordinates. To solve the diffusion equation, we have to replace the Laplacian by its cylindrical form:

cylindrical coordinates - 3D, 2D

Since there is no dependence on angle Θ, we can replace the 3D Laplacian with its two-dimensional form and solve the problem in radial and axial directions. Since the flux is a function of radius – r and height – z only (Φ(r,z)), the diffusion equation can be written as:

solution of diffusion equation - cylindrical reactor

The solution of this diffusion equation is based on the use of the separation-of-variables technique, therefore:

separation-of-variables - diffusion

where R(r) and Z(z) are functions to be determined. Substituting this into the diffusion equation and dividing by R(r)Z(z), we obtain:

separated diffusion equation

Because the first term depends only on r and the second only on z, both terms must be constants for the equation to have a solution. Suppose we take the constants to be v2 and к. The sum of these constants must be equal to Bg2 = v2 + к2. Now we can separate variables, and the neutron flux must satisfy the following differential equations:

separated diffusion equation2

Solution for the radial direction

bessel functions - J - YThe differential equation for radial direction is called Bessel’s equation, and it is well known to mathematicians. In this case, the Bessel’s equation’s solutions are called the Bessel functions of the first and second kind, Jα(x) and Yα(x), respectively.

For r > 0, this differential equation has two possible solutions, J0(vr) and Y0(vr), the Bessel functions of order zero, which give a general solution:

solution for radial direction

From finite flux condition (0≤ Φ(r) < ∞), which required only reasonable values for the flux, it can be derived that C must be equal to zero. The term Y0(vr) goes to -∞ as r ➝0  and therefore cannot be part of a physically acceptable solution. The physically acceptable solution must then be:

R(r) =  AJ0(vr)

The vacuum boundary condition requires the relative neutron flux near the boundary to have a slope of -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 it can be written mathematically as:

radial direction - boundary condition

If d is not negligible, physical dimensions of the reactor are increased by d, and extrapolated boundary is formulated with dimension Re = R + d, and this condition can be written as Φ(R + d) =  Φ(Re) = 0.

Therefore, the solution must be  R(Re) =  A J0(vRe) = 0. The function of J0(r) has several zeroes. The first is at r1 = 2.405, and the second at r2 = 5.6. However, because the neutron flux cannot have regions of negative values, the only physically acceptable value for v is 2.405/Re. The solution of the diffusion equation is:

solution for radial direction - diffusion

Solution for axial direction

The solution for axial direction has been solved in previous sections (Infinite Slab Reactor), and therefore it has the same solution. The solution in an axial direction is:

solution for axial direction - diffusion

Solution for radial and axial directions

The full solution for the neutron flux distribution in the finite cylindrical reactor is, therefore:

full solution of diffusion equation

where Bg2 is the total geometrical buckling.

The constants A and C must be added that they cannot be obtained from the diffusion equation because they give the absolute value of neutron flux. The neutron flux can have any value, and the critical reactor can operate at any power level. It should be noted this flux shape is only in a hypothetical case in a uniform homogeneous cylindrical reactor at low power levels (at “zero power criticality”).

In a power reactor core, the neutron flux can reach, for example, about 3.11 x 1013 neutrons.cm-2.s-1, but these values depend significantly on many parameters (type of fuel, fuel burnup, fuel enrichment, position in fuel pattern, etc.).

The power level does not influence the criticality (keff) of a power reactor unless thermal reactivity feedbacks act (operation of a power reactor without reactivity feedbacks is between 10E-8% – 1% of rated power).

Nuclear and Reactor Physics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Diffusion Theory